ML20207A457

Provides Notes from Review of Investigation of Steam Explosion Loadings W/SIMMER-II. Objections Ctr on Listed Considerations,Including Substantiation of Simple Scaling & Role Propagation Plays in Large Mass Explosions

ML20207A457
Person / Time
Issue date:	06/26/1986
From:	Kelber C NRC
To:	Telford J NRC
Shared Package
ML20207A169	List: ML20207A164 ML20207A187 ML20207A192 ML20207A196 ML20207A211 ML20207A230 ML20207A272 ML20207A279 ML20207A289 ML20207A329 ML20207A334 ML20207A356 ML20207A367 ML20207A457 ML20207A517 ML20207A558 ML20207A563 ML20207A598 NUREG-1116, Responds to FOIA Request.App F Documents Available in Pdr. App G Documents Encl & Being Placed in PDR
References
FOIA-86-678 NUDOCS 8611100304
Download: ML20207A457 (531)

Text

_______ ___.

1. 'o, UNITED STATES o  ! n NUCLEAR REGULATORY COMMISSION h $ WASHINGTON, D. C. 20555 k . . . . + p' June 26, 1986

=

• MEMORANDUM FOR: John Telford W4 1 FROM: Charles Kelber '

SUBJECT:

STEAM EXPLOSIONS REVIEW, J 2, 6

]

I thought the review of the LANL report, "An Investigation of Steam Explosion Loadings with SIMMER-II," was very well done and informative. This memorandum is a record of my notes from that review.

1. There appeared to be unanimous agreement that the report was technically )

very well done; most, but not all, of the claimed deficiences arose from deficiences outside the scope of the technical effort.

l

2. Most of the reviewers, especially Anderson (and Spencer represented by Anderson), Ginsberg, and Merilo (EPRI), gave first priority to establishing the coarse-mixing and explosion phase of the event; that is, the initial conditions.

The LANL staff used existing SIMMER code to fit a particular SANDIA test-MD 19-and then assumed that fit could be scaled up to full size to represent an event in a reactor. No propagation model was used-progation was assumed instantaneous.

The objection appears to center on the following considerations:

1. It is very difficult to substantiate the simple scaling. Anderson in I particular pointed out that the constraints coming about from the sheer f expansion of size ought to make the explosion more vigorous. 1 l 2. Propagation is not actually instantaneous and could play a significant role in large (20 T) masses as compared with explosions at 1/1000 the mass.

l Note, however, as I did not at the meeting, that the length scale only goes up by a factor of 10. If the propagation time is of the order of 1 to 10 l

microseconds at 20 kg., it will still be less than 100 microseconds at 20 T.

Such a time scale is effectively instantaneous regarding the movement of large masses. Thus, I believe the criticism of scaling of initial conditions should center on the exclusive use of MD-19 as a model and the question of whether l

the model of particle break-up to one size-300 microns-is useful. There were some, especially Anderson and Berman, who appeared to feel that larger tests might yield smaller final particle sizes, and Berman and Corradini indicated l that there was a large population of much smaller particles to be examined.

It appears to me that the consensus would be that the LANL staff chose an optimistic set of initial conditions. The remarks on scaling appear to be compelling. During the discussion between LANL and the reviewers, I got the i 8611100304 861030 PDR FOIA CURRAN 86-678 PDR

[ ]Of - - _ - _ _ _ _

r-N 's 2 i

impression that LANL was inhibited in doing a better job of modeling initial conditions by a lack of a three-fluid model in SIMER. I believe that some of the reviewers-Merilo, Young, Marshall, Bennan-agree with this perception. I know that Zuber and Anderson feel it is more important to do more experiments on the initial phase first, then do such analysis as appears indicated.

My view is that a multi-fluid method is needed to model non-zero relative velocities between liquid drops in the same Eulerian cell. Such a method could be incorporated in the Advanced Fuel Dynamics Model, currently restricted to LMFBRs being licensed abroad. (The AFDM is financed by foreigners.) With that method available it might be possible to do a much better job of analyzing the first phase of the process.

Merilo pointed out a number of difficulties with the model such as: high packing of the fuel drops at the initial stage; liquid-liquid drop collisson model is deficient at low diameters; the particle volume appears to have been neglected, and there is no accounting for slug break-up.

Finally, a number of the reviewers urged that attention be given to steam explosions from stratified geometries.

3. There were some doubts expressed about the nature of the loadings on the head, and any failure criterion, but nothing was substantial enough for one to rely on. Again the questions appeared to arise about scaling by a factor of 10 to 1000.

4 Hopenfeld and Kelber raised the question of steel properties since the head is expected to be at over 1000*F. for most of the accident.

5. Bernan and others appeared to be confused over the results of the various cases; this was cleared up through discussion, but the thing that impressed me was that while the changes in kinetic energy delivered to the head were smaller than expected but in accord with conventional " engineering judgement,"

the chan in peak force on the head (apparently the basis for a failure )

estimate)geswere often unexpected in sign as well as magintude. This calls'into severe question, in my mind, the validity of the technical basis for engine-ering judgements in this area.

6. Berman referred to the fact that he was doing some work similar to our  !

steam explosion work, but for D00, and that work is classified as Secret.  !

Obviously we could not discuss such work at the review meeting. '

My recommendation is that we arrange for a briefing for a very small group on that work pertinent to our interests in steam explosions. That group should decide whether the technical content of the work is pertinent to our problem.

If so, that group should attempt to negotiate with 000 a basis for a suitably expurgated unclassified report to be made available. Dr. Ross would be the appropriate level to initiate contact.

7. There was unanimous feeling that the "probablilities" section did not belong in this report. In view of the many limitations expressed, I did not

s O

see what the technical basis was for the subjective judegments of the LANL staff; indeed, this report could serve as a basis for the following type of judgement: with a high probability, a steam explosion in a molten core case will lead to some form of vessel failure. Such a failure may, or, more likely, may not involve a n'issile failure, but will involve expulsion of a diffuse jet of steam, water and corium into the containment. Hence, the contingent failure likelihood appears to depend most heavily on the treatment of direct containment heating, at least for early failure.

8. There were a number of allusions to some considerations by Theofanous. A review of the papers by Theofanous and his students by this same group would beneficial. It appears unlikely that this could be done in advance of the presentation on this work that is apparently scheduled for July 22 next, but that should not interfere with a deliberate peer review.

A Charles Kelber cc: DFRoss,RES MErnst, RES MSilberberg, RES TLee, RES TSpeis, NRR CAllen, NRR TKing, NRR

Peer Review Meeting for the Report "AniInvestigation of Steam Explosion Loadings With SIMMER-II" Phillip Building, Room P-422 .

June 25,1986 .

8:30 am Welcome, Introduction, and Objectives.........'.. Robert Curtis 8:40 am Summary of LANL Report, SIMMER-II, and Response to Submitted Questions-in-Comon....... Charles Bell Y

9:40 am Questions and Comments.......................... Richard Anderson \ T

\

V 10:00 am LANL Response................................... Charles Bell )

v 1n.14 = Quest 15Wud Comet 1ts . . . . . . . . . . . . . . . . . . . . . . . . . . -Bruce _ Spence r

< 10 15 am 1 AN! Resoonse.. ............................ Cheeles-Bel p e d 10:50 am Questions and Comments......................... ,Ted Ginsbe , /

11:10 am .LANL Response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cha rl es Bell ,

11:25 am Questions and Comments......................... M 5ehgal Q .

'/ ~

lg) 11:45 am L ANL Response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cha rl es Bell 12:00 LUNCH

[ m sti an Com ... ...................... Han W

/ pm es

)1: 4 ... ..... ..... ... ...... ii

),p 1:35 pm Questions and Comments.......................... Charles Kelber )

1:55 pm LANL Response................................... Charles Ball 2:i0pm Questions and Coments . . . . . . . . . . . . . . . . . . . . . . . . . . W ii i iam Lamp' W h 2:30 pm L ANL Res ponse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cha rl e s Bell -- _

2:45 pm Questions and Coments . . . . . . . . . . . . . . . . . . . . . . . . . . Ma rshall Beman s

,, /

N %-QS h

2 3:05 pm LANL Response................................... Charles Bell 3:W uestions # C; vTonous '

..... , -ww~,

3:55 pm QuestionsandComments......................(...MikeCorradinifl 1 4:15 pm LANL Res ponse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cha rl es Bel l 4:30 pm Statement of Suma ry. . . . . . . . . . . . . . . . . . . . . . . . . . . . All Participants Conclusions _

Il 5:00 pm Adjourn 'l -

\

'/

r AN INVESTIGATION of STEAM EXPLOSION LOADINGS WITH SDNER-Il by V. R. Bohl Contributors C. R. Bell T. A. Butler L. M. Hull J. G. Bennett P. J. Blewett i

l l

l 1

TABLE OF CONTENTS LIST OF FIGURES ................................................................

LIST OF TABLES .................................................................

AB ST R A CT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 1 )

1. SUM 51ARY OF THE REPORT ...............................................(2)

A. Introduction ...................................................(2)

B. Modification of SlHN1ER-Il ......................................(3)

C. Ca l i bra t i on t o SNL St eat-Expl os i on Expe r ime nt s . . . . . . . . . . . . . . . . . ( 5)

D. Calibration to Los Alamos Shallow-Pool Experieents .............(9)

E. SlHN1ER-Il Reactor Case Calculative Results ....................(14)

F. Containment Failure Probabilities .............................(20)

G. Conclusions and Recommendations ...............................(24)

II. SlhMER-Il MODIFICATIONS FOR THE MOLTEN-CORE /COOLAhT INTERACTION PROGRAM.

A. Introduction ..................................................(25)

B. Summary of Modified Equations .................................(29)

1. AE05 Modification ........................................(29)

2. He a t -Tra n s f e r Mod i f i c a t i on s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 32 )

3. Vaporization / Condensation Modifications ..................(33)

C. Discussion of the AE05 Modificatior.s ..........................(37)

1. Review of Difficulties in the VERSION 10 SlHMER-Il ...........

Fo r cu l a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 3 7 )

2. Examination of Proposed Solutions to AEOS Problees .......(39)

3. The Vapor Heat Capacity Modification .....................(40)

4. Modification of the Gas "Cor.stant* at High Pressures .....(45)

5. Definition of the Vapor Teeperature for Single-Phase Cells and Other Modifications ......................................(46)

6. Discussion of Results ....................................(49)

7. Suggestions for Further leprovement ......................(66)

5. Co n c l u s i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 6 5 )

D. Modification of SIMMER-Il Liquid-Liquid Heat Transfer for .........

Water .........................................................(71)

E. Vaporization / Condensation Model Changes .......................(52)

F. Miscellaneous Corrections for the Molten-Core / Coolant Interaction..

Prograt .......................................................(94) 111. LOVER-HEAD MODEL FOR SlHN1ER-Il .....................................(96)

A. Introduction ..................................................(96)

B. Input .........................................................(96)

C. Head Failure .................................................(100)

D. Lo w e r - H e a d Mo t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 101 )

E. B o u nd a r y Co nd i t i o n s a nd Ed i t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 104 )

F. Correction Set ...............................................(105)

G. Satple Problem ......................................'.........(105)

IV. CORRELATION TO SNL STEAM-EXPLOSION EXPERIMENTS . . . . . . . . . . . . . . . . . . . . ( 117 )

V. ' ANALYSIS OF LOS ALAMOS EXPERIMENTAL DATA FOR SHALLOV POOLS ........(132)

A. Introduction .................................................(132)

B. Experieental Correlation .....................................(132)

C. Hypothetical Cases with Nonuniform Irterfaces ................(149)

D. Experieental Comparison with a Pressure Ratio of 50:1'........(169)

E. Experitental Comparison with a Nonuniform Initial Interface ..(177)

F. Conclusions ..................................................(191)

VI. AN ALYSI S OF STEAM EXPLOSIONS WITH SIH4ER-Il . . . . . . . . . . . . . . . . . . . . . . . ( 195)

A. Case 1 - 20 Percent Preeixed .................................(195)

1. Investigation of water surface area ...................(199)

2. Comparison with the ZIP study using the new codels ....(200)

3. Case 1 update..........................................(206)

B. Case 2 - Conservative Premixing. Explosion, and Expansior. ....(226)

1. Explosion After 1.0 second of mixing .......................

(second calculation) ..................................(225)

2. Explosion after 0.7s of mixing .............................

(secor.d scoping calculation) ..........................(245)

Ca s e 2 - F i n a l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 2 4 6 )

C. Case 3 - 75 Percent Premixed .................................(256)

D. Case 4 - Simulation of Incoherent Explosion ..................(250)

E. Case 5 - Upper Bound .........................................(289)

Vll. PROBABILITY OF CONTAINMENT FAILURE ................................(293)

Vill. RESEARCll PRIORITIES ON STEAM EXPLOSIONS ...........................(309)

APPENDIX A THE SIMMER-I l MANUAL AEOS TREATMENT . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 315 )

1. MATER I AL TEMPERATURE S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 318 )

11. TWO-PHASE FLOW PRESSURES ...............................(325)

III. SINGLE-PH4SE FLOW STATES ...............................(325)

IV. LIQUID MICROSCOPIC DENSITIES . . . . . . . . . . . . . . . . . . . . . . . . . . . (332 )

V. AE05 INPUT .............................................(333)

VI. DETERMINATION OF COMPONENT TEMPERATURES ................(343)

V!!. DETAILS IN THE EVALUATION OF SINGLE-PHASE LIQUID STATES (347)

A. SINGLE COMPONENT SYSTEMS ...........................(347) .

B. MULTI COMPONE AT SYSTEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 3 50 )

APPENDIX B CFS N0DE / VAPOR /WRBAEOS ( EOS CORRECTION SET) . . . . . . . . . . . . . . . . . . (354)

APPENDIX C REVISED Sitt1ER-Il INPUT DESCRIPTION FOR THE AEOS ..............(365)

APPENDIX D AE05 SIMULATION PROGRAM .......................................(370)

APPENDIX E SUGGESTED VALUES FOR THE WATER AEOS INPUT .....................(350)

APPENDIX F ST AND ARD Siht1ER-Il LIQUID-LIQUID HEAT-TRANSFER MODEL . . . . . . . . . . (351 )

APPENDIX G CORRECTION SET FOR MODIFIED LIQUID-LIQUID HEAT TRANSFER .......(359)

APPENDIX H VAPORIZATION' CONDENSATION CORRECTION SET ......................(392)

APPENDIX 1 Siht!ER-Il ht4NUAL TREATMEhT OF THE VAPORIZATION / CONDENSATION MODEL . .

MODEL .........................................................(397)

APPENDIX J M I SC E L L ANEOU S COR R ECT I ON S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 41 1 )

APPENDIX K SUNtt4RY DESCRIPTION OF THE HEAD FAILURE MODEL SUBROUTINE ......(414)

APPENDIX L DEFINITIONS OF VARIABLES IN THE PLUGW CORRECTION SET ..........(416)

APPENDIX M LOWER HFAD DYNAMIC ANALYSIS ...................................(415)

I. IhTRODUCTION AND SU4tARY OF RESULTS OF ADINA STUDY.......(415)

11. THE FAILURE MODE AND CRITERIA............................(421) 111. BASIS FOR SIMPLE MODEL TO ESTIMATE REQUIRED FAILURE ..........

IMPULSE (COLUMN 1 0F TABLE M-I I ) . . . . . . . . . . . . . . . . . . . . . . . . ( 421 )

IV. A SINGLE DEGREE OF FREEDOM MODEL THAT APPROXIMATES FAILURE . .

TIMES ...................................................(423)

V. CONCLUSION ............................'..................(427)

APPENDIX N CONSIDERATIONS RESULTS FROM THE SDOF. LOWER HEAD FAILURE ............

MODEL .........................................................(430)

APPENDIX 0 A LIMITED REVIEW OT SNL STEAM EXPLOSION EXPERIMEhTS . . . . . . . . . . .'(432 )

APPENDIX P Slbt!ER-Il INPUT FOR THE ANALYSIS OF THE MD-19 EXPERIMENT USING . . . . .

A UNIFORM MIXING ZONE .........................................(442)

APPENDIX Q Slbt!ER-Il COMPARISON WITH THREE FIELD MIXING CALCULATIONS .....(446)

APPENDIX R SIkt!ER-Il INPUT FOR THE ANALYSIS OF THE 60-19 EXPERIMEhTS . . . . . . . . . .

USING A NONUNIFORM MIXING ZONE ................................(472)

APPENDIX S EFFECTS OF CHANGING THE AXIAL AND RADIAL CONSTRAlhTS IN THE ........

SlHULATION OF MD-19 ...........................................(450)

APPENDIX T EXPLOSION CALCULATION STARTING WITH A STANDARD SDNER-Il . . . . . . . . . . . . .

CON F I G U R AT I ON . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 4 5 5 )

APPENDIX U S1451ER-Il PREMIXING WITH HIGH STEAM PRODUCTION RATES ..........(457)

APPENDIX Y DETAILS ON THE UPPER BOUND Siht1ER-11 STEAM EXPLOSION CALCULATION ...

CA LCU LAT I ON . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 4 9 5 )

REFERENCES .....................................................................

4

FIGURES

1. Comparison of water chamber base pressures in experiment MD-19.

2. Shallow-pool experieental apparatus.

3. Comparison of P4 pressures in the shallow-pool experiments.

4. PWR structural representation for SIHHER-Il steam-explosion calculations.

5. Integrated upper-head loading for case 1.

6. Integrated upper-head loading for case 2.

7. Integrated upper-head loading for case 3.

S. Integrated upper-head loading for case 4.

9. Integrated upper-head loading for case 5.

10. Sample problee results.

11. Triple-valued "F" function for water vapor.

12. Temperature-averaged vapor heat capacities for water.

13. SESAME /S'IHMER-II comparison plot for fitting eg .

14. "F" function for the revised AEOS.

15. Steam-table coeparison with saturated density and temperature inputs

16. Saturation pressures with internal energy and density input into SIMMER-II.

17. Ir.ternal er.ergies of vapor with density and temperature input along the 22 MPa isobar.

15. Derivative of Eq. (25) as a function of ag,g. with Tg just above TCrt f r water.

19. Recair.ing triple-valued nature of the "F" function near the critical point.

20. A typical equation-of-state comparison at low steam densities.

21. Additional comparison of equation-of-state data at low density.

22. Equation-of-state comparison at high temperature.

3

23. A SESAME /S!HHER-II comparison at cg - 316 kg/e .

24. Pressure as a function of liquid interr.a1 energy taken from the density and temperature describing a ficticious saturation curve above TCrt'

25. An example showing that p > p* is not permitted.

26. Pressure in th: saeple problem with New heat transfer.

27. Pressure in the sample problem with original heat transfer.

25. Water teeperature in the sample problem with new heat transfer.

29. Water temperature in the sample problee with original heat transfer.

30. Pressure in the saeple problee with new revised heat transfer.

31. Water temperature in the sample problet with new revised heat transfer.

32. Mesh and structure geometry for the quasi-eechanistic reactor i

--y- e- y-- e - - -

steae-explosion problee. (Note: Nodes are expanded in the radial direction.)

33. Geometric indices for a radial node in the SI21ER-II coving lower-head model.

34. SIW1ER-Il treatment of the fluid dynamics at the lower-head interface.

35. EXO-FITS a ppara t us.

36. Typical instrumented water chamber.

37. Initial configuration for the S121ER-Il representation of test E-19,

35. Base pressure of the water chamber in experiment MD-19.

39. Best fit for a uniform interaction zone (water-droplet size 75 um; heat-transfer cultiplier 1.7).

s

40. Initial thermite distribution for the three-field calculation compared with the reported experimental results.

41. Ser.sitivity of the peak pressure to the heat-transfer cultiplier for a 0.3-me fuel droplet and a 0.075-cun water droplet.

42. Best fit starting f rom a non-unifore interaction zone with a 0.3 c:c-fuel droplet and a 0.075-ce water droplet.

43. k'ater phase pressure for experiment E-19.

44. Pressure at location (r.2) = (0.27 c, 0.30 m) eensured from the bottom of the Lucite container.

45. Experimental setup for the shallow-pool tests.

46. S141ER-Il cesh using equal area radial nodes and a pressure vs time bottom boundary condition.

47. Experimental pressure recorded at location Pl.

45. P1 recorded pressures over the time range of interest for SI41ER-II calculations.

i

( 49. Experimental P4 pressure trace.

50. Calculated pressure for coc:parison at the P4 transducer.

51. Liquid-volume-fraction contour plots produced by SI41ER-II.

52. Contour plots with time-step control revisions.

53. Revised calculation for coc:parison with the P4 transducer.

54. Liquid-volume-fraction contour plots. Concave initial conditions.

[

55. Pressure trace at the center of the upper endplate for the concave case.

56. Liquid-volume-fraction contour plots. Convex initial conditions.

57. Pressure trace at the center of the endplate for the convex case.

58. Liquid-volume-fractions for the plate obstruction case.

59. Pressure trace at the center of the endplate with a permanent obstruction.

l

60. Reference case pressure integral.

l 61. Concave case pressure integral. .

I i 62. Convex case pressure integral.

63. Plate obstructier. case pressure integral.

64. Center of tube pressure trace for the scaled-up case.

65. Liquid-volume fractions for the scaled-up case.

66. Upscaled case. pressure integral.

67. . Experimental pressures. recorded at location P1 for the 50:1 pressure ratio test.

65. P1 record over the time range of interest for SIMMER-Il calculations in the 50:1 pressure ratio test.

69. Results at location P4 for the 50:1 pressure ratio test.

70. Experimental results in the 50:1 pressure ratio test.

71. Pressure ir.tegral in the 50:1 pressure ratio case.

72. Contour plots for the 50:1 pressure ratio case.

73. SIHMER-Il representation of the experimental configuration with a plate.

74. P1 pressure in test 16 with a stationary plate.

75. P1 pressure in test 17 with a stationary plate.

76. P1 pressure in test 15 with a stationary plate.

77. P1 pressure in test 19 with a stationary plate.

75. P4 pressure in test 16 with a stationary plate.

79. P4 pressure in test 17 with a stationary plate.

50. P4 pressure in-test 16 with a stationary plate.

St. P4 pressure in test 19 with a stationary plate.

52. Calculated P4 pressures for test 19.

53. Calculated pressure integral. test 19.

54. SIHMER-Il liquid-volume-fraction plots for comparison with the experimental plate case.

65. S!HMER-Il results for an idealized expansion in a multinode environment.

56. Configuration model for case 1.

87. Initial core material distribution.

SS. Initial water distribution.

89. Liquid-volume-fraction plots for comparison with the ZIP, study.

90. Pressure at the inlet plenum bottom for comparison with the ZIP study.

91. Pressure at the downcomer top for comparison with the ZIP study.

92. Pressure at the top of the vessel for comparison with the ZIP study.

93. Pressure on the head at 30* to the vertical for comparison with the ZIP study.

94. Pressure on the head at 70* to the vertical for comparison with the ZIP study.

95. Force on the head for comparison with the ZIP study.

96. Total fluid kinetic energy for cocparison with the ZIP study.

97. Upward fluid kinetic enerEy for comparison with the ZIP study.

96. Kinetic energy of materials not in the core and above core regions for comparison with the ZIP study.

99. Pressure differential across the lower head in case 1 (update).

100. Lower-head displacement before failure in case 1 (update).

101. Liquid-voluee-fraction plots for case 1 (update).

102. Pressure at the inlet plenum bottom for case 1 (update).

103. Pressure at the top of the downcomer for case 1 (update).

104. Pressure at the top of the vessel for case 1 (update).

105. Pressure on the head at 30' to the vertical for case 1 (update).

106. Pressure on the head at 70* to the vertical for case 1 (update). '

107. Force on the head for case 1 (update).

106. Total fluid (SIMMER-II liquid and vapor velocity fields) kinetic energy for case 1 (update). ,

109. Upward fluid kinetic energy for case 1 (update).

110. Downward fluid kinetic energy for case 1 (update).

111. Downcocer and inlet pipe kinetic energy for case 1 (update).

112. Outlet pipe kinetic energy for case 1 (update).

113. Ir.itial configuration for the SIHMER-II premixing calculation in case 2.

114. Ir.itial conditions. SIMMER-II conservative premixing calculation (case 2).

115. k'a t e r-f ue l contac t . SIHMER-II conservative premixing calculation (case 2).

116. Fuel contact with the support forging. SIH4ER-II conservative premixing calculation (case 2).

117. Initial contact of-fuel with the lower head. SlHWER-Il conservative premixing calculation (case 2).

lib. BeSi nning of fuel pool breakup. SIHMER-Il conservative preeixing calculation (case 2).

119. Two-phase fuel pool forced. SIHWER-II conservative premixing calculation (case 2).

120. Initial expansion instability, fluidized explosion conditions.

121. Instability development. fluidized explosion conditions.

122. Venting beginning under fluidized explosion conditions.

123. ConfiE uration just before impact. fluidized explosion conditions.

124. Material impact beginning, fluidized explosion conditions.

125. Material impact eceplete and rebound beginning. fluidized explosior.

conditions.

126. Final configuratior.. fluidized explosion conditions.

127. Inlet plenum pressure, fluidized explosion conditions.

125. Pressure at the top of the downcomer. fluidized explosion conditions.

129. Pressure at the top of the ve,ssel, fluidized explosion conditions.

130. Head pressure at 30* to the vertical. fluidized explosion conditions.

131. Head pressure at 70* to the vertical, fluidized explosion conditions.

132. Force on the head fluidized explosion conditions.

133. Total. fluid kinetic energy, fluidized explosior. conditions.

134. Upward fluid kinetic energy, fluidized explosion conditions.

135. Miscellaneous fluid kinetic energy. fluidized explosion conditions.

136. Corium below the bottom of the initial corium pool. SlHHER-II conservative premixing calculation (case 2).

Liquid-volute fractions. explosion at 0.7s.

137.

135. Inlet plenum pressure, explosion at 0.7s.

139. Pressure at the top of the downcomer. explosion at 0.7s.

140. Pressure at the top of the vessel, explosion at 0.7s.

141. Head pressure at 30' to the vertical, explosion at 0.7s.

142. Head pressure at 70' to the vertical, explosion at 0.7s.

143. Force on the head, explosion at 0.7s.

144. Total fluid kinetic energy, explosion at 0.7s.

145. Upward fluid kinetic energy, explosion at 0.7s.

146. Miscellaneous fluid kinetic energy, explosion at 0.7s.

147. Liquid-volume fractions during the expansion process, case 2 (final).

145. Inlet plenum pressure, case 2 (final). -

149. Pressure at the top of the downcomer, case 2 (final).

150. Pressure at the top of the vessel, case 2 (final).

151. Head pressure at 30' to the vertical, case 2 (final).

152. Head pressure at 70' to the vertical, case 2 (final).

153. Force on the head. case 2 (final).

154. Upward fluid kinetic energy, case 2 (final).

155. Downcomer fluid kinetic energy, case 2 (final).

156. Downward and outlet pipe fluid kinetic energy, case 2 (final).

157. Total fluid kinetic energy, case 2 (finil).

155. Configuration codel, case 3.

159. Initial expansion of fuel in case 3.

160. Head loading and reflection in case 3.

161. Pressure at the inlet plenum bottom for case 3. J 162. Pressure at the top of the downcomer for case 3.

163. Pressure at the top of the vessel for case 3.

l l

l

164. Pressure on the head.at 30* to the vertical for case 3.

165. Pressure on the head at 70' to the vertical for case 3.

166. Force on the head for case 3.

167. Total fluid kinetic energy, case 3.

165. Upward fluid kinetic energy, case 3.

169. Miscellaneous fluid kinetic energy. case 3.

170. Configuration model, case 4.

171. Driving pressure for' lower-head failure. case 4.

172. Liquid-volume-fraction plots for post-explosion / expansion, case 4.

173. Inlet plenum pressure, case 4.

174. Pressure at the top of the downcomer, case 4.

175. Pressure at the top of the vessel. case 4.

176. Head pressure at 30' to the vertical, case 4.

177. Head pressure at 70' to the vertical, case 4.

176. Force on the head, case 4.

179. Total fluid kinetic energy. case 4.

180. Upward fluid kinetic energy, case 4.

161. Miscellaneous fluid kinetic energy. case 4.

182. Diagram of an accidents progression focused on the alpha mode of direct containment failure by a steam explosion.

163. Visualization of the state resulting from a failure of the core

-barrel prior to penetration of a coherent colten mass through the below-core structure.

164. Melt flow into the lower plenum by sideways penetration of the core barrel.

185. Proposed schematic SEALS experiment configuration.

A-1 Pressure-volume relation for the single-phase EOS.

F-1 Physical representation of liquid-liquid interaction.

F-2 Collision configuration for liquid m and liquid k.

F-3 Heat-transfer treatment during contact.

F-4 Assumed contact interval.

M-1 Case 6: Regions Used to approximate the spatial variation of the M1 S!HMER transient with four separate pressure-time histories.

M-2 Failure initiation region from ADINA parameter studies.

M-3

• Expanding Vessel" mode.

0-1 Effect of pressure on explosion yield.

0-2 FITS containment chamber.

Q-1 Initial volume fractions at the start of the three-field calculation.

l

Q-2 Three-field calculation following icpact with water.

Q-3 Three-field calculation showing initial total removal of water before the advar.cing celt.

Q-4 Three-field calculation after the celt density is reduced sufficiently to stop total water vaporization (recoval) at the celt front.

0-5 Three-field calculatior. when the container bottom is being approached.

Q-6 Three-field calculatior. at the time of a presumed steam explosion.

Q-7 Initial voluee fractions for the standard SlHMER-il calculation.

Q-$ Melt-water contact with standard SlHWER-Il modeling.

Q-9 Melt penetration of water with standard SlHMER-Il modeling.

Q-10 Start of low vapor production with standard SIMMER-II modeling.

Q-11 Results with standard SlH4ER-Il modeling as the container bottom is beir.g approached.

Q-12 Results at the time of the steam explosion with standard flHMER-Il modeling.

Q-13 initial conditions with revised SlH4ER-Il heat transfer.

Q-14 Melt-water contact with revised SIH4ER-Il modeling.

Q-15 Iritially strong vapor production with the revised SIH4ER-il modeling.

Q-16 Beginning of telt breakup with revised SlHAER-II modeling.

Q-17 Melt-breakup with revised SlHMER-Il modeling.

Q-IS ConfiE uration at end of calculation for cocparison with Figs. Q-6 ar.d Q-12.

5-1 -Results startir.g free a non-unifore interaction zor.e with a rigid-wall radial cor.straint.

5-2 Initial conditions for the case with axial constraint.

5-3 Pressures below the slug in the axially / radially constrained case.

T-1 Explosior. result- starting with standard SIMMER-Il calculated premixing.

U-1 Initial conditions, high steat-production precixing case.

U-2 Initial flow, high steat-production precixing case.

U-3 Initial cixing, high steam-production premixing case.

U-4 Precixture forced above support forginE. bi hS steam-production premixing Case.

U-5 Couterflow beginning. high steam-production premixing case.

U-6 Pool breakthrough close. high stent-production premixing case.

U-7 Corium pool fluidization beginning, high steac production precixing cas,e.

U-5 Fuel spray reachir.g the top of the vessel. hiEh steat-production premising case.

U-9 Final conditions. high steam-production precixing case.

V-1 Geometric setup for boundits case illustratior..

V-2 Compariser. of an Iser. trope from the SESAME EOS with an adjusted SIMMER-II AEOS usir.g the same i r.pu t der.sities ar.d ir.ternal energies.

V-3 -Pressure at the bottoc of the inlet plenue for case 5.

V-4 Pressure at node (11,60), where the head curvature begins for case 5.

V-5 Pressure at r. ode (6.65), the ciddle of the head curvature regior. for case 5.

V-6 Pressure at the cer.ter of the head for case 5.

V-7 Ir.tegrated force or. the head for case 5.

V-8 Liquid volure fractior.s for case 5.

~

V-9 Upward fluid kir. etic energy for case 5.

V-10 Downward fluid kinetic er.ergy for case 5.

V-11 Downcoter and inlet pipe fluid kinetic er.ergy for case 5.

V-12 Outlet pipe fluid kinetic energy for case 5.

I TABLES 3

1. IIIGHEST F ZERO WITilp c = 171.4 kg/rr AND eg = 3.197 MJ/kg.

II. AEOS/ STEAM-TABLE COMPARISON FOR THE 22 MPa ISOBAR.

III. AEOS/ STEAM-TABLE COMPARISON FOR THE 50 MPa ISOBAR.

IV. AEOS/ STEAM-TABLE CB1 PARIS 0N FOR THE 100 MPa ISOBAR.

V. AEOS/ STEAM-TABLE C31PARISON FOR THE 22 MPa ISOBAR IF ag,y IS SET TO 75 K.

VI. SESAME RESULTS FOR FIG. 24.

VII. LOWER-HEAD MODELING INPUT.

VIII. INPUT EDIT FOR THE SIMMER-II LOWER-HEAD-FAILURE PROBLEM OF FIG. 32.

IX. SAMPLE OUTPUT EDIT BEFORE LOVER-HEAD FAILURE.

X. SAMPLE OUTPUT EDIT AFTER LOWER-HEAD FAILURE.

XI. DEFINITION OF THE PXPLOT ARRAY.

XII. CORRECTION SET FOR THE SI WER-II LOWER-HEAD MODEL.

XIII. INITIAL CALCULATIONS FOR FITTING TEST MD-19.

XIV. SI21ER-II INPUT FOR THE REFERENCE CALCULATION.

XV. FLUID KINETIC ENERGY PARTITION.

XVI. DEFINITION OF PROBABILITY SPLIT LEVELS.

XVII. FEATURES THAT COULD BE USEP IN SIW1ER-II PARAMETERS OF STEAM EXPLOSION / EXPANSIONS.

XVIII. PRESEhTLY INTESTIGATED IMPROVEMENTS COMPARING THE CURREhTLY PROGRA41ED THREE-FIELD FLUID DYNAMICS ALGORITHM AND SIMER-II.

E-I. WATER AEOS INPUT.

M-I. PRESSURE TRANSIENTS USED IN LOWER-HEAD DYNAMIC ANALYSIS.

M-II. SU414RY OF LCVER-HEAD FAILURE STUDY FOR A GENERIC'PWR VESSEL.

M-III. COMPARISON OF SDOF MODEL AND ADINA RESUI.TS.

1. INTERMEDIATE SCALE FITS TESTS.

0-1. STANDARD SI41ER-II INPUT FOR THE MD-19 PREMIXING CALCULATION.

.. m. _. . . . . _ _ _ . . _ _ . . . .- . - . _ ___ _. . . . _ _. _. - . . . _ _ . .. _

1-ROUGH DRAFT .

.) AN INVI.STIG ATinN '0lL STI el-I:XI>LOS10N LO4lilNGs WITil $14NI.k-Il

by 1

W. R. IlohI '

Cortributors

, C. R. Bell T. A. Butler L. M. Hall -

<- J. G. Benrett l'. J. Bl eve t t 4

ABSTRACT The S14NFR-I l cede ' wa s used t o. provide esticates of the

, taxitut ar.ticipated leads or the upper head of a pressurized wa t e r- rea6 t er f e l l ow i r.g at it-vessel stent explosion. The ,

S INNI.R- l l equatiot-of-state ard heat-transfer codels were upgraded for this purpese. A calibration to Sandia National Laboratories" s t eat-e xples ior. da ta ar.d a comparison with Los Alamos shallew-pool experiter.ts were perforced. A lowcr-head lialute ard tetsor rede; was developed. Ar.alysis of 4

.pararetrat . cases suggest, that the upper bour.d on the cerditieral probability of alpha-tode tailure. giver core relt, should be o. 01 if the. vessel's upper head.ard bolts are

. r. car r. areal operatiry ter.peratures. Suggestier.s are made for activities that could lead to increased ca r.f i d e r.c c in

g. understardirg ar.d quartifying steat-explessor. issues.

.i :

5 I

I l

1

I In m..

RlRuo J l i

1. St'At1ARY of Till RI' PORT A. I r

• r od a t ' . -

The purpose of this werk was to provide a reasonable estinate el the noxitut loads that right be e xpec t ed a t the upper head of a pressurized watrr reacter ( PVR ) followiry .r in-vessel stest explosion. Using the de t e rti r.e d rarpe of leads. ,iudgre r.t s were to be made on the potential for containtert failure by missile production resulting from a steam explossen. cottonly called alpha-mode failure. assuming core telt had occurred.

The SihtiER-ll code was uied to calculate the estimated loads. S ikt1f.R- I l i s a cultiphase material relocatior, code orginally developed for fast reactor disrupted ccre accidents. A brie f sutta ry de sc ri pt icr. of SIht1ER-Il is given in Ch p. 11. A previous study of this type. herein called the ZIP study. was perferred ir 1950 To provide some basic Slkt1ER-Il capabilities lackir.g ir. the 1950 study, a brief preg ran of code modificat ier. was undertaker, as part of this w.rk. Th e *, e rod i f ic a t iers are discussed ir Se r. . 1-R. Tn calibrate the revised n0Jelir; te available steat explessor experiter.tal data. Sardia ha t s er.a l Laheratcries" (SNL) arterned: ate-s ale thertite-water experiter.ts were assessed, ard ere particular well-ch:.racterized experitert. 41D-19 wa s calculated with SIttlik II. This d i s c u s a. i c t is i r. Sec. I-c. The adequacy cf the SIkt!!:k- I l pes, t e xpl es i e r 'e x pa r.s i c n dvtati c i, wa s investigated by comparir.g calculatier.<. with scaled shallow poe! experiter.ts. The e x pe r i te r.t a, and the analysis perforted are giser in Sec. I - D. As the studv proceeded. nurre r ou s scopir.g calculatacts or a full-scale PWR representatien were perforted: the results of the e c a l c u l a t i e r.*,

are ir Sec. 1-E. These reactor result *, were ir.tegrated irto z.r everall schete for . iud g i ng the probability of certaitrert failure, preser.ted in Sec. 1 - F.

Tirally, c or s l u s i er.s ard reccener.dations from this study are giver. ir Sec. 1-G.

Te leep the volute of this suttary chapter from beir.g excessive. rest of the a bcs e outlined s e t t i e r i. present summary meterial only. Details are giver in subsequent th pters el this report.

T1. G. St everser. "kercrt of the Z i e r. ." I r J :. t Peirt Studv. Velure 11. L. .

Alates Sstertiti. Laberat erv repor t L A - 6 3o6 -515. NURl:G/rR-1.Ill ( Apr i l 198o i, 2

T R4I(O A]l Qf lpg &

l'.

. M-Jifica*ier o f s i tt.11'k- l l The Sistifk-ll sede was rodified ti hetter represert a ceriur-w.ite: s e:

arJ to ir:p l er er t a lower-heaJ lailure ard notion rodel suggested bs the Zil>/ s t udv. The detas!s el these charges are giver i r. Chaps. 11 ard 111 and A p pe r.J i s e s A to h.

The leur ta ,i e r area of code mod i f ic a t ian for t r ea t i r.g the corict-water systet are ar irpreved equation of state (LOS . a more appropriat e t rea trrent of rcrequilibriut heat trarsfer to liquid water. revised a s surpt i or.s ir the vap0rizatser/cordensatser model regarJiry erergies at wh i c h wa t e r is va po r i ze d and (ordersed. and tiu e llar.eeus c ha r.p e s .

The test s i g r i f i c a r.t I:0S modificatier. ir. terns of accuracy was to take the infinitely dilute. water-varer's heat capacity temperature d e pe nd e r.t . which better represerts the arcreasing degrees, of freedet of the water tolecule a r.J the hydroger-oxygen systet at high temperatures. For 3 000 K infir.itely dilute steat. the rew heat ca pa c i t y r e l a t i e r.s h i p gives 1.62 tines the value for infinitely dilute 400 K steat. ,

The ca .i o r t,dificatior te the LOS ir teres of coding was to i r.s e r t a r e l a xa t i er. corstar.1 se that as stear beteres more superheated. Its arternal crergy becctes sleser t( the v.. l u e fer nr. finitely dilute water vapor. This is t.. s t important for stability because it limits the t ri ple-valued rat ure of the va p.a r terperataic as a furctior of der.sity, it is aisc irportart for accuracv.

retevir; a s p t. r i e u s p r e s a. u r e depender.ce for superheated 5. t e a t . as well a6 allowiry the irclusser et the -830 kJ/kg crergy differerce that appear *. betweer stear ;. the critica: p a i r. t a r.J irfirstely dilute stear at the critisal t empe ra t ur e .

A fittiry proteJure was applied to the r.e w LOS r e l a t i er.sh i ps to c ompa r e ther with both the steat tables ard the Lo<. Alates SESAME tables.' Agreetert i i.

withir :20c. over the targe of interest and peterally is much better. (ii ve r. t he J. II. Ke e r a r . I' . (i. Keves. P. (i. 11:11. a r.J J. (i. Maare. St ear Tables

~

(Johr. Wiles and Sons, hew Wrk 1978 ).

'Kathleer S. Ilo l i a r . Ed. 'T -I lia rdhook of Ma t e r ia l l*roperties flata lia s c a. .

Vcl. 1- i.q u a t i t r.s of State. Les A l a te a. ha t n er.a l La bora t ory report LA-1oler.'.h t Ac ve r.be r 14b4 7.

3

'lR R.QW i no i

F.erererological l ur.t e r t a i r t i e s ir stearc e xpl os t er.s . these results e u ced the tc. ai red a,s urais. 11 t urt he r i mp r es c r e r t i, te the 1.0% are desired, t h: r,'

s, t -el: es t n e revis..r w, . l J be a better algoritht for the gas prartttr.

k.

heth rear the critic ] poirt ard at high e, tear der.sities ard terperaturci..

T1.e 514til k 'I hea t -t ransf e r probler wi t h t he coriur-wa t e r syst em re la t e s to the low therrai sorduttivits of water. As a consequence of codel developtert based e r. liquid sodiut. the s t a r.da r d S I At11.R- I l code trarafers heat te bu!L liquid. which t h e r. vaporizes. fer water. both e xpe r iter t s and theoretical c or.s i d e ra t i e r s suppert the view that steat-explesier liquid-liquid heat t ra r.s f e r etcurs to the kater surface. Sare heat then is conducted i r.t o t he wa t e r : the ret ircer c t. u s e i, vaparizatior, lepletertatier of this revised concept allows s i mu la t i or. of ste r explesions it water-rich s ys t emi, with tor.equilibriur v;. po r i z a t a c t.

T1.e rodiatict heat tr:.rsfer occurring durir.g the course prerzi xing phase c; a s t e a r: explesser was ra t todeled explicitly. Boundirg cases that ervelop a va r t e t y o f b e a t - t ra ti f e r rod e i, ard ra pitudes were evaluated i r.s t e a d. Modeliry a f i l t-bo i l i r.g flew repite with r diatier and film layer conductier. i r. these chaotu entirenrents is bevend the scope of the Sikb1ER-il formalist at t h i a, time a nd wou ld :.dJ on ly ta r g i na l improvetent to the overall treatter.t by itself. The lack of physics :or f el-strear bres.Lup and wa t e r ir.terreretration of the fuci are probab!v rcre irportant defscier.cies i r. the Sist1LR-11 code. We found that itslusier el kcter subcoolir.p t r.t o an effective heat of vapor:2atset was recessary tc 4tain r e a a.cr.a b l e results free the revised hea t . t ra r.s f e r -r0J e l .

Past esperierse i.uppested that the cetplerettary prosess, putting varer superheat irts the effettive c e r.d e r s a t : O r. crergy. aise would be desirable.

Although certertually straightforward. these charges did reiult n r. screwhat comple> coding redificatteri.

The miscellareous charges i r.ve l ve d pregrattity the liquid-water's t he rr:a l torductivity as a f urt t ier. of temperature. c ha r.g i ry the taximut water-dreplet size algoritht, n r.s e r t i rg ar. add i t t ora l t ite-st er tort tel for the vapor-crergs we: terr. and terrectarg el two mi r.or 10RTR3 c cJ i rg e r ror s. _

The reteusts to treat structural effects car be introduced with the c hi. e r y;. t i o r that erly energetic stear: e x p l o i. t or s are a c o r.c e r r. ir evaluctir; tertairrent failure. These erergetis explestors tecid sause a rupture ir the lowsr vessel's head are . s ich (ou:d be expested te citigate upwardis dirested kireta crergs siprilia r. tis.

4

O 9 ===

l (I l -

VW U id, L i l i r. i t e - e l e r e r.t , a l t u l a t i c r.s ind uated that vessel failure could be expn ed

.. the radius where the cuterrw t vessel penetratiers were .leso i, 4 s i r c-: e r t r t i I split ws e x pe, t ed t . rreteed areurd the head at th:s rad;L- ard

t. leave the i r.r e r pa r t i or. el the head as a free body. A sirr!e.

s iry l e-deg r ee -ef -f re eder sprirp-ross tedel was torrelated to these

!irite-eletert reults. This rodel ac6epts the average pressure loadi r.g ove r the :ailure regier. erd furrishes the tite a r.d dowr.wa rd head velocity wher the head diserpages fret the vessel.

41 tier ei the head septert arte the lower-vessel cavity was treated by a modificatter ei the S i kMfis- 11 plug e .ie . t i er ecJ e l . The failed po r t'i or. of the lowe r head bec eres a iree body whose acceleratier. is computed frot its mass ard the n r.t e r r a t e d forces acting e r. it. A t wo-d ite r.s i ona l representatier. of the (avit\ was used it this studv: the volute available tc the ser. ting caterials a s sured the below core cavi t y w;.s half-filled with water.

C. rat ihratior to NNL 5 tear-fsploster fsrevirer's The selection of SlhMER-Il input paraceters for reacter s t e am-e x p l o s t er.

tralysis (ar best be utsorplished with referer.te to the experitertal data base.

Two difficulties exist. First. any SINNER-I l toJel will be simplistic. The abilit\ to p epapute a recharistic f ra gt.e r.t a t : O r (deteratior1 wave leavirg behird a distributicr ei particle sizes does r.o t extst ir the code. ]r fast. a geretally agreeJ upor. theoretical f ragnertat ier techarist dees r.c t exist fer stear espiesters. Sc 6 e r.d . the reported experitects have rarder aspects. Later experirettal re wits base tended to attrad at earlier e .s pe r ite r t s ard espetzally tertradict early simplist a theories. Alse, the experitertal i r. l o rta t i er is sul f u n er.t iv ambiguous that ar.v unique simulatsen is impossible, lle c a u s e of these preb ers. ere typical experiter.t was selected for temputatietal sirulatior usirg a simple SikNl:R-l l representatier. Details for this (alibration as well us a brief review of SNL stest-explessor data are pasen ir Chap. IV t.rd ApperJices e to T.

The esperirert selected wsa ShL test W-19.' it which' I. l l Lg et iror-alutira t h e rr.i t e su dropped inte 22-3 kg of ambiert temperature water

'l). I:. Niit hell. kl. L. rerradiri. ard W. W. Tarbell. ~1rterreJiote ka:t St ear. 1:x p l o s i . r l'h e r ar e r l.sperirer - ard 4raissis, sendia ho t i e r .. I Lah.tateries report $ 4hM 1 - o l 2-1. h0RIL FR-21-85 ( 1%11 5

RC3H MA:T tortaireJ :r a square Lucite bn. Iritially. a unifert int erat t ier zere if a size reperted bs the experiretta.ists was used. It was reported t. h.. v e a t he rr.i t ei s t ean 'ka $ e r volure r .. t ' el 0,04'o.45/o.45. A twe-phase. equol s.;ec trastier steat/ uter chitrev 'a assumed to exist above the interactter zere.

The better of the bes was rssured to be rigid and the sides ard top to be at c oni t a r.t pressure. These sateral boundary c or.d i t i or s assume the Lucite has to strergth. Alterr.tive beer.da ry conditions are discussed in Apperdix 5.

lie s t results ir th SikNER-Il sinulation were achieved by haviry a water-drorlet size at a fractieral cultiple of the fuel-droplet size. This relattership u s cc t ide red reaserable because the surface ter.sion of water is about ar orde r of tagritude lower thar. that cf irer.-alue r.a thernite or ceriur.

Ilew e v e r , the fit was less that satisfactory, with a spurious high-pressure tail r e <,u l t a rp f rom bu l k h e a t i r.g o f wa t e r i n t he ir.teractior zone.

A i. e c o rd fit t h e r. was nade using a previously calculated fuel / water distributier. based er a three-velocitv-field film boilirg nodel.' lie r e the outlire of the calculated rixirg 7 ore was correlated to the vague pre-esplosior, experitental cerfiguratier. The rode with the peak thermite dersity had a thertite/ teat < water volume fractier el o.27/o.37/o.36. The best fit achieved was with a fuel part ale diateter of 300 st. a w t e r d r c,p l e t diameter of 75 um.

arJ c l i g t. i d - l i q u i d heat-trarsfer rultiplier of 0.2. ( Ne t e that the r.e w liquid-liquid heat-trarsfer rodel was used with the 0. 2 nultiplier. This rultiplier erly uts or heat trarsfer te the water's surface.) A c etpa r i s er.

with the bai.e pressure transducer i<, shown ir. l'ip. 1.

The fira: exercise usiry SI ANER-I l was te perfore sete pre-explosser coarse-rixiry calculations, with thermite and water (iritially s e pa rt. t e d ) .

Although the furstieral f o rni, of some of the phv a. i c s redeled i r. 514NER-I l is less thar desireJ. obt a i r. i r.g a r. a pp r e c i a t i er o f the qualitative errors invelved w a i, desirable. Th e a, e re wits are used ir Sec. 1 -1: ir ruccir.2 bourding cases for the probler el reatter reltdowr. The first calculatier. assuned a unifort 15-nr dreplet diare'er with the corr.i na l ( urmod i f i e d ) S14N1:k-ll heat-transfer teJel.

Steat productier ard the extent of therrite dispersior were urd+restirated.

This calculatter wa s retur. with the heat-trar.sfer modele, of Sec. 1 - 11. 1:xc e s s i ve radial thermite dispersal resulted. ard the interastier zone did rot rea6h the botter of the Lusste certairer ir the required time. 'Th e sorriguratier e i, t a b l i s h e d w i t !. the Isrst preristry sal 6ulatior was espleded with the sare redels atJ parareters used i t. the fit showr in Iag. I te see the elles1 el 6

4 R0i 3H DRAFT 200 , , , , , , , , ,

l l

1 E

<t 100 - -

CD Q -

~

~

klk  % -

0 - - -

a e 1 l 1 I a l 1 0.0 0.00200 0.00400 TIME (s)

Fig.1(a). Comparison of water chamber base pressures in Experiment MD-19.

7 l

- - . . _ - . .- _. .- - . - . - _ _ _ _ . - - . . . - _ - - . - _ - _ - _ . _ _ . - - - _ _ - , _ - - _ - - - . _ - . . - . . . _ . . l-.

.i l J3 :3 - ' . -

A0t' _

bDTH .

- _ _ _ _ _ _ _ _ - - - _ ~ - _

9 1

D

, ' M t

n e

' i m

r e

p x

E n

i 0 s e

i r

2 u s

s

) e r

S p

. ' Ms

( b e

a r

e

. ' Eb m a

Mh I

c r

Tt e a _

0 w _

i I f .

. 1 o n -

o _

_ s

. ' i _

r a _

p m

. ' o C .

)

. ' b

(

1

, ' g i

F

. _ - - - - - - - - ~ - _

0 0 5 0 5 0 2 1 1

^Ogyv( wE3OMWTL C

.  :,i:i;.i .tl l ;i : l l l,. l!' 4i  ! ,ji'

r-ud Jitterert prerisity or. the characteristics of the e x p l o s i e r.. The reuk rressure irsreased t. 2o.F hil'a : there was a lerper tire at high preasure. but t i.e wid*h ei the pulse at b 41Pa wa. alvu- the s a r:e . These results are torsisttr* w:th what would be expectcJ with less local va po r volure fer early exparsier ard therefore pressure redustrer. Although the calibratior exercise was unable to provide a strorgly supported, urique set of calibratior. parameters. it did provide sore cerf Jente that the Slkt.11:R- i l treattent w a s, qualitativelv rea50rable. Also. Se obtained kr.owledge about the choices of parareters that would envelop the real behavior.

A a, a caveat. details f ror other fully i r.s t r uee r.t ed test series ( l'IT S )

experiterts, iuch as those with double explosiors or those producing s ipri f icart !v tere efficient explosiens because of rigid walls. cannot be toJeled metharistically with S I At.1LR-l l . Parametric studies coupled with -

additieral rodel d e ve l o pme r.t are required to provide high c o r.f i d e r.c e that c or.s e qu e r t e s of all possible s t e a rr -e x p l o s i o n phenomena are ir.cluded in arv study.

ii. ('a l i b r a t f o r to :es Alanas Sha l low-Peel 1:ste r ite r's To irvestig.te a basis to .r ud g e the ability of S i tt.11:k - I l to c a l i. u l a t e pest e xpls s acr slup breakup. a s e r i e *. of shallow-pool atteleratier experiterts was corJus ed arJ ther ara lyzed w i t h 514t.11:k - I I . Detail, of theae e x pe r ire t t i, ard their ar.alssis are provided ir Chap. V.

4 shenatit of the test apparatus is showr n r. fig. 2. The depth of the poel and the height of the tree space above the poel are scaled using the attua; reatter vessel's d i rie n s i e r.s ar.d the total a mo u r.t of fuel available. As irdicated in the schematic, a very t h i n ( 0. 006-er ) diaphrage supports the pool i r. a 102 -rt:- i . d . Plexiglas tube. The pool is 50 er deep. ar.d there are 165 ric of free space above the pool. The bottom of the tube is separated fret the

r. i t r e g e r. d r i v e r gas by a 0.1-mm Mylar diaphragm. The e x pe r irre r.t is st a rted by cuttirp the lower diaphragt; the pressure rises ir. the bet tee t ube and rupt ures the thin diaphragt that supports the water. The irsreased pressure t h e r. irJases the motier. of the water that is of i r.t e r e s t . Three sets of data are cellected T. R. llehl. "4 ('a l s u l a t t e r.a l Advarse i r. the AlJelity of l'ael-('vlart Irter tastier.s. Preseedir u <f the L At

. 1. h. K. s.i t e t y Tarisa! W etire t ee r i s a r a rJ 1.u rope a r. hut i c a r Sos t e t i e s . Lves. Irarse. 1 % 2 ,. pp. 111-?$7 .500.

9

l e.-

j i

P.

d b l

V4 d )

[N l l

4d .P. ,

\

M E _u

~

ma g , , , - . . ,

w NN=mr

,,,,, ,n _ _.

e v -

r - ,,.~ ,,

Y l i p. 2. Shallow-pac 1 expe rimental apparat us, durir; the test: t iie preuure histerv .t u s t under the top diaphrage ( l'1 ) . the preuure histerv at the erdplate (P4 . arJ a high-s pe ed revi e. Although sore rardor behavier was obs e rve d . t ile results of these tests were u<.eful.

Three types o f Sitt1f.le- I l c a l c u l a t t o r *, w e r e perforred. simula t ir; bot h real and hypothet:6al experaterts in this peoretrv. These were (1) (alibratier ca*,es relatir; te one experitettal test. (2) hypothetical cases with postulated curvature in the thir Mvlar diaphrage support ing the pool, and (3) calculative coeparisors with other shallow-pool experiterts using the calibrated paraceters.

Tcr the calibration cases, the initial gas pressure of the nitrcger driver wa s o. 56 Mi's , ard the evacuated space started at -5. 6 k i'a . Consequertiv. these tests are said to have a 100: 1 pressure ratio. The high-speed modes i r.J i c a t e d progressive slug breakup as a function of distance. with complete breakthreugh cu urrirp at about the tire of erdplate sepast. A re: sonable simulatier. of the exretirettal preuure trace was ebtairable w i t h SILT 11:k-Il bv us t rg a sr.all drop dioreter. 100 ur, ard corsequertiv maintairirp (lese souplirg betweer the liquid and varer lield . A!*e. the value el c.

y . the vapor volume tractier where the 10 i

't RCUGH ogjg l

~

two-phase to sitt e-phase trar.sitiet occurs, was ad.ius t ed t o 0.026 t o at t erpt te

~

rairtain Ltwe-phas e : p r'e s s t.r e s as lory as pouable as the water ircp s t cJ the s erdriate. The best s otf a is sor w i t h e xpe r ite r.t a ; d;.t a is shown.it l' y. 3.

Two types of hypcthetical situations were e xaci t ed.' l'irst. the inrer or.e-ha lf of the lower wa t e r 's interface area was raised or lowered I t r- to evaluate the e!!ctts of surface shapes or pool breakup. Se c o r.d . or.e of these rotur.ifort situatiers was . staled up by .a l a s- t o r of 10 to evaluate the scalability of the: treatrett. The rotuniform interface resulted in an-instability growth that definitely made sepact appear to be more of a two phase spray, but ' ac t ua l ventirg of the gas through the water was not achieved. A further ca l c u la t ion wa s' performed with half of the dis t ort ed lower interface actually blotted by a rigid disk to asseu an experitertal concept for testirp _

the effects of i r.t e r f a c e distortion. The results suggested that the disk did rot charge the qualitative difference of the pool's behavior s i gr.i fi c a n t l y.

Therefore, a distorted interface experiment was platred. Ir.c rea s i ng the scale f and drivirg preuure t did produce the ' expected ircrease in peak trpast pressure by a : actor of in. bi. t the upscaled case resulted in a nore coherent impact.

The twe recairity calculatiora cade for ' comparison with experiments were

- tai one with . a 60: 1 pre uure ratio ar.d (b) one with a depressed lower pool

-i r.t e r f a c e by a sentrti disk. The calculated sepact preuure for the 50: 1 case, which was obtained with the 'c a l ibra t ed' ' pa rame t e rs . had peaks nare that twice the esperirettal values.

The character of the 60: 1 calculat ior was tuch like that at 100: 1. Although slug breakup in the high-speed movie of the 60: 1 experirert was similar to that for - the 100:1. a sagrificantly more diffuse sepact w s. s sept ied b3 the preuure trace. ~ k's t h a d i s k to lower the ter.ter of the lower' water i r.t e r f a c e by 3 cm. both the experirettal data ard the cair.ulation showed gas breakthrough before impact. Ilowever. now the calculative impact was signifit.artly more: diffuse that that measured.

The followiry conclusions were reached it-this brief 5141ER Il exatiratier of shallow-pool dv'namics.

(a ) The $1411:R-Il simulation of shallow-pool breakup provides gercrally correct global behavior er terms of ditfuse vs slug lepact and overall impulse delivered bv the fluid to the upper closure, it $arret, ard 1 probably reed rot , matsh the very shert -lised preucre spikes r:ea s u r ed in the experieerts.

l 11 l

20.0 -

PRESSURE RESPONSE FOR TEST SP7014 6

16.0 -

t 7

a.

p. 12.0 -

ca Ct: -

D -

i <n Ea' 8.0 -  !

ct:

a.

m 4.0 -

~

\

\(yQ w  %

a c-0.0 -

. . . .-..,... 1 .

O.0 1.0 2.0 3.0 i..~.i..!,.i....,....,

4.0 5.0 6.0 7.0 8.0 O r

TlME (ms) g C Fig.'3(a).

j Comparison of P4 pressures in the shallow pool experiments, >

m M

R:'l3F CliAq

_o

.D

\ -

T l O

-- p.'

o D- -

C -

CD F

-o a Cn

- cd 5e M 7- i E- g

-e o

% - t d e O -

- a h -

V-E .e a:

~

~

_Q  ;

m. _v e .5 Z _ 2 2o O -

pM5 C- -

a .5 V. 9 Cm E - C E v .

=

a.

~Q

- - C\i E Y=' ,.

m 7

o m

" 9 I Z .~ E 1 -

3

-Q

i i i

i , i i

. . . i i i i

. . i o

Q Q Q Q Q Q o

py e

o]

e v o (UdN) 383SS3Bd 13

=

R0l'GH DRAFT x ~

b t b 1 k'i t h - a n iritially ur.iform pool, the use of stall drops ir Slitti:le-l l probably would enggeret e upper-head loadirgs ir the r e a 61. r s.: e.the (er*,e rva t ive i.

( c ) 1:ver with exaggerated liquid /vaper couplirg. the lack of a model ter turbulent rixiry lead- $14til:R-il to overly acc e le ra t e . s j ug breakup in the case where srall tedes are used to simulate a notur.ilort interface ir. this experirent.

(d) Ilecau e the tede size at reactor scale is euch larger than the turbulent mixirg lergth. exclusior. of a turbulert mixity model appears justified.

Also, the r.e r. l i r e a r u alir.g of the S14t11:k-l l liquid / vapor drag relatiership wao a ta,ter c ort ributor to ir. creasing the impact coherence

.ef the upscaled calculation, s

s ieilletause of~ urresolved .scalirg questiets, further aralysis of these esperirects was ,iudged less cost effective that paratet ric calculations irvelviry rea tor teltdowr sequer.ces.

I:. 514t1!'k-ll Ku. +er rase rala;lative Re s u l t i, The targe .1 urcerteirty i r. the e xpec t ed c or.di t iors resulting i r. a stest explosion durity a core teltdowr is large.' If all possible iritial conditions ar.d rodeling utc er t a ir t ie s we re conside red, a comple t e parame t ric st udy of st est e x p l o s ion s ar.d their effects seuld be an enormous effort. The ef fert ir.velved

~

l in even cre Sist!!:R ll reactor calculatier is appreciable. k'ith the limited sccpe of this studv. the litited understar.dir.g of steam explosions that exists, ar.d the limited extent te which this understarding currer.tly has been prograrred into Slktil:R-II . a rassive Farametric study was ret appropriate. Rather, live s c o p i r.g c t. l c u l o t t e r.- were pe r formed with $14til:k-ll to address the upper beurd questier, l>etails of these c a l c ul a t i or.s . as well as some add i t i er.a l results that help te irtegrate these cases i r.t o the c or.t e x t of a meltd7wr u. sider.t

'41 lie rra r . 11. V. Swersor, a r d 4. .l . kickett. "At Uncertainty Study of INK St e a r; l.sple ters.' Sandia hatioral La bor a t e r i e s re po r t $ 4hD53 135 .

NUki:(i/rk 3369 ( 195-1.

l i

14 1

RCt'GH DRAFT I

sequert e . are giver it chsp. VI ard Apperdues U and V. The results are l surra r :2cd n r t hi s sestier.

ihe rtaster s stri.stur.il represertatior. for these tases is simkr nr. l y. ..

r The ir-veuel 6erfiguratier is the sare as ir. the previous Zili study. A reval+1e

. lower strutture has beer added for the lower-head failure model. The ftve

$1411:k-Il tu se s can be suenarized as follows. -

Ir. case 1. we us e d t he s arre Sl*11:k-I l t o r i um a r.d wa t e r g eome t r y a s it c a i, e s 1 and 2 of the 1950 211' study. This was a premixture accluding 2M of the total I toriue below a single-phase colter corium pocl. The cor ium/s t eat /wa t e r volute fractiers were 0.50/0.25/o.25. The fuel and water were assumed to be fragtetted l t o 3no ur r. d i ar g lobu le s within the preeixture. In additior. 35 000 Lg of steel f

j partitles were added between axial code -16 ar.d rode 60 to represent the ca u .

but rot the strength, of the upper core strutture for this case. The new

. S1411:k Il lower head failure. EOS, and heat-transfer models were used.

The results of the calculatier ir.dicated a substantial reductier (compared l

wi t h t he 211' s t udy ) in the likelihood of crerget ic rrissile . product ior. The rew j

rodels elitirated the spurious high interactier-zore. single phase pressa'es I

- previousiv .btained. Lower-head fail.re was calculated to occur at 3. 5 es . ' a r.d

! about f E of the k i r.e t i c energy w a i, di rec t ed dowrwa rd as a result. Upwardly direc ted Linetit crergy at head stract was -650 11.1 Vith the assistar.te of the dowrward vent ing of (cre mat erial 10!!owing lower-head f ailure , the peak. forte er the head was reduced from 2. 6 GN (giganewtons) to 1. 0 (ih. A plet et the ir.tegrated upper-head leadirg is showr ir flg. 5. This head Icadity is rear the i threshold f or t ailure if the head is near the normal operatir.g temperature.

1 Ir same 2, we used rote techanistic but arguably conservative initial corditiens ter a large-scale c ohe r e r.t stear. exploster. Startirg with a torpletely rolter core at 3 100 K. a 1.55 t-diae coriue st ream was allowed te pour into 15 000 kg of water in the lower p l e r un. St .i r.da r d $1411;k Il heat i trarsfer was used with 20-m-dian, pref ragmert ed tor tue gleys. Itased er the resulta, ir. Set. 1 C. these assumptions should urderestseate stent productier ar.d thereby pernit extensive c:ixing. Mixing occurred arourd the edges of the (criut l

s t r e ar:. ard was reasonably extensive because of the large mesh size used .

(greater thar lo cel.

Ar explosser of the mixed ceriue ard water was triggered 0. 7 s after irittation of t he pour. wher. to (H'o kg of t or t ur- had e r t e red the lower p l e rz .

and tortur flew reversal was startsrg as a result of vapor produstier ard pierur-15

~

l

i n . q i ' N ,.] 7diW Q i_L

(\'J kd ! k s l 0 &

l 40 c OUTLET PFING I

I 50 SOLIO STRUCTURE j l

12.54 m i INLET PIPING 40 DOWNCOMER l

i l

30  !

CORE SUPPORT i 20 FORGING  !

l l ', ~13

[FbEPbhIll] iI1 11 MOVASLE STRUCTURE h I N E tr n i i i 11 M 21J f 4 'PH I I I I ll 5.20 m ,3457 , g gr; i i i ii

. 'LT I _'I"L'LI I I I II i ,U .

'14 4r~ f 4 4 4 i I I IJ fl e k19"] III II iITMDIll

• b I I I 11 Ma ' Y ,Ud.1111 11 1' 1 'L '- "4 WL I I I I "  ; OUTLET TO 1 5 10 15 KEYWAY a- 1.613 m -*

• --- 2.5 3 7 m -

l l )

Fig 4. PWR structura! representation for SI W ER 11 steam-explosion calculations. j

12.0 h ['

FORCE ON HEAD 11.0 :

10.0 i .

9.0i 8.0 ~

z 7.0 --

d 6.0i x  :

2 5.0i 4.0 i 3.0i 2.0i 1.0 i 0.0 ~ . . . . i . . . . i . . . i . . . . i . . . . i . . . . i . . . . i . . . . i . . . i . . . . i 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 TlME (s) l i p. 5. Irtegrated upper-head loadiry fer case 1.

p r e s s u r i:..t t er. This tire was t heser because the rixirg corfiguratier appeared te have the rotertial te produ6e re s s eur. head loads. E x a ni r. i r g details of tirirg sersitivits was bevord the scope el this studv. but ar. explosier af t er 1

• o f re i x i r g i s dis 6ussed ir Chap. VI. The ShL calibrated explosior parareters were used w i t i, the rew 51A M R ll rnod e l s . Steel particles were troluded to r e p r e s e r.t the upper irterral structure, ecu t 6 2 '. of the L i r.e t i t erergy was directed desrwcrd, lipwardly directed kiretit er. ergs at head impact was -600 Al l .

The pc.ik utter head for6e was 0.81 UN. as shewr ir lig. 6.

Ir (ase 3. we used worst case tritial corditiors as irf erred f ror a resert Shl s t uds. ' lie re 75: of the sete f94 000 kg) was assured 10 ni x wit h 2n 0 H' kg of w .i t e r . Is lirst the helpht of the preristure to that el the erigtral 6.rlur p00). a IC aritial s t e u r: velure fra6tior was a sured. The $rriur plcha:e si:e was assured te be firels fregrerted to 100 g ir diat e t e r. 4!theuh h t l.e rew w

17

RC 3 CRL-T "6-

• 10 0 l FORCE' ON HEAD 9.0 -i 8.04 7.0 -i

-- 6.0 -i za'  :

x 5.0 -i O .

h- 4.0 -i 3.0i 2.0-i 1.0-i "r" r "i" ' " r ' "r r 'i 0.0- ...r. r.r -

r 0.70 0.71 0.72 0.73 0.'M 0.75 0.76 0.77 0.78 0.79 0.80 0.81 TlME (s) l' i g . c' . Irtegrated upper-head leadtry fer case 2.

Sltt.ill ll reJels were used, ri heat trarafer frer treter. (cri.r te stear was talculated, lititiry stear terreratures durirg the exparsler te 922 r.. h. -

lower head failure was auumed, and the rerair.iry (cre ruternal ard upper structure were assu eJ ret te irhibit e x pa r s i e r.. The upwa rdly dire 6'ted L re t is erergs was 1 63o 41 1 Ilawever, as a corsetiuer.c e of the diffuse nature el the exporsier, the peak upper head force was erly H.75 (;h. as showr in l'ig. 7. This result shows that a (crplex relatiership exist s betweer the esplesier ard its delive red l< ads te the s i s t e rn. Str.ple correlatters are ret relenbte.

Ir case .l . we t e( L the tritial scrditiers I rcr. c a s e 2 bu t herugerized t he A la

.lo oo" kg et 6eriur ard 10 twHi Lg of water ir the lewer pierur.

s rulated nr whish dri plet sites were irtreased at srder :

estl isier' ther wa s r a g r i t t.de ever the SNL ser re la t ed va lue s. This redused the heat trar ter r te l's 2 crders vi ragritude. TI e idea was to strulate at irscherett. r t. i t s p : e 18

n ,

,\ -..

RmvJ s b- 9.0-i FORCE ON HEAD 8.0 2

~

7.0 .

6.0 2 n .

Z -

'5.0 .

ta3 .

t.) -

ca.

4.0 i.

3.0 .

2.0 .

1.0 -

0.0 ' ........ ........., .................. .........i 0.00 0.01 0.02 0.03 0.04 0.05 TIME (s) l' i g . 7. Irtegrated upper-head loadir.g for case 3.

explessor e r s i r ecre r.t that sculd be core r e p r e s e r.t a t i ve of the reacter situatier. It case .t . the lower plenue was r.o t calculated to f ail. Upward!v Jarected kiretis energy at head i rr.pa c t was -760 bl.l . The peak upper-head f orse was 1. 52 Gh. as showr in l'ig. 6. This case had a significartiv rete (eherert upper head irr pa c t than cases 1, 2. er 3. ever though the esplesser was relat ive!v berigr. A g a t r. , the complexity of re la t ir.g e xplos ior r.ogritudes and characteristics to loads en the head is evidert. lle r e the nest he r. i p t espla ier produced the largest cht.llerge to the head. -

Ir case $ we used the case 3 preristure but wit h u riur and wat er therralls equilibrat ed bef ore the e x pa r s i e r.. The 1.o w A l o r.w 51.se11. t a b l e s were used to give a startirg pressure of 900 kil'a . Rel i t t ed pa rare t e r s ter the SI'.f'.il k-ll 111%

were derived to obtair. these aritial terditiers. The terairirg 25' .I the core was f in ed er ter of the p r e r.i s e d regier. IL t h lower-he.d toilure ord stee' 19

R:t& DRAF b-r 16.0 .

FORCE ON HEAD I

14.0 ,

12.0 .

10.0 -

Z w

d 2

8.0i.

O .

t.a .

6.0 .

4.0 .

2.0 .

0.0 '....,.........,....,...,,.........,....,...., ..., ..., .

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 TIME (s)

Iit. 6. Ir.tegrated upper-head loadirg for case 4.

partitles tfer structure 1 were included i r. the calculattor. The rasieur upwardly directed Liretit eterp was 7 250 M.I . The peak upper head f orce was 12.4 O f. . which was obtair.eJ iust before the calculation w e r.1 u r.s t a b l e . The forte plct is shewr ir l i g . 9. This result cor. firmed clear!v that our calculative apprea h would produce the expected diasterous results for the upper l iru t assurptiets ard idealized physics.

l. Certairrett lailure Pr r.ha b i l i t i e s To obtair ar est irat e let the probability of t or.t a i r.re r.t failure f r er e r.

it vesse: stear explessor, giver sere relt. we r.u s t f .i l ludge how l'Lels the statial terditiers ter .i large stale s t e ar; e xples act are. (br e s t i r .. ' - t ).e 20

~~

u" _" ". t .

FCvJ3t 2,Ms

'b

. 16.0 .

FORCE ON HEAD 14.0 -'

~

12.0-10.0-'

z w

d 8.0 -'

O .

Ca .

6.0-

i 4.0 .

2.0 ..

0.0 " .

0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 TlME (s)

Fig. 9. Integrated upper-head loading for case 5.

in-vessel energetics and loadings produced by the spectrum of possible steam explosions, and (c) evaluate the consequences of such loadings. A brief study attempting to make such judgeents is presented in Chap. Yll. Here we summarize some conclusions from that study.

First, although the best estimate probability is judged to be low (betweer.

10~3 -

10" ) , this is simply a guess. We currently lack the technology to l construct a reliable probability density function for core melt consequences that would port ray a best estimate or most probable value. Indeid., performing any "best estimate" mechanistic core meltdown calculation, with models l representing consensus phenomenology, was not possible because consensus phenomenology did not exist. The Sil44ER-Il calculations performed were for addressing the upper bound question.

~

21

q m - -

( n\ -

i Vw \) s Se s or.d . any large-scale steat explessor or sequerte et explosiirs leadir; te a a rtaarrert ch llerpe a ppa r e r t l v tust a rvolu a sustaintd supersriti,a!

pressus.. Al t h.-upi the si.datrv degraded sore rulerakiry pregran i ll* 0k r .. s e r t s the ra x i ru-- te a r : r.g f u l preuure d u r i r.g the e x pa n s i o r. ph.n e of a r. explossor is about half the thertodvratis critical preuure.' steat-explosion experaterts exist with expansior. Thase preuures, greeter than 34 MPa sustained for tore thar 1 s.'" "Thus. cortaittert challenge free large-u ale steat explosiers carrot be ruled out en the basis of litated pressures.

Third, with the dinipatier. cultulated by the SlhNER-Il code l , to obtain sufficient crergy a large-scale stear explosion a ppa r e r.t l y cust involve e f! i t s er.t erergy trarsier frot a c or.s i d e r a b l e iractior of a colten core. 2W or core, i r. a few ters ef rillisecords. This leads to such requireter.ts as f orni r.g a large te!ter s o r t ur. peo:, having a ceherent pour. r.o t obtanting early t r:ggerirg of at esplesion. ar.d lititing st eam generat ion so as to pertit the e x t e r.s i ve pretixing of (oriur and water. All of these c o nd i t i e r.s are utlikels, but rot outside the spectrut of reaaor. For example. tripperirg could be delaved by aaturated water or elevated atbient pressures. or the correnly quoted steady-state l i t id i za t i on a r gute r.t s , which litit tixing. could be defeated by the irertial effects i r.vo l s e d i i. a large corium paur. Ilecause of t he edge-of -

spectrut character of these ph e r.ote r a . a 0.1 probability was judged to be proper upper limit to assoc ia t e wi t h obt a i r.i rg initial corditiers resultiry ir suffitier.tly ererpetic stear expiossors. Further explar.atior as to the tearirg of this prebability value is ir Chap. Vll.

'lauske and Associates. Inc. "Techrital Report 14.14: Kev Phercrerolorical 4'dels fer Assessire lirlosive Stear Gercratier kates" (The 3 rdust ry llegraded Lore Rulemakir.g Prograt t llX OK ) llurr kidge. ]Ilinois 1953).

' Michael F. Sheridar. ar.d Kenr.eth 11. k'ohletz. "Ilvd r ovo l c a r. i st ; liasis ron ,ide ra t nors and Review.' Journal of Volcar. ology and Geothertal Research E.

1-29 (1953).

" I nf e rra t i er. provided by K. ll. k'ob l e t z , Group 15S-1. 1.05 Alamos !.:. t i ora l 1.a bo r a t o r v . f.o vembe r . 1954. -

22 J

o R] UGH DRAFT leurth. obtcarirt the e xt reme condit ions presert ed in tase 5 appea rs to bc sepou able. 11:xirg 75c of .the core requires water to rise threugh a nolter 6ernut p x ! ir . tilt bcilirg repite because ol '. t he receuity to a,tetroJate the vepor volure prodused. Ir. any rearirgful salculatior. attempting te atta:r. this pre-ixed (erfigurat ion. satisfactior of fluidizatict requirements resultirg ir coriue dispersal wauld be auured. The anurpt irr. of . ins tantaneous terperature equilibra t ier. ' ar.d pessicistas LOS parameters are . alse u r.r e a s ora bl e . Case 5 terely demonstrates that if arbitrarv tixity is allowed, n r.d the explosier efficiency is allowed to approach the thermodvratic limit. S!W1FR-Il obtains the expected uracceptable result.

l'i f t h . if the vessel head is to become .o large missile, an i r.t e g ra t e d -

thresheld head loading cf approximately 1 GN i<, required if the head ar.d bolts exist at operating t empe r a t 'ir e . Case 2. r u r. with edge-of-spectrue initial corditacts for a s i r.g l e large explosier. was below this level. Case 1 produced 1 GN. although its postulated pretixing as well as its explosion /extarsior sequert e irvolved edge-of-spectrue considerations. Case -1 achieved more that 1 UN. Ilow e s e r . true tultiple explosters are not calculated. The initial r

conditiera are idealized (fer example, with ur.ifort material distributiors radiallv); dissipat ion of ene rgy ir the upper core structure is neglected (with 900 til of fluid kir. etic e r.e r g y . this could be signi ficant ); and the tire d e pe rd e r.c e of the r.cnuti form uppe r hecd - J oadi r.g sugge s t s formation of multiple risa.iles free the head apex. which could be stopped by the tissile shields. I r.

brief. 'cass '

i+ an edge-of-spectrue e x pa ns i on (altulatior. llecause 1 GN is exce;ded criv with edge-of-spectrue calculations. if the head ard bolts exist at operating temperature. a 0.1 prebability is judged appropriate te assign to ve ue l loadity, of corcert. given initial tenditior.s resulting ir e r.e rge t i c steat ext: osiers.

5:xth. ans a t t empt to establish a simple alpha-mode failure threshold tendition is f t.zzv. At impulse limit is useful erly if the impact i r.vol ve s single phase liquid. Ca <. e 3 shows that high erergies do r.o t necessarily give u r.a t c e p t a b l e leads. if a highly two-phase sprav is the i mpa c t i eg . fl.u i d. The 1 GN. forc e liest is fuzzy because of the irfluerce ' of the spatial leadity

'd i s t r i bu t i er , the tenperature of the ve uel. ard the respor.se of the contaittert t o-  ;. n v tioile. Ilead loadargs deperd on the auurptiers made ir the

. e x pl es i on/ eipa ri, s er calculations. le this studv. the preserse e: stee!

particles tended to (orcentrate loads. while lower-head tailure raJe u pw.. r d i v 23

h*f,N' Wal; 9D;Mr fg I~T directed impact loads less severe. Thus , consistent analysis must be perforced to link initial conditions, mixing characteristics, explosion characteristics, expansiot behavior, head-failure characteristics, and missile dynamics.

G. Conclusions and Recoerenda t ions This study has resulted in significant progress over the ZIP analyses. The SIH4ER-II water EOS and beat-transfer mechanics are more credible. The significance of lower-bead f ailure can be evaluated consistently and shown to result in considerable mitigation for the constrained (by the corium pool) reactor celtdown case. The initial conditions used for a steam explosion can be related to a mixing calculation. Although SI M R-II models can only parametrize the physics of stene explosions, the correlated models now can simulate better the pressure-pulse ticing for water-rich systems. Further, parameters can be chosen to provide a calculative simulation of shallow pool behavior. Using steel particles for the upper structure reduces fluid breakup and dispersal, which were nonconservative in the ZIP analyses.

Because of the edge-of-spectrue character of both the calculative assumptions leading to a large-scale stream explosion and those required to obtain significant head loading, the upper limit for the containment failure probability given core melt is estimated as 10~' if the upper vessel head and bolts exist near -500 'F. This probability must be increased for higher temperatures. The importance of the laigh loadings in case 4, which were obtained free slow heat transfer simulating incoherent explosions, cannot be ignored. There is no method to quantify a true best-estimate probability, but a qualitative judgment suggests that values of 10~3 to 10" might be reasonable.

Possible future research to obtain increased confidence on the stcae explosion issue is discussed in Chap. Vill. Beyond an extension of this study to better quantify the consequences of the calculated upper head loadings, the proposed priorities for additional research on the alpha-mode failure issue are (1) meltdown process modelingt (2) large-scale steam explosion experiments with associated model development for test analysis, test interpretation, and reactor application; and (3) smaller scale experiments to address specific mechanisc:s.

The reasons for this order of priority are as follows:

23a

,h RCUGH DRA:T

1. Urderstardirp reltdcwn behavior would help place the alpha-code of sbrt.irrer* failui< :n its prope r pe r spec t ive ar.rry ot he r sa f et y i s s u e <. .

2. Ibth the s. t t e <f arv cer tur poel before contact with water in the lower plerar ard the tede el certatt de per.d or t he re lt dowr. sequer.c e whic h is ursertair.

3. The s.te of the vessel's i r.t e r na l structures as well as the temperature of the s e s s e l 's head and head bolts depend or. heat trar.sfer durir.g the' tore heut up t re l t dowr i pha se of ar. ac c ider.t .

4. Ilat h Sitt.1Ek-Il salculatier.E and the t heor ie d of Corradir.i . l'a'u s k e .

a rd Theof ar.ous lead te the concluster that large-scale e x pl o s i or.s will possess characteristics that differ free the staller scale IITS tests. These differer.ces should be observable with currert technology ard their experiter.tal c harac t e rizat ion would help to resolve sore irpart art. urce rtart ies.

5. -Ultinateli. increasirg w ier.t:I n Lt.ow l e d g e . rot pe r f erti rp pa race t r i, studies, vill redute urc e rt a a rt y in the vapor-e xplesion field.

l'i ta l l s . the teierable level of residual urcertainty r.c e d s 'te be detertired. Otherwisc. stear-explosier issues will ter.d to retair. oper-er.ded.

'Ths- is showr when the calculations for test @-19 a re compared with case 2 of Sec. VI. lurther discussier. is ir. Chap. VIII er. research priorities.

• Norta r A. I:va r s . "The Effect of Core Alelt-roolart I r.t e ra c t i ers or. Seve re Ac c i d e r.t Risks i r. Light Water Reactors." Proc. I r t e rr.a t i ona l Meetire or Licht Water Reactor Severe Accidert Ivaluatior. ( Are r a sa r. Nuclear Soc s e t v. La Grarge l'a r ) . IIIinios. pp. 16.1-1 to 15.1-b. I19531.

24

9 " " "

liw J Ji 9}

J in i

11. Slitii'K-ll 4)lill !" ATlohs 10K Till 4 LTi r.-s OKI l ooL4hT lhTERTTina l'Rt xik el A. ItttoJusti r h $ 1 st11 K- 1 I i, . J e w.: s s. s e d ier ti. rurtr:sa a. l s u lo t n ors pe rf erred ir t h i +. studs. The sist11 k asrorst stards ter a S.

r Irp l i c i t J,,a 1 l t i l i e l d .

kla t i i. ce por e r.t . luterior pesritisality cede. The toJe was designed to calculate the ruitiph se estier ei liquid metal fast breeder reactor ( D1Filk ) core r terials d u r i r r .: 6ere di.ruptive a u ide r t . The c a pa b i l i t i c a, of Sist1ER-ll to 1.a r.d i e 5tructure. rapiJ heut trar ter, ard r.or-equi l ibr iue va por i za t ior suggea.1 that the code ceuld be u s.c f u l for calculating the coriequerces of steat e x p l e s i e r. it water reactor degraded core acciderts. The first modelity of stean e xpl os t er.s with S iht11'k- l l was reported i r. the Los Alaces ar.alyses surpartirg the US hu lear Regula t ory Comi s sior. Zier./Indiar Poir.t Project, the

/IP study. The currert reference SI At1ER-l l code. Versior 10 has or.ly tiner charges frer the $l ttil.R- I l code used i r. the ZIP study ir areas relevant to a r.a l yz i n g i,teur expiosiera. Problems observed in the ZIP study suggested rodificatters. lirst, ir the ZIP i,tudv. the EOS had a spurious deperdence er pressure for a.urerheated s t e a r- rear the crit ual p r e i. 5, u r e . A triple-valued turctier hJ te he irverted. Also, sore steat pressures were observed to be high (by a i r. s t a lastor et 2 at i, et e terperatures a rJ d e n s i t i e s. ) . whereas the irterral crergy of superheated steat was gererally too low tachievirg er.ly V of i t a. requi red va lue at 3 om L t. Secord. heat t r a r.s f e r to liquid water wa .

through a bulk heatirg p r e 6 e i. that was int oepa t ibl e w i t h wa t e r s low therral corductivity. This recharisr ceuld r.c t be used te c a s. i l y ieulate a tear explesier ir ars experirer.t lly ob erved water-rich syster. Third. nedificatior cf the s e r i ur., ... t e r heat-transfer procesa, te ebtair. surface vaporizatier requ; red a sharge to the va po r i za t i er./c orde r sa t i er. rwdel. with the liquid s ubo m l i r.g artiuJed ir ar effective heat of vaperizatier to avoid spurious (oelirg el the huik water. l'ou r t h , the high pre a.ure observed ir any stear e xplosier ,uf f u s er.t iv ere rge t ic to cause cortainnent failure suggest n r.c l ud i ng a lower-heaJ lailure urd motion model. Ir p l eree r t a t i er of such structure codit16atiers are giver ir. Chapter 111. .

The above preblers led to the rodificatiora, described in this cha pt e r.

Thest irfreverent- are supplererted by sore ristellarecus charges developed as the wi rk progreased arJ are des ribed ir se6 l. oserall. the redificatier.s de sorrest rea r v previeus prchlers ard allow a better a.a.essrett to be made of the questier of upper beurd energetiss. Ilewever. the phs u s irvolved ir atear 25

, _ = .

s RCU3H DRMT expjessors is still arterpletelv urJerstood. ard. ever with the ird nated r od i : n .. t i r s . a best estirate talculatio  := r* pauible usirg the S ist.11:k- l l soJs.

i Ir charter 11. a suttarv el t he - tod i t n ed equatnors for the aralvtit equa t s or. of state I AIDS ). the coriui kater heat trarsfer. ard the vaporizatiorn or.dersatser rodei are giver nr. Set: i er !!. A detailed descript n r et the AI.os red:l ua iors as well as a c erpa r i ser wi t h s t eat-tabl e da t a ar.d t he 1.o s Alar.os SI.Wil library are giver in Sectier. C. Details of the heat-t rar.st e r todificatiors are ir. Sectier 1). and Sec. 1: illustrates the irrpletert a t ier. of the

. va por iza t ier n or de r sa.t s cr. tode l c ha r.ge s.

Ir. t he AIDS a rea , preuures car. be calu. lated to withir :2M of steat-table er SI:Wil: 1:05 values over the rar.ge of ir. tere t for water /stean and are usually ruch better. This agreetert is considered acceptable because the amour.t of rolter ceriue available. its heat cortert. ard the degree t o whi c h rr.i x i ng ha s occurred are core . urt e r t a i t. _ The calculation of pressure lor high-temperature liquid water results ir larger discreparties. but with steam productier fror rorequilibriur heatirg ef liquid water, this repite ger.erally is avoided. The improved A105 algerathr has three coupled fitting parameters i mpo r t a r.t ter obtairity rear s rit isal ard supert rit ical stear pressures. This study suggests ere set t h.. t is a qualitative sorprecise betweer accuracy and stability.

The reviseJ heat-trar.s:e! .jp orithe simply todifies the liquid-liquid heat-trars:er e spre u ior so that erergy is trar.sferred to the water'5, surtase.

presured te eust at the terperature of sat ura t ed wa t e r. Bulk water heatir; i, row based orly er the ditlererte between the liquid-water terperature arJ the saturati r terperature. The energs diu reparcy betweer. heat trarsferred te ard recoved Iror: the w:.ter/steat i r.t e r l a t e' va po r i z e s wa t e r ( t he wa t e r subcool a r.g i s ircluded withir. the el:ective :at of vaporizatior).

The p r i r:.. r v t ha rg e s to the va por iza t ier./conder.sa t ier. nede l we re to recove durity vapori:atier liquid from the liqun! lield at its true erergy, ret the tor.Jersate crer;v. ar.d te recove during cordersat ier. vapor f rom the vapor field at the ma u -average vapor energy. not at the saturatnor value. The Londersatnor todificatior was rot truly required for this program. but it corrects note lorg-starJ:rg 51kt.11:k-I l problers. ar.d irtroJuses deairable syrectry to the 1ortulatnor.

26

r ROUGH EPAFT

. TI.e - r.i st e l lar eous corre6tiera irsoise sbarres i r. the liquid thereal

.. .:u. i n i t o t er wat e r, t i:s v.. t e r -d r o p l e t -::r. heu-da ry . cell pressure *. .:

t i r st e p s er t r. ' b ed .r

• hi ~ w .ri ter- sr th: sap.r-erergs equatier. ard a tirirur t er a r -irt e rra l l os pa rare t e r.

The AIDS ecJ ficatner.- stard alore. The vaperizatnor/cordersatior charges rela:re the AIDS redi f it a t iers. IL.t h t he s e t o r r e t t_ n e n. sets are-suitable to be i r i o r pi r a t e d ' i r.' , Ve r s i c r. ' 1" o f Si ht11:R- I I . This is particularly true if future

. Sl%til:K I l s t ud i e s are to consider rear tritical pressures. for exatple -22 MPa 1.r water. The heat-trarsler modifications require the vaporizattor/cordersatser charges, and are limited to the assueption that water

- liquid crergv'corporert 3. The eistellareous correctior.s are a rde re r. der t _ of the other rodifuatiors and car be inserted int o t e rs ier.10 of Simil:K-Il also, although percralizatter. bevord the water data base needs to be perforced.

The ca i r. rurer nal dif ficulty ercountered was _the triple-valued r.ature of two of the implant functiers. Although the improvecerts sigr.ificantly reduced the severity of these preblers relative to Verisor 10 of SlW1ER-II. ar. obvious modificatior that would ' e l i ti r.a t e these problems was r.o t a ppa r e r.t . Dr.e dili n ultv- ir.volves the calculation of the va por . tecperature ' i r. the AEOS u r r e s t i ar. . The secer.J cotes from the expressior. for the var.ri: tior/, order.satier. rate. The cethod of treattert in Versior 10 of Sl\tti:k-ll is awkward. iref:itiert, and . inef fect ive, because the true source of the p r 41 e r., was ret reu grized. Ir these correctiors a bi sect ior. _ i t e rat ier.

prosedcre was ' developed t .- stop the Newtor-Raphson algorithm frot rovirp away tror a dor. air where a s e l .: t i e r. r u s t occur.

A- at initial example problen. cre of the previous Z11' study SNL

- e xpe rir ett a l 3 irt.la t s er was run. Calculat ed pre ssures rear the tark wall are

- shown n r. l i g. lo. Although ur. exercise ir fit t irg was tor.cluJed te be required.

ti:e tode d:J esesute ard produse plausible a r.ss e r s (giver the i r.p u t ) with r.o irdia.1ser. ef trouble.

d 27

i i

)$  ;

30 , , ,

g l

- - i

~ .

e 20 - . -

e 2

we W - -

E D -

M M

g - -

10 - -

' I ' '

0 '

O.000 0.002 0.004 0.006 TIME (s) 4 l i; . l a '. %rrie prel.'er results.

28 4

S -u-RVh3 [

~

!! . Su a v . ! M ditied louatiors he s . s r ths Jet ileJ dissus i - 1 ..Jc r d i l ; J a t ; . r a. i r. Sestiers 0 1:

arJ 1. are rather ext er ise. a refererse s ur.na r y of the basic equatiers modified has beer insluded. Tin s settsor also : lustrates the level of sophistication of the codiry sharges rade ler this program. 1.x c e p t where i r.d i c a t e d . the tertirelegv is that of the SIAt11.K-ll ra rua l. lor corvenience. Apperdices A. f.

and 1 preside the Sl5NEk-ll. Versior IP t r e a t te r.t s for the ALOS. liquid-liquid heat t r a r.s i e r . a r.d va po r ::a t i or./corde rsa t i er.. These .ipperdices also cortair de!aritiers f or t he t e rr s talogy.

The three se pa ra t e a reas addressed he re a re (li Analvtic Equatter of State (2) Li qu id- t o-Wa t e r llea t -Tra rs f e r ar.d St eam l>roduc t ion 131 Vaparizatior/forJersatior M del Madificatior.

1. Al'os M'Jificatier . l'i r s t , the vaper'i, heat capacity for infiritely dilute stear wa* c.ude terpe rat ure deperder t. A lir. car rela t iership seered sat isfac torv iror the data available and the terperature range required. The expressior, used is c ygq = ay + hq fT; - 273.16I g (1) where ay = ar input parameter (1 346 .1/ky-K for water 1. and bq = . r. 3rpet pa race t e r ( 0. 3302 J /kg-K; f or wa t e r ).

Secord, the rerlinear increase ir the vaper's heat ca pa c i t y as vapor becores rore .uperhe ted was modeled. A re la xa t ier f orrrula bayed on the terperature Jaftererse trot the saturation line was u ed. The modified heat L . 11. Lusk. " $ 1'NI.1:- 1 I : A rerputer l> rop r or 1or LMFilR Disrupted Cere Aralssis." L, s Alar.ss Tsa t i er.a ! Laboratcry repert LA-7515M. Rev. 2 ( Se pt er he r th. TWklE rk-oa$.i.

29

F q S ---

\u\ .

\

J J L saraiits. s y,,,q. ws* t !.c r uhststuted .rto the inf licit $1\f.ti:R- l l e x p r e s s i e r f o r 1he s .. ; t per..tarc. Ti.t c s p e s s i. r- .i r s

'ti ~ -

'r 'ti . 4 l' = _

-T;=o g (2).

g 'r ' ui.M where e(;, y = e y 7, q - c (.g , y T s3g,y F" ' '"

hj.,y + et q,y - cyty T y ,j,,y - eG.M.cid - 5M-

' ( G . M * ' v(i . M + -

,1(>. -1 Sat.M +

~ '

JM T 3ag,q

'(i M. e l J " "Va p . M - ' vG y Jq = a r i r. p u ' p..! reter t'N' k ler water), and sq = an irput parareter (-5 x l'i' .1/ k g f o r wa t e r ).

Third. at high stear deruties a r.d supercritical temperatures. the pas parareter, kq . used to evaluate the i. teat pressure. increases sipriticantly over the value let trfir.itely dilute steam. A s ir ple f o rr.u l a i r.t r e a s i r.y Ry under these c e rd i t i e r.s ws added. The expressiers (erbirirp this additier. with the sharpes in Ky r. car the c r i t ica l t ertpe ra t ure ( a l ready i r. Ve r s i er. 10 o f SikNI:K- l l '

result ir 1r Ky - pq 1r ktr*M*'I ~ EM ' I' N:4 ' I

• C'M '

30

Ql Iq -..

W UI h \f j ter i g ,7 T;) Is'ter. T(;)g rt g

1r K.,3 = Eq 1r t y+(1 - py l 1r ',y R , i l + l yc.q '

I c r ( J , T(; ),.(

g 4 (;.. Tg ; < t c(;r T(; )g r - I33 wherc

1. (ir Tg ; - ( c(;r T(; i e tM i; pr 1(. i - ( 4-('.. T('. >e e(r* '-

c,q = ar irput p rareter ( 'O. 5 f or wa t e r 1 dy E q " T(, - T( r t . M + a .1.- " Iti > Itrt.4 sq = 1 fc r T(; E T rt.4 g ag = ar irput pa rate t e r (-200 K used li r wa t e r ).

i l

l l

31 l

L 5 4 a

~

OUGH DR/E I's r * . 4 N -

a r t .M "

• 4 T.4 i 1i t . '.1

( * .n T(; 6 g = the produc t of the vapor _ dersity ard temperature at

.T% t . 4 = 0. % TCrt.4 -

E:M * ' v(iq . i IIM~II - C v(i4. i . = i n p u t value of heat capacity. and f y = notir.al_ lit tot kg whe r T(; <Trrt .M -

,, fourth, f or c ons i s t er.c v. these charges require a-corsistent todificatier of the codirp defirirg -the iritral t er pe ra t ure for sirgle-phase cells. The appropriate impin.st foreula is h'.1;T - 1 -

-- i v M ' vti4 T % t .M T M

• I (i =

N* 1;l -T=0 l' . ( -t i

$ Fv4 ' v(;M ' kM

_ r = 1_

where pyy = liquid sapor pressure at Tgg ,y ( fixed ).

TLt.M'= TLr. - superheat (1 i xed ).

but q.(;q a r d kg de perd or T(;.

J2. Ile.i t -T ri.r r e r M. J i l i t a t i on s. The tortal toJet for liquid-liquid heat trarster betweer water ced arv other liquid cerperert is turr.ed off d erever the water is teider that the irdicated -liquid compotents, and the

' va p. r i:a t t or /s or.de :sati r rvdel is used. Ileat t r a r.s f e r frot water to celder liqbids- is.still litsted by the ti e rra ! t erdut t :vit y et water.

32

4

=

a ROUGHDRAE v

[ lleat t rcr '.t he Lli

• t e r 'l igt.id s cepc r.e rt .is ther used to vaporize team fror

- thsi.. e- art.ar . c. d g -

i 15 kLe '

Le '6L3

' ' pc + r g 3 ' .

' ' Pr- * 'P3 r; (T Sat.3 - Ttn>

41Lrti3 1 "I t ,' b r pr. ( 1 - c 5 r pc r p.,-

1 .

. (5)

L Lr I qllLdi3 I *CE IIlr ' v k .s/ t . 15c c3 'Ln) (Tht .3 - Tte). .

rax pc t.here r t

--r p 3 il r 3- p < rg .

rg =r il r pe r3p p7.

' cy g. is the heat capacity el. liquid er.ergy corporer.t t. and ktg. - is-the therral corductivity of liquid'erergy comper.ent t, ard

'C g3 is the input heat-trar.sfer ruitsplier.

. This r. e :. r s . L'q . (5) ^de f i r.e s a r. effective heat -t ransf e r coefficient for- hes.t

-trar.sfer-fret wa t e r va por t o the hotter l i g t. i d s . This c oe f f i c i e r.t . ligg;3. c;.n 4 . be '. ir.l e r red t rer ygitg;3 = lit , g;3 (Tgt.3 - TLr. i . The special cases are li,3g3 g = ,

- 0: - llLdi3 = o i t TL-., < Tg. ,: a r.d Illd;; = 0 if sirgie-phase.,

4

3. \arort atter'fordersatior Odifitattor . The* revised e qua t i er.s for th

s l taJe l r.ew a re i

1 4-

Va rat -1

re rev ' l'aua t ti.r

^I n+1 r -

1 T-1 e("; . cendensation eO 1 , 7.b ,b , 33 r+ r.

3 g.w + 1 ,

O. - -

m GLt -

m=1 e,.3rgn,y + hy j jg,y . Vaporization M1;T-1 2 i ,- - a+1 -

a.1 - --

j. + .t _ -xn qyg;nte + _ q j g;eg

• t6) ,

i- t=1 L-1 l=

M1;T-1 bflI  !

{. r -r+1 -

+

j- +

[. [t1 -x iI ty e g. - a' [ [I l - x3) g"jd.3, j )

p r=3 r=i  :

1 2

i t

}- 33 v- +ww,.ve-.,w,myyrw- .mmw vm grrr +-~W-- * **Wrv- vy=,-'y- -P- - - -

ROUGH DRAF-l l 'im tJ -l'r e re s I c. - -

I t .,r.l. r s.i t t e r

,r 1

7) , ef.-"* r .1 .

' Lr ' L- ^,'r r,t,Lr 3

• Lr ' Lr ,.3 } (71

'Lr . va per i za t s er.

r - r. + 1 -

+ at

'r Hilt-(ir ' (I ~ *r' L: ' Lre.

• O' 411Lbi3 m = 1.2.3.4 Vapnt-rertiret s l ous t : ,r .

r+1 r fa1AT-1 TNAT-1 ,.

[

~

E;g = !(; - at xt. T(";*L) ^ -

II - *r I tr.

• IbI r;= 1 r:= 1 L i ou id-for * : r u '

• v i:au.: t icrs r*1 r ~ r i

t7

=l t7 + at x y rf,[I.-(1 - x.il 7 t n=1.2.3.4 (9)

Stru6 tere-frer;v lou.it:ers TNAT-1

1. • I Z

gt eg = !st e( + $t=1 [ q}^gf(;7 L = 1.2 (101 A10%

T31AT

=+1 "

x+1 a+1 I '

N.n + 1 ' L p .\1 * ' v(ist a r.d

't, -

1 T(x ; + 1 - Sa T"+t.M -

rc= 1 (11) x+1 _

"+1 - , 1 -

ru  % k.+1 u T(=

, + 1 en "ri Tx.i g , , g9 _

L e-r-!rer;v Ilea'-Tr.,r fer <oe 'it ierts

+1 1

- r <j (,r ty 6- pt;

= a r. ,.

li=g, t7

• e xp -r}(;f.Lr Ep(i'Il ,Lr t

- I 34

RCl3F DRAr

'12 i

~

.-1 .

!M ':i-II.o -%.1 =

• 'l ~I.i lM 'pt. II uM

~

• • l I I 1"S t . A1 - T"* 6 l

=

whe r e (IllitLr ll"Lr O

'l tr4it " IIL (in . T[I ,y - Tt;

=+1 1 4 ] tir. 5L " II=+%

G *Tn.1S t.M -gT=+1 x+1 .

- r.+ 1 ti] 3L t;r. = Il (;t s : T=+a 3 ,1 ,g - T3g X"g*

s I - Mau 1r.atier. of r.aternal M at iteratict x+1.

x e -1 is the rer.ir.al case.

>. n -o: alI e: erergy c er pe r.e r.t t is vap:4rizirp and the yllL <i3 a re giver by the heat-trarsfer todiritatier fortalist: the retair.iry teres are de: ired by the S I AN1:k - l l ta r.ua l .

AdJ i r.; the va p. r . ]> quid. a r.J structure energies. 1. q . (6). 1.q . (7) ard I:q. I lo J. arJ .ctttry h('ll~ 2 yr+1 ,br+1 ,

b

-r o ,6, r

pr+1 ,r+1 Lr Lt

, yrLt, r.Lt . - 7 Sk g,r+1 ,e r ,,g SA SL

'r=1 k=1 l er cor.s e r va t ier of crergy. we f i r.d ( f or t he r.ortral case where the x e a r'e 1) 2

+1 n+1 x+1 + =+1 4=10tte + 'il LtG ' - I'll SL(it - 'll b t % '

• I3 6c3 f(";[r " -

~

(138

<p.M.ei,  ;

1 I

l l

l 35 l

RC J3H ClAr frLI:

x-1 where 13= _

(IllLif i3 L-1 l 3 is t iit I.r.'r e s i t v de 11

^I "'

e (",- - c ( .i r. . M li ' r d e r s i r t' arJ hx.1

j. y,,,r3 = ' . .

n-1 +1 *1 f aporizing i i r . M . }, .,; . 4 - e t"r

• M 36

F RC2 C3tg

7. loscuu ier'sf the AFos M,difi6atnors The Sist11:R-i l AI.OS ard its usage is desc r ibed in the Slkt1EK-l l r:a r ua ! .

lu v. 2 in 1.qs. 5111-31-71). l ys. (IV-2-331 I:q s.

41V- 35-4.'s. a rd i r Arre rd i x .in Tia s rateria' is i r.t i t.J e d er Aprerdix A- i .this re; rt for relercitc. Thl -

• ,e c t ior de sc r i be s the details of how the Siht1Ek-II ' Al o$ wa s, cod i'f i e d f o r this stear-explosi..r study.

! .1. Keview ; lii! t ic u !t i e s ir the Versior In SIW.iER-II Forculation. The basit Al.Os prehlet s i, deterrir.irg of vapor states for rear-critical ar.d above-critical pre uures. Irot App. A. the "F- function. I:q. (A-74). whose zero defir.es the s a p, r terperature, is triple-valued ir t h i s, r e g i or.. This is a c or.i.e q u e r.t e o f the pressure depender.cc it hfg,y. the heat of vaporizatier.. which

~

ir the Aros forculatier is a c orpor e r.t of all vapor i. t a t e s . The rarid decreaie of hj g,q rear the c ritical preuure car lead to three pos,s ibl e va lues of T. fcr g ore value of eg; unleu the forralist is tedified. At example of f b a f ur.c t ier of T;-se, giver. it fig. 11.

g lie t e rti r.a t i on of a s o l ut i er. with a pure hewter-kaphsor procedure inretires will tot occur.

13esides the p ri.b l e ni, with pressure d e pe r.d e r t e wher Ts3,- T Crt. the vaper energy shown ir I i g. 11 is a r. eta l ou s i v l ow. The SESAME tables gave a

?

r.ca r -c r i t i c a l preuure of 24 MI'a for stear at tc; - 15.5 kg/t a r.d T(; = 3100 L but the required vapor crergy was 10. 0 M.1/kg . whe r tornalized to the Sikt1ER-l l 2ero e r.e r g y relertr,e potrt. Siht!ER-II only required 64C cf this vapor er.ergy level.

Arother vap r state problet is that the gas "constart." R y. which gives, the partial preuere of a corporert Irot pr " #(i- N T(;.M d e pe r.d '. or pressure ard t er pe rature, but the preuure deper.derce is or.lv effe<tive when a < pCrt.M. and terperature dependence er.tv escurs wh e r T ; > TCrt.M. This fore ha s ir.sufficiert g

flexibility to represent a ll va por stat es.

De f i r. i t i o r. 01 liquid state ali,o invelves sore problers, lirst, there is a r. ir.cor.sistencv betweer what S14t1ER-Il regards a i, a micrescopit liquid der. sits ard the equilibrium vapor der.sity prodused hv the vaporizatior.%ordensaticr codel wherever TL>TCrt. le this r e g i or. the mistostepi. liquid dersity is a corstart.

4(rt.L. hut the equilibriur va pe r dersity will follew a pseudosaturatnor Itre esterdtry hetord the triti6a1 cordstsor.. A secord probler is the irappropriate derer.Jer e el preuures i r. the liquid regime er d e ri. i t i e s t hrot.ph t he use of a sorstart .orpreuibilitt for each caterial taken from the irput scrit velocities. Third. there are trassurasie- ir the liquid's i r.t e r ra l 37

I v 30 , , , , ,  ;  ;  ;

8 25 -

Po= 15.5 kg/m _

og=6.425 x 10e J/kg 20 - -

15 -

Q 10 - -

w (L

5 -

0 -

-5 -

WATER VAPOR -

ea-e*

F: -To _

-10 -

l I I I I I I

-15 3010 3030 3050 3070 3090 3110 3130 3150 VAPOR TEMPERATURE (K)

I ig. 11. Tr i ple-va lued "1" f urc t ier f or wa t e r va per.

3e

L m tq m --

RIVdd-i -

i ererpies due te the rurerical silution w here, fourth the liqu:d st.*e tor-ula' o- is intinatelv tied te the hs d r ed vrar.: t 5 f err:a 1 n a,t. Ext ra6 t i. r .u r e r l .a ere r t hv a better presedare usaid invelve widespread charges ir rethods, l i r.a l l y , the SikNER-Il AI315 rega rds all material corporents as c er.p l e t e l s intinible. For cost liquid retal t a *, t breeder reactor ( 131Filk , applia. tiers this appears adequate. but ser e preblems car arise ir e xpe r a te r.t a l situlatiors wher. this cav net be true.

2. Exariratior of Proposed 531utiors to AEOS Problers. Formulating a r.d writirg a rew, therredyr.atically c o r.s i s t e nt EOS for SikNER-ll is beyor.d the scope el this pregrar. Direct use of the SESAME EOS is also not feasible. The recci,sary syrbiosis betweer the SikNER-Il use of a sudden cor. version froc a twc-phase state te a s i r.g l e - p ha i, e liquid state based o r. a particular void tractner. o p. and the SESO1E tables has not been developed. The need for a T

Sat >T rrt w uld have t be elitir.ated frot the SIhMER-II code because such artificial quantities sculd ret be available from SESetE. The Van der Waals loc ps ir. the SESO1E re pre se nta t ion cean t ha t with a s i r.g l e ca t e r i a l t h e d e n s i t y can be triple-valued as a fur.ctier cf pressure. Thus there would be regires where ep/ccG is small. caussrg the present pressure iteratior to have difficulties, finally, pieces of the AEOS exist :n cary places in the cedirg.

keteval of such scplied AEOS as sumpt ions is a sigraficant task at the presert state of cede naturity.

The litited scere of this prograr percitted only tirical cediry nodif aatiert 13e c a u s e of the characteriat us of the s t e at-e x p l es i er. pr<blet.

c o r ' e c t i r.g the vapor energy away frot the sa t u ra t i er. line was iudged to be the test itys brt rodificatier. The procedt.re was to defire at effective variable vaper heat capacity that will relax the vapor's internal energy to the infinitely dilute value away frot the saturatier lire. The relaxation rate depends or. Tg-TSat. r.o t the absolute value of the pressure. This algerithe corrects the triple-valued nature of the 'f" f unc t ier. shown in Fig. 11 at high values of Tg and reduces the triple-valued extent of 'T' when both Tg and TSat are near TCrt. The pas " constant" calculation was revised by adding a r.o t he r tere at high pressure. This allows R te y i r.c r e a s e above cv GM U t - 1) at high stear densities and pressures, as it should. Icr sir.gle-phase cells, the de rition of the vapor terperature was sharged te be c o n s i s t e r.t wit! tre variable heat capacitv. This pert:1s the s i rg l e- pha a.e e t wo- pha se t r a r a, i t t e r t.

39

n 'N 99 a=

RJ.O- t s.1 he per:orred cersistertiv. The other problets anotiated with defit:rg the liqu .  ?.te listed ir Se, were

(-1 irr. red. Wher the new heat-trars: tr a lpr : hr is used, stry:e-phase inquid u tes with Tt > T( r t should geretalls be avoided.

3. The Va tw r lle a t -Ca ra c i t y Madificatier. The f i r e, t and most s i g n i f i c a r.t tod i f i c a t t er. wa s a c ha r.g e it the vaper's heat capacity. An ir;itial correctier was ferrulated by exatir.ir.g the value of T.

I' T6 c yc' =

./ T=T cygdT / 'l T=TMe1t dT; (14:

Me1t where c, y is the vaper's heat capacity at low pressure frot the stest tables.

The averaged cg y is plotted as a functior. of T g i r. Fig. 12. Th e . l i r.e a r relattership abave T Crt suggested that t

y(;q = ag + by (Tg - 273.16) (15) would be s u f f i c i e r.t for S14NER-11. For water ay = 1346 J/(kg K). and by =

0.3302 J/(kg E J.

Ar.c t he r part of the energy discrerency was a consequer.ce of the low e r.e r g i e s of saturated vapor at near critical pressures. Awa y from the saturation lir.e these eterpies increase rapidly. To simulate this effect ard te recove the pressure dependerce as vapor becotes increasir. gly superheated. the SlhMER-Il f errula f o r t he va po r 's arternal energy was examined. This formula is

'G " *m I'Cer.M + hj.,y - povg i + c ygq( Tg-That.M 1 . (16) t The goal was te have eg agree with stear-table values at least in the infinitelv dilute lirit. where T5at.M = 0 Ir. this litit e cg n , y = e li @ ~_ b LM M. T .t.M-h fp,y = h j ,q. and pavy = o. Because. in ger.eral, eggq,y at ySqT Melt.M + h3g,q.

ir. the infir.itely dilute l i r.i t

'G * *e (CvSM ~ 's Ly 1T y ,;;,q - h, ,y+h) y+c ygqTg .

. (171 e

40

1.8

'?n"!',H 1 Pact i v () 3 g , 7, , .

1.7 -

l ImI l

0 p

6

% STEAM TABLE VALUES iX 1.6 -

(SEE Eq.14) -

H W

Z 9

l E

o W

a.

1.5 - -

. /

/ .

' /

EXTRAPOLATED LINEAR

/ RELATIONSHIP 1.4  ! -

/ --

/  !  !  !

o 400 soo 1200 TEMPERATURE (*C) l'i g . 12. Teeperature-averaged vapor he.:t capacities f o r wi., t e r 41

F RN JU J bl $ I 1 r l'q. r17! the values of tysq ard hfg,y are i r r e l e va r.t because et,q,y res- he

.dded ter terr l:2a'avr te 6erpere with the stear tables. The value et t ygy ,-

4..

liver bs I;q. ( If l. A least-squ res 1st t. h j,,

q= h'.,y j

f1 - TSat.M'Tirt.M was perferned. givir.g hjp ,q = 3. 09 M.3 / k g . The values of Ty ,j,,y = 273 K are cy ty = 1217 J/Lp*Ki. the beat capacity of water at low temperatures. do ret allow for euch adjusttert. Te fit the s t e ar. tables. a material-dependert c o r.s t a r t was added to Ly. (17). This rear.s the i r.f i r. i t e l y dilute vapor e r.e r g v for material M. eg,,,g. is defined by

'G.=.M " v5M ~ 'vly1TMelt.M + hfg,y + h). g, y + cy gq T g - ly (16) where 39 is a r. input paracete r with a repat ive sipr. so that i r.pu t is positive.

To get eg,,,g te ag ree with the steam tables at 1573 K and low pressure. 39=

0.05 MJ/Lp.

A todified heat capacity for vapor. c(,gy . is now defined to force eg,y to eg,,,y for all vapor conditier.s f a r f rom the sa turat ier. lire of cor.porert M. Hv defirirg

'Va p . M " 'Co r. . M + hg j , y - pavy (14i the requirerert car. be stated in cathetatical teres as

'G,M " II - I MI 'Va p . M + ' vGMfTg-TSat.MII + I'M'G.=.M (20)

' Va p , M + ' vGM' Tg-T33;,y) ,

where f'y = 0 wher TG=Tsat.M ""d I'M 4 1 as Tg**.

Eq. (20) can be selved t er (ygy to obtair.

U.=.M ~ 'Vap.M I - 'vGM(Tg-T5at.M'

'vGM " 'v0M + I.M' (T6. - T Sat.M 1

( 1' One simple fore for fg is a ra t i er.a l f ur.c t i or. Therefere. f'g g was defired bv Tg-That.4 f ( .3 , 8 q=Tg-TSat.M

• dM 42

N k!'e r e Jy i s a r. i n pu t r e la xa t ier 6.,ri t a r t . This give.

. ' G . = . M ' * \ p . M ' ' ' v04-Tg-TStt.M'

+ 'VGM * 'vGy

• I23 Tg- 1 Sat.M ' JM . .

1 r. . u s i r.~g ' Eq. ( 2 3 s . two further adjustrerts were four.d to be desirable. Iirst.

if we are to solve e g',y e yy 7 ,y + ( (.(;qfTg - T Sat M i for T.g.t(.gq tust be t er.t a r.uous for all T ha t .M. This is best accomplished by writir.g.the der.ceir.ater of Eq. (231 as ITg -T Sec r.d, we wa r.t ( (.gq ) c ygy.

5at.M ' hi. Therefore.

1. q . (231.is writter as i

G =.M ~ 'Vap.M I ~ (vGyf T(; - Tg33,y)

Cv0M " CvGM + E3X 1 (' . ITg7 }. (2Je T Sat.M + 41 Tne mair. problet in the impl eme nt a t i or. of Eq. (24) is that a new T' fur.ction is new required. or t

'G ~ -- % 'G . M l' = _"' . - Tu=0 -

(26) g Ac 'vGM where eg,y = eyap,y .,ygy T hat.M -

The Newter-Raphsor procedure to extract Tg requires-that we evaluate di/dT g . or deh,y dF' E ,C dT(; (eg ~ x ,ch,y) ,, d( (.GM . ~~

dT G th -1 - (2o r

- h e (.gq e dT g t

f. x cy,(.gq } -. e

' The easiest eethod d:scovered se far tc evaluate the derivattves is to see that 43

~

R:l3F DRAE 3.M + ' Liq.M ~ 'vlM Ty, g , , y - e[i . M. c. l d * $ M ..

'v0M " 'vGM

• Eu\

1 T b Sat.M + "'M whe re eh M.old " '\ap.M ' 'snMT g3;,y . Irotherv.1rd5.cygy is defir.ed ir teres of tecperature a r.d the va por 's true heat capacity. Because; the derivative de[i M old dcy(;q is giver. by Eq. ( A-75 ; wi t h the addi t ier. of t he t e re -TS g ,g . ard becat.se c ygg is giver. by Eq. (151 we car. derive ( f or c(.gy 3 c ygy1 dT

'vGM =b, T Sat.M CvGM - CvGM + gSa t .M( TERM )

dT g M 1+Tg-That.M + OM

-( G

)

TG-TSat.M + 4M for Tg>That.M . ar.d (26)

T 'vGM ' 'VGM

• TG

"! TERM + 2(c(gy - cygq 1 d ' vG'1

= b M' 1 at.M ,; ,

+_T S t.M - Tg + Jg'

.di g 15 .t.M - Tg + Jy for 1 .:5 1.M > Tg where ,

hj Ty-TSat.M TERM =yc gy - cy gg - g,y !!q iT

'+

Tg Crt,M - TSat.M 2

for T Sat.M 8 T a r.d .

3 Cr t .M hjp,q ty Th-2T sat.M TERM = c;,ty - c(.(;q - ,  :;-l }

• 1 Ty TCrt.M - Tsat.M 44

c ,.

RDL 3H [RAl~

for T g .;,q > Tr: t .4 - .

.d.ygq dc(7gy, g 6 is obtatr.ed.

Obviously. for the case of c.ygg = t,ygg. =b.y Orce 6

deg ;,y dT 33,,y d c (.G4

_ g3, di g di g Sat.M dT g where TEk4 i s' defired as . above. This completes the e qua t t or.s r.ecessary to perfort the iteratner.. Eq. (A-75).

at M is taker frer Eq. (A-74).'

[

dT g

4. Modificatier cf the Gas "rcrstart" at Hirh Pressures. -The secord modificatior. cetes frot the desire to calculate the vapor state variables ir. the high t erpe rature cr.d high pressure regime. for example hundreds of regapascals.

Both the Houger. Ltsor.. a r.d Ra ga t z . c ha r t ar.d the steam tables suggest that the effecttve gas c ora tar.t is higher thar. the ir. fir.i t e ly dilut e value ur. der these cor.d i t s or.s. The e x pe d i e r.t modificatior. for T-> o T ert was to c ha r.p e the relaxatier. equatter. for the gas c o r.s t a r.t . Eq. (A-24). to be I r. Rg = gg I r. RCrt.4

• II ~ SM) in (Rg (1+og)}. (30) for age Tg3 (e ge Tg )Crt. ar.d. for cor.titusty.

I r. Rg = gy I r. fg + (1 - gMII"IN Mi II*I4M MO - (3II for (c g Tgg i ( p ge Tg < (pgyTt g ,

'Gr Tg - Icom To \

where (q' =

(# 6t gT 1 -is. T i o o ,,

45

N &' m -y l l j y j] j a rd o.9 i s a r.

irrut c er.s t a r t . The two derivatives required for ger.ero: trarsier*

'Or Ek g Tg '2Rq e p e r .:

• i o r s are - ard These derivatives i rvo l u cr;s rir 4 ETg .

E M #*6r N c ha r.g e s Iror the procedure used to obtair Eq. (A-49) and Eq. (A-50. *

.- r Eq. I 30 s. Ear i q. ( 31 i t he expreu iors are

1. Gr. ON M #r 0I G y OM# Gt gT f 1 - gg)

(32)

-N M 0# Un = ty' I M O'Gm I#Gr3IG Crt

~

/

I#GeT g), J M, TgBR y t gy 2R y - /t f y

+ g T ., Ir(Ry;sg) (33)

RM ET g Rg Es gt y k: Tg-TCrt.4 + *G.M

1. GeT g - (c g7 T1 gp shere Jg = 1 + I# T J p g.

6t g Crt - I'br TgI g These e y r e u ier s have been sirplified through the defir.itier. ef fy which allews

1. rg G T 2:q t gy ?f y Tg Ef y

=-

IM #I # UcTg ; ig

-. (34 ci g7 I g &lg The og wa s selected as 0. 5 f or wa t e r by f i t t i ng SESO1E da t a a t 3 000 K shown ir.

Eig. 13. The fit was for the case of ag,y - 200 K. This choice is explair.ed ir _

Sectser. 6.

5. Defirsticr of the Va rer Terrpe ra t ure for Sincle-Phase cells ard Other 4sdificatiers. The t hi rd modi f ica t ion was i r. the d e f i r. i t z e r. cf a vapor temperature for <.ir.gle-phase cel:5. Eq. (A-40) r.cw becomes ar. impliwt equatior for a fur.ctier.. G. defined by 46

1 i

i 5 l 600 To = 3000 K e i i i i

! E ^

l h 2

[500 -

) 2 SESAME -

v 4oo _

w -

E *'

3 300

{ i -

! s M l CO 200 - '-

SIMMER-H i

i 3

W E -

l a 100 -

i E

I

)

o I I I

4 o 50 100 150 200 m DENSITY (kg/m a )

o

, ' c ca

X i ,

m b

i

! -- H

?

u h\1;T-1 R0lGH EW7 S FvM 'vGy That.M'NM G=~ -Tu=o (35 gg .

EvM ' v0M 'NM r=1 l'or the Nester-Raht sor proc edure . T[,* I = Th - Gr/idG/dTg }E. The derivative se reed to' evaluate is h%1 AT- 1

- .- ' d6EGM dR M

M ~ ' GM' E 6.) /Rjg; T

l Fv4 Sa t .M #N d b, r=1 dT.6

%~ h%14T - 1

~

g Fvy;vGM /R M h%14T-1 -

dR

, d c (*GM M - I (G + Tg) 4-_

pVM ' ~

""I d T '-

M 'G E '

- 1 . (36) hT14T - 1 ,

EvMC vGM #N M e=1 Ter this s i t ua t i e r. . - pyy and cor.sequently TSat.M are c r. starts. This simplifies j the eva lua t tor. cf dc(.gy/dTg from Eq. (26). The other derivative. dRg.edTg . car be evaluated f ret Eq. (33). This rather l or.g iteratier. is or.lv perforced uper. l begir.r.ir.g a~problet. During a t ra r.s i e r.t . values of the vapor der.sities ard l l

varer tenperature from a previous tire step are used to define c(.gq ar.d kg in the a ppl i c a t i er. of 1.q . (35). However, when a 5:rgle-phase cell beccees t w e-pha s e . - or.l y t i.e previous tir e step Tg is used to defire c',gy and Ry for use i r. Eq. (35). A separate iters.tior. irvolving a functior. G'. defited by G'=pg-rgyy RTgg=0 (37) is used to make Rg cor.s stent with at updated cge. .

The iterationisdefir.edbyc[$3Tg=chyg-G T /(dG /d(AgeT g) f. Because Tg is constar.t. the required derivative of Ry car be four.d usirpg Eq. (32) in the e r.c case that is more conplex thar. that described ir. the SlhMER-Il ma r.ua l . Upd:. i e s have beer. included to cover all brarches of the kg e va l ua t i er.. The icy:-

-itt eter.ted l car. be pursued by exatir.ir.g the IORTRAN todirg.

48

RCGH DRAE l eur ot her nirer todificat ters are a rc luded -ir. t he correction set !<r the 1:UN s i. ric . ' WRil;i:US. isce Apperdi x li e. Tiie s e a r e a s toiloss; (a. A bisection algersthe wa*. supe r iepesed or. the Newt er-ka phsor i t e r:. t i er

-to selve- Eq. (25). This prevents stall values cf dF/dT ; free causir.g ar g

e x t ra po l a t i er. te a T ; out side a docain where a solut ior is knowr. to exist f rer-g previcus iteratsera.

(b) The gas ' c o r.s t a nt " calculation was updated in the r eca i r. i r.g input optior.s to be . cors i st ert with the new forculation. The main change was to revise the selutsen of Eq. (371 te consister.tly initialize two-phase cells with the irput opt ier Ilhi' = 1.

( c ) The i t e ra t ier t o solve 1:q. ( 25 ) was removed for single-phase cells. It was replaced by a once-through pa. to defir.e the. gas constant. The s t e ra t i er.

is ;.o t required, and, as is shewr. in Sectier 7 i f cag is decreasing drastically.

a solution te Eq. (25) cav rot exist with previous time-step vapor densities.

(d)- The r.e s variables were assigned to coccon blocks. giver. i r.p u t s t a t e te r.t s . ar.d foreat t ed with out put edits. The rew input descript ion for the AEOS is given ir Apperdix C.

6. Discussior of Results. Now that the codifications to'the AEOS have.

been described se will discusi. how well the AFOS works. First, convergerce et the "f" f u r.c t i en. E t; . (25). around the critical pressure at -3 000 K is re lorger a problet. Fig. 14 shows a plot of a typical case. The pressure at the conve rged terpe rature (3 024 K) i s 22.1 MPa. The SESAME tables give 25.2 MP.i at 3

this densiiv (16.6 kg/r 1 ard teeperature. The SESO1E vapor energy is predicted to be 9.73 MJ/kg. Iloth the type of convergence and values obtair.ed by SISt1ER-Il are at least r e a s or.a b l e for this case.

We aise cade a cor parison alorg the liquid and vapor saturatier. lines. The arternal er.ergies calculated for density ar.d teeperature i r.p u t are compared with the steat-table values in Fig. 15. There is s orre distortier. s r. Slkt1ER-II .

because of the functional form assuced fer the liquid energy at teeperatures above twe-thirds of the critical temperature (431.5 K or 155.4 C). _ A pes s ibl e correction is to subtract one-h.lf the internal crergy of vaporization rather t l.c. c ore-half the enthalpv cf vaporization it the f err ula for the liquid's saturat ion t erperature ( Eq. A-6). Ir. arv case. if t he sa t ura t ed va por der.s i t i e s ard irterr.al energies are input, the result ing plet el pressure as a functier cf ter pe ra t ure is quite reasorable. is showr in Fig. 16.

49

RCLGF DRAT I I I I eo= 9.6 x losJ/kg 80= 16.5 kg/m 3 15 - WATER VAPOR og-e*

F:

10 -

c,o 4 G S -

w a 0 -

-5 -

-10 -

-15 -

-20 -

1 I I I 3010 302o 3030 3040 305o VAPOR TEMPERATURE (K) g ig, 11. "l" furctnor for the r e vi s ed AI'()h.

50 -

F 3 RfV Y VI,' IAL' i\

rr i f;

i I I I e

m 3.0 -

o cn STEAM TABLE VAPOR

( SATURATION ENERGY 2

v e h 2.0 -

w -

Z W

g e = SIMMER- H Z

1.0 h

H STEAM TABLE LIQUID SATURATION ENERGY Z

I I I O

O 100 200 300 400 TEMPERATURE (*C)

Tip. 15. S t e ar - t ;: b l e s orp r i ser w : t h sa t u ra t ed der.s i t y and terperatu e irputs.

51

J '{ h l , ,'

l I I I STEAM TABLE 10.0 -

SATURATION LINE -

A W

Q.

2 v

W 4

C 3

y 1.0 -

e = SIMMER-II -

W K

Q.

O.1 I I 100 200 300 - - 400 TEMPERATURE (*C) 1 ig, l o, Saturatier. pressures with irternal energy and density input inte Sihtdil-II-52

r f ] l mee p RiVw J \ - 1 -

t b the situatier exi<ts, ar acceptable value fer p.

g the relaxatier ccrutart i r. E y . ( 2. i . is 5. . K . Willi pg = 5 . K. l ig. 17 s imw s a t erpa r i wr si crergy along the r e e. r critical 22 MPa asebar above the critica! tetreraturc.

The agreenert is r e a s e r.a b l e . To get a value for the gas c or.s t a r.t ' relaxatier parameter. ag,g. the worst case lor TS ,, > T Crt. was selected. The water's critical dersity and arterral erergy were irserted i r. t o the AEOS. and a trial terperature of Tg=TCrt + 10~ K was used. With pg = 50 K and og = 0.5. plot of df/dT g as a functier of Tg is giver. it l ig. 15. A value of 200 K (approxitately ore order cf tagr.itude above that used in the ZIP study)' a ppe ;. r s e te give or acceptably regative derivative.

Orfertunately. I car still be triple-valued if f g in Eq. (20) is sufficier.tiv stall. and we are close to the critical pressure. A plot of the "I~ f unc t ier (ag,y = 200 K. Og = 50 K. and og = 0. 5) for such a case is shown ir.

Fig. 19 The idea was to produce saturated cor.J i t i on s at 20 MPa. The high SlhMER-Il result of 22.42 MPa is 135 more thar. the low Siht1ER-ll result or 19.53 MPa. As night be expected. this t riple-valued behavior beceres verse if sq is increased. in other werds. the approach te eg,,,y is slowed. Values for the highest zero of f as a functior of sg are shown in Table 1. A lower value of p g will reduce the problet: however. with aG.M f 200 K. the stear pressures abcVe the critical tetrerature are already too low. Tables 11. 111. a r.d IV contair calculated pressures for 22, 50. ard 100 MPa steam table isobars, with density ar.d terperature input. ar.d ag,y of 200 K. and a og of 0.5. Ir. these table <,. all nass is assured to be vapcr. ard the calculat ier. is only perlotted f or a densit s less that -500 kg/c3 liipher stear densities are not anticipated it the nelter-core /ceolant-interactior prebiet. The maximur. deviatien is about -153 below the desired result, if pg is reduced, the SIANER-Il arternal energies rece< sary te achiese these temperatures will be i n c r e a <. e d . it other words, the poirts ir. Fig. 17 will be raised. Consequently. we do r.ot wish to reduce /g. A reduced ag,y wil: improve the pressure relatiorships. as shown in Table V.

giving the calculated 22 MPa isobar with ag,y = 75 K. Now the maximum error is only 4C. Ilowever. lowering aG.M increases instability when T,g, TCrt- "' -

implied by fig. 15. Formulation of a multivariate least-squares fittirp procedure to fir.d the best values for ag,g. pg. and og sects desirable. Ther.

we could detertir.e if the precedure is sufficiertiv stable. or if reformulatior is required. In the reantice, the arpurert car he tade that currert values are 53

4 l

j'01L

AF-~

D("

l\ l JIni>

f l l l l l r

} \

l l

1 5.0 -

STEAM TABLE CURVE l

m i cn . 1 x

N g4.5 v

0 l E '

W l 2 ,

W 4.0 -

J 4 e = SIMMER- H l

. Z l E

W H

Z 3.5 -

I o

, 3.0 -

I 400 I

600 l l [

800 1000 1200 TEMPERATURE (*C)

Isp. 17, I r.t e r r.a l e r.e r g i e s of vap<r with dersity ar.d terperature srput alerg the 22 kil'a t w b.r.

54

RMH DRAp I I I 1

2.0 - -

1.0 - -

pO a t

N LA.

_ D 0 - - -

N 4

- 1.0 - -

1 I i 100 200 300 S G,M(K) --

l l

Fig. 16. Derivative of I q. i251 as e turctior of a g,y.

w i t h T(; .i u s t .ibn e T rt for water.

g 55

l l

l l

O!In m3 .-.,.

I m a 2A ,

1 1

1 I I I I 20 -

og=3.1973 MJ/kg 15 -

W 'd t ', , Pf =171.4 kg/m 3 -

. _ , u 6 -

10 -

5 -

e

• k O - -

4 L&.

-5 --

-10 -

-15 -

I I I I 630 650 670 690 ,

TEMPERATURE (K) l~ s g . 19. Ker.a i r i r.g t r i pl e -sa l ued r.a t u r e of the "I ' l ur(t ier nea r the c rit ica l pirt .

56

'S m. r- - -

RPVw), ',) ,

i I

TABLI: 1.

1110111.5T l' Zl:RO k'ITil e , = 171. 4 kp/r' ANil eg= L 197 41/kg g

i kelaxatier Terrerature Pressure j i

iror i 1. c . 22i e syfLi p( MPs )

50 22.42

)

75. 23.56 100 25.01

! l' 150 26.77 f

200. 28.05 l

TABLE !!.

AEOS/ STEAM-TAllLE COMPARI5ON FOR THE 22 MPa ISoll AR i

Dersity ( L c /r 2 i Terrerature (K) AEOS Pressure ( MPa )

207.3 645. 21.1 121.2 673. 19.4 101.4 695. 19.2 90.05 723. 19.0 76.25 773. 15.9 6' . 5" 523. 15.9

~

61.2" 573. 15.9 5ti. 32 923. 19.0

$2. 35 973. 19.1 46.25 1073. 19.3 41.65 1173. 19.5 35.02 1273. 19.6 32.45 1473. 19.0 30.25 1573. 20.0 acceptable for reasonably low densitt stear. g:ver t he o t he r ucc e r t a i r.t i e s (see Fig. 20 ).

The greatest errers it a two phase cell wtth the preposed AI.05 ferrulatier are with hath stear dersitses as sugge s t ed bs lig. 13. Urfetturatels, the cr!\

real corra r t ser ava ilable is w i t h SE501E arJ as fig. 21 shrws. ser.e deviotur 57

R:GH CR/;7

-TABLE 111.

Al:OUNTret-TaltLE nn1 PAR} son FOR Till. So MP. Is011;k l>e r s i t y (Le>r', 'Terrerature (K) Alus Pressure t M P.. '

577.7 673.2 51.9 495.3 695.2 -

54.7 402.3 723.2 52.3 256.9 773.2 44.5 195.4 .823.2 -44.4 163.6 873.2 44.3 143.6 923.2 44.4 12 .' 4 973.2 44.6 110.2' 1073.2 45.1-97.25 1173.2 45.S 67.63 1273.2 46.4

$ 0. 03 1373.2 47.0 73.74 1473.2 47.4 65.42 1573.2 47.5 TABLE IV.

ALOS/ STEAM-TABLE COMPARISON FOR THE 100 MPa ISOBAR Der.sity (kr r')

e Terrerature (K) AEOS Pressure ( MPa i 524.0 773. 90.4 445.2 823. 94.2 374.4 873. 94.4 321.0 923. 93.6 262.0 973. 93.6 230.5 1073. 94.5 175.7 1273. 96.7 133.9 1573. 104.5 of SESAME- f ret steat-table values occurs ever at ir.tereediate demaities. of course, deviation of SESAME tables fret the AEOS is worse at higher der.sities ar.d temperatures, see rig. 22. However. in the colter-core /coolar.t interastier.

problet. (ases with true high density 11 0 7 car be bour.ded by ruer.ing a paracetrit where pressures are obtuired by heatirp liquid water rather that torpressari steat.

For Tt) T Crt. the ticroscop ( der.sity of li quid wa t e r is assured te be 58

rjn n. r,,

DY di l M [4 TABLE V.

410VSTEet-TGLI nNI'ARlW, lok Tilf 22 Mi% 150Bak 11 a(;,g 15 SET TO 75 L.

lie r s : t s ( L c 'r ' ) Terr *e ra t u r e (Li Alos l>ressure ( MI a .

207.3 645. 21.2 121.2 673. 21.9 101.4 64b. 22.1 40.05 723. 22.1 76.25 773. 21.5 67.50 523. 21.6 61.20 873. 21.4 56.32 923. 21.3

$2.35 973. 21.2 46.25 1073. 21.1 41.65 1173, 21.1 35.02 1273. 21.1 32.45 1473. 21.1 30.25 1573. 21.I f(. rt . This s i, ~317 kg/r'. Rapidly heated water will only pc two-phase ar.d corsequently wi!! be able to vaporize when it s effective tacroscopic density is less that this d e r.s : t s . A plot of pressure as a functier of terperature .: t a 2

dersity of 316 kg/t :s shour ir. Fig. 23. In the single-phase liquid regir e.

appivirt a low single-phase cor.pressibility a r.d adding of a fictitious pg, above the critical pressure ccre that coepersate for r.e g l e c t i rp the sonic velocity's inc r ea se wi t h de ns i t y ar.d t he low vapor pressure upor er.terirt this repite. It is true that pressures along a fictitious equilibrium TSat l'E' above T Crt. which goes deeply ir.to the sir.gle-phase regime. are comparable with SE501E values (see fig. 24). At e - 3. 75 M.1 (Tal 175 K) the maxieur error is 21.50. However. a s i r.g l e - pha s e cell in SlkNER-Il could have the pressures indicated it fig. 24 for much lower water densities. At most, water only has te occupy a volute f ra c t ier. equal to o g i r. the single-phase re g ime_wh e n other liquids are presert. For e xar pl e . 1 000 K water it a single-phase cell could produce a pressure in excess of 2. 57 GPa i f mo r e thar 41 kg/e' exists. The liquid in the rerairg volure is assured to be compressed to the sate pressure ir this example, iAs another arterestirp observatier.. the S1:5 011: curve in fig. 24 is also screwh.' strarge. Ier a r.v irterested observer, the values used tr 59

I I I I I I j

/*

i T= 1573.2 K '

\

/ '

/ -

l 90 -

SESAME ' -

y

! n /

to /.

il~ G.

i 5Y 2 /

, v 70 -

/ -

j i4 W /

-E E /

/

D l

zi ME M 50 W

/

/ STEAM TABi_ES ei w / -

l / * = SIMMER- II i

Ei g /

E g /

/

• I '

o' . :0

! 30 -

,', O C

ol

. /

p * C O

3 i

i 20 I

40 I

60 80 I I 100 i

120 g

! s VAPOR DENSITY (kg/m3)

_ ___ - _ _ _ _ - _ _ - - - - -- - - - - - ^ - - - - ^ ^ ^ ~ ^ ^ ^ ^ ^ ^ ^ - ^ ^ ^ ^ ^ ^ ~ ~

i 1

y 5.6 i i i i (3 i I

! E ________________ ___________

i - W i ?r Z SIMMER-H

gr W 5.4 -

! j'; -

• yn O

i:

e Ax Zs STEAM TABLES i "i m E7 52 i

i c W2 -

FV

_ !! Z SESAME -

y: -

w: E E: O 5.0 -

T= 1573.2 K

! A -

rj

< C)

> I I I i c

20 40 60 80 i i t.3 i

i 100 120 I VAPOR DENSITY (kg/m3)

-3 h  ;

, i H '

i

i I

1 > 98 l O I I I I i - E i .:; W T = 3000 K SIMMER-II p z g - __ ____

i I

r J 9.4 -

i .!I E i ,-

< ^CD i

! ji i Zx EN W2 3g p 2 9.0 -

1

?: Z"

~

3 l 3~ E SESAME O

! O 8.6 -

l < :3p

, > I I I I

u..

! , 50 100 150 200 250:

C.

) VAPOR DENSITY (kg/m3) a

.! .ro

$ V:2 9

j -J

}

.i __ _ _ _ _ _ _ _ _ _

RD;3F JRAF~

1 l

l l

I I I e

SESAME e e

e e

e e i n e ,

W '

O.

• 2 w 100 - -

D e = SIMMER M

C0 W

C" A.

<> l l

1 l

l i I I l 800 1200 1600 2000 l TEMPERATURE (K) l' 5 23. A $1:W1E 'S14N1:k-Il c orpa r i ser a t t'g - 3 H, L g ,er'.

63

i i

l 1

104 l

SESAME y '

/

[

/ 1 e#

aa 103 '

/ -

G.

I f

• /

SIMMER-II w

C '

,/

o /

m .

e i w

e /

a /

100 -

/

/ i I

L 10 i 3.0 4.0

~

LIQUID INTERNAL ENERGY (MJ/kg) fig. 2 3 Pressure as e f ure t ier of liquid n r.t err.a t er.e rgy t a ke r f ror- t he de rs i t s a rd t er.pe ra t ure desc r ibi ng a f i t t :( nous saturatier curve abese T Crt' 64

r TAllLF VI St.W1C Rt.SULTs foR flG. 24 THE DSPLX CODE. LASL T-4 11-27-79 VERSION 2 1 EOS 7150. 07546.600 P E DP/DR DP/DT DE/DR DE/DT 1.079E-02 P

2.194E+00 1.296E-02 6.478E-05 -1.124E+01 4.861E-03 T DP/DR DP/DE DT/DR DT/DE 1 172E-02 6.132E+02 1.594E-01 1.436E-02 1.150E+03 1.923E+02 SOUND SPEED = 4.347E-01 KM/SEC 2 EOS 7150. 3170.647 P E DP/DR DP/DT DE/DR DE/DT 2.555E-02 P

1.730E+00 2.701E-02 2.565E-04 -9.379E-01 4.834E-03 T DP/DR DP/DE DT/DR DT/DE 2.210E-02 6.472E+02 7.554E-02 4.561E-02 1.853E+02 2.193E+02 SOUND SPEED = 2.925E-01 ud/SEC 3 EOS 7150. 3435.700 P E DP/DR DP/DT DE/DR DE/DT 4.121E-02 P

1.947E+00 7.449E-02 2.833E-04 -1.086E+00 4.356E-03 T DP/DR DP/DE DT/DR DT/DE 3.663E-02 7.028E+02 1.401E-01 6.386E-02 2.608E+02 2.413E+02 SOUND SPEED = 3.999E-01 NW/SEC 4 EOS 7150. 4593.800 P E DP/DR DP/DT DE/DR DE/DT 9.433E-02 2.187E+00 2.628E-01 4.203E-04 -1.317E+00 3.319E-03 P T DP/DR DP/DE DT/DR DT/DE 9.178E-02 8.078E+02 4.405E 01 1.454E-01 3.972E+02 3.265E+02 SOUND SPEED = 7.097E-01 KM/SEC 5 EOS 7150. 6216,900 P E DP/DR DP/DT DE/DR DE/DT 2.446E-01 P

2.252E+00 9.901E-01 7.643E-04 -1.361E+00 2.443E-03 T DP/DR DP/DE DT/DR DT/DE 2.396E-01 9.033E+02 1.338E+00 3.179E-01 4.943E+02 4.040E+02 SOUND SPEED = 1.239E+00 KM/SEC 6 EOS 7150. 8136.1000 P E DP/DR DP/DT DE/DR DE/DT 6.804E-01 2.270E+00 2.737E+00 1.296E-03 -7.482E-01 2.463E-03 P T DP/DR DP/DE DT/DR DT/DE 6.766E-01 1.001E+03 3.198E+00 5.354E-01 2.780E+02 4.044E+02 SOUND SPEED = 1.935E+00 KM/SEC 7 EOS 7150.1.023.1100 P E uP/DR DP/DT DE/DR DE/DT 1.733E+00 2.512E+00 6.080i+00 2.018E-03 4.953E-01 2.822E-03 P T DP/ i DP/DE DT/DR DT/DE 1.734E+00 1.100E+03 5.700E+00 7.089E-01 -1.715E+02 3.554E+02 SOUND SPEED = 2.622E+00 KM/SEC 6 EOS 7150.1.242.1200 P E DP/DR DP/DT DE/DR DE/DT 3.866E+DO P 3.015E+00 T

1.165E+01 2.823E-03 1.220E+00 3.139E-Q3 DP/DR DP/DE DT/DR DT/DE 3.644E+00 1.200L+03 1.071E+01 6.375E-01 -3.761E+02 3.222E+02 500fT) SPEED = 3. 576E+00 Kht'5EC 65

r-k' pererate the curve are giver i r. Table VI. hate that tht water's liquidt.s e r.e r g y . U.9043 M.l i k g . cust be added to these values for comparisor w i t ). the

$14NER-II ALOS.

Ir. conc lus s or. for Hy o the revised AEOS produces pressures in the va pc r field that are high for sa t u ra t ed wa t e r be low the critical ten:perature (because of the low i r.t e r na l crergy for the liquid water). perhaps -15G high near the critical terperature (because of the triple-valued characteristic of the f ormu l a t s er. ), up to 15C low or. ssobars above the critical pressure when coepared with stear tables (because of the value of ag,y choser. to increase stability). a little high at high pressure and temperature but low der.sity as shown by Table IV. and perhaps as much as 20s low (for practical problems) at high vapcr der.sities and high pressures. Greater vapor-field accuracy can be achieved in the future if instability problets do not affect practical SlhMER-I l runs.

Sirgle-phase cells with supercritical liquid water still may possess unrealistically high pressures.

Apperdix D gives the progran te produce the results displayed in this section. Appendix E gives the proposed fitting parameters for water, correspor. ding tc the ir.put description of Apper. dis. C.

Succestions for further lerroverert. Except fer the po t e r.t i a l of additier.a1 error correctior.s. no further AEOS improvements were possible withir t he sc ope of t hi s tol t er-c ore /c oolar.t -ir.t e rac t ior.s program. Ilowe ve r . for future reference. we are providing a discussion of some items that deserve additieral attention. These iters are as follows:

(a) A ru!tivariate f i t t i r.g procedure ard graphics capability would be useful not orlv for firding ar.d displaying the best possibilities for the water parameters, but also for obtair.ing SlhMER-Il ir.put for other caterials.

(b) A slightly tedious correction (fror at algebraic viewpoint ) to the liquid's internal energy would be to subtract ere-half the interral e r.e r g y el vaporizatier. rather thar. one-half the erthalpy of vaporizatier. in Eq. (A-6i and (A-13). This should sigr.ificantly improve the apreceer.t ir fig. 15. _

(c) The formalist to define the pas "c ons t a rt ' . g could be made more K.

a ppe a l i r.g from a dimersional standpoint. Takiry the logarithm of a quattity possessing units is not desirable.

66

(

(d) The increase in the gas "c er.s t a n t " upon departure fret the saturainer lire ne<Js tc deperd on pressure as well us temperature. Ur.f o r t ur a t t i v . .

little aJditiir to ar.y rela xat ion equa t ion can add s agr a f ictr.t complenty to the numerical algoritht. An additional i nr.e r iteration to cotpute a corsister.t T

Sat.M M and Tg needs to be considered but cust be desigr.ed carefully to N. .

avoid excessive costs.

(e) Ir. single-pha e states. the effective sonic velocity could depend both on liquid energy a r.d the e x t e r.t of departure frot the saturation line. er density. Because liquid crergy and der.sity are irdependent variables, a properly cer.sidered schete would be straightforward to impleter.t.

(f) We should eliminate the i r.c c n s i s t e n c y in using liquid and saturation temperatures with value*, above the critical temperature. Besides the Eos codificationi, to get a preper pressure at high vapor densities, significant ruterical tethods c ha r.g e s are desirable to implement such ideas correctly. For example, we should corsider incediate t r a r.s f e r of liquid to the vapor field wherever i t a, terperature exceeds the critical t empe ra t u r e. This t rar.s f e r would require codificatter.s to subroutire EXFL1 to stee the possibility that liquid eight corvect free a cell situltaneously with this i r.t e r fi e ld ta s s t ra r.s f e r.

Also. wherever a hypothetical liquid / vapor interface tecperature exceeds the critical tecperature. o r. l y heat transfer and vaporizatior, not c ord e r.st t i en .

should be cor.sidered. It other words, all preser.t use of situations with That >

T Crt w uld have te be removed.

(t) leproveterts tight be considered to take the gas ' c o n s t a n t *' core responsive te der.sity changes at high pressures. However, if highly supercritical regic:es are truly required fer an analysis. continuatior. ef ad hoc codificatiers rm y net be pruder.t. Modi fica t ion of both 514NER-Il ar.d the SESO1E EOS to allow use of SEhetf tight be desirable to permit calculations with a more complete represertatier of t he rmodyr.ati e states. The problem here is efficiercv. At high pressures some degree of seplititress ir. solving the energy equatier is desirable because of the importart contribution el the work tert.

This implies the desirab ] ty of having updated LOS variables for every cell or every pressure iteration. Such an approacn could be exper.sive. particularly for ca l cu l a t s or s wi t h ta r.y te sh poi r.t s that run for more than a few milliseconds el real tite, 67 i

r 1

%k[

R gyQ ']d i

,,/ , 5 b i E r er. the above s ug g e s t i er.s . clearly additional t ure and ef f ort could be exper.ded or. the $1!.t.1! k-Il Los.

Ur f e r t ur.a t e l v. the list of possible ar.d ever desirable i mp r e ve ce r.t s cav be erdless. Decisier.s cust be made to focus efterts for imp r ovete r.t o r. problets that are . iud g e d te offer real benefits for aralysis in terms of credibility.

flexibilitv. robustness. problem resolut:or. ar.d physical reasonableness. As a r.

example, one of the interest ir p situations that arose with the current AEOS is giver in Li t. 25. Ile r e the desired converged pressure is above the p' value, te a r. i r.g that a positive T hat c a r.no t be defir.ed, and no s ol ut i or. exists.

Ec r t u r.a t e l y , this situt. tion is phys ica lly unrea sor.able. It arose in S14t.1ER-I l when a sir.gle-phase rode was rapidly cetpressed to an og of 10' while r e t a i r.i ng the previous time-step vapor der.sity. The SESO!E tables predict Tg=7 100 K and p = 275 01% for the c ord i t i or.s of Eig. 15. Nature is far teo nenlinear to pe rr.i t success over all possible culticompetent EOS regimes with a few simple aralytit formulas that possess densities ar.d internal energies as the i rd e pe r d e r.t variables. These difficulties n e '. d to be e l iti r.a t ed if they continually frustrate the analyst in usir.g of the code.

6. Corclusiors. Part of the de fic iency with the 1950 ZIP study was the iradequacy i r. the S lkt.1ER - I I . A series of c o r r e c t i er.s to eliminate or reduce these AEOS problets were made for this to l t e r.- c o r e / c oo l a r.t - i nt e r a c t i or.s pregran as described it this report. The ma i r. tod i f i c a t i or. is a variable vapor heat capacity to take the vapor state to the ir. finitely dilute e r.e r g y far free the saturatier. curve. Ir the detain of ir.terest for the this prograt. five rew AEOS irput paraneters allow pressures to be calculated te within :200 of steat-table values er SESe11: E05 values. Usually the agreetert is much better. Eurther i r.c r e a s e s, i r. accuracy i r. some regimes clearly are possible but at a cost of decreased stabilits. Hewever, we believe the current approach is sufficiertiv accurate for the reactor celtdowr, calculatters ir Chapter VI because other modelirg problets lead to greater uncertairty it the calculated results. Tliese problets irclude but are certainly not limited te -

( a ) The ir.itial conditions assumed:

(b) The heat-transfer algorithe empleved:

(c) Structural ard three-ditersional effects igrered: and l (d) The othe r approxirat iers tade ir the Sl\t.1EK-Il solutier algorithr.

68

i ROU3V CRIFI 04 l T O + 80 10s _

^ 100 -

E ~

u. ..

, ,a

-104 -

P.-4s00 heim* ~

og=4.50 x IOS J/kg

-108 -

g g r TO -** -

100.0 1000.0 10,000.0 10s_

TEMPERATURE (K)

Fig. 25. An exaeple showing that p>p is not permitted.

! 69

r-~

RiJw U\s i

101 The c or r e c t i er. set forculated c e r. t a i r.s 527 l i r.e s ard - pi(cr ir Apperdi x 11. Suppested irruts I e r wa t e r are giver tr A ppe r.d i s. l. . Kevisi.r- +-

provide impreved Al.0$ preperties fer coriut will fequire additioral ettert.

9 J

m 70 i,

.-_-.--,~__cr,_...--._---, - _ , , - - ____,,___._,,g,,. _

_ - _ , , _ _ . _ , _ . , _ . .. ____ ,__,__,____._7 _ _.__,y_ ,__ __m.~,_ , - . . , _ , _ ,-

w. 3 RCUGH DRAFT  :

i.

- II. kl.id ii i t a t v or of sl*1I'R-il Liqu id-Li ou ld llea t Trarster for-Lter l

?

t f ir e !;. r, e:r qu;. ! : t a ; i ve pr.5-ler with the S1411R l! Z11' studvwas !!a s er: t i .r el using < a wat er-leor r iiiry zert ir tols ulat e r, espe ricer ta l s i r u L. : n r - w

' avoid inrediate t he rci t e ove rquert hing and ir.sul l it iert pressure developeert. A j, i s e t or.d qualitative problet was'the sorrewha t slow rise t irr.e in the calculated

[_

[ pressures. SI';L e xpe r iter.t s - suggest at ove ra ll - wa t e r-r ich fue l-wa t e r-st ear mixture. They also give ar almost i r s t a r.t a ne o u s pressure- rise t irre in sete-_ r s i t ua t s er.s. [

l. 130th problems appear to origir. ate with the S141ER-Il liquid-liquid heat-trar.sfer  !

model. The thermal corductivity of water-is low, for example. -0.65 W(m*K) at about 4(o K. leadiry to a rather low the rcal dif fusivi ty. say ~1.7x1tt m /s. -I

[ . lr u-typical Sl*1ER-ll ' t ite step. -lo " s or shorter. the thermal penetratiot j distance is orly a few micrors. irstead of bulk liquid water heating. we should .[

he. vaporizarg of f ,the surjate in a corequilibrium manner. The- la rge heat 1 ca ra t. i t s presert ;in the liquid water should rot hetone effective, until af ter pressures.have developed. causir.y ar irsrease in the wat er 's surf ace area.

The preser.t liquid-liquid 5141ER-ll heat-trar. ster rodel is described in Apperds.x

~

r i treproJuted iror Apperdix D of the revised Sl*1ER-!! car.ual l. As at initial nodi f i c a t wr., t he fellewing ideas have heer impl errert ed. ,

(a ) The re were 'll" staterer.ts ir.serted to turn off 1. quid-liquid heat t rar.sl e r

~

to water l' r et another liquid wher.ever the other liquid is botter. . Th e-t a l c u l a t_ s ve flow was diverted to t he vapor izat ter/cordensat ier, mode l whe re hea t

. t r a r.s f e r trot the hottei liquid is performed. Ilowe ve r . we do r.o t modify  ;

stryle-phase tells or the situations where Ttn. In

• 31 < Tt3 (water L i (b) At the begiering of subreutine Pilul:. we set up heat-transfer coellicients I hased or the liquid-liquid forma lise modi f ied so that the resistante te heat t

transfer is only that of the hot droplets iluel. steel. or zircerium oxide i.

('or s e q u e r ' l y . l.q. (1-9) it Appendix l' . del a r.iry the overall hea t - t ra r.s t e r toeffissert. is row approximated hv i

l 71

.m. .- . . ., . ,,,m_., _ , - . , . _ , , . . . _ - . . . , _ , . _ . . , ~ . , , , . . . . - . _ ,_ , , - , . , _ . . . , _ , , . . , . . . , . . . , . . . _

,,,,_,_,_,..._,.m..,,_,,,_,.,m,

~

. ROUGH DRUT l(Let3*kne 1 0. 2 r pe .

a3s.

The' water terperature :- .t lia t el t he wat e r suri..se or 1 % t .3* I 4- 'l-12' he s or:e s 45 LLt "Le 6 L3 (rje + r[3) (' r ,( TSat 3 - Tte) 411LHi3 " I.c3 5 r pt (1 - o5) 3 Pm * # P) r pm 3 t'3 p

= ligg;3 (That.3 - Tte) . (39) where-4lltr(;3 is erergy flus going.icto vaporization off t he wa t e r n'ir f a c e ,

ll Ldi3 - is the effective produ61'of-the surface area ar.d the heat-trar.sfer

~

toefficiert. ard l'e3 is ur input coefficiert. .the liquid-liquid heat transfer multiplier.

The quart i t y ligg;3 's I

i zero wherever r - 3. rp , or r p3 are zero. or'Tte < Tt3 The first two r e s t r a t i er.*, are obvious. The last cordition is simply a r e s t a t er e r.t el the idea i r. (a) that c o r.t a t t with a colder liquid should r.o t result.it irstartarecus c or.d e n s a t i o r . Wher r p3 < rpp,. . II Ldi3 w i l l . be limited by 15r.3cgtL g

Ln/rh based or the available surfne area of hot material. Because the liquid-crergy equatiora are t rea ted e xpl_ic it iv. a setend limit on li gg;3 is giver by the heat ir. the hot liquids er ligg;34 na x =1Le'vLM ' * (4")

The problet of vaporization of the available water is discussed it items (c) arJ (1) below. . _

tt) Sc6tior f. gives the 6ceplete forculas to irtegrate these heat-trarsler rodifisatsers withir the sortext et res ised vape r:2a t ior/t ondensa t ser rode l .

liere we sirply review those r.otl i t :( a t i e r s that a rre a r er t he torres t ior se t ter 72 4

r ROUGH E E these heat-trarster revissors. Apperdix (i. The ea u (ortir.uity e.;aa t i or is still appr. sir a t ed bs 7,n 1, r . "=1.r =

2 . r:Tha xt .k1

+1 ,

h ,, , gg , , 7 3 With the additier.s. aj,3andaj,3nust be redefir.ed by T ll 2

-n+1 ,n ) '

a$,3 = t lildL3g";+li T gg3L3+2T II Ldi3TLt + 2 (II("iSLT" (; + li s(;g T3g l .

c=1 L-1 ar.d NCLI. 2 "2.3 " IIIbl3

• IIUi3 + - IILdi3 + III(iSL + IIS(ik ) ) . (421 m=1 k=1

-r+1 where T ty is the result of the liquiJ-liquid heat-transfer model.

(d) The reu redificatior to the va rar i za t ier./cordensa t t or. treatnert is i r. the a d .t u s t re r t of the vaper-side he a t - t r a r.s t e r t oe f f i c i e r.t based or the r.e w eass flux at the water surface. This ad tus t re r.t is described by 1:q. (1:-11 1 er the SI AMl:R- ! ! ma nua l . The r:odi f ied ad tus tr e r t to be evaluated ir this s t e r a t i e r. is giver by (for water, corporert 3)

. hCLI:

[ pt - x I* gi lI Ui3 fTSat.3- Tt3) + _ yllLdi3 + . (Tha t . 3- T(;) + x=o. (43) eg.3.eIt rc = 1 x /ll ut3 e -1 whe n x = -spg;rgg;3t3 ,

or the r.egative et the product el the heat tapatsty ar.d the mass-trarsler rate.

The iteratier is s k+1 - x k - I /tk dl eds i k

. The produit 91 the surface area arJ Ul3 ti e rew vapor-side heat-trarsler toeilisiert is s e c -

1 .

The srr$ 'ol tuses ter terverger.6e et this equ. tier set are the sar e a. ir the Sl\til k Il 73

RCO3H cm ma r.u a l . Ve r a. i or. 2. The heat-trar. ster rates yllLdi3 P IV PI "Y " I*' s n e role sirilar te the lit(;3( T s .,, ,3 - it3) tern.

te) The vapor-energy equatier also cust he modified. first, the steat's r..a u NCLI.

Ilux must he i rc r erze n t e'd hv l [yyjg;3 / h 4.3.cli Se mr.d. when all the water i.41 vaporizes ir ore tirre step the r a i. i. flux is known; however, the add i t i er.a l NrLI:

crergy trarslerred iron the other l i qu i di, o r A5 [ q);tg;3 tust he giver to the re= 1 vapor. The equatters and other details are described ir the vaporizatsor./corderi.atier. todel revisions ( Sec t ior. L ).

(f ) Af t er (orvergerte of t he mode l . heat cust be taken away from the liquids.

Additiera. irclude updat iry the 10RTR AN variable OL by yllLdi3. and the FORTRAN variable $1ELN by at M"o transfer is ac courted f or automa t ically 411Ldi3'Ib

-r+1 with the vapor-der.sity t e rra. 15 and E(ir. Whe r. a ll t he wa t e r va por i ze s o r e x t ra N(LI tern. - yllLHi3 is added to the FORTRAN variable t.Ki.

r=1 (g) Tiie tiral 6errectior ir the correctier set r e s u l t i. Ircr the effett of these heat-transler redi r it at iers er the e!!ettive heat 01 vaporizatier for the vaperizatior case. This, correctier is described ir hection I:. but is retaired nr. this terrectier set to isolate all heat-trarsler model sharges.

This cortludes the deu ript ier of the liquid-liquid heat-trar.ler codel charges,.

The torrectier set ( Appe rd i x (i). without the last nire lines. was eerged with the Sl411:R- l l tode, cetpiled, and leaded. The irput let the SNL experitert taitulated ir. the llP s t udy was upda t ed ard rur. The vaporizatior rate of water was tourd to he extremelv rapid. l'o r a sase it which the hot liquid to water he a t - t r a r.s t e r rui t iplie rs were set to 0.1. the preuure ir the rode at the batten certer of the arteractior. zere rose to 20 MI'a (nearli The ct,tisal preuurei i r. o. ot es. i see l'i g . 2 ti ) . T h i a, rise is much laster thar ir a tervertioral t urrodi f ied ) terrariser suse rur with all the liquid liquid rtu i t i p l i e r s at urity s showr ir i s y, 27. Ir the torsertseral s a i. e . $1411 k I l a t h i e ve J 6 MI'a bs -1 ri.

9 74 l

RIGH DRAFT T

oO I I I I I I I I I I i iI 07 6

- o.

l

- Q i

- o e

o G

• E I:

o e

- < - og 1 - , _ ct 1

i I I i i i t i i i iiiii o 4

o e o 8 8

.i (edN) aanssald 1 Fig. 26. Pressure in the sample problem with new heat transfer.

1 i

75 i

_ _ , _ , , ,,___,..,,,_.__-_______._m-,.

OniIPL DD A t I l\UUUI l UJ\nt i i i i i i i i i i i i i i i ob op

+

o 3

l

_ _ o e

l

_ .x, _ o ,

e- l o

e

+

3 i,

_ cs" _ o e X- @ . c r" i.6 C 1

o l

_ _ o n

- _ o N

l ., _ o. (

l l 1 l i l l l l t I i i I I i o f C '

e e v N o (edin) aanssaJd 1

Tig. 27. Pressure in the saeple problem with original heat t ra r.s f e r .

76 l

_ _ _ _ _ _ _ _ _ . . l

R:l3H [HAr l 1

Urlotturatelv. the vaporizatter formalist initially used (that deu ribed bs the l l

Sl4Mi k- I l rv r u. l i required water to leave tht liquiJ lictd at the t e r d e .1 t t l

crer3s. The eterp difiercrte f.etweer the t orde r ..i t e crergy ar.J the trut 1 : i. .a ,

{

erergy 4 liquid is suhteeledi c arre from the liquid field. With arterlase l 1

va por i za t ion. the liquid-side heat-transfer coefistient for heat trarsler n r.t o l the bulk water was i r.s u f f i c i e r.t to t rarsf er enough crergy from the water'*.

surface to make up for the losses. A plot of the water temperature of the rode in questier. is shown in l ig. 2b. A decrease of more than 20 K hat. oc curred by l 4.05 rn . The temperature from the c onve r.t i ona l comparison case is showr. i r.

fig. 29 lie r e the water has p a i r.e d mo r e t ha r. 220 K i r. 1 es. Although the energy gair i r. the conver.tional case i a, u r.r e a l i s t i c , the e r.e r g y loss ir. the r.ew fornalistr was aise incorrect. I

{

1 Several eptions presented themselves to deal with the problem. These . opt ior .

were as t e l l ew a. : it) Ignore the problet. l'e r ha p s the S!hNFR-Il results could still be argued to be qualitatively reasonahic. (2) Try to put it sore liquid-liquid heat t r a r.s f e r to corpersate for the problem. (3) Rewrite a f tirical vaporizatier/cordensatser noJel. ( -1 ) lle ca u s e arv r e wr i t i r.g i r.vo l ve s significant work, develop a core realistic ru ! t i c orpar.e r t vaperization/cordersatnot codel.

llesau.e of the linited icope of this prograt, ept ior ( 3) appeared to be the best torproria.e with the goal of at t erpt ing to preserve <.cientific i r.t e g r i t y. This revissor is dea.oribed in Sectior. L. It assures that for cor. der.sation. vapor i a, reroved Iror the vapor field at the average vapor crergy and appears ir the liquid Ineld at the condeniate er.ergy. For vaporization. liquid s a. reroveJ frcr the appropriate liquid conponer.t at the l iquid er.e rgy and appear. in the vapor liclJ at ar crergy equal to the erthalpy of saturated vapor. For equertis.

sub.ooling arcreases the effective heat et vaporization. A retur. of the sare sample problet with the revised vaporizat ier/condensa t ion todel tard the Al'OS charge ) produced the results s h ow r. ir lips. 30 ard 31. The

~

liquid-te-water-surfate multipliers were set to 0.1. The pressure ri es to a peak value e 60 Ml'a or a submillisecord t ir e scale ( -0. 25 es l ard the bulk liquid water erly i r c r e a a, e . ir temperature by a moderate amount. The t orr i:

this resper c thus g i v e a, protise for 6alibratirg to ShL vapor-explesion pre u ure data ir a water-rish envirertert.

77

M

--,7,.

.'.J '..! b T'. 'i J  ? -

i e i l I i1 i i i i iiiiiii ob sp

_ o o

_ o e

_ o e

3

_ o o E

1:

_ o n

_ o N

i

_ o e e e ie i i i iiie iiie i i e 6 l

$ $ $ $ $~  !

t (3) aangeaadwal 1 Fig. 26. Water temperature it the sample problem with new heat transfer.

78

I I i i i i i i I 0 _

L

,U _

kO -

E 400 l

l i i i i t I i I i 1 g

~

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

• 10-'

Time (s)

Fig. 29. Water temperattire in the sample problem with original heat transfer.

nuc bur.ti rrj psvVuye nor ,

i i iiiiiiiiiiiiiiiiiiiiiiiiiii e7 9O

- 6 O

_q O

g o

t.

g 3

_ Q D E

F

_ Q e

_ Q w

_ Q cv l11119I1111111111111Ill1ltl Q l

o o o o l e S

, n 8 e

- o t

~

.i (edIq) aanssead l rig 30. Pressure in the sample Problem with new rev'.4ed heat transfer.

80

i i i iiiiiiii,,iiiiis heh bkk.f[o'o e7 i

o 2

o a

o e

3

_ o e

~

E 1:

_ o e

_ o '

, i i

_ o N

i e i i e ie i i ie i e i e i i i i O, O

,e ,8 e

$ @ an

(>I) aargeaadwal Fig. 31. Water temperature nr. the sample problem with new revised heat t ransf er.

81

Rle [RAr

~

l'. Vaperezation/rordersatior A1 dei 8 h.. i r r

• This settnor gise- t ht' t.. t t i v . ' .. ? e qu.. t ion- that are wlved it the va re r i z a t i er / s i rd e r.s.: t 'ini - i . ds ! .id ti r reas o r.s for the codir; ir d i f f i cu l t -t e t or.ve rge -situatiors. 1. r the purposes of brevity. the equat iors p r e s e r.t ed irtiude the liquid-liteid heat-trarsler modifications. This is the marrer i r. which the tede is- expected to be rur. ir. the mol t en-core /coola r.t -i r.t e r a c t r or. prograe. Ilowever, the heat-transfer charges'do

. cor.stitute a separate correctier set'. /VAlmR/WRBilLAT ( see Appendi x G). and the~

va po r i za t i er./conde r.sa t i or. modifications are equally applicable to the- old heat-transfer nodel.

The use of the modified liquid-liquid heat-transfer model to cause vaporizatior directly from a water surface followir.g liquid-liquid contact suggested that water's subcooling energy be included . in an effective heat of va po r i za t t er.. Experierce with runcir.g the SlhNf.R-Il code has further sugges?cd that it the case of tordensatior, the vapor superheat-should also be included ir an e f f e s t ive condensat ier er.e rgv. These two changes have been ir.corporated in a -

r.ew correction. set / valor /WRill'Ilu. l i s t ed i r. Apper.d i x 11.

The differential equatier.s' treated ir the vapor izat ior./condensat ion mode l are taken frot the basst torservation e qua t t er.s by igrorir.g convection. These are (a ) Vapor-1:re rev 1:ouat ior r , hWT- 1 ef;+1 . condensatior.

E(r+ 1 'b = f(; et - At

[ s m Ifi I }

n:= 1 e[.j,,y+h y . vaporizatior.

hmT-1 2

+ at- [ ix,qjdf,,tg+

p [ gjjr\sk) 84-3) e=1 k=1

N414T- 1 n - rH -

NrLI:

+ [

r;= 1

[(1 - AnlELm 'Lt 3 - O' .- !I1 ~1 3

I 4"thii3) 1 c-I 82 n ED' "*N?.--?e -*a**'=7 4 ' - * *9* e' ~ ' - - - - - f-- p 4 --'~

?

R3 UGH DRAH i

(b) Li ou n d-l're rrv l'ntaa' iors

't. '.1 -

. s i r.d c r sa t s er.

i

,r+1 'r+1 "Le tre.

" "r'LeLr.* -

r- #.- I.-I -

'r tiLr. } .3 5 :

'gj Le . vaporizatier. ,

-- n -r.+1 -

(I ' *n i tr: 'Lm + At 'lil lbi3 m = 1.2.3.4  :

+ St x g, q g().(ir. ~

(c) Vapor-forttruttv l'ouatior NT1AT-1 h\ TAT-1 7 r.+ 1 r ~

(46)

~

E(;' =E;-at g _

x n f(";') + y (1 - xg) Egn  :

t=1 n=1 (d) Licu id-ror.t i r.u i t v f oun t iori. .

P r+1 r - r E, t

= E Lt

• OI *e f(",*d - ( 1 - a n iE tc .

n = 1.2.3.4 . t .t ? )

. (e) St ruc t ure-l:re rrv l aua t iers

. hT1 AT- 1

' lESktir: t45)

ISk'kk

• ISk'$k* -

r=1

. L = 1.2 -

p 5

!. (!) AltiS i

h11\T .

+ -

Eu+1 x+1- =+1 -

l ' (=i+1 * - k1

' Va p k1 * ' v(i'1 ,.g T=;+ 1 - Tx+1 hat.k1 J , ar.d t=1 s

.T' I j T",*I I = p'g g exp[

Ph!= + 1 , 7(,,t 3 R"g* g

• I 5' T"Sa t .k1 4

~

i i 83 ,

j

? ,O

[="

INW w DIl hl\ l ,

( p 1 Va por -1.r e

• rv fle a t -T ra r t e r i .ie ! ; i , i r r

• s 41 p*1(c L - [ rh.

li<ilr t " d' d

'

• P ~I$( .Lr; E p (i'U(iln ~ I (50)

+1 -

~ I=I SL ' p(i U.41 liSL " .

exp.-fy.1j gt pc (;/ll'isk t '~I n+1 where g l(ir:Le " U(s+1 iLe :Tn+1sa t . M - T(n+ , 1;.

~

n+1 41 Letir- " II L(i-. .Tn+1 hat.M - tT ; .

u+1 +1 1 ,

4 )(itSL Il=iSL l T=+at.M h - Tx+1 g;

g.1 .

-r+1 US(ik .Tn+1ha t .M - T3g 4]SL(ir x

7

= 1 1. the tor:i ra l s a a c .

x ,. , = 0 if all of energy c orpo r.e r t t i a, vaporizar.p. and the qHLr-(i3 a re giver. bs the heat-transfer nod i f i c a t i er.s of Sec t ior 13. and the r e t.a i r. i r y t e rms are defired by the SikNER-il na rua l. No ger.eralizatier. of the t o r r.a l l s r: te i r.c l ud e erergy t r a r.s t e r at surlates without toss t ra r.s l e r i r. the va po r -e r.e r g y equation has yet be er a c cerr.pl i a hed, l'or e xatpl e , solid particulate carrot sirply heat superheated steat. The available structure surfacea, stil!

cor.t r i but e to the vapar-erergy equatier ever in the cases where all the apropriate liquid vi. p ' r i z e a. . although the T"sj,qu,ed for the,e ,ituatiers s i.

based o r. a tenperature vieldir.g balarced crerry transfer to a structure's surface. A l i. o . 11 , ore el liquid c ompor e r.t 3. water, is presert at the be g i r r. i r g of the tire step. qHLdi3 wi l l always be evaluateJ for each relevart liquid-erergy c orpor.e r t .

The total phase-transitior. rate is obt a i ne d bv surni rp I.'q. (44). (45 . ard (45: ard equatiry the separate corpererts. This gives 84

=

h 9 "

RRV lllJ" l 4 J I i

=+1 x+1

+1 .

G o r. . M ,

t -

. I=ilt =+1

,ien.M 3,*+1 , , , -l

>g.% 1/

2 NCLI:

x+1 =+1 -

x+1 +1 -

x+1

'lltitLr + 41 Le(ie + - 8 4)(itSL + 4=] Sk(itI+Om3 -

'll lLkti3 - I61) k=1 k=1 where En3'is'the Kronecker delta ard xn is assumed to be 1.

n if ryhat < -ALn. . a l l the liquid of cocponent e vaporizes and this equatior. is presumed r.o t to appls. If a r. effective ' heat of vaporization, h} y,',f y . i s defined as c(";+I - e("+d , y if corder. sing h} y,,fr = { ..

+

. (52) ej,y+h y - e "Lr, if vaporizir.g then Eq. (IV-9n) of the SikNI:R-Il canual gives.r$+d'withhj y,,yf substituted for h ". g. (The SlkN1:K-i l manual equations- referred- to- are giver n r.

Appendix 1.)

The first Ne w t or.- ka ph s e r iteration procedure. ~ l:9 (IV-97-102) 'ot the Sitter-Il manual. is now .the solutter. for I:q. (lV-96) of the car.ual written as 2 NCLI:

x+1 x+1 j4 x+1 =+1 x+1 -

1 Ix+1 glr- " 1 (irle + 41LcGr + -- I'lltir SL

• 41SL(ir ) + 6 g3 _ qlllk(i3 'I9' h .j k=1 k=1 j, y,,fg The required mod i f i c a t i e r.s are ir t h e' e va l ua t t er. of h ". y,,pr ar.d Ph} y,,7p/PTSa t .M a u umi r.g t ha t T(; is c or.s t a nt at T"; g .- To obtain the derivatire tert. we must differentiate Eq. (521. llecause se are solving the' coupled continuity /LOS relatiership. eg; is given by 1:q. ( .89 ) where Xg is the updated mass f rac t ior of each c ompor.er.t. Ile ca u s e a t uli evaluat ion of eg; is only made at the end of the iteration. eg' "; l is determired tren 85

F j m==

l'i V w di L fj h'.1;T - 1 p e'x

. e"

~ '

e(i

".+1 ti 4

- - i T *S. ,*1..- T '% . '1 4 -

t . '54>

M.1 '.] %+.M 0 *Ei 'Da p.M x O'(Ein t.here = X, A I(gi ~ Iysat.MI,;

• 2T q CI .

~ 'V(iM + 'T C

L t .M  %'t . 4  % t .4 L '(i . = . M ~ ' Va p . M I ~C v(iM(Tg ; - TS.i t .MI .

Because ({.g;q = cy (;q + ca x { ti. ) .

. T(, , TSa t .M, i

gM

.=-

' v(iM i f c y(;q = cy(;q . -

CT

~ 8* l c '(IM > c V(;q . ther. we obtair.

Sat.M O'[i 2c(g7,y ,, Jq y ,

" Xn ~ ' V(iM -

(i Sat.M BT Sat.M M 21 Sat.4 T=; - TxSat.M + EM g

(551

& c("; - l g"3 4 2ttyg;q - c(.g";q i

" XM ty + t T(; - Tgha t . M ) ,. Tghat.M > T(g; eT S t.4 T $.i t . M - T("; + Jg

-K x #' Var.4 .x where ty g -t y(;q b t .M

-u C

liie e x p r e s s i e r.s f or .%p.4 are eT

% t .M

.x .

I'"~ . 4 ' IT"-% :.M '~

f *Va p.M li"ip . M

= s C Ix%t .M cl Sat.M y ty + . CT

% t .M Ty Ty 4 [2Crt.M ard (56)

' ap,M B h g"j , q j T

• z- hat.4) - h$r.M T

- c . pq + c,T - ht.M > 3 Tt rt.M t

eT hat.M Lt.M - Tg Ty The next term ir. 1:q. ($2) is e"$,q.

g It is giver hv l'y. . (A-6) ar.d$pdatedeach irrer i t e ra t ier.. Its derivatives are 86

e RC AH DRW

-n

ro r . 41 '

BT

• ' V tgj T

% t . 41 EjT r tr . \1 -

L t . k1 and .<-

1. 't er . k1 . 1 N \1 *.

Ts3, , gg > Trrt.k1 -

33 " ' vl41 ~ Y - 37 1 .

Sa t . 41 Sa t .41 Obvious!v. e( or . 41 as well as its derisatives riu s t be updated each i r.r e r steratior. The l i r.a l rew e x p r e s s. i e r is that for e[+1 k1u l t i p l i c a t i or. of 1:q. (471 by e[*I ,

arJ subtituttor n r. I.q. (45) gives 1

r. r. '

gLt',Ltj "gLt'rLe

• 01*r r=+1 * (=

GLri o + n . k1Lr.

~'x+1 -

! + at xnql Ldie 0

-E 'E* g+]

-(1 -x n Ivtc.t e Lr. ~

'g+])

Lt + a't gilLdi3 . (55i Only the x t

= 1 (a e er.d the va po r i z i r.p op t i er. r.e ed be cor.sidered te obtair a r.

I e[$ e st irca t e fer I q. ( 52 ). l'o r t h i a, case, we obtair

""" I e[+l =

e{r+at + aI (59,

• tr

• tr If the e[n used -in this equatier . adiusted to be t or.s i s t e r t withhte (delircJ by 1.q. tIV-9t-921 el t he SitNI.R-! ! ta r.ua l ) . t her 1:q. (541 betores e Lr x+1 "

~

' Lt. + at II LG-(T"*t.k1 s ~ tr ' + IILMi3( T"*,1 ,3 -

I h'.

Lr (6oi Lr:

Pe((I li tg7 a r.J = at ( ei)

. 'I I

1. T"% t.\1 A, g where li gg;3 is ze r. 11 r = 3.

87

RGH Dyr

'ThC completes ~t he resessary rew' e s p r e s s i er i, to evaluate I:q. (52i a nd . i t s derivative .

A secord hewtor-Raphsor s t e r a t t o r. precedure is used to solve l e r. the vapor-side hea t -t rars t e r toefficierts. I:q. ( 50 ). Eq. (50) is substituted into the e x p r e s a. n or s for the na a. s - t r a r l e r - ra t e , at a giver. surface. This give.

expressiors like 1:q. (lV-104) and Eq. (IV-105) of 'the Slbf.1ER-I l manual. The f ur.c t ion for the he w t or -Ra ph s e r. i t e ra t t er. is ther defined by Eq. - (1:-11 ) of the 514f.1ER-I l narual. This i .. s t ra ight f orward . t o modify. The FORTRAN variable.

CPDill'(1. which i s (i'p/hj y i. is simply replaced by (cp /hyp y,,rf ). llecause eore than one surface c a r. engage in mass transfer, for example, droplets can be vaporizing while structure is corderiirg the same coeperent, the object is to tair.tain hy py,,yj c o r.s t a r t throughout this iteration. Alse, f or wa t e r-d repl e t surfaces. the yllLr-(13. t e rms cortribute to the nass transfer. For such water surfaces.'the functier. Ier the Newtor-Raphsor iteration si, giver. by Eq. (43) of Sestior D.

The solutier to the vapor er.ergy equatier is more c or.p l e x . The startirg

• I point-is to rult iply Eq. (46s by e g"; ard subt rat t each tere- of the resultity equ:. tier fror 1.q. (441. Using the fast that the converged e.olutier, e g";+ l . w i l l e qua l eg';* I . thi. operotter. sields Nst;T- 1 0

~

+1 I-7(i e6".+ 1 = 7(i eE 6 - At -

x r r("itt i-

,,j y)

F"I 'Cor.M + hy 3j. , q - e(;

NMAT-1 2

+ At [ lxe gjbr Lr

• 4$bkSk ) (62) e=1 k=1 NMAT-1

_ r -r+1 .

NrLL tm egg -e[p-at y

+ _ il l - e x lh _ ,(1 - x3)

g";gmg;3, j .

r=1 n=1 The obiective is to determire T";+ g 1 from Eq. (62) with the o t.h e.I variables corstart. The value of r";$

g doea, r.o t eatter with a condersir.g c ompo r e r.t .

Cor equentIV. b;.,q,,,y car he a*.sumed ass e"(,y+h". g - eyI (fror Eq. (52i which has re direst deperderte er Tg ;. 'We are assumirg the saturatier temperature 1. - Ie terstart while selvirg 1:q. ( o2 ). Ther makiry the 88

n-icdU6ti DRAFT a ppr ox ica t i er. . 'e";^I = e f; + ~ ".* I ( T";' g l - T",s. ard puttir.g ir. the vslues ser the-hea t - t ra.r s f e r t e rres. I:q. 8e.2 i b es: r e ,

Eg l e g"; + Eyg; I T[;+ 1 - T[; ) = T[; e[;-

h\14T- 1

-A' S

• r: III Uit(T[j ,q - T Lr.' + II[i[j.(T((j ,y - T[;+I) r=1-

• 1-

-r+1

+ H(y T,,t3-.M - T,;

j ( ha g j (3

+

Y 3

-n+1 s t . 3 - Ttq ))

)

- .U[ StiLIT=+t.M S - T 3g iSk _

U tqc;3( T,,a k=1 q=1 O cor.densing

• I .x+1 I

'[' ..M + h M ' ' '[i ~ Ev (; (T[; - Tf; ) . vaporizing f g 1 +I

,(=,+

o r. . M + h ".+ IegLM - e"e )

(63) hT1;T- 1 h;1AT-1 2

+Ai

~

+1 l ~

1

.itl11"itt1 T"*a S t .M - T(".+ s- 1 i - + ai -

~ 11

- S (ik ( Sa T"*t . M - T(".+ 1 )

s rr= 1 t=1 L1 hCLl'

-r+1

, - at(1 --x3) ~ ' ~ llgg;3(T$]j,3 - T te )

rt= 1 h\1 AT- 1 7 ,7 3 _,,

+[' (1 - A g.)h getn l - ef; - C (;fy T";*1 - T"; ) . g =0.

r=1 To pla c e 1:9 (63) i r a na na pa bl e f o rma t , duerv va r i a bl e s are defir.ed as

, , 2

- r.+ 1 -

hCLII

-r+1 ag = - At . Lll (itfT((j,y-T Lr I + Z IICtik(T((j ,y - 1 3g i

) + ~ iltqc;3(TS t.3 - Tty )' .

L-1 q=1 2

b.r - - At (li";[j. + [ ll[;()

g

, a r.d --

L-1

• • corder.sity d, = l ..j l .

r[ f.y + hl- q - ef, s'yt; (Tf,d - T[; ) vap.r:2 r;-

89

R:c3- DRAE Ther. we collect terr or the s.ri.ics p.iwer of T0+1.or NMAl-1 o t.r. der ir;

( Tb".' l a- [ -

s r

s41 .-]

r=1 l'r v(i /h} y,,y; va po r i z a r.g f31;T - 1 hT141 - 1 2 ll((j, - [ i At il{;f

~

T[*l [ - E( C(.[; -[ xy at t=1 rc=1 k=1 f314T- 1 o c erd e rs i r.g

~

• r I -x+1 1 F"I I"tE v(i +bTd,gy.x+1 E (; ) / h"' M.eri vaporizirp (64)

PS1AT-1 t31AT-1 .j n

-[ xn.( mr b d .1/h" y,,rr-[ Cy (; AltII ~ *tI3 t=1 r.= 1 PS14T-1 r x*1 ,

+ [ -;(;(e" - e(; - C y(; T("; i +[ xc.at ll(";[j. T((I ,y t=1

?314T- 1 2 ,

NCLI: -n+1

[at II's*[T"s$t.4~0LI 3 I Z IILt(i3( T((I ,y - T t , )

+[ ~X t=1 k=1 t=1 f31 AT - 1 ad + b T"*l""J. t31AT-1 -r+1 "

. x+1T(;" ) ] = 0 .

+[ xI r '+[(I - *r IELtLe ~ '(i + VCi r=1 h q.n.er: r-1 Eq. ( 6-8 ) has the tort et a quadratic AT[; + BT(; + C = 0. As writter. A<0 se that the regative square root of the quadratic forcula gives the higher terperature ar.d furnishes, the first choice for a correct s o l u t i or..

The special cases adopted ir thi. revistor. of the vapor iza t ion /c ordensa t ion raode l a re a s follows:

(a) le the first Newt or-Ra ph sor i t e rat ion a ll previous special cases ter treatirp a near-critical pressure deu rthed by Apperdix 1: of the 5thNf.R rarcal have been elitirated. Irdeed, with I .q . ilV-looi of the $14MI:R-! ! r;a r ua l ri t writter. hv 90

n q m -- -

RIu llJ-s i

I-l, = ,r .s , .'1.r

• 2.r h:. .M

• (,

+ =0 ' ve ,

- *i; hj ,y,,j3 the I furction car be triple-valued wherever h ,y,,77 is small, not just near the critical preuure. Also. we east avoid oscillations bet weer va por iza t ier. or one i t e ra t i er. and cordersatier. on the r.e x t . To resolve these problems the solution logic contair*, the following elements:

(l')

hh.M.cff p 5 limited to a tiricum value of 100 J/kg.

(2 ) A variable ICOUNT( N ) is defined for each component. Normally. ICOUNTtN) =

0 lor the ne rr.a t case. ou illations are limited by usirp Steffersen's rae t heJ . 1:q. (lV-116, of the SIM'.ll:K-I l ma nua l , a c c e l e ra t i ng t he converger.se to ar. average value. -

( .1 ~ i Ar. abnormal ta e is delsred by IniUNT(N) = 1. This occurs if T(; ) T(rt.4 a r d T,.

> T( r t . 4 or if h (. , y , , g 5 1 000 1/Lperdi'"/dy,n.>-0.1 . Urder t he se wrd i t ier . ta x itur a rd rgi riru pe rti sa. i bl e der.s i t ie*, a re upda t ed ea c h iteratier based er whether 1" is repative or positive i r. the previous 1*

c a l s u l a t i e r *. . A bisestier. procedure is used if the New t on-Ra phwr i t e r a t t o r. tries to extrapolate outside the rarpe where a solution is krewr to exi t. The bisestier a l go r i t hr: is also used every 20 iteratter. il 1('0VNT t N ) = 1 to <. t o p the rare pathological case of oi.cillations .t u s t withir the lower ard upper hourds. l'i r.a l l y if T(;. T g. a r.d T$at.M 3

above T. gr t.y arJ 'he value of T(at.M implies va po r i za t i er. . the riritut i va por dersit y is itialized so that the conve rged value oft [yg,y will be above T('r t . M. I'. rtial solutier.s with a large value of h",, y,,py ,

are l torsidered spurieus for such a case. j i

l f -I i Huillatior. betweer vaperiziry ar.J sordensirp from iteratien to iteratier l l

are turther l ir:i t e d by derardirt that the l' i r s t term ir bratket- ir j l.q. #ofa be p.sitise while the sesord 1 err be r. epa t s ve for cerJer atior. l l

ler va pe r i z.. t i e r . the lir t terr r.u t be repative, a r.J the a.c ord t e rr l 91 l l

)

r i

RC 9 3RI C positive. Orte a selettior is r;.i J c . ..

swi t ch ir the carne r of ca lc ula t iry b .,q,,jg- r e g t. : res s atisiv.q.' h. t i .pp.siry corditiors.

(5 ' Although the spe, sal tighterirt of the torvergente criterior. with T s3, rear Trrt has been elitinated. a r.e w criterior. has been added. how. the relative difference. betweer successive density iterates cust be within lo

in addition to the standard criteria. See the discussion followirr Eq. (IV-102) of the Siktil:R- 1] Manual in Appendix 1. This allows the necessarv extra iterations when eacroscopic densities are stra l l .

(b) In the seterd hewtor-Raphson i t e r a t i or. . the special cases described followir.g Eq. ( 1:- 12 ) et the SlkNER-I l catual remain ur. changed with ore e x c e p t i e r. : arv species that corrplet e ly vaporizes i r. a time step is a ssur ed r.et to contribute te Fj( ir Eq. (50) at a structure surface. The reasor. for this is ( er s i s t e r.( v w i t h the treattert of T[*I ,y ir the vapor-er.ergy equation for such a case.

(t i The spettal cases in the vapor-energy equation associated with the discussier et 1 4 ( l: 15 ) of the 514NI:R-Il tatua l also renair the same. with one charge. Ile c a t s e the structure is row presumed to transfer crergy ever for the case el all the liquid vaporizing. a spesial value for T[j ,y is used ir calculating structure, vapor heat trarsler. Consequentiv. wher. all the liquid is vaperizing, the value c! T[},qusedir.theterell[$(T[j,y-TS*I) is 2 2 1 t 3( T(=;+1 + _ lig g Tr

_ ll=+;g

+1 k=1 k=1 T= Sat.M * . (661 2

1

- ( Il=(+it s + IIrn) k=1 This delir.itier teans that the erergy flows ba l a r.t e . so no mass traEsfer occurs

~

at the structure's su*1 ate. l'o r other uses when all the liquid of a giver 1

comperert vaporizes. TC t .4 is detired so that I:q. (51) is satisfied, which is higher that the true satur. tier t er pe ra t ure.

92

ROLGF [RA-T The retairiry (b.:rgt is ti t nt t. pJ a t t el the liquid erergie*. followiry the vaivrizatier. serdtrs.iior ri..v s 1.. i 11t reitiry-freeziry taltulat i>r.

1.q . (%) is used l- th;- p. , s. l ni t t are ro problets for the va lo r i z i ' ,

tuse. The q[g[7h;3 terr is evali.ated ard added separately. The case where x n is

.c ro is uncharged. The torderairy case. with x g = 1. is evaluated implicitiv f r on-

-r b q] Le(;r 45 f()Lr- r+]

1. Le ^ r r. 'Cor.t

-r+1 I I Lr Lr. -r r e Lr-at- T;g m + 'La~ 'Lt . (67) g 1+

E S

"Le

-r+1 where e te, is the temporary updated liquid energy.

-r egg i r.c l ud e s liquid-erergy updates frot previous energy t ra r.s f e r s . and r -

eLn is the 1 cuid er. ergs at T g. .

This cer ludes the de s6 r i pt ior of va por iza t ion /cor. der.*,a t ion nede l c harge s.

Their irproverert ever the versior in the SlhMI:R-l l tatual ii. derorstrated in Settier 1). Dif fic ult ies arise where the varor-energy equatier ard vapor-t o r.t i r.t. i t y equatiers are closely coupled: ir other words. where T; . a strerp g

! u n s t t e r. of T ha t .M. Ther evaluatier. of T(; hold ir.g TSa t .M c r s t c r.t and T Sat.M he l d i r.g T(; c o r.s t a r t is a poor algoritht. Ilowe ve r , solvir.g a linearized versier of the coupled 1:qs. 1441 to (-19 ) each i t e ra t ion wa s judged to require exceuive prog ramr.i ty a r.d wa s bevord the scope of this prograr. Pe rf orraing ext ra outer iteratiers for sush tases r.av he unpleasart, but are r.e c e i. s a r y.

93

'q ...

RI W 1

l. 'O ss c ' f a r e.'u s t a r r e

• t o r i, the A1 '*er-i..re ( '. i . art I r t e ra t t i or 18terrar f.'-

A sti ta s t .r re t ' i ii se* w.: - terst..'ed * .! e i ' w i t i: proflers rot treated ir the i

.s- L. s Al11% 1e..' 'r.rs: c' .. r . s. i i. . *

• 6 rdirsatnor d i st uu ier s. Th -

scircitier *, e t is piser ir Apperdis 1 4 unr a r y el the serrettnrs is .o l o;1..w :

tal 4 crude lit to the stean tables for the liquid water's thermal conductivity was raJe. The Fregrartiry auure, that liquid water is described by eterpy t emper e r t 3. lle c a u s e the t he rra l c or.Jat t i vi t y of liquid wat er is so much lower ther that e! the tuel. it terds to be one of the key parameters contro))irg w;. t e r terperature and water exparstor..

tb> C.trettieri, were ir. erted allowirp the maximut water-dreplet <ize to be a rultsple o! the i r. p u ' raxieur. This allows bot h f ue l and water to have fixed but ditterert droplet sizes ir the i. a r e tell. Nuch input was used to fit the WL \ti-19 test data, ici The IMI LUli < u h r e t. t i r e was leurd te igrere bourdary cell preuures after servergerse et the p r e s s.u r c iter tio. Thi, om i s s, s or car l e a.' to spurious re uits ir the Weher rurber cal 6ulatior. Icr the liquid's droplet size. A terporary terrestier t ,. relle,' preuures er the left, top, a r.d right-hard bt.urJ . r i e s was :rserted te pertit Ntter calculatter of the shallow-pool experirects.

(di 4 tire-step sentrol allowiry the vapor-erergy work tert to affect the everall t ire-s t e r t ert re) er irternal-erergy s h a r p e i. . in vapor DTL (11). was i r. s e r t e d . Thi, tertrol is 94

c I

(O m y.

~

W 'W j s 3i [

.'-1 ..,

'-1 t-l '

_. . .]

i.1. o.j .  : e,s .

g, i.. k wl.e c

+

g 4, <c ;r i Un >r+1 g , + a,<otrI l. t > ', *I a,<a.V;>[*I+A,<otL" gg V L, = p +

rfar, "' .i crd the r.o t a t i e r. is th t ef the $ 1'411 R- I l r.a r u a l . This t ine -s t e p cha r.ge w a i.

rade retes ary becau e et the c% illatters that otherwise developed ir the aralvs s of the hallew-pool e x pe r ir:e r t s ard because the e r.o rto u s p r e s a.u r e s of ar ir-vessei tc r expl.sier sur readily lead to spurieus vaper-erergy va l t.e s urles she - t:re steps .: ve taker. A better procedure, although bevord the w epe ..! t;is studv. is te irslude the warL terr implicitiv within the preuure iteratier.

tes lit s .. .s e J c r i t i c a.

. at differert tires are used ir its cor.s t ruc t ion. the'

({ L.a r I i I v .

1 ' Lr-

. ~ .

ti l 'l" wh:t h Je t e rra r e s the t i r. i r ce- vaper p r e s i.u r e for a c ocpe r e r.t ir a s i r p l e - ph.. s e cell, wa.

• curd to ye sp.iriou lv regative in sore situatior . The invers- et this uaartity t a rra.t t or t ri.! the pressure ir these s i t ua t i er.s . and a lo'; bourd was trierted.

4 95

Rl {

RUL diI L/ I Ii1. L( Al:R-ill AD WD1 L I OR' s1st.11 R. I 1 A. I .Jc,tn-The lower-head rodel developed for the s t e at-e x p l os i or, study is a teditstatiir o! the plup redel develeped to tiack of raterial frot a reator vessel to the t o r t a i r.re r t building during at Lhil llR rore Disruptive Accider.t t rDA ). That plug model is desc ribed by lle l l . ' ' A descriptier is also p l a r.r.e d for revisior 2 of the Siht1LR-I l ra r.ua l . The modificatier.s cade to this plup rodel to create the preser.t lower-head redel are (a) t rar.sforming the bourda ry treated fror the upper to the lower interface. ( b) takiry t he bour.da ry of the novir.g structure returif ort as a f urt t ier. of radius. (c) i r.s e r t i r.g a siryle-degree-of-freedor head-failure model based on a correlation te results calculated with the ADlh1 code. (d) generalizing a radial c or.t i r.ua t i ve i r.f l ow/ou t f l ow hourdary conditior. for caterial release in the vessel's p i p i r.g n r.d ir. the savity below the ve s s e l , a r.J ( e ) ir.serting r.ew output s t a t ete r.t s for forces over the rerurifort upper head anJ for plottir.g other transler.t details.

11. Irrg, A Jew ription of the todel is ylver through ar explaration of the i r. p u t for . particular p r o b l er:. The probler e x a t i r.e d is the Siht1ER-I l struttural setup for lower-head failure s h o w r. i r. l'i g . 32. with the i r.p u t description t r.

Table Vll. I'rer axial redes 13 te t.n the struttural s e t u p wa s taken free the Zi o r / I r.d : a r.- Po i r t studv. The seature- represented are the vessel's i r.t e r i o r ,

the core support f orgiry. the radial s h i e l d a r.d tore barrel. the vessel wall.

and the irlet ard outlet pipiry. The extra input te represert the failure and notier. of the lower heaJ repl.nes the PLUG IMTA. c u r r e r.t i v described by S 14t11'R- I l irput cards 55-55.

1r this problet. the novable lower structure is defined by setttry ISO - lo. JSo = 12. AhD J PLll(it I ) = 0.0.0.0.1.1.2.3.3.5. The lower head thui. defired is modeled by hewtor's secord law with a c o r.s t a r.t mass, giver as PLU(iA1. or 11 000 kg, on card 55. To roritor the ferces o r. the upper C. R. lle l l . "41od i f i c a - sr e f Sitt.11:R-l l t e r 41.. t e r i a l Tracking irto the Reacter rottairrent llu i l d i r g lA.rirg Pou disasserbls I.xpor ier.~ urpublished Los Alara Lo i. Alares % sertitis L.. b.. r a t o r s .. r ! t t e po r t t 14N ).

96

l R  %. .,-=

=

RP, W ) ,  ;

head. an outlir.e el upper-head structure i' also irput. Ier this case. I VIT = }

II- l h0I = t . a r d .II'lllGTi I i e.- -... < , . ...;.-l. .,..,..4,.5.-6.

i l

1 I

i

)

97 1

f

RCUGH DRAM LOWER HEAD FAILURE PROBLEM 180T JSOT

• a 60 +---OUTLET PIPING 50 l l l ll l l-l ' SOLID STRUCTURE 12.54 m INLET PIPING IIIIIII !

DOWNCOMER 40 i

30 CORE SUPPORT 20 FORGING i i

JS O --*- l 13 i,;_,,,u g hi i i i ii i

, l I r kLPbh 11 t J I I I II i.11.Irl'J' N J'-' MOVABLE STRUCTURE l I r brJ L b L T F1 1 I I il I'l J 9 J t t 'PE b 1 1 I I ll ,

5.20 m lao 4vrdLt2III II

, Pk L PEL 17"L1.1 I I I ll LULI14 4, f 4 4 4 I I I II I C L L P f 3

• k19*l I I I II

, [*k & 41Ll'l 111 I I I 11 19 *1'll h L ?)*Li i l I 11 1' 1 mLtd t i bl'k I I I I !"  ? OUTLET TO 1 5 10 15 KEYWAY

+- 1.613 m + .-

• -- IS O *

+-- 2.53 7 m  :

ig. 32. Mesh and structure geometry for the quasi-mechanistic reactor steam-explosion problec:. (Note: Nodes are expanded in the radial direction.)

98

R3lGH C E TABLI Vil LOVI.k-Ill All %>DILING l hl't;T Card h.. 85 il ok'1;T: 316 ) Iso. Jso. IsoT. JSOT Colurrs Yariable lleu riptnor 1-6 150 Right-hard boundary of the lower head.

7-12 350 Top hourda ry of the lower head.

13-15 lhoT Righ -har.d boundary of the upper head.

19-2-1 JSOT Bot t o:: bourdary of the upper head.

Card No. 56a ( f0R$14T : 10161 (JPLUG(1). 1-1, 150)

Colurrs Variable De se r i pt s or.

1-60 J PLLIG Radicily deperdert addition to 350 defining the top bourdary of the lower head.

Card No. 86b ( FORMAT: 1116) ( J PLUGT( 1 ) . l=1. ISOT)

Colurrs V4. r i a b l e Descriptier.

1-66 JPLUGT Radially deperder.t aJdition to J SOT de f i r i r.g the lower beurdary et the upper head.

Ca rd he 57 iFORst;T: 1216) (JCONT4J). 3-1. JBAR)

Celurrs Variable 11e u r i nt v o r 1-72 JCONT JCONT=0 Rigid radial boundary conditior at this axial location.

JCONT-1 fortirustsve irflow/ outflow bour.dary condition at this axial locatier.

rard No 68 (TOR \itT: E12.5) PLUGM Colurra. Variable Descriptior 1-12 PLUGS 1 Mass of lower head. ;g.

5.

99

-k i

RIGH DRAFT 1r this rodel, the mova bl e "s t ruc t u re ~ tus t he input eairiv as "no-flow ~

velure ir a special reg ior. pa racel e r set. The "r o - I l ow'" vo l ure - tru s t exceed the irtins*e drag licit. A LI)RG. Th n. rt r.s a sr .t ! ! a m..u r t - ef claddity tu - be presert. e xc eed ir.g RsCI AI). whic h sh. uld. he set to a small nurtbe r. Ile ca use the l expar.diry fluid f rom the vessel wn!! he highly two-phase (mostly vapor), erly minimal liquid-volute tractior is desired ir the initial r.' ova b l e " structure" region. A saturated stean er.viror. cent was considered reasonable for this claw of problems.

The JCONT4 3 )' va r iable a l lows out flow f ir.f lew ) at specified rodes along the radial bounda ry. This is more completely explair.ed in the- sec t ior. or boundary c or.d i t i er.s . In the current probjec. the three axial locations where JCONTijs equals ore and consequer.tly allows notion are 1.15 and 66.

C. Ilead railure lie f e r e the lower head can move. it mnt fail. Fir.ite elenert calculatters (see Apperdix M1 irdicate that failure car be expected at the radius where the outercost vessel penetratiers are located. A circumferential split would he expecteJ te proceed arour.d the head at this radius ar.d leave the i r.r.e r por t i er.

of the head as a free body. A simple. sir.gle-degree-of-freedot spring-n ss codel:was correlated to these 1 : r.i t e -e l eme nt results. This model accepts the average pressure load ir.g ove r the the lowe. head. and furnishes the time ard dowrward head selocity when the head diserpages from t h'e vessel. A surnary descriptier of this nodel is giver ir. Appendix K.

Other irf orcat ior. useful for understar. ding the programming used it callir.g the failure codel is as follows:

(a) The pressure differer.t al is corputed assuring o.1 MPa e .s i s t s outside the vessel.

(b) The average pressure used by the rnede l has units of psi.

' ~

displacetects are in inches, and velositses ir ir.ches/s.

(c) The failure flag is hl% I L. Wher. blTIL = 1. lailure is indicated.

The initial veiecity et the rwviry head is the final velocity corputed by the failure nosie alter serversier. to metervs.

100

R:tGH DRA:T D. Lowe r-licad 41 t n or As irdicated ir the i r.pu t deu ript ien. lower-head nat ion following f ailure is computed by hewt er. 's secord law. The au elerat sen is c omput ed f rom the t.:ss

. ot . the head and the integrated ferces acting e r. it. SIWER-ll . cott ut es the pressuie at the upper surfase of the lower head. This is the pressure in the axial r.ed e where 11 surface actually exists; or, to avoid over-expansior.

problems. cre node above the surface level if the head occupies more than 6m of the local cell volute. An opposing pressure of 0.1 MPa is assuced to act or the lower surface of the head. Gravity is arcluded. The initial velocity comes from tie head-failure model.

The variable ZPLUG. defined at head failure as zero, is used to defire the lowe r-head di splaceter.t. Each radial node is given a bias. ZBI AS( l ) . relative to ZPLUG. -from the input atrav JPLUG(1I a r.d the axial mesh spacing. This locates the structure loca t ion. ZPLUG + ZBI AS( l ). between axial interfaces giver.

by ZTOPfl! a r.d ZBOT( I ). Wher. the quant it y ZPLUG + ZBI AS(l) is detertired to pass ar irterface, a' fla2 (JPLUGsti)) is - set to begir a new ride. Upwa rd a s well as downward notier. of the lower head is permitted. although only dowr. ward motior is articipated. Besides charging the interface indices ZTOP(l) ard ZBOT( 1 ) . . downwa rd notion requires the fluid velocity at a forter ZBOTil) i r.t e r f a c e to be set equal to the lower-head velocity when the head passes that i r.t e r f a c e . The array J PLUG ( 1 ) is redefited in the transient to represent the actual axial rode occupied by the lower-head ir.terface. To avoid problets with zero'irdices. JPLUG(1) is not allowed to be decreased below 1. The lower-head structure simply disappears when it reaches the bottom of the mesh. Figure 33 gives a pictorial view of this arrangeeer.t.

As can be i r.1 e r r ed f rom t he above discussion. the lower-head dynamics are coupled explicitly to the SI WER-Il fluid dyr.atics by havir.y the no-flow volute time deper. dent. The SIWER- 11 fluid dyr.atic s then follows a natural course it which the fluid taittains contact to c er t i r.u e the acceleration. Nuterical problets have beer. articipated when the head has a high velocity and passes into a rew cell. A t ime-step cont rol is applied to Ielp stabilize this problet.

Usir.g the quantity I DZt l ) defined in Tig. 33. a t ir e-s t ep cut back IIrevert s the fluid-volume fraction charge caused by the cosity head trot beirg greater thar it< in ore t ite 's t e p. ( ors e qu e r.t l y. wher. the fluid-volute fraction is stall.

the tite step will terd to be stail.

. 101

R:03F DRAr~

gypp(i)

FWID i P = IDWER HEAD 11""1""" BOUNDARY

=

(ZPWG +

ZBIAS(l)) ,

SIRUCI'URE (NO-FIDW VOWME)  !

ZBUI(l) l I-1 I I+1 ZPIIIG + 7RIAS(D 'ZROITD FDZ(1) = .

Z'IDP(l) - ZBOI(l)

= LOWER HEAD VOWME FRACI' ION fig. 33. Geometric indices for a radial rode ir the Sihttf.k-Il moving lower-head codel.

l'u r t he r c ha r.p e s to the SIh61ER-I l pressure iteration (the s'o lEt i or of the tere r.t ur equa t i er s ) are required to limit urphysital pressu:e reductions at the head-lluid irterlase. Ir the f ornula t s er. of the norenturc equations, the area

over wh

6h the pressure pr diert acts is the ritirur area lor the tse hel: sells that fert the corertu. t e 11. This is shosr ir l' a g. 34. SI AB1ER- 1 ] would use A 3 102

l RJU3H DRAFT for this area. As a result, the force driving fluid into cell .JPLt)G(I) is continually too small.

The physical situations should be as shog on the left side of Fig. 34 in which Ay equals Aj . The changes made in SiteER-Il provide for the maximum area from either cell JPLLIG(I) or cell JPLtlG(I)*1 to be used.

E i 1 BOUNDARES 1 1 1 l 1

\ NI i I l

\ . . ,

p---------- , 7-------_-_,

- 1 I I I 1 1 1 1 I I I I 1 l l 1 1 I I I b k I

---

, g I i u..

l N N

~

IN I i- 3 -

k L -_ A \

LNN  !  !

Fig. 34 SIl44ER-Il treatment of the fluid dynamics at the lower-head interface.

Finally, with reference to Fig. 32, sideways venting to the lower cavity-keyway volume starts when the lower head is displaced by one mode.' The total mass of materials moved downward into the cavity volume can be obtained by analyzing of the results with the SDelER-Il post processor.

103

RMH W

l. Itourdary Cord i t i ori, a rd 1:J i t s The belew-reastor cavitv-Leyway v. lure is higl-IV t h r e e -d ir.e r s i ora l and carrot be represented by Sl4t.li.R-II. I' l ow reliet c .s. i s t s . so a rigid wall hourdary cor.dition is incorrect. llowever. as a cor. sequence of flow restrictier.s into the containter.t. a constart (low) pressure boundary condition will allow toe tuch relief to take place. The opt ier. selected was the use of a cor.t n r.ua t i ve outflow /intlow boundary c o rd i t i or.. This cor.d i t i or / opt i or. was applied in three places on the radial bourdary as shown in Fig. 32. These places are at the end of the inlet piping, the end of the outlet pipir.g. and the outlet to the keyway.

The c:.r.rer in which this radial boundary conditier. was defined ar.d applied is as follows. An input variable. JCONTtJ). is read for each axial node. A value of zero tears a rigid wall. A value of one means continuative outflow / inflow. When JCONT(JJ = 1. the pressure in the radial boundary cell.

pb. i . is fixed over each time step to the beg i nr.ir.g-of-t ine -s t e p va l ue in the cell to the left. or py,j,). Ir. other respects the solution can proceed as, if pb.) were fixed by a r. input table. This device allows acceleratien or deceleratier of ou t f l ow / i r.f l ow toter. tut deper. ding on the quar.t i t y (p ,, - p[, ) ). If this d i f f e r e r.t e is unchanged, the outflow / inflow torentur.

will or.lv charge as a consequence of ur.ba lanc ed torent ur. c onve c t i er. . frictier.

or ether teres i r. the tore r.t t.t equatior. For.sequently, once a high interral j pressure ha , built up withir the cavity (er p i p i r.g l . accele rat ior. of fluid out I

el the raJ i.:1 bourda ry wil l c ea se.

Iive edits a uociated with the lower-head failure codel deserve tentier.

First. there is the edit of the input data. For the example problet i n l'i g. 32.

this edit *, given in Table Vlli. The r.urber*, correspond to those described ir the irput s e c t i e r. . Secor.d. before lower-head failure. one extra lire is printed each t ire step. At example is showr ir Table IX. This line mainly contair, data fret the head-failure model, as XPLUG .= displacement. VPLUG is the head velocity. PAVERG is the average pressu-c . t c, over the failure region. FPLUG is the force ( i r. n e wt oni, ) on the fa=?s e e on. ard FillAD is the force (in

~

rewt or.s I on the upper head. Accordirg to the current criteract. lower-head tailure occuri, wher the dia,placetert. XPLUG. re .aes 5. 0 i r. . The third edit i, that otsurrirp atter failure. An exatple is giver. i r. Table X. This edit c o r i i t *, of two l i n e *, . The first lire gives JPLUG(li. the axial nodes where the head intertase is leiated. The sesord lire gives additieral insertation or head 104

TAllt vill INI'lli I:DIT l'OR TIll: SIKR-li 3.gsg:p gig: 43, g 3, g g,,,g

,,pg,g,,, g gp 7,,;, y)

PLUC DATA POR DOTTTIN/ TOP MEAD CALCULATION MASS OF LOWER READ FOR PLUG CALCUIATION RIGNT NAND BOUNDARY OF TME PLUC (Pt,UCN)= 1.1000E404 TOP BOUNDARY OF THE PLUG (IS0)= 10 RIGNT MAND DOUNDARY OF T1tE UPPER NEAD (JSO)= 12 BOTTt)N BOUNDARY OF TME UPPER NEAD (1507)= 11 (JSOT)= 67 AXfAL BIAS NODES AS A FUNCTION OF RADIUS OOOO112345 '

AX1AL BIAS NODES FOR THE UPPER NEAD 0 0 0 ,0 1 3 5 -6 tiADIAL CONTINUATIVE OUTFLOW (INFLOW) BOUNDART NODES I 2 3 4 5 6 7 8 1 0 0 0 0 0 0 0 9 to 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 2R 29 30 31 32 33 34 35 000000000100000000000000n00non0 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 46 000000000000000000000000001 * '..(J C.)

1 C_T' G7 D~

I:_.7

~7 g 3 l w

'T1

.4

4 TF IX SAMI't.I: IMITI' TIT lillT lil: lori: IJnd:R-Ill:All 17All.URI:

- 7.01259E-01 CTCLE- 30 DT= 4.9952 R-05 NilNINC= 109 ITT= 3 CEttp 0 ITR0= 0 ISTPP= 3 CT= 5.5407R+00 DTL( 1)= 1.00000E+20 DTL( 2)= 7.91107E-04 DTL( 3)= 4.99523E-05 I= 7 J-26 DTL( 4)= 1.44739E-04 I= 5 J-23 i

DTL( 5)= 1.03536E-04 DTL( 6)= 1.00000E-03 DTL( 7)= 6.90241E-05 I= 4 J-27 DTL( 8)= 6.1377AE-04 I= A J-24 DTL( 9)= 1.00000E+20 DTL(10)= 1.00000E+20 DTL(11)= 1.20962E-04 I= 5 b23 DTL(12)= 5.08280E-03 I-11 J-23 PLUG DATA: XPLUO= 4.35992E-01 IN. VFLIN> 1.04296E+03 IN/S PAVEltC= 1.01982E+04 PSI PPLIN> 5.74870E+08 PNEAD= 1.05574E+06

'- 7.01300E-01 CYCLE = 31 DT= 4.89130E-05 NUNINC= 110 ITP= 3 CEtte 0 ITR0= 0 ISTEP= 3 CT= 5.35297E+00 DTL( 1)= 1.00000E+20 DTL( 2)= 7.41155E-04 DTL( 3)= 4.89130E-05 I= 7 b27 DTL( 4)= 1.45508E-04 I= 3 J-21 1

DTL( 5)= 9.99045E-05 DTL( 6)= 1.00000E-03 DTL( 7)= 6.66030E-05 I= 5 #24 DTL( 8)= 4.08316E-04 I= 9 J-25 DTL( 9)= 1.00000E+20 DTL(10)= 1.00000E+20 DTM11)= 3.84647E-04 I= 9 b25 DTL(12)= 6.61452E-03 I= 1 J-21

! PLUG DATA: XPLUO= 4.90652 -01 IN. VFLIN> 1.14557E+03 IN/S PAVERC= 1.0349X+04 PSI PFLIN> 5.8338X408 PNEAD= 1.05591E+06

- 7.01354E-01 CTCLE- 32 DN 4.59729E-05 NtstINC= 110 ITP= 3 CEL1r 0 ITR0= 0 ISTEP= 3 CT* 5.38411E+00 DTL( 1)= 1.00000E+20 DTL( 2)= 6.92242E-04 DTL( 3)= 4.59729E-05 I= 7 J-27 DTL( 4)= 1.45776E-04 I= 3 J-21 i DTL( 5)= 9.78261E-05 DTM 6)= 1.00000E-03 DTL( 7)= 6.52174E-05 I= 1 J-22 UTL( 8)= 5.04902-04 I= 9 b25 DTL( 9)= 1.00001R+20 DTL(10)= 1.00600E+20 trfL(11)= 3.97204E-04 I= 5 J-22 DTL(12)= 9.04537E-03 I= 7 J-20

PLUG DATA: XPLuo= 5.4919dE-01 IN. VFLUO= 1.24819E+03 IN/S PAVERC= 1.06422E404 PSI PPLUO= 5.9989R+08 PNEAD= 1.05596E+06

- 7.01398E-01 CTCLE= 33 DN 4.40846E-05 NtstINC= 112 ITP= 3 CELIr O ITRO= 0 ISTEP= 3 C W 5.39615E+00 l DTL( 1)= 1.00000E+20 DTL( 2)= 6.46269E-04 DTL( 3)= 4.40046E-05 I= 7 J-27 DTL( 4)= 1.45981E-04 I= 5 J-22 i DTM 5)= 9.19459E-05 DTL( 6)= 1.00000E-03 DTL( 7)= 6.1297X-05 I= 5 J-23 DTL( 8)= 4.3398R-04 I= 7 D24

DTL( 9)= 1.00000E+20 UTM10)= 1.00000E+20 DTL(11)= 4.09790E-04 I= 7 J=26 DTL(12)= 1.10611E-02 I= 2 J=21 PLUG DATA

XPLUO= 6.0886M -01 IN. VFLUt> 1.34769E403 IN/S PATERC= 1.10244E404 PSI PFLUO= 6.21440E+0A PNEAp= 1.05590E+06

+

= 7.01441E-01 CTCLE= 34 DN 4.27969E-05 Nt981NC= 112 ITP= 3 CELL = 0 ITRt> 0 ISTEP= 3 CT= 5.29R15E+00 DTL( 1)= 1.00000E+20 DTL( 2)= 6.02184E-04 DTL( 3)= 4.27969E-05 I= 7 J-27 DTL( 4)= 2.20904E-04 I= R J=1 A j

DTL( 5)= 8.81692E-05 DTL( 6)= 1.00000E-03 DTL( 7)= 5.87794E-05 I= 6 J-23 DTL( 8)= 5.3372HE-04 I= 7 J=74 DTL( 9)= 1.00000E+20 DTL(10)= 1.00000E+20 DTL(11)= 4.09971E-04 I= 7 J-26 DTL(12)= 1.12226E-02 I= 2 J-21 FIEC DATA: XPLUO= 6.70467E-01 IN. YPLUO= 1.4469R403 IN/S PAVEllC= 1.15262E404 PSI PPLUD= 6.49726E408 PNEAD= 1.05592E+06

- 7.01483E-01 CTCLE= 35 D1w 4.19142E-05 NUNINC= 112 ITP= 3 CELtp G ITR0= 0 ISTPP= 3 CT* 5.455fWR+00 l DTL( 1)= 1.0000m+20 DTL( 2)= 5.59387E-04 DTL( 3)= 4.19142E-05 I= 7 J=27 DTL( 4)= 1.47607E-04 I= 5 J-22 1 DTL( 5)= 8.55939E-05 DTL( 6)= 1.00000E-03 DTL( 7)= 5.70626E-05 I= 6 J-23 DTL( R)= 4.94355E-04 I= 9 J=1 A l DTL( 9)= 1.,00000E+20 DTM10)= 1.00000E+20 DTL(11)= 3.5584X-04 I= 9 b25 DTL(12)= 3.94931E-031= 5 J=23 PLUG DATA: XPLUt> 7.34561E-01 IN. VFLIN> 1.54830E+03 IN/S PAVEIIC= 1.2185m+04 PSI PPLDO= 6.8686 M408 FREAp= 1.05593E+06

= 7.01524E-01 CTCLE= '36 DT* 4.13572"-05 NUNINC= 112 ITP= 3 CELtp 0 17110= 0 ISTEP= 3 CT* 5.37331E+00 DTL( I)= 1.00800E+20 DTti 2)= 5.17472-04 DTL( 3)= 4.13572E-05 I= 7 J-27 DTL( 4)= 9.16051E-05 I= 1 A21 DTM 5)= 8.3828M-05 DTL( 6)= 1.00000E-03 DTL( 7)= 5.58855E-05 I= 8 A18 DTL( 8)= 6.47546E-05 I= 1 J-21 DTL( 9)= 1.00000E+20 DTL(10)= 1.00000E+20 DTL(11)= 4.13887E-04 I= 9 J-19 DTL(12)= 4.86724E-03 I= 4 J-16 PLUG DATA: XPLUO= 8.01657E-01 IN. VFLUC= 1.65325E+03 IN/S PAVERC= 1.28992E+04 PSI PPLUG= 7.2 712E+08 PHEAD- 1.05592E+06

ROUGH DRAFT

i TAlli SAMPLII (MITI' tit 1:IIIT AITIER ItMICR-lil:AD l'All.tlRI:

T= 7.03641E-01 CYCLE = 87 DT* 2.54010E-05 Nt9ft9C= 121 ITP= 3 CEtte 0 1790= 0 ISTPP= 7 CT= 2 35751E400

• DTL( 1)= 1.00000E+20 DTL( 2)= 3.84229E-04 DTL( 3)= 7.40338E-05 i= 7 J-27 DTL( 4)= R.28694E-05 DTL( 5)= 3.81015E-05 DTL( 6)= 1.0000nE-03 DTL( 7)= 2.540lnE-051= 3 J-14 DTL( 8)= 3.42714E-05 i DTL( 9)= 1.00000E+20 DTL(10)= 1.00000Et20 DTL(11)= 1.02821E-04 1= 3 J-14 OTL(12)= 3.1595pE-04 != 5 Jat ?

PLUG DATA

JPLUCI= 12 12 12 12 12 12 13 14 15 17 .

PFLUO= .14261E409 APLUt> .12974E+05 VPtst> .1796R403 T= 7.03671E-01 CYCLE = ZPLUO= .16358E400 PNEAD= .10430E407 88 DT* 2.97100E-05 NtStINC= 122 11T= 3 CEtte 0 1TR0= 0 ISTFP= 8 CT= 5.71277E400

! DTL( 1)= 1.00000E+20 DTL( 2)= 3.5882R-04 DTL( 3)= 3.02444E-05 I= 7 J-27 DTL( 4)= 5.42986E-05 I= 5 J 1 DTL( 5)= 5.00020E-05 DTL( 6)= 1.00000E-03 DTL( 7)= 3.38680E-05 I= 5 J=30 DTL( 8)= 2.9710nE-05 1= 3 l PLUG DATA: DTL( 9)= 12 JPLUCt= 1.00000E+20 12 12 12 12 12DTL(10)=

13 14 15 17 1.00000E+20 DTL(11)= 7.31387E-05 1= 5 J-28 DTL(12)= 2.63906E-03 I PFLUO=. .17688E+09 APLDi> .16090E405 VFLDO= .18004E403 T= 7.03701E-01 CTCLE= ZPLUt> .16815E+00 PNEAn= .10427E+07 89 DP= 3.02892-05 NtBitNC= 122 ITP= 3 CEtte 0 17R0= 0 ISTEP= 3 CT= 5.56999E+00 DTL( 1)= 1.0000R+20 DTL( 2)= 3.2911R-04 DTL( 3)= 3.02892-051= 8 J=27 DTL( 4)= 1.04711E-04 I= 5 J DTL( 5)= 5 94200E-05 DTL( 6)= 1.00000E-03 DTL( 7)= 3.9613X-05 i= 5 J-29 DTL( 8)= 4.52857E-05 t- 3 J-1 i

PIEC DATA: DTL( 9)= 12 JPLUCI= 1.00000E+20 12 12 12 12 12 DTL(10)=

13 14 15 17 1.00000E+20 DTL(11)= 1.48342-041=10 J-17 DTL(12)= 1.1844R-03 I= 7 PFLUo= .24538E+09 APLUt> .22317E+05 VFLDO= .18070E+03 j T= 7.0373tE-01 CYCLE = ZPLUo= .17351E+00 PNEAl> .10425E407 90 DT= 3.03061E-05 NOMINC= 123 ITP= 3 CEL1r 0 ITR0=

l 0 ISTEP* 3 CT= 5.56712E+00 DTL( 1)= 1.00000E+20 DTL( 2)= 2.9882R-04 DTL( 3)= 3.03061E-05 is 8 J-27 DTL( 4)= 1.0577X-04 1= 5 J-l DTL( 5)= 6.05789E-05 DTL( 6)= 1.00000E-03 DTL( 7)= 4.03859E-05 i= 5 J-28 DTL( 8)= 3.62297E-05 t= 9 I

PLUG DATA: DTL( 9)= 12 JPLUCI= 1.0000tE+20 12 12 12 12 12DTL(10)=

13 14 15 17 1.00000E+20 DTL(11)= 2.13984E-04 I= 9 J-31 DTL(12)= 2.44220E-03 I i PFLUD= .42679E+09 APLUt> .38809E+05 T= 7.03751E-01 CTCLE= VPIAO= .18188E403- ZPLDt> .17900E+00 PNEAD= .30422E407 91 DP= 1.93693E-05 NtBIIN0= - 122 11T= 3 CEtte 0 17R0= - 0 157EP=

! 8 CT= 5.480lflE+00 DTL( I)= 1.00000E+20 DTL( 2)= 2.68522E-04 DTL( 3)= 3.03411E-051= 8 J-27 DTL( 4)= 1.26732-04 t=10 J-DTL( 5)= 6.06122E-05 DTL( 6)= 1.00000E-03 DTL( 7)= 4.04081E-05 1= 5 J-29 DTL( 8)= 1.9369X-05 I= 5 PLUG D4TA: DTL( 9)=121.00000E+20 JPl#Cf= 12 12 12 12 12 13 DTL(10)=

14 15 17 1.00000E+20 DTL(11)= 1.58564E-04 1= 9 J-16 DTL(12)= 1.23800E-0 FPLUD .5590R409 APLUO= .5083M+05 VFLOO= .18342E403 T= 7.0376M-01 CTCL8= ZPLDo= .18454E400 PNEAt> .10420E+07 92 DT= 1.45636E-05 NOMINC* 122 ITP= 3 CEtte 0 1TRf> 0 1STPP= 8 CTh S.5930AE400 DTL( 1)= 1.00000E+20 DTL( 2)= 2.49152-04 DTL( 3)= 3.03772E-05 i I

PLUC DATA: DTL( 9)= 1.00000E+20 DTL(10)= 1.00000E+20 DTL(11)a 5.13908E-05 I= 2 J-21 DTL(12)= 1.71904E-03 i= 3 J=l4 JPLUCt= 12 12 12 12 12 12 13 14 15 16 PPLif0= .27910E+09 APLUO= .2538X+05 VFLIN> .1839tE+03 ZPLif0= .19809E400 PHEAf> .10419E+07 l

e A' &

m q mm Rj W l u J i, to t i o r. . I r. this case, the signs indicate the d i r e c t i e r. . ard 51 urits are er:p l o y e d . The quantitici are FPLUU. the ret forse (ext ludir.g gravity) at irg er the lower head: Al'Ll!G . the lower head's acceler.t irr: VPLllG. the lower h e a d ' i.

ve loc i1 v: ZPLUG. the lower head's displaceter.t ( i r.i t s a 1 ized to zero at head f a ilure ): a nd Tile AD. t he force on the upper head.

The fourth edit is to a binary file. TAPL3. before lower-head failure. The output line is NIAIL. T. PXPLOT. XPLUG. VPLUG. P AVI:RG , filLAD, and fPLUG. The qua r.t i t y NIAll is the failure flag, which is zero for this case. The quantity T is the time ir s e c e r.d s . The quar.tity PXPLOT is a r. array of 13 pressures fer future plotting purposes. These pressures have beer set by data s t a t e r.e r.t s to correspond to a.pecific r.od e s in the prebiet of lig. 32. These nodes are listed ir. Table XI. The reta ining va riables are a< defir.ed urder the prefailure prirt edit. The fifth edit is to a bir.ary file, TAPE 3. after lower-head failure. The output lire here is T. PLPLOT. File AD. FPLUG. ar.d ZPLUG. These variables have all beer previously d e f i r.e d.

F. rerrectier Set Table X11 c or t a i r.s a copy of the correctier. set that er. bod s e i, the modificatier.s described. Wher the program library is updated usir.g this ccrrectier set. the PLUG opt ier rrust be defined to activate the codificatior. .

If the PLUG optier. is r.o t defined. the standard SlhNER-II code will be obtair.ed.

Va r i a b l e i. added te c etter. blocks are defir.ed in Apperdix L. Other variables used are t. K. llell's origir.al correctier. set. and are defir.ed i r. the r.e w S14N1:R-1I tar.u:1.

G. Sart!e Problen fluids have beer. added to the configuration described in the structural diagram in rig. 32. The r e s u l t i r.g problet was run. Output figures and a distuasion are giver by this author in Apper.d i x r of the SERG report.' Ref. ]

12. the steat explo<. ion review group ( SERG ) report is a tor;1:ation of opir.ior.s writter by various

• teat explosion " experts". Con t r i bu t i e ni, a r.d correporderie reproduced n r. the SI:RG report and referer.ced here will be attributed to the l l

s a l i e r. t expert wher poi.sible. Addi t iona l di st ua s ier rega r 'i rg the out put et the i todel are ir Apperdices M ard h. l'u r t he r a ppl i c a t i er. of the todel is dea.6tibeJ ir Chapter VI.

l l

108  :

R:U3H DRFT TAllLI: XI DErlNITION OF Till: PXPLOT ARRW PXPLOT (1) r e pr e s e r.t s (15,1) Cavity bottom (2) ~

(1.13) Inlet Pier.uc bottom

~

(3) (9.17) 45 along bottom (4) (15,15) Inlet pipe end (5) *

(1,22) Abgve core support

16) (13,25) 90 alor.g bottom

• ~

(71 (13.36) Side of vessel at 4.4 M th) (11.47) Shroud bour.dary (9) "

(13,47) Downcomer er.d f191 (11.60) Vessel Flar.ge

~

(111 (6.65) llead Curvature

~

(12) ~

(1.66) Top of Vessel (13)

~

(15.66) Outlet pipe end (i.j) are exterral dimer.sier.s from Fig. 1.

The St ear Explosier Review Group ( 51:RG). "A Review of the Currer.t Urderstar.dity ef the Potertial for rer.tuirmer.t Failure Arisir.y I rem Ir-Ve s se ! $t ear. I:xplos t er s . " Ur.: t ed St a t e s Nuclear Regulatory Cettissier report. NUREU- 1116 ( l e brua ry 1985).

l 109

T4HLI XII CDRRECTION SET FOR THE 514fER-11 LOWER

• HEAD ICDEL 1 'IDENT BAA4 2 '/

3 */

4 '/

CHANGE THE 0040N BLfk'K5 FOR THIS PROBLEM 5 'D E1M9.4.5 6 ,

7

• PLUGM. ZPLUG . APLUG. TPLtXi. VPLtlG. FD2( 11 ) . ZBOT( 11 ) . ZTDP( 11 )

• ZB14S(11).FHEAD.XPLUG.PAVERG

& 'D E1M9.7.6 9

• 10 * . J 50.150. J 507.150T. J PLUGl 11 ) . J PLUG 5 t 11 ) . J PLUGT( 12 ) . XX1hT( 70

.NTAIL ll */

12 */

CH4NGE THE INPUT FOR THE PLUG PROBLEM 13 * /

14 *D E1M?.19 15 'IF DEF. PLUG.11 16 'D E1M9.24.25 17 -

.' RE AD (NINP.831) 150.350.1507.J50T .

15 IsoP=l W 1

• 19 ISOTP = ISOT + 1 _

20 READ iNINP.631) (JPLUG(1).1=2.lSOP) 21 READ INihP.831) (JPLUGT(1).1=2.150TP!

22 RE AD NINP.831) (JODhit J ) J-2.JP11 23 * **

24 '/ E114NGE THE OUTPUT EDIT

- 25 */

26 'D Lit 49.32.34 27 WRITE (KT.7803 ) 150.J50.150T.J5nT 26 150P = 150 , 1 29 WklTE (KT.7604) (JPLUG(1).1=2.150P) 30 15nTP = 150T + 1 31 WRITE (KT.7805) ( J PillGT( ! ). I.2.15(ITP 32 WRITE (KT 78u6) (J.J-1.39 ),(J(18his J ).J 2.40) 33 WRITL (KT.7807I (J.Jm40.JRAR 44 WRITE (KT.7607) ( JtIJhTl J ). J-41. JP1 i 35 WRITE IKT.7606 )

46 7606 FORMAT (1HO.1X//)

37 36 'D7602E1M9.35.43 FORMAT I //19X.41HPLUG D4T4 Fi* 110TTOM/ TOP HEAD CALCULATION //)

39 Jo 76o.1 T(*44Ti .1X.31HRIGHT M4M) BrMM)4R) 0F THE PLUG.39X.Ml(150) 112/

41 '3X.2411 Tit kitW)4RY of THE Pitki.46%.6H(J50i=112/

J2 '3\.3?HklGHT HAND Br49RJ4R) of THI UPPER HE4D.32X.TE(ISOT)=-112/

'3%.33HW TTIM BrafM)4R) of THE UPPl.R Hl.4D.36X.TH(JSDT). 312//)

43 'D 1.149.44.55 44 45 Thua Fl*44T e 3X.4(41AX14L B145 NNDI % AS A FUNIT10N OT RADIUS.3X.1013//1

46 76u$ Ft*44T (3A.35H4XI AL BI4s MIDI.s Ft* THE UPPER READ.tX.lll3/// l 4* 76061//3X.3913/3X.3913/<e Ft*MATi12X.52H RADI AL tishTlhU4TIVE (RTTFLOW(INFLOW) SOLDOARY NUDES

4b 7607 FORMAT I3X,4u131 49 * / .

So * / INITIAll24TI(* 20111t*471:55

, 51 **

110

. - , -- . . - - - . _ ,_ . . - ---__,..-,,..,____..__--,,.__,,.,_,_.,._,..,..m y -

y _. . _ . ___ , ,,,-,. - -

-r- ---- ---r~ --m-m

r t r- 1 -r. -

RD dd1 uI& j 52 *D E1M9.61.62' 53 150 = ISO + 1 -

54 JSO - JSO + 1 -

55 ISOT = ISOT + 1 56 JSOT = JSOT + 1 -

57 NT41L = 0 58 'D E1M9.63.73 59 'D PC13.7 60 D E1M9. 80 61 XPLUG = ZERO 62 *D PC13.8 63 'l 5E71.324 64 alF DET. PLUG 65 */

66 ' / INITIALIZE WESH DEPEICEhT PLUG VARIABLES 67 */

65 DO 20$5 !=2.I50 69 JPLUG) = JPLUGt1) 70 ZBIASI11 = ZERO 71 IF ( JPLUGI .EU. 0) 00 TO 2045 7 2 .' Do 2050 h=1.JPLUG1 73 2050 AI4Sil e = ZBI Asil) + DZ(JWN) 74 2045 00hilNUE 75 ZTOPtIs = ZBIASI11 76 ZBOTlI1 = ZTOPtI) - DZ(J W JPLUG1; 77 JPLUG(1) - JSO + JPLUG1 75

• FD2t 1 ) = ONE .

74 JPLtKiSt 1) = 0 Eu 2055 CONTlhTL

• 61 DO 2056 !=2.150T 63 JPLUGT(I) = JSOT + JPLUGT(1) -

63 2056 00hTINUE 64 *ENDil' 65 */

66 * / XrH4h) W Oh 6 ? ' .-

66 'e 2 Oll'ICATI 1NS 70 PLUG KITION bo * :

9i' 'D 0149.214.215 41 IT t 1 . GT. 150i 00 TO 2(x12 42 If a J .NI.. JPLUG(1 )) 00 TO 2(N>2 41 '

ll PC13.33.34 9J IF f rDIs I I .CT. CP6) PPLlki = PilJP I - EP5 45 IF f rIJLilJ) .GE. AKINTI) PPLUG = PflJP) - EP5 96 ' D PC13.39 . _.

97 IT (NF 41L .FU. I l 00 TO .kwil l 96 PAVl(IlkPAVLkG '

94 P4VI ku = TPLtti/t P]E*RIPi150l'RIPt 19H i 1m 'PAVEkb = PAVERG'1.450740-<i4 i In1 11 Nt'W .EU. ni 00 TO 3m2 In2 r ALL SMLD XPLtti. VPLUG. P4VERG . P4VLOD.DTK4CH.NFAI L ) l In3 If fNFAll .EU. on Go TO 2inM. )

l 104 VPLUG = -VPLUG/39.371 in$ 3' ail (11hilhWE lim 'B LIM 9.226

,--n- - - - - . . _ . _ . , . , , . - . . - - --. , ,,-.-,-.,--- , - . . , . . - .

Sn' Jr !

T U A . ~;

r.nn L f W.,"

107 FPLtG = -FPLUG 104 "D E1M9.233.245 .

109 C DO EACH R4DI AL lESH SEPAR4TELY -

110 C '

111 "IF DEF. DBL.1 .'

112 VPLl01 = ONE/ DABS (VPLLE) 113 'IF -DEF. DBL.1 114 VPLUG1 = ONE/ ABS (VPLt0).

115 DO 2010 !=2.150 116 IF (ZPLWZB14S(l) .LT. ZBOT(11) 00 TO 2003 117 IF (ZPLWlB145:I) .LE. ZTOP(I)) GO TO 2001 116 ZBOT( 1 ) = ZTOP( 1 ) '

119 JPLUG1 = JPLtC(l) + 1 120 JPLtGt I) = JPLUG1 121 ZTOPt I J = 2 BOT (1) 4 DZ(JPLUGI) 122 JPLUGS(I) = 1 123 Go 70 2001 124 2003 JPLUGI = JPLUG(1) -1 125 J PLtGI = NAX0( J PLUGI .1 )

126 JPLUG(I) = JPLUGI 127 .~ ZTOP( 1 ) = ZBOT( I i 126 2 BOT (I ) = ZBOT( 1) - DZ(JPLUG1) 129 JPLtGS(l) = 2 13n 2001 CONTINUE 131 *D LIM 9.250 132 JPLUG) - JPLt0(1) 1.13 FDZ(lI = ZERO ,#

134 II ( JPLt01.GT. 1 ) .

135 'FDZ 1 = (ZPL W ZB1ASt11-ZBUT(l11'RDZiJPLUGIi 136 *lF DEF. DBL.1 1.4' FDZ(li = DEAX1(FDZ()1.ZEROJ -

13b *IF -DEF. DBL.1 134 FDZ( l a = AMAXIIFDit i 1.lElto n 14" 'D PC13.41.45 141 'O LIM 9.252 142 DTNN = (UNE-FDZ(l l' ALMIFLt IREG))*EM1*VPLUG1 1

143 'IF DEF. DBL.1 144 DTN = DMIN1(DTN.DTNN) 145 alF -DEF. DBL.1 146 DTN = AMIN 1(DTN.DTNN) 147 2010 CONTINUE 146 */

l 149 */

EXFLUD MGDIFICAT]ONS IN DEFINING A NEW PLUG INTERFACE NODE 150 */ --

151 'D E1M9.124.125 152 IF ( J PLU(it 1 ) . EU. 1 . APO. J . EU. 2 ) AM)FLt !J ) = ZEB0 153 IF t1 .GT. 150: 00 TO 1s a w

• 154 IF tJ .NE. JPLLUt1is 00 To ItNw' 155 'D PC13.12 150 *D E1M9.127 157 *D Pt'13.13 15b 'D E1M9.129.13t' .

159 AMil Le 1 JM i = UKl.

• I toi AmtrLe ! J $ = Flite 1 i 161 AM)FL I J P s = ZEko II2

h ,~,r i e g ; e . ., e ,_ ,

s M. ,o (i ; ...:

.. t .' L i .l , '.li/, ,

i 162 IF (JPLlKiS(l) .EU. 0) 00 70 l(Nxi 163 IF ( JPLUGS( l ) . EQ. 2 ) GO 70 1(k:1 164 *D PC13.14 .

165D E1M9.134 166 JPLUGSt 1) = 0 167 *D E1M9.136 168 ANOTL(1J)=ONE+VPLUG'DT*RDZ(J) 169 JPLtLSt1i = 0 170 '/

171 */ IMFLUD E 01FICATIONS 172 */

173 'D EIM9.146.147

- 174 IF (1 .GT. 150) GO TO 1002 175 IF tJ-1 .EO. JPLUG(1))

176 'D E1M9.154.155 177 I F (1 .GT. 150 ) GO TO 1(M11 176 IF ( J .EO. JPLLKit i 1) 179 'D EIM9.159.160 16n IF (I .GT.150) GO TO 1(m 161 IF (J .EO. JPLUGtI i

. 162 'D E1M9.163 7

183 *D PC13.27.29 164 *B IMFL.1559 165 *lF -del ~. PLUG.1 166 'I.IMFL.1559 167 *11 DFF.PLLU.1 166 IF r J .NE. JP1) GO TO 3030 160 *D EIMu,369,177 -

190 *11' DEF. plt 0 191 C

• 192 C CALCULATE FORCE ON THE BEAD

• 193 C 144 3039 r11hi1 HUE 195 AREA = R4REA(Il'DT 196 IF i1 .GT. ISOTI 00 TO 3040 197 IF ( J+1.NE. JPLlKiT 1)) 00 TO .kun 196 *ENDIF 199 'D E1M9.176 2(Ni *D E1M9.180.205 201 'l E1M9.207 202

• IF -DEF.PLtKi.1 203 'l IMFL.1603 204 *lF DEF. PLUG.1 205 PTPNP = PTPTMP/(RIP (150T)"2* Pill .

206 */

28i7 * / NYDRo w 01FlrATiONS 206 */

1 2n9 'D EIMO.91 210 *lF D!.I .PLtKi.20 211 *D EIM9.95.99 212 IF (KF41L .00. I I (in TO 41%

21.1 WRITE (NL)FS.4174 i XPLtU.VI'LtNi.P4VI:kti.FPLt0.FHEAD 214 4174 FOIMAT (12H PLtti D4T A : . ell Xi'Lth=.1PL12. 5.4H lh. .

215 1 9H VPLUU=.1P112.5.5H Ih %.1"H P4VLRU=.1PE12.5.4H PSI.

216 2 9H FPLt0=.1 PL 12. 5. 9}i Illl 4!= . l PI12. 51 113

1. zss- .

e,-.,. -

H . ..O l'.',,'*

,s,s . . . . u.. .

4 217 GO TO 4178 215 4176 Q)hilhVE 219 DO 4172 !=2.150 22o J PLUG 5( 1 )=J PLlKit 11-1 221 4172 CohTINUE 222 WR I TE ( NOF S . 4171 )( J PLUG 5 ( 1 ) .1 =2 .150 ) . F PLUG . APLUG .VPLUG.2 PLUG . FHE AD 223 4171 FORouT(12H PLUG DATA:.10H JPLUGI=.1013/

224 'D E1M9.101.1oS 225 .

• E12.5.9H

. FHEAD=.E12.5) 226 4176 (MTINUE .

227 DO 4173 I-2,150 226 JPLUGM 1) = 0 229 4173 COh7thUE 23n r 231 */

232 */ CH ANGE CohTINtKR'S lhTLOW/0UTFLOW 800!OARY CDIOITIONS 233 */

g 234 'B IMFL.70 235 alF -DEF. PLUG.1

- 236 'l IMFL.70 237 TiF DEF. PLUG.1

,.' 2.4 1F ( ! . EU. I P1 . APO. KDhTI J ). EV. 0 ) GO TO 2015 239 *B IMFL.499 240 alF -DEF. PLUG.1 241 'l IMFL.499 242 'IF DEF. PLUG 1 243 I F ( l . EU. I P1 . AND. KOhT()). EV. 0 i 00 TO 2230 244 *B B1ro.26 s 245 *lF -DEF. PLUG.1 246 al Biro.26 247 'IF DEF. PLUG.1 246 I F ( 1. EV. l P1 410. J(IlhT( J ) . EV. 0 )

249 *B IMFL.864 250 a lF -DfT. PLUG.1 i 251 *1 IMFL.864 252 IT DEF. PLUG.1 ,

l 253 I F ( 1.EV. lP1. AND. JC0hT( J ).EQ.0) ROLBRT(WNV)=lt0LBRT(N) l 254 *B IMFL.870 255 'IF -DEF. PLUG.1 256 al IMFL.870 257 'IF del. PLUG.1 l 256 IF (1.Ev. lP1.470. J0nhTt J ).EV.o i m1GBRTt MNVi=EIGBRT(N) 259 *B IMFL.906 l 260

• lF -DFl .PLiki.1 _

261 *1 IMFL.906 262 'lF DF.I .PLUti.1 ,

l 2e3 I F ( 1. EV. l P1 . 00. K%Tt J ). FQ. o i ROLBRT t l 3 P+Ny i=BOLBRT(1JP)

)

26J 'B IMFL.913 1

265 'IF -DI F .PLtti.1 l 266 *) IMFL.913 l

207 *1F DEF. PLUG.1 266 I F ( 1. EV. l P1 . 470. JCM7 818. LV. 01 EtiBRTi l J P+NV ).klGBRT( l J P ) l I

269 'M B1Co.26 .

2'o 'll -DEF. PLUG 2 271 'l IMFL.922 ))4

v .s? . a. .. , ,

.i

!- ,.I s

~ca..

-272 *lF DEF. PLUG.2 -

273 IF (JCX)hT(J) .EQ. 0) Pt IPJ ) P(lJ) 274 IF (JCDhT( J ) .EQ. U) 00 TO 255o .

275 *B 88D4.2 .

276 *lF -DEF. PLUG.1 277 *1 88D4.2

'278 *IF DEF. PLUG 1 279 IF (1.EO. lP1. A!O. JO0hT( J ) .EQ. 0) P(IPJ ).P(IJ) 260 *B IWL.1523 281 *lF -DEF. PLUG.1 282 *1 SW L.1523 283 *II DEF.PLLU.1 264 IF (JCX)hT( J ) .EQ. 0) GO To .4006 285 *B DEL.1503 266 *1f -DEF. plt 0.1 287 *1 IW L.150.1 288 *lF DEF. PLUG.)

269 IF t lVIS.EQ.01.OR. (JCDhT(J ) .EO.1)) 00 TO 3015 290 *B IW L.1512 291 *lF -DEF. PLUG.!

. 292 *1 IMTL.1512

- 29.1 *lF DEF. PLUG.)

294 IF ( t IVI S. EQ.0 ) .OR. ( JC0hT( J ). EO.1 )) 00 TO 3018 295 *B EXFL.1395 296

• If -DEF.PLlG.1 297 *1 EXFL.1395 -#

295 'll DEF. PLUG.1 290 1F t JCDh'T( J ). EQ.1 I 00 TO 1504 Ami '8 RINP.566 i

het all' -DEF. PLUG.1

>>2 *! RINP.566 Ai3 *lF DEF.PLlKi.1 A4 IF ( JCI)hT( J l .EO. 1 ) Go 70 .442

) 5 *B 5ET1.364 306

• lF -DEF.PLtti.1

. Ai7 *I SET 1.364

• >$ *lF DEF. PLUG.1 3'*9 1F ( JCX)hi( J 1. EQ. 1 i R(11hi( J ).ONE 310 */

311 */ ADD SPECI AL VAR 14BLES FOR PLOTTING 312 */

313 al XOl4.26 314 alF DEF. PLUG.2 315 DlW NSION FXPLOT(13) 316 DIW.NSION IXPLOT(14).JXPLOTL141 317 *B XOl4.31 ,

316 *lF DEF.PLtKi.2 310 DAT A IXPLOT/15.1.9.15.1.13.13.11.13.11.6.1.15.0/

32t

• DAT4 JXPLOT/1.13.17.15.22.25.38.47.47.60.65.66.66.0/ ,

321 *1 XOl4.16 122 *11 DEF. plt 0.1 123 f f'PLtKi = 1 '

324 *1 B989.172 325

• 11 DLi .PLtKi 115 126 1F (l-1.NE.1XPLOT ICPLUG) . (* . J -1.NE. JXPLOTI ICPLlKi))

%e.e 327 100 TO 4170 328 PXPLOT(lCPLUG) P(1J) -

329 ICPLUG=lCPL W 1

• 330 4170 CDhTINUE '

3.11 *EPOlF . -

332 *1 OV00.7 333 *lF DEF. PLUG.1 334

• TAPE

. 3 335 *1 E1M9.253 336 IF (hTAIL .EQ. 0) WRITE (3) 337 1 NFAIL.T.PXPLOT.XPLUG.VPLUG.PAVERG.FHEAD.FPLUG 336 IF (NFAIL .EQ.1) WRITE (3) T.PXPLOT. FEE 4D.FPLUG.ZPLUG 339 .VK)2 CDhTlhME 340 *D EIM9.26 141 READ (NINP.811) PLUGM 342 'l XCHA.632 343 'lF DEF. PLUG 344 */

345 */ PUT IN SUBROUTINE TO CALCULATE FAILURE 346 */

347 .' SUBROUTINE SM00fX.V.FTN FTO.DT.NFAlL) -

346 DATA SKO. AM).PSKO AK)P /33694. 0.01456.187. .U.00996/

349 35o C THIS SUBROUTINE ASSUMES NO UNLO4 DING AM) ZERO INITIAL CDIOITIONS CNF2=SK0/AMD 351 XOLD=X 352 VOLD V 353 IF(XOLD.GT.O. 2 ) SKO PSK0 .#

454 I F( X0LD. GT. O. 2 ) AWi- AK)P 355 IF(X0LD.GT. O. 2 ) CNF2=SK0/AMO 356 IF( AB5(X0LD).GT.S.0) 00 70 2005 357 AE V=-DiF2*X0LD+FT0/AMD 356 AOlb ANEV 359 1001 AIPl=ANEV

.%o VNEV.VOLD+(AO W AJP1)*DT'O.5 361 XNEWX0LD+VOLD*DT4. 25 ' AOLD'DT" 2+0. 25 ' A I Pl *DT"2

.%2 ANEV.-CNF2'XNEV+TTN/AMD

.% 3 IFt AB5( ANFW-AIP1 ).GT. I. E-7'ABht AIP1 )) GO TO 1001 364 C UPDATE VARIABLES 365 V=VNEk 366 X=XhTV 367 RETURN 366 2005 NFAIL-1 369 V=VOLD 37u X=XOLD ~ ~

371 RETURh 312 LND 373 'DOf f '

3'4

• 1 DENT BCXJ 355 */

~

376 */ ADDIT 10N4L PR1hT STATEMEhT 377 */

376 *B BuJ.15 .

3'9 WRITE IKT.7894) PLOGM 360 7609 FORMAT (34.39HR455 0F Lih1R HE AD FOR PLUG CALCULATION.29X. 116 3b1 1 8H(PLUOMJ .1PL12.41

p l

. .. r

_ 3 ,,. *,.4 . s t

IV. 00RRl:LATION 10 SNL STEV1-I:XPLosloN L\Pl.klMENTs Th r e e . i r.i t i a l obse rvat wrs car be r.. d e regarding the SNL tests. First.

ra r v experiter.ts were terdusted. There arc r ire that 75 I ITS tests at i r.t e reed i a t e uale. Secord. there is a sign licant stochastic c otper.e nt present. Lat e r ' test s have rot ~ terded to reproduce earlier - results. Third, detailed tire-dependert data or. raterial volute fractions and size distributier.s are bes nd the capability of the instrumentation employed (cameras and pressure

.transducersb Because of the lack of precision and reproducibility in the experiter.tal results as well as the lack of models known to r e p r e s e r.t the correct

~ '

ph e r.oce ro l og y, the 4.hility.to correlate to. e xpe rin:e r.tal da ta is limited. for the purposes of 'this study test hEl-19 was selected as representative. Some characteristic results of other terts are discussed in Appendix 0. Test kli- 19 produced well-characterized p r e s e.u r e data and has ' previously been aralyzed by Mi t c he l l ' u s i r.g t h e SNL wa ve c ede CSV. wh i c h c a l c u l a t e s ca s s . meter. tut. internal crergy. and Liretic crergy fluxes between Euleriar. cells. Ca:tulations '

sitalating test 51410 were also performed by this author

• usirg a three-velocity field algorithe, and by Oh usirg a paratetric rorequilibr iut. mode l. The present latited corpariser with Slkt1Ek-Il results is judged to be useful !ct obtair.irg initial estitutes of codel parateters, ar.d for'gair.ir.g at appreciation of.the type and tagnit ude of disc repanc ies betweer. 5141ER-ll calculations ard experirer.tal data.

The overall geometric setup f or this test is shown it' Fig. 35. This was a test it which 5.11 kg of irer.-alumina thermite were dropped into 224 kg cl water cortanced it a Lucite tark. showr. it lig. 36. The cor. figuration at the time of the e x p l o s i o r. (the iritial 51st11:R-l l c or.f i g u ra t i or i was t a k e r. from the experitental report by Mit che ll. ' lie reports a mixture volute of 0.032 e 3. a ri x t u r e d i ate t e r o f 0. -12 t. ard a rixture re.t density of 159 kg/rr' at the tire

~

My utt -1)e on. "Therrt.)-livdraulic Medeling and aralvses for La rge-St a t e Vap:$r

. E x p l e s t e r.s . " Ph. D. Thesis tiniversity of Viuensit hu e l e s. r Lr.gineerirg Departtent. Aopst 1955.

" M. L. corradiri. " Limits to fuel-roolart Mi u rg . " from hl.Berrat ard i R. K. rete. S t a t u s ef fore Melt Prog r ats--! vembe r -Dec embe r 1953.' Memorurdat to J. L. Telferd and S. B. Bursor.. (SNkt (41Fuquerque. h\i. february 29 105l'.

117 P

P w- . _ _ _ _ . _ _ _ _ _ _ - - _ _ _ - _ _ _ _ _ _ _ _ _ _ . _ _ _ _ _ _ _ _ _ - _-_-- -_ _ _ _ . . _ _ _ _ _ - _ _ _ _ _ -

I'.:.A. y. .  :.

of the e x p l o s i e r.. As suti r.g t he t (1) e qt.a l vel ute i, of s t eam ar.d wa t e r ex i s t ir

. t he e xpledir.g ' ri xt ure av ha- beer p.w i t. i .i t e d bs fe r r a d i r. i " , ar.d that (2#

twe-phase t$< stear'$m katers-shitrev exists obive the ir.t e rac t s on zor e . tbe assumed l coarse-tixed cor.figuratier. for thia, test is shown-in Fig. 37. The water t en pe ra t ur e wa s ' t a k e r. a s 299 K. the thertite t emperature was taken as 3 000 K.

ard cor.s tar.t -pressure bourdary c ordi t ier.s ' of 53 kPa were used at the top and r i gh t -ha r.J baur.da r i e s. The thereite drop size was taker. as 0.3 et ir. diameter corresponding te the peak i r. the debris distributior. from tests FITS 34 and FITSSA. Also, this debris size is c er s i s t e r.t with that used in the previous ZIP study.'

The na i n da t a ' u s ed for compariser we re the water chamber's base pressure r e produc ed _'i r. Fig. 35. The locatier of the transducer is direct ly below the explosior.." The two paraceters varied in the calculations were-the water droplet radius ar.d the heat-trar.sfer coe f f i c ier.t tul t i pl i e r gove rti r.g t he rci t e drople t t o wa t e r surface heat transfer. The wa t e r -d r o pl e t size cor.t rol s the surf ace area of the water. For each droplet size the goa l was to reach an approximate peak pressure of 17.5 MPa ir. the base r.od e f l .1 ). Oric e this was obtained. the calculatsers we re e xamir.ed f or the desired width of -the pressure pulse at 5 MP of apprcxicatelv o.35 es. The results of these pa rame t r i c va r ia t ior.s are shown nr. Table XIll. These c a l c u l a t i er.s suggest that a best fit right be achievable for a wa t e r-drople t - d iar.e t e r betweer. o.02 te ar.d 0.075 cm. Ur.f o r t ur.a t e l v , a r.

equilibration effect i r.f l u e r.s e s the results at these low water-droplet sizes.

Fig. 39 pises.the pressure trace for the 0.075-me case. The high pressure tail resulting fret bulk water heatiry does r.o t appear ir. the experimental results.

Thir tail betstes worse for the 0.02-me case because tecperature equilibratier. s is t h e r. tuch faster that arv e xpar s t or. e f f ec t. The SI41ER-l l i r.pu t for this

- case is giver ir. Apper. dix P.

A partial improvemert on the abeve results is to specify a norur.ifere fuel d i s t r ibu t t er.. Ar. e x p e r i me r.t a l distributier is r.o t a va i l a bl e . Cor.s e qu e r t l y , a calculatier. tust be used. The computaticr. selected was that dote ir. a previous M. l.. Corradini. University of Viuertir. Jure 4. 1054.

118

l 9

.?'. y)

-M , -

p ~ . ..

1 I

TEST STAND Els l/

FM N [ CRUCIBLE a

/ / STRIP PLANE L

.[

,  ; E 4.[&

- i WATER CHAMBER ,

h.'5 2

melocsi, " "' / \  :

.' v) N 1 a / STADIA I

") , ' l s, .'

WM '.^ * $ A ' I m3

" A "' Mu au

,, l k , ,,) *

• i * *

• m f

y;-{- '

.4 * "

m 4 mm ,

l Y, a ,

, .ag g ph

"' = == m. ,

a, p ar"* p . 'j'IA

' um m. ,

t',.,#"p-s[,

I I n aslll

, ,< l .

~ ,.-

% - ~

I FROM S AND 81 - 012 4

NU R EG/C R -2145 f I q . M. 1

\o-l ITs a pra ra t us.

l 119 j

--..,--._--c.-_ - _ _ -, , - _ ._. . _ . _ - . - - - - - . - - - - - - . _ _ _ - . . . -

y

\v i

L yJ e

,my--

j Tnt

~

MELT ' CAMERA REF.

F ,, , (STADIA)

MELT VELOCITY A M

SENSORS '

TRANSPARENT '

CHAMBER 'q WATER I

.) TEMP T.C.

WATER LEVEL" '

, 1 -'

SWELL ME AS./

(OPTICAL)

s N I'

l i '

sj ~ -

l WATER PHASE N ,-% j PRESSURE TRANSDUCER

,)'

N,

' WATER LEVEL ss

,s'  %

,/ 's N ERNAL's s TRIGGER RIGID SUPPORT (SE-1 DET FOR WATER CHAMBER (in some tests) or quartz pressure guage k CHAMBER

jCHAMBER BASE (PLEXIGLASS) l 12.7mm

, y -

3 / WATER CHAMBER SUPPORT DETONATOR -

l DETONATOR MOUNTING (NOT USED IN ALL TESTS) l lip. 36. Typical i r s t r uee r.t ed i.a t e r c h..rbe r . l 120

I ROUSFCRAT l

13 12 11 i

10 50% WATER 9

1007 50% STEAM 2007, 8 WA M WA M i 7 E  ;

a 10

, < 6 e R

< 5 4 ,

I 48% WATER 3 E l 48% STEAM -

l 2 R 1 4% THERMITE o '

1 JL JL

• -- 021 m -

0.34 m

i 1 2 3 4 5 6 RADIAL NODES

.i rig. 37 Irit ial (orf iguratier. f er the SIWER-ll represertat ier of test MD-19.

121

RL3H DR/3 200 . . ., i iiii E

< 100 - -

E Q

1 Q

N ~

l O

i e il i iil i O.0 0.00200 0.00400 , ,

TIME (s) l ip. 36. IL e pre m te water charber ir experieert ht) 19 122

^

m; e

.i 1( -;,

T Al LI: NIII INITI AL t AUTL;TIO!.' - l ol ilT11N(i TI AT MI)-19

- E t er-llrople t lleat-Trarsler Peak Pulse Tidth in are t e r Mu!*irlier l're s s u r e - (MPa t at 5 MPa 300'ut 1 24,3 H.I b.1 O. $ 19. $ 1.2 ts

'154 4e 1 26.7

0. 6 13.0

0. 7 lb.9 1.0 es 7$ un ').O 10.6

$. " 27.1

2. 5 2 3. u 1.5 21.4 1.7 17.4 U.47 es

.20 ur' O.$ J.4

, l u. O 19. ti - O.10 ms l

stuJv ' detetstre.tstg the feasibilits er a three-field 6alsulation of the te.irse pretixarp phase of a s t eat e xplos s or.. Addittorally. sete Sid11:R-Il pretistep calculetters were perlerred. These calculations should be helpful it.

(. extrapolatirg to the reat t er t er.l igu ra t ion and la rger . stale experitects. Thev.

are reported n r. Apperd:s V ard corpared with the three-field ard experimental

.resultC. The the rrite dist ribut ier f ree the - three-field calculat ser as well' as sore e xpe r a r.crt a l results are showr it l'ig. 40 The node with the peak' fuel dersity is ter.siderably more f uel rich ( 270 the rnit e. 360 wat e r, and 370 vapor )

than the seca red dist ribut ion of l~ig. 37.

l~c r th. explesirr calculations a o.3-cr fuel-droplet diameter ard a 0.075 tr water-droplet diameter were used, l'e a k pressures talculated as a i

furstior. of the heat-trarsfer cultiplier are showr. it lig. 41. A cultiplier ef

0. 2 a ppea red t e be adequate. A plot of pressure vs tire in the base rode for this case is s how r. in rig. 42. The width el the pressure pu l s'e ~a t 5 MPa is L. - approximatelv o.39 rs. whic h is reasonable. l)a t a fror the water phase pressure gauges is s how r. it l i g. 4 3. The calculotter at a 5.cilar locutier gives ar l ir.itial pres ure rise of about the iare magritude, see l ig. 44. Tine delavs are l

l l 123 L ,

l l

)

l R:03H DRAFT 20 0--

PRESSURE AT THE BASE 16 0 .

7 c.

l 2 12 0 - i EJ: -

W -

D '

b m

80 -

c.  :

.I 4.0 ..

0.0 .. ......i .....i.. .

...i - - - - . .

......... i 0.0 10 20 30 4.0 50 TIME (ms) l'i g. 34 Ilest fit f er a uriferr interatier 2cre (water-droplet si:e 75 ur . heat-trarsler rultiplier 1. 7 ),

124

J l

1 I --

~

l i' ML WATER LEVEL

\.

• 1

-T -

Y n

' WAVE l 7i .

CAOE P3 (

N W i; 0.67 M LOCATION 4 N /, . ATER CHAMBER Ih l MELT-WATER MXTURE OUTLM j

OAGE P2 "y f k C, _k 0.6I N LOCAT80N /

f ~~ ' CHAMBER BASE

,2 f 4

Q2 #

e a- /

2 ". Original E$' Wter /  % TR100ER SITE level \ n f a

-3.3 l

4~ ,;  ? sy 9 1

2 jY r .o.2 16 fl MD-19 f

gg 1 Thermite

,: vo r .,m, ll1 24 e-rraction

/ ' -0.I J{ ,

p 3 , ,

Height -

i.n C3 t

e

'L O.

i Q

f ' s i .

• x ~

2::=

l '1- m

.x j . - ..

M i, _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __

W j l 1 I I l 70 - -

60 - -

n W

n 50 - -

2 w

W g 40 - -

3 W 30 - -

g a.

20 - -

10 - -

1 I I I I o

0.2 0.6 1.0 1.4 1.8 HEAT TRANSFER MULTIPLIER --

l'I p . -1 1. bers it ivs t s el the peak preuure te the heat-trar'te:

rNltiplier for a o.3-r luci dreplet and a 0. 075-r. . ws t e r d r or l e t -

126

i i

' RC03H. DRA:T O a N

~

~

A Q l

[ d" .f

_ 2 V

y .

w

_ 2._

g O.

T=

. l .

m M

I t t , , ,

a e l t t t l l t I g 0 to O LO O m ,- v-(BdW) BBOSSEWd . _

f i g . 12. Best fit starting fror a rer.uriferr nr.teractier. Zer.e with a

o. 3 er: fue; dreplet a r d a 0. t 0 5 et- water dreplet.

I 127

R:03F [gg i i i i i i i i i i i i .. i i i i i 75 - -

75 - -

50 - -

50 - -

m - -

m . _

g _ _

g - _

m - - m -

25 - -

25 -

0 -

0 -

1 I R t i i i t i 1 1 I i i i i i I 0 .00200 .00400 .0 .00200 .00400 TNE N SECONDS TWE N SECONDS (a) GAGE P2 (b) GAGE P3

)

i l

l l

i l

l I r. 8 3. Kater phase pressere f i.t e x pe r ine n q).j o, l

l 128

4 l

I I I I I I I I I I g g I I

^

-}

6 to a _

w 2 _

g _

E 4 -

D fn -

w/ -

W g _

i 0- 2 - - -

I i 1 i-- i i l i I l 1 0 1 2 3 TIME (MS) lip. 44. l'r e s s u r e a t lot.atter (f 21 = 10.27 r:. O.30 t-) measured frer the hi.t t er: of the Lusite 6ortoirer.

4 129

,- -- ------ ,,-...n. _ _ - . , . _._ _,_. _ _ - - - . , . . _ . _--,. .. - , - , - _ , . - - . . , - , , - , - , . . - . . . . - . -

- - _-- _ ~ _ -

~ - .. . ,- ,

L 's 9

. . s  : ~ : g irrelevart , because ne atter pt wa, made to calculate propagatior of a deteratier wave.

The rep.rted ,on e r s e ~ t. i n e tir.ite ( 1. 3- 2, ir. ) is ret direst!s comparable to the S1411 k-ll sa! ulat sers (o.$'. f or the nonuniform case ). l a ri t .

5141ER-ll used a constar.t-preuure' boundary cor.dition at the sides of the tank.

' fluid could not be nt elerated at te r e sit irg - the problem boundaries. Secord.

or.e - re port ed tethod ef ebt a i r.i rg - t he_ experieental conversion ratio was to i r.t e g ra t e the base-pressure transducer surve. The experimental value tirot fig. 36) is at most 6 (Hui t r a s )/t' . The ca lculat ed value (frot fig. 421 is

~

about 6 700 (r'si/t#.

If the experieertal value ii, cultiplied by the entire base area ( 0. 36 c? ) .

which is unrea lis t ic- because the measurerer.t represents only one poirt, the axial impulse is leu t ha r. - 3< H m r.s . The value reperted' by the experimentalists doing this exercise is 10 000 ns. liecause the kiretic energy goes as the square of the impulse, the r e po r t ed k i r.e t :( energy based on this ca lct.la t ier. a ppea rs suspicious. The S1411'R ll irfut for the tase shown ir rigt. 42 and 44 is giver

, it Apperdix R.

This f i t t i r.g e xe rc i se - sugge s t s that r e a s e r.a hl e base-case parameters fer ,

use it a reactor study sheuld he a f u e l-d r o pl e t diameter of 0.3 te. a water-droplet diate t e r of 0.075 t- a nd a h e a t - t ra r.s f e r cu l t i pl i e r o f 0. 2. The case with the unifrrrly l e a r. fuel da tribution was judged to be less useful as compared with i r.pu t t i r g the' cerputationally (three-field) detertired fuel density distribution. The results of using these fitting parameters on a secer.d nonutifort scarse pretixture. that calculated by 5141ER-II ( two-f ie ld ). are in Apperdix T.

If a vaper explosior is pou i bl e only with such water-rich or fuel-lear e rvi ror.ne tt s , any it-ve nel stent explosion car. net develop sufficier.t energy for a containtent challerge to occur because of the querching co pa i t y of the abundart water u r.d simply the lack of suffic ier.t volute within the venet to allow a nigrificant fraction of the fuel te participate. If a urifort.

- f ue l-ric h ervi rontert is assured, evet a multirlier of 0. 2 tay e xagge ra t e the rate of heat transfer with the new heat-trar.sfer assumptions. A calculation of test 43 of liustor'i, SNL series used f or salibrat ion ir the old 211' study with 50:2$:25 ( l ue l :wa t e r :va po r i prerised regler ard the presert fitted input.

130

produces a peak preuure of 12.6 hors at the kall. about twice the desired value f or that test.

Twe . fir ) el s e n.a ie r s s !. . . t. l J 1r raJt. Iarst. ir reality eris sor e fractior of the i t.e l a r t e r a s t a. efittier.tiv te produce stear.. This fractier. will f ra gr.e r.1 te sizes s r:. l l e r that o.3 me. lie s a u s e r.o me thod of calc ula t ir.g this fractier is kr.ovr., all prerixed fuel is used to transfer energy to the water ir our calculations. Sete paratetrat varietiors are desirable to allow for the 3rplied ur.certaarties in the reacter c a l c u la t i or.s . Se c o r.d . the previous Zil' '

study' suggested that the reacter case should be nere highly constrair.ed t ha r.

i the heretolere e x pe r ite r.t a l s imu la t i er.s. l'o r r e f e re r.c e , the effects of ir.treasing corstraint ir. this situlatier. ef test MD-19 are showr. in Apper. dix 5.

131 I

V. ANALYSIS OF LOS ALAMOS EXPERIMEhTAL DATA FOR SHALLOW POOLS R t3H DRIE A. In t rodu c t i er.

A brief SIWER-Il study exacined the shallow pool experiments performed at Los Alamos. 'The objective of the SIWER-II calculations of the experiments was to compare calculated and actual slug breakup dynamics. So that judgments could be made on whether the slub breakup calculated in reactor accident situations is exaggerated or underesticated.

This chapter is organized following the time sequence used in performing the ' calculations. First, we made a correlation to a reference case. Second, some hypothetical cases were run with nonunifore interfaces. Different breakup cod e s we r e obtained. and the question of SIWER-Il scaleup to the reactor was exactned. Third, an experieent with a different pressure ratio was calculated.

Fourth, the e x pe r icie nt a l apparatus was modified to present an initially nonunifort interface. A calculation of this modified configuration was the performed with SlWER-II. The final section treats conclusions.

A schematic of the test apparatus is shown in Fig. 45. The depth of the pool and the height of the free space above the pool are scaled using the actual dimensions of the reactor vessel and the total amount of fuel available. As indicated in the schecatic, a very thin (1/4-cil) diaphrage supports the pool in a 102-mm-ID Plexiglas tube. The pool is 50 en deer and there are 155 cm of f ree l

space above the pool. The bottom of the tube is separated from the nitrogen driver gas by a 4-cil Mylar diaphragt. The experiment is started by cutting the lower diaphrage; the pressure rises in the bottom !ube and ruptures the thin diaphrage thet supports the water. The increased pressure then induces the motion of the water that is of interest. Three pieces of data are collected during the test: the pressure history just under the top r'iaphrage (P1), the pressure history at the endplate (P4), and a high-speed movie.

B. Experimente! Correlation In the case used for correlation to SlHMER-II, the water was to be accelerated by an expected 70 psig ($1.3 psia ) nitrogen gas pressure. The space above the wa t e r wa s at -0.82 psia, with the partial pressures o f wa t e r and nitrogen being equal. This gives a pressure ratio across the water slut of 132

kcl)3iiORAU SHALLOW-POOL APPARATUS .

P4 0 .

W m ill fg bragm 4 mill pi hrogm l

WHHH// Y//H//H /A I

]

4 ,hrasm nitrogen l= # 'I H/H/HH//4 ' /HHHHMA

. T l' i p . -15. 1.x pe r a r.e r.t a l s e t u p f o r the shallow-pool tests.

l 133 l

RD3H CWT 100*1 and a resultir.g acceleration scaled to that expected fcr a core pool followir.g a stene exploster..

The desire to examine water slug breakup in various initial configurations under rapid acceleration dictates the use of as many radial nodes across the water as possible. To limit the expense of running these simulations, only the area above the P1 pressure trar.sducer was modeled, and the experimental P1 data were input as a pressure vs time boundary condition. The initial S141ER-Il setup choser. cor.tair.s 10 radial by 47 axial nodes and is shown in Fig. 46. The actual output, from the P1 transducer is shown in Fig. 47. To avoid the calculation of costly irrelevant but time-consucing high velocity gas flow, tiec zero for the purposes of 5141ER-II was defined as 10. 9 c:s on Fig. 47. The resulting boundary cor.dition for SDNER-Il is shown in Fig. 46. This curve was sleulated by 23 ir.put points. By linearly interpolating between these 23 points the Sl*1ER-II approxicat ed tiee integral of Fig. 48 f rom 0 to S es is 4.155 kPaas. The actual-tire integral of Fig. 48 is 4.157 kPa as. Ccnsequently, the average inlet pressure was 75.4 psia over these 6 es.

The same defir.ition of time zero was used to graph the results of the P4 pressure transducer, showr. ir. Fig. 49. To fit this response, close coupling of the liquid and vapor fields in SIW1ER-II was required to hold the slug together as long as possible. A constant 100 ye droplet diameter was chosen to achieve this objective. This is as small as we can reasonably _ expect the droplet diameters to be ir. the molter.-core / coolant-interactior.s problem. We recognize that this drop diaceter is approxicately an order of eagnitude smaller than the Taylor wave length of the fastest growir.g instability, which is 2r(30/a(O,-c g )) - 900 ut. ICPlications of such required close liquid-vapor coupling are discussed in subsequent sections.) A second adjusted paraceter was e4

o. the vapor-volume fraction at which the two-phase to single-phase

' t ransition occurs. This was lowered to 0.02f to t-s t o ca i r.t a i n cells in a two-phase condition as long as possible during liquid impact with the upper endplate. The calculated pressures at the center of the upper erIplate. r.od e (1.47), are shown in Fig. 50. The peak eagen tude and sepact duration compare favorably to the P4 data. Tte e x pe r a rce nt a l impact occurs slightly earlier thar.

that of the SIW1ER-Il calculation. The water probably begins to cove before the 134

RX 3F [WT SI41ER-I! tree zero. This type of delay is also consistent with previously reported SIW1ER-Il calculations of SRI experieents.

To further check the $141ER-!! results, we constructed a simple slug model.

In th:s model the water was driven upward by the pressures shown in Fig. 45, reduced by the 0.52 psi above the water. lepact occurred at 5. 99 es at a velocity of 58.69 c/s. The peak S121ER-Il iepact pressure was at 5.87 c:s. The slug model should calculate a later time because the effect of any decrease of the P1 pressure is felt testantly. Also, in the slug model, all the water iepact s sic:ultar.cously. The S14fER-Il peak occurs at more of an averaged tice.

Another calculation of interest is the tice integral of the P4 pressure trace. The integra) under the pressure trace of Fig. 49 is 4.34 kPa's. The pressure tire integral averaged over t h'e top boundary in the SIW1ER-il calculation is 4.42 kPa s. Again. this is favorable agreceent. (Both nutbers are larger that the P1 integral. At the end of the calculation the water is coving downward.)

Finally, we cust exac:ine slug breakup. Both the experiment and calculation

- show significant water dispersal before impact. The water's hancer pressure.

pcv. from the simple slug codel is -88MPa. Less than 16 MPa are ceasured and calculated. However, this S141ER-Il calculation does not model the proper physical ecchanism for slug breakup. i.e., the classic Taylor instability obtained when a li ht F fluid accelerates a dense fluid. and the 5141ER-Il result is therefore artificial. in this small-droplet SIW1ER-Il calculation the slug disperses as a consequence of numerical diffusion. caused by ic:pli c i t first-order donor-cell mass convection, and from numerical instability. caused by the explicit evaluation of the work tere coupled with the small cesh and augeented by the reduced oo . Liquid-volute-fraction contour plots obtained froc:

SIW1ER-Il are shown in Fig. 51. Both axial spreadinE, or diffusion, and radial instabilities ir. the se:all outer meshes are shown. ._

' P. E. Rexroth and A. J. Suo-Anttila. "S141ER Analysis of SRI High Pressure Bubble Expansion Experieents." in " Specialists ' Vorkshop on Predictive Analysis of Material Dynamics in LMTBR Safety Experiments." Los Alamos Scientific Laboratory report. LA-7935-C (March 13-15. 1979), pp. 305-319.

135

i m m sn l--

RV '

J ,

) <-

SHALLOW POOL SETUP n 47 a 40 l

185 30 LOW P GAS 20 ~

n 13 sxxxxess3,;s n n 12 xNxNse wecessa n NNNNN ('3 333:ss l C\wTY XN <Ws'sS? i NWNW N20? ?2 Css 50 exmws sN're WATER

\wwwee cess (WN NO N N sN32s x\NN ATN eccCss3

" S '"" ^*"";' '

10 2 HIGH P GAS i I 12 10 ^

= 51 l Isp. 4 ti . Sl ANI R- 1 I te <.h us t ry e uuo 1 e r ea radia1 rodes ard a preuure vs tire bctter boundars uirJitier.

136

m R: 4- DRAF~

0.7-f 0.6 -

0.5- _w,~

0.4 - &

f 0.3 -

02 - i O.1 -

i 0.0

)

0.0 5.0 10.0 5.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0 Time, ms I:y. 4" liste r ir et t a l pre ssere recerded at locatier lit.

137

I e , JnA 08.

PRESSURE FOR SIMMER INPUT SP7014 07-f b3 06 ' [ \

i 5 N.

? ]

c. 0.5 ' . F. \w 2 -

,f.srL s A(%f w i e g

.t!

0.4 - I

$ 0.3-x C-  ;

i 2

0.2 . -

0.1 .

0 0 ', .. ,,, . . ,,,,,, ,,, ,

0.0 1.0 20 30 40 50 60- k0 80 TIME (ms) l'i g . 45. l'1 re,orded pre sures over the time targe of i t.t e r e s t f or SI ANI:K-il t a lt u lat ier.s.

138

ROJ HFT 20.0 -

PRESSURE RESPONSE FOR TEST SP7014 16.0 -

T .

i 12.0 -

2

)

8.0 - ,

d -

l . , ,-

WA j(u l hl!g 4.0 - -

(,

p

(

- \1\ Ihe f h.wv.,._

l 0.0 ....i.-..i....,. ,i. -

i.. ..i....i....i 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 80 l

TlME (ms) l' i t . 89. 1.xperiterial 14 pressure trace.

139

1 R: J C uFT l l

o

[d l 1

\ \

v

- .. c l

o ~

N  !

I b

F =

V. -

.. c F [o 1 T.

_ ]L i

i 1-

._A ,

v c' \

o  : e.

s 1

_r. .

~

r.-

~ c.

v. .v Z e

e Z -

O -

i~

n

- , ] r/..]

- +c V. ,, ,

1

_r.- /

%~ i tw -

,L -)

~

_c

.. 1 i

x

.N V. -

V.

e.-

-_o

_v .-

1 o

3 i

i . .

i i

.e o e o P Q c d o N o v c m - i (edN) 38.7SS38d l i g. 40 ('a l c u la t ed pr e s'ure !*r terpariser at the ly trarsdaser.

140

R0111 EWT I

l I

i s

i l l' l

l l

' M 1m (a) Time = 0.0 ms (b) Time = 2.0 ms ki{M .v r j

~ ,f-NM

,g .'

S

[vL M :)'i\

?

s ~--

"_,%)~  ;

?s ,

(c) Time = 4.0 ms (d) Time = 5.0 ms Fig. 51. Liquid-volume fraction contour plots produced by SIMMER-11.

141

, rru er

\::,- =: ,

(.%e<i

.ib

~: ?,\ r _jf,C

,l

A hi ,0 0' (',;

. ?w;N (e) Time = 5.5 ms (f) Time = 6.0 ms t;' '. 8

-r y 4 'y 2 /E'~~ ' '$.

l C Qi

!  ;;= = = 5;;

~

,  ! 1: ' y

! I i i

! . i i

I  !

L _~J -

.- e D ej

.s s. 4 (g) Time = 7.0 ms (h) Time = 8.0 ms Fig. 51. (continued).

142

r i

To correct the instability problec, an additional t ic:e-ste p cont rol based o r. the work tere was inserted. This control is described in the coding codifications write-up in Section F of Chapter 11. The resultir.g liquid-volume fraction cor. tour plots are shown in Fig. 52. The numerical instabilities have indeed beer. elitir.ated except for a minor residual problem following impact. A revised pressure trace is giver i r. Fig. 53. Although the peak value is no l or.g e r as high with the more stable calculation, the comparison is still reasonable, given the experimental ur.certair.tiet. the pressure-tice integral is r.ow 4.32 kPaas. A listing of the SIkS1ER-II input for the problem is given in TABLE XIV. The pressure-time integral is now 4.32 kP a s. A listing of the SikS1ER-Il ir.put for the problem is given ir. Table XIV.

TABLE XIV SIkt1ER-Il INPUT FOR THE REFERENCE CALCULATION 1 0 -105007VRB AIR / VATER SLUG BREAKUP BASE CASE 2 1 0 0 0 0 32 1 1 200 0 3 AIR / WATER SLUG BREAKUP BASE CASE 4 .0080 10.0 5 10 47 6 FLUID DYNellCS INTEGER INPUT 7 3 47 -23 0 1 0 1 5 0 9 93 1 1 1 0 1 0 1 1 1 1 10 1 1 1 2 1 3 1 4 1 5 1 6 11 1 7 1 5 1 9 1 10 1 11 1 12 12 1 13 1 14 1 15 1 16 1 17 1 18 13 1 19 1 20 1 21 1 22 1 23 1 24 14 1 25 1 26 1 27 1 25 1 29 1 30 15 1 31 1 32 1 33 1 34 1 35 1 36 16 1 37 1 38 1 39 1 40 1 41 1 42 17 1 43 1 44 1 45 1 46 1 47 2 5 15 3 5 4 5 5 5 6 5 7 5 8 5 19 9 5 10 5 2 6 2 13 3 13 4 13 20 5 13 6 13 7 13 5 13 9 13 10 13 21 2 23 3 23 4 23 5 23 6 23 7 23 22 5 23 9 23 10 23 2 40 3 40 4 40 23 5 40 6 40 7 44 5 40 9 ~40- 10 40 24 2 47 3 47 4 47 5 47 6 47 7 47 25 5 47 9 47 10 47 26 4 10 500 50 20 4 0 -1 00 6 6 1 27 PROBLEM DIA1 ENS 10NS AND OPERATIONAL CONTROLS 25 0.016125 1 0.0066503 2 0.005126 3 l 29 0.0043214 4 0.0035072 5 0.0034420 6 30 0.0031652 7 0.0029461 5 0.0027671 9 31 0.0026172 10 l

143 1

-j

ROJCi-! DR/FT ! I VC-)~E'"#CyD.CrLIg!O ygLuyg 7g,;;;gy gr gjggjg TIME 2 000*S i

i .

' l I

f

! -1

[L '

. p

/L' h-

. /),

-N unnummene==

MIN: 9 SSE-06 MA1: 9 89E-01 CI: 9 89E-02 MIN: 9 58E-06 Max: 9 89E-01 CI: 9 89E-C2 v0LuwE FEA '!ON Or L10g:0 VOLE *E FEA 'IO*. Or L10 !O TIME 5 000"!

TIME e ::0**

l l

I i

i I

( 4)

(. .-

MIN: 9 58E-06 max: 9 89E-01 CI: 9 89E-02 MIN: 9 58E 06 MIA- 9 89E-01 CI: 9 89E-02 I r ;. . .n. . t ort eur plot s w i t ): tine-step tertrol revis ('rs.

144

Al & =

== ,

v0;uuE FRa:TICN Or L10u!O v0;UwE FRACTION Or LIOul,0 TIME 5 500"5 i 11pE 6 000pr I

l

. I LO -

MIN: 9 58E-06 Max: 9 89E-01 CI: 9 89E-02 f)

MIN: 9 58E-06 max: 9 89E-01 Cl= 9 89E-02 v0.uuE rRA: TION Or L10g!O v0;UwE rRACTION Or t]O.;1; TIME 7 0;;ws TlwE B OD0ws

. J, . J I

6 i

l MIN: 9 58E-06 max: 9 89E-01 Cl: 9 89E-02 MIN: 9 58E-06 max: 9 89E-01 CI: 9 89E-02 Fig. 52 t e or t irued ).

145

n n , - -

wwJ, \

1 1.. -

11.0 ,

i PRESSURE RESPONSE TEST SP7014 10.0 9.0i 80i 3 ?O-'

[ 60i '

e  :

E 5.0 -i

c. 4.0 ;$ '

il 3.0i

[\ (

2.0-2 )

1.0 V

3 0.0 J.....

i i .i .i .

.i .

O.0 1.0 2.0 30 4.0 5.0 6.0 7.0 80 i TlME (ms) l Fig. 53. Revised calculation for coc:parison with the P4 transducer.

146 l

l

(O %5 --.

O d i J,T [ ,

32 0.005 12 0.0052557 47 33 0. 5 0. 0 - 9. 5 34 .000001 .0000010 .000001 .000001 ,

35 1.0E-12 1.0E-12 1.0L-12 1.0E-5 0.1 36 .025 .99 0. 5 10.0 0.1 100.0 37 EDIT CONTROLS AhD POSTPROCESSOR CONTROLS 38 1. 5 1.0 2.0 0. 0 4.0 39 0.0010 0.0005 40 41 0.005 2. 0 42 43 0.00025 O.00001 0.00005 44 45 0.0055 0.0065 0.0080 46 47 .0020 .0010 45 49 0.004 2. 0 50 51 0.0 52 0. 0 53 0. 0 54 0. 0 55 0. 0 56 0. 0 57 0.0 58 VIEW PolhT PARB1ETERS 59 60 TIME STEP CONTROLS 61 0. 0 1.00000E-06 1.00000E-09 0.100 62 1.0E-4 0.25 10.0 1.0 1.0 1.0 63 64 STRUCTURE AND FAILURE PARAMETERS 65 0. 0 0.0 0. 0 0. 0 0. 0 0. 0 66 0. 0 67 1. 0 1.0 1.0 1.0 65 1.0E10 1.0E10 1.0E10 1.0E10 1.0E10 1.0E10 69 1.0E10 1.0E10 1.0E10 70 0. 0 71 WATER PROPERTIES AND EQUATION OF STATE 72 913.02 1926.0 273.13 3.337E+5 2.22 73 997.61 4217.1 0.0757 6.0E-1 1.789E-3 74 2.26756E+10 4.70579E+03 0.0 3.22689E+06 647.286 0.390597 75 1402.0 1.329 3.737 2.93390E+6 18.0 32.0 76 316.957 95.77 316.957 200.

77 1346. 0.3302 5.0E+4 50. 0. 5 75 O.00000E+00 0.00000E+00 0.00000E+00 79 0.00000E+00 1.0 000E+00 0.00000E+00 SO 1.00000E-30 1.00000E+00 14.9800E+02 51 STEEL PEOPERTIES AND EQUATION OF STATE

$2 8001.0 712.00 1700.0 2.47200E+05 25.0 83 6100.0 750.0 1. 6 20.0 5.36000E-03 54 1.33800E+11 4.33700E+04 0.0 8.17000E+06 10000.0 0.360 55 492.0 1.26 1.64000E-00 0.0 56.0 7700.

56 147

R:UGF [RAr 57 65 O.00000E+00 0.00000E+00 0.00000E+00 59 0.00000E+00 1.00000E+00 0.00000E+00 90 SODIUk1 PROPERTIES AND EQUATION OF STATE 91 0.0 .

92 705.0 1300.0 999.9 50.0 1.50000E-04 93 3.76000E+09 1.21300E+04 10.0 4.81600E+06 2503.0 0.341 94 1460. 1.650 3.56700E-00 0.0 23.0 1375.

95 96 97 0.00000E+00 0.00000E+00 0.00000E+00 95 0.00000E+00 1.00000E+00 0.00000E+00 99 CONTROL kMTERI AL PROPERTIES AND EQUATION OF STATE 100 2520.0 1593.0 2623.0 3.00000E+05 83.74 101 2520.0 1590.0 1.0 80.0 1.00000E-03 102 4.25600E+14 S.36800E+04 0.0 5.00000E+06 7107.0 0.350 103 500.0 1.50 1.46000E-00 0.0 55.3 5472.

104 105 106 INJECTOR GAS PROPERTIES AND EQUATION OF STATE 107 0.0 105 1.0 109 1.00000E+12 4.00000E+03 0.0 0. 0 1.0 0. 3 110 741.0 1.4 3.798E-00 3.0E06 28.0 55.0 111 112 113 'CalPONENT PROPERT1ES 114 913.02 913.02 913.02 913.02 8001.0 8001.0 115 2520.0 0. 0 0.0 116 997.61 997.61 6100.0 705.0 2520.0 913.02 117 913.02 997.61 0.0 115 14.9500E+02 14.9800E+02 14.9500E+02 14.9500E+02 14.9800E+02 14.9800E+02 119 14.9500E+02 14.9800E+02 14.9500E+02 120 HEAT TRANSFER CORRELATION DATA 121 1.0 1.0 1.0 1.0 1.0 1.0 122 1.0 1.0 1.0 1.0 1.0 1.0 123 1.0 1. 0 . 1.0 124 2.30000E-02 5.00000E-01 4.00000E-01 0.0 125 2.50000E-02 8.00000E-01 5.00000E-01 5.0 126 2.50000E-02 5.00000E-01 8.00000E-01 5.0 127 2.30000E-02 5.00000E-01 4.00000E-01 0.0 125 0.000 0.8 0.4 0. 0 129 3.7E-1 0. 6 0.33 0. 0 130 DRAG CORRELATION DATA 131 1.0011 60- 12.0 1.3E-5 1.0E-6 1.0 132 4. 0 1.0 0.5 1.0 .95 133 0.046 -0.20 1.E-5 0.046 -0. 2 0- 1.E-5 134 4.1736E+05 4.2500E+05 4.7963E+05 5.4044E+05 6.4119E+05 6.8182E+05 135 6.2585E+05 5.4856E+05 4.9178E+05 4.4380E+05 4.1061E+05 4.0610E+05 136 4.0336E+05 4.3065E+05 4.3176E+05 4.5320E+05 4.6710E+05 6.0425E+05 137 6.3590E+05 6.2566E+05 6.3923E+05 6.1407E+05 5.4537E+05 135 'O. 2.0000E-04 6.0000E-04 1.0000E-03 1.4000E-03 1.8000E-03 139 2.2000E-03 2.6000E-03 3.0000E-03 3.4000E-03 3.8000E-03 4.2000E-03 140 4.6000E-03 5.0000E-03 5.4000E-03 5.5000E-03 6.2000E-03 6.6000E-03 141 7.0000E-03 7.4000E-03 7.8000E-03 5.2000E-03 8.6000E-03 148 o

,0 O Vw I ".

< j 142 REGION 1 EVERYTHING (No STRUCTURE IN THIS PROBLEM) 143 7. 0 0. 0 0. 0 0. 0 0. 0 0.001 144 0. 0 0.0 1.0 0. 0 0. 0 1.E-5 140 0.10200 0.10200 0.10200 1.E-5 1.E-5 1.E-5 146 195.00 0.64 .00 2.3E-5 1.0E-17 .00005 147 0.5E-4 1.E19 145 VAPOR AND LIQUID VELOCITIES ON THE BOTTOM BOUNDARY 149 0. 0 0. 0 0. 0 0. 0 0. 0 0. 0 150 0. 0 0. 0 0. 0 0. 0 151 0. 0 0. 0 0. 0 0. 0 0. 0 0. 0 152 0. 0 0. 0 0. 0 0.0 153 INLET (VARIABLE PRESSURE) 154 1 2 1 10 1 1 0 0 1 155 0. 0 0. 0 0. 0 0. 0 0. 0 0. 0 156 0.0 0. 0 0.0 157 0. 0 0. 0 0.0 0. 0 0. 0 155 0. 0 0. 0 0. 0 0. 0 0. 0 0. 0 159 0. 0 .01 160 0. 0 0.0 0.0 0. 0 0. 0 296.

161 0. 0 0. 0 0.0 0. 0 0. 0 4.75709 162 296.0 163 0. 0 0.0 0. 0 0.0 .0001 .0001 164 WATER 165 3 12 1 10 1 1 0 0 1 166 0. 0 0.0 0. 0 0.0 0. 0 0. 0 167 0. 0 0.0 0. 0 168 0. 0 0.0 0. 0 0. 0 0. 0 169 995.0 0.0 0. 0 0. 0 0. 0 0. 0 170 0. 0 0. 0 171 296.0 0.0 0. 0 0.0 0. 0 0. 0 172 0.0413950 0. 0 0.0 0. 0 0. 0 0. 0 173 296.0 174 0. 0 0. 0 0.0 0. 0 .0001 .0001 175 UPPER GAS COLUMN 176 13 47 1 10 1 1 0 0 1 177 0. 0 0. 0 0.0 0. 0 0. 0 0. 0 178 0.0 0. 0 0.0 179 0. 0 0. 0 0.0 0. 0 0. 0 160 0.0 .01 0.0 0. 0 0. 0 0. 0 181 0. 0 0. 0 182 296.0 0. 0 0. 0 0. 0 0. 0 0. 0 153 0.0206994 0.0 0. 0 0.0 0. 0 0.0322125 184 296.0 165 0, 0 0. 0 0. 0 0. 0 .0001 .0001 C. Hvrothetical Cases with Nonuniform Interfaces Three additional calculations were perforced with distorted lower interfaces to assess water slug breakup under differing initial conditior.s. The goal was to develop recorrer.dations for a revised experitental configuration.

149

l N[fOh&y Ar.y steac explosion occurring as a consequence of core celt will not possess the flat ir.terfaces of Fig. 52.

First, a concave surface was modeled, where the center was raised 1 ce relative to the side. The split involved one-half the area, and the total mass of water was conserved. The liquid-volume-fraction contour plots for this case are shown in Fig. 54. The instability rapidly grew, and much of the liquid was deposited along the wall. Because no friction was modeled in this problem, the liquid continued unic:peded to the top of the tube. The impact pressure at the center of the endplate was significantly reduced. The impact dynamics was more representative of a two-phase spray. The pressure plot for the center of the endplate is giver. in Fig. 55. The peak pressure occurred earlier, because the scaller amount of water in the center was given a higher acceleration. The pressure was sustained as a consequence of water moving ir. ward from the sides of the tube.

A second paratetric modeled a convex surface, with the water level depressed by 1 cm ir. the center of the tube. Again, one-half of the area was i r.vo l ve d. The liquid-voluce-fraction contour plots are given in Fig. 56. Here also an instability grew. For this case, much of the liquid was calculated to stay in the center of the tube. Because of the concentration of the liquid toward the center, the peak pressure at this location was relatively high, as shown ir. Fig. 57.

The third parametric was to place soce structure (a plate) below the depressed cer.ter liquid in the convex case. This condition was felt to lead to a more achievable expericental situation. The liquid-volume fractions for this case are shown in Fig. 55. With the plate, a volume of liquid perhaps somewhat unrealistically stayed in the centerline node. A turbulence codel might be desirable to better calculate this case. The predicted P4 pressure response is shown in Fig. 59. A reduction from the case without the plate is_ observed as expected.

These three auxiliary calculations indicate that the pressure distribution on the top plate is a function of the initial shape of the lower water 150

t t r, 3 --

R,F VwJ 1

l m

,s, i 1

-

we- rea:*:!'. Cr ::.

-. we.-

s a. ~ . ; . :: ..;.:: i  ;

.v...~E

.u

...v._

g I I I

i  ; l I i ,

1 I i.

t

, I i

• i

• i l l I

i t

' i i

1 1 1 I

m

!. M.. -

l

@- s- s I

i (

'- -c --::. -

C.. --

y s ....

.- 2 'er-06 - M'*: 9 ...at -*

v;s: 2 7EE- t Max: 9 79E-:; 01: 9 79E-02 y- . . . . .i. . ..

,...v. ec : . , ... .. ..

7 .v:.. u.r ::.: . - .v-_

. .. .v:

.  : .- we_

p w

.- ~ -:- .

1 i

, s, ,

)

"v v.., 4 =5 -sN,i - ! , . , i i

s. : . /;:. m.

.?

nr s ., s . y, 1

l..

gz ' -

(

.v < : ' ' /, -m

. .,L ,

F>.

.w .

t.-

d --

• IN: 2 7EE-09 van: 9 9E-c; c1: 9 79E-:2 *:s: 2 sE- s wax: 9 9E-:, ::: 3 ,g.:;

Fig. 54. Liquid-voluee-Fraction cor. tour plots.

Concave initial conditions.

151

i b'IO I mr RDw ~

v..

e v: r:s--.n.u or

. - . .ge..-

-

- : .: J.- ';~i 6 ;;; -

.v:  : : .. -

3 -

s

,~ m; ,'.

^

./ m P. 3  ; L '.:. E

<:G fS s... -

s , U s V I

/

,I e c

./ -

s. v s .

'w V i  !

i i

i - x i p; ,: 2 ., E .: .: "An.1

<tL' cr.., w!N: 2 iSE-C6 Max: 9 79E-0. ~I: 9 '9E-::

.. .. .. . - v: :21--.

- - ~w N n: w .e...

k *. . . y :

• • • ..,s.. * * -

7 ....y*

..y.. %4....

• • ve-o y:.

... - g

's f( =A

.ev -

y .

g . b ,

~ - r.- (_ .

F8 '- [N .

1. r w. ). A ]'

,,. .  ;'s

- .; i

.e- .v - 6 l l' p . _ -

s v .) ,

, -.ig, - ,

v ,, - ,

. / ,s . ,,

g e s %i s

. I i

j e

b l MI : 'BE-06 max: 9 79E-C; C:: 9 iE-::

• N- 2 7EE-Oi *A>: 9 '9E-:: 01: 9 79E- 2 i

i i

I l' I E . 5-3 ( c o r.1 t r u e d ). '

I 152 I

I l

Fn,,

J J IJi ' il,k, j 50g d PRESSURE CONCAVE INITI AL CONDITIONS 1

4.03 1

]

=  :

E 1

z

- 30d e f e -

F ui 2.0 m e  :

C-  :

1.04 v'

0.0 ... ... i .i .i .

, ...i .

i 0.0 1.0 2.0 30 4.0 5.0 60 7.0 80 TIME (ms) i l' : p . 55. lat e s t: r e trate at the s ert er of the uppe r erdplat e scr t i.c tercave sase. I 1

153 I

1

F'!n 3 -y RNwDIi LEj lj v0LuwE FRACT10s or L10 AID v0LuwE rRAcTION Or L;gg ,,

TIME OCD-I TIME 2 000MS l

1 .

i l

t i

f () M El kb)

MIN: 5 77E-06 max: 9 83E-01 CI: 9 83E-02 MIN: 5 77E-06 MAA: 9 83E-01 CI: 9 83E-C2 VOLewE rRACTION Cr L10u:;- v0LuwE FRACTION or L10U10 T!wE 4 CC0ws T!wE 5 000ws

> J s' 'e 1

}\

j i '

M tc3 m)

MIN: 5 77E-06 Max: 9 83E-01 CI: 9 83E-02 MIN: 5 77E-06 max: 9 83E-01 CI: 9 83E-02  ;

rig. 56. Liquid-volute fraction contour plots. convex initial conditions. )

1 154 l

~

30*G'4' 1 Wj l v0LUwE FRACTION Or LIQUID VOLUwE FRACT10N OF LIQUID T!*E 6 000*!

T!wE 5 500 s

,l/ } f, ,

! U ,. j

'li '

l iI i j ili: E

[k!I l l' 1 ,

....,1,. ) gy ,

j p*, y .

iFT b MIN: 5 77E-06 Max: 3 83E-01 CI: 9 83E-C2 MIN: 5 77E-06 max: 9 83E-01 CI: 9 83E-02 VOLUwE FRACTION Or L;ggge V0'.UwE FRACTION Or L10u!;

. T!wE 8 000w:

T1wE 7 000ws Qh t

(

?

fl> ' {I'

. i Y, hI V i

'n l

( h')

MIN: 5 77E-06 max: 9 83E-01 Cl= 9 83E-02 (t:)77E-06 max:

MIN: 5 9 83E-01 C1: 9 63E-;;

lip. 56 (certiruedt. l 155

et !n m, ---

1 -

wd l 1 t 11.0 g 10 0 ;1 PRESSURE IN THE CONVEX CASE 909 8.0 2

- 3 5 7.0$

v 2  :

g 6 0 ;.

e  :

C 5.0 2 d@

2 4.0 :

c.  :

30i .

2.0i 3

104 '

0.0 ', , ., .

00 ,,

1.0 2.0 3.0 ,

4.0 5.0 6.0 7.0 80 TIME (ms)

Fig. 57. Pressure trace at the center of the endplate for the conv'X C85

156

Rn Inq gMFT YOLUME FRA: TION Or L10ulD TIME 2 000MS VOLUME FRAC 110N Dr tigg:D TlwE 000*!

M (f)

M P MIN: 3 81E-06 max: 9 76E-01 CI: 9 76r.02 Mis: 3 81E-06 max: 9 76E-01 Cl: 9 76E-C2 VOLUME FRACTION Or L10ulo vo;UwE roA:T!DN Or L10u:D TIME 5 000*i TIwE A C0:wf

~

l p [5'l .

i t

\ ,I

- s J w

% h'p

} ,

'1li i l

,.. .. (

h (

• it (d )

(g .

MIN: 3 BIE-06 max: 9 76E-01 CI: 9 76E-02 MIN: 3 81E-06 max: 9 76E-01 CI: 9 76E-02 l ig. $5. Liquid-v lure fr.atiors for the plate obst ruc t ior use.

157

T l *V V i

Vi : L

)t

(

l VOLUME FRACTION OF L1' QUID TIME 6 000vs v0LewE FRACTION Or LIGUID --

TiwE 5 500ws, _ ,

/

y 11

"' {

\ i '

')  %

i n

, U)

(d.

MIN: 3 81E-06 wax: 9 76E-01 CI: 9 76E-02 IN: 3 81E-06 ux: 9 76E-C. ::: 5 ~6E-::

v0 uwE FRA TION Or t!og;c T!wE 6 500*i f ,

i.

\0 i;L, 3 l:

I. h. '

'f.;

'i:

! )

an

(#

MIN: 3 8tE-06 max: 9 76E-02 C1: 9 76E-02 l' ii . 55. ( c or.t t rued i I

i 158 l

R:03H DRAT 60, .

] PLATE OBSTRUCTION CASE 5.0- .

q C- 4.0 'd l a J L:

e 30- i D - f B  :

m -

=- 2.0 - \

10- .

3 00 l ... .

... i ... i....i 0.0 1.0 2.0 30 4.0 5.0 60 7.0 80 TIME (ms) l Fig. 59. Pressure trace at the center of the endplate with a percanent obs t r uc t i or..

l 159 1

R: 3H [ W  !

l l

irterface. Also meaningful for comparison with the reference case is an 3 integral quantity, the force on the head. This is defined by SlHh1ER-Il as F( t ) = f ( p - 0.1 MPa ' d A . (69)

Area The results for the four cases discussed so far are given in Figs. 60-63.

Initially, the force is negative because the uppe r chamber is subateospheric.

Further, as a consequence of the conservation of comentum, the integral F(t) will be apprcximately the same for all cases if the integral is extended far enough in time, e. g. , 5 es. This impulse will not compare exactly because of differing degrees of reflection following inpact. The cases do significantly differ however. Without the initially flat interface, the impact incoherence means that the peak force is reduced by a factor of 2 or more, and the I previously narrow peaks are spread out i r. tiee. Consequently, these calculations with an initially notunifere interface imply a significant increase in the two-phase character of any " slug" impact.

In examining the effects of a nonunifere interface, the influence of scaling is of sete importance. The scaling was checked by increasing both the dimensions and the diiving pressure by a factor of 40 for the convex case. This in_ creases the tube radius to 2 ceters or to reactor scale while taintaining the same acceleration. The impact pressures at the center of the tube are shown in Fig. 64. Peak pressure has increased by a factor of 40 over that in Fig. 57, as expected. Of some concern is the configuration development during the expansion. The liquid-volute fractions for this case are shown in Fig. 65.

After dividing out the expected v40 in the time scale, we see that the upscaled case is considerably more coherent.

In teres of the calculation, the biggest difference compared with the small tube results is in the slip between liquid and vapor. A simplified SlHLiER-Il comentue equation for either the liquid or vapor field is a fp*y 's .< .s

+ V * ( Ovv ) = -oT p + g Ivr'V

-. (70) r .

~

where # is the cacroscopic density.

4 v is the velocity.

160 1

l

ROUGE E

'O 80-7.04

,1 60A '

3 50- i

.' n._

E 4.0 --

S  :

E g 30 2 i i

si 20- ,

a j f 10-[

'h[' v 4

0.0 4

?

)

-10' ..

s. .

s- -

s -

00 1.0 2.0 3.0 40 s -

n- s -

s 5.0 60 7.0 80

~~

TIME (ms) rig. 6..

Ke:ererte case pre ut.re ir.tepo!.

161

A R:0GHDRg i

'O

- 50-

+ .

' 4.0- .

i 30- .

- 1

5. 'l - fI b

a:

2.0 -

o ta.

1.01 .

i 0.0 - .

+ -

4

-1.0 .i, ii i- i i. i 00 1.0 2.0 3.0 4.0 5.0 60 7.0 80 >

r TIME (ms)

Y 4

I it. M. t' ors a ve u s e pressure i r.t e g r a l .

162

.---s. - , . , , . - g .-,-.-----y,-,.w -.e,+, - - ~ . . - ,.ya, -,w-g --,e_,, . , - , , - - . , _ , , , - - , ,,,,,,,,.,w-n-, , - - , - , , - - - , , - - - - -

1

,'  ? G n {\ c .

Pf.ULd.L)1"thi

/

.u

-. 4 .0 ,

i

FORCE ON HEAD

.3.0i .

~

l l

j

- 20 2 ~

E  : i ta '

o  : I' Ct: .

O .

L 1.0-.

00' '

- 1.0 ~, ..

....i .

.i ...i. .

i -

i 0.0 1.0 2.0 3.0

...i 1.0 5.0 6.0 7.0 80 TIME (ms) o

.; l it. p- - ('O r v e k case pressure i r.t e[ Ta l -

163

ROUGH DR/

"O 20.0 -

FORCE ON HEAD J

16.0 - i

. e 1

12.0 1 1

z x

$ 8.0 I

~

O

• t,a .

4.0 -.

0.0 -

-4.0 '

i -

i '

'i i ' ' '

i ' '

i i' 'i 'i 0.0 1.0 2.0 30 4.0 50 60 7.0 8.0 TlME (ms)  !

gg, g}, f ) i. ) { gibklId4$III Ed 164

y 'n, , ..

g,g i 500 0 -

UPSCALE OF THE CONVEX CASE 400.0 -

7 g 300.0 -

w .

m -

o ~

b z 200 0 -

c. .

100.0 -

. q

~

, , -~,....,

0.0 . ,

0.01 0.02 0.03 0 04 0.05 0.00 TIME (s) l'i p . 64. Ce r t e r of tube pressure trate for the scaled-up u se.

165'

RQ13H g g VC.vvE rs a;3 j g, Or g ;g ..-

7;M t -

VOLUME TIME Faa: TION Or ticy "-

l 12 650Ms 1

l b

t i

f j%}

maamme l I MIN: 5 53E-06 Max:

h-- t "l j oog.oc CI: I 00E-0: MjN:

(a M 553E-06bx: l CCE 00 J cI: ; g;g,..

v 0,.

T!":g' uE25res:

3;;w: ;;., 3r ;.!; :: VOLU"E FRACTION Or tigg;;

ilWE

' 31 620u5 e

'1 i l,',

i I L' ' '

l

1. f 6 b j, ',;)

i -

i , ;

. n  ! y k k; R 'r (w ,n 1 )

N. ?

7 I $!,

'lI fs $g', l.

/

(? ,\ ~~

MIN: 5 53E-06 Man: 1 00E*00 CI: 1 00E-0: MIN: 5 53E-06 Max: 1 00E*00 CI: 1 OCE-::

l i g. 65. Liquid-velute t rac t iers for t iie ualeJ-Lp case.

166

RCt 1 3 g v 0. vu.- '

T l *E ' r c ,- . g.[ ~I 3- 7E0",. -

v VOLU"E TIwE FRACTICN Or LIQUID

,r 37 940u5 b ff I

"}' .N -

i.f 4

f  :

I: )

I

,A!

W .

i (f)

FIN: 5 53E-06 was:

1 00E 0; 01: 1 00E-0: MIN: 5 53E-06 Max:

1 00E 00 01: 1 OCE-01 tO.uus ret: 17., ;r g:;, : vo-uwt raA;'lON or L10g!;

71*:

1:*I *- 2E:$ 50 000*I

,f

. < '*N 9:

g-.~ : .q:

?R : . .

a y

j)

N[Wo/9 l j i!6N,.  !

,'dv : I

' I wA' I  ;

i l

,)

~

(6)

~

.' /

MIN: 5 53E-06 MA>:

pgs: s 53E-06 M'a: 1 00E*00 CI' 1 00E'Cl 1 00E+00 01: 1 00E-0:

l~ ; p. 66 (sortiruedi.

l l

l 167

RE3- E RA:T o is the volute fraction, r p is the pressure, f is a function of der.sities. volume fractions and the drag coefficient, r is an effective droplet size, and vr is the relative interfield velocity. E Because the velocity scales as v40, all teres scale (retain unchanged) except B for interfield drag.

If the effective droplet size is assumed to be unchanged, because the overall pressure gradient r en:a i ns the same, the force-per-unit Less liquid-vapor volume from interfield drag will increase by a factor of 40.

Further, a larger droplet

' lip is apparent in the results shown in Fig. 65.

size than required to achieve comparable results with those from the small scale would not be justified beccuse any ter.dency toward increasing interfield velocity differences should decrease the droplet size producing less slip.

a In teres of a more theoretical approach, Te nr.a n t " has performed dieensional analysis to obtain (for a flat interface) an equation of the fore (71) 1 - K (//dla (p,)b (p,jy,)c (Fr)' ,

J where s is the breakup distance, 4 is the initial length, of the pool, K deper.ds or. the experimental procedure, d is the diameter of the pool, Re is the Reynolds r.ueber.

We is the Weber number, Fr is the Froude r.ueber, and

a. b, c, and e are constants.

Over the range of parameters investigated experimentally by Tennant. he Three concluded that "a" should be unity, and b, c and e appeared to be zero.

results suggest that breakup should occur over the same scaled distance.

f roe U. 5. Departtent of

'*R. A. Tennant, " Liquid-coluen breakup experiments,"

Energy Fast Reactor Safety Progress Report, April-June 1983, Argonne National Laboratory report, ANL/TMC 1953 p. 102.

166

R:U H CRAr Three additional considerations deserve attention. First, the upscale in the SDNER-Il calculatior. was performed by simply increasing the cesh dieensiens. This approach increases the effective numerical diffusion in the radie l di rect ior.. Slug breakup would probably occur faster if it were possible to use the same size r. odes at both scales. Second. a factor producing less of a liquid slug impact in the calculatior. (as well as decreased slip) was the i r.c r e a s e d nitrogen vapor density used to accelerate the water. For proper scaling. the nitroger.'s i r.t e rna l energy should have beer. increased. Because such an er.ergy increase would have unrealistically produced steam, the density was changed instead. This change ceant an increased proportior. of the kinetic energy went into a acceleration of the nigtrogen. whose density was now an appreciable fraction of that of liquid wa t e r. Third, a model by Corradini

gives the a rate of entrainment frot Taylor instabilities as dV/dt = 4.6 A.Y a7,.

• where 7 is the critical Taylor instability wavelength defined as 1/ 3 times the wavelength of the fastest growing instability given in Sec. V-B, V is the volume entrained and Ap is the area of the pool or slug. For a one-dimensional case V

- Ap z. This model predicts coeplete entrainment of the 50 c:m experimental slug at 4.5 es, which is approximately correct. It also predicts only a weak l#4 deper. dent, a . frot acceleration en the rate of ent rainment. but it does suggest less e r.t r a i nme n t at large scale. In other words, at constant acceleration. ir:reasing the dieensions by a factor of 40 only scales the e nt ra i r.ce r.t rate by 40 based on tle Corradini model.

The lesson from all these scaling censiderations is that a combination experimental /ar.alytical prograc is desirable if truly precise statements are to be made e r. scaling effects i r. a unique physical systee. For the current calculatior., Fig. 66 gives.... Fig. 66 gives the force on the head for the upscaled case. The magnitude is semewhat greater than might be expected based on a scaling f actor of 40 or 64 000, probably as a consequence of the more 2

coherent iepact.

D. Experimertal Cenparisor. with a Pressure Ratio of 50:1

Michael 1. . Corradini. k'a r r e n M. Rohsenow, and Niel E. Todreas. "The effects of sodiue Ent raineent and Heat Transfer wi th Two-Pha s e U0 3, during a Hypothetical Core Disruptive Ac c i d e r.t . ' Nuclear Science and Er.gireering 73 242-255 (l9501.

169

m n,

n< m. --

u s'J s "o

35 0 -

FORCE ON HEAD 30 0 - .

250]d [

i E 20 0 '

15  :

a:

2 15.02, i 1

10.0 -

f. fl 50f .

0.0 ,

)

0.00 0 01 0 02 0.03 0.04 0.05 TIME (s) i,t. oo. or o iea < e tre ~ re interru -

170

T-RDM E WT The next activity was a calculation of an experiment run with a 50:1 pressure ratio using the_5141ER-Il code as previously correlated to the 100:1 pressure-ratio data. Here times should scale as v2. The P1 pressure, starting at 11 ms into a typical transient, is shown in Fig. 67. For SI44ER-Il input.

time zero was defined as 1.3 es into Fig. 67 (12.3 es into the transient). The resulting inlet pressure. Pl. which can be compared with FiE. 48, is shown in

~

Fig. 65.

The previously _used t ic:e -s t e p c or.t rol was insufficient to rec ove instabilities generated by the work tert. so this problem was simply run with a maxicue tace-step size of 10 us. The resulting impact pressure at the center of

< the e r.d pla t e is shown in Fig. 69. The timing was about right. The maxieue pressure occurred at 6.27 es. The simple slug model obtained impact at 8.44 es with a velocity of 43. 05 c:/s. This was reasonable given that the driving pressures were not quite reduced by a factor of 2. The magnitude of the calculated pressure response was as expected. A reduction of a factor of v2 to 2 f rom Figs. 50 and 53 to Fig. 69 was anticipated and observed. Because the magnitude of the peak pressure was somewhat a consequence of instabilities, accuracy was limited. Of concern was the reduction seen in the experimental P4 response, shown in Fig. 70. k'hile the timing of the peak was reasonable, a much more twc-phase response was-observed. The 5141ER-Il impact' characteristics were more like those at the 100:1 pressure ratio. as could be inferred f ree the head-force plot. Fig. 71.

Further study on the difference between the 100:1 and 50:1 pressure ratio cases lead to the following:

(a) The numerical dif f usion of the S141ER-Il code does scale as expected.

Fig. 72 shows the liquid-volume-fraction plots at scaled times. At least the first four plots are very similar to those in Fig. 52. Although the pressure was modified by changing the gas density rather than its internal energy. all t e re:s in the momentum equation. Eq. ( 70). do scale by approxima'tely the same factor.

l 171

ROJGH DR 0.4 -

O.3-e Q. .

I 2

co 2

a 0.2-b z .

A.

0.1-0.0 . . . . , . . , ,

0.0 5.0 10.0 15.0 20.0 25.0 4

30.0 35.0 TIME (ms) i lit. C l.xte r ir e c t a l preuures reu rded at loca t t er l'1 f or the $o:1 preuure ratic test.

l 172

ROMggq 04- '

PRESSURE RESPONSE FOR TEST SP3052 1

l i 3

If'.k'r, i 03 ' / \ Ivy 7

c  : a 3 b

')' #

ty 2

0.2 j" b

c:

c.  :

0.1 -

0.0 '

iff'O 80 I00 12 0 TlME (ms)

} j p, p, til ret.rJ <ver the tare rarpe of i r.t e r e s t for SlANI.K-ll talsulatiers ir the $4>: 1 pressure ratio test.

173

fha 10.0 -

3 CALCULATED P4. PRESSURE SP3052 90j 803J, 4

7.09.

c .

c.

2 60-;

to  :

m 50-o  :

40

% 4 C- '

1 -

3.0 Li

l it :

20 5 Sk 1.0

'h J

0.0 . -

s- s- -

s n- ns l 00 2.0 4.0 60 80 10.0 12.0 TlME (ms) i l'ip. 64 Results et Im rt i sr 19 for the So: 1 preuure rat io test.

174

~

RCUGH QVFf 4.0 -

EXPERIMENTAL P4 PRESSURE TEST SP3052 3.0g 2

CL "E

w 14 ce 2.0 -

D '

l CE '

j c-  :

1.0 ,

I t

l i i

. }4

~

00,........i..................~ . . . ..

0.0 .. . .......i 2.0 4.0 6.0 80 10.0 12 0 TIME (ms) l ig. 70 1.sperirertal ewlt- in the 50: 1 pressure ratie test.

1 175 !

R31Gy DRAF "O

+

7.0 - -

6.0 .

5.0 . -.

4.0 ..

5. -

S m 30i o ,

a -

4', !

i:

20 - , ;lk

]

m n'

iif r!i 1.0 .

• J 0.0 .

J

- 1.0

i

if ' ' ' ' 6b 8 b ' ' ' idO 12 0 TIME (ms) _

l' i g . 71. l>re u u re irtegra! ir the $":1 pressure rat m case.

176

s RM [HAr

1. IT pe r ire r t a l rarpa r i snr w i t h a haruriterr Iritial 1rterface Irscrtier .i .- plate .. s ar obstrusttor is e x pe r i te r.t a l l y f e a a. s h l e . T!.s tesults a. hour in lig. 55 sugge t it tav also provide at i nt e r e s t i ng (or.pa r i a.or.

with a S IStil.R- I l calculation. 1.x pe r i te r.t a l l y , a 2.5 in. diameter plate was iraerted. whuh lewered the terter el the t hin upper diaphrage by 1. 3 t r.. With zero delir.ed as the irterfase betweer redes 2 ar.d 3 in l'ig. 46. the SI4b11:k- 11 repreaentatior et the i r. i t : a l e x pe r irre t t a l configuration is showr. in lig. 73.

The total ruther et axial redes was retained at 47 by i r.c rea s a r.g the tesh spa 6:rg above the water.

v Data frot four pre utably ider.t nal tests were examined. The P1 p r e i. s u r e traces with a zero tire el 10.9 ts are shown in Figs. 74-77. These curves have relstivelv sitslut shapes. ligs. 76-61 give the P4 pressure response curves.

pletted on eitilar scalea.. Ilased or. the previou'. hypothetical calculations. the data of test 19 i. h o w r ir l' i g . 61. were selected as representative. For the purpese of defirity a SINt11:k-l l i r.pu t pressure, a zero time of 12.3 es (or 1.4 es or Iit. ??) was selected.

The calculated pressurt at the (er.ter of the topplate is showr. ir. fig. 52.

The timing of impast serpared with experiment is acceptable, r.o t i r.; t h a t Tag. El r e e d a. to be shifted to th: left by 1.4 rs because of the differente in t i r.c zero. Ilos e ve r . the peak pressure poirg up to 3. 6 MPa is tissing in the calculated pressure rei. peri.e. The integrated force over the area of the head :-

• hown i r. fig. 53. Ir.t r ea s ed snoethiry oser previou, cases is a P r a r e r.t . The calculated pressure-tite ir.tegral oser t l:e top hourdarv i<. 4.01 LPa s. This s i, withir. 9m of the value calculated i r. the refererse c a a. e ( Se e l i g . bo n . The pressure-time artegrals for the four experiterts are 4. 25 kPa s for te t l ei .

7.26 Lpa's for test 17 4.22 LPa- for test 15. ard 4.49 k Pa s 1or test 10 Th e a. e i n t e g r a l s, are for 6 es with zero tire defired a a, the tonert the l'1 t r a r.a d u c e r reaties o.04 411' . In view I their urcharacteristic quality, the data frer. te t 17 are prc % hly s pu r i o u i. . Otherwise. the calc $ lated time artegral is sirp!v a Inttle low. The $14til K-l l calculated liquid volute i

Irattnor plet, are hewr ir I i g. b4. Ir (etpar *.or w 6h the movie 1 les.

?

breakthrough of the slug etsurs tv r e rapidiv. Also. the files are r.o t axisveretrat. and the strev et water lett behirJ the disirtegratirg ilug terds i

I 177 L

i RDJ3H JRAT  ;

u LOW PRESSURE 4.. ,

' l

' l 3-2-

WATER

[ -

l

& 0- - ,

c _ ,

g I

j

_1 .

-2 - ,

-3 -

PLATE --------------------- * ------ INLET P(T)

A A

& A a

i 2 3 4 6 RADIUS (CM)

I f 73- $141TR-Il representat ion of the experimental configuration with a plate.

178

W 0.8 -

P1 PRESSURE TRACE FOR TEST SP7016 0.7 - .

0.6- .

$ 0.5-E  : h to ce 0.4 - -

D

~

b ce 0.3- l o-  :

0.2-

(

0.1 - .

0.0 - . . . i . i 2.0 i ' ' ' ' i ' ' ' ' i ' '

30 4.0 5.0 '6.0

' ' ' ' ' 7.0

' ' ' ' '8 0 O.0 1.0 TlME (ms)

Fig. 74. P1 pressure in test 16 with a stationary plate.

179

. t 08-

P1 PRESSURE TRACE FOR TEST SP7017 0.7 - .

"n 0.6 - ,-

-  : , N 2, 0.5 - . .

(

j w -

c's e 04-R -

v.- .

l

' h 03-x c-  : P g

0.2 -j .

0.1 - .

^

0.0 - , ..., , ,- \

0.0 1.0 2.0 3.0 4.0 50 6.0 7.0 80 TIME (ms) l' i p . 75, l'1 preuure ir t e <.1 17 w i t h .i s t .: t i , r.a r v p ! .i t e .

180

R0G3M gyg 08-Pl PRESSURE TRACE FOR TEST SP7018 07-c

'j-A -

g 05-g 04- 6 8 03-

[

g 0.2 -

01- ,

00' . , , , , , , ,

00 10 20 30 40 50 60 70 80 TIME (ms)

T it. 76. I'! pressure ar. test 15 with a stationary plate.

1 81

l 1

I NO33H DM 0.8 ,

g,7 _-

P1 PRESSURE TRACE FOR TEST SP7 0.6 -

O.5- .

0.4 - .

0.3- .

0.2- .

0.1 -

0.0 n i i -

i 0.0 1.0 2.0 i i i 3.0 4.0 i'i 5.0 6.0 7.0 8.0

! TlME (ms) f l

182

j 1

10 0 -

PRESSURE RESPONSE FOR TEST SP7016

90. .

8.0 .

7.0 -

7

c. -

x 60- ,

ta -

e 5.0 -

D -

cd 4.0 - '

m  :

c. .

3.0 - J 2.0 .

1.0 -

0.0 ... - i -

i ... -

.. N i i i i. i- i 0.0 1.0 2.0 30 4.0 5.0 6.0 7.0 8.0 TlME (ms) l' i g . 75. I'-1 p r e s s u r e ir test 16 with a a stationary plate.

163

RCU3H pyq t

10.0 -

PRESSURE RESPONSE FOR TEST SP7017 9.0i -

8.0-7.0-c -

c s 6.0-- -

t.Q m 5.0--

D - 1 f

b z

4.0 -

I a  : W 3.0 -

2.0 7 1.0 0.0 -r---r-,--- ,r nn sn 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 TlME (ms) l'i g. 79. I'4 pr e s s u r e in test 17 with a stationary plate.

184

4 1

l l

l

, RQuGHEgjp l 6

10.0 -

i PRESSURE RESPONSE FOR TEST SP7018 9.0 .

80-7.0-c -

c. .

m 60-te .

m 5.0-D -

b m

4.0

-i

c. 2 3.0 .-

2.0 4 1.0 y

~

0.0 -

i. .. ..

0.0 1.0 20 3.0 4.0 5.0 60 7.0 80 TlME (ms) l i p. 60. I't pressure ir test 16 with a r.tationary plate.

185

1 l

R3JGH DRIR 10 0 -

PRESSURE RESPONSE FOR TEST SP7019 90-8.0 -'

.~

7.0 -

c -

~

I 60 C,Q -

m D

5.0- -

4.0 -

x -

c.  :

30- l 2.0-I

)

1.0 ,l O z 0.0 - -

7 ,... --g-., 'r' i' i =3 i -l 0.0 1.0 2.0 30 4.0 5.0 6.0 7.0 8.0 TlME (ms) r i g , 81, 12.1 pressure it test 19 with a stationary plate.

186

ROUGHDRAr' 6

4.0 -

CALCULATION P4 PRESSURE TEST SP7019

, 3.Si 3.0-7

n. 2.5-E  :

co  :

m 2.0-D  :

b m 1.5-

c. .

1.0-d 0.5-0.0 . s s n- -

s n- n- s 's 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 TlME (ms) fig. 52. Calculated l'8 pressures for test 19.

187

ROUGH DRA-

'D

- 12.0 -

11.0 10.0 -

9.0 8.0i n 7.0 '

z r.o 6.0 ;:

u -

0:

O 5.0 ' -

4.05 3.0i 2.0i 1.D -i i

0.0i

- 1.0 ; . . . . . . . . . , .

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0

! TlME (ms) 1 -

I 1

g ig. 53. Calculated pressure ir.terval, test 19.

188

RC>UM Dyr 1 r VOLUME FRACTION,0F LIQu!D TIME 2 000Mt I

\ /

Ld MIN: 2 92E-06 Max: 9 19E-01 CI: 9 79E-02 LO LJ MIN: 2 92E-06 Max: 9 79E-01 CI: 9 79E-02 I

VOLUME TIME FRACTION OF LIQUID 3 500M5 VOLUME FRAC 110N Or Llou!L TIME 3 000M5 f i l

/

l

~ ,y ),  !

r 4 I

d.) I

} Q MIN: 2 92E-06 Max: 9 79E-41 CI: 9 79E-02 ,

MIN: 2 92E-06 MAm: 9 79E-01 CI: 9 79E-02 l j

.i Fig. 54. 5!411:k-Il liquid-volume-f rac t ion plot s for t he cor pa risor. wi t h the e xpe r imenta l pla t e case, j 189

RMH D1/F~

VOLUME FRACTION Or LIQUID' TIME 5 000MT vomuwE FRA 110N Or LIQUID 11ME 4 000MT. f'f t

0

$,/\

f j

( ,

l 1

($) ,

4g> MIN: 2 92E-06 max: 9 79E-01 CI: 9 79E-C2 MIN: 29 -06 Max: 9 79E-01 CI: 9 79E-02 I

v0LUME FRACTION Or LIQu!D VOLUME FRACTION Or LjQu}p TIME 6 000Mr TIMc 8 000MT ' i l ' f '? , \

L il l l

4l fI l

! ./ -

l l

l U MIN:

( $'- ,

MIN: 2 92E-06 max: 9 19E-01 CI: 9 79E-02 2 92g.06 max- 9 7 % g 9  !

Fi g . 64 ( c er.t i r.ued ). ,

190

R:tGH DRE

.t o obs6ure further detas.ls. Iloweve r havir.g wa t e r cer.c er.t ra t ed a'lery the axis rav hr es.qgerated by \ l'411:k'- 1 1.

1. Sr.It >-r-41: Re a a.o ra b i e .. g r e c re r t w a i. chtaired to the refererted shailow p..e experirert (pressure ratie el Imn1 a r.d a f l a t lower interface for the slLg h A s' ' in presicus sirulatiers el this type el experitert. this agreetert was-obt a i r ed ai, a 6ersequette of the ruterical approximatters. rathe r , that fror attual tcJelir; of physical irstabilities.

I2i The serstart. K. ir the forcula used by Te r.r.a r t i r. Eq. I71) is stall

fer thei.e shallow-pac! tests. The correlation ir. Ref. Io would give a r.

• s' el 6.5 cc. with at '/" of 5 cr . and a "d" of to cm. Ir the current experiments, breakup does rot cctur ur.til the approximate irpact distance er 15 cc. Also.

t he- sare SlLNER- l i a s s ur.pt i ons carrot be u ed to catch both the tests of Ref.'15. which also obtair. tore rapid breakup. ar.d the currer.t e xpe r ite r.t s.

I.11 The sitNER-ll results apparer.tly scale ~ reasonably" in goir.g from a 1mi:1 ,ressure rat u to a 50: 1 p:es ure ratic. Ilowe ve r . the e xpe r ite r.t a l pre- wre sigrature charges s i gn i f i c a r.t i v. Currer.t opir.icn is that the dil l e r e'r s e ir : t he l'4 pressure t ra r.sdat e r respetse i t. tot the cor.s e qu e r.6 e o f ;.

's.f i l'l e r e t t rrJe il impact. but rather a characterie.ti. of the

..a ppa ra t u -l i c id pre s s u r e-t ra r.sdus e r t o u p l i r.g . The a ppa ra t u s is not rigid and its.asseieratior would i nf l ue r.6 e the results ir a sufficiently e r.e rg e t : 6 t a i. e .

Also, irdividua' water-droplet impacts could affect the readir.ps obtaired frer 2.

-ere-halt it. .dr. transducer. Corsequertiv the prea.sure peak reasured er the

- Imn 1 test ras be high. The a. call droplet i.ize urd tight liquid-vapor (oupliri required te eatsh th;* peak would ret be required il the calibration could base beer made to data without such problet . This also agrees with the idea en idettiivir; the drop et radius an ore-half the Tavler wave lerpth of the fastest

' ~

g rowi r.g t r s t abi l i t s.

' ' ' .l . F. Ileariry. "SihNER- i l Aralysis el Short-Rise-Tire. Two-rorporert.

T r a r. s i e r. t .lkiled-Up l'oe l sirelatier Tests. Le s. Alanos hatiora: Laheraterv report. L A 47.;o-415. Ma y 1451 191

h

~

RCUGH DRUT 4

t -l i l'.s e r with cu,essive 1: quid-vaper tetentum coupling. SINt.11:R- I l slug M e .. L * ; .surs to- quisily s.h e r torp. ired with the presert e $ re r iter.t e ' da n s rtig s. 1 aim uri. Jet irters.uc.

With a st.: l l - s c a l e e x pe r ite r.1 h:.vir, r rt e t ' t l.c !lew area chstrt.sted hv a plate. breakup is a consequence of the lask vi turbulert tsu rg- e! esti, ir the $14t.11:k- l l formalist. A r. idealized exatple e's this probler is shawr. it ing. f>$. lie r e at area expansior occurs it a cultinode

~

structure. 514t11.R- i l is centent to have the e t t e r i r.g l l u i d ta i r.t a i r. t h e sure ve l oc i t y t ra r.s t e r r i r.g r.o. cetent um t e t he s t a 2r.a r.t !!uid along the area blocked by the wall.

(5) A r.orurifort int e rfac e clearly citigates the peak impact pressures i r.

this experimental c or.f i g u ra t i on ard spreads eut. forces over time. More work must be perferred ir e xami r. i r.g e xt e r.s ier-t o-rea c t er scale. With reacter scale j r.ed e t. ard similar fluid acceleratior.s. _ Sibt1ER-Il indicates a significart reductier. it the rate of *, lug breakup. Although this reduction does have s'ere theoretical. support. srall redes ta v he required to obtain. the proper developter.t of the large-scale irstabilities.

fol With the stall r s.d i a l redes used in this problet. a r. inst abiin t v developed resulting fret the currer.t algorithm for treatir.g the work terr in the sap.r-erergy ceuatic . Ar. ad hoc t ite - s t e }- c ent rol was s u f f i c i e r.t - t e suppres, these irstah: lit ies ir ore case. J r. - c ha r.g i r.g the drivity pre sure while

- ta i r. t a i r. i r g recretry. this tire-step c er.t ro l scaled as v~'. similar t o . t he

$14t.11:R-ll DTLi 31 a nd DTLt -i l time-step certrols. This was i r.a d e q ua t e a r.d . ra r e study'is required. A,a cer.*,equer.cc of the difficulty ir.volved ir revisirp the Slkt1LR-il a lger i thm t o treat the work tert itplicitiv. a formulatier for a r.e s

't ite-s t e p cor.t rol ba sed or. t he r. ode size ard 1:0s propertici, is indicated.

(7) Preserting the detailed results el all tests would be too volucirous for this report. llowe s e r . some degree el rardot behavior was cbi.erved. Wher

' this behavior ii. coupled t o the numerica l approxicat ier+. t ha t cust be cade it it l

ary cultiphase r. uter cal fluid-dvr.atits algorithut. exact 'agIcetert or l- correlatter te this tyre of experittert tav be wis,hlut thirking.

192 i

I j.

0 , o ROJGH DR 11 Il h h i

% 1 8 0 0 $ 1 0 0 1  %

'g 6 1 J . L .l. J . L 1 J -  %

s I 8 I I e l I i 1

's

, r =g= 9- r =g= 5 =

r =g= 7 = s .

s '

g i i

. . . . .l 1 1 1 1 1 0

. g g 1 l l l 1 1 i l I  %

=

L 1 J . L 1 J . L 1 J . 'g 4 I I I I

• I I I I i g g g F *1" 9* F "l" 7

• P *4* 9

• i i l g

.l 1 1 8 1 1  %

'. 8 I I I l-1 1 1 I g

g

'g 6 1 J. L .l. J . L .l. J .

g s g i I I I i 1 1 i 'g 1

s r =g= 9= r =g= 5 = r =g= 5 =

g g i i l l l l l 9 9

• g

% 8 1 i l l 1 8 I s

% O g s

g s

s g .

s

. .l. J. g .

g . 'g

. ', s 's s 8 ' s s

's s s', .

s .

', 1 s

- s )

s .

1

. .l .

g g

g i 1 M '

% . .l. J .

's M ,'g i I 's

'_)

s s

g  % 1 5 g s . . . . .

% 0 $

g g . .l. J .

s,

. , , I s

- s .

y s -

1 's  ;

g s

s . .I .

i 1

1

's, s

. .l. J . 's

' s 1 O a g l s 4 >

._ i Tig. 55. S141ER-ll results for ar. idealized expansion ir. a mult inode ervirerrert. l l

193 i

4 i

1 i

i

~

RCUGH DRA7

, . i I

Overall. current. results suggest. that cr.e-d iner s iona l' s lug . e spa r ier ,

s a t e u ! .. t r J hv' %34N1:R 11 with t s pl:t liquid'varer roter. tut .sourl:rg til:  ;

is.+ggirait. the rapritudt . ! eser t ca lL i la:d inepas t pressures. Tw.-ditsr i-rc

+- l uj b r e.i k u p w i l l scar the arrai t es e r ' t ime. At sna l l g s t a l e ' $l4NI:k- l l - w i l l esaggerate su6 h~ sneariry trer lask ef turbLlent c:i x i r.p . At 'large scale with ',

.large r.eJes. SikN!:R-11 r.av underestic te the growth of large-scale i r.s t a bi l i t i e s i t

if'strorg liquid /vL por roter.t ut couplity as presert. The prope r conc lus ior. .to i

draw . if weak liquid / super toner. tut u uplirg is present'. or large droplet sizes

' a re _ u ed. is less clear. As ir. ethe r a spec t s of the s t eareexplos ier. problet.

' enther larger scale tests are required or ciod e l s / r.ute r i c a l techr.iques t. . t. be 4

developed to represent more of the dotirart physics if the ever.tual clasts of

• high cor.fider.cc are to'be ,ius t i f i ed.

4 i

e t

4 i

l I

i 4

t

(.-

t 6

b I

t t

. =

• I

! i r

194 j l

i

?

, L i~,- , --,,--n ';,.n.__~.<-,-_..,---,---,.._-,,,,,.,_.,-a,.....,,--,.,,,_,.~.w,.,....-,.n,-w..,,.- _ .,.nn., , ---

R:H ERFT VI. ANALYSIS OF STEAM EXPLOSIONS VITH S141ER-11 This chapter presents the details of the SIW!ER-11 calculatior.s for the reactor (ase. For each of the five cases summarized in Chapter 1. a discussion is giver.

to put these calculations in context. In the first two cases. several additional 5121ER-Il calculations were perforced. The results of these additional runs are discussed to show the significance of the single calculatier, quoted for each case in Chapter 1.

A. -Case 1-20 Percent Preeixed The first case examined was based on the 201 preeixed configuration used in the 1980 SIW1ER-Il ZIP s t udy' . Cases 1 and 2 were conceived in the ZIP study as appropriate for assessir.g whether steam explosions represented a serious threat to containeer.t integrity. Mixing ~20% of the coriue with water was ider.tified as a einical r e qu i r e ce r.t in the 1975 Reactor Safety Study for an interactior.

leading to contair. cent failure.

The initial configuration for the present case 1 is shown in Fig. ' 6. . The structural setup is that described in Chapter 111. The cylindrical t. ur.g zone is showt ecie clearly in Fig. 87 reproduced from the ZIP study and here represer.tieg axial nodes 13 to 66. This is a volume fraction plot with the radial direction to the right and the axial direction upward. A stall vapor buffer laye r was assueed to exist betweer. the core pool ard the caterial below because of the ir.itially large temperature differences. i.e., the water could not have been in contact with the pool of colten-core caterials. The ei x i r.g region was 1.55 e in diameter and the colten-core pool was about 2-e deep. The i r.i t i a l distributior. of liquid water is shown in Fig. 85. The lower head was assueed to be filled except for the vapor buffer region. The volume of wat er displaced by the core eaterial and the film boiling vapor layers in the eixing region was transferred to the downcome r. The core's caterial temperature was

' ' W. A. Carbiener. P. Cybulskis. R. S. Detr.ing, J. M. Genco N. E. Miller.

R. L. Ritzcan. R. Sicor.. and R. O. Wooton " Physical Processes i r. Reactor Meltdowr. Accider.ts'. Appendix Vill to Reactor Safety Study United States Atotic Erergy Coc=isstor. report WASH-1400. NUREG-75/014 (October 1975).

l' 195

s

r. ' - . , . . .

I I L' U d; a u , ,j ,j' f l

n -

60 OUTLET PIPING i -

u l 50 l 2

l l llll l l l ll - , , SOLID STRUCTURE i

~

-lNLET PIPING 40 -DOWNCOMER MOLTEN CORE WATER 30 PREMIXED CORE SUPPORT REGION 20 FORGING

" '- - ^ -

n 1 3 t. m" r e t t'i i .1' 3 I I I l P l= L P '= b " l4Jl I I I.T 1.1 P 'J- *l J W 'l l MOVABLE STRUCTURE li'Hil' _ l* ,f n i I l'l)9J8 (l kl l I 5.20 m g .L W 'r. 'L '[ U l i I*laLPF .I* " L1. 8 I I W LTA< % J44f I I ICLLP ']. .1 *l' I i l*L H 1_ l?.iII; 11 19 l'PL = L FJ.L I I U 1 1'l I,N " t ? *L'L I P OUTLET TO 1 5 10 15 KEYWAY

+- 1.613 m -* l

~ 2.5 3 7 m  :

1

l~ i g . 66. Corfiguratior eedel fer cai.e 1. j 196 l

~

1 l

1 Y l .

~

E i

s T l

I

- E c u C-E t

0 E

S

\\ )

\-

\ _

1

/_

/

a W

/ j f

\

~

\

> d

, f, E f l'~

E / -

I g i L f

.*-jg i,.

u y l_

d E '/

'5 c 9

?

5 5 I 1 E. V. J r s t t a l c o r e r:. t e r s _ ; g,,,7,yg,,(7, 197

I l

l l

l

'0 { D p l~ ~

'l]'0 I\V 0{1 I i!!YiI l

l E i E  !

E )

L i

$ l 3

l w

$ f g E

8 1 l l .

l

\

1 i

/

\

/ I I

I i

1 i l n

I t

]

I

.. : /

L. - E

/

t'

/'

s -

[s .

s. _

I E

t 2 l' i p . 65. Ir.itial water distributror.. .

198

l assumed to be 3 100 K, which is sicilar to that assuced in the 1975 Reac tor Safety Study. The water was assured to be saturated at 10' Pa. The c: xture region was assured to be coeposed of 505 fuel, 251 water, and 251 stene by volume and to be fragmented to 300 ue diaceter globules.

Three separate investigations were carried out starting with this initial configuration. First. ser.sitivity to the water-droplet size was examined, because of the difficulty in justifying the water flow re g ic:e as s ua:ed in 5141ER-11. Second, a calculatior. without lower-head failure was run to compare results from the new S141ER-Il codels with the ZIP calculational results.

Third, the cor. figuration was codified to determine how cuch dissipation eight be anticipated if effects omitted in the ZIP study are included.

1. Inve s t ira t ion of k'a t e r Surf a ce Area. I r. Chapter IV where the SNL e x pe r ite r.t s are evaluated, there appears to be a codel sensitivity to the surface area of water, (actually water-droplet size in the codel). The quenchir.g effect of heat transfer into the bulk water was important in the expericents for latiting the duration of the pressure pulse.

A The kinetic energy produced from previous reactor scale SIW1ER-Il calculatier.s (the ZIP study) possessed an insensitivity to beat transfer. I!owever, part of this result could be related to the heat-transier-codel/EOS combinatier.

employed.

The standard fuel-droplet size (300 ue) was used in two S!$1ER-Il cases that were run only through the early part of the interaction. One used the standard (ZIP study) water-droplet size of 300 ut. The other case used a water-droplet size reduced by an order of eagnitude. This reduction applied both within the ir.teractior. zone and in the adjacent water.

Although the initial vaporization rate within the interaction zone increased by a f actor of 2.7. the peak pressure in the two cases was within a factor of 1.1 after 2 es (125 MPa vs 115 MPa). By this time the water in the interaction zone was all vaporized in one case and nearly s ir. the other. Conse r:ue nt ly, in this reactor corium/ water configuration, the water-droplet size is not a sensitive paraceter.

199

1 j a u i l ww ,1;<d 1i

2. Compariser with the ZIP Studv Usinr the New Models. To provide a basis for 1 c ocpa r i s o r. . a c oepl e t e recalculation of the ZIP case was made with the new cedels and with or.ly cir.or adjusteents in input. Explicitly, the changes were as follows:

(a) Tne water EOS was updated and the new heat-transfer and "aperization/ condensation codifications were included.

(b) All droplet sizes were set to 100 ue diaceter based on the investigation with the shallow pool experiments (Chapter V).

(c) The freezing point of coriue was lowered by to K (to 3 090 K). Unless fuel vapor exists and unless fuel is core than 1 degree above the melting temperature. heat transfer froe fuel to vapor does not occur (in SIhNER-I I ).

This change guarantees such a heat-transfer path.

The liquid-volume-fraction plots for this SIhNER-il calculation are shown in Fig. b9. Each contour line represents a 109 chanEe in liquid-volute f ra c t i or..

Because the lower head was assumed not to fail. axial nodes 1 to 12 were clicinated frot the calculation. The characteristics of this expansion are similar to those the ZIP study with the water proceeding up the downcoter and cown the inlet pipe, while the corium eakes a two-phase impact concentrated toward the cer.ter of the head. The maior qualitative difference i r. the e x par.s ior. cha ra c t e ri s t ic s is the increase in the vaporization of water, in the vessel. The ZIP case had a water-rich coriue/wate r liquid eixture along the sides of the expar. dang coriuc. Its abser.ce here is mainly the consequence of the wa t e r 's sur f ace vapor izat ion. Also, the ZIP case assumed larger droplets, up to 1 c:e in radius, above the preeixed region. Pressures produced at various locations withir. the vessel for this calculation are shown in Figs. 90-94. The total force on the upper head is showr. in Fig. 95. (As in the ZIP study these curves have a resolution of 1 es af ter the first tillisecond.) , A,1 the i t.l e t pler.ue bottoe, the spurious single-phase peak pressure of the ZIP study is clicinated; howe ~r. the pressure remains above 100 MPa for about the saec tiee.

-15 es. The urper-head iepact pressures are higher toward the center of the vessel. The increased water vaporization and the smaller droplet size during the expar.sion c: ore than coepensate for the excessive steam pressures present in 200

l me iqs -. 3 -a- 1 I d jl

~

5 ,

V .

l, ,

E 4 o l v o  % o 1 g

E I

8 g EE "y go .

' ue

=-

^e O y -

o

_ au ,,

Y o $1 E O 5y

' sw ". . w >~ z g

>- E, 8 -

ll _

v e4 8 8 E C h 'u^

' b 4 -

e4  ;

%  : ze - -

r I SE no

• i ao  ?

8 8, Wo a~ w u ^ *  %

4@ y 6 p a -e ' /

' Oe y /,

w

• mu , <

di ' E Ew ,,

n hb h

'N (A/ j 5 g '

, /

L

, b ' t 8 ),/ e O

E e

mO s. 8 O .

e , _

8 55 8E e 8 8 8  ? V, ma y O *- b$

a ^e -

s c: ,

g .. Y E

-y E_

E z

$: i

>==

l'ig. 66 Liquid-velure-::actier picts Ier cerparisar with the Z11' studv.

201

5 O

l.O ' l O! [ p' --.

Nv

~ a 4:~i L:[hh 1

. O o

+

o 8

2 a .._' ] ( ~

u Vf "

o. -'

E

?

u m o 8 o, o v.a o w -

a=

" NV -

w r "

w Em 5 5  ?

h E 9 W D 8 o

S '

)] )

~

5 5 b D -

• :1 o

,r

. o Oo e

'I L 8  ?

9 >-

8 Yo ^

• 3 P j 72'3 _ E* 4

• v -

W-a . x Sw 2 W w o <

• dE

>- 7 5E'

-o g

-o h g h

u

- e a

o M

W - ..

Sw E dE K "

- a o -

e

?

$o 9, }7 ,,

m o

35

.s

- a - "

u

Ew -- .

e 8

~

T e

b y y -

r gY e 3E c-E"

-o 8 o. >-

2 u

?*

e.y t~ cr.

.$ E dE I

Tip. 59 ( c or.: i r.c e d ).

202

500 0 -

PRESSURE AT INLET PLENUM BOTTOM 4000 , o\5 m I' !

3 ,f ,

,I S 30004 E 1 w  ;

x -

=>-  :

@ L x 200.0 t

c. -

{ U 100.0 i .

5

i

' ' 'i ' ' '

00,. -

i

- - - 'i 002 0.03 0 04 0 05 0.00 0.01 TlME (s)

-Fig. 90. Pressure at the i r.l e t pier.ut bottet for cocparison with the ZIP study.

2000 -

3 PRESSURE IN DOWNCOMER m rs

'i 160 0 I

4 '

I

\

l l

,l ,

2  : .\ : \

g 120 0 - J'I j i

E i D 4 \

3 i b

e 800 -

i- .

' S h

N' -

400" 't 's -

l ' %Q , v,s-3

.N.

00 i i - ,- ... .,

0 00 0 01 0 02 0 03 0 04 0 03 TIME (s)

Fig. 91. Pressure at the dowr. core r top f or coc:pa rison with the ZIP study.

203

700 0 :

i PRESSURE AT TOP OF VESSEL

^ I il hi y Inbr" 3

600 0 : I

'u J Os i L i\ril i

500 0 d2

?

7 c_

i E 4000j r.o  :

a:  :

=> -

$ 300.0 j a: -

• - 5 3

200.0 3 i

100.0 2a 3

5 0.01 , . -

s ?s- s- n 0.00 0.01 0 02 0.03 0.04 0.05 TlME (s)

g. 92. Pressure at the top of vessel fer cor pariser with the ZIP study.

G000 :

i PRESSURE AT HEAD CURVATURE 5

50004 3

3

_ 4000 'i 2 j E <

E 300 0-!

,, a a b 4 E

200 0 i

~

100 0 i --

s 00:i- i- i' 'i' 'i' '

'i 000 0 01 0 02 0 03 0 04 0 03 TIME (s) l' i g . 9 3. Pressure on the heaJ at .V t o t he ve r t s c a l f or c or pa r i ser with the llP studv.

204

2000g PRESSURE AT \'ESSEL FLANGE 1600 1 2

i 1

~

7 c- 120 0--

2-- .

w -

cc s -

$ 8004 3

N 4004-1 0 l 1

/

00 3

. . ..i 002 J. i 0.03

.. .. i 0.0-1 i

0 05 0 00 0.01 TlME (s)

Fig. 94. Pressure on the head at 70 0 to the vertical for comparison with the ZIP study.

c,

- 33 0 :

i FORCE ON HEAD 30 0-i. -

i i l

250j j 1

E 20 0-3 i l

U 5 1 l 2 15 0 ~ j l 10 0

/ ,

)

- t _

3 50{

/

5 ' i i 00- 0 03 0 01 0 02 0 03 0 04 0 00 TIME (5)

Fig. 95. Force on the head for comparison with the ZIP study.

205

RC A E1AF case 2 of the ZIP study. A stror.ger loading bias toward the apex of the upper vessel head was produced ,in this calculatior. with the peak pressure in the cer.te r be ing 550 MPa (vs 375 MPa i r. the ZIP study) while the integrated peak force is 2.6 GN (vs -2.6 GN in the ZIP study).

The total fluid kinetic energy is plotted in Fig. 96. As expected from the hi Eher impact pressures, the peak value of 3 820 MJ is higher than the 3 400 MJ of the ZIP study. Of interest is the fact that only 88% of this peak kinetic 9~.

energy (3 350 M]) is associated with upward moving fluid, as shown in Fig.

(For the preser.t purpose, upward coving fluid is defined by that fluid existing in axial nodes 25 to 66 ar.d radial nodes 1 to 12 in Fig. 86). The recairir.g kinetic energy. shown in Fig. 98, is ma i r.l y associated with water in the.

downcomer and inlet pipe. If detailed ar.alysis of loads on the head structure is desired, the difference i r. timing between loads on the bolts produced by water in the downcoter ar.d loads delivered directly on the vessel head by coriue iepact requires consideratior..

3. Case 1 Update. The comparison with the ZIP study shows that the new results that are heat-transfer and new EOS modifications give overall comparable. Within the limitations of the SibbiER-Il formalise, reduction in head loadings can be achieved or.ly by consideration of lower-head failure and other input changes. The char.ges made for the updated case are as follows" Peak inlet plenue pressures (a) The lower-head failure model was i r.s e r t e d.

exceeded the static failure pressures by core than an order of magnitude i r. . t h e reference case 1. Therefore, lower-head failure should occur.

(b) The droplet sizes were changed to those of the ZIP study. Calculatior.s showed that these sizes overpredict the SNL experiment used for calibration in the ZIP study when the new heat-transfer formalise is used. Consequently, they should still be conservative. ,,

(c) Steel particles were inserted in axial nodes 46 through 60 to represent the structure above the reactor core. Although residual strength of such a structure c a r.r.o t be accounted for in Slh61ER-11, inclusion of 35 000 kg of steel will ir.troduce inelastic collisions and change the character of the expansior..

206

} fb MF dW ) {," 1

/

4 Ot+09 ,

3 CC+09 -

P w 2 Ot+09 -

X t.0C+09 -

I 0 ' ' ' '

C OC 02 04 06 TIME (S)

Fig. 96. Total fluid kinetic er.ergy for (OCPariset with the ZIP study.

l 207

l b'lO i

1

% =P mm NM ,J ,

e ot+o9 . . . . - . .

3 i 2 CE *C9- -

2 CE*C9 - ~

x , .

1 Ot +09 - "

^

e , ,

C c co c: os c6 TIME (5) rig. 97. Upwa rd fluid kine t ic e r.e rgy f or coc:pa ris or. wit h the ZIP s tudy.

20E,

RGHEWT 8 0(+C8 -

( -

6 CE.C2 - .

y .

4 Ct+Cf -

2 C(+08 C

' ' l ' ' I '

O CC C2 04 06 TIME (S)

Fig. 95. Kinetic energy of caterials not in the cere and above core regions for .otparison with the ZIP study. _

209

R:1 7 CIAFT (d) The solid corium heat capacity was changed to 618.5 J/(kg*K). the UOy value

,iust below the eciting temperature. because excessive quenching of solid coriue2 does net eccur with the water-lean pretixture in this problem. The UO2 celting temperature was somewhat arbitrarily set at 3 000 K.

Scee discussion of the results of this case is given in Chapter I. The averaged pressure differential over the lower head is shown in Fig. 99. The static failure pressure quoted in the ZIP study. 44 MPa (6 400 psi). was exceeded relatively early (0.7 es). Calculated vessel displacement is given in Fig. 100.

The failure value, 5 ir.. was exceeded at 3.5 es.

The liquid-volume fractions are given in Fig. 101 for different tices.

Following failure. the main cass moving downward was the failed part of the lower vessel head (11000 kg for the failure radius at 1. 6 c). The fluid that coved downward was a two-phase spray. Compared with the results of the reference case 1 (see Fig. 59). wa t e r wa s vaporized more slowly from the sides of the upward coving corium, less water was forced up the downcocer, and the steel particles made the impact of the upper-head more coherent radially and ir.

"o

- 25 0--

PRESSURE DIFFERENTI AL

_v 2005 d.

E d 15 0 --

[

3 ,

cc 4 /

E

~ 5 /

b 10 0 '~ j' x

c.  : /

3 /

b /

50 /

/ - -

00' i i i i i i i 00 05 10 15 20 25 30 35 TIME (ms)

Fig. 99. Pressure differential across the lower head in case 1 (update).

210

00 DISPLACEMENT OF llEAD .

R, )nJ , a, ,A -

50

.5 102 f--  :

z -

w  :

2 w 302:

U -

J -

c.  :

$ '20-~

Q  :

1.0 i 00~ .

i i i i i i i-0.0 0.5 1.0 1.5 2.0 2.5 3.0 35 TIME (ms)

Fig. 100. Lower-ticad displacement before failure in case 1 (update).

tiee. However, a lower tail of liquid developed from the existance of the relief path, ir. the downward direc' tion increasing iepact incoherence.

Pressures (forces) produced at various in-vessel locations to cocpare with Firs. 90-95 are showr i r. Figs. 102-107. (These plots have the resolution of eve ry 514t.ER-ll t ime step.) The lower pier.ue pressure is drastically reduced f ollowir.g lowe r-head failure. The downcocer icpact pressure was only ~50 MPa.

because of the reduced driving pressures. The magnitude of the peak upper-head loading was n: ore biased toward the center, although the relative loading at 70 to the vertical was increased at the time the peak center loads were obtained.

The kinetic cr.ergy produced is divided into two parts, the lower-head's kinetic energy and the fluid's kinetic energy. The lower-head's kinetic 'enIrgy reached a taxieue of 640 MJ at 24 es. At this poir.t the last radial node of the head reached the bottom of the calculative mesh, and the =^ving head no l er.g e r appeared i r. the calculation. The fluid's total kinetic energy is shown in Fig. 105. Hevever, because kinetic energies appear in various parts of the problet and then are dissipated. this total is not a good indication of the 211

- r.

leg' >>

Rlp eJ J ^d, .)f\ C ,

g;,.;wg e t-t::s or ,;;g::

y*E F R A ' joy CW LIC !: T:vE 5 C :w:

'- 0-- -

I I

i l  !

l

=, i O

) \ b \, l l

M!N: 7 7BE-05 MA>: 1 00E.c; Cl: 1 C;E-01 1

w;s: 7 7EE-C5 MA>: 1 C;E-;; CI: . ::E- .

-?or~ er

-- s ~r '~*~..n '~

va

.~..vE r: 1--  ; _ . . ..

-.ng .. ,

i l ii .

!l , .

( l I

t .

' I l

\\  !  ;

s i

'l l xf.e!

t ,

! M l l l 4

4 i i I

l

' l$ /

l

.- l w:N: 7 7EE-;5 wa== 1 ;;E-:: ::=  ::E-:

\'

":': 7 7Er- ---

, va,-- .

...r.$. ..

i l l' i g . 101. L a ut. id-vc l ure-f ra c t r or plot s f o r us e 1 ( updat e ).

l 212 l

l l

1

, - - - - - - - - - - - - , - - - - - - , , , , - - - - - , , , - . - - , - - - , - . - . - - - , - .-.-g

l

\

I"

• 1 L hnt ,"

1 i

.:-.~E r= : ::s or u ov:o-s,-: - ,e c. ...

.._or .::.::.

.u: - z -c---.- o

• r

w. I I, - t

' :I i i.

i i l - l w

,' 1 m pn

'1 y I ' %.: ..

A _,

I a.-

e-I $t l

I q J l Ill

/

g -j ll ,

hg l ,

! i it l

' ' f

(' ,  !

! r.- . -

s ... .

.. .. .. .- .4 r . r. . . M,N: ., .S.

. t .: .. wax: . .. . . . . ... wa. . . ...

y ; ., . ; gg.;s wA :

i l

.......- . . . w r. r : A .a

. .rr- .. . a . -.. ..- ..

s. .v.;

.e

y.  ;.

-'v7-

- 1. :. (*.*w..

3..- .. r. *. *. w .. *

' g t ,

( I ,

Ii

, f. I.

i.

I' lI i i t - " .yr _. .?, 1 g' g,eg ',y ,'l n

A, ,

.n 5, '

.p t 9,.

.:.h t.-v,

^c y

I e

I,

, l - -

l

Q}.) l

_'I','.  ; E.-

~

w : t.: 7 7EI-;5 wAm: , c g.c :::

- sE.';s van:  :,E-;. ::: 1 **E v:. :.

l' t E . 101 ((crt3 rued).

213

I i

l Ol I

,I '

9 3 = a m.

i - -

f Jw .J ,

l

,---..-*  :. ..ON C5 LtQatt

• -- ,e- g.

. - ": - CCCP5 .. e oc .....'e.c

.*

• w m..

. i

'+ ' '

i  ;

l, ,

l'. I

[' f %. . 4 -)

.) *

l , . r. i. .*. '

l

.6

- - it g ,g .~- #, . . a ,

t 1

-. ...a,.

,. . i ,

. i

- , b' l

' 5::b ' . ',h k; l

, . . , '

• v' l l .

...Jl! l

• • .. )' (t- e . *,

i I h -

M J

I

. i. .

1 i

. I

-:s= , s -:e ~t.: l l ..t. : ::: , :: - _.. .. . ... . . , , ...

I 1 1

l

s. w. ..

t ..e-: r -A .- ..at. v- **

. v .-

.  :-.. -..v.

.w-.

.  :: ?e. u-.

1

~ ~

(. ~ ,, \

n

{i. (c; -

. i . 's . .e t 6 4 IM *.* 1 t \.,

1 .e n a.

t, y,/

. . , i.

Jf' s ;i

! \h,. ,l!

i

_- 4'l l = .n

( . ,

, v i% .* '

t I

i

' j i . I i ,

, U' I rc) -

w : e.: - sE.:i was: .:E-:: :: 1 0*E *' w;.. : -' 7sE. :5 wa . : . ::E-;; :: : ::E-:

t I

l l

Fig. 101 ( c or.t i r.ued ).

l I 214 i

i l

i l

l 1

l 4

l l

1 et 'qt p..

33 *g-i \ '% . . *JSs L,,e l \ g  ;

..vt

.v.:  : : ::. :r .::..-

0..;-E

w

.

rat: ::s er a :.::

c. :..v. ....,a

, _ . 1 l

. -.4

., v-s

) n.8 ,. '.

i, , e ,

L' y ,,

I ..

"^....

, W ':r.l.t -

< l

. ll.: s i

\\  ;

,[

k: l

\;. i >dt 5 l y 1

. .s I

s f

.r,

(/0 j r

• .: - EE-:s - .:  : 5.:: ::: :g.:; ":' : 7 7sE- 5 * =

t.:: :::  ::t.:

. , - v:

. . . vt : : ! :,,

vt

- ... ,

wm

- I I _ .

i

'l5%-e)_

h a I

gn, f

[' < (! l l

.} s. f ;w;l ie " ,'

f g

i 1 1

g 'a l' l 1

> k.

e -

l f

I I

l I

e ',

3.4....-

... .. .. .... ]

• : .= ?

sE-:s -ta .  :.. ..

1 i

L , \

215

p.

I

!Of I  %

' 'N i

l jUJ F"F J jj] y l

.n ,; c : , . . . .. . ... ,.

liy ~ E:'053*~7

~~~~

fA ;9. vi"E ~*

. E:* 'i...v4 E, O' 'I O-' I l l

N3 i

-} R:

4,'vy .a.

j

'I

. .; # .- e ,s

. . I,  ; '

, )[T w/,-Q,1,y .

q iw .

\

\ , 'r h

V 1

\

u ,

. , \ l 1 4 . .

9 i

0 I

t s .

. ... .. . - .gr.  : ... .. ...

)

. . s. , ,i,r.,..

s:. +E r::: ::. :r a::.::

.. .. .. . v:.4 ~E r54:'.:g..*0..N. tr LI; I' a . .. .. . 3.. . . .

W%

f

,\c.n i i

J.ih\ !

lV e 4 8 t ,

9 T ..

l l i i t) (3 'l

'  : E'-

v: .: - EE.:5 ut.: . :::.- -. - t.:: *:N: 'EE-:5 wA*: .  :: -:. ~::

FI6 Ici CCnti ntu oL' M6

pf,I 2500,l R

PRESSURE IN PLENUM

{v < Ji n v:,ni ;

w n ,q ; - .

200 0 5  ; }.

3 -

a A

d4

$ dl 2

~

ta:

' 150 0 3 *l U

D d bm 1000 k,l

c. -

-4 l

5 50.0h

- b \'

00' .

000 001 0.02 0.03 0.04' O.05 0.06 0.07 0.08 0.09 0.10 TlME (s) rig. 102. Pressure at the n r. l e t pler.ue bottee for case 1 (update ).

60 0 :

3 END OF DOWNCOMER 3 ~

, c ,<. .' ?

500j '

2 /

s ^

I' O 400$ '

}/ } ~' l . l '

N $

l .

j _

IE 300 j '

3

~/, ?

h L e

c. l f,' :. ',

200 }l g 3

, _ - Nee g-10 0 r.

3

'\ -

a /tg.

i- .

00* i-

.- ,- i-000 001 0 02 00'l 001 0 05 0 06 0 07 008 0 09 0 10 TlMC (s)

Tag. 103. 1* r e s s u r e at the tep of dewrtorer for c.ise 1 (updatei.

217

~350 0 -

i TOP OF THE VESSEL

-f n'tm* r fi h {l 1 V Jdi t' L/ i 200 0 1 1

2 150 0 -

s L:

m  :

o -

tr. -

@ 100 0 d e i, c.

3 a

1 50 0 -j i

00 l. , .

0 10 000 001 002 0.03 0.04 005 0.06 0 07 0.08 0.09 TIME (s)

F i g . 1 t t 1. Pressure at the top of the vessel for case 1 (update).

1750 2 PRESSURE AT THE HEAD CURVATURE l

15004 125 0 2c. 3 3 100 0 5 te 3 E

a 3

{

ct:

750 500 $

a .._

250j i

00 i- i-i-

i- i- i- .i. .

000 001 0 02 0 03 0 04 0 05 0 06 0 07 0 08 0 09 0 10 TIME (s) f E. 1"f. Pressure er the head at 3# to the vertical for case 1 tupJate*.

I 218 l

l

7002 3

PRESSURE AT VESSEL FLANGE

!. ! h h f s00}

sa

'JUIIUnfr[

50 0-3

- E h

E 40.02!

t.a 3 -

= 3 s

{

2-30.0 ]a a

C. A 20.0 -!

a 1005

~

0.0 1 . . . . . i i'i'i'i ''''''''

O 00 0.01 0.02 '0' .03 0.04 0.05 0.06 0.07 0 08 0.09 0 10 TIME (s) l' i g . 106. Pressure er. the head at 70" to the vertical for case 1 (update).

O 12 11 0 _

0) FORCE ON HEAD 1002, 90 802 E

- 70h

$ Soj 2 50-j b

40: s 30i \

N 20 '

30 00 i- i- i- I' i i i- i ii 000 001 0 02 0 03 0 04 0 05 0 06 0 07 0 08 0 00 0 10 TIME (s)

} , 3. , p .7, lerce er the head ler case 1 (upJatei.

219

__, . . _ - . . . ~ . _ . - - .-. .. --- _. .__. ._ ._ __ __

.r

.'_y-I t.9t+09 i i

. . . g . - i g i i = '

l ,'

8 Ot+Ce -

~

~

~

6 Ot+08 .

~

e of.ce ~ ""

~

~

2 Ot+08 -i -

.=

l l f , i > f 0 00 0- 04 06 Of ' 10 TIME (5) i v

rig. 105. Total fluid (5141ER-II liquid and vapor velocity fields) kinetic energy for case 1 (update).

220 1 l

4

r-m dNNr, F

TABLE XV FLUID KINETIC ENERGY PARTITION Recion Peak Label i = 1 to 12 647 MJ Upward j = 25 to 66 i = 1 to 15 373 MJ Downward j = 1 to 16 and i = 1 to 12 j = 16 to 24 Downcoter ar.d Ir.let Pipe i = 13 t o 15 104 MJ j = 18 to 47 Outlet Pipe i = 13 to 15 5 MJ j = 49 to 66 total techanical energy produced. To obtain a reasonable total. the kiretic energy was divided into four parts shove in Table XV. Plots for these four regions are in Figs. 109-112. Succing these maximums gives.1 130 MJ for the fluid's kir. etic energy or 1 770 MJ for the total kinetic er.ergy. Consequently.

for case 1 (update) approximately 57% of the total kinetic crergy was directed dcwr. ward and 37i of the total kinetic energy was contained in the upward moving two-phase " slug."

Finally. actual coherenev of icpact car be judged by the peak force 1uared divided by the peak in the upwardly-directed kir.et ic energy. F'/KE p. The force represents the time rate of charge in impulse, and the kinetic energy is related 2

to the square of the impulse. Consequently. F /KEpis a measure of the maxicut fractional char.ge in impulse over the~ period of fluid ittpa c t . The reference case 1 for comparing with the ZIP study gives 2.3 GN/c. With the modifications, case 1 (update) gives 1.6 GN/m. The fluid tail shovr. ir. Fig. 101 does result it a less coherent impact. ,_

221

, . . . . . - , . . . . . -~. . . . - . - . . - . - - . - . . . - . - .. - , . .- . . . . - - . ~ . . . - . . - - . . . . . . . . .

. ~ -

~

f. ' , L

. i 1% j n..,

, y y1 .

11 gy i_ -

.c . 4 .

A tr 5

0 Ot+00 , , ,

g

. i i g i i i l

' =

~

~

.r tl.'

=,. s

~

/ - '6 CE+08 -

~

~

~

~

._ y 4 CE+00 -

2 DE+09 -

./: .

, , ,  ! , , i < l . . 1 i . .

0 00 C2 04 06 os . to '

TIME (5)

Fig. 109 Upward fluid kir. etic energy for case 1 (update).

222 ,

. - - - . .m .

_ - . -____m..

9: -

,  ; fO i 5 m pam Nd a1 ihj s

3 3 I E E V T U 5 I E y u

3 Ct+00 -

I -

f i

h y 2 Ot+08 - -

1 . .

s f

l. t Ot+00 - -

i 4

~

1 4 -

g f i i  ! t i i f , i i ,

0 00 C2 04- 06 Ce 10 TIME (S)

.I.

1 i

Fig. 110. Dovrward fluid kinetic energy for case 1 (update ).

223 l i  !

4-d-

--, -- ,-n . , , , .-- . - , - . _ , , . . . . , , , . _ , , . _ , _ . , _ . , . , _ _ _ _ _ . , . -_,,,_,,,, ,_,. _ _ _ _ , _ _ , , _ , , _ , _ _ _ _ , , _ _ _ , _ _ _ _ _ _ _ __ _ _ _ _ __ _ _

. . - , _ _ _. . _ -- . ..- . . = . . . . . , - - . , ,,

r

D, I I O ' ] >M T- =

WWY i ( gj 3 u .:

+

4 i

8 M *08 , , ,

y

~- '- . . l t

~

1.; '

v

< 1 0(*00 - -

. i

~

4 W

g. .

a, S Ot+C7 - -  !

l 4

.1

- a I 1 p B 1 i 1 f

0 00 02 04 06 08 10 TIME (5) t- ,

n 1

J

f .

b,'

Fig. 111. Devr:cor e r ar.d ir.le t pipe kinetic energy for case 1 (update).

224 i

.y..... ~ . _ , _ _ . . , , . - _ _ . . . _ , . . - , , , - - _ _ . - - - , _ . . . . , , , , . _ , _ . . _ ~ . - _ , . . _ . _ . ,...-_-._....,.,,-m . , . . . . , , , , . ,

e

+

a

_- f j

Y', , z 3 j -

t i

e oc.of , ,

i.

i .. . .

f

. _ - ts p- s f .

a f1' s Ct.0f -

t- _

.i .

w ,

A l

2 ot.os -

1 -

4 f

-w 1

-{

1-

, a , i , t . t . .

o oc c os ce ce i; TIME (5) 4 k

T 4_

4 1

I 4

4 Fig. 112. Outlet pipe kir. etic energy for case 1 (update.l.

a 1

e 225 I

~ cr y y- yw--%--+ v- gw,wrm,,y..,,-yy,w. ,.w,-~-n-,,--.,,,,w-<,,m., , ,,,-e%,-~+,mwww--,----,-%-_

F' n! l  % 3

• • j,) {

B. CASF 2 - Conservative Premixinc. Expsosior, and Expansion Ne t ra ns pa r e r.t argutent exists to quar.tify the degree of conservatise to associate with the postulated conditions in case 1. Case 2 is an attecpt to produce a plausibly conservative case, starting with an unmixed configuration se that the effects of fluidization may be consistently included.

The initial configuration used for this case is shown in Fit . 113, although axial nodes 1 to 13 used for modeling lower-head motion and venting were eliminated from the pretixing calculatior.. The corium temperature was 3100 K, the coriut's heat ca pa c i t y wa s 0. 54 J/(ge K), the corium's heat of fusion was 276 J/gt, and the coriut's melting temperature'was assumed to be 2 700 K. This eay not be the upper limit for the corium temperature, which might be as high as the saturation temperature of steel at some elevated pressuret however, this temperature is -higher than those calculated as "best-esticates" for a molten pool in MELPROG. ' The amount of coriut was assumed to be 131 760 kg, in other words, the entire core. Although crusts must exist so that radiation losses do not lower the temperature. and consequently the entire core cannot be molten at er.ee, a r. unknowr. amount of steel could have celted in from the upper core structure. Alsc, it is probably the inertia of the movable caterial, not the actual accunt of ec l t e r. corium, that is impertant. The 18 000 kg of wa t e r assumed to be preser.t in the lower plenue was assumed to be saturated at a r.

assuced vessel pressure of 1 ate. Elevated pressures could well be present, which would a: low less steam production and core cixing during a pour of corium i r.t o t h e i c+ e r p l e nut, but with I att, questions regarding the lack of a trigger are reduced. A partial blockage was assumed leading to a pour diaceter of 1.65

m. This is not as large as allowed oy SNL ir. recent study' but considerably larger than judged to be preser.t by most experts in a recent review of steat explosions' .

Untodified SIkNER 11 heat-transfer assumptions and codels wereiTsed with a droplet size of 20 cr in diameter for the corsut and the water, These heat-trar.sfer assutstions were shown to underestimate steam production i r. a Informatier. provided by V. J. Catp. Sar.dia Natier.al Laboratories. 1954.

226

l ~^;'

, r- r. n n.

J si L.ji/,~[

m 60

}G -OUTLET PIPING SOLID STRUCTURE p if lNLET PIPING ..

12.54 17 I

F DOWNCOMER i

40 $

,x

\

CORIUM "

h BLOCKAGE  !

30 E r;

VAPOR WATER =

E CORE SUPPORT 20 _

FORGING ll 13 i

=xam m

l 5.20 ,-

MOVABLE E23 ____ STRUCTURE l ____

l .

1 2 _!-* OUTLET TO KEYWAY l 10 15 l

+ -1.613-*l l 2.537  :

l 1

l ip. 113. Iri'iol certipuratier for the S1411:R- 1 ] prer. i x i rg u l c u ls t ior n r. o s e 2.

227

RF'jn Y i S1hb1ER-ll calculation of a SNL test that was run with 15-ne diaceter droplets see ( Apper. dix Q). Leeressing heat transfer further should assure a low steam produc t ier. ra t e here. also thereby taxicizing the extert of mixing. The cesh size _in the Fig. 113 eixing region tends to be greater than 10 ce. This may be considered a . SihB!ER-Il mixing length scale for this problee. Theofanous ' has argued that 10 ce is an upper limit over which precixing can occur.

Contour plots of fuel and coolant (water plus steam) der.sities over the first second of fuel (corium)/ water contact with these heat-transfer assumptions are shown in Figs. 114-119. The coriue is seer. to push the water away and up the downcomer. In contrast to the e xpe riee r.t - s r. which all the telt eized with' water, in this calculation cixing only occurred around the edges of the downward pouring corium. Not until 1 s did enough vapor pressure develop to cause cour.terflow through the corium. In addition to the licited heat transfer and the downcomer escape path, this time delay is a cor. sequence of the inertial constraint posed by the coriue pool. This inertial effect is i ripor t a r t . It is not considered in coccon quasi-steady-state fluidization argument s. ' The first prelicir.ary 51kt1ER-Il explosion and postexplosion calculatior. was started af ter 1 s of mixing.

1. Explosier After 1 Secord of Mixine (Scorirr Calculat ior).

At 1 s ar. explosion was assumed to occur. The Slh61ER-II calculation was rezoned so that the new SlhS1ER-Il heat-transfer and va por i za t ion /c or.d e nsa t i or. codels could be used. The droplet sizes of 300 ue diaeeter for coriue and 75 ue diameter for water, obtained f ree the correlation to SNL experiments, were used everywhere. The steel blockage restricting core motion to the 1.85-e diameter value was changed to adiabatic steel particulate at 300 ye d i ac:e t e r. No

J. E. Kelly.. "HELPROG: An Integrated Model for ira essel Melt Progression Ar.alysis." Proceedings of the 13 Water Reactor Saf ety Research Inforestion Meeting or. October 22-25. 1985 Sandia National Laboratories reprot 1AND55-2026.

NRC Proceedir.gs te be published.

'T. G. Theofar.ous and M. Saito. "LVR and ilTGR Coolant Dynamics: The Contair.eent of Severe Accidents." Purdue University report. NUREG/CR-2316, 1961.

226

q) tp[; p n ,3 . .

. UUJi j j .' c.:3 }-

1 0 0 0 M ',

! ! F/E ST E Av E,rtcs iord [ VEtJ T SE cut f act rUEt er t;c i 3 cooterg orrciTy l

M A A t/ i t; i v av 1 0 0 E O 5 000-0:

~~

7 41E 4 C 3 1 07E+C3 l

r,Ot4 T OUr- itai[ L . AL 7 41[402 1 O'[+C2 l G[ G e QtJ-G AD I AL ( 1,15) A>lAL i 1.5:'

l'i g . 11-3. I r i t i a l c ord i t i er.s . Slatil:Is'- l l c or s e rva t i ve preri x i r.g c a l c u la t i cr

( 6 .: s e 2 ) .

229

r' Rb J{UJ'dIknf,s, i r' g r- - '

TIME 300.000 MS 51EAv EXPLOSION EVENT SEOUENCE rUEL DENeiTy COOL ANT DENSeTy n =-

vit.iv.v 1 00E+00 6.00E-02 7 4YE+03 1.07E+03 r' yt;T 0vo INTCPvAL 7.41E+02 1.07E+02 5-[ r,10tJ-P AL I A L ( 1,15) ANIAL ( 1.64 )

Fig. 115. Vater-fuel centact. SikNER-Il conserva t ive preeixing calcula t ion (case 2 ).

230

n !O '( 9m "

1

- ., F. "

vw U l J\ l I A!/ 50 . .M.:

STEAV EsPt05 TON EVENT SEQUENCE re st DENS is COOL ANT DENStTY I

l s

M s

m)c c

,s,,.,,,,,

! V 1 00E+00 5.00E-02

,st.1 s ,s .;e!

7. 41E + 0 3 1.07E+03 r eje cu . r i t. r p y t.t i 4IE+02 1.07E+02 cr','0'4-PADit.L f 1,15) A*lAL ( 1.54) flg. 116, Fuel contact with the suppert forging.

SikNER-!! corservative pree.ixing calculation (case 2).

231

l l

l mn'fn'r sn .

Ed ,

i l

l I

pf.4 00 _ /q ,, h . l' ' '

. t ..t.' ta*LuslVN EvlN1 SEQUENCE r etr prNciis COOLANT CiEN0'T' I

i i

I l

i r . .

6 1/

x hw.c ))[ ==

Vit4ivsv 1.00E400 5 00E-0 VA ' V'JV 1.07E+03-7.41E+03 - . -

'ONi0ut i t4T E 7

' RV4t[+0p AL 1 07E+02 stciot; NAE AL ( ).15) AniAL ( 1.54)

/ ,

~,,.'

l' t y. 117. Iritial c e r.t a c t of fuel with the lower head.

SI Atli k - l l t er.s e r v a t i s e pretix ary taltulat ter ( 6ase 2 ).

232

'n sn Lmuc BA-T 1 l

I 1

TIME 900.000 M1

]

S T E A'/ E>PL C S i O'4 E VEtai SEQUEt4CE rq. rr '* COOL At4T DE ta? ' T f f s

I

\

' l

! l I

I I

fI . . _

if i.

• 1 m q ,

S s .% 3 e

\ -

/

i T _- -_

$.00E-02 1 00E+00 ef . . i v ;t/

7 41E403 1 07t+0.t - -

. j , - ,, un i t.i t s v t.t 1 07C+02 7 41E402 st- j . _ c A t. . A , ( 1.15; A> AL ( 1. $ .1 )

l~ i g . 115. Begirriry cf fuel peel breakup.

S ly.11'.K - i l c o r s e r v a t i v e prerixing ta hulat sen (case 2 ).

233

l i

"'IG { " r p F""

i NV w d ) [\nl J l l

l l

l I'UL 1000 000 r,' ;

I

e. . ' ' t . t ,- . . t , r r, -

..t c 1.; g L_ a Illi ' ' ' , .16 /' l ! ', , . ;, _ .

I i

l

, I i I

i

. .l Ii I

=E jF .,

Lt'ni14:

.. .'?

i!

l ' It -

.-h-<g$b al - /P

.* 9s f,. .. < > l;

-\w.-z/

. _ _ _ m. . .

.i

'r

- 4r, r

.I{* k 5 4,

• 4 1; ,

. L t. ; .Ai ' -  ;< '

A. f, , i,..,

rig. 116. Begir.r.ir.g of fuel peel breakup, SisNER-Il cor.servative prec.ixir.g calculatier. (case 2).

234

pr jl q Idu s

r. 4 U l ..)hhl~

lower-head fa: lure was allowed for this scoping calsulation. and this part of the mesh was ignored.

Contour plots for the expansion fuel density and coolant density are given i r.

Figs. 120-126. The expansion appears to be similar to the shallow-pool calculatior simulating a hypothetical plate obstruction (see Chapter V). Corium that poured into the lower plenue but did not cix with the water had only lieited motion. Fluid was primarily accelerated up the center as well as up the sides of the vessel and then coalesced at the top. Pressures and forces for comparison with those of other cases are given in Figs. 127-132. (With the e x c e pt i or. of the first tillisecond, these plot have 1-es resolution.) The pressures at the inlet plenum again would be sufficient to result in lower-head failure. The peak iepact pressures are reduced compared with case 1. However.

although the sepact is diffuse, the pressures at various spatial locations on the upper head are much core unifore. leading to a peak force of 0.94 GN.

T WE 1020.000 MS Si[av tir.01 TON CSEN1 SEOutN;E rtet tr : +- .

col aN- *tN,5 e

(Qyp' se >!

[ e--- / f,'

'.: [h l L.i.. $ '

I l, ' t .]

't I s-jl

) f, >M il l li 9 '

f Es'

,I (

' 1

^ ^ ^

v t.svav 1 00C+00 S 0;[-C:

varivJV 7 41[*03 1 D i E

• 0.'

r.0t.t ove s tat t a va t 7 41L*02 1 C'E.0:

Et LiGN pa: iat ( l ,18, ) anist ( l,S4)

Fig. 120. Initial expar.sier. instability, fluidized explosion conditions.

235

l i : */[ 10 ' ? Ok f/ '.

S'tav (it.* 0'. ( = f '. t[ 0.-t '.: t rts, fr.. .

(* .a.  !'-

• I l

I i

! 8

g. r ~ %,, ,

.f lll,  ! !  !

l' q-g

N ,

.{ *'

-. , .  %^^ 'n ,

I,( . .

w 4 i

I v . *. v e u

' COI.*. t C:( ~.

4,[.;y 1 c -[ . 7 t , .t ogs i t.' t s . a .

7at.. t C *[ . :.

s[, t... s a; a, e ti an.a. i*a fig. 121. Irstability develorrer.t. fluidized explosior. cor.ditier.s.

, I i P/ t 1s t \(i Ofi9 t/ .

%1[av ier6. ON ( v [ t< ' !.( tA t e.a t 2 .t3 qr e >s. trat av ' t't %* ' ' =

s.

l f

f' It j} :' ).,,p\. !c. ,

ft l In 1

, . a ,

'e- l(L

'l lt;  ;

i -

N '

nn{

'~

I  ;

$ ~-

s,\.ltj0.h)D.

sg s.2. < .-

Np y.s'f .j i I

v en ,v v ~~~ " '

t ( >,,t . i r e i.,y c; vae,o u

• a is .c ? t t g . g.

r, r o. s ry ,s n o. ' r sna_

s t , . r,,. . s a : .a.' a',tof, i..,..,,iy, .

t 0*[.c; lag. 122. Verttry begieritt urJer fluidized expln ier corditnors.

236

i 1

D P.t .f) t i t. a .

t

&l &a p. r,bb l' 5

ni ( J 1'v[ 1040 000 v'

• t t au ( r t 05 t ou i vf N1 StoutN!!

_s.. g r1%sni, coot a%' fir N' e *

• j 'l ',il ed

" . 'd p l L< @"(;

.L E, =.\ lV . 1

' i l' I!{ll l

' ,) p i l!

' L ,1 4

l ., i 6

(

Lf 0i i

f i 0 le e i i~ q' u

i 3

j s , s.gr ". p

(

e' v '4 v v N.

w. ' - __,

l i et t . r.< b 00t.C; w a . . v .,v

' e '[

• 0 3 1 0 7( + 0.%

....**,.f6 . tt6.a( i 03g.e; a if . r,7 s r *, i e,,e, . s a; i s. i e is , a s i s, ( i cta i l ay. 12's. Co r.! : p u r a t i e r .iu s t before irpact. fluidized explosicr. corditsor.s.

TiUE 1050 000 US Sitaw EnF.C$iO% tvth' StostN*t te;r et%t.t. coota%' tt%t.

.I ,f 3

! n

't 4

~.

• i ,l'

t. . i l

i p- I dj , ,

\

w *. v.w

'% s 'J\i 1 t 00t.*' S tr[.c; use w w 1 t, * ( . 0 2

• s't.C2 e a.

• La 6%'t#.a.

a'[.C; i c -( .0 s g *, i 3 . s a ; ,'s . t * *t an a. t

• Se l ig. 121. M.2 ' e r i a t rr.u t begirrirg. f l u id i z e d e x p l o s t er. t e r.d t t i er s .

237

yntlR** ~. 7 r m

~

i 88 s . , t 11ME 1060.000 US Sitau tsPLO5 ION t vt NT StovtNet t et t 5th'at* C00 tan' OfNi

~'

y e

. _~R Q r .'.

)-

d L..

is '

I V

(Id4*L s/

O) .

4 4 i p j u n N e u.,v i CCl*0C S OC I - 0.'

+ a it .c,3 p .t .c i cr.e. 00s e%'r6.a. ,

ait.0; i e #.r; si t i O%. s a r. i a, e i tie a.ias ( e Sa) fig. 125. M terial ir pa c t c er p l e t e a r.d r e bo u nd be g a r.r. i r.g .

f l u id i zed e x p les s er. t or.d a t i er.s.

1 i vi i i nii opo y t' tau rie .~. tite.' i t o. v. t

, e .. - ..

r no . a . . i ,, .. g

_rp?$ > ._, .

f. p. -

r,;\ ',$ I ' _ ' l .

i t i /. I  ! :I

.I6 l ., .l i . -

' .g ;

- ), '

y v 1

t l 4

.. l

.t l ~t . 7 q ,,0 c. , * *

\'

l M. -

.E' s

}

,,.,,,,.w - . - - -

t 8 va.,o .

e o . r. . , . , . ..

. . . f ,s

. . . . i. 6 a ..,

6 . <*

• a : a, . .i ..a , i .a i it. 1.' t . I t r.a t cor.figur tser. fluidized explosier terd:tior.s. 238

c.

RCU3H CRAF7 l PRESSl'RE AT INLET PLENL'M . BOTTOM 200 07, ,

e t

i \

$ 150 0 i j u

\.

100 0 i k

=

'\i /

,n 50 0 i

\ ./ N. , '

j 00'

,00 ibt ib2 ib3 ibi ib5 ib6 ib7 ibe i5 ii0 TIME (s) fig. 127. Inlet plenue pressure, fluidized explosien conditions.

iso l - PRESSURE IN DOWNCOMER 150 0 ".

i j I e

q 1200h fi

)

J 4

li-

$ 90 0 5 a

y 4, .l , ,

E 60 0 , 4 i r.

30 0 j

e /h* *. [ !k -

. i' wn.

00 .,. ... ,- ,

,99 ,bi tb2 ib3 ibi b5 lb6 107 los I M 110 TIME (s)

Fig. 12b. Pressure at the top of downcomer, fluidized explosion conditions.

239

p 10l'3H DEFT l PRFS81'RI: AT TOP OF VESSEL

~

120 0 , - i,

- \

d \

~ IOO O - i 7 \ ,

N 80 0 - .

u \

e ,

, N g 600 ,

E , l 400) /

f a 20 0 ;

00 -

i- T- i- i- i- i- i-'

100 1 01 102 101 101 105 100 107 108 109 110 TIME (a)

Fig. 129. Pressure at the top of vessel, fluidized explosion conditions.

120 0 :

} PRI:88URE AT llEAD CURVATURE 100 0 $.

I (

3 , .\'

? 80 0 !

./

t  :

E  !  !

w I p 60 0 i ,e g

3 -s E

r 400ig x l / \

200 2 s

i N~

/

~

00' i- ... 7,I .,. .,. .,. , .,. . f- l 100 1 01 102 101 104 105 106 107 106 lb9 110 )

TIME (m) I Fig. 130. Head pressure at 30 to the vertical. fluidized explosion conditier.s.

243

R:@' DRAE 100 0 ;

PRESSitRI: AT VESSEL, I'LANGE 0005, --

80 0 ] / '\

\

s 70 0 / \

c s s 600;: I' g

. l' s.

c.: 1 x %D::, x

- .I s

b 400 5 f e J

c. N 30 0 ~ N -

3

[

2005 /

/;

/

100 00 b ,- ,-

J',- ,- ,- ,- ,- , .,. ,

100 1 01 102 103 104 105 106 107 1 08 109 110 TIME (s)

Fig. 131. Ilead pressure at 70 to the vertical. fluidized exP osion l e nditions.

cn 11 0 i l'OR('l: ON lil:AD 10 0 i 90-j 80i; 70i E

- E 60 F 1 E 504 C

40i 30j .

203

i 104a -~

(O . , #.,:,..,.,....,.-,r-..,-r-,,

100 1 01 102 101 101 10') 106 107 1 08 IM l lC Tl MI: (*)

Fig. 132. Force on the head, fluidized explosion conditions.

241 I

'1:L3H DWT 2 Ot*09 , ,

g g

g

,,g , ,

~

I St*09 -

g 1 0(

• 0 9 - -

1 .

5 Ot+08 - -

U '  ! I ' '  ! ' ' ' I ' ' '

C t CO 1 0; i Os t 06 i C8 1 to iIME (5) rig. 133. Total fluid kinetic energy. fluidized explosion conditions.

242

t N 'IO -

WW d . i, t 5[*09 I

l '

'6

• C[ + 0 9- -

M L.)

a -

~

5 O!+0e - -

,1 %g

!  ? I i  ! ,

g > 1 t 00 1 C .' 1 Ca 06 1 CS ' 'O IIVE (5) 1 l

l l

l

. _ \

l rip. 131. Upward fluid kir. etic er.ergy, fluidized explosior, conditiens.

243

R: M [RA:T 3 Cl*Ct i i l l t

e -

2 Ot +0e - _

• Ot

• Ce - _

i i

"g / -

P

' ' 'I l i iI C

' CC 1 C. I Ca 1 Ot i ce i ic T(ME (5) l I

l .-

I I

t Fig. 135. Miscellaneous fluid kinetic energy. fluidized explosion conditions.

I L

244 l

l

R': & ERA r

The fluid's kinetic er.ergy plots are in Figs. 133-135. Total and upward kir. etic s r.s ip 1. . . . ilic . . n.c c e s r. i ng as in Table XV. Tne eisceliancous category includes everything else but the upward kinet ic energy. Except for the early peak c omi r.g free the acceleration of materials in the lower plenum, it represents the kinetic e r.e r gy of wa t e r in the downcocer and inlet pipe. For this calculetion. the coherence parameter for fluid impact, F /KE p . is 0.59 ,

GN/t, reflecting the diffuse nature of the impacting spray on the upper head.

2. Explosier. After 0.7 s of Mixine (Second Scopir.c Calculation).

By 1 s into the preciming calculat ier., the fuel began to fluidize. Fig. 136 shows the corium cass that moved to an elevation below that of the bottom of the initial corium pool as a function of the premixir.g time. A value of 40 metric f tons was reached at about 0. 7 s. Beyor.d this tiee, counterflow occurred and little additional coriut was added to be lower plenue. As shown in Fig. 117, a 60.0 . . .

i i

i . . .

i . . i l

n g .

l Z ~

1 0 40.0 f '{u2 04y l  ! jf G Iq 1 0 ,

l g - .

r e l -

\

>- 9 m - -

2 w - - l 2 - -

3 20.0 g - -

O .

9 -

. i I i i i l i i i l i i i 1 e i 0 0.2 0.4 0.6 0.8 1.0

, TIME (s)

Fig. 136. Coriue below the bottom of the initial corium pool.

SikNER-Il cor.servative precixir.g calcula tion (case 2).

245

R 'fS 5 =,

NWd j[ [

time of 0.7 s is also ceaningful, because that is when fuel initially contacted the 'uu l i uc. vi ihe lower pienue and could be associmied with iriggering the explosion.

The assumptions made in this calculation are the same as those in the scoping calculation made after I s of eixing. The one exception is that the water-droplet diateter was set to the saec value as the fuel diaceter (300 ut) through ar. oversight. Again, the most important assumption was the exclusion of lower-bead failure.

Liquid-volume-fraction plots are given in Fig. 137. As in the previous calculation. corium that poured into the lower plenue but was not mixed with water had only limited motion. Ilowe ve r , unlike the explosion at I s, a single tongue of liquid was accelerated up to the top of the head. Plots for comparing pressures and forces are giver in Figs. 135 to 143, with the same resolution as before, 1 ms. In general, the driving pressures are reduced, because of the reduced extent of cixing. The icpacting fluid produced higher pressures, but a narrower pulse width. Fluid motion was less diffuse and the peak pressures were more concentrated toward the center of the vessel bead. Kir. etic energy plots are given in Figs. 144-146. The total kinetic energy was reduced, although energy partition remained about the same. For this second scopir.g calculation F'/KEp = 1.8 GN/t, comparable to the coherence obtained in case 1.

3. Case 2 Final.

The case in which precixing is calculated, although including cany conservative aspects, resulted in reduced stcae-explosian energetics compared with case 1.

The evaluation of this case was cocpleted by making the following modifications to the scoping calculation that started at 0.7 s.

(a) The lower-head failure model was inserted. Unfortunately, the dynacic amplifitation f actor of 2 was not included, the factor of 2 was omitted in the stiffnsss coefficient, Eq. (M-7) in App. M so failure cay be slightly early.

Rerun of a corrected version of case 2 was not performed, because of the bounding character of case 4.

246

Rf fG't 3 .

W )) g p' T M'E 70~ CO v1 1fvE 710 000 US s'tav t 8.:t :. t t.' st:.t :t Utav Enc.:s.;. t t'.' st .tt.:t

• -: -re: r4 .st . -

r e _r, ,,s l

I p

, [

t t .

' ..  ; lj i

?

l'. (*

'ji .j

i. 4..

l

\ r -

}:

' C*E 2 1 0;[- 2 g.) ' C t-:: y1 C;t.::

9 9it-:: 9 99t.*:

. ,,,. ... , ,. ,,e I'VE 73~ C0~. Vi

• ~ ~' '

1*tav t,:.:3.c. gyg.,

. cr- r ., g 5't av t es.1 : . [ .:'

• E! " . E *.' : *

. _- . 4: r, .: * . ~ - ' -_r a -r4 .:

i

,e,

(

- ll/C' i

ei

[ 't

" ,- l'

.. t E...Q':

i Q N' H. - ,

!/

,t ),

~

g,

' p' . ,

l I

'WnW sx 1 C:t-;2 ' COE ~2

,'.)*  :::-:: ()'COE-

, g; :: 9 9EE :

Fig. 137. 1.iquid volume fractior.s. explosior, at 0.7s.

247

. r d i ,

. ,, .... ..... ,,-. i tf E 7!,0 000 W s : . t <t . st;.E'.;t

, , I ., ; . :: : :.:. :::.: :t t*" r,-

,... . ..,.r, .; : n-:_' -r2* _

r4 .  ::

)

/

• \

i l I'. '

.\ '.

. //$

. p <\

. o

. f,W i

_o\\ (; -

tI,. ,

ll i- , 1 J l l

0

.y 1 .,

\'

,'\.' ' -)I i

. o

'( '.

i l

\i l l

rj .-

{<

. k '0' V/ L S ,

.I .

i j *

, Vp, f 4 Vi

• I

t. i i a a

..g .y 1 C*t *2

• i c;t :

(( j ::t-:: 9 99t ~;

,9;;_::

I

,. .-. ,,, e-

~ ".' E -: ::: .,- .-

st;.t ::

'I'*  ;* t t

• It;.: .:t t et g.:.:t : t.t.

6 -: ra7 r4 ,:: v. _ : . .,. ., ,

j I -'

. i i

i . . ( ,;

\' I i

l l [.e 'f,.? *.:

3, n. ,

l b Y g,l

'J -

i l-' '

e Il~ l' b '

r

. . . 3, i .

"'1 , l [.

l!

- s ./. , . . ,

t$g'i i

(;.} , '

g.

-. $w! V ,

V+ + '

)

I . .

t-: i c:t-:3 C;t-::

( ~J 3) 1 (h') 1 0:t-::

9 99t ~; 9 95 -::

1ig. 13~ (torttruedi 248

R D DRAFT

.,,,- g: ::: v; I'U~ ~'~

C~; v1

' '" I'* :t ~ c<c * !!;.t.:t 5 t t... t.:.:1 :., t.t .- st;.: .:t

,, _:.  : - : .: - r- .:. :. _..

. . M. ~, .

6w..i. n.. .

l g pe, . .

, j, i.,

._, Ifi '

i

.., ,p . 1 i )n$. U- .

I i>iJBi,Jf, y 5.i .

i

\

l

.I

,1 f; .  !'

i, .

.. .. . . .. 1.

I iw \ v /.

'V'

' l. \ l s / ,

. \ / * - * ,

j ,

' *:t ~2 1.0;t 2

t-:: tc:c.::

g :9ic-:: 9 Q)9 99t-::

1 v 5:: ::: v- TivE 5 g o;;. v; s ttv t.:.:1 :. t.t. st;.t.:: 5 C'v c ". 5 :. Evt'. st:.cs:t

. -:. .t : r: .:: a -: -r:- r: . ::

}, ,

- l

/ i '. l ,i.pt l

. , . 1) o '

- i I . . , ,

4 g; ' ,

s

.  ! I.  ! , I, 8-o p h; p, l

{ !,

I ') .

,j -

[.' . ._

I I , --

f.' ' .

"  ; i. i . '

i. i y ' ,l/ ,'. '. \, ,,

t l\2

\v l

1 I l

. - t, 1 o .s .

t 0 ! *2 1 C*t-02 Igs )

w.r.

('y1 11 o;t. .- .

9 95t-:: 9 95t ~;

E . r,. ;1 i ( . Je i t 4 c t. ., l 249

I l

R: 2 JRAFT l

i IBO O -

1 PRF.88tiRI' AT INI.!T PLl;NilM 130TTOM 1,

I">0 0 - ,

1 i

120 0 - ,

E l E  ! 1 E 90 0 - ,

l b i E '

00 0 - ,,

I -

\l ~

30 0 - ,

. ~'-~ -

00 i- i- i- i- ,- i- i-i-i-m 070 071 072 071 0 71 075 076 077 078 070 080 081 TlW E (s) r i g. 135. Inlet pletur. pressure, explosic.n at 0. 7 s .

800 -

] PRFS8t'RI: IN DOWNCOMER 700j '

1 0003 ,

1 \

7 5004  !

i i \*

E 4004 b 4 A.' ' '

E 30 0 2 -

3  !

20 0da b } '--L's - v /\ss -

10 0 lI(, ..

~

00 - ? -r m ~ n - r m -, 3 . . i,, , ,, , . i 070 071 072 073 071 075 076 077 0 7H 0 70 0 80 0 81 TlW E (s)

] ip. 139, l'r e s u r e a t the tep of tlie downtere r . explesser et U. 7 % .

250

m ,

6 RCUMERAFT  :

. m_ , um o -

E'~ l' l'RIF81'RI: AT TOP Ol' VI'881:1.  ;

1 A 2200 , -n p

a . .

L -

I 200 0 - '

T.. T L

1 4

E J 4

E 150 0 -

g 1  ;

= 1

= <

2-1000 ' \, .

i-  !  ;

i.

1 1

< ./, \-

50 0--. / 's L

,. s__

d {

00 ., ., .<,i""r"m'~i- i- 'T'vm ~!

! 070 071 072 073 071 075 076 077 07H 079 080 0 81 1

TIME (s)

I.t p. 1-10 Pressure at the t er of ves se l , e xplos i or. - s t 0.~ h.

t i.

1, 2300 3 ..

PRIF8URP. AT lit:AD cllRVATURl:

i 2000 * \

1 i  :'

I  : .

T' 3 g 150 0 ; ,

1 '

2 y  :.

-

I i I

\

~~ { 100 0 ,

=

]. I k

, I -

)

50 0 i f x '

i j.

'/

-s m. _. _

3 00 i- i 4-- ~i" m "w" v i - i- ~ '1 070 071 072 073 07 075 076 077 078 070 080 081 i

TIMF: (=)

4 l'ir. 141. Head preuere a* h t o the sertital explo ier. t ti. 7 s .

251

.) ^

I:

I *

- , , . , , . _ . _ _ , . . . . _ , . _ , . , , , , .,,_._..,,..,_.;,,m.._,.__,,. _,, _ , , . , _ , _ . _ _ _ _ , . , _ , _ _ _ . . . , _ . _ . . . . , _ _ . , _ , , ._ , _ _ _ . , , , , , , _ . ,

i 4

T.

RCUG!iDRAFT

,. . .nn--

7" ~~4 Imlwritt: 'AT YlF81:1. I'l AM;l: .

1-4  ; -

. 1000 g i

. : Room i-6 .-- .. 7 c

1 l

2, 3 (.

1  ; i E

~

rt 60 0 I \

$ ] i g

_J . / .

40 7 /

J l N

[

~

20 0

~ ~'

i

? /

a .

~

00 i">' 'i' ~ i i-

. - F- - r" i -

l' 070 071 072 07.1 071 0 7F 076.077 076 079 080 081 TIME (s)

l. l'i g . 1 12. Head pressure at 7m" te the vertical, explosier. at 0.7s.

I

% 18 0 -

i l'Oltci: ON lil:AD

, l' .

. g *, 0 -  !.'-

4

\

i 12 0 0 ,

, - j  ! I c -

\

E 90q '

(

= ,

E i

\ s 7 J \

4

.coJ. .[ a 1 i 3  :

. - 1 103 4

i 00 - , -.'"i'"i-i"'i'i""v"rm 070 Oil 07J 07) 0 71 07Pi 07G 0 77 07ti 070 0B0 081 4

Tiur (-)

J 1, rte er. the he.J. exp' stor at O. .<*

l' l' i g . 1-13.

252 3

i i

i R:lGH [RA:T k

2 _. i W

4 1

2 CE+09 i , , , , , ,

g 3 l

~

l t SE*09 -

i' 4

l g 8 0(+09- -

l * -

l 1

I t

$ Ot+08 - -

t

. I

. r

. i

, t . . I , , ,  ! , f 70 72 7e 76 78 80 TIME (5) 1

(

-1 1 ,

I 4

I I

I fi g. 144. Total fluid kinetic er.ergy, explosior. at 0.7s. j e

i t 253 i t

l {

4 .

l

.. . .. .- .. . .- . . . _ - . . . . . - . . .. . . _ _ - - - _ - - . . . _ ~ _ . . ~ .--.-.~ ~ , _ - . .

/-

OO i  % ..

'W j P

r

.:a.

p p <.a-l

• Stees ,

4 b 1

'A t

i -

l i 6

.. t 4'

I OC+09 - -

I w

s /

/ -

! t

• I I

1 .S 0(+06 - -

i t

1 =t

< . i h$ .

! 3, ,  ? t I , . I I i C

ve 72 7. te te se i 4

t TIME (5) I i

1  !

i t

i t

4.,

c i

. t e

4 fig. 145. Upward fluid kir. etic energy, explosion at 0.7s.

1 1 -

254 A ,

4 =.

[

l- . . __ ........ ...~. - ... _- L _ - ... _ ,,_ _ . _ . ~ ,,_._ - _. _ .._,.... _ . _ .._ ,_ _,_, _ - _ ...._ _ . _ _. ~ ~

' " m 1 ')

~

R1 h 1

4 i ih j 3 Ot C9 , , , , , 3 g  ; g ,

2 Cl*00 -

- 2 w

a -

\ -

t Of*CF -

- 1 0 -

] I

~

g . . 1 . , t i t . I ' '

O 70 72 74 76 78 80 TIME (5) i l

l l

Fig. 1 -t c> . Miscellar.cous fluid kir. etic er.ergy. explosien at 0.7s. 1 255

{

mt - r- -

0 Y ih (;

(b) The oversight in the water-droplet size was corrected. The water size was unif ormly set to 73 ue c:aceter.

~

(c) Steel particles were inserted to represent upper core structure as in case 1. The d-nsity of particles was adjusted so that the total of 35 000 kg included the steel particles forced by breakup of the core-supporting blockage.

Lower-head failure for this case occurred at 2.7 es. Figure 147 gives the '

Inquid-volute fraction during the expansion process. More liquid progressed downward in this case compared with case 1. The pressures and forces are given i r. Fig. 145-153. (These plots have the resolution of every SIWER-ll time step.) The peak pressure at the apex of the vessel divided by the peak pressure at 70 from the centerline is about 2.4 compared with about 3.7 in case 1. The lower-head's maxicum kinetic energy is 613 MJ. The four coeponents of the fit.id's kinetic energy are given in Figs. 154-156. The sue of their peak magnitudes gives 1 052 MJ for the fluid's peak kinetic energy. or 1 665 MJ foi the kir. etic energy produced by the explosior. Consequently, for case 2,62% of the kinetic energy was direc ted downward while 31% of the kinetic energy was cor.tained in the upward coving two-phase " slug. " FiE.157 shows how cuch of the fluid's total kir. etic energy is realized at any one time. The coherence paraceter for this case. F /KE p. is 1. 3 GN/c:. Reduction from case 1 cight be anticipated because more fuel coved downward.

Finally, we should eo:phasize that the explosion at 0. 7 s followed a preeixing calculaticr. in which steam production was einimized. A complementary calculation. in which a comparison with the SNL e x pe r ie:e nt s suggests steam production to be exaggerated, is given in Appendix U. At high steam generation rates, the calculated extent of precixing of coriue and wa t e r in a reactor situation was very limited. With such configura~ ions. large steam explosions do not seem possible.

C. Case 3 - 75 Percer.t Preeized The explosion assumed to occur at I s in case 2 suggests that. if the corium on top of any preeixture is fluidized, upwardly-directed kinetic energy will be less effective in causirg significant head loadings. Case 3 takes this 256

RM 3G Human (a) Time = 700.00 ms (b) Time = 705.00 ms l r li '

=::- ,- l fUh/

3 L l

(c) Time = 720.00 ms (d) Time = 735.00 ms I

Fig. 147. Liquid-volume fractions during the expansion process, case 2 (fiaal).

257

_l

I R: 2 DRA57

.s

' K ',

(. . .

' ., I:h t b,i i

.. b,:.  ;

q  ! ,

s l (e) Time = 750.00 ms (f) Time = 760.00 ms l

l l

S3 3s\n ,

I f, l h . I','t;f i "s'-l, ', ,

{

n

'n ".

o,  !' . ,

i l *:. I tl t

'n. .

.g.

.t>s. .

o 7 v y I (g) Time = 770.00 ms (h) Time = 780.00 ms Fig. 147. (Continued).

258

l N  !$ t. .

R'Vw UII l h" I l l

, \q b r.'.

j,; &

w.-

t

\

l . .

u (i) Time = 795.00 ms (j) Time = 810.00 ms Fig. 147. (Continued). j l

l 259

j 2'>0 0 Piiil44Uiii i N Pi.E N U M i

2000f O 1 N 150 0-!

% 4 E 1 r

't c 100 0 -

.LC 50 0 - ;,

00-~ , ,-- ,-- -- s~"ar-~t- -r ~. r-mi-- r~~i 070 071 072 073 0 71 075 076 077 078 079 080 081 TIMI: (-)

l' i p . 145. Irlet pl rur preuure. case 2 (ftr.al).

60 0 ) 3 END OF DOWNCOMER 500 3 i

a g 4002 ,

c.  :

2  !

U a: 30 0 .

4  !

j Ei  !

L c: \

C-200 j (

1004 i

(,'%qd,,fv ft.,j\'.-

r 00 ' , , -

070 071 072 073 074 075 076 077 078 079 080 081 TIME (c) l ip. 144 I'reoute at t iie top el the d wrcorer, case 2 ( f i r.a l 1 260

f RCCG!EMFT 2000 .

PRtySl'ItC AT 'lOl' Ol' Tlit. [t-S. .A.

f .j 4

3 -

100 0 ;

J

<! 4 4

'i 2 h 1200 $,

c m d E'

Q l

d 800m

+ F

~

I 1

3 400$

i

/ t

! l '

/

4 00 , . , , , - - , - -----t----'i 070 071 072 073 071 07507607707H 079080081 TIME (s) rig. 150. Preuure at the tep of vessel. case 2 (fir.al).

t 200 0 - ,.

l Pitr.SSt'IRC AT }ll'AD CURVATl'ill' g

1 160 0J' 1-i ,

'2

r 12004A 3

i. i w

l  !

3 i /

E i d 80 0 *~

F

- \

4006 N, _ '_

i i

,- 00 . . > > - ,-- > 'ar""-"T"-~~r- ai l' 070 071 072 073 071 0 75 076 0 77 07H 0 79 0 00 0 81  ;

TIME (:)

lie d pre uure at V te the vertic.i! case 2 ( I s r.a l l.

lsp. 151.

261

/

- , - - . n.,- - - - - - , .--. - , _ . , , - . , , - , , , , - , - , - - ,.,,w _ n ,_ _ . , ~ , . - - , , , - ,,. ,-,..,e., --,-,,,,w.,,... , - , - - ., . - , - . , , - , , , , , , , - , . . ,

R M DRU7" 200 03 PRESSURE AT VESSEL FLANGE 160 0 -

T .

I 12004 -

M L

b 8004 ~

E ,

400 ,

\,

00' i- , ,- i-

,< -,- -i,--,

070 071 0720730 Al 075 076 077 078 079 080 081 TlME (s) l'i g . 152, llead pressure at 70 0t o the ve r t ic a l ca se 2 ( f ir.a l ).

b- 10 0 ,

FORCE ON HEAD 90-

~

80,

70 7 60

$ 50-8 6 40, 30-20, 10- /

070 071 Oh2 073 Oh Oh5 Oh6 Oh7 Oh8 Oh9 Ob0 081 TIME (s) 13 p. 153, l'e r t e e r tlie head. toe 2 ( fira l ).

262 l

ROUGH DRAT

& Ot.08 , ,

g e Ot+0e - -

A 2 CI *0P - -

, , t i , , , t , , , I ,,,

to 72 74 76 78 80 1IME (5) l i

1 Tig. 154. Upward fluid kir. etic er.ergy, case 2 (fir.al).

26?

1

1 y,

ROUGH DiA4 T tac x.a - -

7-g I I I I 1 I T y u y g g y 3

~

8 Of+08 -

1, 2 g -

S 0(+0, -

-i I

g:

y 0 - I I ' ' I ' ' I '

0- ,, ,. ,, ,,' ,,

TIME (S) l-

/

Fig. 155. Dowr. cec e r fluid kir.et ic enerry. case 2 (fir.al).

1 1

264 l 1

R t t 3t= i

=

l =

R1dW l i

y 3 I T T 3 3 3 3

I I I g l e IC se -:

go? -

5 IC I

N , 4 6 , to 1C g c .

10 D -

"li tC w -

=

x  :

8 10 "E 10 10 3 - - 10 iO 2 s le

,,i . t . , t . . . t . . t . .

ie so 12 se is se ao TIME (S)

Fig. 156. Dowr.vard and outlet pipe fluid kir. etic er.ergy, case 2 (final).

265

Ft !q' in, r--

u 1 w Ji .sl \ ,

4 MISCELLANEO'JS FLU:D KINETIC ENERGY FOR CASE 2 REGION SETS 1to,T O 3 3 3 j . .

.g . . ,

i . . .-

! DOWNWARD E

,  % ~*

A

'o'B DOW N COM E R s g 5 7  : .

n 106g  :

E,'

g 9 t

u x

10 5{  :

b -

E 10" OUTLET  :

E

- PIPE E 10 3 p

= g

=

2 10

?

E -

,p' , , , i ,,,t , , , t , , , I , ,

7C .72 .74 .76 .78 .60 TIME (S)

MINIMUM VALUE= 7.25921E+01 MAXlMUM VALUE= 4.19841E+08 IG Fig. %

Downward and outlet pipe fluid kitetic energy. Case 2.

266

O,' D  % *~m.

Yu U f,

' Of*09 i i g i e i g . . i g i i i g i . .

n 8 Of*08 - -

6 0(*08 - -

~

La e Ot*00 - -

2 0(+08 - -

e 70 72 74 76 78 80 TIME (5)

Fig. 157. Total fluid kinetic energy. case 2 (final).

267

R:b3F [iAF assumption to the extreme case of total fluidization. A recent report by SNL ' 3 proposed that the upper licit on corium-water premixing was 7M of the core, or 94 000 kg. In case 3 all this corium was premixed with water.

The cor. figuration assumed is snown in Fig. 158. The mass of water i r. the precixture was 20 000 kg. An extra 8 000 kg of water was assumed to exist uncixed in the downcomer. The remaining solid corium (250 was assumed to cor.sist of unmovable crust and was ignored. The c or i nci's teeperature was assumed to be 2 500 K. Its beat capacity was 500 J/(kg *K), with no beat of fusion so that its er.ergy above the w. ster temperature of 400 K was 1. 2 MJ /kg.

the assumed best-estimate energy in the SNL study. To limit the height of the preeixture, the initial steat-volume f raction was assumed to be 0.19 in the corium/wate r mixture. This led to a precixture height equal to that shown in Fig. 113. used for the initial conditions in case 2. The corium was assumed to exist as 100 um diam solid particles and the water was assumed to exist as 100 ye diae drople t s. The new models for SIWER-Il heat transfer and water vaporization were employed. These assumptions cause all the premixed water to vaporize withir. the first tillisecond. producing 194 MPa steam at 922 K with a steae-volume f rac t ion of 0.69. Because the SIMMER-II models do not have heat transfer betweer. solid particles and stent, the particle-stene expansion is adiabatic (except ir. the downcomer) following the initial steam productior.. The lowe r head wa s a s sured r.ot to fail in this calculatior..

Der.s it y profiles of corium for the expansion are shown in Figs. 159 ar.d 160.

Initially a spray developed, then caterial decelerated and collected near the head. a r.d f i na l l y a smeared shock front delir. eating upward moving coriue from downward moving caterial developed. k'ater movemer.t in the downcocer limited the su e of corium penetration ir. that directior. Pressures and forces are shown ir. ,

Figs. 161-166. Although loading on tae head was fairly uniform as i r. the previous fluidized case, the peak force o r. the head stayed below 1 GN.

Lower-head failure also should occur. The fluid's kinetic energy plots are giver. ir. Figs. 167-169. Although the upward fluid's peak kinetic energy (1634 1 bid. Ref. 6.

268

RDi W u

6

= OUTLET PIPING 50 SOLID STRUCTURE

• \

12.54 p lNLET PIPING PREMIXED

• l REGION WATER  !

30 CORE SUPPORT I 20 FORGING 13 MOVABLE 5.20 STRUCTURE JL 1

+ OUTLET TO KEYWAY 1 5 10 15

+-1.613%

e 2.537  : --

l 1 it. 158. cor1iguratser r.,,Jei c a s e 3. l 269

ROUGH DWT t

101:L Dr 'r. i . , , r i ,ir t to14t eten11, f il i UE L 1 !'a: L :'_ 1 1 Mi 5 000

l _l l r ,I I

s s.

't Y l i

1 A

MIN: : v4 .

N iti . , _, i 1 : , i c i . 0,- M!rn 0

'x _/t M t, . - .

Ir<t.C:,ci- , Ic .;;

le'4L L L '. .1, v. I ut ; 101:L DE.NL 11, or r et t l II" I L 0::*- 11ML 15 000MS ,

i

-f-- y- -- \

N -

l l I

> l

'\

f e x 1 . 7 Ie if ' .

i  :

i ) ,

I i l{

p' I

i ,

f .

I i i d

w l

I 5

s Mi :- e s.. ,m .,.z,,. , ,,<.c, M i .e O M. i m . t. , : .

Fig. 159. Initial expar.sion of fuel ir. case 3. 270 l

l 1.i r i p[ t; ] 1, Di i til. i. .

-* r.t, ', r: . t o sr,y -

1 g v.

• • ' ) ~,

1

~ -~m- - , .

Ir ,5- _-' U=)y i

b zb'

.n.

' i l

J) ./ \  !

]j f \ j' -k 3

m Ir

-l i 's }-) -,

\=,

k \I ,

r )

q l

l

's.

-f k' _. ' L .' ...

'Q .. , i Y ;~ - .

( % . ; , ' ), - --

2

. __h _ _ /

v'*. ' J T .*

\  ; n. .

'.k ,.

,, vits C v

. . l .- <

1.:

1. . i n i, pi ra,1 t 3 Os i v[;

T*._ (('T 5 ' '! t

e- 1 I M. '"C'>'"

1 &ar s

f , k

( ,

( 0) , ]'

w ss l ..

t

(/ -

Yj

. d.2-b W _- -

aw

-i ,

i .. '

'q ~~ 11 (- 3~

s ,,-% 9 7

x

,I ,

i ,

- - . h  ! .

i t l

, lt o' 'A l  !

j:!

i ," I  !

.l ;- .

l J

%s J . -l i .. , .

j

- -11 , n- h ~ - \ p q ,-) .

N...'-

t c' / .

. y.., i..

av. . .1,.. . ./ _3i! . n. , . wira. p l e p, r. , ,

Fig. 160. Head loading and reflection in case 3. 271

I R0JGFI DE 220; .

1 PRI:SSt'RI: AT INI.CT PLENUM BOTTOM 2000i 7 .

I . 1500 i ,

E \

i 100ci \

l I

500- ~

N

\ -~

00' . -

0 00 0 03 0 02 0 03 0 04 0 05 TlW E (s) l i

l l

Iigi'161. I'ressure at the inlet plenum bottom for case 3.

180 0 -

P9CSSttRC IN DOWNCOMER l- 15004 l l

12004 ,^

I E 9004

=>

b E 6004 IY 3004 00 , i- , i- .

0 00 00: 0 02 0 03 0 04 0 05 TIME (s) lig. 162. l>reuure er the top of the downcomer for use 3.

272

v t

wn gson.,J,RIF7 P

• ~ #

PRI:S8t!Rf:, AT Till: TOP Ol' Till: VI:SSI:1, ,

3-800 3

-j ,

t #

/  %

', k. 600 j W '

a b~ 4002 '

t .

l 20 0 /

i /,

3 /

4 3 ,

., 00 . .,, - ,,.......,,,,,,,,,,,,, ,,,,,,,,,,

0 00 0 01 0 02 001 0 04 0 05 TlW E (<)

i Fig. 163. Pressure at the top of th'e vessel for case 3. -

100 0 -

j* PH F'.SSt'RI: AT lil:AD Ct'RVATtlRI' 1

800j 3

's.

T l ~ '

$ 60 0 --

/ -s, E .

\

N Y,-

i.!

2 l. '-

t 400I - '

4

/

200 f 00 .i....,-,r,,,,_r,,,,,,,,, .,

0 00 0 01 0 04 00.1 003 0 05 TIME (s)

Fig. 164. Preu ure or,the head at 30" to the vertical for case 3.

}- 273

- - ,.., . , , - -n---,n- n,, - , --".,,y ,,e.,,,,,n-, ,,a,, , , ,,,,m,w.,, ,.,,.e,,,.-e- , , y,. , , ....-v..,- , , _ ,p. ,w..-,,.-,,, ,,,,w, -

400 07-

' ' PHl:SSlfRE: AT Till: TOP OF' Tile VI;SSI:1.

80 0 i

/

Y ,

x ,

I

- 60 0 i

'\

E /

9 \

b. 400i / \'

4

/

/

20 0 -. ,'

/

/

,  ?

00 .-,,,,.....,,,,,,,,,,,m,,,,,,,.,,3 0 00 0 01 00J 00.3 0 04 ' 0 05 TI M F' (x)

Fig.'163. Pressure at the top of the vessel for case 3.

100 0--

PRF'SSilRI: AT lil:AD CllRVATilHE 8001

-l ' n

< N '~

7 'N s I 60 0i

[ '

X t.

\~%

j

' t.! 4001 ,

E j 20 0 i 3

L -

00 ' i ++ e** ++' er* * ' ' - i- i- _

, 0 00 00I 00J 00.1 008 0 05 TIME (s)

Fig. 16-1. Pressure or. the head at 30" to the vertical for case 3.

274

ROJ3H D?kFT 100 0 -

PRESSURE AT VESSEL FLANGE 800-7 .

60 0 -

to .

m D

(n V

{3 400 2 m .

n. -

20.0 -

00' ... .T... ...... ......... ...... .......... .i 0 00 0.01 0.02 0.03 0.04 0.05 TIME (s)

Fig. 165. Pressure on head at 70* to the vertical for Cese 3. .

275

ROUGHc..WT

• o 90~

r .

FORCE ON llEAD 80-70- N 6.0 -

b 5.0-W U

@ 4.0 -

m 30_

20-10 00 -

i '

b2 O 0.h Ob TIME (s)

Fig. 166. Force on the head for Case 3.

\

276

\

l '&.

k - o

' I ROUGi DRAT 1

!  ?

t, i .I 1

1-' ~ ~

3 of*C9 ,

g ,

r . ,

4 -

!? ? "t i' ' .

'i 2 Ot*09 - -

].

I ~'

.- t 1 a; x .- -

.. t og*09 --

1 -

r-i .,

3 E

i.

i

.~.

i t

4 0 ' ' ' ' '

C 00 02 04 Ct i

T WE (5) c.

j.

i-E s

i i

i Fig. 167. Toial fluid kinetic.er.ergy case 3.

t j.

L 4

277 4

4

, , , , , ,.w. + --,--_-,w<- , . . , , , - . , - , - - - - . , -,n, .-.-.,,,,na..,, ,n-n,.,n,4,,,,,.w_,. ,.-.,.,,-.a- ,.,m,-,-,,. . , . ,, w .m .m m e . .- w,er,

t P

l.-

ROU3H DRAFT

• i l

s 2 0(+C9 , i g

r 1

I St+09 -

~

g .I Ot+09 -

E -

.~

S Ot+08 -

7 -

0  ! I I '

O 00 ' 02 04 .C6 TIME (5) ,

1 i

1 1:

1

?

1- ,_

4 5

, -Fig. 165. Upward fluid kir. etic er.ergy, case 3.

1 i'

i J

278 i

.- , , . , , - - , - . . , - , _ . , - - _ , - . . _ . . .._.,..m__,.. , . , _ - _ - - . . . . _ , - . , , , , , _ , _ . - , - , - . - , _ . , . . . _ . - - - - - - , , . , - _ - - - - . . . . . - -

/

ROU3F DRAF-4 g

) ~ t Ot*00 ,

i

.k . r 4

4 4

~

1 6 Ct.Ct -

1 .

2 ~

i w . et ce --

2

.x . l l

~

' . 4

~ ^

2 cc.ce -

l 1

, ,e O o. 06 e 'o: o2 TIME (5) i

+

L.

t t

1 -

a i,-

I i

(

4 Fig. 169. Miscellaneous fluid kinetic energy, case 3.

i l

279 .

i i

a

=r',--* - e %-r,e-.- * - . - - ,,e ,-.*----ee-,s, s.,m.,%-,--r,...------,-,,eew-ee-wm,%-e,.- .-wwt-e*,s---+--W+v-" -t r *-rv 'r -D ^ e '-P*r-'-9ev-'*F'*v'v---**'-FA v v v **-

R O3H : E

~

MJ) significantly exceeded that of Case 1 (647 MJ) and Case 2 (516 MJ). the cohere.,cy factor. F /KEp . is 0.33 GN/m. Ar. increase in pre-explosion fluidizatior apparently does reduce effectiveness of energy transmittal into head loadings.

D. Case 4 - Simulatior of an incoherert Explosior One of the questions raised in the ZIP study was whether a large ci~ssile could be generated from a stene explosion by in-vessel acceleration of liquid at driving pressures below those required to fail the lower head. Case 4 attempts to obtain a conservative licit on upper-head loads that might be anticipated if pressures ere not sufficient to cause lower-head failure.

Figure 136 free case 2 provides the starting point to formulate initial conditions. The 40 000 kg of corium that fell below the bottom of the initial

~

corium pool was assueed to be homogenized with the remaining water (10 000 kg) in the lower plenut at 0.7 s. A slow explosion" was then calculated in which droplet sizes were increased an _ order of magnitude over SNL experimentally correlated values. This reduced the heat transfer by two orders c' magnitude.

The objective was to simulate an incoherent, cultiple explosion environeer.t that could be representative of the reactor meltdown situation. As described in Appendix D of the SERG report . ' coupling length considerations prohibit such homogenization from producing a highly efficient explosion when the precixing calculation showed most of the bulk corium separated from the bulk water.

However, the chaos introduced by incoherer.t expidionsmight lead to progressive mixing. Although the e sherence introduced by the assumed homogenization is exaggerated, the large fuel constraint on the system would favor progressive ,

mixing before the water separated beyond recovery.

Figure 170 shows the starting configuration. The preeixed ret on i was assueed to occupy the entire space below the remaining single-phase corium pI>oi. The steel particles that represent the broken-up core blockage were sceared across the radius of the core to make the expansion more one-dimensional, and consequently conservative, based on the SIWER-II analysis of the shallow-pool experieents.

Preeixed corium and water were includ,ed in these steel particles te generate pressure. Steel particles were included above the corium pool to r e p r e s e r.t 283

m p_ m p' RV al d , , J)AJ l

=

l RnW 201 myi g/I\ng L

~

u

'60 OUTLET PIPING STEEL

PARTICLES SOLID STRUCTURE 50 INLET PIPING 12.54 ,

40 DOWNCOMER CORIUM = ^

30 PARTICLES &

PREMIXED ___ MIXTURE REGION 2 CORE SUPPORT FORGING 1

-- memElill l

- -== sum sal l ll I m uassamm I1 I I I I m:eusa n u i w u--

MOVABLE 5.20 5 Z".$U STRUCTURE

-- m 33 R RBil%E==

n .

rus i =

OUTLET TO KEYWAY 1 5 10 15

• - 1.613 -*

2.537  :

Fig. 170. Co.-figuratior. model. case 4.

281

q l RlF w ' a= f; upper internal structure. The -8 000 kg of water in the downcomer permitted the water mass to be conserved.

The avgraged pressure differential across tha* part of the lower head used in the head-failure calculation is shown in Fig. 171. The loading time corresponds to roughly quasi-static loading in terms of the model, meaning the failure threshold is below but close to 6 740 psi (the failure pressure from the model if the driving pressure rises with infinite slowness). Although the failure applicability of the failure model in this situation could probably be improved, the no-failure prediction is reasonable.

Liquid-volume-fraction plots are shown in Fig. 172. Because the lower head does not fail, only axial nodes 13-66 are plotted. The expansion was ,more one-dimensional than case 2. Also, a single-phase liquid region developed at the corium-pool / premixed-region interface. In this region, water must undergo bulk heating (surface vaporization cannot occur in the current single-phase model). When the " water" vapor pressure beceme sufficient to disperse this 65000 -

30909 j l'Rl'88t 'RI: DIFFERENTI AL 2

'6000-i .

5000 0 1 -

? i '

450002 l

} 40000- ,

a:

3500 0 2 - . \ '.

/

(.

~ ~

\ ..

g 30000 a: 25'10 0 c.

2000 0 3, 15000ff 10000f 50002 ,

00)+ i=w"T"'i"" ''i"i""i "'i"i 'n w 1

000 001 002 00'l 001 00"> 0 00 007 0 0H 0 09 010 O il TlMF: (*)

Fig. 171. Driving pressure for lower-head failure. case 4. ,

282

in [ mpp--

N ~l Nbb o

r L!c.::

,s.. ,.

. . ..r......

..... So..u.-E rat:  ::'.

i i

I .

l  ! l i

e I

i e

t

~~ ' n N

,. i

(. l;,' hf I (0) p..... .- . :. . c. -c c. :..: .. . ... c. c. :. :. . ,

MN: 2 CiE-:5 "A*: 9 95E-C; CI: ' 'EE-02

.. v.' . . . . . . . . .. ..

q . .

' : : :J. ' - * - -..

v 0.u-E rot:* : . :: g ;,::

,g 3; c;;w

t

i l 1 i lI I

l ,o

/

. [t/ i -

l j

. s M

j u)/

C r

I]J . ,,

f n' ~- ~

.1 ee es

, 'v .

-Ql Q}f ; '

l ,

W' b,:*)

• .:  : : E-:5 va : - 9si-;.  !: 9 9EE-:2 (0) l

, ., N .., c c.ee.

. . yi,.. . c.....,..

.: ... c. c. :. :. . ... .

l l

Tag. 172. Liquid.voluee-fractier. plots for post-explosion /expansior.. case 4.

283

m n ~, r~

-~-

'd Q' O NN <

,s.,

., . ..w r. r:.--

. .. . . .. -.. . s . _ t.w r. r e s . - . m. '. . ......

, y-. . . ...r.. s. . u. c. r. . .R .* w *-

l

! t l l i

l l o i

+

L. . .

_~ l

'A W

w f i p

)

/ ~ ,fli i l

e 1

• Q$.% u /.-

N

, ku m--ceW, l

(e w: .:

)  : : E E ves: 9 tii-; . :::: 9 95E-:2 w : *.: 2 C'E-05 wen: 9 tii-;:

+

9 95E-::

M. "- -

" E i; # ~:.;v--

' ' ' ' ' '. :: .:;.;; s. ;. ,,:;ur r .e...

--  :-.,w-

0N tr c

;.::

~-

D

, 'A' I , 'I

, i I ~ --

vJ i I .- f Imm, d

! l!

-J I

. . r. <

s) . l,1 I

1 i

- -:=> 1

_ i I ,

s.

l . \*, W

. wy/:* . ./ .' UN h';.

W s/-

L, . - > . /*

i l

. J.... . . .. .. J l \n / i,

.: "A>: 9 955 0; Cl: 9 95E.;; w N: 2 CiE-05 "An: 9 95E ^: ~:: ? ?:: --..

Fig. 172 (Cortinued) 284

RDJM DRE" st.cv; rear- - ,.: ..........

$ n. .,.,wr. r e a . v , s .- e .*.. :.

E; :::v- v}pt 9; ; ;wr l

d%

)Y -

llfh  ; W h ll' k k e.-,*,

'.,,j

. /

. w .~ .

(. Y #s)

! O ';

'\ /) s. -

7

• N 7/

i \ (j)

P '. : 2 0'E-Os wa>: 9 ssE.;; ;= 9 93g... w : *.: 2 C E-:: wa>: 9 tit ;  :: 9 9sE.:-

v

._c ut r::--,; .-

v.0...wE .. ... ras.- : , ;r :,,;;

w.,.

j g, ,,,

• • . . ... 4.. -

(

. . s

/ \ ( 1 a e',

I

\

f u O

• VM f (M-  ;

l  ? i f l

-- 1

= .

• l 3 l

• .:  : t-:s w.,: c c. :. :. . . . ,1. . ... .,,. " : '. : 2 C E-ti ~4a: ' 95E-:: :::

• i!E-::

.. 6 . . . :-

l d

>" . l7s :e-0v.7.)

286

. - , - .,,- --. - , - . ,..__,,,..,,.m.-- --.,,,,.-.-,-~-,.w- _ ,c,. - - _ , ,~e,- , , , , e - ~ , . - - - - - - , - - - . - --

Nm ,

L[ w[O'l m,AtT zone, starting at -50 es, an extra boost was giver. to the upward movir.g coriuc.

The effect of the-buildup of this pressure car. also be seen in Fig. 171. The liquid-voluee-f ractior. plots exhibit cor.s id e ra bl e impact coherence-at the upper head.

Figures 173-175 give the calculated pressure and force transients for comparison with other cases. The resolution is that of every SIWER-II time step. The peak pressure at the bottom of the lower plenue was slightly higher than the averaged value used to calculate lower plenue failure, showing the effect of relief up the dowr. comer. Downcocer impact pressures were relatively low. The peak pressure at the apex of the vessel divided by the peak pressure at 70 froe the cer.terline.is about 2.7. greater than case 2.

The kinetic e r.e r g y plots are given in Figs. 179-181. The upwardly directed kinetic er.ergy reacbed a eaxieue of 765 MJ. leading to_a coherence paraceter.

T'/KEp . of 3.0 GN/m. Case 4 is further considered in the discussior. on the probab.ility of cont a i r.ce nt failure as the upper bound on plausible head l oa d i r.g s .

60 0 -

l'iti'.88t ' RI: I N l'LI:NL'\1

>0n-

• f 10 0 --

c E 100-7  :

l 20 0 - p i i 't...

100-

,,. e , - ,- , - ,.

00 ...3- n .3. -.i 000 001 0 02 001 001 0 0"> 0 00 007 006 009 010 O tt TIL1M (s)

Fig. 173. Inlet pier.ue pressure, case 4.

286

EW DRAFT 60 0 j END OF DOWNCOMER 500-1

)

g 400- I, t -

t 3

~~

E 3001  ;

" '/

)

t

{ t \ \

5 20 0 - )

4 100 J p, L

00 i- i- -i- i- i- i- > i- i- > i-000 001 002 003 004 005 006 007 008 009 010 011 TIME (s) l' a g . 174. Pressure at the tcp 6 I the dowr. comer. case 4.

130 0 -

T01' Ol' Till: YESSI:L d

300 0 i I, r>0 0 - t.

i

ji i 4 i E 200 0 2 fI l

E 150 0 2 i d  ; .

1

=  :  ; i 100 0 - , j 1

50 0--

00 ~ i 1 , ,- i ...,- m , w .. ,,,,,,,,

0 00 O DI 0 02 0 0 3 0 01 0 0~2 006 007 008 009 OIO Oil Tl MI: (<)

I ig. 17f. lir e s u r e a t the top of vessel case 4.

287

R:di DRA I 2 .0 0 -

~ l'ltl'8Fl'lti: AT Till lli AD (' tilt \'ATl31tl' 4, .

200 0-- f' i'

f O 1 C * 'i p I">0 0 -

)

.5 1 l E -

~

1. -

d 1% 0 - I

?.

4 9

'>0 0 --

j o

00- 1- ,- . . . .. ....,....i....i.... *...i,.

0 00 0 0! 0 02 0 01 0 01 0 0"> 0 06 007 0 08 009 010 O l t TlMI: (<)

l' i g . 176. Ilead presstre at 300 to the vertical. case 4.

I200-

~

l'Ifl:88t'Iti: AT \T881:1, I'I.AN(;F' in0:

1200 j f,

10502

, l!

4 li 5 90 0 i I\

7.

E 750 i fI z {

U

b .

60 0A \

F.

. - f l

450f

'3005 -

s 150 $

2 00~ ,- ,

i,+,.i...,3+,..,. ,.i..,,,,,..,..

000 001 002 001 00 0 0"> 0 06 0 07 0 08 009 010 Ott TiMI: (<)

I i g. , 177. liead pressere ai 70" te the vertioa1. tase 4.

288

P

?- O R0lGE ERAF~

"o

- 16 0 ,

FOltCC ON lil:AD ,

~

14 0 - \

12 0 100 1 E '

H 80- '

E .

O 603 i s.

402 \

\

20 d b

00' i- i- i- i- i T-' ./>- i- i- i- i-000 001 002 003 004 005 006 007 008 009 010 011 TIME (s)

Fig. 178. Force on the head, case 4.

E. Case 5 - Upper Bourd If limits to mixing are ignored, and large premixed regions ur.dergo er.ergy equilibrat ier. before expansior. (as predicted by detonation theory), much more energetic e ve r.t s can be calculated. In case 5 such excessively pessicistic assumptions are made. The expected result, a containment challenge. is obtained. Details of case 5 are in Appendix V. Case 5 is simply a reminder that physical limits or arguments on the assumed initial conditions'must be made if the t r e a t te r.t of post-steam-explosion /expansior.s is to be consistent with ma i n t a i n i r.g c on t a i r.me r.t integrity.

289

. ._ . _ . . _, .. . _ _ . . . m . .._-m . _ _ .. - _ _ - _.,m . 4 ...

l

. Ell ORAFT 1

j i Ot+09 , ,

g g

s 1 8 Ot+08 -

2 r =.

6 Ot+08 -

t y

> x J.

e Ol+0n  :

)

1 f

2 Of+08 -

c -

C 00 0; 04 06 08 to t

~_ TiWE (5) i E

, r F

e f

1 i

1 5

1 A

i T

5 Fig. 179. Total fluid kinetic energy. case 4.

i. -

I 29C i

4

.,-....,,..-......_.-,_..,_.-.c.--__.,__.--....-- .-,. _ - - _ _ - - - - _ . - _ _ . _ _ _ . - , - . , . _ . _ . , _ . . - - _ _ , .

.... . , . , _ y. . .. . . . . . . ~ . . . ..~ . . . _ .m.. -

--..---..-_,..,.4 , , - .-_ . ......_m ...m._.-. .

+ -

?

~

h i

. h s s. r

. . = ea

?

e U H DI. .

h s .

j'j  ?

a A I

e 44 a

7 t  ?

B OCA00 , , , , , ,

g  ; , ,

y a

1 f.

f i. )

v 6 Ot+C0 - ~

s~

~

V'

~

i. - >

- +

j -

j- 4 Ot+08 --

y -

f-1 *

t. ,

t

• e 2 Cl*C8 - -

t t.

t 0

" - I I I I '

C CC 'C2 04 06 i De .to '

itut (S) .

4 i

i

+

1 1

s i -

i u 2:

i I

i 1

4 4

.'d i.

i I

j. Fig. 180. Upward fluid kinetic energy, case 4.

I i

!- -l i-4

i a 291 4

i

~

i

~

i..'

L.

ROUGH [ RAFT 1

2 c(.ce . . . ' ' '

l ' '

g g i

l' '

I

1. .

1 St 00 -

.y .

t Ot+00 - _

X . _

$ 0(+01 - _

f-t

, , , , i . I - -

0

, ,e a: 0 06 De. 'O TIME-(5) r Fig. 181. Miscellaneous fluid kinetic energy. case 4.

292

p ,

9 n.-

~

{J l

vw m V11. PROIMlllL1TY Ol' roNT4Ih\ TENT l'A1LURI:

Ker rding the possibilitt of cortastrert failure I r cr- a steam explosion.

' Prof e s ser T. o. Theof arous has stated in App. l' of the SI:RG report ' that the word "urlikelv te a r.s nothir.g." - To provide something other than additional phraseology that could be misinterpreted. this chapter presents,a formalism to quantify the word "unlikelv." k'e then offer. using this formalism. what the c or.d i t i ona l probability of cor.tainment failure might be. given core melt. The caveat that must be - made before beginning is that the conclusions will have ur. certainties. Krowledge of t he ' physics and chemist ry of severe (core melt) accidents is limited. Analysis tends to be a bootstrap process. The more you know. the more qu e s t i or.s you can ask. although hopefully you car. better delar.eate and resolve the importart ones.

The method used to address the cor.ditieral probability of cor.tainment failure by-a missile from a steam explosion is an exransion of the approach formulated by the riirch River Ilreeder Reactor study' d re for the USNRr on core disruptive acciderts. A r. initial presentatior. and use of this method was prepared for the st eam explos ion workir.g group ar.d is documci.t ed ir. Appendix C of NUREG-1116 ( see hef. 12). This chapter is a revision of that appendix because it includes a distussion of all results obtaire'd in the Los Alamos-molter-core / coolant-interaction program. A generic progression diagran was devised. and each branch was assigned a probability as defined in Table XVI.

The algorithm to assign the probabilities was as follows:

(a) As s un e that an i r.t e g ra l . accider.t-analysis computer program exists with models that describe the correct meltdowr. phenomena to the precision expected from our currer.t uncertainties.

(b) llo ge.anken for theught ) computations or this program varying the input purameters.

~. G. Theofarous and r. k. llell. "An Assessnent of CRilR Core Disruptive Accidert 1.r e r g e t i c s . " Les Alamos hational Laboratory report. NUREG KR-3224.

L A-9716 415. (W rth 1%4 ).

293

.1

c- ,

n q ===

1 \

t a w lEI TABLI: XVI lilTINITloN Of PRolMRILITY SPLIT LI:VEls 1/10 lle ha vi o r within k nowr. trends but obtainable only at the edge-of-spectrut paraceter values.

1/100 llehavior c a r.r.ot be positively excluded but outside the spectrue of reasor..

1/1000 Physically unreasonable behavior violatir.g well-known reality ar.d 'ts occurrer.cc car be argued against positively.

(c1 liased or. the results of these computat ions. a ss igr. probabili ty rar.ge s from the defir.itier.s ir. Table XVI.

A prelatirary diagram of ar. accident s progression. focused on the alpha mode of c on t a i nme r.t failure. is given i r. Fig. 152. This diagrat assuces a ZION-type PWR. The effetts of differing accider.t s e qu e r.c e s . for example, high pressure vs low pressure. are discussed where appropriate. The brarch points are anrotated on the diagrat for r e f e r e r.c e . The r e a s or.s for the indicated chences are as lollows:

1. A l e s s -o f - c oo l a r.t a c c i d e r.t (or LOTA) leading to core celt was postulated. A station blackout or pipe break are possible accident initiators. Although the details of the a c c i d e r.t 's progressten are seportant in assessing the likelihood of ar ever.tual stean explosior.. they were beyord the scope of this investigatior.

except as discussed in b r a r.c h 2. In this study we did r.o t include the possibility that a g iver. a c c ident init iator tight not result i r. c o r e me l t . or that operator arterver. tion might stop the accident's progression.

2. A rubble bed leadirg to pool formatier. seces likelv. This was The expect ed behavior ir the 1975 Reactor Safety Study. No prototypic experimertal evider.ce apparentiv exists that justifies use of alternative assumptier.s. Eventual poel 7.

t R. I:ida rc. "TMI-2 Reactor Cha rac t e r i za t ior. Preg rat. " Trans. Ac. hu6 506.

fl. p. 305 51954).

294

l ROUGH DRAT i

1DFA l

. l o

V10 CORE MELT (V10-1) o 1

, CORIUM POOL (1-VO) 1 UTHER APPROPRIATE PHENOMENOIDGY o o o -MOLTEN 00RIUM INCOHERENT,

-A LIMITED, OR NO RE EXPlhSIONS l

1 o (Vo-Voo) h  !

! BIG STEAM EXPIDSION wWER HEAD (VO-VOO) j EFFECTIVELY y3co SIGNIFICANT UPPER NO LARGE (1-VO)

i VENTS  : =

HEAD IDADING ENERGETIC AND/OR MISSILE i EFFECTIVE SWG BREAKUP o (yg_g @

j OCCURS LARGE ENERGETIC W-! '

MISSILE MISSILE FIDPPED j

('-VS) l , m @ ( .

!- CONTAINMENT FAILS d\ t 5# -

j'h ?

l 9 o L) -

/4 o CONTAINMENT INTEGRITY -e 7 .7/. ASSURED j

4

) l' i g . 162. Diagrae of ar accidert's progresstor. focused or the alpha mode of i direct c o r.t a i r.re r.t failute by a steam explosjor.

!.f j

& (; .-

295

forratier is t e r.s i <. t e r t w i t ]. the behavior observed ir the Three kii l e Islard ord the results reparted for Power ilurst Facility tests SID-ST.;'

accadert.

Sit 1-3', and the dataged fuel test Dl - 1. Iret papers by lli s a nz ; '

Slb 1-1.

ard Schtidt. ~ the German best-estimate core meltdown code. KESS-2 with ,

I k11:LSIM-3. a ppa r e nt l y s l umpi, fuel rods ir each zone once a slumping temperature l is reached, assures blockage forr.atien i r. disrupted core regiors, and then allews c oh e r e r.t downward motion into the lower plenum once failure of the core-e.uppor t irg st ruc t ure occurs. IDCOR apparently assuces intact geometry ar.d the cer. sequences of conduction limited freezing to perform calculations with the KtAP code;' irdicatir.g i r.c oh e r e r.t fuel rre l t ou t . We cannot rule out this possibility. It may be reachable with edge-of-spec t rum assumptions. Ilowe ve r .

the j ud gr.e r.t here is that IDCOR's assumption, that as fuel rods melt they drep l

'B. A. Cook. S. A. Ploger. R. E. klason, and D. M. Tow. " Severe Fu e l ' Dama g e Sc o p i r.g T e <. t Pestirradiation Exatir.ation Results." Proceedines of the Irterratioral kieetire or Licht Water Reactor Severe Accident Evaluatier.

( Ace r icar Nuc l ea r Soc i e t y. La brar.ge Pa rk. I l l i r.o i s , pp. 1.4-1 to 1.4-4 (1983).

"

• P. E. kl.i c Do r.a l d . C. L. Nalezv. a r.d R. K. kkCa rde l l . ' Severe fuel Damage Test 1-1 Results." Trans. Ar. . Nuc. Soc. 46. p. 475 ( 1984 ).

'

• Russell C. Smith ard Johr. E. Kelly. *k1ELPROG Analysis of the DF-1. Severe Core Damage Ex pe r ine r.t . " Proceedires of the Irternational ANS/l:NS Tepical Meet ire or The rma l keac t or Sa f e t y ( Ace r ican Nuclea r Society. La Grange Park. Illiness. 19b6), pp. XIX.2-1 to XIX.2-b.

R. Ilisatz. W. Scheuercarr. and 11. Unge r . ' Analysis of Fuel Rod a r.d Burdle Behavior u r.d e r Boildowr. Condition <. with SSYST. EXMEL and MELSIW Proceedires of the Interr.atieral Meetire or Licht Water Reactor Severe Attidert I s a l u a t i o r. (Americar Nuclear Society. La Grarge Pa r k . Illinois, 1953).

pp. 2. 2-1 t o 2. 2-b.

'F.Scheidt and R. Ili s a r.z . "The Core Meltdowr. Computer Program Syster Kl.SS- 2 --Re c e n t I)e ve l o pme nt and Perspectives". pp. 53-1 to 5.3-8.

'M. A. Ker. tor ard R. I:. lie r.r y, "The kRAP-PWR Severe Accident Analysis Code' pp. 7.3-1 to 7.3-b.

" M. L. Rva r . 1:d. . ' Consequences of Two Accidents St i l l Da vide NRC-and IDCOR.

Ir. side N.k.C. 6( 18 ) . 4-10 ( Se pt embe r 3. 1984 ).

'Ir.ferratior er severe fuel damage test 1-3 provided by R. J. Ilent.irger. Les Alatos hational Laberatorv Group (J-4 ( Novembe r 1984 ).

296

R:D CWT a- bl.A irto the better of the reactor ve uel. is core of a hypothesis thar a ser:! cd redel based o r. - w ier.tifs, esiderce ci corium's- behavier. l>..  ;

Ictn t(er - also cor.s stent with LMi'BR integral Lol te<ts with much shorter fuel pits at euch higher power levels. although steel does not wet u r a r. i ur dioxide while colter zirceriut r.o t only wets the fuel, but dissolves it. Aucu t.* studies dore with ANCilAR ( ;NL/NSAC ) . ' ' CORMLT ( SAI/EPRI ). '- atd MELPR(Ki iSNL/NRC).' should clarily the core teltdowr. picture ir the future. At presert, our gedar. ken prograt needs quite improbable a s sumpt s er.s . at least edge-of -spes t rur.. m te fort a pool.

3. Several other characteristics in the teltdowr. sequence cust be present besides pool formatior. for a dangerous stent explosior. to be likely. Ir. view of the urcertairties. these prerequisites can only be discussed qualitatively.

They include:

(a) Molter coriut is ir the poel. At cor.conly quot ed corium temperatures.

e.g.. 2 533 K ir. the c or t a i r.re r.t loads wo r k a r.g g r ou p PWR- l a r g e -d r y- c on t a i r.ee r.t problet tush of the urar.ium dioxide may still be solid. Dr.e reasor. speculated ler weak or r.o r.e x p l o s i ve behavior of some fuel /coolar.t interactier. (ICl) tests w i t h c or i um s ir ular.t s has-beer the presence of solid eaterial. Theoretically a 2'C. II. Bowers. R. P. Hostery. and G. R. Thetas. Status of Major Modelarg Pherotena i r. the ANL/NSAC Core Heatup Ar.d Red i s t r i but t or. (ANCHAR) Code.~

Proceedires of the Irt err a t ional Meet ire or Therr a l Nuc lear Reac t or Safet y i An e r i c a r. hus i e a r Soc i e t v. NUREG/CP-0027, 1982), pp. 1209-1221.

2; Verr.or E. Der.ry. "The CORMLT Code." Science Applicatiers. Inc., sal-361-53-PA.

14 b-i .

W. J. Catt ard J. B. Rivard. " Mode ling of Light -Wa t e r-Reactor Systets During Sesere Au iderts: MLLPR(Ki Perspective." Sardia Na t i or.a l Laboratories report.

SANllb3-1205 (1983).

' 'Cor t a i r.te r t loads workirg group, "Esticates of Ea r ly Cont a ineent loads free Core Mcit Accider.ts." U. S. Nuclear Regulatory Cottissior. d ra.f t report.

NURL(i-1079 i 1985 '..

297

IR ~

RP wd ~

stear explosior involves rapid f ragte r.t a t ier. ir a liquid-liquid systet. rot a liquid-solid e rv i r otr:e r t . A sigr.ificart qua r.t i t y of corium above the liquidus t er pe ra t ure appea r , essent ial.

(b) Sufficiert water is i r. the lower plerut. A small, pr e l a ti r.a ry explosien must r.o t blow the wa t e r awa y f rom the mixing region if this necessary cor.dition is to be met. Sete water nust vaporize frot dowr. ward heat transfer, but IlX'OR s arguter.t of exteraive downward thermal radiation that boils water away seems implausible if a solid crust is supporting a colten pool.

(c) Core barrel or molter. corium openings to the downcoter cust not lead to i nc ohe r e r.t cerium / water c e r.t a c t . Water would be expelled and any subsequent explosier. would be stall. Core barrel teltthrough is a possibility with the large potential for thertal rad ia t ier. hea t transfer from tolten corium. Also, radial heat transfer frot convec t ier. wi t hin the pool is apparently larger than downward heat trarsfer ir L41FHR rolten pool problems, and sore of the sare phenoterology should occur i r. thermal reactor cornut pools. One cor. sequence could be collapse of the core as shown in l i p. 163 fret the SNL ZIP" studv.

althcugh this ignores the cere's secondary support system. A s e c o r.d . tore 1ikeIV. possibility is shewr. in Fig. 154 free frot a more recent SNL study bv lle rta t. ' l.s ta j a t i or to a big steat explosior becomes far more difficult in these situatiers.

(d) The failure of the structure to hold up the pool must be s u f f i c i e r.t i v coherent to perrit the iritiatier of large scale liquid-liquid contact. The explesien free a small peur is arcensequential to the alpha mode of certair.mert failure u r. l e s s it can cause mere c ch e r e r.t contact.

The Los Alates tolter-core / coolant-interaction program does r.o t have a task to provide quartitative calculations on these ph e n ot e r.a . It is our understandity that c u r r e r.t M1.Ll'R(KP' results are tending toward a large "sem b molter." poel with temperature, below the telting temperature of urar.iur dioxide but with a rather coherer.t failu e of the lower grid plate. " I r. o u r o p i r. i o r. , t he WASil-1 loo

' ' k. 11. 41a r f i r ( 1:d 1 " Report of the /ior/lediar l'e i r. t Study - Vol. I. Sardia ha t ier.a l Labora t or i e s repert. NURlW CR-1410 SANDho-06171/. (1480s.

298

corclusier. that up to bo'. el the cere cav be molten before a massive pour eu urs does rot seer u r r e a ...r o ble at thi, tice. Therefore, our current gederker calculatier would satisfy the r e y t. i r e d conditions for edge-et-spectrur parameters. and perhaps u s i r.g best estimate parameters in some accidert s e q t. e r c e . A 1 to 1/10 probability range was consequently selected for t h i s, brarch. although the vaiue el unity may be a result of ignorarce of progress i r.

other research programs.

4. A discua ion of brarch 4 requires defining of a big steam explosion. A big s t e ac: explosior. is defir.ed as an explossor. possessing the potential of sustained supe rc rit s ca l pre ssures, that drive the expansion until head impact. The reasor.s for this defirition are as fellews: The critical pressure o f wa t e r is 22 MPa.

The pctential e x pa r.s i er. vo l ume ir the vessel is approximately 50 t'. As stated in Chapter 2 of the SERG repert.' c or.t a i nee r.t failure limits lie in the range of 1-3 GJ of er.ergy released ir a steam explosion. The mediar. value. 2 GJ. can he obtaired hv e x pa r a. i o r with a c or.s t a n t 25 MPa pressure differential. I r.

practice, because of slug breakup ard reduction of pressure as a consequer.ce of expansier.. higher iritial pressures are required to obtair. 2 G .I i r. a 50 t' expansion. Usir.g the words " pa t e r.t i a l of" in describir.g the sustair.ed supercritical pressures is a c o r.s e qu e r.c e of the pressure relief :.va i l a b l e through failure of the lower head. This pressure relief is a mitigation feature considered separately in branch 5. The occurrence of lower-head failure also is judged to satisfy the requirerents for a big steam e x p l o s i er.. Provided the initiatior of a c oh e r e r.t pour or molten corium/ water contact can be obtair.ed.

the stardard arguments against a large steam explosion come frot f l u i d i za t i c.:.

pherotera. If fluidizatien turr. shes the only argumer.ts that can be made to limit coarse mixirg. ard therefore big steam explesions. then a 1/10 probability would be appropriate.

k. J. Ile nr. i g e r arJ 11. E. Boyack. " A r. Integrated TR AC/MELPR(iG ~ analysis of core damage from a Severe feedwater Transient in the Oconee-1 PWR .

l' rec e ed i r ri of the Irterratioral Ahs/ ins Topical Meetire or Thermal Reu,ter s.i r e t v (Ancritar he., l e a r Sn i e t s . La Grar.ge Park. Illinois. (19hn).

pp. XX111. $- 1 te KNill.5-5.

l I

299 1

i

I FROM SNL ZIP STUDY l

STEAM FRACTURED l

. ._ : -:-:( c CORE BARREL

]

y)..m.,$ MELT %:

y I o s ' %-i i i i _ :-:il,

\

I I

,a g

WATER

' 0, h/d .'.

l FRACTURED

CORE BARREL j ,, f i

k)/

I T.

1 Nf VIGOROUS M BOILING ' BOTTOM SUPPORT PLATE RESTING ON BOTTOM HEAD I

Visualizatier of the state resultirg fror a failure of the core Tag. 183.

barrel pr mr te peretr.itier et a coherert melter r.au through the befew sore structure.

300 l

1

ROUGH 3Ri\F:

1 NUREG/CR - 3369 '

r -2 UST, SINTERED RUBBLE f 3 ,-

~

l -

. FRACTURED FUEL, ZrO2 t.s .  ! INTACT FUEL RODS ?

g ,

h

/r  ! '/ l I MELT '

j (g

I

! h ,_, ,, WATER LEVEL  ;

- l L,i 3. , , -

. .. I. l MELT FLOW /

MdTO WATER 35 5 M A_}~y M @ gjI j j k -

g6 WATER NNwe i

rig. 15-1. Melt flew arte the lowe r pl e r ur: by sideways peretre, tier ef the tore barrel.

301

m 1 r\ m ~~~

\ - \

' 9 ~

w w' Cl, .s < L li tust rat ier of a calc ulat ier us s rg edge-et -spect rue parate ters, but achieving a big steat explosion is giver in case 2 ef Thapt er VI. Start ing the prerixiry c a l c u la t : en w i t h a r e r t i te l y ru i t e n c ore a t 3 100 K is c learly edge-ol-spe c t rur :

tinatizing steam produc t ion during pretixing by the heat-transfer models empleved is also edge of spec t rue. llowe ve r . the resul t ing e xplos ion us a r.g SNL correla ted paratet ers g e r.e r a t e s sufficient pressure to satisfy the lower-head's tailure criterion in both the 0.7 s and 1.0 s explosions. Also, when lower-head failure was igrered, the e x p l o s i o r. at 1.0 s generated sufficient pressure to tair.tain the bottom of the lower plenue at supercritical pressures until head impact.

A big steam explosion should still be obtainable e ve r. if some of the edge-of-spectrut characteristics of the case 2 pretixing calculation are relaxed. The inertia of a large tass of corium is not accounted for ir. standard fluidizatior arguterts. Nevertheless, the calculation with alternative high heat trarsfer shows the potertial for corium dispersal by steam gereratier without extensive mixir.g. Additional help to decide whether extensive eixirg is possible starting with a large melten pool could be obtained by const ruct ing a to r e c er.s i s t e r.t calculational capability with three-velocity fields. Trot past e x pe r i e r.s e . ' edge-of-spectrut parameters would be o r.e expected r e q u i r ete r.t to obtain a big steat explossen from such a computational tool. Our gedanken results should be c o r.s i s t e r t with these expectatior.s.

Ar a l t e r r.a t i ve scenarie car be censidered. Recent SNL tests have showr a tendency toward early detoration. Although little is known about how to model triggerirg phenomena in the context of a integral computer program, if confirmatior of this early triggering tendency could be obtained from larger scale tests, achievetert of a 1/1m) probability would be possible. The presence of structures in the lower plenut should act as further potential triggering sites. Irdeed. cur gedar. ken prograr suggests that a likelv outcome is a series of inccherert explosiors. Although the possibility c a r.no t be ruled out, that a lit t le explosier tav be just the correct magnitude to be the p r e c u rTo r o f a big ^

explesien, rodeltr.g such a sequence with consist ert a s sumpt i ons ir a reacter cor.figuratier sects dallisult. A big steat explostor could well lie outside the spectrum of reaser.

302

1 n (\n' l

d" U V, We observe that the core probable a u i d e r.1 sequences ( TMLil' and the stall break U K'A ) tav be speculat ed to lead te a somewhat elevated achient preuure at the time el a steam exple ior. Al e w.ter ir the lower plenat will 1:Leiv be saturated. Ilo t h the existerce of high pressure and the presence of saturated water ircrease filn: boiliry stability and decrease the tendency toward early triggering. If the resulting falt boilir.g stability means that a steam explosion canr.ot be triggered. the problem is eliminated. However, because the volute of steam is reduced at high pressure, higher concentrations of fuel ar.d water can exist at lower vapor-volute fractions. If a stent explosior. is triggered, it could be more efficiert. Also, increased ini x i ng could simply result from the time delav furrished by trigger suppression. In any case.

triggering requirements are still net clearly understood, and maintaining a probability range of 1/10 to 1/100 sects reasonable at this time.

Finally, a correlatier exists between branch 3 and branch 4. The larger arv molter pool becones ard t he rmre purely liquid characteristics it exhibits. the greater the likelihood of a big steat explosion. for an upper limit discussion.

an edge-of-spectrut character should probably be associated with the brarch 3-4 c omb i na t i e r.. Multiplyirg independert numbe r *, may excessively lower arv probebility estitate.

5. A discu u ier. el brarah 5 requires definirg of significant head loading as ar averaged forte over the head of greater t ha r. -1 GN. Some minitut impulse is also required. but this impulse is small given the duration of the expected two-phase l ea d i r g i. . The value of 1 GN cotes, from the f o l l owi r.g :

(a) The recent SNL uncertainty study' suggests that the force reuuired to exceed the bolt-failure tersion is 1 170 MN. cor respondirp to a static-failure pressure of 50 MPa.

(b) The Los A l a nn e. ZIP study' using less conservative a s s ump t i er.s fourd a

~-

hydrostatic pressure capability of 100 MPa before bolt failure wou1d occur.

tc) In a computer simulatier. the Les A l a to i. /IP study also obtained suba,tartial p l a <. t i c d e l o rt.. t i e r. o f the upper head at a ur.ilore pressure load of 70 MP .

303

RMF [RAr (d) Because the thermal conditior.' of the upper head are so ur.certain and depend er the accident 5,equence. orly a rough ru-ber is pou ible. A 1 GN failure should be pooible in sore case . but without the exceu metertut presert te form an energetic large tissile. If the temperatures are significar.t ly highe r than auuted i r. either Ref. 1 or 6 further considerations are necessary as discussed in branches 6 and 7.

With this d e f i r. i t i e r. . the S141ER-Il cases support the results of the gedanken c a l c u l a t i or.s s h ow r. i r. l' i p . 152. The case run for comparison with the ZIP study is outside the spectrum of reaser for not allowing lower-head failure ar.d ventirg. The postulated exFarsier conditions in case I are edge of spectrum in lititing 1.,;al cetpliarte for the steam explosion. ar.d in assumir.g the exploster.

could occur throughout such a water-lean system. The explosior./expansier starting from fluidized pool c or.d i t i ons (case 2) exhibited effective slug breakup. Both it ard the 0.7 s explosi.r. scoping case would have been mitigated by irclusior of lower head vertirg. Case 2 "firal." although usirg ar edge-of-spectrum p r e r i x i r.g calculatiet. did use best estimate post e x p l o s t e r.

assumptions; however. the peak force was 0.51 GJ. Case 3 used pretixiry assumptions that are beyord the spectrum of r e a s e r. . but the possible influence of lower-head ver. ting car. be auuted as limited. and the expansion assutpt iors are net extreme. In case 3 ef f ec t ive slug breakup limited the peak force to

0. 75 GN. Case 4 is the plausible bounding case in this studv. The postulated pretixirg is edge of spectrut. ard the exparsson car be claimed to be edge of s pe c t rur.. The problem was deliberately made as one -d in ens i ona l as possible te avoid spurieus slug breakup. The late expar.sion of supercritical " water" provided as tr uc h mainter.arte of high driving pressures as car probably be achieved without artifically increasirg surface area as a function of time. The upper-head loading wert to 1.52 GN. but this might be expected for a edge-of-spectrue case. Case 5 u'es pretixiry assumptions that are bevond the spec t rue of rea'or, as ir case 3. Although both ef fective lowe r-head vent irg and effective slug breakup are calculated, loadiers are severe. This is a corsequence of the thermal equilibration at constant volute thIoughout the premixed region and of pessimistit 10% assumptions used to expard the premixture. These exparsior conditiors are thought to be beyond the spectrun ef reaser. The torbiratier of beverJ-reaser pretixirp a r.d exparsion assurptiers gives this case the flaver el phssical impenibility.

304

R]LGH DWT Thus, the conclusion 'f or branih 5 is a 1/1o to 1/1<He probability range. -This thuse carrot be made with high 6ertiderae besause of t he di f ficult y in def inir.g what should he ~ edge tof.-s pes t rue sl411:k ll e xpor* ion auuept ions. the urcertair t he rea l - cond i t ier.s of the upper head, and the correlation to branch 4 which deals with how much the definition of a big stean explosion has been exceeded.

6. Ilecause a quantitative ar.alysis of the path of an energetic missile through the containeert was beyond the scope of the current study, a large, energetit tissile was defir.ed as a cissle in which the vessel head is given an initial velerity of greater than 25 m/s. This is the velocity under which a missile could reach the top of the c or.t a i nee n t with only gravity as a l imi t a t i er.. The effects of_sepeding structures for a large missile are cor.sidered i r. b ra r c h 7.

The initial kinetic energy associated with 25 m/s is not large. For a 65 (HHi kg head with associated structure. 1/2 ev - 20.3 MJ.

Auurirp that t he . t wo-pha s e force on the head is greater than 1 Gl. so that failure occurs before din ipatien of the impacting eccentum. the principal concern is the mode of failure, r.o t satisfaction of soee energy requirerent.

The. charact er of head l oa d i r.g ir case 4. where edge-of-spectrue assuertiers produced a peak load of.1.52 GN. was to Iirst produce a peak load in the area of the bolts (50 MPa at 75 es h The r.. raa t e r ia l wa s concentrated at the apex of the veuel produc irg 310 MPa at 55 e s. The peak force followed soon af ter wher liquid and sclid particulate collected around the surface of the head. Iloweve r .

the sesordary peak at the bolt area was or. l y 115 MPa at 65 es. This loadiry bias ef approxicately 2.7 favors heavy shraprel being produced rather thar. the intact head becotirg a tiulle. Of course, a detailed quantitative analysis is desirable. At sore large loadir.g level, the spatial distribution of the leadtry is of little importarse.

Addittoral urrettainty is int roduced f ron the thereal conditions, fluid impact in t he dowr.c orce r , and va r ia t ions in the shape of the expandir.g two-phase " slug. "

At higher ter pe ratures, a force of leu thar 1 GN could p rod'uIe a 25 t/s eiutle. The area of t he dowra ct-e r i s such that ar approximate 400 MPa pressure at the top of the dewrcerer is required to produse a 1 GN force on the bolts, l.x, e pt for case 5. taltulated pre uures ir all sases were ecte thar a factor el 2 helow Joo N1Pa. Ir tase 4 the peak dow n t orte r preuure was -60 MPa . A 305 l

L R:UI J M i~ r e re. i r i r y corsert was eether the tir rg ci the dowrcocer loadiry would tre r J te th* cl the fluid leadirg en t he uppe r head. for the Sitti:k-I l rurs periorred se f .: r . th:s was not' t he situat ien wher pretixity was either calculated or assured to nethanist isally f orce wat e r partially up the dowrcoce r betere the e x p l e s t o r. . ' The dowr6ere r pressure pulse occurred much earlier than the direst fluid loadirg el the upper-head. Finally, although the apex leading bias sects to be a rather pereral characteristic resulting from a center pour of coriur inte the lower pler.ur, a s i g n i f i c a r.t variat ion . in loading pat terns was observed it the $1411'k- I l c a s e s. Dogratic s ta t etent s . r ega rd i r.g the bereticial effects of such a bias are unwarranted at this time, frot the above discussion. our gedankan calculation is seer to be indetereinate at this time about the path at - br a r.c h 6 that requires edge-of-spectrut parameters. A 1 - 1/10 est_imate was consequently chosen for both paths.

7. The assumptior was that once the vessel head or the vessel's head bolts failed. all or prt of it ( di f f e re r.t pieces of the failed head) would cove vertically ard strike the missile's shie ld assembiv. (The term vertically is used rather looselv here because. as discussed in the SNL uncertair.tv studv.'

differences it belt failure tires would scuse some asymmetries). The equipmer.t o r. top of the head (control rod drise eethanisms, coolir:g syst ee, etc.I would contact the tini;e shield first and absorb some energy as it crushed. The head itself would cortact the two, shield-supportir.g 36-in. wide flange I bears before it serta6ted the concrete tissile shield sections. Again, deformation of

-4 these strustur.1 rerbers and the head at the contact area would absorb s or.e crergy er plastic defercation. Connectier. of the concrete eissile shield slabs Ithree 6 It by IF-ft b) 3-ft thick slabs with a 1-ir.. thick steel plate on the bot ter s to the I beats is with relatively small bolts that er.sure th'e slabs do rot move during *eistic events. These blobs would rapidly fail and allow the two outside slabs ard the I beats to slip off the upward moverg head. The

'Ot he r c or.c e rr.s resultir.g f rom differity post uisted initial cordit io s could rot be considered within the scope of this studs. fer example, the downceeer could be coepletely full of water at the tire of the exploster. A pressure pulse trar.sr.itted directly through the liquid 6euld strair the belt s erough to reduce the required failure terse tc he assosiated with later fluid irps61 er the upper head. Sush a case requires a rvre detailed structs.al aralysis.

306

A a 4 g A. ._J-__.

RCUGH DWT eiddle slah could stav in place urt:1 the head / slab combiration contacted the t polar crare structure. The tiss:le shield strutture probably varies  !

c ens id e r.:bl y irer reactor to reastor hat .. l rn t ull l>WR a rd ilWie de s ig n s include a polar crare that will be positioned direttis,above the center of the reactor ve uel. The mass of the polar crar.e is typically two to three times the mass of the head of the reactor vessel. This structure may de f orm conside rably. but could brirg the velocity of the head down to levels that will preclude significant damage to containeert. Because of the potential stoppage of even a large, er.ergetic cissile by the polar crane. a 1 to 1/10 probability was adopted for the contair.eert-f ailure path. A realistic esticate requires knowing by how much the initial head velocity exceeds 25 t/s. The SNL uncertainty study suggests that contair. rent failure will occur if the head becomes a missile with a ve loc i t y g reat e r t har. 50 n/s. Because of poter.t ial containment failure free those s l a b e, of . the tissile shield that do not proceed straight up, this assueption is reasorable. An indica t ier. of the energy requirements ir. such a case car he rade as follows. If a 50 000 kg ' slug" has an inelastic collistor with a 65 000 kg head. the reeulting unit will have a velocity of 50 m/s if the kinetic energy is 14.1 MJ. The 50 0o Lp rust have had an energy of 330 MJ (after all losses from upper internal structure and from the head-failure techanics1 before the inelastic collisor. With case 4 produc i r.g 760 MJ in upwardly di rect ed e r.e r g y. absorpt ior. of energy in de f orma t ior. ar.d failure by raterials of utkrowr temperature are important. In any case, for high ecough [

velocities actual stoppage may prove to be outside the spectrum of reason. so a 1/100 to 1 probability was selected for this alternatiw. l l

It cerclusior, we suggest that with current uncertainites, the corditional probability of direct cortairrert failure free a steam explosion car rar.ge f ret  ;

essertially zero to 0.01. The higher turber is obtained by assutir.g that only achievement of (a ) a big stent explosien ar.d (b) sagraficant upper-head loadings justit, edge of-spectrum parameters in the gedarker code. The unlikelihood of a big stear explo ior is based primarily or coarse mixing difficulties, while SlhNi k Il cases irdicate edge-of-spectrut parameters are required ir at explosior/exparsier sequerte to generate the 1 (iN loads that are regarded as -

signifitant. lin gh confiderce is d i f l i s t. l t because of the la(L of k r.ow l e d g e [

conterr.irg teltdown sequerces, the la6L of data on large-ssale stene explosier.s.

e -

307 i

RIGHDRAFT the .nability (limited scope of the study) to perform extersive SimiER-Il pa rar et r u s.

the difficulity in quartifvirg the distortion that rav be caused by Slk N il. Il ' me d e l a u urptiers a r ,' input parareters, and the absence of a qua r.t i t a t ive a na lys i s of the actual calculated Icads. or even the temperatures er structures to which these loads should be applied. Because of these dilliculties ard the existence of some correlations among the branches of our diagram of at accident's progression. a 0.1 limit might be proposed as a more ,

conf der.t upper bound, llowe ve r , failure of containment by an i r.-ve s s e l steam explosion is alrost by definition an edge-of-spectrum phenomecom. Some events-

of our actident-progression diagrae may be more improbable than we assumed. l'o r i

l example. (atast ropic - failure of the boundary of a large. hot, molten pool of l

corium may be un r e a s o r.a bl e . If a best estimate probability is desired, the l current 0.01 bourding estimate oc the containment failure path might be expected to be reduced at one or two brarches by an additier.al order of magnitude. Care is required here. because the combinatson of three edge-of-spectrum calculatiets does not necessarily teply physically urreasonable behavior. Again, we riust not permit our probab;lities to becore too low simply through branch proliferation.

Ilow e v e r , a best est imate proability is judged to be it the range of 10'8 to lh*'. meaning that containment failure from a in-vessel steam explosior. is physically unreasonable.

l f

l 8 N

~

l l

' 308 i

- - - - - .-..-_,_m- . - . . - . _ . , - - _ -

f'fOI i

I m l mF l

VWdi v. [

Vill. Kl 51 Akt il PKlokITilA ON ST):At1 INPLUSIONS h t t. *:.

a! probler ir dissui, sing the t e r a, e q u e r t e i, of

• tear esp'-si. - i-t i.s i s in aasrtairis. TI t questior that reeds te be adJressed i,I+ i .. . t residual un ett.irts is assertable. To artroach this questior.. we irt r Jus e heir a s irpl: t ied (ors ept deliriry levels of k r.ow l e dg e . l'i ve l e ve l a, that a, c e r apparert te thei. ebserver follow.

Level 1 - The problera are not discuued seriously. This seems to have beer the poa.itier of the ruclear i r dui. t r y or c l a u 9 a c c i d e r.t e, before the "safetv-relateJ ou urrerce" at Three klile Islard. Defer.se ir-depth was used as a philor.ophy bl. t rever realls had a quant if ied or urique meaning. Ever, today. those atterdirp plcrarv ee u sori, at i.evere accident tor.ferences are likely to hear that r e a s t e r i.

are inheter.tiv sale, c a t a s t roph u au idert s cannot ha pper., and t ha t we must step t h i i, i. elf-defeatirp inves t iga t ion of the science fiction involved in meltdowr behavior.

Lesel 2 - lJrert epinion dorir.atei, the distusi. ion el <,evere acciderts. At level 2 irple q u a i, i - s t e a d y - i, t a t e rodels are introdu6ed. and experieertal data are uttritically preserted te support the cerclusters drawn. Accident s t e r.a r i o *, a r e s ors t ruc t cJ us s rg eng :r ee rirp judgr:ert ; or simple todels t, r e connected together er a terputer pregrar. as for exarple i n t he 4 HAP c ode. T h i a, is the level at which the i lirth Rive r lireedei keacter proiett presented its transition-phase aralvssi,. It is also the level at whic h t he <,t eat explosion review proup ( SI:R6 :

eperated. The resert ML study' argued that an approach to steam exploi.rors .i t this level still poueues too c:us h unt e r ta i nt y. The current Los Alaros a, t u d s s u g g e ai t s th t the a l l - t r.pe r t a r t quasi steady-state level 2 fluidizatier a r g ure r.t s lititirg r i x i r.y ir large i, t a l e systers are flawed besause el ret corsiderirg el trertio.

Level 3 - lie t a i l ed t r a r i. i e r t calculations are performed or p i e t e i, et the prehlers. These salculatiors are supported where ;ouible with data. The data are r ore quar t ita t ive t h.: r a t level 2 ard it tar. be u tied r:o r e d i r e cIl v t e the6L theoretital spesula iers or what the phenorerology *, h e u l d b e .

i Alse at lesci 4 there is core e ar appresiatior that direct extrapelatier el e x p e r i r e r t .. !

resu'ts 1. r e .. . t o r assiacrt situatiers rasbe misicadiry. Serie a spests e the 309

a RIGH DRUT hR( assessteri of CRilk core disruptive attsdert energetits ~ were perforced at lesri 3.

Le s e l As t u.: 1 detailed. irt egra t ed tode systers exist to calculate serplet e at6ident sequerses based or atterted pherocenology. These code systems presuccdly would be ascurate crough to pre 6alculate confirmatory ir-pile experiments, or relevant out-of-pile experiments. This appears to be the level that t he - hkr was at ore tiec targeting in programs to obtain a predictive capability for large-scale, pipe-break accidents in LWRs.

- Level $ - A l a s t - r u r r i r.y code (or codes) exists that is suitable to detereir.e all the sersitivities requi red ir. a risk assessment. This calculational syster would sir.ulate the dominart phenomera involved in a severe accident so the ,

required source tert d i s t r i bu t i or.s as a furctior, of time, space, and input torditions would be readily obtainable. Untertairty would be given a precise i

rathreatital defiritior.

l how it should be a p pa r e r.t to ever a casual observer that dismissal of torcerns regardirp uttertairties abeut steat e x ples s or.s or a level 1 basis is r.o t a viable lerg-tert solutior. The SI:RG - suggestcJ it its report that further researth was required bevord the level 2.argurerts and experimer.ts that existed for their review. This Les A l a n.i s study, and arv future S141f:R-l l studs er steam explessors 6arried out with or.lv turrently available knowledge. is flawed by the recessary applicatter of s i g n i f i c a r.t erparcerirg .ludgeent to the reanity el t he S1411:k-!! talc ula t s or.s.

ClearlV. a level $ uppreath is desirable for a future study. Assumirg level $

taltulattors esist, a6turate statistital results could be obtained as a l'urt t ler l of attident initiators, teattor characteristics, ard other iritial and boundary l torditiors. Urlorturately, while there have been clains of approathes that will I vield level $ results for the steam-explosion probier. all of those knowr to

, this author deserve to be labeled overly optimistit. wi shf ul "t h"t rLirt. As i

M. Ilertar points out ir Apperdix 1: of Ref. 12. "The hist ory of steae explosior i aralyses is replete with exitples of results el currer.t thinkiry wh:6h have later-beer showr to be false." Without attually knowirg the dottrar.t phe rercer .

310

RCUGH DRAFT how c a r. w e- as t ura t t iv include ther in a computer c a l c u l a t i er. l'a rare t r u esersese- are r. ub titute for trowledge, llesause the narrer o: leapiry to level $ is urkrowr. we propose in this i ha pt e r .

that level 3 r e s e a r i. h activitie, receive future emphasis. Beca.se .a partial level 3 approas h was ludged to be satisfactory on L elass 9 accidents in the Citoch kiver Hreeder kractor luensing proceedirgs, activities of this type sheuld be appropriate for reducing residual uncertainties sufficiently in the stear-explo ion area.

The tirat level 3 suggestior. is to improve modeling the ineltdown process. Table XVil lists au ide r t features that could be used as a basis f o r . Sl*1F.k-I l stear-e xplosier paratet ric s. Many of these features relate to how the init ial phases of ar. accident occur. lot example, when and how does a high-pressure l

accident sequence becore a low-pressure ' acc ident sequence f rom failure of the pr ira ry sys t er a Vhat does the stear c i rc u la t ier r:eari, while the syster t i. at i

i high pressure. to the teereratures of the upper core structure. the vessel head, and t he head *

• holddowr. bol t s Turther, reliable mechanistit calculatiora el

! the reltdewt prosesses tFat are currently under NRC investigation could be-u s e t t.1 ler other a *. p e t t i, el sourse term characterization. For exacple, the arount el urosidized /r is important for determining of the exter.t of direst

( o r.t a i nte r t heatiry following vessel lailure, i:stimates of unoxidized /r at veuel f ailure are currently made with the MARCil code. The results of the MAkcli tode have beer quest iored it application to crucial high-pressure sterarios le. ding ir the direct ior of core reeltdowr.

L The secord level 3 suggestion is for iarge scale steam-explosion e x pe r i re t t i. .

Sore Sl*11:k-ll sterirg ca lc ulat ions on proposed SNL la rge-scale t es t a,. the SI;ALS

! t e *.1 settes. are reported ir. Appendix r of the SI:R(i report ( Re f. 12 ). To save spase. the r e a u l t i, of these calculations are not reproduced here. The

' ' i sortiusiers t ror t he se s!211.R. Il ca lc ula t t ora a re a s follows:

' 't I:. lie r r y ard 11. R. Schgal. "Analvtital Irediction of ore Proteedires of t he Irternational Meetir, 6

! llea t up/Li que l a s t i et /Sl urpi r.g .

l or Licht W., t e r Re.a 1.r %evere hiidert Ivaluatier ( Ame r i c a t huslear Soiletv. La l

brarge Park. Illirios p;. $.4 1 to $.4-9 i l963 i.

I 311 l

l

y T AllLI XVil l'l: ATURI:s Til;T ( OULI) Ill USI:1) IN S14NI:R-il PARAMl:TRICS 01 STIAM I:XPLosloN/l:XPAhS10NS (1) l) f f e r iry c ore . vessel, ard tortaartett designs

-PWR

-HWR

-1)cersional and geor.etric details (2) Diller a ry re ltdown sequertes

-Accident n r. i t i a t o r

-lxtent of pool beIore the pour (upper crusts, side trusts, ard st eel me lt ed ir. f rom t he uppe r int err.al st ruc t ures )

j -1:f f ec t o f Cd , It, and Ag control red meltdowr.

-Melt toepositter. and property urcettair. ties

-Melt t erpe ra t ure

- Ar b i e r.t pressure at the tire of at interaction

- Arc u r. t of water (3) Ib i t c riry paur ing redes irto the lower plenut

-terter

-Sides (through the tore harrel)

-Prelinirarv explosier causing rassive collapse 14: Triggeriry tirc ard locat ion ( e t ret t s of structure)

($) Preparatier. velocity of explosion (6) 1:l l e c t i ve re s s of explosier Surlate area of fuel and water as a functior. of time ar.d locatior

-Malt iple e xplosion possibilit ies

-l'ragtertatier and heat-trars!er characteristics (7) Mode of lower-head failure and consequerces

-Tice ard le6atior

-Verting ard exparsson characteristics (b) Size distributior of f ragments durirg expansier

'~

(4) State of upper core arternals (los Terrerature ard gercral condition of upper head of the reactor vesseI (111 Modelirg of the t or t a i r.re r t it the upper head is proietted to bet er e a rissile, 312

hF I m

1. W w )QlI h{ }\ [

(ai SI:4LS a r.J the full-scale reactor case both have signifscantly augretted f l u i d i :.. ' i .> r ro'ertia!s relative t i- esiatirg SNL te ts (flTS). if coarse l i c y~ r r t .i t s e r h it res e s. t e r s i s e . luel rav be levitated by the s t e u rc produced betere at e xples ier occ ur s.

Ib n As stale s i, artre.iscJ beverJ the ilTS level, rore fuel is starved of water.

1. x p l e i, i v e arterattiers that car occur face increased cor.straint.

(t 1 There is a water quer.chirg effect ir. the f!TS tests. This quenching effect is reduced i r. the SI: ALS s imulat ions. and is further reduced ir full-scale calculations. Simulat ier of the querching ef f ect in calculations of l'I TS testa, deperds or. cor r e la t ier to data. SEALS could significantly influente the choice of parameters.

(d) The fluid " slug' resaltirp from a steam explosion is calculated to differ betweer the reacfor case ard the experimental <, i rr u l a t l o r s . A water *, lug is predicted for l'ITS arJ Sl;LS experiter.ts. A fuel slug is evidert ir the f u l l - *, t a l e reacter taa.c.

The critical heat flux theory of llenty anJ l'auske ' states that or.ly about 100 kg (order of tagritudes of relt could mix with water without violating their fluidizatior criterior. The e f a r oui. ' ha' Troposed that a linear diner.<,ior of to t re s i, t h e upper limit over which teari,e prerixing 6an occur that will support a stear e x p l o i i e r. . Large-scale tests wi t h 2 000 kg of melt would allow these qualitatise arguter.ts te be e x a r i r. e d . as well as significantly increase the level of confiderte i r S!kNI.K-Il e xt ra pola t ior.s to reactor scale.

Two dalierertes frem the previous SliMIR Il write-up o r. the SI:ALS p r o poi,a !

should be roted, lirst. support for SI:4LS was previously tenative, it r. e w appears t t- he a J e i, i r a b l e experimental attivits. Setord. the previous write-up ir Rel. 12 proposed that t r a n <, i e n t 5, t e a r - e x p l oi, i o r preissures of several hurdred ee ga ra i t a l i, are required to obtain t o r t a i r.ee r.1 lailure dire"tIV c f rer. ar i r -ve s is e l tean e x p l oi, i on. This i i, still the tase if lower-head failure arJ ellcotive vertiry ea ur. II .we ve r , tase 4 el ('ha pt e r VI achieved sigrilitart upptr-head loads through s ui, t a i rcJ s u re r s t i t i s a l p r e s s u r e a, that were below t he 313

ROUGH DRAFT

-$o b1Pa level requirtd ler lower-head failure, l'u r t h e r investigatior of the torsequeries o: .ase'1t. !...i.teJ loads is r e s orre nd e d.

'The pressous retor.rerdatn rs to.the SI:kb are repeated here. The confinemer.t i r.

the $1:4LS tests should he artreased, and a serious modeling program tr.a itta i red.

l' i g . lbf reprodu.cs eur suggestion or. ar axial constraint. Although 2 (n) kg of thermite does tot simulate the ir.ertia that is important in the course premining phase of a reactor stale torium pour. both SNL and Los Alamos experiments as well as S1411.k-Il t u lt ula t ions irdicate confinemer.t does increase the energetits involved in a stear explosion.

An init ial modelity program using three velocity fields and applied to analyre s SNL e x pe r ite r.t a l data was reported at Lyon.' leprovements achieved by this program are shown ir Table XVill.

TAllLI: XVill l'RI.Sl:NTLY INVI.STi(itTI:1) th1PROVl31ENTS ON1%RINti Till: CURRENTLY PROGRet.11:D Tilkl'.I l ll:LD I LUID DYNeilrs ALGORITibi AND SIWER-11 S i tt'.tr R- I l New Alcor i t hr-

1. Two velocity fields Three velotity fields

2. One flow regir e Three flow regimes alh6persed Ilows -Ilubbly flow Churr-turbuler.t flow

-Dispersed flow

3. I're u u re iteratier performed Pressure iteration accommodates with a constart preuure 1:05 ron linearities resulting from gradiert ter each cell tell compositional changes and the twe-phase / single-phase transition.

4 1: quat ion of state and the 6, t or.t e p t expanded in the~ equation pre uure iteratier tannot oI state. Irrplement ed consist er.t iv at t erroda t e gas /va por a t the in the pre u ure iteration to remove two-phase / single-phase trarsitier problems with material conservatier.

$. l.r e rp t equatter explititl\ lloth mass urd trergy equattors ftrite ds!!erer6ed with begieriry- torsist ert ly fi.ite-defterenced el -t trze-s t e r ve lot i t le s. 41a s s with a semi irap] cit u ! go r i t he.,

equattors implit it ly dillerett ed 314

ROUGH DRA3 I

L t.t n  :

F ACTUATOR CONFINEMENT  ?

BODY SAND h b

' UNATTACHED

.; /

CRUCIBLE GUIDE '

SKIRT l

.-0.203 m

( RETRACTABLE cRUcmLa K ' ,, ;;I l '

h" s

+ 0.85 m 1.6 m

111 1 1

Q \ ,

l MOUNTING +  % h l v N

, BRACKET ,/ s )r-

]  ; ._-_ ',

, s "

8 TEEL CRUCIBLE J/

N ,

s ,

SHELL RELEASE CABLES kt ^h A

l d THERMITE LEVEL

($ M T ) s { <

RIGID NTERACTION I ...

MELT LEVEL WATER REFRACTORY

\N - -

s o LINER i

! REPLACEARLE LOWER VESSEL RING SECTtON

\ .

MgO REFRACTORY LINER  % -

N I M FINAL 25 cm OF LOWER \ s -

STEEL BASE VESSEL RING AND BASE 0.78 m PLATE  !

A

t. CONCRETE BASE l

• • ~

\\\\\\\\\\\\\\\\\\\\\\\\\\\b PAD l A'- Y .s %Y

• h 31 5 rig. 165. Propmed scherns t ic $1ALS e xpe rimenta l coni ipurst ier.

R:lGH DRMT usiry er.J et -t ir e-st e r se lo i t s es.

s ii. I~t.ll J-r t sell ilis.icrers *, A ster derer sell opi nor added with t he as % i.ited wrt..ris, redut irg turerical dif l usion.

the er ly usable opt ior. .

'. Dr.]v a r.or.-equilibriur An opt iora l ( f a t.t e r ) quas i-equi libr iur vaporizatior./terderutnor redel heat-trarsler-limited vaporizat ion / condensation algorithm

6. ' Exchange rates 6ar.lorte Algorithm written to accept t ime-s t ep s Ze reduc t iors arbitrary exchange rates.

This prograr was car 6eled in 1951. A renewed three-field c od e -d e ve l o pme r.t effort was started in FY 1966. Its limited scope, duratier., and resources may be insuffitiert to reduce the steam explosion ur.certair. ties as desired. If the Sl:4LS tests are performed, some further improvemer.t and use of the three field .

.t capability should be considered. Other a6ceptable modeling prog rams are at expar.Jed MIMW (Ml:LildXi) effort; or writirg a two-dimensional TEXAS code as has beer. proposed by Youry. '

The third level-three s ug ge'. t i en is to sortirue more form of smaller $6 ale testirg. Although a contiruatior of exploratory testing where thermite is simpiv peured atto water to see what will happer. would not be very helpful (as is explained ir. Apperdix 0), well plartcJ tests could be very useful to confirn or disprove theories or stear; explosions, or to provide ideas for modelity. The ,

coupling lergth uale proposed hv Theolanous and Corradini ir, the - SERG report sects amerable to experiter.tal irvestigattor.. 1:nt e r ir.g solid sphe res into water and a llow i r.g fi tr: ' hoi l a ry to take plate in a two-Jirer.sioral geometry could he used to the6L !!ow repites ard romertum toupling. Mare information is needed or the time-deperder.1 breakup of therr ste ard surlate area formation durir.g coarse pretixity, llesides a " bu s i r.e s s as usual" continuatior of flTS tests, cont inuat ion of sete other previous attivitles may not be cost effective. Some e xamgle s are the following. Setting up additional committees of experts to merely mar.ipulate i nc orpl e t e and insulliciert data can he expected to produce simply rrore werd

4. l. Yourg. "The TIT e rede for luel-feelart I r t e r a c t i or. Aralysis". Imd.

Kei, b.. pp. l i l .* i? t e 111 .476.

316

~ .

RCUGH DRIE er.gineering, not credible issue resolution. Performing further extensive l pararetric studies w i t h Nl AMi:l?- l l r ar.v ether code ir . which the cortiusions l' reas hed de pen.! so s rit is t h .-r the tudgrert of the ir.vestigators will rot be-cor.va nc a rg. Level 2 qu.isi-steady-state arguments, which claim to eliminate this iuue. apparently have not gained wide endorsetert for the complex area of steam explosions. Their unwrstical e ndo r s erte n t will lead to loss-of credibility as other natnons centtrue to ir.vestigate this area and perhaps disprcve simple models.

Finally. we suggest that the response of the vessel head to case -t loads be evaluated i r. detail. 18 the head-f a i j ure ' mode s do not appear to suggest a c o n t a i r. rte n t threat at various " reasonable' vessel head temperatures, the l proposed 0.01 upper bour.d would have considerably ir. creased confidence. Brar.th i

b tr. Chapter Vll weuld be less i r.d e t e rti r.a t e .

Although it is obvious that nuch additional work' could be dore ir the areas ar.alysed in this studs, we cannot guarantee that perfornir.g this work will allow clesure or the steam-explosion issue. Additior.al knowledge of c:e l t d owr.

sequences, of what novel characteristits stean explosions might exhibit at large scale ar.d of how these characteristics are to be included in a future analysis seer te be prerequisites f or t ruly reducing the outstanding uncertainties, l

i*

l t

I 317

V k Alt! N1'IN A :

Till S1st.11 K-11 k14NUAL Al:US TKl:AThil:NT The r teria! T:c tre sJ it thi- Apr.erdis has heer reprirted trar th S I Nt.11.k - I l tatual. kts. 2. Othcugh the r.or e ri l a t u r e has beer retaired. the equatnor numbers have beer cade tor.s tert with the currer.t report.

The equatvers ol rotnor preserted ir. the Slkt.11:R-l l ma rua l a r e not corplete without the IOS relatiry the entrescopic dersities ard internal energies of the t empor e r.t s ir the three fields to the pressure and temperatures. lurthermore.

1:US arlorrnation is required for the phase-trar.sition models to predict the mass-trar.sfer rates. Two types of EOS are available ir Slkt.1ER-I I . First. the AEOS of Slkt1Ek-l has been extended to include centrol material and fission gas as well as fuel. steel, ard sodium. Secor.d. a tabular EOS included as an optier to provide realistic material properties that can be revised as r.ew support irg e x pe r a te r.t a l data beceme available. Currently, the tabular approach is r.on f urc t iona 1.

The Al:US used ir Sl%t.1ER- I l c o r.t a i n s simplified codels that are not sufficiertiv pert r. I te appis te all thermodvratic states. Several of the simplifications arise I rer. a u urp t i or.s made ir the developter.t of the fluid-dvr.atics rethods. Other simplifications are caused by the ar.alytic nature of the 1.0 $. This r.a t u r e a u i s t a. in urJe r s t a r.d i r p the thermodynamic phenoters occurrirp durity ar acsidert calculatior. M.i r e corplex EOS do not have this advantage.

1. kia t e r i a l Terrerature .

The t he rnodvr.a r.i t relatier betweer the terperature a r.d the erergy cf a caterial is ob t a i r.e d by i r. t e g r a t i r y the spec i f ( heat of the tra t e r i a l temperature trot zero K to the material terperature. lot the solid state, the spetslit heats a r e a u uteJ t or.s t a nt . Thua.

e sy -t ysq Tsr t - 1. .... NChr . (A-11 where --

t ygg is the c or a. t a r t - vo l ute *, p e t ifit heat for the solid phase of raterial M.

4 - Mi r i is the 10 $, r a ' e r i .. i r ur be r for s t r u s t i. r e - 1 : e l d energy comporert

r. arJ 318 l

R: A DRAF-T sy is the terperature of stru c ture-field energy coeperert r.

At the reltiry t e r r e r .. t u r e . I q. iA-1 sirids the deliritier of the s c l i d t. -

crergs.

' So l . M " ' v sM Ty , ;, ,y M = 1. .... M AT . (A-2) where eggg,y is the solidus er.e rgy f or ca t e r ia l M.

T Melt.M is the reitirg terperature of material M. ar.d STAT = 5 is the r.uther of taterials.

During the reltirg precess, the material teeperature is assuned to r eca i r.

c e r. s t a r.t as the crergy equivalert to the material heat of fusior is absorbed.

Corpletion of the rettirt process defires the liquidus erergv.

' Liq.4 " 'vsM TMcit.q + b ,y g 4 = 1. .... N'1AT . (A-3) where eliq.4 is the liquidus energy of maternal M. and hf ,y is the heat of fusior of caterial M.

The t r a r s i t i er. fret the liquid to the vapor state is sitilar to the solid-liquid trarsition. llav e ve r , the trar.sitier temperature is the saturatier temperature lor the material ard the saturation temperature i s d e pe r.d e r.t er the local pressure. I r S itt.111- i l , t h i s vapor pressure-teeper ture relatier. is

- T g, 5

1 p.y g = pq e Sat,M 1 = 1. .... M1AT - -( A-4 )

where 319

s; L R:lGH DRA!T .

i.

! pyy is the' vapor pressure fer.caternal M.

phar.dT' are irra? r.i arctets l'e r r t c r ia l M. a nd T

ht.M. 's i, t he w t u ra t ien t er pe ra t ur e not caterial M.

i' 1:qua t i er - ( 4 1 ) is ca ily ir.verted to yield the saturation temperature, i

Ty j TSat.M " -

• ( A 6) 1 F

/r ( M . '

i FvM .

l J

J

} The crergy for each liquid eaterial at the saturation temperature is i obtained usirg a cor. star.t . liquid-heat capacity until the saturatier. temperature

! reaches twe-thirds. of the critical temperature. For higher t empe ra t u r'e s . the average erthalpv betweer the liquid ard vapor phases of a caterial, neglectirg 2

pressure-volue- effects. is approximated by extrapolating linearly to .the assuced critical pe i r: t . e r e r gv. Thus. the cer.densate crergy is obtair.ed by 1

subt ract ir.g er.e-hal f t he heat of vaperizat ion f roe the average i n t e r r.a l e r.e r g y ,

extrapolated to the critical energy. The cor.deraate er.ergy is defir.ed by i

'Cen.M " ' Liq.M * 'vlM 'TSat.M - TMelt.M I

• t ,

j TSa t .M ( j TCrt.M

• i 1

' Con.M * ' Liq.M + CvLM !T 333,y - T Crt.M I + "L.M - h yg,q (A-6)

I

] TSat .M > j TCrt.M i'

t M = 1. . . . . IMAT.  :

i i

where 1

et ,y is the liquidus crergs of caterial M.

I-i 320 I

1'

_. _ . , . , _ . . . . , . . - , . . . . , - . , _ , , _ , . , . _ _ - , _ , , , , , , , . , _ _ _ . . , , _ ..--..m_--._

R: A DWT b ,.g,y is the erthalpv os vaperiz tier of raterial M.

s y[q is the 6erstar*-selutt srestlit heat for the liquid p h.: s e !.r

t.

• t 7 i .i l M .

T is the sat urat ien ter pe rat ure for raterial M.

hat.M T

Crt.M is the critital terperature for caterial M.

T is the telting tenperature ter material M.

Melt.M at,y is deleted to obtain cor.tinuity ir e Co r. . M t. t T hat.M

= 2/3 T( t t . M . a r J (yty is defir.ed se that the input critical point energy. e(rt.M- 3' obtaireJ.

The saturated varer er.ergy ther. is defined for each caterial by adding the heat of va por iza t ier. to the cordersate e r.e r g y a r.d subtracting the work ir.volved in the expansion process. or eyyp,y = erer.M + hig . M ~ F'1VM , M = 1. . . . . M1\T . ( A-7 )

Ur. l i k e the selidus ard, liqu Jus erergies. the cordersate energy ard saturated vapor crergs vary with preuure b e c t. u s e T Sat.M deperds er the pressure. I r.

addition. the heat of vapetization is a functior. ef the saturation temperature.

T s3 y (q hj,q=h[,yil - T Crt.M g f . M= . ... T . (A-6 where hj , y ard (y are i r.pu t parameters for raterial M.

Thus, at,y ard t yty are easily evaluated as 1

• 1I '

"L.M " 3 ig.M ') M * 'vl4 jTirt.M - T y ,,,,y h . and (49i 321

RC M D W

( 'c: t . k1 * ' L s c ' M * ' L . M '

(sty = 3 M - 1. . . . FN AT. (A-tor irt.M The pre u ure-volure sert tern i r. 1:q. (A-7) is obt a ined by di f f e rer.t ia t ir.g the saper pre uure-terperature r e : .i t i o r. . I:q. (A-4). a r.d ther using t he - Cla pe vror.

equatier.

dpy y h j . ,y

= -

. (A-11) dT S t .M ISat.M0'M Cersequer.tiv. we obtair Saf.M (A-12) povy = , hg ,q .

Ty The e q ua t i e r.s for the i r.t e r r.a l crergies of the liquid c orper.e r.t s are evalua,ted i r. a f a s h i o r. sirilar to these equations i r.vo l ve d i r. o b t a i r. i r.; the (erders te erergies. Thev .re

'Lr " ' Liq.M + 'vl4 iT Lr -T MeIt.M i TLr ( T Crt.M - (A*I3I 1

' Lr. " ' L i q , M + ' v LMLr( T : -jTCrt,MI + "L.M 3 hlg .y .TLr>jTCrt.M -

r = 1. . . . 4 .

where 322

O, f9 '

S S -==

- t w

M Y \> \

T

=

te y,9 h+j,,,

y = hj.,y ,1 -

TLr < Trrt.M -

(rt.M h!, y = o T Lr 3T rrt.M , a r.d T

t is the terperature of liquid-field crergy c orpo r.e r t t.

Dr.lv four liquid-field crergy c orpor.e r.t s are indicated for Eq. (A-13) because er.lv the first feur are i r. the liqu J state. The solid-fuel and solid-steel e r.e r g i e s are deterrared fret Eq. ( 4-1 ). Add i t t or. of er.ergy to the varer n r.c r e e s e s its erergy above saturatier. accordirg to e(;,y = ey,p,q + t y(;q I T,; - Tbt.M) . M = 1. . . . . PM;T , ( A l )

where t y(;q is the torstart-vs lure specific heat of vapor phase of material M.

ard T;g i i, the va rer t ercpe ra t u re f or t he va per r.i xt u re.

ler I i u t o r. g a s (M=$). the e qua t ier. f or the vapor crergy is revised to regle61 pre u ur e de pe r.dert e er the nr.terral crergy. The a u ured relatior. ship is

'(i . $ * 'C r t . 6 + ' v(iff T(; - I rt.$'

g ' ( A'I* I 323

Iq 5 -= =

RRV w l3 s This lerr is ebtaired by always evaluatirp eg,3 Irer 1

'U.5 * ' Liq.5 * 'vL5 ,TSat.$ ' I t rt .5 * "L.5 # I 'I

.5 3 t

- POV3+Lygg iTu - T s ,,,,3 1 (A-16) and presettnr.g the paraceters ir. this equatior witn the cor. star.ts

'vl5 "

'vG5 1

I "L 5 *

'Crt.5 ~ ] 'vG5 TCrt.5 ~ 'Lig,5 -

h j p. .< = 0 a r.d T33 ,3 = lo' I qua t ier: (A-14 is writter fer each raterial. I r. general. SIkNLR-Il cerputes a tist ure of vapors havirp a sirgle energy and temperature. To obtain a relattership between the vapor t empe ra t ur e ar.d vapor er.ergy. we defste

, 'G 4 " 'G.4 ~ 'v04 To . M = 1. ... ?A1AT . (A-17) l Ther we assute that the tixture's total is the sum of the er.ergies of the mixture's corpore r.t s , or l tat \T l

}

'(; " - i n '(;, y (A-15) r=1 where i 7 xy =$ is the mass f ract ion of va por cat e rial e ir. t he vapor mixt ure.

G

! Ile c a u s e the sare LOS is used for fertile-luel vapor a r.d fissile-fuel vapor. ,

their rass tract Ors have beer torbited te t e r r. x. j Subs t i t u t t or. of 1:q. LA-17) {

ir: Eq. (A-15) results ir r

I l 324  ;

> 1 I i

1 y -==

m ,q RJJU j \\

? MAT iMAT

• e;= T(; ([

g xy c y(;q ' +[ sp . c(;,y . (A-14) r:= > n=i I:quatior ( A-1 ) is inverted e.n i l s ter the structure and solid pa r t i c l e terperatures.

eg.

T r = 1. .... M si.. ard (A-20) 3e = 'v5M

~

T ty = t = 5. 6 . (A-21)

'vs4 Below 2/3 T(.rt,M. Eq. (A-13) car. be inverted for T Lt "'

I'Lr - ' Liq.M' .t =1.2.3.4 (A-22)

TLr = TMelt.M +

'vl4 Above 2 /3 T('r t .q. Eq. ( A-13 ) is solved for T ty, by a fiewton-Raphsor procedure as is explaired n r. Se6. VI. Eq ua t i or. ( A-191 ( o r.t a i n s the vapor temperature both explisitiv and implicitly (through the pressure ). Ag a i r. , Sec. VI details the algorithm fer solvirg for the vapor temperature.

11. TVO-PilASE l~LOV PRE 5sURIs.

lor two-phase flow. the local pressure is obtained from the vapor EOS.

lle ca u s e the vaporization and cor.densation models described n r. App. I are r.o t equilibrium models. the pressure is for the vapor a l or.e , rather t ha r. for the vapor in equilibrium with its liquid. Thus. the vapor pressure equatior.

1:q. (A-4), is not evaluated for the vapor pressure using the liquid tenperature.

ard instead. each vapor material is assumed to obey a modified polytropic gas relationship with pressure giver. as ptir * #(in MNTg; e= 1. .. fN \T (;-231 325

EUGH DRAFT d

where p6 -- I' 'I'e partial preuure of s a po r n.a t e r i a l r .

/(,,.; as the ticroscop a der. sits; el va po r ca t e r i a l r:. a r.d Ry is a variable gas " constant ~ for vapor caterial M.

The nes t ieportant parateter in this equatier. is R. y The algorithm chosen to evaluate R q is based o r. the observatier. that in the analysis of LMfBR hypothet ual core-disruptive accident (HCDA) situations. high vapor densities and significant dev:attens free ideal behavier are geretally calculated near saturated corditions. For example, it appear's very difficult to obtair. highly supe rhea ted sodium vapor except at low density and pressure. However, in some cases SlhNER- 11 has been erployed with vapor temperatures well above the critical temperature. For exaeple. ir. the expansior. phase of hypothesized steat explosiens, large arounts of fuel can exist at 3 000 K. but the critical temperature of water is 647.3 K.

Therefere. va r i a t i er.s ir R g are assumed to be based on saturatier.

c or.d i t i ons with a correctier. for vapor t empe ra t u re s above the critical temperature. lor T gbelow TCrt.M NM is defined by a f unctior. fy. where fy is a simple er.p i r i c a l fit based o r. the product of.the vapor der.s i t y and pressure.

Also, e a c h c orper.e r.t is assured to be fit irdependently using saturated vapor conditiers. l'er T(; chove T(.rt.M. the Rg is assured to approach the ir.finitelv dilute value conotonically. The ferculation choser. cav be suncarized by in Ky = gy ir. fg + (1 - py) 4r. Ry, . (A-24) where f y - 1. Tg(Ter* .M

• Tg>TCrt.M - -(A-2 M VM"80.M UTg-Trrt.M + "G.M). .

i 326

m, 'q' 3  % ==

l , -

aU- J \,\

I q =k, g t gT(; < Rtx;rDT R(MirVT ( r ggT g ( (cGmT g )g

'EfM"UM10 + D M20 #El # GnT g > + ay3n ( cgeTg .

1. EfM * "M11 + "M21 #Gt T}g + ay3j pggT g . (c GmT g ), < p gcT g ( (o ggT g)rrt -

I M

" N Crt.M - #GtTg > (9gmT g)Crt -

a with k g3 =( y -1 )c ygg

• Prr t.M Crt.M " I#

Crt.MTCrt.M I .

I# GtTg)Crt *#

Crt.M TCrt.4 *

(t ggTg)g = the product of the va p.,r d e r.s i t y a r.d temperature at Tsat.M

• O.95 T Crt.M '

'M1o "M20- 'M30- D M11 "M21. and ag33 are fittir.g constarts.

) 'pGM = the ratio of vaper specific heats at i r.f i n i t e dilutior for y = 'vGM toterial M.

plrt.M = the critical pressure for caterial M.

= the critical density for material M. and (Crt.M R(X; RUT a nd ag , y = SIkt.1f.R i n pu t variables.

The six f i t t i r.g c ons t a r.t s are detercited along the saturation curve with T g E T ert,M fret the followir.g conditions.

(1) l or cont inui t y. Rg=Ryg at c g7 Tg = R(x;rVT.

(2) At infinite dilution. Rg should be constart: therefore. .

d(Rg1

=o at cg,.Tg = R(x;rtiT d( g_Tg i 327

RJJ3H DRAFT (3) At TSa t .M = 0. 95 T r t .4- g N M

" N Mr "h N-M ' ' d ' I I I d " ' i E -Ih' va po r - p r e s i. u r e relationship and (FGrTi g g.

1.1 : Values for Ry tu a. t he ser.tinuou._at 'v brIti'o*

(5) l'or continuity. Rg=krrt.M *I T

( #(it G 'Or t -

(6) Wher. a c oepo n e r.t ' *. partial pressure equals the critical pressure, irfirite compreuibility ii. a uuted : therefore, B(cgeRTgg)

- o wher. AgyTg = (ognTg)Crt -

-. 6,n Aga i r.. t he e r.e exceptior. to this fittirg procedure is the fissior.-gas c orpor.er.t . whe r e R3 is assumed to be R5i, or R3 = (13 -11 c yg3 . (A-26)

The total pressure now car. be expressed as the sue of the partial pressure -

TAMT TAMT P=2 P(;r = 2 #(;ekT y (, . (A-27) n=1 n=1 The pas c o ra t a r. t . pressure. and vapor teeperature are obtained f rom Eqs. (A-10) a nd ( A-27 ) by c. Newton-kaphsen procedure outlined in Section VI.

111. SINGLI:-PIMSE FLOW STATES.

For sir.gle-phase flow. the local pressure is obtair.ed from the EOS for the liquid components. This pressure should be continuous at the trar.sition betweer.

single-phase and two-phaa.e flow. For single maternal systers. suc h cor.t i nu i t y is obtained by requiring equilibrium conditions between the liquid and the vapor c ompor.e n t s . Multimaterial syst ems a re rnere complicated and requfri additioral

a. utptiens to obtain cor.tir.uity. In SIS 11:R-ll . t he ra t e rials a re a ssur.ed t o be striscible; a r.d thus. at equilibriur. the vapor pressure for the caterial -

tixture is the .e of the vapor pressures Ier the i r.d i v i d u a l caterials. b derived in Se6. Vll. the resultirg preoure for sirgle-phaa.e flow is given bs 326

-1 . _ . . . . . . .. ... __ . - . ._ . _ _ _ _ _ _ . - . _ . . . . - . . . _ . _ - . _ _ _ _ - - .. -..__

4 e .

{;

RC AH ERUT .

E F

~

?. r

$- 2 i

~

l ot' ogg l- p pt + 1-0 .E L. (A-25) '

4 .

3- og

!- 4

~

j where

• a .

i

.g.

. hl ." 5 PvM(rI

. (A-29)

- m=>

t p

hvM(e)"'4IE!PvM(Tte), pT,t)

~

' IA~3"I~

1 4:

,I s PT .c " E Le - A g,e *55 Gra + ELt h'Mg)/cg} T (A-311

.L i-i i  !

4 .

4 i

hCLE t- 1 -

aL-(A-32) t =

j c* ~r=1 6L Le C I

i *

j. -t 4

i t

• C

( ) .aS ))

. , (A.33)

!- pt[k*c- c: a(c {t QLL a , ,

.i - ( 1 - o 5- ) "e' i* k=1 k *n.

6L 'LL

't

j. --

j and where i i -

l

.- i ogg is the liquid-16 eld-volute fractier at the a.ssured trarsition peint between sirgle- and two-phase flew, t

2 l- 329 4

u - __.~- . . . _ . . ____ _ _ _ . _ __ _ __ _ _ _ ._.-

'T!

R91y 1 i iM i c- is the square of the effective senit velocity for the 1 squid-!seld tixture.

pl is the sur el the iequid-field varer pressures.

Pv4( t i is the effective vapor pressure for liquid-field e r.e r g y correr.e r t t. .

p.,,q( Tg,m ) is the vapor pressure [Eq. (A-4)] for liquid-field e r.e r gy c oe pe r c r.t t evaluated it t he compor.e r.t temperature and is the liquid sonic velocity for component t.

cte k'i t h the decrease in liquid fuel der.sity as a function of temperature.

surgle-phase liquid pressures can setetites develop to limit the erergy prcJuctier. in a fast reacter disassetbly calculatior with S DNER- I I . These pressures tav be ur. realistically high if a constant ser.ic velocity of fuel is employed. Cor.s e q u e r.t l y . the actual decrease in the sor.ic velocity of fuel is progratted via the algorithe

. p c L1 - ta .s A (1 -T L1/Trrt.I I C.-

TL1 < TCrt.1 -

(A-34)

'L1 = 0 T t3 3T Crt.1 -

where A. 11. a nd C a r e i r.p u t constar.ts.

T is the teeperature of liquid fuel.

L1 Trrt.1 is the critical teeperature of liquid fuel, and

' L1 is the seric velocity of liquid fuel.

A c or.t i r.ueu s thermodynamic state at the t r a r s a t n o r. from single- to two-phase flow impeses furtherconditior.s en the vapor state at t he_t r ar.s i t t en.

These cor.ditiers are relatively trivial for s i rg l e -c orpo r.e n t systems. that is, the liquid and vapor conf'nent s have the same pressure ar.d temperature. lor the A131S t he single-cetrerert (onditiers are expressed as 330

m n 9 7 """

M5 I L Tg=Tt . ard ( A.35) p(; =c c R T(; = pt = pytTt ). (A-36) for cultic eponert systets. the situatior. is somewhat d i f f e r e r.t . In SI At.1E R - i l , the state of each vapor caterial is assumed to be i nd e pe nd e r.t of other va por raterials: ar.d each vapor material is assured to be in saturated equilibriut with its liquid cour.terpart. This implies that 'he vapor's internal er.ergy should be defir.ed by

?A1AT PA1AT ,j

'O * $ {# Ge va p , ki l' ._

f #

Gt' - (A-37) t=1 t=1 ler the ALOS. Eq. (A-37). Eq. (A-19). a r.d the defir.ition ofekggare used t<

elitir. ate ey,p,gg and give Tg as TA1AT PA1AT ,j Tg=_ p # (A-38) gg c yg.,$Tha t .51' - Ge' vG'.1 '

t=1 t=1 In Eq. (A-35). O gy, ar.d T5at.k1 retain arbitrary. The cor.dition of saturated equilibrier requires that the saturatior temperatures equal the respective liquid temperatures. The vapor density is choser. by assuming Pv411 t ) = Pct =# Gt gj F Tg . (A-39)

Wher. T(; is below TCrt.ht RAI is oly a function of the product p gy, Tg. and Rgg car be calculated iteratively from Eq. (A-39) for known values of p gg(n). y If at e s t ica t ed va l ue of T g is above T Cr t ki, the iteration for R gg i r.v'o Tve s ggg fret Eq. (A-251. I r. the present versier of Slkt.1ER- I I . ggg is estimated u s i r.g a Tg calculated from Eq. (A-191 using the vapor d e r.s i t i e s and interral energy fret the previous evcle. Atter Rgg i s obtaired. Eq. (4-39) is solved fer c(; . which is substituted ir.te Eq. ( A-3b i t e obt a ir Tgfree kr.cwr quartities as 331

Rbdll l

~

Li ,

h'.1 AT- 1 '

- Pvs1 r i 6 vcs1 T R

Lr)

""I T . ( A -lo )

c' = hMAT- 1 , Pvsit r ) ' vG41

. i p .

r=1 61 ,

This suttat ion doe s not include the fission-gas component, m - hWT. In single-phase cells, the - f i s s ior.-ga s 's vapor density is assumed to be zere, further, this result does not guarantee thereal equilibriut conditions at the single-phase /two-phase transition. The vapor-tixture t empe ra t u re obviously canrot equal all the liquid temperatures simultareously: hence, temperature differences will exist to drive phase transitior.s. For a given EOS and a single vapor temperature. r.o other retourse seems available; hopefully, the temperature differences will rot produce large phase-trarsitior rates-In the previous discussions, the vapor pressure. p gg. y was evaluated at the liquid temperature. Tt,. . To account for superheat, which de l a ;s the single-pha*,e/two-phase transition urtil the liquid temperature reaches the rertal boiling temperature plus the superheat temperature, the s a t u ra t i c r.

temperature and vapor pressure actually are evaluated at the liquid temperature tinus the superheat-temperature increment. When the switch from single to two phase occurs, the pfessure is cor.tinuous. However, because the saturation temperatures are lower than the liquid temperatures from the supe rhea t -t empe ra t ure i r.c r eme nt , rapid vaporization and pressurizatier of the two-phase regier. can occur.

IV. L10011) MirROSCOPlc DENSITIFS.

The use of constant ticroscopic densities does not allow the calculation of many single-phase e x pa ns i or effects, which are important in determining the neutronic behavior of reactor t ra r.s i e n t s. However, calculation of variable tieroscopic dena.ities for all c ompon e r.t s would involve extinsive code codifications to be e f fic ient. As a comprotise, temperature-dependent. liquid.

l ricroscopic densities are calculated for the first three liquid crergy s etpor.e n t s , or fuel. steel, and sodium. The formulas used are 332 l

l

_g.. , , _ _ , - . . _ . _ _ _ - . , _ . . . . - . . , -.-. -r. .,-.4-_y

% ~==

lR o

RlP W w'd=

l

,d, j

, 3

1. Le " "1r + D2r gn T + a3n Tp+aac g Tgg. TM It.M ( Tgn ( 23 TCrt.M-b,. ,

1. r*#Crt.M L II

• h1 r. It ~ + I'3e (;]. TCrt.M < Tgn sT Crt.M-(A-41)

1. e*#Crt.q L Ttn > TCrt.M -

where i n =1 -T Lt/TCrt.M '

ACrt.M = the critical der.sity, and a j n. . ayn. a3n. a an, b3n. b y r, . and b 3n are input parameters.

The corporent temperature for Eq. (A-41) are obtained from the i nt e r r.a l crergies by irverstor. of Eq. ( 4-13) a s described by Sec. VI.

V. AFOS INPUT.

The vapor st at e is determind in one of four ways, depending on the value of the i r. p u t variable l l NP. In the first tethod (IINP = -1 ) . the liquid temperatures a:e specified a:J or.e vaper-field cetpor.ent is selected through the i r.p u t variable IC(P.1PO to be i r. equilibrium with its liquid-field cour.t e r pa r t .

Because the twe fuel vapors, fertile a r.d fissile, are assumed to have the sate EOS. the variable Ir(P.1Pu refers to the first four liquid-field erergy c ompor.e n t s : fucl. steel. sodium, and control. in that order. Thus. the vaper-field terperature is set equal to the temperature of liquid caterial

]COMPO ar.d the cell pressure. vapor d e r.s i t y , ard vapor er.ergy result from the LOS. l i r.a l l y . the total c ompo n e r.t mass is consersed by subtracting the vapor mass ger.erated fror the liquid fiel.

For the AEOS. the calculatior for IINP = -1 is dor.e as follows. First. the vapor temperature ard saturatior. terperature for comporent t = ICmiPO are s determined froc _

Tg-Tsat.H = TLt (A-42' where 333

n =

RlP l l .

W W ,

j M = 4( t ) It = IO P11'01 is the caterial r.utbe r for vapor-field taterici c orpor e r.t m.

Th:-  ! ! !.a s the tell pressure te he detertired trot

-T(9 (A-43) p = ph e xp 1 i .

7 smi.n The vapor's ir.ternal energy is r.ow evaluated frot

'G " 'vSM T Melt.q + hp ,y + c yty (T5at.M - TMelt.M) + hj,,g - pavy . if T r That.M ( Crt.M *

'G * 'v5MT Melt.M + hf ,y + cyty 'Tg31,y - T(, r t . M ;

1

+ at,q + - hj,,q - pavy . if T3 ,,, q > (A~44I T rt.M g

The initial titroscopic d e r.s i t y for vapor-field material c ompo r.e r.t ICOMi>0 is obt a ired f rer-I ( ;g,.T(; ) =

p - (Gr NTMg=0 . (A-45)

This is selved usir.y a Newtor.-Raphsor. iteration where the iteratior. schete is 7x

( sg ; T(;1"*l - I cgn.T ;g 1" -

. (A-46)

'd i(ir. Tg r 334

R2 3F DE The derivative has the fort dl' l (A'47}

dia g g>

T ~ ~kM ( 1 + l }M

• where fg Tg; dR g Dy= kg d(A gnT(; )

. (A-48) i s, evaluated from Dy=0 a geTg < R(KiCUT ,

y - agyg + 7ay3n(OgnT g);

1

.D ROGCUT ( v geT g ( (pgeTg )g .

• M21 I# Ge T g) + ay3j cgnT g g )g <cgeT g (

( vg =T ( pg UM" a r.d

,7 Tg)Cr t .

Dy-o A geTg > ( c gmT g)Crt * (A-49I In the c u r r e r.t ve rs ier. of SlhMTR-Il the optior. has net be e r. p r og rarr.me d t o i r.pu t the vapor temperature above a component's critical temperature with ag,y fret Eq. (A-25) staller t h a r. a large rutber, e.g.. 10. To detertire the new c om po r.e r. t dersitles. we first cor. serve volute.

l cl', + a[n = ( 1 - og - a[) + Ah(1 -o g )/pge . (A-$o; and ther. cor. serve tass.

6{' Atir- * '[t #Lt' ' I - 'o I "

• bt i4-51 '

335

i===

g, g i

,3, J j \

where o[ is the i r i t i.: 1 va por-ve l une f r u t t i e r. ,

o[n. is the iritial volume fraction for liquid-field componer.t t.

og is the structure-field's total volute fractior.,

o[ is the input liquid-field's volute fraction, which necessarily accour.ts for the retaining liquid components.

E[e is the irput macroscopic density for liquid component ki, O is the microscopic der.sity for liquid componer.t ki, and te op is the fraction of the flow volute fraction below which sirgle-phase calculatiors are performed.

These equations car.besolvedforo[as p

(1 - og - of) t Le o p' = -

. ( A-52 )

.# Le - # tic t II

~'o IJ lle r.c e . the ini t ia l nac roscopic densi t y of va por f or corpor.er.t e is E[e=c[t g , (A-53) ar.J the initial liquid macroscopic der.sity becores E[n=E[7 - E[ , . ( A-5-1 )

If the vapor- and liquid-field compor.ert specified by ICall'O is fuel. ther the densities cust be determined for the f e r t i l e -a r.d f i s s i l e -f u e l ;va po r. The cass fractions of the fertile fuel and fissile fuel in the vapor field are assuced to be the same as the ca s <. f r a c t i er.s .r the liquid fuel. These ma u fractzor.s are j 1

336 1

l l

l

F i& -..

l V w'),"

l i

m i

• i '

• 1 xjjtj = --- ard

" L1 E, i

= IA-ff' x(2L1 * # L1 where xjjtj is the cass fraction of fertile fuel in the liquid fuel.

xjytj is the cass fraction of fissile fuel in the liquid fuel.

Ejj ard E j3 ~

are the tacroscopic densities of the liquid fertile-fuel and the liquid fissile-fuel density competents, respectivelv.

and A

LI is the eacrescopic der.sity of the liquid fuel energy t otter e r.t .

Then, the initial fuel's varer densities for fuel are calculated as E ,7 =

x /t L 1 >(ii . r = 1. 2 . (A-56) for the second rethod ( llhl' = 0) . the vaper-field pressure is i r.pu t and this pressure again is assigned to a single vapor-field corporent through the i r.pu t variable 1('(NPO. The vaper-field energy cetrener.t. ther. is calculated to have a va po r terperature equal to the saturation teeperature for the input pressure. 1 r. a dd i t i o r. , the tecperature of the liquid-lield c r.e r g y componer.t also is set te the sace saturatior temperature, thus satisfying Eq. (A-42). I'o r the a r.a l v t i s LOS. the tajor differente between this c pt ior. and the llNP = -1 path is that we now solve for the inverse of Eq. (A-431 or

(^'57' Tsat.4 " .

. ,ru e r. --

p 337 l

l

F tq '

l -

I VEE l I J where p

i is t l.c ir.put varer-1teld preuure.

l'o r the third methed (llNP = 11. the vapor state is specified by the vaper temperature and the microscopic densities input for the vapor-field cor.porerts.

Ilv u s e of the 105, both the initial vapor energy and the pressure ther. are ob t a i r.e d . lir the ALOS. the desired pressure is assumed to be obtained by usirg the gas c or.s t a r.t at infinite dilution or P(it "#n( $1 ~ } ' VG\1Tg . (A-55) where v[n is the i r. p u t tieroscopic vapor density, and h*T p=[ pg7 . (A-57) t=1 The actual ticroscopic vaper der.sity is ther det e rtinet! f ret FUt *# Gt Nh1T(; (A-60) usirp a Ne w t o n -Ra ph s o r. algorithe as was done to solve Eq. (A-45). The partial pressures of the vapor-field corporents determine the corresponding saturatier.

temperatures Tsa t 31 " .

- (A-61)

P --

/r. ( si )

PGt tror which the vorer's arterral erergy car. be evaluated by 338

• R'lq m -.-

ikJ $Ul, .

j ~

FN;T

'ti " -

• r: ' ' t i . '.1 * ' vt N T ;)

g

. (A-62) r=1 lle ca u s e the i r. p u t d e r.s i t y has be e r. c ha r.p e d . the legitimate question to ask is whether ma t. s cor.servation should be attempted. In this s i t ua t i or. , settirp 1r(Mio to ore will force suc h ar. a t t empt for two-phase cells. The equatior.s that must be selved are c0 + [ o[n = of; + [ ofn. . and (A-63) e c

• (i #G e * *Lt #L c'Il ~ 6 o 3"*i #be+*be#Lt'Il ~ "o) . (A-64) where c(3; is the input vapor-volune fraction before tass adjusteents. theo[e are the irput liquid-field c ompo r e r.t volute fractions. a r.d we consider the four er.ergy c or por.e n t s that can vaporize. Fortally. Eq. (A-63) and (A-64) c a r. be selved to ribtair.

c ;; ..1

$# ic ( 1 ~G o)  ;

0 C"I '

c('.

. (A-65) d P(;e ( 1 -o p)!

,1

( _

p t=1 Lt Ur.f o r t ur.a t e l y, a r.egative og may result if i r.s u f f i c i e r.t liquid der.sity is available. for example. in a cell that contair.s sodium vapor but r.o liquid sodium. Therefore. Eq. (A-64) is valid for these compor.ents where ,

og'; ( 0(;c - Of;n ) < o[e t te 'Il ~6

0) . (A-66)

T!' i s still allews errirs. because 339

O Fm RhIa aH &N-o[r#Li: I ~'e'  ? 'i>Mi ~ 6ji#jir-is not guaranteed. llow e ve r , an i t e r a t i er. is avoided, and this restrictier is be.'ieved adequate for the llNP = 1 opt ior.. Application of Eq. (A-65)-therefore deficesc{.andther.

E(;n = a[Pgn , t = ,1 .... 5 . ar.d E

tp. = cax (0, h[n. - Ign) , t = 1. ... 4 . (A-67) for those cells initially havirg single-phase flow, the initial pressure is detercioed ir. the above manner, but the single-phase state is calculated using the sir.gle-phase EOh. For a cell to have single-phase flow, the user must have specified sufficient structure ar.d liquid for the vapor-field volute fractior. og to be less thar. the s i ng l e - pha s e / t wo- pha i> e transition volute fraction, ogg =a p (1 -o s ). Then the vapor pressure at the liquid temperature is calculated for each r.on- pa r t i c u l a t e liquid-field e r.e rgy compor.er.t present in the cell. That is.

- Tg'g (T Lt -TSu p .k1) 4 (A-65) py.,gg n j = p39 e . t = 1. .... .

where T

Sup.M is the superheat of material M.

The total vapor pressure at the transitior. is formed from these liquid pressures 4

PL* - Pvy . (A-69) r=1 '

Equatters (A-25) and (A-32i are written in teres of dens i t v coeperer.1 s ard are 340

n R ~= -

1 i n -

l V w J j ther solved for the liquid-fie ld volute f ra c t i or. that yields the desired cell pressurr (j . a j NCLR c.i

~

o{= ole + I E ~ P) L

• IA-7"I E3 .c= 1 tj; t

where is the ir.itial liquid-field volume fraction, o{

ogg =

1 - o3 - o(;g is the liquid-field volure fractior. at the transition betweer. single- ar.d two-phase flow.

7[ is the total macroscopic density, input for the liquid-field o '.n. is the irput volure fraction of liquid-field density comp 0nert t.

c ;,n is the square of the s o r. i c velocity for liquid-field d e r.s i t y c orpor.e r.t t. a r.d p" is the initial cell pressure d e t e rti r.e d fret Eq. tA-59).

The adiusted liquid-field volute fraction is assumed to apply uniformiv to all liquid-field comperer.ts so that c"

A')n - Ejn . t = 1, .... NCLR , (A-71)

't where F~,r' is the ir.itial macroscopic der.sity of liquid-field density corpor.er.t

t. ard E3 is the i r.p u t ea c rou epi c der.sity of l iquid-fi e ld dens i t y corpor.er.t E.

341

a ,

^

R: UGH X T pL " Fv " P(; #(;RnT(; (A-87) ard=

.Tt - T(; .

(A-56) where pt is the single-phase liquid pressure.

py _is the vapor pressure, p(; is the var *r state pressure determined from the EOS, ,

p(; is the citroscopic density of the vapor-field Ry is the= ratio of the specific heats in the vapor-field T(; is the vapor-field temperature, and ,

TL is the liquid-field temperature.

In SDNER-ll the vapor pressure py is given by the two-parameter approxication py (Tsat) = p'e -I I sat .

(A-89) where p' and T' are input parareters deterr ired f rom a fit to appropriate two-phase data and T sat is the saturatior. temperature for the pressure py.

To develop the s i r.g l e - pha s e EOS. a single-component systee is considered first arJ the t r e a t te r.t then is extended to a culticoeponent systet. Equatier.

(A-67) i r.d i c a t e s the appropriate liquid pressure at the sir.gle-phase /two-phase transition. that is.

pL " Fv (TL ) wher. a(; = agg =a p ( 1 - ca g ! (A-90)

Te develep the I:OS further. liquid must he corpressed to add more liquid'te the mesh cell. thus increasirp the pressure in the tell. This path is illustrated in Fig. A.1.

342

I l

Vwl The adjusted liquid-field volute f ra c t i er. also leads to a r.e w i r. i t i a l vapor-field vt lure frattser.

f agj = 1 -a s - o[ . (A-72)

The i r. i t i a l vapor 5. tate for the sir.gle-phase cells is conputed in the car.rer discussed ir. Set. 111.

The fourth method .IINP = 2) is similar to llNP = 1; however, it permits c or.s e rva t i er of ta u to be taintair.ed with significantly less effort in two-phase cells.

Jr. this case. the pas c e r. s t a r. t . R. gg is computed directly froc pgeTg. The pressure thus is cetputed frem pgn. = (gn.k T39 g (A-73) without a r. iteratior. Because the microscopic va po r density is ur. charged i r.

two-phase cells. re nodificatior of the liquid der.sity is performed. Otherwise.

this ortior preceeds i d e r.t i c a l l y to the llNP _= 1 a pproac h.

VI. DETERA11NAT10h 0F GNPONFNT TEA 1PERATURES.

Before the excharge functiers car. be evaluated. ar. EOS call is required to evaluate t he c orpor.e r.t temperatures. The simple cases f o r t h e AEOS a r e g i ve r. i r.

Eys. ( A-20-22 ). The more complex cases involve solvir.g Eq. (A-19) for the vapor temperature a r.d solvirg Eq. (A-13) for T tn. above 2/3 TCr t . ht. Equatior. (A-19) can be rearrarged as h\iAT

'O~- *t'G . k1

"*I F( T(.' i

= -

TI '. - 0 . (A-74)

NstAT h C

_ r vG'.1 ra= 1 343

R1 'R m t

m m=,

1 1 1

OW i jMj j The Newter.-Raphser algcrith~ is l'

T";*l=T[;-

g g * (A-76)

_dT -

g where fat \T d'G.M

- ,e dl .

di r=1 0

-- -1 . (A-76) dT;g fMt1 h

- mC vGM r=1 The explicit e x p r e s s i or.s f er e[;,y a re

' i . M * ' L i q . M + ' vl4 ( TSat.M - TMelt,M I+h /g.M

~PVM S - 'vGM TSat,M , T Sa t .M I I Crt.M '

2

• i . M * ' L i q . M + ' vlM ( TSat.M ~ 3T Crt.MI + "L.M 1

1 i

+ 3}hj ,y - pavy - c GM T v Sat.M .

TSa t .M > j TCrt,M '

l l

(A-77)

Usir.g Eq. (A-12) f or the work term expresstor: pav e, .

l l

l l

344 l

RPM'jO Amk-..

d e[;,q dT s ,,,y

~

1. g.,, ,y , , Th - T gag,y ,

dT b- LM VUM ~ - - '+ '

dT(i Tg 'M'Ttrt.M - TSa t .M T Ed for Tsat.M ( rrt.4 -

de(;,y

~

dT 33,,y hjp ,y ,ty , Th-2T S3,,y , ,;

dT(; dT g MM ' W ~ ' '

Th TCrt.M - TSat.M for That.M > T Crt.M - IA~78) where dT sat.M -T s3,,q (1 + Dy 1 . and (A-79) dT g Gr ETM g, T(; <, r. , A , ,

FM T(; dry 1~R g dT g Due to the fern adopted fer Rg. Dy is the sare as Dy giver. in Eq. (A-49) if Tg<

T Cr t .M. If T(; >T Crt.M-Dy = yy iDy + Tg iln Rg, - /n f )/(Tg-TCrt.M y + "G.M IJ - IA-60) where gy. R g; fy and ag,q are defined nr Lys. (A-24-25), and Dy is giver. by

1. q . (A-49).

345

RClGH DRUT The two diffiiulties with this . iteration are that converger.ce is slow initially. ar.J if T g ,q = Tn, ,y ft.r all coeponents, then

~

dr

-=0 .

dT g

.The first problet is avoided by storing the value of the vapor temperature obtair.ed on the previous call to the EOS as an additior.al cell variable and by using this value as ar. init ial est icat e . f or Tg. The secor.d problem is avoided by defining a. minicut value of -cygy + 10 J kg K-' to (de" M/dTg ). .

This.

ir.sure s _ tha t (df/dT g ) recair.s sufficiently far froc zero to avoid numerical

-difficulties.

3 In the Tte case, wher. TLt > jTCrt.M. we define a similar function. T. by

["I'Lt * ' Liq.MI - 'vtM ( Tte - T Crt.MI - "L.M + h)gM**0 *IA~blI The derivative is evaluated as dF 'IM

• I^'82)

" h+jg ,y - c LM v

• di Lr- 'tTCrt.M - TLe) and the initial esticate of Tte.T[e,forthe steration is TCrt.M)I (A-83)

T[e-ein'Ti Crt.M - 2 K. cax (TSat.M.

hecause the iteratier. is done only if e te < c('r t .M. the iteration is only applicable wher. Tte < TCrt.M. I f a ny T" d l > TCrt.M (because of a p r i r. i t i a l I esticate perhaps). the algorithe becomes 346 '

R '

q t 5 ---

1 V w J

[1% J, l -

Tyl - f (T[r. + TCrt.M' ' IA-b4) until the Newton-Raphser. iterationcar.obtainaTyl below TCrt,M' Vll. DETAILS IN Tilf EVALUATION OE SINGLE-PH A5E L1001D STATES.

In SlkNER-ll the transition from a single- to a two-phase flow cc]culation occurs when the vapar-volute fraction exceeds a specified percer.tage of the available flow-volute f ra c t ior.. The available flow volute fractier is given by 1 -o s..where c3 is the st rut t ure-fie ld 's total volute f ra c t i or.. The fraction of t _h i s that detertires the transition between single . and two-phase flow i s, denoted by o g, that is, o '; > go II - og, g indicates a tesh cell having two-phase (A-55) flow, and og < og il - og I ir.dicates a tesh cell having single-phase (A-56) fIow.

For tesh cells with two-phase flow. the cell presssure is determined from the vapor EOS for exemple. the modified pe rfect-gas LOS if the AEOS is used. If the vapor-velure f r a c t i o r. cutoff. o g, is very stall, the pressure calculation

! rot the va por LOS car be ve ry ser.s it ive to char.ges in the vapor-volute fracti'n as the transitior be t w e e r. s i r.g l e - and two-phase flow is approached. Thus o g e q u.: 1 to zero is r.e ve r used and most often a value of 0.05 is input to the 51\MI:R-Il c ode.

A. Strele corporent s y s t e r% .

At the single-phase /two-phase t ra r.s i t i on it is necessary to have a -

c or.t i r.uou s therrodynamic 5 tate. Otherwise. a cesh cell could oscillate betweer sir.gle- and two-phase flow. Because the pressure determines the f l u i d -dynac:i c s state. pressure cor. tar.uity cust be maintair.ed. Ir. addition. there:a l cont inuity is necessary so that large temperature g ra d i e r.t s do r.o t induce high phase-transition rates. Cor.tinuity in pressure and temperature implies that the liquid ar.J vapor are in equilibrium at the t ra r s i t i er. from two phase to sirple pha e. That is. f or a single-corporert system.

347

uR'[*O' '

A--

pn Po B

1 l

I l

Pv(Tt) - - - - i- A I I

I I

I I

I I

I I

I I

I I f I Vt Vw V fig. A-1. I'r e s s u r e -vol ute relation for the sir.gle-phase EOS.

Ir this pressure-volute diagrat. point A indicates the pressure-der.sity state at the sirgle-phase /two-phase t r a r.s i t i o r.. Poir.t B is the state achieved through cerpre uior. of the liquid. That is EFL (A-91) pl

• Fv (TL) +

6vt or ETL (A-92) pt = py (T L) + --- 6 c l .

c" L The r ic roscepit dersits is used irstead of the specific volur.e because SlkNER-Il cPt uses d e r. s i t i e s . The derivative -- is the square of the liquid-field's seric cVL velotitv. ct. and is assuted constart n r. SlhMER- I l except for case of liquid fuel where it d e p e r.d s or the liquid terperature. Thus I:q. ( A-921 bet ore s 348

m q 3 r- - -

~

d=

1 l

'W , j pt = yy (Tt i + t[ t t L.e:: ~ #

L ID 33 where a t,,77 is the effective microscopic de r.s i t y of the liquid field and at is the microscopit density of the the liquid field at the single-phase /two-phase t.ra ns i t i er. Because the norr.al microscopic density used in SikNER-Il detertires

'the sargle phase /two-phase trar.sition po i r.t . P L is the r.o rta l ticroscopic dersity. Because the r.o rta l cieroscopic der.sity is generally desired to be associated with a liquid's density reported i r. the literature. for example, as in stear tables, we delar.e the liquid volute fraction, og, by

--L (1 - o g = og (A-94)

• L where Et is the tacrostopic deraity of the liquid field. Consequer.tly, at the single-phase /twe-phase trar.sition E becomes L E gg. which is giver. by i

tg = (1 - og i t (A-95) and the ratroscopic density of the liquid field is the same value as if it occupied all the volute available for fluid ir a cell. This de fir.it ier cear.s that r t,,g, is giver. by E t/: 1 - 0 3,' or the mass per unit of available fluid volure. Cor.sequentiv, we may write

- D

-# (A'96I pt = py (TL I + 'l [3 L)

• Using the d e f i r. i t i e r of the vo l ur:e fraction at the single-phase /two-phase truraitson, ole' "' D'"'"

349

m ,n -..

3" 1 o t e. = 1 -o g - og;,

=1 - c y -

6,, 1 - <. 3 (A-47;

=1 1 - o g 1 -o s} .

which gives by usir.g 01 Eq. ( 4-9-1 )

cl

• PL " Is (T L'*1 -o s .L~*Lo)I i

og L - IA'9b) er

([ ,0 e Il-os ) - o(;.

pt = py (T L) + -

, Et . (A-99) ot (1 - o s Ir additier.. the SI ANER-I l fluid-dynamics tethod r.c e d s the char.ge ir pressure with respect to r.a c r e s. c o p i c d e r.s i t y. This charge is obtair.ed fret Eq. (A-90) as A= ,

(A-100)

Bi g I ~ *$

li. k1u l 1 i c or por e r t %vster.

ler a cu l t i c orpore r.t system the corporerts are assumed to be ine.i u i b l e .

lie r.c e the resh tell pressure at the t ra r.s i t i on is the sur of the liquid-field comparert Pressures as giver. by the comper.er.t vapor pressure curves. Additier.al mass added to a resh cell agair causes compressier. of the liquid. arcreasirg the pressure. lor a giver li qu id- t s e ld c orpor.e r.t t. the pressure ir, that c orpo r.e r t is 350

R 3 m. I~ ~

m l) 1 w

Pr=_

t pyg (Ttt i + cte 6c c (A-101)

L wlfe r e p tn. is the pressure i r. l i qu id- f i e l d c ompor.e r.1 m.

cg is the square of the soric velocity for liquid-field comporert c, a r.d 6% is the charge in micrescopic der.sity from the state at the single-phase /two-phase trar.sition to the current liquid state.

To obt a i r. the single-phase pressure. all liquid-field c ompo n e r.t pressures are assumed to be equal and the sur of the macroscopic densities is assumed to be the total tac roscopic densit y f or the liquid field. First pt = [ pyg (TLL) (A-102)

L is d e f i r.e d . Ther.. by assumirg that the actual volute occupied by each compor.er.t is propo r t ier.a l 10 its volume fractror, the multicompor.ert equivalent to Eq. (A-96) is p = pte = pt + (Le [ - P tn} , fA-103)

Ln

- (1 -os:

't where the r. subscript has beer. added to indicate compor.er.t deper.Jence. Equa t i or.

( A-103 ) is rult s plied by 'Lt be ar.d sur:ed over m to yield

/

(p-pt) [ ' =[! -og P Lr ' )

• IA-l"4I h

t (Lr F The multicerperent verstor of 1:q. IA-44i is 351

t3 -- -

I)

RmW l i

'Lr r,

g = og .

(A-105i

• Lr Substitutler. of 1:q. ( ;-105 i n r.1:q. (A-10-1) gives l p = pt + - (1 -o g.) [ Etp". . ( A-lO6 )

_ 6 Lt [I - 6s n F 'Le I 1

Use of the definit ier. for a(; allows fq. ( A- lob ) t o be wr i t t e r, a s j ,ap il - a s)-n<;)# L p=pl+ IA-107I

_ 6 1,e 1 - 6s

_ ~

C C Lt which is sitilar to Lq. ( A-94 ). By usir.g the definition of agg. Eq. (A-97). i r.

I:q. (4-106) we obtair. Eq. tA-251, where the square of the effective s e r. n velocity is giver by 1=[

c'

( A-108 )

e at C Lt-In the currert version of SlhMER-II . the pressure i t e ra t i or. is perferred l assutir.g all sir.gle-phase cells r e t a i r. c or.s t a r.t t empes i t n or.. By assutir.g aLr.'"L is a c or. s t a r t . Eq. ( A- 10-1 ) can be differer.t ated to obtain f

_I.E. = .

( A- t ou .

1-a s 352

R !R R =

".~=

Yw b= ,J l

\

This is evaluated ir the wo e by ussrg the defir.itier. of aln.. Eq. (A-105), and c y, I.y. f A-97 s . ti V:cid E = og (A-110) ci L 'Lo . E, t t #Le ' Le Ur. der sete conditiors 51.4NEk'-Il will compute that a small amou r.t cf caterial at a high temperature is present nr. a tesh cell. Hence t_he sue of the liquid pressures. pt ir. Eq. (A-102). car. be due primarily to a stall amour.t of liquid. To offset this anomaly, we submit the following argutents.

The high pressure of a small a mou r.t of material will compress the other material n r. a cell. If this pressurizatier. were considered to come froc additior.al sirgie-phase tass added to the mesh cell (that is, not f rom t he vapor pressure of the sr a l l a r o u r. t of r.a t e r i a l ) , ther the c o r r e s pond : r:g der.sity i r.c r e a s e could be four.d irer ryt tTu) =

,r

• L 6E * (A-IllI L

Ce g kheit i 7,g = 1 ;I -a sj .

C"L y,( 'l ' Lr lf the tacroscopic density A Lk f the small amour.1 of material is less than 6?g ir Eq. (A-111), then the stall amour.t of caterial must vaporize to occupy the velure, d e c r e a s i r.y its c or.t r i bu t i e r. t o the total vapor pressure. Consequertiv.

for the Al OS t h e p r e a. s u r e u s e d i n 1:q. (A-lo21 for compor.ert k is the tirirut of e

353

I pvk (TLk)=p[e'I/I Lk , (A-112) the relationship for the vapor pressure, Pvk (TLk) " #Lk )e*k , (A-113)

BI L for replacement by compressed liquid, or Pvk (TLk) " { YGk + # Lk ) R M (k) Tg ]/ag , (A-114) for complete vaporization, where Ag+Etc is the total cacroscopic density of component k, Rg (g) is the gas constant for component k.

Tg is the vapor temperature, and M is the material corresponding to liquid energy component k.

Finally, we note that SI41ER-II FORTRAN treats the liquid's sonic velocity in the form of a density component rather than an energy component. The original intent of this division was to allow different sonic velocities for fertile and fissile fuel. However, because the most recent version of SletER-II forces one temperature-dependent function for the fuel's sonic velocity, the density subdivision has becoce irrelevant, and Sec. VII has not been coeplicated by these considerations. Equation (A-70) illustrates how Eq. (A-106) could look when written in terms of density cocponent s and solved for an at based on an assumed pressure, p.

APPENDIX B CFS NODE / VAPOR /WRBAEOS (EOS CORRECTION SET)

The following is a listing of the actual codifications to the 5141ER-Il pregrae library required to impleeent the equations of Sec. II-C. This correc-tion set is stored on the Los Alamos common file systec: (CFS).

354

ROLE [lAr APPENDIX 11 (Ts: NODI./VAloluWRil Al os/f l.os roRRITT10N SET )

1 'IDI:NT 11714 2*/

3*/ CORRITTION SET FOR Till: AI DS 4 */

$*/ ACVG is Tile FIRST COErl IrlENT ON Till: VAPOR llEAT CAPACITY 6*/ Br\G 15 Tile TLMPERATURE del'ENDENT COEITICIENT ON TllE VAPOR 7/ IIEAT CAPACITY 5 */ CORR lh Tile CONSTANT CORRECTION TERM FOR Tile VAPOR ENERGY IN 9 */ IN Till: DILUTE DENSITY LIMIT lti / BPAR A IS Tile RELAL\ TION CONSTANT FOR Tile IIEAT CAPACITY 11 / CPARA IS Tile MULTIPLIER TO ADJUST Tile lilGli DENSITY GAS CONSTANT 12 */ 4trVG2 1 S L(Xil CVGGM1 * ( ON!:+CPAR A ) )

13 */ DENARG lh AN INTERPOLATION VARI ABLE USED IN COMPUTING RMAT 14 * /

15

• 1 LOS.10 16

• ACVG( 5 ) .BrVG( 5 ) .CORC( 5 ) BPARA( 5 ) .CPAR A( 5 ) . ALCVG2( 5 ) . DEN ARG( 5 )

17

• CVGR( 5 ) .TSTARM 15 I B959.29 19 RE AD ( NINP 611 ) ACVGIM).BCVG(M ).CORC(M).BPARA(M).CPARA(M )

20 11' (ACVGiM) . LV. Zl:RO) ACVG(M i=CVG( M) 21 11 (llPAR4(M) . EU. ZERO) BPAR4(M)=EP10 22 "D Al.05.21 23 *D AIDS.25 24 '/

25 * / Cll ANGl:S TO MODifi Tile ITERATION FOR Till VAPOR TEMPER \TURE 26 * /

27 *I B4K3.304 25 TGMIN = EM10 29 TGMAN = TSTARM 30 *1 A105.43 31 CVGf M i= ArVG( M l+TG

• ltrVG( M i 32 *D BAK9.9 33 1 41.05.41 34 SUNS X'O = ZERO 35 SUNU)l: = ZERO 36 D ll4K3,304 37 r DRDRG= ( k(XiPM/RMAT )-(DEKlV ATIVI: OT RMAT VITil RESPECT TO ROGPM) 35

• D 114 73. 5 b . 61 39 *11 -Di T . Dill.1 40 RMAT(M) = EXP( ALRCRT(M)*GTUNC+ALCVG2(M P (ONE-GTUNC))

41 *II DEI.Di!L.1 42 RMAT( M t = DEXP( ALRCRT(M PGFUNC+ALOVG2( M)* (ONI:-GFUNC))

43 *D B473.63 44 DRDTG(4) = GTUNC'TG"( ALCVG2(M )- ALRCRT( M) )* RGTUNC(M) 45 *l 11473.7o.75 46 COEl r = ONE + (( ARG-T95k(Xi( M ))'DENARGt M))*CPARA(M) 47 11 -del . Dill. 2 48 00114 = AL(xii r0El t i 49 RM AT( 4 > - EXPi ITUNt "Gr0Nc+t ALcVGM( M i+rol:rM P ( ONE-GFUNt ) )

So 'II DI.I'. Dill 2 51 00Ll 4 - DL(Ki( col.i t i 355

F'I& g--

} ~ 1 ~

52 RM AT( M ) = DEXP( f rutir

• GTUNC+( ALCVGM( M )+COETM ) * ( ONE-GFUNC ) -)

53 DRI)RGt 4 i-G1 UNr i ARGDUM*llALF+Al.0S31( 4 P ARG )

54 1 +t t UNI.-GrVNr PCPAR A( M > >l>l:N ARGt M )* ARG 1/COEFC 55 DRDTG 4,=l:RDkut 4 wGI UNr TG'( ALcVGMt 4 i+ Col:1 M-ITUNr )

• RGTUNr( y -

56 *I B473.92 57 ESENT = SENT( M P ( ON!:-TS AT( M )

• RTSTAR( M ) )

55 D Al:05,56 59 *ll -del . Dill. 2 60 CVGR(M >= CVG(M i - AMIN 1( ZERO. ((CVL(M)-CVG(M))*TSAT(M) + ESENT 61 1 - IISTAR(M) + CORC(M))/( ABS (TG-TSAT(M))+BPARA(M)))

62

• lr DI:1. DBL.2 63 CVGk(M>= CVG(M1 - DMIN1( ZERO. ((CVL(M)-CVG(M))*TSAT(M) + ESENT 64 1 - IISTAR(M) + CORC( M))/(DABS (TG-TSAT(M))+BPARA(M)))

65 SUMDI: = SUMDI: + CVGR( M P Y(M) 66 ESTAR = SILLS (M) - CVGR(M PTSAT(M) + ESENT 67 *D AE05.91.93 65 CVGR( M )= CVG( M i 69 DTSDTG(M) = ZERO 70 ESENT = ZERO 71 SI ELS( M ) = S I Ell G( M )+ AEOSLM( si )+CV LP( M ) * ( TS AT( M )-TWO3RD

• TCR I T( M ) )

72 GO TO 447 73 445 CVS AT - CVLP(M u (TS AT(M )-TWO3RD'TCRIT(M))+AEOSLM(M) 74 SIELSCM) = SIELIV(M) - IIALF* SENT(M) + CVSAT 75 *lf -DEF, DBL.3 76 CVGK(M )= CVG(M) - AMIN 1( ZERO. (CYSAT-CVG(4)*TSAT(M)+CVL(M)*TMLT(M) 77 1 + SENT( M P (II ALF-TS AT( M P RTST AR( M ) J -H ST AR( M)+CORC( M) )

75 2 /( ABS ( TG-TSAT( M ) )+BPARM M) ) )

79 11 DEF . Dill. 3 50 CVGR( M J. CVG( M )-DMIN1( ZERO. ( CVS AT-CVG( M )* TSAT( M)+CVL(M)* TMLT(M) 51 1 + SENTI M P ( H ALT-TS AT( M P RTSTAR( M) )-ilSTAR( M)+CORC(M) )

52 2 / ( D Alls ( TG-T S AT( M ) )+BPAR M M i ) )

53 447 CONTINUE

$4 SUSU)I - SU\D)E + CVGR( M )

• Y( M )

$$ ESTAR = SIELS( M s-CVGR(M)*TSAT(M l+ESENT

$6 *1 ALOS.99 57 CVS ATF = CVLPP - SENT(M)'FACTok'RTSTAR(M) 55 lx'VGRT = 11CVG( M )

59 I T ( CVGR( M 1. LI. CVG ( M )+EM10 ) 60 TO 455 90 1 F (TG .Gl.. TS AT(M))

91 1 DCVORT = -( ( CVGR( M 1-CVG( M )+DTSDTG( M P ( CVS ATF-CVGR( M) ) )

92 2 / t TG-TS AT( M )+BPAk At M i ) 1+11CVG( M P ( ONE+TS AT( M )

93 3 / ( TG-TSAT( M )+BPAR A( M ) ) >

94 11 (TG .LT. TSATIM))

95 1 DCVGRT = ( ( CVGR ( M )-CVG( M )-DTSDTU( M ) * ( CVS ATf+CVGR ( M )-TWO' CVG( M ) ) )

96 2 /( TS AT( M )-TG+BPAR A t M ) ) )+11CVG( M P ( ONE+TSAT( M )

97 3 / ( TS AT( M )-TG+BPAR A( M i ) )

95 455 CONTINUl:

99 *D Al:05.101.106 100 SU\t)rG = SUAEx G + Y(M PIX'VGRT 101 SUMNU = SUM';U + Y( M )* ( DTSDTGt M P ( CVS ATF-CVGR(M))-DCVGRT'TSAT( M )

102 1) Al:05.105 103 RSUNU)E = ON13SUMDI 104 I COLI = ( $11 GL - SUMI P RSUAS)E 105 i = 1001:1 - TU 106 SUMNU = SUMNU + 1001:1

• SUNEX'O 356

R ~~

Rg ' G RJ y- l -

107 *I AEos.109 105 'll -del' . Dl!L .1 104 Dfl TG = AMINii -bl6.1)fl>TG >

110

• 11 I)I I .1)llt.1 111 DFDTG = DMIN1 t -D16.DFDIG )

112 *I ALOS.110 113

• lf -DEF. Dill.2 114 II (l . LT. Zl:RO I TG414X = AMIN 1( TG.TGh1AX )

115 II (f .GT. ZERO) TGMIN = A51AX1(TG.TGMIN )

116

• 11 DEI . Dill.2 117 If (l . LT. ZEROI TGi.14X - DMIN1(TG.TGMAX )

116 If (I .GT. ZERO) TGHIN = DstrX1(TG.TGMIN )

114 If (TGN .LT. TGylN .OR. TGN .GT. TGhiAX) 120 1TGN = (TUh1AX + TUMIN)*llALF 121 */

122 */ ril ANGl S IN Tile ENTRY TO COMPUTE h1ATERI AL DERIVATIVES 123 */

124 *D B473.100.103 125 *1f -DEF.DllL.1 126 RMAT(NZ) = EXP( ALRCRT(NZ PGFUNC+ALCVG2(NZ P(ONE-GTUNC))

12"

• If DEF. Dill.1 126 RA1ATI NZ ) = DEXP( ALRCRT( NZ )*GFUNC+ALCVG2( NZ)* (CNE-GFUNC))

129 *D 11473.110.114 130 00Eli = ONI: + f t ARG-T95ROG( NZ I P DEN ARGt NZ ))*CPARA( NZ )

131 *If -del. Dill.2 132 Cof fM = AL(Xit COEFC) 133 RMAT(NZl = LXP(FFUNr*GFUNr+(ALCVGM(N/)+00ETM P(ONE-GTUNC1) 134 " If Df f. Dill . 2 135 COEf4 - DLOGtCOEfr) 136 RstAT( N/ i = DEXPt FFUNr*GFUNr+( ALCVGM( NZ )+COEl M ) * (ONE-GFUNC))

137 DRDRG( N/1=GTUNr* ( ARUDUM II All +Al:0S31 ( NZ )

• ARG )

135 1 +( ( ONE-GFUNr )* rPAR A( NZ P DENARG( NZ P ARG )/COEFC 134 */

140 */ CllANGl3 JN Till: ENTRY TO DErlNE Tile VAPOR ENERGY 141 */ DURING VAPORIZATION / CONDENSATION 142 */

143 D 11473.133.136 144 *11 -DEI . Dill.1 145 RMAT(M) = EXPT ALRCRT(M)*GTUNr+ALCVG2( M P ( ONE-GFUNC) )

14o *lf DEF.Dl!L.1 147 RMAT(Mt = DI:XP( ALRrRT(M)*GFUNC+ ALCVG2(M P LON!:-GFUNC))

148 *D 11473.143.147 144 COEfr = ONI: + (( ARG-T95K0G(M) P DENARG(M ))*CPAR A(M) 150

• I f -DET . Dill. 2 151 COElM = ALOG(COEfr) 152 RstAT( M i = EXP( f fUM:'GFUNr+( AlrVGM( M )+rOEfM P (ONE-GFUNC))

153

• 1f DEF. Dill.2 154 Col:FM = DLOGt COEl C) 155 RstAT( M i = DEXP( f FUNr* GFUNr+( ALOVGM( 4 i+r0El M P ( ONE-GFUNC ) 1 156 DRDRUt M )=GTUNr* t ARGDUM*ll All + A1:0S31( M P ARG )

157 1 +t ( ON1:-Gf UNr P fl%R At 4 i

• DLNAkU( M )

• ARG )/COEl r 155 *] Al:05. 285 150 CVGktMi = ZEKo too CVG( 4 ) = ArVG Mi + lirVGiMi'TG lol *1 A105.334 357

h'I jK}\ [

I r=,

i J J US ,

162 *ll -DIT . Dill . 3 163 CVGRi M i- rVGt M i - A41N11/l:RO ( (rVL( M )-CVG( M) P TS AT( M )-ilST ARi M 164 1 + SI:NT : 4 ' i ONI.-T s ATI 4

• RTST AR( M ) i+r0Rr( M ) 3 165 2 it 4115 TG-1 MTt 4 i 1-ilP AR M M i ) i 166

• 11 I)1 I .Dilt.3 167 CVGk( M )= CVGI M ) - IA11 N1( ZERO. f (CVL( M )-CVG( M) )* TSAT(M)-ilSTAR( M )

165 1 + SENT ( M ) * ( ONI - TS AT( M i

• l:T STAR ( M ) )+CORC( M ) )

164 2 /( DAllsf TG-TSATi M ) 1+BP AR4( M ) ) )

• 170 *I AI05.335 171

• 11 -Di I . Dill. 3 172 CVGRiMs= CVGiM) - AMI N1( ZERo. ( SIELS( M ) - SI EL10( M ) + CORC(M) 173 1 + SENTI M P ( ONE-TSAT( M )

• RTSTAR( M ) )+CVL( M )

• TMLT( M )

174 2 - IISTAkt M J-rVG( M PTS AT(M1)/t ABS (TG-TSAT(M))+BPARA(M)))

175 *lE DI:1. Dill. 3 176 CVGRfMi= CVG(4) - ININ1( ZERO. ( SIELS(M 1 - SIELIO(M) + C0kC(Mi 177 1 + SI:NTI M >

• f 0NE-TSATt 4 )

• RTSTARI M) 1+CVL(M )*TMLT(M) 175 2 - IIST Ak ( M i-CVG( M )

• TS AT( M ) ) / ( DABS ( TG-TSAT( M ) )+BPAR M M ) ) )

174

• D A1:US. 341 150 SU41 = SU41 + ( SIEGS(M)-(CVGR( M)+SENT(M P RTSTAR(M))*TSAT(M))*X(M i 161 *D AE05.347.345 152 CVUllAR = X(M PCVGR(M)+ CVGBAR 153 CPGink = X( M P (CVGR(M)+RstAT( MI) + CPGBAR 154 */

165 */ INPUT. PRINTING.AND ADJUSTMENTS 156 *,

157 *1 B959.52 155 VRITE ( NOUT.652 ) ( BPAR A( 4,,M-1. 4 )

154 VRITE ( NOUT.653 ) ( CPAR A( M ) . M= 1. 41 190 VRITE ( NOUT.654 ) (ArVG(M).M-1.5) 141 VRITE ( NOUT.655 ) ( BrVG( M r . 4= 1. 5 )

192 VR1TE ( NOUT.656 ) ( CORR ( M 1,4-1. 4 )

193 *1 B954.63 194 652 FORstAT ( 36X.611 BPAR4.4( 3X.1PE12. 5 ) )

195 653 FORMAT ( 36X.611 CPAR A.4( 3X.1 Pl:12.51) 196 6b4 FORMAT ( 36X.611 ArVG.5.3X.1PE12.5)1 197 655 FORA 1AT ( 36X.611 IlrVG.5 3X. I PI:12.5))

195 656 TORM AT ( 36N. ell COR(.443X.1PL12.5;)

199 *I B954.72 200 VR1TI: (NF1LM.652) ( BP Ak u 4 ) .4=1. 41 201 VRITI: (NFIL4.653) ( rP Ak u y ) .M-1.4 )

202 VRITI: ( NI I L4. 654 ) ( ACVG( M ) . M-1. 51 203 VRITE ( Ni IL4.655 ) ( BrVG( M ) . M-1. 5 )

204 VRITl: ( N1 1 LM. 656 ) (CORR (M1.M-1,4) 205 *D B473.2 206 *II' -DEI. DBL.3 207 ALrVG2 ( M i - AL(xi( CVGGM1 ( M P ( ONI:+0P AR u 4 ) ) 1 205 'll P473. 5 209

• 1i Dl:1. DBL. 3 210 ALrVG2fM) = DlJXi( rVGG41 ( M ) * ( ON1 +rP Ak U M ) ) 1 211 'll B4L3.56.55 212 r 213 r 130 Till N111.ss ARY ITl:R ATIONS ALLOWING 10R TG GKl:ATER TIMN TCRIT 214 (~

215

• 1 1.IN1'.145 216 11 :TG .LT. TrRITt4ii Go To 574 358

217 GFUff = APAR Af M o( TG-TCRIT( M)+APARA(M))

215 *11 ' DI:f. Dill.1 214 RM ATt M i = ' l XPt ALKrRTi 41 *GI Uff+ AU'VG2( 4 ) * ( ONE-Of Uhr i) 22g

• 11 DI ! . Dill.1 221 RM AT( 4 ) = DI:XPi ALRURT( 4

• GFUhr+ALCVU2( M P (ON!:-GFUNC))

222 574 roNTINUE 223 *l LINP.209 224 11 ( TG . UT. TCR IT( M ) ) GO TO 552 225 *1 EINP.220 226 552 RGrUtr(M) = ONE/(TG-TCRIT(M)+APAR A(M))

227 GFUff = APAR4(Mi'RGTUTC(M) 225 *lf -DEF. DBL.2 229 FFUfC = AEOS10(M )+AEOS20(M)* AUXi( ZIP)+AEOS30(M)'SQRT(ZIP) 230 RNEW = EXP( FFUNC*GIUNC + ALCVGM( M)'(ONE-GFUTC))

231 'If DEF. Dill.2 232 fFUNC = AEOS10(M)+AEOS20(M)'DUXif 21P)+AEOS30(M)*DSORT(ZIP) 233 RNEW = DEXP( f fUNC*GTUNr + ALCVGM(M)* (ONE-GIUNC))

234 ZIPNEW = ZIP + (PNI-RNEW' ZIP)/

235 'If -del. DBL.1 236 1( RNEV'( ONE+GTUNr*( AEOS20(M)+11ALF* AEOS30(M)* SQRT(21 P) 237 'If DEF. Dill.1 235 1( RNI:W-(ONE+GfUff *( AEOS20(M)+11Alf

• AEOS30(M)'DSVRT(ZIP) 239 2 )))

240 'If -l>EF. DBL.1 241 If ( AllSf (ZIPNI:W-ZIP)/ZIPNEW) .GT. EM10) GO TO 550 242 *1f DEF. DBL.1 243 IF (DABS ((ZIPNEW-ZIP)/ZIPNEW) .GT. EM10) GO TO SSO 244 GO TO Sg 245 'l EINP.224 246 i L ( TG . GT. TCRIT( M) ) 60 TO 557 247 *l EINP.235 245 557 RGTUNr(M) = ONE/(TG-TCRIT(M s+APAR A(M ) )

249 GTUhr - APARA(M PRGIUNC(M) 250

• 11 -DEI . Dill.4 251 COElf = ONE + AMAX1(ZERO.((ZIP-T95ROG(M) PDENARG(M))*CPARA(M))

252 001:1M = AuXia COEFr) 253 fl Uf.r = AEOS11( M )+ AEOS21(M)* SORT ( ZIP )+ AEOS31( M)* ZIP 254 RNI:V = EXP( ffUNc'GIUNC+( ALCVGM(MidOEfM P(ONE-GTUNC))

255

• 11 DI:1. Dill.4 25o 001:Ir = ONE + DktAX1(ZERO.((ZIP-T95R(KF M))*DENARG(M))'CPARA(M))

257 C01:14 - DL(Xif COEFC) 255 fl Uhr - Al:0511(M)+AEOS21(M)*DSQRT(ZIP l+4EOS31(M1' ZIP 259 RNFW = DI:XP( fl UFC'GFUNC+( ALCVGM( M)+00[fM P (ONE-GfUNC))

260 ZIPixn1 -

261

• I f -del' Dill.1 262 I f RNI:W'(ON1:+GFUff *( AEOS21(M Pil Alf' SVRT( /IP)+ALOS31(M)' ZIP 263 'If DEF. Dill.1 264 1( RNI:W' ( ONL+GfUNr' ( AEOS21 ( M )

• ll4LL

• DSVRT( Zl P )+ALOS31 ( M )

• Z I P 265 2 ))1 266 11 (COEFC .GT. UNI:)

267 121 Pix A1 = ZIPik A1 + ((UNI:-GI UNr PCPARM MPDENARG(M)* ZIP)/COEfr 265 ZI PNI:V = /IP + ( PNI - RNI:W' ZI P )//I Pixn1 264 *11 Dl:1. Dill.1 270 11 ( Alls ( ( /I PNfW-il P )/ lI PN! W ) .GT. EM10) GO TO 555 271 *11 I)E L . DBL.1 359

P'lRt ~ I e==

i Vwd ~

272 II ( DAlist ( ZI Phl:V-ZI P >/Z1PNI k') .GT. EM101 GO To $55 273 II (llPNrk .LT. T95ktki M)i f Go TO 55o 2?4 Go To bu,,

275 1 1:lNP.1b 276 DIMENSloh 1:sTU 41.Rui UNn 5 2.DRDTGf 51 2'7 *1 LINI.53n 275 ESTM 4 > = Sil.LSr 4 6 + SENTtM P(ONE-TSAT(M)*RTSTAR(M')

2'4 1 -

IISTARf M i - SIEllV( M 1 + CVL(M PTMLT( M) + CORR (M) 250 5900 CONTINUI -

251 C 252 0 ITER ATE To CALCULATI: A CONSISTENT VAPOR TEMPERATUR!:.

253 C 254 NITNo = 0 255 TGN = TG 256 420 CONTINUl:

257 NITho - NITho + 1 255 IF (NITho .GT. 50) Go To 759 259 SUMNU = ZERO 290 SUsE)l: = ZERO 241 SU4DR1 = Zl:KO 292 SUMDR2 = Zl:ko 293 SUMDR3 = ZERO 294 SU4DK4 = ZEko 295 Th TGN 296 Do 460 M = 1.NrLl:42 297 ARG = P ARPLt M)/RMAT( 4) 295 CVG(Mi - ACVGtMi + BrVG:M PTG 299 11 (ARG .LT ROGrVT*LM3) Go TO 435 300 11 (TG .GT. TCRITIM), RG F UNC( M l =0NE / ( TG -TCR I T( M )+ A PAR A ( M ) )

301 IF ( ARG . LT. T95R(kif 4 ) ) Go TO 430 302 IT ( ARG . LE. TCRIT( M P ROGCRT(M i ) GO TO 425 303 I T ( TG . GT. TCR I T( M ) ) Go TO 423 304 Rst;Tt 4 )=KCRITi M )

305 DRDTG(M1 = ZEKO 306 Go TO 440 307 423 GrUNC = AP ARU M)* RGTUNC(M )

305 'll -DEF. DBL.1 304 RMAT(M i = EXP( ALRCRT(M)*GTUNr + ALCVG2( 4 P (ONE-GFUNC ) 1 310

• ll DET. Dill.1 311 kMAT(M i = DEXP( ALRCRT( M PGIUNr + AtcVG2(4)'(ONF-GI UNC ))

312 DRDTGr41 - GTUNr TG- AtrVG2( M i- ALRCRT( M ) P RGFUNC( M )

313 Go TO 44o 314 *lf del' Dill.3 315 125 AkGDUM = A1:oS21(M)*DSQRT( ARG) 316 Il t TG.GT.TrklT(M)) GO To -125 317 R4 AT( M i =DI:XPt AEOS11(M)+ARGDUM+ Al:oS31(M P ARG )

315

• If -DEI . Dill.3 319 425 AKGDUM = Al:oS21(M)'SQRT( ARG )

320 11 ( TG . GT. TrR I T( M I ) Go TO 425 321 RstAT( 4 - EXPi A1:0511(M#+AkUDUM+AI.os31:M) ARG) 322 DRDTGtMi=ARGDUM IIALI + AIDS 311 M P ARG 323 Go TO 44o 324 425 Gl Uhr - APAl:M M P KGl'UNrt 4 !

325 fl UNr = Al.oSil t M + ARGl>U4 + 41.0531( M )

• Akb 326

• 11' -DI:1. Dill. 2 360

R F q -

1 i [d w (

ColTM = AUxb roElr) 327 325 RM AT( M i = 1:XPi l l VNU* GFUNt +1 ALrvat 4 )+cOEfM ) * (ONE-Gf 0Nr) )

326

• ll 1)lT.Dl!L. 2 330 rol 1 M = litt kii 01 I r i 331 R4 AT( 4 i - DI:NP( l i UNr* GI UNr+( AtrVG4( M j+r01:14 ) * ( ONE-GFUNr 1 )

332 DRDRGt M >=UI UNr* ( ARGI)U4*ll ALI + A1.0531( M l' ARG )

333 1 +f ( UNE-GFUNr 1 *CPAR 4( 4 )

• DI:N ARGl M l' ARG ) /00EfC 334 DRI)TG( k' >=I ARDRG t 4 )+0F Utr

• TO' ( ALCVU4( M >+COEFM-TFUNr )

• RGFUNC( M )

335 GO TO 440 336 ' ll DI.F.DIIL. 3 337 430 ARGDU4 = AEOS30(M)*DSVRTL AkG) 335 If(TO.GT.TCRIT(MI) G0 TO 432 339 Rst4T( M ) =DEXPl AEOSlo( M)+ALOS20(M)*DLtXit ARG )+ARGDUM) 340 *1f -DEF.Dl3L.3 341 430 ARGDU4 - ALOS30(M)* SQRT( ARG) 342 I F ( TG . GT. TCRI T( M ) ) GO TO 432 343 RAtATf Mi - EXP( AEOS10(M)+AEOS20( M)* AL(Xi( ARG 1+ ARGDUM i 344 DRDTG(M) = ALOS20( M ) + llAlf*ARGDUM 345 GO TO 440 346 432 GFUNr - APAR A(M)* RGFUNC( M) 347 *1f -DEf DBL.2 345 fFUhr = ALOSlof M1+ AI:0S20( M 1* ALOG( ARG ) + APGDUM 349 RMAT( M ) = EXP( f f UNr

• GI VIT + ALCVG4( M ) * ( ONL -GFUNC ) )

350 *IT DEI. DBL.2 351 f f UNr = A1:0510( 4 >+ Al 0520( M 1 *') LOG ( ARG I + ARGDUM 352 RA1AT( M ) = DI:Xpt f f 0Nr*GFUNr + AlrVG4t M)* (ONE-GTUNC))

353 DRDRUC M i = UfUIT* ( A1.0S20(M )+11 ALI

• ARGDUM) 354 DRl1TG(Mi - DRDRG:M) + GI UNr* TG* ( ALCVGMC M)-FFUTJC )* RGfUNrt 4 )

355 GO TO 440 356 435 R4AT(4 =CVUG41t4) 357 DRDTU(Mi = ZLko 355 Ga TO 46" 359 440 CONTINUE 36o DRDTGt M ) = DRDTGtM)/TG 361

• 1 f -DEf . Dill. 2 362 CVGk( M ) = CVG(M) - AMI N1 ( Zf RO. ( LST A( M )-CVG( M )

• TS AT( M ) )

363 1 / t Alls ( TG-TS AT( M i )+BPAR At M ) ) )

364 *If DEF. DBL.2 365 CVGk( M i - CVG( M ) - Dyl N1 ( ZERO. ( EST Af M i-CVG( M )

• TS AT(M ) )

366 1 / ( D Alls ( TG-T S AT( M ) )+BPARA( 4 ) ) )

367 FACTOR = ARG 365 I art 02 = TACTOR

• TS AT(M )

369 DrVGRT = BrVG(M) 370 If tCVOR(M) . LI:. CVG(M)+EM10) GO TO 455 371 I f ( TG .01.. TS AT(M))

372 11x TURT = lx'VGRT+(CVG( M )-CVGR( M )+BrVG( M )* TS AT( M I )

373 2 /( TG-TS AT( M )+11 PAR 4( M ) :

374 If (TG .LT. TSAT(M))

375 1lX:VGRT=lx TORT +( rVOR( M I-rVG( M )+BrVG( M i TS AT( 4 ) :

376 2 /(TSAT(4 -TG+11 PAR;tyii 377 455 CONTINUI 375 SU4NU = SU4NU + lAUT02*rVGRt4i 379 $UAD)! = SUMDL + l ArTOR rvuki q i 350 SUAD)R1 = SUMDK1 + 1 art 02*lxTURT 351 SUMDR2 = SU@R2 + l art 02 *r\GR( M i'DRDTU( M )

361

R:tGH DRUT 352 SULS)R3 = SUst)R3 + F ACTOR'IrVCRT 353 $U49)R4 = SUst)R4 + I ArTOR'rVGkf M )*DRDTU( M) 354- avio CONTINUl 365 RSUSEll: - Ohl:, SUst>l' 36t> 1 = SUMNU' kSUNU)l. - TG 357 Df tiTU = ( SUWR1 -SU@R2-( 1 +TG )' ( SUMDR3-SU@R4 ))* RSUMDE - Ohl:

355 TGN = TG - f/DFDTG 354 ' I f -DEF . Dill.1 390 I F ( ABSt ( TGN-TG )/TG ) .GT. cal:05) GO TO 420 391 *lf DET. DBL.1 392 II' ( DABSI ( TUN-TG ) /TG ) . GT. CALOS ) GO TO 420 l 393 (H) TO 740 394 'D li4K3.71.74 l 395 't> LINP.537.573 l

396 'D 1:lNP. 575.552 '

397 *D li4K3.75.77 396 'D EINP.553 399 TG = TGN 400 'D EINP.557.555

.401 M=lLOSLE(N) l 402 Rt XiTN( N ) = PARPL(M)/RstAT(M) l 403 If ( PAkPL(N) . EU. ZERO) GO TO 5950 404 /

405 */ Tills PAKT IS TO CORRECT Tile INPUT INITI ALIZATION WilEN 4o6 */ ON1: 00MPONI.NT EXISTS AND TO EVUALS TSAT l 407 */

l Jos 'l) B4K3.43

404 C ll AS NOW llEl:N CONSISTENTLY PROGRAWED l 410 'D 1:lNP.93 l 411 *!) 11303.26 i 412 Gr0NC - APARAf 4)/(TG-TCRIT(M)+APARA(M))

413

• If -DEF. Dill.1 414 RNEW = 1:XP( AlkCRT(M)*GrUtr+ALCVG2( M)* (ONE-GIUNC))

415 'If del . Dill.1 416 RNEW = DEXPt ALRrRT( H)*Gl'UNr+ALCVG2(M P (ONE-GFUNC) 1 417 ZIPN!:V = PN!/RNEW 415 'e 419 */ FINALLY WE MUST CORRI:CT TIIE TRANSIENT 42o */ Tills IS Till. SINGLE Pil ASI. CALCULATION IN LXFLUD l

'421 */

422 *D 1:XfL.547 423 'l 1:XI L. 56o 424 CVU( M ) = ACVG( M ) + liCVGt M )

• T(i 425 ESTAR = Sil.LSt M) + SENT( M P (ONE-TS AT(M i'RTSTAR(M)1-rVGIM P TSAT(M) 426 ' I f -del' . Dill. 2 l 427 CVGR(M)= CVG(M) -

AMI N1 ( Zl:RO. ( -IIST ARI M )- S I EL I Q( M )+CVLI M )' TMLT( M )

l 426 1 +(ORC (M) + ESTAR)/tTG - TSATtM) + BPARAIM))i l 424 'If der. DBL.2

43o CVGR(M)= CVGIMI - INI N1 ( ZERO. < IIST AR( M I- SI EL IVi M )+CVL( M ) ' TMLT( M )

l 431 1 + Cokr( M t + ESTAR )/( TG - TS AT( M ' + BPAR A(M)))

l 432 ZIP = ( PARPLt N i CVGRt M i ) 'RstAT M i 433 'I) 114 73.13.1h 434

• 11' der. Dill.1 435 RM ATI M i = INI'( ALRrkT( M PG' Ut.r+ ALrVG2< M P ( ONI:-GrUNr # )

43o 'I f DI:I . Dill,1 l 362 ,

l l

9O l II d f I{

ft 437 ' RMAT( M i - DI:XP( ALkcRT( M PGI Utr+ 4LrVG2( M P (ONE-GI UNC) )

435 'D 11473.33 439 2 )>>

440 'l 11473.35 441

• I f -DI.F. Dl!L. 2 442 001:Ir = ONE + AMAXI(Zl:RO.(( ZIP-T95R(Ki(M s)'DENARG(M))*CPARA(MI)

! 443 COEl4 = AL(xit 00Elr 1 444 'If der. Dill.2 445 COLI C = ON!: + DM AX1 ( ZERO. ( ( Z I P-T95R(Kit M ) )

• DEN 4RG( M ) )

• CPAR A( M) )

446 001 l M = DLtxit 001.1 r )

447 'D 11473. 41 445 RNEW = EXPi fl UTC'GFUNC + ( ALCVGM(M)+COETM)*(ONE-GFUNC))

449 'D B473.44 450 kN!:V = DEXPt ffUNC'GFUNC + ( ALCVGM(M)+COETM)*(ONI.-GfUNC))

451 'D 11473. 45 452 ZIPIXB1 -

453 *D B473.So 454 2 )))

455 IT (COETC .GT. ONE )

456 121PDOM = ZIPIXA1 + ((ONL-GFUNC PCPAR A(M)*DENARG(M)* 21P l/COETC 457 ZIPNEW = ZIP + ( PARPL(N) - RNEV ZIP)/ZIPDOM 456 IPATil - 2 459 *I EXTL.990 400 IPATil = 1 461 'l EXFL.992 462 11 ( IPATil . LV. 2 . AND. PARPL(N)/RNI:V .LT. T95ROG(M)) GO TO 760 463 'D EXf t.1(H)1.1(wn7 464 'D EXIL.1010.1912 465 CVG( M > = ACVG( M i + 11rVG(M l'TG 466 i ST AR = SILLS (M i + SENT(M i'(ONE-TSAT(M)* RTSTAR(M))-CVG(M)*TS ATf M 6 467

• 1T -DEI . Dill.2 465 CVGKt M)= CVGt M i - AMIN 1( Zl:RO (-ilSTARf M)-SIELIOf M)+CVL(4)*TMLT( M) 469 1 + CORrt M) + ESTAR)/(TG - TS4T(M) + llPAR4(MI))

470

• If der. Dill.2 471 CVGRt M )= CVGrM) - IE11 N1 ( Zi ko. ( -ilSTAR( M )- $11: LIQ { M )+CVL( M )

• TMLT( M )

472 1 + CORC( M ) + ESTAR #/t TG - TSAT(M) + BPARM Mi))

473 ZIP = ( PARPL( N PCVGR(411/ KNEW 474 SILGTE = SIEGTI: + TSAT( N PZIP 475 R(KiTLM = R(XiTL4 + ZIP 476 2410 CONTINUE 477 TG = SIEGTE/ROGTEM 475 TI:MGtSt 1) ) - TG 479 SIEGTI: - ZERO 450 Do 2415 N = 1.NCLEM2 451 IF (PARPL(N) . EO. ZERO) Go TO 2415 M = IEOSLI:(N) 452 483 'l P5EO.3 454 DEN ARG(M) = ONE/(ROGCRT(M)'TCRIT(MI-T95ROG(M))

485 '/

456 '/ lui NOT WANT TO IX) Till: VAPOR ITER ATION IN SINGLE Pil4SI: CELLS 457 '/

455 *ll AIOS.14 459 Dl41:N510N ALPilGf 11. ALPilsi 1 )

490 i VUIVALI.NrE ( ALPilGt 11. A4srt 1 ) 1.( ALPilst 1,, Arsr( 2 ) 1 491

• 1 Al:0S.107 363

7

~

ROUGH DR\FT 492 1T (ALPilG(lJ) . LT (UNI:- ALPilsi i J ) r ALPlio1 GO TO 450 443 *l EXfL.1146 444 ALPilG I 1 i = AGS AVI:

jus ALPils' 111 - Assel.

Jun *I EXi'L.1144 447 AGSAVI: = ALPilGI]J) 495' ASSAVI: = ALPilhi ! J 1 449 ALPils'1J = ALPilsh Sun ALPilG(lJ1 - ALPil Sol */

502 */ hilSCELLANEOUS INITI AllZATIONS 503 */ ^

504 *1 EXFL.555 505 TEHliAS( l J )' = TG

$06 *1 AEOS.31 507 CVGR(41) = ZERO 505

• 1 132B0.15 509 TGMAX=EP10

$10 TG411N=Eh110

$11 *1 B473.91 512 SU@E = SU@l:. + CVG( M )

• Y( k1) 513 *1 DENT BCil4 514 */

$15 */ MAKl: AIX)lTIONAL 00kRECT10NS 516 */

$17 */

$15 *D ALOS.34

$19 NITNO = -6 520

• 1 SI:T1.135 521 *11' DEf.ALOS.1 522 TSTARA1 - TSTAR(1) 523 *1 SET 1.156 524 'll Def. Dill.1 525 TST art 1 - D41N1( TSTAR\1.TST AR( M i )

526 *lf del. Dill.1 527 TSTARM = AMIN 1(TSTAR4.TSTAktM))

364

i i

RIGFDMF' Al'I'l:NDI\ s' RIT I NI:1) SI4N! R-il lhl>l:T li! v'Ril'Tioh lok Till: Al'0%

Cards 32 -17 spe6: 1s the raterial properties a r.d 1 05 data for each of the raterials: fuel. M = 1: steel M = 2: wdium. M = 3; cort rol, M = 4; and finier gas. M = 5. A conplete met of these cards cust be input for each caterial it the above order. Additional cards are required for fuel. steel, and sodiu as ardicated.

n Card No. 42 ( FORst\T: 1644) (llf4R!hl). 1 = 1. 15 )

Columrs Variable De s c r i M i cr 1-72 IlX'ARD( 1 ) Message id e r.t i f yi ng the material whose properties and eq6attor of-state data are to be imput.

Card ha. 4 3 ( 10RM AT : SE12. 5 ) ROSEfM). CVS(M). TMLT( M ) . HFUSlM). THCON5(Mi Colurrs Vartable Deserirtnor 1-12 ROSE ( 4 ) The solid tieroscopic density of material M (kg/t').

13 24 CVSL M ) The selid corstart-volure specific heat of caterial M (J/kg*L1 25 36 TMLT( M ) The melting temperature of material M (K).

37-45 IITUS( H ) The heat of fussor of ca t e r ia l M ( J/kg ).

49-60 THCONs(4: The therral conductivity of material M in the solid phase i W/t

• K ).

rard No. 44 (FORMAT: $1:12. 5 ) ROLL (M). CVL( M ) . S l(i( M ) . TilCONL( M ) . X4UL(M)

Colurrs Vartable Deu rirtior 1-12 Roll:(4 ) The liquid microuopio dens it y of mat e r n a l M ( kg/t').

13-24 CVLfui The liquid-rhase constart-volute specific heat of raterial M (J/kg'L).

25 36 S IG( M ) The surface tensier of caterial M (kg/s ).

37 45 TlKONL(Mi The thettal conductivits of caterial M i r. the lig-id phase (W/r 'L ).

49 60 B1UL( M ) The vsu osits el r:a t e r i a l M i r the liquid phase ( l'a *

• 365

RNGF [3A:T rard Nc. -15 (I ORMAT: 61:1 2. 5 PSTAktM). TsT4kt M s. TsVP(M). IIST AR( H s . TCR IT( M i .

/I:T At M i r.lurr- V.. r i .ih t e lar sriptior 1-12 PST AR( 4 ) The p* paraneter in the va por pressure-temper;ture r e la t i on f o r ma t e r i a l M ( Pa ).

13-2-1 TSTARfM) "The T- parameter in the vapor pressure-temperature relation for material M (K).

25 36 TSUP( M i The superheat of material M (Kl.

37-46 ~ !! STAR (M) The h* pa rameter in the hea t -of vaperiza t ion equct ier.

for ma t e r n a l M ( J/kg ).

40-60 TCRIT(M6 The critical temperature of ruterial M (K).

61-72 ZETA (Mi The exponent parameter in the heat-of-vaporiz;. tion equation for material M.

Card No. .t6 ( FORA 1AT: 6E12.5) CVG(M). Get MJ. ATOM (M). ENCRIT(M). kTMOL(M).

EPSK(M)

Colurrs Vartable Descriptlor.

1-12 CVGCM) The vaper-phase corstant-volume specific heat of maternal M (J/kg

• K ). If ACVG(M) is defined (Card 47a). CVG( M ' is orly used to compute the infinitely dilute gas corstant.

13-24 Get: M ) The ratio of the constart-pressure specific heat to constant-volume specific heat for material M in the vapor phase.

25-36 ATOM'MJ The molecular diameter of vapor material M (A).

(Suggested values: fuel. 4.4; steel. 1.64 s od i u.r .

3.567: control. 1.46: fission gas. 4.047; nitrogen.

3.796).

37 .15 ENCRIT(M' The irternal crergy of material M at the critical reirt (J/kg). Default: I NrRITIM) = max [llSTAR( M) . TMLT(4:

• CVS(H) + lil'US( M i + CVLl M ) * (TCRIT(M) - TMLT(M))}.

49-60 k'TWIL( M e The molecular weight of material M._.

61-72 EPSKfM: The molecular forse constant (th.,) .-of material M ir the '

vaper phase. ( Sugge s t ed values: fuel. 6465: steel.

7700: sodium. 1375: tontrol. 5472: fi s s ion ga s, 231 ).

366

l'

. t Card N.. 47 (IORMAT: 4112.5: Ri xirRTI M I . Rf XiP95( M i, ROLrRT( M ) . APAR A( M) r.. l ar r s Variabit lie c r i rt mr 1-12 RtxiURitMi The crit ual der ity ( kg /m' ) of ma t e r ia l M. [ Default:

O. 375 ROLI ( M I).

13-24 ROGP95( M ) The sa t ura t ed va por de r.s i t y (kg/r 3 ) of ma t e ria l M whe n TSAT( M i = 0. 95'TCRIT( M). [ Default: 0. 25' ROGCRT( M )] .

j 25-36 ROLCRTIMi The liqu id c r i t ica l der.s i ty ( kg/m') of ma t e r ia l M. If I

a c o r.s t a r.t liquid density is desired for c on s i s t e r.c y with previous versions of S141f.R. ROLCRT(M) should be set to RHOA1(Mi f rorr Card 46. [ Default: ROGCRT(M)).

37-46 APAR A( M ) The parameter used to increase the gas c or.s t a nt above the critical temperature. This is ag,y in Eq. (A-25).

[ Default: 10 ' ' ) .

Card No. 47a ( FORN14T : Se 12. 5 ) ACVG( M ) . BCVG( M ) . CORC( M) . BPARA(M). CPARA(M)

Colurrs Variable Descriptier 1-12 ACVG(M) Coefficient a ir the Eq. cygg - ag + b g(Tg - 273.101.

[ Default: CV (M)]

13-24 Coefficier.t bgin the c gg y equatior..

BCVGtM) 25-36 CORC(M ) Correctior tere for the vapor e r.e r gy (J/kg) i r. the l dilute vapor density limit.

37-45 BPAR A(M r Relaxatsor. cor.stant (K) for the vapor heat capacity.

19-6u CP AR A( M ) Maltiplier to adiust the gas c on s t a r.t at high varer densitses.

I

ra rd Na, 45 ( FOR414T

4E12. 5 ) FHOA1(M). kHOA2 ( M ) . RHOA3( M ) . RHOA4( M )

l

[ Note: This card orly read for materials 1. 2. ar.d 3 i.e.. fuel, steel, ard sodium.)

! Colurrs Variable De sc r i pt t er 1-12 RH041(M' first coefficiert for the density vs temperature equation at low temperature. Variable a in i Eq. (A-41). [ Default: ROL( 1 ) for M-1. ROL%ge ) .i fer

! M21. See Card No. 53 for the definition of ROL.)

13 24 Ril042 ( M . Secord coefficiert for the dersity vs t err.pe r a t u r e

l. equatier a low teeperature. Variable a 2r-3'

I.y . ( A 41 1.

367

ROUGH E WT Colu rs Variable De s e r i nt i er.

24 36 Kil'I A U 41 - Tlii r J s oel1 it s er1 for the dersitv vs temperature e 6p.a t i o r at 1.n, temperature.

(A-41).

Variabale a3m ir Lq.

37-45 Rili AJ i Al i leurth coefficiert for the der.sity vs teeperature equatnor at low temperature, variable a 4e ir 1.q .

(A-411.

Ca rd No. 4o (FOR41;T: 3L12.5) Kilolll t k1) . Ellol!2( k1) . Ril0B3(41)

[ Note: This card er.lv read for materials 1. 2, ar.d 3. i.e., fuel, steel, ard sodium.]

Colu-rs Variable Deseriptnot 1-12 Rilolli t ki t first coefficiert for the der.sity vs temperature equatser at high temperatures. Variable bj ,. ir Eq. A-41).

13-24 RildB2 t k1 E x po r.e r.t for the der.sity vs temperature e qua t i er. at high temperatures. Variable b 2m in Eq. (A-41).

25-36 Rilo!!3th1 Last coefficient for the density vs temperature equatson at high temperatures. Variable b 3n ir E ti .

(A-41).

Card ha. So ( f0R41;T: 3E12. 5) A50UND. B50UND. CSOUhD

[ Note: This card cr.ly read for material 1. fuel.)

Colu rs Variable De s c r i pt ion 1-12 ASOUND Coefficiert to compute velocity of sound as a furctier of temperature. Variable A i r. Eq. (A-34). [ Default:

2212,339 m/s).

13 24 ilSOUNI' Exponent for the velocity of sound ecuntior.. Variable B in Eq. (A-34). [ Default: 0.3539176;.

25-36 CSOUND bit r. i mum fuel velocity of sound (m/s l. Variable C ir Ly. (A-34). [ Default: 400 m/s}.

card he 51 ( f0Rs1 AT : 15A4) ( llrARD( 1 ) . 1 = 1. 15)

Colu rs Variable Description 1-72 llCARlu l ) kie s s a g e identifyir; c e r.po r.e r. t properties to be inpct rext.

368

ua l l o

V ra rd No. 52 (l' ors 14T: 61.1 2. 5 ) (RtN N1 N = 1. FCSR ro ; u- s V.i t i .i b l e Deu riptior 1-72 ROSi N > The rstroscopic density of the solid phase of st ruc t ure-f ield der.sity component N (kg/m3 ). NChk = 9.

N=1. labricated fertile fuel.

N = 2. fabriated fissile fuel.

N = 3. frozen fertile fuel.

N=4 frozen fissile fuel.

N = 5. cladding.

N=6 can wall.

N = 7. control.

N = 8. intragrar.ular fission gas, and N = 9. intergranular fission gas.

Card No. 53 (FORN14T: 6E12.5) ( ROL( N ) . N = 1. NCLR)

Coluers Variable Description ,

1-72 ROL(N) The microscopic der.sity of liquid-field density coctor.ent N ( kg/m'). NCLR = 6.

N = 1. liquid fertile fuel.

N = 2. liquid fissile fuel.

N = 3. liquid steel.

N=4 sodiut.

N = 5. control.

N = 6. solid fertile.

N = 7. solid fissile fuel, and N = 8. solid steel.

For liquid fuel. steel, and liquid sodium the ROLs are to be used as default densities as indicat ed in Ca rd No. 48. Also, for liquid fuel and steel they furr.ish the cicroscopic densities while ree l t i ng structure or particles.

For example, see Eqs. (111-5) and (Ill-6) of the $1hNER-Il manual.

Card No. 51 (FORkt4T: 6E12. 5 ) ( SVEL( N ) . N = 1. NCLR )

Coluers Variable Descriptior l 1-72 SVEL(N) The sonic velocity of liquid-field density componer.t N l

1 (m/s). NCLR = 6.

1 I 369

E UGH DRAFT 2 APPEN111N la AE05 SIMULATION PkfxiRAM i

This is a stard-alore FokTRAN prograr:'to . eva luat e wate r propert ies Ier the SIW11:k-l l Al:oS so that' c ompa r i s or.s with other ' data may be performed ard properties of the solutier. a l go r i t hr: evaluated. This program was used te produce the r.uebers f or Sec. 11 0. It is stored or. the Los Alamos c ontor. f i l e sys t em (CI S ) as r.ede /VA10R/ALOS3.

! 1 $rilAT 4ME.ATVST.BTNS. LIST.t#>7.5.S..r la 2 CALL Cil4NGE( 511+FVST )

3 PR(KiRe1 NB1Et TAPE 55. TAPE 221 4 DIMI:NSION PV( 2 )

5r 6r Tills IS A PR(xiRe1 TO SIMULATE T}lE S141ER AEOS FOR VATER (ll20).

7C Tile NOTATION RI3141NS AS IN Sleil:R-ll. '

5C 9 2ERO = 0 +

L 10 TG = 373.

11 Coke = 5.01:+4 12 R(xirUT = 1. l:-lo 13 1:43 = 1.0-3 14 ArVG = 1346, 15 BrVG = 0.3302 16 ONI: - 1.

17 1.M2 = 1.E-2 15 EP2 = 100.

19 IF = 1. l:-7 20 NOUT = 22 21 CLiv = 1495.

22 NIN = $5 23 rV6 - 2151.6 24 T95ktXi = 5. 8591961:+4 4-25 TrRIT = 647.29n 26 RTrRIT = ONE/TrRIT

27 Al:0$11 = 6.35525 1 25 Al:0521 - -3.125081'-3 29 Al:0$31 = -1.42444E b 30 A1.0$10 = 6.13434 31 A10$2" - 3. 533491:-6 32 Al:0$30 = -2.424$10-3 33 R(xirkT = 316.957 .

34 ROLrkT = 316.957  !

35 RrRIT = 107.6264 i 36 il All = . 5  !

37 TVo3R!i = 2./3. .

{ 35 , SII. LIV = 4.u43041:+5  !

39 TMLT = 273,16 i

40 1.P4 = 1.144 l' 41 TKO = 2.

42 1:l'1 = 10  ;

. 43 lil'US=3. 3341 +f 1 l 44 rYS=2ouo.

} I 370 l l I i '

I

, + - ,_.,-,,,---_-.__.,__.~,__-,.___-_,.-_e_m .

,. w w . .- -,-w,-..- - ,~...-m,,,-,,-,,--~- - , - . - . .,

i l 3H DRA:T 45 RrVS = 4.764690 4 40 Sil: SOL = 5.709041:+5

,47 Sil:LrR = 1.572164).+o 45 k(TL = 2. 37131. 4 av ENrRIT = 2.93341.+6 ,

So EM10 - 1.l:-10 51 EPlo=1.1:+10 52 CP9 = . 9 53 EMI = .1 54 1:45 = 1. E-6 55 rVGGM1 =~461.405 56 PSTAR = 3.177711:+10 57 RPSTAR = ONE/PSTAR 58 TSTAR = 4. 705791:+03 59 RTSTAR = ONE/TSTAR 60 PCRIT = PSTAR*EXP(-TSTAR*RTCRITi 61 CP95=o.95 62 IISTAR = 3.05.9505E+06 63 ZETA = 0.3660361 64 AE05LM = 1.70124E+6 65 CVL = 4217.1 66 CVLP = 1521.62 67 ROLE = 1050 65 kHo41 = 636.607421 64 Ril042 = 1.3453123 70 Ril043 = . 00274914042 71 Rilollt = 2.65005286 72 kiloll2 = . 351221451 73 Rilull3 - .0746002905 74 I:PsoK = 32.

75 kT4Le 15.

76 kkTWIL = 5.555555556l.-2 77 UAM = 1.2115 75 R ATMst) = 7.1606*2E-2 79 CP9244 = 0.92495 60 r20743 = 2.0736bl:-3 51 iP7192 = 0.719265 62 01P151 = 1.15044 63 r$46M2 - 5.46-1520 2 54 063142 - 6.314E-2 65 C267Mo = 2.6711:-6 66 01P32 = 1.32 67 02P5 = 2. 5 65 ALRCRT = AL(Mi(krRIT) 59 ALITGM = AL(Xi(CVGGM1) 90 til:h 4RG=0hl:/( Rt XirRT

• TCR I T -T95 R( Ki l 41 ( Al:Us = 1.1-10 42 i= 1.

43 e 44 r h is Till: Mi@l:k 01~ V4k!411LI 5 TO INPET 45 r 10Pih = 1 k'lLL OllTAIN l>KOPl RTil:s I R( A1 Tilt VAPOR 1)l:hs1TY AN1r 4n e' Till V4Piik Ahli LitJt'lli lhil:kNAL 1:NI RGil:s 47 r 10PTN = 2 OllTalhs $4TitK4Tl0N PROPl: Rill s l'kt At TS AT 45 r 101'Th = 3 OllT4ths S ATlIR4TloN Pkol'ERTil.s I kt*1 Tile PRI:sstiki . Phi 49 r 10PTh = 4 OllT4ths TIII LitJtill) MirRusr0Pir DENslTY l'Kint TL 371

Zlt Cfr 01 on #113J1 *19' 91) 11 til I OMY sidos.orsuit = h0(193% of t til ott 01 09 Cit

.W0J03.(JN01.l'h l:l(U+h91Gi 1 01..Will9 + DMtlM(l = !)1(IM(I til J 1:10J/ t 93Y.9MVN Id.M{t dJ. t . Nil.19 .WO e 1+ l ofI 19Mt . t f 50 lt + 11til.b009Mt ) ..W019 = DMUMd 6tl

((.WA.191NO).(b.l:10J+b!) int i + .Wh.19..Will i bl\:1 = DhM itt .

i .)1.1031901% = h 1:10J Lri VMtil.). I OMtN:l(i.(!w nl961 9MY l } + INO = J.l:10J 9r!

9MY. lfsn'lt + h009MY + l(NO:lt = .Will.1 itt JN0J9M.Ulbli = WA.19 SCt trl i ott 01 09 fri  !

91tlM(1 = DMUMG Cr! j 9MY. !CSO:lY + 11til.h0(19Mt=91(IMO tri

( DMt. IC50:lt+b000Mt+t !SO:lt blN:l = DhM ori S C r 01 09 ( 113.11 *19 ' Di l .11 60t ,

(93Y )1Mos.1C50:lv = h0fl0Mt VCt 901  !

Ort 01 09 LC! ,

JNA.193.(1MJM1Y C9AJ1Y ).01..~WQJ9 = DidMU 90t l 0317 = 9M(IMd 90L

((.N019 3N01.C9AJ1Y + WOJ91MJMit idX3 = Dh3 til

.WQJ9M.WitdY = .W019 (Cr Erl ott 01 09 CCt 03:17 = 94dMU !it ml17 = 91dMG ofI 11MJM=D hM 6CL fCr 0109 v 11331 *1!)' 01) .11 SCI 1 iCr 01 09 (1MJ!nt.11MJ1 "ll' 9Mt) li LCl l oft 01 09 800Mi61 'll' OMt) .11 9tl (VMYdV+11MJ1-91)/:lN0=.WQ.19M (11MJ1 '19' 01) .l ! iCl if t 0109 i th:1 10Jtet '11' OMt ) Il tCt 91.h<l!v1M = DMY fCI i

( 91 'fl.C-91) .9AJ11 + DAJV

• DAJ CCI N91 = 91 1C!

031Z = ANh0S OC1 ,

0M17. = lh0S 61! J OL t 01 09 ( S L ':19' tW11N ) .! ! 9it i I + ON11N = (Wi!N 41! I

10N!1NOJ Oft 9iL 91 = N91 it!

Old] = MhD1 t!!

Oth1=Nih91 f1I O = (Wi!N Cl!

Oli 0109 (OM1Z 'ol' 19315 ) JI tLI li'Ciltdt ) DHM0J C ott

( f 'o t:19d 1 ) DHMO.I C 60L Hd'10M'Tl3IS'1931S *Hd!nf ( f *NIN ) UY3M L Sol 11Yd010 TlV3 (il Trl* A7N) .l! Lot N ' (=67N tt E x1 90t

10N!1N(D 9 iot Nido!*tol'6*L'L'91 01 09 tot t ' Mihil+ Wo I . th9%U'u Ot=C9A OY fot Mihl.)'Utbill'tMhlt *Nidol"N I f C *111) t it.lM Col All%0JSla N0dtA (Mt All A11J0(NOJ J tol lihM3111 M0dvA :llli .10 N0!D11)J1YJ M:llt!S :lill Snit 1N(U i

• Nidol J 001-I M10 HE E

CLC a MY1513.MODY.l.LN35 M!)AJ ilillA.)).91tlS1(i. A + ONh0S = 0'.h0S nos

5. Mils'l + lh0S = th0S soc 10111NOJ ist Lot

( f tMbill+91 DS1)/DSi+'INo s.9tul+ ( i t Mbl11+91-1M11/ t not

(

• Mt1SIM.MODYJ.1N.ls-DAJ.lM1*M9A.1+<lillAJ a .910S10 9AJ MDAJ ) i=1M9A.Xlt ioC tin i *11' 911 11 tos

( OMWl8+1YSI 91)/DSi+:lN01 9A.nl + t (VMbill+1M1911/ t fus

((MY1 SIN.MODY.l.LN:lS-M9AJ sidlAJ).910S10+9AJ MDAJl1- = 149A.Xl1 Col (DS1 ':19' Dil 11 tot ist 0109 (OIKi+0AJ *:11' M9AJ) Il nos 9AJtl = IM9A.XI h61 30NILIOJ oir 161 illAJ = <lellAJ 461 l :lNo L 9oL

+(DSi-11831)/(DS10.u-MY1S1) .11%ll.Y13Z = MODY l! 96L (11MJ1 'll' iMil JI r61

OM3Z = MODYJ 06L (MY1S181YSI 3NO).1N3S+1M1.M9AJ S13IS = MY151 toI j M9 U/

lNo = l ldh 0SM tet

( ( VMhltl+f itSI-91 sst!Y ) /. f 06L (JM0]+MY1 Sit - 11H1 1AJ + ( Mils 1M.DSI .llt11) .113S + C 691' DS19AJ h1503Y+(11MJ1.tlMfoal-DS1).11AJ)'OM3Z)INihY DAJ-39AJ 99( J

( 1(MJ1.dMfo.u-DSil.<llAJ + t LSI l LN:IS..llvil-h150lt+n!1:llS=S131S 9tt 991 l 1

rill = lisi f tt 991 oit 01 (M) r91 1AJ = <!illAJ 091  ;

3NO+(1YS1-11MJ1)/(1M1 Mt151).Y13Z a MODYJ C91 l

( MY1SIM.DS1-:lNo LLTIS+1M1.M9AJ S131S=Mt153 L9L  !

M9AJ/:lNO = Rlh0SM 09L

( tiMhill + t IM1 - 911SilY l/( JMOPMt1Sil - 1YSI.DAJ- C 64I (NY151M.1YSI-3N01 1N3S + DSI.1AJ POM:1Z ilN!bV- 9AJ = M9AJ 941 ,

'IA). tom 3Z'11hi 1YS1)!XthY +611315 = N1]IS 441 l itt 0109 (11MJ1.UMfo.u *19' 1YS1) .! ! 941 l (91.htN:ld)/( t 3N0+910MG).1v51)- = 910510 it!

Y137. .(11MJ1M.DS1-3N0hMY1Sil = IN35 1 plt .

(11M31 TI' DS1) .11 (41

. OM3Z=LN3S CL1 NN30/MY151-=DS1 1L1 (MY15 elm.DhM.93Y)!x)1Y = NN:10 041 30N!1NOJ ott 691 09t 01 (M) 99t OM37 = DMGMd 49I OM3Z = 910M0 991 thD9AJ=DhM ift 191 ott 01 (M) t9L JN039M.(JN0J.1-hDAJ1Y b91. WQJ9 + DMGMG = 91dMG C41 t h009MY..lltil+0CSO:1Y ) ..WA.19 = DMUMd 29L

( ( JNA.10-:lNO ) .h9A. fly + ' .WQJ9..WQ.l.I blX3 = Dhd 191 h009MY + t 9Mt )!XrlY.oCSO:lV+otSO:lY = aN0J.I 091 ')

JN A.19M .t MblY . = WA.19 Cfr ni1 -l ott ni 09 ' sil

, 91tlM(I = HMd'id Lit i h009MY .lltil + "CSO:lV l =- 91d*l 991 t

( h009Mt+t 9MY )!xrli .oCS03t+o tS03Y bik:1 = DhM ;iit -

]

l 9

.. M0HW8

- . . - - . -- r.- y + . , . , , . . - - - - , . -yc- - , - -- ,,e_ , . . . - - , ri,r.' m- --

tlE o=<WilN 09C oi 0109111MMl"l9'11:lls UI fut

10N 11No.) If C9C 09 01 09 toc

( 00llM.11+CtollM ).11 + liollM = YN10llM o9C

!AN!1N01 SC 6it 1 A.11.d l 2+11H1=11 ist if 0109 (MJ'1315 '19"l1:11S 111 Lit

.10N!1NOJ os 9ic 9C 01 09 iiC l'lH1=11 tic OC 0109 f OM3Z '19'dl2 ).11 fit A113tS dl7-dI7 CSC 09 0109 (10s'ils ':11 '<l17 > 11 Iis

110M = YN10lld - oit 11:lls =<ll7 6tt

• S AJM .11:llS=11 StC Nd!X)M = hd10M (OM32 'A3

• bd10M ) .!! 4tC

91. ( ! I Ad=( l 1 Ad 9tt 91.( C lad =( C I Ad itt

10N11NOJ ooc trC i it 51- 11 M31 # /1N:IS . t 1:12 - =s 105110 f tC ..

Y13Z. .(11MJ1M.1Ysi-:lNO).Mt1511 = IN:IS Ctt ooc 0109 f 11MJ1 *:10' ItN1) .11 Its

! h l!x)M/t Ds11.DS11 =9MUsid Ort D S 11.MY151-=D S L 6fC

( MY1SdM.91.DhM.h<!!X)M I!Xni/:lNo = DSIM SCC out 0109 (OM:12 'h:1

• bd!X>M ) 11 VoL 4fC l o0C 01 09 9fC l

( DS1 IIM31)/(1N:lS.Y1:1Z )- = %1(151101 ifC 11[M31 'll' D51) .11 tit i Hd 904 . NON:Irl i / ( ( :lN0+9MrlMrl i . D si ) - = 980 slo ffs

( MY1Sdd.DhM.Hd!X)M.91)!XMY = h0N:10 CfC )

961 01 i 9 ( thD9AJ 'ol' DHM ) .!! IfC DhM .NdDOM+( t ) Ad=( I ) Ad ofC i

DhM. A.( CH3'3N0+9MUMU)IXThY + ( C )Ad = ( C ) Ad 6CC Ch3- = S105110 SCC OH3Z = 930S10 LCC OM:1Z=( C ) Ad 9CC OM3Z=( 1 ) Al SCC N91 = 91 tst TlONI1NOJ 09P CCC QYi!096 TitJ Ott CCC 09t 01 09 LCC Ott 0109 ( S03Y3 *19 * (91/(91-N91))st!Y ) .II oCC

.11Yll. I NihD1+Dh91 s =N911 - 6IC

( XTh91'19 'N91 *MO ' NIh91'1'l ' N91) iI 3iC i (N!h01'91) LXThY=NIhD1 (OM:17 *19' l) .! ! itC

( \th91*91)INihT=MhD1 (OM:17 'll' .I) .11 9iC 910JU/.I - 91 = N91 i!C

, ( Dit1.10 ' CN:1- INIhi=oltI.lfi t1C

lNo-:ltthSsM ANhos- = ole lId i!C i 91-:lt 340SM.( lhd5-19:11% * = .I C!C I

IdNI1NOJ 09t TIC

lGiD5H.81M9A.Al.( A.MY153 '19:llS ))+DSI.1MDA XI.A- t OLC 1W80 H0008

l sa

( 9 't4 C 91 ) .9 uti + 9A. Y = 9AJ nif

't=X itf EKl* =sidslid .IC OM:17=9Musitl 9If OM:17=1%S1 V!f OM]7= Mill 9:l) rIf OM:17= Mil 19AJ fif OM:17=IN05 CLC 01 ( f'NIN ) (IY:ld IIf

10NI LNOJ uts uti dois oof'

( C 'O t.lf

• 9IC ) ithM01 f t 900

10NILNOJ tt 400 J 900 0:IMIS:Id .!! ril1NIMd 39 AW S3111YIMYA :13(4% J $00 A11SN]G JIdOJSOM3!H U10011 Jill Si tN10lld a rof

.IMaltM:IdH31010011 :lill SI 11 a fof

IM01tM3dKl1 NolltM01tS :lill Si ltS1 J Cof i IM01tM:Idh31 ModVA :liti 5191 , a 100 I Nolit71MoelYA JO 1Y:lH Jill SI LN:15 a utf A11SN30 01 IJ3dS3M lilla :IMSSS:IMd Jill .10 :lA11YAIM3(I :lill SI ( C 1Ad J 66C

IMASS:IMd ModVA Jill SI (I)Ad J 16C 3 46C Al'id

• t N10HM * *II*It 51' 91'1N:lS * ( C 1 Ad * ( ! ) Ad ( C *100N ) ll!M5 96C 10N11NOJ 04 96C llid1Y/hd10M.(i6 'u-111d'lV).I C. 0!'13) + Alld = 0114 tot OL 01 09 ( 96 'O 'll

• llid1Y ) .11 06C tN10HM/hd103.i6 *O = 'llid1Y C6C (I1Ad = n!'Id 16C

10NILNOJ 09 06C IMJION = VN10HH 09C

10N!1No 1 19 99C 11M31+d1AJ/(11MJN3 1131S >=11 49C

10N11NOJ 09 99C or 0109 iSC SYdS9S TlV3 rSC i

10NILNOJ St ESC ]

09 01 09 CSC t C . . ( 11 MJ13 11-:lNO ) .f H0HM+CUolid . 1 L9C (11M31M.11-3NO ) . IUOllM+:lNO ) 1MJ10M=YN1ol'M OSC Sh3-11MJi=11 ( Sh3-11MJ1'10'11).!! 6LC

10N11NOJ Or SLC 90 01 09 LLC I 1+0N11N=0N11N 4LC

'11 = 111 ilC 9t 0109 ( 09 '0:1'ONilN ).11 TLC ,

111. IKI + 11M316d3 = 11 (11MJ1 *:19

• ll) .11 f4C or 01 09 I oth3 *11 *(11/(111-11) ) SHY ).11 CLC

( d'I AJ-1710JG ) /.I-l'11=~Il  !LC 11:lG+( 11331.UM f 031-111 ) . el'I AJ-d ! 7=.1 ott I i 111-11 MJ1 ) / ( l Tid . Y1:171 1 - =1110.10 n9C Y 1:17. . I il MJIM .111-:lNO 1. MY1Sil . llill= I1:lt I i9C-10111NOJ if l,9 C h' ISO:lY ell 7=,117 99C 11MJ1.ffffff9*o=111 i9C k

>,m - - a - ---e. ~ - 9

i 9R im e=

s b(nl j f, 32o if ( RtMila.1. LV. Zl:Rtn GO To $50 321 A RG= kt xil".1' TG 322 11 sARG .LT. kix.rVTI Go To $35 323 11 tiu .61. TtRIT4 Ruf 0ff =ONI./ t TG-TrR I T+ 4P AR A i 324. If 4 Aku . LT. T95Rixii GO TO $30 325 11 ( Aku . LI:. T( klT'R(xirRT i 60 TO 525 326 11' (TG .GT. TrRIT) GO TO 523 327 Rs14T = krRIT 325 DRl*G = Zl.ko 324 60 TO 54" 330 $23 GIUtr = APARvRGfUNC 331 RMAT = EXPT ALRrRT*GfUNC + ALCVG2'(ONE-GFUNC))

332 DRDkG=ZEko 333 60 To Sao 334 525 ARGDUh1 = AEOS21*SQRTIARG) 335 11 (TG .GT. TCRIT) 60 TO 526 336 RMAT = EXPT AEOS11+4RGDUhi+AEOS31

• APG) 337 DRDUG=4RGDUs1*ll4LF + Al:0S31' ARG 335 (X) TO 54o 339 525 UTUNr - APAR4*RGTUNr 3to FFUNr - AEOS11 + ARGDUN1 + AEOS31* ARG 341 COEIC = ONI: + (( ARG-T95ktxi)'DENARG)*CPAR A 342 COETM = AUXH COErr) 343 R414T = EXPiITUM 'GIUtr + ( AtrVGM+roErM)*(ONI:-GFUNC))

344 DRDRG = GFUNr*( ARGDUM*ll4Ll +AEus31' ARG )

345 1 + f ( UNE-GTUNC )

• CPAk A

• DEN 4kO' ARG ) /COElr 34o GO To Sao 347 530 ARGDUM = AEOS30'SVRTtARG1 345 11 (TG .GT. TrRIT) Go TO 532 344 RMAT = EXIhAl:0510+Al.0S20* AL(Kit ARG)+ARGDUM) 350 DRDKG = ALOS2o + llall'" ARGDUM 351 GO TO $4" 352 532 GrUf( = APARU RGTUNr 353 iFUNr - Al:USlo+AEOS20' AL(xi( ARG 1 + AkGDUM 354 R441 - EXPT ITUNr'GFUtr + ALCVGM*(ONI:-GFUNr))

355 DRDkG = GTUNr=( AEOS20+11ALF' ARGDUM) 356 GO To $40 357 535 RMAT = CVGGM1 355 DRDRG = Zf.R0 359 Sao ARul= AKU'RMAT*RPSTAR 360 11 (AkUP.LI..ZERO) Go TO 545 361 til: tar.1 = Alexit ARUP) 302 TSAT= TST AR/DEMA1 363 DTsliRG = -t TS AT'(DRDRG+0NI:))/( DEMM* R(xiPM )

364 SI:NT=Zl:RO 365 11 iTS4T.Gl:.TrRIT) GO TO $41 366 SI:NT = list 4R'(ONE-TSAT*RTrRIT 1"21:7A 367 DilSDTs - -1 ZETA

• SENT l/(TCRIT-TS4T )

366 541 00NTitjUl:

369 11 (Ts4T .GT. TV03Ril'TrRIT1 GO To $42 37o sil:Ls - Sillit.4444N1r TS4T TMLT.zi ko i'rVL 3?) rVuk=rVG AMif.I t Zl:RO. i rVL'Ts4T + SI:NT'( ON1 -TS AT' RTST AR )

372 1 -rVG'TS4T - lisTAR+r0Rr 1/t Allst TU TS AT i+ llPAR A > )

373 GO TO SJJ 374 542 Sil:L5 - Sil: Lit.4rVLl"i TS AT Tv03RD' TrRIT I 376

LLC 0b109611t') 619 oCr on9 01 (H) SCr iS9 01 09 i n th:1 '19' (M.IN II7/lil!7 M Will/ s istit i il .Cr i t .Il7. lf so lV+i dl7 :1Mos ..llvil. L Cso:lV+ No i .3:WM) L 9Cr e s il!7.M:lNM-INil)+ril7=3:Wd!7 VCr f dI7.lfSult+1 dl711 MAS.Itso:lt+1tso:lY blX:1 = 3:WM rCr 3:lNdl7 = d!7 fCr o99 0109 (ui *:19' ONilN) .11 CCr i + (WilN = INilN (Cr

10N11No.) iS9 ot t-oh8 01 09 btV 099 01 09 hilh:1 *19' (M:lNd!Z/(d!Z-M:WdIZ)istIY i 11 9tr I ( ( d IZ '1Mos .of so:lt ..lltH+0CS01Y+:lNo .3:lNM 1 1 4Ir

/f dlZ.M3NM INil)+dl7=3:lNdl7 9tr

( ( d ! 711MoS .0f SO:lV+( d I Z )D01Y .0CS03t +01 su:lt id\:l=3:lNM VLP M:lNdlZ = dl7 rir 699 0109 (Oi *:10' IN11N ) .!! Ctr

! + aN11N = (WilN 091 Ctr 999 01 09 (IM)i6d *10' INdI .11 (tr t (11MJ1.i643)/MY1SI-idXl.MYisd = INJi64 oIr 10N!1NOJ 949 60r 069 01 IM) Sor M3NM/INd = M:lNil!Z Lor

((JN039*3N01. COA.)1Y + JNG.191MJM14 hlN3 = 5:lNM 90r G MYelV+11MJi-itsi)/VMidY = JNA.10 Vor i49 01 09 111MJ 1 *11 ' INd ) .11 tur 069 0109 (103!X)H *11' M:lNd!Z) Ji for o = ON11N Cor 3:lNM/INd = 3:Wd!Z 10r thD9AJ = M3NM our

10NILN(L) o99 onf (Nd (f*NIN) flV3M 96f

10N11NO.) 9 4hf  ;

099 01 (M) ofo (itS1/MY151-)<lX3.MilSd = INd iot itsi ( C'NIN ) (IV:IM rof )

9 01 09 ( f *h'l ' NidOI ) .! ! f60 N't=nZN 969 Exl Cof

10NILNOJ 4 160 09r 01 (M) 06f 91 = N91 690 19315 '9MGMG'ithM '91'hdfx)H ( C *10td ) :11133 990 th0S+91. Mil 19AJ"'l0:ll5 410

10N!1NOJ oi; 99C-Mil 19d.)+(ithM+MDAJ).X = MillDil] 99f Mtil0AJ+MDAJ.X=MillDAJ r1C

'InNILNOJ iti ESC (OM37'1N3S-Ed:1)tXthY + s911s = strils Cif X.I l%1. ( MVis 1M .LN3S +MOAJ )-5911 S ) = lhns 190 iri 0109 410]!x)M 'll' 94Y a li oif-11:ls + sTils = strils tri 6.0 t ( VMbill+t 1%1-91 Wtit 1/ C 9.f

( JMOJ+Mtisil - 11N1 1AJ + ( Mils 1M.1%1- iltili .1N.ls + 1 ..f l 1%1.!)AJ- h"lS03Y + (11M.11 llMf051-itsi).<llAJ) *0M:171 LNiht 9AJ=M9AJ ' ' N. C j LN35.11%11 - hisoit + 1 iC i 1

.]

i IMG M8 1 a

i 94E i I

90th'!)NO.}ll1* P)1 i C *10011 ~111MM t9r l 9066.( t :lNO*lW91.!)AJ . fit l

. iil C. MW 9. 9AJ t . C fil l d+9NO. llll=9NOllli Cir MolV IM . I . M ll .1( 411M l lM AS . oh& 9 CJ=90hX (9r i 10h0M.( J01 114413M 61MOS. CHIC 9J=9NOJill 09r l I t %1'l(N111MOS .Ch9ti.)* . 6Lt

( ( L $ 1 <l 1.)- ) . .%l:10M1 ) . C6 L Lil)+%1:lt dl .Ch4of.Por Chil.18, Oshit M =1thV pl 9Lr Mo%I:l/J01=%l:l(M1 L&r '.

J01 t (*NIN I (IV:IM 9&P j N't=A7N 916 (nl itt

10NI1NOJ of plt elois fLr iN10llM'll (C*100NI 111M3 016 CLr ,

( 11 113]M 1/111/ MY1SI- hlX:1.Mt1%I=tN10llM ttr I

10N!1No.) r06 04r 016 01 (PJ 69r

( f tolid.11+00llM ).11 + lioll3 = VN10lld 99r

10N!1NOJ fo6 49P 016 01 09 99r  !

( C . (11 M31M .11 3NO ) . Et!OllM+ C!!O113. . t $9r

( 1! 331M .11-:lNO ) . lliollM+:lNO ) .1MJ'X Rl=t N 10llH r9r to6 01 00 f 9 :l ' t -11 MJ1 '19 ' 11) .11 E9r  ;

foe 0109 (11MJ1.UMf051 *11' 11) .11 C9r '

016 0109 a 11h1 'll' 11) .1! 19r l

1103 = YN10llM 09r )

11 ( f'NIN ) (IY3M 6ir I N't=0ZN oth (xl Sir R

10NI1NOJ 6 Lir ,

il015 9ir 1 30N11NOJ i69 ist 59:1iS *S13IS

• tN10llM

• IN<l'3:lNM'DS1' hit!X Rt ( C *100N I 'll!MM rit I f Mt1SIM.itSI-:lNo n.1N35 + S1:llS = 59:lls 094 fir l 0!1315 + 1 Cir I (11MJ1.UMfoal DS1).<llAJ + IN:IS..llill - h150'lY = S1:115 itL lir oil 01 09 oir 61131S + 1A3.(11bi-itS1) = S1315 6tr .

Vr4 01 09 (ll331.UMCOM1 '19* D S1) .I! 9tt Y13Z..(11M31M.DS1-1NO).MY1 Sit =1N3S (11M31 *1'l

• DS1) JI trr OM32 = IN:IS. 9tr i

10N!1NOJ OIL Str I

( D SI .11 M3M ) / ( D S1/ MY151- )dX3. MY15<!= t N10llM ttr

10NI1NOJ t0L frr 014 01 09 str J (f Yolid.DSl+CYOHM ).DS1 + ttollM = YN10ild tre

10N!1NO) COL Orr utL 01 09 6ft

( C . . ( 11 MJ1M .it SI-1NO ) . fIlollM +CilollM . . I 90t i 11 MJ1M .DSI :lNo ) . I fiollM+:lNO ) .1M.PioM=WlullM Lfr toL 0109 ( 9-1

• t-11MJ1 *19' DSL ) .l ! 9ft l COL 0109 (11M31.(IMf031 *11' Dsi) .l ! ift 014 01 09 ( 11h1

• 11

• D N 1 ) .! ! tfr

'lloM = V11ollM fit 1 Dsi .. TIN 1 l ' IN.I = h<190M Cfr

( ( NYl%IM. IN<I exnt L Mt151- = DSI . Ift 30NI1NOJ 069 Urr H/h'0 EDCh

.] Jl q.1 3 '

V J ~< 1 465 91f n)NTINlfl' 886 STol' 467 I:fd r I

l 1

I

)

379

S 'n -

~n p==

RV Ji JI5 f APPENDIX E:

SUGGESTED-VALUES l'OR Till: VATER AE05 INPUT

. A surrw rs el t hh irput suppested by the evaluatter ir, Sec. Il-C is given ir. l Tshle. E-1. The rurbers sortespord to the tornat and definitions of Apperdix t~.

~ The water preperty d ta has been substituted for the. sodium data. M - 3.

TAllLE E-1 VATER AEOS INPUT WATER PROPERTIES ANI) EQUATION DE STATE li H M. 2090. (H s 273.16 3.334E+5 .68 1(H 81. 75 4217.1 0.0727 .65 1.00-4 3.17M11:+10 4. 70579E+03 0. 0 3. 0898051:+6 - 647.296 0.3660361

-2151.6 1.2115 3.737 2.93390E+6 lb. 32.

316.957 95.77 316.957 200.

1346. O.3302 5. I:+04 50. .5 535.607921 1.34531230 .0027491404 2.55605256- .351221451 .0745002405 i

\

380 r . -' . .... . .. . -.

l a APPill)lX l~

STAN11ARD SlWl:R-Il LIQUID-LIVUID lil.AT-TRANSrl:R W)DI L The liquid-liquid heat-transfer model is based on a droplet-droplet collisier. cor. cept similar to that assumed for coalescer.ce. The total collisier.

rate per unit volute is derived in the followir.g discussion along with the heat-t rar.s f e r area. overall heat-transfer coefficitr.t. a r.d contact tice per collision. The collision rate. between liquid droplets of components e ar.d k is-forculated free the physical representation shour. in Fig. T-1. Each droplet is assumed to have ar. average randort velocity given by g E and k Y that differ's froc the' liquid-field velocity. Differing accelerations resulting from both droplet-size distributions and der.sity variations might be expected to produce a spectruc of liquid velocities. As a result each droplet sweeps a volume of space per u r.i t tire giver. by the product of the randot velocity and the pro,iected area of the droplet. The volute swept per ur.st time and volume'by all droplets or particles of corponent m is given by VOl g =n e e r jg., Vg. . (f-1) where r e is the r.utber der.sity of droplet component m. The-number of collistors with droplets or particles of c ompon e r.t L i r. VOL c is sicply VOL c tires the r.utbe r der.sity of type k. The colli s ior. ra te per unit volume of coepor.ert r wi t h corpor.e nt k caused by the movemer.t of corpor.er.t e is given by E E . (1-2)

Zm- k " "T he m e k Similarly the droplets of component k sweep a velure resulting in collisions with type c. This rate is given by Z ge=rrhk krg rt . ( F-M l

381 l

lqf(9'! Nh """

% J W s Ji1 ~,

The total collisier. rate is approxstated by 0.5 times the sum of the rates giver by I qs. (l-21 a rd f I'- i.

ZLelk " "It rg irhr e*'hk k "* 5

• f 'J The factor of 0.5 accour.ts for cour.t ing each collisier. twice. The r.utte r de r.s i t y of droplets of each c ompo r.e r. t is related to the respective volure fractier.s. c., ar.d radii by r

n = 3ct7/4rr'n (F-5) r.g = 3c.ty /4rr h .

Vi -

Vm Vu yk

.Vs Vm

-=2r n --

!" . yot, k

4 k k* -

y,

~

V. Vol. )

_ l 1

l l' i g . 1-1. l'hys i c a l represertatior. of liquid-liquid irteractior..

382

e f#II r W b'h Jifin[g-T Substituting 1:q. (F-Si in ry. (T-4) gives the collisior. rate in terms et the va r i a bi c s s erpu t ed hs Sittil.k- I l i n e a c h re sh c e l l . Thus.

4 ' Lc L L - -

Z LeLL

• E r p.

f - Y

_ r pk 'pe e + 'pk - Ik) . fr-6)

The collisic, rate does n r.t behave we': . in the limit of the radii approaching zero. In the following this situation is considered below in terms of physical licitatiors or. the heat-transfer rates.

We consider the area of c or.t a c t or area for heat transfer tha t , may be expected for each collision. Fig. T-2 shows the assumed collision cor.figuratior.

fer droplets of componer.ts m and L. In general. the contact area. AcLelk' E8V bc highly variable due to its dependence.on surface tension, velocity of impact.

relative densities of the two cocpone r.t s . and angle of sepact. Though the techar.ics of the impa c t may be complicated by the above factors, a reasonable assumptter. is that the sentact area i. tlie order of the cress-sectional area 'of the staller droplet. Theref ore we def ine the cor. tact area to be Ac tr:LL " C1"The 'pc < r pk a r.d (T-7) r pm Brgp AcLtLL " fl"Thk The value of r j ma y. range from 0.1 for g l a nc i r.g collision or low icpact

. velocities to thg/rh. if r pn < rp k o r rjn/r[g if rpm ) r pk. This upper licit corresponds to the situation in which the smaller droplet deforcs and covers an area equal to the cross-sectional area of the larger droplet.

The heat-trar.sfer rate per unit area durirg the collisior. is assumed te be quasi-stat a. This is c oni. i s t e r.t wi t h othe r heat -t rans f e r code lir.g in Slktil'R-il th t approxirates the process reasonably well for a millisecend. I r. t h i a-333 v* . , - - - -- .w e , - , , , , , , , - , v., ,

y ,--. , _ . ,

ROU3H CENTROIDAL DE

~

- 0.2 r ,m SURFACE r

0.2 r , ~ -- ..,,

s ...

, i 5

t f

LlOUIDu ,

a - .

CONTACT AREA .. .

N LIQUID.

fig. F-2. Collision cor. figuration for liquid e ar.d 1 quid k.

parti 6ular case of Spherical dreplet i n t e ra c t i or. . it can be argued that because cost of the heat capacity in a sphere is near the outer surface. the effective corduction path l e r.g t h is gercrally very stall (about 0.2 r p ). lor droplets with radii et about 1o' n the thermal respon<e tire car be approximated bv ro, tr-61

= 1 10.2 rp r where is the thertal diffusivity of the liquid. Fuel. steel. and s od i urz

't droplets (10- ' r radiu ) respond thermally in t ime intervals of 0.070 s. o.005 ,

and o.0005 s respectivelv. The quasi-static approximation is therefore not goed 3

v for fuel ard steel droplets of 10' n radius but would be adequate for these with 10' r radi s. For short c or.t a c t times ard these relatively large droplets.

the average heat-transler rate during a tollistor will be s ubs.* .r t i a l l y undere<.tstated by this codei.

384

F. e r 7-. , ,

)1j g The overall heat-trtr.sfer coefficient is based or conduction between tw slabs shown ir l' i g . I -I and assured to have thicknesses of 0.2r pr. ard o.2rg.

The e thickresses represert the distanse t ror the outer surface of a flattered or deterred spher- to the centroid surface (one-half of the mass is irside this sur f at e ). The overall heat-transfer coefficier.t is approximated by U *

(F'9) delk

• m 2 r pr . Rh pk k gg L gg The contact tire per collisier. also is uncertain due to the nature of the co l l i s t or. : i.e. the velocity of collision, the angle of collision, surf ace teraion, a r.d thermal int eract ion at the contact surface (film boilir.g. etc.).

k'e corveniertly estirate the contact time on the basis of a mutual residerce time, cor.sidered to be p r o po r t i or.a l to the tire that the two i nt e r a c t i r.g A B l I

+ r g)

~ 2( r ,m Vg . .

l Q ,

~

CONTACT k...CONTACT BEGINS '

ENDS

=

Imp l'ip. f-3. bsur.ed cortatt interval.

385

n n ?

i.r* :. } I'. ". r-= p ii V Oi l Lj;l f j dropicts or particles are i r. possible c or.t a c t . Figure F-4 illustrctes the s i t u.. t i .r it which dreplet L is rwviry relative to droplet m with ar aver,qt velesity 5. The tertast seuld begin wher. dreplet k is at pos i t i er. A ard er.J when it is at po s i t i e r. 11. The distar.cc droplet k travels is about 2rp t + 2rpn.

The cor.t.at t ir e . t tntg. is giver by t ctett 2('y .' r pn . + rt i p

( ,

= ,

The c or.s t a r.t ('y r e p r e s e r.t s the fraction of the residence time that would characterize the average c o l l i s i e r.. The average velocity is divided by two because ir g e r.e ra l droplet m will be movir.g. and as a result, the relative velocity will be reduced.

ASSUMED TEMP.

PROFILE TRANSIENT TEMP.

PROFILE w

u

[ T, t a:

W .

LIQUID. E LIQUID.

~ 0.2 r ,.  : - 0.2 r, - E

~

fig. 1 - 4. lleat-trar.sfer treatnert durir.g contact.

386

"* f . i i p 4 s

[.r.= -

LUGliL)hk[

The urce rtairt y-~ in the average contact time is related to the unt e r t a i r.t v ir the be.t trars!er per collision because of the thermal tire-corstart tor.sideratiors discussed Previousiv. To sore extert the uncertainties tartel.

The error g ir. .the heat-transler rate -during a collision, qctntg. will be inverselv proportional. to the contact time due to the transiert t e epe r a t u r e -p r a d i e r.t effect. Iletause the total heat t ransferred per collisior.

is the product of qq tett and t cLtLk the tagr.itude of the error.in the total heat transf erred may be considerably less t h a r. the individual errors i r. qc LtLk a rd -

'cLcLL' The'overall heat-trar.sfer rate per unit volute is given by the product of the heat t rar.sferred per collision and the number of collisions per unit volume ar.d tice.

Z

'411LeLL

• S LeLk A c Lr:Lk tcLtLk TLk - Tte LtLk

• II'llI Substituting 1:q s. (T-4), (F-7). (F-9). and (F-10) it (F-11) gives the heat-trar.sfer rate.

1 'Le 'Lk (r{g+ r{g '

qIlltLL = L, 45 b r pg /L t + r tp /Ltg ;) - og j r pe rg

'r pn

+ rg)rj(TLk p - Tte i .

(F-12) where ,

ry = rtp if r t p< r pn ry = r p7 if r pt > r pn C = O gr y .

In Eq. ( D-12 ) Eg and E t have beer approximated by E. Thus, the rardor velocities of . va r ious types of droplets or particles are characterized hv a s i r.g l e average value E that car.tels ir 1.q . (D-121 The i1 - c.3) tern due to 387

RD M CRA:T structure is the ret result of cor.verting the liquid-field-volume fractier.s fret a t o t e.1 6 e ! 1 vo l t.re basis to o flow 6harrel basis, but keeping the heat-trar.sfer rate or a tetal 6 ell v, l u e basis upprepriate to its defir.ition in the lenservatson equatfor.s.

As. the radius' r pe of e r.e of the droplets approaches zero. Eq. (D-12) r e d u(. e s to k o te, o gg 45 tg '

j 41LtLk 1 =C 7- ,, - (Ttt - Tte)_ . (F-13)

The heat-trar.sfer rate approaches i r.f i r. i t y a s r pe approaches zero, resultir.g from the product of the nurber density of competent m ( Eq.- D 5 ) ari t he cor.t a c t area _ per collision. e rP (. Physically, the average instantaneous contact area exceeds the total surface . area of all droplets of c ompor.e r.t k. This effect-gives a heat-transfer rate greater thar. that which could occur with zero tlierr al resistarse ir the l i q u i d t otte r.e r.1 r;. The limitir.g heat-transfer rate for this latter case is given by k

LL

( qllLtLL ~Ca x " 10 C 'Lk 7 ,TLk - Tte l as r pe approaches zero . (F-141 pk or if r t pappron bes zero.

(411LtLL _ " 10 I' 'Lt (TLk - Tte j . ( D 151 pm 388

D r. ! ! n ' m j,

I\ v J ,

APPENDIX G nlRkl.CTION SET FOR W)DilIED LIQUID-LIQUID llEAT TRAN511:R 1

• Ilil.NT 11514 2*

3* n>RkirTloN si:T To TokCE VAPORIZATION 10R LIQUID /LlQUID 4 */ lil:AT TRAh5fEk 5 *i 6 */

7 */ Till: ilEAT TRAN5fER IN TSilTR IS W)DIFIED FIRST 5*/

9 *I T511T.333 10 IT (ALPilG(lJ) . LT. ALP 110* ( ONE- ALPilS( I J ) ) . OR. NCVC . EQ. 0 )

11 1 GO TO 7704 12 I T ( N . EQ. 3 . AND. TL( M ) .GT. TL(N)) OllTC=ZERO 13 if ( M . EU. 3 . AND. TL( N ) .GT. TL(M)) OllTC=ZERO 14 77o4 r0NTINUl:

15 *D TbilT.423 16 IX) 627 N=1.2 17 *1 T511T.443 1b IT (M . EQ. 3 . AND. TL( N) .GT. TL(M) . AND. NCVC .GT. 0 19 1 . AhT:. ALPilG( l J ) .GE. ALP 110'(ONE- ALPilS(1J ))) OllTC = ZERO 2n * /

?! */ INITI ALIZATION Of VARI A13LES FOR VAPORIZATION / CONDENSATION 22 */

23 *ll PilW.25 24 DIMINSION llL3(61 25 ! P!!M. 46 26 IIL3T = ZERO 27 HL3TN = /Eko 25 10 lof 4 h=1. hT LI.

29 llL3( N )=Zl.ko 30 11 (N .1:U. 31 GO TO 1955 31 i f - ( TLt N i . LT. TL( 3 ) ) 60 TO 1056 32 IF (RPtN1 .LE. ZERO) GO TO 1055 33 If (RPt 3) . LI:. ZI:RO) GO TO 1055 34 kt1=l EOSLE( N )

35 TilCON = TilCONL(ht1) 36 CVLL=CVL( hti) 37 1F ( N .GT. NCLEM2 ) TilCON-TilCONS(hti) -

38 IT ( N .GT. NCLEM2 ) CVLL=CVS(ht!)

39 OllTCL2 = ROLP(N )*CVLL* RDTT 40 illt = TilCON*llX1(N) 41 A12 = IIL1 *llALPS*VOLFX(N)*VOLfX( 3) 42 1 * ( ( RPt N 1' 2+RP( 3 )"2 )* (RP( N)+RP( 3 ) )* RP( N i'

• 21/( RP( N)" 3* RP( 3 )*

• 31 43 OllTCL = C15P*llL1 *VOLTX(N)/RP(N) -

44 IT (RP( 3) . LT. RP(N))

45- 1 A12 = A12 * ( RP( 3 )* '2/RP(N)*

• 2 )

46 1057 CONTINUE 47 RLMULT = RLL(N.31 45 II ( h . GT. 3) RLMULT = RLL(3.N) 44 ' ll -Dl:1. Dill .1 50 llL3( h i = AMIN 1( A4th1 t ollTCL. 412 P RLMULT.OllTCL21 51

• 11 I)l:f.Dl:L.1 52 IIL3( h - ININ11IX11hl t ollTCL. A121* RLMULT.OllTCL2 )

53 IlL3T = IIL31 N)+1tL3T 389

m a' uun J L M 54 IIL3TN = HL3TN + llL3(N)*TN(NCSE+N) 55 1055 CONTINUl:

56 IlL3T = IIL3T 1)TT 57 IlL3TN = llL3TN 1)TT 56 */

$9 */ FIRST. VI: MODIFY Tile CONTINUITY EQUATION 60 '/

61 'l Pil AS. 982. -

62 11 ( N . NE. 3 ) GO TO 7704 63 Al Rllot N ) = AlRil0CN1 + llL3TN 64 A2Rilo(f.I = A2Ril0(Ni - HL3T 65 7704 CONTINUE 66 '/

67 */ NEXT, VE W)DIFY Tile VAPOR SIDE IIEAT TRANSFER COEFFICIENT 68 */ ITERATION 69 */

70

• I P}lAS 1322 71 FOSUM = ZERO 72 I F ( N . NE. 3 ) GO TO 34 73 IX) 35 M-1.NCLE 74 35 FOSUM = FOSUM + HL3(H)*(TTSAT(N)-TN'NCSE+M))

75 ARGG= AKGG- CPDilFG(N)*FOSUM 76 34 CONTINUE 77

• I Pil AS.1332 78 FO = 10 + CPD}lFG(N P roSUM 74 */

50 / TillRD. VE PUT THE MASS TRANSFER INTO THE VAPOR ENERGY EON.

b1 "/

52 *I PilAS.666 '

53 FOSUM = ZEko -

64 11' ( N . NE. 3 ) GO TO 36 65 IX 37 M-1,NCLE

$6 FOSUM = FOSUM + HL3(M)* (TS AT(N)-TN( NCSE+M))

57 37 CONTINUE 88 36 CONTINUl:

54 "I Pil45.855 90 AR1 = AW4 + foSUM 91 1 PilAS.905 92 SUW = SUMP - FOSUM*DTT 93 */

94 */ ATTI:R CONVERGENCE VE TAKE THE ilEAT AVAY FROM THE LIQUIDS 95 * /

96 *1 PilAS.1525 47 foSUM = ZEko 95 lxl 35 N-1.NCLE 94 11 (llL3(N) .EU. ZERO) GO TO 35 100 UL( l J+N-1 ) = QL( l J+N-1 ) + HL3( N P ( TTS AT( 3 )-TN( NCSE+N) )

101 SI ELN( N 1=SI ELN( N)+( HL3( N ) * ( TTS AT( 3 )-TN( NrSE+N i ) )/ ROLPP( N )

102 l osVM-105USH L3( N ) * ( TTS AT( 31-TN( NCSE+N i )

103 35 CONTINUE 104 *I PilAS. loo 4 105 11 ( N . EU. 3) 100 1(na 13 J-(Kit I) )-FoSUM 107 */

106 '

MODIFY Tile llEAT OF val 0Rl/ATION RESULTING FROM I:N!:RGY LOSS 390

piq g -

m , - -

d w' . ,

109 */

llo ' l 11754. 67 111 11 L . NI:. 3) 112 1 SI:f;T t i; - SENT t h i - 1;TT IIL3t N ) * ( TS AT( 3 )-TN( frSt.+N ) 1" Rkotl'( h 1 113 *1 11754.109 114 11 ( f; . NL. 31 115 ISENT( N i - SENT( N) - DTT* llL3( N )' ( TS AT( 3 )-TN( NCSE+N ) )

• RROLP( N )

e f

391

5m oq g 3m p=-

Vw 1 ./ b APPL:hl)lX 11 VAPORIZATION /rohlil.hSATloh CORRirT10N SET 1 IDENT B754 2*/

3'/ Till5 CORRECTION SET W)Dil li.S Till: PilAsl. TRANSITION MODEL 4*/

5 'D PilAS 983 6 'D B1EO.25' 7 'D PilAS.984 5 'D B4K3.277.260 9 *D PilAS.965.1046 10 *1 PilAS.962 11 IVPP = IVPALL(N) 12 Ir0UhT(N) =0 13 'D PS43.15 14 C 15 C IDIR(N) = +1 IMPLIES CONDENSATION 16 C IDIRlN) = -1 IMPLIES VAPORIZATION 17 C 16 IE (IVPP .EO. 1) GO TO 443 19 SETDR = AIRH0(N) + A2Rilot h)'TSAT(N) 20 IDIR(N) =+1 21 IE ( SETDR .GT. ZERO) IDIRf N) = -1 22 443 CohTINUE 23 R(KiPPP( N) = R(XiPPt N) 24 IE ( NITNU .EV. 1 i GO TO 7705 25 II ( TG. GT. TCRI T( N ) . Ahl). TLIV' N ). GT. TCRIT( h ) i ICOUNT(N) =1 26 I E - ( ICOUNT( N ) .EV. 01 GO TO 7705 27 XLR(Ki(N i - EM10 25 11~ ( IDIR( N) . EV. -1 .AND. TSAT(N) . GT. TCRIT( N I) 29 1XLk(Xi( N ) = ( PCR1T( N l' ALPilGi IJ ))/( RMATt N)* TG) 30 XilR(Kit N ) = EP10 31 'D Pil AS. 1051:1053 32 'D B7SO.1,3 33 'D PilAS. 1060.1071 34 slEGLN = SIEGL 35 10 447 NN=1 NCVC 36 f= ICGC(NN) 37 SIEGLN = SIEGLN + A3RH0( N)* (TSAT( N)-TTSATP(N))

35 447 CONTINUE 39 'D PilAS. 1076.1077 40 *D B7G9.56 41 'D Pll AS. 1096,1112 42 'D P240.26 43 'D Pil45.1114 i 44 'D P240.27 45 'D PilAS. 1116.1120 46 'D P240.28 47 'l Pil4S.1140 48 DCODTS = CVLtN) 49 11 ( TSAT( N).GT.TWO3RD TrRIT( N )) IX'0DTS = CVLP( N) - 11 All 'DilSDTSt N i 50 SETDR= A1 Rllor N )+ A2Rilot N )'TSAT( N I 51 II ( SETDR.UT. ZERO . AND. Rt Kil'P h ). GT. R(KiPt N ) ) IDIRt h i=-1 52 1E ISETDR.LT.ZERO . Ahl). kt KiPPt h 1. LT. Rt KiPt N ) i IDIRth>=+1 53 IE (IDIR(N) .1:V. +1) GO TO 635 392

'[O  %

g r-- -

[w J d j }\X ,

54 SENT( N ) - Sr.NT( N ) + Sil:Ls(N ) - Sil:LNPIN) 55 1 - DTT*ltl>(N)*kROLPtN "(TSATtN -TLlV(N))

56 1)llsDTSf N ; = lillsDTsi N + lx'01)Ts - DTT lilit N 1 RKOLP( N) 57 (,0 To 64o

$s 635 00NTINT).

5- SENT(N i - SIEGLN - Sll:LS( N) 6o DilSDTSI N ) = -lit ODTS + A3Rilo( N) 61 640 CONTINUl.

62

• 1 Pil%.1144 63 I F ( SI:NT( N ) . EU. LP21 DilSDTh(N) = -1:M3 64 'D Pilu.114o 65 FMED - SETDR*RSENT(N) 66 *D Pilu.1152.1150 67 'D AEOS.275 68 11 (TSAT(NZ) .Gl:. TCRIT(NZ)) GO TO 342 69 *1 Al:05.277 70 1F (TSAT(NZ) .GT. TV03RD TCRIT(NZ)) GO TO 342 71 'll' -DEF. DBL.1 72 S I ELS( NZ )= S I E L I V( NZ )+ AstrX1 ( TS AT( NZ ) -TMLT( NZ ) . ZERO )'CVL( NZ )

73 *IT DEF. Dill.1 74 S I ELS( NZ )= S I EL I Q( NZ )+DstrX1 ( TS AT( NZ )-TMLT( NZ ) . ZERO )'CVL( NZ F 75 GO TO 340 76 342 SIELS(NZ) = SIEL10(NZ )+CVLP( NZ )* (TSATINZ)-WO3RD*TCRIT(NZ i) 77 1 + Al:USLMt NZ 1-il All

• SENT( NZ) 78 *D Pil%.1224.1242 79 *D Pl!u.1227 bo *D l'i ss.1595 bl *D PilAS.572 52 GO TO 3365 53 *D Pilu.443 64 *! Pilu.1160 55 R(Xil'2t N ) = R(KiPPP( N )

56 I Pilu.1162 57 11' (IRilolT . I:U. 01 (in TO 77032

-85 11' ( Ir0VNT( N 1 .1:U. o s GO TO 77031 89

• If -Df f.Dl3L.2 90 77024 11' (1 .LT Zi:RO X11R(Ki( N ) = AMI N1 ( R(KiPPP( N ) . X11R(Ki( N ) )

41 II' ( F .GT. ZERo e XLR(Ki( N ) = .etAXI ( R(KiPPP( N ) .XLR(Xi( N ) )

92

• lf del Dill.2 43 77024 1r (I . LT. ZERo n XllR(Xi(N) = DMIN1(R(KiPPP(N).X11R(Xi( N))

94 11 ( T . GT. ZERO ) XLR(Xi( N ) = D41AXI( R(KiPPP( N ) XLR(Xi( N) )

45 IF ( R(KiPP( K) . LE. XLI:(x;( N : .OR. MOD (IRl101T.20) . EU. 5 . OR.

96 1 R(XiPP( N) .GE. X11R(Xi( N > ) R(XiPP( N ) = ( XllR(Xi( N ) +XLR(xi( N ) ) *ll ALF 97 IF (IRllolT .EQ. 50) PAUSE 98 Go TO 77032 99 77031 CONTINUE 100 if (IR110T2 . EU. ZERO) GO TO 77032 101 DR(XiP2 - R(KiPPP( N1 - R(XiP2( N) 102 DR(kip = R(XiPP( N ) - R(XiPPP( N 1 103 11' (DR(kip 2'DR(XiP .61:. ZF.Rol Go To 77032 104 DENOM = R(KiPP( N i - TVO R(XiPPPi N i + R(XiP2( N )

105

• I f -DI:f. Dill. 2 106 I F ( Allsi Dl:Nr P41. GT. I:44.

• R(XiPPI N i = R(KiPPt N 1-( DR(XiP'DR(XiP)/DE9P.1 107 I M AX = AMW1( I MG. Alls (11R(kip i' RTil l lob 'll DE F . Dill. 2 393

%n 3 3 ---

&lq ~

J {\

109 Il' f DABSt DEh0\1).GT. [416 ) R(XiPP( N i = Rt XiPPi N )-( DR(kip

• DROGP )/DEfdr.1 11" f 41W = D51u1( l k16.1411S(I)R(kip P RTil i 111 77o32 rohTINUl:

112 *1 PilAS.1222 113 SENTPI N i - SEhT( h 6 114 11~ ( ll)lkt N) .IO. +1) 00 TO 645 115 SENT( N ) = SENTI Ni + SIELSt h) - Sil:LNPt N1 116 1 - DTT*llD( N P RROLP( N P ( TS AT( N)-TLIV( NJ )

117 GO TO 650 115 645 SENT(N1 = SIEGL - SIELSt N) 119 650 CONTINUE 120 *D PilAS.590 121 I%ti = ZERO ~

122 11 (IDIRIN) . EQ. -1 ) D461=SIEGS( N )- SIEGL+CVGB AR* TG 123 "D B151.3 124 IF (IDIRfN) . LV. +1 ) GO TO 5564 125 *D PilAS.902 126 $UsD1 = SUkB - (( Ahlbti TSAT(N))*CVGBAR+BNtt'Dkti)*RSENT(N) 127 5564 CONTINUE 125 SU4tB = SU4E + B\t1 129 *D B7G9.49 130 5565 SUNUl = SUNUl + B\t1 131 1F (lDiR(N) .EU. -1) 505111 = SUhm - (R(XiPP(N)-ROGP(N))'CVGdAR 132 1 Pil AS. 906 133 I f ( NCSC , EV. 01 GO TO 5575 134 bat 1P = ZERO 135 IX) 5573 kl = 1.Nche 136 B4t1P = B5t1P + llVS(kl.N)

- 137 5573 CONTINUl:

13b SU!.1B = SUNDI - DTT* Int 1P -

139 SUW = SUW + DTT* Int 1P TS AT(N) 14o 5575 CONTINUE 141 *D PllAS.27 142 D141ENSION 1 DIRT 4 ). SIELNPt 4 ).RROLP( 4 ). A3R110( 4 ).RtXiP2( 4 ).

143 1 XLR(Ki( 4 ) .XilR(Xit 4 ) . SENTPt 4 )

144 al PilAS.764 145 SIELNP( N ) = SIEL(IJ+N-1I + CVL(h)*(TLIUCN)-TL(N))

146 RROLP( N ) - ONE/ROLP( N) 147 SENTP( N) = SENT( N) 145 *D PilAS. 1122.1136 149 *D PilAS.972.973 150 *D Pil6.964,965 151 "D Pil4S.975 152 "D PilAS.516 15 3 *l) Pilu.1191 154 *1 PilAS.976 155 SI:NT( N i - SENTP(h) 156 A3Rilo! N ) - CVLt N) + DilSDTS( N "( ON!:-TTS ATi N P RTSTAR( N1) 157 11 iTSATiN) .GT. 1YO3RD*TrRil(N11 155 1 A3Rllot h i = CVLPt N) + DilSDTSt N P ( ll ALI -TTS ATt N P RTST AR' N ) 1 159 A3Rllo! N i - A3Rllo: N : -SI:NTt N P RTST AR: h > -rvGki N :

16o 11' . ( CVGRi N ) . LI:. rVGt h ) + EA11o i Go To 445 161 11 i TU .Gl:. TTS AT( h >

162 1 A3Rllot h p = ( A3Rilo, L P BP AR AI N r 1/( TG-TTS ATt N i+11P AR4( h 11 163 IF ( TG . LT. TTS ATi h a s A3Riloi h i - A3Rilot N i 394

)\v lO' Jd' L A f7 ~

~

164 1+ t ( - A3R110( N )+TWIP ( CVGt N i-rVOR( N 11 ) * ( TG-TTs AT ( N ) ) 1 165 2 / f TTS AT( N )-TG+IIPAR A( N t )

166 445 43RilotN = ( A3R110( N )* RtxiPPa N1 RTil1/R(Ki 1 6 7 *l) Pil u.157 7.1579 165 11' ( IDlk( N i . I'U. 1 ) GO TO 7735 169 (. Kit 1 J l = Q(i( l J ) + (.Kirt N : - (iet'l:( l J NL)* S1 EGS( N )

170 QLt 1 J NL i=VL( l J NL 1+VLC( N )+G A49 'E( l J NL ) * ( SI ELNP( N #+DTT* RROLPI N )

171 1

• VLC( N i l 172 SIELNt N I = SIELN(N) + QLC(N)*DTT* RROLP(N) 173 GO TO 7740 174 7735 (. Kit I J ) = (.Ki( l J ) + QGC( N) - G AMCI:( I J NL)* SIEGL 175 VL(lJNL1 = QL(IJNL) + OLrt N) + U AMcE( lJNL )* S1ELSI N) 17o SIELNtN) = SIELN(N) + ( SI ELNP( N)+

177 1 ( ( QLr( N )+G etCE( l J NL )

• S I ELS( N ) )'DTT

• RROLP( N ) ) )

175 2 /( ONI +G AMCE( l J NL )

• DTT

• RROLP( N ) ) - S I E LNP( N )

179 7740 CONTINUE 160 *D PilAS.785 161 *D 117G9.33,36 182

• 1 Pil6.960 163 IRil0T2 = 0 184 *1 Pilu.1205 155

• If DEI . IIN.1 156 IRil0T2 = kW)D( lRl10T2.21 15 7

• l f -DET . IIN.1 155 IR110T2 = IRilolT. AND.1 159 *D Pilu 1196.1195 190 *1 Pilu.1263

- 191 IE (NITral .Gl:. 50) PAUSI:

192 *1 Pilu.1535 193 IT (IVPALL(N) .1:U. 0 ) GO TO 670 144 SU'!) = 2000 195 SUM 2 = Zl:RO 196 DO 665 M-1.NCSr 197 II' ( ISTRilT( M.N ) .EU. 1) GO TO 670 19b SUM 1 - SUM! + 11St M. N )

• TST( M > + llVst M. N )

• TGN 199 $UM2 = SUM 2 + llS(M.N) + llVst M.N )

200 665 CONTINUE 201 TTSAT(N) = SUMI/ SUM 2 202 670 C()NTINUE 203 *D Pilu.1416 204 11 ( I STRilT( M.N ) . EV. 1 . OR. IVPALL(N) . EU. 1) 00 TO 34055 205 */

206 */ AVo!D TROUllLI: BY CORRECTING PRI:VIOUS IDENT 207 *,

20b IDI:NT 11514 209 *1 I1754.54 210 1 F ( SI:NT( N I .GT. EP3 .AND. Dft)kG .LT. -EMI) GO TO 454 211 XilRt Xi( N ) - EPlo 212 XLR(Ki( N > = EM20 213 IrOUNTIN> = 1 214 GO TO ~7024 215 454 CONTINU!:

216 *1 11754.56 217 I F ( X11R(Xi( N ) . LT. I:M10 ) IrOUNT(N) =o 218 *1 11754.16 395

p n== *

• g ng j gummm W

• RI6 d Ul '

I i <

214 A41P = ZERO ,

220 11491 = ZEko 221 *1 11754.1b

,222 ant 1P = A41P + 11S( kl.N J TST( N1 s 223 B491 - B4HI + ~ 11S( kl. N )

224 *D 11754. 20.21 225 1F ( B4t1P+B4111. LT. ONE ) GO TO 8575 226 SUNEl = SUkDI - (DTT*ll41P IPtill1/(list 1P+1BtBI) .

227 SUht' = SUk1C + (DTT*BNt1P Akt!P)/(B41P+B4DI) 226 *D B754.113 229 *D B754.67 230 *D PilAS.1150 231 *1f -DEF. DBL.1 232 DFDRG = Ah11N1(DFDRG.-D12 )

233 *lf DEF. DBL.1 234 DFDRG = Dk11N1(DFDRG.-D121 235 *D PilAS. 1185.1166 236 IF (lRilolT .GT. 0 . AND. W)D( lRil01T.20) .N!:. 5) GO TO 77039 237 IRiloIT = IRilolT + 1 235 D PilAS. 1199.1204 234 'If DEF. DBL.3 240 1 F ( ( DGSt TTS AT( N)-TS ATt N ) 1. LT. D15 . OR. D1AX. LT. FTEST )

241 1 . AND. DGSf R(KiPPPt N)-ROGPP( N))/ROGPPt N) . LT. D16 )

242 2 00 TO 77052 -

243 "Il , -Dff.D!lL.3 ,

244 . If ( ( ABS (TTSAT(N)-TS AT( N i ). LT. Dib .OR. DiAX. LT. FTEST) 245 1 . AND. ' ABS ( R(XiPPP( N #-k(KiPP( N ) )/R(XiPP( N 1 . LT. Dio )

246 2 GO TO 77052 247 *D B754.76 245 *D PilAS.1161 249 *1 PilAS.1147

'250 R(XiP2( N) = ROGPPP( N )

251 R(XiPPP( N) = ROGPP(N 1 252 *IDENT llrla 253 */ '

254 */ kitKE ADDITIONAL CORRECTIONS 255 */

256 *I 11754,2 257 Dlh1 ENS 10N TSATill4) 255 *I PHAS.969 '

254 I F ( I VPP . EV. O ) GO TO 424 260 TS AT( N ) - TS ATil( N )

261 TTS AT( N ) = TSATil( N )

262 424 CONTINUE 263

• 1 Pil4S.122' 264 TS ATili N1 = TSAT(N) 265 *1 PilAS.1314 266 1I ( lVPALL( N ) . EU. 1) GO TO 34001 267 I PilAS.1314 268 TSATIN) = TTS AT( N )

264 TS ATil( N : - TTSATiN) 395

.i

RO3ll [RAF Al'l>l:hl)1X 1:

Slkt1ER-Il MANUAL TRI:AT4ENT l'OR Till: val'ORIZATION/(UNDLh5ATION MODI:L The roterial ir this a rre nd i s is t aker. di rec t iv t ror t he Slbt11:R-ll canua l .

It has the sare sestien r.uthe r s and equ.. tion nutbers. References to other

~

appendices within App. I are to appendices of the S121ER-Il manual. This raternal has beer included both f or convenient reference by Sec. Il-E, ar.d for the material ir. this report to be as self-contained as possible.

7. Methods for the Sierle Vaporization /Condersation Model. In S141ER-II .

the eethod for determining the vaporization and condensation rates differs considerably frem that used ir. Slkt1LR-1. To obtair. a phase-trar.sition rate that autecatically accourts for the c ha r.g e in saturation properties from phase t ra ns i t ier.. the vapor c orpen e r.t s are treated implicitly. Dr. l y the liquid ar.d structure corporer.ts are treated explicitiv. It also differs f rom t he - Sikt1ER-1 t e c hr. i que in that the sol ut i er. is obtained iteratively rather t ha r. i r. a sirgle step. Cetpared with the 514t1ER-1 method. the iterative solutior. method-provides more conW stert a r.s w e r s a r.d nr.terrediate results that are easier te ir.terpret when debuggirg. Ihwever. the iterative t e c h r.s qu e is more coeplicated to describe ar.d the solution requires core computer time.

The equatior.s are preser.ted i r. t e rr.s of the iteratier. values of the vapor properties. These values approach the end-of-time-step values as the iteration converges. The heat flows fer phase transitien at a liquid-field energy-ccmponent interface are gj((tn= lid $,'TCt.M-T[*l. (IV-56) for the heat flew to the vapor field f rom tl e interface ar.d 45LUt " II LGe Td,y-i lt

' IIY-b7) for the heat flow to the l i qu id-t s e ld cre rgy t orporer.t trot the n r.t e r f a c e . liere 397

f_

ROUGH [ RUT 4

x _is the s t e ra t ier. index for heat flow, and the heat-transfer coefficients and surface areas h.a e heer combired into the a. i r g l e va r i a bl e 11.

4 The s e l u t i r. re t hod assunes that'the liquid ' possesses sufficier.t thertal i r.e r t i a so! that liquid temperature

• are not- affected significar.tly 'hy vaporizatior or corder.satier ir any ore time step. This is. not a good a ppr ox ita t i er. for stall arounts of a liquid component possessing limited heat ca pa c i t y. 1:xami na t.i on of implicitly formulated liquid-component er.ergy

equatier.s reveals- that e ve r. for infir.ite liquid-side, hea t -t ra r.s f e r coefficients. the er.ergy-trar.sfer rate to the liquid should r.ot be larger than I r.+ 1 Lt' 41 Lect

=

'vl4 (TSat.M-T[e)

, . (IV-SS) at lle c a u *. e other processes can, char.ge the liquid temperature, experience has ir.dicated that sor e fractier. of this maxitut rate is desirable. Therefore. the maximum liquid-side heat-transfer coefficiert is restricted-to IC L~ vl4

-(hJLr Att 'r a x * "9 -

• IIY~89) at The va po r - f i e ld t empe ra t ure i n I:q. (lV-56) is evaluated at the advanced time. The liquid-field. energy-corporent temperature also could be evaluated implicitly but this wuld complicate the algorithm further. To partially a c c ou r.t for the thereal response of the liquid-field energy component, the temperature used in 1 9 (IV-87) deperds o r. the change in the liquid-field e r.e r g v-c oeper e nt temperature durir.g the time step from other heat-transfer mod e a. . This sharge i <. predicted by the implicit heat-transfer calculation. The basis for the cheite of temperature is that the liquid-side, heat-transfer c oe f f i c i e r.t is ua.ually significartiv larger than the vapor-side, heat-transfer c oe f t u s e r.t . Thua . compariaer. of the liquid-1:elJ. energy-competent temperature with the saturatier temperature detertirca, whe t he r va por iza t ier. or cordensa t ier.

398

. 7m D [

J Jil n Lt' 'MIn r',- -

is likely te occur. Also tor.sidered in the cheice . is -that the coepnnert t er pe ra t ure p r ol'a b l y till arcreast d u r i r.g tordersation a r.d desrease durirp vaparization. These 6ersiderat kr result ;in the icllowiry selettior. f. r' 11.6 liquid-field energv-corporent temperature to be used in Eq.-(IV-57).

-t+1 -r+1 Tg=

g Tty . -if T[ t .k1 > Tte >T[e

-r+1 or T[at.k1<Tte <T[e  : (IV-90)

-n+1 if Le = T[ra ,

T[at .51 > T[e > TLn

- r.+ 1 or T[a t .51 < T[n < tlc IIY-913

-r+1 T

Lt = T[,, , g3 . if T tn >T[,,,g,>T[e

-r+1 er Ttn < T at.hi<T[n , gjy,933

- r.+ 1 -

Jr. general. T Ln 'ard T,n [ differ little and the above choice has -lit t le effect compared with usiry the advanc ed-t ime -t eepe ra t ure. Ilow e ve r , wher. the thermal r e s pe r.s e t irce of - t he liquid-field er.ergy component is stall. this approach provides stability by a r.t i c i pa t i r.g . t he c ha r.g e i r. the liquid-field component teeperatures resulting fron vaporization or condensatior..

In a - s it:l; r ca r.n e t . the heat flows from the interface- of .the

/

structure-field er.ergy cstporent are evaluated-as

+1 1 I 4"1GnSL . 611"+5L : T"s*a t 6. 51 - T".+1 1 (IV-93) for heat, flow to the vapor field trot the arterface. ar.d 399

lO I N' J h h*f c.

n+1 -r+1-

'II5kGr

• USGL t,Ts+1ha1.y Tgt ( l \,-94 )

for the heat Ilow.te the 'st rut ture-I s eld energy component from the arterface.

Aga in. - t he product s of the heat-transfer toefficient a r.d ' t he surface area are .

c omb i r.e d. The c'otpo r e r. t temperature used i r. 1.'q. (IV-94) is the advarced-tite temperature because the frozen caterial's resistances are added to the resistarce fer the underivirg comperent when the thermal response of the frozer.

caterial is rapid.

To obtair. the expected charge ir. the saturation cor.ditions durir.g the time step, the cass-conservation e qua t i er.s for the vapor-field caterials a r.d the vapor-tixture e r.e r g y equa t ior. a re used in a t r ur.ca t ed fort that ir.cludes only terms from phase t r a r.s i t i o r.s . After differencir.g n r. time, the t rur.ca t ed ma s s-cor.s e rva t ior. equa t i er.s a re 7[+l=y ;y - td r[;h t = 1. .... hm T-1 . (IV-95) s.

where f(";[I is the tatroscopic der.sity of vapor-field caterial c cepo r.e r.t c after s-1 sterations.

$ is the estnr.ated tsee step. ar.d li";h, is the total tass-trar.sfer rate frot vapor-field caterial componer.t e after =+1 sterations.

This total phase-trarsitzer. rate is giver. by 3

1 =+1 x+1 I s+1 +1 I=+Lt1"+

G .416ttr + 41LmGr + - {41btSL + 'l=l5kGta) * (I5'"I h,,3 j ,q L-1 shere 1 h" q is the heat of va r.ir i za t ior f or r:a t e r ia l M a l t e r =+1 sterations.

400 i

ROH DRA7 1:qua t i er.s (IV 05 : arJ (IV-4bi are the first e qua t i or.s solved ir the phase-transitier riid e l because a paed e s t i na t e of the cass-t rans f e r ra t e is requi red ,to solve t hi var r-energy e c u.. t i . . r. . To evaluate the heat flows i r.

1:q. IIV-96). T";*g I . Il((,l . a rd il(";'d a r e a p p r o u ta t e d by T(";. Iljt,,,,. a nd il(";sg. Ther..

we rote a strong deperder.te of vapor-fie ld densit ies on sa t urat ior. condi t iers.

t'or.sequentiv. a s imple - eva lua t ior.. of saturation properties from the I:OS a r.d their i r.s e r t i er. i r.t e ' t h e e qua t i er.s gererally overest icat e the degree of cau t ra r.s f e r. An iteratier. is required to couple the nonlinear relatior. ship betweer.

f(";y,l ardTd,y it the EOS with Eq. (IV-95). The iteratior. proceeds by writirp the differenced, truncated, mass-cor.servation equation for each vapor-field c orpor e r.t as x

8 Tn+1 7=+1 , yr 1.r. + "x2.r 33 3 , gg bc .Gr # . (lV 97) n.3 i (p . k1 where

"$ . r: = at lll("ilr T"; + litg. i Lr:

g *A lII[iSk T[; + ll3gt T-r+1}

3t .

k-1 (IV-95) ard 2

D3,e--at (ll{t.+II(i-+ L Z III[iSk + IISUL I * 'IIv'99) k=1 To fir.d the saturatier t empe ra t u re , heat of vaporization, a r.d va po r -f i e l d (otporer.t der.sity that are (onsister.t for s a t i s f v i r.g 1:q. (IV-97) and ' the EOS relation betweer. the saturation temperature ard the der.sity, the Newt on-Ra phsor.

t e c hr. i qu e is emplored. Itera'iors ute performed over e a c h ena t e r i a l usirp the vapor-lield density as i r.d e pe nd e r t variable. The furttien used irvolves a rearrarperert et I.q . IIV 4 as 4 01

tq O l 93- I" "

V ;O)

J \ u

~l' I ' =og;[f;,,,, r g p(=; ,- .a

+a", T(t.M i ""

• IIY'I""'

U I"p.M where $ is the ir.re r i t e ra t ier. i r.de x. The. iteration equatior. ther. is 0 ;[I = c ;e - ( T" J /; } .

. ( IV-101 )

dc(";n where dT" j , ET 33, ,y ,; . (a".r + a".r T(; ,n ) , Bhjp ,y . ;,

m -

u3 . . - G ', ,

dt5g'r. ~

C ll xig.M ISat.M' '

h"Ig.r (IV-102) where the partial derivatives o r. the r i gh t -ha r.d side of Eq. (IV-102). the saturatior. temperature, ard the heat of va por i zat ier. a re obtair.ed f rom the im after each i r.r e r i t e ra t i on.- Durir.g the _ iteration, the caterial-comperert density of the vapor field is restricted to be betweer. zero ard the total liquid a r.d va po r ma s s of the caterial i r. the cesh tell. If the component der.sity is computed to'be less than zero-in ar. i t e ra t ior.. the c omper.e n t density is set to 0.001 of the value for the previous i t e ra t i er.. The cor. vergence p r e c i s t or.

achieved ir. this iteratier is crucial for oserall (orvergence of the codel. The criteria are

a. A caritut of_ the i nr.e r i t e ra t t er.s cust be perforced because consistetty betweer 1(;n a r.J T Sat.M is ret guaranteed when the first M(;. ,

calculatnot is pe r i e rr.e d .

b. I:i t he r the saturatior terperature must be tor. verged te lo K. or 'he s ha r.g e i r e a c h r..a t e r i a l 6 orpar e r t M(ir. . eva l ua t ed hv l'q. I lV-lol l. rust be less thar 6 where 402

~., .. . .- .. . _ _ _ _ - . _ _ _ -. . . .

4 4

y [ny y -- -

6 J JI I .ill -

h-

[

t

z

. 4 m -

o..

.) ,_- ,

h. .-a

. ig = l l'Il%!. ar irputiparareter. ehert that

[  ;. 6, = .' > Iot,, it TCr t . h1 ' "' "('S K < That.k1 < Tr r t .51 . .

. i r. ' s cr:e u.ses. e.g.. ter Tha t 31 rea r T the derivatives-charge so rapidly CrtJ1 that cervergente sii! r.e t occur u r. l e s s the e s t i tu t e d ' i r. i t i a l conditions are.

" ~

sufficiently c'l e e ' t e the cerverged solutier. Me timd s for t h e *. e cases are dis.assed it Apperdix 1. of the Sl%NI:k- I l ' ma r.ua l . Also. a quasi-equilibriur y.

approx:tatier for very la rge hes.t.-t rars t e r coe f f icier.t s has bee r.. e l iti r.a t ed t rer thi*, versier el SI'41FR-Il due . t c .d s f f ic ul t ie s s'i th er.e rgy cor.se rva t ion.

l l

The seterd set of e qui. t i ens corrects for t he .va por-s ide heat-trarsfer-teefficiert- ' based e r. . t h e tass trarsfer at the various surfaces ecdeled tv i

L I:a. I IV-% ). As s h o w r. t r Set. I l l-li. . ~ e f the SlhNr.R-II earua l ~ t he . cor r ect ed heat-trar. ster toeff usert 10: u rdersat ior en liquid-field erergy cetper.ert r is l; - Fj (;tt7' sp,;

litiL " IIY'I"3' t-r (;_ty 3 .pg;/il";Lr.' ,j 6

i i

where i

l'. -

L ll(;g i s' the torrested predat of the . vapor-side he a t - t ra r.s t e r

[- (tie fl i t i e rt a rd i r.t e r f a t i a l 2: r e a pe r' ur i t volute.

I' I

llittra is the corder.atter mass-trarster rate at the surface.

Cp(; is- the average .tarstart presaure. specif't i heat of the' vaper

j. r Sture. are l .

j. Il(;t, is the product el the- vapon-side heat-transfer c o e f f i c i e r.t ard L

the int e rl as ia l area per ur i t ' volur.e in the a ba.e r.s e of phase i

• r e r. s i t i o r. s .

l-

.Il^ vapor 2a t ier , rather t ha r. t er Je r sa t s er., is etturriry, t h e r. Eg(;etp. ir t

,1 ~q .- fl\-lo34 is repla,eJ hs -f l Lr(i, .

lo- the s t r t.t t u r e - f i e ld surlate , the

[

, tass-trarsier ra*e tr I q. ilk-lo3, is th- total tasa.-transler rate at the 1

surlate 1 e r a l 1 v.. pe r -1 i e le s ery .-r c r t s . Ag it. . high degree el seuplir.y exists

[

403 I

i i

. . . , . .,-- . . , -,. , , , . -., , . , , , - a w, ,.,,,,-n ,~,,n.-- -

n . -r-,,, , . , , - - . ,

r__

%p lPJ fl f'sr Jr(Ar, -

betweer the heat-trarster $ ettit ier- .: r d tht ph.ise-trar. ition rates, arJ ir t. 5 : !: ties sar r se il 1 4 IV-1"i - 'udt' cither directl\ tr the e re r; s e .: L. i 1. r

• i t a" ' 'r - .- '

. L.

• t r :teratier. T, pr s:ct ta!21:ts. we Le the dt!it;cr i i t he rh.. t r..r itten rate.

fu+1 l br; L " ' UL (ir Tv41g, , , ,,j - Tty, h .j

....s

• g.

, - +1 I .< - 1IIeLe

'=pG

. .j ITs+1S..t.M b- Tx>>

'I .lt,r) Lr.' pG ' h,o tr .

e

-1

( IV-It:4:

whii.h p v e =, t !'c additiora! c nupiig betweer the h e a t - t r .: r a. f e r coefficiert ard the pha6e-trarsit er rate. 1.q ua t i er 'lV-lo4 is writter for the t o r.d e r s a t s e r rate er iicaid .ariases. The e q u.i t i e r f .4r va per :za t ier -

T j'Lr b-j ' UL (i- T"sj,q-T,t n,. g . .i .

1 -=+1

+

I<] LHir 'pei l  !

-I pild,r ' rG- = + 1,gi T'^t.M sa - T(' i e 'b -Lr . ;

( IV-lof i

irilar prctedare i, u.eJ ter itrus tae s u r i .. t e s . A hewter-14aphser proteJL c is uscJ to selve 1. q s . t lV-1 >4 ) and (IV-105) li r the -hase-transitier rate.

takiry the values T[,1 ,y C" ;lgardh y ' r er t !.e iirst :teration. (~e r. e q u e r t i s .

the tendersation rate.

If("kr'l%*.

i the v.. rer i za t : er rate If"*l LHi- 'I Sk

!arriihe. t i: e s:rgle i t e r .. t i s e s.tiabit. l ie t ... : Is et the prosedure. ir .ther werds. wher t, take t l.e c c a . s .. l e : ' '

l.. i l \ -It >4 and wher t L*e the equisalent et 1. q . t I V - I t '5 - o r ,: w h.: : p re r p' i . i t. I e l l o t. ii r. s. l u t 1. r 404

'Oll 3 ~ - ---

TUvaf~. h[ AA.

esists are it Apperdix 1: . Atter a rte is ihtaired. the corrested hea+-

trarsfe! 6ee!!: 6 .crt is ev lua ed *cr t hi equis. ei' .! l.y. (lV-1031 .* its s a p r ~. i er T. 4 $ ps.

I': r.: l l y , the vaper-crergy equatier i, solved. The t r urt a t ed va por -ri s t ure energs equatt r is hT1;T y(s.- 1 ,b a+1 , -r. er , ;{ -

i (i b - p(s ilt- 1 , ,(s + 1o r . 41 + h"q* I41 )

r=1 N'.14T 2

+

AI ~_ 4}hf1e + 2 (l}[d u (IV-106i r=1 L=1 5he:e h'.1;T

?[,* l =

[ 2((I i. the total ratrossopic dersity of the va po r -t i e !J r=1 c or p. r.e r t s atte.r evaluating Lq. (IV-95).

• I e[; i, the spe, itis interral e r.e r g y cf the vapor r.i s t u r e ter whith thi egoatsur i, solsed, e[

• f , ,,9 is the spes il is i n t e r r.a l crergy of liquid-ticid raterial c ar p r e r t r at the saturatier t e r:pe r a t u r:

de t e rr.i red i r eva lua t t ry 1:q. t I V 4 5 ).

Dr. ; v the crergs-trarsfer terrs frer rw . t ra r.s f e r and the a. esiated he.t tratster are irsluded i r I:q. ( IV-lo6 : T!ie phase trarsitior is assured to oscar at the 1:c id-field ard strusture-field erergy c er:pe r.e r.t irterl.ces, whach a e assured t. bc at the . a t u r .. t t i r terperature. Therefere. the ta s , trars: erred tret the vapor licld is a st. red te ; eave the field at saturated tordit Ori..

slaltiplisa* c- sf 1. q . Ily.uci hs e,s+1 ure.a t t er over t. and subt ras t ier

! ret I q. tIV-106* sseld 405

r FOf %ll  % p A ;- --

i LUail JMr; ISitT

-r -1 -- r - --.?

1 .1 -1 -1

'b t ,

• t, 'b .

r l' fissict pas r:e r e l v add s to the vapor dersits.

h e .: t capa,its. ard codifies the vapor's thertophysical preferties tard t erseque rt iv t r.11 uers e t he vaper-side hea t-t rar sf er coef ficiert s ).

(2- The case ir which . liquid (orp.irerf car va p.-r i z e totally is hardled bs irsertir; el t e rr:s that piase the : quid ereryv directly i r.t o the varer

i Id.

• t3: We wish to .else 1. q . iIV-lo'i terTf,*l a rd e(*;* l . Ve appro. sir *e e f,'l by

= cb5 + ~^*

cN+I 0 $ vb,I (T"^l b - TN)b iIV-lobi This stelds f3121 - 1

,r -s+1 iTS+1 - bT5 i = bf ,tirr,,;{

1 x+1

,0 ,5t,. vb b -

3 r ;~p=

G +Lr -,tsr.h., + h"'Ie g . N1 - t e 'i r=1 2

-s+1 -

+1 -

• +1

'vc tTa+1 g; - T5gs ,- -

( q=ig7 t7 + _ ti j ay g g k=1 406

{fI Ifi ! i )hbl

~ ~

p ' 'T } I~ ~

nL U06 N.1;T - 1

,1 . xr i 7Lt ' f Lt*1 .

,b

. ;vn+1 iT**I b - T(*,t',

r=1 tIV-104i kherc 4

x.=1 r

is the r.or. i r.a l c a <, e . ar.d s.=o r w h e r. a l l o f Qr.: V"P3T32

T. plate 1. q . flV-lo4) i r.t e a raragable ferrat. du=v va r iabl e s , a e. b. y ard d e are detired as

~

~

~

'r = t ,H L (ir (T"s,g3-Tte ' + ,_ H y;t (T"3l} ,33 - T~r,1 3t [ . ( lW11b # L-1 2 1.- 7

                .I ll";*Lr!*         II"t *5 x          'rd                                                         (IV-111 k=1
               "'I                    I                      '

T5 (IV-112' d - e(,r.A1 + I"e'g.A1 - e" 6 + i'b.I v b r The qaartitie' "r- b. r a r d rJ . .: r e regarded as corstarts durirg the seluti.r o: the var r-erergs equatien. Thes use the updated saturation quantities frer the first he s t e r -l<a phs er s t e r .. t i e r . ard updated. vaper-side. heat-transter tre:iisiert- trer the sesonJ steratien. Ther it is p.. sible to show that h'.1;T - 1 8 =91 r[vb ( T'?I P(T,*I s r 0 F i vti hy.) t=1 j , gj R.1-\T - 1

                                                                                              . .i a rsst,    + l' ',

• hr Ty.3 s, . ,

                                                                                                                                   . .i ggs st,
                                                           .                                                  r
                                                                  -          g      -l' .

r=1 a-1 gi . p . 41 407

   < ./ -
                                                                                                                                                                                                      -).m ' [  r, g p 7 r"   ~

i N jJ 1. UMij  ; fa1AT-1

                                                                                                                     . -                  - =* 1 ;.r vb, . Lr il x t,-     T'*Iis r=1
                                                                                                                                          =
                                                                                                                    .( i -y 6r, , ,b . . ,tir , ;v+istT".                       1 b) -
                                                                                                                                                                                                           ~

fol;T-1 1 td g.,

                                                                                                                                 ~
                                                                                                                                                ;bgTn.1                               "            + bmT[t .M dr -

yj

• c.J sa t .M ~ yj .

it.M f31 AT - 1

                                                                                                                    +            [        [7gn ($I - ef; + i"(+;ITf;1} [1 - xe }}
                                                                                                                              .r=1
                                                                                                                    =0              .

(IV-113) This is a quadratis et the ferra f J ATj;. + 111(; +r-o (1V-111) solved u s t r.g the quadrati, fortula. assurirg Tf;+ I is the e r. ! v u n k r.ow r.. The quartity A.< o by definiticn. so the r.e ga t i ve' square root gives the higher terperature. ard ir tost cases. f u r r.a s he s the torrest i.o l u t i er . The exceptiers a r.J the presedure tellowed it ll -3 M' is le .s than zere -are discuued in

          -Apperdix E.            The e r.e ca .ie r sre,ial case requiring i rcre d i a t e discussier occur 3 wher. we hese ; large, vaper-side. heat-transler coefl uient. In.this case. the
          .va per cont i nui t v' ard vapor-energy equatiers are stror.gls coupled. 1:x pe r i e nt e has. irJita t ed that tl:e best approach te this prehlet is to evaluate .it Pf;h.
           'directiv' hv u ing the previou expressior for f[jy - If;+,I . This reduce, 1:q. :(VI-113 ) to a linear equatier int [;'I. The criterion for this tase is that n-1 U
                  ->C          I
                                     > lini     .

(IV-115i

                       ' h "* 'g , c
                                                                                                                                                                                                                                ~

l e r a r.\ serperert t. 40S

5 p

                                                                                                            , ,    b  tff   ,   l- %. m        g.
                                                                                                         -i YYD h Three spesna1 O t t.a t i or.s                   s.hould be = ter.t iored regardir.g ihe .overa11 it s ta* h r.          Itrst.        1 1.. alI the I ni;uid - of           a g ivs r. s crper ett var r:ri .              the-c.R     q .r r e @.-r d ; c 'T a t u r a t s i+. etrerature returred ' iran t i:c rmdr i is -rrerer. cJ: .                                          .

t h.: t ; r[;'b, s t i l l - ca r b'c . a l d l a. t ed t er r e t t i v i r on t empe ra t u r e d i f f e r e r.c e s. Thiv artisitial sat urat ior t emperature appears et the SlW.1F.K- I l f u l l - p r i nt . Se drd.

         .consergerse cf the t.dcl either car occur with the variables ou illatiry about ar apparert (cr.ve rged va lue from iteration t o i t e ra t i e r. or with ' the variables ghargir.g torotenitally' i r. - a                           g i ve r. .- algebra n - direction.               Converger.ce        .is accelerated-ir the eb.illatory                           case by 5teffensen's method.where for~a giver variable.-y.

y x-1 ,g x .- IV* - v=-1 l'- . (lV-116)-

      ,                                 ga . yyk-1 + y a-2  .

Third. it cotylete cervergence does not occur. T sat.M is set to -a {.sa y ard a r.euage - :. printed. Th:s errer exit elitirates va por i za t ion /c onde r.sa t s er. for suc h- # te)) fer the t ite s t e r s i. questier. (ie ne ra l l y . t hi s ! wa r r.i ng te u a ge stditate. a p r ob l e r:. and: the i r. pu t ardtor coding should be investigated it enessive-warr.st; teuages are given. 1 r . s urra r v. t he - t echnique to solve t he cer. densa t ier./ var. r iza t ion tede! t or. s t o a. ~of three i r.r.e r iteratiers contair.ed within an outer iteration. 1.ach c the.three strer s t e ra t ier.s results it a phase-trar.sitier rate that . cor.sistert with ore c! three varer properties: the saturation temperature. the sp-tetterature. Or the varer-side heat-t ransf er coef f ic ient. The steps arvolved tr obt a i r. i r g ::: f i r.a l pha s e'- t r a r.a. i t i on r a t e a r e (11 estitate the liquid-field. erergv-torponert temperatures fror the herirtsrg-o!-tite-step'.terperatures. the advarted-time temperatures er the sa t t.ra t ic r t erre rat ure s us i ng 1:qs. flV-90) through ilV-921: r (2) pe r t o rn. the inrer iteratier between the saturation temperature and

         -the ra u          erservatier equatters t er the va por - f ie ld ca t e r ia l s u a. i r.g f.y.                         (IV-1"l' and the equation e: state:
             'I'. lit t r u i .      "I.l ete r t s of f;aterisal Aralvsi ." Jchr Wilev ard sors. Jr.                                    hew
        -ierk. l u6-I . pr -             91                                                                                                                                                 409

31 J r. -3 --- IV J'J .)i\ , 31 if the otter iteratier is cerverped. evaluate the terverped h e .i t tiuse- .rd r h.: s e - t : . r s i t z e r rotes a r.d esit t r er- the Tahse-trars: .i.-

..isoi.i t -l ' peris tr         the         irrer    iteratier    betweer the corrected vap?r-s Je b..t-trarste g tre!! tierts arJ                      the phase-trarsitier-rate defir.itier ustr; i.q s . (l\-lo4a. I lV-lof t. arJ the associated equatiers for strutture surfaces, il required; (5i     s.. i s e the varar-lield ereryv equation for the new vapor-field temperature usiry 1:q. (]\-113); ard i t> )  returr to ht t p 2.
       $'ep 45I is pe r!( rr:ed a t least erte.                  The convergence criteria required ir St ep ( 3, a re tht the tass-trarsfer tractieral c ha r.p e betweer successive outer iteratiers asse(iated w i t h Il[;[r r.u s t he cerverped te withir ar. irput pa rar.e t e r (1:VAh ik i, arJ that the vaper-side heet-iransler coefissierts in Step (41 tust cervenge withir o.1: betweer suttessise euter iteratiers.

Ti'e saperizatier terdersation tedel has carv spesial cases that arise wher . st.!! a < .: r t i: .. liquid-field crergy cetporert is presert er when the liqu;d .telJ e t e r p s - c or:re r e r t terperatures appreach er exceed the critis ! terpera ure t i- r the raterial. These spesial cases a r.d their treattert ae det. led ir Apper. dis I. . t 410

R]UGHDRAFT APPI:N11!X J M!sel:LL Ahl.ot's CORRECTIOhS 1

• 11)! h1 !! 74 ~

2-3-' ril Ah il. Till LIVUll) Till:RMAL roNDurTIVITY 4 ?i 5 *I Xril;.113-6= 'll iTLi3i .LT. 445.1 Go TO 4100 7 IF e TU 3 i .GT. TCRIT( 3 ) i GO TO 4150 5 Til"oNU 3 >=o. 714 73- 1.1751 E-4

• EXP(o. 012693* TL( 3 ) )

4 Go 10 4160 to llo TitroNU 3) = .650 11' Go TO 4160 12 41fo TliroNU 31 = .275 13 416.. CONTINUl: 14 */ 15 /. PUT IN Tlli. DROPLET RADIUS RATIO lo * -17 *1'il7G4.I 15

• RI'R AT l'-*1 SETU.174 2o
• RPRAT

21. .

li'. ( RPR AT . 00. ZERo l RPR AT = ONi: 22 *1 T$11T.212 23 RPRATN = ONE 24 I f l h . EV. 3 ) RPRATL = RPRAT 25 *D B511.13 20- RPi h t = At11 N1 ( RP1. RP2.RP3. RP4. RP5. RPh1AX( IREG 1' RPR ATN ) 2 7

• D 11511.17 25 RPi h n = D'.11h1( RI'1.RP2.RP3.RP4.RP5.kP'. TAX ( IREG )* RPRATN t -

24 'D INPh.17 3o -

• 0.1P.RoGrUT. ALPilo. AllikG. AU:sro.kOSI AL.RPR AT 31
• I Il7Gu, j 2 32
• RPk;T 33 'It P.704.15 34 2411' I V APok != .1 PE12. 5 / 3111 Till: WATER-TUEL DROPLET RATIO.30N.

36 3411f RPR AT >=.1 Pl:12. 5 > 3 t,

• IDENT 11504 3.,

35 */ TEMPORARY cokRECTION FOR Tile IlOUNDARY CONTilTIONS 34

• Jo
• 1 Kril A. 5o 41 11-(1 .l:U. 21 P( IMJ ) = Pt I J l 42 11' ( l .1#. IP1 ) P(IPJ ) = I4131 43 If (J .IN. JP11 P(ljP) = 1913 )

44

• IDENT 11514 45 * /

Jn , l'UT IN AhoTill:R TIME STEP cohTROL ON Tile VAPOR ENERGY a. 45 *1 IM)L.124o

u r 50 C PUT lh AhoTill.R VAPOR I.hl'RGi TIMI: STI:P roNTKol 51 0  ;

1 52 *11 -DI:l' .1 : lit .1 53 PTLA = Allst PTI. i 411 .! s

                                                                                       ~ _ , _ , - . , ,   -     ,. .-

2 l*J ofo *1tX:1 (l. 501

                                                                                                                 . L.sl ilM.IN'l 'j! 'ioMM:1 'it LJ I'f M'u                     . 90L
                                                                                                                 . iot r f sil 1.'1.1:11. rol "C<!'l = L ilit]                 fot ift'll.15 I. Col I t a f ' i 13[ 'b lil'il ' -              tot si' !hil d. i'o!

l iliti'll<lltitl' . 6o oc ' ihli (1. So llMr = ( l t il \f 46 [l31 = ' (( il Al 96 Cof'MCL\ll !. i6 1-f = li3f r6 l-! = tt3[ f6 trC'frC'hYifl. C6 II\f = (lI)fAf to il\1 = (ILllA! "6 t ri 'M(It,il [ . ni

                                        . !)<! bid / I '11t!.h lh10 )

• l tll(li l[bri = til10 s3 l 'liid ' .l:lt i II . 19 i !)ilh1GM 'iltl.ltih10 i
• I Ilid ) I'iIhY = Illid 93 1 'litt l'- 1 Ill- 1I ?S of 'M(L(il ti. rs Iill(I = 1 I ilit i i3 C L L l(L\ll (l. C3 ll3f = iltif\f L3 ItMI = ! I t '[ A! 09 ifi'M(L\ll != 64 f11ifAf = LINf S; i11ilAl = IINI 11 oCi "id\ll I . 94 i!)fAf = t(Mf ?1 II)lAl = LIMI ri o_rIfit.fi !. fi attl:1 = L iliti Ci of'llNI I. 14

ll'ill.'itu r?9C of shf =tILelAf 69 shl = l t t I \1 39 D IMit! = Illit! 49 ti9C 01 o') (ll11(I *19* 1YI'f1(Ii .I[ '*9 V31<l/ ( ( M 'silM . ( Ih:l' ( f l 1911<lD J L'J Ih(1 ) = lt !'ild i9 L lifti' 1:lt! II . r9 V:llcl/ ( i M silM.( th:l'( f I !)licllt > (*1[hY l = lt !M1(I f4 i'littl* l Kl II C9 ti9C 01 o') Ol'1 L1

• t 111 11 19

( I i'IMo3!Y: = t 11,1 .9 ti9C 01 (10 1 ( ( f I isil,1 It - INO ) .ollellY '11

• i i ! i!)ll<lD i iI ni ir r I 'l lhi I.W t i t,I = fI colo t ti .i Cnc t 'l th! I. of i 11<l Nitfl = t'll:1 si l 'lilil' .1:l'l (( . ri

   -         1 yly 1n  1

--- Vj u

m q  %, - - - 1 - -, VEd l

                                                                        ,\     a   l 1<'9 'll -DI'.l . Dill .1 11o            ATI'.All' = V.1;\1 An r.1!' - KLn11'. I120 )

111  !! I;1.1.1)iiL . I 112 ATl?11' = f ri;\1 At(r.1" - KL n11' ' 12 ' ' - 413

R:)3H DRAFT t APPf.fC)1X K NU41;16 lif.srkiPTION 01 Till: 111:AlbIA1LURI: 4)l;I:L SUllROUT!f.1: The *, i n g l e -J e r r e e et freed,t ( SIUl i head-f a ilure code l integrated arto the 514N!:k-ll code-ir thi study is based o r, a series of c a l c u l a t i e r.s usiry the rer.l a rca r f i r. i t e e l ece r.t (11: 1 terputer prograt AI)l fM. A r. axisyttettu fl: represertz, tier of the reactor ve uel was developed that included the lower head. the tvlirdr aal tore barrel, the stifier.ed pipe leop ar.d support r e.g i or. . a r.d t he reatter head. This 11: redel was sub,1etted .to five differert. 't rans iert . ut il ere.-i r t e r ior. pre u u re -t ime histeries that rarged from very large impulse ' spikes to a rest quasi-static loadiry that would produce failure. In additior.. a sixth time hasters that approximated the spatial distribution of the pulse was used. Ala,o a triasial a, tress failure criterior. was implemented ir.to ADINA based. or the work tited it Re f. -to. A: 1 Al)lfM failure tedes begar. ir. the lower head rear the d i s c or.t i r.u s t y radius ict the lowe r-head per.e t ra t ions. Exatira t ier of. .the ~ st ress stat es ir ' the 'talled" elecerts irdicated that a t i r.;utt e r e r.t i a l split of this lower head a r e u r.d this radius would be the failure teJe. thu-allewiry the lower cap to cove away free the vessel.ar.d vent the preuure. The equ. tier e; rvtser for the simple. s phe r ica l-head Slo! tede! is given by ex + lx = Pi t n (K-1) where n is the equivalert cass per ur.it area. K is the stiftre n per ur.it area ar.d P i i, i l: e pre,sure. The equivtlert ca u was established bs salculatirg tlie frequertica, ar.d rede shapes Iret the Alilf; redel tsir; rirst. elast:s properties and se ..rd. F.; r t t a i plasti. prepertie . The stiftreu ter the better head is taler te be 414 ,

O 1 = ~

                                                                                  \      IlR  -    I Ow     b. Y,                ,

, 'I' L = ~-- T Ih-2) A k-where 6 1: is either the elastic (29 x 10 psi) or the plastic strait harder rg d (lei.2 x lo psi) Neung % tadulus. T is the t h i c k r.e s s (5.375 in.). and R is the average radius ( 96. 2 i r.. ) . A failure "d i s p l a c ere r.t

• fer this equatier of totier is calculated based er the equivalert uriaxial feilure criteria calculated from the ADlf4 triaxial failure redel ard the reridioral ard hnap strait espressior. The equatior .1 rotter car ther he irtegrated rurericalls te approxicate the ADlhA result.

T. A. It.: t l e r . "Resiew et l .1 ard l'r e po s e d failure Criteria for Duttile Alterials. L( s A : a ru - ha t n er.. ] La be ra t o ry report LA-10007-MS. fiUki:G/rk-3oaa ( April, 19 4 >. 415

                                                          .w  ,                            ,.        , - - ,           -,

M in 1 ,n. art IW JuIl J M I s APPL:hT)!X L ) 1)l:1 lhlTlohs 01 VARI ABLI.s th Till PLlKiV CORRECTION STT PLUtN . Lh er-head ta k te he coved. i r put -_ i r. kg. 4 ZPLUG Di s t ar.c e (te t e rs ) .f ree ref e rer.ce that the lower head has coved.

          -APLUG:                   Au e le ra t'ior. (te t e rs/< 2 ) of the lower head.

FPLUG ' Terse (rewtors) cr the lower head:as a consequence of ur.ba la r.c ed preuures. Stored as a positive r.utbe r before head failure, a r.J as a -- r.e ga t i ve r.uthe r t he rea f t e r. -

VI' LUG .Veltuity of the lower head. VI' LUG is a positive r.urber it

irahews before head failure a r.d a r.egative nutber ir. t/s after C head tailure.

i FDli l l The tr ct_ict cf the current asial node occupied by the lower head I- at the head's i r.t e r f a c e with_ the fluid, as-a fur.ctiet of .. the radit.1 subscript 1. Z110Tf l e The axial Insi On tir. reters) of the bottoe of the ir.terface r.ede I betweer. the 1 ewer head ar.d the fluid. as a furctier of the radial .I subscript 1. ZTOPi l ) The axialLinatier (nr teters) of the top of the ir.terface r.od e betweer the lower head ar.d the fluid. as a furct ier. of the radial substript 1. a Zill W 1 1 The axiol laatier (in reters) of the lower head at radial l a a t i v r. I relative t. / PLUG. i I

!                                                                                                                                              416
        .           _ _ _ . . _ .    . _ . . _ . - - _ __       --  = _._.._ .        --_._ ,_ __ ._ _          . _ . - - . _ . . _ _ . ~._ -_

o- , I

                                                                                                                     %              O                  F.P        "* "

B Ms 'l J W{ )l_ Jiid[ f ," l l ill Ali lorce ( r.estons i int egra t ed ove r the'~ upper head. Presertly fill:Ali i corputed ord output. f or ir.fortat ive and plot t irp purr, se er h. j: . _ XPLllii l)i s p l a c er.e r t i i r.- i r.s h e s a calculated -by the failure subroutare 5% )\ i. i l l'AVI:kli Average pressure ( i r. psi) at the er.d of ' the time step used for

i r.p u t into the failure s ubrout a r.e. 5ICI).

350 ' Input integer for the top boundary of-the lower hea'd. 150 I r.pu t ir.teger for the right-hand boundary of t.he~ lower head.- J50T I r.pu t integer for the lower boundary of the. upper head. - 1%oT 1rput for the right-hard bour.dary of the upper head. l l-Radiallv:derendert irteger added to J50 defir.itg the top bourdarv 31'lll(it ! ) of the leser head (irput 1. L , J PLllG5i l l Tetporary variable to t rar.s f e r inforcatier. Meit use is it INFLI)l . detertitirp whether the lower head has crossed an axial cesh cell hour.dary at a given radial locatier.. The-choices are JPLllG5rIi = 0 No bour.dary crossed JPLllGS'li = 1 Upper boundary crossed J PLllG5: I ) =2 Lower bour.dary crossed l J I'LilGT i l l Radially de pe rd e r.1 a r.t e g e r added to 350T defining the bot t er bourdarv of the upper head (ir.put>. J('ONT( J ) Asial index for the radially c or.t i r.ua t i ve inf low /out flow bourdarv ( ord a t i er.. 11

                                                                                                                                                                         ,1 JroNT(Ji = o Radial bourdarv is rigid                                                                                   ,

JroNTt J l =1 kadial beurdary has cortirustive i t s 1..w '. u t i 1,+ 417 1

, = - 5.l

                                                                                 ^

l& .. I& 'w h E jdj APPENDIX M LOVI:R-llLAD DYNAMIC ANALYSIS

1. IhTRol>PriloN AN!: st414RY of RESULis 0i AlilNA STUDY Ar ADlh4 taJet was developed ar.d was used to run parateter studies and to develop a reasonabic single-degree-of-freedot ( SDol' ) model that could be used ( wit h ser e ' judgrer t ard care) in the Sikf61ER-Il code. Thus, a coupled analvs:s ~ couli he rade vi the pressure relief caused by lower-head failure and consequert verting. The pressure trarsier.ts were selected to cover a spectrut of steat-esplossor. intersities ar.d characteristics. Six cases were calculated. It cases 1-5 the pressure was applied unifortly as a tiee histart - over the entire i r.t e r i o r surface of the axisyttetric fir.ite element nodel of the vessel. Pressure histories for these cases are giver in Table M-1. Case 6 used the results fret the SINt.1ER-ll scoping calculation that was exploded alter o.7s of rixing (see Sec. VI-B.2). It case 6. the sp..ce-time effects were ' approximated by applying four dif f erent pressure-tir.e histeries over four of the teridional sections of the vessel as shewn schetatically or the geometry ir. Figure M-1. Table M-Il suttarizes-the ca.ior results of the study with ADINA. ar.d of a simple specific-icpcise calculation.

The first colurt gives the tire at which enough specific impulse has beer. delivered to a spherical segtert to exceed a fracture straar. (for this hard calculatier. set a t 0. 050 ). The reason for the double er.try for cases 1 and 2 is that the first ts of loading for these cases was deemed r.egligibly stall in sottributir.g to the spesifts impulse. ar.d the har.J computatier. teek the raer pulse to begir at 1 ts. The secord c(luer gives the time at which ADIN4 computes that the failure criterior has beer exceeded. It all cases, exatirir.ation of the stress state at the iritiatier poirt of failure i r.d i s a t e s a split of the vessel alor; the c i rc utt e re r t ia l di rec t ier ( i. e. , ac ross a teridiar.) with a portier. of ti.e better 1.ead cetirg off irtact ard beity accelerated destward. The third celuttgives the approxitate ir.itial velocity that this lewer-head cap would have us a rigid body. The followiry sections discuss the ar.alyses and assumptions that were used ir arriving at the Table M-Il results ard a simple SD01 todel that car approxicate the result. TABLI: M-Il SUsMARY Of LOVI:k-IIEAD ITILURE STUDY foR A GENER!r PWR VLSSI:L Tite (es ) ADlha Approx. lepulse - I tit. Tire at velocity of te lat) whuh failure lower head initiates ers) (tr./s) Case 1 0. $ 1 1. f i 1.5 11 000 t ase 2 0. 5 + 1. 5 ) 1.5 11 000 t ase 3 1.43 2.1 f 2$o 418

ROUGH DRA:T e!a e k C C ID O I

    .d Io e
                 *2                                                                                                              e 2 .i e,            w r         We mo                                                                                                                                                         ~

l oe Iw. .@ '~ E *1 g O. .

                                                                                                                                -g                                      .-
                                                                                                                                                                            ~ c                              '

b C A - . O. b ,

                                                                                                                                                                             ,?                             4 r ,.                          -    ,
                                                                                                                                                                                                  '-o t/1                                                                                                                           o LJ                                                                                                                                                                              .

e-2 .", -

   ~

cc

                                                                                                                                                                             ?s.c O

• J s w w *

                                                                                                                                                                                                           "3 cc                                                                                                                                                                        ""<~

L-

                                                                                                                                                                       , i ' .%.

j g O. e* ** h ...,....,0.",--,- - Tt- ~! ~~ a . w

                                                                                 .           I    /                                                                           , . _

a g .= c > v .

• M -

                                                                                              -                       il         o i
                                                                                                                              -6                                         __

cr . y -

  'I                                                                                                                             O T~

i i . . . . i 7 I - O*009 0

• E5 0'0% 0*00t O*002 0'001 0'O 0'001-Z 419 l

w. R:tGH DR/fi Cirse 4 2.863 3. 0 3 460 ( ia c 5 3. s .. . 9. o 2 250 fcse~6 -. 2.1 3 0646 TAllLI: M-1 l'ki:55URI:.TRA%il'NTS USI:D IN LOWl:R-ill:AD DYNAMIC ANALYSIS

    ' Tire                                             1)r e s s u r e ( Mi>a l fr*I          fase 1            CLse 2             fase 3              ('a s e 4   Case 5 o             o.1               o.1                  0.1                 0.1       0.1 1              b                   6                l'd)                0.1       0.1 2'           ]000              1600 ~                200                 100        $O 3             150               15o                  200                 100        54 4              9"                90                  200                 100        50       '

5 150 luo 200 100 54 - 6 170 luo 200 100 50 7 leo too 200 100 50 5 150 100 200' 100 50 9 150' ?ou 200 100 50 lo 15o 1on 200 100 50 12 120 100 200 100 50 14 110 luo 200 100 50 16 100 ho 200 .100 50 lb 90 60 200 100 So 20 80 50 -200 100 So 25 oo ao loo 50 25 30 46 35 loo 50 25 35 42 35 100 50 25 Jo 40 35 100 50 25 50 35 3o 1..o 50 25-60 35 25 loo 50 25 420

R I jR I R =" l u ., J w J J

11. Till: 1 AILURI 41oDI:L AN1: rkITI:Kl A T! c !.tlure r. J:  : h ed t 'br r e t core rJ.i t wr < ! lu ard liutler' ard this a r r r o.u i: h.. s sigriti,at- e x pe r ire r t .21 serilisatien, l'ror; t he stateif streu at eat l. i rt e g ra t ier p< int. a triasialits fact.sr (TlI is computed.

The Tl is delir.ed as a dilational stress (o,,1 (sur on repeated indices 1 Javided by the equivalert stress e i .1 1 .1 I 2* # 1 '# 2) (#2-#)3 3~#1 * ( h1- 1 1 with 5,3 being the deviateric stress ard c, the prir.cipal stress. For a TF greater'that er equal te crity, a dama ge f a c t e r 1 ( Ti' l is cotruted as sirh 1 - r f Tl i = . . ( ht- 21 sich E(1 - r.) TI 3 . Ir this studs. we to-ak r = 0.2. The triaxial fracture strain is ther sceputeJ as i, = 0. 7 i ! ( Tl 1 where i fu is the ur.iaxial true failure strait. ir this case a u ured to be 20 t . When the taxieur prircipal strait, at th?s p o i r. t ex eeds t i.e triaxial tr.ature strair ther a ductile raterial failure is de61ared. Ir the case fer TI' less thar. unity. failure is declared in the elletti\e s t reu exceeds t h e c r i a x i a ! u l t ir:a t e s t r e s s . All of these criteria were progratred irt o t he ADIN A code ar.d eva lua t ed a t eac h t ire. s t e p a rd each a r t eg rat ier. paint. (ie r e ra l l y the trissiality factors at failure ranged tror 1.5 to 2 ard the tracture strains were about 5*. Study el I it. 15. (Re!'. -t o ) . reveals that the criteria were prograrr:eJ irto Al>lNA correctiv. 111. IM51s IOR 5151PLl: MEL To ESTIkt4T!: RI.VUIRI:D I AILURE 141PULSI: (COLUN1h 1 of T AllLI: V I 1 1. Ter llu t l e r . (irou p V-13 a t Los Alamos suggested ar. icpulse failure model and it wa- used te salculate the times giver ir reluer 1 e f Ta b l e 61- 11. l'er ar experJirp sphere urder a u r. i l o rt impulsive pre uvre loadirp. the sepulse per unit area is i = rs ' A1- 3 ) 421

n [q -== 1 \ . 1 Vw h, \ i j where r is the re u per ur.it area and v is the velocitv of the spherical segrert. Thert: .re.

              ,          , 3 i-         r v-
           - =                                                                        ( M -1 )

2 2 or

              ,       _1Mv-

_i- = 2

                                =

Eiretit Energv 2r  ; lt r i t Area where M is the total reving ra u. If we assure that erough irpulse is delivered to just fail the sphere at zere velesity. t!er all Lirei n erergy is abwrbed by the s t r a i r. e r.e r g y. Thus 2 i t., tiil U. V

                                =-                                                    (M-5) 2r               4 where  l', is the strair crergy per urit valute a' failure and V is the velure. Icr er expardirg sphere r                     =o pp i=0) ard
                           't                 'f             '1 V, = [ o,p d4              +

[o,,de =2 [ ode ,

s. e ,

wi.i.h is two tires the are urJer the uriasial st reu-st r ir curve tc the biaxi ) tut el! la: Jure strair. lor blaxi 6erditiers a nd A53311 St e e l . s g 422

m , R:20WT

        < g J ' O. oS a r d l'g ' i s easily (orputed to be lo $74 (it.            lb/in.3'. Thus ter u
      - 1 squ..re irsh segrert et ar r>parJirg si be rs , the jailure.itptise is iga,;  -- 2 tUg T 1/2 .                                                   (M-6) where T is t he t hic kr.eu of,t he sphe re , ar.d a value fer 2 7,;j can be corputed as 14.47 lib s/ar..* . Ih.t the impulse per ur.it area is the area under tlTe rreocre vs time diagrar. Thus. by solvity for the' tite from each p-t trar.siert at which the impulse per area giver by Eq. M-6 has been delivered, the results ir. Celutr 1. Table M-Il were determined.

I V. A Sif.GLL DEGREE 01~ TREED (r.1 MODEL Til4T Al>PkoX1mTES TAILURE TIMES A study of the Alllia results reveals that for these severe trar.sients. failure alwass initiated ir. the hetispherical portion of the bottot. head in the regler tarked or lig. M-2. Study of the actual vessel indicates thai there is a distertitusty ir the bettor. head geometry because of the bettet head's peretratters at about k = 64 ar. Whe r t he dyr.atic s t re u state is

      ' perturbed by this gecretry charge. it is'likelv. then, that the actual vessel failure will be arour.J this discertituity. This failure ende wa s suggested for the S14t.!Ek-I l c od e . that is. a splittir.g of the better head
      . along this coordir. ate with verting of the vessel eccurir.g as this cap res ed dowrward. licweser a todel is r.eeded fer the 5141ER-Il code that gives a r e a u r.a b! c predittier cf failure time nr.d ir.itel velocity of the cap.

The equatser o! r. * : e r s e r a r. e x pa r.d i r.g s phe r e ur.de r ur. i f o re i nt e r na l pressure l't t i is 21:T iT' k + "-- p- Uk"h1'

                                                       '                                        'M'7I where c is the mass dersity:

T is the t h i c kr.e u . R is the average radius. Up is the radial displaceter.t. I is the redi.lus of elasticits. and 1 is tire.

 ,     ard o d.

• ird 6etes the tire derivatise. It is er this pr i r.6 s pa l t ha t we based the Si n i! r J e ' s ugg e s t gd r e r sl At.11.k- l l, it we let r = tT be the taas per urit crea 4.rd L = 12iT/k be the sti!!ress per crit erec. ther the ratier c: a s phe r n I s e gre r
• t a r be r e p r e s e r t ed. w i t h l'p = x . M 423

! o g- LOWER HERO DYNAMIC FAILURE INVEST. * "' m *** * * "- m GRIO T- 0.001000 i

o STEP- 6 j g. '

DSr. 5.000 Sj o I 1 I o e

                            ~

y N l' k~ 1 1 et I 8- 3 1 N l o

                                             /     General renton for ci-                /       failure initiation                                        C Q

in ADINA studies.

 \

I 8

                        #             y         W     "

C W F 5 5 W

;                          -sos.o    o.o     son.o      ano.o   300.0   4oo.o     s00.o      soo.o I                           .=-u w ..erm                      R W

r, M .A ' ,, . IM 3 :,udt,,e udmt,n.o),cv [ic ,[ /lh I f / t po u: en t li , , A . il , , l O

     ~

1

s - RCl3H E WT ^ r$ + b = l' t - . (M-bl l'o r a r. e l a s t :( / Fl 6 ast n r. oJ e l . I: iseithertheclustjcnodulus(20x10 ps i ) or plastic strain hardenir.g modulus (16.2 x 10 psi). Because the nedel 15. te represert the totior. of the lower head as it relates to far.d is allected by) the entire vessel._(i.e., it should approximate the ADIh4 result ). Eq. M-6 was treated as follows. The stiffness-per unit area. K. was four.d as ahese. The cass per unit area, c. was taker. to be the-modal mass as derived from K = c,t . (M-9) P where the frequercy used corresper.d*, te the frequer.cy for the mode shape of the " expand rg veuel" as ir. l'ig. M-3. This frequer.(v turns cut to iorrespord to the secer.d code. Thus, the ADINA model was used wi.t h t he sate beutJars 6trditier.* as used ir..the t r a n s i e r.t t u r.s to determir.e the first tour mede shapes ard frequercies. The secord mode, which sortesponds

     - to the "e yardir.g veue! rode, gase u = 1521 /s which gives ar. equivalert ca u ior la.. 4-6 of Ib s 2          3 t= 0.01156               / i r. . -    .     <

it Note that this eau is rearli twice the actual ma n per ur.it area that a

       $.375 in,         thi(L head would ha.e.             Ar. elastic 14 tit (yield) displacenert based or H.2          strair. ard c y ,,g7        60 .400 lheir.. gives x y eld = 0.2 ir..
        !ct 1 y. (4 61.

A failure displateeert value of 5.0 it. was set based on a triaxial tailure strait et 6.31 1:q. M 8 wa s ir.tegrated ruretically INewr. ark beta

     - rethod). ard the f ailure t ires were toepared to Al)lNA predittions of Table 4 11. These teeparisors are shown it reluer 2. Table M-Ill under "clastit'

• r.a u . The tailure tire terrarlser.s or this case are tot (orsidered goed.

but iritial selecities are worse. Ir. an ellert to irpreve this codel. lurther charir.atiet el the AlslNA results rescaled the S101 redel to be quite usturate up to 1 rst sielding of the vessel. lurthernere, large pertiers el the vessel's bottor and top heads vield litst with the thitker (slirdtli.al partiers retairtrg clastit. Ilaud cr. this ebservatier, a new . 425

4 f i i o l l- LOWER HERO DYNRMIC FAILURE INVEST. *"'"*****"-m . G't 10 T- 0.001000 I o STEP- 6 g- ' DSr- 5.000

                                                         \

j o- 1 g- i e i g- 1. N I o initial  % g- vessel shape l i

• l 8- l

                  ~

I [ P i s p l a r.ed j N shape 3- CD c-- o s D i 7 . . - l -100.0 0.0 100.0 200.0 300.0 400.9 500.0 600.0

                      --n w ..orm                                               R                                                        C Z

3> j li '- T (s i .g i (? <f.ys i s \ t .. .l 4 s .A l' s 1.'i r.rI t . y m Os

4 .

         +

R:lMH DE I:rgervalue prob!ct was rut. for whist. the pla' tic mat erial nwdulus was used ir pertier.s at t he cesh appreura'e ls sortesp.*rdity to the plast: 6 regler

t ah' ' a ppear . Jerirg t! t t ir s _ t r. r ':r - sie;J urtii lailure is pred uted tr the Al)lfA rede ! . I r. thi rus. a riw ircquerts 6.tresp-rJiry to the ex pars ior rmJe was. idertitned d a'r ~egaivalert plasti 3" cass was derived

      ._fer the Six)) todel based or this va l u e ( 1.17 r a d / ). The SIMil mede l was retur f or all case usiry the value of the tass parameter in the plastu
       .regare     i.e..           displasetert bev..rd o.2 it.s.             The results are shewr nr. Table
      ' M- l l ! p r(.J thee r  plast *. cau, telur.r 3.                 Sigrii nar.t i mp r ove te r.1 s ir. tice to l ailure i.rd t r it ia l se let it ie s were noted.                   A v. a sider.ote. ore tight be terpted to use the astual tasseurit area of the spherical head ir the SIX)!

todel. This cetrarsier.is also i.howr ir Table M-Ill ar.d though failure tires are'probabis r i too bad for the larger impulse. cases, iritsal veletsty pred utier. are vers peer. Dr. the whole. SIX)F Model 2 with a set of c lu s t n 'ard pla s t i s s t i f f r.e u a r,J ca n parateters was judged to give the best approxit. tier e: the AlJlfa rea.ults. With respett to ca e N we teted that it is approxicately the static, ve uel desigr. pres ure case. applied dyratically. Sirce the dyratic load

       .latter a r t r e a a. e s the resperse bv rearly two for this shape of pulse, the

j. fact that the veu el f ails is r.e t supristry. Ilu t it should also be expected that ditterert po r t i er.s of the vessel will go plastic at i

sagr.ilicartly di!!crert tires (i.e.. spatial respor.se-time effects are itportantI rei.ultirp ir a deviattor from the Sixtl codel. This (orclusior is retlected in Table M-Ill. It fact, the pritary reaser that the model doe sr ' t give as good a result for case 5 is that s i g r i f i c a r.t spatial l cfle6ts Jo begir. te ppear. Cast 6 also shows this trerd. The results

fret Sl\t11

k-l l u s i r.g t h i a, redel cust ther. he ir.terpreted with sete cautier.

If *igrafstart spe.t al variatter 1r pressure tagratudes for " lor.g" periods of tire are expested. ther the predittiers will ret be very good. For I stear-exploster werk. pressure tagr.itude di!!ererces disappe r rapidly he ause the pul e propagatier speed si high. It should be roted that the ! lailure tite *, are slightiv torservative. with the exceptior. of (ase 5. Arether res'ru sor er the Slall rede l as suggest ed f or SILT 11:R-Il is that there sat be re urleadirg et the veuel i.e.. trcrecertal dia.pla6ererts tu t be a l w;. v- pe s i t i s e i s i r.s e the ve nel would ur.lcad elastscally. Ile t a u s e the stear esplosiers urder study have irvariably resulted ir. vessel l failure. th:s restri tier is not deeted to be s i g r. i f i c a r t , but 514til:k-I l results it ar.v parareter .tudy should be exatir.ed with this restration ir. t rd. V. CofrLUSION The Altifa *,t ud ie s are believed te give a fairls atturate puture of both failure tires ard tcJe. The automatie- et the tailure rodel free Ret. 40 is largely resporsible for allowirg a s ortidert predittier. of the strair ! ard rode of ve nel tailure. Strair rates. by the wav. were estir.ted ts be about 44 . stase 11, which l tears the caterial pr pertie- used .,te 'il: as urate si.e. strair- rate

_ el!etts are r.eglspihic i. Ths si n) r Je1 Lyrc ted Iet tse w i t h Sl%til R-il

( t r be used te s t i.J s the ser*ir, c:'t,'- }.. arettisai:v with at least as l rush tortiderse ir the result a- *here . ir the tedei t the phys u s of c the st eam e xples icr. l 427

O 'nt S. G . r" '- V $3[ '

                                                                                                             .\ \f ,

Le s eterte 4-- 1. I: lo ., T. A. I'. *lt: " l: s . e t ' l'r ; e d l o r i t. r e criteria for l i.. . t ; c M.ier:.Is. L Alcr - f..

• i s r i. L.. h. r a t i.t v r e pt r t LA-looo7-Ms.

fXKI t,' Ek .k.44 i\pr 1 1954-l 428 l

TABl.E M-III COMPARISON OF SDOF MODEL AND ADINA RESULTS ADINA RESUI.T ELASTIC MODAL MASS PLASTIC MODAL MASS ACTilAL MASS T fail I""} T fall I "") Velocity T fall I"*) I"8} (In./s) Y"I".CIEY (in /s) Y'I".C (in /s)I " Y T fall Y" (inI ".C

                                                                                                            /s) I ' Y Case I    1.8         II 000      2.5       8 841              2.3       10 920           1.9       16 360 Case 2    1.8         II 000      2.5       8 84I              2.3       10 920           1.9       16 160 case 3    2.1          5 250      3.3       4 364              3.0        5 275           2.1         7 581 Case 4    3.0          3 460      4.3       3 048              3.9        3 721           2.8-       5 815 Case 5    9.6          2 250      6.I       2 140              5.4        2 582           3.9        4 107 Case 6    2.1          3 060      3.4       5 317              3.0        4 503           2.0        6 664
                                                                                                                      ,O___

I O '

 .                                                                                                                      3D
                                                                                                                     '3 >

r'

V O Al'I't.f.DI A h I o'.sil l hT it f.S kl(> Aki:!Ni Till Kl'st'LTs l'IM.1 Till SIXil Lir,ill-ll! Ali 1 AlLl'LI. 411'l L Ti,c c ; L.. t i t r 1. r th: h! Nil ~ roJ e l . 1:q. ( 45 ), is s + s- x = lo t ) r (h-1, Three cases are u rsidered. lirst. i f lat t i is assur ed te be a constar.t. ar.J > rd err zere .: 1 = 0 the selLtier te this e q ua t i er. is l' (h-2

              \ = ~             l - sir. (wt +re)*

L where h = r w-The r: css re t ii.

• car be arrlied is related te the tire the vessel goes plasti, ir ether words. wher x = 0.2 tr. V:th the eerstarts g i v e r. . further displacerert 1, t e:;e t ir the p'astis regire is altest assured, lle r e t he ti r itur. I' t o c a e s s ic :J ii, .* .
• t =. (r l'r r
• h2
• Ib'3' Sessrd, it the loadirg b IJs up produclis se that i is small. the lead t, prcJ .e 1.i!Lic is ,

l'l a : , = Lx (h-4' T!. i r d . : leadiry hs the ferr af I'( t , = l> sir (wt). arJ the s a r.e iriti l sercitiers are ap; lied as with the corstart leaJ situ tier. the selutier is l' ( h .4 : s=- siri ot ) - wt t e s t s. t i . 2h it:c r:.; sir r ;orsity lursticr is Ateired wher t = e'2. There!ere. the r.xtr.- [!cssL?t *I.* sed.G he e\erled h rt \ : e 'i d ; T 1 estLrs is 430

R:L 3F 0 RAr l'r e s

• 2 E' ( N-6 3 s

Tin s ., I t. t i ,

                                          ,i     .'--  4;7 y   .-  3.; n u     -i p i i r. . Wher  x-o.2           it. we i h t .. i r 3 3nu. J ps:            1. 19 r hs I ij . th-34
                                              .                     o  735.b pii li r l*f 3 ; j by I:q. Ih-4?.

.rJ 13 475 rsi

- P., , bs I .. t h-o s. These s e l u t i or.s irditate the rarpe over thish the upr;ied pressare differertial right he saried ard produce failure.

The statis l oad t., c eu ,e 1,wer-head failure it the ZIP study was est irated at 44 MP or b Joo rst. t, 431

RCUGH DW Al'l'1f.P1N O. 'b L . 'A l I 14!TLli R!Tlik o! V.L s.!!.V.1-ITi>LosinN INi i.'Rihli.KTs This apperdix reflects-experiterts der.e at SNL ar.d re port ed a t , the t ime et this re viei. Se pt er+e r 1954. Two types i! experatents had beer. conduc.ted at S',L

                                                                                                                                    -i by that date.

These -were stoll-wale e x pe r iee r.t s irvelving 0.05-te 15 graes ef . telt lockirg at the behavier ii surgle- drops- ard "ir.termedihte-scale" experater.ts firvelsity o.6 te 20 kg.

                -The      stall-wale          experaterts' "             examined         such      ph e r. vee r.a    as stear-explosion triggerabilitv. droplet f rageent a t ior.. cor.verster ratie. debris                                    -

size distri utters a r.J characteristics. a r.J hydreper gereratien rates. Sete

qua'litative c o r r e l a't i e r.s are available. see

• l'ig. 0-1. Alse. the careful u settif u r ture of theie e h re r s te r.t s has led- to some detailed models of the fragtertatict proccu,. A recer t ' paper by llarkef f provides a revi.w of these teJeling efferts as well as presents his multiple bubble f ragterta t ier . eade !

o based c r. the experiments in Ref. 43.  ; I r.c l u s i or. of the level os detail represerted by these experiterts ir $ 3t.11:k- I l is bevord the ucre nt' cur prograr. llewe ve r , the experimental i n f o rca t t er. obtair.ed dees ret prestude c sign lisant s t e ur: esplester. ir a reactor reltdest e rs i rer.rce r.t .

                -Intermediate-w ale experinerts are divided arte two typesi partial!v

f. i r.s t r ume r t e d tests and fully instrurerted tests (l lTS ). The partia!!v

[ i r.s t r ure r.t e d t e s t s were pe rictred hv lluxton ard lier.edic k. The first 45 tests irvelveJ 1, 27 kg es irer alurtro thertite poured inte a r. open tark of water. The pratarv t ort !ui ser draer was t he. t ar erergetic explosior. car cccur under these sirsurstarte . Other arterestirg results were that saturated water appeared te ha. v e to eifect ard tha t the highest reported efficier.tv was with a cover plate providir; a s or t i r.er e r t eficct. The rext 11 tests itselved 13.6 ' or 19.4 kg of i r er -a l t.t i r.. therrite e r s e r i ur - A+R t he rei t e . which certaired f 3 w/o Voy . 17 w /o Zroy arJ 30 s 'e stairless steel. The rre l t was ;sured n r.t o a larger tark of water btried i r. the greurd. The coriut was foutJ to produte i l either re explesiers er tild explosiers senta red wi th the iter-alutira tixture. i

" l'i g a r e presideJ hv M. Ilert r..

S r.Jia f.atter ] Lahoratarles tAugust lo. 1954'. i ! 432

RCcGi DRAF Og I I I ol

                                                                                            /

0

                                                                                     /

7 - 4.0 -

                                                                                  /
                                                                               /
                                                                         ,!                O
                                                                       /                     -

9 3.0 -- j E

                                                                /

5 ' O' O t O ' -

n. 2.0 -

f

              ^
                                                /

O ,/ O o,/O - 1.0 /

                              -8 8

I I I I l o e.e o.s 1.0 1.2 o 0.2 o.4 AMBIENT PRESSURE (MPa) 4 l Ils.*). I:l l e 6 t cf pressur e er emples ser vie ld. i 433

n e RCUGH DRUT This 'was spesulated te .bc . t hi ser equer6e of either t e r. - t o r d e r i. i b l e gas L_ pe re ra t t er . cr . n 're sa t ' : a. t- r i n . t ! t s o e m.e r . e e l exide', solidi::. t hr. Nutervus i Il s t e s t e !..a t hst rat. A a rr.+ t i s ertpj e t e l i s t ' i C r e pt. Ju. ci! l- in Table 0-1. Tae 1irst- srwiler te ts. irvolvirg 1-5 kg of iror-alutira

i. , ' _

l thertite it the celt . drep f kBri .ard ilTS4 series. a re the most. interest:rg f rJe the;'viewperrt er ara!vsis. .The' lD) . tests were performed. ir the llo-TITS apparatus- showr-ar rig. 35 et the cast body of this report. The-IITSA series was perforced it t cort.intert chaeber showr. in rig. 0-2. Both series were run arJ repc rted by tt6he. .' Although sote rardet behavner is preser.t. a ._ ( e rr.c t

                      - result was triggerir;                              a steat ~ explosion . at the base of the wat e r - cor.ta n te r                                                                        .

follewed bv a prqagat ir; wave, similar to a deter.ation wave, at a velocity of-2$o te f oo t % threugh the relt-coolart mixture. This is similar to'the idea prepesed by Bnt<r that s por.t a r.c ou s initiattor of .an- explosion probably j i nvolves wall er stri.ar other solid surfase contact by t he cie l t . - J o Not ruth insirratier. was found er the flTh(i series tests. f.v i d e n t i v.' '(i' stards for stea- generatter. TITS 1(i irvelved 21. 4 k g o rd FITS-2(i.13. 5 kg of irct alue r.a ther n t e p; ured ir.t e saturated water.'I The water tasses.werc relatively stall. 4 -3 kg nr i lTS lii ,and 65 kg it f l T S- 2(i. ha explosiers , os c u r r e'd. It was rep <rted t hs. t the u t u r a t ed wa t e r peel and fog t reat ed bv fue!-water tent st rode visu ! ebservotiet impe u i b l e . The fact that there were to explos sora w; t h s t ure.t ed wa t e r c ert radic t s the results of the earlier Buxter tests. The st X -Se r i e s ' ' trvolved 17 tests. rot 7 as reported in Table 0-1 ard these were Cinilar te the ht!-Series. but invelved coriur rather that ' irer. alut:r. thertite, leur of the experieerts had explessors, each el thet triggered at the base ef- the tark. The cerversion ratio ir the tests that exploded was reported te be sitilar to that of i r o r. -a l uc i r.a . A r. idea was advaried that a tirirur telt volure (0.5 liter or 1 pint I is required te surpress a r.cru rdersible gas effect and tern ar explesive mixture. he further dissussier er tl e previ.us difficultses et explodirg ternur it the ilu s t en ' "

                     - tests was p t v e r. . altheLgh these earlier tests did use more than 1.5 liters el 6ortur.

434

      + - . - , -

y 9 .r-+>.--+.-s-- y - n-,w w , . - - , - .,--n-r,- f.4 +3m --g - g -m n m 9 -, y-,.er,y.,-w,,--ep,y- p,----y,,,,%,w ,- y - v,w.-..ww--www--,,-,,-r

R?lnH D?AF'  ! IhTERMEDI ATE-SCALE FITS TESTS I.OEPF'OENT VARIABLES MEGURBe, s  ; N $ N $ $ $ $ E K EXPER. .W EER h  ! $ E l

• F E P s Q

{g * { b I SERIES OF

                              ^

5 t- ~ E

                                                                                '  p,      gd       E h
                                                       $         b   M 71.575        %  g          g a        ;

5 y.  ;; w Lg E g E = g g u e E

                                            $ E        5     ;-             12   m                                  E g                  E         $      g                                 5 k          a          e     c                   2         2        -

E 5 s s 2 @ g d e c 3 E s g e c c g _ c v. k 5 C c' E E

' w -

E O 5 c E - E i E 1

• O O o O O

    .'"           19

'

• O O O O
• e FITsA s
• O e e FITSG  : l I -
• O O 3rr --

1

• O O O O O *
• FITss 9

FITSc 5 O O 8 O O O e

• O O i

O O  ? O O e e 7 O \ cu 3:

                                           .              . O             . .. .                  . . ..                ;

e e e' e' pc 7, e er e- e- e- e-

        .....         2, 1

e PRIMARY VARIABLE l 0 SECONDARY VARI ABLE

• PRELIMINARY SCOPING TESTS 435 t pov yrT MEASURED

g ROUG- DRAr p . .,, ,..ne ...,

                                                       -         a o ,sehet. Crucini.                            Y
                                 , Cha.,e              7_                                     n, .. . . ..         ..e....
                                                          ,k c

• Ci.eu,. vsi..

l

  . Actwater N                                                      l     ~
                                                                            -W.it Chamb.r l            pud"'                                                        32 cm Chamber Static.                                                     Hi-Cam 8000 fas Pressore & T.mp.
                                                                    ,                         p .mov.abl.H.ec R

Acc.es Walk g / , 3

                       ..,,, h.,,   .,   LJ,       entcon.c \ /                        \    %,,               , ,,

184 cm l ( / Jl Chamb.' Pr.s o u,.

                                      )LU g.- 'x Chemb.,

Pr.e,e,,,uria ation 1.Sm DIA

                                                                                           .5:'                                        .

l l%, Q' Wein Chamb.e W E A MPs W.P. l/ p S.8 M Volume 3 He-Cam 6000 fos instrum.nt /

                    , . e ,-t h,u     gg i

Porte ,, - HbCam 8000 fps lLl Luett "

                                                             /      i Chamber       -                              I   /                         -

d

p. -

P,.eso,. - asymanim

                                                                                           .9 l
                                                    /               !

N l

                                                                              )             '
                                                                                                       -Drsin &Purg.
                                                                               \

1)f 806 cm l' E. O.2. TITS cert ir.rer- charber. 8lrer kt: -t ' 436 l - - - - - - - -

ROUGH DRA:T The flTSli t e s t s .' ' certuse the pisture ehtaired fror earlier results. Tnev

      'used a noti r.a l 15. 7 kg c f ir.t a urir. therr;te wit h va rsirg aeourt s of wates.

Iritiatier sites te>p!.sier t sprerss s. c r e eNe ved t rardot locatiers ir the re l t -wa t e r mi xt ure , as .cppased t e the rore tornor ha t.e t rigge rirg pheroters in earlier tests. Scre el t he. expe ritert s ir.dicated a wave propaga t a ry away fret the trigger sites. llow e ve r . t he - wave - f rer.t c ould r.ot be dist ir.guished clearly in these experiterts, a r.d . ir. sore cases, could erly be observe'd'as a charge ir. light ir.t ens it y trar.statted to the c ar.e ra s . The experieetts with the largest ateurts of wa t e r gave double explosions. with ar. i r.c r e a s e ir. explosive vield. Two experiter.ts were dote it saturated water, resulting in r.o explesions. Ar.o t h e r level of cent us s or. wa s added by the llTSC'* series. The pratarv purpose of these e x pe r ire nt i, was to compare t_h e cerium A+R ard irer.-alucira therrites it. teres of explosiver.ess ard resultirg debris f o rca t i or.. A ni t roge r. atmosphere was ua.ed se-that hydroger gererat ion (culd be deternired. The two cerium experiterts produced weaker explosters that ir the hilr tests. The speculatier was that . . . wi t h c o r i ur: ir air. the atrospheric oxvger reacts with the absorbed hydroger t -2m ppr i it the startir; t or.s t i t u e nt s : t h i a. liberates heat and e rha r.t e s the thertite reaction so that the temperature of the- firal

     'telt stays above the liquidus peirt until attes the telt is released."'*

i r The 01 s e r i e s consist of tere resert tests ( FY53-rY5- :. Results. are I reparted i n SNL bitor.t bl y progre w reports.' "- '; The purpose of these tests was te d i a. t i r.g u i sh betweer existing redels for (carse tixity, although the docurentatier does ret explant how the test data would be used te a c c etpl i s h this ob iet t ise. Twelve tests were run i nvolvirr .3-15. 5 kg of i r er.-a l uti r.a . Both saturated and subcoeled water were used. The utexpected results were water-surface everts of sulticient violer.cc to expel telt free the water. as well as prevent some relt tret er.terir.g the water. which occurred in every test. Why t hese - evert s caurred i r. these tests atJ rot ir earlier cres is ret explaired. One ' spec ula t ive hyp0t hes i s is that sore charpe ir. telt totposition. prepa rat ier. or d* ;ive rv tay have been resparsible. "-

       ' I n f e r r.a t i c t prov Jed by 41. lle rta r. . Sardia Natier. l Labort.t or i e s ( Augus t 16.

14b-3). l 437 l l 1-

RME [4KI

               'I b e    tr1    test J' trselved or exide. relt. using the reastior.

31e. t LClo a a l'e3"a + Erl. w:1h th6 M1 prida61 driver 'of f as a va r. r. Ar

't r a t ial test ussrg t in s tra,11.r. 41-1. pe r f ortced earlier. had de l n e red
 -boo y of rater al tc . the water with to explesser.
                              ~
                                                                                       ^

Ir the new tests. a b. . .: t 20 kg 'of t he rti t e wa- loaded wi t h abot.t er.e-hair this arourt delivered te the water. Violer.t arJ unambigueus explesiers eu utred tr. all f ou r (P.1.t e s t s rur. although the experitental details i r. each test were u r. i q u e , it was -corcluded that rapid oxidat:(r cf the retallu c ompe r e r.t in the 01 series was strongly ir.fluenc irg the' t rigger try and/or propagat ion el the explesive ar.d r.cne xplos ive interactier.s that were cbserved. A firal level el cerfusier. is provided hv the RC' and AGl* tests. The .two rigid cor.firerent tkfl tests ir.volved a ( 61 -c rr-o. d. , schedule-60 steel pipe 0 ct i.d. ) 'irst ead 0: a Lucite box for the water container. The telt was 20 kg of Te304 Test RC-1 did ret expleJe. Test RC-2 produced an explesser qualitatively tore er.erpet u t ha r. ir the previous 01/O'.1 tests. This was urexpected. The suggestier was rade that corfirerert was a llowirg rare of t he telt to participate it heat transfer. The twe alterrate c or.t a c t mode (Anti tests i r.vo l ve d pouriry water er top of s t or.-a l uci r.a. In the first test a vielert explesien 06 curred. !r. the secord test, r.o explosier. occurred, llei a u s e water should'ex st it file boilirg unJer these cordstions, the violert explesser was ret art ic i pa t ed. f.o speculatise explaratier. is giver. by the experiter.ters. The-actual final tests rur (bv the sutte r of 10541 have beer the I I T M )" series, which are te a rvest igat e - cha rac t e ri st ics of at TC) that potentially coulJ lead to i t.d i r e 6 t certainrent failure. such as s t ear gene ra t ier.. hydreper g e r e ra t i er. . ard dire t tartastrert heatirg. Ilow these questions are te be addressed by peuring 20 kg of irer-alutira thertite i r.t o water is unclear. Ir arv case. ef the e n gl.t tests rur hv August 1954. the only results possibly rear.ingf ul te 'he Los Alates talter sere ieclar.t irteract ion pregrar were that r.o s per.t a r e o u s explestors at high achtert pressure or low water subicelity were observed. It c orc ui' s i or. . test MI)-19 does appear to represent a typical stear explosior. Its selection in Chap. IV for correlation with Sikt.11:R- I l seers

,t u s t i f i e d .      Arv explanatior of all the test data is probably rot possible k a t h .' u t corside rirg r rv e xpe rirerte l deta:Is that are net pertare to the rait i

l 438 l

eem 9 i

                                                                                                    . O'   !

a}'W. 1 - QQQt t issues o!~ stear ehple sluis tr i t .. s i rs. her'e test CharatterlstIs s k }. i s h are r'e r t u r t ve r s.. ) r .i v l' . d.: t' - * .1 6 s t' t tht sast i1 r e a t t e r r'e l t d,'b r . Tiie s e ir s ludi

1. Sore terderty ter e s p i e s a c r.s t tripper e r. walls, other u lid. e-liquid sur ases.

2. kedesti.r el vield wher. selids rav be preser.t ir. the telt. ard

3. A terJern 1( r ererpetit s t e ar: explaier.s net to develop tr saturated water.

We here that if i t. t u r e tests are perforced. they c a r. be dore se as to raise fewer questters ard provide r. r e a r. s w e r s . ' I r l e r r .. t i t r. provideJ Fs M. lir tr .. r . Sardia f.a t i e r a l Laheratories ' August l e. 1954s. 1 439 l l I i l

mq m3y -

                                                                                                      '               -         I        -

i Vwa, \ J, 6 \. RI 11:ki f,r l v

41. L'.u

          .        S.       T.t
                                 '   '   . J L.wrerst 1 ;.             li.. s t . t    " Stear I:x p les s. r Triggtr:rg I'l.e r rir          St . i r . c s - stee! ard toriur-1 Sirulatts stuJied with a fl. eda'le
       ;r6 W :tirg Arpar..tus." SarJ:a hatseral Laimratories report. NURl:G/rk-ol22.

S AN):T1 -"u95 Ma s 1 "' 8 .

42. L'evd S. he: ss r. . Lawrerte 11 Iluxter, ard llarry N. Pl a rr.e r . ' Stear la r ! es s er Tri,perary Ph e r. ore r o . l'a r t 2: roriun-A a r.d Cornut-E Stralarts ard Oxides i J r. r ard Cebalt Studied with a floodable Arc Meltirp Apparatu . sardia hatiiral Laboratories report. NURI.6 /CR-0633. S Ahl)79-02tm aM s l ogo ;,

43. L:evd S. heiser. 1%tricia M. Inda. "St ear laplesier Expe rirect s With sirple 1)r s p , t ! Iri r Ox ide W l t e d w i t h a 00 Laser. Sardia Na t i ere. ! Laborateries repert. NURl h 'rR-2295. S AN1:51 -134n ( se rt erbe r 19811.

44. L!csJ S. helser. " S t e a r. 13plesser Studies with Single Drep- of reriur-kelated Wit a : l e r r e.i s Wtals a r.d U- and Zr-Cer.tairsty Oxides.

l'reseedircs ,

• l.e I r t e r r a t i s r .. ! W e t : r r c r Licht Water Reac*or severe Atsidert I'v.. l u a
• i . r ( Are r i t a r N sicar Sn iety. La Grange l'a r k . l i l i r.c : s .

i lus3 i, 77, p,7 1 te tv . 7 - 3. 4f. S. G. IL-L.It. I. kcv rikt. ard 3 . W. T a r.y . A Mod e l for fragter.tatier ' 4 Iter Wta! Oxide. ir fortact with Water.~ l>ro. c e J i r e s " the Irterr..tieral Meetire er Licht Water Reactor Severe At t ide r ' I u l ua' i. r ( te r i s ar ha leur $miety. La Grar.ge l'a r k . I 11 t r.o i s . 1 l t1953i pp. 6. e. - 1 1e o.6-5. l 1 1 l Jo. L:.w r e r.t - 1.. Ilu s t er ard W: 11iar 11. llered:cL. "Steat E x p l o s i e r. I:f fi c s er t v i Studies. Sa r.d l a hatisrai Laberat, ries tepert. NURl:G/CR-0947 S Wl)79 - 13vu (lie i erbe r l u7u ,, 440

p rq g ... w') l W J, l . 4~ L. 12 lk s t V. I '. . liered: L. a r.J M. L l'orradtri. " stear I.splesact I It.:tr,s S .. J c : P.. i 4 11 t , i .:r Isrt: ir:rts. S. r ,' . . Nat:<r.' L.A - - - re: -

                                            *i.'kio ak-1 4n s.\N1:5o.132a e o,
                                             .                                                    ..he r lobo:

45. Marsh ll llc rr a- 'Ligh- Vater kta te- Salety Research I'r og r a t Se r s a r.r u a l ter.-t 0 tober 1451 -M.. r t h 1952. SarJia hatioral Laborateries kepert.

NURi t, 'rk- 2 b 41. S ANiiF2-1572 t j oct erher 1952i. 49 Marshall lie rr a r . " Light Water Reacter Safety kesearch Prograr Set: a r.r ue l repert. April-Septerher 14b2. Sandia h tier.cl Laborateries Repert. NUKl b 't L- 34o7 $ 2.N1:5 3- 15 7n iuttober 19831 50 f. A. I'.va r s and K. K. Cele. "St a t us of fore Melt l>rog r ats AL r t h Apr:' 1953." Meri te T. J. Valker ard S. 11. Ilu r s e r. . USNRr ( Albt,querauc . f?1. .l u r e 20 14531. fl. M. Ilerr - ard R. K. Cele. "$tatus cl~ l ere Melt Progrars May June. 1453. Mer ' te T . .l . W.lker ard S. 11 . ILr ser . 05hK( ( Albuquerque. TN. October - 19531

52. M. lle r r..: r .rJ R. I,. relt. " Status o l' Core Melt l'regrars --

herterher 0 teher 1453. Me r .' tc .l . L. Telferd a r.J 5. 11 . Itu r s e r. . UBKr Albuquer .e. h'.1. lebruert 22. 1954,

53. M. lirrner arJ K. K. reit. Status et fore Melt l'r e g r a rs --

    ' l u 1 v A .. g t. s t      957-           Me e        te      T.    .l . Va1Ler       atJ       S . 11 . Ilurser.             USNkr tAlbuquerque. TN. O,'.ber 15. 19632.

441

2 ROUGH [ RAF~ APPIhlilX l' . S I'Sti.k - 1 1 Ihl'UT IOR Till: AML) sis 01~

                                                              .Till 'A-lu l \li l.IMI LT l' sit.G A Uhllolat Mi\lt o Bes!'

W .r.  % 1 'h -loff.k ?WRi! AMLYsis 01 1:XPI klMlIT 591-14 TRY 1 2- . 2 - o. --

a. o o 32 1 :1 100 o 3 SI'411.R-2 STUlri 01 Till 'salil A VAPOR I:NPuisloh TI:STs 4 AhAlisis ol 1:\Pi ki41.';T st >-19 TRY 1 5 o. n u s4 5 1o

                                  .6                  6             13'                                                                                                                            -
                                     ~

l'LL'Ils 1:YhrylrS If;TI:GER INPUT b- 3 o o- -2 5 0 3 o o 0 -2 o , 9

                               .lo                   3o                1         1             1   0          1    0            1                       1          1  1

, 11 1 1 2 1 3 1 4 1 5 1 6 1 1 12 1 2 1 3 1 4 1 5 1 6 1 7 13- 1 5- 1 9 1 to 1 11 1 12 1 13 14 4 4 5 4 2 5 3' 5 4 5 6 5 15 2 4 3 4 4 9 .5 9 6 9 5 5 In- 4 lo $* ni. -50 2" 5 0 --1 2 t. 6 1 i 17 PRollLI:4 I;1411sloNs Ahl: OPi:RATinh;L roNTROLS

                              - 16                       n.0525-                              4    o.06426                      6 14                   0.05 " 4                                4    n.i5575                  13

20 o. 5 o. " 4. 6 1. 01

- 4 o 25

                              . 21                          .0091                   1. 01:- 6        1. of.- 4       1. 01:- 3 22                    1. 01:- 1 2               1.ol-12            1. ol'- 12        1. 01:- 5                                 o.I 23                            o.05                      o.40               o. 3         100                                    o. I          loo.

24 E!)lT cohTkots Ahl POSTPRori:Shok roNTkoLS 25 1.5 1.0 3. o 0. 0 2.5 26 o.Ooo2 o.0o05 o.001o 2; o. ' 26 o.onlo o.0035 9.0045-

29 o...

38, , E H .* H v 1 . t h M H )5 o. ()(H )] 31 o. o

• 32 o.noog o ool o.0045 -

33 o. o

,                              34                          4 s. 8 1 o                   o.n2 i'                              35                               a .. o 36                               4. o                      5. o 3;                               o...

s 1 35 o, o . t 34 ...o

                             - ao                               n.e.

41 o. 0 42 0. ' 43 o. o 44 e t. o 45 VilV IArTORS

       ,                     ' 46                                                                                                                                                                          -

_ 47 TIMI'-STl:P rof.TROLs as o. o 1. t .) - 6 1. 01. 4 o. 2 4t. 49 .ooolini o. $ lo. o 1. t i 1.o 1.o 442 't

                   ~r--ws,--    ,,,w,-  .- , . .   -
                                                                                                                                 ,_,,,,,..,,--.,-----..---,-g           ,w-->----,----,v--+,-,-,----

R1@ mar 50 1.o o.46 o.02 O.02 O. o o. o

$1 STI:UCTUI,l AND 1 AILURI l' AR AMI.Tl:Rs 92              o. o
%               i.. ..

54 1.o 1.** 1.o 1.0 55 1.14 t o 1. I + 1 ' ' 1.14lo 1.141o 1.1.+ 10 1.l.1' Sei 1.1+1o 1.l+1' 1.l+10 g- o 56 l'UI L PkoPERTl! s ANU 1:Ul%TloN 01 STATI:

$u            1oi,~            2 090, no                273.1e          3.3341 5         1. 01:- 1 2 bo         1001.'s               4217.1                 o.072?            1. 01:-12        1. ol -4 61   3,17771)+1o 4,705791.+o3                                0, o 3. 226591:+06          647.2b6        O.390507 62            14'2.                1.320                 3.737 2.93390146                        18.          32.

63 316.057 95.77" 316.957 64 65 6o 67 2212.339 o.3539176 400.00 66 IRoh PkuPEkTll.S AhD EQUATION 01 STATI: 64 73o$.o 634.o 1700.O 2.600"OE*05 25.o 70 6100 o 75ti. o 1.6 30.o 5.36oooE-03 71 1. 335001 All 4.337001+o4 o.O b.17000E+06 10000 o o.36' 72 492.o 1.26 1.64 o.000000000 5 ti. o 7700 73 74 75 70 7' WATER PROPERTIl s ALU I UUATloh 01 STATl

~5            1000             2090.oo                  2 '3. l e.      3.334E+5                 .65 To         1001.75               4217.1                 0.072'                  .65        1. 01:-4 bo 3.17771141o 4. 70579) +o3                                 o.o 3.Ob9505E+6             647,29o       0.3660361 bl           21b1.o              1.2115                  3.737 2.93390E+6                        15.           32.

b2 316.957 95.77 316.957 200. b3 134n. o.33"2 5. I:+o4 So. .5 64 535.6o~421 1.3453123o .0027491404 F5 2.55005250 361221451 .o746002905 bo IRON- ALLMINtN tiler 41TI' PkoPERTIES AhD EQU ATION DE ST ATI: b~ 4000 loo". 200. t i 2. 760"ol:+of 22. bb 3 bo' *. o l' en' t. 1. oi s 22, 4.3000 *l'-03 59 1. 4 Joool + 11 5.17 obol +o4 o. o 2. 620o01:+o6 6400.O o.597 90 442.i. 1.26 4.4 o.00000000o 75.o 6465. 41 92 93 IlWROUl::. ias PROPl.RTil:5 AND Till: 1:UUATloh ol STATE 9a o.o o. o oo 1. ol .4 o.o 95 1. 96 1.01412 4.OE+03 0. 0 0.0 1.0 0. 3 9' 10164. 1.405 2.915 O. o 2.016 38. 98 90 100 uni >oNI:NT PROPl: Rill:s 1o1 oso n .o o s o. ,. o uso . o . 954".o 7365.o 7365.o 12 4 no. . o o, o Ic3 1. r .o. o 1. no. i, o 610 ..o 1 o00, o 350o. o 9suo, o 1o4 45uo... 7 3c.5. o o. o 443 l l

RCCGH DW~ 105 2, u i. 2 0. .o. 200o. 1a48, 2000, 2 o. .. . 1..o 2.nono..I.o3 2 o.i,ou1+oy 2.non. .! o1 1'- 111 AT TR Af.sil L t s RKl L AT lo'. I:AT; 14 1.o 1.' 1." 1. . 1.o 1. lo" 1. 7 ". . 1.Too 1."o" 1. 7" 1.70 . 1 . " '. . 11" 1." 1.0 1. " 111 0.021 n. h o. 4 0. 0 112 0.023 n. b o.4 5. 0 113 a. 4)23 is. $ o. 4 o.se 114 i.023 o. b o. 4 0. 0 115 o.023 o. 5 o. 4 o. o 116 o. 6 o. 5 O.33 o. 0 11'  !)R AG CoKki:L AT lo*.1) AT A 116 1.o 12.0 2. o1. 3 9. 21:- 7 1.0 114 4.

• 1.0 0. 5 001 1.0 1.f+20 120 o.o40 -o. 2 0.001 0.046 - 0. 2 0.001 121 l'RI ssVRi lloU!,UAki ro'.blTloM 122 o. 5 31 + 5 o. 531 +5 123 o.o 5. 0 124 0.53f+5 0.53145 125 o. o 5. o 126 1%R AN11 TER KrGlo?.1 PRD11X KEGION 127 7. o. 0 1. 01:+ 5 0. 0 0. 0 0.1 125 o, o 34. 34. oo u.125 o.75 129 o. 3 o3 o. 2 1000 1. 01.+ 4 1000 130 1950. o.64 1.014 5 2.3L-5 1.1.-17 00o15 131 1. 51:- 4 132 1%RV.1ETl:R Kl Glo!.2 Clllh1N1:Y 133 7. O. o 1. ol + 5 o. o o.o 0.I 134 0, o 34. 34. o. o 0.125 o. ~5 135 o. 3 o. 3 o. 2 1000 1. ol + 4 looo.

130 105o. o.64 1. ol + 5 2. 31. - f 1.1.-17 .oo'15 13- 1. 51 -4 135 1%KV11 Ti.R ki'Uloh 3 VATl:R 13u . o. o 1. 01:45 o. o o. o o.1 14o o. o 34. 34, o. o n.125 o '5 141 o. 3 o,3 o. 2 1000 1. 01.+ 4 1000 142 l uso, 4., c4 j , o1;+ 5 2. 31 _5 1,1.j ,ooo35 143 1. 51.-4 144 LOVI:K lioUht;KT INITIAL VI:LorlTIES 145 o. o o. O o. o o. o o. o o. o 14 e. oo n. o o. o o. o o.o o. o 147 h11.511 Sl:T 1 n r.11%f'TEI) CORl: 145 1 4 1 4 1 1 o o 1 144 15o 151 152 o. o 0. 0 0. 0 475.5 159.0 153 154 o. O o. o 204, 3: > w .. 155 o. o o. o o. o o.4414125 15n 36t,. o5 51 15- o, o o, o o, o o. o ,oolo .ool 155 41Esil si T 2 rillt17.n 15u 4 j3 1 a j j o o 2 444

N h 5 r -- Vwd=l i i L

                                                                                                      ,s     i 1*.'<

1e 1 1e : 1e ',

                                     ...**               4 t' $ . 4 I r.4 1 t.$              o. o        o. o           2 usa , , ,

I t , i.o s '. o o. O o.4914125 lo- 300.8*551 10$ o.' o. 0 0. 0 0. 0 0.001 0.001 I f,u \11 All St T 3 k'ATI R I T* ' 1 11 5 6 1 1 O O 3 171 O. o 4. o 0. o 0. O U.O O. 1 ., 173 O. o n. o O. O O. O O . .o 4. o 174 0. " 4. o O. O 994 17f 176 o. " 4. 0 249.0 177 O. o **.O O.0 O.4914125 175 300.of51 1 ' O. o o. O O. O O. O o.401 O. o 1 1 1 l l l l l l 1 l 445

7-_ i R0JGH E WT Alti h'alX V . n r.ll Akl W. of Till:Fl -l Il th A'J' 'il A1'.1ri:-II CAlrU!ATIONS of t on's! l>Kl?.11Nihti A cuartitative 1:rit te i t.e ar urt~cf telt that car (oherer.tly participate i r. a - s t e a r : e x p l e s s e r. is ore of the cere i cpe r t a r.t areas of' disagreetert arerg certerporary investigsters. Where quar.t i t a t ive a rguter.t s have heer. developed t o give litits to p r e ti x : r.p . :they have utilized a r.uthe r of simplifyirg a s s utp t i er.s . rotahlv-these of a s t eadv st a t e a r.J a cr.e -dite r.s ior.a 1 flow pa t t e rr. This a ppe r.d i > diwusses $cce rur.e r i c a l calculatier.s of the coarse pretixity . process in t!>e (criest of expe r itent !@-19. Although .these ruterical s itul a t i or.s are both t r a r '. n e r.t - and t wo-d ite rs i or.a l . ro ( la ir is cade that the (orres t f o rt:s of the er.ergy ard roter. tut teurlir; terts a r e _ p r e s e r.t - ir the t er.s e rva t i er. e qua t i or.s. To scre extert these te c hk r. : s t i s .lortulas are still speculative, for e x a r.p l e . for the rate o! tragtertatict c: the telt. We helseve that c o r.s i d e r a b l e progress could be cade in todelir.g these relatiersh:ps: however, t h s '. type of irvestigatier'is bey 0rd the siepe el the surrent pregrar. Whot is effered here a re sete calsulatiers wl. i s h r:a v heir to suggest ourd . wher extrapelated to a larger scale. The first calcelatier is ar espletatery. three-field calculatier. with separate velo;itses -f or vapor-. water. ard thermite, i t . wa s first reperted at Lvor.. " The. initial volute t ras t iers are s how r. ir rig. U-1. The telt is e l or.pa t ed ar.J steared ir:tialls. The elergat ser appears to be real- f ror the files. Te wl.a f. extent the dcrsits is reduced hv retairity gas is ret k r.ow n . The degree ci steariry ad.*rted here was ir.terded to obtair. a r. ore terresertative ci.lcula t t er erse the telt s ortacted the water by avoiding excessive heat transfer. The iritsal J. wrwa rd ve loc i t y of the relt was f.9 t/s. ligures V-2 through U-6 shew t he dowrwa rd prag re s s ier el the thermite. The notion is quite t wo-dite rs iera l . w i t h wat e r t er.d i rt t e neve l rt o ard outwa rd around t he thertite at the-leadir; edge ard radii.11v irward arte the steam shierev at the top. This

        .calculat ier used a radiat ior hea t -t rars f e r fortalist with a reduced etissivity te athieve the des: red rate cf peretr..::ct.                            The t he rru t e wa s prefragrericJ to a 15-rr d 2.:r e t e r .

446

vuO t M-Y N,I-

                's WATER
                                                                        's t                                             N
    \
                         ,   1                                               ph
      \      , .a N

a vrooc TIME 0.000 5

         ~~J s'
           %   \
                    /
                        /

THER*!?E # 4,

                                      *>y*/
                                     /

rig, y.1. Ir.:tia! velure fractiers at the start of the three-field c a l s u la t s er.. 447

RCtGF 0 yq Ys WATED

                                           . c.
                             { IN x                  ,

s4

                                          ,s n p . ..

voou 4> 7 3 TIME o.050 S e s Ns s 1 T w- t.* ; T E

                                                   / /
                                                  /wk    u
                                                      ,, ,>*w k"

l'ip. O-2. Thre'e-l'reld calculatier f e ll owi rt ir. pac t with water. 448

i t I

           ,-                                                      WATER 4>.\'                                      o l
                \ $
                       \i                                                 /,
                      , x                                l}         /
                  ./-
                                .p v t.0 0C             ?
     \        s                                       u te            > 1. s M

zgecu11E I k"Q&k=

                                 +

Thr e e -f ie ld ca lc ulat ior show ing ir i t sa l t ota l t ermval of l i p . (J- 3. w.ter bef,re the adva r4 r.p r:e l t . 449

                                                                      \

J [ wA1EF

                 /\,\\                                                  'K
                          $ \ \ '.
                             .\                             N,        /x
                                .g                                           '% x
                                                      ~

N

                                .Y
                                                          -                   'fg x                                  -
 .                               /-                           -ll
                                         /

g

                                           ./

yA;OG . [ 3.150 5 T 114I:. N zgravilE 7 Fig.0 -1. Three-field calculatier after the c.elt der.sity is reduced sufficier.tiv ti ster total water vaperszattor t r ecove l ) e t the rcit :r r'. 450

R R -.. RIW I I J" I WATEF

                  ~

s

                    "\                                        .

Il /

                                                                    ,           /
                                                                    '0l'flll: !!

varn N TIME 0,200 S s

               \

b 1 TWEPMlTE , ., jx N"~ l I l l' : g . V-5. Ih r e e - f i e ld c a l t u ;.i t i s r wiie r t he c e r t .; i r.e t botter is he:r, i appriatheJ. 451

                                                                                      'O ]
                                                                               }"
                                                                               ,\uo    d1  '}G i[T
                                                                                           -   si 1 WATER 11

\'\ *

                                                                            ~~ N
                                                                        /,

q, '

                                      . '                       ,           //

l

                                                                               /

VAFC: (( TIME 3.224 S s T HEnu. ! T E rig. Q.6. Three-field calcu;atier at ti:e tire of a presur.ed stear espl.sier. 452

n Rj Vud ,'s,)\n.. " k ",, " TABLE Q-1 SI MER-11 INPUT FOR THE @ -19 PREMIXING CALCULATION 1 0 -105007h*RB ANALYSIS OF PRE-MIXING IN @-19 2 2 0 0 0 0 32 1 1 100 0 3 SIMER-2 STUDY OF PREMIX]NG IN EXPERIMENT @-19 4 ANALYSIS OF EXPERIMEhT @ -19 5 0.225 1.0 6 12 26 7 FLUID DYNAMICS IhTEGER INPUT 8 7 26 0 -2 5 0 3 0 0 0 0 0 9 10 78 1 1 1 0 1 0 1 1 1 1 11 1 1 2 3 4 1 1 -1 5 1 6 1 12 7 1 8 1 9 10 1 1 11 1 12 1 13 1 2 1 3 4 5 1 1 1 6 1 7 14 1 8 1 9 1 10 1 11 1 12 1 13 15 1 14 1 15 1 16 17 1 1 18 1 19 16 1 20 1 21 1 22 23 1 1 24 1 25 17 1 26 1 27 1 28 2 2 3 2 2 3 18 2 17 3 17 2 18 3 18 2 19 3 19 19

• 2 20 3 20 2 21 3 21 2 22 3 22 20 2 23 3 23 2 24 3 24 2 25 3 25 21 2 26 3 26 2 27 3 27 2 28 3 25 22 3 3 4 4 5 5 6 6 7 7 8 8 23 9 9 10 10 11 11 12 12 4 2 2 4 24 4 10 500 50 20 5 0 -1 2 6 6 1 25 PROBLEM DIMENSIONS AND OPERATf6NAL 00hTROLS 26 0.0282 12 27 0.0305 21 0.075 28 28 0. 5 0. 0 -9.8 1.0E-4 29 .0001 1.0E-6 1.0E-4 1.0E-3 30 1.0E-12 1.0E-12 1.0E-12 1.0E-5 0.1 31 0.05 O.90 0. 3 100. O.1 100.

32 EDIT CDhTROLS A@ POSTPROCESSOR 00hTROLS 33 0. 0 1.0 2.0 3.0 4.0 34 0.0225 0.0225 35 0. 0 36 0.225 0.50 37 0. 0 38 0.003 0.003 39 0. 0 40 0.225 0. 5 41 0. 0 42 0.010 0.02 43 0. 0 44 4. 0 8. 0 45 0. 0 40 0. 0 47 0. 0 45 -

0. 0 49 0. 0 50 0. 0 51 0. 0 52 0. 0 453

m RDJ3H DRE~ 53 VIEW FACTORS 54 55 TIME STEP CDNTROLS 56 0. 0 1.0E-5 1.0E-9 0.249 57 .001000 0.5 10.0 10. 10. 10. 58 1.0 0.96 0.02 0.02 0.0 0. 0 59 STRUCTURE AND FAILURE PARAMETERS 60 0. 0 61 0. 0 62 1. 0 1. 0 1. 0 1.0 63 1.E+10 1.E+10 1.E+10 1.E+10 1.E+10 1.E+10 64 1.E+10 1.E+10 1.E+10 65 0 66 FUEL PROPERTIES AND EQUATION OF STATE 67 2000. 2090.00 273.16 3.334E+5 1.0E-12 68 1001.78 4217.1 0.0727 1.0E-12 1.0E-4

   .69 3.17771E+10 4.70579E+03                          0.0 3.22689E+06                647.286        0.390597 70            1402.           1.329            3.737 2.93390E+6                         18.             32.

71 316.957 95.770 316.957 72 73 74 75 IRON PROPERTIES AND EQUATION OF STATE 76 7365.0 639.0 1700.0 2.60000E+05 25.0 77 6100.0 750.0 1. 6 30.0 5.36000E-03 78 1.33800E+11 4.33700E+04 0.0 8.17000E+06 10000.0 0.360 79 492.0 1.26 1.64 0.000000000 56.0 7700. 80

• 81 82 83 WATER PROPERTIES AND EQUATION OF STATE 84 1000. 2090.00 273.16 3.334E+5 .68 85 1001.78 4217.1 0.0727 .68 1.0E-4 86 3.17771E+10 4.70579E+03 0.0 3.089805E+6 647.296 0.3660361 87 2181.6 1.2115 3.737 2.93390E+6 18. 32.

88 316.957 95.77 316.957 89 838.607921 1.34831230 .0027491404 90 2.85605286 .381221451 .0748002905 91 IRON-ALUMINUM THERMITE PROPERTIES Af0 EQUATION OF STATE 92 4000. 1060. 200. 2.76000E+05 22. 93 3800.0 1060. 1.00 22. 4.30000E-03 94 1.44000E+11 5.17080E+04 0.0 2.62000E+06 8400.0 O.597 95 492.0 1.26 4.4 0.000000000 75.0 96 6466. 97 AIR PROPERTIES AND THE EQUATION OF STATE 98 0. 0 0. 0 0. 0 0.0E+4 99 0.0 1. 100 1.0E+12 4.0E+03 0. 0 5.0E+06 4.0 0. 3 101 716.67 1.4 3.796 4.E+6 28.967 65. 102 103 COMPONENT PROPERTIES 104 9890.0 9690.0 9590.0 9890.0 7365.0 7365.0 105 4000.0 0. 0 100 1000.0 1000.0 6100.0 1000.0 3800.0 9690.0 107 9690.0 7365.0 0. 0 454

y 105 2000. 2000. 2000. 1498. 2000. 2000. 109 2.00000E+03 2.00000E+03 2.00000E+03 110 HEAT TRANSFER CORRELATION DATA 111 1.0 1.0 1.0 1.0 1.0E-10 112 1. 0 1.0 1.0 1.0E-10 1.0 1.0 1.00E-10 113 1. 0 1.0E-10 1.0E-10 114 0.023 0. 8 0.4 0.0 115 0.023 0.8 0.4 5.0 116 0.023 0. 8 0.4 0.0 117 0.02.2 0.8 0.4 0. 0 118 0.02.'s 0. 8 0.4 0. 0 119 0. 6 0. 5 0.33 0.0 120 DRAG CDRREI.ATION DATA 121 1.0 12.0 2.0E-4 9.2E-7 1. 0 122 4.7 1.0 0. 5 .001 1.0 123 0.046 1.E+20

                                   -0.2          0.001         0.046           -0.2       0.001 124 PRESSURE BOUNDARY Q)NDITIONS 125        0.83E+5         0.83E+5 126              0. 0            5.0 127 PARAMETER REGION 1 AIR 128                T.            O. 0       1.OE+5            O. 0          0. 0         0.1 129 <             0. 0            34.            34.           0. 0 130                                                                       0.125         0.75

0. 3 0.3 0.2 1000. 1.0E+4 1000.

131 1950. 0.64 1.0E+5 2.3E-5 1.E-17 .00750 132 7.5E-3 133 PARAMETER REGION 2 THERMITE 134 7. 0. 0 1.0E+5 0. 0 0. 0 0.1 135 0.0 34.

• 34.

136 0.0 0.125 0.75

0. 3 0. 3 0. 2 1000.

137 1.0E+4 2000. 1950. 0.64 1.0E+5 2.3E-5 1.E-17 .00750 135 7.5F-3 139 PARAMETER REGION 3 WATER 140 7. 0. 0 1.0E+5 0. 0 0. 0 0.1 141 0. 0 34. 34. 0. 0 0.125 0.75 142 0. 3 0. 3 0.2 2000. 1.OE+4 1000. 143 1950. 0.64 1.0E+5 2.3E-5 144 1.E-17 .00750 7.5E-3 145 LOWER BCUf0ARY INITIAL VELOCITIES 146 0. 0 0. 0 0. 0 0. 0 0. 0 0.0 147 0. 0 0. 0 0. 0 0. 0 0. 0 0. 0 148 149 150 MESH SET 1 W/.TER 151 1 20 12 0 1 1 1 0 3 152 153 154 155 0. 0 0. 0 0. 0 996.00 156 0. 0 0. 0 1 i 157 0. 0 0. 0 299.00 155 0. 0 0. 0 0. 0 0.601619 159 299. 160 0. 0 0. 0 0. 0 0. 0 0.0075 0.0075 161 Ml:Sil SET 2 AIR BETVEEh THERMIT). AND WATER 162 21 21 1 4 1 1 0 0 1 455

I j 163 164 165 166 167 0. 0 .1 168 0. 0 0. 0 0. 0 0. 0 0. 0 299.0 169 0. 0 0. 0 0. 0 0. 0 0.0 0.0965112 170 3000. 171 -5.90 -5.90 0. 0 0.0 0.0075 0.0075 172 MESH SET 3 THERMITE RING 1 173 22 27 1 1 1 1 0 0 2 174 175 176 177 0. 0 0. 0 0. 0 0. 0 673.3743 178 179 0. 0 0. 0 0. 0 3000. 180 0.0 0. 0 0. 0 0. 0 181 3000. 0.0 0.0965112 182 -5.90 -5.90 0.0 0.0 0.0075 0.0075 183 MESH SET 4 THERMITE RING 2 AND 3 184 - 22 27 2 3 0 1 1 0 2 185 186 187 185 0. 0 0. 0 0. 0 0. 0 336.6871 189 190 0. 0 0. 0

• 0. 0 3000.

191 0. 0 0. 0 0. 0 0. 0 0. 0 0.0965112 192 3000. 193 -5.90 -5.90 0. 0 0. 0 0.0075 0.0075 194 MESH SET 5 TERMITE RING 4 195 22 27 4 4 0 1 1 0 2 196 197 198 199 0. 0 0. 0 0. 0 0. 0 168.3436 200 201 0. 0 0. 0 0. 0 3000. 202 0. 0 0. 0 0. 0 0. 0 0. 0 0.0965112 203 3000. 204 -5.90 -5.90 0. 0 0. 0 0.0075 0.0075 205 MESH SET 6 GAS OUTSIDE OF THERMITE ~ 206 21 25 5 12 1 1 0 0 1 207 208 209 210 211 0. 0 .1 212 0. 0 0. 0 0. 0 0. 0 0. 0 299. 213 0. 0 0. 0 0. 0 0.0 0. 0 .96534 214 299. 215 0. 0 0. 0 0. 0 0. 0 0.0075 0.0075 216 Mr.Sil SET 7 G4S ABOVE THERMITL 217 26 26 1 4 1 1 0 0 1 456

i W.f p' -

                                       \\
                                           \                          '
                             /
                                      /a x                                 -
                                                                                      /
                                                                                         /,
                           //
              \r b C D*

vs /

                                              /

f,e f

                 /

x\ w ,z o oo "

                          ~\   .
                                           /*

3 &Y s - r ,v 7

                                               * ,s,/>yy,
                                       =
                                                         >s a T 1 g . U * .-       1r.11l'3 volV"T 3,actier*                 ,, , , , ,.e , e s e,   - 1 " > " " " " ~

457

ROUGH DRAFT o 1

 \                  ,
                      /
                 \'                           '

g' f C 2

  /
                                                  , ,           3.oso s
         ~

N , . p 3p(PVill >

                             .3 l
\ l i' -        Q;11*bd           ori Ii * ,g 3 7 J 51MI'E ' I 458

ROj3H DRAFT U hb ~ e

                  /,'
  /
                                      ,1,        0.100 s
  ,omu                       '

yng. U-9 Melt pereu d ier e , ,,33, star.dard SikM 3'I 459

10;3- DWT W

            - ~ ', \
                                                                         ^

vae

                          ,,  7
  /

n u- 0. iso s

          \
  ]Q    $                             h rig. U 1"-       start of IO*  I    roductier *'lth star.dar $} hidk.

• r.isde l l t V '

450

ROJGF [RAr s WATER

              #          ,\
                     ' f a,
                        , \                                      '

W

                                                                      ,);;
                                                                     ,f vArOR k-
                                                                            .  -n,    c T]Mt             J. m   w  a
                         ,-              ~

x- , TsEnv1TE 4

                                    $o,:f*

g,v 1 1 lsp. U-11. kesults witi s t a rde r d SI ANI'K- i l red e l i r g a s the c er.t e i r e r h, ,, r- is be iry apprca6 hed. 461

                                                                        &   'N
                                                            ],n                }   { ~~

iiV wU s \1ti wt.TER

              \s \
              .\
            ; u*

3 i s

                                                                 ~
                                                                       \~ '
              \                                          1, x                                            y ,,

l' VAPT y TIME 0 22a S

                'x
                        'e

(

                          .e 1wtRM]TE                en           " s~
                        +

A , , "* ~ ~ z.2 lie. U-12. Reults at tirce of the stear explosier with stardard 514Nik-Il r.delir.g. 462

m q 1 h I" l - h (d ( i l\bi The s e t . r, s o ; , u ! .1 1, used the s'ardarJ S!!Ni k'-l l t ode withec* ars r Jel si ar: t- 1: t NI'Sli k ' I - : s jaser ir Ta bl e V- 1. Irit:a1 t . r.J t , - * - T r.: J wru..rd seit pr.gres s .: r e .- :r  ! er- , ,. . r. i.r - - s: ir lig- U t i.r ugi U-12. lneit is a sirrificant sir ilarity to the three-tield results. a1though it.e va re v. lure Irottnor is obstousiv low toward the hetter e- the prerietre at* the erJ el the calculatior. The a- c er:e r t i r. the salcul ti.'rs r.. \ he .. sensequette of the bett transfer assurptier . 11 the fuel s surlate aree certro: leJ the heat transler, a radiatier heat ilux per unit \ c l ar e t r I.e expressed b\ 3r ty . 4 h= c T, -l g e (U-1s khete c, t , is *he t ut l-s el ur e trattier i is the Iut! Jr.p r J Ls. e is 5. i, - 1. - k <r L' T 3' the tut! sc-1 se terperature. 1 Tg . is . w .. t e r terrerature. water surface terperature. e- stect t er pe ra

• ur e . arJ

   <            is s ar e % r*               e' criss!vits view-laster produst.

1: T, -3 im K ard T 37 .too L. we sur ! tter T, - T Sur quartity arJ obtair 3 r. t ,

           }: -                  1. e+          lo'       T,   -T,g      <                                            iU-2 463

H lO P=) e) Af

               ~

Or the ot he r ~ h:.rd. the b dt-irarsler rate per urit valute with th'e a. t a rda r d S lit.1: K- I l l i c u i d ' ! . qu e i

                                                  .-r.:                   r. w i t h ' s .p..: : radii. i-3'L: - 1.'       ' '% '      ,

gji = r

                                 ' ;'                      1T' ~T S                                                        '

tU"3) _i- + _r.

                                       -L 3

L, where

c. , is the water-volute'fra(tier-.

        'L,         is the thert.:te's thertal cor.du6tivity. 22 W/(c-k).

L, is the water s thermal cerductivity. t e. 65 W/t t-L ) . ard

         .T y       i i. the water terper:.ture.

If we a u ure T, - Tsu r I'l* '"-21 i ar.J 10-31 wi!! be equel i f-

                  <            3' s
                 .(
                         .M.   --- = 1. 7ts = 10 2

( U -1 ) I f r = 7. .* r.r . i = o. 3 7-seg. Consequent ly, although the S int.11:R-! ! f o rta l i s t does ret pow e u the prefer t ur.t t iera ! fert ler filt-beiling heat transfer Jriser by radiathr. the water's l es. thertal corductivity dee, lead te heat-trar. !er rates sitilar te these with radi tier. usir.g a reduced etissivity. The third calculatior used the new Siht1f.R- l l codeling of this study. Spesit ually. the he t trarsler w a *- revised to salculate both the er.ergy t r a r.a. f e r t c the water a.urlase tellowirr.tertatt with a fuel dreplet as weil as t he. erergy trar., er ir.te the bull water. The diflerence betweer these two e r.e r g v- t r a r s l e r rates was used to prode 2 tear e r. the surface of the water d r tip l e t . ' l.l l es t i ve l y. the stear p'os . .i-. . rate is irc reased by a fatter el a ppresir.a c l v 20 ese that inpli . r. L,. 10-3 ). The results of thi. calculatter are shewr ir l ips. t0-13 te t V- 15 ). lie r e t he .va por produstier is 464 1

N s

  ~                                 '

ROUGH DE so'strerg tha- t he r:$ :

• d..e s ro- reath t he . bot t er of the certairer dur ry the.

                 - s.iarse .: the s . r p .: t . *                        .

De a.!!. ~th: t i.ri r - :r i. . 1. t. i .. ' : i r pp h.. bis ;urrishes the best .q r e e v r t to experirert as i- 3h s.r'by I ; g . - -it s it'the tait text. Ilowe ve r . its sur he-de ve l opt e r

• ard use is besord the acore el;this pregram. Th e s t rd a r d 51'.t.11.k- l l ialculatact tlear!v does rot .produse enough vapor, while the rev: sed d.

heat.-transfer rodel produses eue ve va pe r . This suggests that litits. tar te obt ired en the extert c: precisity b larger u;.le, if triggerir.g is igrered, l'i ra ! !.v . these t Isulati r.s a l so. suggest a directior. ftr the test cost-effective ruterital code l i r.g I c rt:. l i st - for coarse pretixirg. if future

                 . develop er.ts ir t h i s'     area are                         .udged

te pes sess t ec htica l - te r i t. _The followit;

                  . facts-are available.           Ti.e t we- t i e ! d S Dt.11.R- 11 c a l c u la t i ons . e ve r. wi t h ' wa t e r a r.d tel t .ecviry wit h t he         sate l e t .: ) velocity, probably could . be made to                                     tsechle closelv the t h r e e - 1 : e l d s a l o t.1:: t i e r s w i t h ad ius,t te r.t s of mode l pa raee t e r s.                  The experimental irrerratier                                  is trude for d e t a i l ed med e l i r.g purposes.             Improved rodels for crergy ard rorertur c ouplieg t errs will rot he sirple.                                                A three-field calcula* t er wil: be s igr a t is a r lv tor e e xper.s ive than a two-field calculatier.

With thiy knowledge. t. rew code with tw. velocity fields. ore fer the rett a r.d ere tor the w.ter-stear-hydroger. cs re presert the appropriate approach !c this p r e h ; e r:. 465 +. . .... .

                                                                                          ==*  P
                                                                           \
                                                                              \

N \ t

          $                                                               Mk    '

W~ 7. 9 . 4 (Y

                      / ,a, yAPCC                                 :

[ 4 g 0 000 M

      ,,ce D E                   

v *?E

                                    ,<j($',
   , Q * \ ~,' ' jygtis          331ser' th       T'Vt*2 spML.K.51 heu\ gyarsiCI' 466

R:JGF DRAr 4

                                       /            N          wi.iER e"

og\\ - p 3, . VAPO x TIMc- ".050

                                                                  ;         S 1sto#l'E           }"      s l

s , ,3(<86' 33;. U-l#' g 31.watti COT'" , 3 g d. r e v t " 3 p9gy.11 nod e l i r.g - 467

M

                                                                                 \        .

4.c qikg\, ,, e -

                           \

N > A x y. I,

                                                                           /
                                                                                      /

v oo" -

                                            /

7 s d ms- -.1e0 s A A V

                        \-   ..

y~ c m t *=

                                    , V' 11E  g.13.       it,1,4 g,stiefE               yoduct2Cf with the           T'I' si e ,g gedtIIIt 468

R @

• y "

h D.l vw '] w ' WAIER t n x s

                         \.

i

                                                 ; \c; 4

av

                                          /

we s w_ "

                  .A 7

M

   ,grem TE
      ~                                #   ~
                              "/, [,%,'>>* ; -

rig. V gegstring g 7,,1 t breakur 55 th revi" ggggy.I1 g.edeiiEE-469

                                                              \
                                                                          \

Z

                  .s                                        .

s p. .

                                                     .     (
     \                                              :    y s A, y/ ..

I

                                                       /

vAPOS 7 u st 0. m s

            \

M pfUY b 1ig. V hie l t bre3 M th T'V'**d Sibbill'II *

• 470

R3.GH DRA T WATER

                                             \
                    /df                         \      ,>~ 'N
       \

N

                                             ,       ,         \

N ,

                                                       'u /

vacce 4/ / TIME D.22a s s

                            "sh THEcu!TE
                        ++

y-y' rig. Q-16. Corfigurstier at erd of calculstion for cor pa risor w:th I'ig. O-ti ard fig. V-12. 471

ROJ3F DRAFT Al'Pl:NlilX k Siti.1!'k-ll INPUT lok Till. ANALYsl$ 01 Till @- 19 iTI'll1 All.NT US1Nb A NUNU'.1I OR'1 tilXlhG toNI. 1 o -lo5:J '7kkli AN;Li st s of I'.XI'I.kis11.NT m-19 TRY 2 2 2 o o o o 32 1 1 100 0 3 SI ANI K-2 STUin ol' Till' S AfJ11 A val'UR IT!'LosloN TESTS 4 ANAlisi s 01 INPl.kl'.11'NT Lil- 14 Tki 2 5 .oo4500 1.o 6 12 2h 7 l'LUID DYNA 411r$ INTLGI k INI'UT 5 1 o 0 -2 5 0 3 o o 0 -2 8. 9 to 54 1 1 1 0 1 0 1 1 1 1 11 1 1 2 1 3 1 4 1 5 1 6 1 12 7 1 b 1 9 1 10 1 11 1 12 1 13 1 2 1 3 1 4 1 5 1 6 1 14 1 E 1 9 1 10 1 11 1 12 1 13 15 1 14 1 15 1 16 1 17 1 lb 1 lu lo 1 20 1 21 1 22 1 23 1 24 1 25 17 1 26 1 27 1 26 2 2 3 2 2 3  ! 15 3 3 4 4 5 5 6 6 7 7 6 5 19 4 4 to 10 11 11 12 12 4 2 2 4 20 4 to 50" So 20 5 0 -1 2 6 6 1 21 PRollLI31 l'I A11:N$10NS AND OPERATIONAL CONTROLS 22 o.o252 12 23 o.o305 21 o.075 25 24 o. 5 0. o -9.5 1. 01:-4 0.25 25 0001 1.oE-6 1. 01 -4 1. 01:- 3 1.0E-5 1.0L-4 26 1.oE-12 1. 01:-1 2 1. Ol' - 12 1. 01.-5 0.1 27 0.05 0. 9 . o. 3 l oo. 0.1 l o' - 25 EDIT CONTROL $ Af3> POSTPRori:ssoK roNTkols 20 o. O o. 5 1.o 1. 5 2. O 3o 0,oo.,2 o.ono$ o.onto 31 o. o 32 o.0010 o.0035 O.0045 33 0. 0 34 .ooool 00005 0. t H H i1 35 0. 0 30 o. ( H H p5 O.Otil (l.0045 37 o. o 35 O.01o o.02 34 o. o die J.(* 6. t l 41 o. o 42 o. o 4I ( s. (' 44 oo 45 oo 46 o. o 4" o. E ' 4s o. o du Vllk ] art 01.s g., $1 Tltti. STI P roNTkoLS 472

R:DiDRA:T

 $2                   o. '          1. o! -n         1. ol c              o.24u
 $3             o-.o!-r.                 o. 4           1. . . o              i n,          10               jo,
 $4                    1.o             n.un            o,i.2               o, o2            0. O             o.

f5 sll.t . ~Tt i.I M.1 i Alti isi PAks.4LTIKs Fb o. ' g- o, o

 $5                   1.o                1.o              1.o                1.o 54             1. l.+ 1 "          1. I -l o        1. l.-l o           1.1.+ 10        1. l.+10          1.1-lo 60             1.1-lo              1.1.-10          1.E+10 61           0 62 EUEL PROPERTil.S AN1: IUUATION 01 STATI:

63 lo"o. 2000.00 2"3.16 3. 3341:+5 1.01-12 64 loo 1.7b 4217.1 0.4727 1."E-12 1.OE-4 65 3.177711:-1o 4.7of"4L+o3 0.0 3.22659E+ob 647.256 0.390f07 66 14o2. 1.324 3.737 2.93390L-6 15. 32. 6' 316.45- 95.77o 316,957 65 60 70 71 72 I Rof. PROPERTIES Ahb EQUATION OE STATL 73 73ef." 639.0 1700.0 2.60000E+05 25.0 74 6100.o 750.o 1.6 30.o 5.36000E-03 75 1. 3 3 8. u sE.1 1 4.13 7. rol +o4 o. i t 6.17000E+ob 10(Hio. O O. 36 76 492.O 1.26 1.64 o.000000000 56.o 7 70u. 77 75 79 go 51 VATEK PROPERTils AND EQUATION DE STATE

 $2               1000            200" . 0"         273.16           3.334E+5                 .65 63          1001.75               4217.1           0.0727                   .65         1.0E-4 54 3.17 7 711:-1" 4. 70f 741.+03                        0. 0 3.05050$E+6              647.246       0.366"301 55             2151.6             1.2115             3.737 2.93390E+6                        lb.             32.

66 316.057 45.77 316.957 200 67 1346. o.33o2 5.E+o4 50 .5 65 535.6o7421 1.34531230 .0027-1914o4

 $4     2.556o$256 . 3512 214 f 1 . o"'46002905 9o Ikoh- ALUMINUM TilERMITI: l'ROPERTil:s AhD EUUATION OE STATE 91               anno.              1o60.             200 2.76oooE+05                        22.

92 380o. o lo6o. 1.00 22. 4.3o0001:-03 93 1.4.80001 11 f.170boE+04 0.0 2.62000E+06 6400.0 0.f0" 94 492.o 1.26 4.4 o.000000000 75.o 6-105. 95 90 97 IlYDimiEh G As l'KuPl.KT]l:N AhD Till EUVATION OE STATI 95 O. O o. O o, o 1. Ul+-1 0.0 94 1. loo 1. 01~- 12

                    .            4.01+ot                 o. o               0. 0              1.0             o. i 101              1o164.               1.Jof            2.415                 o. o          2.016               35.

lo2 lo1 104 ro4Po'.lET l'kol'i:RTII.S tof 989... . . 9 5 u. .o 459...o 9s vo. o 7365.o 7 3cf . o lon .: . ,00. . . o... 473

R:v3H DRAr te.7 18uv. o

             .             1000.o            b l oo. o           t oon, o          3500.o          4b9". o 105 9 hoo. o               7305.o            0. o joo              2o0..             2 o. .. .            2. .oo;            laus.          2000             2 . .. .

11' 2. ""no'I -" ; 2. no' a 'ot +' i 2. ooooal +"3 III 111 AT TL A*.si l k rokki.L ATlot, ll;T A 112 1.0 1.0 1.0 1.0 1.OE-10 1.o 113 0. 2 0. 2 1.01-10 0. 2 0.2 1. 0 01.- 1 " 114 1.0 1. 01.-1 0 1.01-10 115 0. 023 0. 5 0.4 0. 0 116 0.023 0, b O. 4 '5. 0 117 0.023 0. 5 O. 4 0. 0 115 0.023 0.5 O. .t 0. 0 114 0.023 0. 5 0. 4 0.0 120 0. 6 O. 5 O.33 0. 0 121 DRAD CORKELATION DATA 122 1.o 12.o 2. 01:-4 9. 21:-7 1.0 123 4. 7 1.o 0. $ .001 1.O 1. l.-2o 124 0 . 884 6 -0.2 O.001 O.04b - 0. 2 U. 8 H P1 125 PRESSURE Bott;DARY CONTi!T]ONS 126 0. 5 3l A f 0.53E+5 12' O. o 5.0 125 0.53E+5 0.53E45 120 0. 0 5. 0 130 PARE 1ETER kIGION 1 PRl311.\ RTGION 131 7. o. o 1.014 5 O.O O. O O. I 132 0. o 34. 34. O. o O.125 O. 75 133 O. 3 o. 3 v. 2 1000 1.OE+4 10"o. 134 145' O.04 1. Ol> 5 2.3L 5 1.E-17 .0""15 135 1. 51 -4 136 PAReil:TLR ki:GIO';2 Cil!41';EY 137 7. o. O 1.OE+f O. O O. O O.1 135 O. o 34, 34. O. O O.125 O.75 134 O. 3 0. 3 0. 2 1000 1.OE+4 looo. 14o 1450 O.64 1. 01.+ 5 2.31:-5 1.L-17 .Doo15 141 1. 51:-4 142 PARE 1ETER RI'GION 3 WATER 143 7. o. o 1.01:+5 O. o 0. O o.1 144 O. o 34. 34. O. O O.125 O . 5 145 O. 3 0. 3 0.2 1000 1.OE+4 1000 146 195's. O.64 1. 01.+ 5 2.30-5 1.E-17 .Ooolf 147 1.51-4 145 LOWl:k BOUtill;K) If ITI AL VI:LOCITILS 144 6. (

• O. O O. O O. O O. O o.i' 15's o. o 0. 0 o. o 0. 0 0. 0 o."

151 152 153 kil:sil S1 T 1 nr.11'ArTl:l> rORI: 154 1 2A 1 12 1 1 0 4 1 155 91 b2 5t. 57 15o 15' 155 15u o... o.i. o. o 475.5 154.i. lo" o. o.1 1 (31 o, b oo )OU. 3000 o,o 2 UU. U 474 l 1

REGH DRA:T 162 o.O o.O o. O o.4914125 103 3ta."$'l 1oJ o. o O. o H. O oo .0010 .o'1 1of M i'ul; 71 Mly la Tl'l.1 100 7. 4 hoobl "2 4. 702551 -o2 3.132451402 3. 0o4 721 +02 3. (HH111E+02 3.(HH v a.1 '2 107 3. oooo'l -o2 3. s a n n 'ol +o2 3. o' n H H )l +"2 3. (H H H H)l + 02 3. 0(H)00E+02 3. (H H eo"l -o2 los 5.2. soul..,2 7.617o31+o2 4.3517514o2 3.057941>02 3.00251E+02 3.0000611o2 jou 3. o, H ,nol +o2 3, o000. .l .o2 3. oo. H .olwo2 3. t H H H H ilgo2 3. (H H H H)E+02 3. t H H H >or-o2 170 5. 654131 -02 5. 244 7el4o2 6. 056581402 3. 44 7491402 3. 022 391.+02 3. (HH+of E-o2 171 3. ooool l +o2 3. on.H n .1 +02 3. 8 H)ooolao2 3. (H H H Hil A02 3. (H HH Hilho2 3. (HHu il o2 172 5.5o465i+o2 5.323521+o2 6.535551.+02 3.955511A02 3.07540l>02 3.003101 "2 173 3. o H a >$1 +02 3. om a n #1 -o2 3. ooooolf n2 3. 00oooE+02 3. 00000E+02 3. 000001: 02 174 5. 4o4241 Ao2 5. 2 to971s"2 7. 23943l +o2 4. 465451>02 3.15950L+02 3.00590E-02 175 3. t H H'311%o2 3. 0000114o2 3. om kiol>02 3. oooo0E+02 3. 00000lkO2 3. 00000E-02 176 5.60051L+02 5.273101 "2 7.72032iko2 4.927921A02 3.28597E+02 3.02091E 02 177 3.ooo46L.02 3.oooo31*o2 3.oooo0E+02 3.00000L+02 3.00000E+02 3.00000E-02 175 5.620691402 5.495541+o2 7.94709t+02 5.66259E+02 3.55657E+02 3.05424L+02 179 3.oo304L+02 3.000121402 3. 000001Mo2 3. (HH600E+02 3.00000E+02 3.00(HooE+02 15o 5.377651+02 5.59655L+02 5.non24E+02 7.29257E+02 4.45674E+02 3.22456E-02 151 3. 013951402 3. 000551 +o2 3. m H H 2E+02 3. 00000E+02 3. 00000E+02 3. 000ooE 02 152 5.413621>o2 9.122761402 5.21516E+02 7.50720E+02 6.39179E+02 3.99465E-02 153 3.04954L+02 3.oo417E+02 3.ooo13E+02 3.00000E+02 3.00000E+02 3.00000E+o2 154 9.19076E+02 9. 532321bo2 5. 59173l402 6. 65754E+02 7. 53621E+02 5.460591. o2 155 3.514441402 3.o27621402 3.ooo97L+o2 3.OOOO2E+02 3.00000E-02 3.oooooE-o2 156 1.045561403 1. 3432bL+03 1. 064521s03 5. 55007E+02 6. 95054E+02 7. 20252E-o2 157 4. 435721 +o2 3.130431402 3. 00540E+02 3. 000160+02 3.00000E+02 3.oooooE-o2 155 1.1144 l +03 1.17159L+03 1. 25410E+03 9. 21361 E+02 6.16095L+02 7. 976451:-o2 159 5.56do2i+"2 3.40140E+02 3.015771302 3.00070E+02 3.00002i>o2 3.Do000E-o: 14o 1. 4554o1 +o3 1. 9735oL+o3 1. 450131ko3 1. 21236E+03 5. 50999E+02 7. 32646lNo2 191 6. 6"21olso2 3. 514241%o2 3. 04160E+02 3. oo191 E+02 3. 0000$E+02 3. (Hwioot-o2 142 1. 506551403 2.160951403 1. 75552E+03 1. 37371 E+03 5. 536091302 6. 62959L-o2 103 7.19t,71) ,02 4.17529L+02 3.06595E+02 3.OO346E+02 3.0001oE+02 3.000001so2 194 1. 9162 3f Ao3 2. 247711403 2.14022E+03 1. 52139E+03 5. 991351.402 6. 241791 o2 145 7. 2011oL+02 4. 205351302 3. 07570E+02 3. oo4441kO2 3. 000111>02 3. oooooE+o2 146 1. 911591443 2. 26415!403 2. 405551>03 1. 723791 +03 1.02975E+03 5. 49353E o2 197 e. 45 341402 3. 547751 +02 3. 06309E+02 3. 004071>02 3. 00015E+02 3. 000ool o2 195 2.194061 +03 2. 332551403 2. 66559E+03 2.007561M03 1.05251E+03 4. 555221402 109 5.1495o1402 3. 35 5161302 3. 042 5 7E+02 3. 00245E+02 3. 0001J E+02 3. oooo0L+o: 20:' 2. 55395L+03 2. 725351.+03 2. 545631403 2. 421241403 1.093651A03 3. 76354L-02 2o1 4.0$$$7lMo2 3.176121402 3.026271402 3.oo164E+02 2.999951.402 2.99945E o2 202 3.<HHil.E+03 2.935991+03 2.952151+03 2.630651403 1.129251M03 7.522501 "2 203 3. 36300L+02 3.155511 +o2 3. 009411)o2 2. 994921402 2. 993591M02 2. 946011 "2 2o4 3. 00o1114o3 2. 443751 +o3 2. 99252L+o3 2. 754141403 1. 217691403 1.175461a"3 205 6. 755501402 4. 575121 +o2 3. 357521402 3. 002691 +02 2. 94560lko2 2. 916551102 206 3.mpilolwo3 3.000071403 2.999511303 2. 926151403 1.90$$$1403 1.1715114"3 207 1.193561 +03 5. 545651402 6. 76654E+02 4. 655591 +02 3. 63.1251302 2. 933541 '2 205 3.oo00514o33.000061+033.fHHiO51%032.994631fo32.66195E+032.45434E-"3 209 1. 5 24451 +o3 1. 22052iko3 9. 7.15641402 6. 351551402 t. o.16501302 3. 34430! "2 21o 3. 000051 +o3 3. oooo51%o3 3. oooo5 Iso 3 2. 964471403 2.126131903 1. 742001 "3 211 1. 351461 +03 1. 210491 +o3 1.114f 2E+03 1. o70571 +03 9. 51625E+02 5. 555111 "2 212 3. oooo31 +o3 3. oo"o214o3 3.

• H H H eol +o3 2. 769241403 9. 44930E+02 6. 34 34 1 <

213 5. u973214o2 6. 65 3691 +o2 5. 505451 +o2 9. 255541 02 6. 465451%o2 4. 056151 "2 214 3. ooooll +o3 3. ooooll Ao3 2. 994861 +o3 2. 4 305o1 +o3 5. 550251,o2 4. 55u41 "2 215 5.944721-02 6.544231+o2 .21oobl+o2 o.515151+o2 4.41102E+o2 3.4 ?151 "2 216 3. 00o0114o3 2. 04044L+o3 2. 444oll -o3 2.191051 +03 4. 950451a"2 3. 51 541 -o2 475

R03 DRE 217 4. 23o761 +02 4. 962 301:.o2 5.12 501 +o2 4. 654021 +o2 3. 79407E+02 3.1931ol +02 215 3.onoool+o3 2.ovuo 1+o; 2.quf211+o3 2.79039).+o3 7.347661+02 3.551021 *2 219 3.553441+02 3.545101+"2 3. Hobbul+"2 3.6677314o2 3.255571ko2 3.065 31 e2 22o 3. o..o.iol .o3 3. o. l ..1 2. 99b 111 +"3 2. 52 7b113o3 5.171621 +02 3. 44625l + 0 221 3. 321961 +o2 3. 3542o1 +o2 3. 34 773I.+o2 3. 234341Mo2 3. 091351ko2 3.021obl m2 222 VATE.R Mi>0K IT MIT) 223 2.4o2721-01 3.777o31. 01 5.741641.-01 5. 956701:-01 5. 49592E-01 5. 996131 -01 224 5.996131:-01 5.99613E-01 5.99o131>o1 5.99o131-ol 5.99613E-01 5.996131.-o1 225 2.150651 -01 2. 361001:-01 4.133551 -01 5. 525371,01 5. 99053E-01 5. 996921.-n1 226 5. 996131.-01 5. 996131:-01 5. 996131.-01 5. 996131:-01 5. 99613E-01 5. 996131 -o1 227 2. 075591:-01 2.165651:-01 2. 964910-01 5. 217521.-01 5. 95172 E-01 5. 994 541,o1 225 5. 99611 E-01 5. 996131:-01 5. 99613E-01 5. 996131:-01 5. 99613E-01 5. 99' 13E-o1 229 2. 04 3051:-01 2.161151 -01 2. 6314 7E-01 4. 513111.-01 5. 549131:-01 5. 95994E-o1 230 5. 99597E-01 5. 996131:-01 5. 99613E-01 5. 99613E-01 5. 99613E-01 5. 996131 -o1 231 2.020211:-01 2.15250E-01 2.45475E-91 4.025320-01 5.69343E-01 5.97539E-01 232 5.99551E-01 5.99612L-01 5.99613E-01 5.99613E-01 5.99613E-01 5.996131>o1 233 2.091551.-01 2.17432E-01 2.33001E-01 3.650300-01 5.46930E-01 5.954631-01 234 5.99421E-01 5.9960?E-01 5.99613E-01 5.996131.-01 5.99613E-01 5.996131-o1

 . 235 2. 050651:-01 2.11732E-01 2. 26352E-01 3.176541:-01 5. 01549E-01 5. 859661.-01 236 5. 99007E-01 5. 995901:-01 5. 99613E-01 5. 99613E-01 5. 99613E-01 5. 99613E-01 237 2.14 715 E-01 2. 02194E-01 2. 24 54 5 E-01 2. 466571.-01 4. 00924E-01 5. 578561:-01 235 5.9e533E-01 5.994951:-01 5.9961oE-01 5.99o13E.-01 5.99613E-01 5.99613E-01 239 2.13501E-01 1.97151E-01 2.155560-01 2.396}}