ML20003B673

Forwards C-E Responses to NRC Questions Presented in Re Topical Rept FIESTA,CEN-133 (B)

ML20003B673
Person / Time
Site:	Calvert Cliffs
Issue date:	02/09/1981
From:	Lundvall A BALTIMORE GAS & ELECTRIC CO.
To:	Clark R Office of Nuclear Reactor Regulation
References
NUDOCS 8102250152
Download: ML20003B673 (39)

Text

{{#Wiki_filter:_ BALTIMORE G AS AND ELECTRIC COMP ANY P.O. B O X 14 7 5 B A LTIM O R E. M A R YL A N D 21203 Antwum E.LuosovaLL.Ja. "'"[*"J.*"' February 9, 1981 Office of Nuclear Reactor Regulation U. S. Nuclear Regulatory Comission l Washington, D. C. 20555 ATTENTION: Mr. R. A. Clark, Chief Operating Reactors Branch #3 Division of Licensing

SUBJECT:

Calvert Cliffs ?!uclear Power Plant Unit t'o.1, Docket l'o. 50-317 Responses to t!RC Questions on FIESTA (CEt'-133(B)) REFERENCE (A): R. A. Clark to A. E. Lundvall letter, dated October 24, 1980 Gentlemen: Reference (A) presented several NRC staff questions on Combustion Engineering, Inc.'s topical report, FIESTA. Enclosed is Combustion Engineering's responses to those questions. Very truly yours, 4, ,.7 1 1.x D sa:: f BALTIMORE GAS A ' mEC IC C0!'PAf'Y { l g._ cv .-r $ i,):5 E.L l0:5 i =56 a11 m A. E. <u val, Jr.' I l M, ' ~ ' g, Vice Pres dent - Supply A r AEL/WJL/djw Eclosure (25 copies) Copy To: J. A. Biddison, Esquire (w/out Encl.) G. F. Trowbridge, Esquire (w/out Encl.) E. L. Conner, Jr., NRC P. W. Kruse, CE 8102250 15 1

ENCLOSURE Question 1. C-E has stated that the effects.of feedback on the time variation of the spatial flux during a reactor scram can be neglected.

However, in the HERMITE code topical report (CEMPD-188), there is approximately a 3% increase in reactivity worth a central CEA at full power neglecting thermal feedback as compared to including feedback.

Please explain why feedback has an appreciable effect on ejected CEA worth but not on scram worth.

Response

The bas's for this question appears to be the numerical data of Table 5-1 of CEHPD-188. This table is potentially confusing for the following reason. Reactivity data comparing rod worths calculated by different means are usually given in units of ap. This table, however, used %Ao as the unit of reactivity. Consequently, the apparent 3% reactivity discrepancy between rod worths calculated with and without thermal feedbacks is only a 0.03% ao difference. Thus, the HEliMITE topical data tends to show that feedback has no more appreciable an effect on ejected CEA worths than it has on scram reactivities. Feedbacks can be neglected in the calculation of scram reactivities because the temperature distributions of the fuel and the moderator are effectively " frozen" during the scram rod motion. The time constants associated with heat removal from the fuel rods are on the order of several seconds greater than the scram time. Consequently, the influence of temperature changes on the flux distribution during the scram rod motion would be smali. i I i l

i Question 2. In addition to the axial shape index, it seems that the space-time scram shape should also b sensitive to the type of transient. For a reactivity-initiated tran;ient, for example, the scram would begin in a situation where much more of the neutrons are prompt than would be true in a steady state. Have situations been investigated where the delayed neutron background is small relative to the prompt flux when the scram starts because the reactor has been on a large positive period?

Response

Situations have been investiaged in which the scram starts during a large positive reactor period. It is correctly pointed out that as delayed neutron effects becomes smaller, the difference between static and space-time calculations of scram reactivities also become smaller. A study of a prompt neutron insertion event was made. Results are shown in Figure 2.1 for a set of investigations in which varying amounts of reactivity are injected in 0.1 seconds and scram rod motion occurs at 1.0 seconds (feedback is neglected). This figure shows that as the ratio of inserted reactivity (g) to 8 approaches unity, the scram reactivities approach the values calculated by static methods. The conservatism associated with the use of FIESTA-generated scram data for accidents with positive reactivity insertion prior to the scram would depend upon the relative influence of the scram and other phenomena (e.g., Doppler l coefficient) on the outcome of the accident. Consequently, C-E does not use FIESTA-generated scram data for the analysis of accidents having a positive reactivity insertion prior to scram. 1

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Response

Results from positive reactivity insertions indicate that the use of static reactivities is usually conservative. fio credit for this potential conservatism is taken in the safety analysis. For the roo ejet.i. ion analysis, the rod is ejected at such a rapid rate that the reactivity at intermediate insertions is of little consecuence. It is only the fully ejected rod worth which is of interest. tievertheless, a consistent comparison of the rod worth at intermediate rod positions indicates that space-tirl.e reactivity calculations during ejection are conservative relative to static reactivity calculations. This is shown l in Figure 3.l' The fully ejected rod worth is calculated consistently with both analysis methods. For the rod withdrawal analysis the difference between static reactivities and space-time reactivities is very small. The slow rod withdrawal rate allows the neutron precursors to redistribute so that they are essentially in an asymptotic equilibrium with the flux shape. The use of static reactivities is then a very good approximation for the rod withdrawal l accident as shown in Figure 3.1. For the negative reactivity insertions associated with a reactor scram, the difference between space-time and static calculations is much larger than for positive reactivity insertions. This is shown by the difference between space-time and static reactivity evaluations shown in Figure 3.1 for positive reactivities and Figures 3.2 and 3.3 for negative ones. These differences increase as the strength of the rod bank increases. l

For negative reactivities at a given rod insertion. as the strength of the rod bank is increased, the static reactivities increase less than the space-time reactivities. Because of the nature of the static solution, very black rods provide little additional negative reactivity insertion. Since the space-time solution accounts for the delayed neutrons as well as the prompt neutrons, blacker rods continue to provide additional negative reactivity insertion. At full rod insertion both analysis methods predict the same reactivity. For negative reactivities, as for positive reactivities, the reactivity versus rod insertion is sensitive to the rod insertion rate. This sensitivity has been accounted for in the development of conservative space-time scram reactivities used in the safety analysis. The results shown in Figure 3.1, 3.2, and 3.3 were developed for the same initial power shape (EOC saddled power shape). For other initial power shapes, a different set of curves would be calculated. However, the above conclusions are not expected to be sensitive to the initial power shape :: sed in the analysis. l l { j ~ L

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c. n. e n .n FIGURE 3.3 Comparison of Static Reactivity and Space-i Time Reactivity During a Reactor Scram for / Various full 1.ength Rod Worths and Various /

• 'I '

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Question 4. Since FIESTA is only a one-dimensional code and the CEAs are not unifomly distributed in the raf al direction, how are the group constants obtained for each region in a (Z, t) calculation? How is a stuck rod accounted for and is it included in the scram curves shown in the topical report?

Response

The coding of FIESTA is based upon the combination of space-time equations and the neutron diffusion and themal feedback equations used in steady state two-group, one-dimensional codes. FIESTA, has the capacity for representing the cross sections for the distribution of materials, control rods, dissolved poisons, and moderator and Doppler feedbacks. As used in FIESTA, the axial distribution of fuel, fission products and structural materials are represented by the sut of local macroscopic cross sections at a fixed moderator temperature and fuel temperature in each axial region, and incremental macroscopic cross sections representing the local differential l effects of distributed moderator and fuel temperatures. Both the local macroscopic cross sections and the incremental cross sections are derived from steady state 3D power distribution calculations. The materials having spatial distributions which may change rapidly (dissolved boron and water) have macroscopic cross sections calculated by functions which account for the instantaneous material distributions. The macroscopic cross sections representing Doppler effects are represented by functions reflecting the instantaneous fuel temperature distribution. Because feedback effects do not impact the scram results generated with FIESTA, these latter functions are only.used to set initial conditions. For consistency with other one-dimensional codes at C-E, each of six groups of rods is represented by a different set of macroscopic cross sections. However, all rod groups must have the same ratio of fast cross sections to thermal cross sections. Within this limitation, each red group's cross sections are adjusted so that sequential full length insertion of rod groups has the reactivity effect ascribed to the superposition of the rod groups i l in a 3D calculation. In the representation of the scram bank, the cross sections are adjusted so that at full insertion the bank has a reactivity effect equal to the calculated worth of the scram bank with the rod having the highest worth stuck out. l 1

The scram curves in the report represent different values of the scram minus stuck rod worth, so that the effect of different strength banks can be evaluated. O t 4 9 1 . 0 4 1 f a 4

Question 5. Although it is stated in the report that feedback effects are not needed for scram reactivity calculations, there is only a minimal amount of infomation in the report describing the themal-hydraulic model. For completeness, provide additional information on the themal-hyraulics including: the heat conduction model in the fuel pellet, gap and clad; the coolant enthalphy and density solutions; and the physical properties of the fuel, gap clad and coolant.

Response

The detailed description of the feedback modules which are available in the FIESTA code were not included in the FIESTA reportII) because they are not used in the scram reactivity calculations. The details of these modules are therefore irrelevant for the current applications of the FIESTA tbde. For your infomation, we have attached these portions of Refer'nce 2 e which describes. the feedback modules of the FIESTA code (Section III-8, References, and Appendix A). Please note that Reference 2 is not part of the FIESTA report but has been abstracted for your infomation only. (1) FIESTA-A One-Dimensional, Two Group Space-Time Kinetics Code for Calculating PWR SCRAM Reactivities, CEft-133(B), flovember,1979. l ) (2) Ulrich Decher " Light Water Reactor Accidents: A Consistent Analysis of the Importance of Space-Time Effects" Carnegie Mellon University, PhD Thesis, April,1975. l l l

III-B Thermal Hydraulics The thermal-hydraulic model that is used in the reactor model is based on a closed channel flow model. The basis for this model is outlined in the following three sections. The first section presents the thermally c:enandable flow model which considers the mass and energy conservation in addition to the Martinelli-tielson separated fica model (M-4). The second section presents the heat trancfor equation for con- ~ duction in the fuel pin. The Boiling Crisis cocrelations that are used in the reactor model are shown in the last sec-tion. III-B-1 Thermally Expandable Flow Model a) Conservation Eouations The theoretical basis for the hydrodynamic flow model is based on the Momentum Integral model developed by Meyer (M-1). The equations of conservation of mass, momentum and energy ~. are solved for a fluid flowing parallel to a heated surface l l in a closed channel *. An approximation is made which neglects l pressure propagations through the fluid. This approximation has been shown to be numerically stable and time steps are not restricted by the sonic velocity through the fluid. The model has also been extended to account for two-phase flow (M-2). i

• The momentum equations are not solved in this thesis since the inlet flow rate is always assumed to be known.

- yt

The conservation lows of mass, momentum and energy are respectively: + = 0 III-B-1.1 E N DI OC E + b (o ) pg III-B-1.2 = at az az 20h ( )+ h+( ) h(pH)+h(GH) t = p III-B-1 3 where: 0 = G(z,t) is the mass flow,, rate at position z and time t (1b/hr-f t' ) P = P(z,t) is the fluid pressure (lb/ft2) H = H(z,t) is the fluid enthalpy (BTU /lb/hr) p = p(z t) is the fluid density (lb/fr.3) is the fluid friction factor (dimenatonless) I f is the gravity acceleration component in the g negative z direction l l is the hydraulic diameter of the channel D l h (4 x flow area / wetted perimeter), (ft) & - $(z,t) is the heat flux into the fluid from the channel walls i l 1 is the flow area divided by the heated 7 perimeter of the channel (ft) The mass velocity, the pressure, and the fluid enthalpy are calculated with the above equations using the fluid proper-ties and boundary conditions. A direct numerical solution to the compressible flow equa-To avoid these tions may result in numerical instabilities. instabilities, the time step size must be restricted as follows: --

At 4 e+ lul where: Az is the mesh size c is the velocity of sound in the fluid jul is the absolute velocity of the fluid For a mesh size of 5 cm. this equation restricts tJme steps ~3 to about 10 second, where c = 5600 ft/see in water and u = 15 ft/see for a 2WR. This time step size is overly restrictive since it is desirable to investigate accidents of several minutes duration. The Momentum Integral method makes the following assumptions fb f(H,P*) III-B-1 5 where: P* is a reference pressure at which all fluid properties are evaluated independent of spatial position s A further approximation is made that the pressure changes and dissipative forces result in a negligible enthalpy rise. The first approximation (Equation III-B-1 5) neglects the finid expansion caused by local pressure variations. It treats the acceleration of the entire fluid channel as a rigid body. In this thesis, problems consisting of pressure boundary conditions will not be investigated. It will always be assumed that the channel inlet flow rate is known and that it is positive. This assumption has the effect of replacing the momentum equation with the inlet flow rate boundary condition. Therefore, only the conservation of mass and energy needs to be considered. The model also assumes that changes in pressure are predetermined. + = 0 III-B-1.6 ( p II) +h(GH') = where two phase flow variations are introduced through the variables p, if and H'. These are defined as follows (M-3) h I p dA P = h / pHdA III-B-1.7 II = pA A hI Hpu dA H' = g a, N 1/ pu dA G = A z g where A is the flow area l u,is the fluid velocity f The energy equation may be re-written in a simpler form. H' 3H III-B-1.8 p" ,g = 3g 9 where the enthalpy is now a flow-average enthalpy and a new density p" is introduced. This density may b: evaluated by: III~0~1*9 = p + (II - H') _ pa

• This model accumes that dencity variationc are caused only by enthalpy changes. The effect of precsure changec on the time-rate of doncity changen ic neglected.

The model ic valid for constant or for slowly varying preccure. During more rapid b The conservation of mass equation may be written in a more convenient form: ff = - fy (f,)(f - G lE ) III-B-1.10 f b) Two_ Phase Flow The thermally expandable model is extended to two phase flow by assuming a separated flow model developed by Martinelli-Nelson (M 4). the model assumes that the vapor and liquid phases are completely separated and that each has constant properties in the direction perpendicular to the flow (velocity gradients normal to the flow are neglected). The following definitions apply for separated flow (M-3): mixing cup or flowing enthalpy H' = ~ III-B-1.11 H (1 - x) + H x H' = g y volume weighted mean or static enthalpy II = {p H (1-R) + pHR } III-B-1.12 g 1, gg y y p' where: x is the mass flow fraction of vapor pVa y y x = G E is the void fraction A v R = g , a y g pressure variations, the validity of this model needs to be investigated. 30 -

l V is the vapor velocity y are the cross sectional area occupied by the A,Ag vapor and the liquid, respectively y The density p" may be written for separated flow as follow's (M-3): {pg x + p (1 - x) } hh III-B-1.13 p" = y This equation defines an effective slip-flow-density used for energy considerations (M-3). A void correlation is used throughout this thesis which relates the mass flow fraction of vapor and the slip to local channel conditions (T-1). (H - BH ) / (H - SH ) s x' g y g = III-B-1.14 Pi vapor slip S = = p 1 .15 4"/0 where: S = inlet enthalpy to the channel (BTU /hr) H = in saturated enthalpies of vapor and liquid, H,H = y g respectively (B1U/ib) 2 heat flux (BTU /hr ft ) q" = exp f4.216 ((y - 8 353)2/(8.353)2 _ 1)1/2) Y = En(P/3206) y = pressure psia P =.

Constant slip between the vapor and liquid phases is assumed with this model. The average density is calculated as a function of local fluid conditions. y) yx'(p, -p III-B-1.15 = pg y, x, (Y-1) p Similarly the density in the energy equation may be written in terms of local fluid conditions. {pt** *Pv(1-X')}Y III-B-1.16 p" = {1 + (y-1) x')2 To evaluate the conservation of mass (Equation III-B-1.14) an additional term is defined in terms of the local fluid conditions. s IP -Pt} 1 Bo v III-0-I*I7 p (gg,) (H = p,(1-x ' ) ) H 3)( p x' t g y The last three equations when used with the void correlation, i which defines x' and y, completely define the parameters needed to solve the conservation of mass (Equation III-B-1.14) j and the conservation of energy (Equation IV-B-1.8). j III-B-2 Transient Thermal Enuntions The radial temperature in the fuel pin is calculated using the time-dependent heat transfer equation. l t (k(T)r ' ' I ) + q '" ( r, t ) PC (T) aN Q) = p III-B-2.1 40 -

5 where: p is the material density C (T) is the material heat capacity p T(r,t) is the temperature k(T) is the material conductivity q'"(r,t) is the time and position dependent energy deposition rate in the fuei pin pellet and The fuel pin is assumed to concist of a U02 the clad is assumed to be zircon)um. The temperature dependent properties are as follows: For UO heat capacity (3-2), (H-2); 2 3 lil for T < 2240*F .076 + 3.33 x 10-6.p, 4.74 x 10 .g C (T) = P (T + 460) 2 - 1.8426 + 3 8303 x 10 T - 2.0447 x 10-7'r C (T) = p 4 +h.657x10-ll3 - y 62S9 v 10-15,p for T > 2240*P T For U0 conductivity (L-1): 2 [92 9bt r + 4.7475 x 10-12(T + 460)3 (ft br*F k(T) = l III-B-2.2 For For Zirconium heat capacity (B-2), (11-2 ) ; 6.80 x 10-2 + 1 33 x 10-5.r T < 1376

• F C (T)

= p 8.60 x 10-2 7 > 1376

• F C (T)

= l p 41 - i L

For Zirconium conductivity (T 4), (H-2); 4.14 + 1.044 x 10-2T - 5.276 x 10~0r 2 k(T) = + 1,536 x 10~9 3 III-B-2.3 r The gap conductance (H-2) is evaluated by considering the varying gap thickness due to differential expansion of the UO2 pellet and the Zirconium clad and by considering the temperature dependent conductivity of the gas inside the gap (assumed to be Helium). and Zirconium is: The expansion coefficient for UO2 3,72 x ig-6, 1,737 x 1g-93, a g0 2 3 10 x 10-6 + 0 975 x lo-9 r T < 1584*F a = e c 2r 5 4 x 10-6 Tc 2 1584=F = ~ III-B-2.4 where T and T are assumed to be the volumetric average clad c p and fuel temocratures, respectively. The conductivity of Helium is: I 0 5374 JII-B-2 5 3 126 x 10-3 (T + 4601 k, = g g where T is the temperature of the Helium gas. l The assumption is made in Equation III-B-2 5 enat f(T(pelletsurface)+T(cladinsidesurface)) T = f c III-B-2.6 IC -

The gap conductivity is E III-B-2.7 h = gap hr-ft - F D in( ) p f 2 R (1 + a l where: D = p p UO I 2 (T - 681) 2(Rp+Tgap)(1 * "Zr D = c z fuel pellet radius (ft) R = f T,p gap thickness (ft) = g An upper limit of 1000 BTU /hr f t2 *F is used for the pellet-clad contact conductance for the reactor models in D. this thesis. This limit prevents division by zero when D = 7 p The heat transferred to the coolant is calculated based on various experimental correlations. \\ For forced convection heat transfer, the Dittus Boelter I correlation (J-1) is used: Re.8 III-B-2.8 Pr.4 0 o h, = 2 e ft -br *F conductivity of water (BTU /ft hr 'F) where:' k, = thermohydraulic equivalent diameter (ft) ~ D, = Prandtl number of the coolant Pr = Reynolds number of the coolant Re = For boiling heat transfer, the Thom correlation is used (T-2). 05 0*o77 0 Tsat + exp(P/1200) T = e l In.

clad surface tempe ratu re (*P) khere T c saturated water temperature ('F) T,,g surface heat flux (BI'U/hr f t2) Q P Pressure (psia) The heat Eeneration rate is assumed to be distributed among the fuel pellet, the clad, and the coolant. I9 9 mod m0 (1 - f,)q0 gr E p P 9 clad clad 9fp where: q0 the total power generated the power deposited in the mcderator qmod the power deposited in the fuel pin (pellet qp P plus clad) in the clad qclad - the power deposited The above equations define f, and Pclad' The power generated in the fuel pellet is assumed to be dis-tributed parabolically q (1 - Pdip) + 2 Pdip ( q"' (r) = the averEEe volumetric heat generation rate where: q in the fuel pellet P the fractional power dip i the center of the dip pellet (t _ que (center)/q) For the reactor models presented in this thesis, no heat is 1 assumed to be generated in the clad. The following values are assumed for the power deposition characteristics.. A

_ - ~ _ _ _ _ _ PWR BWR P 0.0 0.0 clad r, 2.6% 4.0/ P 3*3N N 05 dip The power generated in the moderator is obtaintd from FSAR data (D-1) (D-2). These values are expected to chnnge slightly as How-the local moderator density changes during a transient. ever, the core-average, full power, beginning-of-life (BOL) values are used throughout this thesis. The power dip in the fuel pin is obtained by a unit cell calculation of a PWR and a BWR lattice with the LASEB code (P-1). These power dis-tributions will vary with exposure as plutonium accumulates .in the fuel rod and central voiding occurs due to pore migra-tion (R-6). In addition, fission gas buildup in the gap and fuel pellet swelling will change the gap conductance, and fuel pellet cracking will change the effective conductivity of the fuel. The model presented in this thesis does not conside r these various irradiation ' effects. This assumption is judged to be adequate for the purposes of this thesis. The time dependent heat transfer equation (Equation III-B-2.1) is solved numerically. The details are presented in Appendix A. III-B-3 Boiling Crisis Correlations l To evaluate the Departure from Nucleate Boiling (DNB) i several margin which occurs during the accident analyris, l l, l

correlations are presented. For Boiling Water Reactors, the Janssen Levy correlation (T-3) is used, and for Pressurized Water lleactors, the W-3 correlation (T-5) is used. The Janssen and Levy correlation is as follows: III-B-3 1 DNB = loc critical heat flax (BTU /hr-ft ) where: Q = crt 1 cal heat flux (IAT0/hr-f t2) Q = 1oc 41+ 0 (1000-P) Q crt 0.705 x 106 +.237G X<X Q 1 = 1 6 6 1.634 x 10 .270 4.71 x 10 X X <X<X Q 2 = 1 0.605 x 10 .1640 . 4 53 x I L,6 6 x x,x Q = 1 s 6 .197 .108 G/10 where: X = 1 6 .254 -.026 G/10 X = 2 the local quality of the steam X = mass flow rate (1b/hr ft2) G = pressure (psia) P = The range of parameters 13 600 to 1450 psia P = 6 6 4 x 10 to 6.0 x 10 lb/hr ft G = negative to.45 X = De = .245.to 1.25 inches 29 to 108 inches l L = The W-3 correlation is as follows: - 4 f, -

I \\ [(2.022 .0004302P) * (0.1722 -.0000984P) a"DNR.EU = 6 0 III-B-3 2

• expI(18.177

.004129P)x) i [(0.1484 - 1 596x + 0.1729xlxl) c/106, 1,g)7) [1.157 - 0.869x1 * [0. 2664 + 0.8357 exp(- 3.141De)1 [0.8258 + 0.000794 (H, -H II in where: P = 1000 to 2300, psia 6 6 0 = 1.0 x 10 to 5 0 x 10, Ib/(he rt2) De = 0.2 to 0 7, in. X1oc = -0.15 to +0.15 H, > 400 BTU /lb 1 L = 10 to 144, in. heated nerimet g = 0.83 to 1.00 wetted perimeter \\ heat flux is in BTU /(hr ft ) Tong extended this correlation to channels with a nonuniform axial flux distribution by 9bNB,N 9bHB,EU/E III~0~3*3 = DNB heat flux for the: nonuniformly heated where: q NB,N D channel qDNB'EU = equivalent unir rm DNB flux from equation III-B-3 2 and DNB gt_fxpg_ggDNB)j Cf dz q"(z) e xp[-c( bNB-* I l F = qlocal I *9 7 (1 - X DNB in.-1 0.44 C = (c/10')l 72 le n n

~ axial location at which DNB occurs, in. E = DNB The local heat flux used in tr.ece correlations consists of the surface heat flux plus the direct coolant heat flux. (It is an open question whether the direct coc. tant heating should be included since it is not a rod surface phenomenon. Its inclusion, however, is conservative.) The reduction of the film heat transfer coefficient due to burnout is not modelled in this thesis. Tr.e DilB analysis, therefore, has no consequence on the transient behaviors pre-sented in this thesis other than on the evaluation of safety margins. i \\ f - 4a -

w = l REFERENCES A "The Calculation of Thermal Amster, H. and Suarez, R., A-1 Constants.sveraged Over a Wignne-411 kins Flux Socctrum: Description of the SOFOCATE Code", JAPD-1h-39 (January 1957). P.,

Horowicz, J., "New Method of

Amouyal, A.,

Benoist, Determining the Thermal Util hation Fact;r in a Unit Cell",

A-2 Journal of Muclear Enem, Volume 6 (195? ).. E Barry, R. F., " LEOPARD - A Soectrum Denet; dent Non-Soatial 5-1 Depletion Code for the IBM-7094", 4;AP-3269-26 (September 1963). B-2 Brassfield, H. C., 'dhi te, J. F., Jodahl, L. S.,

Bittel, Kinetics Data J.

T., "decommended Prooerty and deactios. 'for Use in Evaluating a Light-suber Cool. d deuctor Loss-Of-Coolant Incident. Involviny, a.irc 4 or 304SS Clad", GEMP 482 (npril 1966). s. Bahstad, P., et al, "RAMONA1 - A Fartren Code for Transient B-3 Analysis of Boiling water he ictors and bciling Loons", PB185277, Institute for Atomenergi, Kjeller Research Establishment, Kjeller Norway (November 1966). S Caldwell, 4. R., "PDQ-7 Reference Manual", WAPD-TM 678 C-1 (January 1967). E Docket-50344 " TROJAN NUCLEAR PLANT - Final Safety Anal-D-1 -ysis Reoort', (June 25, 1969). l Docket-5c259, "Brcwn's Ferry Nuclear Plant - Fir.a1 Safety D-2 Analysis Reoort", (September 25, 1970). D-3

Denning, R.S., "ADEP, One-and Two-Dimensional Few Group Kinetics Code", BM1-1911 (July 1971).

Denning, R.S., " Space-Time Kinetics Studies for Commercial D4 Power Heactors", Transactionn of t.ha American Nuclear Society, Vol. 13,2

REFERE*!CES (cont'd.) E Ehrenoreis, S. N., McAdoo, J. D., "lienctor Safety analysis", E-1 WCAP-3269-60 (Aoril 1966). E-2 Eisenhart, L. D., Poncelet, C. C., "quasistatic Applica-to [!uclear tions to Parabolic Equations with auplication Reactor Kinetics", Proc. Int. dymoosium cn Namerical Solutions of Partial Ulfferential c;uations, University of Maryland (1970). E Ferguson, K. L., " Development and E:<amina tion of Data F-1 Handling Schemes for ileactivity Measurements in the Pres-ence of Sostial Ef fects in Large Nuclear 1-)wer deactors", PhD. Thesis, Carnegie-Mellon University (1973). G E H-1 Henry, A. F., " Computation of P3rumenters.socearing in the Kinetics Equations", unPD-142 (Daceaber Iv55). A H-2 Hocevar, C. S.,.Jir.einger, f. a., "*LilE h l a computer Code for Nuclear Reactor Gore 'ihermal analysis", Idaho Nuclear, IN-1445, UC-80 (February 1971). Henry, A. F., "A Theoretical Method for Getermining the H-3 Worth of Control Rods", WAPD-218 (August 1959). H4 Hansen, K. F., "GAKIN - A One-Dimensional Multigroup Kir.etics Code", GA-7543 (August 24, 196?). Henry, A. F.,

Vota, A.

V., "dlGL2 - A Program for the H-5 Solution of the One-Dimensional Two-Grour, Space Time Diffusion Equations.wccounting for Temperature, Xenon, and Control Feedback", WAPD-TM-532 1 L l J-1

Jacob, M., " Heat Transfer", Vol. I, New York, diley and Sons (1957).

E - :", : t -

_

REFERENCES (cont'd.) L L-1 Lyons, M. F., et al, "UO2 Pellet Thermal Jonductivity From Irradiations with Central Meltind",.1EaP 4624 (July 1964). L-2 Lamarsh, J. R., "Intrcduction to Huelcar iteactor Theory", Addison-desley (1966). 5 M-1 Meyer, J. E., " Hydrodynamic Models for the Treatment of Reactor Thermal Transients", n2E-10, pp. 269-277 (1961). M-2 Meyer, J. E., Williams, J. S., Jr., "A Momentum Integral Model for Treatment of Trabstent Fluid F1;w", JAPD-BT-25 (May 1962). M-3 Meyer, J. E., " Conservation Laws in One-Dimensional Hydro-dynamics", WAPD-BT-20, pp. 61-72 (September 1960). M-4 Martinelli, S. C., Nelson, D. B., " Prediction of Pressure Drop During Forced Circulation Boiling of Water" Trans. ASME, Vol. 7C, op. 695-702 (1943). s M-5 McClintock, a. e., silvestri, u. J., "Fcruulations and Iterative Procedures for the Caletdaticn ,f Prooerties of Steam", The american Society of.9pchanical nngineers, Nev: York (1967). M-6 Meyer, J. E., Smith, R. B., G el ba rd, H. G., George, D. E., Peterson, W. D., " ART - A Program for the Treatment of Reactor Thermal Transients on the IBM-704", WAPD-TM-156 (November 1959 ). M-7 Meneley, D. A., Ott, K. O., diener, E. S., " Influence of the Shape-Function Time Derivative un Soatial Kinetics Calculations in Fast Reactors", Trans. Am. Nuc. Soc., Vol. 11, No. 1 (June 10-13, 1968). M-8 Meneley, D. A., Ott, K. O., diener, E. S., " Fast Reactor Kinetics -- The QX1 Code", ANL-7769 (March 1971). E 0-1 Ott, K. O., Meneley, D. A., " Accuracy of the quasistatic Treatment of Spatial Heactor Kinetics", HgE-36, po. 402-411 (1969). P P-1 Poncelet, C. G., " LASER - A Depletion Prwrsim for 1.nttica Calcula tions Based on c'.i.. 'I nnd THEi1MOS",.;C a P -6.7 3, Westinc. house Electric ...poratti.n (april 1966). en ~

u -= REFERENCCg (cant'd,) .S E R-1 " Reference Safety Analysis Report - RESAR41", Westinghouse Nuclear Energy System.s (Deccmber 1973). R-2 Riese, J. W.,

Collier, G.,

Rick, C. E., "'!ARI-QUIR' - A Two-Dimensional, Time Denerdeat, dulti-Grc.up Diffusion Code", WANL-T!!R-133 (December 1965). " CHIC-KIN - A Program for Intermediate Redric1d, J. A., B-3 and Fast Transicnts in a.iater Moderated Reactor", daPD-TM-479 (January 1965). R4 Redfield, J. A., Murphy, J. H., Davis, V. C., " FLASH-2: A Fortran IV Program for the Uigital Simulation of a Multinode Reactor Plant During Loca of Coolant", WAPD-TM-666'(april 1967). B-5 RettlE, W. H., Jayne, G. A., Moore, K. V., Slater, C. E., Untmor, M. L., "RELAF) - A Ccanuter Program for Reactor Bloudown Analysic", IN-1321 ( J un.: I??O). B6 Rim, C. :3., "Neutrcnic an.1

me. ii analy..i s i.f Euclear s

Fuels", Thesis, Macsachucet,tu I n s t,i tu tt-ef n-chnology s (August 1969). s Stern, R. C., Wocley, J. A., Cheng,!!. S., "The Modeling S-1 and application of the adiabatic Annroximation for Anal-yzing had Droo Excursions", Prc.ceedings of Conference on New Developments in deactor nuthematic; and aoulications, CONF-710302, Vol. 1, Idaho Falls, Idsbo (narch 29-31, 1971). .T. T-1 Tong, L. S.,

Welcman, J., " Thermal Analyals of Fressurized Water Reactors",,AEC Monograph (1970).

T-2 Thom, J. R. S., et al, " Boiling in Subcooled Water During Flow Up Heated Tub 2s or Annu11", Proc. Inct. Mech. Engrs...Vol. 180, Part 3C, pp. 226-246 (1966). T-3 Tong, L. S., "Bo'11ng Heat Transfer and Two Phase Flow", Wiley (1965). T4 'Touloukian, Y. 3. (ED.), "Thermonhysical Prorc2rties of High-remperature 3o113 iuterialc", '/cl. I, Part 2 McMillan, New York (l967). , :10 -

REFEREMCES (con t'd. ) T (cor.t'd.) T-5 Tong, L. 5., "Predic tion of nei.arture from nuclear soiling for an axially Hon-Uniform ileat plux Listribatica", Journal of Nuclear Energy, Vol. 11, pp. 241-248 (1967). _U 1 W d X Y-1 Yasinsky, J. B.,

Matelson, M.,

Hngema L. A., "TdIGL - A Program to Solve the Two-Dimensional, Two-Grouo, Soace-Time Neutron Diffusion Ecuations with Temoerature Feedback", WAPD-TM-743 (February 1968). Y-2 Yasinsky, J. B.,

Henry, n.

F., "Some !!uaerical Experiments Concerning Scace-Time Reactor Kinetics Bel.avior", N5cE 22 (1965). e Y-3 Yasinsky, J. B., "On the Uue of Point Kir.: Lies far the Analysis of Hod Ejection accidents", jt s!:li J p, ( 1 9 7 0 ). Y-4 Yasinsky, J. B., " Notes on Nuclear Reactor Kinetics", WAPD-TM-1960 (July 1970). - 201 -

APPENDIX A NUMERICAL SOLUTION OF THE TRANSIE;;T HEAT CONDUCTION EQUATIONS IN THE FUEL PIN The transient heat ecnduction equation in the fuel pin is solved numerically in the FIESTA ccde. The fuel pellet, gap, clad and moderator are considered explicitly. The equation solved is .pC(T)ff(r,t) = f h-rk(T) JT r,t) , g,(r,t) A-1.1 o p The conductivity and heat capacity are temperature dependent and the heat rate is distributed radially in tne fuel pin as described in Chapter III. ~ To perform the numerical solution, the fuel pellet is divided into several numbers of concentric rings whose areas are equal. The came is done to the clad. The boundaries between the concentric rings are the solution points for the temperatures, and the temperature is ascumed to be constant up to the mid-point of each ring. The numerical model is shown in Figure A-1.1 for three fuel pellet and ti.o clad intervals. The heat transfer equations are integrated from the mid-point of each mesh interval to the midpoint of the next interval. r+AE r+AE AE v4 f rdr pC f=f - (kr f) dr + f rd e q"' A-1.2 p r-f r-f r , ;L

~-. T1 T2 I I l 3 l l l 1 l "1 i 4 T) l l l l l 1 T 5 l l I T6 l I T l l l w I l l i I 1 I l W s n Fuel Pellet 11adius Gao ulad Moderator Figure A-1.1 Numerical Model of the Fucl Pin Each ring interval is assumed to have uniform properties at the average temperature of the boundaries C (T ) k(Y ) C, = k p 1 = l g 1 f-(Ty+Tg) where T = 1 f Since discontinuities occur at mesh boundaries, the integrals in Equation A-1.2 must be divided into two integrals. The first term of Equation A-1,2 is evaluated as follows: P> 3 -

Ar Ar t t r+2 r+2 r 1 i =[i 3,p ST BT rdr PC g+[ rdr PC f rdr PC, g p p p Ar _1 Ar,7 r g t i 1 2 1-2 BT (pC )1_7 (3r 1 r _g)(r - r _y) 1 g 1 g 1 g at 2 4 (PC )1 (3r + r,7)(r ,7_ - r )1_ ] t g p 1 2 4 Similarly, other terms are as follows: 1 k,7 1+ gf (p _p ) f ,h(kr ) dr g ( = Ar,7 1 1-1 g

1. '1 2

t A l k r g 1 + y (ritl - r )(Tid - T ) 1 i Ar g

1. 1 2

g y '_7 (3rg + r,7)(r; - r _7) g 1 f rq dr = 2 4 Ar _1 i

1. ~1 2

9[' (3ri + r,7)(r,7 - r ) i y g 2 4 l These equations may be written in matrix form. -ET+3 A-1 3 0 = l - :m -

9 where T is a temperature solution vector of N clements 6 is a diagonal matrix (N x N) I is a tri-diagonal matrix (N x N) S is a fixed source vector N equals number of fuel pellet intervals plus number t NC + 2) of clad intervals plus two (N = Np The elements of G are g g 0 .0 gy 982. 5= 0

• gg

+a{(pC) = 0 gy p ~ (PC )1 1 + a*(PCp) = a Et s p ENF+1 hF+1(PC)pp& 0 p F+2 (PCg)NF+2 ~ 0 +a ENF+2 = + a (pC )i 1(PC )1,7 =G gi g p g 1 l + G pN N l r l r _1) (3r + r _1)(r 1 1 1 1 1 where a = 4 r,1)(rgy-r) l (3r 4 g g g a = 4 .n

. ~. a The matrix I is a symmetric tri-diagonal matrix. a ,-b , 0 w0 y y 4) a ,-b , O e0 g 0 ,-b , a -b, g .. -l - 0,-b,y,a;. 0 y 0 +k B a = 1 y y 8 k,y 8,y + kg 1 a = g 1 g k ONP + 2 RG g b a = s NFt1 NF 2R h +k O a = NF+2 G 6 NF+2 NF+ 2 S,y + ki y B k,y a = g g 1 i k,7 SN-1 + 2 Sp h a = l N g P +1 + Pi i where B = y r +1 - r1 1 gap radius R = G gap conductance h = g b rilm conductance = y fuel pin radius R = 7 l l l - ::06 -

~ b = k 8 y g 7 k 8 b = y 1 1 2H h b = NFt1 g G b k 8 NF+2 IIF+ 2 NF+2 k 0 b,7 = g N-1 11 - 1 The elements (s ) of the fixed source vector 5 are y + a{ q{' 0 s = y 1 1q{'y+a1 q{' s = a yp,y NF+1 95'F + 0 s = a + a*NF+2 9 h*F+2 s = 0 NF+2 i q {' g y q((y +a s = a i N 951 4 +2S h T' ~ s p g I where T is the temperature of the coolant y The matrix equation A-1 3 is backward differenced to f ensure stability. 31, j;t-1 39, A-1.4 = g g At I l . ?.7 -

.5 WT .~ This sporoximation leads to the fullvwing result A I' T 5' A-1 5 = where I' is a symmetric tri-dingonal matrix uhose elements are a{ and b{ 5' is a fixed source vector whos elen.ents are s[ t t+g t a{ t = a At g b At b{ = y A+g T -1 A A s{ = s At 1 The elements a% b, g% l and s are not knawn cince they are calculated from the temperatures at the new tiue level. These elements are assumed to be equal to those of ttie previous time icvel (1-1) for a first guess. An iterstive procedure is pro-gra:r.med in FIEST.4 such that the atore ele: men tr: are uv. luted. This yields a true backward-diff:rencing, and the method con-verges rapidly (usually in two iterations since the material properties are not a etrong function of temperature). The general transient heat conduction equ.ition ( A-1 5) may also be used to evaluate the steady state temperatures. The steady state condition is obtained by setting gy = 0, which reduces the matrix elements as,follows a{ g b{ b s{ g = s = a = y - 288 -}}