ML20012C614
| ML20012C614 | |
| Person / Time | |
|---|---|
| Site: | Vermont Yankee  |
| Issue date: | 06/29/1989 |
| From: | Cacciapouti R, Cronin J, Slifer B YANKEE ATOMIC ELECTRIC CO. |
| To: | |
| Shared Package | |
| ML20012C608 | List: |
| References | |
| YAEC-1694-A, NUDOCS 9003220283 | |
| Download: ML20012C614 (81) | |
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_ . _ . _ - . . - _ ~ . . 6 I1 -< S- s Is g , METHOD FOR GENERATION OF ! ONE-DIMENSIONAL KINETICS DATA FOR RETRAN 02 i June 1989 ; i' by James T. Cronin r' Papared By: # 0 I ' gmes T. Cronin, Senior Nuclear Engineer Reactor Physics Group Nuclear Engineering Department (Date) Approved By: M 6h[P9 I . Cacciapp()tf, Manager eactor PhyMs Group Nuclear Engineering Department (date) Approved By: - M , B.D.Slifer,Dimetor U (Date) ' NuclearEngineering Department I Yankee Atomic Electric Company ,. I, Nuclear Services Division 580 Main Street Bolton, Massachusetts 01740-1398 LI rI m . . - . - - - -
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E p gDClou G ; 4 ; f, ff = UNITED STATES p g ILUCLEAR REGULATORY COMMISSION 5 W ASHINGTON, D, C. 20555 ,
!e 'E . December:11,1989 :
hy * . . . f Mr. R. W. Capstic8
= Licensing Enginesr t I Vermont Yankee duelear Power Corporation -
Engineering Office r 580 Main St. - Bolton, MA 01740
Dear Mr.. Capstick:
I
SUBJECT:
~ ACCEPTANCE FOR REFERENCING OF TOPICAL PEPORT YAEC-1694
" METHOD FOR GENERATION OF ONE DIMENSIONAL KINETICS DATA i
FOR RETRAN-02" ! The staff has completed its review of the Topical Report YAEC-1694, " Methods i for Generation of One Dimensional Kinetics Data for RETRAN-02" submitted for ' NRC review by Vermont Yankee Nuclear Power Corporation (YAEC) by: letter dated June 30, 1989. This report (YAEC-1694) describes the methodology used to 3 produce one-dimensional (1-D) cross sections and kinetics parameters for input
-to'l-D space-time kinetics RETRAN-02 calculations. The methodology uses j l SIMULATE-3 to characterize the neutronic response of a BWR and a. linking code gh SLICK to functionalize the neutronic data in terms of the RETRAN-OP feedback variables.
L gWe find the' application.of the l'-D space-time kinetics model acceptable for use with:the RETRAN-02 code under the limitations delineated in the associated . NRC technical evaluation. The evaluation defines the basis for acceptance of = this topical' report.- j L We do not intend.to repeat our. review of.the matters found acceptable as , L3 described in YAEC-1694 when the report appears as a reference -in license '3 applications, except to assure that the material presented is applicable to the specific plant involved. 0ur acceptance applies only to the matters described in the application of YAEC-1694. In accordance with procedures established in NUREG-0390, it is~ requested that the Vermont Yankee Nuclear Power Corporatforf publish an accepted' version of L: = this topical report, within three months of receipt of this letter. The L accepted version shall include an -A (designating accepted) following the L report identification symbol.
l> .: L Ji I Mr. R. W. Capstick December 11, 1989 1 Should our criteria or regulations change so that our conclusions as to the I acceptability of the report are invalidated, Vermont Yankee Nuclear Power Corporation and/or:the applicants referencing the ~ topical report will be expected to revise and resubmit their respective documentation, or submit - 'l justification for the continued effective applicability of the topical report without revision of their respective documentation. l l Sincerely, ) l J I A j
./6AshokC.Thadani, Director Division of Systems Technology
$ Office ;
of Nuclear Reactor Regulation ,
Enclosure:
I YAEC-1694 Evaluation l
- II a I
I .. I I
- I; I .
l +f p air o g g UNITED STATES NUCLEAR REGULATORY COMMISSION
]r,,
I % is s* ij- WASHINGT ON, D. C. 2055$ ENCLOSURE 3 > l , SAFETY EVALUATION FOR THE TOPICAL REPORT YAEC-1694, "METH_0D FOR GENERATION OF ONE' DIMENSIONAL KINETICS DATA FOR RETRAN-02"
1.0 INTRODUCTION
4 By letter dated June 30, 1989, the Vermont Yankee' Nuclear Power Corporation submitted the topical report YAEC-1694 for NRC review (Ref. 1). Additional information was submitted on November 6, 1989 (Ref. 2). This report (YAEC-1694) describes the methodology used to produce one-dimensional (1-D)- cross sections and kinetics parameters for input to 1-D space-time kinetics RETRAN-02 calculations. The methodology uses SIMULATE-3 to characterize the
-neutronic response of a BWR and a linking code SLICK tc functionalize the
~
neutronic data in terms of the RETRAN-02 feedback variables. Restrictions to be observed in the application of this topical report are listed in section 3.1. 2.0
SUMMARY
OF TOPICAL REPORT
'I First, the report adopts the standard industry practice, using coolant density, fuel density and control variables as independent parameters to functionalize the kinetics model. This functionalization is given in tems of quadratic polynomial fits and are used for coolant and fuel density while the effect of R control is accounted via different sets of similar polynomials. The kinetics parameters include two group fission and absorption cross sections, diffusion f -
coefficient, buckling, group velocities and total delayed neutron fraction. LI L b
3 2 SIMULATE-3 is used to predict the global core (3-D) response and then perform a homogenization to 1-D parameters. A discrete ordinates solution is performed _for the power distribution, which is consistent with the methods employed by both RETRAN-02 and SLICK. As with the neutron kinetics data, the thermal-hydraulic parameters are estimated for the average 1-D channel. This , involves the solution of the RETRAN-07 energy and momentum equations, the I . neutron flux and the fuel heat conduction equation. The analytical model has been adopted to the geometry of the core and includes the homogeneous ? equilibrium and the algebraic slip models which are supported by RETRAN-02. I For the solution of the coupled energy and momentum equations SIMULATE-3 provides the needed boundary conditions. For the neutron flux (and the corresponding power density distribution) RETRAN-02 uses the Le11ouche-Zolotar drift _ flux parameters and a profile fit for flow quality. Finally, for the fuel heat transfer, RETRAN-02 uses a finite-difference scheme for solving
.the conduction equation. To demonstrate that the SLICK and.RETRAN models are indeed the same, comparisons were run and the results showed no discernable difference.
The 1-D kinetics data are fitted to the RETRAN-02 polynomial form. To do this SLICK uses a standard least-squares fitting to determine the polynomial coefficients. The final chapter of the report deals with the verification of the model which includes extensive comparisons to SIMULATE-3, 3-D calculations and to static control rod calculations. 3.0 EVALUATION
-As pointed out in the proceeding section, the 1-0 kinetics data generation is based on matching the results of a SIMULATE-3 calculation to the homogenized 1-D geometry. The RETRAN-02 uses a polynomial format for the kinetics parameters; SLICK supports this format and functionalizes the group constants in terms of the fractional change in coolant density and fuel temperature change. During a transient calculation, the cross sections and kinetics I
{
I parameters for a given region are evaluated by interpolation. For transients
. experiencing a reactor scram,,the axial control distribution in the RETRAN-02 calculation is based on the assumption that all control rods move at the same ,
velocity to the end of their travel. This approach is standard in.the industry, is known to produce accurate results and is acceptable. g The spatial homogenization from 3-0 to 1-D neutronics is performed by 5= SIMULATE-3, however, the- power distribution used in the thermal-hydraulic j - routines in SLICK are based on a finite difference 1-D two group neutron diffusion equation solution. This is necessary in order to retain consistency with the RETRAN-02 method. Comparison of the all-rods-out and all_-rods-in axial power distributions shows that the SIMULATE-3 and the SLICK results are almost identical. I . As in the neutronic variables discussed above, the thermal-hydraulic variables require a transformation from the'3-D SIMULATE solution to a 1-D ' representation. RETRAN-02 uses a 1-D average channel to represent the core, ,
. while the 3-D, SIMULATE-3 model explicitly represents each channel. To reconcile the different approaches, the proposed method functionalizes the SIMULATE-3 output in terms of the RETRAN-02 average channel model parameters i.e. calculate the response of the RETRAN-02 average channel for each SIMULATE-3 case. To accomplish this, the steady state form of the RETRAN-02 model' requires . calculation of the neutronic feedback variables which have been-included in SLICK. This requires the solution of the RETRAN-02 energy and momentum equations, the neutron flux and the fuel conduction equation which I have been described in theLpreceding paragraphs. To demonstrate that indeed the PETRAN-02 and SLICK results are the same, a series of calculations have been documented which show that the results are almost identical. In the
' range of. the variables built in the model, the RETRAN-02 neutronics calculation is in excellent agreement with the 3-D, SIMULATE-3 calculation; therefore,.we find the model. acceptable.
I: The next level of verification is the degree to which the RETRAN-02 (SLICK) represents the SIMULATE-3 results. Two types of calculations were performed l,
_ - - - _ . . .- ~ _ _ _
. : c I' .
a I ' j for this verification; one set compared the neutron flux and fuel temperature distributions with the thermal power and- thermal-hydraulic boundary l conditions. The other set of comparisons is made to static control' rod worths ; with all other (except control) held fixed at the initial conditions. Because i the thermal-hydraulic variables are held constant, the' static worth _{ calculation involves only neutronics, i.e. interpolation of the cross sections - (for the specified control state) and subsequent solution of the diffusion equation. Comparisons of the results showed excellent agreement between the ; 3-D, SIMULATE-3 and the SLICK calculations. ( I The comparison of core state parameters involves two types of calculations: changes from an initial critical to a final critical state and changes to a f non-critical final state from critical.- For the first type, system pressure, L core input temperature and core flow were examined for Vennont Yankee cycles 9 and 13.at end of full power life and cycle 9, 2.0 GWD/ST.before the end of full power life. The same cases were then independently evaluated using SLICK. Comparison of the results showed differences from +2 to -12 pcm. This represents excellent agreement. Comparisons for states away from critical > lg3 also sh' owed good agreement i.e. 7% discrepancy in the eigenvalue for an equivalent eigenvalue change of 440 pcm. The axial power distribution was also in good agreement with a maximum error of about 5%. These differences show excellent agreement for this type of calculation and are acceptable.
;I 3.1 Restrictions 1 I The proposed model shall be used for licensing applications only in the context of one-dimensional kinetics for BWR transient analyses in conjunction with pg !W YAEC-1693, and RETRAN-02.
l 4.0
SUMMARY
AND CONCLUSIONS f I The staff reviewed the YAEC-1694 topical report presenting a method for oeneration of one-dimensional kinetics data for use with the RETRAN-02 code. LI LI l
5 ,
.The results of the calculations of the proposed method 'were compared to the
~
l equivalent (1-D) results of.,the SIMULATE-3 code andlshowed excellent agreement.
- I >
.There is no absolute basis for the review of this code except in comparison -l with the results of SIMULATE-3. In this context, the results of the proposed l
one-dimensional model with the 3-D SIMULATE-3 are excellent and, thus, are
.l ' acceptable. The use of this. method is to be limited in providing one-dimensional kinetics data for RETRAN-02. The application of the RETRAN-02
~
-transient analyses.for BWRs described in YAEC-1693 has been approved by the staff in a separate action.
; 5.0 REFERENCE 1.. Letter from R.W. Capstick, Vermont Yankee Nuclear Power Corporation to
.USNRC, " Method for generation of One-Dimensional Kinetics Data for RETRAN-02 " June 30, 1989.
- 2. Letter from S.P. Schultz, Yankee Atomic Electric Company to USNRC,
" Request to chan9e the ' Method for Generation of One-Dimensional Kinetics Data for RETRAN-02,' and ' Application of One-Dimensional Kinetics to Vermont Yankee Transient Analysis Methods' Reports to Generic
' Reports,'" November 6, 1989.
I I IL
l I. I DISCLAIMER OF RESPONSIBILITY I This document was prepared by Yankee Atomic Electric Company (" Yankee") The use ) ofinformation contained in this' document by anyone other than Yankee, or the Organization for which this document was prepared under contract, is not authorized and, with resnect to any l
- l. l 5/ unnuthorized use. neither Yankee nor its officers, directors, agents or employees assumes any i l
obligation, responsibility, or liability or makes any warranty or representation as to the accuracy j or completeness of the material contained in this document. 1 I I g I I
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E ABSTRACT .g +5 - This report describes the methodology used to produce one-dimensional (1 D) cross sec-tions and kinetics parametersfor input to the 1 D space time kinedes model ofRETRAN-02. This method will be used to support core wide transient sinadations performedfor BWR reload licensing. The methodology makes use ofSIMULATE 3 to characterize the neutronic response I ofa BWR core. A linking code, SUCK, has been developed tofunctionalize the 1 D neutronic dataproduced by SIMULATE-3 in terms of the thermal hydraulicfeedback variables used by i RETRAN-02. The role ofeach of these codes is explained and results ofcalculations vertfying the methodarepresented. I . I I I . II I I u I I: I -lii. I EI l
1, l> 'I l ACKNOWLEDGEhENTS I The author wishes to acknowledge the contributions of Michele Sironen in performing the SIMULAE3 depletions associated with the verification cases of this repnrt and Michael ! Tremblay in assisting with SIMULATE-3. Also,I wish to thank Kord Smith of Studsvik of America for his work or. the homogenization of the kinetics data in SIMULATE-3 and for his help in defining an interface between SIMULATE 3 and SLICK. Finally, I thank Sam Forkner of the Tennessee Valley Authority for his ideas and encouragement on this project. I ! I I I I I ' I I I I < I I I -iv-I I
1 I i l i TABLE OF CONTENT _ S 1 DISCLAIMER OFRESPONSIBILITY .... . . .. .............................il i I l ABSTRACT ....... - ............................................................. . ...... ... iii j i ACENOWLEDO EMENTS ... .......... ..... ............ .. ... ........ ....... _.. ............ ..... ........ ... i v j LIST OF MOURES .. ......... ... ........ ... . ..................-........Vi LIST OF TABLES .. ..... : ..............................vil I
1.0 INTRODUCTION
..- . . . . . _. ...........I 1.1 Purpose.........................................................................................................1 : I 1.2 1.3 Overview..........................................................................................................I Organization of Remainder of Report ................ ............................................ 3 .
'2 0 RETRAN.02 CROSS SECrlON MODEL ..... . ......................................4 I 3.0 ROLE OF SIMULATE.3 . . . - - . . . . . . . . . . . . - - - - - . -----.....................6 3.1 Spatial Homogenization From 3.D To 1 D Neutronics Data ............................... 7 4.0 FINITEDIFFERENCE SOLUTION OF 1.D 2.OROUP DIFFUSION EQUATION I1 6 5.0 TRANSFORMATION OF THERMAL. HYDRAULIC VARIABLES ........ ..... ..... .. 1 3 6.0 HTTING OF 1.D KINETICS DATA TO RETRAN 02 POLYNOMIAL FORM ............ 20 I 7.0 VERIM CATION ..... .... ...... ..............- - -
7.1 7.2 Comparisons of State Calculations ................................................................... 26 S tatic Scram Wonh Comparisons ..................................................................... 2 8
.. ..... ....... 25
8.0 REFERENCES
... . - . . . . . . . . ......... 43 l
APPENDIX A SLICKTHERMAL.HYDRAULICMODELS
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1I I l'I l V-1
i I LIST OF FIGURES I i Number Tide Ease r SIMULATE 3 and SLICK Power Distributions at ARO .................................... 9 g 3.1 3.2 SIMULATE-3 and SLICK Power Distributions at AR1.. ................................ 10 4.1 SLICK and RETRAN-02 Power Distributions ......................................... ..... 12 5.1 Comparison of RETRAN-02 and SL1CK Void Profiles ................................... 15 5.2 Comparison of RETRAN 02 and SLICK Density Profiles ................................. 16 13 Comparison of RETRAN-02 and SLICK Average Fuel Temperatmes .............. 17 5.4 Compaision of RETRAN-02 and SLICK Fractional Density Change ................ 18 5.5 Comparison of RETRAN 02 and SLICK Fuel Temperature Change ................. 19 l 7.1 3 D and 1 D Axial Powers for +30 psi Pressme Change .................................... 31 . 7.2 3 D and 1-D Axial Powers for -4.8 F Inlet Temperature Change ....................... 32 7.3 3.D and 1-D Axial Powers for +7% Inlet Flow Change ..................................... 33 7.4 3.D and 1 D Axial Powers for TIWOBP Conditions ........................................ 34 7.5 3 D and 1-D S tatic Scram Worth at EOFPL ....................................................... 35 7.6 3 D and 1 D Axial Powers After 2 Foot Rod Move (EOFPL) ........................... 36 , 7.7 3 D and 1-D Axial Powers After 4 Foot Rod Move (EOFPL) ........................... 37 7.8 3 D and 1 D Axial Powers After 6 Foot Rod Move (EOFPL) ........................... 38 7.9 3.D and 1 D Static Scram Worth at EOFPL - 2 GWD/ST ................................. 39 g 7.10 3 D and 1-D Axial Powers After 2 Foot Rod Move (EOFPL - 2) .................... 40 7.11 3.D and 1-D Axial Powers After 4 Foot Rod Move (EOFPL - 2) .................... 41 l 7.12 3 D and 1-D Axial Powers After 6 Foot Rod Move (EOFPL - 2) .................... 42 I I I
.vi.
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I 1.1RT OF TABl EE Numher .Titic East 3.1 SIMULATE.3 Versus SL1CK Eigenyalues ............. ....... .............................. 8 6.1 Perturbation Cases Used in Generation of 1.D Kinetics Data ............................ 23 6.2 I Co"Larison con of 1.D Neutronics Calculations with Fitted Data to 3.D Pe s.........................................................................................................u l 7.1 7.2 Differences Between 3.D and 1.D Eigenvalues for Critical State Calculations Differences Between 3 D and 1-D Eigenvalues for'ITWOBP State Calculation 30 29 I I I - j I I I - I I I I I
.vii.
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I 1.0 IN T R O D U C TI O N 1.1 Pumme - I ' This report describes the methodology used to producc one-dimensional (1 D) cross sec-tions and kinetics parameters for inptit to the 1 D space time kinetics model of RETRAN 025 . g This method will be used to support core wide transient simulations performed for BWR reload licensing. 1.2 Overview The general approach taken in developing 1 D kinetics data for RETRAN 02 is very simi-lar to Yankee's previous methodology
- and another method which the United States Nuclear Regulatory Commission has found acceptable *, The aim of the method is to take the neutronic l
effects as predicted by the three-dimensional (3 D) nodal simulator, SIMULATE 35 , and func-tionalize them with the feedback variables used by RETRAN-02. The ideal goal of the method is that any perturbation should yield the same 1 D and global effects whether calculated with the 3 D SIMULATE-3 or 1 D RETRAN 02 representation of the core. SIMULATE 3 is used to predict the 3 D neutmnic response of the core and to produce spatially homogenized kinetics data suitable for the 1 D two-group kinetics model of RETRAN 02. SIMULATE 3 producer all the cross sections and kinetics parameters used in the RETRAN-02 kinetics model. The 1 D cross sections and diffusion coefficients produced by SIMULATE-3 effectively preserve all planar reaction rates, planar interface currents, and the reactor eigenvalue. One set of spatially homogenized kinetics data is produced per SIMULATE-3 solution. Our method requires multiple .'iIMULATE 3 solutions to the 3 D two-group neutron diffusion equation and subsequent spatial homogenization calculations. I I 1-4 8 ;
. . , , ,.+ , . . . . . , , . . . . . .. . . . . , . . . . . , . . . .. .
4 I i The 1-D kinetics data produced by SIMULATE 3 are pure diffusion parameters. No I attempt is made by SIMULATE 3 to functienalize the data in the form required by RETRAN 02. l The SLICK (SIMULATE 3 Linking for Core Kinetics) computer code has been developed at ! Yankee to perform this task. To get an overview of the method, we will present the steps i involved in generating data for a given exposure point. The example below is for an operational i transient where a reactor trip is expected to occur. (1) The SIMULATE 3 model is brought to the exposure point of interest. : I (2) SIMULATE 3 perturbation cases are run for the initial control state. The penurba- t tion cases are selected to produce conditions representative of the transient to be ana-lyzed. The perturbation option of SIMULATE-3 allows for independent variation of either the 3 D density distribution or fuel temperature distribution. All other variables normally associated with the SIMULATE 3 cross sections (exposure, con-trol history, fission product inventory, etc.) are held constant. The perturbations are from the reference state of step 1. SIMULATE 3 produces spatially homogenized l_ I kinetics data for the reference case and all perturbation cases. I (3) To obtain the effect of the control rod scram, a SIMULATE-3 reference state is achieved with all rods-in. The density and fuel temperature distributions (as well as l the other variables associated with cross sections) are the same as the initial control state case. Identical perturbations to those used in step 2 are run. Thus, the only l- difference in the resulting homogenized kinetics data is that of control fraction. (4) The result of steps 2 and 3 are separate sets of homogenized neutmnic data for the
~
I initial control state and the all rods in control state. SLICK processes this data one control state at a time. For each case within a control state set, SLICK: L I 2-LI i'
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- a. Calculates the power distribution by solving the finite-difference form of the 5 1 D two group neutron diffusion equation and
- b. Calculates the thermal hydraulic variables used by RETRAN-02 in evaluating the kinetics data. This is accomplished by solving the steady state form of the :
RETRAN-02 thermal hydraulic equations at the same conditions used in the SIMULATE 3 case. g.I After processing of all cases for a contml state, the kinetics parameters for each neu- ; p . f tronic region are fit against the RETRAN-02 thermal hydraulle variables of neu-tronic density and fuel temperature to produce the polynomial form required by - , RETRAN-02. g 1.3 Orannlintion of Remainder of Reoort The above has served as an overview of the method. The remainder of the report presents the main elements comprising the methodology and verification of the method. These are: , I - the RETRAN-02 Multiple Control State cross section model and its data require-ments, i
- the s atial homogenization from three dimensional to one-dimensional neutronic models,
- the finite-difference solution of the 1 D two-group neutron diffusion equation employed in SLICK, 3
a
- the transformation of thermal hydraulic feedback variables between the 3 D and 1 D models, the fitting of the kinetics data to the RETRAN 02 polynomial form, and
- verification of the method. -
I LI I 3-c r
I 2.0 RETRAN 02 CROSS SECTION MODEL ; I SLICK generates data for the hiultiple Control State (htCS) model of RETRAN 02. This is the model used in the core wide transient model. As in most transient kinetics models, there l are three variables associated with the evaluation of the transient cross sections and kinetics ! parameters. These are coolant density, fuel temperature, and control. In the hiCS model, the effects of coolant density and fuel temperature are represented by polynomial fits of the regional l cross sections and kinetics parameters in terms of these two variables. The polynomial equation is, I t uu (2'1) l I EEp I Z=E.v.n.cxf'xf'x3 i where, Z = any 1 D neutronic parameter xi = p(r) ~ p(0) fractional change in density p(0) (22) 22= VT/t)-VTf0), change in fuel temperature x3 = T,,(t)- T,(0), change in coolant temperature The maximum order of the polynomial terms is limited to two by RETRAN-02. SLICK supports the above form, but only functionalizes the group constants in terms of the fractional change in h coolant density and change in fuel temperature. This approach is standard in the industry. The hiCS model introduces the effect of control via different sets of these polynomial fits, one set for each control state. During a transient calculation, the cross sections and kinetics parameters for a given region are evaluated by interpolation of the control state sets (evaluated at the region's current thermal-hydraulic conditions) which bound the existing centrol state of the I 4 I I . --
y , region. For transients experiencing a reactor scram, the axial control distribution in the I RETRAN 02 calculation is based on all control rods moving at the same velocity until their physicallimit is reached. The two-group cross sections and kinetics parameters required by the 1 D RETRAN-02 model for each neutronic region are listed below. , I 1. Total delayed neutron fraction, [ 2.- Group 1 absorption cross section, E.,i
- 3. . Group 1 radial buckling,B'i
- 4. Group 1 diffusion coefficient,D i
- 5. Group 1 down scatter cross section, E, i
- 6. Kappa times group 1 fission cross section,xEti j
- 7. Nu times group 1 fission cross section, vEti
(
; 8.- Group 1 neutron velocity, Vi ;
- 9. Group 2 absorption cross section,I.,2
- 10. Group 2 radial buckling, Bl .
I. ' 11. Group 2 diffusion coefficient,D:
- 12. Kappa times group 2 fission cross section,xIts
~ '
- 13. Nu times group 2 fission cross section, VIts
- 14. Group 2 neutron velocity, V2 f
Additionally, the model requires the relative yield fractions of each of the six delayed neutron groups and their effective decay constants. RETRAN-02 assumes that these variables are spa- i dally and temporally constant. I I 5 I I
I ! 3.0 ROLE OF SIMULATE 3 In our method, SIMULATE-3 is used to predict the 3 D neutronic response of the core to .j global penurbations. Additionally, SIMULATE-3 performs the spatial homogenization of the ; 1 3 D data to 1 D and core integral data required by RETRAN-02. The approach of using full 3-D I solutions to changes fronithe initial state was also used in Yankee's previous methodology * (point kinetics). It has severalimponant advantages: SIMULATE 3 is an advanced nodal code that has been validated against measure-I ments in operating BWR cores". SIMULATE 3 provides a practical means to j evaluating the 3 D neutronic response to core penurbations. 1
- SIMULATE-3 penurbation cases have the capability to separately penurb either the density or fuel temperature distributions. This is important to the functionalization i process. By isolating the penurbation to one variable, we are able to capture the individual reactivity components.
E - SIMULATE-3 has the ability to isolate the effect of changes in control state. Thus, 4 1-D parameters based on identical penurbations to the 3-D density and fuel tempera-ture distributions may be produced with the only difference being that of the control distribution.
- Since the pnurbations are made fmm the initial state and are representative of the conditions encountered in the transient, the use of SIMULATE-3 eliminates arbitrary specifications of changes in the 3-D density and fuel temperature distributions.
I I. I I I
I , 3.1 Snatini Hnmngeni7atinn From 3 D To 1-D Neutronics Data ) 8 , SmiULATE 3 produces all the 1 D and core integral kinetics data required by I RETRAN-02. The details of the homogenization process used by SBiULATE 3 are described I in Refe ence 7. Both RETRAN-02 and SLICK use the finit> difference method to solve the neu-tron diffusion equation. The 1-D cross sections and diffusion coefficients produced by ! SIMULATE-3 preserve all planar reaction rates, planar interface currents, and the reactor eigenvalue when solved with the SmiULATE 3 polynomial model. As long as the 1 D polyno- , ) . mial and finite-difference solutions are spatially converged, then the finite-difference solution i will match the axial and global results of the 3 D model. In practical use, this is the case. l I l Figures 3.1 and 3.2 present a comparison of the SIMULATE-3 and SLICK axial power B a distributions for all-rods-out (ARO) and all rods in (ARI) conditions at a typical Haling-produced end of full power life (EOFPL) condition. The models use 27 neutronic regions,25 regions to mock up the 150 inch active core and one region each for the bottom and top reflectors. This degree of nodalization is typically used in the SIMULATE 3 and RETRAN 02 core models. The finite-difference solution uses 12 mesh intervals per region. As can be seen from the figures, the axial power distributions are in excellent agreement. The calculated eigen-values for these two cases are presented in Table 3.1. The difference in the calculated eigenva-lues for the ARO case are within the convergence limits of the calculations. The ARI cases yield a slightly greater difference. The ARI results are acceptable particularly when they are viewed as changes from the initial ARO control state. I I - I I ; B I .
, j I~ 1 Table 3.1 I-SIMULATE-3 Versus SLICK Eigenvalues l Control Difference
- Sag _ SIMULATE 3 SLLCK (ncm1 I ARO 1.001360 1.001362 0.2 l
ARI 0.802620 0.802685 6.5 .
= ;
- l.
- Difference in pcm = (bsus - bsmuun)
- 105 i
I . I i I ; I l I . Li I LI l I 8- , I .
+
\: { l ll SLICK vs. SIMULATE-3 Power -- ARO Case < l l 27. 3....i....i....i....i....i. I g- 25 _ ' t E . . 23 _ _ l- 21
-I 19 _ 3 _
g 17 - t) l g 15 _ a I w . . o 13 - t) - Cr 3 ,
'A-11 - o _ i
. o - SIMULATE-3 .
g g_ o - SLICK g _ 7_ l 5_ _ I 3_ l-th-T Ti ,,,,,,,,,,,,,,,,,,,,,,,,
.0 .2 .4 .6 .8 1.0 1.2 l Relative Power I
I Figum 3.1 SIMULATE.3 and SLICK Power Distributions at ARO E 9
ll 4 l SLICK vs. SIMULATE-3 Power -- ARI i 27 6... . i . ...i... '.... l 25 - I r 43 _ 23 43 _ I 21 43 l- 19k 3 _ l SIMULATE-3 D- > 17 .- : o - SLICK - c 15 _ _ l
-l h e 13 _
cr I . . 11 _ _ g 9 _' _ 7_ l 5_ _ o I 3_ - ; ,l 1I:,,,,,,,,,,,.,,,,,,,
.0 1.0 2.0 3.0 4.0 ll i
i-Relative Power l' Figure 3.2 SIMULATE.3 and SLICK Power Distributions at ARI I LI 1 1I
E ! 4.0 MNITE DIFFERENCE SOLUTION OF I D 2 GROUP DIFFUSION EQUATION l t The power distributions used in the thermal hydraulic routines in SLICK are based on a l finite-difference form of the 1 D two group neutron diffusion equation. This is done to make the l . SLICK solution consistent with that of RETRAN 02, which also uses a finite-difference solution. Additionally, the finite-difference solution is used in SLICK for verification calculations. The level of discretization employed in the model is identical to that of the RETRAN-02 I model. The core is modeled as a set of uniform material regions with uniform mesh spacing I within a region and mesh points at region interfaces. As is the case with RETRAN 02, the l model can use zero flux or current boundary conditions. A comparison of the power distrib-utions as calculated by SLICK and RETRAN-02 for a typical Haling depletion is shown in Fig- l l ure 4.1. As expected, the models produce identical power distributions and the calculated eigenvalues agree to within the convergence criteria used in the calculations. I 1 l I . p lI L I l LI Li
l SLICK vs. RETRAN-02 Power -- ARO < l l 27 ... t . . ..e ....k....s ....
, 25 _ _ ;
i 23 _ _ I 21 _ l 19 _
]
)
g' 17 _ n _ 15 _ g n I a
.o 13 .
(E a _ i I 11 _
. o - RETRAN-02 a
I 9_ o - SLICK g _ l 5_ I g 3. l 1
,: m ,,,,,,,,,,,,,,,,,,,,,
.0 .2 .5 .7 1.0 1.2 l
l Relative Power I l- Figure 4.1 SLICK and RETRAN.02 Power Distributions I 12- !I g
5.0 TRANSFORMATION OF THERMAL HYDR AlfLIC VARI ABLES I The thermal hydraulic variables used for neutronic feedback by RETRAN-02 are the frac-tional change in coolant density and change in square root of fuel temperature from the " time-zero" conditions. Since our method relies on 3 D SIMULATE 3 neutronic solutions, there is a need to transform the thermal hydraulic conditions associated with the 3-D conditions to a 1 D representation of the above variables. RETRAN-02 uses a 1 D average channel to represent the core, while the 3 D u SIMULATE 3 model explicitly represents each channel. In general, the volume averaged 1 D ; density and fuel temperature distributions from the 3 D model will not be the same as that pre-dicted by the 1 D average channel model even though the same axial power distribution and boundary conditions are used in the 1 D model calculation. Since RETRAN 02 evaluates the kinetics data using the thermal-hydraulic results of the average channel model,it is appropriate ; that the kinetics data produced by SIMULATE 3 be functionalized in terms of the RETRAN 02 average channel model. The approach used is to calculate the response of the RETRAN-02 aver-age channel for each SIMULATE 3 case. l To accomplish this, the steady state form of the RETRAN 02 models required tc :alculate t the neutronic feedback variables have been included in SLICK. This involves the solution of the RETRAN-02 energy and momentum e51uations, the neutronic density calculation, and the fuel
- l. heat conduction equation. These models are described in Appendix A.
The thermal hydraulic calculation in SLICK is consistent with SIMULATE 3. As men-tioned previously, the SIMULATE-3 perturbation calculations separately perturb the density or I I 1 I -
l fuel temperature distributions. For perturbation cases, SLICK similarly calculates the I appropriate distribution while holding the other distribution constant. For normal state calcula-tions, SLICK performs consistent calculations for the density and fuel temperature distributions. 1 As a demonstration that the SLICK steady state equations are indeed the same as l I RETRAN-02, we present a comparison of results for 100% power / flow conditions in Figures 5.1
]
through 5.3. The thermal hydraulic modelis the algebraic slip model with the profile fit model , used to predict voiding in the subcooled portion of the channel. These models are employed in the core-wide transient model. As expected, the predictions of the two codes are almost identi-cal. Differences in the two predictions are mainly attributable to slight differences in the water I propeny routines. As mentioned above, RETRAN 02 uses the fractional change in coolant density and change in square root of fuel temperature from " time zero" conditions as the independent vari-I ables in the polynomial form of the cross sections. A comparison of these variables as predicted by the two codes is presented in Figures 5.4 and 5.5. These figures are for the changes in the steady state conditions going from 100% power / flow conditions to 104.5%/100% power / flow
- conditions. The agreement between the predictions of the two codes is excellent.
I I I . I I 1
I I ; l RETRAN-02 vs. SLICK Void Fraction l .8 i i i ' ' ' ' ' ' ' 3 !
- ~
O- RETRAN-02
.6 - o - SLICK -
I 'D c O o I u. 8 .4
~
.9 I- >
O . -
.2 --
~
g : :
.0 9 , , , , , , , , , ,
1 2 3 4 5 6 7 8 9 10 11 12 l Core Volume
.I Figme 5.1 Comparison of RETRAN-02 and SLICK Void Profiles t-. 15-
)
l I ;
I r l , RETRAN-02 vs. SLICK Neutronic Density : g - 50 i ' ' ' ' ' ' ' ' ' ! p
- ~
O- RETRAN-02 40 _ o - SLICK - m
.n - -
I x h . E : E :9 30 _ !W v o
,e
- - v) . -
C D . - O 20 _ - l . - j l 0 , , , , , , , , , , 1 2 3 4 5 6 7 8 9 10 11 12 ,l Core Volume I 5 Figme 5.2 Comparison of RETRAN-02 and SLICK Density Profiles E- , I. I . . -
llb b Il l lll l l h I il4 I u u u u IE la E uEuI i iIiiIi u lu m luI E d Im I m Is u l lll h d I l RETRAN-02 vs. SLICK Fuel Temperature l 1200 i i A: i i i i i i i i l - 1 l 1100 l C v ci. E . . l f1000 m 3 C . 7 . D- RETRAN-02 . I 900 0- SLICK
-l . .
I 800 1 l 2 I 3 4 I 5 i 6 i I 7 i 8 I 9 I 10 11 i 12 l Core Heat Conductor I Figure 5.3 Comparison of RETRAN.02 and SLICK Average Fuel Temocratures I I , I: !
I l Comparison of Fractional Density Change l .05 i i i i i i i i i i g : :
.04 _
l l_ .03 _ e I 01 . . C o - -
.c l
3 0 .02 - - p . . m . . C
'I. e . .
a . .
.01 _
l
~
C l
.00 _ _
o - RETRAN-02 l 3
. o - SLICK .
1 2 3 4 5 6 7 8 9 10 11 12 g Core Volume I Figure 5.4 Comoarison of RETRAN.02 and SLICK Fractional Density Change I -18 I g
3 i l RETRAN-02 vs. SLICK Delta SQRT T-Fuel g' .50 ' ' ' ' ' ' ' ' ' I , w w g .45 _ - l I T) o '. - I 1*- p . - O' O ~ . y) . - I - o T)
.35 _
a - RETRAN-02 O o - SLICK - D
-I . -
g .30 _ - I l-g .25 , , , , , , , , , , D-1 2 3 4 5 6 7 8 9 10 11 12 l
)
Core Heat Conductor l \. 1 l-Figure 5.5 Comparison of RETRAN-02 and SLICK Fuel Temocrature Change l t I 19 lI
I 6.0 FITTING OF 1 D KINETICE DATA TO RI'TR AN-02 POIXNOMI AL FORM The polynomial equation supported by SLICK is, I t u z= I Icg xf' (6-1) I where, s.u.s e 2 = any 1 D neutronic parameter , I xi = p(t)- p(0) fractional change in density p(0) x2 = VTf (r)-VT f(0), change in fuel temperature SLICK uses standard least square fitting techniques
- for determining the polynomial coeffi-cients. This requires that enough SIMULATE 3 cases be run to have an overdetermined system ofequations.
SLICK has several options for fitting to this polynomial. Either the actual cross sections or the changes from the base cross sections (coefficient ci,i) may be fit. The set of polynomial basis functions may be the complete set as shown above or may be a subset which does not include cross terms involving both density and fuel temperature. I In practice,it has been found that fitting changes from the base cross sections with a set of l - basis functions not including the cross terms yields good results. Fitting changes from the base cross sections assures that the initial power distribution is the same as predicted by SIMULATE-3. As a demonstration of the method and for verification purposes, kinetics data sets have been generated for two different cycles using the Vermont Yankee SIMULATE 3 model*. The I 20-I il
I Cycle 9 core had four types of fuel bundles, while Cycle 13 was comprised of two fuel bundle types. The exposure and power distributions associated with the initial state of these data sets l are based on Haling depletions. The 3 D penurbation cases used in the generation of all data sets are presented in Table 6.1. The cases have been selected to cover the range of density and fuel temperature experienced l l during the imponant ponion of a turbine trip without bypass transient ('ITWOBP). The set con-sists rnostly of pressure and fuel temperature increase penurbations. These penurbations are considered representative of the early portion of the transient. The remaining penurbation cases are included to cover conditions funher out in the transient well after the neutron power has peaked. This set of perturbations is representative of what would be used in a reload licensing l calculation. t The " goodness of fit" is assessed by performing a 1 D neutronics calculation for each of the cases used in the fitting and comparing the results to the 3 D calculation. Table 6.2 presents the results for the three data sets. The change in k.n from the base condition for each control state is used as the figure of merit. The mean and standard deviation of the relative errors in the change of k.n are presented. The fitted data matches the 3 D results well. The calculations for the all-rods in (ARI) control states have a higher standard deviations than those of the initial con-trol states. However, it should be noted that the effect of controlis not included in the compari-son. The major impact of the ARI data on the transient calculation is that of control. The type of data presented above is pan of the normal calculation in SLICK. The statis-tical data is of great value in looking at trends from cycle to cycle (note the similarities between the Cycle 9 and Cycle 13 EOFPL data). However, this type of comparison is not the only basis l for confidence in the fitted data. Independent verification calculations using the fitted kinetics I 21-I I - . . . .
data can be performed and compared to a 3 D calculation. These provide a more meaningful I way of evaluating the kinetics data. Verification calculatior.: associated with these data sets are {
; presented in the following section, I :
I ! I: I ' I I 1 I : I- : I l I 22- . g l
. - . . - ~ . .. . . . .. . .. . - . . , . . . .. -1
i I I ! l Table 6.1 I j i Penurbatinn Cases Used in Generation of 1 D Kinetics Qgg
]
I Perturbation Perturbation j Quig Tyne Marninirle I 1 Base Case' - - - -
)
2 FuelTemperature +100.F 3 FuelTemperature +200.F ; 4 Pressure ' +50. psi Pressure +100. psi I i 3 i 6 Pressure +200. psi
'7 Pressure +300. psi j 8 Pressure +400, psi i 9 Pressure 75. psi 1
10 Pressure -150. psi 11 Inlet Temperature +9.F 12 Inlet Temperature +18.F I
- Time zero power, flow, pressure, and inlet temperature, i 1
I ) I 1
- I 23-I I -. -
I : Table 6.2 1 enmnarison of 1-D Neutronics Calcuintinns with Fitted Data to 3-D Penurbation Cases i Ficenvalue Difference Error (%f 1 Control Standard 1-D Kinetics Data Set For: STATE .higgg, Deviation I Cycle 9 EOFPL Inidal ARI
+1.6
+2.1 2.9 12.9 Cycle 9 EOFPL- 2 GWD/ST Initial +2.2 4.2 I
.I ARI +2.7 13.7 ] l Initial I Cycle 13 EOFPL +1.5 2.5 l ARI +0.9 9.1 ) h ! I #
. Eigenvalue Dirrerence Error = # "'j' I
I ; I : I I I < I ; I : I I I
7.0 VERIFICATION The ideal goal of the method is that any perturbation should yield the same 1-D and global
- j effects whether calculated with the 3-D SIMULATE-3 or 1-D RETRAN-02 representation of the _
core. To assess the degree to which this goal can be approached, we make comparisons between - ; 3-D SIMULATE 3 results and results produced using the polynomial representation of the kinet-ics data and the RETRAN-02 thermal-hydraulic model. Two types of comparisons are made below. One set of comparisons is made to SIMULATE-3 state calculations. These calculations
; are performed with the neutron flux and thermal power in equilibrium. The resultant density and fuel temperature distributions are consistent with the thermal power and thermal-hydraulic boundary conditions. The other set of comparisons is made to static control rod wonh calcula .
h- [ tions. These calculations are performed with all variables other than comrol held fixed at the j initial base state condition. The ability to perform the above type of comparisons has been incorporated into SLICK. The state calculation requires the coupled solution of the neutmn diffusion equation and the I- RETRAN 02 thermal hydraulic model. Since the solutions of the neutron diffusion equation and the thetmal hydraulic equations in SLICK are effectively the same as the RETRAN 02 solutions, the msults of the SLICK state calculation may be viewed as equivalent to the RETRAN-02 cal-culation. Since the thermal hydraulic variables are held constant, the static worth calculation is a pure neutronic calculation. The calculation involves interpolation of the cross sections for the i I> ; Ei
- gg
1: - - i i i specified control state and subsequent solution of the neutron diffusion equation. Again,it has been previously demonstrated that the SLICK and RETRAN-02 neutronics solutions are equiva-lent. The results of the comparisons presented in the following subsections show that the - RETRAN-02 neutronics calculation (using the kinetics data generated by our method) is consis- .j tent with the 3 D SIMULATE-3 calculation. We emphasize that the SIMULATE-3 cases com-pared to are independent of the perturbation cases used to generate the data. The close . .] agreement between the 1-D and 3-D results provides a basis of confidence for the metixxi. 7.1 Comparisons of State calculations L
.- To assess the method, two types of state calculations are presented. The first type involves -
changes from the initial critical state that result in a critical final state. The second type of state - calculation involves changes to the initial state that result in a final state far from critical. ,.f For the critical type calculations, perturbations for three variables are examined: system . pressure, core inlet temperature, and core inlet flow. Each of these perturbations were run at EOFPL conditions of Vermont Yankee Cycles 9 and 13 and at Cycle 9 EOFPL - 2 GWD/ST, - Thus, nine state calculations were evaluated. , The SIMULATE-3 calculations were made by changing the single thermal-hydraulic con-
- dition (pressure, inlet temperature, or inlet flow) and increasing power to counter the increase in reactivity caused by the perturbation. In all cases power was increased by 5% to arrive at the final critical condition. The final eigenvalue was very close to the initial state eigenvalue (typi-( cal agreement was 4 pcm). These cases were then evaluated using independently generated kinetics data with SLICK. For the ideal case where SLICK exactly reproduces the l
j
SIMULATE-3 eigenvalue, the SLICK solution may be regarded as the state to which the 1 D RETRAN 02 model would steady out if subjected to the same penurbation. The degree to which the SLICK and SIMULATE-3 solutions of the final state differ is a measure of the reactivity l errorin the kinetics data. , The results of the comparisons are provided in Table 7.1. The errors in the prediction of 1 f the final state eigenvalue range from +2 to -12 percent milli k (pem). This is considered very good agreement. To put thit ieo pagedve, consider that a difference of 12 pcm in reactivity may be accounted for by a di!ference of 0.2% in rwd power. Since the change in power for the cases was 5% rated, a 0.2% difference corresponds to a maximata clative error of 4% in the I. power change. The final state axial power distributions for the Cycle 13 cases are presented in Figures 7.1 through 7.3. The axial power distributions are in good agreement with'the maximum error being about 3%. The sgreement of the power distributions for the other cases am quite similar to these. To further assess the accuracy of the method for generating 1-D kinetics data, comparisons l are made to a state calculation far from critical. The state calculation chosen is representative of the core conditions at the time of peak mactivity in a turbine trip without bypass transient. The final state results from a change of +70 psi in reactor pressure, +5% in core thermal power, and a 1.5 ft. insertion of all control rods into the core. Comparisons are made at EOFPL for Cycles 9 and 13. The results of the comparisons are provided in Table 7.2. The error in the final eigenva- _l lues is about 30 pcm. The SLICK results overpredict the change in eigenvalue. The increase from the initial eigenvalue to the final eigenvalue is about 440 pcm for the SIMULATE-3 cases. Thus, the error in the change in eigenvalue is about 7%. A comparison of the axial power dis-
.I tributions for the Cycle 13 case is show in Figure 7.4. The axial power distributions are in good agreement with the maximum error being about 5%. =
I-I g
l Static Scram Worth Comparisons j I 7.2 Static scram worth comparisons are made at Cycle 9 EOFPL and EOFPL - 2 GWD/ST (
,W- ' conditions. The control rods are fully withdrawn in the EOFPL case. The EOFPL - 2 case is initially rodded. Figures 7.5 and 7.9 pmsent the scram worth as a function of distance for the -
- first six feet of rod insenion. The 1 D msults underpredict the 3-D scram worth in both cases.
l The EOFPL has a maximum underpmdiction of 3%, while the EOFPL - 2 GWD/ST underpre-dicts the wonh by about 6%. The axial power distributions with control rod moves of 2,4, and 6 feet am shown in Figures 7.6 through 7.8 for the EOFPL case and Figums 7.10 through 7.12 for the EOFPL - 2 GWD/ST case. As seen from these figures, the 1-D and 3 D axial power distrib-utions are in excellent agreement.
- I LI-I .
I I LI s 'I I -28 I IL
l 1 d, Table 7.1 Differences Between 3-D and 1-D Eigenvalues for Critical State Calculations g Perturbation- ~ Error in Final Einenvalue (ocmf (; h Magnimde EOFPL9 EOFPL9-2 EOFPL13 - 1 L ') Passun +30 psi -5.5 -5.4 -6.2 , InletT -4.8F -0.6 -7.7 9.0 Inlet Flow +7.% +1.8 -12.0 -10.7 l
= -
~
5
- Difference in pcm = (k.n.suex - k,y.smuun)
- 10 I-I ;
5 3 I I I I
.g
-29 I .
I m.- .g--
I 4 Table 7.2 J- : 4 .( Differences Between 3 D and 1-D Eigenvalues for TTWOBP State Calculation 'l Change in Eigenvalue (pcm)* i ' (Final-Initial) Error in Eigenvalue Change Absolute Relative I: Condition - R ,1.Q (pcm) A Cycle 9 EOFPL 441 472 +31. +7.1 h Cycle 13 EOFPL 444 473 +29. +6.6 u I 5
- Difference in pcm = (k.n.ru - k,u.wu)
- 10 I
I l
'I I <
I I
'I:
lI I 30-a
e .
.I Ll) ..
Cycle:13 EOFPL-30 PSI Increase' # 27 , . i....i.... lh . 25 _ N;;, _ 23 _ 21 _ _ l 19 _ 17 _ I c 15 _ ll _h o- 13 _ O' .
, 11 - _
o - SIMULATE .
.g g_ o - SLICK ~ 7 _
!. 5: , _ 7_ l i- _ ; L. 5_ _ l-.. .
- 1. - -
'3 _
l.
- l. . .
i
.il i i l
i e i i '4-
.0 .5 1.0 1.5-l Relative Power Figme 7.1 3.D and 1-D Axial Powers for +30 psi Pressure Change g
l: l.
l' t E Cycie .13 EOFPL 4.8 F inlet T Decrease 4
- 27. p , , ' - t . . , , , . . . ,
25 _ .
~
.l 2 3 .'.
~
E 21 _' ~ f E 19 _ '~ - 17 _' ( _ _ll - . g 15 _ _ I E(D x 13 - E 11 _ - 0- SIMULATE-3 . I 9_ o - SLICK _ w 7-l 5_ _ l 3_ E 1 n 3,g 0 .5 i
- l. Relative Power
=l. Figure 7.2 3-D and 1-D Axial Powers for -4.8 F Inlet Temnerature Change I I
- g;
I . I s LCycle :13.EOFPL-7% Flow Increase l 27
, . . . s . . . . n . , , ,
25 _ _ 23 _ _ ! l 21 _ s I
, ~ 19 _ _
l 17 _ _ j g 15 3 _ ( '5 . . e - 13 _ _ cc I 11 _
. o - SIMULATE .
3 E - 9- o - SLICK , 0 - ?g . . l g- -7 _ j _ ,3 , 5'_ _. I'- 3_ _ 1' . , , ,
.0 .5 1.0 1.5 l Relative Power I
g Figure 7.3 3-D and 1.D Axial Powers for +7% Inlet Flow Change I I
+
ll} Cycle 13 EOFPL +70 psi +5% P Rods 1.5' vl -27 ,, . . . i' . ...i...;i.... m 45 _ _ i 2 3 _' _ I 21 _ l 19 _ _
. 1 17 _ '
I c 15 - I .o._ m (D tr 13'_ I 11' _ 3 _ _ o - SIMULATE-3 . 9_ o - SLICK >,I- _ 7_ _ Ll E 5_ I 3_ L. . - ll 1 L , , , , L' .0 .5- 1.0 1.5 2.0 Ll Relative Power sI Figure 7.4 3-D and 1-D Axial Powers for TFWOBP Conditions ,. Lij I -
If 1..
~
. SCRAM Worth Comparison.at EOFPL-r
'l . .0 t
, . i . '- . i - i . i .
g l c . .
,j_
,5 _ _
l m v I- -5 L. . - o . g . . 1 l
- 1,0 _ .
lI g . . I - O- SIMULATE-3 o - SLICK ; !
. . ]
- 1.5 L-(l ,
0- '1 2 3 4 5 6 l. lf l. Rod Move (ft.) l l Figure 7.5 3-D and 1-D Static Scram Worth at EOFPL i 1l LI;
f f I 7l.U Relative Powers After 2 ft. Rod Move l
;el 2 7 - _ :, . . r. -
. . ...i... .i ....
25 _ I i 23 _
? )
- l[
- 21. _
f l
- ~
19 _ > - c 15 _ - o
' En . - .
o 13 _ - E. 0:. y su 11 _ _ Ll 9._ _
.g 7_ _
- g. . .
5_ _
~ -
h O- SIMULATE-3 3- o - SLICK - l l. i' 1C , , , ,
.0 .5 1.0 .1.5 2.0 Ll Relative Power 1 4
1 Figure 7.6 3-D and 1-D Axial Powers After 2 Foot Rod Move MOFPL1 c ~E 1 . .. ._. ..
l l .
~
Relative Powers After 4 ft. Rod Move 2 7 1, . . . . t ....e ' l-}
. .. . n . . . e . . . .
E 25 _ _ ,
3 23 _ _
+
LI 21 _
-l; 19._ _
]
17 .-. _ c 15 _ _ E- .9
- g-. .cn - -
o 13 _ _ 11 _ _
-l -
7_ _ _l. . . 5 .43 _ O- SIMULATE-3 3 -E] o - SLICK - 4 4 4 I g i I 5
.I g i 8 5 I g 5 8 5 5 g
F 3 5 4
.0 . 5 1.0 - 1.5 2.0 2.5 l Relative Power $l Figure 7.7 3.D and 1-D Axial Powers After 4 Foot Rod Move GOFPL)
I - E .
'j p ;
l 1 l' , Relative ' Powers After 6 ft. Rod Move i 27 : ...i....i....i....i....i.... q j 25 _ _ 23 _ _
'g 21 _
_ .I j 1 19 _ _ l 17 _ _ I' . . c 15 _ _ g- .9 . . 7 e 13 _ _ Oc 11 _ _ ,g 9 3 l 7 43 _
- l 5 43
~ -
lt- O- SIMULATE-3 3 43 o - SLICK - 1
-c......... ....
.0 .5 1.0 1.5 2.0 2.5 3.0 l Relative Power j!
Figure 7.8 3-D and 1-D Axial Powers After 6 Foot Rod Move WOFPL) l -3 8-LI o - - - -
L
! SCRAM Worth at EOFPL-2 GWD/ST 1
l
.0 , . i . i . i . i- . i . ;
l i l- . . I
,5 _ _
i
- 1.0 _ 1 m '
M - - I v 2 1: - - l 0 . . 1 I 3 ' - 1,5 _ _
\
5
-2.0 _ _
- 1 o - SIMULATE-3 -
o - SLICK
- 2. 5 . , , , . , ,
0 1 2 3 4 5 6 Rod Move (ft.) t Figum 7,9 3.D and 1.D Static Scram Worth at EOFPL - 2 GWD/ST
k l s . i Relative Powers After 2 ft. Rod Move. , I1 27 p . . . . i . . . . i. . . .i . . . .
~
1 25 _ _ I, . . 23 _ l 21 _ - 19 _ j _. 17 _ _ c 15 _ _
;l- $ . .
c) ~13 _ x . .
- 1-1 _ _
9_ - 7_ _ 5_ _ l D- SIMULATE-3 3- o SLICK
'll- 1
.0 .5 1.0 -1.5 2.0 Ll. Relative Power .
t
-e .
r Figure 7.10 3.D and 1-D Axial Powers After 2 Foot Rod Move MOFPL - 2)
-40 I
1
- 3
'l ..
~
Relative Powers After 4 ft, Rod Move
-27 t ,.. ..i....i. . i....i.... -
. . j 25 _ _
23 - _ g . . 21 ._ 3 _ 19 _ _ 17 _ _ c 15 _ _ I
.{ .{c) '1 3 _
- 1.1 _ _ .
;l 9- -
7- _ 5 43 _ 3 O ' SIMULATE-3 L 41 o SLICK H3 3 -t: $y ,, l ..
.0 .5 1.0 1.5 2.0 -2.5 Ill ,b-Relative Power Figure 7.11 3.D and 1.D Axial Powers After 4 Foot Rod Move mOFPL - 2) l~
l l 41-
. . . .._.._.,q I
l ~ Relative Powers After 6~ ft. Rod Move l' 27 t,....i....i....i....i....i.,..
. j
'g. 25 _ _ B . . 23 - __ 7 21 L .19 ' _ _ 17 _ g.. .
-g.. -c o -
g '5 . . o 13 _ _ tr I.-- 11 _ _
-l 9h _
g 7 43- _ E - . 5 .g _
' ~
O - SIMULATE-3 42 o - SLICK - 1 y....i,,,,,, ,,,,,,,,,,,,,, _
.0- .5 1.0 1.5 2.0 2.5 3.0
- lL Relative Power-
;l Figure 7.12 3-D and 1-D Axial Powers After 6 Foot Rod Move GOFPL - 2) 3
I
8.0 REFERENCES
- 1. EPRI, RETRAN-02 -- A Program for Transient Thermal-Hydraulic Annivsis of Comnlex Fluid Flow Svstems. EPRI NP 1850 CCM-A (November, .1988). ;
- 2. J. M. Holzer, Methods for the Annivsis of Boiling Water Reactors: Transient Core Physics, YAEC-1239P(August,1981). s
- 3. Letter to Mr. Hugh O. Parris, Manager of Power Tent,essee Valley Authority from Mr.
Domenic B. Vassallo, Chief Operating Reactors Branch #2 Division of Licensing, "TVA RETRAN TOPICAL REPORT," Docket Nos. 50 259 50-260 50-296, (April 7,1983). J
=
4.' D. M. VerPlanck, K. S. Smith, and J. A. Umbarger, SIMULATE-3P: Advanced Three. Dimensional Two Group Reactor Analysis Code. Studsvik/SOA - 88/01 (February,1988). -
- 5. A. S. DiGiovine, J. P. Gorski, and M. A. Tremblay, SIMULATE-3 Validation and Verifi-sation, YAEC-1659 (September,1988).
- 6. R. A. Wochlke, MICBURN-3/CASMO-3/I' ABLES-3/ SIMULATE-3 Benchmarking of Vermont Yankee Cveles 9 Through 13. YAEC-1683 (March,1989).
- 7. K. S. Smith, SIMULATE-3: 1-D Model and Kinetics Edits, Studsvik/SOA - 89/01.
- 8. William H. Press et al., Numerical Recipes! The Art of Scientific Computing (New York:
-Cambridge University Press,1986).
L LI L RI I LI l LI
- I I .
w -+ ey7 y - e- --
t I APPENDIX A i SLICK THERMAL. HYDRAULIC MODELS I A.I.0 INTRODUCTION The purpose of this appendix is to describe the thermal hydraulic models used in SLICK. , The aim of the SLICK thennal hydraulic calculation is to reproduce the results of RET RAN-02W
- for any given set ofinitial conditions. The modeling encompasses the solution of the fluid mass,
- fl. energy, and momentum equations, the subcooled void model (used for neutronic feedback), and the fuel pin heat transfer model.
A.2.0 FLUID CONSERVATION EQUATIONS E , E The fundamental forms of the mass, energy, and momentum equations are taken directly from RETRAN-02. We are only intemsted in the steady-state solution as applied to the active section of the core. The solution of the mixture mass conservation equation for this case is trivial. The energy and momentum equations are derived below.
'A.2.1 Energy Eauntion
.- : For the case of control volume k of height g with a single inlet junction i 1 and outlet ,
junction i, we write. r 3 r
-l y2 y*2s ,
Qa + W,,i-1 L ' s h, + 2 ,,.i- W,,i sh, + 2 ,i (A-1) r 3 / y2 3-y2
+ W,i-n - W ,i h; +j si-n <
hi +k si - W8 z = 0 n I l L Solving the above for W,,, yields, 1 I l l A-1 l l L k.
I y e c < .c Da - W g z, +i h ,s ++f,o W,,s- qh, + ,_, + Wo.u-nJhi +,_,
, W,,, = r ye (A2)
< ,-hi + , -k; h
where we have used the relation, . iI. Wu,i = W - W ,,, (A-3) in arriving at the above form.
, Note that the right hand side of the energy equation has terms dependent on W,,,, Namely, V,,, .
and V,,4, V ,,u = ,
,_, (A 4) ,
Yt .'
- p,,Ao k)
I "
; I _~ The velocities are also dependent on the void fraction. Given that the inlet variables and the thermodynamic properties are known, the solution of the energy equation for W,,, requires a.
void-quality relation (i.e. slip model). RETRAN-02 has three distinct models for slip: the homogeneous equilibrium model (HEM), the algebraic slip model, and the dynamic slip model. SLICK supports the HEM and algebraic slip model. The treatment of these models in SLICK is functionally equivalent to RETRAN 02. SLICK uses the above form of the energy equation rather than recasting the energy equation in terms of RETRAN's slip variable. For the HEM and
! algebraic slip models, the void-quality relation replaces the RETRAN-02 slip relation. The Zuber-Findlay void quality relation is, xf (A-5) et= i Co,xf +k(1 -xf) + p,h f
A-2 I.
L where the flow quality, xf, is defined by, W x;=j (A-6) JB and Co and Vg are the drift flux parameters. The void model of Lellouche and Zolotar* is used to evaluate the drift flux parameters for the algebraic slip model. For the HEM model,' the drift flux parameters reduce to,
- I Co = 1 (A-7)
Vg =0 For the case of algebraic slip, there is a point that should be mentioned. RETRAN-02 uses the Zuber Findlay void-quality relation in evaluating the slip velocity at junctions. For volumes,- the slip velocity is evaluated at the average of the upstream and downstream junction slip velocities. Thus, equations A 2, A-4, and A-5 are the set of equations evaluated forjunction properties. For volume properties, the Zuber Findlay relation must be replaced by an equation relating the void fraction to the RETRAN 02 sli.p variable. The slip velocity is defined by, V, = V - V, (A-8) This may be expressed in terms of the superficial velocities and void fraction as, u, u, (A-9)
- y' _ (1 - 0:) or where, I u; =
W - W' (A-10) W, u* = 9,A
- l.
^~'
I: . l 1 l 1
Solving the above for the void fraction and taking the appropriate root yields,
= I _-
(A-11) 1 (ui + u,) + 44V,u, + (V, -s u - u,)2
""2
~
2V, 2V,
' In evaluating volume properties, the above relation along with the definitions of the superficial l velocities replaces the Zuber-Findlay relation.
The above energy equation is written for the case of two-phase inlet conditions. For single phase inlet conditions, there air two cases to consider. These are single phase and two phase out- l Ict conditions. For the single phase in, two phase out case, equation A 2 reduces to, Da-W grn + hi ,i +h-h;.s - TV,,, = r q q, (A-12) l gh , - hi + 3 3 ,, j I. For the single phase in, single phase out case, equation A-1 is solved for the outlet enthalpy, l p f y,2 (A 13) h; = g, +%h + y2' - g nr -- di-1 - u- and equation A-4 reduces to, tv (A-14)
'I- V'.=
p(pi,h i)A; A.2.2 Momentum Eauation - The starting point for the derivation of the momentum equation used is the steady-state
; form of equation II.3-48 of Reference 1. For the control volume bounded by the centers of vol-umes k and k+1 and connected byjunction i, we write, l-LI l
LI I A4 I I
< W: W:. , ,
a
; E:' Ps , s
- Ps + --
2
<P&A? Pa+sAs+s,
'u suspaps ' W, Wi ~ ~
<=pA* ,, g as p, a pa,, s ;
'a,a;p,p * ' W, W, '
s PA* ,s+srusPs aiPs,s.s gg . 4* h h 4* W,2
~
~
+ 24,1
/
<Ds,,,pi,, Ds,,,, spa,,,,, 2A i W,' ' '1 1' ';
I
+2.1. # W' --
<a,p, + a:Pt,i<AE+ s AI,
- }>
1 W*i . 1 1 l e z 2(pA ) s 3P,z,g 5p,.i ,,,g 2 l To express the momentum equation in the desired form, we make use of the following relations, which may be derived from the definition of the slip velocity and continuity conditions: f ' W* a ,p, W, 5 a,p,,,
= -A, V,,,
(A-16) ; t (A-17) W=- Wa,p, - V,A a,p,a pi
^
s 8 p- 1 Wa spi + V,A a,p,a;pi W= P (A-18) l
. where, L p = a,p, + a,p (A-19)
. Substituting the above into equation A-15, yields the desired form of the momentum equation, l
I ' A-5 I . .
. . . . . . . . . . . . . . . - ..._.....=.....;._
Ps . s = Pn - Pst s8 - Ps.nta+s8 I ri/2 r, . i/2 W/
- 4f,4 D ,,4 p,,, +D q .,,4 ip,, . i, 2A 2 W' Vla,a,p,pA 2V 3 (A-20)
.pA'q W ,.,
I W'
~
2 V,'a,a,p, p,A '
- 1 W?
1 -K PA*< + W ,-r+n 2 PiA? 1 Vla,G,P,PA* ' I ___ 1
,2 pt q W' <An*+ 1 A n* ,
I Note that if the slip velocity i zero, the momentum equation reduces to the HEM momentum equation (equation II.3-26 of Reference 1). The momentum equation contains the Fanning friction factor,f., and the two-phase fric-tion multiplier, $,. SLICK uses the same relation as RETRAN-02 for the friction factor. The Baroczy model is used for the two-phase friction multiplier. This is the same model used in the I core-wide transient model. The above momentum equation yields the volume pressures. However, the junction pres-sures are also needed to evaluate junction properties. RETRAN-02 evaluates pressure at ajunc-tion based on the pressure in the donor volume corrected for elevation and wall friction pressure drop in the donor volume. 1 ' r/2 ' wl (A-21) Pi = Pi 3 p,r, - 4f ,,$' ' D ,p,A ,, 2 ( I I I A-6 I I
I A.2.3 Coupled Solution of Energy and Momentum Equations The energy and momentum equation are coupled. The momentum equation provides the u' pressure distribution of the 1-D core model. The pressure distribution is used in the evaluation of the thermodynamic and transport properties required by the energy equation and void model. To start off the solution, the initial pressure distribution is assumed uniform. The control volume energy equation is evaluated starting at the center of the bottom reflector and proceeding up the com. After the thermodynamic states of all volumes and junctions am known, the momentum equation is evaluated for the pressure distribution. 'Ihis iteration is repeated until the
~
pressum and density distributions converge. The SIMULATE 3 case provides the needed boundary conditions for the system of equa-tions. These are the inlet flow rate, inlet enthalpy and core average pressure. The energy depos-ited into the coolant is consistent with the power of the SIMULATE-3 case. The distribution of energy is based on the finite-difference solution of the neutron diffusion equation and the energy
.I. deposition model used by the core-wide transient model.
A.3.0 NEUTRONIC DENSITY RETRAN 02 uses the drift flux parameters of Lellouche and Zolotar and a profile fit rela-tion for flow quality to determine the neutronic density. The profile fit relation used is, r { - x, -x,s 1 - tanh 1 - , v
> l"f xf =
(A-22) 1-x,s 1 -tanh 1-- s
< a.
s I A-7
, . . - - . _ . . . - - - . - - ~ . _ - - ., s .
The departure point quality,x a,is determined ~using the model of Lellouche and Zolotar.
- RETRAN-02 depans from the conventional calculation of x, in two phase control volumes. ;
~
. RETRAN 02 defines x, as,
~
I h-hf' (A23) > t x' = his Lwhere h is the contml volume enthalpy. For a two-phase volume, [ ap,h, + (1 - a)p,h, (A24): 3 ap, + (1 - a)p, . For a single-phase volume, h comes out of the solution of the energy equation.
. With the flow quality determined, the void fraction is evaluated using the Zuber-Findlay relation and the drift flux parameters of Lellouche and Zolotar. The drift flux parameters are modified by RETRAN-02 to smooth the subcooled void prediction into the vold fraction pre-
- dicted by the thermal hydraulic equations. The smoothing relations are, C;=%+(1 - y) (A25)
V[= yVd
+ where, x'
y= 1 lxal- (0 <x, < lxa l) (A-26) y=1 (x,s;0) Forx, ;t lx l,a the neutronic density is equal to the thermal-hydraulic density. l l I f A-8 I i
_7 f' 6 i I ' A.4.0 FUEL HEAT CONDUCTION Ig . RETRAN 02 uses a finite-difference scheme for solving the conduction equation. Our goal is to solve the RETRAN based fmite-difference equation for the case of a cylindrical fuel - rod comprising three materials: fuel, gap, and clad. The one-dimensional heat conduction equa-
. tion for the system is, Id dT' - (A27). ,
o . yk(T)rp =-q (r).
- h. t where q~(r)is a step-wise function.
The finite-difference mesh contains nodes at the material boundaries and uniformly spaced nodes within a given material. The temperature at the outside of the clad is considered known, ,
- Thus, finite difference equations are required for the interior and center line nodes. For an inte- ,
e . rior node i, we define,- 4 r, a radial distance to node i from center-line. h.,, a r, - r, . i l l: '
; h,,, a r, . i - r, .
h.,i r.,, a r, a T (A-28) r ,, a r, +
't ~ ~
q ,, a the value of q (r) between r, i and r,.
~ ~
i r . q , a the value of q (r) between r, and r,.i. o
. Integrating the conduction equation over an interior control volume bounded by r.,, and r,,,
and using a central-difference formula to approximate the temperature gradient yields, s A-9 I - .
i a,T,.. + b,T, + c,T,, i = S, -(A 29) wheir, ; I 1 k,,,r,,, .
,' , j
- h. ,,
k,,ur,,, Cg = be,i b, = -(a, + c,) (A 30) r,* S, , _ ' r, ,, - r * ' - r' ,, i t 2 s ( 2 , J I k,,, = k(T)lr , .r , . r, s k,,, = h(T)Ir . r, 7, ,, l , Si.nilarly,!:stegrating the conduction equation over the control volume associated with the center line node ud noting that the temperature gradient is zem at the center line, we rirnve at I the finite-difference ectuation for the center line nMe
% + c,Ta u t, (A37) , ..# where, 3' ;
c3 =
S h,,i
-l bi = -ci (A32)
.I S=-
~
q, i i I:
- The equations for the center line and interior nodes along with the specification of the ;
.g surface temperature form a tridiagonal system of algebraic equations. The equations are l nonlinear because of the temperature dependence of thermal conductivity. However, the system A 10 I
I'
I of equations is solved using a standard linear equation algorithm
- by evaluating thermal l conductivity with an assumed temperature distnbution. To start off, the temperature disuibution :
l is assumed to be uniform. All coefficients of the finite-difference equations are evaluated and the linear tridiagonal system is solved. This process is then repeated with the coefficients being > , calculated from the latest temperature distribution until convergence of the temperature disuibu-tion is achieved. ;
. A.4.1 Mad Surface Temocrature ;
- I
!. As mentioned above, the temperature at the surface of the clad is determined from the con- [ vective boundary condition, (A-33) T* * " g *
- h, where, q',4 surface heac flux.
? h, a heat tmnsfer coefficient, T, a volume bulk fluid tempe:ature. ,I For sir.gle phase conditions in the associated control volume, the heat transfer is based ! either on forced convection or subcoole4 nucler.te boiling. For forced convection, the Dittus- - L Uoelter correlation is used, k 'GDu * 'c,p * (A*34) h' = 0.023 Du p ) g k, with properties evaluated at T3. I I A-11 I . .
I' i For subcooled nucleate boiling, the Thom correlation is used to predict the clad surface temperature, . T,,, = T,,(p)+ 0.072cemp (A-35) where English units are used in the above dimensional equation. The lower of the two temperature predictions is used for the surface temperature. For two phase conditions either nucleate boiling, forced convection vaporization, or a combination of these two heat transfer modes is used. For volumes with void fractions less than t 0.8, nucleate boiling is assumed and the Thom correlation is used to predict the clad surface tem- , perature. For volumes with void fractions greater than 0.9, the following correlation attributed ; to Schrock and Gmssman is used, k I ms (A 36) , h, = 0.0575g, [Re,(1 -x)]"Pr," where the Martinelli parameter is given by, r I _1_ n x' V ,*I4._,Y'
.1, p
<1-x,gp, , ps) g (A'37) i For voltunes with void fractiou between 0.8 and 0.9, the surface temperature is based on e -
weighted average of the nucleate boiling and forced convection vaporization heat transfer heat fluxes. In this range, RETRAN-02 ass,umes that l . 0= (a 0.1
- 0.8) Gra<-i + (0.9 - a),
0.1
~
(A-38) The appropriate correlations and relations are substituted into the above relation and the resulting equation is solved for the surface temperature. b I A 12 E
I ! AJ.0 REFERLhCES 4
- 1. EPRI, RETRAN-02 -- A Program for Transient Thermni Hydraulic Annivsis of Complex !
Fluid Flow Systems, EPRI NP 1850-CCM A (November,1988).
- 2. EPRI, Mechanistic Mndel for Predicting Two-Phase Void Fraction for Water in Vertical Tuhes. Channels and Rnd Bundles. EPRI NP 2246 SR (February,1982),
- 3. William H. Press et al., Numerical Recipes
- The An of Scientific Onmnuting (New York: l Cambridge University Press,1986).
i I I I I I : I - I : i A-13 I
- _ - _- . . .}}