ML20245L182
| ML20245L182 | |
| Person / Time | |
|---|---|
| Site: | Vermont Yankee  |
| Issue date: | 06/30/1989 |
| From: | Cacciapouti R, Cronin J, Slifer B VERMONT YANKEE NUCLEAR POWER CORP. |
| To: | |
| Shared Package | |
| ML20245L154 | List: |
| References | |
| YAEC-1694, NUDOCS 8907050397 | |
| Download: ML20245L182 (64) | |
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METHOD FOR GENERATION OF ONE-DIMENSIONAL KINETICS DATA FOR RETRAN-02 June 1989 by James T. Cronin Pmpared By: # b gmes T. Cronin, Senior Nuclear Engineer (Date) Reactor Physics Group Nuclear Engineering Department Approved By: If .Cacciap , Manager h!M!M (date) eactor Phy s Group Nuclear Engineering Department Approved By: . M B.t.Slifer, Director lf (Date) Nuclear Engineering Department l Yankee Atomic Electric Company Nuclear Services Division !' 580 Main Street } Bolton, Massachusetts 01740-1398 }' { f l 8907050397 890630 PDR ADDCK 05000271 P PNU e
i p l DISCLAIMER OF RESPONSIBILITY
-l This document was prepared by Yankee Atom ?lectric Company (" Yankee"). The use -
E a
. 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 respect to any unauthorized use, neither Yankee nor its officers, directors, egents or employees assumes any obligation, responsibility, or liability or makes any warranty or representation as to the accuracy or completeness of the material contained in this document.
l
-ii-5
ABSTRACT This report describes the methodology used toproduce one-dimensional (1-D) cross sec-tions and kinetics parametersfor input to the 1-D space-time kinetics model ofRETRAN-02. This method will be used to support core-wide transient simulations performedfor BWR reload licensing. The methodology makes use ofSIMULATE-3 to characterize the neutronic response of a BWR core. A linking code, SLICK, has been developed tofunctionalize the 1-D neutronic
' data produced 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 of calculations verifying the method represented. i I l ) l 1 \
-lii-l 1:
, .4 e. )b
.t. t.
ACKNOWLEDGEMENTS h The author wishes to acknowledge the contributions of Michele Sironen in performing the~ u
. SIMULATE-3 depletions associated with the verification cases of this report and Michael
~
g -
~
Tremblay in assisting with SIMULATE-3. ' Also, I wish to thank Kord Smith of Studsvik of . I
- America for his work on the homogenization of the kinetics data in SIMULATE-3 and for his '
h help in defining an interface between SIMULATE-3 and SLICK. Finally, I thank Sam Forkner J - of the Tennessee Valley Authority for his ideas and encouragement on this project.- n-h l
,\\
j s
.iv-L 1
i i 1 4 k 1 3 TABLE OF CONTENTS ' DISCLAIMER OF RESPONSIBILITY L.. .. . . . . .. . . .. ..... . .... . .. ... .. ... ii , j t a- . . . . ABSUMCr ..
... .. . .............................................m
]1
' ACKNOWLEDGEMENTS ....... .... ...... ........ ... . . ...................................iv
.I b LIST OF FIGURES .. ...... ... ..... . ..... . .. ... . .............................Vi l LI ST OF TABLES ....... . .. ...... ... ... ... ... ..... ..... . ... ... .. .... .. . . .. ... ... .. . ... yli l
' I.0 INTRODUCTION . ... ....~. ...................................................1 l 1.1 Purpose.......................................................................................................1^ i
, 1.2' Overview.........................................................................................................1 i L 1.3 Organization of Remainder of Report .............................................................. 3 2.0 RETRAN-02 CROSS SECTION MODEL ..... .... ..... ...........4 ~ l 3.0 ROLE OF SIMULATE-3 ..... .... .. . ......................... . ......................6 H
' 3.1 Spatial Homogenization From 3-D To 1-D Neutronics Data ............................... 7 4.0 FINITE. DIFFERENCE SOLUTION OF I.D 2-GROUP DIFFUSION EQUATION .. . ..... ... .. 11 5.0 TRANSFORMATION OF THERMAL-HYDRAULIC VARIABLES ..... .... .. 13 6.0 FITTING OF I.D KINETICS DATA 'ID RETRAN-02 POLYNOMIAL FORM ...... .....~.. ............ 20 '
. 7.0 VERIFICATION .. ... .. ...... . . . ... ..... .. .. 25
! 7.1 - Comparisons of State Calculations ........ .......................................................... ' 26
' 7.2 S tatic Scram Worth Comparisons .............. ..................................................... 28
8.0 REFERENCES
... . ............ ............................~....... . ..... 43 APPENDIX A SLICKTHERMAL-HYDRAULICMODELS i
g l s 1 i I l j' l
-v- l
_____i______.____.________m____m_ _ _ _ _ _ . . . _ _ _ . _ _ . _ _ _ . _ . __ ._ _ . _ _ _ . _ _ _ _ _
l LIST OF FIGURES Number Ijile Egge 3.1 SIMULATE-3 and SLICK Power Distributions at ARO .................................... 9 3.2 SIMULATE-3 and SLICK Power Distributions at ARI ..................................... 10 j 4.1 SLICK and RETRAN-02 Power Distributions ................................................... 12 5.1 Comparison of RETRAN-02 and SLICK Void Profiles ..................................... 15 5.2 Comparison of RETRAN-02 and SLICK Density Pmfiles ............. ................... 16 5.3 Comparison of RETRAN-02 and SLICK Average Fuel Temperatums .............. 17 5.4 Compaision of RETRAN-02 and SLICK Fractional Density Change ...... ....... . 18 i i 5.5 Comparison of RETRAN-02 and SLICK Fuel Temperature Change ................. 19 7.1 3-D and 1-D Axial Powers for +30 psi Pressure 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 TTWOBP 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 i 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 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 l I l ) l 1 1
-vi- i
) ___________ a
i 1 TRT OF TABLES Number - 3 g 3.1L SIMULATE-3 Versus SLICK Eigenvalues ........................................................ 8
- 6.1 Penurbation Cases Used in Generation of 1-D Kinetics Data ............................. 23 i- 6.2 Comparison of I-D Neutronics Calculations with Fitted Data to 3-D Perturba-tionCases............................................................................................................24 l
.7.1 Differences Between 3-D and 1 D Eigenvalues for Critical State Calculations 29 7.2 - Differences Between 3-D and 1-D Eigenvalues for'ITWOBP State Calculation 30 i
P t
. I 4
6
-vii.
i I i
3
1.0 INTRODUCTION
1.1 Pumose i 1 This report describes the methodology used to produce one-dimensional (1-D) cross sec-tions and kinetics parameters for input to the 1-D space time kinetics model of RETRAN-02m, l 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 effects as predicted by the three-dimensional (3-D) nodal simulator, SIMULATE-3*, 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. I SIMULATE-3 is used to predict the 3-D neutronic 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 produces all the cross sections and kinetics parameters used in the I RETRAN-02 kinetics model. The 1-D cross sections and diffusion coefficients prodred by SIMULATE-3 effectively preserve all planar reaction rates, planar interface curmnts, and the reactor eigenvalue. One set of spatially homogenized kinetics data is produced per SIMULATE-3 solution. Our method requires multiple SIMULATE-3 solutions to the 3-D two- )' t group neutron diffusion equation and subsequent spatial homogenization calculations. i l ) I t J
)
) The 1-D kinetics data produced by SIMULATE-3 are pure diffusion parameters. No j attempt is made by SIMULATE-3 to functionalize the data in the form mquired by RETRAN-02. t p The SLICK (SIMULATE-3 Linking for Core Einetics) computer code has been developed at Yankee to perform this task. To get an overview of the method, we will present the steps j
-{
. 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 modelis brought to the exposure point ofinterest. l l (2) SIMULATE-3 perturbation cases are run for the initial control state. The penurba-f tion cases are selected to pmduce conditions representative of the transient to be ana-lyzed. The perturbation option of SIMULATE-3 allows forindependent variation of either the 3-D density distribution or fuel temperatum distribution. All other variables normally associated with the SIMULATE-3 cross sections (exposure, con-r trol history, fission product inventory, etc.) am held constant. The perturbations am from the mference state of step 1. SIMULATE-3 produces spatially homogenized kinetics data for the reference case and all perturbation cases. } (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 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 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 neutronic data for the initial control state and tN all-rods-in control state. SLICK processes this data one control state at a time. a > N ch case within a contml state set, SLICK: i i )
h i, j? h, . a.1 Calculates the power distribution by' solving the finite-difference form of the' d l-D two-group neutron diffusion equation and
- b. . Calculates the thermal-hydraulic variables ~used by RETRAN-02 in evaluatingl the kinetics data. This~ is accomplished by solving the steady-state form of the n RETRAN-02 thermal-hydraulic equations at the same conditions used in the -
SIMULATE-3 case. ; L After processing of all cases for a contml state, the kinetics' parameters for each neu- : tronic region am fit against the RETRAN-02 thermal-hydraulic variables of neu-tronic density and fuel temperature to produce the polynomial form requimd by
- g. : RETRAN-02.
l 1.3 Orannization of Remaintier of Renort L. The above has served as an overview of the method. The remainder of the report presents - f -- the main elements comprising the methodology and verification of the method. These are:
- 'the RETRAN-02 Multiple Control State cross section model and its data require-ments, L --
the spatial 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,
-: the transformation of thennal-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' ) I L
' 2.0 RETRAN 02 CROSS SECTION MODEL 1
SLICK generates data for the Multiple Control State (MCS) model of RETRAN-02. This ' u is the model used in the core-wide transient model. As in most transient kinetics models, there am three variables associated with the evaluation of the transient cross sections and kinetics parameters. These are coolant density, fuel temperature, and control. In the MCS model, the effects of coolant density and fuel temperature are represented by polynomial fits of the agional cross sections and kinetics parameters in terms of these two variables. The polynomial equation is, t uu
^-' (2~1)
Z =i=1j=1&=1 E E E c gxl-'x!-8x3 where,
' 2 = any 1-D neutronic parameter xi = p(t)-p(0) fractional change m. density p(0)
(2-2). 22= VT/t)-4Tf0), 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 coolant density and change in fuel temperature. This approach is standard in the industry. ,
.l l The MCS model introduces the effect of control via diffemnt sets of these polynomial fits, one set for each control state. During a transient calculation, the cross sections and kinetics f 1 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 control state of the
)- 1 e
i I e region. For transients experiencing a mactor scram, the axial control distribution in the - - RETRAN-02 calculation is based on all control rods moving at the same velocity until their . L ' physicallimitis reached. The two-group cross sections and kinetics parameters required by the 1-D RETRAN-02 model for each neutronic region are listed below.
- l. Total delayed neutron fraction,
, 2. . Group 1 absorption cross section, E.,i 2
- 3. ' Group.1 radial buckling, B i
- 4. ~ Group 1 diffusion coefficient,Di .
- 5. Group 1 down-scatter cross section,I,
- 6. Kappa times group 1 fission cross section,xI,3 f
- 7. Nu times group 1 fission cross section, VE f,3
- 8. Group 1 neutron velocity, Vi '
9.. Group 2 absorption cross section,I.,2 2
- 10. Group 2 radial buckling,B s- 11. Group 2 diffusion coefficient,D 2
, 12. Kappa times group 2 fission cross section,xE f,2
- 13. Nu times group 2 fission cross section, VI,2 f ]
1
- 14. Group 2 neutron velocity, V2 I Additionally, the model requires the relative yield fractions of each of the six delayed neutron
} 1 groups and their effective decay constants. RETRAN-02 assumes that these variables am spa-i tially and temporally constant. I l l l 1
3.0 ROLE OF SIMULATE 3 In our method, SIMULATE-3 is used to predict the 3-D neutronic response of the core to ) global penurbations. Additionally, SIMULATE-3 performs the spatial homogenization of the 3-D data to 1-D and core integral data requimd by RETRAN-02. The approach of using full 3-D solutions to changes from the 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-ments in operating BWR cores". SIMULATE-3 provides a practical means to evaluating the 3-D neut onic response to core penurbations.
> -- 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 captum the individual reactivity components. t
- SIMULATE-3 has the ability to isolate the effect of changes in control state. Thus, 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 penurbations are made from the initial state and are representative of the conditions encountered in the transient, the use of SIMULATE-3 eliminates arbitrary l specifications of changes in the 3-D density and fuel temperature distributions. I 1 i i j I I
)
) l
I j h j
. 3.1 Spatial Homogenintion From 3-D To 1-D Neutronics Data SIMULATE-3 produces all the 1-D and core integral kinetics data required by !
1 !' RETRAN-02. The details of the homogenization process used by SIMULATE-3 are described ' j l in Refemnce 7. Both RETRAN-02 and SLICK use the finite-difference method to solve the neu- 1 i
. tron diffusion equation. The 1-D cross sections and diffusion coefficients produced by J
' SIMULATE-3 preserve all planar reaction rates, planar interface currents, and the reactor l eigenvalue when solved with the SIMULATE-3 polynomial model. As long as the 1-D polyno- j mial and finite-diffemnce 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.
Figures 3.1 and 3.2 present a comparison of the SIMULATE-3 and SLICK axial power ; I 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 I regions to mock up the 150 inch active core and one mgion each for the bottom and top reflectors. This degree of nodalization is typically used in the SIMULATE-3 and RETRAN-02 l core models. The finite-difference solution uses 12 mesh intervals per region. As can be seen t fmm the figums, the axial power distributions are in excellent agreement. The calculated eigen- i values for these two cases are presented in Table 3.1. The difference in the calculated eigenva- f 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 , i as changes from the initial ARO control state. i 1 )
~' Table 3.1 -
SIMULATE-3 Versus SLICK Eigenvalues
. Control.. Difference
- State SIMULATE-3 SLICK ' (pcm)
ARO 1.001360 1.001362 0.2 ARI- 0.802620' O.802685- ' 6.5 , Halcl i 5
- Difference in pcm _= (k.a.sucx - k,a.sen) _* 10 i
i t t, )- I l l )
SOCK'vs.: SIMOLATE-3. Power -- ARO. Case
.P5 t 'i i f ! I 'I f I ie I e ! t t a l ! t 1 I I I I e 25 _ _
23 _ _
+ . .
- 21. _ _
19 _ 3 _ f 17 _ tr _
,co. 115 _ c) _
y . . o 13 _ c) _ of 11 _ . c) _ o . SIMULATE-3 .
- 9. .- o - SLICK -
t]- E , , 7_ _ i 5_ _ 3_ _ l y.,,.i.,,,i...., ,, i,,,,.i,,..
-4
.0 .2 .4 .6 .8 1.0 1.2 '
) Relative Power ? p Figure 3.1 SIMULATE.3 and SLICK Power Distributions at ARO L
-9 i.
)- E
t ,
,,j SLICK. vs. SIMULATE-3 Power -- ARI' 27 a . . . . . ...I . . . . I . . . .
'25 43 _
23 43 _
'21.43 _
19 q3 _ SIMULATE-3
, O-17 - o - SLICK -
c O' 15 _ _ 3 . e 13 _ _ m . . 11 _ _ 9_ _ f 7_ _ 5_ _ l-3_ - 1 c c, . . , . . . ,
.0 1.0 2.0 3.0 4.0
! Relative Power 1 1 F Figure 3.2 SIMULATE.3 and SLICK Power Distributions at ARI L 1 I ! _ _ _ _ _ _ _ _ - - _ _ - - - - - _ . - - - - - i
l l' , t i f: 1 uJ .
.4Al FINITE DIFFERENCE SOLUTION OF 1.D 2-GROUP DIFFUSION EQUATION i; . The power distributions used in the thermal-hydraulic routines in SLICK are based on a finite-diffemnce form of the 1-D two-group neutron diffusion equation. This is done to make the SLICK solution consistent with that of RETRAN-02, which also uses a finite-difference solution.-
i Additionally, the finite-difference solution is used in SLICK for verification calculations.- F The level of disentization employed in the modelis identical to that of the RETRAN-02 model. The com is modeled as a set of uniform material mgions with uniform mesh spacing within a region and mesh points at region interfaces. As is the case with RETRAN-02, the model can use zero flux or current boundary conditions. A comparison of the pov'er distrib-t
.utions as calculated by SLICK and RETRAN-02 for a typical Haling depletion is shown in hg-um 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-4 1 I l 1 l r. l l I
p.; $ . li[
~
>% SOCK vs.. RETRAN-02 Power --! ARO. '
+.
L
- - 2 7 v. . . i . . . .i.i
.i'. ..i....
25 _ _ 23 _ _ 21._ . l f .\ ; ,
, 19 _ - _
L- 17 _ o _ i c 15.- u- -
.o _
.c), '13 _
g _ 0: m 11 _ . a _
. o - - RETRAN-02 .
9_ . O . SLICK g _ 7_ 3 5_ _ 3_ _ 1 m m .s . . ,..,,, ...,....i.... .
.0 .2 .5 .7 1.0 1.2 j L Relative Power
!. l i I a ).. Figure 4.1 SLICK and RETRAN-02 Power Distributions { i u
-]
L i
-12 I y;
1 t_- f .' l
H 7
. 5.0 TRANSFORMATION OF THERMAL HYDRAULIC VARI ABLES 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, them is a need to transform the thermal-hydraulic conditions associated with the 3-D conditias 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 SIMULATE-3 model explicitly apresents 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. To accomplish this, the steady-state fonn of the RETRAN-02 models required to calculate the neutronic feedback variables have been included in SLICK. This involves the solution of the RETRAN-02 energy and momentum equations, the neutronic density calculation, and the fuel heat conduction equation. These models are described in Appendix A. p 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 r f L 1 l
fuel temperature distributions. For perturbation cases, SLICK similarly calculates the appropriate distribution while holding the other distribution constant. For normal state calcula-- 4
.I tions, SLICK performs consistent calculations for the density and fuel temperature distributions.
{ I As a demonstration that the SLICK steady-state equations are indeed the same as j RETRAN-02, we present a comparison of results for 100% power / flow conditions in Figures 5.1 through 5.3. The thermal-hydraulic model is 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 property 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-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 l i l' 1 I 1: [ '. l l: u 1 RETRAN-02 vs. SLICK Void Fraction
.8 i i e i l' i i i i i 3
~
o - RETRAN-02 l .6 - o - SLICK _. 1 c . _
.9 a
g . . 2 .4 _ _ 12_ - p
.,o - - -
o l
.2 _ _
.0 4 , , , , , , , , , ,
i 1 2 3 4 5 6 7 8 9 10 11 12 ! ) Core Volume i i l K Figum 5.1 Comparison of RETRAN-02 and SLICK Void Profiles I l i i
- t. j
)
I _=___-______. _ _ _ _ -
r 1 RETRAN--021vs. SLICK.Neutronic Density 50- i i i i .i i i - i i M. . o - RETRAN-02
~ ~
40 _ o - SLICK _ m g . .
.L N - -
E v B 30 _ _ y - -
- E) .
c a) . .. . 20 _ _ 3 10 i i i i i I I I I I 1 2 3 4 5 6 7 8 9 10 11 12 > Core Volume L j L Figure 5.2 Comparison of RETRAN-02 and SLICK Density Profiles I )
j a 1
. RETRAN--02 vs. SLICK Fuel Temper'oture-1200 i i i i i i i -i i i l
1100'_ _ 1
, C 1 1 g
E - . e t-- 1000 _ _
~6 . .
C . f . a - RETRAN-02 . o - SLICK 900 _ _ d o 800 l I I I I I i I I i 1 2 3 4 5 6 7 8 9 10 11 12 Core Heat Conductor ! Figure 5.3 Comparison of RETRAN-02 and SLICK Average Fuel Temperatures {. 1 1 L
f Comparison of Fractional Density Change
.05 i 'i' i i i i i i- i
.04' _ _
.03 _ _
y . -
- 07 . .
C
.o . .
.C . .
U .02 _ _ h . . _s . -
- C . .
O o . .
.01 _ _
~
C
.00 _ _
. o - RETRAN-02 3
. o - SLICK .
.01 i i 1 I I i I I I I 1 2 3 4 5 6 7 8 9 10 11 12 Core Volume Figmt 5.4 Comparison of RETRAN 02 and SLICK Fractional Density Change l
h , i
1 RETRAN-02 vs. SLICK Delta SQRT T-Fuel
,(
.50 i- i i i i i i i i i
. . i
- m m m -
.45 _ _
i 1 . 3 . . u_ .40 _ _ i g . . p . . OC . . 9 v) . .
~
o .35 _ _ 3 - O- RETRAN-02 - O g o - SLICK .
.30 _ _
[ .
.25 h
r I I I I I I I I I l F 1 2 3 4 5 6 7 8 9 10 11 12 Core Hect Conductor I Figure 5.5 Comparison of RETRAN-02 and SLICK Fuel Temperature Change i i
t.-
- s. ,
i- 6.0 FITTING OF 1 D KINETICS DATA TO RETRAN-02 POLYNOMIAL FORM h The polynomial equation supported by SLICK is, l t-u (6-1) z = E E cg'4~'
- 4. i, . i i
where, Z = any 1-D neutronic parameter (6-2) xi = p(t)- p(0) fractional change in density p(0) ) x2 = VTf (t)-VT f(0), change in fuel temperature SLICK uses standard least-square fitting techniques
- for determining the polynomial coeffi-cients. This mquires 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,5) 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. In practice,it has been found that fitting changes from the base cross sections with a set of basis functions not including the cross terms yields good msults. 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 1 __._______.______.m.____.___ _ _ _ _ _ _ - . _ . _ . _
' Cycle 9 com 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 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 during the important ponion of a turbine trip without bypass transient (TTWOBP). The set con-sists mostly of pressure and fuel temperature increase perturbations. These penurbations are considemd 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 penurbations is representative of what would be used in a reload licensing calculation. 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 Iqu 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 ofIqnam 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 pmsented above is pan of the normal calculation in SLICK. The statis-tical data is of great value in looking at tmnds from cycle to cycle (note the similarities between l the Cycle 9 and Cycle 13 EOFPL data). However, this type of comparison is not the only basis i l for confidence in the fitted data. Independent verification calculations using the fitted kinetics l l 1 I I
.g o o
p , :<
. data can be performed'and compared to a 3-D calculation. These provide a more meaningful.
way of evaluating the kinetics data. Verification calculations associated with these data sets are presentedin the following section. i-1 9 2 l l
\b '
~
Table 6.1 , Perturbation r' aces Used in Generation of 1-D Kinetics Data Perturbation - Perturbation Quig Tyne Maaninwie 1 Base Case * ---- - 2 FuelTemperature +100.F 3 FuelTemperature +200.F 4 Pmssure +50. psi - 5 Pmssure +100, psi 6 Pressure +200. psi 7- Pressure +300. psi 8 Pmssure +400. psi 9' Pressure -75. psi 10 Pmssure -150. psi 11 Inlet Temperature +9.F 12 Inlet Temperatum ~ +18.F-E0%
- Time zero power, flow, pressure, and inlet temperature.
l l p s t l l
7 @ n ! Table 6.2 - l2 Comnarison of 1-D Neutronics Calculations with Fitted Data to 3-D Perturbation r' aces
- . Eigenvalue Difference E ror (%Y
. . Control Standard 1-D Kinetics Data Set For: STATE Mean Deviation Cycle 9 EOFPL Initial +1.6 2.9 i 1
ARI +2.1 12.9 Cycle 9 EOFPL - 2 GWD/ST Initial- +2.2 4.2 ARI- +2.7 13.7-Cycle 13 EOFPL Initial +1.5 2.5 ARI- +0.9 9.1 HQ1C
- Eigenvalue Difference Error =
## ~"" ~ *~"
# .5-D 4
1 l l l l l
l
\ '
7.0 VERIFICATION The ideal goal of the method is that any perturbation should yield the same 1-D and global l: 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 cata and the RETRAN-02 thermal-hydraulic model. Two types of comparisons are made below. One set of comparisons is made to SIMULATE-3 state calculat'ans. 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-tions. These calculations are performed with all variables other than control held fixed at the initial base state condition. l The ability to perform the above type of comparisons has been incorporated into SLICK. The state calculation requires the coupled solution of the neutron diffusion equation and the RETRAN-02 thermal-hydraulic model. Since the solutions of the neutron diffusion equation and the thermal-hydrauhc equations in SLICK are effectively the same as the RETRAN-02 solutions, the results 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 l !
1 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-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 method. i 7.1 Comparisons of State Calculations To assess the method, two types of state calculations are presented. The first type involves changes from tl.: initial critical state that result in a critical final state. The second type of ', tate i calculation involves changes to the initial state that result in a final state far from critical. 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 wen: 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 f
1 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 perturbation. The degne to which 1 1 the SLICK and SIMULATE-3 solutions of the final state differ is a measure of the reactivity error in the kinetics data. The results of the comparisons are provided in Table 7.1. The errors in the prediction of the final state eigenvalue range from +2 to -12 percent milli-k (pcm). This is considend very good agmement. To put this into perspective, consider that a difference of 12 pcm in reactivity may be accounted for by a difference of 0.2% in rated power. Since the change in power for the cases was 5% rated, a 0.2% difference corresponds to a maximum relative error of 4% in the power change. The final state axial power distributions for the Cycle 13 cases are presented in i Figures 7.1 through 7.3. The axial power distributions are in good agreement with the maximum i error being about 3%. The agreement of the power distributions for the other cases are quite similar to these. To further assess the accuracy of the method for generating 1-D kinetics data, comparisons are made to a state calculation far from critical. The state calculation chosen is representative of I the core conditions at the time of peak reactivity 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 1.5 ft. insertion of all control rods into the core. Comparisons are made at EOFPL for Cycles 9 and 13. The results of tlm comparisons are provided in Table 7.2. The error in the final eigenva- q lues is about 30 pcm. The SLICK results overpredict the change in eigenvalue. The increase l l- 4 l 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-tributions for the Cycle 13 case is show in Figure 7.4. The axial power distributions are in good l agreement with the maximum error being about 5%. l l [
I 7.2 Static Scram Wonh Comparisons Static scram worth comparisons are made at Cycle 9 EOFPL and EOFPL - 2 GWD/ST conditions. The control rods are fully withdrawn in the EOFPL case. The EOFPL - 2 case is j initially rodded. Figures 7.5 and 7.9 pasent the scram worth as a function of distance for the first six feet of rod insertion. The 1-D results underpredict the 3-D scram worth in both cases. ) i The EOFPL has a maximum underprediction of 3%, while the EOFPL - 2 GWD/ST underpre- l dicts the worth by about 6% The axial power distributions with control rod moves of 2,4, and 6 feet am shown in Figures 7.t' through 7.8 for the EOFPL case and Figures 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.
k ) 1 1
-Table 7.1 t-Differences Between 3-D and 1-D Eigenvalues for Critical State Calculations Perturbation Errorin Final Eigenvalue (pcmf ,
Type Macnitude EOFPL9 EOFPL9-2 EOFPL13 Pressure +30. psi -5.5 -5.4 -6.2 l Inlet T -4.8 F -0.6 -7.7 -9.0 Inlet Flow +7.% +1.8 -12.0 -10.7 N_01C
- Difference in pcm = (k.n.sucg - ke rossutan)
- 10'
(. d l [ L
Table 7.2 Differences Between 3-D and 1-D Eigenvalues for TTWOBP State Calculation Change in Eigenvalue (pcm)* (Final - Initial) Error in Eigenvalue Change Absolute Relative Condition 3. D. .1-D_ (pem) (%) Cycle 9 EOFPL 441 472 +31. +7.1 Cycle 13 EOFPL 444 473 +29. +6.6 EDic; 5
- Difference in pcm = (km.m.i- km.wu)
- 10 l
I t l' l 1 \
t' ~ Cycle 13 EOFPL 30 PSI Increase 27 & . . . . I . . . . ' . . . . 25 _ _ 23 _ _ 21 _ _ 19 _ _ 17 _ . _ c 15 _ - o 5 . - e 13 _ _ o' . r 11 _ . o - SIMULATE-3 . 9_ o - SLICK _ ) . 7_ _ l . 5_ _ 3_ _ 1 . . . , y . . . ,
.o .5 1.0 1.5 Relative Power I
1 1 Figure 7.1 3-D and 1-D Axial Powers for +30 psi Pressure Chance \ ( . I
Cycle 13 EOFPL 4.8 F Inlet T Decrease 27 &, . . . i . . . . i . . . . 25 _ _ 23 _ _ 21 _ _ 19 _ _ 17 _ _ c 15 _ _
.9 -
, cn - c) 13 _ . _ Cr . 11 _ _ o - SIMULATE-3 . 9_ o - SLICK _ ) 7_ _ 5_ . 3_ _ 1 y . . . . , . . . ,
.0 .5 1.0 1.5 i Relative Power i
l Figure 7.2 3-D and 1.D Axial Powers for -4.8 F Inlet Temperature Change Cycle 13 EOFPL 7% Flow increase 27 & . . . . * . . . . t - . . . 25 _ _ 23 _ _ 21 _ _ 19 _ _ 17 _ _ c 15 _ _
. 9 m - ~
3 o 13 _ _ Cr 11 4 0 - o - SIMULATE-3 - 9_ o - SLICK g _ 7_ j _ 5_ _ 3_ _ 1 y . . . .
.0 .5 1.0 1.5 Relative Power i
I Figure 7.3 3-D and 1-D Axial Powers for +7% Inlet Flow Change l
) -....m..... ... . . . ..
Cycle 13 EOFPL +70 psi +5% P Rods 1.5' 27 &....'....n . . . - ' . . . . 25 _ _ 23 _ _ 21 _ _ 19 _ _ 17 _ _ c 15 _ _ o 5 . - e 13 _ _ Cr - 11 _ 3 o - SIMULATE-3 . 9 o - SLICK 7_ _ 5_ _ 3_ _ 1 h... ,....,....i.... 1.0 1.5 2.0
.0 .5 Relative Power Figure 7.4 3-D and 1-D Axial Powers for TTWOBP Conditions l
t 1
SCRAM Worth Comparison at EOFPL
.0 & . I . I - I - I - I .
\ .
.5 _ _
S v . . E - - 8 3: . -
- 1.0 _ _
l
~
0 - SIMULATE-3 o - SLICK ;
. ] . j
, - 1.5 , O 1 2 3 4 5 6 l Rod Move (ft.) l ) Figure 7.5 3-D and 1-D Static Scram Worth at EOFPL l !
Relative Powers After 2 ft. Rod Move 27 '....I 25 _ - 23 _ - 21 _ _ 19 _ 3 17 _ _ c o 15 _ _
~ ~ ~
g . a) 13 _ - Cc 11 _ - . 9_ - i - - 7_ - 5_ -
~
O- SIMULATE-3 . 3- o - SLICK 1 b, ,,,,,,.,....,....
.0 .5 1.0 1.5 2.0 l Relative Power l
) Figure 7.6 3-D and 1-D Axial Powers After 2 Foot Rod Move mOFPL) 1 \
Relative Powers After 4 ft. Rod Move 27 p . . . t ... 1 ....I ....'.... 25 _ _ 23 _ _ 21 _ _ 19 _ _ 17 _ _ c 15 _ _
.9 y . .
e 13 _ _ o' . . 11 _ 9 _ 7_ r _ 5m _ O- SIMULATE-3 3 -El o - SLICK - 1 g....,.... .... . .. ....
.0 .5 1.0 1.5 2.0 2.5 Relative Power I
I Figure 7.7 3.D and 1.D Axial Powers After 4 Foot Rod Move MOFPL)
)
Relative Powers After 6 ft. Rod Move 27 t....'....t ....n....'....'.... 25 _ _ 23 _ _ 21 _ _ 19 _ _ 17 _ _ c 15 _ _
.9 e 13 _ _
Cr 11 _ _ 9 4 - 7 43 _ 5 +3 - O- SIMULATE-3 3 42 o - SLICK 1 p....,,,,,, ..,,....,....i....
.0 .5 1.0 1.5 2.0 2.5 3.0 l Relative Power l
Figure 7.8 3-D and 1-D Axial Powers After 6 Foot Rod Move mOFPL) r._.___ l
i SCRAM Worth'at EOFPL-2 GWD/ST
.0 -r - i - i . I . I - i -
.5 _ _
- 1.0 _ _
m m c g . - o _ - 3 - 1.5 _ _ 5
-2.0 _ _
~
]
~
O - SIMULATE-3 - o - SLICK
-2.5 ,
0 1 2 3 4 5 6 .* l Rod Move (ft.) 1 ) Figure 7.9 3-D and 1-D Static Scram Worth at EOFPL - 2 GWD/ST
Relative Powers After 2 ft. Rod Move 27 p....I . . . . I - . . . ' . . . . 25 _ _ 23 _ _ 21 _ _ 19 _ j _ 17 _ _ c 15 _ o o 13 _ _ Cr 11 _ 9_ _ 7_ _ 5_ _ o - SIMULATE-3
~
3- o - SLICK 1
-y.,..,..,,,... ,....
.0 .5 1.0 1.5 2.0 Relative Power Figure 7.10 3 D and 1-D Axial Powers After 2 Foot Rod Move MOFPL - 2) l !
Relative Powers After 4 ft. Rod Move 27 &....'....'....I ...I ... 25 _ _ l' 23 _ _ 21 _ 3 - 19 _ _ 17 _ _ c 15 _ _
.9 I w - -
l e 13 _ _ 1 t T . 11 _ _ l . l 9_ _ 7_ _ 5m o - SIMULATE-3
~
3 43 o - SLICK 1 y,.. ,,...,...,,...,,....
.0 .5 1.0 1.5 2.0 2.5 l
) Relative Power l f Figure 7.11 3-D and 1-D Axial Powers After 4 Foot Rod Move OFPL - 2)
Relative Powers After 6 ft. Rod Move 27 .+..,.'....'..> . 25 _ _ 23 _ _ 21 _ _ 19 _ _ 17 _ _ c o 15 _ _ 5 - - e 13 _ _ - Cr - 11 _ _ 9 4 -
~
7 43 - 5 .e - o - SIMULATE-3
~
3 -O o - SLICK 1
-y,...,... ,... ,....,....i....
.0 . 5 1.0 1.5 2.0 2.5 3.0 Relative Power -
Figure 7.12 3-D and 1-D Axial Powers After 6 Foot Rod Move mOFPL - 2) l - 4
- _ _ _ _ - _--___-_____._____-__.__________m_-____________-m_ _ _ _ _ _ _ _ _ . . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ - _ _ _ _ _ _ -
8.0 REFERENCE 5;
- 1. EPRI, RETRAN-02 -- A Procram for Transient Thermal-Hydraulic Analysis of Comph Fluid Flow Systems. EPRI NP-1850-CCM-A (November,1988).
- 2. J. M. Holzer, Methods for the Analysis of Boiline Water Reactors: Transient Core Physics, YAEC-1239P (August,1981).
- 3. Letter to Mr. Hugh G. Parris, Manager of Power Tennessee Valley Authority from Mr.
Domenic B. Vassallo, Chief Operating Reactors Branch #2 Division of Licensing, "TVA l RETRAN TOPICAL REPORT," Docket Nos. 50-259 50-260 50-296, (April 7,1983).
- 4. D. M. VerPlanck, K. S. Smith, and J. A. Umbarger, SIMULATE-3P: Advanced Three-Dimensional Two Groun Reactor Analysis Code, Studsvik/SOA - 88/01 (February,1988).
- 5. A. S. DiGiovine, J. P. Gorski, and M. A. Tremblay, SIhilJLATE-3 Validation and Verifi-cation, YAEC-1659 (September,1988).
- 6. R. A. Woehlke, MICBURN-3/CASMO-3/ FABLES-3/ SIMULATE-3 Benchmarking of Vermont Yankee Cycles 9 Throuch 13, YAEC-1683 (Marth,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).
APPENDIX A SLICK THERMAL-HYDRAULIC MODELS A.
1.0 INTRODUCTION
The purpose of this appendix is to describe the thermal-hydraulic models used in SLICK. The aim of the SLICK themial-hydraulic calculation is to reproduce the results of RETRAN-02 W for any given set ofinitial conditions. The modeling encompasses the solution of the fluid mass, 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 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 Equation For the case of control volume k of height zg with a single inletjunction i-1 and outlet junction i, we write, f y2 T r y2 Qa + IV,,i 1 h, + 1 - W,.i h, + 1 ( 2 ji. ( 2 ,e (A-1) r r T y2 3 y2
+ Wi,; .1 hi + '- -W hi + '- - Wg r = 0 i,i s
\ Ji-1 % di l
Solving the above for W,,i yields, \ A-1 1
I r 4> r q Qn-W g r, + hi .s + f + W u.1-1 h, + 3q + Wi.i -1 hi +3,_, 1 W,,, - yy yp (A-2) ; h, - hi + 3 3, l
)
where we have used the relation,- I l W ,,, = W - W ,,, (A-3) j in arriving at the above form. <i Note that the right hand side of the energy equa: ion has terms dependent on W,,s. Namely, V,,, and V,i, i i w,,, i 8 ' " e,,,dA l (A-4) j w-w I.i pg,,4,0 ) l 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. l SlJCK suppons 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 Itcasting 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, ) J ! xf (A-5) l a =Cfxf +f(1 .ry)+ p,h A-2 l l 1 f \ \ i i i
u. s j
- c. ,r s
.p 1
l,. .
; where the flow quality,xf,is defined by, I
W,.
.e
.x,= W (A-6) and C, and Vg are the drift flux parameters. The void model of Lellouche and Zolotar*is used 4
to evaluate the drift flux parameters for the algebraic slip model. For the HEM model, the drift flux parameters reduce to, 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 as 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 1
relating the void fraction to the RETRAN-02 slip 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) V' = (1 - a)a where, W-W, u; = PA (A-10) w, l u'= p,A- ' I l l A-3 l t
Solving the above for the void fraction and taking the appropriate root yields, I (u, + u,) 44V,u, + (V, - u, - u,)2 (A-11) )
""2
~
2V, 2V, In evaluating volume properties, the above relation along with the definitions of the superficial 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 are two cases to consider. These are single phase and two phase out-let conditions. For the single phase in, two phase out case, equation A-2 reduces to, 2% Da -\V g z, + h,,; +-h.,-h,2 i M,,,i = r v2 q' (A-12)
;h, - h, + 3 - ,,
For the single phase in, single phase out case, equation A-1 is solved for the outlet enthalpy, T vf (A-13) h = C, + rh + y2 - g z, -- i g 2 ,;., ~ and equation A-4 reduces to, Mr (A-14) V'.= p(P;,h;)A; A.2.2 Momentum Ecuation - 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, i L L A-4 l l
' W* - -
- W,2. , . a Pa + s = Pi + .
<PsAE ' Pa+ NAE +1,
'a,uip,p,' ' W, W,
+
~
r pA* ' ,, gu,p aip,,,
'a,aipap,' ' ' ' W, . W, .
( -PA' s4si tu,P, a,p,,, 3 - 4* 5 4* h W,'
/de,i (D*f.iPe.a '+ Day.A + ips.a 24,2 + 3, 2 %r s-l'We + y,2 3 3
+2(a,9, apij,gA?+1s An',
I W[ e
. I 1 r
2(pA ); , 5pizig. 5p,.3 ,.3g 2 To express the momentum equation in the desired form, we make use of the following relations, 1 which may be derived from the definition of the slip velocity and continuity coaditions:
' W, W, ' (A-16)
-A,V,',
r u, P, aPiss (A-17).- g';Wa,p,-VAa,p,aip, P Waip, + VA a,p,afp, W, _ E (A-18) ' where, p = a,p, + a,p, (A-19) - Substituting the above into equation A-15, yields the desired fonn of the momentum equation, I lL ( A-5
I i 1 1 t 2 Ps + n = Ps 5Pa a8 ' 5Pa +1 s+ n8 1 ri/2 23/2 'W2 4 l
- 4f,$ Ds,,npi,1+ Ds,,n + 1ps,s . ,,
< 2Af )
1
. W*' Vla,uip,p;A' .
4 (A-20) '
. pA' <9 W ,.s i
2Y I W*' Vla,aip,p;A 1 W? 1 --K
.pA's + W- s.4+, 2 p,Al 1 W?'1 V+ ,a,a p,p;A' '1 s -
1*
+ _2 pi ( W 1
,, (A *+ , Al, Note that if the slip velocity is zero, the momentum equation reduces to the HEM momentum equation (equation 11.3-26 of Reference 1).
The momentum equation contains the Fanning friction factor,f , and the two-phase fric-tion multiplier, Q,. 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 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 pressum at ajunc-tion based on the pmssure in the donor volume corrected for elevation and wall friction pressure drop in the donor volume. 1 r/2 3 wf (A-21) (p,,p,4,,y Pi = P4 5 P4 42 -4/ ,4&r,4 l l A-6 1 1 h 1
A.2.3 Qupled Solution of Enerev and Momentum Equations The energy and momentum equation are coupled. The momentum equation provides the I pressum distribution of the 1-D core model. The pressure distribution is used in the evaluation of the thennodynamic and transport properties required by the energy equation and void model. To start off the solution, the initial pressum distribution is assumed uniform. The control volume energy equation is evaluated starting at the center of the bottom reflector and proceeding up the core. After the thermodynamic states of all volumes andjunctions an: known, the momentum equation is evaluated for the pressure distribution. This iteration is repeated until the pressure and density distributions converge. The SIMULATE-3 case provides the needed boundary conditions for the system of equa-tions. These am 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 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, ! I
< v x, -xa 1 - tanh 11 xf =
7 1-xa.1 -tanh s
)
1 [f (A-22) J l l i 1
\
1 A-7 i > 1 l l
The departure point quality,x,,, is determined using the model of Lellouche and Zolotar. RETRAN-02 departs from the conventional calculation ofx, in two phase control volumes. RETRAN-02 defines x, as, h-hf (A-23) x* = hi, where h is the con:rol volume enthalpy. For a two-phase volume, h =ap,h, + (1 - a)p,h, (A-24) 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 l 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 void fraction pre-dicted by the thermal-hydraulic equations. The smoothing relations are, C; = % +(1 -y) (A-25) Vg= yVg where, x y=1-lx,',I (0 < x, < lx,,l) (A-26) y=1 (x, s 0) Forx, 2 lx,,l, the neutronic density is equal to the thermal-hydraulic density. I t A-8 I
AA.0 FUEL HEAT CONDUCTION RETRAN-02 uses a finite-difference scheme for solving the conduction equation. Our goal is to solve the RETRAN-based finite-difference equation for the case of a cylind-ical fuel rod comprising thme materials: fuel, gap, and clad. The one-dimensional heat conduction equa-tion for the system is, 1d' .
= -q (r)
(A-27) r dr . k(T)rdr.dT' 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 consi6 red known. Thus, finite-difference equations am requimd for the interior and centur-line nodes. For an inte-rior node i, we define, r, a radial distance to node i from center-line. h,,, a r, - r, . 3 N,,, 5 rg 4 3- r; h a ,, ! r.,, e r, T (A-28) h,,, r,,, e r, + -
~ ~
q ,, e the value of q (r) between r, 3 and r,.
~ ~
q , a the value of q (r) between r, and r,.3 Integrating the conduction equation over an interior control volume bounded by r.,, and r,,, i and using a central-difference formula to approximate the temperature gradient yields, j l, t 1 l A-9 : I l l t
a;T;_ i + b,T, + c;T;. i = S; (A-29) where, k,,,r,,; a; = h ,,,- k,,,r a,;
" h,,s b, = -(a; + c;)
(A-30) S; =
'' ~ ~
q, ;
q,~,,;
< 2 , < ~2 ,
k.,; = k(T)lr , .r . .r, k,,, = k(T))7 ,r.7, . i Similarly, integrating the conduction equation over the control volume associated with the center-line node and noting that the temperature gradient is zero at the center-line, we arrive at the finite-difference equation for the center-line node, b T + c3T2 = Si i (A-31) f where, ! k,,ir,,, i c = h ,,i ; b = -ci i (A-32)
~
S=- i q, i 1 1 The equations for the center-line and interior nodes along with the specification of the 1 surface temperature form a tridiagonal system of algebraic equations. The equations are I nonlinear because of the temperature dependence of thermal conductivity. However, the system 1 l I A-10 I i ____________A
3 ;,, of equations is solved using a standard linear equation algorithm
- by evaluating thermal
- conductivity with % assumed temperature distribution. To start off, the temperature distribution
, : is assumed to be uniform All coefficients of the finite-difference equations are evaluated and .
the linear tridiagonal system is solved. This procns is then repeated with the coefficients being calculated from the latest temperature distribution until convergence of the temperature distribu-tion is achieved. 1 A.4.1 Clad Surface Temocrature f
' As mentioned above, the temperature at the surface of the clad is determined from the con-vective boundary condition, '
T,,,= + Ti where, i
~
q m surface heat flux. h, a heat transfer coefficient. 7 3m volume bulk fluid temperatme. For single-phase conditions in the associated control voltme, the heat transfer is based either on forced convection or subcooled nucleate boiling. For forced convection, the Dittus-1
- Boelter correlation is used, h, = 0.023 ,
k 'GDu* 'c, ,* (A-34) , Du pi , k, with properties evaluated at Te. l l l A-11 l l
-- - _ - - _ - - _ - _ _ _ _ _ _ - _ - _ _ _ - - - - _ _ _ = _ _ _ _ _ _ _ _ _ _ - - - - _
For subcooled nucleate boiling, the Thom correlation is used to predict the clad surface temperature, T,,, = T,,(p) + 0.072e v eg (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 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 cormlation attributed to Schrock and Grossman is used, f w5 (A-36) k I h, = 0.0575 , [Re,(1 -x)]"Pr," D,, gx, a where the Martinelli parameter is given by, r y 3ep, ws rp,m (A-37) x, g 1 -x, (p,, p, , For volumes with void fractions between 0.8 and 0.9, the surface temperature is based on a weighted average of the nucleate boiling and forced convection vaporization heat transfer heat fluxes. In this range, RETRAN-02 assumes that
- (A-38) 0= (a0.1 - 0.8) -#*"' + 0.1 (0.9##~- a) -
The appropriate correlations and relations am substituted into the above relation and the resulting equation is solved for the surface temperatum. A-12 _-- __ i
f .. [ f!" ,< l
- AJ.0' REFERENCES
- 1. '. EPRI, RETRAN-02 -- A Program for Trancient Thermni-Hvdraulic Analysis of('nmnlex Fluid Flow Systems. EPRI NP-1850-CCM-A (November,1988).
2.- EPRI, Mechanistic Model for Predictina Two-Phase Void Fraction for Water in Vertical - Tubes. Channels and Rod Bundles. EPRI NP-2246-SR (February,1982).
- 3. ' William H. Press et al., Numerical Recines: The Art of Scientific Comnuting (New York:
Cambridge University Press,1986).. A-13 - i i
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