skinath: a computer program for solving the reactor point
TRANSCRIPT
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S K I N A T H - A Computer Program f o r Solv ing t h e Reactor Po in t
K i n e t i c s Equat ions wi th Simple Thermal-Hydraulic Feedback
H. L. Dodds, Jr. Nuclear Engineering Department
The Unive r s i ty of Tennessee Knoxville, Tennessee 37916
and
R. M. Westfall COMPUTER SERVICES
a t Oak Ridge Na t iona l Labora tory
Post Office Box x Oak Ridge, Tennessee
Date P u b l i s h e d - S e p t e m b e r 1984
Martin Marietta Energy Systems, Inc. o p e r a t i n g the
Oak Ridge Na t iona l Labora tory Oak Ridge Gaseous D i f f u s i o n P l a n t
O a k Ridge Y-12 P l a n t Paducah Gaseous Di f fus ion P l a n t
under Con t rac t No. BE-AC05-840R21400 f a r t h e
DEPARTMENT OF ENERGY
T a B L E OF CONTENTS
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ACKNOWLEDGMENT. . . . . . . . . . . . . . . . . . . . . . v
A B S T R A C T . . . e . e . V i i
1 . I N T R O D U C T I O N . . . . . . . . . . . . . . . . . . . . . . . . 1
2. T H E O R Y . . a . . * 1
3 . RESULTS AND CONCLUSIONS . . . . . . . . . . . , . e . . . . 3
4 . RECOMMENDATIONS FOR FUTURE WORK . . . . . . . . . . , . . . 7
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . 8
APPENDIX A . . . . . . , . . . . . . . . . . . . * . * . 9
APPENDIX € 3 . . . . . . . . . . . . . . . . . . . . . a . . 15
iii
ACKNOWLEDGMENT
This work was supported by the Oak Ridge Gaseous Diffusion Plant a s p a r t of a m u l t i s f t e s tudy on t h e p l a u s i b i l i t y and consequences of low- power c r i t i c a l i t y e v e n t s . The authors e x p r e s s t h e i r a p p r e c i a t i o n to D. J. Hamby for t h e preparat ion of the manuscript .
V
ABSTRACT
A FORTRAN computer program named SKINATH has been developed for s o l v i n g the r e a c t o r p o i n t k i n e t i c s equa t ions with s i m p l e thermal- h y d r a u l i c feedback. The program u t i l i z e s t h e o r d i n a r y d i f f e r e n t i a l equa t ion s o l v e r , LSODE, developed by Hindmarsh a t the Lawrence Livermore Nat iona l Laborat,ory. Based on comparisons w i t h bo th a n a l y t i c a l and o t h e r numerical s o l u t i o n techniques , the program g i v e s a c c u r a t e r e s u l t s w i th impress ive ly s h o r t r u n times.
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1 . I N T R O D U C T I O N
SKXNATH aimpic t i c s and Xhermal-&ydraulics, is a FORTRAN computer program developed f o r t h e IBM 3033 (double p r e c i s i o n ) f o r s o l v i n g t h e reactor p o i n t k i n e t i c s e q u a t i o n s ' coupled wi th a s imple thermal-hydraulic feedback model. SKINATW c o n s i s t s of a main r o u t i n e which e s s e n t i a l l y sets up the i n p u t for c a l l S n g LSODE, a n o r d i n a r y d i f f e r e n t i a l equa t ion s o l v e r developed by Hindmarsh3 a t t h e Lawrence Livermore Nat iona l Laboratory, and one s u b r o u t i n e which i s r e q u i r e d by LSODE. The e n t i r e program c o n s i s t s o f about one hundred FORTRAN s t a t emen t s , exc lud ing LSODE, and u s u a l l y e x e c u t e s i n less than two CPU seconds (IBM 3033) for most problems of i n t e r e s t .
S e c t i o n 2 of t h i s document p r e s e n t s a brief d i s c u s s i a n o f the equa t ions which are so lved and P few comments about t h e methodology employed i n LSODE.
I n S e c t i o n 3 r r e s u l t s ob ta ined wi th S K I N A T H are compared wi th r e s u l t s ob ta ined a n a l y t i c a l l y and with other numerical r e s u l t s .
A listing of t h e program is presented i n Appendix A which inc ludes the i n p u t for a sample problem. The ou tpu t f o r t h i s sample problem is presented i n Appendix B.
2. THEORY
SKINATW s o l v e s the fo l lowing equa t ions :
1 . 2 5 - - dT I E P ( t ) - AHD.75 [ T ( t ) - Tc] /T(t)*25) dt C - - t P
dCi B i W - - - d t A - hiCi(t), i = 1, ....., M ( 3 )
Equation (1) r e p r e s e n t s a n energy ba lance f o r a r e a c t o r I n t he shape of a long h o r i z o n t a l c y l i n d e r which is cooled e x t e r n a l l y by an idea l gas v i a n a t u r a l convection.2 T is the reactor tempera ture , Cr, i s t h e heat c a p a c i t y of t h e r e a c t o r , H and D are t h e r e a c t o r height and diameter, r e s p e c t i v e l y , Tc is the c o o l a n t tempera ture (which is cons tan t w i t h r e s p e c t to time, t ) , and A is a c o n s t a n t determined by
2
where K i s t he c o o l a n t thermal conduc t iv i ty (J/m-sec-cc), i s t h e coo lan t is the coolant v i s c o s i t y (~g/m-sec) , and eP is the C O O I ~ ~ L ~ p o ~ c i r ^ i c heat. (J/Kg-"C). The va lue af A (17.52) l i s t e d in t h e sample problem I n Appendix A i s for air a t s tandard temperatuse and pressure . The E i n Equat ion ( 1 ) i s t h e f r a c t i o n of t he energy from f i s s i o n depos i t ed as heat i n the! system.
pc 2 d e n s i t y (Kg/m3), g is t h e g r a v i t a t i o n a l c ~ n s t a n t (mlsec. ),
Equat ions ( 2 ) and (3) are t h e s t anda rd p o i n t k i n e t i c s equa t ions where P i s the neut ron d e n s i t y (neutrons/cm3), which is p ropor t iona l t o the reactor power, p is t h e system r e a c t i v i t y ( d o l l a r s ) , A i s t he system generatj-on t i m e : (seconds) and Xi, C i , B i ii1-e t h e decay cons tan t ( s e c o n ~ s ) , precur so r concen t r a t ion ( n u c ~ - e i / c m ~ 1, and delayed neut ron fraction, r e s p e c t i v e l y , for pi-ecursor group i. The number of delayed
p recu r so r groups is M and B Pi. S i s an ex t raneous neutron
SQU~TB neutrons/cm3) OP known magnitude.
M
i=l
The system r e a c t i v i t y , p, c o n s i s t s o f an e x t e r n a l d r i v i n g functioPi which may be ei ther a s t e p , linear r m p , or a q u a d r a t i c i n time p l u s a l i n e a r feedback component as shown below:
where t is i n S ~ C Q ~ ~ S and the r e a c t i v i t y feedback c a e f f i c i e n t , R'HOCOF, has UnitS Of dollars De?." " c .
Most oi' t h e a p p l i c a t i o n s cons idered t h u s fa r have used s i x (M = 5 ) delayed neutron p recu r so r groups 90 that eight equa t ions are being solved by LSODE. The energy balance equa t ion i s t h e first equation fa l lowed by t h e neut ron ba lance equation and t hen t h e s i x precursor ba lance equations.
LSODE s o l v e s i n i t i a l va lue problems for e i t h e r stiff or r im-s t i f f s y s t e m of first o rde r ordinary d i f f e r e n t i a l equat ions, either l i n e a r or non-linear. I n t e g r a t i o n time s t e p sizes m e selected au tomat i ca l ly by LSODh". accord ing t o t o l e r a n c e parameters s p e c i f i e d by the user which con- t.rsl the accuracy of the r e s u l t s . M O D E r e q u i r e s the Jacobian matrix i n some form which may be e i the r user supp l i ed via an a d d i t i o n a l u 3 e ~ w r i t t e n sub rou t ine OF LSQDE w i l l compute i t i n t e r n a l l y us ing d i f f e r e n c e
3
q u o t i e n t s . SKIMATH also assumes tha t t he i n i t i a l c o n d i t i o n o f t h e r e a c t o r is s t e a d y state delayed cs i t ical w i t h no thermal-hydraulic feedback, I n o t h e r words, the feedback i s not i n equ i l ib r ium w i t h t h e n e u t r o n i c s a t time zero. The u s e r s p e c i f i e s t he i n i t i a l neut ron d e n s i t y ( o r power), t he coo lan t tempera ture (which is set equa l t o t h e i n i t i a l r e a c t o r tempera tuse) , the e d i t times (i.e., when r e s u l t s are desired), and v a l u e s f o r the o t b e r c o n s t a n t s appear ing i n Eqs. ( 1 ) 4 4 ) which model t h e system o f i n t e r e s t ,
3 . RESULTS AND CONCLUSIONS
Eva lua t ion of SKINATH has been accomplished by comparing r e s u l t s ob ta ined w i t h S K I N A T H w i t h a n a l y t i c a l r e s u l t s and wi th o t h e r numerical r e s u l t s .
The first problem cons idered i n t h e e v a l u a t i o n is a problem tha t can be so lved a n a l y t i c a l l y . S p e c i f i c a l l y . a s t e p change i n r e a c t i v i t y (7 c e n t s ) w i t h 0ne delayed neu t ron group (M = 1, h l = 0.1, B 1 = 0.0065) and no thermal -hydraul ic feedbaok (RHOCOF = 0.0). The i n i t i a l r e l a t i v e power is 1 .O and A = 5.OE-05 see.
1
The numerical e v a l u a t i o n o f the a n a l y t i c a l s o l u t i o n f o r t h i s problem is shown i n Table I along with SKINATH r e s u l t s and also r e s u l t s ob ta ined v i a a f o u r t h o r d e r Runge-Kutta p r ~ c e d u r e . ~ The Runge-Kutta s o l u t i o n employed 0.01 sec time s t e p s and r e q u i r e d approximately 8 CPU seconds on t h e IBM 3033 for the 15 minute t r a n s i e n t . The S K I N A T H s o l u t i o n r e q u i r e d on ly 0.12 CPU seconds f o r 15 minutes of t r a n s i e n t time. Thus, i t is concluded t h a t SKINATH produces a c c u r a t e r e s u l t s w i t h ve ry s h o r t r u n times for s imple k i n e t i c s problems wi th no thermal- hydrau l i c feedback.
The nex t problem cons idered i n t h e e v a l u a t i o n of SKINATH is a step change i n r e a c t i v i t y ( 4 . 3 c e n t s ) wi th s i x delayed neut ron groups and thermal-hydraulic feedback p r e s e n t (kWQCOF = -0.306 c e n t s pe r degree Ce lc ius ) . The i n i t i a l r e l a t i v e power and t h e i n i t i a l r e a c t o r tem- p e r a t u r e are 0.01 watts and f o r t h i s sample problem are i s presented i n Appendix B. so lved i n Reference 2 which t h e power, (2) i gnores t h e and ( 3 ) u s e s E u l e r ' s method
20 *C, r e s p e c t i v e l y . Other inpu t parameters presented i n Appendix A. The problem ou tpu t
T h i s problem was o r i g i n a l l y de f ined and ( 1 ) assumes a pure exponen t i a l behavior f o r prompt neut ron c o n t r i b u t i o n t o r e a c t i v i t y , f o r the time i n t e g r a t i o n , A f o u r t h o r d e r
Runge-Kutta procedure was also used to o b t a i n a t h i r d s o l u t i o n t h i s problem ( w i t h A t = 0.01 sec). P l o t s of the power v e r s u s time obta ined by each of these three s o l u t i o n approaches are presented i n F igu re 1 . Corresponding tempera ture p l o t s are presented i n F igu re 2. Addi t iona l S K I N A T H r e s u l t s f o r t h i s problem ou t t o a f i n a l time of 1000 minutes are presented i n Appendix B.
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1 .o 1.687 2 -649 4.154 6.510 10.21 1s .02 25,111 39.47 67.91 99 02 152.2 238 97 374.6
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0 R N C- D W Q 8 3C - 16676
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0 20 40 60 80 100 120 140 TIME (mln)
Figure 1. P o w e r V e r s u s T i m e f o r P r o b l e m No. 2
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F i g u r e 2 . Temperature Versus Tiine € o r Problem N o . 2
7
Note t h a t t h e SKINATW r e s u l t s are similar t o t h e r e s u l t s of Reference 2 whish employed sevwal s i m p l i f y i n g assumptions. The Runge- Kut ta and SKINATH r e s u l t s are also q u i t e similar. More impor t an t ly , t he Runge-Kutta r e q u i r e d 6 minutes o f CPU time on t h e IBM 3033 for a t r a n s i e n t time Qf 130 minutes; whereas, SKIMATH r e q u i r e d on ly 1.8 seconds of CPU time for a t r a n s i e n t time of 1000 minutes. The Reference 2 r e s u l t s were obta ined us ing a d i f f e r e n t computer.
c a l c u l a t i o n
Based on these r e s u l t s and o t h e r comparisons t h a t were performed bu t are no t presented i n this document, w e conclude t h a t SKINATH is an a c c u r a t e and e f f i c i e n t program for s o l v i n g p o i n t k i n e t i c s problems wi th s imple thermal-hydraulic feedback.
4 . RECOMBNDATIOMS FOR FUTURE WORK
The c u r r e n t v e r s i o n of S K I N A T H employs t h e LSODE o p t i o n t o c a l c u l a t e the Jacobian i n t e r n a l l y by d i f f e r e n c e q u o t i e n t s . An improvement i n accuracy and/or computing e f f i c i e n c y might be realized i f the Jacobian is c a l c u l a t e d by t h e u s e r as recommended i n the LSODE documentation. The improvement may be s i g n i f i c a n t p a r t i c u l a r l y f o r stiff problems.
The thermal-hydraulic model i n SKINATH i s r e a l l y q u i t e crude. Many improvements could be made depending on the system being modeled. One improvement t h a t could be made is to i n c o r p o r a t e an energy ba lance f o r the coolant . Cur ren t ly , t h e coo lan t is modeled as an i so the rma l heat sink which is u n r e a l i s t i c for most a p p l i c a t i o n s .
As i n d i c a t e d earlier i n t h i s document, e q u i l i b r i u m between the n e u t r o n i c s and thermal -hydraul ics a t time zero does no t ex is t i n t h e c u r r e n t v e r s i o n o f SKINATH. Such e q u i l i b r i u m is desirable, a l though no t necessary , f o r most practical problems of i n t e r e s t and should be implemented i n a la ter v e r s i o n of SKINATH.
F i n a l l y , comparing SKINATH r e s u l t s w i th t he r e s u l t s of s e v e r a l t r a n s i e n t exper iments should be performed i n order t o better e v a l u a t e both the models employed i n SKINATH and its s o l u t i o n methodology.
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REFERENCES
1. A. F. Henry, Nuclear-&eactor A n ! , NIT Press, Cambridge, Chapter 7, P. 308 (19755).
2. C, E. Newlon e t a l . , wAn assess ent of the Minimum Criticality Accident of Concern," N d: R Technical Associates , H ti R 82-2, Oak Ridge, Tenn. (1982).
3 . A, C. Hindmarsh, "LSODE and LSODI - Two New I n i t i a l . Value Ordinary Differential Equation Solvers," ACN-Signum Newsletter, Vsl. 15, No. 4 (1980).
4 . R. L. Burden e t a l . , Numericah , Prindle, Weber and Schmidt, Boston, p. 244 (1978).
APPENDIX A
Program Listing and Sample Problem Input
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APPENDIX €3
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PRECURSLiR C3NCENTRATIONS
4623.
3990.
3446
2979.
2579.
2233.
1937.
168 1
1461.
1272.
I 1 0 8 .
966 5
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3 .10570 35
9130.
7893.
6030.
5917.
5131.
4455.
3872.
3370
24 36
2562.
2237.
1957.
1714.
1502-
1319.
I 1 59.
1021.
899.4
793.7
701.3
620.5
549.7
487.6
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2 1 1 1 .
1826-
1580.
1370.
1188.
1032.
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781.7
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594.8
519.9
454.9
398.6
349.7
307.1
270.1
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209.5
185.2
163.5
145.0
128.5
114. I
101.4
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80.37
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1008.
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7.769
5 .453
5.736
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4.127
7.329
6.397
6.506
5.237
4.572
3.997
3.498
3.565
2.690
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1.531
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