ANL-6798 Reactor Technology AEC Research and Development …/67531/metadc871660/m2/1/high_re… · ANL-6798 Reactor Technology (TID-4500, 27th Ed.) AEC Research and Development Report

ANL-6798 Reactor Technology (TID-4500, 27th Ed.) AEC Resea rch and Development Report

ARGONNE NATIONAL LABORATORY 9700 South Cass Avenue Argonne, Illinois 60440

A FAST REACTOR EXCURSION SIMULATOR

by

Lawrence T. Bryant Applied Mathemat ics Division

and

D. V. Gopinath* Reactor Phys ics Division

*A.I.D. Par t i c ipan t at Reactor Phys ics Div., p resen t add re s s : Health Phys ics Division, Atomic Energy Establ i shment

Trombay, Bombay-74, India

December 1963

Operated by The Universi ty of Chicago under

Contract W-31-1 09-eng-38 with the

U. S. Atomic Energy Commission

DISCLAIMER

This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency Thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.

DISCLAIMER Portions of this document may be illegible in electronic image products. Images are produced from the best available original document.

T A B L E O F CONTENTS

P a g e

A B S T R A C T 3

I. INTRODUCTION 4

II. S IMULATOR 6

A. T e m p e r a t u r e - t r a n s i e n t E q u a t i o n s 7

1. A s s u m p t i o n s 7

2. E q u a t i o n s 7

3 . S p e c i a l C i r c u i t s 11

(a) P h a s e T r a n s f o r m a t i o n 11

(b) Coo lan t L i m i t e r 11

B . R e a c t i v i t y F e e d b a c k 12

C. T h e R e a c t o r K i n e t i c s E q u a t i o n 13

III. CONSTANTS FOR E B R - I I A C C I D E N T ANALYSIS 16

A. C o r e Da ta 16

B . S o u r c e T e r n a Da ta 16

C. R e a c t o r K i n e t i c s and F e e d b a c k Data 17

IV, DEFINITION O F S C A L E F A C T O R S AND MACHINE EQUATIONS 18

A. S c a l e F a c t o r s 18

B . M a c h i n e E q u a t i o n s 19

1. T e m p e r a t u r e T r a n s i e n t 19

V. ANALOG C O E F F I C I E N T S 20

VI. COMPARISON O F ANALOG AND DIGITAL R E S U L T S 27

VII . A C K N O W L E D G E M E N T S 29

LIST O F F I G U R E S

No. T i t l e P a g e

1. S i m u l a t o r Block D i a g r a m 6

2. R a d i a l Node P o i n t s 8

3. F i n i t e D i f f e rence A p p r o x i m a t i o n C i r c u i t 9

4. Analog C i r c u i t D i a g r a m for T e m p e r a t u r e T r a n s i e n t 10

5. P h a s e - t r a n s f o r m a t i o n N e t w o r k 11

6. H e a t Conten t of U r a n i u m - 5 % F i s s i u m Alloy 11

7. Coolan t Output L i m i t e r C i r c u i t 12

8. R e a c t i v i t y F e e d b a c k C i r c u i t 13

9. N o n l i n e a r R e a c t o r K i n e t i c s S c h e m a t i c for r)j 14

10. N o n l i n e a r R e a c t o r K i n e t i c s S c h e m a t i c for rj = £n 7]j 15

11 . C o m p a r i s o n of S i m u l a t o r and Dig i t a l T e m p e r a t u r e C a l c u l a t i o n s for I n s t a n t a n e o u s F low B lockage 2 7

12. C o m p a r i s o n of S i m u l a t o r and Dig i t a l T e m p e r a t u r e C a l c u l a t i o n s fo r P u m p F a i l u r e 28

13. P o w e r E x c u r s i o n due to C e n t r a l S u b a s s e m b l y D r o p 28

LIST O F T A B L E S

No. T i t l e P a g e

1. N o m e n c l a t u r e 3,4

2. M a t e r i a l P r o p e r t i e s of the C o r e 16

3. R e l a t i v e P o w e r in the Axia l Sec t i ons of the P i n 17

4 . D e l a y e d - n e u t r o n Da ta 17

5. Weigh t F a c t o r s a n d R e a c t i v i t y F e e d b a c k Coef f i c i en t s 18

6. Analog Coef f i c ien t s for T e m p e r a t u r e T r a n s i e n t and

F e e d b a c k 2 0 - 2 3

7. Analog Coef f i c ien t s for N e u t r o n K i n e t i c s (T)J) 24,25

8. Analog Coef f i c ien t s for N e u t r o n K i n e t i c s ( T] = £ n T]j) 26

3

A FAST R E A C T O R EXCURSION SIMULATOR

by

L a w r e n c e T. B r y a n t and D. V. Gop ina th

ABSTRACT

Th i s p a p e r p r e s e n t s a t echn ique for the s tudy of f a s t r e a c t o r a c c i d e n t s by u s e of an ana log c o m p u t e r . The t e c h ­nique is app l i ed to r e a c t o r p r o b l e m s involving p u m p f a i l u r e , o r i f i c e b lock , and f a s t e x c u r s i o n .

T h i s m e t h o d a l l ows a good flux plot o r t e m p e r a t u r e -d i s t r i b u t i o n a n a l y s i s to be m a d e by the use of only a 5 x 5 m e s h of an a v e r a g e c h a n n e l in the c o r e .

A s a m p l e p r o b l e m is w o r k e d by th i s m e t h o d and c o m ­p a r e d wi th a d ig i t a l c o m p u t e r so lu t ion . All c o m p u t a t i o n s , c i r c u i t s , and o p e r a t i n g i n f o r m a t i o n n e c e s s a r y to dup l i ca t e the so lu t ions a r e g iven . R e s u l t s a r e inc luded and c o m p a r e d w i th the d ig i t a l so lu t ion .

Tab le 1

N O M E N C L A T U R E

effec t ive a r e a b e t w e e n the i - t h and i + 1-st c e l l s

spec i f i c hea t

d i s t a n c e b e t w e e n i - t h and i + 1-st node poin ts

h e a t t r a n s f e r coeff ic ient

r a d i a l c o o r d i n a t e of the m e s h

a x i a l c o o r d i n a t e of the m e s h

conduc t iv i ty b e t w e e n i - t h and i + 1-st node p o i n t s

s u r f a c e conduc t iv i ty b e t w e e n two so l id r e g i o n s

d e n s i t y

p o w e r g e n e r a t e d

Ai, i+i

Cp

d i , i+ i

h

J

Ki, i+i

K s

P

q(v, t )

Tab le 1 (Contd . )

T

V

V

Az

t

n

C;

K ex

^i

^ = 2 / 3 .

£

^ i

ni =-

Cii =

AK

Keff

K^iex

^i ^ l i

t '

Kiex

r) '

Vi

n(t) n(0)

C i ( t )

Ci(0)

^ e x ~ |6

^

t e m p e r a t u r e

v o l u m e

coo l an t flow r a t e

a x i a l m e s h s i z e

t i m e

p r o m p t - n e u t r o n flux d e n s i t y

flux d e n s i t y of i - t h d e l a y e d - n e u t r o n g r o u p

e x c e s s r e a c t i v i t y

the f r a c t i o n of p r o m p t n e u t r o n s e n t e r i n g the i - t h d e l a y e d - n e u t r o n g r o u p

the f r a c t i o n of p r o m p t n e u t r o n s e n t e r i n g a l l d e l a y e d g r o u p s

p r o m p t - n e u t r o n l i f e t ime

d e c a y c o n s t a n t of n e u t r o n s in the i - t h d e l a y e d g r o u p

the r a t i o of the p r o m p t - n e u t r o n d e n s i t y a t t i m e t to the e q u i l i b r i u m v a l u e , n(0)

the r a t i o of the d e n s i t y of n e u t r o n s in the i - t h d e l a y e d g r o u p to the e q u i l i b r i u m v a l u e , C;(0)

f e e d b a c k r e a c t i v i t y (in d o l l a r s )

Kiex - AK

e x c e s s r e a c t i v i t y (in d o l l a r s )

p r i m e s deno te the c o m p u t e r v o l t a g e s w h i c h r e p r e s e n t the u n p r i m e d v a r i a b l e s

I. INTRODUCTION

A p r i o r i a n a l y s e s of h y p o t h e t i c a l f a s t r e a c t o r m e l t d o w n a c c i d e n t s a r e of i n t e r e s t b e c a u s e of t h e i r p o s s i b l e " a u t o c a t a l y t i c " n a t u r e . The fuel

5

occupies approximately 30% of the core volume, and the remaining 70% is made up of s t ruc tu ra l m a t e r i a l s and coolant. The fuel configuration is not optimum with r e spec t to react ivi ty .

If the fuel were to mel t and become r ea r r anged into a denser con­figuration, la rge amounts of excess react ivi ty would become available. Thus, failure of a fuel e lement in a fast r eac to r might lead to large r e l e a s e s of nuclear energy. In such cases it is not the total excess react ivi ty ava i l ­able, but the ra te at which the excess react ivi ty is added, that de termines the ult imate sever i ty of an accident,-^

The ra te of addition of react ivi ty in a meltdown accident is de­te rmined by the t ime sequence of fuel e lement fai lure, which is dependent upon the t empera tu re t r ans ien t s of the fuel e lements .

In o rde r to study the dynamics of a fast r eac to r accident, a detailed knowledge of spat ial and t ime dis tr ibut ions of t empera ture in the fuel e l e ­ment is neces sa ry . This is n e c e s s a r y to calculate the react ivi ty feedback effects, which de te rmine the pre-mel tdown trend of the accident. It also de te rmines the coherence of pin fa i lure , which de termines the meltdown effects of the accident .

In view of the impor tance of t empera tu re distr ibution in a fuel e l e ­ment in fast r eac to r accident ana lys i s , an IBM 704 FORTRAN code "Argus" was p repa red at the Argonne National Laboratory to calculate the t i m e - and space-dependent t empe ra tu r e distr ibution in mul t i - reg ion cylindrical bodies. The code accommodates twenty-five concentr ical cyl indrical regions (solid or liquid) with or without heat generat ion. The coolant can be static or flow­ing, and there is a provis ion for var iable flow ra te .

"Argus" is a t e m p e r a t u r e - t r a n s i e n t code and does not calculate the react ivi ty feedback and subsequent change in reac tor power.

Greeb le r e_t aL developed a digital p rog ram, " F o r e , " for T r a n s a c -2000, which takes into account the react ivi ty feedback effects on reac tor

• W. J. McCarthy, R. B. Nicholson, D. Okrent, and V. Z. Jankus, Studies of Nuclear Accidents in Fas t Reac tors , P roc . 2nd UN Intn'l . Conf. on the Peaceful Uses of Atomic Energy, Geneva, Switzerland, 12, 242(1958).

^D. F . Schoeberle, L. B. Mil ler , and J, Heestand, A Method of Calculating Trans ient T e m p e r a t u r e s in a Mult i -region, Axisymmetr ic Cylindrical Configuration. The Argus P r o g r a m 1089/RP248, wri t ten in FORTRAN II, ANL-6654 (1963).

3p , Greeb l e r , D. B. Sherer , and N. H. Walton, "Fore , ' ' A Computational P r o g r a m m e for the Analysis of Reactor Excursion, GEAP-4090.

p o w e r . T h i s p r o g r a m is not a s v e r s a t i l e a s " A r g u s " in t e m p e r a t u r e c a l ­cu l a t i ons b e c a u s e of the r e s t r i c t i o n s on m e s h s i z e and b e c a u s e coo lan t flow r a t e m u s t be c o n s t a n t .

Since s u c h p r o g r a m s c o n s u m e c o n s i d e r a b l e t i m e on l a r g e d ig i t a l c o m p u t e r s and due to the a m e n a b i l i t y of the e n t i r e p r o b l e m to ana log s i m u l a t i o n , th i s p a p e r p r e s e n t s an ana log c o m p u t e r s tudy of a f a s t r e a c t o r e x c u r s i o n .

The s i m u l a t o r t a k e s into a c c o u n t the t e m p e r a t u r e dependence of the coo lan t p r o p e r t i e s , p h a s e change of the fuel , f e e d b a c k effects on r e ­a c t o r p o w e r , and v a r i a t i o n of coo lan t flow.

T h i s r e p o r t d e s c r i b e s the s i m u l a t o r and p r o v i d e s a l l c o m p u t a t i o n s , c i r c u i t r y , and o p e r a t i n g i n s t r u c t i o n s for d u p l i c a t i o n of an E B R - I I - t y p e a c c i d e n t . O t h e r da t a , for e x a m p l e , for a l a r g e c e r a m i c r e a c t o r , can be u s e d w i t h a p p r o p r i a t e input c h a n g e s .

II. S IMULATOR

The s i m u l a t o r c o n s i s t s of t h r e e p a r t s , a s shown in F i g u r e 1,

T e n n p e r a t u r e T r a n s i e n t

R e a c t i v i t y F e e d b a c k

F i g . 1. S i m u l a t o r B lock D i a g r a m

The s y s t e m is d e s i g n e d such tha t the fol lowing s t u d i e s can be m a d e :

1) Obta in the d e t a i l e d t e m p e r a t u r e d i s t r i b u t i o n in an a v e r a g e fuel e l e m e n t due to a change in any of the o p e r a t i n g cond i t i ons of the r e a c t o r .

2) Ca l cu l a t e the f eedback r e a c t i v i t y due to t e m p e r a t u r e d i s t r i b u ­t ion in the fuel e l e m e n t .

3) D e t e r m i n e r e a c t o r p o w e r change due to f eedback r e a c t i v i t y and e x t e r n a l l y i n s e r t e d r e a c t i v i t y , if any .

7

A. T e m p e r a t u r e - t r a n s i e n t Equations

1. Assumptions

In the t e m p e r a t u r e - t r a n s i e n t calculat ions, a fuel element is taken as the represen ta t ive unit of the r eac to r . The fuel element is s imu­lated by a m e s h of point in tegra to rs for which the following assumptions a r e made :

1) The fuel is cyl indr ical with clad and coolant as concentr ic cyl indrical she l l s .

2) Axial heat conduction in the fuel is negligible.

3) Tempera tu re gradient in the radia l direct ion in the coolant is ze ro , and only one radia l mesh point need be considered in the coolant region.

4) There is axial heat t ransfe r in the coolant region due to coolant movement .

For each cel l in the fuel e lement m e s h we have a heat balance express ion:

Rate of change of heat content = (heat flowing in - heat flowing out + heat generated) per unit t ime. (l)

2. Equations

Referr ing to Figure 2 and considering the assumptions made above, the heat balance equation (l) for solid regions can be wri t ten as

(2)

for all i and j / 0. Rearranging the t e r m s and integrating, we obtain

Ti(t) = Ti(0) +J{Ci , i - iT i - i+Ci , i+ iTi+ i -CiTi + q(i,j,t)}dt, (3)

where

KA ""'•'-' ^ Uyi,i-i(vpCp)i

C; i,i+i KA

i,i+i VpC P ^ i

^ i ~ ^i, i+i ''" ^ i , i - i

(4)

with the following r e m a r k s :

(1) ^i , i+i / Ci , i - i .

(2) If Ci and Cj j a r e in two different regions , K^ ^^^ a lso takes into account the in tersurface r e s i s t ance and is given by the relat ion

1 ^i di+i + ^ ^ +

Qi. i+i Ki,i+i - Kg • Ki • Ki+i' (5)

where d^ i+j and K^ i+j a r e the distance and effective conductivity between Ci and Ci^-j. Here di+j and Ki^i a re distance and conductivity, respect ively, in the region of the (i+i) point.

o r

Fig. 2. Radial Node Points

For i = 0, there is no heat flowing in; therefore ,

( v p C p ) o ^ = - ( - ^ ) ^ ^ (To -T i ) + q(o,j , t) ,

To(t) = To(0) + j {Co,iTo-Co,iTi + q(o,j,t)}dt.

The heat balance equation for the coolant region is

, ^ , dT dT , , ^ ^ ^ S ) ^ = - V — +q2(v.z, t ) .

(6)

(7)

(8)

where q2(v,z,t) is heat flowing into the coolant from the solid region.

In the f in i te d i f f e rence a p p r o x i m a t i o n , equa t ion (6) can be w r i t t e n a s

dT iJ _ V h A ;

dt Az ^-"^'J ^i>J-i^ ( p C p V ) i j + ; ^ ^ ^ ^ (T- • - T- •) (9)

w h e r e the i - t h poin t is in the coo lan t and the ( i - l - s t ) point is the b o u n d a r y po in t of the so l id r e g i o n . T h e r e f o r e ,

Ti , j ( t ) = T i , j ( 0 ) + / (C*ATj_,^j + Hi^jATi_^_i)dt, (10)

w h e r e

C* = v / A z ;

Hi, j = h A i ^ i _ y ( p C p V ) i j

^ T j - i , j = Tij( t ) - T i j - i ( t ) ;

ATi.i- T^. i - i , J • i . J -

F i g u r e 3 shows a t y p i c a l ana log c i r c u i t d i a g r a m for the solu­t ion of a p a r t i a l d i f f e r en t i a l equa t ion by finite d i f fe rence a p p r o x i m a t i o n . The output vo l t age ei of the i n t e g r a t o r i is g iven by

-i = e - ( 0 ) + j (Ci_ie i_ i + Ci+jCi+i - C i e - + q i ) d t . ;H)

ei+i

1+1

Fig . 3. F in i t e Di f fe rence A p p r o x i m a t i o n C i r c u i t

C l e a r l y , equa t i ons (3), ( lO), and (11) a r e equ iva len t . Hence , an ex t ended c i r c u i t of the type shown in F i g u r e 3 can be u s e d to s i m u l a t e the t i m e and s p a t i a l v a r i a t i o n of t e m p e r a t u r e in the fuel e l e m e n t . The a c c u r a c y of s i m u l a t i o n wi l l depend on the f i n e n e s s of the m e s h , i . e . , the n u m b e r of i n t e g r a t o r s u s e d .

F i g u r e 4 shows the c o m p l e t e n e t w o r k which s i m u l a t e s the t e m p e r a t u r e t r a n s i e n t s in a fuel e l e m e n t by m e a n s of a 5 x 5 m e s h . All

o

-n , +n|

• 4 ^

I — @ -P'n^z.

7^&-

-I oov-

• - ^

'2,2

-T3,2

-» @-

I — ^ '

-@-

T3,3

' < feq

, — ( g a — I

"—^4-

T3,4

J^^"c. ^ ^ " - ^ ^ " - J^^" - Lra...-<^Vfli> " <D ' O ' 0 0 0 ^

^ c*0

< ^ c*0

• ^

+ I O O V

-^3,5

0"" 0' "v v

^"0 ^ c*'0 c*'0 ^ c*"0 ^ 0 c* .

•TC5

c*0 • ^ ^

F i g . 4. Analog C i r c u i t D i a g r a m for T e m p e r a t u r e T r a n s i e n t s

11

of the coef f ic ien t s by which the t e m p e r a t u r e s a r e to be m u l t i p l i e d , excep t in the c l ad and coo lan t r e g i o n s , a r e a c c o m p l i s h e d by l i n e a r p o t e n t i o m e t e r s .

The "h" w h i c h e n t e r s into the coef f ic ien ts of v a r i a t i o n of c o o l ­an t and c lad t e m p e r a t u r e s is c o o l a n t - v e l o c i t y dependen t . A c c o m m o d a t i o n for c o o l a n t - v e l o c i t y v a r i a t i o n in t h e s e p a r a m e t e r s is a c c o m p l i s h e d by the u s e of s e r v o - m e c h a n i c a l m u l t i p l i e r s for coeff ic ient m u l t i p l i c a t i o n in t h e s e r e g i o n s .

In o r d e r to d e c r e a s e the effect of i n h e r e n t e r r o r of the s e r v o -m e c h a n i c a l m u l t i p l i e r s , t e m p e r a t u r e d i f f e r e n c e s a r e t aken in the c lad and coo lan t r e g i o n s be fo re m u l t i p l y i n g by the coo lan t ve loc i ty v a r i a b l e coef f ic ien t .

3. Spec ia l C i r c u i t s

ma ,^

6—o—*6 To A +

o—o—*o

rf-

+ Tu

REL

i j

^ -10

^ ^

iOOV -

F i g . 5. P h a s e - t r a n s f o r m a t i o n N e t w o r k

I D -

EQUILIBRIUM-

APPROXIMATION USED-

a + y+Uz

400 600

TEMPERATURE,

1200

F i g . 6. H e a t Conten t of U r a n i u m - 5 % F i s s i u m Al loy

a. P h a s e T r a n s f o r m a t i o n

S imu la t i on of p h a s e t r a n s ­f o r m a t i o n in the fuel i s a c c o m p l i s h e d by the u s e of h i g h - s p e e d d i f fe ren t i a l r e l a y s in the fuel i n t e g r a t o r n e t w o r k s . The r e l a y c i r c u i t i s shown in F i g ­u r e 5. The r e l a y i s se t to o p e r a t e at 1010°C, the p h a s e - c h a n g e t e m p e r a t u r e of the f u e l - r e g i o n node. At th i s tena-p e r a t u r e t h e r e is a sudden change in the en tha lpy vs . t e m p e r a t u r e c u r v e ( see F i g u r e 6) for , say , the u r a n i u m -5% f i s s i u m al loy. Th i s i s equ iva len t to a change in the spec i f ic hea t of the a l loy , which c h a n g e s the v a l u e s of the coef f ic ien ts in the fuel e q u a ­t i o n s . Upon o p e r a t i o n of the r e l a y s a l l inputs to the node i n t e g r a t o r a r e m u l t i p l i e d by C l^vii' w h e r e Cp^ is the spec i f ic h e a t of the f i s s i u m a l loy p r i o r to p h a s e change and Cp^ is the effect ive spec i f ic hea t of the a l loy a f t e r p h a s e c h a n g e .

b. Coo lan t L i m i t e r

A s e r i e s output l i m i t e r i s u s e d in the coo lan t i n t e g r a t o r n e t ­w o r k s to l i m i t the p a r t i c u l a r coo lan t node to i t s boi l ing point . The ana log c i r c u i t r y u s e d for th is p u r p o s e i s sho'wn in F i g u r e 7.

0 -eo

X - ^ i

F i g , 7. Coo lan t Output L i m i t e r C i r c u i t

The p e r f o r m a n c e equa t i on for the l i m i t e r is g iven by

1 / eo \ 1

Ri + Rf 0 (12)

when the diode is not conduc t ing . Since A i s of the o r d e r of 10^, e^/A can be n e g l e c t e d . T h u s ,

Rf (13)

When the d iode c o n d u c t s

RzRf E -

RzRf (R2+Rf)Ri (R2+Rf)Ri (14)

w h e r e R2Rf/(R2 + Rf)Ri i s the f inal s lope of the l i m i t e r c h a r a c t e r i s t i c s . The va lue of R2 i s c h o s e n such tha t i t is m u c h l e s s than Ri a n d the final s lope i s a p p r o x i m a t e l y z e r o , thus l i m i t i n g the coo lan t t e m p e r a t u r e to the boi l ing point .

B . R e a c t i v i t y F e e d b a c k

G e n e r a l l y , t e m p e r a t u r e coef f i c ien t s of r e a c t i v i t y a r e a v a i l a b l e for the r e a c t o r a s a whole r a t h e r t han for any p a r t i c u l a r fuel e l e m e n t . In the r e a c t o r a c c i d e n t a n a l y s i s it s o m e t i m e s b e c o m e s n e c e s s a r y to s tudy o t h e r t han the a v e r a g e fuel e l e m e n t . In s u c h an even t the a v e r a g e fuel p in is s i m u l a t e d to ob ta in the r e a c t o r p o w e r t r a n s i e n t . The p o w e r t r a n s i e n t thus ob ta ined , w i th the p r o p e r m u l t i p l i c a t i o n f a c t o r , is u s e d a s a s o u r c e t e r m in s i m u l a t i n g the p a r t i c u l a r fuel e l e m e n t of i n t e r e s t .

The r e a c t i v i t y f eedback s i m u l a t i o n i s such tha t the t e m p e r a t u r e a t e a c h po in t is m u l t i p l i e d by the r e s p e c t i v e t e m p e r a t u r e coeff ic ient of

r e a c t i v i t y and s u m m e d . T h i s s u m a long w i th any e x t e r n a l l y i n s e r t e d r e ­ac t i v i t y goes into the r e a c t o r k i n e t i c s s i m u l a t o r a s Kgff. F i g u r e 8 shows the c i r c u i t r y for Keff.

J z J S ^ I Tcj ora .^ I 1 ^ ^

*rz, ^^^>^

F i g . 8. Reac t i v i t y F e e d b a c k C i r c u i t

C . The R e a c t o r K i n e t i c s Equa t i on

R e f e r e n c e is m a d e to A N L - 6 0 2 7 , Analogue C o m p u t e r Solution of the N o n l i n e a r R e a c t o r K i n e t i c s E q u a t i o n , by L. T. B r y a n t and N. F r a n k M o r e ­h o u s e , J r . The e q u a t i o n s and d i a g r a m s u s e d a r e r e p e a t e d h e r e for c o m p l e t e n e s s .

The r e a c t o r k i n e t i c s e q u a t i o n s a r e

6 £ dn i

^ ~ d F = 1 Z Pi — C i i

1=1 ^

a n d

d C U d t

^ i ^ K i e x '^i + ^^i^i - ^ i C i i ( i = 1 , 2 , . . . , 6 ) .

(15)

(16)

L e t

T) = Jin n^

e - ^ Q i > .

Gi - 1 = ^ i

S u b s t i t u t i n g e q u a t i o n s ( l 7 ) i n t o e q u a t i o n s (15) a n d ( l 6 ) , w e o b t a i n

dTl d t £

V /^i ( l - ^ ) K i e x + Z -g -^ i

i = i ^

(17)

; i 8 )

a n d

14

''*' = X i P K . , . - ^ - * i L.^).U-U2 6). dt dt dt

(19)

It is noted that in o rde r to study the sys tem for a large excursion, it is nece s sa ry to solve the kinetics equations in t e r m s of £n Uj. The antilog is taken by uti l ization of a diode function genera to r for inser t ion of nj as power into the t empera tu re equations.

Figure 9 and Figure 10 show the analog c i rcu i t d iagrams for the r eac to r kinetics equation.

Fig. 9. Nonlinear Reactor Kinetics Schematic for rji

15

Fig . 10. C i r c u i t D i a g r a m for rj = i n Uj

III. CONSTANTS FOR EBR-II ACCIDENT ANALYSIS

The constants given below a re based on the values used in the "Argus" calculation.

A. Core Data (see also Table 2)

Length of the fuel pin 36. 12 cm; Radius of the fuel region 0.183 cm; Externa l radius of the clad + bond 0.221 cm.

Two types of pins a re cons idered for the coolant region:

(1) in ternal pin;

(2) pe r iphera l pin.

Effective external radius of the in ternal pin coolant channel 0.2895 cm; effective external radius of the pe r iphera l pin coolant channel 0.326 cm.

Table 2

MATERIAL PROPERTIES OF THE CORE

Fuel

Clad + Bond

Coolant

pCp(cal /cc-°C)

1.061

0.633

0.239

k(cal /cm^-sec)

0.1118

0.065

0.157

Remarks

(pC ) of fuel includes en­ergy of t ransformat ion at 630°C. (pCp) of coolant is taken at 650°C.

Az = 7.22 cm (Ar)f = 0.0915 cm (Ar)cl+bond = 0.038 cm (Ar)coolant = 0.0685 cm (for internal pin)

= 0.105 cm (for per iphera l pin)

B. Source T e r m Data

Operat ing power = 62.5 MW

Average (axial average) power of the in ternal pin of the 1st row = 816 .72-ca l /cc -sec (see also Table 3).

T a b l e 3

R E L A T I V E P O W E R IN T H E AXIAL SECTIONS O F THE PIN

Sec t i on

R e l a t i v e p o w e r

T e m p r i s e ( ° C / s e c )

1

0.92

662.9

2

1.15

828.7

3

1.20

864.7

4

1.15

828.7

5

0.92

662.9

Avg.

1.068

The p h a s e t r a n s f o r m a t i o n for the fuel i s s e t at 1010°C.

C Pi 4 .437

Cp2 19.41 0.2286

Coo lan t v a p o r i z a t i o n i s s e t at 1020°C.

C* = B 0 ( t ) / A z

Hij = K2 + K3 0(t)

K2 = 148.8 for i n t e r n a l pin

= 88.88 for p e r i p h e r a l pin

K3 = 0.219 B 0 ( t )

C. R e a c t o r K i n e t i c s and F e e d b a c k Da ta

£ = 8 X 10' s e c jB = 0.007347

T a b l e 4 shows the d e l a y e d - n e u t r o n da ta .

T a b l e 4

D E L A Y E D - N E U T R O N DATA

i

1

2

3

4

5

6

h 0.0127

0 .0318

0 .1153

0.3110

1.40

3.870

^

0.000232

0.001423

0.001333

0.002956

0.001121

0.000282

The f e e d b a c k coe f f i c i en t s ( f r o m A N L - 5 7 1 9 A d d e n d u m , p. 85) for fuel t e m ­p e r a t u r e c h a n g e :

= -0 .50 x 10-5.

F o r coo l an t t e m p e r a t u r e c h a n g e :

= -2 .22 X 1 0 - ^

R e a c t i v i t y f e e d b a c k due to c l ad and s t r u c t u r a l t e m p e r a t u r e c h a n g e s i s i n ­c luded in the c o o l a n t coe f f i c i en t s on the a s s u m p t i o n t h a t the coo lan t h a s the a v e r a g e t e n a p e r a t u r e of s t r u c t u r e and c l a d .

T a b l e 5 g ives the w e i g h t f a c t o r s and the coe f f i c i en t s u s e d for the r e a c t i v i t y f e e d b a c k of the s e v e r a l r a d i a l r e g i o n s .

T a b l e 5

WEIGHT F A C T O R S AND R E A C T I V I T Y F E E D B A C K C O E F F I C I E N T S

R a d i a l R e g i o n

1

2

3

c o o l a n t

Weigh t F a c t o r

l / l 6

1/2

7 / 1 6

1.0

Coeff ic ient

3.1 X 10"^

2.5 X 10-^

2.2 X 10"^

2.22 X 10-5

As the r e a c t i v i t y coe f f i c i en t s a r e g iven for the r e a c t o r as a w h o l e , they a r e a p p l i c a b l e only for the a v e r a g e fuel pin . T h e r e f o r e , the a c c i d e n t i s s i m u l a t e d for a p in w i th a v e r a g e p o w e r and i t s p o w e r t r a n s i e n t i s o b ­t a ined . F o r a l l o t h e r p i n s , t h i s p o w e r t r a n s i e n t m u l t i p l i e d by the r a t i o of t h e i r r e s p e c t i v e p o w e r to the a v e r a g e p o w e r w a s u s e d a s the input p o w e r .

The r a t i o of r a d i a l m a x i m u m to m i n i m u m p o w e r for E B R - I I -was found to be 1.6.

IV. D E F I N I T I O N O F S C A L E F A C T O R S AND MACHINE EQUATIONS

A. Sca le F a c t o r s

a t = t ' ; d iC i i = C l i , o r d i ^ j i = ^ l i ;

b m = n l ; f T i j = T l j , (20)

eKj ex = K i e x '

19

w h e r e the p r i m e d v a r i a b l e s r e p r e s e n t m a c h i n e v o l t a g e s , and a, b , e, d^,

and f- a r e c o n s t a n t s .

B . M a c h i n e E q u a t i o n s

1. T e m p e r a t u r e T r a n s i e n t s

(a) F u e l R e g i o n s

= 1(^^] (T! - T ' ) - f l f ^^ ] ( T ' - T ! ) a l^VpCpdy/i^i./ 1-1 i a\vpCpd;i^i^./ 1 iW

dTj j _ j ^ / KA

dt ' " a \VpCpd

f q ' ( i . j , t ) abVpCp •

(b) Clad + Bond

(21)

^Ik - i. / KA \ dt ' a \VpCpd,

(c) Coo lan t

( ^^ ) (T! -T!) +-^(^^] (T'-T! ). WpCpd/i^i . j ^ - ^ ' a VP^P^/iM^ i +1

(22)

dT!- hA. . i i = _ _ I _ f X ' - T ' ) + ^'^~' . (T ' - T ' )

d t ' aAz^ i,J i , j - i^ a (pCpV) i j ^ i - i j i . j ^ ' (23)

(d) R e a c t o r K i n e t i c s

d t ' 2i£ ( l - ^ ) x . ,

e 1 e x 1 V ^i

^ 1 - ^ 1 + b A ^ ^ \ i 1 = 1 ^

(24)

dCl i _ djXi^Kl exn i _ diXinj Xi ^

dt ' " a eb ab " a li ' :25)

dT]' _ j3

dt ' a i be 1 ex JLV A ^ '

^ h ^ d.^ ^i 1 = 1 ^

(26)

df[ diX. ^Kl d t '

ex d ^ d V ^ i , ly^i dT)' ae ' b dt ' " a ^ i ' b dt '

(27)

V. ANALOG COEFFICIENTS

The potent iometer settings a r e l isted in Tables 6, 7, and

Table 6

2Ugonnt Bational l^oratorg APPLIED MATHEMATICS DIVISION

ANALOG COMPUTER

POTENTIOMETER SETTINGS

" B " T E M P E R A T U R E T R A N S I E N T . H I G H S P E E D *

POTENTIOMETER NO.

DRAWING

J = l

J = 2

AbC-ZC (8

MACHINE

2 1

2 2

2 3

2 4

2 5

2 6

2 7

2 8

2 9

3 1

3 2

3 0

4 2

3 3

3 4

3 5

3 6

3 7

3 8

3 9

4 0

4 1

4 3

' ' F o r I I

MATHEMATICAL

VALUE

C o o / a

f k i / a b

C i o / a

C o i / a

C i i / a

f k i / a b

C z i / a

C j z / a

C a z / a

C s z / a

C z a / a

C a s / a

k 4

1 0 - 1

1 0 - 1

1 0 - '

1 0 - 1

1 0 - '

1 0 - 1

1 0 - '

1 0 - 1

1 0 - 1

1 0 - 1

1 0 - 1

1 0 - 1

1 0 - 1

C o o / a

f k i / a b

C i o / a

C o i / a

C i i / a

f k j / a b

C z i / a

C i z / a

C z z / a

C s z / a

^ o w e r T r

l i g h S p e e

l o w S p e e <

1 0 - 1

1 0 - 1

1 0 - 1

1 0 - 1

1 0 - 1

1 0 - 1

1 0 - 1

1 0 - 1

1 0 - 1

1 0 - 1

a n s i e n t

i

1

VALUE

. 5 0 3 4

. 3 3 1 5

. 0 5 9 6 8

. 5 0 3 4

. 2 3 8 8

. 3 3 1 5

. 2 2 5 1

. 1 7 9 1

. 6 2 9 3

. 6 2 8 5

. 4 0 4 1

. 6 2 8 5

. 1 1 6 3

. 0 7 0 9

. 5 0 3 4

. 4 1 4 4

. 0 5 9 6 8

. 5 0 3 4

. 2 3 8 8

. 4 1 4 4

. 2 2 5 1

. 1 7 9 1

. 6 2 9 3

. 6 2 8 5

a = 102

0 . 1 Mfd

1.0 M f d

CORREC­TION

, " B " V

F e e d b a

F e e d b a

SETTING

0 . 5 0 3 4

0 . 0 6 6 3 (

0 . 0 5 9 7

0 . 5 0 3 4

0 . 2 3 8 8

0 . 0 6 6 3 (

0 . 2 2 5 1

0 . 1 7 9 1

0 . 6 2 9 3

0 . 6 2 8 5

0 . 4 0 4 1

0 . 6 2 8 5

0 . 1 1 6 3

0 . 0 7 0 9

0 . 5 0 3 4

0 . 0 8 2 9 (

0 . 0 5 9 7

0 . 5 0 3 4

0 . 2 3 8 8

0 . 0 8 2 9 (

0 . 2 2 5 1

0 . 1 7 9 1

0 . 6 2 9 3

0 . 6 2 8 5

[ a c h i n e

i c k C a p

Lck C a p

561

5

5

5

5

11

a<

ac

)

)

1'

1

: i

Bi

t o

t o

o

r

r

w

PROBLEM NO DRAWING NO. DATE

PARAM

f o r P o w e r

T r ^ n f l i p n t

. 6 6 3

. 6 6 3

C e n t r a l P i n

P e r i p h e r a l

P i n

. 8 2 9

. 8 2 9

S p e e d

ETERS

a - 10

f = 5 • 1 0 - 2

b = 1.0

V = B 0 ( T )

Table 6 Contd.

SUgonne Bational Xaboratorg APPLIED MATHEMATICS DIVISION

ANALOG COMPUTER

POTENTIOMETER SETTINGS

PROBLEM NO. DRAWING NO. . DATE

POTENTIOMETER NO.

DRAWING

J = 3

J =4

MACHINE

4 4

54

58

4 5

46

47

4 8

49

50

51

52

53

55

56

62

66

57

59

60

61

63

64

MATHEMATICAL

VALUE

C j j / a • 10-1

C j j / a • 10-1

k4 • 10-1

Coo/a • l O ' i

f k i / ab • 10-1

Cio/a • 10-1

Coi /a • 10-1

C u / a • IQ-i

fk i / ab • IQ-i

C2, /a • 10-1

C i j / a • IQ-i

C22/a • 10-1

C32/a • IQ-i

C j j / a • 10-1

C j s / a • 10-1

k^ • 10-1

Coo/a • 10-1

Cio/a • 10-1

Coi /a • 10-1

C i i / a • 10-1

C^i/a • 10-1

C i j / a • 10-1

VALUE

.4041

.6285

.5034

.4324

.0597

.5034

.2388

.4324

.2251

.1791

.6293

.6285

.4041

.6285

.1163

.0709

.5034

.0597

.5034

.2388

.2251

.1791

CORREC­

TION SETTING

0.4041

0.6285

0.1163

0.0709

0.5034

0.0865(

0.0597

0.5034

0.2388

0.0865(

0.2251

0.1791

0.6293

0.6285

0.4041

0.6285

0.1163

0.0709

0.5034

0.0597

0.5034

0.2388

0.2251

0.1791

• = - = 1

SET

5

5

)

)

PARAM

for P o w e r 11 T r a n s i e n t

C e n t r a l P i n

P e r i p h e r a l 11 Pin

.86 5

.865

C e n t r a l P i n

P e r i p h e r a l P in

AM1.2C ( 8 - 5 7 )

Table 6 Contd.

2lrgonnt Bational Xaboratorg APPLIED MATHEMATICS DIVISION

ANALOG COMPUTER

POTENTIOMETER SETTINGS

PROBLEM NO. DRAWING NO. . DATE

POTENTIOMETER NO.

DRAWING

F e e d b a

MACHINE

8

9

10

11

12

13

14

15

16

17

18

19

20

6 1 A

6 2 A

6 3 A

6 4 A

6 5 A

;k

7 7 A

7 8 A

79A

8 0 A

MATHEMATICAL

VALUE

fTjKO)

fT4J(0)

fTsKO)

(1 /3 .5 ) C p , / C p ,

(1 /3 .5) C p / C p ^

(1 /3 .5 ) C p / C p ^

(1 /3 .5 ) C p / C p ^

(1 /3 .5 ) C p / C p ^

(1 /3 .5 ) C p / C p ^

(1 /3 .5 ) C p / C p ^

(1 /3 .5) C p / C p ^

(1 /3 .5 ) C p / C p ^

(1 /3 .5) C p / C p ^

(1 /3 .5 ) C p / C p ^

(1 /3 .5 ) C p / C p ^

(1 /3 .5 ) C p / C p ^

(1 /3 .5 ) C p / C p ^

(1 /3 .5) C p / C p ^

C , e / B f

C^e/Bf

C j e / B f

C^e/Bf

VALUE

30.00

27.50

25.00

.8001

.8001

.8001

.8001

.8001

.8001

.8001

.8001

.8001

.8001

.8001

.8001

.8001

.8001

.8001

.0169

.1361

.1198

1.209

CORREC­

TION

Volts

Volts

Vol ts

SETTING

.3000

.2750

.2500

.8001

.8001

.8001

.8001

.8001

.8001

.8001

.8001

.8001

.8001

.8001

.8001

.8001

.8001

.8001

.0169

.1361

.1198

.1209(1C

SET PARAMETERS

C, = 3.1 • 10

C2 = 2.5 • 10

C3 = 2.2 • 10

C4 = 2.22 • 10

AIC-2C 18-571

Table 6 Contd.

Argonne Bational Xaboratorg APPLIED MATHEMATICS DIVISION

ANALOG COMPUTER

POTENTIOMETER SETTINGS

PROBLEM NO. DRAWING NO.. DATE

POTENTIOMETER NO.

DRAWING

J = 5

O t h e r I

MACHINE

65

67

68

70

74

69

71

72

73

75

76

77

79

80

71

1

a r a m e t

2

3

4

5

6

7

MATHEMATICAL

VALUE

C 2 2 / a • 1 0 - 1

C 3 2 / a • 1 0 - 1

C 2 3 / a . 1 0 - 1

C j j / a . 1 0 - 1

k^ • 1 0 - 1

C o o / a • 1 0 - 1

C i o / a • 1 0 - 1

C o , / a • 1 0 - 1

C i i / a • 1 0 - 1

C 2 i / a • 1 0 - 1

C i 2 / a • 1 0 - 1

C 2 2 / a • 1 0 - 1

C j j / a • 1 0 - 1

C23 /a • 1 0 - 1

C j j / a • 1 0 " '

k^ . 1 0 - 1

e r s

T c m • f

B / l O ^ A Z a

kj B * ( T ) • 1 0 " / e

k z / a

fTiJ (O)

f T 2 J ( 0 )

VALUE

6 2 9 3

. 6 2 8 5

. 4 0 4 1

6 2 8 5

. 1 1 6 3

. 0 7 0 9

. 5 0 3 4

. 0 5 9 7

. 5 0 3 4

. 2 3 8 8

. 2 2 5 1

. 1 7 9 1

. 6 2 9 3

. 6 2 8 5

. 4 0 4 1

. 6 2 8 5

. 1 1 6 3

n 7 n q

1 8 . 6 0

1000

. 2 1 5 9

1 4 . 8 8

3 0 . 0 0

3 0 . 0 0

CORREC­

TION

V o l t s

V o l t s

V o l t s

V o l t s

SETTING

0 . 6 2 9 3

0 . 6 2 8 5

0 . 4 0 4 1

0 . 6 2 8 5

0 . 1 1 6 3

0 . 0 7 0 9

0 . 5 0 3 4

0 . 0 5 9 7

0 5 0 3 4

0 2 3 8 8

0 . 2 2 5 1

0 1791

0 . 6 2 9 3

0 . 6 2 8 5

0 . 4 0 4 1

0 . 6 2 8 5

0 . 1 1 6 3

n nvriq

. 1 8 6 0

. 1 0 0 0

. 2 1 5 9

. 1 4 8 8

3 0 0 0

. 3 0 0 0

SET

C e n t r a l P m

P e r i p h e r a l P i n

C e n t r a l P i n

P e r i p h e r a l P m

AkC-2C 18 57)

Table 7

Slrgonne Bational Xaboratorg APPLIED MATHEMATICS DIVISION

ANALOG COMPUTER

POTENTIOMETER SETTINGS

PROBLEM NO. DRAWING NO. . DATE

"A" REACTOR KINETICS TI, POTENTIOMETER NO.

DRAWING MACHINE

21

22

23

24

25

26

27

28

53

54

55

56

33

34

35

36

37

38

39

40

41

42

4 3

4 4

MATHEMATICAL

VALUE

xya

X ^ / a

X j / a

K/^ X s / a

\/^

ftb/^di

P^b/Pd^

^3b/i3d3

/34b/|3d4

ftb/Pds

P 6 b / P d ,

X j d i / a b

X j d ^ / a b

X j d j / a b

X . d / a b

X j d s / a b

X . d ^ / a b 1 0 - 1

X i d j / a b

X2d2 /ab

X s d j / a b

X 4 d 4 / a b

X s d g / a b

X . d ^ a b • 1 0 - '

VALUE

. 0 0 1 3

. 0 0 3 2

. 0 1 1 5

. 0 3 1 1

. 1 4 0 0

. 3 8 7 0

. 0 3 1 6

. 1 9 3 7

. 1 8 1 4

. 4 0 2 3

. 1 5 2 6

. 0 3 8 4

. 0 0 1 3

. 0 0 3 2

. 0 1 1 5

. 0 3 1 1

. 1 4 0 0

. 0 3 8 7

. 0 0 1 3

. 0 0 3 2

. 0 1 1 5

. 0 3 1 1

. 1 4 0 0

. 0 3 8 7

CORREC­

TION SETTING

. 0 0 1 3

. 0 0 3 2

. 0 1 1 5

. 0 3 1 1

. 1 4 0 0

. 3 8 7 0

. 0 3 1 6

. 1 9 3 7

. 1 8 1 4

. 4 0 2 3

. 1 5 2 6

. 0 0 3 8 ( 1

. 0 0 1 3

. 0 0 3 2

. 0 1 1 5

. 0 3 1 1

. 1 4 0 0

. 0 3 8 7

. 0 0 1 3

. 0 0 3 2

. 0 1 1 5

. 0 3 1 1

. 1 4 0 0

. 0 3 8 7

SET

0 f

PARAMETERS

i - 8 • 10"* j3 = 0 . 0 0 7 3 4 7

i Xi

1 . 0 1 2 7 2 . 0 3 1 8 3 . 1 1 5 3 4 . 3 1 1 0

5 1 .4000 6 3 . 8 7 0 0

/3i/p = . 0 3 1 6 /^z /P = . 1 9 3 7

/BJ/A = . 1 8 1 4 jSy/i = . 4 0 2 3 iSg/ZS = . 1 5 2 6 / 3 y p = . 0 3 8 4

1- /3= . 9 9 2 6 5

a = 10

b = d i = 1.0

e = 20

«||<>.2C IB-57 )

Table 7 Contd.

argonne Bational ILaboratorg APPLIED MATHEMATICS DIVISION

ANALOG COMPUTER

POTENTIOMETER SETTINGS

PROBLEM NO. DRAWING NO.. DATE

POTENTIOMETER NO

DRAWING MACHINE

45

46

47

I .e. 1

" 2

" 3

" 4

" 5

" 6

" 7

MATHEMATICAL

VALUE

(1-P/e) • 102

(3 • loVe

(aV/3) 10

.01 d i

.01 d j

.01 d3

.01 d4

.01 dg

.01 d^ • 10-1

.01 b

VALUE

4.963

.03674

.00109

1.0

I I

1!

n

"

0.1

1.0

CORREC­TION

Vol t

..

I I

11

M

Vol t

Vol t

SETTING

.4963(1

.03674

.0011

.0100

.0100

.0100

.0100

.0100

.0010

.0100

S b l

0 )

h ' A K A M t T t K ^

'

AI>C-ZC (8 -57 )

Table 8

argonne Bational Xaboratorg APPLIED MATHEMATICS DIVISION

ANALOG COMPUTER

POTENTIOMETER SETTINGS

"A" REACTOR KINETICS in.

PROBLEM NO. DRAWING NO.. DATE

POTENTIOMETER NO

DRAWING MACHINE

21

22

23

24

25

26

3

4

5

6

7

8

1

2

MATHEMATICAL

VALUE

X / a

X , / a

X j / a

X^ /a

Xs/a

^ e / a

P i b / P d i

^2b/ i3d2

P j b / ^ d j

/3 4b/pd4

P s b / ^ d j

jS 6b/ i3d,

d i / b

dz /b

d j / b

d^ /b

dg/b

dfc/b

Ob(l-i3)/a£e)-10-'

(/3/ad) • 1 0 - '

VALUE

.0000

.0000

.0000

.0003

.0014

.0039

.0316

.1937

.1814

.4023

.1526

.0384

1.0

1.0

1.0

1.0

1.0

1.0

.4558

9.184

CORREC­TION

SETTING

.0000 1

.0000

.0000

.0003

.0014

.0039 1

.0316

.1937

.1814

.0402(1

.03052{

. 0 0 3 8 ( |

.4558

.9184(J

SET

f T T

h

1 r

T

PARAMETERS

M u l t i p l i e r

d a i n s T a k e

C a r e o f

Tb f i se

C o e f f i c i e n t s

a = 10^

b = d i = l.C

e = 20

6

AWI-ZC ( 6 - 5 7 )

27

VI. COMPARISON O F ANALOG AND DIGITAL RESULTS

The a c c i d e n t to be s tud i ed for c o m p a r i s o n is as follo"ws: it is a s ­s u m e d tha t a c o m p l e t e b l o c k a g e o c c u r s i n s t a n t a n e o u s 1-y for the c e n t r a l c o r e subassembl -y , r e a c t o r p o w e r r e m a i n s c o n s t a n t , and h-ydrod"ynamic effects of the b lockage can be i g n o r e d , so tha t the hea t t r a n s f e r p r o b l e m is tha t for a fuel e l e m e n t in s t a g n a n t s o d i u m .

R e s u l t s f r o m the s i m u l a t o r a r e c o m p a r e d wi th output f r o m " A r g u s " in F i g u r e 11. T e m p e r a t u r e t r a c e s a r e shown, as a function of t i m e a f t e r flo-w b l o c k a g e , for the fuel s u r f a c e a t t he ax i a l m i d p l a n e .

1200

UJ 1000 <

800

2 § 600 CO

400

ID

2 200

• SIMULATOR OUTPUT ARGUS OUTPUT

J_ 0.2 0.4 0.6 0.8 1,0

TIME AFTER BLOCKAGE, sec

F i g . 11

C o m p a r i s o n of S i m u l a t o r and Dig i ta l T e m p e r a t u r e Ca lcu l a t i ons for I n s t a n ­t aneous Flo"w Blockage

1.2

The effect of m a t e r i a l p h a s e c h a n g e (at 1010°C) is shown c l e a r l y a t 0.8 s e c . Coolan t boi l ing f i r s t a p p e a r s at 1.0 s e c , at a t i m e when a l l coo lan t node poin ts a r e r i s i n g s lowly , so tha t the outputs do not sho"w any l a r g e effect f r o m the l i m i t a t i o n of coo lan t r i s e .

The r e l a t i v e l y l i n e a r t e m p e r a t u r e r i s e be tween 0.1 and 0.8 s e c d i s p l a y e d by the s i m u l a t o r output is due to the u s e of a s ing le a v e r a g e va lue of pCp o v e r th i s t e m p e r a t u r e r a n g e . " A r g u s " output in th i s r e g i o n i s n o n l i n e a r , b e c a u s e the l a t t e r c a l c u l a t i o n inc luded t"wo d i f fe ren t v a l u e s of fuel p C p b e t w e e n 600°C and 1010°C, and one fuel p h a s e change with a p p r e c i a b l e h e a t of t r a n s f o r m a t i o n a t 630°C (see F i g u r e 6).

A g r e e m e n t in F i g u r e 1 1 is s a t i s f a c t o r y for c o h e r e n c e of f a i l u r e s t u d i e s ,

28

A g r e e m e n t is shown in F i g u r e 12 for an a c c i d e n t in which coo lan t flow^ i s i m p o r t a n t . In t h i s c a s e pumping pow^er is a s s u m e d to fai l , but r e a c t o r power r e m a i n s c o n s t a n t .

SIMULATOR OUTPUT X ARGUS OUTPUT

8 10 TIME, sec

F i g . 12. C o m p a r i s o n of S i m u l a t o r and Dig i t a l T e m p e r a t u r e C a l c u l a t i o n s for P u m p F a i l u r e

The da ta shown a r e fuel s u r f a c e t e m p e r a t u r e s for the top (hot tes t ) ax i a l p o s i t i o n of the c e n t r a l fuel e l e m e n t in the c o r e . In t h i s c a s e , coo lan t flow d e c a y s exponen t i a l l y wi th a t i m e c o n s t a n t of the o r d e r of 10 s e c .

Note tha t both s e t s of output show r e l a t i v e l y l i t t l e effect of the p h a s e t r a n s f o r m a t i o n at 1010°C. T h i s i s p r o b a b l y due to the r e l a t i v e l y s low r a t e of o u r t e m p e r a t u r e e x c u r s i o n .

F i g u r e 13 p r e s e n t s the p r o m p t - n e u t r o n r e s p o n s e of E B R - I I due to a loading a c c i d e n t . The S - s h a p e d r o d w o r t h c u r v e p e a k s out a t 2%.

20

15

c

P 10

5

n

-

-

/

K

— - H — r " 1 1 1 1 1 1

F i g . 13

P o w e r E x c u r s i o n due to C e n t r a l S u b a s s e m b l y D r o p

80 100 TIME, ms

VII. A C K N O W L E D G E M E N T S

T h e a u t h o r s g r a t e f u l l y a c k n o w l e d g e the a d v i c e and d i s c u s s i o n s with D r . C. E . D i c k e r m a n of the R e a c t o r P h y s i c s Div i s ion and M r . N, F r a n k M o r e h o u s e , J r . of t he App l i ed M a t h e m a t i c s D iv i s ion .

The a s s i s t a n c e of M r . F r a n k M a l e t i c h and M r . W i l l i a m E . Scot t for o p e r a t i o n of the c o m p u t e r i s a l s o acknow^ledged.