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N U C L E A R S C I E N C E A N D E N G I N E E R I N G : 3 7 ,
1 8 6 - 1 9 1 ( 1 9 6 9 )
Discrete Kinetics for Periodical ly Pulsed Fast ReactorsG . B l
a es s e r an d J . A . L arr i m ore
Reactor Physics Department, EURATOM C.C.R., Ispra, ItalyReceived
Novem ber 27, 1968Revised March 17, 1969
A d i s c re te n eu t ron k ine t i c s fo r pe r iod ic a l ly
pu l s ed fa s t rea c to rs i s fo rm ula tedin which the t ime
behav io r o f the de layed neu t ron p r ec urs o r concen t ra t
ions i scons ide red exp l i c i t ly on ly jus t b e fo r e and
jus t a f t e r e ach power pu l s e . T he powerpu l s e i s rep
re s e n ted by a de l t a func t ion and a gene ra l in teg ra t
ion o f the p rec ur s o requa t ions be tween pu l s e s i s us ed
. T he d i f fe ren ce equa t ions ob ta ined a r e we l ls u i t
ed fo r us e in d ig i t a l s imula t ion of pu l s ed rea c to rs
. An " inho ur equa t io n" fo rpu l s ed re ac to rs i s de r ived
f r om the d i f fe r ence equa t ions and i s s hown to reduceto
the re l a t io n ob ta ined f ro m the pe r iod -ave rage d k ine
t i c s equa t ions , i f the dev i -a t ion f ro m pu l s ed c r i
t i c a l i ty i s s ma l l .
I. INTRODUCTIONIn period ical ly pulsed fas t rea ctor s (such
asIBR, SORA, etc.) the repetition frequency is of theorde r of 100
cps while the pulse width is
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a r e m u l t i p l y i n g a s s e m b l i e s w i th p e r i o
d i c a l l y , o rr e p e t i t i v e l y , m o d u l a t e d r e
a c t i v i t y w h i l e " a c c e l e r -a t o r p u l s e d r e
a c t o r s " a r e t h o s e w i t h m o d u l a t e dr e a c t i
v i t y a n d s y n c h r o n i z e d s o u r c e i n j e c t i o n
f r o ma n a c c e l e r a t o r . W h i l e w e c o n s i d e r i
n t h i s p r e s e n tw o r k o n l y s y s t e m s o f t h e f i
r s t k i n d , t h e i n c l u s i o no f a t i m e - d e p e n d
e n t s o u r c e i n o u r e q u a t i o n s a l l o w sa n e x t
e n s i o n o f t h e p r e s e n t m e t h o d t o " a c c e l e r
a t o rp u l s e d r e a c t o r s " t o o .
II. DISCRETE KINETICS DIFFERENCEEQUATION FORMULATIONWe denote (
s ee F ig . 1 ) :
Ci(p) = i't h p r e c u r s o r c o n c e n t r a t i o n a t t
h ebeg in ning o f the powe r pu lse in thep' t h p e r i o d
Ci+(p) = i ' th p r e c u r s o r c o n c e n t r a t i o n a t
t h e e n dof the power pu lse in the ' th per iod .
W e d i v i d e th e p e r i o d i n t o t w o p a r t s : d u r
i n g th ep u l s e , an d b e t w e e n p u l s e s .A. During the
Pulse
T h e i n c r e a s e i n p r e c u r s o r c o n c e n t r a t
i o n d u r i n gthe p' th pu lse i s
b e g i n n i n g o f t h e p u l s e . T h e n
AQ = Ci+(p) - Ciip) = liiUFJp) (1 )w h e r e Fp(p ) i s t h e n
u m b e r o f f i s s i o n s i n t h e p'\hp u l s e a n d f tf i
s th e z' th d e l a y e d - n e u t r o n p r e c u r s o rp r o d
u c t io n p e r f i s s i o n .
I n t h e p r e s e n t f o r m o f th e d i s c r e t e k i n e
t i c sm e t h o d r e a c t i v i t y f e e d b a c k d u r i n g
t h e p u l s e w i l lb e n e g l e c t e d ; t h e r e f o r e ,
Fp(p ) i s p r o p o r t i o n a l t ot h e t o t a l n e u t r o n
s o u r c e s t r e n g t h S ^ t d u r i n g th ep u l s e . N e g
l e c t i n g t h e s m a l l e f f e c t of t h e i n c r e a s ei
n Q d u r i n g t h e p u l s e , w e e v a l u a t e S t o t at
the
C,tt)
PERIOD p
T 2T
Time3T
Fig. 1. Schem atic Variation of Pr ecu rso r
Concen-trations.
w h e r e( 2 )
M : p r e c u r s o r p r o d u c t i o n d u r i n g p u l s es
o u r c e n e u t r o n p r o d u c t i o n r a t ean d
S = e x t e r n a l n e u t r o n s o u r c e ( w h i c h i s p
r e -s e n t f o r s t a r t - u p ) .
T h e p a r a m e t e r M h a s b e e n i n t r o d u c e d i n
t h em e a n p o w e r k i n e t i c s f o r m u l a t i o n s . 2
' 3 It can beo b t a i n e d b y i n t e g r a t i o n o f t h e n
e u t r o n k i n e t i c se q u a t i o n s d u r i n g a p u l s
e , i n t r o d u c i n g t h e a p p r o -p r i a t e r e a c t i
v i t y p u l s e .
Com binat ion o f Eq s . (1 ) and (2 ) g i ve s a d i f -f e r e
n c e e q u a ti o n f o r t h e p r e c u r s o r c o n c e n t r
a t i o n sa t the end of a pu lse in t e rm s o f the va l ue s be
for ethe pu lse and M(p)} the value of M c o r r e s p o n d i n
gto the react iv i ty pu lse in the p ' th per iodC {+(p) = C {(p )
+ ( f t / 0 ) M(p) [EXjCj(p) + S ] . (3)i
M(p) i s a f u n c t i o n w h i c h i s c o n s i d e r e d t o
b ek n o w n a s f a r a s t h e d i s c r e t e k i n e t i c s a
r e c o n -c e r n e d . I t c a n b e d e t e r m i n e d b e f o
r e h a n d b yone o f the usu a l m etho ds o f t rea t ing rea ct
iv i tye x c u r s i o n s . F o r in s t a n c e , a s a n a p p r
o x i m a t i o nw h i c h i s q ui te s a t i s f a c t o r y f o
r s m a l l p u l s e d r e -a c t o r s th e p o i n t n e u t r o
n k i n e t i c s f o r m u l a t i o n w i t h -o u t f e e d b a
c k m a y b e u s e d f o r t h e d e t e r m i n a t i o n o fM. T
h e e q u a t i o n f o r t h e f i s s i o n r a t e w i s
dw _ ew_ Stotdt T + TV (4 )
wh ere r i s the prom pt gener a t ion t im e and e = p -/3 i s
t h e p r o m p t r e a c t i v i t y . I t i s u s u a l l y a s s
u m e dthat the peak of the rea ct i v i t y pu ls e has a pa ra -b
o l i c s h a p e e x p r e s s e d i n t h e f o r m= 6m - avzt2
(5 )
with em t h e p e a k p r o m p t r e a c t i v i t y , a t h e
p a r a b o l i cc o n s t a n t a n d v t h e v e l o c i t y o f
t h e m o v i n g p a r t .A n a p p r o x i m a t e i n t e g r a
t i o n o f E q . ( 4 ) u s i n g
E q . (5 ) y i e l d s a n a n a l y t i c r e l a t i o n f o r
M , v a l i dw h e n em i s w e l l a b o v e p r o m p t c r i t i
c a l
M = (av2em e x p4 ; /2
3 Tiav2) /2 ( 6 )T h e r e l a t i o n M(em ) can be obta ine d
for the who lerange o f em, i n c l u d i n g t h e p u l s e d s u
b c r i t i c a lr e g i o n , b y n u m e r i c a l i n t e g r a
t i o n o f E q s . ( 4 ) a n d( 5) u s i n g a s t a n d a r d r e
a c t o r k i n e t i c s c o d e . T h e nthe va lue M(p) i s o b
t a i n e d i m m e d i a t e l y u s i n g th eva lue o f em at
the p'th p u l s e .
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The se m ethods for determ ining M(p) have beenr e c a l l e d f
o r r e f e r e nc e pu r po s e s . Ho w e v e r , w h i l ean
expl ici ty expression [of the type of Eq. (6)] isrequ ired for the
ap pl i ca t ion o f the di sc rete k i -net i cs method, i t s
determinat ion i s independentfrom thi s method.As ment ioned ear l
i er , the l inear dependencebetween Fp a n d S t o t use d in Eq.
(2 ) i s not a ne ce s -sary fea ture o f the di scr ete k inet i c
s method.Exten s ions can be made to treat acc e ler ato rpu l s e
d r e a c t o r s o r s y s t e m s w he r e r e a c t i v i t y f
e e d -back during the pulse is impo rtant.B. Between Pulses
We a ssum e that between the pu l ses the promptreact iv i ty (p
- 13) has the constan t value -e 0 . Thef i s s ion rate fo l lows
the source wi th neg l ig ibledelay, and from Eq. (4) ,Mt) = |~E +
S
The pre cur sor equat ions become= ( 8 )
with boundary conditions C,(0) = C, + .We see fr om Eq. (8 )
that the pre cur sor conc en-trations at the end of the period Ci
(T ) can beexp res sed in the ter m s o f the in i t ia l va lues
and S ,for a given set of constants ft, A,-, e 0 , T. To
obtainthese exp res s ion s w e f i rs t cons ide r the cas e wi
thno e x t e r na l s o u r c e .No Externa l SourceWe w rite the
so lution for C,-( /) in ter m s of theGreen's function G,-?(f)
which gives the number ofi'th pre cur sor s at t ime t r e s u l t
i ng f r o m o ne j ' t hpr e c u r s o r at t i m e z e r o . The
n ,
Ciip (T)Cj+(p) , (9)/wh ere we se e fro m Eq. (8) that G t-,(f)
for 0 < t < Ti s the solution to the equa tions
dGif/dt = - A G if + (h/e0) E ^ mGmj (10)msub jec t to boundary
con dition s G,,(0) = 6,y.S ince G-y (:T) = Gij is a fun ct ion on
ly of A,-, /3;, e 0and T, which are o f ten cons tant for a whole
ser iesof pr ob lem s, the form ulation in Eq. (9) is ver ycon ven
ien t. A G,-? matrix can be readi ly ca lcula tedby num erica l
methods from Eq. (10) .
With External SourceTo obtain the solution of the inhom ogene
ousEq. (8) with external source, we use Eq. (9) assolutio n of the
ho mog ene ous equation and obtain a
solution of the inhomogeneous equation by intro-ducing a
function g{{t ) equal to the numbe r of i 'thpr e c ur s o r s a t
t i m e t resul t ing from one sourceneutron introduced at t im e
ze ro . Then the inho -mogeneous so lut ion may be wri t tencjnh
(T) = f : gi (T - t) S(t) d t s f j g. (t) dt , (11)where S is the
average value of the externalsou rce during the per iod. (Thus, the
solutio n isa l so good for a sour ce s trength which var i ess
lowly in t ime. )
Fro m Eq. (8) we s ee that g { i s the solution ofthe equationW
= - + T0 S ^ (12)
with boundary condition
Comparing Eqs. (10) and (12) defining Gij and gi,we see
thatgi(t) = Gi, (t) . & . (13)i e
ThusCinh(r) S s (14)
/where we define
Aa = fo GijiDdt . (15)The complete solution for C,- (p + 1) with
externalsource is the sum of Eqs. (9) and (14)CAP + 1) = E Gi,, C/
(p) + f E P>iAa (16)
7 fc jC. Com plete Precursor Difference EquationFormulation
We now have obtained the diff ere nc e equations(3 ) and (16)
exp ress ing the change in pre cur sorconcentrations over one
period as a function of theini t ia l va lues , M (em), S , T, e 0
and the kinet i c sp a r a m e t e r s Pi a nd A I f d e s i r e d
, a n e x p r e s s i o ncon tain ing on ly C,- or C,+ can be
obtained by com-bining the two equ ations. In pr act ice , i t i s
con -venien t to make the ca lculation in two step saccording to
Eqs. (3) and (16).D. Other Quan tities of Interest
N um be r o f F i s s i o ns Be t w e e n P u l s e sThe number
o f f i s s ions between pul ses in thep' th period Fb(p ) i s
calcu lated as \
Fb(p)=f?w(t)dt (17a)
/ ( w o ) . ( 7)
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and us ing Eq. (7 ) th i s bec om esF ^ ^ ^ ^ i f o d t C M
)
Using Eq. (16) and the definit ion Eq. (15)JJ dtCi (t) =
AijC+ip) + 1 0/ >t o
3
(17b)i n s t e a dy s t a t e /
M/T + j3/e 0 = 1 . (23 )If we con side r Eq. (22) as a se t of l
ine ar eq ua-t ions , we may formal ly express the so lut ion
forthe pr ecu rso r c oncen trations at the end of thepulse in s
tead y-s ta t e operat ion C? us ing m atr ixnotation asw he r
e
a{j = f*dt .Aij{t) ,and thus,
( 1 8 )Cf- = Bi w
with
ve 0X
STve 0
fyvXi ""i
(24)
(25)where B i f are e l e me nts o f the inve rse of the B
{jmatr ix which has matr ix e l ements( i ^ s ^ ) ( H ^ ^
v fo vn n 1 +V a c an rtn /4 f o yvyi rr> t k o ^ 0 /
(26)With no externa l sou rce , the second term on ther ight s ide
i s zero .Mean Fi s s ion Rate or Power in the Per iodThe mean fiss
ion rate in the period is
w(p) = (l/T)[Fp(p) + Fb(p)] , (19)w h e r e Fp(p ) an d Fb{p )
are given by Eq s. (2) and( 1 7 c ) , r e s pe c t i v e l y .
Pr ecu rso rs Concentrat ions in Steady Sta teIt i s of in ter
est to know the values of C,- (p )}Cf(p) in s teady-s ta te condi t
ions , in part i cular , tohave s tart ing va lues for a d i sc ret
e k inet i cs ca lc u-la t ion. We cons ider two ca se s : pu l sed
cr i t i ca l
(no source) ; pul sed subcri t i ca l wi th source .Pulsed
Critical:
In this ca se , we us e Eq. (19) with Eq s. (2) , (9) ,and (17c)
to exp re ss the mean pow er in ter m s oft he pr e c u r s o r c o
nc e nt r a t io ns . A f t e r s o m e r e a r -ranging, we
obtainw (20 )The m ean (per iod-a vera ged ) value of the
z'thprecursor concentrat ion in s teady s ta te i s ob-ta ined from
the neutron precursor equat ion as
Ci = & vw ( 2 1 )and thus, j3vw = J^ X {Ci, so we may wri te
from Eq.(20)
C , . = s ( f ' r e l i c t . ( 2 2 )We m ay introduce in Eq.
(22) the pu lsed c ri t i -ca l i ty condi t ion, which expresses
that product ionand decay o f precursors during a per iod are
equal
The prec urs or concentrat ions a t the beg inningof the pu lse
in st ead y state C,- can be obtainedusing Eq. (9)
C, = E Gij C* (27)The inverse matr ix (B {j) _1 can be eas i ly
ob-ta ined us ing s tandard matr ix invers io n cod es .
Pulsed Subcritical With SourceSteady -s ta te condi t ions in
subc ri t i ca l s ta te canbe obtained by using the condition
Ci( p + 1) = C i(p ) = Ci . (28)Fro m Eq s. (3) and (16), we may
write after som ea l g e br aCi(P + 1) = E G {j C , (P ) + J Pi Gij
| e X , Q ( / > ) j
The equ i l ibrium condition, Eq. (28), g iv es fromEq. (29),
the set of equations
w he r e
an d
Z/ Pij Cj - RiS )i
MPi j = 6 - Gi j - Xj E f ikPikp k
(30)
(31)
We can form al ly exp re ss the so lut ion of the seequations
for C { a sCi = ( p Pik1 -R k)s , (33)
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w h e r e P , / " 1 a r e t h e e l e m e n t s o f t h e i n v
e r s e o f th eP i ] m a t r i x w h i c h m a y b e e a s i l y f
o u n d b y s t a n d a r dm a t r i x i n v e r s i o n m e t h o
d s .E. Some Simplifying Approximations
W h e n \{T 1 , t he e x p r e s s i o n s o b ta in e d a b o
vec a n b e s i m p l i f i e d b y u s i n g a n a p p r o x i m a
t i o n o f t h eG r e e n ' s f u n c t i o n G ij i n w h i c h w
e c o n s i d e r o n l yd i r e c t z' th p r e c u r s o r p r o
d u c t i o n f r o m t h e j ' t hs o u r c e p r e c u r s o r a
n d w e n e g l e c t a l l p r e c u r s o rd e c a y d u r i n g
t h e p e r i o d . 3 T h i s i s e q u i v a l e n t t oc o n s i
d e r i n g t h e r i g h t - h a n d s i d e o f E q . ( 1 0 ) a
sc o n s t a n t a t t h e i n i t i a l v a l u e . I n t e g r a
t i o n o v e r t h ep e r i o d t he n g i v e s
% =3i + j (34)
wi th A/ = A, (l - /3,/e 0) . (35)W i t h t h i s a p p r o x i
m a t i o n , A t j a n d b e c o m e
P i X j T 2
e 0 2
(36)ir i* J
/3A/ T 3e 0 6
(37)i * j
With this ap pr ox im ati on fo r a,-,-, the las t ter m int h e
b r a c k e t i n E q . ( 1 7 c ) b e c o m e s
1e o T ^
T
2e 0
h ? ftX w h e r e t h e l a s t t e r m i s ju s t t h e f i s s
i o n s d u e t oaImpro ved ap proximations can also be used. For
ex -ample, a l lowance for decay of source precurso rs g ives
Gn = e x p ( - X j T ) ,while allowance for decay of one
precursor (7'th) gives
ftX/,T r i - exp(-A;T)"| ^ 0iXjT / X'fT \~ e0 L A)T J e0 V1 " 2
/ *
These relations are useful for improving values for theshortest
l ived precursor.
p r o m p t m u l t i p l i c a t i o n o f t h e s o u r c e n
e u t r o n s e m i t -t e d d u r i n g t h e p e r i o d .F.
Relations for One-Precursor-Group Model
A o n e - p e r c u r s o r - g r o u p m o d e l i s s u f f i
c i e n t f o rs o m e k i n e t i c s a n d d y n a m i c s s t u
d i e s . T h e r e d u c -t i o n o f t h e p r e v i o u s m u l
t i g r o u p e q u a t i o n s t o o n eg r o u p y i e l d s
:
Fp(p) = [M{p)/M (AC +S )C+(p) = [1 + AM(p)] C(p) + M(p) S
C(p+1) = GC(p) + (p/e0)ASFb(p) = [AC +(p)/ve0]A + (ST/ve0)
G = e x p [ - A ( l - / 3 / e 0 ) T ] = e x p ( - A T )A = ( 1 -
G ) / A '
T h e s e e q u a t i on s r e d u c e t o t h e n o - s o u r c
e c a s ew i t h S = 0 . T h e s t e a d y - s t a t e c o n d i t
i o n s r e d u c e t ot h e f o l l o w i n g :
(38)
no source
C~ =ft vw p vwA
1 l + A Te 0 2(1 - AT)(39)
with sourceGM + A - tLc~ sL " 1 - ( 1 + X M ) G
PX'TK +2 e 0 ( l - A T )A T SA (40)A T
w h e r e , i n t h e a p p r o x i m a t i o n s , E q s . ( 3
4 ) a n d ( 3 6 )h a v e b e e n u s e d , an d t h e v a l u e of
t h e s u b c r i t i c a lm u l t i p l i c a t i o n f a c t o r
h a s b e e n i n t r o d u c e d
K=M/T+j3/e0 . (41)W e n o t e , f o r c o m p a r i s o n , t h
e r e l a t i o n s f o r t h e
m e a n ( i . e . , p e r i o d a v e r a g e d ) p r e c u r s
o r c o n c e n -t r a t i o n s :no source
with sourceC = fivw/X (42)
C = {S/X)K/{1 -K) . (43III. "INHOUR EQUATIO N" FORPULSED
REACTORS
A. General FormulationIn ord er to obta in an ' ' inhour equ at
io n" for the
b e h a v i o r o f Q(p), w e c o n s i d e r a s t e p c h a n
g e i n
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reac t iv i ty from pulsed cr i t i ca l i ty and see k s o lu
-tions to the difference equations of the formC t{p + 1 ) = e x p (
u > r ) Ciip) . (44)
Introducing this into Eq . (16) and usin g Eq . (3) toel im ina
te C}, we obtaine x p ( w r ) C ^ p ) = E G i j C ji p ) + j Z G i
j p j
[ E HCkiP)\ (45)Th is i s a s et of hom ogen eous equations
which canbe wri t ten in matr ix form as
(F ) [C,-] = exp(cor) [C{] (46)wi th the matr ix e l ements
Fi j = G ( 7 + ^ X ; - E foGih (47)P k
Sett ing the determinant /F - e x p ( w T ) E / = 0 , where is a
unit ma trix, y ield s the inhour equation forthe decay constants w
k .B. Simplification for Slow Transients
For per iods where wT 1 , w e m a y a p p ro x i-m a t e e xp(
co r ) = 1 + u T . We m a y a l s o us e t heapproximate Eq. (34)
for G i;-, valid for X,- T 1 .Introducing these approximations into
Eq. (45) andcarry ing through the summation over j y i e l ds- ( .
. x a r C f ^ j M ^ r /Xf - o 1 T
x e ^ C i ( p ) = o
ftAf h M1 (w+Xi) Ie0 T 1
We ne gle ct [X,- - (l /e 0 ) S fo>k\T compared tounity and
introduce the multipl ication factor Kfrom Eq. (41); then
Multiply by X,/(w + X,)T and sum over i; cancel thec o m m o n t
e r m Z Xm Cm (p ) to find
1 = ^ E P Aij3(w + Xf) (48)
Th is m ay be rear ran ged to give the inhour equ a-t ion prev
iou s ly determ ined from the mean powerk i ne t i c s , 2 ' 3
K y |3(w + Xf ) (49)
The pulsed crit ical i ty condition K = 1 is see n toresul t
from thi s equat ion.
I V . C O N C L U S I O N
The d i scre te neutron kinet i cs model for pul sedreactors
developed in th i s paper i s an extens ion o fthe m ean kinet i cs
model which a l lows accuratetreatment o f pul sed reactor kinet i
cs for muchf a s t e r t r a ns i e n t s .An inhour equation for
pulsed reactors hasbeen developed from the di sc rete k inet i cs
equa-tions and shown to r educ e to the inhour equationdetermined
from the mean power kinet i cs in thel imit of periods much longer
than the pulse period.The di f fer ence formulat ion developed i s
wel lsuite d for digital calculation . The multigrou ptrea tme nt
has b een include d in a digital pulsedreacto r s imulat ion code
inc luding therma l andcontro l feedback loops . The one-grou p
treatmenti s be ing incorpo rated in a d ig i ta l -ana logue pul
sedreacto r s im ulato r . These app l i ca tions wi l l bedescr
ibed in future publ i ca t ions .
