physical meaning of kinetics parameter “lifetime” used in the new multi-point reactor kinetics...

~ ) Pergamon 031)6-4549(95)00061-5 Ann. Nucl. Energy Vol. 23, No. 10, pp. 827-841, 1996

Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved

0306-4549/96 $15.00 + 0.00

PHYSICAL MEANING OF KINETICS PARAMETER "LIFETIME" USED IN THE NEW MULTI-POINT

REACTOR KINETICS EQUATIONS DERIVED USING GREEN'S FUNCTION

KEISUKE KOBAYASHI

Department of Nuclear Engineering, Kyoto University, Yoshida, Sakyoku, Kyoto, Japan

(Received 16 May 1995)

Abstract--The reactor kinetics parameter l which is used in the new multi-point reactor kinetics equations derived using Green's function is shown to have a physical meaning of neutron production time which can be calculated by the Monte Carlo method, although the corresponding reactor kinetics parameter used in the conventional one-point reactor kinetics equations derived using an adjoint function as a weighting function has, in general, no clear physical meaning.

1. INTRODUCTION

Usually one-point reactor kinetics equations are derived using an adjoint function as a weighting function, and reactor kinetics parameters like A appearing in these equations have, in general, no clear physical meaning (Lewins, 1978; Henry, 1986).

Nelson (1971) derived equations to calculate a neutron mean production time and destruction time analytically which are the mean time for one neutron to produce next generation neutrons and the mean lifetime for one neutron to be removed from the system by absorption or leakage, respectively, and he showed numerically that these times agree well with those calculated by the Monte Carlo method. He also showed that these times do not agree with those used in the conventional one-point reactor kinetics equations derived using the adjoint function as a weighting function.

Recently, new multi-point reactor kinetics equations have been derived rigorously using coupling coefficients calculated by making use of Green's function (Kobayashi, 1991a,b, 1992). If we use these equations as one- point reactor kinetics equations, these equations have a similar form as the conventional one-point reactor kinetics equations derived using the adjoint function.

The purpose of the present paper is to show that the kinetics parameter l used in the new one-point reactor kinetics equations has a clear physical meaning. Namely, it is shown that the kinetics parameter l used in the new one-point reactor kinetics equations is equal to the mean production time by making use of the equations derived by Nelson (1971).

This may be due to the fact that in the conventional one-point reactor kinetics equations the dependent variable weighted with the adjoint function has no clear physical meaning, whereas the dependent variable used in the new multi-point reactor kinetics equations has a clear physical meaning of the number of fission neutrons produced.

Marotta (1981) discussed the difference between the production time and destruction time. In the present work, a simple expression for the ratio of the production time and destruction time is given from which we can see when both quantities become the same or how the ratio depends on system properties.

2. KINETICS EQUATIONS DERIVED USING GREEN'S FUNCTION

Rigorous multi-point reactor kinetics equations have been derived using a transport equation and a Green's function for a perturbed system (Kobayashi, 1991b), and using a diffusion equation and a Green's function

827

828 Keisuke Kobayashi

for an unperturbed system (Kobayashi, 1992). Here, we apply the later results to the transport equation using Green's function for an unperturbed system, and we consider here only one-point reactor kinetics equations.

In the following, equations (H.1), (KI.I) , (K2.1) and (N.I), for example, mean equation (1) of references of Henry (1986), Kobayashi (1991b), (1992) and Nelson (1971), respectively, and only necessary equations will be quoted.

We assume that a system is critical with a criticality factor k, and the angular flux fx(r, 12) for the direction 11 and g-th energy group satisfies the following steady state multi-group transport equation in region V,

Aft(r, a ) = 1 Bfg.(r, 113, (1)

where operators A and B are defined by

Af~(r, l'~) = ( a . V + Y.~)ft(r, n) - E f4 d11" Z~(g, a ~ g', n3fg,(r, 1"~, g ' ~t

(2)

and

Bf~,(r, [¥) = 4~ S(r), S(r) = Ffg,(r, 113. (3)

S(r) is the fission source, and F is the fission operator defined by

Ff~,(r, n3 = E v Xf,,(O f, dt)%,(r, n'). g ' x

(4)

5ztg, v Eft, Xg and Z~(g, £1*-g', ~ ' ) are the total cross section, fission cross section multiplied by the number of fission neutrons, the average fission neutron spectrum for prompt and delayed neutrons, and the scattering cross section from g'-th group and direction fl" to the g-th group and direction fl, respectively, which can be space dependent.

The boundary condition of no incoming neutrons is used at the external boundary S of the system V, namely

ft(r,11) =0 , forn-11<0 a t r e S , (5)

where n is an outward unit vector normal to the boundary surface S of system V. Equation (1) can be regarded that the system is just critical with a fictitious fuel which has a number of

fission neutrons ¢ = vlk. The angular flux is expressed using the neutron density n~(r, ~) and velocity vg as

fg(r, 1~) = n~(r, 12)v r (6)

Using the adjoint operator A t of the operator A of equation (2),

.4t = - a - V + E ~ - , ~ f , , dl ' l"~,(g ' ,~ '~g, l ' l ) , (7)

Kinetics parameter "lifetime" used in the new multi-point reactor kinetics 829

a function Gg(r, II) is defined by the following equation [equation (Kl.13)],

AtGg(r, ['~) = v ]~fg(r). (8)

This function must satisfy the following boundary condition,

Gg(r, fl) =0 , forn-[~>0 atrES . (9)

The function Gg(r, [1) has a physical meaning explained at equation (43) and can be called as an importance function (Kobayashi, 1991). However, this function can be calculated from Green's function by equation (Kl.12), and we call it here simply as Green's function to avoid confusion with an adjoint function used in the next section, since the adjoint function can be also called an importance function (Nelson, 1971).

We assume that a perturbation is added to the system of equation (1) and the operators change as A" = A + fiA, B' = B + fiB, and the angular flux changes by the equation

l dfg(r'~'t)-(-A'+l(1-fl)BP')fg,(r,~',t)+lzz~A,C,(r,t ), (10) vg at i

where operator B p' for prompt neutrons is defined by

Bp,fg,(r, n , , t ) = 1 x~F,fg,(r, Fl,, t ) = 1 Xpg(F + 6F)fg,(r, n , , t). (11)

C,(r, t), 21 and ~, are the density of the delayed neutron precursor, its decay constant, and the energy spectrum of delayed neutrons, of i-th delayed neutron group, respectively. Z~, v~ and fl are the prompt fission neutron spectrum, the neutron velocity of g-th group and the total delayed neutron fraction, respectively.J/

Multiplying equation (10) by the Green's function of equation (8) as the weighting function and integrating it over the whole region V, we obtain the one-point reactor kinetics equation

l ( t ) ~ t ) = ( P ( l ) - l f l ( t ) k P ( t ) ) S ( t ) + ~ k ~ ( t ) 2 , C , ( t ) , (12)

where S(t) and C,(t) are the fission neutrons produced in a unit time and the delayed neutron precursor of i-th group in the whole system V, respectively, defined by

S(t)-- f vS ( r , t )dr , C,(t)= fv C,(r,t)dr. (13)

Here, the time dependent fission source S(r, t) is defined using the perturbed fission operator by equation (K2.17),

S(r, t) = F'f~'(r, ~ ' , t) = (F + 6F)fr(r, 0", t). (14)

The reactivity due to the perturbation is defined by

p(t) = 1 kP(t ) + AkF(t) _ AkA(t) _ 1, (15)

The criticality factor k in equation (10) is unity, if the system described by equation (1) is critical with k = 1.


with coupling coefficients for prompt neutrons,

47tl£dr£ df~Z Gg(r,~)x~,S(r,t) n g

kP(t) , (16)

fv drS(r, t)

and with the direct changes of coupling coefficients due to the perturbation of 6A and tSF defined by equations (K2.22) and (K2.23), namely

fvdr £ df~ Z Ge(r' a)6Afg(r, U, t) g

Ak~(t) _" g drS(r, t)

(17)

fvclr £ df~aFf~,(r,.q', t)

A/~(t) - "

fv d r S ( r , t )

(18)

The coupling coefficients for delayed neutrons of i-th group, the effective delayed neutron fraction and kinetics parameter l are defined as equations (K2.18)-(K2.21), namely

4 1 £ d r £ d f ~ Z Gg(r,~)z{C,(r,t) k~ (t) = g , (19)

f vdrCi(r, t)

fv dr Z Gg(r, ~)flz~S(r, t) fl(t) = * , (20)

f v. dr Z Gg(r, al,~PS(r, t) g

dr £= dn ~' %(r, a) 1 Off(r, ~ , t) g v~ Ot

Z(t) = (21) fvdr OS(r, t)

Ot

In the asymptotic time region, the flux and precursor can be assumed to change with a single period as

fg(r, ~, t) =fg~(r, ~)e '°', C,(r, t) = Cio,(r)e ~', S(r, t) = S~(r)e ~,. (22)


Using equations (22), l of equation (21) becomes

l = (23)

fv drS~(r)

If there is no perturbation, co = 0, fgo(r, ~) =fg(r, ~) and Sgo(r) = S(r). In this case, l can be expressed as

l - , (24)

n g

where the following relation [equation (K1.15)] is used

n g

(25)

3. KINETICS EQUATIONS DERIVED USING THE ADJOINT FUNCTION

Conventional one-point reactor kinetics equations derived using the adjoint function are quoted briefly in the following (Henry, 1986).

The adjoint equation to equation (1) is

Atftg(r, I~) = I Btf~,(r ' IY). (26)

The boundary condition for ~(r, fl) is the same as equation (9) for Gg(r, ~). Operating on equation (10) with

f f4 an 'fi(r, n g

the conventional one-point reactor kinetics equation is obtained (Henry, 1986);

dt~(t) p t ( t ) - _fit(t) t~tD dt - It(t) "" + Z ).,(7,(t), (27)

i

where the dependent variables are defined by

g g

(28)

C,( t) dn Z B(r, g

(29)


Here, the kinetics parameters are defined using the loss rate Afg,(r, fl ' , t) in the denominator as equation (N.4) by

p * ( t ) =

fedr; d~f~g(r,~)(--A'+lB'~g'(r, ll', t)

fzdr ~ dD ~f~(r, n)Afg,(,, n', t) x g

fedrf4d~J~g(r,~)(-fiA+lfB~g'(r,~', t)

;dr ~ dn E~(r, l~)AA,(r, f~', t) g

(30)

P(t) = " , (31)

fvdr f4 d'Q Efg(r,~)Afg'(r, Fg, t) z g

fv drf4 d'O Ef~g(r, fl) ~-~ ]~,B~i f~' (r, 1"~', t) x

/~(t) = g ' , (32)

fvdr ~ d~ ~(r,n)Afg.(r, lY, t) n g

where fli is the fraction of the i-th delayed neutrons, and

1 d = ~ z~F. (33)

The advantage of using the adjoint function is that the error in the reactivity of equation (30) is the second order. However, the dependent variables of equations (28) and (29) have no clear physical meaning, and there are differences in their definitions of the kinetics parameters, since the definition of kinetics parameters by equations (30)-(32) has an ambiguity in which quantities are used for the weighting functions and for the denominator. Namely, Henry (1986) used an arbitrary weighting function Wg(r) and also the same adjoint fluxFAr, f~) in his equation (H.8.3.44) as in equation (26). Lewins (1978) and Nelson (1971) used the adjoint neutron density n*g(r, FZ) instead of the adjoint neutron flux fg(r, f~).

Thus, the magnitude of the kinetics parameters depends on the definition and has no unique physical meaning, and only the ratio p*(t)/P(t) of the form in equation (27) has a meaning as discussed by Henry (1986).

4. M E A N P R O D U C T I O N AND D E S T R U C T I O N T I M E S

Nelson (1971) showed an equation to calculate a mean production time analytically. We quote here his equations in a multi-group form instead of his continuous velocity variable.

Let Pf(t, r', g', ~',-r, g, f~)dt be the probability for a neutron born at time t = 0, position r, energy group g and direction f~ to induce a fission between t and t + dt at position r', energy group g' and direction ~ ' .


The expectation number of fission neutrons produced between t and t + dt by a neutron born at time t = 0, position r, energy group g and direction f2, Pf(t, r, g, 11)dt is given by equation (N.7), namely

Pf(tr, g,11)dt=fzf4~v~.(r')Pf(t,r',g',11"*--r,g, 11)dtdr'd~', n g"

(34)

where vr(r') is the mean number of fission neutrons at position r' and energy group g'. The mean production time zp for a neutron from its birth to the production of a next generation neutron

is given by equation (N.6), namely

frf4~Bfs"(r, 11")(fo~tPf(t,r,g, 11)dt) drd11 Zv - ( 3 5 )

~ Bf~,(r, 11") ef(t,r,g, 11)dt dfl n g ~ J O - -

This production time can be calculated by the Monte Carlo method as follows. Neutrons are lost from the system by fission, capture or leakage. For a neutron which induces a fission, let the flight distance between (n - 1)-th and n-th collisions be 2,, and its velocity be v~. Then the time between collisions is z~ = 2Jv~. If the neutron makes a fission at N-th collision, the mean time ~ for a neutron from its birth to the next fission is given by

N Zviy . T , z~= , (36)

i

where i is a suffix for histories and vj is the number of fission neutrons produced at N-th collision. Repeating many histories for neutrons born with the probability Bfg.(r, 11") and calculating the average value of z, we can obtain an expectation value z v of equation (35). In such a Monte Carlo calculation, we should not use any biasing method.

The probability P~t,r',g',11"*--r,g, 11) is shown by Nelson (1971) to be given by his equation (N.13), namely

aP~t,r',g',11"*--r,g,11) t , • dt - - A vtP~t,r',g ,11 ,-r,g, ll). (37)

The initial condition at t = 0 is given by equation (N.14),

e ~ 0 , r', g' , 1 1 ' , - r, g , 11) = v , Zf,(r)~(r' - - r )~ , ,~(11" -- 11). (38)

The boundary condition is given at the outer boundary S of the system V by equation (N.15), namely

P~t,r',g',11",-r,g, 11)=O, for n '11>0 at reS. (39)

It is assumed that the following condition, equation (N. 16), holds at t---, or,

lim tPf(t, r', g', 11' , - r, g, 11) = 0. (40)


Multiplying equation (37) by vg'(r'), integrating it over r ' and fg, and summing up with respect to g', we obtain

OP~t, r, g, ~) AtvgP~t, r, g, ~). (41) Ot

The initial and final conditions for probability Pt(t, r, g, ~) are obtained using equations (34), (38) and (40) as

P~O, r, g, ~) = %vg(r) Zfg(r), lim tP~t, r, g, fl) = 0. (42)

According to Nelson (1971), a function Gg(r, ~), which expresses the expected number of neutrons produced by a neutron born at position r in energy group g and direction fl is given by equation (N.20), namely

Gg(r, ~) = vg,(r')P~t, r', g', ~" ~ r, g, ~) dr 'dlY dt n g"

= Pf(t, r, g, ~)dt. (43)

Integrating equation (41) over t from 0 to oo, and using the definition of equation (43), we obtain

t = o o

Pr(t, r, g, ~) ,=0 = -- Atvgag(r' ~)" (44)

Using equations (42), equation (44) becomes

AtGg(r, [~) = vg(r) Efg(r). (45)

From equation (39), the boundary condition for Gg(r, ~) is

Gg(r,~) =0, for n - ~ > 0 at r~S. (46)

Equation (45) and the boundary condition (46) are equal to those of equation (8) and equation (9) which are used to derive the multi-point reactor kinetics equations.

Since the numerator of equation (35) can be rewritten as equation (N.22), namely

f v f 4 ~ B f , ' ( r , " ' ) ( f o ° ° t P ' t , r , g , i ' ~ ) d t ~ rdf~

= k f z ~ , ~ f ~ ' ( r , IY)(~o~tAtP~t,r ,g,~)dt) drd[2


n g g

(47)

% of equation (35) becomes

fv ~ ~ 6~(r, Et)n,(r, n)d rd f t n g

(48)

We can see that the production time % of equation (48) is the same as l of equation (24) for the case of to = 0 without perturbation. Namely, the kinetics parameter l used in the one-point reactor kinetics equations, equation 02), has a clear physical meaning of the mean production time.

Using equation (25), Zp of equation (48) can be also expressed as

f v ~ d rdn 2 Gg(r, ft)n~(r, n) n g

~P ~V S(r) dr (49)

Writing the right-hand side of equation (26) explicitly, we obtain

1 Btf~g,(r, ~ ' ) 1 ;(g" I4 ' ' = ~ zfl ,)~ ~ - ' ~ daTA,, a ) . g"

(50)

If the adjoint flux is independent of position r, and we normalize it as

_1 N~ Xg' ( dEE fret, EE) = 1, kZ-/4rc J4 ~

g

(51)

equation (26) becomes the same as equation (8). Namely, if the system is homogeneous and infinite, and the adjoint flux is independent of the position, the adjoint function ftg(r, fl) becomes the same as Gg(r, ~) of equation (8). In this case,/t of equation (31) is the same as I of equation (24).

In Hayashi's discussion (1992, 1994) about the lifetime and production time, he claimed that the kinetics parameter of equation (31) with a weight of the adjoint function has a physical meaning of the production time and can be calculated from the Monte Carlo method. It is clear, however, that this is valid only for an infinite system.

Nelson (1971) gave an equation for calculating a mean destruction time for a neutron from birth to disappearance from the system. Let Pa(t, r, g, Et)dt be the probability for a neutron born at time t = 0, position r, energy group g and direction 1~ to be lost by absorption or leakage from the system between times t and t + dt. Then the mean destruction time Zd from birth to death is given by equation (N.5), namely

(52)


Integrating equation (1) over the position and direction, and summing over energy groups, we obtain

(53)

The left-hand side of equation (53) means the number of neutrons lost per unit time, namely the loss rate, and the right-hand side the production rate.

Using a similar formulation to derive equation (48), Nelson (1971) showed that the mean destruction time Td is given by the inverse of the left-hand side of equation (53) with the normalization of the density to unity, equation (N.3), namely

* (54)

x g

From equation (25), the criticality factor k is expressed as

k - n g

fv ~ Z~S(r) dr (55)

Now, replacing zgS(r) by nz(r, 11) in equation (55), let us define the following quantity c,

c -

dr 1 (56)

The k of equation (55) is an average value of the expected number of fission neutrons Gt(r, fl) produced by a neutron born at the phase space (r, g, I1) with a weight of the fission source S(r), whereas in equation (56), the average is made with a weight of the neutron density n~(r, ~) , and in general, x,S(r) ~ n~(r, 11) and then c ~ k.

Using equations (55) and (56), we can see that the mean production time Zp of equation (48) has the following relation with the mean destruction time Td of equation (54),

C *p=~za, (57)

from which we can see that in the case of c = k, zp = Za. If we use a weighting function of unity instead of the adjoint function in deriving equation (27), the

dependent variable and kinetics parameters have an obvious physical meaning: the dependent variable of equation (28) becomes the integral of neutron density, and r(t) of equation (31) becomes equal to the destruction time *o of equation (54).


5. SIMPLE EXAMPLES

We will consider here some simple cases explicitly where analytic solutions can be obtained.

5.1. Infinite homogeneous system with one group

In this case, from equations (1) to (8), we obtain

A=Y~a, B=vY . f , k vY.f S = 4 n v Y . f n v ' G = V Y . r = k = y----f, ~.~ •

(58)

Then, from equations (49) and (55), Zp and Zd become

v Ef - - n

4nGn Ea 1 n 1

zp S v Ernv v E, ' ~d = Ya vn = v Ea" (59)

In this case, the destruction time Xd is equal to production time %, since c = G = k holds in equation (57). If we rewrite % of equation (59) as

1 1 % = 1 vOY-f =zd' (60)

v-~v~,r

we can see that this is the reproduction time defined by Lewins (1981) for the system which is critical with a fictitious fuel that has a value ¢ = vlk.

5.2. Infinite homogeneous system with two groups

In the case of a homogeneous infinite system with two groups, equation (1) becomes

1 E,O~ = ~ v ~.r,.~, E~f2 = p E~tf~, (61)

where Y-r~, Y-s, and p are removal and slowing down cross sections and resonance escape probability, respectively, and ft and f2 the angular fluxes of the first and second group, respectively.

From equation (61), normalizing the flux of the second group as S = 4nv Y-o_f2 = 4n, we obtain the flux of the first group and the criticality factor k as

Ea2 1 k = p E~l v Ef2 (62) f l = n l V t - p E s , vEr2' f 2 = n z v 2 = ~ n ~ ' Y-,I Y-~2"

Equation (8) for Green's function becomes

(63)

Solving these equations, we obtain

v ~-f2 '7- --P ~'al v ~'U = k, G2 - (64)

As expected, G~ and G2 express the expected number of neutrons produced by one neutron of the first and second group, respectively, where GI is a product of a probability for a neutron of the first group to be

838 Kcisuke Kobayashi

Table 1. Production and destruction times for infinite systems with two groups

E,2 (cm ~) 2.4000 1.9200 1.6000 k equation (62) 0.8000 1.0000 1.2000 k +_ Ok 0.8015 _+ 0.0014 1.0012 _ 0.0015 1.2015 _+ 0.0017 % (10 5 S) equation (65) 1.208 1.260 1.313 % ± % (10 -~ s) 1.208 __4- 0.002 1.261 ± 0.002 1.313 _+ 0.002 za (10 5s) equation (66) 1.167 1.208 1.250 zd ± a~ (10 5 s) 1.168 _+ 0.001 1.209 ± 0.001 1.251 + 0.001

slowed down to the second group and an expected number of emitted fission neutrons due to an absorbed neutron of the second group.

Using equations (62) and (64) in equations (49) and (54), we obtain

1 I. (65) "gP = ~ "[- 'U2 ~'~'a2 '

1 1 1 ) 1 1 za = k ~ V ~ - r + V Z ~ = - V ~ + V z f E f 2 '

(66)

which are easily confirmed to satisfy equation (57). In this case, % # Zd even if k = 1. Since p ~'~,sll/~rl < 1, and ¢ E ~ > Z a , then Tp•Z d holds independent of the value o f absorption cross section

~a2. In Table 1 some numerical results are shown for this two group problem. Cross sections ~sl l ~ ~ r l =

0.1 cm 1, Y.f2 = l cm -j , a n d p = 0.8, v = 2.4, vl = 106 c m / s , V2 = 2 x 105 cm/s are used. Monte Carlo calculations are performed using random numbers and equation (36) etc. by making a simple For t ran program. In all cases for three different absorption cross sections, 100,000 histories are used, and standard deviations tr for each quanti ty are also shown.

It is seen that the production and destruction times calculated by equations (65) and (66) are in good agreement with those calculated by equation (36) etc. by the Monte Carlo method.

5.3. Finite slab with one group

In order to calculate a solution analytically, we use a diffusion equation. In this case, using the diffusion coefficient D, equation (1) for one group becomes

-- D ~ + Ead~(x)=lv zf~(x). (67)

We assume that the thickness of a slab is a including the extrapolation distance, and we use the zero flux boundary condit ion at x = + a12.

The solution o f equat ion (67) is

~(x) = nv cos(Bx), k = k~ 1 + L2B 2' koo = v ZJEa, (68)

where B = r~la. Green's function of equation (8) is given by equation (K2.46), namely

a(CO h cosh - -

2

(69)

where x = IlL = x/EalD.


Using equations (68) and (69) in equations (49) and (54), we obtain

;i ~2 G(x)n cos Bxdx a12 l® = k 1 Zp- j- 2 1 + L2B 2 vv Zf v17 Zf' (70)

v Zrnv cos Bxdx a/2

f f2 n cos Bxdx o/2 1~ 1

- D ~ + Z~ nvcos Bxdx ~2

From equations (70) and (71), we see that Zp = Zd holds independent of k. This is because the equality k = c holds independent of the magnitude of k. ~d of equation (71) is the same as that obtained for the simple homogeneous system by the analytical method (Lamarsh, 1966).

Since the one group diffusion equation of equation (67) is self-adjoint, the solution of the adjoint equation of equation (26) for this problem is proportional to ~b(x) of equation (68), which is different from Green's function of equation (69) even for a one group problem in the case of a finite system. Green's function of equation (69) is more flat compared to the adjoint function of equation (26), and in general % may be different from % as discussed by Nelson (1971).

6. R E L A T I O N W I T H O T H E R R E S U L T S

Using the eigenfunction expansion method, Henry (1983) derived an equation to calculate the generation time which is given by

f dp ~ f. ~ v Yfg(r)qJ.(p)

f dp ZLv. (72)

if we rewrite the continuous energy variable by the discrete group form. Here the notation p is used to express all dependent variables (r, g, 11). The 2. and qJ.(p) are eigenvalues and eigenfunctions of the operator A defined by

A ~b.(p) = 2. ~O.(p), (73) %

and f . is the expansion coefficient of the angular flux f(p) of equation (1) expanded using the eigenfunctions ~.(p) as

f(p)=Zf.~.(? ). (74) n

Let us show that the generation time given by Henry is mathematicaUy equivalent to the production time given by equation (49). We can see that the denominator of equation (72) is equal to that of equation (49).


We define the eigenfunctions for the adjoint operator A t by

At@~(p) = ~ @~(p), (75)

and we assume that the eigenfunctions are normalized as

f dp~(p) ~ ) = t~,. (76)

Multiplying equation (74) by the adjoint eigenfunction ~,~(p), the expansion coefficients are obtained as

f.=fdpl¢.tr)fCp). (77)

We expand the Green's function of equation (8) using the adjoint eigenfunctions as

G(p) = E g"~'~(P)" (78) n

Substituting equations (78) into (8), multiplying the eigenfunction ~bn(p) and operating S dp, the expansion

coefficient g. is obtained as

Using equation (79) in equation (78), the Green's function is expressed as

Substitution of G(p) of equation (80) into the numerator of equation (49) yields

from which we can see that the production time given by equation (49) is equivalent to the generation time of equation (72) derived by Henry.

Henry (1983) gives simple examples for generation time and lifetimes of one and two group problems with the diffusion approximation. We can see that the production time and destruction time for one group problem given by equations (59) are the same as those given by Henry. We can see also that those by equations (65) and (66) for two group problems are the same as those given by Henry (1983), if we put B 2 = v Zn = 0.

Lewins (1960) gives the generation time for the homogeneous infinite multiplying medium as

1 % = (82) vv ~.f"


However, if the system is not critical and the non-critical system is assumed to be expressed by the fictitious static equation of equation (1) with the criticality factor k, the production time is given by equation (60), namely, the generation or production time depends on the criticality factor k and it is equal to the lifetime.

7. CONCLUSION

We can summarize the above discussions as follows:

1. If the adjoint function is used as a weighting function as in equations (28)-(32), the dependent variable and P have no clear physical meanings. However, the reactivity given by equation (30) has an advantage that the error to use the unperturbed flux is the second order.

2. If a weighting function of unity is used in equations (28)-(32) instead of the adjoint function, the dependent variable and ! have the physical meanings of the number of neutrons and destruction time Td, respectively.

3. If the Green's function of equation (8) is used as the weighting function, the dependent varible ~(t) and I have the physical meanings of the number of fissions and production time %, respectively. Only using this Green's function, the multi-point kinetics equations can be derived rigorously.

If we use Green's function for problems of multi-regions of 235U fuel and 239pu fuel, for example, we can derive multi-point reactor kinetics equations whose dependent variables and kinetics parameters have physical meanings of the fission rate, the effective delayed neutron fractions and generation times for each region.

Acknowledgement--The author wishes to express his sincere thanks to Dr E. Kiefhaber of Forsehungszentrum Karlsruhe, Germany, for his many useful comments.

REFERENCES

Hayashi M. (1992) 1992 Fall Meeting of the Atomic Energy Society of Japan B29. Hayashi M. (1994) 1994 Fall Meeting of the Atomic Energy Society of Japan G72. Henry A. (1983) Adv. Nucl. ScL Technol. 15, 55, Henry A. (1986) Nuclear-Reactor Analysis, Chap. 7. The MIT Press. Kobayashi K. (1991a) Ann. Nucl. Energy 18, 13. Kobayashi K. (1991b) J. Nucl. Sci. Technol. 28, 389. Kobayashi K. (1992) J. Nucl. Sci. Technol. 29, 110. Lamarsh J. R. (1966) Introduction to Nuclear Reactor Theory, Chap. 9. Addison-Wesley. Lewins J. D. (1960) Nucl. Sci. Eng. 7, 122. Lewins J. D. (1978) Nuclear Reactor Kinetics and Control, Chap. 2. Pergamon Press, Oxford. Lewins J. D. (1981) Nucl. Sc£ Eng. 78, 105. Marotta C. R. (1981) Nucl. Sci. Eng. 78, 106. Nelson P. Jr. (1971) Nucl. Sci. Eng. 43, 154.

physical meaning of kinetics parameter “lifetime” used in the new multi-point reactor kinetics...

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