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The time-dependent multigroup transport equations in reactor kinetics
Ann. nucl. Energy, Vol. 16, No. 6, pp. 279-291, 1989 Printed in Great Britain.All rights reserved
0306-4549/89$3.00+0.00 Copyright © 1989PergamonPress plc
THE TIME-DEPENDENT MULTIGROUP TRANSPORT EQUATIONS IN REACTOR KINETICS G.-S. CHEN Department of Nuclear Engineering, National Tsing Hua University, Hsinchu, Taiwan, R.O.C. 30043
(Received 8 December 1988) Abstract--A rigorous description of nuclear reactor kinetics usually invokes neutron transport theory. If the multigroup transport theory is considered, the description of space-time nuclear reactor kinetics consists of partial integro-differential equations coupled with ordinary differential equations. The purpose of this paper is to give an iterative scheme for the construction of the solution of the coupled systems and to establish some qualitative analysis of the coupled system. The iterative scheme consists of two monotone sequences, which converge monotonically from above and below to a unique solution, respectively. The qualitative analysis includes the existence and uniqueness of a positive solution, explicit upper and lower bounds of the solution, and sufficient conditions in terms of reactor parameters for subcriticality and supercriticality of the time-dependent system.
l. INTRODUCTION It is well-known that the one-speed transport equation has been extensively studied in neutron transport theory as an idealized model of nuclear reactor system. The principle deficiency of the model is the assumption that all neutrons can be characterized by only a single speed or energy. However, the neutrons in a reactor have energies range from 10 MeV down to less than 1 ¢V. Moreover, the neutron-nuclear cross sections depend rather sensitively on the incident neutron energy. Hence, in any accurate description of realistic reactor system, it is necessary to consider the neutron energy dependence in more detail. The multigroup transport equations is the most popular model in which energy-dependent effects are included. In this article, we will study the time-dependent multigroup transport equations for reactor kinetics with the form (Lewis and Miller, 1984; Duderstadt and Martin, 1979; Williams, 1971):
Vg ~i ~g(r'fLt)+f~'Vd~g+%(r)~g =
g'=t
aog,(r, fV,~)dpq,(r,~',O d~'
dS
"
j=l
and for the delayed neutrons ~C(r,t)=
£a ~,
vg,trfa,(r,n')qbg,(r,~)',t)df~'-2jCj(r,t ) j = 1. . . . . 6,
(t>O,r~D,n~S).
(2)
The quantity va is the average speed of the neutron in group g, while tra, afa represent the macroscopic absorption and fission cross section, respectively, in group g. The macroscopic cross section tra.q, refers to a scattering interation which changes the neutron in group g ' with direction fl' to group g with direction f~. The quantity vg, is the average n u m b e r of neutrons produced in a fission induced by a neutron in group g'. The angular flux in group g is denoted by ~bg, which is the neutron speed vg times the n u m b e r of neutrons at point r with direction l) at time t. The quantity Cj represents the concentration of delayed neutron precursors of type j with decay constant 2j and delay fraction flj (fl = X j=, 6 flj). The fraction of the neutrons produced directly by fission and by precursor decay that have energy within group g are denoted by 7~ and ZJa, respectively. The quantity D is the reactor region, which is assumed to be a bounded convex domain in R ~ (m = 1,2, 3), S is the unit s0here in lks, and V is the gradient operator with respect to the spatial variable r ~ R =. ASE X6".6-e 279
280
G . - S . CnEN
In addition to equations (1) and (2), we require that there be no incoming neutrons for all positions r on the boundary surface : ~ba(r, t , t) = 0,
(t > 0, r e t~D, t incoming)
(3)
and we consider the initial condition ¢ , ( r , t , 0) = ¢g0(r,t), Ci (r, 0) = Go (r),
(reD, tieS),
(4a)
(r e D),
(4b)
where ~D is the boundary of D. The time-dependent neutron transport equation has been treated by many investigators and a number of analytical and numerical methods have been developed for the determination of the solutions. (For example, Case and Zweifel, 1963 ; Richtmyer and Morton, 1967 ; Vidav, 1968 ; Belleni-Morante, 1974; Pao, 1981, 1983 ; Vida, 1968 ; Ganapol, 1980 and Wilson, 1974.) When neutron energy dependence is taken into account, it is often investigated in the framework of either the multigroup diffusion approximation or the multigroup transport method. (For example, Kastenberg and Chambrt, 1968; Stacey, 1969; Chen and Leung, 1985; Larsen, 1979; Lewis and Miller, 1984; Hill and Reed, 1976; Lathrop et al., 1971.) Some attention has also been given to the problem of the multigroup time-independent transport equations. (For example, Williams, 1985; Lee et al., 1985; Wood, 1985; Wood and Williams, 1986; Kobayashi et al., 1986; Sanchez and McCormick, 1982 ;Chen and Leung, 1989.) The purpose of this paper is to present a monotone iterative scheme for the construction of the solution, and to study some qualitative property of the system (1-4). This includes the existence and uniqueness of a solution, upper and lower bounds of the solution, and the asymptotic behavior of the system. In Section 2, we use the monotone iterative scheme and the notation of upper and lower solutions to establish an existencecomparison theorem (Leung, 1984; Pao, 1983; Ladde et al., 1985). Through suitable construction of upper and lower solutions in Section 3, we obtain some explicit upper and lower bounds of the solution and some explicit relationships among the reactor parameters so that one can determine the sufficient conditions for subcriticality and supercriticality of the system (1-4) without explicit knowledge of the solution. We also give an application to the linear slab problem where a more explicit estimate is given.
2. M O N O T O N E SEQUENCES OF ITERATION
In this section, we apply the monotone iterative scheme and the notion of upper-lower solutions to construct two monotone sequences for the system (1-4). Throughout the paper, we assume that ag, afg, age are piecewise continuous nonnegative functions on D x S and D x S × S. The quantities Xg, Zj~,/~,/?j and vg are nonnegative constants and 0
< ...
0<~1 <~2<...<~6, 0~
and
0~
We write the vector function (~bo,Cj) to represent (4',62 . . . . . 4'~, C, . . . . . C6) and (~g, ~'j)I> (~bg,Ci) to represent :
$1 ~ ~bl,$2 ~ (#2. . . . . $G ~ (~G, el ~ Ci . . . . . C6 ~ C6We let Qo = D x [0, T], Q ~ = D x S x [0, T], where T is finite hut can be arbitrarily large. L e t / ) = D w 3D, where 6D is the boundary of D, and let Qo, ~1 be Qo, Q1 with D replaced by/). For convenience, we write equation (1) as follows :
0 ~-tdp.(r,t,t)+vgt'Vdp. +vaa.(r)¢g = v. .
aag,(r, t', t ) ¢¢ (r, t', t) dn"
+x,(l-t~)~Vefsajg,(r, g'=l
fl')dpq,(r,t',t)d,'+~Xjg2jCj(r,t)}. j=l
(5)
Multigroup transport equations in reactor kinetics
281
In order to describe the process of the monotone iterative method and to ensure the convergence of the sequences to a solution, problem (1--4) is reduced to an integral equation :
¢~(r, fl, t) = ¢.o(r-v.Slt, gl) exp --
Vg%[r-Vq~l(t-~)]d~ +
jo
exp
Vgffg[r-Vgl'l(t-~)] d~
-
X[Fg(d~¢,Cj)][r-vgQ(t-z),tl, z]dr, y = 1. . . . . G, [(r,Q,t)~Q1], Cj(r,t) = Cjo(r)exp(-2J)+ Here
exp [ - 2 j ( t - r ) ] ~ . ( ¢ ¢ ) ] ( r , z )
dz, j =
1,2 . . . . . 6, [(r, 0 e Q o l .
(6) (7)
Fg(d~g,,Cj) and fj(¢¢) are defined as :
[Fo(O,,,Cj)](r,Q,t) = vg
ago,(r, Fg, Q)dpg,(r, ffg, t)dFg+(l-fl)Zg g
vg,
1
c%,(r,Q')dpg,(r, ffg, t)dg~"
g'=l
j=!
~(c~g,)](r,t)= flj ~ v¢ fsarg,(r,Q')dpg,(r, tl',t) dQ'.
(9)
g'=l
F o r the sake of completeness, we give a brief derivation of integration equation (6). Using the new variable r' = r - v~llt, z = t and (8), equation (5) reduces to :
~ ~pg(r'+ Vg~Z, ~, r) + va% (r' + vgflz)c~g(r' + vgflr, fl, z ) = [Fg(~g,, C )](r' + vgflr, fl, "c). Multiplying by : exp
fo
Vg%(r' + v,f~) d~
and integrating from 0 to t over z, we obtain : exp
Vg%(r'-bVgf]~) d~ dpg(r"+Vggtt, gt, t)-¢o(r',gt, O) =
jo
VgeTg(r'-lLvg~'~)d~
exp
x [r. (q~.,, Cj)] ( r ' + vgf~z, f~, r) dr. Applying the initial condition (4) and substituting r' by r-vcf2t, we obtain equation (6). Similarly, an integral equation (2), using (9) and the condition (4), yields the integral equation (7). In the integral representation (6) it is defined that for g = 1, 2 . . . . , G : Cgo(r,n)=O,
%(r)=O,
Fg((a¢,C~)(r, tl, t)=
whenx¢/).
Clearly, every solution of the differential system (1-4) is a solution of the integral equations (6), (7). Conversely every solution of (6), (7) is a solution of (1-4) when (¢g,+vgtl'Vq~) is considered as the total derivative
(d/dt)¢,(r + v~t, tl, t). We now apply the monotone iterative scheme to construct the solution of the integration (6) and (7). We construct the sequence { ¢~*), C) *)} from the iteration:
4a~k)=C~go(r-vgt'lt, Q ) e x p { - f o ' V g g g [ r - V g t l ( t - O ] d ~ } + ~ ' e x p { - f ' v g % [ r - v g Q ( t - O ] d ~ x [Fg(t~g(k
C¢k)(r,t) = Go(r) exp ( - 2 i t ) +
}
1),C)*-l))][r--vgtl(t--'c),~l,'c] dr,
exp [ - ~ j ( t - r ) ] ~ ( ¢ ~ , k " ) ] ( r , r ) dr,
(10) (ll)
282
G.-S. CHEN
by a suitable choice of the initial iteration ($g(0), C~O)). In order to ensure the convergence of the sequence, we require the existence of an upper and lower solution which is defined as follows.
Definition 2.1 A continuous vector-valued function : (¢g,Cj) ~ (¢1,¢2 . . . . . e G , C l , C 2 . . . . . C6) is called a lower solution of (6-7) if it satisfies the inequalities :
dp9(r~fLt)<`q~.~(r-vg~t~I))e~p{-f~tvg~g[r-vgf~(t-~)]~}+f~`exp{-f~v~g[r-va"(t-~)]d~} x [Fg(dpg,,Cj)][r-vafl(t-z),f~,z]dz , g = 1..... G, [(r, fl, t)~Ql], q ( r , t ) ~< Go(r) exp ( - 2 i t ) +
exp [ - 2 j ( t - z ) ] ~ ( 4 ~ g , ) ] ( r , r ) dr, j = 1. . . . . 6, [(r,t)eO0].
Similarly, a continuous vector-value function (~g, C~)= (g,, ~ . . . . . ~a, C,, C~ . . . . . C~) is called solution of (6-7) if it satisfies all the reversed inequalities in (12-13). Let (~g, C~) and (4~g, C~) be a pair of upper and lower solutions such that: (q~g,(7~)~>(~g,q)~>O,
~g>.d?g>~O,C~>~C~>~O,
i.e.
g=l
(12) (13)
an
upper
. . . . . G,j= I ..... 6.
By choosing (q~(O).Qo))= ( ~ , (~) as the initial iterate in (lO-ll), we can construct a sequence which is denoted by { q~g(~),C~)}. Similarly the sequence obtained from (lO-1 l) with (~bg(°)' C~°)) = (q~g, C~) is denoted by {q~(~),_C~)}. These two sequences, referred to as maximal and minimal sequences, possess the following _g monotone properties.
Lemma 2.1 The maximal sequence {q~a(~), CJ~)} is monotone nonincreasing, i.e. (bg(t), C) ~)) 1> (bg(*+ '), C) ~+ ')) k = 1, 2 . . . . . while the minimal sequence ~.g ,~ (~), C) ~)} is monotone nondecreasing, i.e. (~g(~), _C~~+')) ~> (~j(~), _C~~)) k = 1,2 . . . . . Moreover, d) (k) (~(k) ]
~g , ~ j
-(k) - ( k ) ~ ~< (~bg ,Cj )
f o r e v e r y k = 1,2 ....
Proof--We first show the monotone nondecreasing property of ,f~(k) c(k)~ by the induction principle. It is obvious from (I0-11) with k = 1 and (_$~0), _C)0)) = (~g, Cj) and definition of a lower solution that :
~ o ) _ ~ , ) = ~pg-q~go(r-vgf~t,f~)exp {-- £'va~g[r--vaf2(t--~)] d~} - joI' exp { - f ' Vg~g[r-Vgf~(t-~)l d~) x [F,(4,~,, Cj)][r--van(t--r), n , d dr ~< O, g = 1,2 . . . . . a ,
C_~°)-C_~') = C j - C j o ( r ) e x p ( - 2 j t ) Assume, by induction, that (~b~ _ k- ~), _C) k
exp[-2j(t-z)][fj(~g,)](r,~)d~>~O,
J)) ~< (,~(k) ,~_g , C!k)~. _ j ,,
~_(qk'--~_~k+l)=fo'eXp{--f'vg%[r--Vg~(t--~)]d~}[Fg(~.(k,
then using (10) and
(11),
j = 1,2 . . . . . 6. we obtain:
'),C} k '))--Fg(_~g(k.),C}k))]
x [r-vgf~(t-z),~,z]dr,
g = 1,2 . . . . . G,
(14)
Multigroup transport equations in reactor kinetics
fo
(:,(k)- - _.~j (7(k.,) = .=_j
exp[-2/(t-OlD~j(q~.k-t))-~(~b~,k))](r,~)d~,
283
j = 1,2 . . . . . 6.
(15)
Since for each (r, O, t) e Q ~,
(k ., ,C/(k ~,)-F.(cp¢(k, ,Cj,k,)](r,a,t)=v~ ~ ~ [Fo(q~o, .
.
.
.
.
.
.
[
+(l--fl)Zu ~ v.q, fa g'=l
(.h(, ,)
Jlb"
W g'
--
f a~.(r,.'.f~)(~p(q k, ')-~q(O)(r.f~'.t)df~'
q '~= 1 ,,]S
.
.
.
.
.
.
~b(,))(r f l ' , t ) d D ' + i '
g'
j=l
_ _. [f(qbg(k,~))--f~(~b~k))] (r,/) = f l j g ~ vg,
~ ' " ~ ( ' ')
C}'))(r t)} ~<0 (16a)
aSg,(~ ,~-L) -_qSg, ck)~. df~' .N<0,
(16b)
we see from (14-16a,b) that ~ba(k)-~b~k+ ') ~< 0 and (,(k) c '(k+ ~) ~< 0. Using the induction principle, we conclude that ,:,(d)(k), -~JC'(kh,~< -'~g(~(k+,), ~JC'(k+~)~,for each k = 0, 1,2,. .. This _~(k), ~J (.(k)~,- A similar argument leads to the conclusion proves the monotone nondecreasing property of {.~a • ~ /> (~g '), 6"J~+ ')) for every k = 1,2,3 . . . . (~*', C k)) (*+ (,6(o) , ~i ¢(0)~, = (4~a,C~) ~< (43~, ~'~) = (4Sq(°), (7/°)) and a similar argument as before, we Finally, using (0, 0) ~< .'c_o _
-
-~J
--
~ j
obtain : d~(~)--~ (') = Xa
,q
C ('' -.i
C(" = --
/
exp
--
v.a.[r-vo~(t-~)]d~
r~" t.h(o) ('(oq_t~ bg(o) ' ff;)O))l[r_v~fl(t_t),f~,~]d ~ <~O, 1 "Ot'e'.q' l a g s . W 0" , ~ - j
do
f,
Jo
exp
[-2At-Ol~(~°b-~($~°')l(r, Od~
~< 0.
Suppose (~b ~)) ~< (q~J~), ~J~)). It _ g '~ ~', _C~~- ')) ~< (q~*-'), 6"~-'))' ~ then a similar argument leads to (~b _ g (k), _C~ (~(k) ' -~J c(kh" ~< ($(~), follows by induction that ,xa ~ ~j(k)) for every k = 0, 1, 2, ... This proves the conclusion of the lemma. For convenience, we define the least upper bound (supremum) and greatest lower bound (infimum) of %¢, a~. and au o n / 5 x S x S and/5 x S, respectively, as follows : 6~,, = sup { Jsraq¢(r'f~"fl) d ~ ' ; ( r ' " ) e / s x S } ~o.(~') = inf
6. =
max
'
{%¢(r, ft', fl) ; (r, ~) e/5 x S},
{6gg. ; 1 <~g',g <~G},
0z = max {6zo ; 1 <~g<~G}, v = max {vo;1 ~
max {~ ; 1 ~< g
~< G}.
Based on the results of Lemma 2.1, we can now establish the following theorem for the system (1-4).
Theorem 2.1 Let (q~, (T~), (4~, C~) be a pair of upper and lower solutions such that ($~, ~ ) / > (4~, C~)/> 0. Then the minimal sequence { ~b(~), C~(*)} converges monotonically from below to a unique solution (4~, C~) of (1-4) and the maximal sequence (q~(~), ~'¢~)~,converges from above to the same solution. Moreover,
284
G.-S. CHEN
(o,o) < ( ~ , c~) = (_~(o,, _c)o,) < (_~,,,, _c?,) < ... < ( ~ , cj) < . . . < (~,(',, c~',) < ($,(o,, Co)) = ($~, ~). (17) Proof--In view of Lemma 2.1, the minimal sequence (_~o(k), _C~k)) is monotone nondecreasing and is bounded from above by upper solution (~g, Cj) while the maximal sequence { q~k), 6"~)} is monotone nonincreasing and is bounded from below by lower solution (4~g,C~). Therefore the pointwise limits : lim (,4, (k), _C(k)) = (4)g, _Cj)
k~oo
~-g
-
lim ((~g(~), ~j(k)) = ((~g, (~j),
and
k~c¢
exist and satisfy the relation : (~g,Cj)
<
(d)(l) C{l)~< \'~_g , - - J ]
•. .
<(~)g,C_j)<(l~g, fj)< --
". .
<
(~(l),~(i))< g
(~g
~j).
By letting k ~ oe in (10-11) and applying the dominated convergence theorem we conclude that both (q~g, Cj) and (¢g, Cj) are solutions of (6-7). The equivalence between the equations in (6-7) and (1-4) ensures that they are also solutions of (1-4). In order to show that (~,q, _Cj) = (q~g, Cj), we let: Cj*=(C)-Cj)e
¢ * = (q~q-~g) e -~',
-bt,
g = l . . . . . G,
j = 1. . . . ,6,
where b ~> 0 is a large constant so that (18)
b > v,G(~s+26)-l-(GVl +6)v6f. Then (q~*, Cj*)/> 0 and satisfies the equation :
dp*+vof~'Vc~g+(voag+b)¢* = vg
g
I JS
, , ,t) d f ~ ' + Z g ( 1 - f l ) agg,(r,D , ,f~)dpg,(r,~
g'~l
vg,
+ i ZJg2jC*}=[Fg(d)g*',C,*)](r,",t),
afg, ¢ ,g df~' g= 1,...,G,
j=l
£c, +0,j+b)Cj* Ot j
= flj ~ v,, ~afg,dp*d~' - [fj(~b~)](r,t), j = 1,2,... 6, js g'=l dp*(r, fl, t) = O, (t > O, reD, D incoming),
q~*(r, fLO) = O, (r6D, DeS), C]*(r, 0) = 0,
(r e D).
Using the same procedure in obtaining (6) and (7), we have :
c~*(r, fl, t) =
;i{i exp
-
}
[b+vg%(r-vof~(t-O)]d ~ [ F g ( ¢ ~ , C * ) ] [ r - v f l ( t - z ) , f l , z] dr,
C~(r, t) =
fo
exp [ - (2j+b) (t - 0 1 [ £ (4~)l(r, 0 d~.
(19)
(20)
Define : 114"11 = s u p { l ¢ * ( r , ~ , t ) l ; ( r , f L t ) e a l } ,
IIQ*II =sup{lCj*(r,t)l;(r,t)eQo}.
Next, we show that II¢~* JJ = 0, IIQ* II = 0, which implies that ~g = _~g, ¢2j = _Cj. By (19) and (20), the definition
of fs, ff, v and O <~ ~jg, ~g <~ l, O <. fl, flj <~ l, O < vg <~va, O < 2j <~26, we obtain :
Multigroup transport equations in reactor kinetics
r
114,gll. < vo(a~+v~:) t,_
}
114"11. +v.
kg~l
x (e.+vef)
IIQ*ll
114.*11 +,~
IIG*l[
I
}];;
285 vg
e x p { - ( b + v g t r o ) ( t - z ) } d z < b~+- v- q_~. g
~<~-(e~+vef+,~,)
IIG*II , (21)
114"11 +
j=l
I
j=l
(22)
since 2,. > 0, Vgag >~ O. Addition of (21) and (22) from g = 1,2 . . . . . G , j = 1,2 . . . . . 6, leads to : /= i
riG
_
_
6v5/
,
,
(
<~(Gv,(ff,+26)+(Gv,+6)v6:}"
6
0=l ~ 114"11+~,.= IIG*ll
}
• (23)
Since b is chosen as (18) such that: vl G(ffs q-J.6)q-(Gv ~+ 6)v0/
<1,
relation (23) cannot hold unless G
6
Z IIq~*ll + ~ IICTll =0. g=l
j=l
Hence, we obtain H4* ]l = 0, I] C* I1 = 0, g = 1,2 ..... G, j = 1,2 . . . . . 6. This implies that q~g = _qb~, (~j = C/, g = 1. . . . . G, j = 1. . . . . 6. Finally, if (q~g, C/) is another solution such that (4g, Cj) ~< (4g, Cj) ~< (-q~g,(~j) then it is also a lower solution and an upper solution. Using (4g, Cj) as the initial iteration in (10-11), the same argument as the p r o o f of Lemma 2.1 implies that ( ~ k , , _C}k)) ~< (q~g, C/) ~< (q~k), C,)k)) for every k = 1,2 . . . . Letting k ~ oo, it follows that :
(~,, _c:) ~< (4,, c,) ~< (~,, Cj). Since 4o = q~g, C: = C:, we obtain
(~,, c,) = (~g, cj) = ( ~ , ¢,). and (q~q, Cj) [or (~_g, _6",.)]is the unique solution of (1-4) satisfying the relation (17). This completes the proof of the theorem. Since Fg(0, 0) = 0, f:(0) = 0, it is easily seen from the definition of the lower solution that (4g, Cj) = (0, 0) is a lower solution. The result of Theorem 2.1 implies the following. Corollary 2.1
Let (q~g, C'j) be a nonnegative upper solution and (4g, Cj) = (0, 0). Then the minimal sequence/d) t ~ g (k)~ tT(~)~ "~j J and maximal sequence {q~(k),~)k)} converge monotonically from below and above, respectively, to a unique solution (q~g, C~) of (1-4) such that (17) holds with (4g, Cj) = (0, 0).
3. CONSTRUCTION OF UPPER AND LOWER SOLUTIONS
In view of Theorem 2.1 in the previous section, the existence of a solution of (1--4) is ensured if we can find a suitable pair of upper and lower solutions. The aim of the section is to construct some explicit upper and lower solutions, so that these functions not only serve as upper and lower bounds of the true solution, but also they can be used to determine whether the solution decays to a steady-state or grows unbounded as t ~ oo. This decay or growth property of the solution depends on the reactor parameters of the system without explicit knowledge of the solution.
286
G.-S. CHEN
Theorem 3.1 There exist positive constants A, b such that a unique solution to problem (1-4) exists and satisfies the relation : 0 ~< ~bg(r, t2, t) ~< A e b' g = 1. . . . . G, (r, t), t) e Q t,
(24)
O <~Cj(r,t) <<.A eh'
(25)
j= 1..... 6,(r,t)eQo,
whenever (0, 0) ~< (~bgo, Cj0) ~< (A, A), where b is any constant such that : b > v. [Gff, + 626] + (v. + 1)Gvfff.
(26)
Proof--In view of Corollary 2.1 it sufficient to show (~g, Q) = (A e b', A e b') is an upper solution, since ($g, Cj) = (0, 0) is a lower solution. In order to show that (A e b`, A e b') is an upper solution, we observe from the equivalence between the integral representation (6-7) and the differential system (1-4) that (~, ~j) is an upper solution if:
O_c~t ~g(r'"'t)+v°~'VC~g+Vg~rg(r)~g>~Vg { ~ fs a°g'(r'f~''fl)~a;'(r'"''t)d"' g'=l
=°;v/(rfg(r,fl)dpg,(r,D,t)dD'+
+Xg(1 - f l ) g Z
' ~
'
1
d~
~tCj(r,t) >~flj
g'=l
Zj,2jej(r,t) j=l
~ afg(r,D')C~g(r,D,t) ~ " dD-2jCj(r,t), j= vg. js ' ~
}
, g = 1,2 . . . . . G,
1,2 . . . . . 6,
~a(r, fL t) >tO, (t > O,reD, Q incoming), q~g(r, f~, 0)/> 0, C/(r, 0)/> 0,
(27)
(28) (29)
(r e D, f~ e S), (r e D).
(30a) (30b)
The boundary and initial requirements (29-30) are clearly satisfied by (q~o, C'j) = (A e b', A e b') with (A, A)/> @go, Cjo). Hence, we only need to verify that :
(b+vgag)Aeht>~Vg(~ Iegg.(r,D',D)Aeb'df~'+(l-fl)7~ ~ fsVg. fstrfo.(r,fl')Aebtdfl' g'=l Js
g,=~
+~Xjg,~jAebtt, j~l
~ va, f c%,(r,D')Aeb'd,O", j=
(b+2j)Aeb'/>fl/
9'=1
Since
Vgrg1> 0, 2j >
)
g = l . . . . . G,
1,2 . . . . . 6.
JS
0, the above inequalities are satisfied if:
b>~vu{~ ~agg,(r, fV, fl)d~'+(1-B)L, ,q'~l ~ v~,fsalg,(r,Q')dO'+~Zjg2j }, g'~l j=[ g'=l
JS
Using 0 ~< ~ig < l, 0 ~< fir ~< l, 0 ~< 1 - fl ~< l, the above inequalities are satisfied by :
b ~ vu b >~
g ' = l j,Is
~gg"d~t"~
g'~l
Vg" afa. drY+
2/ ,
~ Vg, ~ (Tfg,d~-~'. g'=l
In view of definitions of
5gg,,6I~,, (31-32)
(31)
j=
are satisfied, if for each g = 1, 2,..., G :
(32)
Multigroup transport equations in reactor kinetics
g'=l
g'=l
287
j=l
G
b >/ ~ v~.~y~,.
(34)
g'--I
Using v~ > v 2 . . . / > va, 26 > )-2. • • > ~.l and the definitions of v, 6s and 5I, (33-34) are clearly satisfied if:
b >1 vl(Gfs+Gvff+6).6), b >1 Gv~y. The above relation follows immediately if we choose : b > v 1(G~., + 626) + (v i + 1)Gvff. This leads to the choice of b satisfying (26). The existence of the solution and relations (24-25) follow from Corollary 2.1. Before we study the asymptotic behavior of system (1-4), we seek a solution u= for the boundary-value problem :
~'Vu+(~+a/va)u= 1 [(rxf~)~DxS], u(r, fl) = 0
(re OD, ~ incoming),
(35)
where c¢ is a positive constant. The solution u~ is given by :
u~(r,n) --
( r , n ) exp { - ( 6 + a / v a ) r } dz = { 1 - e x p [-(6+e/va)s]}/(6+a/va).
(36a)
When e = 0, we define :
uo(r, n ) = {1 - exp ( - Os)}/&
(36b)
where s - s(x, f~) >1 0 is chosen such that for any fixed r e D, Q e S, the point r* = r - s Q is on the boundary OD of D. The boundary point r* is determined by extending a line r in the direction of ( - ~ ) until it is intersected by the boundary. Hence, the value of s is s = I r * - r l / I~ I = I r* - r [, by If~ I = 1.
Theorem 3.2. Supercriticality Suppose
.q'~l
[ eag. (n'){1 --exp [-- #s(r, fl')]} dn'/5 > 1, g = 1, 2 . . . . . G,r e D .is
(37)
and assume ~bgo/> 6Uo, Cjo/> 0 for some 6 > 0, g = 1,2 . . . . . G, j = 1,2 . . . . . 6, where u0 is given by (36), then there exist positive constants • and e such that the solution (q~g, Cj) of the problem (1-4) satisfies: ~bg(r,Q,t) 1> ee~'uo(r,f~),
g = 1,2 . . . . . G,[(r,Q,t)EQd,
C/(r,t) >>.O, j = 1,2 . . . . . 6,[(r,t)eQo].
(38) (39)
Moreover, lim q~(r, ~, t) = oo as t --, oo for g = 1,2 . . . . . G. P r o o f - - T h e function g(y) s > 0, since :
[1 - e x p (-ys)]/y possesses the nonincreasing property, g (y) .~ 0, for y > 0 and
g'(Y) = e_(y.o (1 + y s ) - e x p (ys) y2
~<0.
According to the hypothesis (37), there exists ag > 0 such that for each g = 1,2 . . . . . G, ANE 16: 6-D
288
G.-S. C ~ N
~s #g,.Cft'){1-exp g'=l
[-(6+~/vc)sl} dfl'/(6+%/vc) >t 1.
Let ~ be the smallest value of %, g = 1. . . . , G. Hence for g = 1,2 . . . . . G :
.q'=l
[-(6 +~lvG)s]} dfl'l(6+~tvu) >1 l, (xeD).
fd~g,(ft')(1-exp 4s
(40)
Since ego i> 6uo t> ~u~ for some ~ > O, we now show that (~b~, Cj) = [s e~u=(r,~),O] is a lower solution. We observe from the equivalence between the differential system (1-4) and the integral representation (6-7) that (~g, Cj) is a lower solution if:
O~'.o+vgf~'Vdpg+vgagdPg<~V.q{~ ~a,qg,(r, ft',ft)dpg, dft'+(l-fl)Zg~vg, fa]g,(r,f~')dpg,(r, IY, t)d~' b' ' = 1
.q'=l
+ ~ZjgmiCj(r,t)},
°f
Cj,+i.~Cj~
dp.q(r,fLt)
g = l . . . . . G.
(41)
j=l
Vq,afq.(r, ft')~g,d.Q',
j = 1. . . . . 6,
(42)
I
~< 0,
(t > 0, re(~D, flincoming),
(43)
4~g(r,O,O) <~Cg0, (reD, tieS), Cj(r,O) <~Cjo, (reD).
(44a) (44b)
Since the initial requirement Cg(r,Q, 0) = Cz0 ~> 6u0 ~> eu= = ~bg(r, ft,0), Cj(r,0) = Cjo >>.0 = Cj(r,O) and (~bg, Q ) satisfies the boundary requirement, we only need to verify that (~bg,Ci) satisfies (41-42). The inequality (42) is easy to establish. Since (~9, Q ) = (s e='u~, 0), we have: q,+4Q
= 0 ~
g i.]s
v,'%'(r, fl')ee='udft'=l~j
,=
Using the definition of 6, 6, v~ > Vm... >
va, (35) and
a I
v¢ay¢,4~g,dft'.
(40), we obtain that:
4~,+ v~ft" V4~+vga. 49. = ~ e" [~u~+v.ft" Vu. + v.a.u.] <<.v.s e='[~ • Vu~+(6 +~/v.)u.] <<.vuse~'[Ft.Vu. + (6 + ~/vc )u.] = Vgee"-
ao.' { 1 - exp [-
1 ~< vu~ e =' kg'=
(6 + ~/VG)S]}d~'/(6 + ~/va)
1
=vuse~'{ ~=~fsa~(fg)~=dD~}<``v~g~ fsagg~(~ft~F~)s~'~u'dD~}=v~{~fsa~4~gdft~ } <~Vq ~'=,~ a,.q.(r,f~',ft)4~o,(r,tT, t) dfg+(1-fl))G.q,=,~ vg, ayg,(r,f~')dp,.(r,n',t)dft'+ ~l~,~2,C~ . Hence, we conclude that (~b~, Q ) is a lower solution. An upper solution is taken as (q~, Cy) = (A e~', Ae ~t) as in Theorem 3.1 with the sufficient large constants A and b so that (q~o, ~y)/> (~b~,C~), i.e. (.4 e~',A e~'`)>>. (s e=tu,, 0). Using Theorem 2. l, we obtain that :
This completes the proof of the theorem.
¢~1>se"u,,
g = 1. . . . . ,G,
Q~>0,
j=l
. . . . . 6.
Multigroup transport equations in reactor kinetics
289
Theorem 3.3. Subcriticality Let q~oo~< M, C~o ~< M, for some M > 0, g = 1. . . . . G, j = 1. . . . ,6, and assume that : a~ > ~o'=, 6g~, + (1--fl))~ "u.=, v°'ff/~'+,= ~
'
(45)
g = 1. . . . . . G
and ).j >/~j ~
vg, fit:q', J = 1. . . . . 6.
(46)
h,'=l
There exists a constant p > 0 such that a unique solution (~b., (7,) of problem (1-4) exists and satisfies: q~.q0(r-voDt, D ) exp (-vuffut) <~ c~,(r,D, t) <<.M e Qo exp ( - 2 i t ) ~< Ci(r,t ) <~ M e
t,,,
(47)
~".
(48)
P r o o f - Multiplying (45) by vu, there exist small positive constants pg, g = 1. . . . . G, such that : voa o - p g t> v.
I 5gu' + (1 -
9(~g,~i .
vo, 5yg,+ •
'=
l= I
Using the definition of 5u.q,, 5~o, we obtain :
g'=l
g'~l
]=1
Similarly, there exist small constants ?j, j = 1, 2 . . . . . 6, such that :
2j-ej >~lJ~ ~ vo,~co,>~llj ~ v~, faj~,dta'. g'=l
(50)
g'=l
Let p be the smallest value ofpg and ?j, 9 = 1. . . . . G , j = 1. . . . . 6. We now show (q~, Cj) = ( M e - P ' , M e -p') and (~o, Cj) = (0, 0) are upper and lower solutions, respectively. It is clear that (0, 0) is lower solution. Since ~g(r,D,O) = M >>.dpgo, Q(r,O) = M >1 Go and q~g(r, t'l, t) ~> 0, ~j(r, t) >t 0 satisfy boundary requirement (29). We only need to show (~o, Q ) satisfying the differential inequalities (27-28). Using the definition of p, a u, (49), (50), we obtain:
otc~o+v.D'V~+voag~ = ( V o a g - p ) M e P' >~ ( v . a g - p . ) M e ~' >t t'g
~ t.g'=
aog,M e - "
vg,a~g,M e-"' d D ' +
I
'=
=
~jg2jM e -°' j= !
E
dn'+
k. #'= I
,q'= I
x.Z,q j= I
O~j c~t
I
}
dD'.
'
g'=l
This shows that ( M e -p', Me -p') is an upper solution. By Theorem 2.1, a unique solution (~bg,Cj) exists and satisfies :
O < d p o ( r , ~ , t ) < M e p, O ~ C j ( r , t ) < M e - P , . Using (~o(°), C~ °)) = (~bg, Cj) = (0,0) as the initial iteration and observing that Fg(0, 0) = 0, ~(0) = 0, the first iteration in recursion formulas (10-11) is given :
dpl'~ = dpgo(r-vg~t,D)exp _g
-
v~ag[r-v~t'l(t-~)]d~
>~ dp~o(r-VgDt, f~)exp ( - V g f fl),
(51)
290
G.-S. CrmN _C~t)(r, t) = Go exp ( - 2 i t ) .
(52)
In view of Theorem 2.1, (_~at ~), C)~)) is a lower bound of solution (q~g,Cj). Hence relation (47-48) follows from (51-52). This completes the proof of the theorem. At the end of this section, we study the asymptotic behavior of the solution for the slab by making use of the results developed in Theorem 3.2 to give a more explicit condition for the growth property of the solution. We consider a slab with length l and its faces perpendicular to the x-axis and located at x = 0 and x = l. The general system (1-3) in a slab model is reduced to :
g
I% , (x, P', #) ~bg,(x,#', t) d#'
I
+(1-/s)z.g'Ev.f (53) =lAx,.')O.,(x,u'.Odu'+Zzj.zjCj(x,t)}, j=l NG(x,t) =/~j
g'=lv,,
%,(x,#')g~g,(x,#',t)dl~'-,~jCj(x,t), (t>O,O
(t>0,0
~b~(/,p, t) = O,
(t > 0,--1 ~ < p < O )
c~g(x,p,O)=dpgo(X,#),
(54)
l)
(O~
Here p = cos 0 and the initial and boundary conditions for Cj. as (4) with replacing r by x. The boundary value problem (35) in the slab becomes :
tmx+(ff+ot/vo)u= 1, ( O ~ < x < / , - - 1 ~<#~< u(O,#)=O
forO<#~
and
u(l,/~)=O
1)
forO~<#
Using (36), the solution of the above equation is : [ { 1 - e x p -(6+a/va)x/#}/(5+~/v~)
0 < # <. 1,
u,(x,#) = I 1/(5+c~/vo), p = 0, [ { 1 - e x p [(6+a/v~)(l-x)/#]}/(6+e/va),
(55)
- 1 <~# < 0,
since s = x/# for 0 < p ~< 1 and s = ( l - x ) ~ ( - # ) for - 1 ~< p < 0. Using the value o f s given as above, the condition (37) becomes :
~ { II #gg,(p') [ 1 - e x p ( - 6 x / p ' ) d p ' + f)
g'=l
do
ds¢(#') t
] 1-exp[~(l-x)/I/]d#' } >&
(0 ~< x ~
If we let :
then (56) becomes :
g'=l f01do,q,(p'){2--exp( - 6 x / p ' ) - e x p
[-6(l-x)~#']} d#' > 6.
Let
g(x) = 2--exp ( - - # x / # ' ) - - e x p [--6(l--x)~#'], we have
(57)
Multigroup transport equations in reactor kinetics 5 g' (x) = :7; {exp ( #
~xlu')-
291
exp [ - 6 ( l - x)t~'] }
and g'(x)>0,0~
g'(//2)=0;
g'(x)<0,
l/2<.x<~l.
Hence, the function g(x) has m i n i m u m at x = 0 or x = l. The inequalities (57) are satisfied if for each g = 1,2 . . . . . G :
fo' 6 g g , ( , ' ) { l - e x p ( - # l / p ' ) } a t + ' > 5.
(58)
g'=l
In view of T h e o r e m 3.2, we have the following result.
Theorem 3.4. Supercriticality Assume ~bg0 f> 6u0, Cjo/> 0 for some 6 > 0, g = 1,2 . . . . . G, j = 1,2 . . . . . 6, where u0 is given by (55) with ct = 0. T h e n u n d e r c o n d i t i o n (58) for g = 1,2 . . . . . G, each ~bg,g = 1. . . . , G, grows to ~ as t ~ oo.
REFERENCES
Belleni-Morante A. (1976) Nucl. Sci. Engng 56, 59. Case K. S. and Zweifel P. E. (1963) J. Math. Phys 4, 1"376. Chen G. S. and Leung A. W. (1985) J. Math. Phys 15, 490. Chen G. S. and Leung A. W. (1989) SlAM J. Appl. Math. 49. Duderstadt J. J. and Martin W. R. (1979) Transport Theory. Wiley, New York. Ganapol B. D. (1980) Transport Theory Statist. Phys 9, 145. Hill T. R. and Reed Wm. H. (1976) TIMEX: a time-dependent explicit discrete ordinate program for the solution of multigroup transport with delayed neutrons. Report LA-6201-MS, Los Alamos Scientific Laboratory. Kastenberg W. E. and Chambr6 P. L. (1968) Nucl. Sci. Engng 48, 211. Kobayashi K., Oigawa H. and Yamagata H. (1986) Ann. nucl. Energy 13, 663. Ladde G. S., Lakshmikantham V. and Vatsala A. S. (1985) Monotone Iterative Techniques for Nonlinear Differential Equations. Pitman, Boston. Lathrop K. D., Anderson R. E. and Brinkley F. W. (1971) TRANZIT : a program for multigroup time-dependent transport in (p, z) cylinder geometry. Report LA-4575, Los Alamos Scientific Laboratory. Larsen E. W. (1979) J. Math. Phys 20, 1776. Lee C. E., Fan W. C. P. and Dias M. P. (1985) Ann. nucl. Energy 12, 613. Leung A. W. (1984) J. Math. Analysis Applic. 100, 583. Lewis E. E. and Miller W. G. Jr (1984) Computational Methods of Neutron Transport. Wiley, New York. Pao C. V. (1981) Prog. nucl. Energy 8, 191. Pao C. V. (1983) J. Math. Phys 24, 132. Richtmyer R. D. and Morton K. W. (1967) Difference Methods for Initial-Value Problems. Wiley, New York. Sanchez R. and McCormick N. J. (1982) NucL Sci. Engng 80, 481. Stacey W. M. Jr (1969) Space-Time Nuclear Kinetics. Academic Press, New York. Vida V. I. (1968) J. Math Analysis Applic. 22, 144. Williams M. M. R. (1971) Mathematical Methods in Particle Transport Theory. Wiley, New York. Williams M. M. R. (1985) Ann. nucl. Energy 12, 167. Wilson D. G. (1974) J. Math Analysis Applic. 47, 182. Wood J. (1985) Ann. nucl. Energy 12, 217. Wood J. and Williams M. M. R. (1986) Ann. nucl. Energy 13, 479.
0306-4549/89$3.00+0.00 Copyright © 1989PergamonPress plc
THE TIME-DEPENDENT MULTIGROUP TRANSPORT EQUATIONS IN REACTOR KINETICS G.-S. CHEN Department of Nuclear Engineering, National Tsing Hua University, Hsinchu, Taiwan, R.O.C. 30043
(Received 8 December 1988) Abstract--A rigorous description of nuclear reactor kinetics usually invokes neutron transport theory. If the multigroup transport theory is considered, the description of space-time nuclear reactor kinetics consists of partial integro-differential equations coupled with ordinary differential equations. The purpose of this paper is to give an iterative scheme for the construction of the solution of the coupled systems and to establish some qualitative analysis of the coupled system. The iterative scheme consists of two monotone sequences, which converge monotonically from above and below to a unique solution, respectively. The qualitative analysis includes the existence and uniqueness of a positive solution, explicit upper and lower bounds of the solution, and sufficient conditions in terms of reactor parameters for subcriticality and supercriticality of the time-dependent system.
l. INTRODUCTION It is well-known that the one-speed transport equation has been extensively studied in neutron transport theory as an idealized model of nuclear reactor system. The principle deficiency of the model is the assumption that all neutrons can be characterized by only a single speed or energy. However, the neutrons in a reactor have energies range from 10 MeV down to less than 1 ¢V. Moreover, the neutron-nuclear cross sections depend rather sensitively on the incident neutron energy. Hence, in any accurate description of realistic reactor system, it is necessary to consider the neutron energy dependence in more detail. The multigroup transport equations is the most popular model in which energy-dependent effects are included. In this article, we will study the time-dependent multigroup transport equations for reactor kinetics with the form (Lewis and Miller, 1984; Duderstadt and Martin, 1979; Williams, 1971):
Vg ~i ~g(r'fLt)+f~'Vd~g+%(r)~g =
g'=t
aog,(r, fV,~)dpq,(r,~',O d~'
dS
"
j=l
and for the delayed neutrons ~C(r,t)=
£a ~,
vg,trfa,(r,n')qbg,(r,~)',t)df~'-2jCj(r,t ) j = 1. . . . . 6,
(t>O,r~D,n~S).
(2)
The quantity va is the average speed of the neutron in group g, while tra, afa represent the macroscopic absorption and fission cross section, respectively, in group g. The macroscopic cross section tra.q, refers to a scattering interation which changes the neutron in group g ' with direction fl' to group g with direction f~. The quantity vg, is the average n u m b e r of neutrons produced in a fission induced by a neutron in group g'. The angular flux in group g is denoted by ~bg, which is the neutron speed vg times the n u m b e r of neutrons at point r with direction l) at time t. The quantity Cj represents the concentration of delayed neutron precursors of type j with decay constant 2j and delay fraction flj (fl = X j=, 6 flj). The fraction of the neutrons produced directly by fission and by precursor decay that have energy within group g are denoted by 7~ and ZJa, respectively. The quantity D is the reactor region, which is assumed to be a bounded convex domain in R ~ (m = 1,2, 3), S is the unit s0here in lks, and V is the gradient operator with respect to the spatial variable r ~ R =. ASE X6".6-e 279
280
G . - S . CnEN
In addition to equations (1) and (2), we require that there be no incoming neutrons for all positions r on the boundary surface : ~ba(r, t , t) = 0,
(t > 0, r e t~D, t incoming)
(3)
and we consider the initial condition ¢ , ( r , t , 0) = ¢g0(r,t), Ci (r, 0) = Go (r),
(reD, tieS),
(4a)
(r e D),
(4b)
where ~D is the boundary of D. The time-dependent neutron transport equation has been treated by many investigators and a number of analytical and numerical methods have been developed for the determination of the solutions. (For example, Case and Zweifel, 1963 ; Richtmyer and Morton, 1967 ; Vidav, 1968 ; Belleni-Morante, 1974; Pao, 1981, 1983 ; Vida, 1968 ; Ganapol, 1980 and Wilson, 1974.) When neutron energy dependence is taken into account, it is often investigated in the framework of either the multigroup diffusion approximation or the multigroup transport method. (For example, Kastenberg and Chambrt, 1968; Stacey, 1969; Chen and Leung, 1985; Larsen, 1979; Lewis and Miller, 1984; Hill and Reed, 1976; Lathrop et al., 1971.) Some attention has also been given to the problem of the multigroup time-independent transport equations. (For example, Williams, 1985; Lee et al., 1985; Wood, 1985; Wood and Williams, 1986; Kobayashi et al., 1986; Sanchez and McCormick, 1982 ;Chen and Leung, 1989.) The purpose of this paper is to present a monotone iterative scheme for the construction of the solution, and to study some qualitative property of the system (1-4). This includes the existence and uniqueness of a solution, upper and lower bounds of the solution, and the asymptotic behavior of the system. In Section 2, we use the monotone iterative scheme and the notation of upper and lower solutions to establish an existencecomparison theorem (Leung, 1984; Pao, 1983; Ladde et al., 1985). Through suitable construction of upper and lower solutions in Section 3, we obtain some explicit upper and lower bounds of the solution and some explicit relationships among the reactor parameters so that one can determine the sufficient conditions for subcriticality and supercriticality of the system (1-4) without explicit knowledge of the solution. We also give an application to the linear slab problem where a more explicit estimate is given.
2. M O N O T O N E SEQUENCES OF ITERATION
In this section, we apply the monotone iterative scheme and the notion of upper-lower solutions to construct two monotone sequences for the system (1-4). Throughout the paper, we assume that ag, afg, age are piecewise continuous nonnegative functions on D x S and D x S × S. The quantities Xg, Zj~,/~,/?j and vg are nonnegative constants and 0
< ...
0<~1 <~2<...<~6, 0~
and
0~
We write the vector function (~bo,Cj) to represent (4',62 . . . . . 4'~, C, . . . . . C6) and (~g, ~'j)I> (~bg,Ci) to represent :
$1 ~ ~bl,$2 ~ (#2. . . . . $G ~ (~G, el ~ Ci . . . . . C6 ~ C6We let Qo = D x [0, T], Q ~ = D x S x [0, T], where T is finite hut can be arbitrarily large. L e t / ) = D w 3D, where 6D is the boundary of D, and let Qo, ~1 be Qo, Q1 with D replaced by/). For convenience, we write equation (1) as follows :
0 ~-tdp.(r,t,t)+vgt'Vdp. +vaa.(r)¢g = v. .
aag,(r, t', t ) ¢¢ (r, t', t) dn"
+x,(l-t~)~Vefsajg,(r, g'=l
fl')dpq,(r,t',t)d,'+~Xjg2jCj(r,t)}. j=l
(5)
Multigroup transport equations in reactor kinetics
281
In order to describe the process of the monotone iterative method and to ensure the convergence of the sequences to a solution, problem (1--4) is reduced to an integral equation :
¢~(r, fl, t) = ¢.o(r-v.Slt, gl) exp --
Vg%[r-Vq~l(t-~)]d~ +
jo
exp
Vgffg[r-Vgl'l(t-~)] d~
-
X[Fg(d~¢,Cj)][r-vgQ(t-z),tl, z]dr, y = 1. . . . . G, [(r,Q,t)~Q1], Cj(r,t) = Cjo(r)exp(-2J)+ Here
exp [ - 2 j ( t - r ) ] ~ . ( ¢ ¢ ) ] ( r , z )
dz, j =
1,2 . . . . . 6, [(r, 0 e Q o l .
(6) (7)
Fg(d~g,,Cj) and fj(¢¢) are defined as :
[Fo(O,,,Cj)](r,Q,t) = vg
ago,(r, Fg, Q)dpg,(r, ffg, t)dFg+(l-fl)Zg g
vg,
1
c%,(r,Q')dpg,(r, ffg, t)dg~"
g'=l
j=!
~(c~g,)](r,t)= flj ~ v¢ fsarg,(r,Q')dpg,(r, tl',t) dQ'.
(9)
g'=l
F o r the sake of completeness, we give a brief derivation of integration equation (6). Using the new variable r' = r - v~llt, z = t and (8), equation (5) reduces to :
~ ~pg(r'+ Vg~Z, ~, r) + va% (r' + vgflz)c~g(r' + vgflr, fl, z ) = [Fg(~g,, C )](r' + vgflr, fl, "c). Multiplying by : exp
fo
Vg%(r' + v,f~) d~
and integrating from 0 to t over z, we obtain : exp
Vg%(r'-bVgf]~) d~ dpg(r"+Vggtt, gt, t)-¢o(r',gt, O) =
jo
VgeTg(r'-lLvg~'~)d~
exp
x [r. (q~.,, Cj)] ( r ' + vgf~z, f~, r) dr. Applying the initial condition (4) and substituting r' by r-vcf2t, we obtain equation (6). Similarly, an integral equation (2), using (9) and the condition (4), yields the integral equation (7). In the integral representation (6) it is defined that for g = 1, 2 . . . . , G : Cgo(r,n)=O,
%(r)=O,
Fg((a¢,C~)(r, tl, t)=
whenx¢/).
Clearly, every solution of the differential system (1-4) is a solution of the integral equations (6), (7). Conversely every solution of (6), (7) is a solution of (1-4) when (¢g,+vgtl'Vq~) is considered as the total derivative
(d/dt)¢,(r + v~t, tl, t). We now apply the monotone iterative scheme to construct the solution of the integration (6) and (7). We construct the sequence { ¢~*), C) *)} from the iteration:
4a~k)=C~go(r-vgt'lt, Q ) e x p { - f o ' V g g g [ r - V g t l ( t - O ] d ~ } + ~ ' e x p { - f ' v g % [ r - v g Q ( t - O ] d ~ x [Fg(t~g(k
C¢k)(r,t) = Go(r) exp ( - 2 i t ) +
}
1),C)*-l))][r--vgtl(t--'c),~l,'c] dr,
exp [ - ~ j ( t - r ) ] ~ ( ¢ ~ , k " ) ] ( r , r ) dr,
(10) (ll)
282
G.-S. CHEN
by a suitable choice of the initial iteration ($g(0), C~O)). In order to ensure the convergence of the sequence, we require the existence of an upper and lower solution which is defined as follows.
Definition 2.1 A continuous vector-valued function : (¢g,Cj) ~ (¢1,¢2 . . . . . e G , C l , C 2 . . . . . C6) is called a lower solution of (6-7) if it satisfies the inequalities :
dp9(r~fLt)<`q~.~(r-vg~t~I))e~p{-f~tvg~g[r-vgf~(t-~)]~}+f~`exp{-f~v~g[r-va"(t-~)]d~} x [Fg(dpg,,Cj)][r-vafl(t-z),f~,z]dz , g = 1..... G, [(r, fl, t)~Ql], q ( r , t ) ~< Go(r) exp ( - 2 i t ) +
exp [ - 2 j ( t - z ) ] ~ ( 4 ~ g , ) ] ( r , r ) dr, j = 1. . . . . 6, [(r,t)eO0].
Similarly, a continuous vector-value function (~g, C~)= (g,, ~ . . . . . ~a, C,, C~ . . . . . C~) is called solution of (6-7) if it satisfies all the reversed inequalities in (12-13). Let (~g, C~) and (4~g, C~) be a pair of upper and lower solutions such that: (q~g,(7~)~>(~g,q)~>O,
~g>.d?g>~O,C~>~C~>~O,
i.e.
g=l
(12) (13)
an
upper
. . . . . G,j= I ..... 6.
By choosing (q~(O).Qo))= ( ~ , (~) as the initial iterate in (lO-ll), we can construct a sequence which is denoted by { q~g(~),C~)}. Similarly the sequence obtained from (lO-1 l) with (~bg(°)' C~°)) = (q~g, C~) is denoted by {q~(~),_C~)}. These two sequences, referred to as maximal and minimal sequences, possess the following _g monotone properties.
Lemma 2.1 The maximal sequence {q~a(~), CJ~)} is monotone nonincreasing, i.e. (bg(t), C) ~)) 1> (bg(*+ '), C) ~+ ')) k = 1, 2 . . . . . while the minimal sequence ~.g ,~ (~), C) ~)} is monotone nondecreasing, i.e. (~g(~), _C~~+')) ~> (~j(~), _C~~)) k = 1,2 . . . . . Moreover, d) (k) (~(k) ]
~g , ~ j
-(k) - ( k ) ~ ~< (~bg ,Cj )
f o r e v e r y k = 1,2 ....
Proof--We first show the monotone nondecreasing property of ,f~(k) c(k)~ by the induction principle. It is obvious from (I0-11) with k = 1 and (_$~0), _C)0)) = (~g, Cj) and definition of a lower solution that :
~ o ) _ ~ , ) = ~pg-q~go(r-vgf~t,f~)exp {-- £'va~g[r--vaf2(t--~)] d~} - joI' exp { - f ' Vg~g[r-Vgf~(t-~)l d~) x [F,(4,~,, Cj)][r--van(t--r), n , d dr ~< O, g = 1,2 . . . . . a ,
C_~°)-C_~') = C j - C j o ( r ) e x p ( - 2 j t ) Assume, by induction, that (~b~ _ k- ~), _C) k
exp[-2j(t-z)][fj(~g,)](r,~)d~>~O,
J)) ~< (,~(k) ,~_g , C!k)~. _ j ,,
~_(qk'--~_~k+l)=fo'eXp{--f'vg%[r--Vg~(t--~)]d~}[Fg(~.(k,
then using (10) and
(11),
j = 1,2 . . . . . 6. we obtain:
'),C} k '))--Fg(_~g(k.),C}k))]
x [r-vgf~(t-z),~,z]dr,
g = 1,2 . . . . . G,
(14)
Multigroup transport equations in reactor kinetics
fo
(:,(k)- - _.~j (7(k.,) = .=_j
exp[-2/(t-OlD~j(q~.k-t))-~(~b~,k))](r,~)d~,
283
j = 1,2 . . . . . 6.
(15)
Since for each (r, O, t) e Q ~,
(k ., ,C/(k ~,)-F.(cp¢(k, ,Cj,k,)](r,a,t)=v~ ~ ~ [Fo(q~o, .
.
.
.
.
.
.
[
+(l--fl)Zu ~ v.q, fa g'=l
(.h(, ,)
Jlb"
W g'
--
f a~.(r,.'.f~)(~p(q k, ')-~q(O)(r.f~'.t)df~'
q '~= 1 ,,]S
.
.
.
.
.
.
~b(,))(r f l ' , t ) d D ' + i '
g'
j=l
_ _. [f(qbg(k,~))--f~(~b~k))] (r,/) = f l j g ~ vg,
~ ' " ~ ( ' ')
C}'))(r t)} ~<0 (16a)
aSg,(~ ,~-L) -_qSg, ck)~. df~' .N<0,
(16b)
we see from (14-16a,b) that ~ba(k)-~b~k+ ') ~< 0 and (,(k) c '(k+ ~) ~< 0. Using the induction principle, we conclude that ,:,(d)(k), -~JC'(kh,~< -'~g(~(k+,), ~JC'(k+~)~,for each k = 0, 1,2,. .. This _~(k), ~J (.(k)~,- A similar argument leads to the conclusion proves the monotone nondecreasing property of {.~a • ~ /> (~g '), 6"J~+ ')) for every k = 1,2,3 . . . . (~*', C k)) (*+ (,6(o) , ~i ¢(0)~, = (4~a,C~) ~< (43~, ~'~) = (4Sq(°), (7/°)) and a similar argument as before, we Finally, using (0, 0) ~< .'c_o _
-
-~J
--
~ j
obtain : d~(~)--~ (') = Xa
,q
C ('' -.i
C(" = --
/
exp
--
v.a.[r-vo~(t-~)]d~
r~" t.h(o) ('(oq_t~ bg(o) ' ff;)O))l[r_v~fl(t_t),f~,~]d ~ <~O, 1 "Ot'e'.q' l a g s . W 0" , ~ - j
do
f,
Jo
exp
[-2At-Ol~(~°b-~($~°')l(r, Od~
~< 0.
Suppose (~b ~)) ~< (q~J~), ~J~)). It _ g '~ ~', _C~~- ')) ~< (q~*-'), 6"~-'))' ~ then a similar argument leads to (~b _ g (k), _C~ (~(k) ' -~J c(kh" ~< ($(~), follows by induction that ,xa ~ ~j(k)) for every k = 0, 1, 2, ... This proves the conclusion of the lemma. For convenience, we define the least upper bound (supremum) and greatest lower bound (infimum) of %¢, a~. and au o n / 5 x S x S and/5 x S, respectively, as follows : 6~,, = sup { Jsraq¢(r'f~"fl) d ~ ' ; ( r ' " ) e / s x S } ~o.(~') = inf
6. =
max
'
{%¢(r, ft', fl) ; (r, ~) e/5 x S},
{6gg. ; 1 <~g',g <~G},
0z = max {6zo ; 1 <~g<~G}, v = max {vo;1 ~
max {~ ; 1 ~< g
~< G}.
Based on the results of Lemma 2.1, we can now establish the following theorem for the system (1-4).
Theorem 2.1 Let (q~, (T~), (4~, C~) be a pair of upper and lower solutions such that ($~, ~ ) / > (4~, C~)/> 0. Then the minimal sequence { ~b(~), C~(*)} converges monotonically from below to a unique solution (4~, C~) of (1-4) and the maximal sequence (q~(~), ~'¢~)~,converges from above to the same solution. Moreover,
284
G.-S. CHEN
(o,o) < ( ~ , c~) = (_~(o,, _c)o,) < (_~,,,, _c?,) < ... < ( ~ , cj) < . . . < (~,(',, c~',) < ($,(o,, Co)) = ($~, ~). (17) Proof--In view of Lemma 2.1, the minimal sequence (_~o(k), _C~k)) is monotone nondecreasing and is bounded from above by upper solution (~g, Cj) while the maximal sequence { q~k), 6"~)} is monotone nonincreasing and is bounded from below by lower solution (4~g,C~). Therefore the pointwise limits : lim (,4, (k), _C(k)) = (4)g, _Cj)
k~oo
~-g
-
lim ((~g(~), ~j(k)) = ((~g, (~j),
and
k~c¢
exist and satisfy the relation : (~g,Cj)
<
(d)(l) C{l)~< \'~_g , - - J ]
•. .
<(~)g,C_j)<(l~g, fj)< --
". .
<
(~(l),~(i))< g
(~g
~j).
By letting k ~ oe in (10-11) and applying the dominated convergence theorem we conclude that both (q~g, Cj) and (¢g, Cj) are solutions of (6-7). The equivalence between the equations in (6-7) and (1-4) ensures that they are also solutions of (1-4). In order to show that (~,q, _Cj) = (q~g, Cj), we let: Cj*=(C)-Cj)e
¢ * = (q~q-~g) e -~',
-bt,
g = l . . . . . G,
j = 1. . . . ,6,
where b ~> 0 is a large constant so that (18)
b > v,G(~s+26)-l-(GVl +6)v6f. Then (q~*, Cj*)/> 0 and satisfies the equation :
dp*+vof~'Vc~g+(voag+b)¢* = vg
g
I JS
, , ,t) d f ~ ' + Z g ( 1 - f l ) agg,(r,D , ,f~)dpg,(r,~
g'~l
vg,
+ i ZJg2jC*}=[Fg(d)g*',C,*)](r,",t),
afg, ¢ ,g df~' g= 1,...,G,
j=l
£c, +0,j+b)Cj* Ot j
= flj ~ v,, ~afg,dp*d~' - [fj(~b~)](r,t), j = 1,2,... 6, js g'=l dp*(r, fl, t) = O, (t > O, reD, D incoming),
q~*(r, fLO) = O, (r6D, DeS), C]*(r, 0) = 0,
(r e D).
Using the same procedure in obtaining (6) and (7), we have :
c~*(r, fl, t) =
;i{i exp
-
}
[b+vg%(r-vof~(t-O)]d ~ [ F g ( ¢ ~ , C * ) ] [ r - v f l ( t - z ) , f l , z] dr,
C~(r, t) =
fo
exp [ - (2j+b) (t - 0 1 [ £ (4~)l(r, 0 d~.
(19)
(20)
Define : 114"11 = s u p { l ¢ * ( r , ~ , t ) l ; ( r , f L t ) e a l } ,
IIQ*II =sup{lCj*(r,t)l;(r,t)eQo}.
Next, we show that II¢~* JJ = 0, IIQ* II = 0, which implies that ~g = _~g, ¢2j = _Cj. By (19) and (20), the definition
of fs, ff, v and O <~ ~jg, ~g <~ l, O <. fl, flj <~ l, O < vg <~va, O < 2j <~26, we obtain :
Multigroup transport equations in reactor kinetics
r
114,gll. < vo(a~+v~:) t,_
}
114"11. +v.
kg~l
x (e.+vef)
IIQ*ll
114.*11 +,~
IIG*l[
I
}];;
285 vg
e x p { - ( b + v g t r o ) ( t - z ) } d z < b~+- v- q_~. g
~<~-(e~+vef+,~,)
IIG*II , (21)
114"11 +
j=l
I
j=l
(22)
since 2,. > 0, Vgag >~ O. Addition of (21) and (22) from g = 1,2 . . . . . G , j = 1,2 . . . . . 6, leads to : /= i
riG
_
_
6v5/
,
,
(
<~(Gv,(ff,+26)+(Gv,+6)v6:}"
6
0=l ~ 114"11+~,.= IIG*ll
}
• (23)
Since b is chosen as (18) such that: vl G(ffs q-J.6)q-(Gv ~+ 6)v0/
<1,
relation (23) cannot hold unless G
6
Z IIq~*ll + ~ IICTll =0. g=l
j=l
Hence, we obtain H4* ]l = 0, I] C* I1 = 0, g = 1,2 ..... G, j = 1,2 . . . . . 6. This implies that q~g = _qb~, (~j = C/, g = 1. . . . . G, j = 1. . . . . 6. Finally, if (q~g, C/) is another solution such that (4g, Cj) ~< (4g, Cj) ~< (-q~g,(~j) then it is also a lower solution and an upper solution. Using (4g, Cj) as the initial iteration in (10-11), the same argument as the p r o o f of Lemma 2.1 implies that ( ~ k , , _C}k)) ~< (q~g, C/) ~< (q~k), C,)k)) for every k = 1,2 . . . . Letting k ~ oo, it follows that :
(~,, _c:) ~< (4,, c,) ~< (~,, Cj). Since 4o = q~g, C: = C:, we obtain
(~,, c,) = (~g, cj) = ( ~ , ¢,). and (q~q, Cj) [or (~_g, _6",.)]is the unique solution of (1-4) satisfying the relation (17). This completes the proof of the theorem. Since Fg(0, 0) = 0, f:(0) = 0, it is easily seen from the definition of the lower solution that (4g, Cj) = (0, 0) is a lower solution. The result of Theorem 2.1 implies the following. Corollary 2.1
Let (q~g, C'j) be a nonnegative upper solution and (4g, Cj) = (0, 0). Then the minimal sequence/d) t ~ g (k)~ tT(~)~ "~j J and maximal sequence {q~(k),~)k)} converge monotonically from below and above, respectively, to a unique solution (q~g, C~) of (1-4) such that (17) holds with (4g, Cj) = (0, 0).
3. CONSTRUCTION OF UPPER AND LOWER SOLUTIONS
In view of Theorem 2.1 in the previous section, the existence of a solution of (1--4) is ensured if we can find a suitable pair of upper and lower solutions. The aim of the section is to construct some explicit upper and lower solutions, so that these functions not only serve as upper and lower bounds of the true solution, but also they can be used to determine whether the solution decays to a steady-state or grows unbounded as t ~ oo. This decay or growth property of the solution depends on the reactor parameters of the system without explicit knowledge of the solution.
286
G.-S. CHEN
Theorem 3.1 There exist positive constants A, b such that a unique solution to problem (1-4) exists and satisfies the relation : 0 ~< ~bg(r, t2, t) ~< A e b' g = 1. . . . . G, (r, t), t) e Q t,
(24)
O <~Cj(r,t) <<.A eh'
(25)
j= 1..... 6,(r,t)eQo,
whenever (0, 0) ~< (~bgo, Cj0) ~< (A, A), where b is any constant such that : b > v. [Gff, + 626] + (v. + 1)Gvfff.
(26)
Proof--In view of Corollary 2.1 it sufficient to show (~g, Q) = (A e b', A e b') is an upper solution, since ($g, Cj) = (0, 0) is a lower solution. In order to show that (A e b`, A e b') is an upper solution, we observe from the equivalence between the integral representation (6-7) and the differential system (1-4) that (~, ~j) is an upper solution if:
O_c~t ~g(r'"'t)+v°~'VC~g+Vg~rg(r)~g>~Vg { ~ fs a°g'(r'f~''fl)~a;'(r'"''t)d"' g'=l
=°;v/(rfg(r,fl)dpg,(r,D,t)dD'+
+Xg(1 - f l ) g Z
' ~
'
1
d~
~tCj(r,t) >~flj
g'=l
Zj,2jej(r,t) j=l
~ afg(r,D')C~g(r,D,t) ~ " dD-2jCj(r,t), j= vg. js ' ~
}
, g = 1,2 . . . . . G,
1,2 . . . . . 6,
~a(r, fL t) >tO, (t > O,reD, Q incoming), q~g(r, f~, 0)/> 0, C/(r, 0)/> 0,
(27)
(28) (29)
(r e D, f~ e S), (r e D).
(30a) (30b)
The boundary and initial requirements (29-30) are clearly satisfied by (q~o, C'j) = (A e b', A e b') with (A, A)/> @go, Cjo). Hence, we only need to verify that :
(b+vgag)Aeht>~Vg(~ Iegg.(r,D',D)Aeb'df~'+(l-fl)7~ ~ fsVg. fstrfo.(r,fl')Aebtdfl' g'=l Js
g,=~
+~Xjg,~jAebtt, j~l
~ va, f c%,(r,D')Aeb'd,O", j=
(b+2j)Aeb'/>fl/
9'=1
Since
Vgrg1> 0, 2j >
)
g = l . . . . . G,
1,2 . . . . . 6.
JS
0, the above inequalities are satisfied if:
b>~vu{~ ~agg,(r, fV, fl)d~'+(1-B)L, ,q'~l ~ v~,fsalg,(r,Q')dO'+~Zjg2j }, g'~l j=[ g'=l
JS
Using 0 ~< ~ig < l, 0 ~< fir ~< l, 0 ~< 1 - fl ~< l, the above inequalities are satisfied by :
b ~ vu b >~
g ' = l j,Is
~gg"d~t"~
g'~l
Vg" afa. drY+
2/ ,
~ Vg, ~ (Tfg,d~-~'. g'=l
In view of definitions of
5gg,,6I~,, (31-32)
(31)
j=
are satisfied, if for each g = 1, 2,..., G :
(32)
Multigroup transport equations in reactor kinetics
g'=l
g'=l
287
j=l
G
b >/ ~ v~.~y~,.
(34)
g'--I
Using v~ > v 2 . . . / > va, 26 > )-2. • • > ~.l and the definitions of v, 6s and 5I, (33-34) are clearly satisfied if:
b >1 vl(Gfs+Gvff+6).6), b >1 Gv~y. The above relation follows immediately if we choose : b > v 1(G~., + 626) + (v i + 1)Gvff. This leads to the choice of b satisfying (26). The existence of the solution and relations (24-25) follow from Corollary 2.1. Before we study the asymptotic behavior of system (1-4), we seek a solution u= for the boundary-value problem :
~'Vu+(~+a/va)u= 1 [(rxf~)~DxS], u(r, fl) = 0
(re OD, ~ incoming),
(35)
where c¢ is a positive constant. The solution u~ is given by :
u~(r,n) --
( r , n ) exp { - ( 6 + a / v a ) r } dz = { 1 - e x p [-(6+e/va)s]}/(6+a/va).
(36a)
When e = 0, we define :
uo(r, n ) = {1 - exp ( - Os)}/&
(36b)
where s - s(x, f~) >1 0 is chosen such that for any fixed r e D, Q e S, the point r* = r - s Q is on the boundary OD of D. The boundary point r* is determined by extending a line r in the direction of ( - ~ ) until it is intersected by the boundary. Hence, the value of s is s = I r * - r l / I~ I = I r* - r [, by If~ I = 1.
Theorem 3.2. Supercriticality Suppose
.q'~l
[ eag. (n'){1 --exp [-- #s(r, fl')]} dn'/5 > 1, g = 1, 2 . . . . . G,r e D .is
(37)
and assume ~bgo/> 6Uo, Cjo/> 0 for some 6 > 0, g = 1,2 . . . . . G, j = 1,2 . . . . . 6, where u0 is given by (36), then there exist positive constants • and e such that the solution (q~g, Cj) of the problem (1-4) satisfies: ~bg(r,Q,t) 1> ee~'uo(r,f~),
g = 1,2 . . . . . G,[(r,Q,t)EQd,
C/(r,t) >>.O, j = 1,2 . . . . . 6,[(r,t)eQo].
(38) (39)
Moreover, lim q~(r, ~, t) = oo as t --, oo for g = 1,2 . . . . . G. P r o o f - - T h e function g(y) s > 0, since :
[1 - e x p (-ys)]/y possesses the nonincreasing property, g (y) .~ 0, for y > 0 and
g'(Y) = e_(y.o (1 + y s ) - e x p (ys) y2
~<0.
According to the hypothesis (37), there exists ag > 0 such that for each g = 1,2 . . . . . G, ANE 16: 6-D
288
G.-S. C ~ N
~s #g,.Cft'){1-exp g'=l
[-(6+~/vc)sl} dfl'/(6+%/vc) >t 1.
Let ~ be the smallest value of %, g = 1. . . . , G. Hence for g = 1,2 . . . . . G :
.q'=l
[-(6 +~lvG)s]} dfl'l(6+~tvu) >1 l, (xeD).
fd~g,(ft')(1-exp 4s
(40)
Since ego i> 6uo t> ~u~ for some ~ > O, we now show that (~b~, Cj) = [s e~u=(r,~),O] is a lower solution. We observe from the equivalence between the differential system (1-4) and the integral representation (6-7) that (~g, Cj) is a lower solution if:
O~'.o+vgf~'Vdpg+vgagdPg<~V.q{~ ~a,qg,(r, ft',ft)dpg, dft'+(l-fl)Zg~vg, fa]g,(r,f~')dpg,(r, IY, t)d~' b' ' = 1
.q'=l
+ ~ZjgmiCj(r,t)},
°f
Cj,+i.~Cj~
dp.q(r,fLt)
g = l . . . . . G.
(41)
j=l
Vq,afq.(r, ft')~g,d.Q',
j = 1. . . . . 6,
(42)
I
~< 0,
(t > 0, re(~D, flincoming),
(43)
4~g(r,O,O) <~Cg0, (reD, tieS), Cj(r,O) <~Cjo, (reD).
(44a) (44b)
Since the initial requirement Cg(r,Q, 0) = Cz0 ~> 6u0 ~> eu= = ~bg(r, ft,0), Cj(r,0) = Cjo >>.0 = Cj(r,O) and (~bg, Q ) satisfies the boundary requirement, we only need to verify that (~bg,Ci) satisfies (41-42). The inequality (42) is easy to establish. Since (~9, Q ) = (s e='u~, 0), we have: q,+4Q
= 0 ~
g i.]s
v,'%'(r, fl')ee='udft'=l~j
,=
Using the definition of 6, 6, v~ > Vm... >
va, (35) and
a I
v¢ay¢,4~g,dft'.
(40), we obtain that:
4~,+ v~ft" V4~+vga. 49. = ~ e" [~u~+v.ft" Vu. + v.a.u.] <<.v.s e='[~ • Vu~+(6 +~/v.)u.] <<.vuse~'[Ft.Vu. + (6 + ~/vc )u.] = Vgee"-
ao.' { 1 - exp [-
1 ~< vu~ e =' kg'=
(6 + ~/VG)S]}d~'/(6 + ~/va)
1
=vuse~'{ ~=~fsa~(fg)~=dD~}<``v~g~ fsagg~(~ft~F~)s~'~u'dD~}=v~{~fsa~4~gdft~ } <~Vq ~'=,~ a,.q.(r,f~',ft)4~o,(r,tT, t) dfg+(1-fl))G.q,=,~ vg, ayg,(r,f~')dp,.(r,n',t)dft'+ ~l~,~2,C~ . Hence, we conclude that (~b~, Q ) is a lower solution. An upper solution is taken as (q~, Cy) = (A e~', Ae ~t) as in Theorem 3.1 with the sufficient large constants A and b so that (q~o, ~y)/> (~b~,C~), i.e. (.4 e~',A e~'`)>>. (s e=tu,, 0). Using Theorem 2. l, we obtain that :
This completes the proof of the theorem.
¢~1>se"u,,
g = 1. . . . . ,G,
Q~>0,
j=l
. . . . . 6.
Multigroup transport equations in reactor kinetics
289
Theorem 3.3. Subcriticality Let q~oo~< M, C~o ~< M, for some M > 0, g = 1. . . . . G, j = 1. . . . ,6, and assume that : a~ > ~o'=, 6g~, + (1--fl))~ "u.=, v°'ff/~'+,= ~
'
(45)
g = 1. . . . . . G
and ).j >/~j ~
vg, fit:q', J = 1. . . . . 6.
(46)
h,'=l
There exists a constant p > 0 such that a unique solution (~b., (7,) of problem (1-4) exists and satisfies: q~.q0(r-voDt, D ) exp (-vuffut) <~ c~,(r,D, t) <<.M e Qo exp ( - 2 i t ) ~< Ci(r,t ) <~ M e
t,,,
(47)
~".
(48)
P r o o f - Multiplying (45) by vu, there exist small positive constants pg, g = 1. . . . . G, such that : voa o - p g t> v.
I 5gu' + (1 -
9(~g,~i .
vo, 5yg,+ •
'=
l= I
Using the definition of 5u.q,, 5~o, we obtain :
g'=l
g'~l
]=1
Similarly, there exist small constants ?j, j = 1, 2 . . . . . 6, such that :
2j-ej >~lJ~ ~ vo,~co,>~llj ~ v~, faj~,dta'. g'=l
(50)
g'=l
Let p be the smallest value ofpg and ?j, 9 = 1. . . . . G , j = 1. . . . . 6. We now show (q~, Cj) = ( M e - P ' , M e -p') and (~o, Cj) = (0, 0) are upper and lower solutions, respectively. It is clear that (0, 0) is lower solution. Since ~g(r,D,O) = M >>.dpgo, Q(r,O) = M >1 Go and q~g(r, t'l, t) ~> 0, ~j(r, t) >t 0 satisfy boundary requirement (29). We only need to show (~o, Q ) satisfying the differential inequalities (27-28). Using the definition of p, a u, (49), (50), we obtain:
otc~o+v.D'V~+voag~ = ( V o a g - p ) M e P' >~ ( v . a g - p . ) M e ~' >t t'g
~ t.g'=
aog,M e - "
vg,a~g,M e-"' d D ' +
I
'=
=
~jg2jM e -°' j= !
E
dn'+
k. #'= I
,q'= I
x.Z,q j= I
O~j c~t
I
}
dD'.
'
g'=l
This shows that ( M e -p', Me -p') is an upper solution. By Theorem 2.1, a unique solution (~bg,Cj) exists and satisfies :
O < d p o ( r , ~ , t ) < M e p, O ~ C j ( r , t ) < M e - P , . Using (~o(°), C~ °)) = (~bg, Cj) = (0,0) as the initial iteration and observing that Fg(0, 0) = 0, ~(0) = 0, the first iteration in recursion formulas (10-11) is given :
dpl'~ = dpgo(r-vg~t,D)exp _g
-
v~ag[r-v~t'l(t-~)]d~
>~ dp~o(r-VgDt, f~)exp ( - V g f fl),
(51)
290
G.-S. CrmN _C~t)(r, t) = Go exp ( - 2 i t ) .
(52)
In view of Theorem 2.1, (_~at ~), C)~)) is a lower bound of solution (q~g,Cj). Hence relation (47-48) follows from (51-52). This completes the proof of the theorem. At the end of this section, we study the asymptotic behavior of the solution for the slab by making use of the results developed in Theorem 3.2 to give a more explicit condition for the growth property of the solution. We consider a slab with length l and its faces perpendicular to the x-axis and located at x = 0 and x = l. The general system (1-3) in a slab model is reduced to :
g
I% , (x, P', #) ~bg,(x,#', t) d#'
I
+(1-/s)z.g'Ev.f (53) =lAx,.')O.,(x,u'.Odu'+Zzj.zjCj(x,t)}, j=l NG(x,t) =/~j
g'=lv,,
%,(x,#')g~g,(x,#',t)dl~'-,~jCj(x,t), (t>O,O
(t>0,0
~b~(/,p, t) = O,
(t > 0,--1 ~ < p < O )
c~g(x,p,O)=dpgo(X,#),
(54)
l)
(O~
Here p = cos 0 and the initial and boundary conditions for Cj. as (4) with replacing r by x. The boundary value problem (35) in the slab becomes :
tmx+(ff+ot/vo)u= 1, ( O ~ < x < / , - - 1 ~<#~< u(O,#)=O
forO<#~
and
u(l,/~)=O
1)
forO~<#
Using (36), the solution of the above equation is : [ { 1 - e x p -(6+a/va)x/#}/(5+~/v~)
0 < # <. 1,
u,(x,#) = I 1/(5+c~/vo), p = 0, [ { 1 - e x p [(6+a/v~)(l-x)/#]}/(6+e/va),
(55)
- 1 <~# < 0,
since s = x/# for 0 < p ~< 1 and s = ( l - x ) ~ ( - # ) for - 1 ~< p < 0. Using the value o f s given as above, the condition (37) becomes :
~ { II #gg,(p') [ 1 - e x p ( - 6 x / p ' ) d p ' + f)
g'=l
do
ds¢(#') t
] 1-exp[~(l-x)/I/]d#' } >&
(0 ~< x ~
If we let :
then (56) becomes :
g'=l f01do,q,(p'){2--exp( - 6 x / p ' ) - e x p
[-6(l-x)~#']} d#' > 6.
Let
g(x) = 2--exp ( - - # x / # ' ) - - e x p [--6(l--x)~#'], we have
(57)
Multigroup transport equations in reactor kinetics 5 g' (x) = :7; {exp ( #
~xlu')-
291
exp [ - 6 ( l - x)t~'] }
and g'(x)>0,0~
g'(//2)=0;
g'(x)<0,
l/2<.x<~l.
Hence, the function g(x) has m i n i m u m at x = 0 or x = l. The inequalities (57) are satisfied if for each g = 1,2 . . . . . G :
fo' 6 g g , ( , ' ) { l - e x p ( - # l / p ' ) } a t + ' > 5.
(58)
g'=l
In view of T h e o r e m 3.2, we have the following result.
Theorem 3.4. Supercriticality Assume ~bg0 f> 6u0, Cjo/> 0 for some 6 > 0, g = 1,2 . . . . . G, j = 1,2 . . . . . 6, where u0 is given by (55) with ct = 0. T h e n u n d e r c o n d i t i o n (58) for g = 1,2 . . . . . G, each ~bg,g = 1. . . . , G, grows to ~ as t ~ oo.
REFERENCES
Belleni-Morante A. (1976) Nucl. Sci. Engng 56, 59. Case K. S. and Zweifel P. E. (1963) J. Math. Phys 4, 1"376. Chen G. S. and Leung A. W. (1985) J. Math. Phys 15, 490. Chen G. S. and Leung A. W. (1989) SlAM J. Appl. Math. 49. Duderstadt J. J. and Martin W. R. (1979) Transport Theory. Wiley, New York. Ganapol B. D. (1980) Transport Theory Statist. Phys 9, 145. Hill T. R. and Reed Wm. H. (1976) TIMEX: a time-dependent explicit discrete ordinate program for the solution of multigroup transport with delayed neutrons. Report LA-6201-MS, Los Alamos Scientific Laboratory. Kastenberg W. E. and Chambr6 P. L. (1968) Nucl. Sci. Engng 48, 211. Kobayashi K., Oigawa H. and Yamagata H. (1986) Ann. nucl. Energy 13, 663. Ladde G. S., Lakshmikantham V. and Vatsala A. S. (1985) Monotone Iterative Techniques for Nonlinear Differential Equations. Pitman, Boston. Lathrop K. D., Anderson R. E. and Brinkley F. W. (1971) TRANZIT : a program for multigroup time-dependent transport in (p, z) cylinder geometry. Report LA-4575, Los Alamos Scientific Laboratory. Larsen E. W. (1979) J. Math. Phys 20, 1776. Lee C. E., Fan W. C. P. and Dias M. P. (1985) Ann. nucl. Energy 12, 613. Leung A. W. (1984) J. Math. Analysis Applic. 100, 583. Lewis E. E. and Miller W. G. Jr (1984) Computational Methods of Neutron Transport. Wiley, New York. Pao C. V. (1981) Prog. nucl. Energy 8, 191. Pao C. V. (1983) J. Math. Phys 24, 132. Richtmyer R. D. and Morton K. W. (1967) Difference Methods for Initial-Value Problems. Wiley, New York. Sanchez R. and McCormick N. J. (1982) NucL Sci. Engng 80, 481. Stacey W. M. Jr (1969) Space-Time Nuclear Kinetics. Academic Press, New York. Vida V. I. (1968) J. Math Analysis Applic. 22, 144. Williams M. M. R. (1971) Mathematical Methods in Particle Transport Theory. Wiley, New York. Williams M. M. R. (1985) Ann. nucl. Energy 12, 167. Wilson D. G. (1974) J. Math Analysis Applic. 47, 182. Wood J. (1985) Ann. nucl. Energy 12, 217. Wood J. and Williams M. M. R. (1986) Ann. nucl. Energy 13, 479.