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OPTIMAL DESIGN OF THE CROSS SECTION IN A NUCLEAR REACTOR
atW/(C:-S'cientia
1996,16(2): 229-233
...4
l~tmJl~m OPTIMAL DESIGN OF THE CROSS SECTION IN A NUCLEAR REACTOR 1 Song Degong
(~ttsJJ) Jiang Weihua (j.LJ1..~) Wang Miansen (L~4) Department of Mathematics, Xi'an Jiaotong University, Xi'an 710049, China.
Zhu Guangtian ('*- r tJJ)
Institute of Systems Science, Academia Sinica, Beijing 100080, China.
Abstract The existence of the optimal total cross section is obtained for steady-state neutron transport system in an infinite slab with generalized boundary conditions. Key words Optimal design, neutron transport system.
1 Introduction The power factor optimization for a nuclear reactor acquired a great deal of attention in recent years. The existence and the optimality conditions of the scattering fission section have been given in [1-3] for diffusion approximate equation and transport equation. As for the optirnal design of the total cross section, the result is only obtained for diffusion approximate equation[4,5]. In consideration of transport equation, the difficulty centers on the existence of the optimal total cross section, no paper has appeared on this problem up to the present tirne. It is the aim of this paper to solve this problern for monoenergetic neutron transport in a slab with generalized boundary conditions. Consider the following steady-state anisotropic transport equation
/-'a"t,l') + E(x )'ljI(x, /-,) = t (I)
(1/;(x, ~), h(x, ~))
1/;( -a,~) =
= p,
1:
a(~)1/;( -a, -~), 0
k( z , /-', /-,') 'ljI (x ,Jl)d/-",
<~
~ 1,
1/;(a,-/-L) ='Y(/-L)1/;(a,/-L),O < ~ ~ 1, xEV =
[-a,a],~ E D
= [-1,1],
in which A is a nonzero parameter, 2a is the thickness of an infinite slab with boundary reflection coefficients a(~) and 'Y(~), P is a positive constant, h(x, ~) is a given nonnegative function, E( x) and k( x,~, ~') are the total cross section and the scattering-fission kernel respectively. 1 Received
Nov. 22, 1993. Project supported by NSFC:
ACTA MATHEMATICA SCIENTIA
230
Vol.16
Throughout this paper, it is assumed that 01 ) a(J.-L) and 1(P.) are measurable functions satisfying 0
S r.x(p.) :S
1,0 :S 1(J.-L) :S
1, a( -p.) = a(p.) and 1{ -p.) = 1{P.).
V, X2
O2 ) 0 < k{x, J1" p.') :S ko(const.)< +00. Furtherrnore, there exists an interval [Xl, X2] C > Xl, such that k( z , p., p.') 2: kl (consb.) 0 for (x, u; /1,') E [Xl, X2] x D x D, a.e. {E(x) E Loo(V)la(x) ~ E{x) :S b(x), X E 03 ) The admissible control set of E(x) is U
=
V, a.e.}, where a(x) :S b(x), b(x) E LOO(V), and
ess ~~t a(a~)
>
o.
For any E( x) E U, it follows from [6] that there exists a unique positive eigenvalue Ao of Eq.(I) associated with an eigenfunction 1/Jo
> 0, a.e. Ao and 1/Jo are called critical parameter
and critical angular density of Eq. (I), and they are denoted by A{E) and ',p(E) respectively. Define the cost functional
where G
=V
x D, M is the rnean operator, i.e.,
M 1/J =
1( G)
lues
JfG 1/J( x, JL )dxdJi-.
Set U; = {E E UIA(E) = r}, where r is a given positive constant. Then the optimal design problem we consider is to find
all
adrnissible control E* such that
J(E*) = inf J(E),
(II)
{ E* E ti..
~EUr
2 Existence of the Optimal Design
~*
Assurne U; =f: . It is easy to see that U; =/:
F~· = E{x)·,
1 1
K· =
-1
k(x,Ji-,JL')· dJL', L· =
'j,~, ax
= D(K)
= L 2 (G), D(L) = {1/, E L2(G)IL',p E L 2 (G); ',p(-a,J-t) = O(Jl)',p(-a,-Jt) and ',p(a, -J.-L) = 1(p.)'¢)(a, p) for J-L E (0, I]}. Then Eq.(I) can be written as D(F~)
L',p + F~4' = tK',p,
{ (4', h) = p.
Let {En} C U; be a rninirnizing sequence of J inf J(~). Denote F1l • = F~ ·,'tPn
~EUr
n
011
U«, i.e.,
= ',p(~n),U>n = K-V'n, then r
A(~n)
=
1·
and lim J(~n) = '11.--+00
(1) (2)
Song et al: OPTIMAL DESIGN OF THE CROSS SECTION
No.2
231
Lemma 1[5] {1/;n} and {
~
tpn
Proof It follows from Eq.(l) that (3) Applying the operator (L
+ 1)-1
to both sides of (3) yields
Lemma 1 indicates that both {1/;n} and {
such that ',pn ~ 1/;0 and
+ 1)-1
+ 1) -1
is a compact operator
and K(L + 1)-1',pn ~ K(L + 1)-11/;0'
Consider K( L+ 1)-1 Fn 1/Jn.{ Fn4)n} = {En (x )',pn( X, 11,)} is obviously a bounded sequence in L 2(G), hence a subsequence of {FnWn}, written still {Fn'l/Jn}, can be selected such that
Fn',pn
s. f
E L 2 ( G). Therefore, K(L
+ 1)-1 Fn"pn O.
~ K(L
+ 1)-1 t,
and
> 0, we get Lemma 3 A subsequence of {"pn} exists such that "pn ~ "po.
Noting that
Proof Applying the operator (L
+ Fn )- 1 to
both sides of (1) yields
Since U is hounded in LOO(V), a subsequence of {En}, written still {En}, can he selected such that ~n ~ ~o E U C LOO(V). Denote F(}. = FrJ o ' and define '~(x,J-L)
'~)n ~ 'J. ForrH the following relations, 114)1& - '~II
= :(L+ Fo)- 1
= II~(L + Fn)-l
+ Fn)-lll· I/
1 ~. essinf En (x)
I/(L + Fn)-lll
lim
n-+oo
it suffices to prove
-
+ Fo)-1
11(£ + Fn)-l
As is shown in [7], for any "p E
(L
1
(L
:s; .-essinf a(x)
lI
[~
,
0,
+ Fo)- 1
£2 (G)
and i = 0, 1, 2, ... , n, . · . ,
(6)
ACTA MATHEMATICA SCIENTIA
232
Vol.16
for J-L positive and
(7) for J-L negative, where P i 1, P i 2, ... ,Pi6 are bounded linear operators on L 2(G) defined by
Consider Pn1C(Jo - P01C(Jo
l1
= -1 fX {exp[-J-L
-a
l1
x
En(s)ds] - exp[-s:' J-L
J-L
Denote by T(n,l) the right-hand side of Eq.(8).
t.r ~o(s)ds because of ~n ~ ~o, hence T(n, 1) IT(n, 1)1
--+
r a(s)ds]dx'
~ ~ f'"
x
Eo(s)ds]}C(Jo(x', J-L)dx'
(8)
a:'
For any fixed x, x',
°
1~ ~n(s)ds
--+
for any fixed (x, Jl) E V x [0,1]. Since
E L 2(V x JO, 1]), by virtue of Lebesgue's
control-convergent theorem, it is obtained that
(9) Similarly, nl~~ II Pnj C(Jo - POj
= 0, j = 2,3
(10)
nl~~ IIPnj
(11)
From Eqs. (6)-(11), it is not difficult to verify
which implies 'l/Jn ~ {J,Le., 1/Jn ~ 'l/Jo. Theorem 1 Eo is a solution of problem (II). Proof Consider Eq.(4). Since 'l/Jn ~ 1/Jo and En ~ Eo, it is seen that By virtue of weak convergence, Eq.( 4) irnplies
Fn'l/Jn
= En 1/Jn s. Eo'l/Jo.
Song et 801: OPTIMAL DESIGN OF THE CROSS SECTION
No.2
i.e.,
L1/;o·+ Fo1/;o Eq.(2) indicates ('1)0, h) = p
>
1 = -K1/;o. r
O. Furtherrnore, since
solution of Eq.(I), it follows form [6] that A(Eo) = lim
11,-+00
J(~n) =
lim
n-+oo
~1I1/Jn 2
233
M1/JnW
T,
,,po IS
a nonzero nonnegative
'rP(E o) = 1/;0, so Eo E U«. Noting that
= -2111 1/Jo.
M1/JoW
and {En} is a minirnizing sequence of J(E) on Ur,we getJ(E o )
=
= J(~o),
inf J(E).
~EUr
Remark In virtue of Dubovitskii-Milyutin theorem[8], it is not difficult to deduce the
necessary conditions of Eo by a process similar to [5].
References 1 Feng Dexing, Zhu Guangtian. On optimal design of nuclear reactor. Preprints of 3rd IFAC Symposium on Control of Distributed Parameter Systems, Pergamon, 1982. 2 Feng Dexing, Zhu Guangtian. An inverse problem in power reactor theory. Acta Mathematica Scientia, 1982,2:9-15 (in Chinese). 3 Feng Dexing, Zhu Guangtian. On opbimal design of scattering-fission Gross section. J. Sys. Sci. & Math. Scis., 1985,5:73-80 (in Chinese). 4 Bi Dachuan, Zhu Guangtian. The optimal control of the power distribution of the nuclear reactor. Preprints of Brd IFAC Congress, Pergamon, 1982. 5 Feng Dexing, Zhu Guangtian. Op'timal control for a class of systems governed by eigen-equations and
its applications. J. Math. pure et appl,, 1984,1: 169-186. 6 Song Degong.
On stability of the solution for neutron transport system in a slab geometry with generalized boundary conditions, M.Sc. Thesis, Xi'an Jiaotong University, March 1989 (in Chinese). 7 Song Degong, Wang Miansen, Zhu Guangtian. Asymptotic expansion and asymptotic behavior of the solution for the t ime dependent neutron transport pr-oblem in flo slab with generalized boundary conditions. Sys. Sci. & Math. Scis., 1990,3:102-125.
8 Girsanov I V. Lectures on Mathematical Theory of Extremum Problems. Springer Verlag, 1972.
1996,16(2): 229-233
...4
l~tmJl~m OPTIMAL DESIGN OF THE CROSS SECTION IN A NUCLEAR REACTOR 1 Song Degong
(~ttsJJ) Jiang Weihua (j.LJ1..~) Wang Miansen (L~4) Department of Mathematics, Xi'an Jiaotong University, Xi'an 710049, China.
Zhu Guangtian ('*- r tJJ)
Institute of Systems Science, Academia Sinica, Beijing 100080, China.
Abstract The existence of the optimal total cross section is obtained for steady-state neutron transport system in an infinite slab with generalized boundary conditions. Key words Optimal design, neutron transport system.
1 Introduction The power factor optimization for a nuclear reactor acquired a great deal of attention in recent years. The existence and the optimality conditions of the scattering fission section have been given in [1-3] for diffusion approximate equation and transport equation. As for the optirnal design of the total cross section, the result is only obtained for diffusion approximate equation[4,5]. In consideration of transport equation, the difficulty centers on the existence of the optimal total cross section, no paper has appeared on this problem up to the present tirne. It is the aim of this paper to solve this problern for monoenergetic neutron transport in a slab with generalized boundary conditions. Consider the following steady-state anisotropic transport equation
/-'a"t,l') + E(x )'ljI(x, /-,) = t (I)
(1/;(x, ~), h(x, ~))
1/;( -a,~) =
= p,
1:
a(~)1/;( -a, -~), 0
k( z , /-', /-,') 'ljI (x ,Jl)d/-",
<~
~ 1,
1/;(a,-/-L) ='Y(/-L)1/;(a,/-L),O < ~ ~ 1, xEV =
[-a,a],~ E D
= [-1,1],
in which A is a nonzero parameter, 2a is the thickness of an infinite slab with boundary reflection coefficients a(~) and 'Y(~), P is a positive constant, h(x, ~) is a given nonnegative function, E( x) and k( x,~, ~') are the total cross section and the scattering-fission kernel respectively. 1 Received
Nov. 22, 1993. Project supported by NSFC:
ACTA MATHEMATICA SCIENTIA
230
Vol.16
Throughout this paper, it is assumed that 01 ) a(J.-L) and 1(P.) are measurable functions satisfying 0
S r.x(p.) :S
1,0 :S 1(J.-L) :S
1, a( -p.) = a(p.) and 1{ -p.) = 1{P.).
V, X2
O2 ) 0 < k{x, J1" p.') :S ko(const.)< +00. Furtherrnore, there exists an interval [Xl, X2] C > Xl, such that k( z , p., p.') 2: kl (consb.) 0 for (x, u; /1,') E [Xl, X2] x D x D, a.e. {E(x) E Loo(V)la(x) ~ E{x) :S b(x), X E 03 ) The admissible control set of E(x) is U
=
V, a.e.}, where a(x) :S b(x), b(x) E LOO(V), and
ess ~~t a(a~)
>
o.
For any E( x) E U, it follows from [6] that there exists a unique positive eigenvalue Ao of Eq.(I) associated with an eigenfunction 1/Jo
> 0, a.e. Ao and 1/Jo are called critical parameter
and critical angular density of Eq. (I), and they are denoted by A{E) and ',p(E) respectively. Define the cost functional
where G
=V
x D, M is the rnean operator, i.e.,
M 1/J =
1( G)
lues
JfG 1/J( x, JL )dxdJi-.
Set U; = {E E UIA(E) = r}, where r is a given positive constant. Then the optimal design problem we consider is to find
all
adrnissible control E* such that
J(E*) = inf J(E),
(II)
{ E* E ti..
~EUr
2 Existence of the Optimal Design
~*
Assurne U; =f: . It is easy to see that U; =/:
F~· = E{x)·,
1 1
K· =
-1
k(x,Ji-,JL')· dJL', L· =
'j,~, ax
= D(K)
= L 2 (G), D(L) = {1/, E L2(G)IL',p E L 2 (G); ',p(-a,J-t) = O(Jl)',p(-a,-Jt) and ',p(a, -J.-L) = 1(p.)'¢)(a, p) for J-L E (0, I]}. Then Eq.(I) can be written as D(F~)
L',p + F~4' = tK',p,
{ (4', h) = p.
Let {En} C U; be a rninirnizing sequence of J inf J(~). Denote F1l • = F~ ·,'tPn
~EUr
n
011
U«, i.e.,
= ',p(~n),U>n = K-V'n, then r
A(~n)
=
1·
and lim J(~n) = '11.--+00
(1) (2)
Song et al: OPTIMAL DESIGN OF THE CROSS SECTION
No.2
231
Lemma 1[5] {1/;n} and {
~
tpn
Proof It follows from Eq.(l) that (3) Applying the operator (L
+ 1)-1
to both sides of (3) yields
Lemma 1 indicates that both {1/;n} and {
such that ',pn ~ 1/;0 and
+ 1)-1
+ 1) -1
is a compact operator
and K(L + 1)-1',pn ~ K(L + 1)-11/;0'
Consider K( L+ 1)-1 Fn 1/Jn.{ Fn4)n} = {En (x )',pn( X, 11,)} is obviously a bounded sequence in L 2(G), hence a subsequence of {FnWn}, written still {Fn'l/Jn}, can be selected such that
Fn',pn
s. f
E L 2 ( G). Therefore, K(L
+ 1)-1 Fn"pn
~ K(L
+ 1)-1 t,
and
> 0, we get Lemma 3 A subsequence of {"pn} exists such that "pn ~ "po.
Noting that
Proof Applying the operator (L
+ Fn )- 1 to
both sides of (1) yields
Since U is hounded in LOO(V), a subsequence of {En}, written still {En}, can he selected such that ~n ~ ~o E U C LOO(V). Denote F(}. = FrJ o ' and define '~(x,J-L)
'~)n ~ 'J. ForrH the following relations, 114)1& - '~II
= :(L+ Fo)- 1
= II~(L + Fn)-l
+ Fn)-lll· I/
1 ~. essinf En (x)
I/(L + Fn)-lll
lim
n-+oo
it suffices to prove
-
+ Fo)-1
11(£ + Fn)-l
As is shown in [7], for any "p E
(L
1
(L
:s; .-essinf a(x)
lI
[~
,
0,
+ Fo)- 1
£2 (G)
and i = 0, 1, 2, ... , n, . · . ,
(6)
ACTA MATHEMATICA SCIENTIA
232
Vol.16
for J-L positive and
(7) for J-L negative, where P i 1, P i 2, ... ,Pi6 are bounded linear operators on L 2(G) defined by
Consider Pn1C(Jo - P01C(Jo
l1
= -1 fX {exp[-J-L
-a
l1
x
En(s)ds] - exp[-s:' J-L
J-L
Denote by T(n,l) the right-hand side of Eq.(8).
t.r ~o(s)ds because of ~n ~ ~o, hence T(n, 1) IT(n, 1)1
--+
r a(s)ds]dx'
~ ~ f'"
x
Eo(s)ds]}C(Jo(x', J-L)dx'
(8)
a:'
For any fixed x, x',
°
1~ ~n(s)ds
--+
for any fixed (x, Jl) E V x [0,1]. Since
E L 2(V x JO, 1]), by virtue of Lebesgue's
control-convergent theorem, it is obtained that
(9) Similarly, nl~~ II Pnj C(Jo - POj
= 0, j = 2,3
(10)
nl~~ IIPnj
(11)
From Eqs. (6)-(11), it is not difficult to verify
which implies 'l/Jn ~ {J,Le., 1/Jn ~ 'l/Jo. Theorem 1 Eo is a solution of problem (II). Proof Consider Eq.(4). Since 'l/Jn ~ 1/Jo and En ~ Eo, it is seen that By virtue of weak convergence, Eq.( 4) irnplies
Fn'l/Jn
= En 1/Jn s. Eo'l/Jo.
Song et 801: OPTIMAL DESIGN OF THE CROSS SECTION
No.2
i.e.,
L1/;o·+ Fo1/;o Eq.(2) indicates ('1)0, h) = p
>
1 = -K1/;o. r
O. Furtherrnore, since
solution of Eq.(I), it follows form [6] that A(Eo) = lim
11,-+00
J(~n) =
lim
n-+oo
~1I1/Jn 2
233
M1/JnW
T,
,,po IS
a nonzero nonnegative
'rP(E o) = 1/;0, so Eo E U«. Noting that
= -2111 1/Jo.
M1/JoW
and {En} is a minirnizing sequence of J(E) on Ur,we getJ(E o )
=
= J(~o),
inf J(E).
~EUr
Remark In virtue of Dubovitskii-Milyutin theorem[8], it is not difficult to deduce the
necessary conditions of Eo by a process similar to [5].
References 1 Feng Dexing, Zhu Guangtian. On optimal design of nuclear reactor. Preprints of 3rd IFAC Symposium on Control of Distributed Parameter Systems, Pergamon, 1982. 2 Feng Dexing, Zhu Guangtian. An inverse problem in power reactor theory. Acta Mathematica Scientia, 1982,2:9-15 (in Chinese). 3 Feng Dexing, Zhu Guangtian. On opbimal design of scattering-fission Gross section. J. Sys. Sci. & Math. Scis., 1985,5:73-80 (in Chinese). 4 Bi Dachuan, Zhu Guangtian. The optimal control of the power distribution of the nuclear reactor. Preprints of Brd IFAC Congress, Pergamon, 1982. 5 Feng Dexing, Zhu Guangtian. Op'timal control for a class of systems governed by eigen-equations and
its applications. J. Math. pure et appl,, 1984,1: 169-186. 6 Song Degong.
On stability of the solution for neutron transport system in a slab geometry with generalized boundary conditions, M.Sc. Thesis, Xi'an Jiaotong University, March 1989 (in Chinese). 7 Song Degong, Wang Miansen, Zhu Guangtian. Asymptotic expansion and asymptotic behavior of the solution for the t ime dependent neutron transport pr-oblem in flo slab with generalized boundary conditions. Sys. Sci. & Math. Scis., 1990,3:102-125.
8 Girsanov I V. Lectures on Mathematical Theory of Extremum Problems. Springer Verlag, 1972.