On Optimal Design of Nuclear Reactor

Copyright © IFAC 3rd Symposium Control of Distributed Parameter Systems Toulouse . France . 1982

ON OPTIMAL DESIGN OF NUCLEAR REACTOR Feng Dexing and Zhu Guangtian Institute of Systems Science, Academia Sinica, Beijing, China

ABSTRACT. We present the optimal design of scattering-fission cross section of a nuclear reactor by using the single energe static isotropic transport equation. Such design is more suitable for fast neutron reactors of smaller size than the diffusion approximate equation method traditionally used for non-multiplicative thermal neutron reactors. We show the existence of the optimal scattering-fission cross section and give the corresponding optimality conditions. KEY WORDS. Nuclear reactor, optimal design, optimal cross section, distributed parameter system.

INTRODUCTION

tively. We denote by ~ the strong convergence and bY-4 the weak or weak star convergence.

Generally the diffusion approximate equation is used to design optimal nuclear reactor. This method is quite effective for non-multiplicative thermal neutron reactors of big size, but for fast neutron reactors of smaller size, we should use the neutron transport equation. In this paper we consider the optimal design of scattering-fission section by using the following isotropic single energy transport equation :

(1) S

s.grad

x 2

u + 0a(x)u

2 D(L) = {UEL (D) Is.grad UEL (D), u(x,s)=O, x x E aV, s. nx < O} It is well known that (Vladimirov 1961) the adjoint of L is (3)

s.gradxg + 0a(x)g

2 D(L*) = {gEL 2 (D) Is.grad gEL (D) ,

E S

x

g(x,s) = 0, x E aV, n .s >

(2)

u(x,s) = 0, x E av, n.s < 0 x

x

(4)

o}

2 The bounded operator K in L (D) is defined as

where V is a bounded open convex domain in~3, occupied by the reactor medium, with a sufficient smooth surface av, 0a and Of are the to-

Ku

tal cross section and the scattering-fission cross section respectively. S is the unitary .

Lu

L* g

°

(x) u(x,s) s.grad u(x,s) + x a Of (x) f u(x,s')ds', x E V, 4n s

2

We define in L (D) the linear operator L,

4n

f

u (x, s' ) ds'

S

It can be shown that K is a self-adjoint ope2 rator defined on L (D). Thus the equation (1) (2) can be rewritten as

3

sphere ~n~ , u(x,s) represents the angular flux of the neutrons situated at x with the velocity s, nx is the unitary vector along the

(5)

°,

external normal at x E aV. Suppose that ' a Of E L00 (V), 0a (x) ~ 0 and that t h ere ellt

We now consider the eigenvalue problem

°

constants cl' c 2 such that < cl ~ Of (x) ~ c 2 p.p; x E V. We define the set of the admissible scattering-fission cross sections, as

(6)

It is well known that the reactor system (1), (2) functions well only in the critical state. Mathematically, this means that the largest (in modulus) eigenvalue A(Of) of (6) should be equal to 1, and that the corresponding eigenfunction u(Of) is positive for p.p.x E V,

Each function in U is simply called an admissible cross section.

E S. According to the neutron transport theory (Vladimirov 1961), we know that:

S

Let D = V x S and we denote by < .,.> and (.,.) 2 2 the inner products in L (D) and L (V) re spec-

359

360

F. Dexing and Z. Guangtian

1° The operator T(of) = L

-1

2

(OfK) in L (D) is

compact and positive, 2° The spectral radius r(T(Of» of T(Of) is posi ti ve, 3° r(T(of» is just the largest eigenvalue A(Of) of T(Of) with the multiplicity 1,

(12) By applying the operator K to (11), we obtain gn

= KL-

(13 )

where gn = KU

4° The corresponding eigenfunction u(Of) is almost everywhere positive.

-1

o

1 LU(Of) = A (of) Of KU(Of)

(7)

where c

(KU(Of), 0f/A(Of»

(8)

where A(Of) and u(Of) are the state variables.

n

It is well known that KL is an integral ope2 rator in L (V) whose kernel G satisfies

Let P be the total power of the reactor. Then the state equation is

= p

1 (On f

< G (x,y)sc /

3

·2 1x-yl , x,y

€

V, x

;t

is a positive constant and 3 Euclidien norm in]R3.

y

(14)

1.1 is the

By applying the operator M to (13) and by using ( 14), one has

In order that the reactor operates efficiently and safely, the admissible cross section Of

* = 1 and the should be selected such that A(Of)

c3 m(V)

quadratic functional

n Of (y) fn (y)

vI

dx

I ---2 V

'ITc 3d

(9)

d Y S

1x-yl

n

I Of (x) gn(x)ds

S m(V)

V

* on U under the conattains its minimum at Of di tion that A (0 f)

4'IT2 c

1. In (9), M denotes the

4'IT2 c

mean operator Mf = m(V)

I

f(x)dx

d/m(V)

3

(KUn'O~)

p d/= m(V)

3

where d is the diameter of V.

V

where m(V) is the Legesgue measure of V in R3. In other words, the optimal design problem of the scattering-fission cross section redu-

00

* € U such that ces to find a Of

star closeness of U in L (V), without loss of generality, i t can be assumed that ( 10)

inf °f€U,A(Of)=l

The above indicates that (Mg ) is a bounded n number sequence. And also (g ) is a bounded 2 n sequence in L (V). As a result of the weak

J(Of)

n 0 0f~Of €

U

O~gn

2 KUn-lh € L (D)

=

O~

(n-+ (n-+

oo

)

-1

The function Of € U satisfyind (10) is called

Then by using the compactness of KL , one has

an optimal scattering-fission cross section. In the two following sections we prove the existence of the optimal scattering-fission cross section and give the corresponding optimaliry conditions .

Noting that (KU ) is a bounded sequence and n -1 2 that L is a bounded operato r in L (D), and

1. EXISTENCE OF OPTIMAL CROSS SECTION

by the relation un = L

-1

n

(Of Ku ), we can conn

2

clude that (u ) is bounded in L (D). Because In the remaining text, we suppose there exists at least a Of € U such taht A(Of) = 1. Eviden4 tly, i t is a reasonable hypothesis. n

n

-1

of the compactness of L K, we might as well assume that U -+ u ' (n -+ 00 ) . In addition o n n 0 from Of..1 0 f and un -+ u , it follows that o

Let (Of) € U be a minimizing sequence,i.e. n

A(Of) = 1

Then from ( 11) it results that inf J(Of) °f€U,A(Of)=l

U

n

= L -1

L

-1

0

K( Ofuo)

(15)

By the formula (13), it is not difficult to obtain

n

Setting un = U(Of)' we have u

o

(11 )

p

(16)

361

On Optimal Design of Nuclear Reactor Thus it can be seen that U is an eigenfunco tion of (6) corresponding to the eigenvalue 1. Obviously U is non negative. Therefore, by o the uniqueness of the non-negative eigenfunco tion of (6), one has A(Of) = 1. Finally, let n one obtains

~

o

00

in the expression of J(Of) ,

min ofEU, A (of) =1

J(Of)

then the problem (10) will be equivalent to the following one : find a v J (v)

( 19)

E U such that

o

min

J (v)

VEU and the optimal cross section becomes

Theorem 1 shows the existence of the solution of (19). We now derive the conditions for optimality.

i.e., O~ is an optimal scattering-fission cross section. The following theorem summarises the above results. Theorem The problem (10) admits at least a solution, in other words, there exists at least an optimal scattering-fission cross section.

Note that (u (v) , A (v!.) is a Frechet differentiable mapping from U into L2 (D) x lR', based on the differentiability concerning the simple eigenvalue (Kato 1966). Let v

JS

be a solution of (19). For any fixed

VEL (V), set vt=v +tv,A =A(v) u =u(v ) o t t ' t t

2 . OPTlMALITY CONDITIONS We have shown that the problem (10) is an extremum problem with non-convex constraints. For the sake of simplicity, we transform it into an equivalent problem, but only with convex constraints. First we show that there exist constants c 4 and Cs such that

In particular, A = A(v ), u = u(v ). To sim000 0 plify the notations, we denote dA du Ao = dt tl ,uo = dtt1t=O t=O It follows from (17) and (18) that (20) (21)

In fact, as stated above, the operator T(Of) = L-1(OfK) is compact and positive for all Of E U. Thus it is not difficult to see that

Therefore, the spectral radius r(T(Of» fies (Krasnosel'skii 1972)

-1 By noting the boundness of L and by differentiating (20) with respect to t, one has

A Lu + A Lu = v Ku + vKu o 0 0 0 000

~

satis-

Ao Lu0

-

v Ku + vKu 000

o

Applying the operator LNoting A(O f) conclus i on .

r(T(Of»' we obtain the above

o

A

-1

L

1

v Ku 0

(22 )

0

to (22), one obtains

(vK)u

o

(23)

Now let d

1

= c c ' d 1 4 2

c c 2 S

According to the compact operators theory (Kwan 1958), the equation (23) admits a solution i f f

and


-1

(vK) u

Obviously, for any v E U, one has

(24)

o

where go :~ an eigenfunction of (L = VoK(L*) Therefore , if we rewrite the state equations (7) , (8) as (17 )

(Ku(v), v)

( 18)

= A (v)p

(VoK»

*

corresponding to the eigenvalue

AO ' i.e.

*

Lu(v) = V/A(V) .Ku (v), v E U

-1

VoK(L)

-I-

go = A090

From (25) it follows that

(25)

362

F. Dexing and Z. Guangtian

~ 41T g, A0 LU 0 -v 0 Ku0 > + c(v,KU0 } -

(26) where go that AO

(L*)-1 g

o

- AoCP

Thus we obtain from (24)

(VKU ' Kg -Kg o + O

(27)

Ao /

If go satisfies the following normalizing condition (28) Finally according to (Lions 1968), we obtain the necessary condition as follows

then the formula (22) becomes A Lu

o

(29)

0

and the formula (27)becomes

Moreover, from (21) and (30) we have (31)

(KUO'V) + (KU ,v ) = PA 0

(37)

-Mku , Ku )( (4Ir ) ~ Kg - l}] dx ~ 0 o 0 0 0 p (30)

o

~ (V-Vo}KUo[Kg -
0

0

0

V € U

To summarise, we present the following theorem 'lbeorem 2

(29) and (31) are just the equations satisfied by u ' o

J(V), clearly is Frechet differentiable and J' (v ) (v) o

= J (Ku -MKu ) (Ku -MKU }dx V

0

~ 4n 0 0' 0

0

0

0

(32)

~

0, v

(33)

€ U

Let us introduce the adjoint state g described by the following equation (34) where c is a constant. According to the alternative theorem of Fredholm (Kwan 1958) it is easy to see that (34) admits a solution iff

By

virtue of o

=

A(V }' U = u(vo }, then o o there exists an adjoint function g € D(L*} such that and denote Ao

= v 0 Ku0

A Lu o 0

V €

However, we know from (Lions !968), that J(v} attains its minimum at Vo on U if and only if JI(V } (v-v ) O o

Suppose Vo is a solution of the problem (19),

(KuO'VO) = AoP A L* g = v Kg + (Ku -MKu ) o 0 0 0
1

-(Ku -MKu ,Ku }v p

0

000

=0

J (V-Vo}KUo [Kg -
V €

0

1

-}] dx

~

P

0,

U

where go is solution of the equations

4np, one obtains

c = - _1_ 41TP 0 0' 0 (35) p

Bi Dachican and Zhu Guangtian (1981)

Moreover, we impose on g the following condition (36) so that (34),

(35) have the unique solution.

From (31), (34), (35), it follows that J' (v ) (v)

o

REFERENCES

1

.

4n 1- 1~,\ r.*g _ v Kg - cv ,u• > _

4n

0

0

0

0

The optimal control of the power distribution of the nuclear reactor, Preprints of 8th IFAC Congress, Pergamon . Vladimirov V.S.

(1961)

Mathematical problems in the one-velocity theory of particle transport, Proc. Steklov Inst. Mathematics 61 : English transl. Atomic Energy of Canad. Ltd., Chalk River, Onto Report AECL-1661 (1963).

On Optimal Design of Nuclear Reactor Kato T.

(1966)

Perturbation theory for linear operators. Springer. Krasnosel'skii M.K.

(1972)

Approximate solution for operator equations, Wolters-Noordhoff pub. Kwan Chao Chih (1958) Lectures of functional analysis. High Education press, Beijing. Lions J.L.

(1968)

Sur le contrOle optimal des systemes gouvernes par des equations aux derivees partielles Paris, Dunod-Gauthier-Villars. Lions J. L.

(1977)

Remarks on the theory of optimal control of distributed systems in control theory of systems governed by partial differential equations. Academic press, INC, New York.

363