PERIODICAL SOLUTIONS FOR THE NEUTRON TRANSPORT EQUATION

Olga MARTIN

Department of Mathematics

University “Politehnica” of Bucharest

Splaiul Independentei 313,Bucharest 16

ROMANIA

Abstract: - In this paper we provide an analytical method for computation the periodical solutions of the neutron transport equation in the one-dimensional stationary case. Also, the same problem was solved with a numerical method and the analytical solution and approximate solution was compared.

Key-Words: neutron transport equation, method of finite differences, the trapezoidal rule, series uniformly convergent.

1 Introduction

In a reactor, the neutrons are yielded at the fission of the nucleus and they are named the rapid neutrons with the average speed 2107m/s.

The rapid neutrons are subject to a slowness process, their energy decreasing until these are in an equilibrium state with the others atoms of the environment. When the reactor is in a stationary state, the atoms have the tendency to move from a region with a great density to another with a small density and thus to obtain an uniform density. This process is named diffusion.

The main problem in the nuclear reactor theory is to find the neutrons distribution in the reactor, hence its density. This is a scalar function, which depends on the next variables: the position vector of the neutron in a datum coordinate system, the neutron speed and the time.

The density is the solution of an integral-differential equation named the neutron transport equation. Many authors paid attention to this problem and its applications [1-3, 4-6].

In this paper we provide a periodical analytical solution for the one-dimensional stationary problem (1), where u(x,y) is the neutron density and g(x,y) is the source function.. The neutron moves to a direction, which makes the  angle with Ox axis and y = cos. Using the method of the finite differences and the trapezoidal approximation, we obtain a numerical solution for our problem and this is compared with the analytical solution.

2 Problem formulation

Let us now consider the integral-differential equation of the neutron transport theory:

(1)

We want to know how is the form of function g(x,y), in order the solution u(x,y) to be periodical function with respect to the variable x. If the period is H, then we have

Z (2)

and it is sufficiently to find the solution on the domain D = [0,H][-1,1].

The equation may be written in the form

(3)

where

(4)

Integrating (3) with respect to variable x we obtain

,

(5)

Now we get

(6)

The improper integral, , thus defined has no meaning (diverges), while there exists its Cauchy principal value given by the formula

and .

Replacing v(x) in (5) we have

(7)

and because u(x,y) is a periodically function with respect to x we find need to have

(8)

This result shows that c(y) may be of the form

(9)

or

, nZ (10) or

Z (11)

where D1 is a constant.

Further we consider

c(y) = y (9’)

and let

(12)

For a fixed y, the series (12) is uniformly convergent on any interval (0,x), therefore the series can be integrated term wise from zero to x and the next integral from (7) becomes:

(13)

The last term from (7) will be

(14)

and the periodical condition with respect to variable x leads to

(15)

Then we obtain for a k:

(16)

From the continuity condition of the uk(x,y) in any point y[-1; 1] we have

.

If we choose

Ak(y) = yk, k  1 (17)

then Bk(y) will be of the form

(18)

Finally, from (7), (9’), (16), (17) and (18) we obtain

(19)

for

(20)

Hence, the source function, g(x,y) must to be periodically. Further on we will demonstrate that the functional series (19) is uniformly convergent on (0;1), if H =1, y is fixed in

[-1; 1]. We use Dirichlet’s test:

“If series , fn:DRRis of the form , the sequence Sn(x)of the partial sums for the seriesis equal bounded on BD and n(x)is a sequence which is uniformly convergent to zero on B, then its is uniformly convergent on B.”

If n(x) is a numerical sequence, its uniform convergence means the sequence convergence.

Let

, where >0 is very small in comparison with the unity and n a natural number.

Then, exists M=1/sin >0, such that N,.

Since y is fixed, y[-1; 1], N*, is a sequence which converges to zero and is equal bounded series. It results from Dirichlet’s test, that series (19) is uniformly convergent on B.

Let us now consider in (19) the variable x fixed. The numbers series has the sequence of the partial sums bounded by , and the sequences of functions, N* are uniformly convergent to zero on [-1; 1].

The uniform convergence of the functional series (19) is demonstrated, too.

3 Numerical method

Further, to find the solution of the problem (1) – (2) a numerical algorithm is presented.

Let a domain be, H=1 and a network :

where i = 0,1,.., Nx, j = 0,1,…,Ny -1, x0=0, xNx+1=1, y0= -1, yNy= 1. In the network we have: Nx = 2, Ny = 5, h = 1/3 and  = 2/5 and the coordinates of the nodes P(xi, yj) = ij are presented in Fig.1

Using the method of finite difference and the trapezoidal rule for the integral calculus we obtain a system of liniar equations, which depends of Nx.

Fig.1

1. If Nx is an even natural number we have

- for 1iNx/2 :

- for NxiNx/2+2:

- and the linking relations:

2. If Nx is an odd natural number we have

- for 1i < (Nx+1)/2 :

- for (Nx+1)/2iNx:

- and for i =(Nx+1)/2:

k = 0, 1, …,Ny.

We add at the systems corresponding to 1. and 2. , the limit conditions:

u(x0, yk) = u(xNx+1, yk) = yk. k = 0, 1, …,Ny,

We obtain a system of 12 linear equations in 12 unknowns. The function g is calculated with (20) (summation index k = 1):

.

The approximate values of the solution are compared with theirs of the analytical solution and the results there are in the Table1.

Table 1

uanalytic /
unumeric
u10 / -0.8 / -0.6
u20 / -1.1 / -0.5
u11 / -0.46 / -0.32
u21 / -0.7 / -0.5
u12 / -0.06 / -0.1
u22 / -0.3 / -0.2
u13 / 0.3 / 0.2
u23 / 0.06 / 0.1
u14 / 0.7 / 0.5
u24 / 0.46 / 0.32
u15 / 1.1 / 0.8
u25 / 0.8 / 0.6

The absolute errors in the presented algorithm are of the h2 and 2 orders.

References

[1] Case K.M., Zweifel P.F., Linear Transport

Theory, Addison-Wesley, Massachusetts,

1967.

[2] Davis W.R., Classical Fields, Particles and

the Theory of Relativity , Gordon and

Breach, New York, 1970.

[3] Glasstone S., Kilton C., The Elements of

Nuclear Reactors Theory, Toronto – New

York – London, 1982.

[4] Marchouk G., Méthodes de calcul

numérique, Édition MIR de Moscou, 1980.

[5] Marchouk G., Shaydourov V., Raffinement

des solutions des schémas aux différences,

Édition MIR de Moscou, 1983.

[6] Marciuk G., Lebedev V., Cislennie metodî

v teorii perenosa neitronov, Atomizdat,

1971.

[7] Martin O., Une méthode de résolution de

l’équation du transfert des neutrons,

Rev.Roum. Sci. Tech.-Méc. Appl., Vol.37,

No.6,1992.

[8] Mihailescu N., Oscillations in the power

distribution in a reactor, Rev. Nuclear

Energy,Vol.9, No.1-4, 1998, pp.37-41.

[9] Pilkukn H., Relativistic Particle Physics,

Springer Verlag, New York - Heidelberg-

Berlin, 1980.