One-dimensional time-dependent diffusion equation

Подпись: d'(x, t) v dt Подпись: 2

image074

Slab reactor. The diffusion equation reduces to a one-dimensional equation

image075
image076

where we have used a single absorption cross-section Xa, independent of x, and a plane neutron source at position x = 0. At the boundaries x = ±a/2, we require ‘(x = ±a/2, t) = 0. It is, therefore, convenient to use a Fourier development of ‘ and S,

image077 Подпись: (_-DBn + ^a(ki Подпись: S 1))An(t)+2 a. Подпись: (3.34)

with Bn = ш/a (n = 1,3,…). The coefficients An(t) are obtained by solving the equations

If S = 0, the solution is

A„0) =An(0) exp (^1 — 1 — B £avt.

For кж < 1 + B^(D/St) = 1 + (7r2D/a2Sa) all terms vanish exponentially. For кж > 1 + (7r2D/a2Sa), the first term, and possibly some other low order ones, increases exponentially. The reactor becomes critical for ki = 1 + (7r2D/a2Sa); in this case A1(t) becomes time independent, while higher order terms decrease exponentially. Therefore, the neutron flux distri­bution becomes time independent and is a solution of the time-independent diffusion equation

2

Подпись: (3.35)Ddx1 ‘(x, t) + ‘(x, t)^a(ki — 1)= 0

j2 2

d к

dX2 ‘(x, 0+^2 ‘(x, t) =0

which has the form

kx

‘(x) = A1 cos —.

Simple solutions are also obtained for spherical and cylindrical reactors.

Spherical reactor. For spherically symmetric systems the time-independent diffusion equation reads

image082

IA (r2 d’

r2 dr dr

 

(3.36)

 

R being the radius of the reactor. The solution satisfying the boundary con­ditions is

image083

(3.37)

 

image084

(3.38)

 

image085

Cylindrical reactor. Similarly, for infinite cylindrical systems

г d(r d)+B'(r)=0

Table 3.1. Fission and capture cross-sections (barns) averaged over a PWR neutron spectrum [56].

Nuclide

PWR spectrum

Fission

Capture

235U

40.62

11.39

238U

0.107

1.03

239Pu

101.02

42.23

240Pu

0.44

109.39

241Pu

109.17

37.89

242Pu

0.28

57.55

243Pu

0.462

11.51

243Am

0.092

72.257

244Cm

0.62

29.261

whose solution is

‘(r) = AJ0(Br),

with J0 the ordinary Bessel function of order 0. J0(BR) = 0. This condition is fulfilled for a number of values, the smallest being BR = 2.405.

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