## One-dimensional time-dependent diffusion equation   Slab reactor. The diffusion equation reduces to a one-dimensional equation  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 = Вұa/2, we require ‘(x = Вұa/2, t) = 0. It is, therefore, convenient to use a Fourier development of ‘ and S,    with Bn = СҲ/a (n = 1,3,…). The coefficients An(t) are obtained by solving the equations

If S = 0, the solution is

AвҖһ0) =An(0) exp (^1 — 1 — B £avt.

For РәР¶ < 1 + B^(D/St) = 1 + (7r2D/a2Sa) all terms vanish exponentially. For РәР¶ > 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 distriВӯbution becomes time independent and is a solution of the time-independent diffusion equation

2 Ddx1 ‘(x, t) + ‘(x, t)^a(ki — 1)= 0

j2 2

d Рә

dX2 ‘(x, 0+^2 ‘(x, t) =0

which has the form

kx

‘(x) = A1 cos вҖ”.

Simple solutions are also obtained for spherical and cylindrical reactors.

Spherical reactor. For spherically symmetric systems the time-independent diffusion equation reads IA (r2 d’ r2 dr dr

 (3.36)

R being the radius of the reactor. The solution satisfying the boundary conВӯditions is (3.37) (3.38) Cylindrical reactor. Similarly, for infinite cylindrical systems

Рі d(r d)+B'(r)=0

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

 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.