2.9 Critical Mass of a Cylindrical Core (Optional)
111
x
y
z
= R
L
Fig. 2.24 Cylindrical core of radius R and height L
We begin with the general neutron diffusion equation of Appendix G:
∂ N
∂t
=
v neut
λ f
(ν − 1) N +
λ t v neut
3
∇
2 N
.
(2.147)
The goal here is to apply this to the neutron population within a cylinder of radius
R and length L as illustrated in Fig. 2.24. The bottom of the cylinder is imagined to
be lying in the xy plane, with its center at (x, y) = (0,0).
The separation of the diffusion equation into time and space-dependent parts
proceeds as in Sect. 2.2; the temporal dependence is not of interest to us here as
we seek to determine the threshold-critical condition. The spatial part of the neutron
density N will be a function of the cylindrical coordinates (ρ, φ, z), and is assumed
to be separable as
N ρφz (ρ, φ, z) = N ρ (ρ)N φ (φ)N z (z).
(2.148)
The Laplacian operator in cylindrical coordinates is
∇
2 N ρφz =
1
ρ
∂
∂ρ
ρ
∂ N ρφz
∂ρ
+
1
ρ 2
∂
2 N ρφz
∂φ 2 +
∂
2 N ρφz
∂z 2 .
(2.149)
On substituting (2.148) and (2.149) into (2.147) and dividing through by N ρφz ,
the spatial part of the diffusion equation appears, in analogy to (2.24), as