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ID: past-exam-of-the-mphil-in-nuclear-energy-course-of-the-university-of-cambridge/2021/paper-ne1/3/b/solution

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Past exam of the MPhil in nuclear energy course of the University of Cambridge / 2021 / Module NE1 / 3 / b / Solution by

Codex  0 2026-10-20

For the prescribed homogeneous one-group model, number balance is

\frac{\partial n}{\partial t} = - \nabla \cdot J + (η - 1) Σ_{a} ϕ + S .

Use steady state,

\partial n / \partial t = 0

, and the diffusion approximation

J = - D \nabla ϕ

. A spatially constant

D

gives

D \nabla^{2} ϕ + (η - 1) Σ_{a} ϕ + S = 0.

For a spherically symmetric scalar field,

\nabla^{2} ϕ = \frac{1}{r ^{2}} \frac{d}{d r} (r^{2} \frac{d ϕ}{d r}) .

Dividing by

D

reproduces the requested radial equation. The assumptions are a steady source, spherical symmetry, homogeneous effective coefficients, a single energy-group description and a diffusion closure. In particular, replacing the general production term by the supplied

(η - 1) Σ_{a} ϕ

is a model assumption, not a general identity for heterogeneous reactor materials.

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