= Solution
For the prescribed homogeneous one-group model, number balance is
$$
\frac{\partial n}{\partial t}=-\nabla\cdot\mathbf J+(\eta-1)\Sigma_a\phi+S.
$$
Use steady state, $\partial n/\partial t=0$, and the <diffusion approximation> $\mathbf J=-D\nabla\phi$. A spatially constant $D$ gives
$$
D\nabla^2\phi+(\eta-1)\Sigma_a\phi+S=0.
$$
For a spherically symmetric scalar field,
$$
\nabla^2\phi=\frac1{r^2}\frac d{dr}\left(r^2\frac{d\phi}{dr}\right).
$$
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 $(\eta-1)\Sigma_a\phi$ is a model assumption, not a general identity for heterogeneous reactor materials.
Back to article page
