Spherical reactor

For spherically symetric systems the time independent diffusion equation reads:

$$\ {\frac{1}{r^{2}}}\ {\frac{d}{dr}}\left( r^{2}{\frac{d\varphi}{\partial r} }\right) +\frac{\pi^{2}}{R^{2}}\varphi(r)=0$$ (4.34)

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

$$\varphi(r)=A\frac{\sin\left( \pi r/R\right) }{r}$$ (4.35)

and the critical equation reads:

$$k_{\infty}=1+\frac{\pi^{2}D}{R^{2}\Sigma_{a}}$$ (4.36)