Consider the diffusion-reaction problem represented in the three geometries.

Verify the analytical solutions shown in the text for the three geometries with the Dirichlet condition of \(c_{\mathrm{A}}=1\) at \(\xi=1\) and a symmetry condition (Neumann) at \(\xi=0\).

Note that the solution for a sphere needs a small coordinate transformation \(\left(c_{\mathrm{A}}=f(\xi) / \xi\right.\), which reduces the governing equation to a simpler one in \(f\) ).

Find the average concentration in the system for the three cases which represents the effectiveness factor. Make a plot of the effectiveness factor vs. \(\phi^{*}\) for all of the three cases, where \(\phi^{*}\) is a shape-normalized Thiele modulus defined as

\[\phi^{*}=\frac{\phi}{s+1}\]

Thus \(\phi^{*}\) is equal to \(\phi / 2\) for a cylinder and \(\phi / 3\) for a sphere. Show that the results for the three geometries are quite similar when \(\eta\) is plotted as a function of \(\phi^{*}\), which is referred to as the generalized Thiele modulus.