Consider the diffusion–reaction problem represented in the three geometries

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 cA = 1 at ξ = 1 and a symmetry condition (Neumann) at ξ = 0. Note that the solution for a sphere needs a small coordinate transformation (cA = f(ξ)/ξ , 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. φ∗ for all of the three cases, where φ∗ is a shape-normalized Thiele modulus defined as φ∗ = φ s + 1 Thus φ∗ is equal to φ/2 for a cylinder and φ/3 for a sphere. Show that the results for the three geometries are quite similar when η is plotted as a function of φ∗, which is referred to as the generalized Thiele modulus.