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Consider the diffusion–reaction problem represented in the three geometries

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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.