| Abstract: | Collocation methods are developed for the solution of some differential equation models for transport phenomena problems in one-and two-dimensions in co-axial annuli of spherical and cylindrical shapes. General formulae are developed to obtain orthogonal polynomials over an arbitrary interval using two types of weighting functions. The convergence and accuracy of the methods are demonstrated using two test problems, i.e., calculation of effectiveness factors in (a) a spherical pellet with peripherally deposited catalyst and (b) a Raschig ring type cylindrical catalyst pellet. Comparisons of results obtained from the present methods with analytic solutions for the first-order reactions indicate good agreement. Numerical solutions are also obtained for the second-and the third-order reactions for which analytic solutions are not available. Results obtained in terms of a new Thiele modulus involving the ratio of volume of peripherally deposited part of catalyst to exterior surface area indicate that this normalization brings effectiveness factor versus Thiele modulus curves close together for co-axial spherical and long cylindrical pellets, as it does for these geometries without the inner co-axial portion. |
