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Application of orthogonal collocations to some transport phenomena problems in co‐axial cylinders and spheres
Author(s) -
Goyal S. K.,
Esmail M. N.,
Bakhshi N. N.
Publication year - 1987
Publication title -
the canadian journal of chemical engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.404
H-Index - 67
eISSN - 1939-019X
pISSN - 0008-4034
DOI - 10.1002/cjce.5450650518
Subject(s) - thiele modulus , mathematics , orthogonal collocation , spheres , raschig ring , mathematical analysis , pellets , cylinder , modulus , collocation method , differential equation , mechanics , geometry , materials science , ordinary differential equation , chemistry , packed bed , physics , composite material , chromatography , astronomy , mass transfer
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.