The solution of a boundary value problem in reactor design using Galerkin's method

Abstract

Galerkin's technique for the approximate solution of ordinary and partial differential equations has been described by several authors (1 ) . However, its application to chemical engineering problems has been rather limited (2,3,4 ) . Since the method serves as a valuable alternate to conventional numerical techniques, it is worthy of further popularization and exploitation.In the work presented here, the technique was applied to the problem of a tubular reactor in which axial diffusion is superimposed upon a one-dimensional flow. Several solutions to this problem have been published, providing a basis of comparison of the method with conventional analytical and numerical techniques.The differential equation and associated boundary conditions were first investigated by Danckwerts (5) for the linear case of first-order kinetics. The boundary conditions have also been studied in detail (6, 7 ) . For nonlinear kinetics, no analytical solutions have appeared and recourse has been made to numerical integrations. Bischoff and Levenspiel (8) have calculated results for a secondorder reaction and these are summarized in Levenspiel's text (9). Fan and Bailie (10) have also calculated concentration profiles for kinetics of order n, for n = 1/4, $, 2, and 3. Limited results for n = 2 have also appeared in the work of Lee (11).For nonlinear kinetics it is difficult to account for the effects of the three physical parameters of the problem; the reaction rate group, the order of kinetics, and the axial Peclet number. For nonlinear kinetics one must either numerically integrate the differential equation or interpolate by using the limited curves of Levenspiel or Fan and Bailie.The use of Galerkin's method yields approximate solutions which are both easier to utilize and more clearly show parameter behavior. Simple solutions were found for integral reaction orders. For general nth order kinetics the problem is reduced to the solution of a single nonlinear algebraic equation. These approximate solutions are in excellent agreement with available results for wide ranges of the parameters of the problem. More complex Galerkin solutions in terms of additional parameters converge to published results, but here recourse must be made to digital computer solutions. APPROXIMATE METHOD OF SOLUTIONIn Galerkin's technique a solution, YN, is assumed as a set of prespecified functions and associated parameters which must be determined. The set of functions 4 i ( z ) are chosen and the parameters are related so that all of the boundary conditions can be satisfied for all values of the parameters ui. If the set +i are chosen as ( z -l)i-l the Galerkin approximation in terms of N parameters isThe second boundary condition ( 3 ) is satisfied if a2 = 0 and the first condition ( 2 ) is satisfied if the ai are related bvThe additional conditions needed to determine the remaining parameters are derived by requiring that the error in the fit of the assumed solution to the differential equation, E (u+, z ) , be orthogonal to each of the first N -NBc ap...

Solution of stiff boundary value problems by orthogonal collocation

“…The latter authors used Galerkin's method and were able to obtain solutions for larger eigenvalue separations than was possible with other methods. The exact solution to the problem is f(q) = 1.0 + i q 2 +sinh (q) (3) and is independent of the constant, c. The eigenvalues, i.e., the solutions of the characteristic equation Hence the equation is stiff for large values of c.…”

Applied Mathematics

“…A new book covers both the theory and applications in a superb fashion ( 140), while a second book presents many interesting examples (30). A series of papers on nonlinear boundary value problems has been presented which emphasize Galerkin's method , and an equivalent method has been applied to convection (100) and reactor design problems (120). Other papers have proposed a globally convergent shooting method (10), analyzed the use of contraction mapping (200), detailed the use of invariant imbedding (20) and quasilinearization (790), and suggested the use of cubic spline functions (30).…”