A modified Crank—Nicolson technique with non-equidistant space steps

Abstract

Abstract-A finite difference method with non-equidistant space steps, based upon the Crank-Nicolson techruque is presented. Its prime feature is the automatic positioning of axial grid points at required positions. thus reducing considerably the total number of grid points and hence the amount of computer time.The method is demonstrated for a number of examples of tubular reactor calculations. It proves to be well suited for the solution of all kinds of diffusion type models, especially if steep gradients or moving profiles occur, and can be used even on moderate size process computers.Since many problems in chemical reaction engineering ultimately require the solution of sets of nonlinear partial differential equations, there is a continuing need for the development of numerical methods which can accomplish the task with computation time and storage requirements as small as possible. Among the vast variety of methods which have been proposed, two stand out as having been used extensively and with good success: the classical Crank-Nicolson technique [I] combined with quasilinearization [2], and the method of orthogonal collocation [3,4]. However, in reaction and reactor modelling for highly exothermic reactions in which very steep concentration and temperature gradients are commonly encountered, both of these methods have significant drawbacks. The main disadvantage of the CrankNicolson technique is the requirement of a large number of equidistant space steps (usually 200 or more), resulting in a large amount of computer time. Drawbacks of the method of orthogonal collocation result from the fact that the spatial grid points are determined by the zeros of the orthogonal polynomial which, of course, have nothing to do with the solution profile. This is particularly detrimental in case of unsteady or creeping profiles, where the grid points should be concentrated in the (moving!) region of maximum reaction rates. An increase in the number of grid points (i.e. in the degree of the orthogonal polynomial) again results in a better approximation but at the expense of a considerably increased amount of computer time, since a set of fully occupied matrices has to be inverted instead of a set of tridiagonal matrices as in the case of the Crank-Nicolson method. In reaction engineering this is apparently the reason why the method of orthogonal collocation has mostly been applied to single pellet diffusion-reaction problems or to the approximation of radial profiles in tubular reactor calculations-both cases with relatively smooth profiles.tNew address: BASF AG, Abt. TEl, 06700 Ludwigshafen. 681Figure I shows typical concentration and temperature profiles of an exothermic reaction in a nonadiabatic tubular reactor. It is obvious that in the range of the maximum temperature, the grid points have to be concentrated very close together to obtain accurate approximations of the axial derivatives (and of the source terms), whereas in the fore-and aft section large space increments could be allowed. Thus the density of axial grid ...

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