The model for a chemical reactor with axial diffusion is
The boundary conditions are due to Danckwerts [1953] and Wehner and Wilhelm [1956]. This problem can be treated using initial value methods, also, but the method is highly sensitive to the choice of the parameter s, as outlined above. If one starts at z = 0 then making small changes in s will cause large changes in the solution at the exit, and it may not be possible to satisfy the boundary condition at the exit. If one starts at z = 1, however, and integrates backwards, then the process works and an iterative scheme converges in many cases [Hlavácek and Hofmann, 1970]. However, if the problem is extremely nonlinear the iterations may not converge. In such cases, it is necessary to use the methods for boundary value problems described below.
Computer software exists for solving two-point boundary value problems. For example, the IMSL program DTPTB uses DVERK, which employs Runge-Kutta integration to integrate the ordinary differential equations [Rice, p. 301, 1983].
See also: