This chapter primarily deals with analytical solutions to the diffusion equation, where all that is needed is a piece of paper, something to write with, and knowledge of the solution techniques that are applicable. There are, however, many boundary conditions that cannot be well simulated with the simple boundary conditions used herein and require a digital solution with a computer. The numerical integrals given in equations (2.49) and (2.50) and in Example 2.9 were a simple form of these digital solutions, although they would be classified as a numerical integration.

Computational mass transport involves (1) discretization (division into discrete elements) of the spatial domain into control volumes with an assumed equal concentration, similar to complete mixed tanks; (2) discretization of time into steps of Δt; and (3) computing the fluxes into and out of each control volume over time to determine the concentration in each control volume over time. Once we have the discretization completed, all we will be doing is adding, subtracting, multiplying, and dividing numbers. Of course, for our computational solution to approach the real solution, the control volumes must be sufficiently small and the time steps must be sufficiently short. We will therefore be doing a great deal of adding, subtracting, multiplying, and dividing and would welcome the assistance of a computer.

The analytical solutions are still useful when moving to more realistic boundary conditions and a computational solution.