The Pigford Problem
Many mass transfer operations involve diffusion in fluids in laminar flow—for example, absorption without chemical reaction of a gaseous compound in a thin liquid film of a nonvolatile compound flowing vertically. Another example is laminar flow of compound through the membrane walls of the channel. The first case has applications in chemical engineering and environmental science; the second, in medicine and biomedical engineering. Both cases can be treated by the same equations, based on the solution given by Pigford in 1941.
This Demonstration shows a solution of Pigford's problem using an orthogonal collocation method; the dimensionless equation is
with the initial and boundary conditions
Here is the concentration of compound , is the equilibrium constant, and and are the positions, all expressed in dimensionless units.
 J. V. Villadsen and W. E. Stewart, "Solution of Boundary-Value Problems by Orthogonal Collocation," Chemical Engineering Science, 22, 1967 pp. 3981–3996.
 R. B. Bird, W. E. Stewart, and E. N. Lightfoot, Transport Phenomena, 2nd ed., New York: John Wiley and Sons.