An important chemical engineering operation is that in which a certain fluid id passed through a stationary bed for the purpose of removing from it, or adding to it, one or more substances. The purification of gases by passing them through a bed of activated charcoal, the softening of water by passing it through a zeolite bed, the reactivation of a zeolite or a charcoal bed, the cracking of petroleum in a packed bed reactor are just a few of the many important applications of this unit contains. The excellent review article by Thiele (18) summarizes many of the problems that are encountered in this field. Discussing this operation, most of the researchers have used a practical model with two independent variables--the time and the distance along the axis of a packed bed. This model produces a number of simplifications, in that it assumes plug through the bed, and neglects the variations of the dependent variables like the composition of the solution and the temperature of the radial position. In addition, except in isolated times the heat conduction or mass diffusion along the length of the bed is often overlooked. Even so, this one dimensional model can be very useful in studying, both experimentally and practically, the performance of packed beds and the more , but well worked out ion exchangers.