On the mathematics of exchange processes in fixed columns I. Mathematical solutions and asymptotic expansions

The paper is intended as a contribution to the mathematical discussion of the exchange processes, of heat or of matter (as in ion exchange or adsorption), that arise when a fluid flows through the pores or voids along a column containing matter in the solid state. The exchange is taken to be governed by a linear or bilinear or Langmuir-type kinetic exchange equation. The matters dealt with include the solution of the exchange and conservation equations for any initial conditions, mathematical properties of and asymptotic expansions for the functions arising, and the appearance of the functions in the solutions of other physical problems. The discussion of the mathematical properties of the functions and their asymptotic expansions is needed to establish the relationship between solutions on the equilibrium theory and those on the kinetic theory, and to find closer approximations to the kinetic-theory solutions.