On the mathematics of exchange processes in fixed columns. III The solution for general entry conditions, and a method of obtaining asymptotic expressions

Abstract

The equations for an exchange process between a suitable solid and a flowing fluid in a fixed column—e.g. heat exchange, ion exchange, adsorption or adsorption exchange—are solved explicitly for general entry conditions. It is shown how asymptotic expressions may be found for large values of the ratio of the exchange coefficient to the rate of volume flux of the fluid for the various cases that arise. In part IV, discontinuities arising in the first term of the asymptotic solution will be discussed in detail. In part V the results will be shown to be in agreement with those for the leading terms in the asymptotic solution obtained by considering certain singular and non-singular perturbations of a simplified set of equations (the ‘equilibrium theory’ equations). The equations considered are reduced to ∂u∂x+∂v∂y=0,ε∂v∂y=u−rv+(r−1)uv in x ⩾ 0, y ⩾ 0, with the boundary conditions v(x, 0) = 0 for x ⩾ 0, and u(0, y) = f{y) for 0 ⩽ y ⩽ Y, u(0, y) = 1 for y ⩾ Y, where f(0) = 0, f(Y) = 1, and f(y) is monotonic in (0, Y). Asymptotic expressions of the solution are sought for small ε. For a bibliography and description of previous solutions for particular entry conditions see Goldstein (1953).