Dispersion from a prescribed concentration distribution in time variable flow

Abstract

An exact method for solving unsteady convective diffusion boundary-value problems is illustrated by solving the specific case of dispersion from a prescribed axisymmetric concentration distribution at the inlet of the system. The correspondence between the source density distribution and the concentration distribution at the inlet of the system is established rigorously. This enables one to generalize the method previously developed for initial value problems to apply to practical boundary-value problems wherein the concentration is specified on the boundaries of the system. To demonstrate the method, dispersion from a partial step change at the inlet to a tube in which the fluid is in steady laminar flow, is analysed in detail; no exact solutions of this problem have been reported in the past. In the case which applies to the largest class of practical problems, when the step change exists across the entire cross-section of the flow, the results show that a constant coefficient dispersion model inadequately represents the system for small values of time. A comparison of solutions for different source density distributions shows the regions in which the inlet distribution plays a significant role in determining the concentration distribution in the flow.