Exact analysis of unsteady convective diffusion

An exact solution to the unsteady convective diffusion equation for miscible displacement in fully developed laminar flow in tubes is obtained. The most interesting result of this work is that it shows the generalized one-dimensional dispersion model describes the average concentration distribution exactly for all values of the independent variables if the dispersion coefficients are defined properly as functions of time. The analysis given reveals the precise structure of the dispersion coefficients necessary for the dispersion model to be exact. Also, it is shown that an exact solution for the local distribution can be constructed in which the dispersion coefficients play the role of eigenvalues which are functions of $\tau $.