Dispersion of a non-uniform slug in time-dependent flow

Abstract

Dispersion of a non-uniform slug in time variable fully developed laminar flow is studied by an exact method which in principle is valid for all values of time. A generalized dispersion model, which involves an infinite set of time-dependent coefficients K i ( T ), arises naturally as a consequence of the analysis. Expressions are obtained from first principles for the convective coefficient K 1 ( T ), the apparent diffusion coefficient K 2 ( T ), and the rest of the coefficients K i ( T ). The convective coefficient K 1 ( T ) is a new quantity which is time dependent, even if the velocity field is steady, because the initial distribution of solute is non-uniform. The general method of analysis proposed is applied to the particular case of a slug of material of length x s and radius aR which is initially symmetrically located in a tube of radius R , and then is dispersed in steady laminar flow. Breakthrough curves for various downstream positions show that the mean velocity of the material in the slug is greater than the average velocity of flow for small residence times and, as one would expect intuitively, approaches this value as the distance between the injection and observation points increases. It also is shown that for T ≲ 0.5, K 2 , the axial dispersion coefficient, decreases as the initial slug radius decreases. However, for all values of ' a ', both K 1 and K 2 approach the asymptotic values of —½ and (Pe) -2 + 1/192 respectively for values of dimensionless time T ≳ 0.5. Since the initial concentration distribution of solute in the slug affects the time of arrival of the peak mean concentration at a downstream location, this may cause errors in the estimation of mean flow rates by tracer techniques. For example, if one injects a quantity of material into a stream and assumes it is uniformly distributed across the tube, when in reality it occupies only a part of the tube, this leads to overestimating the average flow velocity. This may be of practical importance in a variety of applications.