Diameter‐dependent dispersion in packed cylinders

The purpose of this note is to present new evidence of diameter-dependent dispersion in solute flowing through packed cylinders. Hydrodynamic dispersion was simulated at the pore scale in cylinders packed with spheres of diameter d, with cylinder radii ranging from R 1⁄4 5d to R 1⁄4 25d. The time-dependent, axial dispersion coefficient, DL(t), was plotted for different cylinder radii. These plots cross over, implying diameter-dependent dispersion. Comparisons with periodic sphere packs suggest the cylinder wall introduces new scales of length and time for dispersion. These results were obtained from threedimensional, pore-scale simulations of flow and transport in packed cylinders. The simulations are computationally demanding and we have been limited to relatively short durations. However, we were recently able to extend the duration enough to reveal the cross-over dynamics. We begin by describing the background for the new results. Taylor dispersion of a solute flowing in a tube illustrates the phenomenon of diameter-dependent dispersion. Taylor observed that radial variation in fluid velocity leads to an axial dispersion coefficient greater than the bulk molecular diffusivity, DM, of the solute, and he showed that this coefficient develops over a time and distance that depend on the tube diameter and average velocity, v. Indeed, this result is general for arbitrary velocity profiles, as shown by Aris. The packed column is similar, but requires the additional consideration of the characteristic length scale of the porous medium. It is generally accepted that the asymptotic axial dispersion coefficient in a packed cylinder depends on the ratio of cylinder diameter to particle diameter, b 1⁄4 2R/d because of nearwall packing effects on the velocity profile. Delgado reviews experimental investigations, dating to at least 1958, that discuss the effects of radial variations in porosity and velocity on dispersion. However, the consensus among these investigations is that dispersion is independent of the ratio for b . k, where k is a number in the range from 10 to 15. Gunn, for example, makes a case for diameter independence based on a mathematical model and earlier experimental results. De Ligny was perhaps the first to propose a correlation for the dispersion coefficient that explicitly accounts for both particle and cylinder diameters. The correlation predicts the effective dispersion coefficient in a packed cylinder from these two parameters and from the effective radial dispersion coefficient. The radial coefficient is assumed to be only weakly dependent on b. A number of authors have developed models of radial variations in porosity and velocity and used them to predict concentration distributions and mass transfer zones in packed cylinders. Perhaps the simplest illustration is given by Rudraiah et al. for a rectangular packed duct; they assume Brinkman’s equation to describe the near-wall velocity profile and further assume the value of a bulk dispersion coefficient that is characteristic of the ! 2007 American Institute of Chemical Engineers *This article ia a U.S. Government work and, as such, is in the public domain in the United States of America.