Dispersion of chemical solutes in chromatographs and reactors

Abstract

The dispersion of a chemically active solute in unidirectional laminar flow in a channel of constant cross-sectional area is considered. Adsorption/desorption of the solute at the wall or the presence of a bulk or surface chemical reaction introduce additional timescales, in addition to the diffusive and convective ones, such that, under certain conditions, the asymptotic evolution of the cross-sectional mean concentration cannot be described by a one-dimensional Taylor-Aris model. We use the centre and invariant manifold theories to establish the proper time and length scale separations necessary for the existence of an effective transport equation and to determine the dependence of the effective transport coefficients on the kinetics of adsorption/desorption and reaction. For the case of classical Taylor-Aris dispersion with no reaction, we derive the effective transport equation to infinite order in the parameter, p , representing the ratio of the characteristic time for radial molecular diffusion to that for axial convection. We show that the infinite series in the effective transport model is convergent provided p is smaller than some critical value, which depends on the initial concentration distribution. We also examine the spatial evolution of time dependent inlet conditions and show that the spatial and temporal evolutions differ at third and higher orders. It is shown that, except for slow reactions with a kinetic timescale of the same order as the transverse diffusion time, fast bulk reaction does not allow an asymptotic axial dispersion description. Slow bulk reactions do not affect dispersion but a correction to the apparent kinetics may arise due to nonlinear interaction among reaction, diffusion and convection. It is also shown that with a slow bulk reaction, steady-state dispersion due to a coupling of reaction and transverse velocity gradient can arise. Although this mechanism is distinct from the transient Taylor—Aris mechanism, the dispersion coefficient is identical to the classical unreactive Taylor—Aris coefficient. Surface reaction of any speed yields the proper asymptotic behaviour in time because the species still needs to diffuse slowly to the conduit wall. In the limit of fast surface reaction, the Taylor-Aris dispersion coefficient is reduced by a factor of 4.2, 7.1 and 4.0 for pipe, plane Poiseuille and Couette flows, respectively, as the slow-moving solutes near the wall are depleted. For the case of a linear surface reaction, we use the invariant manifold theory to derive the effective transport equation to infinite order. We also show that the radius of convergence of the invariant manifold expansion is approximately three times that of the no reaction case. We demonstrate that if adsorption/desorption is as slow as transverse diffusion an adsorption-induced dispersion, distinct from the Taylor-Aris shear dispersion, exists. While the total dispersion may increase because of the contribution of both, the Taylor-Aris component is reduced by a physical mechanism similar to surface reaction. The adsorption/desorption induced dispersion coefficient is shown to have a maximum when the adsorption equilibrium constant is exactly 2. Nonlinear Langmuir type adsorption at large concentration is shown to introduce a nonlinear drift term which causes non-Gaussian pulse responses with long tails These tails are detrimental to separation chromatography since they cause overlaps which increase with the length of the chromatograph