Taylor dispersion in a capillary

On the basis of a number of assumptions, Mazo (1998) derives an expression for the effective dispersion coefficient in terms of the velocity profile, system geometry, etc., that reduces to Taylor s (1953) formulation for dispersion in a capillary (i.e., where the dimensionless velocity distribution is given by Eq. [15]). Using this approach dispersion for other velocity profiles can be calculated, although no other examples are presented. 

Gupta, V.K., and R.N. Bhattacharya. 1983. A new derivation of the Taylor-Aris theory of solute dispersion in a capillary. Water Resour. Res. 19 945-951. 

To the extent that dispersion in an inertia free porous medium flow arises from a nonuniform velocity distribution, its physical basis is the same as that of Taylor dispersion within a capillary. Data on solute dispersions in such flows show the long-time behavior to be Gaussian, as in capillaries. The Taylor dispersion equation for circular capillaries (Eq. 4.6.30) has therefore been applied empirically as a model equation to characterize the dispersion process in chromatographic separations in packed beds and porous media, with the mean velocity identified with the interstitial velocity. In so doing it is implicitly assumed that the mean interstitial velocity and flow pattern is independent of the flow rate, a condition that would, for example, not prevail when inertial effects become important. 

Figure 1.180 (a) Mixing in a capillary tube by Taylor dispersion. 

A last variant we mention is capillary zone electrophoresis (Gordon et al. 1988). It employs an electroosmotically driven flow in a capillary, arising from an electric field applied parallel to the capillary, which is charged when in contact with an aqueous solution (Section 6.5). The flow has a nearly flat velocity profile (Fig. 6.5.1), thereby minimizing broadening due to Taylor dispersion of the electrophoretically separated solute bands. 

Axial dispersion in packed beds, and Taylor dispersion of a tracer in a capillary tube, are described by the same form of the mass transfer equation. The Taylor dispersion problem, which was formulated in the early 1950s, corresponds to unsteady-state one-dimensional convection and two-dimensional diffusion of a tracer in a straight tube with circular cross section in the laminar flow regime. The microscopic form of the generalized mass transfer equation without chemical reaction is... 

Note Since the model is linear for the special case considered, the same equation is also satisfied by the other three variables.) The following observations may be made from Eq. (98) that expresses the dimensionless dispersion coefficient A (i) The first term describes dispersion effects due to velocity gradients when adsorption equilibrium exists at the interface. We note that this expression was first derived by Golay (1958) for capillary chromatography with a retentive layer, (ii) The second term corresponds to dispersion effects due to finite rate of adsorption (since this term vanishes if we assume that adsorption and desorption are very fast so that equilibrium exists at the interface), (iii) The effective dispersion coefficient reduces to the Taylor limit when the adsorption rate constant or the adsorption capacity is zero, (iv) As is well known (Rhee et al., 1986), the effective solute velocity is reduced by a factor (1 + y). (v) For the case of irreversible adsorption (y — oo and Da —> oo), the dispersion coefficient is equal to 11 times the Taylor value. It is also equal to the reciprocal of the asymptotic Sherwood number for mass transfer in a circular... 

G.I. Taylor (1953, 1954) first analyzed the dispersion of one fluid injected into a circular capillary tube in which a second fluid was flowing. He showed that the dispersion could be characterized by an unsteady diffusion process with an effective diffusion coefficient, termed a dispersion coefficient, which is not a physical constant but depends on the flow and its properties. The value of the dispersion coefficient is proportional to the ratio of the axial convection to the radial molecular diffusion that is, it is a measure of the rate at which material will spread out axially in the system. Because of Taylor s contribution to the understanding of the process of miscible dispersion, we shall, as is often done, refer to it as Taylor dispersion. 

The functions M (t) were determined from the complete unsteady axially symmetric convective diffusion equation (Eq. 4.6.7), and M (f) were obtained from the Taylor dispersion equation, which was used as the model equation. The phenomenological coefficients U and in the equation were determined by matching the first three moments of the infinite sequence M (t) to M (t) for asymptotically large times [t>a lD). Applying his scheme to the circular capillary problem, Aris showed that D fj, where axial molecular diffusion is not neglected, is given by Eq. (4.6.35). Fried Combarnous (1971) later showed that the satisfaction of the first three moments for t—implies that c x, t), obtained as a solution of the Taylor dispersion equation with = D + Pe /48), is asymptotically the solution of the complete, unsteady, axially symmetric convective diffusion equation averaged over the cross section. 

Figure 4.6.8 Measured mean concentration distributions at three positions along a capillary tube. Dashed line is distribution that would be due to convection alone for comparison with curve III. [After Taylor, G.l. 1953. Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. Roy. Soc. A219, 186-203. With permission.]...

We derive the Taylor dispersion equation using an area averaging approach as described by Stone and Brenner [8]. We consider axisymmetric flow in a cylindrical capillary of radius a, but will later discuss the application of these principles to other cross sections (such as typical wet etched... 

The chemical diffusion coefficient is a phase property where the reference velocity is chosen in the Fick reference system. Various methods are known for the experimental determination of chemical diffusion coefficients, such as open and closed capillary methods, dynamic light scattering, or the Taylor dispersion method. 

This expression indicates that for a given velocity, the Taylor time scale increases with reduction in the capillary radius. Hence, Taylor dispersion acts slower as the channel radius is decreased. 

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