Taylor diffusivity

Concentration and temperature differences are reduced by bulk flow or circulation in a vessel. Fluid regions of different composition or temperature are reduced in thickness by bulk motion in which velocity gradients exist. This process is called bulk diffusion or Taylor diffusion (Brodkey, in Uhl and Gray, op. cit., vol. 1, p. 48). The turbulent and molecular diffusion reduces the difference between these regions. In laminar flow, Taylor diffusion and molecular diffusion are the mechanisms of concentration- and temperature-difference reduction. 

Equihbrium concentrations which tend to develop at solid-liquid, gas-liquid, or hquid-liquid interfaces are displaced or changed by molecular and turbulent diffusion between biilk fluid and fluid adjacent to the interface. Bulk motion (Taylor diffusion) aids in this mass-transfer mechanism also. 

Glaser and Litt (G4) have proposed, in an extension of the above study, a model for gas-liquid flow through a b d of porous particles. The bed is assumed to consist of two basic structures which influence the fluid flow patterns (1) Void channels external to the packing, with which are associated dead-ended pockets that can hold stagnant pools of liquid and (2) pore channels and pockets, i.e., continuous and dead-ended pockets in the interior of the particles. On this basis, a theoretical model of liquid-phase dispersion in mixed-phase flow is developed. The model uses three bed parameters for the description of axial dispersion (1) Dispersion due to the mixing of streams from various channels of different residence times (2) dispersion from axial diffusion in the void channels and (3) dispersion from diffusion into the pores. The model is not applicable to turbulent flow nor to such low flow rates that molecular diffusion is comparable to Taylor diffusion. The latter region is unlikely to be of practical interest. The model predicts that the reciprocal Peclet number should be directly proportional to nominal liquid velocity, a prediction that has been confirmed by a few determinations of residence-time distribution for a wax desulfurization pilot reactor of 1-in. diameter packed with 10-14 mesh particles. 

With turbulence, there is an effective axial dispersion coefficient 3, called Aris Taylor diffusion, which is driven by the turbulent eddies,... 

Sir Geoffrey Taylor has recently discussed the dispersion of a solute under the simultaneous action of molecular diffusion and variation of the velocity of the solvent. A new basis for his analysis is presented here which removes the restrictions imposed on some of the parameters at the expense of describing the distribution of solute in terms of its moments in the direction of flow. It is shown that the rate of growth of the variance is proportional to the sum of the molecular diffusion coefficient, D, and the Taylor diffusion coefficient ko2U2/D, where U is the mean velocity and a is a dimension characteristic of the cross-section of the tube. An expression for k is given in the most general case, and it is shown that a finite distribution of solute tends to become normally distributed. 

It is well known that for a given pressure drop, the flow is greater in a circular tube than in an elliptical one of the same area, and if in the Taylor diffusion coefficient a2 is replaced by the area of cross-section (na2 for the circle and nab for the ellipse) the constant k is least for e = 0. Thus the dispersion in a circular tube is less than in an elliptical tube of the same area. 

Thus for the parallel walled channel the Taylor diffusion coefficient is... 

This shows that the mean of the temperature wave moves with the kinematic wave velocity and that an apparent diffusion coefficient may be defined to describe the dispersion. This coefficient is the sum of the diffusion coefficients which would be obtained if each effect were considered independently. Such an additivity has been demonstrated by the author for the molecular and Taylor diffusion coefficients elsewhere (Aris 1956) and is assumed in a paper by Klinkenberg and others (van Deemter, Zuiderweg Klinkenberg 1956) in their analysis of the dispersion of a chromatogram. 

The mixing characteristics of the fluid in a spouted bed of solids have not been studied. Since the particle Reynolds numbers involved in both the annulus and the spout are relatively high, the extent of any axial mixing experienced by the fluid in either channel is likely to be small, especially for the more common case of gas spouting. For the bed as a whole, however, it could be significant due to the very uneven velocity distribution as between spout and annulus, which would cause an effect akin to Taylor diffusion. 

G.I. Taylor, Diffusion by Continuous Movemenu, Proc. Land. Madi. Soc. 20,196,1922. O.G. Sutton, A Theory of Eddy Diffusion in the Atmosphere, Proceedings of the Royal Society of London, A135, page 143,1931 and Diffusive Properties of the Lower Atmosphere, MRP 59, 1942. 

G.I. Taylor, Diffusion and mass transport in tubes, Proc. Phys. Soc. London 67 (1954) 857. 

Tayloring a denuded zone depth results in tayloring diffusion length and trap density Ny. This has influence on different aspects for device performance. 

The practical importance of this Taylor diffusion analysis lies in the justification of the effective transport models to take into account complicated velocity and concentration profiles in a simple manner, as well as providing a theoretical framework for the dispersion coefficient, D . Similar results have been worked out for turbulent flow, packed columns, and other situations. For correlations of the axial dispersion coefficients, see Himmelblau and Bischoff [4] and Wen and Fan [2]. 

In many ways, the two models are rather similar, although the mathematical details for the tanks in series model is much simpler than for the axial dispersion model On the other band, no theoretical justification such as Taylor diffusion is possible in general nor are theoretical estimates of the model parameter, n that is, n is strictly empirical The only exceptions to this are the finite stage models for packed bed interstices as briefly discu ed in Chapter 11. 

The first term describes the longitudinal diffusion in the axial direction. The second term is called the Taylor diffusion coefficient and describes band broadening due to the parabolic flow profile and therefore radial diffusion. The height equivalent to a theoretical plate, H, is a measure of the relative peak broadening and is defined as... 

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