Taylor diffusion coefficient

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 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... 

The generalized Stefan-Maxwell equations using binary diffusion coefficients are not easily applicable to hquids since the coefficients are so dependent on conditions. That is, in hquids, each Dy can be strongly composition dependent in binary mixtures and, moreover, the binaiy is strongly affected in a multicomponent mixture. Thus, the convenience of writing multicomponent flux equations in terms of binary coefficients is lost. Conversely, they apply to gas mixtures because each is practically independent of composition by itself and in a multicomponent mixture (see Taylor and Krishna for details). 

Fig. 14. The outer-sphere relaxivity at zero magnetic field as a function of relative diffusion coefficient for S = 1. Reproduced with permission from Kruk, D. Nilsson, T. Kowalewski, J. Mol. Phys. 2001, 99,1435-1445. Copyright 2001 Taylor and Francis Ltd (http //www.tandf.co.uk/journals/tf/00268976.html).

Eq. 7.2, where is then the eddy diffusion coefficient (Taylor and Spencer 1990). The height of the turbulent zone, within the atmospheric boundary layer, is orders of magnitude greater than that of the laminar flow layer, and dispersion of contaminant vapors in the turbulent zone is relatively rapid. 

The mathematical method of Aris assumes a doubly infinite pipe (as does Taylor), with both the velocity distribution and the diffusion coefficients constant in the direction of flow. Hence in any real pipe, the length would have to be long enough so that the buildup of the velocity profile at the entrance would not invalidate the doubly infinite pipe assumption. Thus there are some practical restrictions on the method used by Aris. 

Actually, Taylor originally suggested using this formula in reverse for obtaining diffusion coefficients. Di, could be found simply from experimental data and then the formula could be used to obtain the diffusivity, D. 

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

As already said, Taylor s effective model contains a contribution in the effective diffusion coefficient, which is proportional to the square of the transversal Peclet number. Frequently this term is more important than the original molecular diffusion. After his work, it is called Taylor s dispersion coefficient and it is generally accepted and used in chemical engineering numerical simulations. For the practical applications we refer to the classical paper (Rubin, 1983) by Rubin. The mathematical study of the models from Rubin (1983) was undertaken in Friedman and Knabner (1992). 

Turbulent mass transfer near a wall can be represented by various physical models. In one such model the turbulent flow is assumed to be composed of a succession of short, steady, laminar motions along a plate. The length scale of the laminar path is denoted by x0 and the velocity of the liquid element just arrived at the wall by u0. Along each path of length x0, the motion is approximated by the quasi-steady laminar flow of a semiinfinite fluid along a plate. This implies that the hydrodynamic and diffusion boundary layers which develop in each of the paths are assumed to be smaller than the thickness of the fluid elements brought to the wall by turbulent fluctuations. Since the diffusion coefficient is small in liquids, the depth of penetration by diffusion in the liquid element is also small. Therefore one can use the first terms in the Taylor expansion of the Blasius expressions for the velocity components. The rate of mass transfer in the laminar microstructure can be obtained by solving the equation... 

At sufficiently large Schmidt numbers [as in liquids, where the diffusion coefficient is very small (D 10 5 cm2/sec)], only the region very near the wall is affected by diffusion. Therefore, the velocity distribution can be approximated by the first term in the Taylor expansion... 

Shape factors of a different sort are involved in the Taylor dispersion problem. With parabolic flow at mean speed U through a cylindrical tube of radius R, Taylor found that the longitudinal dispersion of a solute from the interaction of the flow distribution and transverse diffusion was R2U2/48D. The number 48 depends on both the geometry of the cross-section and the flow profile. If, however, we insist that the flow should be laminar, then the geometry of the cross-section determines the flow and hence the numerical constant in the Taylor dispersion coefficient. 

In Reprint C in Chapter 7, the behavior of a tracer pulse in a stream flowing through a packed bed and exchanging heat or matter with the particles is studied. It is shown that the diffusion in the particles makes a contribution to the apparent dispersion coefficient that is proportional to v2 fi/D. The constant of proportionality has one part that is a function of the kinematic wave speed fi, but otherwise only a factor that depends on the shape of the particle (see p. 145 and in equation (42) ignore all except the last term and even the suffixes of this e, being unsuitable as special notation, will be replaced by A. e is defined in the middle of p. 143 of Chapter 7). In this equation, we should not be surprised to find a term of the same form as the Taylor dispersion coefficient, for it is diffusion across streams of different speeds that causes the dispersion in that case just as it is the diffusion into stationary particles that causes the dispersion in this.7 What is surprising is that the isothermal diffusion and reaction equation should come up, for A is defined by... 

It is not unreasonable to use the left-hand side of this equation as the definition of the effective diffusion constant K, the more so as it will be shown that any distribution tends to normality. With this definition K is the sum of the molecular diffusion coefficient, D, and the apparent diffusion coefficient k = oP-U2I 48D, which was discovered by Taylor in his first paper (Taylor 1953, equation (25)). Equation (26), however, is true without any restriction on the value of p, or on the distribution of solute. The constant 1/48 is a function of the profile of flow, and for so-called piston flow with x — 0 this constant is zero and K = D as it should. 

In his analysis of the effect of diffusion on an open-tube distillation column Westhaver (1942) came up with the apparent diffusion coefficient 11 a2 /2/ 48D, and since he assumes a parabolic profile it is at first surprising that this should differ by a factor of 11 from Taylor s result. It appears, however, if the more general problem in which the solute can be retained on the wall be considered, that the value of k varies continuously from to is as the fraction of solute held on the wall varies from 1 to 0. This result is implicit in Golay s analysis of the tubular chromatographic column (Golay 1958). He considers the stationary phase of the column as a very thin retentive layer held on the wall and derives an expression for the dispersion by arguments entirely analogous to Taylor s. He has also discussed the effect of diffusion in the retentive layer. 

It is useful to list certain special results for the case of viscous flow and molecular diffusion. Results for turbulent flow profiles and diffusion coefficients can be obtained by numerical integration as has been done by Taylor (1954) and more recently by Tichacek and others (1958). For molecular diffusion ipj = 1, and for the problem of diffusion only in the absence of a second phase, R = 13 = y = 1 giving a single factor k = - 2kx2 + i3. 

In his analysis of the open tube distillation column Westhaver (1942) goes into a detailed consideration of radial concentration gradients which is very similar to Taylor s approach. His final formula, however, is the same as if he had assumed a constant velocity profile and an effective diffusion coefficient (Dt + llU2r2l48Dt). This is just the diffusion coefficient that we have found for viscous flow in the presence of a film on the tube wall in which the solute concentration is infinitely greater than in the fluid. This is clearly the case for... 

As might be expected, the dispersion coefficient for flow in a circular pipe is determined mainly by the Reynolds number Re. Figure 2.20 shows the dispersion coefficient plotted in the dimensionless form (Dl/ucI) versus the Reynolds number Re — pud/p(2Ai). In the turbulent region, the dispersion coefficient is affected also by the wall roughness while, in the laminar region, where molecular diffusion plays a part, particularly in the radial direction, the dispersion coefficient is dependent on the Schmidt number Sc(fi/pD), where D is the molecular diffusion coefficient. For the laminar flow region where the Taylor-Aris theory18,9,, 0) (Section 2.3.1) applies ... 

The method of Blanc [16] permits calculation of the gas-phase effective multicomponent diffusion coefficients based on binary diffusion coefficients. A conversion of binary diffusivities into effective diffusion coefficients can be also performed with the equation of Wilke [54]. The latter equation is frequently used in spite of the fact that it has been deduced only for the special case of an inert component. Furthermore, it is possible to estimate the effective diffusion coefficient of a multicomponent solution using a method of Burghardt and Krupiczka [55]. The Vignes approach [56] can be used in order to recalculate the binary diffusion coefficients at infinite dilution into the Maxwell-Stefan diffusion coefficients. An alternative method is suggested by Koijman and Taylor [57]. 

This equation allows one to consider the cumulative distribution of small-intestinal transit time data with respect to the fraction of dose entering the colon as a function of time. In this context, this equation characterizes well the small-intestinal transit data [173, 174], while the optimum value for the dispersion coefficient D was found to be equal to 0.78 cm2 s 1. This value is much greater than the classical order of magnitude 10 5 cm2 s 1 for molecular diffusion coefficients since it originates from Taylor dispersion due to the difference of the axial velocity at the center of the tube compared with the tube walls, as depicted in Figure 6.5. 

G. I. Taylor s concept of the effective axial diffusion coefficient, which has proved so useful in combining variable axial advection with radial transfer into one parameter, works best when there is no exchange of a passive tracer with the pipe walls. An analogue of his method, which should be applicable when development lengths are large and there is exchange at the wall, has yet to be provided. It would be of great value. 

If Eqs. (5-200) and (5-201) are combined, the multicomponent diffusion coefficient may be assessed in terms of binary diffusion coefficients [see Eq. (5-214)]. For gases, the values Dy of this equation are approximately equal to the binary diffusivities for the ij pairs. The Stefan-Maxwell diffusion coefficients may be negative, and the method may be applied to liquids, even for electrolyte diffusion [Kraaijeveld, Wesselingh, and Kuiken, Ind. Eng. Chem. Res., 33, 750 (1994)]. Approximate solutions have been developed by linearization [Toor, H.L., AlChE J., 10,448 and 460 (1964) Stewart and Prober, Ind. Eng. Chem. Fundam., 3,224 (1964)]. Those differ in details but yield about the same accuracy. More recently, efficient algorithms for solving the equations exactly have been developed (see Taylor and Krishna, Krishnamurthy and Taylor [Chem. Eng. J., 25, 47 (1982)], and Taylor and Webb [Comput Chem. Eng., 5, 61 (1981)]. 

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