Taylor dispersion effective diffusivity

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). 

The use of the Coanda effect is based on the desire to have a second passive momentum to speed up mixing in addition to diffusion [55, 163], The second momentum is based on so-called transverse dispersion produced by passive structures, which is in analogy with the Taylor convective radial dispersion ( Taylor dispersion ) (see Figure 1.180 and [163] for further details). It was further desired to have a flat ( in-plane ) structure and not a 3-D structure, since only the first type can be easily integrated into a pTAS system, typically also being flat A further design criterion was to have a micro mixer with improved dispersion and velocity profiles. 

As our first application, we consider the classical Taylor-Aris problem (Aris, 1956 Taylor, 1953) that illustrates dispersion due to transverse velocity gradients and molecular diffusion in laminar flow tubular reactors. In the traditional reaction engineering literature, dispersion effects are described by the axial dispersion model with Danckwerts boundary conditions (Froment and Bischoff, 1990 Levenspiel, 1999 Wen and Fan, 1975). Here, we show that the inconsistencies associated with the traditional parabolic form of the dispersion model can be removed by expressing the averaged model in a hyperbolic form. We also analyze the hyperbolic model and show that it has a much larger range of validity than the standard parabolic model. 

The essence of the side walls effect follows. The flow velocity turns to zero at the side walls as well as at the main (accumulation and depletion) walls of the FFF channel. Therefore, the flow profile is nonuniform, not only along the width of the channel but also along its breadth. The size of the regions near the side walls where the flow is substantially nonuniform is of the same order as w. The nonuniformity of the flow in both directions, combined with diffusion of solute particles, leads to Taylor dispersion and peak broadening that could be different from the one predicted by the 2D models. 

The influence of longitudinal dispersion on the extent of a first-order catalytic reaction has been studied by Kobayashi and Arai (K14), Furusaki (F13), van Swaay and Zuiderweg (V8), and others. They use the one-dimensional two-phase diffusion model, and show that longitudinal dispersion of the emulsion has little effect when the reaction rate is low. Based on the circulation flow model (Fig. 2) Miyauchi and Morooka (M29) have shown that the mechanism of longitudinal dispersion in a fluidized catalyst bed is a kind of Taylor dispersion (G6, T9). The influence of the emulsion-phase recirculation on the extent of reaction disappears when the term tp defined by Eq. (7-18) (see Section VII) is greater than about 10. For large-diameter beds, where p does not satisfy this restriction, their general treatment includes the contribution of Taylor dispersion for both the reactant gas and the emulsion (M29). 

Many microfluidic devices involve diffusion of concentration as well as fluid flow. The velocity crossing a plane normal to the flow is seldom uniform, so the diffusion must be examined in the midst of non-uniform velocity prohles. The simplest illustration of this effect is with Taylor dispersion. 

It is interesting to note that the first sample (a) dispersed slightly more than did samples by c, and dy which were stopped in the coil. The decrease in the D value of the stopped samples is caused by radial diffusion, which affects the laminar flow profile while the zone s tands still. (See the discussion of Taylor s effect in Chapter 3.) The principles of stopped-flow technique are outlined in Section 2.4.3 while its practical applications and experimental details are described in Section 4.3. 

Thus for Dm = 7 x 10 , the minimum time for Taylor s effect to begin to develop is 6.2 s for = 0.025 cm. This minimum time limit is just about met in FIA, since tubes of more than 0.5 mm ID are seldom used, while typical residence times are 20 s or more. Further decrease in channel radius and flow rate will lead to progressively more intensive redistribution of sample material in the radial direction by molecular diffusion and therefore decrease in the flow rate in a straight narrow tube will lead to a decrease in dispersion (cf. Rule 3, Section 2.2.3). 

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. 

Pe = 10. The model shows interface width (e.g., the interface between two adjacent species) as a strong function of Pe (and Eq). In the conditions Saville modeled, Taylor dispersion increases the effective dispersion coefficient up to 18-fold over diffusion alone. 

Effects of LE (NaCl) concentration, Cle, TE (HEPES) concentration, Cte and initial concentration of sample (Alexa Fluor 488), Cs,initiai on sample preconcentration are summarized in Figure 38.18. Cle was varied from 10 mM to 1 M to study its effect on maximum focused sample concentration, Cs,finai and concentration increase. Cl (Figure 38.18a). The 5 mM HEPES TE solution contained 1 nM Alexa Fluor 488 as a sample. The focused sample concentration is nearly directly proportional to the concentration of LE, as expected from a one-dimensional nondispersive model (i.e., BCRF theory). However, the nondispersive model drastically underpredicts the proportionality constant the measured focused sample concentrations are 35(X)- to 7900-fold less than that predicted by Equation 38.43 despite all cases reaching fully-focused state. This gross difference between KRF theory and experiments is because the sample is in a smeared region of locally varying conductivity and electric field, as dictated by the effects of diffusion and Taylor dispersion. 

This coefficient is called the Taylor-Aris s dispersion factor. It follows from (6.141) that at t R /D the coefficient of effective diffusion is equal to the sum of the molecular diffusion coefficient and the effective diffusion coefficient given by (6.131). 

In the vicinity of the wall the tracers are submitted to a Taylor dispersion, that is, their diffusion combined with shear enhances the migration speed in the flow direction. This phenomenon seriously complicates all velocimetry methods since to extract the velocity of fluid this effect should be modeled precisely. As shown by analysis of DF-FCS data, large observed values of the apparent slip at the hydrophilic wall are normally fully attributed to a Taylor dispersion of nanotracers (see Fig. 2.6). The data obtained with other velocimetry technique still awaits clarification. We suggest, however, that some very large values of a hydrophobic slip might reflect a Taylor dispersion too. 

Note (1) For Taylor dispersion, there is an effective diffusion coefficient which increases proportionally with square of the velocity of the flow. The spot spreads much faster in the direction of the flow than if molecular diffusion is the only mechanism responsible. 

For miniaturized chemical analysis, there are often samples of given size but not fixed permanent flow rate, where two samples A and B are required to be mixed with each other. A ring-shaped micromixer as shown in Figure 4.8 can be used for this situation, where a peristaltic pump drives the flow around a ring. The Taylor dispersion operates along the ring, and after some rotations, the two spots A and B mix completely. The speed of rotation of the fluids is U, and the transverse distance, that is, the height of the channel, is b. The effective diffusivity is... 

As mentioned in the introduction, microemulsions show non-negligible crossdiffusion due to compartmentalization which can be characterised by means of the Taylor dispersion technique [2,3,14,15]. In a wide range of the ME composition the diffusion matrix D presents a large and positive cross-diffusivity D12 relating the motion of the water (species 1) to the flux of the surfactant (species 2) while a small negative cross-diffusion D21 quantifies the effect of water motion on AOT diffusion ... 

Taylor (T2) and Westhaver (W5, W6, W7) have discussed the relationship between dispersion models. For laminar flow in round empty tubes, they showed that dispersion due to molecular diffusion and radial velocity variations may be represented by flow with a flat velocity profile equal to the actual mean velocity, u, and with an effective axial dispersion coefficient Djf = However, in the analysis, Taylor... 

Taylor (T4, T6), in two other articles, used the dispersed plug-flow model for turbulent flow, and Aris s treatment also included this case. Taylor and Aris both conclude that an effective axial-dispersion coefficient Dzf can again be used and that this coefficient is now a function of the well known Fanning friction factor. Tichacek et al. (T8) also considered turbulent flow, and found that Dl was quite sensitive to variations in the velocity profile. Aris further used the method for dispersion in a two-phase system with transfer between phases (All), for dispersion in flow through a tube with stagnant pockets (AlO), and for flow with a pulsating velocity (A12). Hawthorn (H7) considered the temperature effect of viscosity on dispersion coefficients he found that they can be altered by a factor of two in laminar flow, but that there is little effect for fully developed turbulent flow. Elder (E4) has considered open-channel flow and diffusion of discrete particles. Bischoff and Levenspiel (B14) extended Aris s theory to include a linear rate process, and used the results to construct comprehensive correlations of dispersion coefficients. 

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

If the flow rate is increased so that Peclet number Pe l, then there is a timescale at which transversal molecular diffusion smears the contact discontinuity into a plug. In Taylor (1993), Taylor found an effective long-time axial diffusivity proportional to the square of the transversal Peclet number and occurring in addition to the molecular diffusivity. After this pioneering work of Taylor, a vast literature on the subject developed, with over 2000 citations to date. The most notable references are the article (Aris, 1956) by Aris, where Taylor s intuitive approach was explained through moments expansion and the lecture notes (Caflisch and Rubinstein, 1984), where a probabilistic justification of Taylor s dispersion is given. In addition to these results, addressing the tube flow with a dominant Peclet number and in the absence of chemical reactions, there is... 

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. 

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. 

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