Taylor dispersion theory

Howard Brenner Let me give a simple example of this, that derives from the generalized Taylor dispersion theory references cited in my previous comments. Think of a tubular reactor in which one has a Poiseuille flow, together with a chemical reaction occurring on the walls. One can certainly write down all the relevant differential equations and boundary conditions and solve them numerically. However, the real essence of the macrophysics is that if one examines the average velocity with which the reactive species moves down the tube, this speed is greater than that of the carrier fluid because the solute is destroyed in the slower-moving fluid streamlines near the wall. Consequently, the only reactive solute molecules that make it... 

Four novel approaches to contemporary studies of suspensions are briefly reviewed in this final section. Addressed first is Stokesian dynamics, a newly developed simulation technique. Surveyed next is a recent application of generalized Taylor dispersion theory (Brenner, 1980a, 1982) to the study of momentum transport in suspensions. Third, a synopsis is provided of recent studies in the general area of fractal suspensions. Finally, some novel properties (e.g., the existence of antisymmetric stresses) of dipolar suspensions are reviewed in relation to their applications to magnetic and electrorheolog-ical fluid properties. 

The question is whether there is a time large enough (or alternatively a position z 1 corresponding to z = 0) for which this scaling (and thus the Taylor dispersion theory) is applicable. We will address this question shortly. For the moment, we simply assume that (3-232) is valid and rewrite (3-229) in terms of 0,... 

Naturally, there are two more Peclet numbers defined for the transverse direction dispersions. In these ranges of Reynolds number, the Peclet number for transverse mass transfer is 11, but the Peclet number for transverse heat transfer is not well agreed upon (121, 122). None of these dispersions numbers is known in the metal screen bed. A special problem is created in the monolith where transverse dispersion of mass must be zero, and the parallel dispersion of mass can be estimated by the Taylor axial dispersion theory (123). The dispersion of heat would depend principally on the properties of the monolith substrate. Often, these Peclet numbers for individual pellets are replaced by the Bodenstein numbers for the entire bed... 

Edwards [105] has extended the macrotransport method, originally developed by Brenner [48] and based upon a generalization of Taylor-Aris dispersion theory, to the analysis of electrokinetic transport in spatially periodic porons media. Edwards and Langer [106] applied this methodology to transdermal dmg delivery by iontophoresis and electroporation. 

For axial dispersion in the micro-channel reactor, the usual relationships from Taylor-Aris theory were employed. In order to assess the performance of both reactor types, the widths of two initially delta-like concentration tracers are compared after they have passed through the flow domain. The results of this comparison are displayed in Figure 1.16. 

Howard Brenner has generalized the method to a whole class of phenomena in his magisterial paper, A general theory of Taylor dispersion phenomena. Physicochem. Hydrodyn. 1, 91-123 (1980). 

The theoretical foundation for this kind of analysis was, as mentioned, originally laid by Taylor and Aris with their dispersion theory in circular tubes. Recent contributions in this area have transferred their approach to micro-reaction technology. Gobby et al. [94] studied, in 1999, a reaction in a catalytic wall micro-reactor, applying the eigenvalue method for a vertically averaged one-dimensional solution under isothermal and non-isothermal conditions. Dispersion in etched microchannels has been examined [95], and a comparison of electro-osmotic flow to pressure-driven flow in micro-channels given by Locascio et al. in 2001 [96]. 

The theoretical foundation for this kind of analysis was, as mentioned, originally laid by Taylor and Aris with their dispersion theory in circular tubes. Recent... 

Thus, we recover the Danckwerts model only if no distinction is made between the cup-mixing and spatial average concentrations (with this assumption, the effective axial dispersion coefficient is given by the Taylor-Aris theory). This derivation also shows that the concept of an effective axial dispersion coefficient and lumping the macro- and micromixing effects into one parameter is valid only at steady-state, constant inlet conditions and when the deviation from plug flow is small. [Remark Even with all these constraints, the error in the model because of the assumption (cj) — cym is of the same order of magnitude as the dispersion effect ]... 

In a companion pair of contributions, Mauri and Brenner (1991a,b) introduce a novel scheme for determining the rheological properties of suspensions. Their approach extends generalized Taylor-Aris dispersion-theory moment techniques (Brenner, 1980a, 1982)—particularly as earlier addressed to the study of tracer dispersion in immobile, spatially periodic media (Brenner, 1980b Brenner and Adler, 1982)—from the realm of material... 

The latter mechanism assumed is the well-known Taylor dispersion (T9, TIO, Til), which has been studied extensively (All, G6, L9, T14, S2). High-speed motion pictures taken by Towell et al. (T23) in a 40-cm bubble column (R3) have shown the presence of turbulent eddies, on a scale roughly equal to the column diameter, with systematic large-scale circulation patterns superimposed. Their pictures showed that liquid near the wall flowed downward, while liquid near the center of the column flowed upward, consistent with the flow theory developed earlier and with the Taylor dispersion mechanism. 

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. 

During mixing, the dispersed phase progressively breaks down until a rninimum drop diameter is reached. As the drop diameter decreases, further breakup becomes increasingly difficult. For emulsions, the size of the smallest drop that can be broken can be calculated from Taylor s theory, but experiments have shown that in most cases the equUibrium droplet size is larger than predicted. Furthermore, the deviation increases with concentration of the dispersed phase, ( ) - ( ), where experimentally the smallest value for which the deviation occurs, ( ) 0.005 [Utracki and Shi, 1992]. 

A set of empirical equations was obtained by Wu to describe the dispersed phase average particle size obtained after dispersive mixing in an extruder (13). The equations were based on the case of a Newtonian drop suspended in a Newtonian matrix, that is, Taylor s theory (16,17) with an extension to the case of a viscoelastic drop in a viscoelastic matrix. The empirical data employed were for blends containing 15 wt% dispersed phase and 85 wt% matrix phase. The particle size was found to be critically dependent on the ratio of the dispersed phase to the matrix phase melt... 

In contrast and for comparison, the theoretical equation from Taylor s theory (16, 17) for a Newtonian drop suspended in a Newtonian matrix with the concentration of the dispersed phase particle assumed to be vanishingly small is... 

We have already discussed the basics of dispersion, noting how decreased dispersion improves resolution and sensitivity in separation applications [61], and also yields improved dynamics for concentration and purification applications [4], However, there are some key differences to consider when comparing focusing techniques such as TGF with other techniques. We describe the basics of TGF theory, implementation details, and the modifications to Taylor dispersion required for TGF. Finally, we present tips for empirical optimization of TGF preconcentration factors and resolution. 

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. 

The theory of the Taylor dispersion technique is well described in the literature [11-20], and so the authors only indicate some relevant points concerning this method on the experimental determination of binary and ternary diffusion coefficients (Figure 2). 

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. 

Checks on the relationships between the axial coefficients were provided in empty tubes with laminar flow by Taylor (T2), Blackwell (B15), Bournia et al. (B19), and van Deemter, Breeder and Lauwerier (V3), and for turbulent flow by Taylor (T4) and Tichacek et al. (T8). The agreement of experiment and theory in all of these cases was satisfactory, except for the data of Boumia et al. as discussed previously, their data indicated that the simple axial-dispersed plug-flow treatment may not be valid for laminar flow of gases. Tichacek et al. found that the theoretical calculations were extremely sensitive to the velocity profile. Converse (C20), and Bischoff and Levenspiel (B14) showed that rough agreement was also obtained in packed beds. Here, of course, the theoretical calculation was very approximate because of the scatter in packed-bed velocity-profile data. 

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