Dispersion Taylor model

This result was first derived by Aris (1956) using the method of moments. While the resulting model now includes both the effects (axial molecular diffusion and dispersion caused by transerverse velocity gradients and molecular diffusion) it has the same deficiency as the Taylor model, i.e. converting a hyperbolic model into a parabolic equation. 

Before closing this section, we compare the hyperbolic model derived here with the wave model of Westerterp et al. (1995). For the classical Taylor problem (without axial dispersion), the model of Westerterp et al. may be written in the present notation as... 

Analysis of Eq. 3.6 reveals that the function C =/(f) tends to be Gaussian and approaches the Taylor model when N increases. On the other hand, the validity of this equation is dubious for low N values, as the shape of the curve skews. This means that the model provides good results when applied to unsegmented-flow systems with long reactors but fails to describe sample dispersion in short reactors. This limitation of the model is not as relevant to efficient mixing devices such as the single bead string reactor [5,52],... 

Key words laminar flow, dispersive flow, bulk flow, porous media, permeability, convective dispersion, diffusion, dispersion, Taylor dispersion, convection, momentum balance, Darcy s law, Navier-Stokes model, packed bed reactor, CDE, STM, MIM. 

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. 

Commenge et al. extended the one-dimensional model of reacting flows to include Taylor-Aris dispersion, i.e. they considered an equation of the form... 

Ruckenstein and Li proposed a relatively simple surface pressure-area equation of state for phospholipid monolayers at a water-oil interface [39]. The equation accounted for the clustering of the surfactant molecules, and led to second-order phase transitions. The monolayer was described as a 2D regular solution with three components singly dispersed phospholipid molecules, clusters of these molecules, and sites occupied by water and oil molecules. The effect of clusterng on the theoretical surface pressure-area isotherm was found to be crucial for the prediction of phase transitions. The model calculations fitted surprisingly well to the data of Taylor et al. [19] in the whole range of surface areas and the temperatures (Fig. 3). The number of molecules in a cluster was taken to be 150 due to an excellent agreement with an isotherm of DSPC when this... 

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. 

Our goal is the study of reactive flows through slit channels in the regime of Taylor dispersion-mediated mixing and in this chapter we will develop new effective models using the technique of anisotropic singular perturbations. 

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). 

Even with this enormous number of scientific papers on the subject, mathematically rigorous results on the subject are rare. Let us mention just ones aiming toward a rigorous justification of Taylor s dispersion model and its generalization to reactive flows. We could distinguish them by their approach... 

In this section, we will obtain the non-dimensional effective or upscaled equations using a two-scale expansion with respect to the transversal Peclet number Note that the transversal P let number is equal to the ratio between the characteristic transversal timescale and longitudinal timescale. Then we use Fredholm s alternative to obtain the effective equations. However, they do not follow immediately. Direct application of Fredholm s alternative gives hyperbolic equations which are not satisfactory for our model. To obtain a better approximation, we use the strategy from Rubinstein and Mauri (1986) and embed the hyperbolic equation to the next order equations. This approach leads to the effective equations containing Taylor s dispersion type terms. Since we are in the presence of chemical reactions, dispersion is not caused only by the important Peclet number, but also by the effects of the chemical reactions, entering through Damkohler number. 

In each of the cases we will solve the full physical problem numerically. Its section average will be compared with the solution the proposed effective ID model with Taylor s dispersion. Finally, if one makes the unjustified hypothesis that the average of a product is equal to the product of averages, averaging over sections gives a ID model which we call the "simple mean". We will make a comparison with the solution of that problem as well. 

Mass transport in laminar flow is dominated by diffusion and by the laminar velocity profile. This combined effect is known as dispersion and the underlying model for the theoretical derivation of a kinetic study had to be derived from the dispersion model, which Taylor [91] and Aris [92] developed. Taylor concluded that in laminar flow the speed of an inert tracer impulse initially given to a channel will have the same speed as the steady laminar carrier gas flow originally prevailing in this channel. 

While vinyl acetate is normally polymerized in batch or continuous stirred tank reactors, continuous reactors offer the possibility of better heat transfer and more uniform quality. Tubular reactors have been used to produce polystyrene by a mass process (1, 2), and to produce emulsion polymers from styrene and styrene-butadiene (3 -6). The use of mixed emulsifiers to produce mono-disperse latexes has been applied to polyvinyl toluene (5). Dunn and Taylor have proposed that nucleation in seeded vinyl acetate emulsion is prevented by entrapment of oligomeric radicals by the seed particles (6j. Because of the solubility of vinyl acetate in water, Smith -Ewart kinetics (case 2) does not seem to apply, but the kinetic models developed by Ugelstad (7J and Friis (8 ) seem to be more appropriate. 

A reactive dispersion model based on the Taylor dispersion model was proposed, which predicts a change of speed if the tracer impulse consists of reactants which react at the walls of the channel (see Figures 3.89 and 3.90). 

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. 

We note that this averaged model is hyperbolic. The third term in Eq. (45) represents the Taylor dispersion term (due to velocity gradients and transverse molecular diffusion) while the last term is the Aris-correction term (representing the influence of axial molecular diffusion). In dimensional form, the reduced model may be written as... 

When the Taylor approximation is used in Eq. (46) the averaged model is again parabolic but now the effective dispersion coefficient is given by... 

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... 

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 ]... 

The accuracy of the averaged model truncated at order p9(q 0) thus depends on the truncation of the Taylor series as well as on the truncation of the perturbation expansion used in the local equation. The first error may be determined from the order pq 1 term in Eq. (23) and may be zero in many practical cases [e.g. linear or second-order kinetics, wall reaction case, or thermal and solutal dispersion problems in which / and rw(c) are linear in c] and the averaged equation may be closed exactly, i.e. higher order Frechet derivatives are zero and the Taylor expansion given by Eq. (23) terminates at some finite order (usually after the linear and quadratic terms in most applications). In such cases, the only error is the second error due to the perturbation expansion of the local equation. This error e for the local Eq. (20) truncated at 0(pq) may be expressed as... 

In [64] the possible stable configurations of films in polyhedral foams is discussed from the thermodynamic point of view that any disperse system tends to minimum surface energy. Almgren and Taylor [64] modelled the shape of the films and the angles between them with wire devices and studied several film configurations. They established that only film configurations which obey Plateau laws are stable with respect to minor deformations. 

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