Taylor-Aris dispersion coefficient

This is termed the Taylor-Aris dispersion coefficient, and is simply the sum of the axial molecular diffusion coefficient and the Taylor radial dispersion coefficient. As can be seen, at large Peclet numbers D ffD increases as the square of the Peclet number (the Taylor dispersion limit), and at small Peclet numbers D ifD approaches 1 (the convective axial diffusion limit). 

Hydrodynamic dispersion refers to the stretching of a solute band in the flow direction during its transport by an advecting fluid. Variation in the fluid velocity across the channel cross section leads to such band broadening which is often quantified in terms of the Taylor-Aris dispersion coefficient. 

Diffusion coefficients of solids into fluids can conveniently be measured by capillary evaporation [109-111] or evaporation from flat plates or surfaces of one sort or another. The Taylor Aris dispersion technique [112-115] can be used not only for solids but also for any component which will dissolve in the solvent of interest. The above two techniques are probably the ones most frequently used in connection with near-critical solvents. 

The dispersion coefficient has three physical causes in adsorption axial diffusion, adsorption kinetics, and Taylor-Aris dispersion ... 

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,... 

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 ... 

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 ]... 

Flow in microchannels with diameters between 10 and 500 pm is mostly laminar and characterized by a parabolic velocity profile. Therefore, the molecular diffusion in axial and radial directions plays an important role influencing the RTD. Diffusion in the radial direction tends to diminish the spread of the parabolic velocity profile, whereas diffusion in the axial direction increases the spread [72,73]. Taylor [72] and Aris [73] established the following relation to predict the effective axial dispersion coefficient for laminar flow ... 

The Taylor-Aris result for the dispersion coefficient (Eq. 4.6.35) has been applied to the empirical correlation of measured and calculated longitudinal dispersion coefficients in flow through packed beds and porous media (see Eidsath et al. 1983). Typically, the velocity in the Peclet number of the Taylor-Aris formula is identified with the superficial velocity, and the capillary diameter with the hydraulic diameter for spherical particles. An alternative velocity suggested by the capillary model is the interstitial velocity, and an alternative length is the square root of the permeability. In an isotropic packing of particles is about one-tenth the particle diameter (Probstein Hicks... 

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 microstructured channels, laminar flow can be considered when the hydrodynamic entrance length remains short compared to the channel length. Therefore, the axial dispersion coefficient can be estimated with a relation developed by Aris [20] and Taylor [21] ... 

Laminar flows in microchannels with their parabolic velocity profiles face superposition by radial and axial diffusion, as described by Taylor and Aris [61], introducing a global axial dispersion coefficient Dax which is used as a parameter in their dispersion model [12] ... 

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 functions M (t) were determined from the complete unsteady axially symmetric convective diffusion equation (Eq. 4.6.7), and M (f) were obtained from the Taylor dispersion equation, which was used as the model equation. The phenomenological coefficients U and in the equation were determined by matching the first three moments of the infinite sequence M (t) to M (t) for asymptotically large times [t>a lD). Applying his scheme to the circular capillary problem, Aris showed that D fj, where axial molecular diffusion is not neglected, is given by Eq. (4.6.35). Fried Combarnous (1971) later showed that the satisfaction of the first three moments for t—implies that c x, t), obtained as a solution of the Taylor dispersion equation with = D + Pe /48), is asymptotically the solution of the complete, unsteady, axially symmetric convective diffusion equation averaged over the cross section. 

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