Dispersion coefficients magnitude

Dispersion In tubes, and particiilarly in packed beds, the flow pattern is disturbed by eddies diose effect is taken into account by a dispersion coefficient in Fick s diffusion law. A PFR has a dispersion coefficient of 0 and a CSTR of oo. Some rough correlations of the Peclet number uL/D in terms of Reynolds and Schmidt numbers are Eqs. (23-47) to (23-49). There is also a relation between the Peclet number and the value of n of the RTD equation, Eq. (7-111). The dispersion model is sometimes said to be an adequate representation of a reaclor with a small deviation from phig ffow, without specifying the magnitude ol small. As a point of superiority to the RTD model, the dispersion model does have the empirical correlations that have been cited and can therefore be used for design purposes within the limits of those correlations. 

The dispersion coefficient is orders of magnitude larger than the molecular diffusion coefficient. Some rough correlations of the Peclet number are proposed by Wen (in Petho and Noble, eds.. Residence Time Distribution Theory in Chemical Tngineeiing, Verlag Chemie, 1982), including some for flmdized beds. Those for axial dispersion are ... 

The dispersion coefficients for the mixed-symmetry component 7 5 which describes the deviation from Kleinman symmetry are for methane more than an order of magnitude smaller than coefficients of the same order in the frequencies for 7. Their varations with basis sets and wavefunction models are, however, of comparable absolute size and give rise to very large relative changes for the mixed-symmetry dispersion coefficients. 

The dispersion coefficients are dependent on the character of the media and the flow velocity. It is difficult to predict these coefficients within an order of magnitude, so they are normally measured or fit to measured data in the field. For example, if a concentration variance in the x-, y-, and z-directions can be measured in response to a pulse release, then equation (6.51) could be applied to obtain... 

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. 

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 contrast to the equilibrium-dispersive model, which is based upon the assumptions that constant thermod3mamic equilibrium is achieved between stationary and mobile phases and that the influence of axial dispersion and of the various contributions to band broadening of kinetic origin can be accounted for by using an apparent dispersion coefficient of appropriate magnitude, the lumped kinetic model of chromatography is based upon the use of a kinetic equation, so the diffusion coefficient in Eq. 6.22 accounts merely for axial dispersion (i.e., axial and eddy diffusions). The mass balance equation is then written... 

Three figures of solute transport in a soil column will be discussed to contrast unsteady-state and quasi steady-state behavior, to illustrate the effect of loading frequency on solute distribution in a soil column, and to show the effect of the magnitude of the dispersion coefficient on decreasing the magnitude of the ampli-... 

Diffusion of a solute through immobile water to a reaction site also is affected by interstitial water velocity. If the diffusion rate is slow compared to the interstitial velocity, physical nonequilibrium occurs (5-7). The immobile water can be a layer on the grain surface (film diffusion), in dead-end pores between tightly packed grains (pore diffusion), or within crevices or pits on the grain surfaces (particle diffusion). Calcium and chloride breakthrough curves from column experiments done by James and Rubin (8) indicate that nonequilibrium transport occurs unless interstitial velocities are decreased so that the hydrodynamic-dispersion coefficient is of the same order of magnitude as the molecular-diffusion coefficient. 

The problem is to determine the effect of the excluded volume of particles of radius on the Taylor dispersion coefficient (Eq. 4.6.27). In so doing, note that in the dispersion coefficient, a should be replaced hy a - a, and U should not represent the average translational speed but rather the magnitude of the difference between the largest and smallest velocities in the flow field. The largest particle velocity remains equal to the centerline solvent velocity but due to the excluded volume, the smallest velocity is not zero but is the value of the solvent velocity at a distance from the wall. This represents a reduction in the velocity difference and, therefore, in the dispersion coefficient. Accordingly, U should be multi-... 

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