Aris numbers diffusion coefficient

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 ... 

Using the dusty gas model [5] analytical solutions are derived to describe the internal pressure gradients and the dependence of the effective diffusion coefficient on the gas composition. Use of the binary flow model (BFM, Chapter 3) would also have yielded almost similar results to those discussed below. After discussion of the dusty gas model, results are then implemented in the Aris numbers. Finally, negligibility criteria are derived, this time for intraparticle pressure gradients. Calculations are given in appendices here we focus on the results. 

If the concentration CA influences the effective diffusion coefficient DtA, the Aris numbers can be calculated from... 

Hence, anisotropy can be accounted for in the Aris numbers if the effective diffusion coefficient DtJi is replaced by Dl AH and the characteristic dimension VJAp by... 

Rather than accounting for anisotropy by modifying both the characteristic dimension and effective diffusion coefficient, this is achieved by modifying the effective diffusion coefficient only. Thus, for anisotropic catalyst pellets a modified effective diffusion coefficient D is defined, which accounts for the anisotropy. Hence, for anisotropic catalyst pellets and simple reactions the Aris numbers can be calculated from... 

The procedure given above can be extended for other catalyst geometries. For example, for an anisotropic parallelepiped with the dimensions width length height = W L H, the Aris numbers can still be calculated from the Equation 7.126 and 7.127. However, the modified effective diffusion coefficient DtAnow becomes ... 

First Aris Number Anx for Concentration-dependent Effective Diffusion Coefficients and Simple Reactions... 

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). 

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