Mass transfer versus diffusion

For hquid systems v is approximately independent of velocity, so that a plot of JT versus v provides a convenient method of determining both the axial dispersion and mass transfer resistance. For vapor-phase systems at low Reynolds numbers is approximately constant since dispersion is determined mainly by molecular diffusion. It is therefore more convenient to plot H./v versus 1/, which yields as the slope and the mass transfer resistance as the intercept. Examples of such plots are shown in Figure 16. 

While microscopic techniques like PFG NMR and QENS measure diffusion paths that are no longer than dimensions of individual crystallites, macroscopic measurements like zero length column (ZLC) and Fourrier Transform infrared (FTIR) cover beds of zeolite crystals [18, 23]. In the case of the popular ZLC technique, desorption rate is measured from a small sample (thin layer, placed between two porous sinter discs) of previously equilibrated adsorbent subjected to a step change in the partial pressure of the sorbate. The slope of the semi-log plot of sorbate concentration versus time under an inert carrier stream then gives D/R. Provided micropore resistance dominates all other mass transfer resistances, D becomes equal to intracrystalline diffusivity while R is the crystal radius. It has been reported that the presence of other mass transfer resistances have been the most common cause of the discrepancies among intracrystaUine diffusivities measured by various techniques [18]. 

As the temperature is varied in a reactor, we should expect to see the rate-controlling step vary. At sufficiently low temperature the reaction rate coefficient is small and the overall rate is reaction limited. As the temperature increases, pore diffusion next becomes controlling (Da is nearly independent of temperature), and at sufficiency high temperature external mass transfer might limit the overall process. Thus a plot of log rate versus 1 / T might look as shown in Figure 7-15. 

Figure 7-15 Plots of r versus T and log i versus 1/r. We expect the rate to exhibit breaks on the 1/r plot as the reaction process goes from reaction limited at low temperature, pore diffusion limited at intermediate temperature, and external mass transfer limited at high temperature.

Van t Hoff plots of In k versus the inverse of temperature (generally 1000/T for convenience) are very often linear, especially with monomeric bonded phases. They can exhibit nonlinear behavior, and the transition temperature is often close to the undefined room temperature. Temperature optimization is one trend in LC. A rising temperature increase reduces viscosity and increases the diffusion rate, thereby enhancing mass transfer, which flattens the HETP curve at high velocities (31). Conversely, Sander and Wise (32) investigated the influence of temperature reduction. 

The intra particle diffusion model (single resistance) assumes external mass transfer is significant only at initial stages. The internal diffusion parameter, k, is calculated from the plot of dye adsorbed qt versus square root of time [5]. 

We see that internal diffusion limits the reaction if a plot of r versus dp is linear. Under these eonditions the overall rate of reaction can be inereased by deereasing the partiele size. However, the overall rate will be rmaffected by the mixing conditions in the bulk liquid that would ehange the mass transfer boimdary layer thickness next to the pellet surfaee. 

Temperature Dependence of Fast Reactions It is to be noted that rate constants for fast (diffusion-controlled) steps are also temperature dependent, since the diffusion coefficient depends on temperature. The usual experimental procedure, suggested by the Arrhenius equation, of plotting In k versus /T will indicate apparent activation energies for diffusion control of approximately 12-15 kJ moP. For fast heterogeneous chemical reactions in which intrinsic chemical and mass transfer rates are of comparable magnitude, care needs to be taken in interpretation of apparent activation energies for the overall process. 

Figure 6.6 Comparison of the chromatogram given by the film mass transfer-pore diffusion model of chromatography with a Gaussian Profile. Dimensionless plot of versus f. Solid line Gaussian profile. Dotted line Carta s solution [34]. (a) Nap = N = 25 theoretical plates, (b) Nap = N = 100. Reprinted by permission of Kluwer Academic Publishing, from S. Golshan-Shirazi and G. Guiochon, NATO ASI Series C, vol 383, 61 (Fig. 4), with kind permission of Springer Science and Business Media.

A plot of H/(2mq) versus 1/Ug is a straight line with a slope equal to Di and an ordinate equal to 3f + S )/Sq. The coefficient of external mass transfer is estimated using one of the several correlations available for it (see Chapter 5, subsection 5.2.5, correlation of Wilson and Geankoplis [62], Kataoka et al. [87], or the penetration theory [88]). Correcting for the contribution due to the external mass transfer resistance gives the last term in the plate height equation, 5, hence the intraparticle diffusion coefficient, Dg. 

A plot of b versus (1 + B) is shown in Fig. 11-9. The result at B = 0 is the result calculated in Section C. For B < 0, we see that b (and thus the rate of mass transfer) is increased. The asymptotic behavior is already achieved at (1 + li) 0.2. On the other hand, for B > 0, the rate ofmass transfer is decreased, and the asymptotic behavioris achieved by(l + B) 4. It can be seen from Fig. 11-9 that the rate of mass transfer is quite sensitive to B. In particular, the rate of mass transfer from a body can be much different from the rate of heat transfer, even when the Reynolds number and the Schmidt number for the mass transfer problem are identical to the values of the Reynolds and Prandtl numbers for the heat transfer case. This is a consequence of the fact that the transport process right at the body surface, in the absence of any mass flux across the surface (as in the heat transfer case), is dominated by a diffusion process. Hence even a rather tiny convective transport contribution can be extremely important. 

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