Local mass transfer rate

The local mass transfer rate is related to the concentration gradient by... 

Eqs. (40H41) are obtained from the analytical solution using the first two terms in the 0-series expansion of the concentration profile. As a result, they are accurate only for small values of meridional angle, 8. To correct for large values of 6, Newman [45] used Lighthill s transformation and Eq. (15) for the meridional velocity gradient to calculate the local mass transfer rate as Sc - oo. His numerical result is plotted in Fig. 5 in the form of Shloc/Re1/2 Sc1/3 vs. 8 as the thin solid line. The dashed line is... 

Fig. 5. Local mass transfer rate on the surface of a rotating hemisphere in laminar flow. Here the meridional angle, 9, is given in degrees.

For turbulent flow, we shall use the Chilton-Colburn analogy [12] to derive an expression for mass transfer to the spherical surface. This analogy is based on an investigation of heat and mass transfer to a flat plate situated in a uniform flow stream. At high Schmidt numbers, the local mass transfer rate is related to the local wall shear stress by... 

The Chilton-Colburn analogy can be also used to estimate the local mass transfer rate in laminar flow where the wall shear stress is related to the azimuthal velocity gradient by... 

Of considerable interest is the use of small isolated electrodes, in the form of strips or disks embedded in the wall, to measure local mass-transfer rates or rate fluctuations. Mass-transfer to spot electrodes on a rotating disk is represented by Eqs. (lOg-i) of Table VII. Analytical solutions in this case have to take account of curved streamlines. Despic et al. (Dlld) have proposed twin spot electrodes as a tool for kinetic studies, similar to the ring-disk electrode applications of disk and ring-disk electrodes for kinetic studies are discussed in several monographs (A3b, P4b). In fully developed channel or pipe flow, mass transfer to such electrodes is given by the following equation based on the Leveque model ... 

Overall mass-transfer rates at a sphere in forced flow, and mass-transfer rate distribution over a sphere as a function of the polar angle have been measured by Gibert, Angelino, and co-workers (G2, G4a) for a wide range of Reynolds numbers. The overall rate dependence on Re exhibited two distinct regimes with a sharp transition at Re = 1250. Local mass-transfer rates were deduced from measurements in which the sphere was progressively coated by an insulator, starting from the rear. 

Weder s experiments were carried out with opposing body forces, and large current oscillations were found as long as the negative thermal densification was smaller than the diffusional densification. [Note that the Grashof numbers in Eq. (41) are based on absolute magnitudes of the density differences.] Local mass-transfer rates oscillated by 50%, and total currents by 4%. When the thermal densification dominated, the stagnation point moved to the other side of the cylinder, while the boundary layer, which separates in purely diffusional free convection, remained attached. 

Rao et al. (R7) first made local mass-transfer measurements, by ring electrodes embedded in the perforated plates between which a packed bed was contained. They measured the local mass-transfer rate at ring electrodes... 

Three dimensional packed bed electrodes are generally considered for reactions which operate with low current densities in order to increase localized mass transfer rates and/or increase overall current per unit cell volume. The maximum current density at any position in the electrode structure is limited by the prevailing conditions of mass transfer. The limiting current thus can also have... 

It should be noted that the local mass transfer coefficient can only be obtained experimentally and is case specific. An analytical relationship for the local mass transfer rate coefficient can be obtained if a mathematical expression describing the gradient of the dissolved concentration at the NAPL-water interface is known. Unfortunately, the local mass transfer coefficient usually is not an easy parameter to determine with precision. Thus, in mathematical modeling of contaminant transport originating from NAPL pool dissolution, k(t, x,y) is often replaced by the average mass transfer coefficient, k(t), applicable to the entire pool, expressed as [41]... 

The properties of the surface layers have a strong effect on the deposition process. The driving force of the electrochemical reaction is the potential difference over the electrochemical double layer. Adsorption of species can change this potential. For example, the additives used in electrodeposition adsorb in the Helmholtz layer. They can change the local potential difference, block active deposition sites, and so on. The thickness of the diffusion layer affects the mass-transfer rate to the electrode. The diffusion layer becomes thinner with increasing flow rate. When the diffusion layer is thicker than the electrode surface profile, local mass-transfer rates are not equal along the electrode surface. This means that under mass-transfer control, metal deposition on electrode surface peaks is faster than in the valleys and a rough deposit will result. 

Mixing effects in chemical reactions must be formulated in terms of local mixing rates or local mixing times. The easily formulated global blend time seldom has an effect, while the time constants based on local conditions in the reactor, such as local mixing time or local mass transfer rate can be very important. 

To calculate local mass transfer rates, local mass transfer area is obtained from bubble size distributions. Mass transfer fluxes are calculated in separate subroutine, and the mass transfer rate is obtained by multiplying mass transfer area by mass transfer fluxes. 

CFD has become a standard tool for analyzing flow patterns in various situations related to chemical engineering. In many cases related to multiphase reactors, mass transfer limits overall chemical reaction. In these cases the accurate calculation of local mass transfer rates is of utmost importance. This is best done with the population balance approach, where local properties are used to model bubble or droplet breakage and coalescence phenomena. It has been proven that these rigorous models along with other multiphase and chemistry related models can be implemented in the CFD code, and solved simultaneously with the fluid flows. 

Typical perforaiance values such as Llu, a, d>, and PAf for each geometry are given in Table 13-9. Of interest is that Ll ranges from 0.0001 m/s for the packed, spray, and tray columns to 0.001 m/s for the stirred vessel and jet loop reactors. The pipe and static mixer reactors and the bnbble colimms are intermediate. The result of this observation is that for a given reactor, such as the stirred vessel, the local mass transfer rate is probably approximately proportional to the mass transfer area per unit volume, a, since the kL values are not very sensitive to hydrodynamics. Of course, fluid properties can also affect kL values, and they must be taken into account. 

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