External Sherwood number

Traditionally, an average Sherwood number has been determined for different catalytic fixed-bed reactors assuming constant concentration or constant flux on the catalyst surface. In reality, the boundary condition on the surface has neither a constant concentration nor a constant flux. In addition, the Sh-number will vary locally around the catalyst particles and in time since mass transfer depends on both flow and concentration boundary layers. When external mass transfer becomes important at a high reaction rate, the concentration on the particle surface varies and affects both the reaction rate and selectivity, and consequently, the traditional models fail to predict this outcome. 

FIG. 16-10 Sherwood number correlations for external mass-transfer coefficients in packed beds for = 0.4. (Adapted from Suzuki, gen. refs.)... 

The designer now needs to make some estimates of mass transfer. These properties are generally well known for commercially available adsorbents, so the job is not difficult. We need to re-introduce the adsorber cross-section area and the gas velocity in order to make the required estimates of the external film contribution to the overall mass transfer. For spherical beads or pellets we can generally employ Eq. (7.12) or (7.15) of Ruthven s text to obtain the Sherwood number. That correlation is the mass transfer analog to the Nusselt number formulation in heat transfer ... 

Section II shows that the dimensionless external velocity field uJU, UqIU) is a function of dimensionless position r/a, 0) and k for creeping flow. The dimensionless concentration defined in Eq. (1-45) is a function of these quantities and of the Peclet number, Pe = 2aU/. Hence the Sherwood number, Sh = is a function of k and Pe (with additional dependence on Re unless the creeping flow approximation is valid). The exact solution of Eqs. (3-39) to (3-42) with the Hadamard-Rybczynski velocity field is not available for all values of Pe and K, but several special cases have been treated. 

Fig. 3.16 Variation of instantaneous overall Sherwood number with dimensionless time for spheres in creeping flow with negligible external resistance.

The mechanism of mass transfer to the external flow is essentially the same as for spheres in Chapter 5. Figure 6.8 shows numerically computed streamlines and concentration contours with Sc = 0.7 for axisymmetric flow past an oblate spheroid (E = 0.2) and a prolate spheroid (E = 5) at Re = 100. Local Sherwood numbers are shown for these conditions in Figs. 6.9 and 6.10. Figure 6.9 shows that the minimum transfer rate occurs aft of separation as for a sphere. Transfer rates are highest at the edge of the oblate ellipsoid and at the front stagnation point of the prolate ellipsoid. 

Sherwood number for external resistance based on volume-equivalent sphere,... 

So far we have considered an infinite value of the gas-to-particle heat and mass transfer coefficients. One may encounter, however, an imperfect access of heat and mass by convection to the outer geometrical surface of a catalyst. Stated in other terms, the surface conditions differ from those in the bulk flow because external temperature and concentration gradients are established. In consequence, the multiple steady-state phenomena as well as oscillatory activity depend also on the Sherwood and Nusselt numbers. The magnitudes of the Nusselt and Sherwood numbers for some strongly exothermic reactions are reported in Table III (77). We may infer from this table that the range of Sh/Nu is roughly Sh/Nu (1.0, 104). 

Besides, if SHa — oo, then xa w=i.o = xAb This also corresponds to a negligible external mass transfer resistance. In both cases, that of a finite Sherwood number SHa or for SHa — oo, we get a two-point boundary value differential equation. For the nonlinear case this has to be solved numerically. However, as for the axial dispersion model, we will start out with the linear case that can be solved analytically. 

Considering these Biot numbers, we can observe that they are similar to the Nusselt and Sherwood numbers. The only difference between these dimensionless numbers is the transfer coefficient property characterizing the Biot numbers transfer kinetics for the external phase (a x heat transfer coefficient for the external phase, k ex- mass transfer coefficient for the external phase). We can conclude that the Biot number is an index of the transfer resistances of the contacting phases. 

Cybulski and Moulijn [27] proposed an experimental method for simultaneous determination of kinetic parameters and mass transfer coefficients in washcoated square channels. The model parameters are estimated by nonlinear regression, where the objective function is calculated by numerical solution of balance equations. However, the method is applicable only if the structure of the mathematical model has been identified (e.g., based on literature data) and the model parameters to be estimated are not too numerous. Otherwise the estimates might have a limited physical meaning. The method was tested for the catalytic oxidation of CO. The estimate of effective diffusivity falls into the range that is typical for the washcoat material (y-alumina) and reacting species. The Sherwood number estimated was in between those theoretically predicted for square and circular ducts, and this clearly indicates the influence of rounding the comers on the external mass transfer. 

For nonspherical particles, the equivalent diameter used in the Reynolds and Sherwood numbers is dp = jAp/T = 0.564 J p, where A is the external surface area of the pellet. 

The external mass transfer coefficient kj, can be defined by the Sherwood number Sh using the following equation ... 

The limiting Sherwood number of 2.0 corresponds to an effective film thickness of DJ2 if the mass-transfer area is taken as the external area of the sphere. The concentration gradients actually extend out to infinity in this case, but the mass-transfer area also increases with distance from the surface, so the effective film thickness is much less than might be estimated from the shape of the concentration profile. 

The interphase mass transfer coefficient of reactant A (i.e., a,mtc), in the gas-phase boundary layer external to porous solid pellets, scales as Sc for flow adjacent to high-shear no-slip interfaces, where the Schmidt number (i.e., Sc) is based on ordinary molecular diffusion. In the creeping flow regime, / a,mtc is calculated from the following Sherwood number correlation for interphase mass transfer around solid spheres (see equation 11-121 and Table 12-1) ... 

To determine whether the measured reaction rate is influenced by external mass transfer, the Sherwood number (Appendix 8-14) of the catalyst is calculated (Equation 2.1-17)... 

Smith and Quinn (35) and Hoofd and Kreuzer (46) Independently developed analytical solutions for the facilitation factor which holds over a range In properties and operating conditions. Smith and Quinn obtained their solution by assuming a large excess of carrier. This allowed them to linearize the resulting differential equations. Hoofd and Kreuzer separated their solution into two parts a reaction-limited portion which is valid near the interface and a diffusion-limited portion within the membrane. Both groups obtained the same result for the facilitation factor. Hoofd and Kreuzer ( T) then extended their approach to cylinders and spheres. Recently, Noble et al. (48) developed an analytical solution for F based on flux boundary conditions. This solution allows for external mass transfer resistance and reduces to the Smith and Quinn equation In the limit as the Sherwood number (Sh) becomes very large. 

Although the microparticles circulate in the fluid bulk as a result of external agitation, it is assumed that the particles contained in the fluid elements in transient contact with the interface remain stationary. Thus it is possible to assume a Sherwood number of 2, a generally accepted value for mass transfer to and from a spherical particle in an infinite medium. 

Schmidt number Sherwood number dimensionless time dimensionless fluid temperature initial dimensionless temperature distribution dimensionless soUd phase temperature dimensionless volume-averaged solid temperature scale for temperature change, K dimensionless external temperature rise dimensionless inlet temperature maximum dimensionless temperature in the solid maximum dimensionless internal temperature rise... 

Thus if the values determined by these equations are approximately unity, we can neglect the influence of external mass transfer without any further calculation of the exact value of the Sherwood number, which (in a fixed bed) can only be higher than 3.8. 

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