Average 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. 

The average Sherwood number, Shiv, and average limiting current density, iljmjav, can be obtained by integrating Eqs. (40)-(41) over the electrode surface. The results for a hemisphere, whose entire surface is subject to mass transfer are ... 

Here, the dimensionless thickness of the momentum boundary layer A, and the dimensionless quantity, a, relating to the direction of local flow are functions of 0 they are shown graphically in Fig. 3. The average Sherwood number may be obtained by integrating Eq. (51) over the surface the result for a hemisphere may be given as [22] ... 

Mass transfer can produce films of nonuniform thickness because the deposition rate can depend on the velocity field u over the sohd. Regions with high whl have the highest deposition rates in a mass-transfer-hmited process. For flow over a flat plate of length L the average Sherwood number for laminar flow is given by the expression... 

Combination of Eqs. (1-51) to (1-55) with either Eq. (1-56) or (1-58) yields an equation which may be solved to give concentration profiles from which mass transfer rates may be found. For a solid particle the average Sherwood number is... 

Kim and Chrysikopoulos [69] have shown that for rectangular NAPL pools the average Sherwood number is related to appropriate average Peclet numbers as follows ... 

Yi, y2, and y3 are empirical coefficients that were determined by fitting the nonlinear power law correlation Eq. (91) to 484 average Sherwood numbers computed for 121 different pool dimensions and four different sets of hydro-dynamic conditions. The resulting time invariant, average mass transfer correlation for elliptic pools is given by... 

For a given geometry, the average Nusselt number in forced convection depends on the Reynolds and Prandtl numbers, whereas the average Sherwood number depends on the Reynolds and Schmidt numbers. That is. 

We see from Eqs. 9.4.2 and 9.4.6 that when t oo equilibrium is attained, and the average composition (xj) will equal the surface composition Xjy. The time averaged Sherwood number and mass transfer coefficients for a rigid spherical particle may be obtained directly from Eqs. 9.4.4 and 9.4.5 with F given by Eq. 9.4.6 above. The Sherwood number at time t may be found using Eqs. 9.4.3 and 9.4.6 as... 

Derive expressions for the local and average Sherwood numbers for mass transfer over a flat plate in laminar flow. 

Integrating and rearranging in terms of the average Sherwood number, ShL yields... 

If one divides the average mass transfer coefficient A c. average by the simplest mass transfer coefficient in the absence of convective transport, then the resulting dimensionless ratio is identified as the average Sherwood number. Hence,... 

The dimensionless tangential velocity component at the gas-hquid interface is independent of the Reynolds and Schmidt numbers for creeping and potential flow. The final expression for the surface-averaged Sherwood number in any flow regime where turbulent mass transfer mechanisms are absent is... 

Now begin with the final result from part (e) for kc, struct an expression for the average Sherwood number ... 

Since the product of Re and Sc is LVapproach/ A.mix, the preceding expression for the average Sherwood number reduces to... 

The dimensionless mass transfer correlation, given by (12-1), reveals the complete dependence of the surface-averaged Sherwood number on the Reynolds and Schmidt nnmbers ... 

TABLE 12-1 Effect of Flow Regime and the Nature of the Interface on Dimensionless Correlations for the Surface-Averaged Sherwood Number via Steady-State Mass Tk ansfer Boundary Layer Theory in Nonreactive Systems... 

Fig. 4.3-16 Time averaged Sherwood number of fluid particles depending on the Fourier number with limiting laws...

Figure 4.3-16 illustrates the time averaged Sherwood number Sh depending on the Fourier number of the dispersed phase Fo = t ... 

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