Equation mean Sherwood number

The local and mean Sherwood numbers are obtained from the numerical results using the equations... 

In simultaneous heat and mass transfer in binary mixtures, mean mass transfer coefficients can likewise be found using the equations from the previous sections. Once again this requires that the mean Nusselt number Num is replaced by the mean Sherwood number Shm, and instead of the Grashof number a modified Grashof number is introduced, in which the density p(p,T, ) is developed into a Taylor series,... 

In the interval 200 < Pe < oo, for any values of the phase viscosities, the mean Sherwood number for a drop can be calculated by solving the cubic equation... 

Drops in the entire range of phase viscosities. For low and moderate Peclet numbers in an arbitrary laminar flow past a spherical drop under limiting resistance of the continuous phase, it is expedient to calculate the mean Sherwood number by using formula (4.7.3), where Shp and Shb are the Sherwood numbers for the limit cases of a solid particle and a bubble. These quantities can be calculated by formulas (4.7.9) and (4.7.10). For high Peclet numbers, in the entire range of phase viscosities, the mean Sherwood number can be found by solving the cubic equation [359]... 

Figure 4.6 shows the results of calculation of the mean Sherwood number obtained in [151] by a numerical solution of the corresponding integral equation for various values of dimensionless time and the Peclet number. One can see that, after the inner diffusion wake has been developed, the complete substance flux to the inner surface of the drop decreases rapidly. 

General correlations for the Sherwood number. In [364], the following approximate equation was suggested for the mean Sherwood number ... 

Note that to calculate the mean Sherwood number for particles of an irregular form, the following more general equation must be used ... 

To calculate the mean Sherwood number, one can replace formula (5.3.8) by the cubic equation [359]... 

In equations 11.78 and 11.79 Shx and (Shx)m represent the point and mean values respectively of the Sherwood numbers. 

These equations have been solved for rigid (Nl) and circulating spheres (Jl, K6, W3, W4) in creeping flow. Since the dimensionless velocities within the particle are proportional to (1 + k) (see Eq. (3-8)), F is a function only of Tp and PCp/(l + k). In presenting the results, it is instructive to consider the instantaneous overall Sherwood number, Shp, as well as F. The driving force is taken as the difference between the concentration inside the interface, and the mixed mean particle concentration, Cp, giving... 

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. 

The process of formulating mesoscale models from the microscale equations is widely used in transport phenomena (Ferziger Kaper, 1972). For example, heat transfer between the disperse phase and the fluid depends on the Nusselt number, and mass transfer depends on the Sherwood number. Correlations for how the Nusselt and Sherwood numbers depend on the mesoscale variables and the moments of the NDF (e.g. mean particle temperature and mean particle concentration) are available in the literature. As microscale simulations become more and more sophisticated, modified correlations that are based on the microscale results will become more and more common (Beetstra et al, 2007 Holloway et al, 2010 Tenneti et al, 2010). Note that, because the kinetic equation requires mesoscale models that are valid locally in phase space (i.e. for a particular set of mesoscale variables) as opposed to averaged correlations found from macroscale variables, direct numerical simulation of the microscale model is perhaps the only way to obtain the data necessary in order for such models to be thoroughly validated. For example, a macroscale model will depend on the average drag, which is denoted by... 

Abstract Evaporation of multi-component liquid droplets is reviewed, and modeling approaches of various degrees of sophistication are discussed. First, the evaporation of a single droplet is considered from a general point of view by means of the conservation equations for mass, species and energy of the liquid and gas phases. Subsequently, additional assumptions and simplifications are discussed which lead to simpler evaporation models suitable for use in CFD spray calculations. In particular, the heat and mass transfer for forced and non-forced convection is expressed in terms of the Nusselt and Sherwood numbers. Finally, an evaporation model for sprays that is widely used in today s CFD codes is presented. 

To keep the integration simple, we use the mean integral Sherwood number given by Equation 5.6, rather than the distance-dependent local coefficient listed in Table 5.2. We have... 

Free convection heat transfer as a source of forced convection mass transfer. It has been demonstrated on numerous occasions that the Chilton-Colburn analogy appearing in Table 2.3 is applicable for converting a forced-convection Nusselt number to a forced-convection Sherwood number as a means of converting the imbedded HTC into its equivalent MTC. In the present situation, the thermal buoyant forces provide the momentum source, which in effect provides the forced-convective flow that drives the mass transfer process. In addition, Grj Gta and Sc > Pr. For this case the alternative equation is... 

Trivial algebraic manipulations in Equation 15.38 reveal that only values of L/Rout and / ini// out have to be known and, consequently, there are only two degrees of freedom to determine. In the numerical simulations, the mean length of a tube is given as L = dp/100 and the last parameter Rout is chosen such that the Sherwood number has the same value as in the case of the spherical blobs model by Powers et al. (1994) in the previous section. 

The effects of concentration dependent physical properties on the correlation of dissolution mass transfer data have been reported in some detail by Nienow, Unahabhoka and Mullin (1966, 1968). Mean solution properties should be used for the Sherwood and Schmidt groups in equation 6.119 if the mass transfer data for moderately soluble substances are to be correlated effectively. The arithmetic mean will suffice for viscosity and density, but the integral value must be used for the diffusivity (equation 2.27). Bulk solution properties are used for the Reynolds number. 

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