Mean Sherwood number

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

Experimental mean Sherwood numbers are shown in Fig. 10.2. The asymptotic solution, Eq. (10-21), gives a good representation of the data for large Sc when Ra > 10. For extremely large Ra, a turbulent range is expected where... 

In (1.196) the quantities c,n,m still depend on the type of flow, laminar or turbulent, and the shape of the surface or the channel over or through which the fluid flows. Correspondingly, the mean Sherwood number can be written as... 

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,... 

The main quantity of practical interest is the mean Sherwood number... 

The calculation of diffusion fluxes and the mean Sherwood number is usually carried out in three steps. First, the problem of convective mass transfer is solved and the concentration field is determined. Second, the normal derivative dC /dC) =0 on the surface is evaluated. Finally, one applies formulas (3.1.25)—... 

The mean Sherwood numbers for a disk of finite radius a at high Schmidt numbers can be estimated using the formula... 

In the turbulent regime of flow, the mean Sherwood number for a disk of finite radius a can be approximated by the formula [439]... 

The unknown quantity which is of most practical interest in these problems is the mean Sherwood number, which is determined by (3.1.28) and is related to the mass transfer coefficient ac by... 

From the mathematical viewpoint, the diffusion problem (4.3.1)-(4.3.3) is equivalent to the problem on the electric field of a charged conductive body in a homogeneous charge-free dielectric medium. Therefore, the mean Sherwood number in a stagnant fluid coincides with the dimensionless electrostatic capacitance of the body and can be calculated or measured by methods of electrostatics. 

Tables 4.2 shows the values of II for particles of various shape (according to [94, 166]). It follows from (4.3.5) that the mean Sherwood number can be obtained from the data in this table by dividing the shape factor by the surface area of the particle and then multiplying by the characteristic length.

Sherwood number. For solid particles, drops, and bubbles of spherical shape, the mean Sherwood number can be calculated according to the formula... 

Peclet numbers, the problem of mass exchange between a particle of arbitrary shape and a uniform translational flow were studied by the method of matched asymptotic expansions in [62]. The following expression was obtained for the mean Sherwood number up to first-order infinitesimals with respect to Pe ... 

Circular cylinder. The mass exchange between a circular cylinder of radius a and a uniform translational flow whose direction is perpendicular to the generatrix of the cylinder was considered in [186,218] for low Peclet and Reynolds numbers Pe = Sc Re and Re = aU-Jv. For the mean Sherwood number (per unit length of the cylinder) determined with respect to the radius, the following two-term expansions were obtained ... 

For a spherical particle in an arbitrary linear shear flow (4.5.1), the first four terms of the asymptotic expansion as Pe - 0 of the mean Sherwood number have the form [5]... 

The mean Sherwood number is obtained by dividing (4.6.26) by the dimensionless surface area of the particle (drop or bubble). 

In this section, some interpolation formulas are presented (see [367, 368]) for the calculation of the mean Sherwood number for spherical particles, drops, and bubbles of radius a in a translational flow with velocity U at various Peclet numbers Pe = aU /D and Reynolds numbers Re = aU-Jv. We denote the mean Sherwood number by Shb for a gas bubble and by Shp for a solid sphere. 

Spherical particle as Re —> 0, 0 < Pe < oo. The problem of mass transfer to a solid spherical particle in a translational Stokes flow (Re -f 0) was studied in the entire range of Peclet numbers by finite-difference methods in [1, 60, 281], To find the mean Sherwood number for a spherical particle, it is convenient to use the following approximate formula [94] ... 

Spherical drop as Re —> 0, 0 < Pe < oo. In the range 0 < Pe < 200, the results of numerical calculations of mean Sherwood numbers for a spherical drop in a translational flow under a limiting resistance of the continuous phase is well described by the approximate formula [68]... 

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... 

Spherical particle at various Reynolds numbers. In the case of a spherical particle in a translational flow at 0.5 < Re < 200 and 0.125 < Sc < 50, numerical results concerning the mean Sherwood number (e.g., see [114, 281]) can be described [94] by the approximate formula... 

Spherical bubble at any Peclet numbers for Re > 35. For a spherical bubble in a translational flow at moderate and high Reynolds numbers and high Peclet numbers, the mean Sherwood number can be calculated by the formula [94]... 

Formula (4.7.9) can be used for the calculation of the mean Sherwood number in laminar flows of various types without closed streamlines past a solid spherical particle. The auxiliary value Shpoo must be chosen equal to the leading term of the asymptotic expansion of the Sherwood number as Pe - oo. 

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]... 

The mean Sherwood number for spherical solid particles, drops, and bubbles in a linear straining shear flow (Gkm = 0 for k m) at low Reynolds numbers and high Peclet numbers... 

We assume that the fluid velocity distribution remote from the interface is given by Eq. (4.5.1). The mean Sherwood number for a spherical particle, drop, or bubble does not change if we change all signs of the shear coefficients, that is, Sh(G m) = Sh(-G m). 

The solution of hydrodynamic problems for an arbitrary straining linear shear flow (Gkm = Gmk) past a solid particle, drop, or bubble in the Stokes approximation (as Re -> 0) is given in Section 2.5. In the diffusion boundary layer approximation, the corresponding problems of convective mass transfer at high Peclet numbers were considered in [27, 164, 353]. In Table 4.4, the mean Sherwood numbers obtained in these papers are shown. 

For a solid spherical particle in an arbitrary linear straining shear flow, the following interpolation formula was suggested in [27] for the mean Sherwood number ... 

For a spherical particle in an axisymmetric shear Stokes flow (Re — 0), numerical results for the mean Sherwood number can be well approximated in the entire range of Peclet numbers by the expression (4.7.9), where the asymptotic value Shpoo must be taken from the first row in Table 4.4. As a result, we obtain the formula... 

For an arbitrary straining shear flow (Gkm = Gmk), the mean Sherwood number for a solid sphere can be expressed by the similar formula... 

In the case of an arbitrary straining shear flow past a spherical drop for 0 < PeM 200, the expressions (4.8.5) and (4.8.7) must be substituted into the expression (4.7.3) of the mean Sherwood number. 

This limit property of the mean Sherwood number differs essentially from the corresponding behavior of Sh in the presence of singular hydrodynamic points, where the mean Sherwood number increases infinitely as Pe - oo (e.g., see formulas (4.8.4) and (4.8.6)). 

The analysis of the mass transfer problem for a sphere freely suspended in an arbitrary shear flow at high Peclet numbers leads to the following two-term asymptotics for the mean Sherwood number at low values of the angular velocity [343]... 

Mass exchange between a spherical particle and the translational-shear flow (4.9.1) at high Peclet numbers was studied in [175]. For the mean Sherwood number depending on the parameters... 

The mean Sherwood number for the translational-shear flow (4.9.1) past a spherical drop under limiting resistance of the continuous phase at high Peclet numbers can be calculated by the formulas [164]... 

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