Sherwood-number

Values of the mass-transfer coefficient k have been obtained for single drops rising (or falling) through a continuous immiscible Hquid phase. Extensive Hterature data have been summarized (40,42). The mass-transfer coefficient is often expressed in dimensionless form as the Sherwood number ... 

The values of k and hence Sb depend on whether the phase under consideration is the continuous phase, c, surrounding the drop, or the dispersed phase, d, comprising the drop. The notations and Sh are used for the respective mass-transfer coefficients and Sherwood numbers. 

The constant depends on the hydraulic diameter of the static mixer. The mass-transfer coefficient expressed as a Sherwood number Sh = df /D is related to the pipe Reynolds number Re = D vp/p and Schmidt number Sc = p/pD by Sh = 0.0062Re Sc R. ... 

Correlations for the mass-transfer coefficient, as the Sherwood number for various membrane geometries have been reviewed (39). 

Using this simplified model, CP simulations can be performed easily as a function of solution and such operating variables as pressure, temperature, and flow rate, usiag software packages such as Mathcad. Solution of the CP equation (eq. 8) along with the solution—diffusion transport equations (eqs. 5 and 6) allow the prediction of CP, rejection, and permeate flux as a function of the Reynolds number, Ke. To faciUtate these calculations, the foUowiag data and correlations can be used (/) for mass-transfer correlation, the Sherwood number, Sb, is defined as Sh = 0.04 S c , where Sc is the Schmidt... 

The Sherwood number, Sh, is estimated from Table 16-9, and the dispersion parameters Yi and Jo ffom Table 16-10 for well-packed columns. Typical values are a 1-4 and b 0.5-1. Since HETP -2HTU, Fig. 16-13 can also be used for approximate calculations. 

The mass-transfer coefficient, /c, is contained in the Sherwood number ... 

FIG. 22-66 Characteristic curve for flux as a function of feed composition for cross-flow membrane. Right curve is for a higher Sherwood number than the left curve. 

The dimensionless numbers in tlris equation are the Reynolds, Schmidt and the Sherwood number, A/ sh. which is defined by this equation. Dy/g is the diffusion coefficient of the metal-transporting vapour species in the flowing gas. The Reynolds and Schmidt numbers are defined by tire equations... 

The comparison of the magnitude of the two resistances clearly indicates whether tire metal or the slag mass transfer is rate-determining. A value for the ratio of the boundary layer thicknesses can be obtained from the Sherwood number, which is related to the Reynolds number and the Schmidt number, defined by... 

The dimensionless quantity Sh is called the Sherwood number. The heat transfer factor a is defined bv... 

It is not possible to translate the above reasoning to turbulent flow, as turbulent flow equations are not reliable. However, in practice it is typical to assume that the same analogy is also valid for turbulent flow. Because of this hypothesis level, it is quite futile to use the diffusion factor D g in the Schmidt number instead we will directly use the number D g as in the Sherwood number. Hence in practical calculations Sc = v/D b-... 

Used for mass transfer = Pr number for mass transfer = Colburn number. Sherwood number... 

The average Nusselt number is not very sensitive to changes in gas velocity and Reynolds number, certainly no more than (Re)I/3. The Sherwood number can be calculated with the same formula as the Nusselt number, with the substitution of the Schmidt number for the Prandtl number. While the Prandtl number of nearly all gases at all temperatures is 0.7 the Schmidt number for various molecules in air does depend on temperature and molecular type, having the value of 0.23 for H2, 0.81 for CO, and 1.60 for benzene. 

The correlation studies of heat and mass transfer in pellet beds have been investigated by many, usually in terms of the. /-factors (113-115). According to Chilton and Colburn the two. /-factors are equal in value to one half of the Fannings friction factor / used in the calculation of pressure drop. The. /-factors depend on the Reynolds number raised to a factor varying from —0.36 to —0.68, so that the Nusselt number depends on the Reynolds number raised to a factor varying from 0.64 to 0.32. In the range of the Reynolds number from 10 to 170 in the pellet bed, jd should vary from 0.5 to 0.1, which yields a Nusselt number from 4.4 to 16.1. The heat and mass transfer to wire meshes has received much less attention (110,116). The correlation available shows that the /-factor varies as (Re)-0-41, so that the Nusselt number varies as (Re)0-69. In the range of the Reynolds number from 20 to 420, the j-factor varies from 0.2 to 0.05, so that the Nusselt number varies from 3.6 to 18.6. The Sherwood number for CO is equal to 1.05 Nu, but the Sherwood number for benzene is 1.31 Nu. 

Provided that the catalyst is active enough, there will be sufficient conversion of the pollutant gases through the pellet bed and the screen bed. The Sherwood number of CO is almost equal to the Nusselt number, and 2.6% of the inlet CO will not be converted in the monolith. The diffusion coefficient of benzene is somewhat smaller, and 10% of the inlet benzene is not converted in the monolith, no matter how active is the catalyst. This mass transfer limitation can be easily avoided by forcing the streams to change flow direction at the cost of some increased pressure drop. These calculations are comparable with the data in Fig. 22, taken from Carlson 112). 

The first group is the Sherwood number, the second is the Reynolds number, and the third the Schmidt number. If surfactants are present, their effect on p should also be included [cf. Eq. (66) or (68)]. 

For mass transfer, which is considered in more detail in Chapter 10, an analogous relation (equation 10.233) applies, with the Sherwood number replacing the Nusselt number and the Schmidt number replacing the Prandtl number. 

Experimental results for fixed packed beds are very sensitive to the structure of the bed which may be strongly influenced by its method of formation. GUPTA and Thodos157 have studied both heat transfer and mass transfer in fixed beds and have shown that the results for both processes may be correlated by similar equations based on. / -factors (see Section 10.8.1). Re-arrangement of the terms in the mass transfer equation, permits the results for the Sherwood number (Sh1) to be expressed as a function of the Reynolds (Re,) and Schmidt numbers (Sc) ... 

The point values of the Sherwood number Shx and mass transfer coefficient ho are then given by ... 

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

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