Sherwood number correlation

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

Table 5 Sherwood number correlations for NAPL dissolution...

The nonlinear least squares regression program PEST [79] was used to fit the proposed correlation relating the time invariant Sherwood number to overall Peclet numbers for circular pools given by Eq. (91) to the seven experimentally determined Sh values presented in Fig. 12b, in order to estimate the empirical coefficients fi, y2> and y3. The experimental, overall Sherwood number correlation applicable to circular TCE pool dissolution in water saturated, homogeneous porous media can be expressed by the following relationship ... 

And kg can be calculated from the modified Sherwood Number correlation ... 

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... 

In the absence of any chemical reaction, the Damkohler number vanishes and the steady-state Sherwood number correlation, given by equation (13-18), reduces to... 

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) ... 

For laminar flow adjacent to a high-shear no-slip solid-liquid interface, with one-dimensional flow in the mass transfer boundary layer, the mass transfer coefficient fcA.MXc is obtained from the following Sherwood number correlation (see steps 17 and 18 of Problem 23-7 an page 653, particularly the scaling law exponents a and b) ... 

The true mass transfer coefficients were evaluated and the Sherwood numbers correlated with the Reynolds numbers of the continuous phase, the coefficients being obtained by regression as below ... 

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 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. 

In defining a 7-factor (jd) for mass transfer there is therefore good experimental evidence for modifying the exponent of the Schmidt number in Gilliland and Sherwood s correlation (equation 10.225). Furthermore, there is no very strong case for maintaining the small differences in the exponent of Reynolds number. On this basis, the /-factor for mass transfer may be defined as follows ... 

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 side-by-side diffusion cell has also been calibrated for drug delivery mass transport studies using polymeric membranes [12], The mass transport coefficient, D/h, was evaluated with diffusion data for benzoic acid in aqueous solutions of polyethylene glycol 400 at 37°C. By varying the polyethylene glycol 400 content incrementally from 0 to 40%, the kinematic viscosity of the diffusion medium, saturation solubility for benzoic acid, and diffusivity of benzoic acid could be varied. The resulting mass transport coefficients, D/h, were correlated with the Sherwood number (Sh), Reynolds number (Re), and Schmidt number (Sc) according to the relationships... 

For a free-falling spherical particle of radius R moving with velocity u relative to a fluid of density p and viscosity p, and in which the molecular diffusion coefficient (for species A) is DA, the Ranz-Marshall correlation relates the Sherwood number (Sh), which incorporates kAg, to the Schmidt number (Sc) and the Reynolds number (Re) ... 

The convective mass transfer coefficient hm can be obtained from correlations similar to those of heat transfer, i.e. Equation (1.12). The Nusselt number has the counterpart Sherwood number, Sh = hml/Di, and the counterpart of the Prandtl number is the Schmidt number, Sc = p/pD. Since Pr k Sc k 0.7 for combustion gases, the Lewis number, Le = Pr/Sc = k/pDcp is approximately 1, and it can be shown that hm = hc/cp. This is a convenient way to compute the mass transfer coefficient from heat transfer results. It comes from the Reynolds analogy, which shows the equivalence of heat transfer with its corresponding mass transfer configuration for Le = 1. Fire involves both simultaneous heat and mass transfer, and therefore these relationships are important to have a complete understanding of the subject. 

Correlations for heat transfer coefficient between a single sphere and surrounding gas have been proposed by many researchers (Table 5.2), for example, Whitaker,1584 and Ranz and Marshall,15051 among others. The correlation recommended by Whitaker is accurate to within 30% for the range of parameter values listed. All properties except jus should be evaluated at Tm. For freely falling liquid droplets, the Ranz-Marshall correlation 505 is often used. The correlations may be applied to mass transfer processes simply by replacing Nu and Pr with Sh and Sc, respectively, where Sh and Sc are the Sherwood number and Schmidt number, respectively. Modifications to the Ranz-Marshall correlation have been made by researchers to account... 

The mass transfer coefficient describes the effect of mass transfer resistance of the reactants flowing from the gas phase to the surface of the individual particles in the bed. The mass transfer coefficient can be obtained from a correlation for the Sherwood number (or dimensionless mass transfer coefficient) given by Eq. (7) ... 

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 ... 

This is simply the definition of the mass transfer coefficient km, the subject of mass transfer courses is to find suitable correlations in order to estimate k A (units of lengthAime). The mass transfer coefficient is in turn defined through the Sherwood number,... 

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