Ranz-Marshall correlation

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) ... 

For a reaction represented by A(g) + bB(s) —> produc1(g), derive the relation between time (t) of reaction and fraction of B converted (/B), if the particle is spherical with an initial radius R0, and the Ranz-Marshall correlation for kAg(R) is valid, where R is the radius at t. Other assumptions are given above. 

Assume that the particle is spherical and isothermal, that both gas-film mass transfer resistance and reaction resistance are significant, and that the Ranz-Marshall correlation for k g is applicable. Do not make an assumption about particle size, but assume the reaction is first-order. 

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... 

Keywords Dufour effect Evaporation Frossling correlation Heat transfer Fatent heat Mass transfer Nusselt number Ranz-Marshall correlation Soret effect Stefan flow... 

Note that the Sherwood and Nusselt numbers are obtained from correlations discussed in the previous section, e.g., the Ranz-Marshal correlations given in (12.31) and (12.32). 

Keywords Cocoa butter Crystallization Freezing Latent heat Nucleation Nusselt number Ranz-Marshall correlation Recalescence Schmidt number Sherwood number Solidification Supercooling... 

Roes and van Swaaij [35] (Pall rings) and Verver and van Swaaij [6,37] (double-channel baffle column) experimentally obtained values of the mass transfer rate constant, which were much lower than values calculated from experimental solids holdup data and the well-know Ranz-Marshall correlation [38,39]. The low experimental values are to be attributed to particle-shielding phenomena due to the formation of less diluted suspensions or trickles. 

FIGURE 22.11 Particle Sherwood number (Sh) versus particle Reynolds number (Re). Comparison of results in gas-flowing solids-flxed bed contactor and Ranz-Marshall correlation for single sphere. 

In these equations, Tj is the temperature at coordinates (r, t) in the macrograin, pcp is the average value of the heat capacity per unit volume of the macroparticle, is the effective thermal diffiisivity in the macrograin and (—Affr) the enthalpy of polymerization. In Equation 2.141, the parameter hp is the average convective heat transfer coefficient, usually calculated from a Nusselt number correlation. Early works tended to use the well-known Ranz-Marshall correlation for evaporation from a droplet however, it has been... 

Experimental mass transfer data at low Reynolds numbers show a great deal of scatter, but most of the reported values fall somewhat bi ow the limit of 2.0 predicted from the Ranz-Marshall correlation. For examii>le, expressed in terms of Sherwood number, the correlation of Petrovic and iTiodos for gases is equivalent to... 

Equation (7.16) is similar to the Ranz-Marshall correlation [ q. (7.13)] and shows the sairi limiting behavior at low Reynolds number, but both the coefficient and power of the Reynolds number are somewhat larger. In the application of this expression it is important to note that axial dispersion coefficients mu t be properly estimated [e.g., according to Eq. (7.11) if the isotherm is hi ly favorable] otherwise the overall dispersion arising from external mass transfer resistance and axial mixing may be underestimated. 

Ranz-Marshall correlation for spheres. Assuming a constant Prandtl number for air, the most relevant parameter is the Reynolds number, which is mainly determined by the relative velocity between the dispersed and the continuous phase. The transfer of heat between the phases is given by the product of Nusselt number and the local, driving temperature difference of droplet and gas. 

The parameters bo, at, and Pi are determined from experimental data. One common correlation is the Frossling/Ranz-Marshall correlation. 

The mass transfer coefficient has already been calculated in Chapter 2 using the Ranz-Marshall correlation its value is 9.0 cm/sec. Therefore the mass transfer rate from gas to pellet under conditions of mass transfer limitation is... 

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