Sherwood number modified

The diffusion coefficient as defined by Fick s law, Eqn. (3.4-3), is a molecular parameter and is usually reported as an infinite-dilution, binary-diffusion coefficient. In mass-transfer work, it appears in the Schmidt- and in the Sherwood numbers. These two quantities, Sc and Sh, are strongly affected by pressure and whether the conditions are near the critical state of the solvent or not. As we saw before, the Schmidt and Prandtl numbers theoretically take large values as the critical point of the solvent is approached. Mass-transfer in high-pressure operations is done by extraction or leaching with a dense gas, neat or modified with an entrainer. In dense-gas extraction, the fluid of choice is carbon dioxide, hence many diffusional data relate to carbon dioxide at conditions above its critical point (73.8 bar, 31°C) In general, the order of magnitude of the diffusivity depends on the type of solvent in which diffusion occurs. Middleman [18] reports some of the following data for diffusion. 

As a measure for the degree with which these states are governing the drying process, an extended Sherwood number Sh is introduced ( called a modified Biot number in [95]) ... 

Fig. 7.15. Liquid utilization factor versus < l for an irreversible first-order reaction at different values of the modified Sherwood number.

Reynolds number (-) gas constant (J K 1 mol 1) radial distance (m) modified Sherwood number (-) dimensionless number for particle bath concentration (-)... 

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

Here Sh is the modified Sherwood number defined as Sh = /Csnsdp/fl,D and We is the modified Weber number defined as We = UoLpi.dr/hlci. A graphical illustration of the above correlation is shown in Fig. 6-20. The predictions of Eq. (6-67) also agree fairly well with the data of Lemay el al.so Specchia et al.9i showed that, in a trickle-flow reactor, KLaL and Ksas are essentially of the same order of the magnitudes. They also evaluated the conditions under which the mass-transfer (gas-liquid and liquid-solid) influences significantly the performance of a trickle-bed reactor. 

Most recently, Sano et al.121 derived a relation for the liquid-solid mass-transfer coefficient (or Sherwood number) based on Kolmogoroff s theory for isotropic turbulence. The Reynolds number based on this theory is defined in terms of , the rate of energy dissipation per unit mass of liquid, dp, the specific surface diameter, and vL, the kinematic viscosity of liquid. Thus, the modified Reynolds number Re was defined as Re = Edp/vl. The Sherwood number was correlated 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 Stanton number for mass, Stm = kdvz,av, that determine the ratio between the mass transfer velocity and the flow velocity, a modified Sherwood number. 

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... 

Ron Observed Rate iBol/g cat. a Modified Sherwood Number, k,LID. Modified Nnssett Number hfLIX,... 

Shp = kgpRJDgp = modified Sherwood number for poison Da = [Pg.276]

Sh = k, R iy,A = modified Sherwood number for main reaction = J JKJdIa = modulus... 

Many experimental results and these of Barthole et al. (3) concerning kgS were represented in dimensionless, Sherwood or modified Sherwood s number against Reynolds and Schmidt s numbers. Their values are comparable with those of Goto et al. C14) and the type of correlation proposed by Darwadkar and Sylvester (45) seems the best to correlate their data (Fig. 9)... 

D9. Determine the modified Sherwood number. abPu>i for the gas-side-... 

Experimental results show that the mass Biot number (B()m> which is sometimes called the modified Sherwood number [17], is much larger than unity, indicating that the major resistance to mass transfer resides in the internal pore diffusion process. 

The model was validated on NaCl solutions. In Eq. 6.3, Di is the diffusion coefficient of the solute in the liquid phase, Dg is the diffusion coefficient of the solvent vapor in the gas phase, Qi and pg are the liquid and gas densities, respectively. Sh is the dimensionless mass transfer rate in the vapor phase. This modified Sherwood number, that accounts for the film thinning effect of Stefan flow, lies typically in the range 2 < Sh < 5. The quantity Bm is the Spalding transfer number according to Abramzonand Sirignano (1989) (Sirignano (1999), compare with Eq. 1.66 in Volume... 

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