Mass transfer theoretically-derived correlations

Equation (5b), in Table VII, established by Lin et al. (L9) for coaxial flow in annuli with k = 0.5, was originally taken by them to be a striking confirmation of the theoretically derived correlation, Eq. (27), with 0=1. The latter condition, however, corresponds only to the limiting case of Eq. (27), at /c->0, that is, mass transfer to the wall of an outer cylinder, without an inner cylinder present. On these grounds, and because of other experimental conditions, the correlation of Lin et al. (L9) was criticized by Friend and Metzner (F9), who calculated that the constant in Eq. (5b) of... 

Another survey by Ibl (13) in 1963 listed 13 mass-transfer correlations established by the limiting-current method, only four of which were derived from quantitative considerations. At the time of writing the total number of publications is more than 200. The majority of these concern flow conditions under which theoretical predictions are, at best, qualitative. More recently, an increasing number of publications deal with model hydrodynamic studies of more complex situations, for example, packed and fluidized beds. 

Recently, a set of correlations including the effect of channel shape has been proposed by Ramanathan et al. (2003) on the basis of solution of the Navier-Stokes equations in the channel, with different solutions derived for ignited-reaction and extinct-reaction regimes. The comparison of various empirical and theoretical correlations with experimentally evaluated mass transfer coefficients is given by West et al. (2003). The correlations by Ramanathan et al. (2003) or Tronconi and Forzatti (1992) have been used in most simulations presented in this chapter. 

If there is no mass transfer resistance within the catalyst particle, then Ef is unity. However, it will then decrease from unity with increasing mass transfer resistance within the particles. The degree of decrease in f is correlated with a dimensionless parameter known as the Thiele modulus [2], which involves the relative magnitudes ofthe reaction rate and the molecular diffusion rate within catalyst particles. The Thiele moduli for several reaction mechanisms and shapes of catalyst particles have been derived theoretically. 

If the physical situation closely matches the assumptions made in the development of theoretically derived mass-transfer correlations, these correlations can be quite accurate. If the physical situation is somewhat different than the assumptions, the correlation may still be useful by using a small amount of experimental data to tune the coefficient. Practicing engineers would seldom do the derivations, but since they commonly use the results, it is important to know both the assunptions and the limits of validity of the correlation. The best way to understand these is to closely follow the derivation. 

As mentioned earlier, Pangarkar et al. (2002) have critically examined the available information on correlations for predicting in stirred tank reactors. It was concluded in Section 6.4.2 that an approach based on the analogy between momentum and mass transfer can yield simple yet theoretically sound and reliable correlations for the mass transfer coefficient in several cases. Before this approach is applied to prediction of in stirred tank reactors, it is desirable to reconcile the available literature on the effects of various relevant parameters on The conclusions derived by Pangarkar et al. (2002) on examination of data from their group as well as other literature data are as follows ... 

In theory it is not necessary to have experimental mass-transfer coefficients for laminar flow, since the equations for momentum transfer and for diffusion can be solved. However, in many actual cases it is difficult to describe mathematically the laminar flow for geometries, such as flow past a cylinder or in a packed bed. Hence, experimental mass-transfer coefficients are often obtained and correlated. A simplified theoretical derivation will be given for two cases in laminar flow. 

The design of heat and mass transfer operations in chemical engineering is based on the well-known correlations that use the dimensionless numbers Nu (Nusselt) for heat transfer and Sh (Sherwood) for mass transfer By balancing the acting forces, energies, and mass flows within the boundary layers of velocity, temperature, and concentration, the theoretical derivation of general relations for Nu and Sh is given in fundamental work [35]. 

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