Turbulent flow in mass transfer

The approach taken te describe mass transfer within a phase under these corrqrlex conditions is to erttplqy a mass tranrfer ooelficiem that is defined as the ratio of the molar (or trass) flitx of a species across a particular stttfiice to the composition difference causing the transfer of mass. The coefficient for mass transfer betweoi two locations 1 and 2 can be defined by ariy of the following equations  

The foim involving partial pressures is only used for gases, while those involving mass concentrations or mass fractions are usually employed for liquids. When defined in this way the mass transfer coefficient includes effects of geometry, hydrodynamics, physical and transport properties of the fluid, and bulk flow contributions. 

Although the definitions of mass transfer coefficients expressed in Eq. (2.4-1) are most commonly used, an alternative definition originally employed by Colburn and Drew is useful under conditions of large convective flow in the direction of transport. The flux across a transfer surface at position 1 is given as 

Other driving force units and corresponding coefficients can be used in analogous definitions. Equation (2.4-2) explicitly allows for the bulk flow contribution to the molar flux relative to stationary coordinates. The coefficient or its analogues is thought to exhibit a somewhat less complex relationship to composition, flow conditions, and geometry. Further discussion of the relationships among those coefficients is provided in Section 2.4-1 in terms of particular models for turbulent mass transfer. 

FIGURE 2.4-1 Effect of Reynolds and Schmidt numbers on mass transfer in turbulent flow. Adapted from Data of Refs. 2 and 3. 

The solution of the Aim model is based on ihe steedy-state species balances (mass or molar units) that were developed in Section 2.3 for a binary system. 

The ftua of A is not completely specified by Eq. (2.4-3), even if a film thickness were known, until a separate statement of the value of NAI(NA + WB) is provided. This auxiliary information might he obtained from solubility considerations—for example, the flux of air into water is zero dering evaporation of HjO— or possibly from the stoichiometiy of a chemical reaction. Two common cases are often considered equimolar connterdiffusion end the diffusion of a species A through an inert stagnant mixture (B). 

For the case of NA = -NB, which is sometimes a good approximation duresg the distillation of a binary mixture, Eq. (2.4 3) reduces (via L Hdpitai s Rule) to 

The mass transfer coefficient k is identical to k, in this case since WA + WE = 0. The film theory suggests that the mass transfer coefficient is linearly related to the diffusivity—a result not generally supported by exparimenla] observations of transport in turbulent fluids. 

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The velocity, viscosity, density, and channel-height values are all similar to UF, but the diffusivity of large particles (MF) is orders-of-magnitude lower than the diffusivity of macromolecules (UF). It is thus quite surprising to find the fluxes of cross-flow MF processes to be similar to, and often higher than, UF fluxes. Two primary theories for the enhanced diffusion of particles in a shear field, the inertial-lift theory and the shear-induced theory, are explained by Davis [in Ho and Sirkar (eds.), op. cit., pp. 480-505], and Belfort, Davis, and Zydney [/. Membrane. Sci., 96, 1-58 (1994)]. While not clear-cut, shear-induced diffusion is quite large compared to Brownian diffusion except for those cases with very small particles or very low cross-flow velocity. The enhancement of mass transfer in turbulent-flow microfiltration, a major effect, remains completely empirical. 

C. Some Experimental Data on Mass Transfer in Turbulent Flow... 

In transverse flow along a flat plate the transition from laminar to turbulent flow occurs at Reynolds numbers waL/v between 3 105 and 5 10s wa is the initial flow velocity, L is the length of the plate over which the fluid is flowing. The heat and mass transfer in turbulent flows is more intensive than in laminar. In general, at the same time there is also an increase in the pressure drop. 

An exclusively analytical treatment of heat and mass transfer in turbulent flow in pipes fails because to date the turbulent shear stress Tl j = —Qw w p heat flux q = —Qcpw, T and also the turbulent diffusional flux j Ai = —gwcannot be investigated in a purely theoretical manner. Rather, we have to rely on experiments. In contrast to laminar flow, turbulent flow in pipes is both hydrodynamically and thermally fully developed after only a short distance x/d > 10 to 60, due to the intensive momentum exchange. This simplifies the representation of the heat and mass transfer coefficients by equations. Simple correlations, which are sufficiently accurate for the description of fully developed turbulent flow, can be found by... 

Von Behren et al. (1972) analyzed multicomponent mass transfer in turbulent flow in a pipe. Show that their model is fundamentally incorrect. You may also refer to the paper by Stewart (1973). 

Stewart, W. E., Multicomponent Mass Transfer in Turbulent Flow, AIChE J, 19, 398-400 (1973). Stewart, W. E. and Prober, R., Matrix Calculation of Multicomponent Mass Transfer in Isothermal... 

Temperature profile. Let us discuss qualitative specific features of convective heat and mass transfer in turbulent flow past a flat plate. Experimental evidence indicates that several characteristic regions with different temperature profiles can be distinguished in the thermal boundary layer on a flat plate. At moderate Prandtl numbers (0.5 < Pr < 2.0), it can be assumed for rough estimates that the characteristic sizes of these regions are of the same order of magnitude as those of the wall layer and the core of the turbulent stream, see Section 1.7. 

More detailed information about heat and mass transfer in turbulent flows past a flat plate, as well as various relations for determining the temperature profile and Nusselt (Sherwood) numbers, and a lot other useful information can be found in the references [184, 185, 212, 289, 406], which contain extensive literature surveys. 

This result suggests diet the mass transfer coefficient varies as Djg, a result intermediate to those of the film theory and tha panetration theory. In fact, the j power dependence of the Sherwood number on the Schmidt number is observed in a number of correlations for mass transfer in turbulent flow in conduits aed for flow about submerged objects. (See Section 2.4-3.)... 

Hughmark employed this u to derive a correlation for Son and Hanratty (1967) and Hughmark (1971,1974) correlated wall to fluid heat transfer in pipe flow based on the relatively simple and well-established boundary layer theory. In the case of pipe flow, momentum transfer is solely by skin friction because of the geometry involved. Nonetheless, this approach was extended to particle-fluid mass transfer in turbulent flow. The correlation proposed was of the following form ... 

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