The Mass-Transfer Coefficient

We may define a mass-transfer coefficient in a manner similar to that used for defining the heat-transfer coefficient. Thus 

If one considers a steady-state diffusion across a layer of thickness Ax, 

Note that the units of the mass-transfer coefficients are in meters per second in SI units. 

We have already seen that the phenomenological laws governing heat, mass, and momentum transfer are similar. In Chap. 5 it was shown that the energy and momentum equations of a laminar boundary layer are similar, viz.. 

We further observed that the ratio via, the Prandtl number, was the connecting link between the velocity and temperature field and was thus an important parameter in all convection heat-transfer problems. If we considered a laminar boundary layer on a flat plate in which diffusion was occurring as a result of 

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Equations 11 and 12 caimot be used to predict the mass transfer coefficients directly, because is usually not known. The theory, however, predicts a linear dependence of the mass transfer coefficient on diffusivity. 

Rate Equations with Concentration-Independent Mass Transfer Coefficients. Except for equimolar counterdiffusion, the mass transfer coefficients appHcable to the various situations apparently depend on concentration through thej/g and factors. Instead of the classical rate equations 4 and 5, containing variable mass transfer coefficients, the rate of mass transfer can be expressed in terms of the constant coefficients for equimolar counterdiffusion using the relationships... 

To use all of these equations, the heights of the transfer units or the mass transfer coefficients and must be known. Transfer data for packed columns are often measured and reported direcdy in terms of and and correlated in this form against and... 

Other correlations based partially on theoretical considerations but made to fit existing data also exist (71—75). A number of researchers have also attempted to separate from a by measuring the latter, sometimes in terms of the wetted area (76—78). Finally, a number of correlations for the mass transfer coefficient itself exist. These ate based on a mote fundamental theory of mass transfer in packed columns (79—82). Although certain predictions were verified by experimental evidence, these models often cannot serve as design basis because the equations contain the interfacial area as an independent variable. 

The Mass Transfer Coefficient, Because of the interaction between and a when air is dispersed in the media, the two have rarely... 

The mass-transfer coefficients are typically between 10 to 100 p.m/s, depending on hydrodynamic conditions and the values of D. 

Values of the mass-transfer coefficient k have been obtained for single drops rising (or falling) through a continuous immiscible Hquid phase. Extensive Hterature data have been summarized (40,42). The mass-transfer coefficient is often expressed in dimensionless form as the Sherwood number ... 

The enhanced rate expressions for regimes 3 and 4 have been presented (48) and can be appHed (49,50) when one phase consists of a pure reactant, for example in the saponification of an ester. However, it should be noted that in the more general case where component C in equation 19 is transferred from one inert solvent (A) to another (B), an enhancement of the mass-transfer coefficient in the B-rich phase has the effect of moving the controlling mass-transfer resistance to the A-rich phase, in accordance with equation 17. Resistance in both Hquid phases is taken into account in a detailed model (51) which is apphcable to the reversible reactions involved in metal extraction. This model, which can accommodate the case of interfacial reaction, has been successfully compared with rate data from the Hterature (51). 

Interfacial Contact Area and Approach to Equilibrium. Experimental extraction cells such as the original Lewis stirred cell (52) are often operated with a flat Hquid—Hquid interface the area of which can easily be measured. In the single-drop apparatus, a regular sequence of drops of known diameter is released through the continuous phase (42). These units are useful for the direct calculation of the mass flux N and hence the mass-transfer coefficient for a given system. 

To estimate the slumping motion of the kiln bed which periodically exposes a fresh, vapor saturated surface at the bed—freeboard interface must be considered. Based on Pick s second law in a bed of porosity, S, and for an effective diffusion coefficient, the mass-transfer coefficient on the bed side is... 

The constant depends on the hydraulic diameter of the static mixer. The mass-transfer coefficient expressed as a Sherwood number Sh = df /D is related to the pipe Reynolds number Re = D vp/p and Schmidt number Sc = p/pD by Sh = 0.0062Re Sc R. ... 

Correlations for the mass-transfer coefficient, as the Sherwood number for various membrane geometries have been reviewed (39). 

Sedimentation is also used for other purposes. For example, relative motion of particles and Hquid iacreases the mass-transfer coefficient. This motion is particularly useful ia solvent extraction ia immiscible Hquid—Hquid systems (see Extraction, liquid-liquid). An important commercial use of sedimentation is ia continuous countercurrent washing, where a series of continuous thickeners is used ia a countercurrent mode ia conjunction with reslurrying to remove mother liquor or to wash soluble substances from the soHds. Most appHcations of sedimentation are, however, ia straight sohd—Hquid separation. 

The use of molal humidity as the mass-transfer driving force is conventional and convenient because of the development of humidity data for, especially, the air—water system. The mass-transfer coefficient must be expressed in consistent units. 

The integration of equation 35 requites a knowledge of the mass-transfer coefficient, ky-a, and also of the interface conditions from which could be obtained. Combining equations 27, 28, and 34 gives a relation balancing transfer rate on both sides of the interface ... 

To model convection drying both the heat transfer to the coated web and the mass transfer (qv) from the coatiag must be considered. The heat-transfer coefficient can be taken as proportional to the 0.78 power of the air velocity or to the 0.39 power of the pressure difference between the air in the plenum and the ambient pressure at the coatiag. The improvement in heat-transfer coefficients in dryers since the 1900s is shown in Figure 20. The mass-transfer coefficient for solvent to the air stream is proportional to the heat-transfer coefficient and is related to it by the Clulton-Colbum analogy... 

If requited, the drydown can be hastened by increasing desiccant mass, particle surface area, or mass-transfer coefficient. The mass-transfer coefficient can be altered to some extent by the design of the desiccant container. 

Mass-Transfer Coefficient Denoted by /c, K, and so on, the mass-transfer coefficient is the ratio of the flux to a concentration (or composition) difference. These coefficients generally represent rates of transfer that are much greater than those that occur by diffusion alone, as a result of convection or turbulence at the interface where mass transfer occurs. There exist several principles that relate that coefficient to the diffusivity and other fluid properties and to the intensity of motion and geometry. Examples that are outlined later are the film theoiy, the surface renewal theoiy, and the penetration the-oiy, all of which pertain to ideahzed cases. For many situations of practical interest like investigating the flow inside tubes and over flat surfaces as well as measuring external flowthrough banks of tubes, in fixed beds of particles, and the like, correlations have been developed that follow the same forms as the above theories. Examples of these are provided in the subsequent section on mass-transfer coefficient correlations. 

With complicated geometries, the product of the interfacial area per volume and the mass-transfer coefficient is required. Correlations of kop or of HTU are more accurate than individual correlations of k and since the measurements are simpler to determine the produc t kop or HTU. 

To determine the mass-transfer rate, one needs the interfacial area in addition to the mass-transfer coefficient. For the simpler geometries, determining the interfacial area is straightforward. For packed beds of particles a, the interfacial area per volume can be estimated as shown in Table 5-27-A. For packed beds in distillation, absorption, and so on in Table 5-28, the interfacial area per volume is included with the mass-transfer coefficient in the correlations for HTU. For agitated liquid-liquid systems, the interfacial area can be estimated... 

It is important to recognize that the effects of temperature on the liquid-phase diffusion coefficients and viscosities can be veiy large and therefore must be carefully accounted for when using /cl or data. For liquids the mass-transfer coefficient /cl is correlated in terms of design variables by relations of the form... 

In developing correlations for the mass-transfer coefficients Icq and /cl, the various authors have assumed different but internally compatible correlations for the effective interfacial area a. It therefore would be inappropriate to mix the correlations of different authors unless it has been demonstrated that there is a valid area of overlap between them. 

The unknown intermediate concentration C, has been mathematically ehminated from the last term. In this case, r can be solved for explicitly, but that is not always possible with surface rate equations of greater complexity. The mass transfer coefficient /ci is usually obtainable from correlations. When the experimental data are of (C, r) the other constants can be found by linear plotting. 

The general expression given by Eq. (14-8) is more complex than normally is required, but it must be used when the mass-transfer coefficient varies from point to point, as may be the case when the gas is not dilute or when the gas velocity varies as the gas dissolves. The values of yi to be used in Eq. (14-8) depend on the local hquid composition Xi and on the temperature. This dependency is best represented by using the operating and equilibrium lines as discussed later. 

Solution. By assuming that the mass-transfer coefficient varies as the 0.8 power of the local gas mass velocity, we can derive the following relation ... 

There are a number of different types of experimental laboratory units that could be used to develop design data for chemically reacting systems. Charpentier [ACS Symp. Sen, 72, 223-261 (1978)] has summarized the state of the art with respect to methods of scaUng up lab-oratoiy data and tabulated typical values of the mass-transfer coefficients, interfacial areas, and contact times to be found in various commercial gas absorbers as well as in currently available laboratoiy units. 

It would be desirable to reinterpret existing data for commercial tower packings to extract the individual values of the interfacial area a and the mass-transfer coefficients fcc and /c in order to facilitate a more general usage of methods for scaling up from laboratory experiments. Some progress in this direction has afready been made, as discussed later in this section. In the absence of such data, it is necessary to operate a pilot plant or a commercial absorber to obtain kc, /c , and a as described by Ouwerkerk (op. cit.). 

Principles of Rigorous Absorber Design Danckwerts and Alper [Trans. Tn.st. Chem. Eng., 53, 34 (1975)] have shown that when adequate data are available for the Idnetic-reaciion-rate coefficients, the mass-transfer coefficients fcc and /c , the effective interfacial area per unit volume a, the physical solubility or Henry s-law constants, and the effective diffusivities of the various reactants, then the design of a packed tower can be calculated from first principles with considerable precision. 

Inspection of Eqs. (14-71) and (14-78) reveals that for fast chemical reactions which are liquid-phase mass-transfer limited the only unknown quantity is the mass-transfer coefficient /cl. The problem of rigorous absorber design therefore is reduced to one of defining the influence of chemical reactions upon k. Since the physical mass-transfer coefficient /c is already known for many tower packings, it... 

Estimation of for Irreversible Reactions Figure 14-14 illustrates the influence of either first- or second-order irreversible chemical reactions on the mass-transfer coefficient /cl as developed by Van Krevelen and Hoftyzer [Rec. Trav. Chim., 67, 563 (1948)] and as later refined by Periy and Pigford and by Brian et al. [Am. Inst. Chem. Eng. /., 7, 226(1961)]. 

Note that the product of the mass-transfer coefficient and the interfacial area is a volumetric coefficient and obviates the need for a value of the interfacial area. While areas for mass transfer on plates have been measured, the experimental contacting equipment cuffered significantly from that used for commercial distillation or gas absorption, and the reported areas are considered unreliable for design purposes. 

The mass-transfer coefficient of Eq. (14-139) is carried as a product with interfacial area (giving a volumetric mass transfer coefficient) ... 

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