Binary Mass Transfer Coefficients

The mass transfer coefficient k in phase x for a binary system is best defined in a manner suggested by Bird et al. (1960, p. 639) 

A word now about the units of With the fluxes expressed in moles per second per meter squared (mol/s) (m interfacial area), and in the units moles per meter cubed (mol/m ), the units of kf, are meters per second (m/s). But is it really a velocity To examine this further, we replace the diffusion flux in Eq. 7.1.3 with - w) to 

The numerator on the right-hand side of Eq. 7.1.4 is the velocity of transfer of Component 1 with respect to the molar average reference velocity of the mixture u. For a binary system (but not always for a multicomponent system), the quotient in Eq. 7.1.4 is positive and the coefficient kf is positive. The greatest possible driving force Ax is unity thus, with x = 1 and Xjy = 0, the maximum value of the denominator of Eq. 7.1.4 is unity, that is, lAxi/xJ 1. Therefore, 

Equation 7.1.5 gives a physical significance to the mass transfer coefficient It is the maximum velocity (relative to the velocity of the mixture) at which a component can be transferred in the binary system. The actual velocity of Component 1 relative to the mixture velocity is given by 

Equation 7.1.6 should be compared to Eq. 3.1.1. It appears that may be related directly to the binary Fick diffusion coefficient D. Indeed, this will be shown to be the case when we examine various specific hydrodynamic models for mass transfer later in this book. 

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In this case it is assumed that a pure gas A is being absorbed in a solvent eontaining a chemically inert component B. Both the solvent and B are not volatile and the fraction of A in the liquid bulk equals zero. The binary mass transfer coefficient Kij between A and the solvent in eq. (4) is given a typical value of 1 X lO" m/s, whereas the total concentration of the liquid Cr is set to 1 x 10 mol/m, also a typical value. Parameters to be chosen are the solubility of A, x i, the fraction of B in the solvent Xg, the mass transfer coefficient between A and B, K/ g and the mass transfer coefficient between B and the solvent, Kg. The results of the calculations are presented in Table 1. Since both the solvent and component B possess a zero flux. Kgs has no influence on the mass transfer process and has therefore been omitted. The computed absorption rate has been compared with the absorption rate obtained from analytical solutions for the following cases. 

Figures 7(a)-(c) show a comparison between the numerically computed absorption flux and the absorption flux obtained from expression (31), using eqs (24), (30) and (34)-(37). From these figures it can be concluded that for both equal and different binary mass transfer coefficients absorption without reaction can be described well with eq. (24), whereas absorption with instantaneous reaction can be described well with eq. (30). If the Maxwell-Stefan theory is used to describe the mass transfer process, the enhancement factor obeys the same expression as the one obtained on the basis of Fick s law [eq. (35)]. Finally, from Figs 7(b) and 7(c) it appears that the use of an effective mass transfer coefficient m the Hatta number again produces satisfactory results.

From comparison of eqs (Cl) and (C2) it can be concluded that a better expression is obtained if Kas is replaced by an effective" mass transfer coefficient which takes into account the difference in binary mass transfer coefficients eq. (34). The more general expression for the effective mass transfer coefficient of component i is given by... 

In general, all elements of the mass transfer matrix depend on the process variables, and in particular on the vapor phase composition. The mass transfer mechanisms in membranes can be rather complicated. However, for the conceptual analysis of the considered membrane process, it is not advantageous to go into the details of mass transport. Therefore, in the following the effective binary mass transfer coefficients k,j are assumed to be constants. 

Kij dimensionless binary mass transfer coefficient between components i and j... 

Formulation in Terms of Binary Mass Transfer Coefficients... 

Given the binary mass transfer coefficients and the mole fractions and there are three unknown quantities in these equations the molar fluxes, N-, N2, However, there are only two independent mass transfer rate equations. Thus, one more equation is needed this will be the bootstrap relation ... 

There are a great many correlations available in the literature for estimating binary mass transfer coefficients. It is beyond the scope of this book to review these correlations in detail [the reader is referred to the text by Sherwood et al. (1975) for more information]. For present purposes it suffices to cite only a couple of examples of useful empirical expressions. Other correlations are discussed in Sections 12.1.5 and 12.3.3. 

The binary mass transfer coefficients estimated from these correlations and analogies are the low flux coefficients and, therefore, need to be corrected for the effects of finite transfer rates before use in design calculations. 

In some correlations it is the binary mass transfer coefficient—interfacial area product that is correlated. In this case, then k should be considered to be this product and the that are calculated from Eqs. 8.2.14 et seq are the mass transfer rates themselves with units moles per second (mol/s) (or equivalent). 

This approach is, in fact, equivalent to replacing the binary diffusivity D by the matrix of multicomponent diffusion coefficients [D] and the binary mass transfer coefficient with the... 

Binary mass transfer coefficients for the vapor in the jetting-bubble formation zone may be computed from... 

The binary mass transfer coefficients for the liquid phase may be evaluated with a penetration model... 

The remaining binary mass transfer coefficients are computed in a similar way to give... 

As for the small bubble population, the binary mass transfer coefficients for the vapor phase are calculated using Eqs. 12.2.39... 

Three methods of estimating binary mass transfer coefficients in packed columns are presented below. 

If we use the simplified models discussed in Section 8.8 for the matrices of mass transfer coeflicients we may relate the inverse matrices of numbers of transfer units to the matrices of inverse binary mass transfer coefficients as (cf. Eqs. 12.2.20 and 12.2.21)... 

The nonequilibrium stage in Figure 14.1 may represent either a single tray or a section of packing in a packed column. In the models described in this chapter the same equations are used to model both types of equipment and the only difference between these two simulation problems is that different expressions must be used for estimating the binary mass transfer coefficients and interfacial areas. 

The next step is to compute the binary mass transfer coefficients. For this example we must make use of the Chilton-Colburn analogy as discussed above. The Schmidt numbers for the 1-2 binary pair is computed first... 

The multicomponent mass transfer coefficients may be evaluated with the help of Eqs. 8.3.30 and 8.3.31 using the binary mass transfer coefficients determined above and the bulk gas-vapor composition. The result of the calculation is the following matrix ... 

Repeat Example 8.8.1 (ternary distillation in a wetted wall column) using the linearized equations to calculate the molar fluxes but following the suggestion given in the second paragraph of Section 8.8.3 for estimating the multicomponent mass transfer coefficients in terms of the binary mass transfer coefficients. 

Binary mass transfer coefficients in the liquid phase where 0.2141... 

Correlations of numbers of transfer units developed for binary systems may be used to compute numbers of transfer units for multicomponent systems as described in Section 12.1.5. An alternative method that follows the ideas put forward by Toor in his development of the linearized theory of mass transfer is to generalize binary correlations by replacing the binary diffusivity with the matrix of Fick diffusion coefficients (in much the same way that we generalized correlations of binary mass transfer coefficients in Section 8.8.2). Let the number of transfer units in a binary system be expressed as... 

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