Mass-transfer coefficient for equimolar counterdiffusion

Mass-transfer coefficient for equimolar counterdiffusion. Generally, we are interested in N, the flux of A relative to stationary coordinates. We can start with the following, which is similar to that for molecular diffusion but the termcj y is added. 

Some clarifications on the mass-transfer coefficients for use in the fiux expressions (3.1.139) are also in order. If the mass-transfer coefficient under convection was measured or correlated under particular conditions of A/r (= NiJ(N + Alsz) or A/az/(A/az + AIbz) ), then either of the btisic relations (3.1.126) or (3.1.134) should be used to determine the value of (T>ABC,/<5g) or (DjsCt/Si) as the case may be. These values are simply the mass-transfer coefficient for equimolar counterdiffusion in the presence of convection. For any other value of Ng, the value of k j, kcj or kg in equation (3.1.139) is obtained by using the bulk flow correction factor... 

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... 

Define and use special mass-transfer coefficients for diffusion of A through stagnant B, and for equimolar counterdiffusion. 

Equations 15.9 and 15.10 are empirical with respect to the dehnition of the mass transfer coefficients, but the form of the equations is based on molecular diffusion theory. Applying the theory to a multi-component mixture where each component has a distinct diffusivity is impractically complex and must rely on diffusivity data for all the components in the mixture. To derive usable equations from the diffusion theory, certain simplifying assumptions must be made. The basis for the derivation of Equations 15.9 and 15.10 is to assume that mass transfer takes place either as equimolar counterdiffusion or as unimolar diffusion under dilute conditions. 

The two situations noted in Chapter 1, equimolar counterdiffusion and diffusion of A through stagnant B, occur so frequently that special mass-transfer coefficients are usually defined for them. These are defined by equations of the form... 

In this case, there is no bulk-motion contribution to the flux, and the flux is related linearly to the concentration difference driving force. Special k -type mass-transfer coefficients are defined specifically for equimolar counterdiffusion as follows ... 

A packed-bed distillation column is used to adiabatically separate a mixture of methanol and water at a total pressure of 1 atm. Methanol—the more volatile of the two components—diffuses from the liquid phase toward the vapor phase, while water diffuses in the opposite direction. Assuming that the molar latent heat of vaporization is similar for the two components, this process is usually modeled as one of equimolar counterdiffusion. At a point in the column, the mass-transfer coefficient is estimated as 1.62 x 10-5 kmol/m2-s-kPa. The gas-phase methanol mole fraction at the interface is 0.707, while at the bulk of the gas it is 0.656. Estimate the methanol flux at that point. 

As will be recalled from transport phenomena texts, it is most useful to define a mass transfer coefficient to describe only the diffusive transport and not the total, comprising diffusive plus convective contributions. The coefficients are identical only for the special case of equimolar counterdiffusion, and this is the value of the coefficient A° that is actually correlated in handbooks. 

The definitions for the mass transfer coefficients can be used to theoretically predict them using the diffiisivity, concentrations, length scales, and fluid flow characteristics, thus rendering the two mass transfer approaches equivalent. This can easily be done in the cases of equimolar counterdiffusion (Maz + A bz = 0) and diffusion of A through a stagnant film (Ab = 0) (Hines and Maddox, 1985, p. 140). Also, the theoretical models of film, penetration, surface renewal, and film penetration have been proposed for the estimation of the mass transfer coefficients at a fluid-fluid interface (Hines and Maddox, 1985, pp. 146-151). 

Finally, mass transfer coefficients can be complicated by diffusion-induced convection normal to the interface. This complication does not exist in dilute solution, just as it does not exist for the dilute diffusion described in Chapter 2. For concentrated solutions, there may be a larger convective flux normal to the interface that disrupts the concentration profiles near the interface. The consequence of this convection, which is like the concentrated diffusion problems in Section 3.3, is that the flux may not double when the concentration difference is doubled. This diffusion-induced convection is the motivation for the last definition in Table 8.2-2, where the interfacial velocity is explicitly included. Fortunately, many transfer-in processes like distillation often approximate equimolar counterdiffusion, so there is little diffusion-induced convection. Also fortunately, many other solutions are dilute, so diffusion induced convection is minor. We will discuss the few cases where it is not minor in Section 9.5. 

---