Film Model for Binary Mass Transfer

Let us consider a planar film between the position coordinates r = Tq and r = r. Mass transfer between the two edges of the film occurs purely by molecular diffusion under steady-state conditions. The thickness of the film is — Tq. The equation of continuity 

The generalized Maxwell-Stefan diffusion equations (Eq. 2.2.1) simplify to 

The linear differential equation (Eq. 8.2.7), can be integrated to give the composition profiles 

Comparison of Eqs. 8.2.10 and 8.2.11 with the basic definition of the low flux mass transfer coefficient (Eq. 7.1.3), shows 

The flux can be calculated by multiplying the diffusion flux by the appropriate bootstrap coefficient 

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Using a steady-state film model, obtain an expression for the mass transfer rate across a laminar film of thickness /. in the vapour phase for the more volatile component in a binary distillation process ... 

Fast and satisfactory mass transfer calculations are necessary since we may have to repeat such calculations many times for a rate-based distillation column model or two-phase flow with mass transfer between the phases in the design and simulation process. The generalized matrix method may be used for multicomponent mass transfer calculations. The generalized matrix method utilizes the Maxwell-Stefan model with the linearized film model for diffusion flux, assuming a constant diffusion coefficient matrix and total concentration in the diffusion region. In an isotropic medium, Fick s law may describe the multicomponent molecular mass transfer at a specified temperature and pressure, assuming independent diffusion of the species in a fluid mixture. Such independent diffusion, however, is only an approximation in the following cases (i) diffusion of a dilute component in a solvent, (ii) diffusion of various components with identical diffusion properties, and (iii) diffusion in a binary mixture. 

Single Sphere Model II (Equations 4, 5, 8, 9 and 10 in reference 6) In this model allowance is made for the resistance to mass transfer offered by the surface film surrounding the herb particles. The mass transfer coefficient kf was obtained from correlations proposed by Catchpole et al (8, 9) for mass transfer and diffusion into near-critical fluids. An average of the binary diffusivities of the major essential oil components present was used in calculating kf (these diffusivities were all rather similar because of their similar structures). 

A more comprehensive analysis of constant pattern behavior for a binary system has been given by Rhee and Amundson [3]. In this work, these authors have extended to binary systems the analysis of the combined effects of mass transfer resistance and axial dispersion that they had previously made in the case of single-component bands [4]. Rhee and Amundson [3] assumed the solid film linear driving force model, finite axial dispersion, and no particular isotherm model. The system of equations becomes... 

In Chapter 7 we define mass transfer coefficients for binary and multicomponent systems. In subsequent chapters we develop mass transfer models to determine these coefficients. Many different models have been proposed over the years. The oldest and simplest model is the film model this is the most useful model for describing multicomponent mass transfer (Chapter 8). Empirical methods are also considered. Following our discussions of film theory, we describe the so-called surface renewal or penetration models of mass transfer (Chapter 9) and go on to develop turbulent eddy diffusivity based models (Chapter 10). Simultaneous mass and energy transport is considered in Chapter 11. 

The explicit method of Taylor and Smith (1982) for mass transfer in ideal gas mixtures is an exact solution of the Maxwell-Stefan equations for two component systems where all matrices are of order 1. Does the generalized explicit method derived in Exercise 8.40 reduce to the expressions given in Section 8.2 for a film model of mass transfer in binary systems ... 

He et al used a binary mixture-based film model to perform a theoretical analysis on the concentration polarization in a generic membrane. They defined a concentration polarization coefficient for both the two species involved in the separation as the ratio of the actual flux to the ideal one (without polarization), quantifying the polarization effect by means of the ratio of the actual fluxes of the components. Although this is a simplified approach that cannot be generalized to multi-component systems, nevertheless, under some operating conditions, the authors predicted a significant influence of the external mass transfer on the process. 

Combined solution-diffusion film theory models have been presented already in several publications on aqueous systems however, either 100% rejection of the solute is assumed or detailed experimental flux and rejection results are required in order to find parameters by nonlinear parameter estimation (Murthy and Gupta, 1997). Consequently, it is difficult to apply these models for predictive purposes. Peeva et al. (2004) presented the first consideration of concentration polarization in OSN. They coupled the solution-diffusion membrane transport model, Eq. (16.4), with film theory to describe flux and rejection of toluene/ docosane and tolune/TOABr binary mixtures. This approach was able to integrate concentration polarization and nonideal solution behavior into OSN design models and predict fluxes over a wide range of solvent mixtures from a limited data set of the pure solvent fluxes. The only parameters to be estimated, other than physical properties, are the mass transfer coefficients, which may be measured, and the permeabUilies, which may... 

Propose a model for this system in the form of a BVP, specifying both the set of PDEs and the appropriate boundary conditions. Assiune a film of thickness b = I mm, length L = 50 cm, inclined at 0 = 80°. Use effective binary diffusivities in the film of Dj = 10 cm /s, j = A, B, AB. Compute the steady-state concentration profile of each species within the film and the average absorption rate per unit area. Then, decrease the rate constant to zero to see what the mass transfer rate would be without reaction. 

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