Film Theory Model

Z.V.P. Murphy, S.K. Gupta, Estimation of mass transfer coefficient using a combined nonlinear membrane transport and film theory model, Desalination 109 (1997) 39-49. 

In the first part of this paper, we therefore generalize the analysis of the above-mentioned reaction system. An approximate reaction factor expression is derived without restriction on the reaction regime, and the accuracy of this factor is tested by comparison with numerical solutions of the film-theory model. 

Approximate vs. Numerical Solution. The accuracy of the approximate reaction factor expression has been tested over wide ranges of parameter values by comparison with numerical solutions of the film-theory model. The methods of orthogonal collocation and orthogonal collocation on finite elements (7,8) were used to obtain the numerical solutions (details are given by Shaikh and Varma (j>)). Comparisons indicate that deviations in the approximate factor are within few percents (< 5%). It should be mentioned that for relatively high values of Hatta number (M >20), the asymptotic form of Equation 7 was used in those comparisons. 

Solution The film-theory model described earlier in the chapter can be applied to the solution of the above problem. The solution for the concentration profiles, as developed by Ramchandran and Sharma,139 is... 

SOLUTION A schematic diagram of the temperature and composition profiles in this process was given in Example 8.3.2, where the mass transfer part of the problem was solved. All we have to do here is compute the heat flux from the film theory model in Section 11.4.1. 

In 1924 Lewis and Whitman 1 suggested that the film theory model could be applied to both die gas and liquid phases during gas absorption. This two-film theory has hed extensive use in modeling steady-state transport between two phases. Transferor species A occurring between a gas phase and liquid phase, each of which may be in turbulent flow, can be described by the individual rate expressions bstween the bulk of each phase and the interface. 

Concentration Polarisation is the accumulation of solute due to solvent convection through the membrane and was first documented by Sherwood (1965). It appears in every pressure dri en membrane process, but depending on the rejected species, to a very different extent. It reduces permeate flux, either via an increased osmotic pressure on the feed side, or the formation of a cake or gel layer on the membrane surface. Concentration polarisation creates a high solute concentration at the membrane surface compared to the bulk solution. This creates a back diffusion of solute from the membrane which is assumed to be in equilibrium with the convective transport. At the membrane, a laminar boundary layer exists (Nernst type layer), with mass conservation through this layer described by the Film Theory Model in equation (3.7) (Staude (1992)). cf is the feed concentration, Ds the solute diffusivity, cbj, the solute concentration in the boundary layer and x die distance from the membrane. 

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 [38], or detailed experimental flux and rejection results are required in order to find parameters by nonlinear parameter estimation [43, 44]. Consequently, it is difficult to apply these models for predictive purposes. In OSN, it is also important to account for the effect of different activities of the species on both sides of the membrane. We have proposed a set of equations [32], Eqs. (7) to (13), taking these factors into account We assume a binary system, although the equations could be generalized for a system of n components. In this analysis component 1 is the solute and component 2 is the solvent. The only parameters to be estimated, other than physical properties, are... 

The combined solution-diffusion film theory model (Eqs. (7)-(13)) is used to describe the experimental results. The equations were solved numerically. 

Film Theory ModeL There will be convective mass transfer of solute to the wall of the membrane, vsbich is balanced by divisional transport away from the membrane owing to a concentration gradient. At steady state, assuming the diOudon coefficient D to be constant ... 

Fortunately, however, although both k and kj are strongly influenced by the details of the fluid mechanics involved, their ratio I turns out to be almost independent of it. Therefore, the theory of coupled mass transfer and chemical reaction can be developed on the basis of very crude models of the fluid mechanics involved. The film theory model will be used throughout this lecture. 

The simplest hydrodynamic model proposed in the literature is the film theory model.This assumes the existence, near the gas-liquid interface, of a stagnant film of thickness 6, through which mass transfer can only take place by molecular diffusion. 

Eq.l2 does not have predictive value, since the value of the film thickness 6 (into which the whole ignorance about the true fluid mechanics has been lumped) is not known. However, if a problem of mass transfer with chemical reaction is analyzed on the basis of the film theory model, the value of I will usually turn out to depend on 6, and Eq.l2 can then be used to express... 

The film theory has an important drawback. Although, the value of 6 is not known, one should regard it as uniquely dete mined by the hydrodynamics of the liquid phase. On the basis, Eq.l2 would predict k to be proportional to the diffusivity D. Empiri cal mass transfer coefficient correlations available in the lit rature for a liquid in contact with a gas consistently indicate that in fact k is proportional to the square root of D. Therefore, analyses based on the film theory model are not expected to predict correctly the influence of diffusivity values on the enhancement factor I. Therefore, one is lead to a more complex model of the fluid mechanics involved, the penetration theory model. This model leads, in its several variations, to the correct prediction of the... 

Eq.l6 implies that the rate of reaction is not constant throughout the liquid, since its value depends on the local comp sition of the liquid phase. In the presence of the chemical reaction, the diffusion equation for component A becomes, for the film-theory model. 

An analysis of chemical desorption has recently been published (Chem.Eng.Sci., 21 0980)), which is based on a number of simplifying assumptions the film theory model is assumed, the diffusivities of all species are taken to be equal to each other, and in the solution of the differential equations an approximation which is second order with respect to distance from the gas-liquid interface is used this approximation was introduced as early as 1948 by Van Krevelen and Hoftizer. However, the assumptions listed above are not at all drastic, and two crucial elements are kept in the analysis reversibility of the chemical reactions and arbitrary chemical mechanisms and stoichiometry.The result is a methodology for developing, for any given chemical mechanism, a highly nonlinear, implicit, but algebraic equation for the calculation of the rate enhancement factor as a function of temperature, bulk-liquid composition, interface gas partial pressure and physical mass transfer coefficient The method of solution is easily gene ralized to the case of unequal diffusivities and corrections for differences between the film theory and the penetration theory models can be calculated. 

Four experiments were caried out at different stirrer speeds under conditions as given in tad le 2, Mole fluxes were obtained from measured concentration curves and plotted in figure 8 together with calculated fluxes for both penetration and film theory models. The latter model was derived from our numerical penetration model by zeroing time derivatives and equalizing the penetration depth to the film thickness. 

A significant criticism of the above model is that the high mass transfer flux should influence the effective film thickness, yet no allowance has been made for this in the film theory model. A boundary layer model to be described shortly allows for the effect of the high flux on hydrodynamics. 

As solution passes through a membrane, accumulation of solute at the membrane surface occurs as a result of one or more possible mechanisms, i.e., partial or total size exclusion of solute molecules from pores, electrostatic repulsion of solute molecules by a membrane, chemical reaction, adsorption of solute molecules, etc. The configuration of the eoncentrated solute region contiguous to the solution-— membrane interfaces makes it amenable to analysis by the film-theory model [132]. 

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