The Film Model

In the film model we assume that all the resistance to mass and heat transfer is concentrated in a thin film and that transfer occurs within this film by steady-state diffusion and 

For steady-state heat transfer within a planar film, the energy balance relation (Eq. 11.1.1) simplifies to 

If the reference state for the calculation of the partial molar enthalpies is taken to be the pure component at temperature 7 ref, then we may write 

Equation 11.4.3 is to be solved subject to the boundary conditions of a film model 

In proceeding further, it is convenient to define a heat transfer rate factor 

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The predictions of correlations based on the film model often are nearly identical to predictions based on the penetration and surface-renewal models. Thus, in view of its relative simphcity, the film model normally is preferred for purposes of discussion or calculation. It should be noted that none of these theoretical models has proved adequate for maldug a priori predictions of mass-transfer rates in packed towers, and therefore empirical correlations such as those outlined later in Table 5-28. must be employed. 

Effects of High Solute Concentrations on Ug and As discussed previously, the stagnant-film model indicates that fcc should be independent of ysM and/cc should be inversely proportional to The data of Vivian and Behrman [Am. Tn.st. Chem. Eng. J., 11, 656 (1965)] for the absorption of ammonia from an inert gas strongly suggest that the film model s predicted trend is correct. This is another indication that the most appropriate rate coefficient to use is fcc. nd the proper driving-force term is of the form (y — yd ysM-... 

A more rigorous seheme of gas-liquid ehemieal reaetion and absorption followed by preeipitation is deseribed based on the film model (Englezos etai, 1987a,b Waehi and Jones, 1990 1991a Skovborg and Rasmussen, 1994). The eoneept is illustrated in Figure 8.13. 

Jss Flux obtained experimentally or from the film-model... 

In a process where mass transfer takes place across a phase boundary, the same theoretical approach can be applied to each of the phases, though it does not follow that the same theory is best applied to both phases. For example, the film model might be applicable to one phase and the penetration model to the other. This problem is discussed in the previous section. 

Surface Renewal Theory. The film model for interphase mass transfer envisions a stagnant film of liquid adjacent to the interface. A similar film may also exist on the gas side. These h5q>othetical films act like membranes and cause diffu-sional resistances to mass transfer. The concentration on the gas side of the liquid film is a that on the bulk liquid side is af, and concentrations within the film are governed by one-dimensional, steady-state diffusion ... 

It has been observed that under reaction conditions mass transfer is often significantly faster than would be expected based on the film model. This is modelled by introducing an enhancement factor, E. In case the concentration in the bulk liquid, ca, is zero, the rate of mass transfer of A now becomes ... 

The explanation of the pressure-independent region during the ultrafiltration of macromolecules requires the arbitrary introduction of the concept of a gel-layer in the film model. A more complete description of the dependence of the membrane permeation rate on the applied pressure may be given by considering the effect of the osmotic pressure of the macromolecules as described by Wijmans et alS18 Equation 8.2 may then be written as ... 

Reactions carried in aqueous multiphase catalysis are accompanied by mass transport steps at the L/L- as well as at the G/L-interface followed by chemical reaction, presumably within the bulk of the catalyst phase. Therefore an evaluation of mass transport rates in relation to the reaction rate is an essential task in order to gain a realistic mathematic expression for the overall reaction rate. Since the volume hold-ups of the liquid phases are the same and water exhibits a higher surface tension, it is obvious that the organic and gas phases are dispersed in the aqueous phase. In terms of the film model there are laminar boundary layers on both sides of an interphase where transport of the substrates takes place due to concentration gradients by diffusion. The overall transport coefficient /cl can then be calculated based on the resistances on both sides of the interphase (Eq. 1) ... 

The mass and heat transport model should be able to predict mass and energy fluxes through a gas/vapour-liquid interface in case a chemical reaction occurs in the liquid phase. In this study the film model will be adopted which postulates the existence of a well-mixed bulk and a stagnant transfer zone near the interface (see Fig. 1). The equations describing the mass and heat fluxes play an important role in our model and will be presented subsequently. 

The film model referred to in Chapters 2 and 5 provides, in fact, an oversimplified picture of what happens in the vicinity of interface. On the basis of the film model proposed by Nernst in 1904, Whitman [2] proposed in 1923 the two-film theory of gas absorption. Although this is a very useful concept, it is impossible to predict the individual (film) coefficient of mass transfer, unless the thickness of the laminar sublayer is known. According to this theory, the mass transfer rate should be proportional to the diffusivity, and inversely proportional to the thickness of the laminar film. However, as we usually do not know the thickness of the laminar film, a convenient concept of the effective film thickness has been assumed (as... 

Compared to the film model or the penetration model, the surface renewal approach seems closer to reality in such a case where the surface of liquid in an... 

It can be seen that a theoretical prediction of values is not possible by any of the three above-described models, because none of the three parameters - the laminar film thickness in the film model, the contact time in the penetration model, and the fractional surface renewal rate in the surface renewal model - is predictable in general. It is for this reason that the empirical correlations must normally be used for the predictions of individual coefficients of mass transfer. Experimentally obtained values of the exponent on diffusivity are usually between 0.5 and 1.0. 

The first model, the film model by Whitman (1923), depicted the interface as a (single- or two-layer) bottleneck boundary. Although many aspects of this model are outdated in light of our improved knowledge of the physical processes occurring at the interface, its mathematical simplicity keeps the model popular. 

Figure 20.5 gives an overview of the basic ideas behind these three models. The upper picture shows the situation as depicted in the film model and the boundary... 

In the film model the air-water interface is described as a one- or two-layer bottleneck boundary of thicknesses 5a and 8W, respectively. Thus, according to Eq. 19-9 ... 

To understand the principal idea of Deacon s model we have to remember the key assumption of the film model according to which a bottleneck boundary is described by an abrupt drop of diffusivity, for instance, from turbulent to molecular conditions (see Fig. 19.3a). Yet, theories on turbulence at a boundary derived from fluid dynamics show that this drop is gradual and that the thickness of the transition zone from fully turbulent to molecular conditions depends on the viscosity of the fluid. In Whitman s film model this effect is incorporated in the film thicknesses, 8a and 8W (Eq. 20-17). In addition, the film thickness depends on the intensity of turbulent kinetic energy production at the interface as, for instance, demonstrated by the relationship between wind velocity and exchange velocity (Figs. 20.2 and 20.3). 

Therefore, instead of Sc/a we use the diffusivity ratios to compare v,a of different substances. According to the empirical observations of Mackay and Yeun (1983), the appropriate exponent is 2/3. That is, it lies between the film model and the surface replacement model ... 

Let us again exemplify the theory using the film model. For simplicity, we consider one (water-phase) film (see Fig. 20-12). We adopt the notation introduced in Eq. 12-16, where [A] stands for the aldehyde and [D] for the diol concentration (the hydrated aldehyde), and assume that only the aldehyde diffuses into the atmosphere, and that in the aqueous mixed layer the two species are in equilibrium ... 

In the film model of Whitman the water-phase exchange velocity, v,w, is a function of the molecular diffusion coefficient of the chemical, while in Deacon s boundary layer model v[W depends on the Schmidt Number Sc W. Explain the reason for this difference. 

Hint The process of heat exchange across an interface can be treated in the same way as the exchange of a chemical at the interface. To do so, we must express the molecular thermal heat conductivity by a molecular diffusivity of heat in water and air, Z)thw and Z)tha, respectively. At 20°C, we have (see Appendix B) flthw = 1.43 xl(T3 cm2 s-1, Dlh a = 0.216 cm2 s 1. Use the film model for air-water exchange with the typical film thicknesses of Eq. 20-18a. 

A more general theoretical approach for dissolution modeling called the Film-Model Theory was postulated by Nernst (1904) and expanded upon by Brunner (1904) in an effort to deconvo-lute the components of the dissolution constinIBoth Nernst and Brunner made the following assumptions ... 

Figure 4.2 Fluid flow velocity through the channel of a membrane module is nonuniform, being fastest in the middle and essentially zero adjacent to the membrane. In the film model of concentration polarization, concentration gradients formed due to transport through the membrane are assumed to be confined to the laminar boundary layer...

Figure 4.4 Salt concentration gradients adjacent to a reverse osmosis desalination membrane. The mass balance equation for solute flux across the boundary layer is the basis of the film model description of concentration polarization...

Kenig EY, Butzmann F, Kucka L, Gorak A. Comparison of numerical and analytical solutions of a multicomponent reaction-mass transfer problem in terms of the film model. Chem Eng Sci 2000 55 1483-1496. 

The component fluxes N entering into Eqs. (A1)-(A3) are determined based on the mass transport in the film region. Because the key assumptions of the film model result in the one-dimensional mass transport normal to the interface, the differential component balance equations including simultaneous mass transfer and reaction in the film are as follows ... 

The latter strongly depends on the specific reaction mechanism, the stoichiometry, and the presence or absence of parallel reaction schemes (69). The rate expressions for Rt usually represent nonlinear dependences on the mixture composition and temperature. Specifically for the coupled reaction-mass transfer problems, such as Eqs. (A10), it is always essential as to whether or not the reaction rate is comparable to that of diffusion (68,77). Equations (A10) should be completed by the boundary conditions relevant to the film model. These conditions specify the values of the mixture composition at both film boundaries. For example, for the liquid phase ... 

In rate-based multistage separation models, separate balance equations are written for each distinct phase, and mass and heat transfer resistances are considered according to the two-film theory with explicit calculation of interfacial fluxes and film discretization for non-homogeneous film layer. The film model equations are combined with relevant diffusion and reaction kinetics and account for the specific features of electrolyte solution chemistry, electrolyte thermodynamics, and electroneutrality in the liquid phase. 

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