Effective diffusivity film model

The theory of seaweed formation does not only apply to solidification processes but in fact to the completely different phenomenon of a wettingdewetting transition. To be precise, this applies to the so-called partial wetting scenario, where a thin liquid film may coexist with a dry surface on the same substrate. These equations are equivalent to the one-sided model of diffusional growth with an effective diffusion coefficient which depends on the viscosity and on the thermodynamical properties of the thin film. 

Both questions have been recently addressed via a surface diffusion-reaction model developed and solved to describe the effect of electrochemical promotion on porous conductive catalyst films supported on solid electrolyte supports.23 The model accounts for the migration (backspillover) of promoting anionic, O5, species from the solid electrolyte onto the catalyst surface. The... 

Mooney et al. [70] investigated the effect of pH on the solubility and dissolution of ionizable drugs based on a film model with total component material balances for reactive species, proposed by Olander. McNamara and Amidon [71] developed a convective diffusion model that included the effects of ionization at the solid-liquid surface and irreversible reaction of the dissolved species in the hydrodynamic boundary layer. Jinno et al. [72], and Kasim et al. [73] investigated the combined effects of pH and surfactants on the dissolution of the ionizable, poorly water-soluble BCS Class II weak acid NSAIDs piroxicam and ketoprofen, respectively. 

Figure 9.7 shows concentration profiles schematically for A and B according to the two-film model. Initially, we ignore the presence of the gas film and consider material balances for A and B across a thin strip of width dx in the liquid film at a distance x from the gas-liquid interface. (Since the gas-film mass transfer is in series with combined diffusion and reaction in the liquid film, its effect can be added as a resistance in series.)... 

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... 

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). 

To implement these simulation approaches, the value of the liquid film mass transfer coefficient Kf is required, which for nonporous and porous HPLC particles, can be calculated from literature correlations derived for bath357,400,408 or column models.407,408 For the case with porous particles, the apparent pore liquid mass transfer coefficient Kp can be expressed as an effective pore diffusivity over an average effective diffusion path length, such that... 

A comparison of the film models that ignore diffusional interaction effects (the effective diffusivity methods) with the film models that take multicomponent interaction effects into account (Krishna-Standart (1976), Toor-Stewart-Prober (1964), Krishna, (1979b, c) and Taylor-Smith, 1982). 

In considering very many condenser simulations (not just those reviewed here) we have yet to find an application where the differences between any of the multicomponent film models that account for interaction effects (Krishna-Standart, 1976 Toor-Stewart-Prober, 1964 Krishna, 1979a-d Taylor-Smith, 1982) are significant. There is also very little difference between the turbulent eddy diffusivity model and the film models that use the Chilton-Colburn analogy (Taylor et al., 1986). This result is important because it indicates that the Chilton-Colburn analogy, widely used in design calculations, is unlikely to lead to large... 

Numerical simulations of Sardesai s experiments are discussed by Webb and Sardesai (1981) and Webb (1982) (who used the Krishna-Standart (1976), Toor-Stewart-Prober (1964) and effective diffusivity methods to calculate the condensation rates), McNaught (1983a, b) (who used the equilibrium model of Silver, 1947), and Furno et al. (1986) (who used the turbulent diffusion models of Chapter 10 in addition to methods based on film theory). It is the results of the last named that are presented here. 

Develop the film model for simultaneous mass and energy transfer including Soret and Dufour effects. Use the Toor-Stewart-Prober linearized theory in developing the model. An example of a process where thermal diffusion effects cannot be ignored is chemical vapor deposition. Use the model to perform some sample calculations for a system of practical interest. You will have to search the literature to find practical systems. To get an idea of the numerical values of the transport coefficients consult the book by Rosner (1986). 

Frey, D. D., Prediction of Liquid Phase Mass Transfer Coefficients in Multicomponent Ion Exchange Comparison of Matrix, Film-Model, and Effective Diffusivity Methods, Chem. Eng. Commun., 41, 273-293 (1986). 

Fluid-fluid reactions are reactions that occur between two reactants where each of them is in a different phase. The two phases can be either gas and liquid or two immiscible liquids. In either case, one reactant is transferred to the interface between the phases and absorbed in the other phase, where the chemical reaction takes place. The reaction and the transport of the reactant are usually described by the two-film model, shown schematically in Figure 1.6. Consider reactant A is in phase I, reactant B is in phase II, and the reaction occurs in phase II. The overall rate of the reaction depends on the following factors (i) the rate at which reactant A is transferred to the interface, (ii) the solubihty of reactant A in phase II, (iii) the diffusion rate of the reactant A in phase II, (iv) the reaction rate, and (v) the diffusion rate of reactant B in phase II. Different situations may develop, depending on the relative magnitude of these factors, and on the form of the rate expression of the chemical reaction. To discern the effect of reactant transport and the reaction rate, a reaction modulus is usually used. Commonly, the transport flux of reactant A in phase II is described in two ways (i) by a diffusion equation (Pick s law) and/or (ii) a mass-transfer coefficient (transport through a film resistance) [7,9]. The dimensionless modulus is called the Hatta number (sometimes it is also referred to as the Damkohler number), and it is defined by... 

A detailed mass balance for benzene as well as thiophene over a spherical catalyst particle is given elsewhere (Frycek, 1984). Further simplifications are made here in order to minimize the complexity of model as a whole. Thus, effectiveness factor, ry, was defined for benzene while assuming the transport parameters such as D, effective diffusivity, or kg, film mass transfer coefficient, to be of equal magnitude for benzene and thiophene (Lee and Butt, 1982). The simplified t) is presented as follows ... 

Non-catalytic reactions involving two phases are common in the mineral industry. Reactions such as the roasting of ores or the oxidation of solids are carried out on a massive scale but the rates of these processes are often controlled by physical, not chemical, effects. Reactant or product diffusion is the main rate controlling factor in many cases. As a result, mechanisms of reaction become models of reaction with consideration of factors such as external diffusion film control or the shrinking core yielding the various models. Matters are further complicated by considerations regarding particle shape and external fluid flow regimes. 

In 1958, Toor and Marcello investigated the effect of removing the short residence time constraint in the film that is implicit in penetration models, so that the film model would become a limiting case of the penetration model. Not unexpectedly, the resulting model predicts a dependence on diffusivity of the ki or kg value from D to D ... 

As in any solid-liquid reaction, when the solid is sparingly soluble, reaction occurs within the solid by diffusion of the liquid-phase reactant into it across the liquid film surrounding the solid. Thus two diffusion parameters are operative, the solid-liquid mass transfer coefficient sl and the effective diffusivity D. of the reactant in the solid. A reaction in the solid can occur by any of several mechanisms. The simpler and more common of these were briefly explained in Chapter 15. For reactions following the sharp interface model, ultrasound can enhance either or both these constants. Indeed, in a typical solid-liquid reaction such as the synthesis of dibenzyl sulfide from benzyl chloride and sodium sulfide ultrasound enhances SL by a factor of 2 and by a factor of 3.3 (Hagenson and Doraiswamy, 1998). Similar enhancement in was found for a Michael addition reaction (Ratoarinoro et al., 1995) and for another mass transfer-limited reaction (Worsley and Mills, 1996). 

Along these lines, the vapor-Hquid mass transfer is modeled as a combination of the two-film model presentation and the Maxwell-Stefan diffusion description. In this stage model, the equilibrium exists only at the interface. A reasonable simplification for RD is represented by the effective diffusivUy approach, provided that the effective diffusion coefficients are estimated properly. These coefficients can be obtained, for instance, via a relevant averaging of the Maxwell-Stefan diffusivities [42]. 

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... 

This model has several limitations. The film model assumes that mass transfer is controlled by the liquid-phase film, which is often not the case because the interface characteristics can be the limiting factor (Linek et al., 2005a). The liquid film thickness and diffusivity may not be constant over the bubble surface or swarm of bubbles. Experiments also indicate that mass transfer does not have a linear dependence on diffusivity. Azbel (1981) indicates that others have shown that turbulence can have such a significant effect on mass transfer such that eddy turbulence becomes the controlling mechanism in which diffusivity does not play a role. In most instances, however, eddy turbulence and diffusivity combine to play a significant role in mass transfer (Azbel, 1981). 

The border diffusion layer model was introduced as an amendment to the film model to present a more realistic description. It accounts for an undefined film thickness, turbulence effects, and the role of molecular diffusion. When the flow is turbulent, the flow around the bubble is split into four sections the main turbulent stream, the turbulent boundary layer, the viscous sublayer, and the diffusion sublayer. Eddy turbulence accounts for mass transfer in the main turbulent stream and the turbulent boundary layer. The viscous sublayer limits eddy turbulence effects so that the flow is laminar and mass transfer is controlled by both molecular diffusion and eddy turbulence. Microscale eddy turbulence is assumed to be dominant in the viscous sublayer. Mass transfer in the diffusion sublayer is controlled almost completely by molecular diffusion (Azbel, 1981). 

Film model is undoubtedly the most widespread approach for the rate-based mass and heat transfer through an interface (Wesselingh and Krishna, 2000). It effectively combines species diffusion and fluid flows and is based on the assumption that the resistance to mass and heat transfer is exclusively concentrated in a thin film where steady state diffusion and mass and heat convection take place (figure 2.5). 

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