Modelling interfacial diffusion

The interfacial diffusion model of Scott, Tung, and Drickamer is somewhat open to criticism in that it does not take into account the finite thickness of the interface. This objection led Auer and Murbach (A4) to consider a three-region model for the diffusion between two immiscible phases, the third region being an interface of finite thickness. These authors have solved the diffusion equations for their model for several special cases their solutions should be of interest in future analysis of interphase mass transfer experiments. 

Fig. 15 Two of the simplest theories for the dissolution of solids (A) the interfacial barrier model, and (B) the diffusion layer model, in the simple form of Nemst [105] and Brunner [106] (dashed trace) and in the more exact form of Levich [104] (solid trace). c is the concentration of the dissolving solid, cs is the solubility, cb is the concentration in the bulk solution, and x is the distance from the solid-liquid interface of thickness h or 8, depending on how it is defined. (Reproduced with permission of the copyright owner, John Wiley and Sons, Inc., from Ref. 1, p. 478.)...

The hydraulic permeation model predicts highly nonlinear water content profiles, with strong dehydration arising only in the interfacial regions close to the anode. Severe dehydration occurs only at current densities closely approaching/p,. The hydraulic permeation model is consistent with experimental data on water content profiles and differential membrane resistance, i i as corroborated in Eikerling et al. The bare diffusion models exhibit marked discrepancies in comparison with these data. 

E. Adsorption-diffusion model of interfacial mass transfer 201... 

E. Adsorption-Diffusion Model of Interfacial Mass Transfer... 

FIGURE 17.1 (a) Diffusion-layer model of dissolution, (b) Interfacial barrier model of dissolution. 

Two of the simplest theories to explain the dissolution rate of solutes are the interfacial barrier model and the diffusion-layer model (Figures 17.1 and 17.2). Both of these theories make the following two assumptions ... 

DESIGNER also contains different hydrodynamic models (e.g., completely mixed liquid-completely mixed vapor, completely mixed liquid-vapor plug flow, mixed pool model, eddy diffusion model) and a model library of hydrodynamic correlations for the mass transfer coefficients, interfacial area, pressure drop, holdup, weeping, and entrainment that cover a number of different column internals and flow conditions. 

Figure 5.2 Schematic representation of the dissolution mechanisms according to (A) the diffusion layer model, and (B) the interfacial barrier model.

In the interfacial barrier model of dissolution it is assumed that the reaction at the solid-liquid interface is not rapid due to the high free energy of activation requirement and therefore the reaction becomes the rate-limiting step for the dissolution process (Figure 5.1), thus, drug dissolution is considered as a reaction-limited process for the interfacial barrier model. Although the diffusion layer model enjoys widespread acceptance since it provides a rather simplistic interpretation of dissolution with a well-defined mathematical description, the interfacial barrier model is not widely used because of the lack of a physically-based mathematical description. 

The evolution in the relative interfacial concentration obtained with these diffusion models is shown in Figure 2 for short and long times on a linear and logarithmic scale,... 

Figure 2 Relative increase in the interfacial concentrations with time due to different diffusion models for the growth conditions indicated in the figures. Left short times right long times on a log-scale. The dashed line is the exact solution from referencc . ...

Figure 3 Critical time (left figure) and relative interfacial concentration (right figure) at the onset of convection obtain by combining equation (10) for a rigid boundary with the diffusion model (8) for different solution salinities. The square is a laboratory observation reported by Foster at 25 %o.

It is further noted that the use of interfacial mass flux weighted transfer terms is generally not convenient treating multicomponent reactive systems, because the phase change processes are normally not modeled explicitly but deduced from the species composition dependent joint diffusive and convective interfacial transfer models. Moreover, the rigorous reaction kinetics and thermodynamic models of mixtures are always formulated on a molar basis. 

For solids with continuous pores, a surface tension driven flow (capillary flow) may occur as a result of capillary forces caused by the interfacial tension between the water and the solid particles. In the simplest model, a modified form of the Poiseuille flow can be used in conjunction with the capillary forces equation to estimate the rate of drying. Geankoplis (1993) has shown that such a model predicts the drying rate in the falling rate period to be proportional to the free moisture content in the solid. At low solid moisture contents, however, the diffusion model may be more appropriate. 

A rather important aspect that should be considered is that interfacial quenching of dyes does not necessarily imply an electron-transfer step. Indeed, many photochemical reactions involving anthracene occur via energy transfer rather than ET [128]. A way to discern between both kinds of mechanisms is via monitoring the accumulation of photoproducts at the interface. For instance, heterogeneous quenching of water-soluble porphyrins by TCNQ at the water-toluene interface showed a clear accumulation of the radical TCNQ under illumination [129]. This system was also analyzed within the framework of the excited-state diffusion model where time-resolved absorption of the porphyrin triplet state provided a quenching rate constant of the order of 92 M-1 cm s 1. 

In experiments on nonionic surfactants, namely Triton X-405 Geeraerts at al. (1993) performed simultaneously dynamic surface tension and potential measurements in order to discuss peculiarities of nonionic surfactants containing oxethylene chains of different lengths as hydrophilic part. Deviations from a diffusion controlled adsorption were explained by dipole relaxations. In recent papers by Fainerman et al. (1994b, c, d) and Fainerman Miller (1994a, b) developed a new model to explain the adsorption kinetics of a series of Triton X molecules with 4 to 40 oxethylene groups. This model assumes two different orientations of the nonionic molecule and explains the observed deviations of the experimental data from a pure diffusion controlled adsorption very well. Measurements in a wide temperature interval and in presence of salts known as structure breaker were performed which supported the new idea of different molecular interfacial orientations. At small concentration and short adsorption times the kinetics can be described by a usual diffusion model. Experiments of Liggieri et al. (1994) on Triton X-100 at the hexane/water interface show the same results. 

These two models illustrate how the properties of the compound influence the rate of evaporation from water under static conditions. Environmental conditions such as wind speed and turbulence in the water phase will have a marked influence on rates of evaporation that would reduce gradients and also reduce the width of the interfacial diffusion layers and systematic analysis of these effects have been discussed. Other variables will affect evaporation rates by controlling the actual concentration of the compound in solution. Suspended sediments and/or DOM would act in this manner. Weak acids and bases would only evaporate as the neutral species since the complementary anions or cations would be more water soluble and essentially have no vapor pressure. Consequently, environmental pH relative to pA values will be a consideration. It should be mentioned that compounds may distribute into the vapor phase by other processes than evaporation. Formation of aerosols, for example, can be a factor. 

In the case of composite laminates, new problems linked to the anisotropy of diffusion paths, the eventual role of interfacial diffusion and the role of pre-existing or swelling-induced damage appeared in the mid-1970s. The interest was mainly focused on the effect of humidity on carbon fibre/amine crosslinked epoxy composites of aeronautical interest. For the pioneers of this research (Shen and Springer, 1976), the determination of diffusion kinetic laws appeared as the key objective. Various studies revealed that, in certain cases, diffusion in composites caimot be modelled by a simple Pick s law and that Langmuir s equation is more appropriate. Carter and Kibler (1978) proposed a method for the parameter identification. At the end of the 1970s, the kinetic analysis of water diffusion into composites became a worldwide research objective. Related experimental results can be summarized as follows. 

This model divides the slurry bed reactor into several CSTR. The model can be regarded as a quasi axial diffusion model, and the idea of the model was that the interfacial resistance between the gas and liquid can be omitted fluid particle was considered weU-distributed the exit of pre-CSTR is the entrance of the latter stage CSTR. The model is shown in Eqs. (24) and (25) ... 

The profiles of concentration and velocity of the gas-liquid interfacial diffusion process can be obtained by the simultaneous solution of the model equations. 

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