Permeation Modeling

In order to solve the MDPE in Equation 9.1, it is necessary to understand how y is related to x. There are numerous methods of describing this relationship. Gas separation involves the diffusion of a gaseous mixture through the membrane material. The rate at which a gas, or particular component of a gas mixture, moves through a membrane is known as the flux (total or individual). In order to obtain a general flux model for a gas separation membrane, the fundamental thermodynamic approach incorporating the solution-diffusion model [11] will be used at vacuum permeate conditions (itp 0)  

P is the permeability of component i (which is equal to the product of solubility and diffusivity through the membrane material) itR is the (high) retentate pressure (Pa), 

APPLICATION OF COLUMN PROFILE MAPS TO ALTERNATIVE SEPARATION PROCESSES 

Since the flux of any component is the amount of material that permeates through the membrane per unit time per unit area, one may also write 

It can be shown that by combining Equations 9.4 and 9.5, for a gas separation membrane 

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In the main, to this point, the permeation model represented by Eq. (9) can be considered esoteric. The value of the model to practicing pharmacists stems substantially from what the model teaches us about barrier damage. To measure the result of extreme barrier damage, one only needs to use zero for... 

Perhaps the simplest Fick s law permeation model consists of two aqueous compartments, separated by a very thin, pore-free, oily membrane, where the unstirred water layer may be disregarded and the solute is assumed to be negligibly retained in the membrane. At the start (t = 0 s), the sample of concentration CD 0), in mol/cm3 units, is placed into the donor compartment, containing a volume (Vo, in cm3 units) of a buffer solution. The membrane (area A, in cm2 units) separates the donor compartment from the acceptor compartment. The acceptor compartment also contains a volume of buffer (VA, in cm3 units). After a permeation time, t (in seconds), the experiment is stopped. The concentrations in the acceptor and donor compartments, CA(t) and C (t), respectively, are determined. 

Wilschut A, Ten Berge WF, Robinson PJ, McKone TE (1995) Estimating skin permeation. The validation of five mathematical skin permeation models. Chemosphere 30 1275-1296. 

An external gas pressure gradient applied between anode and cathode sides of the fuel cell may be superimposed on the internal gradient in liquid pressure. This provides a means to control the water distribution in PEMs under fuel cell operation. This picture forms the basis for the hydraulic permeation model of membrane operation that has been proposed by Eikerling et al. This basic structural approach can be rationalized on the basis of the cluster network model. It can also be adapted to include the pertinent structural pictures of Gebel et and Schmidt-Rohr et al. ... 

The hydraulic permeation model in Eikerling ef al. helped rationalize main dependence of fhe critical currenf densify on membrane parameters. A sharply peaked 5-function-like pore size disfribufion. 

The theoretical analysis of the hydraulic permeation model, moreover, provided an expression for the current density, at which membrane dehydra-... 

The diffusion model and the hydraulic permeation model differ decisively in their predictions of water content profiles and critical current densities. The origin of this discrepancy is the difference in the functions D (T) and /Cp (T). This point was illustrated in Eikerling et al., where both flux terms occurring in Equation (6.46) were converted into flux terms with gradients in water content (i.e., VA) as the driving force and effective transport coefficients for diffusion, A), and hydraulic permeation,... 

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. 

Recently, it was shown that the hydraulic permeation model could explain the response of the membrane performance to variations in external gas pressures in operating fuel cells. i Figure 6.15 shows data for the PEM resistance in an operational PEFC,... 

PEM resistance in operational PEFC as a function of the fuel cell current density, comparing experimental data (dots) and calculated results from a performance model based on the hydraulic permeation model for various applied gas pressure differences between anode and cathode compartments. (Reprinted from S. Renganathan et al. Journal of Power Sources 160 (2006) 386-397. Copyright 2006, with permission from Elsevier.)... 

The experimental data (dots) are reproduced very well within the framework of the hydraulic permeation model (solid lines). For the basic case with zero gas pressure gradient between cathode and anode sides, APe = 0, the model (solid line) predicts uniform water distribution and constant membrane resistance at )p < 1 A cm and a steep increase in R/R beyond this point. These trends are in excellent agreement with experimental data (open circles) for Nafion 112 in Figure 6.15. A finife positive gas pressure gradient, APs = P/ - P/ > 0, improves the internal humidification of fhe membrane, leading to more uniform water distribution and significantly reduced dependence of membrane resistance on X. The latter trends are consistent with the predictions of fhe hydraulic permeation model. 

One decade has passed since the parallel artificial membrane permeation assay (PAM PA) was first introduced in 1998 [47]. Since then, PAM PA rapidly gained wide popularity in drug discovery [3, 48-51]. Today, PAMPA is the most widely used physicochemical membrane permeation model. The term PAMPA is nowusedas the general name for a plate-based (HTS enabled), biter-supported (filter immobilized) artificial membrane. Typically, phospholipids dissolved in an organic solvent are impregnated into the filter to construct a PAMPA membrane. 

There are no recent improvement in the paracellular pathway permeation models, probably because there is no specific in vitro or in vivo system to measure the paracellular pathway contribution. The paracellular pathway models was constructed using very hydrophilic compounds [107] or subtracting the contribution of transcel-lular pathway from the total passive permeation [78]. Paracellular pathway was modeled as permeation through a charged aqueous pore. A combination of size sieving function and electric field function was found to model the paracellular pathway [78, 87, 88]. 

Table 8 Summary of the different models for the gas permeation Model Ref Filler Particle geometry Formula...

The experimental results in Fig. 24 suggested that the gel supported on the porous glass had micropores with sizes comparable to those of the polymer solutes. Possibly the number and radius of the pores greatly change with the swollen state of the gel. In our previous work [27], a permeation model was proposed which considered the changes in the number and radius of the pores. 

An important value of a permeation model is not simply its ability to correlate experimental data, but rather to provide a framework for understanding the principal factors controlling membrane performance. The dual mode model is derived from... 

Several assumptions must be made to define any permeation model. Usually, the first assumption governing transport through membranes is that the fluids on... 

The Steady-State Permeation Model for Underground Coal Gasification... 

The steady-state permeation model of in situ coal gasification is presented in an expanded formulation which includes the following reactions combustion, water-gas, water-gas shift, Boudouard, methanation and devolatilization. The model predicts that substantial quantities of unconsumed char will be left in the wake of the burn front under certain conditions, and this result is in qualitative agreement with postburn studies of the Hanna UCG tests. The problems encountered in the numerical solution of the system equations are discussed. 

Gunn and coworkers (1,2) were the first to propose a steady-state model, and their predictions agreed very well with the Hanna UCG test results. In an analysis of the different versions of permeation models that have appeared in the literature, Haynes (3) judged the steady-state model superior for most applications since reaction kinetics are taken into account and only a modest computational effort is required. Despite these desirable features, applications of the steady-state model have not been as widespread as one might anticipate. 

An expanded formulation of the steady-state permeation model has been presented. Two numerical problems - stiffness and an ill-conditioned boundary value problem - are encountered in solving the system equations. These problems can be circumvented by matching forward and reverse integrations at a point near the inlet (n = 0) but outside the combustion zone. The model predicts a... 

Simonsen, L., Fullerton, A. (2006). Development of an in vitro skin permeation model simulating atopic dermatitis skin for the evaluation of dermatological products. Skin Pharmacol. Physiol. 20 230-6. 

Adenot M, Lahana R. Blood-brain barrier permeation models Discriminating between potential CNS and non-CNS drags including P-glycoprotein substrates. J. Chem. Inf Comput. Sci. 2004 44 239-248. 

For the tortuous and irregular capillaries of porous media, it has been reported theoretically and experimentally that a minimum in the permeability of adsorbates at low pressures is not expected to appear. In our study of n-hexane in activated carbon, however, a minimum was consistently observed for n-hexane at a relative pressure of about 0.03, while benzene and CCI4 show a monotonically increasing behavior of the permeability versus pressure. Such an observation suggests that the existence of the minimum depends on the properties of permeating vapors as well as the porous medium. In this paper a permeation model is presented to describe the minimum with an introduction of a collision-reflection factor. Surface diffusion permeability is found to increase sharply at very low pressure, then decrease modestly with an increase in pressure. As a result, the appearance of a minimum in permeability was found to be controlled by the interplay between Knudsen diffusion and surface diffusion for each adsorbate at low pressures. 

As have been seen above adsorption plays an important role in permeation through microporous membranes. So, single and multicomponent adsorption isotherms are required for a successful modelling of the permeation behaviour. An extensive treatment of the recent state of the art of zeolite permeation modelling is given by Van de Graaf et al. [70]. A shortened treatment follows here. 

General permeation model in which the emulsion drop is composed of the inner core of emulsion phase and the peripheral thin oil layer. The model accounted for chemical reactions, diffusion in the emulsion phase, in the peripheral thin oil layer of the emulsion drop, in the external aqueous film. 

Kataoka T, Nishiki T, and Kimura S. Phenol permeation through liquid surfactant membrane—permeation model and effective diffusivity. J Membr Sci 1989 41 197-209. 

Note that diffusion models and hydraulic permeation models have their own caveats the membrane is neither a homogeneous acid solution, nor is it the well-structured porous rock. Critical comparison of the results of the two approaches with each other and with experiments, is of crucial importance for understanding the membrane functioning within the cell and developing the strategies on water management and optimized membrane properties. 

Membrane performance characteristics in the hydraulic and diffusion limits are compared to each other in Fig. 9. Figure 9(a) illustrates that in the diffusion model considerable deviations from the purely ohmic performance of the saturated membrane arise already at small jv/Jj, well below the critical current density. This is in line with the comparison of the water-content profiles calculated in the diffusion model, Fig. 9(b), with those from the hydraulic permeation model, in Fig. 7. Indeed, membrane dehydration is much stronger in the diffusion model, affecting larger membrane domains at given values of jp/./j. Moreover, the profiles exhibit different curvature from those in Fig. 7. 

Fig. 10 Membrane resistance in H2/O2 fuel cell as a function of proton current density. Experimental data, normalized to the resistance 9ts of the saturated membrane at various temperatures have been extracted from Ref. 94. They are compared to the values calculated in the hydraulic permeation model (main figure) and to the results of the diffusion model, taken from Ref. 7 (inset).

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