The Pore Model

The fine-pore model was developed assuming the presence of open micropores on the active surface layer of the membrane through which the mass transport occurs (10). The existence of these different pore geometries also means that different models have been developed to describe transport adequately. The simplest representation is one in which the membrane is considered as a number of parallel cylindrical pores perpendicular to the membrane surface. The length of each of the cylindrical pores is equal or almost equal to the membrane thickness. The volume flux through these pores can be described by the Hagen-Poiseuille equation. Assuming all the pores have the same radius, then we have 

This equation clearly shows the effect of membrane structure on transport, and indicates that the solvent flux is proportional to pressure difference as the driving force. 

Equation (30) gives a good description of transport through membranes consisting of a number of parallel pores. However, very few membranes possess such a structure in reality. Membranes consists a system of closed spheres, which can be found in organic and inorganic sintered membranes or in phase-inversion membranes with a nodular top layer structure. Such membranes can best be described by the Kozeny-Carman relationship  

Phase-inversion membranes frequently show a sponge-like structure. The volume flux through these membranes is described by the Hagen-Poiseulle or the Kozeny-Carman relation, although the morphology is completely different. 

Schulz and Asumnaa (48), based on their SEM observation, assumed that the selective layer of an asymmetric cellulose acetate membrane for reverse osmosis consists of closely packed spherical nodules with a diameter of 18.8 nm. Water flows through the void spaces between the nodules. Calculate the water flux by Eq. (30) assuming circular pores, the cross-sectional area of which is equal to the area of the triangular void surrounded by three circles with a diameter of 18.8 nm (as shown in Eig. 8). 

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To convert the core area into the pore area ( = specific surface, if the external area is negligible) necessitates the use of a conversion factor R which is a function not only of the pore model but also of both r and t (cf. p. 148). Thus, successive increments of the area under the curve have to be corrected, each with its appropriate value of R. For the commonly used cylindrical model,... 

Hint. Use the pore model to estimate an isothermal effectiveness factor and obtain eff from that. Assume le =0.15 J/(m s K). 

Water Transport. The dependence of the diffusive permeability of tritiated water Pq., (HTO) and hydraulic permeability LpAX on the lEC are presented in Figures 4 and 5. Both Pj and LpAX can be seen to increase exponentially as a function of lEC. Since the volume fraction of water, (J>, is also a linear function of the lEC, a similar exponential relation is obtained for these parameters vs. (J>. In terms of the pore model, the increase in either diffusion permeability or the hydraulic permeability may be caused by one or both of the following possibilities (a) an increase in the number of passageways, or (b) by increase in the radius of the pores. This question may be resolved by examining the g factor, defined as a ratio of two permeabilities (15) ... 

Before detailed conclusions are presented with regard to the physical meaning of the present model more fundamental studies are needed. While it is clear that ethanol "induces" new pores or "activates" latent pores in hairless mouse stratum corneum at high ethanol concentrations (10), the role of ethanol at lower concentrations is less clear at this moment. It is well known (12 ) that ethanol at low concentrations, may "fluidize" bilayers, thus leading to changes in both partitioning and diffusivity. Thus a complete description for permeation through stratum corneum will have to consider the effects of adjuvants on the properties of lipid bilayers in addition to the pore model described here. 

We present a kinetic relation, which we derived by extending the pore model derivation of Bhatia and Perlmutter vrith two additional parameters to account also for additional effects, such as the gradual creation of new surface area by the particle disintegration process, respectively, catalyst accumulation or re-activation effects in the alkali metal catalysed gasification. The resulting relation is found to describe our gasification results very satisfactorily over the entire conversion range. 

The pore model is unable to describe this late reactivity maximum around X=0.7 as it foresees a possible maximum reactivity to occur only between 0 X < 0.393. In the literature, the late occurrence has been explained by intercalation of alkali metal species into the carbon stmeture, leading to a gradual release of active centres with conversion. We note, however, that intercalation effects have seldom been reported for charcoals (in contrast to graphite). In our opinion the cause for the "anomalous" reactivity behaviour stems from a combination of structural and catalytic phenomena emerging from the reaction mechanism involved. The most important mechanism proposed nowadays is the oxygen transfer mechanism in which the oxygen is extracted from the reactant gas (CO2) by the catalyst, which then supplies it in an active form to the carbon. 

In the pore model developed by Bhatia and Perlmutter, the rate of the gasification reaction per unit pore surface area is characterised by the reaction rate constant, K,. As the original work addresses structurally based effects only, Kj may well be assumed constant throughout the gasification stage and, under kinetic control, the char reactivity is then a direct measure of the available surface area. To allow the description of additional (i.e., non-porous) phenomena, we follow a semi-empirical approach in which we assume that Kj can vary with time, the cause of which can either be structural or catalytic in nature. Accordingly, we define Ks(t) = KsoucnirtCt) Strictly... 

The matrix models were therefore calculated from a combination of the flat surface and model pore isotherms by the following algorithm Starting with the lowest pressure point, the amount adsorbed indicated by the flat surface model was compared to that of the pore model the flat surface isotherm was followed until the amount predicted by the pore model was the greater, then the pore model isotherm was followed for the remainder of the pressure vector. 

There may also be cases in which the equilibrium is not completely irreversible (i.e., very low solute concentration). A solution to the pore model for favorable equilibria (R<1) is given by Vermeulen (11). Values of X and N (T-l) are tabulated in (7) for various values of R < 1. ... 

Figure 9.35 Three proposed models of P-gp drug efflux, (a) The pore model, in which cytosolic drug is transported through a protein channel, (b) the flippase model, where drug associated with the inner leaflet of the membrane bilayer is flipped into the onter leaflet where it might passively diffuse out of the cell, and, (c) the hydrophobic vacuum cleaner model, where drug associated with the inner leaflet of the bilayer is exported out through the protein. (Reprinted with permission from Varma, M.V.S., et al. P-glycoprotein inhibitors and their screening A perspective from bioavailability enhancement. Pharmacol. Res. 2003, 48, 347-359, copyright 2003, Elsevier).

It is easy to show that this model gives precisely the same formal kinetic predictions as does that of the simple pore. The meaning of the parameters in Q and R of Eqn. 13 in terms of rate constants is different for the two models (the relevant results are collected in Table 1), but the prediction in terms of experimentally determinable parameters are identical. Making the pore model more complex in this particular way does not save it, so that if the simple pore is rejectable so is this more complex pore. 

Figure 4J-1 Schematic representation of the pore model [from Szekely and Evans [16]). 

Figure 4.5-3 Equivalent penetration versus ifCgo for the pore model with the following parameters ...

The pore model of dyeing was first suggested for dyeing cellulose fibers in the thirties... 

The adsorption/desorption isotherms show a mariced decrease in nitrogen adsorption due to grafted carbon deposits. As Cq increases the specific surface area, 5bet. mesopore surface, S,, pore volumes, Vp and V eso. and pore radii, (Table 4.7) decrease nearly linearly. Initial Si-60 and CSG are mesoporous materials (Figure 4.18 and Table 4.7). The deviations (Aw) of the pore model (cylindrical pores in silica gel and voids between spherical carbon or silica nanoparticles with the SCR procedure) are relatively small (4%-13%). 

Mass transport through porous membranes can be described with the pore model. In accordance with particle filtration, selectivity is determined solely by the pore size of the membrane and the particle or the molecular size of the mixture to be separated. This process is driven by the pressure difference between the feed and permeate sides [83]. The processes described by the pore model include microfiltration and ultrafiltration. Whereas membranes for microfiltration are characterized by their real pore size, membranes for ultrafiltration are defined according to the molar mass of the smallest components retained. 

Before proceeding into any detailed calculations, we immediately recognize that our model network (5.1) topologically coincides with the network (4.38) of the model for the enzyme-catalyzed reaction of Section 2.2. This means that we can formally transfer all results from Section 2.2 to the present problem and only have to interpret those results according to the physical and biological context of the pore model. In this way, we derive the steady state flux 1 across the pores from (2.6)... 

First of all, we notice that J as a function of c saturates at high values of c similar to the pore models of Sections 5.1 and 5.2. The difference between the present model and the pore models comes about when calculating the relaxation behaviour. Insertion of (5.30) and X + X = Q into the first of (5.31) gives... 

It is elucidative to compare the carrier flux (5.46) with that of the pore model in (5.5). The results are very similar in their qualitative structure in both cases the flux J is proportional to the difference Ac of the substrate concentration and saturates for c —> > and c = const or vice versa. The only principal difference between (5.5) and (5.46) is the fact that the denominator in (5.5) is a linear function of c and c whereas in (5.46) it is quadratic. This means that if both c and c tend to infinite values such that c c but c + c the pore flux remains finite whereas the carrier flux vanishes. This latter property is due to the fact that c c —> 00 blocks the cycle of the carrier action in (5.41) at the two sides of the membrane in opposite directions. Clearly, a more realistic way to distinguish... 

Similar to the pore models, the flux is proportional to the total amount E of the transport facilitating enzyme. From (5.54) we also conclude that T > 0 is possible even for c < c if A/B is sufficiently large such that our model actually describes... 

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