Models/modeling membrane transport phenomena

Widdas s quantitative model of the simple carrier was able to explain a number of earlier observations and to make predictions about what would be observed in more complex experiments on membrane transport. Thus it was a highly productive scientific insight. One of the earlier, apparently anomalous, results that the theory explained was the dramatic fall of membrane permeability found for solutes which were rapidly transported as solute concentration was increased. For example, in the human red blood cell, Wilbrandt and colleagues had previously measured a permeability constant for glucose which was 1000 times higher in dilute solutions of glucose than it was in a concentrated solution. This phenomenon, subsequently called saturation kinetics, is formally equivalent to the fall, as substrate concentration increases, in the proportion of substrate converted to product by a limited amount of an enzyme. 

This mechanism is important for compounds that lack sufficient lipid solubility to move rapidly across the membrane by simple diffusion. A membrane-associated protein is usually involved, specificity, competitive inhibition, and the saturation phenomenon and their kinetics are best described by Michaelis-Menton enzyme kinetic models. Membrane penetration by this mechanism is more rapid than simple diffusion and, in the case of active transport, may proceed beyond the point where concentrations are equal on both... 

Similarly, Shukla and Cheryan [18], studying the behavior of the permeation of 18 UF membranes, found that 15 of these membranes agree with the Darcy model, in which the permeate flux decreased in linear correlation with the increasing viscosity of the permeation solvent, indicating that in these 15 UF membranes, the transport phenomenon of the solvent was affected by viscosity. 

FIGURE 33.3 Transport phenomenon and membrane orientation of the FO process using an asymmetric FO membrane, (a) PRO mode and (b) FO mode. (Adapted from Desalination, 309, Tan, C.H. and Ng, H.Y., Revised external and ICP models to improve flux piedietion in FO process, 125-140, Copyright 2013, with permission from Elsevier.)... 

As the permeability of the membrane for ions of different charge signs largely varies, salt diffusion through a membrane is accompanied by the establishment of a membrane potential. These concentration or dialysis potentials play an important part in the study of membrane phenomena. With the above described model, the phenomenon of electro-endosmosis i.e. the transport of solvent across a membrane under the influence of an electric field, can easily be explained also. 

Titanium, as an example for the transport model verification, was chosen because of the extensive experimental data available on LLX and membrane separation [1,2,74—76] and of its extraction double-maximum acidity dependence phenomenon [74]. This behavior was observed for most extractant families basic (anion exchangers), neutral (complexants), and acidic (cation exchangers). So, it is possible to study both counter- and cotransport mechanisms at pH > 0.5 and [H] > 7 mol/kg feed solution acidities, respectively, using neutral (hydrophobic, hydrophilic) and ion-exchange membranes. 

Finally, we have not observed a spontaneous transition from a low flux to a high flux state (Figure 24.7 A) with our previous MR-based membranes [3,5]. The fact whether this transition is observed depends on the feed concentration suggests that the transition is a transport-related phenomenon. It is possible that this transition relates to the concept of cooperative (high flux) vs. noncooperative (low flux) dehybridization (Figure 24.9), but further studies, both experimental and modeling, will be required before a definitive mechanism for this transition can be proposed. 

Several mechanisms have been proposed to explain reverse osmosis. According to the preferential sorption-capillary flow mechanism of Sourirajan [114], reverse osmosis separation is the combined result of an interfacial phenomenon and fluid transport under pressure through capillary pores. Figure 5.58a is a conceptual model of this mechanism for recovery of fresh water from aqueous salt solutions. The surface of the membrane in contact with the solution has a preferential sorption for water and/or preferential repulsion for the solute, while a continuous removal of the preferentially sorbed interfacial water, which is of a monomolecular nature, is effected by flow under pressure through the membrane capillaries. According to this model, the critical pore diameter for a maximum separation and permeability is equal to twice the thickness of the preferentially sorbed interfacial layer (Figure 5.58b). 

There are different ways to depict membrane operation based on proton transport in it. The oversimplified scenario is to consider the polymer as an inert porous container for the water domains, which form the active phase for proton transport. In this scenario, proton transport is primarily treated as a phenomenon in bulk water [1,8,90], perturbed to some degree by the presence of the charged pore walls, whose influence becomes increasingly important the narrower are the aqueous channels. At the moleciflar scale, transport of excess protons in liquid water is extensively studied. Expanding on this view of molecular mechanisms, straightforward geometric approaches, familiar from the theory of rigid porous media or composites [ 104,105], coifld be applied to relate the water distribution in membranes to its macroscopic transport properties. Relevant correlations between pore size distributions, pore space connectivity, pore space evolution upon water uptake and proton conductivities in PEMs were studied in [22,107]. Random network models and simpler models of the porous structure were employed. 

The prediction of the existence of a phenomenon analogous to a Bose-Einstein condensation, as in Frohlich s vibrational model, has been confirmed via several other approaches, namely, via a transport theory formalism by Kaiser and a molecular Hamiltonian approach by Bhaumik et and also by Wu and Austinthe basic concept has been shown to be on firm theoretical ground. The difficulty in these ideas gaining wide acceptance is the lack of conclusive experimental evidence to provide confirmation that such effects occur in biological membranes. Furthermore, the theory presented thus far is not at the stage where it can be used in an analytical sense to explain those data thus far reported or be used in a predictive manner. 

Concerning mass transfer modeling across nanofiltration membranes, the transport mechanisms occiuring are convection, diiiusion, and electromigration (when the solutes are charged). Taking into account the polarization concentration phenomenon, which can occm dming the filtration operation (and thus the increase of the concentration at the membrane interface C ), the solute real retention (l reai) can be calculated from the observed retention by ... 

So far, our simple model only describes steady state transport across membranes, but no relaxation. In the network of Fig. 2, the flux J would adjust to a new steady state value without time delay after a change of the potential difference. In order to exhibit relaxation behaviour, the membrane must be capable of storing the molecules which are transported across the membrane. The network representation of such a storage phenomenon is a material capacitance which has to be added to the network as shown in Fig. 3 ... 

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