Concentration polarization model

The mechanism of such UF can be explained by the following concentration polarization model (cf. Figure 8.3) [3, 4]. In the early stages of UF, the thickness of the gel layer increases with time. However, after the steady state has been reached, the solute diffuses back from the gel layer surface to the bulk of solution this... 

The different solute concentrations on the feed and permeate sides are linked to the volumetric permeate flux in terms of the concentration polarization model, which is based on the stagnant film theory [14] ... 

To understand the flux decline in pressure-driven membrane operations, a number of models were developed. Two of the most widely smdied models are the resistance model and the concentration polarization model. The resistance model is the oldest and is based on the cake filtration theory, where it is assumed that a cake layer of rejected particles, which are too large to enter the membrane pores, is formed. The frictional drag due to permeation through these immobile particles leads to additional hydraulic resistance [21]. The cake layer and the membrane are considered as two resistances in series, and the permeate flux is described by Darcy s Law as... 

The concentration polarization model, which is based on the stagnant fihn theory, was developed to describe the back-diffusion phenomenon during filtration of macromolecules. In this model, the rejection of particles gives rise to a thin fouling layer on the membrane surface, overlaid by a concentration polarization layer in which particles diffuse away from the membrane surface, where solute concentration is high, to the bulk phase, where the solute concentration is low [158]. At steady state, convection of particles toward the membrane surface is balanced by diffusion away from the membrane. Thus, integrating the onedimensional convective-diffusion equation across the concentration polarization layer gives... 

For the UF of proteins, the concentration polarization model has been found to predict the filtration performance reasonably well [56]. However, this model is inherently weak in describing the two-dimensional mass transport mechanism during crossflow filtration and does not take into account the solute-solute interactions on mass transport that occur extensively in colloids, especially during MF [21,44,158,159]. The diffusion coefficient, which is inversely proportional to the particle radius, is low and underestimates the movement of particles away from the membrane [56]. This results to the well-known flux paradox problem where the predicted permeate flux is as much as two orders of magnitude lower than the observed flux during MF of colloidal suspensions [56,58,158]. This problem has then been underlined by the experimental finding of a critical flux for colloids, which demonstrates the specificity of colloidal suspension filtration wherein just a small variation in physicochemical or hydrodynamic conditions induces important changes in the way the process has to be operated [21]. 

It was found that the gel layer concentration is not constant, but variable, which can be predicted by using the ordinary concentration polarization model with appropriate mass transfer coefficient. Also this concentration has a particular relation with the gel layer resistance. Using this relation the flux through the membrane can be calculated for the case of gel layer formation. 

The mechanism of such UF can be explained by the following concentration polarization model (cf. Figure 8.3) [3,4], In the early stages of UF, the thickness of the gel layer increases with time. However, after the steady state has been reached, the solute diffuses back from the gel layer surface to the bulk of solution this occurs due to the difference between the saturated solute concentration at the gel layer surface and the solute concentration in the bulk of solution. A dynamic balance is attained, when the rate of back-diffusion of the solute has become equal to the rate of solute carried by the bulk flow of solution towards the membrane. This rate should be equal to the filtrate flux, and consequently the thickness of the gel layer should become constant. Thus, the following dimensionally consistent equation should hold ... 

C.H. Tan, H.Y. Ng, Revised external and internal concentration polarization models to improve flux prediction in forward osmosis process. Desalination 2013, 309,125-140. 

A. Zydney and C. Colton. A concentration polarization model for filtrate flux in cross-flow microfiltration of particulate suspension. Chem. Eng. Commun., 47 1, 1986. 

The phenomenon of concentration polarization, which is observed frequently in membrane separation processes, can be described in mathematical terms, as shown in Figure 30 (71). The usual model, which is weU founded in fluid hydrodynamics, assumes the bulk solution to be turbulent, but adjacent to the membrane surface there exists a stagnant laminar boundary layer of thickness (5) typically 50—200 p.m, in which there is no turbulent mixing. The concentration of the macromolecules in the bulk solution concentration is c,. and the concentration of macromolecules at the membrane surface is c. 

Reviews of concentration polarization have been reported (14,38,39). Because solute wall concentration may not be experimentally measurable, models relating solute and solvent fluxes to hydrodynamic parameters are needed for system design. The Navier-Stokes diffusion—convection equation has been numerically solved to calculate wall concentration, and thus the water flux and permeate quaUty (40). 

J. Siler, "Reverse Osmosis Membranes-Concentration Polarization and Surface Fouling Predictive Models and Experimental Verifications," dissertation. University of Kentucky, Lexington, Ky., 1987. 

By incorporating a flux model [Eq. (20-73)], solute concentration [Eq. (20-77)], and polarization model [Eq. (20-60)] into the area cost... 

The theoretical approach by Samec based on the ion-free compact layer model established that the true apparent transfer coefficient is obtained after correction for concentration polarization effect [1] [see Eq. (14)]. Subsequent studies by Samec and coworkers on the ferricyanide-Fc system provided values of a smaller than the expected 0.5. Preliminary attempts to rationalize this behavior were based on defining effective interfacial charges and separation distance between reactants [79]. The inconclusive trends reported in these studies were ascribed to complications arising from ion pairing of the ferro/ferricyanide ions. Later analysis of the same system appeared to show that k i is... 

Concentration in the permeate is expressed by that in the feed as Equation (5), using solution-diffusion model assuming no concentration polarization. 

Here we are concerned with applying the same model and optical methods to analyze (1) the influence of applied pressure on intrinsic compaction, and (2) aqueous solution effects on Intrinsic transport parameters and concentration polarization. 

In the case of Hollosep, it may be possible to employ a simplified module model because of its low concentration polarization, uniform feed water flow and low pressure drop. 

Assuming that the concentration polarization, pressure drop and back mixing are negligible, module model can be expressed simply in terms of the transport equation and the material balance of each component. 

In this section we shall consider the simplest model problem for the locally electro-neutral stationary concentration polarization at an ideally permselective uniform interface. The main features of CP will be traced through this example, including the breakdown of the local electro-neutrality approximation. Furthermore, we shall apply the scheme of 4.2 to investigate the effect of CP upon the counterion selectivity of an ion-exchange membrane in a way that is typical of many membrane studies. Finally, at the end of this section we shall consider briefly CP at an electrically inhomogeneous interface (the case relevant for many synthetic membranes). It will be shown that the concentration and the electric potential fields, developing in the course of CP at such an interface, are incompatible with mechanical equilibrium in the liquid electrolyte, that is, a convection (electroconvection) is bound to arise. 

We shall remove an important drawback of the polarization model described in Section VI by considering another variant of a composite model than that described in previous Section VILA. We use again a linear-response theory to find the contribution of a vibrating dipole to the total permittivity . We split the total concentration N of polar molecules into the sum Nm and Nv b, where each term refers to rotation of a like rigid dipole (viz. with the same electric moment p) but characterized by different law of motion ... 

Using this model and the assumption that concentration polarization occurs only on the feed side of the membrane, the flux across the combined resistances of the feed side boundary layer and the membrane can be written as... 

The second approach to concentration polarization, and the one used in this chapter, is to model the phenomenon by assuming that a thin layer of unmixed fluid, thickness S, exists between the membrane surface and the well-mixed bulk solution. The concentration gradients that control concentration polarization form in this layer. This boundary layer film model oversimplifies the fluid hydrodynamics occurring in membrane modules and still contains one adjustable parameter,... 

Figure 4.2 Fluid flow velocity through the channel of a membrane module is nonuniform, being fastest in the middle and essentially zero adjacent to the membrane. In the film model of concentration polarization, concentration gradients formed due to transport through the membrane are assumed to be confined to the laminar boundary layer...

Figure 4.4 Salt concentration gradients adjacent to a reverse osmosis desalination membrane. The mass balance equation for solute flux across the boundary layer is the basis of the film model description of concentration polarization...

As described above, the initial cause of membrane fouling is concentration polarization, which results in deposition of a layer of material on the membrane surface. The phenomenon of concentration polarization is described in detail in Chapter 4. In ultrafiltration, solvent and macromolecular or colloidal solutes are carried towards the membrane surface by the solution permeating the membrane. Solvent molecules permeate the membrane, but the larger solutes accumulate at the membrane surface. Because of their size, the rate at which the rejected solute molecules can diffuse from the membrane surface back to the bulk solution is relatively low. Thus their concentration at the membrane surface is typically 20-50 times higher than the feed solution concentration. These solutes become so concentrated at the membrane surface that a gel layer is formed and becomes a secondary barrier to flow through the membrane. The formation of this gel layer on the membrane surface is illustrated in Figure 6.6. The gel layer model was developed at the Amicon Corporation in the 1960s [8],... 

The formation of concentration gradients caused by the flow of ions through a single cationic membrane is shown in Figure 10.8. As in the treatment of concentration polarization in other membrane processes, the resistance of the aqueous solution is modeled as a thin boundary layer of unstirred solution separating the... 

Concentration polarization can dominate the transmembrane flux in UF, and this can be described by boundary-layer models. Because the fluxes through nonporous barriers are lower than in UF, polarization effects are less important in reverse osmosis (RO), nanofiltration (NF), pervaporation (PV), electrodialysis (ED) or carrier-mediated separation. Interactions between substances in the feed and the membrane surface (adsorption, fouling) may also significantly influence the separation performance fouling is especially strong with aqueous feeds. 

The concentration polarization occurring in electrodialysis, that is, the concentration profiles at the membrane surface can be calculated by a mass balance taking into account all fluxes in the boundary layer and the hydrodynamic conditions in the flow channel between the membranes. To a first approximation the salt concentration at the membrane surface can be calculated and related to the current density by applying the so-called Nernst film model, which assumes that the bulk solution between the laminar boundary layers has a uniform concentration, whereas the concentration in the boundary layers changes over the thickness of the boundary layer. However, the concentration at the membrane surface and the boundary layer thickness are constant along the flow channel from the cell entrance to the exit. In a practical electrodialysis stack there will be entrance and exit effects and concentration... 

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