Gel Polarisation Model

Film and gel-polarisation models are developed for ultrafiltration. These models are also widely applied to cross-flow microfiltration. 

In the following section, film and gel-polarisation models are developed for ultrafiltration. These models are also widely applied to cross-flow microfiltration, although even these cannot be simply applied, and there is at present no generally accepted mathematical description of the process. 

This boundary-layer theory applies to mass-transfer controlled systems where the membrane permeation rate is independent of pressure, for there is no pressure term in the model. In such cases it has been proposed that, as the concentration at the membrane increases, the solute eventually precipitates on the membrane surface. This layer of precipitated solute is known as the gel-layer, and the theory has thus become known as the gel-polarisation model proposed by Micii i i.si 0). Under such conditions C, in equation 8.15 becomes replaced by a constant Cq the concentration of solute in the gel-layer, and ... 

This is the same prediction for the limiting slope of a plot of J against In Cf as for the gel-polarisation model. The value of the slope of such plots at all other conditions is less in magnitude than hD. 

The Gel Polarisation Model is based on the fact that at steady state flux reaches a limiting value, where increases in pressure no longer increase the flux. According to the Gel Polarisation model, at this limiting value, the solubility limit of the solute in the boundary layer is reached and a gel formed. For 100% rejection, the expression for this hmidng flux (Jiim) is described by equation (3.10). cg is the gel concentration, beyond which the concentration in the boundary layer cannot increase. 

Moaddeb and Koros (1997) described the deposition of silica on polymeric MF membranes as non-uniform. This means that cake characterisation is difficult as a cracks could vary the results. Meagher et al (1996) stated that attractive interaction between membranes and particles would cause a flux decline, even if the particles were aggregated. Aggregation reduced the flux decline if there was no attraction between the membranes and colloids. The authors outlined the restrictions of the gel polarisation model, as the porosity of the deposit is not accounted for in the model. It was also suggested that the resistance of the gel layer is more important than the particle-surface interaction (what is often referred to as adsorption). 

Concentration profile for a gel-polarised UF membrane Figure 1.7 Concentration/gel polarisation model schematic. In the absence of a gel layer, Cg=Cw. 

According to Equations (1.3) and (1.6), flux is related to the boundary-layer mass transfer coefficient, k. Both k and are calculated from the data plot shown in Figure 1.9 based on the gel polarisation model (Region III in Figure 1.8). Wall concentration,... 

The processing time required for a given UF/MF operation is controlled by the membrane area, feed volume to be processed and solute concentration. Of these, solute concentration is the most important factor as it controls flux and the membrane gel concentration, which is determined by the well-known flux-concentration mass transfer limited gel-polarisation model introduced in Chapter 1 as ... 

The effect of slip coefficient on concentration polarisation (CP) was mathematically modeled for flat membrane and tubular membrane systems [12,13,15,16]. Lowering of CP due to slip coefficient as a function of product water recovery ( ) for different normalised diffusion coefficients (a) is shown in Figure 6.8. The data show that CP decreases both with and a. Since a is a measure of particle diffusion from the membrane surface to the bulk solution, slip-flow possibly augments diffusive back-transport of particles from the membrane surface to the bulk solution. Thus, the slip-flow velocity model possibly accounts for higher or actual UF/MF flux, which is under-predicted by the gel polarisation model discussed in Chapter 1. 

Tu et al. (1997) predicted NF flux by incorporating a gel-layer into the concentration polarisation model. Results corresponded well to filtration experiments with tannic acid. Gill %t al (1988) showed that viscosity effects in the boundary layer are more important than diffusivity. Concentration factors in UF of macromolecules were 40 to 400 times. A similar effect can be expected for large natural organic molecules. Kim et al (1992b) showed cake formation for low initial fluxes, and aggregation for high initial fluxes in protein UF. This demonstrated that solute-solute interactions in the boundary layer are important. 

Concentration Polarisation is the accumulation of solute due to solvent convection through the membrane and was first documented by Sherwood (1965). It appears in every pressure dri en membrane process, but depending on the rejected species, to a very different extent. It reduces permeate flux, either via an increased osmotic pressure on the feed side, or the formation of a cake or gel layer on the membrane surface. Concentration polarisation creates a high solute concentration at the membrane surface compared to the bulk solution. This creates a back diffusion of solute from the membrane which is assumed to be in equilibrium with the convective transport. At the membrane, a laminar boundary layer exists (Nernst type layer), with mass conservation through this layer described by the Film Theory Model in equation (3.7) (Staude (1992)). cf is the feed concentration, Ds the solute diffusivity, cbj, the solute concentration in the boundary layer and x die distance from the membrane. 

Although this model may be considered to be a significant contribution to the theory of concentration polarisation and limiting flux behaviour in ultrafiltraiion, some drawbacks should be mentioned. In literature data have indicated that the gel concentration Cg is not a constant but depends on the bulk concentration and the cross flow velocity [16], In... 

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