Gel layer model

As mentioned above, concentration polarisation can be very severe in ultrafiltration because the flux through the membrane is high, the diffusivity of the macromolecules is rather low and the retention is normally very high. This implies that the solute concentration at the membrane surface attains a very high value and a maximum concentration, the gel concentration (Cg), may be reached for a number of macromolecular solutes. The gel concentration depends on the size, shape, chemical structure and degree of solvation but is independent of the bulk concentration. The two phenomena, concentration polarisation and gel formation are shown in figure VII -12. 

Gel formation may be reversible or irreversible, a very important factor in membrane cleaning. An irreversible gel is very difficult to remove and precautions have to be taken to avoid this situation as much as possible. However, it is not important whether the gel is 

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 

Macromolecules are retained by the membrane in ultrafiltration whereas tow molecular weight components permeate through freely. Because the main contribution to the osmotic pressure of a solution arises from the low molecular weight solutes (the concentration of these being the same in the feed and permeate), the osmotic pressure of the retained macromolecules is often neglected. 

However, for high flux values, high rejection levels and low mass transfer coefficient k values, the concentration of macromolecular solutes at the membrane surface can become quite high and hence the osmotic pressure cannot be neglected anymore. This has been commented upon by several investigators [12-16]. If the osmotic pressure at the membrane surface is taken into account, the flux equation is then given by  

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Womans, J. G., Nakao, S. and Smolders, C. A. Journal of Membrane Science 20 (1984) 115. Flux limitation in ultrafiltration osmotic pressure model and gel layer model. 

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 gel layer model described above is very appealing and is widely used to rationalize the behavior of ultrafiltration membranes. Unfortunately a number of issues cannot be easily explained by this simple form of the model ... 

Formation of these surface complexes occurs in addition to simple ionization of surface groups via Equations 1 and 2. Smlt et al. (14, 15) have recently shown experimentally that Na is specifically adsorbed within the compact layer of silica and alumina surfaces. The experimental evidence supported a site-binding model for the EDL of non-porous oxides (1, 7), rather than gel layer models proposed by other workers (16, 17, 18). 

In the case of the gel-layer model, the theory led to a simple analytical formula for a limiting or critical flux, The flux turned out to be proportional to the cube root of the ratio of the gel concentration to the feed solution concentration, rather than to the logarithm of this ratio, as the simplifled Michaels-Blatt theory predicted ... 

Denisov, G.A. 1994. Theory of concentration polarization in cross-flow ultrafiltration Gel-layer model and osmotic-pressure model. J Membr Sci. 91 173-187. 

How can the gel layer model and the osmotic pressure model, be related to each other In the gel layer model, a plot of J versus IniCj,) gives a straight line with a slope equal to -k. A similar J versus ln(c,) relationship can be obtained from the osmotic pressure model. From eq.vn - 25 the following relationship can be derived ... 

When (Art n)/(q R , k) 1, the right-hand side of eq.VII - 31 reduces to -k. Hence the osmotic pressure model gives a linear plot in the region where R can be neglected with a slope equal to -k similar to that obtained from the gel layer model. 

The considerations given above still leave the question open as to which of the two models is actually valid. A qualitative method of dhscriminating between the two models is provided by the J o versus InfCj,) plot, with the intercept on the absciss providing an answer. If physical-chemical reasons suggest that gelation should occur, the gel layer model may be valid and the intercept gives c = Cg. If, on the other hand, the osmotic pressure at point c is equal to the applied pressure [AP = n(c)], then Cj, = c ,. However, it should be noted that often, in practice, the phenomena are much more complex than those described here. Thus, adsorption and other phenomena (see figure Vn - 2) have not been taken into account. Even the way in which the pressure increments occur can lead to other results which cannot be predicted or even described by the two theories advanced above. 

The flux through the UF membrane can be described with the gel layer model... 

Calculate the permeate flow rate of this module at an inlet pressure of 1.5 bar and calculate the pressure drop (For the flux calculation the gel layer model may be... 

Gas adsorption Gas desorption Casolin Oas permeadon Gas sepatadon Celadon Cel layer Gel-layer model... 

In their 1997 study of the effect of neutral polymer on the ESA signal from silica, Carasso et cd. [1] used a gel layer model to derive an expression for the slip coefficient which has the form ... 

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