Prediction of ultrafiltration rates

There have been many models, both simple and sophisticated, that describe the operating patterns of ultrafiltration processes [4]. Most of these models describe how the rate of ultrafiltration is controlled by the properties of a region of very high solute concentration, a filter cake or concentration polarised layer, close to the membrane surface. Relatively few of these models have a genuinely predictive capability. Remarkably, only a very few [5-7] of these models consider the most important feature of the solutes being separated by ultrafiltration—that they fall in the colloidal size range. For colloidal materials, the properties of the filter cake or concentration polarised layer will be controlled by the interparticle interactions in such a region. The important interactions which need to be taken into account are [8]  

Electrostatic. Virtually all colloids in solution acquire a surface charge and hence an electrical double layer. When particles interact in a concentrated region their double layers overlap resulting in a repulsive force which opposes further approach. Any theory of filtration of colloids needs to take into account the multi-particle nature of such interactions. This is best achieved by using a Wigner-Seitz cell approach combined with a numerical solution of the non-linear Poisson-Boltzmann equation, which allows calculation of a configurational force that implicitly includes the multi-body effects of a concentrated dispersion or filter cake. 

Dispersion. Dispersion or London-van der Waals forces are ubiquitous. The most rigorous calculations of such forces are based on an analysis of the macroscopic electrodynamic properties of the interacting media. However, such a full description is exceptionally demanding both computationally and in terms of the physical property data required. For engineering applications there is a need to adopt a procedure for calculation which accurately represents the results of modem theory yet has more modest computational and data needs. An efficient approach is to use an effective Lifshitz-Hamaker constant for flat plates with a Hamaker geometric factor [9]. Effective Lifshitz-Hamaker constants may be calculated from readily available optical and dielectric data [10]. 

Entropic. The packing of particles at a high concentration leads to an entropic pressure tending to disperse them. Such entropic pressures may be precisely calculated using molecular dynamic approaches, and such calculations may be well-represented for both low and high volume fractions by analytical expressions of which the most accurate is a Pad6 approximation [11]. 

In some cases, it may be necessary to take into account more specific interactions, such as hydration forces in the example of silica colloids. 

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