A unifying model for concentration polarization, gel-layer formation and particle deposition in cross-flow membrane filtration of colloidal suspensions
Introduction
Cross-flow ultrafiltration is now used in a wide range of industrial applications (in the food industry, biotechnology, the pharmaceutical industry, water and waste-water treatment). The nominal molecular-weight cut-off of ultrafiltration membranes (1–) makes them appropriate for processing colloidal suspensions (particles or macromolecules whose size is less than ). Such a process is mainly limited by the accumulation of matter on the filter that includes concentration polarization and membrane fouling (formation of a gel layer or a deposit). With this in mind, the study of membrane fouling by colloidal dispersions is of considerable interest for developing the process. Here, modeling opens up the possibility of better understanding mechanisms that reduce process efficiency, of optimizing the way the filtration should be operated and of creating expert systems for the design of membrane modules and membrane plants processing colloidal suspensions.
The complexity of colloidal matter comes from the presence of surface interaction between the suspended materials. Over the past two decades, experimental observations have revealed the role that colloidal interactions can play in the filtration of colloidal suspensions (Cohen & Probstein, 1986; McDonogh, Fane, & Fell, 1989). Fifteen years ago, the existing models were incapable of quantitatively predicting permeate flux and of qualitatively representing the effect of a suspension's physico–chemical properties, such as ionic strength or pH, on the permeate flux. What was described by Cohen and Probstein (1986) as “a colloid flux paradox” has recently been underlined by the experimental finding of a critical flux for colloids. This critical phenomenon demonstrates the specificity of the filtration of colloidal suspensions and raises interest for modeling in this area: just a small variation in operating conditions (particle size or surface charge, pH, ionic strength, concentration, pressure, cross-flow velocity, permeation rate, etc.) induces important changes in the working point and so in the way the process has to be operated. As detailed in the next section, various models accounting for colloidal interaction have been recently developed to describe colloid filtration. However, the model for a limiting phenomenon in ultrafiltration is often selected according to the application (concentration polarization and gel layer for macromolecules, deposit for particles) and the models differ in their theoretical treatment. But colloidal suspensions often exhibit the behavior both of particles and of macromolecules, thus leading to a delicate choice as to the way modeling should be developed. Furthermore, detailed analysis of the effect of colloidal interaction on filtration is often carried out in a one-dimensional system (i.e. normal to the membrane surface), whereas the design of membrane modules and the definition of appropriate operating conditions has to take account of the development of the mass-transfer boundary layer. For the filtration of colloidal suspensions, this cannot be done using the standard calculation based on purely diffusive mechanisms, as was pointed out by Jönsson and Jönsson (1996). However, these authors suggested to determine the boundary layer thickness using an experimental technique; we shall show below how this thickness can be estimated from a two-dimensional model.
In the present work, we have investigated the possibility of introducing particle–particle colloidal interactions into a two-dimensional analysis of transport phenomena along the length of a filtration device. This allows the specificity of colloid filtration to be accounted for. Also, a phase transition, related to the balance between dispersive and attractive forces, accounts for the passage from the liquid state to the gelled phase. By integrating such phenomena into the description of membrane fouling, the model can depict mechanisms of concentration polarization, gel formation and particle deposition, within a single approach. Important suggestions as to the way filtration should be operated when processing colloidal suspensions are underlined.
Section snippets
Background
Solute accumulation is the antagonistic phenomenon of the filtration process and as such can only be partially reduced (for example, by changing the hydrodynamics) but never totally eliminated. Solute accumulation is a self-regulating phenomenon as it causes a drop in permeate flux, thus inducing a simultaneous decrease in the accumulation rate and so on. Consequently, when operating at a fixed trans-membrane pressure difference (TMP) and feed concentration, surface fouling leads to a
Theoretical development
We consider the cross-flow filtration of a colloidal dispersion in a tubular configuration at steady state (Fig. 1). This system is assumed to have rotational symmetry, thus reducing the three dimensions of the tubular geometry to the two dimensions r and z. The hydrodynamics in this system will be treated as the sum of tangential shear flow and radial flow. This approximation was shown by Berman (1953) to be acceptable in a thin layer near the permeable wall, when he solved the Navier–Stokes
Discussion
Simulations using the model developed in the previous section have been performed for various operating conditions and physico–chemical properties of the media. To illustrate the capability of the model to describe both concentration polarization and particle deposition, simulation results are first presented for two different kinds of colloidal material: small particles ( in radius) which could represent a macromolecule such as a big protein and larger ones () like latex particles, for
Conclusions
The model presented and discussed in this paper is capable of accounting for concentration polarization, gel-layer formation and particle deposition and depicting the continuity between the major fouling mechanisms involved in ultrafiltration of colloidal suspensions. It shows that a single theoretical approach is capable of covering a wide range of suspension sizes and of cross flow situations. Basic transport phenomena, such as convection and diffusion (with colloidal interactions and their
Notation
Hamaker constant, particles or macromolecules radius, L diffusion coefficient, friction factor Happel correction for sedimentation velocity mean permeate flux, Boltzman constant, mobility of particle or macromolecule, pressure, flow rate, membrane channel radius, L hydraulic resistivity of deposit, hydraulic resistivity of membrane, temperature, K axial velocity, local permeate flux at the wall, p particle or macromolecule
Acknowledgements
The authors are grateful to Indo-French Centre for the Promotion of Advanced Research (Grant No 1615-1) for supporting a part of this work (D.S.H.). V.S. and P.A. acknowledge a support of the Royal Society (Grant ESEP/JP/JEB/11159).
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