Modélisation de la filtration de colloïdes mous et déformables: Cas du lait
Résumé
When filtrating a colloidal dispersion, colloids accumulate at the membrane surface, thus forming a polarized layer that can ultimately turns into a deposit (= a "gel") in some extreme cases. Modeling such a filtration has always been a challenge. In recent developments, Bacchin et al. propose a model based on the analogy between permeation in a deposit and diffusion in a polarized layer [1]. The permeation flux is then given by the following equation, which is the analog of the classical Darcy's law: J = k(C)/μ.dΠ(C)/dx, where x is the distance to the membrane, μ is the solvent viscosity, k(C) and Π(C) are the permeability and colloidal osmotic pressure at distance x, respectively. As for now, the model of Bacchin et al. has been used successfully to describe the filtration of impermeable and "hard" spherical objects like latex particles [2]: the idea is to measure the osmotic pressures Π(C) of the latex dispersions through osmotic stress experiments and to estimate the permeability k(C) using the well-known theoretical expressions of Happel. Knowing Π(C) and k(C), the equation is then used to obtain important information like permeation fluxes or concentration profiles in the accumulated layer. The objective of our study is to adapt such a model to the filtration of more complex colloids, namely, deformable and permeable objects like colloidal microgels. Casein micelles, which represent 80% of the proteins in cow milk, are a good candidate for that since they are highly permeable and compressible protein aggregates [3]. Additionally, ultra- and microfiltration are widely used for the concentration of milk and there is a serious need for a model that is able to describe and predict these operations. This paper is dedicated to the dead-end filtration case only. The more complicated case of cross-flow filtration, for which the effect of cross-flow velocity on the build-up of the accumulated layer needs to be taken into account, will be investigated in future studies. The osmotic pressure of casein micelle dispersions is now perfectly known [3]. So the main difficulty in milk modeling lies in determining k(C) since no theoretical expression is currently available for deformable and permeable objects. To get around this difficulty, we followed two distinct approaches to actually measure the permeability k of dispersions of casein micelles: a direct approach, involving osmotic stress experiments, and a "reverse calculation" approach, consisting in estimating k through well-controlled filtration experiments. The reported evolution of k(C) is the first important result of this paper, as such experimental data have been rarely reported until now for such peculiar objects. From the knowledge of k(C), we then construct a filtration model and analyze it through a series of test and "virtual" experiments. We also directly compare the predictions of the model to filtration data obtained in combined filtration/SAXS experiments. The results of those calculations and comparisons are quite unequivocal and totally validate the modeling approach.