The current study applies the Monte Carlo simulation method for microscopic modeling of colloidal transport in membrane filtration. Monte Carlo simulation utilizes a stochastic modeling strategy that first selects discrete particle displacements with statistical randomness and then evaluates each selection on the basis of its physical validity. The model will accept those proposed displacements that qualify as being physically legitimate and disqualify physically unreasonable ones. As the simulation proceeds, the model generates a pool of statistically random, discrete, microscopic particle movements from which it maps each particle trajectory. The Monte Carlo method has traditionally proven itself to be a powerful modeling application across a myriad of scientific and mathematical disciplines [8,9]. The emergence of the Monte Carlo method in membrane filtration research remains in its burgeoning stage. It is our aspiration for this paper to contribute to the sustaining growth of the Monte Carlo method as a viable modeling alternative in the research of membrane separation processes and also to apply it to further understand the critical phase transitionphenomenonoccurringduringcolloidalmembranefiltration.
Fig. 1 shows examples of membrane filtration systems operating in the typical dead-end and crossflow modes along with an outline of the important forces to consider. In the dead-end mode of operation (Fig. 1(a)), the applied pressure within the system imposes a downward permeation drag force that propels particles to accumulate on the membrane surface. The formation of the particle deposit is stratified into two regions, a fluid polarized layer situated above a compact cake layer below. When an additional fluid flow in the horizontal direction is added, the mode of operation becomes the crossflow type as shown in Fig. 1(b). Here, the particles still will accumulate on the membrane surface during filtration, while the added crossflow now serves to afford the particle deposit another degree of movement along the horizontal direction.
Fig. 1(c) shows a more introspective breakdown of the combination of forces acting together in the system. Hydrodynamic drag in the downward direction promotes particle deposition onto the membrane surface to initially form a polarized layer of highly concentrated suspension and then continues to further induce solidification of the polarized layer to form a compact cake layer below it (Fig. 1(a) and
(b)).Conversely,inter-particleinteractions,inparticularelectrostatic repulsion, mollify particle consolidation and create a more porous cake layer. Moreover, particles may coagulate into aggregate form with adhesion to one another prior to their deposition onto the membrane surface. Under these conditions, the cake layer formation dynamics may entail a two-stage progression where the initial deposit on the membrane surface is a porous coagulated substructure which then undergoes a second stage of compression and restructuring to reach the final cake layer configuration. Such a dual-stage formation is observed experimentally in Tarabara et al. [10]. The gravitational compressive force (Fig. 1(c)) occurs as the net resultant of the sum of all accumulated particles positioned above a particular location in the cake layer. Furthermore, particles once deposited may restructure their configuration. Restructuring may be traceable to a number of possible causes. Among them is the deflection of the downward hydrodynamic drag and compressive forces so as to induce lateral movement of particles. The deflection of forces arises due to the complex interactions of many forces operating within a many-body particle configuration. When dealing with a crossflow filtration system, the crossflow shear introduces another source of restructuring dynamics whereby particles flow laterally across the membrane surface and possibly become re-entrained into bulk suspension [8,11–13]. Other important considerations include aggregate breakage, inelastic particle collisions (Fig. 1(c)), and time dependent issues such as the implications of a slow cake development in dilute suspensions with singular particle deposition as compared to the rapid case at high concentrations with particles depositing in clusters [14,15].
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