Here it should be noted that a model that falls under region II does not automatically disqualify it as being incorrect. That is because of the presence of counterbalancing forces in the system that will allow for the particles to restructure into configurations of higher volume fraction. Those forces include crossflow shear and compression by gravity. The formation of the cake layer may involve a two-stage process where particles first readily deposit on the membrane surface at lower volume fractions (region II behavior) and then subsequently restructure or collapse into a final formation of high volume fraction [10]. This trend may be present in dilute bulk suspensions where particles deposit singularly onto the membrane surface. So, in a system involving a dilute bulk suspension in crossflow filtration, the dynamics of cake formation may extend beyond a simple region I relationship between the downward hydrodynamic drag and the inter-particle potential. Therefore, the selection of models to implement should not be aimed solely at seeking a region I behavior.
What Fig. 9(a) and (b) do convey are the repercussions that result from the sensitive choice of mathematical expressions and values to input to define the system. For the model in the current study, two quantities have been identified to be paramount: hydrodynamic drag force and inter-particle potential. Which model to choose and what values to input for its physical parameters? That is among the most important questions for the development of a model that remains responsible to the true situation. The final decision should be rigorously tested against experimental observation of the actual physical parameters. The mandate for judicious selection of the model to implement appears overly obvious, and likewise, the testing of physical parameters may be relegated as mundane detail. However, these simple yet crucial steps of inspection are often overlooked in the development of model simulations.
In Section 4.1, one of the possible definitions for the point of phase transition from a fluid-like polarization layer to a solid cake layer has been given. It relies on the inter-particle separation distance as the defining criterion for phase transition. Phase transition is identified to be the point where particles come into contact with one another (surface-to-surface inter-particle separation equals zero). As mentioned previously, this view is intuitively clear but may not bear practical significance as it does not relate itself to the operational performance of a membrane filtration system. More pragmatic interpretations of phase transition may be formulated by integrating the concept of an observed critical flux during the filtration cycle.
Some ambiguity still remains about the exact interpretation of the critical flux [16,39–41]. The opinions of several authors, for instance Howell et al. [41], Chang et al. [1], and Chan and Chen [42], are converging in agreement to the critical flux concept introduced by Field et al. [16]. Field et al. [16] classifies the critical flux into two subcategories, the strong and weak forms. The strong form of the critical flux states that there exists a point below which the filtration of a colloidal suspension will yield the same flux as pure water for the same applied pressure [16,17]. The weak form relaxes this stringent guideline and casts the critical flux as the point below which a linear relationship exists between the applied pressure and permeate flux. The slope of the linear relationship is allowed to differ from that of the pure-water flux [16,17]. The interpretation of Field et al. [16] will be applied here to refine the simplistic approach regarding the point of particle contact as indicating phase transition so as to arrive at a more holistic set of pragmatic criteria adaptable to various filtration operating conditions.
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