Monte Carlo simulation of colloidal membrane filtration: Principal issues for modeling

Transcription

1 Advances in Colloid and Interface Science 119 (2006) Monte Carlo simulation of colloidal membrane filtration: Principal issues for modeling Jim C. Chen a, Albert S. Kim a, * a Department of Civil and Environmental Engineering, University of Hawaii at Manoa, Honolulu, HI 96822, USA Available online 22 November 2005 Abstract The principal issues involved in developing a Monte Carlo simulation model of colloidal membrane filtration are investigated in this study. An important object for modeling is the physical dynamics responsible for causing particle deposition and accumulation when encountering an open system with continuous outflow. A periodic boundary condition offers a solution to the problem by recirculating continuous flow back through the system. Scaling to full physical dimensions will allow for release of the model from flawed assumptions such as constant cake layer volume fraction and thickness throughout the system. Furthermore, rigorous modeling on a precise scale extends the model to account for random particle collisions with acute accuracy. A major finding of this study proves that forces within the colloidal filtration system are summed and transferred cumulatively through the inter-particle interactions. The force summation and transfer phenomenon only realizes its true value when the model is scaled to full dimensions. The overall strategy for model development, therefore, entails three stages: first, rigorous modeling on a microscopic scale; next, comprehensive inclusion of relevant physical dynamics; and finally, scaling to full physical dimensions. D 2005 Elsevier B.V. All rights reserved. Keywords: Monte Carlo simulation; Colloidal membrane filtration; Cake layer formation; Periodic boundary condition; Random particle collisions; Force summation and transfer Contents 1. Introduction Background Overview of membrane filtration system Governing equations for the system Equations for mass transport Equations for filtration Interaction forces in membrane filtration Inter-particle interaction Hydrodynamic drag force Monte Carlo simulation Discussion Strategic planning Problematic issues when modeling an open system with continuous flow Add-on features to the model Particle deposition and accumulation dynamics Dynamics of particle deposition in an open system Common flawed assumptions employed in model development Spatial variations in cake layer thickness and volume fraction Temporal variations in cake formation * Corresponding author. Tel.: ; fax: address: albertsk@hawaii.edu (A.S. Kim) /$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi: /j.cis

2 36 J.C. Chen, A.S. Kim / Advances in Colloid and Interface Science 119 (2006) Particle collisions and shear-induced diffusion Concentration gradient driven diffusion for interacting particles Cake layer compression Scalability to full system dimensions Computational efficiency Conclusion Acknowledgement References Introduction In our previous work [1], we perform a comprehensive review of the attributes of three powerful modeling techniques: Brownian Dynamics, Molecular Dynamics, and Monte Carlo Simulation [2 4]. In that study, we effectively differentiate the three methods in accordance to their strengths, limitations, and applicability. Spawning from that initial work, we authored a follow-up study [5], in which we select the Monte Carlo simulation method and integrate it into a case study of particle deposition in dead-end membrane filtration [6]. The Monte Carlo simulations generate a key physical parameter, the cake layer volume fraction, which when jointly inputted into filtration theory produces the modeled system behavior in terms of the permeation flux. Using this complementary procedure between Monte Carlo simulation and filtration theory, we are successful in obtaining valuable insights into the phase transition phenomenon of the particle deposit from a fluid polarized layer to a solid cake formation [7,8]. Now, we have arrived at a significant stage of advancement in our focused progression of model development. In this study, we will begin to bring our model presented in our previous work [5] to closer conformity to a true facsimile of the actual physical situation occurring in a long-channel crossflow membrane filtration system. With a more representative model of the actual system comes better clarity of the intricate, multiplexed physical dynamics that drives the system behavior. Our overarching goal is to produce a Monte Carlo simulation model that is a total mirror reflection of an actual crossflow membrane filtration system. With this current work, we will embark by taking certain measurable steps towards that grand undertaking. New features have been added onto our model [5] to bring it to an unprecedented level of representation of the real system. The progressive features of the model consist of rigorous modeling of: & A long-channel open crossflow membrane system with continuous flow & Dilute bulk solution & Singular particle deposition & A full 3-dimensional parabolic flow field & Spatial variations in cake layer thickness, resistance, and permeation flux & Particle collision dynamics and the resultant shear-induced diffusion effects & Concentration gradient driven diffusive flux & Dependence of the diffusivity on concentration and interparticle interactions & Cake layer compression due to layered gravitational and hydrodynamic forces & Capability of the model to be scaled up to large dimensions. Building upon the current model with further add-on features to encompass other dynamics at work in the system as well as scaling up the model to larger physical dimensions will continue the effort towards achieving a total mirror reflection model. Throughout the process of model development, we identified certain principal issues that must be addressed in order for the model to truly represent the system. These issues deal with both the physical dynamics of colloidal deposition in membrane filtration as well as computational attributes of the numerical model that seeks to simulate these behaviors. They originated as a collection of separate issues that arose as we added the aforementioned new features to our model. As each issue was investigated, important new insights were uncovered. The model to be presented in this study addresses these principal issues for effective model development: 1) Overall strategy 2) Problems that arise when modeling an open system with continuous flow 3) Dynamics responsible for particle accumulation by singular deposition 4) Flawed assumptions that are frequently employed 5) Inclusion of random particle collisions and the resultant shear-induced diffusion 6) Concentration gradient driven diffusive flux for interacting particles 7) Compression of the cake layer due to force transfer through particle layers 8) The need to scale the model to the full physical dimensions of the system 9) Practical constraints imposed by computational capacity. These nine issues, once resolved, will position the current model to be ready for scaling up to become a true representation of the physical system. This study is a discourse on these nine principal issues for effective model development. 2. Background 2.1. Overview of membrane filtration system Fig. 1(a) shows an example of a membrane filtration system operating in the typical crossflow mode along with an outline of the important forces to consider (Fig. 1b). During filtration

3 J.C. Chen, A.S. Kim / Advances in Colloid and Interface Science 119 (2006) a b Crossflow z x Bulk Phase Permeate Flux Compression by Gravity Permeate Flux Polarization Layer Cake Layer Membrane imentally in the work of Tarabara et al. [9]. The gravitational compressive force (Fig. 1b) 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, as will be discussed in Section 4.4. When dealing with a crossflow filtration system, the crossflow shear introduces another restructuring dynamics whereby particles flow laterally across the membrane surface and possibly become re-entrained into the bulk suspension [10 12]. Other important considerations include aggregate breakage, inelastic particle collisions (Fig. 1b), 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 [13]. Crossflow shear Particle Aggregation / Adhesion Inter-particle Forces Particle Aggregate Breakage Particle Collision Fig. 1. (a) Schematic of a crossflow filtration system. (b) Outline of forces driving colloidal transport, aggregation, and breakage. (Fig. 1a), 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. Fig. 1(b) 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. 1a). Conversely, interparticle interactions, in particular electrostatic 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 exper Governing equations for the system Equations for mass transport The governing equation for colloidal transport in a crossflow membrane filtration system with only concentration polarization is [14,15]: u B/ Bx ¼ B Bz D ð/ Þ B/ v/ Bz with the boundary conditions: / ¼ / b at z ¼ d cp / ¼ / b at x ¼ 0 D ð/ Þ B/ Bz ¼ v/ m at z ¼ 0 ð2þ where / is the volume fraction of particles, / b is the bulk volume fraction, / m is the particle volume fraction on the membrane surface, D(/) is the concentration-dependent diffusion coefficient of particles [16], z is the axis perpendicular to the membrane surface, d cp is the thickness of the concentration polarization layer, v is the permeate flux, and u is the crossflow velocity in the axial, x, direction Equations for filtration In general, one of the quantities of special interest in applications of membrane filtration is the permeate flux which, for crossflow filtration (after cake formation has occurred), is governed by Darcy s law [8,17,18]: DP v w ¼ v¼ ð3þ lðr m þ R c Þ where v w is the permeate flux, DP is the applied pressure within the system, l is the absolute fluid viscosity, R m is the membrane resistance, and R c is the cake layer resistance further defined as [19] R c ¼ r c d c r c ¼ 9/ c 2a 2 X ð1þ ð4þ

4 38 J.C. Chen, A.S. Kim / Advances in Colloid and Interface Science 119 (2006) where r c isthespecificcakeresistance,d c isthecakelayer thickness, a is the particle radius, and X is Happel s correction factor, defined as [20]: X ¼ 1 þ 2 3 /5=3 c /1=3 c þ 3 2 /5=3 c / 2 c of which its physical meaning is the adjustment made to obtain the true hydrodynamic force experienced by particles within a porous medium of equally sized spheres Interaction forces in membrane filtration Inter-particle interaction One of the principal input parameters to the model is the inter-particle interaction potential. In the current study, the inter-particle potential is modeled by DLVO theory [21,22]: E DLVO ¼ E VDW þ E EDL ð6þ where E VDW is the van der Waals potential and E EDL is the inter-particle electrostatic double-layer potential. The attractive van der Waals potential between equal-sized particles can be modeled by many available expressions. In this study, it is calculated by [23]: E VDW ¼ A H 12 4 þ 4 s 2 s 2 4 þ 2ln 1 4 s 2 where A H is the Hamaker constant and s (=r / a) is the dimensionless center-to-center separation between two particles scaled by the particle radius a. The electrostatic interaction potential between the particles is calculated from the linear superposition approximation of [24,25]: E EDL ¼ 128pn 0ac 2 ja s 2 k B Te ð Þ j 2 ð8þ s c ¼ tanh z sew s ð9þ 4k B T sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j 1 e r e 0 k B T e r e 0 k B T 2e 2 z 2 s n ¼ e 2 ð10þ N A I where j 1 is the Debye screening length, N A is Avogadro s Number, I is the ionic strength of the solution, e r is the dimensionless dielectric permittivity of water, e 0 is the permittivity of free space, n 0 is the ion number concentration, k B is Boltzmann s constant, T is the absolute temperature, z s is the valence of ions, e is the charge of an electron, and w s is the surface potential of the particles, which can be substituted by the zeta potential [7,26] Hydrodynamic drag force The second primary physical parameter inputted into the current model is the hydrodynamic drag. The expression for the ð5þ ð7þ hydrodynamic drag force derives directly from Stokes law for a single particle in an infinitely unbounded medium: F Stokes ¼ 6plav w : ð11þ The crossflow drag is computed in like manner except the quantity v w is replaced by the crossflow velocity (see Section 2.2.1). For fluid flow through a porous medium composed of equal-sized spherical particles, the true hydrodynamic drag force exerted on each particle within the medium is adjusted by Happel s correction factor X, defined earlier in Eq. (5) [16,19,27]: F hydro ¼ F Stokes X: 3. Monte Carlo simulation ð12þ The algorithms for implementing the Monte Carlo simulation are given in the work of Chen et al. [5]. An additional component of the crossflow is included in the model of the current study. The presence of the crossflow drag requires the proposed displacements in the Monte Carlo simulation to be scaled properly. When more than one force is acting in the system, the proposed displacements in the Monte Carlo simulation must be scaled with respect to the magnitude of the forces acting in the direction of the proposed displacement. The procedure accelerates convergence of the MC simulation to the steady state of the system. Fig. 2 provides a simple illustration involving two forces acting in perpendicular directions. In Fig. 2, the force acting on the particle in the x-direction is twice the force acting in the z-direction. The proposed random displacement in the x-direction, Dx, must then be twice as large as the proposed random displacement in the z-direction, Dz. The scaling of the proposed displacements in the Monte Carlo simulation of the current model is performed by computing the ratio of the sum of forces in a given direction to a chosen constant reference force: X! Fx Dx ¼ Ds Dy ¼ Dz ¼ F ref X Fy F ref X Fz F ref! Ds! Ds ð13þ where Ds is a constant random displacement selected for a Monte Carlo step, Dx, Dy, and Dz are proposed random displacements broken down into x-, y-, and z-directions, respectively, F ref is a chosen constant reference force (here the crossflow hydrodynamic force at the epicenter of the parabolic flow field is chosen as the reference), and ~F x, ~F y, and ~F z are the sum of forces acting in the x-, y-, and z- directions, respectively.

5 J.C. Chen, A.S. Kim / Advances in Colloid and Interface Science 119 (2006) F 2 = 2F 1 In the current study, the sum of the forces in the x-direction is: X Fx ¼ F crossflow þ F DLVO;x þ F Diffusion;x ð14þ where F crossflow is the crossflow drag force as given by Eq. (11), F DLVO,x is the force from the inter-particle interactions in the x-direction given as F DLVO;x ¼ de DLVO dr dr dx : ð15þ F Diffusion,x is a virtual force driving the Brownian diffusion of particles given as [28] F Diffusion;x ¼ k BT C BC Bx z F 1 x Similarly, the sum of forces in the y-direction is: X Fy ¼ F DLVO;y þ F Diffusion;y Particle Trajectory Fig. 2. Relative scaling of displacement for Monte Carlo simulations. ð16þ ð17þ with the components of the expressions given in Eqs. (15) and (16) in the x-direction replaced by the y-direction. The sum of forces in the z-direction is X Fz ¼ F Hydro þ F DLVO;z þ F Diffusion;z þ F g ð18þ where F g is the gravitational force acting on the mass of the particle and is equal to the mass of the particle (less the mass of the solvent to account for the buoyancy force) multiplied by the acceleration of gravity. The scaling, as depicted by Eqs. (13) (18), will allow the particles in the Monte Carlo simulation to be displaced by the correct amount with respect to the forces acting in that direction. The main procedure of the Monte Carlo simulation uses the hydrodynamic force bias algorithm to model particle motion under the influence of inter-particle interactions as well as hydrodynamic drag force [5,29]. We have described this procedure in detail in our recent publication [5]. In brief, performing MC simulations begins by casting the process as being Markovian, indicating that the outcome for a particle displacement depends solely on the outcome that immediately precedes it [4,30]. A particle of interest initially located at the old position is proposed to be displaced to a new position in a random manner. The proposed displacement is next evaluated for acceptance or rejection based on the energy criterion of the system stability. Total energy difference between the new and old positions of the particle, experiencing inter-particle and hydrodynamic drag force, is DE ¼ E new E old þ Y FIð Y r new Y r old Þ where E new and E old are the inter-particle potential energies at the new and old positions of Y r new and Y rold, respectively, and Y F is the bias force of which x-, y-, and z- components are described in Eqs. (14), (17), and (18), respectively. If DE is negative, then the proposed move is downhill and readily accepted, but if DE is positive, then the proposed move is uphill and is accepted with the qualification of the transition probability that corresponds to the Boltzmann factor of the total energy difference, exp( DE /k B T). In this procedure, a uniform random number between 0 and 1 is generated and compared with the Boltzmann factor. If the random number is less than the Boltzmann factor, then the uphill move is accepted. Otherwise the proposed move is rejected. If the uphill move is rejected, then the particle remains in its original (old) position, and the non-move is taken to be a new state nevertheless. The simulation procedure continues by randomly selecting successive particle for proposed displacements and evaluating each move. 4. Discussion 4.1. Strategic planning Our first principal issue deals with the overall strategy for model development. This speaks of strategic planning at the big-picture level. Achieving a true model representation of the physical system is a formidable task that exceeds the bounds of any one particular study or paper. A series of cohesive and logically progressive studies must be assembled in order to realize the overarching goal of a full-scale, truly representative model of the actual system. The overall strategy should be aware of how previous works, the current study, and the future direction are inter-linked. Fig. 3 shows a summary of the progression from past works to future study in our model development. In stage 1, we engage in a comprehensive literature review of the theories and algorithms for three microscopic modeling techniques: Brownian Dynamics, Molecular Dynamics, and Monte Carlo Simulation [1 4]. A major goal of the study is to present microscopic-level modeling techniques as advantageous alternatives to traditional macroscopic-level approaches, such as integral methods [16]. Microscopic techniques operate based on discrete individual particle displacements and are therefore capable of breaking down complex physical dynamics into simplified sequences of distinct particle movements. Where macroscopic methods are often thwarted by a rise in complexity in the system, microscopic techniques, in contrast, can readily adapt and include those complications by breaking them down into simple, precise particle movements [5]. Subsequently, in stage 2, we demonstrate the capabilities of the Monte Carlo Simulation method by applying it to a case study of cake formation in dead-end filtration. Through

6 40 J.C. Chen, A.S. Kim / Advances in Colloid and Interface Science 119 (2006) Scaling the physical dimensions of the system to achieve total mirror reflection: future direction 3. Conforming the preliminary model to a long channel crossflow system: Current study Stages of Model Development rigorous modeling of particle motion under the influences of hydrodynamic drag and inter-particle potential, we effectively illustrate the sensitivity of a key physical parameter, the cake layer volume fraction, to variations in the system operating conditions, such as applied pressure, ionic strength, and particle radius [5]. Furthermore, we next link the cake layer volume fraction to filtration theory to investigate the phase transition phenomenon of the particle deposit from a fluid polarization layer to a solid cake formation [7,8]. In so doing, we identify the point of phase transition as being an elaborate concept entailing several options for its definition, including cake layer volume fraction, critical flux, and cake irreversibility. In summary, at stage 2, we implement a rigorous modeling technique and apply it to show the richness of insight that is available through this powerful method. Stage 3 is the subject of the current study. At this juncture, we will begin to bring the model of stage 2 to closer conformity to our system of interest, which is a long-channel crossflow membrane filtration unit [14,15]. We accomplish this task by adding on new features to the model as outlined in Section 1, for instance, a three-dimensional parabolic flow field and spatially varying cake thickness. Thus, the model will become more comprehensive in the scope of physical processes that it envelopes. Also, the model of stage 3 will be connected to the next stage through the process of scaling up the physical dimensions of modeling. Finally, the model will enter stage 4. At stage 4, the model will become a total mirror reflection of the actual system. The model will approach the actual full-scale physical dimensions of the system while remaining rigorously microscopic in its computations. This is the prevailing goal of our model synthesis. In reaching it, we will have encountered and resolved the nine principal issues outlined in Section 1. Here we address our first principal issue with a summary of our overall strategic planning according to Fig. 3: model rigorously (stages 1 2), model comprehensively (stages 2 3), 1. Comprehensive review of literature [1] 2. Preliminary model using small simulation box [5] Fig. 3. Overall strategy for series of four sequential studies. and finally, model expansion (stage 4). Each step is predicated upon a defining question: & Model rigorously: What are the correct and accurate methods for computing forces throughout the system (e.g. inter-particle interactions and hydrodynamic drag)? & Model comprehensively: What other physical dynamics need to be included in the model to make it more complete (e.g. particle collision, aggregation, and breakage)? & Model expansion: Does the model mirror the full-scale physical dimensions of the actual system? A modeling effort seeking to achieve a true representation of the system must continually confront these three questions. In the discussions to follow, we will demonstrate how the model of our current study effectively answers all three Problematic issues when modeling an open system with continuous flow Add-on features to the model In our current study, we begin to make our model a closer facsimile of the actual physical system by adding these features: & Modeling a long-channel open crossflow membrane system with continuous flow & Dilute bulk solution & Singular particle deposition & A full 3-dimensional parabolic flow field. Fig. 4 shows a schematic of these add-on features. The addon features advance the model from its previous state of simulating a small box of concentrated particles with no crossflow [5] to becoming more representative of the actual physical scenario occurring in a long-channel crossflow system. Fig. 4 describes a situation where the system is open with continuous flow, meaning that particles are aided by the crossflow to traverse the length of the channel and can leave the system at the outlet. Additionally, the bulk solution is dilute, and so the deposition of particles onto the membrane surface occurs individually [13]. 3-D Parabolic Flow Dilute Bulk Solution Continuous Flow Singular Particle Deposition z y Long Channel x Fig. 4. Schematic of parabolic crossflow and dilute bulk suspension.

7 J.C. Chen, A.S. Kim / Advances in Colloid and Interface Science 119 (2006) The 3-D parabolic flow field is modeled in two steps. First, the variation of the velocity along the z-direction is considered by computing [31]: V x ðþ¼a z z z 2 þ b z z þ c z a z ¼ 4V mid h 2 ch b z ¼ 4V mid h ch c z ¼ 0 V avg ¼ 4 9 V mid ð19þ where V x (z) is the crossflow velocity of the flow at a given position z along the height of the channel without yet considering the variation along the y-direction, V mid is a given constant value of the crossflow at the epicenter of the parabolic flow field, V avg is the average crossflow velocity in the flow field, and h ch is the height of the channel. Once V x (z) is computed, the variation of the velocity along the y-direction follows as [31]: Vðz; yþ ¼ a y y 2 þ b y y þ c y a y ¼ 4V xðþ z w 2 ch b y ¼ 4V xðþ z w ch c y ¼ 0 ð20þ where V(z, y) is the flow velocity at a given z and y coordinate adjusted for variations along both directions and w ch is the width of the channel. The parabolic flow field exits at the outlet of the channel, creating an open system. The modeling of a dilute bulk phase uses a mixing tank at the inlet to supply feed at a specified dilute concentration and maintains the concentration of particles at the top portions of the system to be equal to the same concentration. Fig. 5 shows a schematic of this system. The situation described by Fig. 5 corresponds to particle deposition occurring at the entrance to the channel. The flow field continues to extend upwards because the height of the Mixing Tank Bulk Phase Membrane Flow Field Fig. 5. Singular deposition of particles onto the membrane from the mixing tank and bulk solution. channel is much larger than shown in Fig. 5. Particles originate from either the mixing tank or the bulk phase, with some proceeding to deposit onto the membrane surface and others exiting the system prior to deposition. The model focuses in on a very specific location within a much larger overall membrane unit. The add-on features shown in Figs. 4 and 5 introduce a problematic issue to the model, shedding new insights regarding the dynamics of particle deposition in an open system. These new findings would otherwise have escaped us under our previous model presented in the work of Chen et al. [5]. We discuss these findings next Particle deposition and accumulation dynamics When particles deposit individually within an open system, the physical dynamics governing the accumulation of particles on the membrane surface becomes much different from those modeled in a small-box simulation with high initial particle concentration [5]. A key difference is that the particles now deposit slowly while traversing the length of the channel to exit at the outlet. Under this scenario, if the rate of deposition is much slower compared to the rate of traverse along the channel, particles will then readily exit the system with minimal accumulation. In fact, in an open system, the continuous exiting of particles will eventually reach a steady-state condition where particles simply undergo sequences of first depositing onto the membrane surface, then traversing along the channel, and finally exiting, forming only a monolayer of particles streaming out of the system. This is precisely the situation observed in the current study when modeling an open crossflow system with a dilute bulk suspension. Fig. 6 better illustrates this point with an example of modeling results (operating conditions and modeling parameters are given in Table 1). In Fig. 6, the particle deposition dynamics have reached a steady state with particles continuously streaming out of the system, traversing along the monolayer on the membrane surface. This is an insight that would not have been seen if modeling immediately a simulation box with high particle concentration without allowing particles to freely exit the system [5]. By endeavoring to model more rigorously the real system using an open channel with dilute bulk suspension, we uncover an important finding. Particle deposition in a model simulation of an open crossflow system is not a given and should not be readily assumed to occur. Instead, the physical dynamics causing particle deposition are themselves objects of modeling that should be included as additional considerations. So how do particles deposit and accumulate on the membrane surface? Afterall, cake formation and membrane fouling are known to occur even at the normal operating conditions for the model results in Fig. 6 [32]. To answer this question, we will examine more closely the dynamics responsible for particle accumulation in an open system Dynamics of particle deposition in an open system The predicament introduced in Section 4.2.2, with particles not accumulating on the membrane surface but readily exiting

8 42 J.C. Chen, A.S. Kim / Advances in Colloid and Interface Science 119 (2006) Fig. 6. Steady state for particle deposition in an open system with only a monolayer of deposit. Operating conditions are given in Table 1. the system, poses a fitting case for demonstrating the capability of the current model to investigate particle deposition dynamics in an open system. We begin by recognizing that the results shown in Fig. 6 point to a very specific location within the membrane channel, which is at the inlet. Particle deposition dynamics at the channel entrance coincides with the situation described in Fig. 5. Under these circumstances, the system draws its supply of particles from both the mixing tank directly adjacent to the inlet and the bulk phase located above the length of the channel. Both sources of supply contain only constant dilute concentrations of particles, resulting in the singular deposition behavior that is central to the problem. To circumvent this limitation, we will shift our attention away from the channel entrance momentarily and instead examine a more manageable Table 1 Operating conditions and modeling parameters Operating conditions Hamaker constant, A H J Valence of ions, z s 1 Dielectric permittivity of water Permittivity of free space C 2 /Nm 2 Temperature K Membrane resistance (1/m) Crossflow velocity, V mid m/s Height of the channel 0.01 m Applied pressure 10 psi Zeta potential of particles 30 mv Ionic strength of solution 10 3 M Particle radius 100 nm Bulk phase volume fraction Particle mass density 2300 kg/m 3 Modeling parameters Length of simulation box Width of simulation box Height of simulation box 60particle radius 20particle radius 80particle radius scenario, which is particle deposition dynamics within the middle of the channel. Particle deposition dynamics in the middle of the channel differs from the case at the inlet in one crucial respect: the source for replenishment of particles. In the middle of the channel, the supply of particles is not derived continuously from a mixing tank held at a constant dilute concentration but rather from directly upstream of the channel where the concentration is variable and in fact increases as filtration progresses. Fig. 7(a) better illustrates this. As can be seen from Fig. 7(a), the area of focus highlighted in the middle of the channel draws its supply from volumes of ever-increasing concentrations upstream. In fact, upstream from the area of focus is a series of volumes with similar concentrations to that of the area of focus. These circumstances can be closely mimicked by installing a periodic boundary condition into the model, as shown in Fig. 7(b) [30]. The periodic boundary condition takes particles that have exited the outlet and recirculates them back upstream where they re-enter the system. By doing so, the concentration of the upstream volume will be pegged to the concentration within the area of focus. The flow conditions now become a close facsimile to that depicted in Fig. 7(a). Results of the model with the modification of the periodic boundary condition prove that particles will deposit and accumulate under these flow conditions (Fig. 8a). The concentration profile within the cake layer of Fig. 8(a) is given in Fig. 8(b). Notice that the cake layer profile is not uniformly horizontal along the x-direction. The entry and exit flow conditions lead to respective rise and decline of the cake layer thickness at the beginning and end of the cake layer. The cake formation is occurring at a location near the middle of the channel. The current model proves itself to be helpful in solving a problematic issue. First, we were confronted with the problem that particles in an open system will readily exit without

9 J.C. Chen, A.S. Kim / Advances in Colloid and Interface Science 119 (2006) a Continuous Flow b Upstream (variable, increasing concentration) Bulk Phase (constant dilute concentration) Area of Focus Recirculation Flow Bulk Phase (constant dilute concentration) Area of Focus accumulating on the membrane surface. By adapting the model to a new area of focus at the middle of the channel, we gain a valuable understanding regarding particle deposition dynamics. An important factor contributing to particle deposition on the membrane surface is the supply of particles from upstream. The middle of the channel draws particles from a series of volumes upstream containing ever increasing concentrations. This scenario has been shown to lead to deposition and accumulation of particles. The solution offered here is helpful but still unsatisfactory. Adaptation of the model to a new location provides knowledge of particle deposition in the middle of the channel, but the initial question remains unanswered: what are the dynamics causing particle deposition at the entrance of the channel where the supply of particles is dilute and constant? We will revisit that question in Section 4.6 in our discussion on cake layer compression Common flawed assumptions employed in model development Continuous Flow Fig. 7. (a) Flow of particles in the middle of the channel with varying and increasing concentrations upstream from the area of focus. (b) Periodic boundary condition with recirculation flow Spatial variations in cake layer thickness and volume fraction Use of assumptions is a common convention in model simulations to circumvent complexities within the system and proceed with expedience. However, the advantage of expedience can be associated with coterminous disadvantages of restrictions to the applicability of the model. So here we will expound on widely utilized assumptions in colloidal membrane filtration that should be removed from any model simulation that seeks to attain a true representation of the system. A fundamental governing equation for colloidal membrane filtration is Darcy s law (Eq. (3)) [8,17,18]. In general, the quantity of interest that defines the system behavior is the permeate flux. To compute the permeate flux, the key parameter to be inputted into Darcy s law is the cake layer resistance. Subsequently, the goal of many model simulations of colloidal membrane filtration targets the cake layer volume fraction as the subject of modeling [5,17,33]. Upon finding the cake layer volume fraction, the permeate flux would immediately follow from Darcy s law. In the step of inputting the cake layer volume fraction into Darcy s law, two major assumptions are commonly made, and they are the flawed assumption of a constant volume fraction within the cake layer which is further exacerbated by another flawed assumption of a constant cake layer thickness throughout the system. The critical parameter, cake layer resistance is a function of both the cake layer volume fraction and thickness. Simply dismissing both quantities as being constant constitutes gratuitous assumptions that greatly detract from the accuracy of representation of the model. In reality, both the cake layer volume fraction and thickness are spatially and temporally varying quantities. The cake layer volume fraction varies along the vertical direction within a section of the cake layer as well as horizontally across the length and width of the membrane channel. The cake layer thickness also varies along the length and width of the channel (Fig. 9). In Fig. 9, the profile of the cake layer is shown to entail a rise in cake layer thickness at the inlet of the channel, a flat plateau along the center regions of the channel, and then a drop in cake layer thickness towards the exit. The reason for this drop in thickness is that at x = L (exit region), the sudden disappearance of the permeate flux causes enhanced backdiffusion of deposited particles into the bulk phase. Within the cake layer itself, a stratification of volume fractions can be seen where the particle deposit transitions from a fluid-like polarization layer of low volume fraction to a compact solidified cake layer at the membrane surface at a high volume fraction of 0.6 (random maximum packing according to Onoda and Liniger s work [34]). The value of the cake volume fraction offers one criterion for delineating the point where cake formation initiates, so it is quite important to account for its variations within the system [5] Temporal variations in cake formation The volume fraction and thickness also change with time as filtration proceeds. If cake layer collapse occurs, the changes may be drastic and immediate [9]. A robust model simulation should possess the capacity to capture the spatial variations in cake volume fraction and thickness as well as their temporal changes. Fig. 10 shows an example of the temporal evolution of the cake formation. The elapsed time of filtration is mimicked by the procession of MC steps in the simulation.

10 44 J.C. Chen, A.S. Kim / Advances in Colloid and Interface Science 119 (2006) Fig. 8. (a) Particle accumulation on the membrane surface modeled with the addition of the periodic boundary condition. Operating conditions are given in Table 1. (b) Concentration profile of the particle deposit shown in (a). Fig. 10(a) (f) shows the progression from initial deposition to the onset of cake formation to the compaction of the cake layer with numbers of steps in the Monte Carlo simulations of 10,000 (Fig. 10(a)), 50,000 (Fig. 10(c)), and 100,000 (Fig. 10(e)). Fig. 10(b), (d), and (f) shows the particle deposition associated with their respective contours, Fig. 10(a), (c), and (e). In Fig. 10(a), the particle deposition is only beginning with very little accumulation on the membrane surface and low volume fraction. In Fig. 10(c) at the onset of cake formation, the layer of particles near the membrane surface (depicted by the contour with range in volume fractions from ) contains average center-to-center inter-particle separations of 2.1 times the particle radius, indicating that the particles are not in contact. Subsequently in Fig. 10(e), the particle layer at the membrane surface compresses to particle contact at times particle radius. The reduction in inter-particle separation is small at a decrease of 0.1 times the particle radius. However, the volume fraction in the particle layer from this compression Fig. 9. Profile of cake layer thickness and stratification of cake layer volume fraction. Operating conditions are given in Table 1.

11 J.C. Chen, A.S. Kim / Advances in Colloid and Interface Science 119 (2006) increases much more disproportionately from a range of in Fig. 10(c) to in Fig. 10(e). An increase on the order of 0.1 in volume fraction occurs with only minimal reduction in inter-particle separation. This suggests that the compression of the cake layer as shown in Fig. 10(c) and (e) is not predominately derived from closer approach of particles to one another. So, perhaps the restructuring of the particle configuration appears to be the major driving force for the increase in cake layer volume fraction after the particles have reached close separation. As can be seen here, much information regarding the dynamics of cake formation is gained by modeling both the spatial variation and temporal evolution of the cake layer. The assumptions of constancy in the cake layer volume fraction and thickness both spatially and temporally grossly misrepresent the true situation and should be avoided. However, dispensing of such gratuitous assumptions does entail consequences. If a constant cake layer thickness is not to be assumed, then the model must simulate the actual physical dimensions of the cake layer with a profile of volume fractions within it. Fig. 10(a) (f) shows the results of small-scale simulations with cake layer thickness on the order of only 10 times the particle radius. When inputted into filtration theory such as Darcy s law given in Eq. (3), such a thin cake layer will not lead to appreciable cake layer resistance. Therefore, an approximation of a constant full-scale cake layer thickness would be necessary, which according to the discussion above, is a major flaw. In order to avoid the use of this assumption, the Fig. 10. Temporal evolution of the cake layer formation: a) cake layer profile after 10,000 steps of simulation, b) particle deposition after 10,000 steps, c) cake layer profile after 50,000 steps of simulation, d) particle deposition after 50,000 steps, e) cake layer profile after 100,000 steps of simulation, and f) particle deposition after 100,000 steps. Length dimensions are given as multiples of particle radius.

12 46 J.C. Chen, A.S. Kim / Advances in Colloid and Interface Science 119 (2006) Fig. 10 (continued). model would have to be scaled to the full physical dimensions of the system. Scalability is a major issue to consider for building an accurate model simulation and will be examined more closely in Section Particle collisions and shear-induced diffusion Another feature of the current model is that it incorporates particle collision dynamics and the resultant shear-induced diffusion effects [35,36]. Particle collision dynamics is highly random and complicated. However, one of the major strengths of the model of the current study is its capability to break down complex physical phenomena into simple discrete particle movements and effectively include them in the simulation. To demonstrate this, we will examine the model s ability to accommodate the highly convoluted phenomenon of random particle collisions and the resultant shear-induced diffusion [35,36]. Here it will be shown that the model can competently capture the dynamics of random particle collisions with acute precision and accuracy.

13 J.C. Chen, A.S. Kim / Advances in Colloid and Interface Science 119 (2006) In a colloidal system, particle collisions occur inevitably, continuously, and disorderly. Fig. 11 illustrates the elaborate nature of collision dynamics. Collisions due to the variance of the fluid speed along with the particle translation will deflect particles from their original intended path, causing a form of diffusive motion termed shearinduced diffusion. From just one original particle collision springs forth a series of chain reactions. Summing the net effect of all particle collisions in the system will quickly compound into an impossible entanglement. The conventional method for coping with particle collisions is to introduce a shear-induced diffusivity into the model [35,36] given by Du; ð / Þ, Bu Bz a2 ˆD ð/ Þ ð21þ Bu where D(u,/) is the shear-induced diffusivity, Bz is the variation of the fluid velocity in the z-direction, and Dˆ (/) is the dimensionless concentration-dependent particle diffusivity without considering shear effects. However, reducing the extremely complex dynamics of particle collisions down to a single defining quantity, as given in Eq. (21), presents quite an extrapolation. Furthermore, Eq. (21) only considers the change in fluid velocity with respect to the z-direction. In the fluid flow of a real system, the fluid velocity will vary in all three directions (x, y, and z). Hence, the model for shear-induced diffusion given in Eq. (21) is highly approximated. Instead, the model of the current study offers a much better alternative by simulating particle collisions and the resultant shear-induced diffusion with exactness. To begin we refer to a stipulation made in our previous work [5] that prescribes for particle displacements in a Monte Carlo simulation to proceed in sequential orthogonal directions (Dx only, Dy only, and Dz only). This stipulation, along with an appropriate model for inter-particle interactions, will precisely simulate every particle collision and chain of reactions occurring throughout the system. First, we use an analogy of a spring to illustrate the role of inter-particle repulsion as shown in Fig. 12. The spring stores the inter-particle potential energy between the two particles in Fig. 12 and propels them to separate to reach an equilibrium separation. Thus, the compressed spring essentially models inter-particle repulsion. Using the spring Resultant chain of reactions Inter-particle repulsion modeled by a spring analogy Fig. 12. Spring analogy for inter-particle repulsions. analogy as the appropriate model for inter-particle repulsion, along with the stipulation of displacing particles solely in orthogonal directions (Dx only, Dy only, and Dz only), the dynamics of particle collisions can be simulated as shown in Fig. 13. Particles 1 and 2 are originally separated by a distance r 1. When particle 1 is displaced by Dz, the compression of the spring will deflect particle 2 a distance Dy to relax the spring to a new inter-particle separation of r 2. The net effect mimics a collision between repulsive particles 1 and 2. Adding subsequent displacements in the other two directions (Dx only and Dz only) for particle 2 will complete the simulation of one collision due a Dz move of particle 1. This technique can be repeated for all particles in the system with various combinations of initial displacements (Dx only, Dy only, and Dz only) that induce consequent reactions in each of the three directions. With each particle displacement, the effect of that displacement on other particles is stored in the inter-particle interactions and then realized as each particle is displaced (Fig. 13). In this way, the random collisions of all particles in the system are precisely modeled. What originated as an overwhelmingly complex network of particle collisions (Fig. 11), when broken down microscopically, reduces to simple discrete Monte Carlo simulation displacements with inter-particle interactions (Fig. 14). Moreover, the inclusion of particle collisions does not require any additional modifications to the model. The displacement pattern shown in Fig. 13 will occur naturally in the model simply by sequentially displacing particles in the system and evaluating the energy difference of each move with 1 Original particle collision Displacement of particle 1 in z only 1 r 1 2 r 2 2 Particle 2 is deflected y due to inter-particle repulsion Fig. 11. Schematic of random particle collisions. Fig. 13. Particle deflection due to inter-particle repulsion.

14 48 J.C. Chen, A.S. Kim / Advances in Colloid and Interface Science 119 (2006) Original particle collision Resultant chain of reactions Fig. 14. Random particle collisions precisely modeled by the spring analogy. consideration for inter-particle interactions (Section 3). The tendency for particles to be deflected as shown in Fig. 13 is contained within each evaluation of the energy difference for a proposed particle displacement in the Monte Carlo simulation. By modeling rigorously each distinct particle displacement, the overwhelming complexity of particle collision dynamics is inherently contained in the current model with acute precision and accuracy Concentration gradient driven diffusion for interacting particles The diffusion of particles propelled by concentration gradients in the system follows the same pattern of particle collisions as shown in Fig. 13. The difference here is that the collisions are induced by Brownian motion rather than a shear force. The random selection of particle displacements in the Monte Carlo simulation directly models the Brownian motion of the particles. Thus, Brownian diffusion is also inherently contained in the Monte Carlo simulation. In fact, the Smart Monte Carlo [37], Force Bias Monte Carlo [38], and Brownian Dynamics simulation methods [1] are known to be equivalent for small trial displacement of particles [29,30]. The current model incorporates Brownian diffusion with precise accuracy without the need to approximate diffusion coefficients. The diffusivity is a highly varying quantity, depending on the particle concentration and inter-particle interactions. In many previous studies, the dependence of the particle diffusivity on the concentration and inter-particle interactions casts it as a difficult parameter to accurately quantify. Continuum level modeling typically utilizes integral techniques such as the Ornstein-Zernike (OZ) method with Hyper-netted Chain (HNC) or Percus-Yevick (PY) closures to compute the diffusivity of interacting particles [16]. However, these integral methods are only approximate and limited by particle concentration. Their implementation generally involves interpolation of data, and if the system conditions exceed the limits of the method, for instance a high volume fraction, extrapolation is required. Furthermore, the OZ method with HNC or PY closures does not work well for cases of strong repulsive as well as attractive inter-particle forces. The current model not only inherently contains an accurate method to model Brownian motion, but the dependence of the diffusivity on particle concentration and inter-particle interactions are also already embedded into the model by Eqs. (15) and (16). Just as in the case of particle collisions, the effects of the inter-particle interactions on the diffusivity are stored in the model through the spring analogy of inter-particle potential given in Fig. 13. Furthermore, there are no limits on the particle concentration because every particle in the system is displaced discretely. This is a tremendous advantage over other forms of modeling which require the input of diffusivities, which are often approximated, interpolated empirically, or extrapolated. By modeling rigorously each distinct particle displacement, the current model becomes more comprehensive as it inherently simulates the effects of concentration gradient driven diffusion for interacting particles with acute accuracy and robustness Cake layer compression The question raised in Section 4.2 regarding the dynamics of particle deposition in an open system with continuous flow can now be revisited using a similar argument to that shown in Section 4.4 for particle collisions. Recall that when modeling an open system with continuous flow and dilute bulk suspension, a problem arises where particles reach a steady state of streaming out of the system with only a monolayer of deposition (Section 4.2). One cause for this is that the traverse of particles along the membrane channel occurs more rapidly than their counterpart downward deposition onto the membrane surface. If the forces governing downward deposition were greater, particle accumulation could conceivably manifest. Again, we will use a spring analogy for the inter-particle repulsion as shown in Fig. 15. Particle A is positioned below consecutive particles 1 and 2 at equilibrium spring compression with particle separations of Dh 1,2 and Dh A,1 that produces spring forces F spring,2 and F spring,1, respectively, as shown in Fig. 15. The terms k 1 and k 2 are the spring constants. At equilibrium, the downward forces (hydrodynamic drag and gravity) acting on particle 2 are 1 2 A F spring,2 = k 2 h 1,2 F spring,1 = k 1 h A,1 Fig. 15. Spring analogy for particle summation and transfer.

15 J.C. Chen, A.S. Kim / Advances in Colloid and Interface Science 119 (2006) balanced by the force F spring,2 acting from the spring connecting particles 1 and 2 (Fig. 16a). Next, particle 1 then experiences its own hydrodynamic drag and gravitational forces along with F spring,2 that is transferred from above it (Fig. 16b). Finally, particle A then will experience the net sum of forces F 1 and F 2 that is transferred from above it through the spring connecting particle 1 and particle A (Fig. 16b). The overall result is shown in Fig. 16c. The net sum of forces that particle A experiences as a result of the two particles above it will be F A : F A ¼ F 1 þ F 2 a 2 F 2 = (m 2 m f )g + 6 F spring,2 = F 2 1 a 2 v F 1 ¼ F Stokes;1 X þ ðm 1 m f Þg F 2 ¼ F Stokes;2 X þ ðm 2 m f Þg ð22þ b F spring,2 = F 2 = (m 2 m f )g + 6 a 2 v w where F Stokes,1 and F Stokes,2 are the Stokes hydrodynamic drag according to Eq. (11) acting on particles 1 and 2, respectively, X is Happel s correction factor defined by Eq. (5), g is the acceleration of gravity, m f is the mass of the fluid, and m 1 and m 2 are the masses of particles 1 and 2, respectively. In this simple case, X is close to 1, and so the Stokes law hydrodynamic drag force can be approximately applied to each particle. This is an extremely critical observation. It states that a particle will experience the net sum of downward forces from hydrodynamic and gravitational forces exerted on the particles positioned above it. Therefore, if a layer of 100 particles were positioned above particle A, the net sum of downward forces acting on particle A would be: c 1 F 1 = (m 1 m f )g + 6 a 1 v w F spring,1 = F 1 + F spring,2 = F 1 + F 2 A 2 F A ;F 1 þ F 2 þ IIIþ F 100 ð23þ In a real physical system where the number of particles reaches well into the thousands and perhaps millions, the sum of forces grows to very large magnitudes: F A ¼ F 1 þ F 2 þ IIIþF 100 þ IIIþF O >10 3 ð Þ ð24þ This consecutive summing and relaying of forces through many layers of particles is more than sufficient to lead to particle deposition and accumulation. Fig. 17 shows a schematic of this dynamic. As layers of particles continue to be added, the cake layer will compress and possibly collapse. Just as in the case of random particle collisions, the situation depicted in Fig. 16 entails tremendous complexity. Fig. 15 shows only a simplified and isolated example of three particles that have reached force equilibrium. In a real system, the configuration of particles numbers into the millions while undergoing constant motion in kaleidoscopic fashion with its network of springs (inter-particle repulsion) being continuously compressed or relaxed. The overall summing and relaying of forces become impossible to characterize mathematically. But just as seen with particle collisions, the precision of the model becomes operative here and readily breaks down the original unmanageable problem into simple, discrete particle movements. 1 A F 2 F 1 In Monte Carlo simulation, the energy evaluation for each proposed particle displacement contains the net resultant of all of the forces that are modeled to act on that particle. The net sum of forces are relayed and encoded into every inter-particle interaction and then overall resultant is realized when the particles are displaced. Thus, the model inherently contains all of the intricate complexities of forces being summed and relayed through the inter-particle interactions, as shown in Figs. 15 and 16. The same advantage reaped in our modeling particle collisions manifests again here. By modeling rigorously discrete particle displacements, we gain both valuable insights pertaining to the nature of particle deposition as well as an 2 1 A F 1 + F 2 Fig. 16. Relay and transfer of forces through the inter-particle repulsion: (a) from particle 2 to particle 1 (b) from particle 1 to particle A. Summary of the overall force transfer shown in (c).

16 50 J.C. Chen, A.S. Kim / Advances in Colloid and Interface Science 119 (2006) exact method of incorporating this phenomenon. Modeling rigorously with precision is the key to unraveling complex physical dynamics in the system Scalability to full system dimensions The issue of scalability has risen on two occasions in this study. In both cases, scalability is seen to be necessary to achieve true representation of the system. First, in Section 4.3, we find that in order for the model to be released from the assumptions of constant cake layer thickness and volume fraction throughout the system, it must compensate by simulating the actual physical dimensions of the cake layer and the associated inner concentration profiles. Next, and more important, from the discussion on the cake layer compression in Section 4.6, we conclude that a key driver of particle deposition in an open system is the summation and transfer of forces through the inter-particle interactions, which can only reach its full magnitude when the model is scaled to the full physical dimensions of the system. Hence, scalability is certainly a necessary component of achieving a total mirror reflection of the system. It is the final step in completing the four stages of model development as shown in Fig. 3. The model of the current study is now ready to be scaled to full dimensions. In its current state, it is a rigorous and comprehensive small-box simulation of an open channel crossflow filtration system. With scaling, it will become a total mirror reflection model Computational efficiency Fz Particle Deposition and Accumulation Fig. 17. Cake layer compression from many layers of force transfer. Our final major principal issue for model development is the mandate for efficient computations. The stipulation for efficiency evokes realism into any modeling endeavor. For a model simulation to be of relevance and impact, its results must be pragmatically attainable and realizable. The defining parameter for gauging the attainability of a model s results is its computational demand. So, to realize our overarching goal of a total mirror reflection model that is scaled to full system dimensions, a major obstacle in terms of its astronomical computational demand must be overcome. Here, we will examine the issue of computational efficiency. Traditionally, modeling efforts have pursued the route of computationally efficient algorithms [14,15,18]. These types of models are, in general, macroscopic in scale and predicated on theoretical development as the main driver for further progress. For instance, a given model would be centered on a particular mathematical expression to quantify the issue of interest, such as an integral expression for the excess particle flux in Arora and Davis s work [18]. Further advancement to this model calls for the derivation of a new, more accurate mathematical expression for the excess particle flux or a more efficient computational algorithm for solving the current one. The emphasis for this modeling strategy rests heavily on theoretical development. As new theories are developed, the model advances. This approach produces models that are computationally efficient, which is a great advantage. However, the disadvantages are that the model often must resort to the use of such approximations as interpolated empirical data to derive parameters such as film coefficients or diffusivities because it cannot rigorously compute them. Furthermore, some of these approximated physical parameters, for instance cake layer thickness and volume fraction as well as particle diffusivities, may then be implemented with gratuitous assumptions such as assuming them to be constant when in fact they are highly varying in the system [5,17,33]. Though easily attainable, the models may not be fully representative of the actual physical scenarios occurring in the system. Improving upon macroscopic approximations, microscopic modeling techniques have begun to emerge as alternative modeling solutions in such fields of studies as colloidal transport in membrane filtration [1,5]. However, due to the much greater computational demand of microscopic techniques, they are often utilized used concurrently with complementary macroscopic theories. The work of Chen et al. [5] demonstrates such a modeling approach, using a Monte Carlo technique to perform an efficient small-box simulation to obtain the cake layer volume fraction and then inputting that quantity into a macroscopic filtration theory to generate a system response. Improvements are indeed made using this approach, as certain parameters of the system are now modeled rigorously. By this method, Chen et al. [5] attain rich insight about the cake layer phase transition phenomenon in relationship to the maximum random packing limit and critical flux. However, this approach is still unsatisfactory because in the transfer of the rigorously computed parameter to the macroscopic theory, gratuitous assumptions are made. In the case of the work of Chen et al. [5], the assumption is that the volume fraction computed remains constant throughout the cake layer. Furthermore, a small-box simulation will overlook important physical dynamics, such as the force summation phenomenon presented in Section 4.6, which only becomes apparent with a full-scale model. So, a hybrid combination of a microscopic model to a macroscopic theory offers only a compromise solution that balances limited rigorous modeling with computational efficiency.