Oil droplet behavior at a pore entrance in the presence of crossflow: Implications for microfiltration of oil-water dispersions
Tohid Darvishzadeh, Volodymyr V. Tarabara, Nikolai V. Priezjev
aa r X i v : . [ phy s i c s . f l u - dyn ] J un Oil droplet behavior at a pore entrance in the presence ofcrossflow: Implications for microfiltration of oil-water dispersions
Tohid Darvishzadeh , Volodymyr V. Tarabara , Nikolai V. Priezjev Department of Mechanical Engineering,Michigan State University, East Lansing, Michigan 48824 and Department of Civil and Environmental Engineering,Michigan State University, East Lansing, Michigan 48824 (Dated: September 5, 2018)
Abstract
The behavior of an oil droplet pinned at the entrance of a micropore and subject to clossflow-induced shear is investigated numerically by solving the Navier-Stokes equation. We found that inthe absence of crossflow, the critical transmembrane pressure required to force the droplet into thepore is in excellent agreement with a theoretical prediction based on the Young-Laplace equation.With increasing shear rate, the critical pressure of permeation increases, and at sufficiently highshear rates the oil droplet breaks up into two segments. The results of numerical simulationsindicate that droplet breakup at the pore entrance is facilitated at lower surface tension, higheroil-to-water viscosity ratio and larger droplet size but is insensitive to the value of the contactangle. Using simple force and torque balance arguments, an estimate for the increase in criticalpressure due to crossflow and the breakup capillary number is obtained and validated for differentviscosity ratios, surface tension coefficients, contact angles, and drop-to-pore size ratios. . INTRODUCTION Understanding the dynamics of an oil droplet at a pore entrance is a fascinating problemat the intersection of fluid mechanics and interface science that is of importance in such natu-ral and engineering processes as extraction of oil from bedrock, lubrication, aquifer smearingby non-aqueous phase liquids, and sealing of plant leaf stomata [1–4]. Membrane-based sep-aration of liquid-liquid dispersions and emulsions is a salient example of a technology wherethe knowledge of liquid droplet behavior in the vicinity of a surface pore is critical for thesuccess of practical applications. Milk fractionation, produced water treatment, and recov-ery of electrodeposition paint are examples of specific processes used in food, petroleum, andautomotive industries where porous membranes are relied on to separate emulsions [5–7].The membrane separation technique can be particularly useful when small droplets needto be removed from liquid-liquid dispersions or emulsions because other commonly usedtechnologies, such as hydrocyclones and centrifugation-based systems, are either incapableof removing droplets smaller than a certain critical size (e.g., ∼ µ m for hydrocyclones)or are expensive and have insufficient throughput (e.g., centrifuges). The early work bythe Wiesner group [8] and others [9] on oil droplet entry into a pore provided an estimateof the critical pressure of permeation; however, the understanding of the entire processof the droplet dynamics at a micropore entrance is still lacking, especially with regard tothe practically-relevant case of crossflow systems where blocking filtration laws [10] are,strictly speaking, not applicable. Crossflow membrane microfiltration is used to separateemulsions by shearing droplets of the dispersed phase away from the membrane surface andletting the continuous phase pass through [11]. In contrast to the normal, or dead-end,mode of filtration, crossflow microfiltration allows for higher permeate fluxes due to betterfouling control [12]. However, the accumulation of the dispersed phase on the surface ofthe membrane and inside the pores, i.e., fouling of the membrane, can eventually reduceefficiency of the process to an unacceptably low level even in the presence of crossflow.Another important application that entails interaction of liquid droplets with porousmedia is membrane emulsification, where micron-sized droplets are produced by forcing aliquid stream through membrane pores into a channel where another liquid is flowing [13].The emerging droplets break when the viscous forces exerted by crossflow above the mem-brane surface are larger than surface tension forces [14]. Membrane emulsification requires2ess energy and produces a more narrow droplet size distribution [15, 16] than conventionalmethods such as ultrasound emulsification [17] and stirring vessels [18].In general, the studies of petroleum emulsions have been performed at two differentscales, namely, macroscopic or bulk scales and mesoscopic or droplet scales [19, 20]. Earlyresearch on membrane emulsification and microfiltration involved bulk experiments aimedat determining averaged quantities and formulating empirical relations [21]. These studiesconsidered macroscopic parameters such as droplet size distribution, dispersed phase con-centration, and bulk properties such as permeate flux [22, 23]. These empirical approacheswere adopted due to inherent complexity of two-phase systems produced by bulk emulsifica-tion, where shear stresses are spatially inhomogeneous and the size distribution of dropletsis typically very broad [19, 24]. However, with the development of imaging techniques andnumerical methods, the shape of individual droplets during deformation and breakup couldbe more precisely quantified for various flow types and material parameters [25, 26].First studies of the droplet dynamics date back to 1930’s, when G. I. Taylor systematicallyinvestigated the deformation and breakup of a single droplet in a shear flow [27, 28]. Sincethen, many groups have examined this problem theoretically [29, 30] and in experiments [31–33]. A number of research groups have studied experimentally how a droplet pinned at theentrance of an unconfined pore deforms when it is exposed to a shear flow [15]. Experimentshave also been performed to measure the size of a droplet after breakup as a function ofshear rate and viscosity ratio [34, 35]. Numerical simulations of the droplet deformationand breakup have been carried out using various methods including boundary integral [36],Lattice Boltzmann [37], and Finite Volume [38] methods. These multiphase flow simulationsgenerally use an interface-capturing method to track the fluid interfaces. Among otherfront-tracking methods, the Volume of Fluid method simply defines the fluid-fluid interfacethrough a volume fraction function, which is updated based on the velocity field obtainedthrough the solution of the Navier-Stokes equation [39, 40]. The Volume of Fluid method ismass-preserving, it is easily extendable to three-dimensions, and it does not require specialtreatment to capture topological changes [41].The drag force and torque on droplets or particles attached to a solid substrate and subjectto flow-induced shear stress depend on their shape and the shear rate. Originally, O’Neillderived an exact solution for the Stokes flow over a spherical particle on a solid surface [42].Later, Price computed the drag force on a hemispherical bump on a solid surfaces under3inear shear flow [43]. Subsequently, Pozrikidis extended Price’s work to study the case ofa spherical bump with an arbitrary angle using the boundary integral method [44]. Morerecently, Sugiyama and Sbragaglia [45] varied the viscosity ratio to include values other thaninfinity (the only value considered by Price [43]) and found an exact solution for the flow overa hemispherical droplet attached to a solid surface. Assuming that the droplet is pinned tothe surface, an estimate for the drag force, torque, and the deformation angle as a functionof the viscosity ratio was obtained analytically [45]. Also, Dimitrakopoulos showed that thedeformation and orientation of droplets attached to solid surfaces under linear shear flowdepend on the contact angle, viscosity ratio, and contact angle hysteresis [46].More recently, Darvishzadeh and Priezjev [47] studied numerically the entry dynamicsof nonwetting oil droplets into circular pores as a function of the transmembrane pressureand crossflow velocity. It was demonstrated that in the presence of crossflow above themembrane surface, the oil droplets can be either rejected by the membrane, permeate into apore, or breakup at the pore entrance. In particular, it was found that the critical pressure ofpermeation increases monotonically with increasing shear rate, indicating optimal operatingconditions for the enhanced microfiltration process. However, the numerical simulationswere performed only for one specific set of parameters, namely, viscosity ratio, contactangle, surface tension coefficient, and droplet-to-pore size ratio. One of the goals of thepresent study is to investigate the droplet dynamics in a wide range of material parametersand shear rates.In this paper, we examine the influence of physicochemical parameters such as surfacetension, oil-to-water viscosity ratio, droplet size, and contact angle on the critical pressure ofpermeation of an oil droplet into a membrane pore. In the absence of crossflow, our numericalsimulations confirm analytical predictions for the critical pressure of permeation based onthe Young-Laplace equation. We find that when the crossflow is present above the membranesurface, the critical pressure increases, and the droplet deforms and eventually breaks upwhen the shear rate is sufficiently high. Analytical predictions for the breakup capillarynumber and the increase in critical permeation pressure due to crossflow are compared withthe results of numerical simulations based on the Volume of Fluid method.The rest of the paper is structured as follows. In the next section, the details of numericalsimulations and a novel procedure for computing the critical pressure of permeation aredescribed. In Section III, the summary of analytical predictions for the critical pressure4ased on the Young–Laplace equation is presented, and the effects of confinement, viscosityratio, surface tension, contact angle, and droplet size on the critical transmembrane pressureand breakup are studied. Conclusions are provided in the last section. II. DETAILS OF NUMERICAL SIMULATIONS
Three-dimensional numerical simulations were carried out using the commercial softwareANSYS FLUENT [48]. The FLUENT flow solver utilizes a control volume approach, whilethe Volume of Fluid (VOF) method is implemented for the interface tracking in multiphaseflows. In the VOF method, every computational cell contains a certain amount of eachphase specified by the volume fraction. For two-phase flows, the volume fractions of 1 and0 describe a computational cell occupied entirely by one of the phases, while any value inbetween corresponds to a cell that contains an interface between the two phases [49]. Inour simulations, GAMBIT was employed to generate the mesh. In order to increase thesimulation efficiency, we generated a hybrid mesh that consists of fine hexagonal meshes ina part of the channel that contains the droplet and coarse tetrahedral meshes in the rest ofthe channel. A user-defined function was used to initialize the droplet shape and to adjustthe velocity of the top wall that induced shear flow in the channel, as shown schematicallyin Fig. 1.As we recently showed, the dynamics of the oil-water interface inside the pore slowsdown significantly when the transmembrane pressure becomes close to the critical pressureof permeation [47]. Hence, the interface inside the pore is nearly static and the pressure jumpacross the spherical interface is given by the Young-Laplace equation. However, numericalsimulations are required to resolve accurately the velocity field, pressure, and shape of thedeformed droplet above the pore entrance. In the present study, we propose a novel numericalprocedure to compute the critical pressure of droplet permeation and breakup, as illustratedin Fig. 2. First, the pressure jump across the static interface inside the pore is calculatedusing the Young-Laplace equation. Second, we simulate the oil droplet in the presence ofsteady shear flow when the droplet covers the pore entrance completely and the oil phasepartly fills the pore. In the computational setup, the pore exit is closed to prevent the massflux and to keep the droplet at the pore entrance. The difference in pressure across thedeformed oil-water interface with respect to the inlet pressure is measured in the oil phase5t the bottom of the pore (see Fig. 2). The critical pressure of permeation is then found byadding the pressure differences from the first and second steps. In the previous study [47],the critical pressure of permeation at a given shear rate was determined iteratively by testingseveral transmembrane pressures close to the critical pressure. Using the novel approach, wewere able to reproduce our previous results [47] faster and with higher accuracy. Moreover,this numerical procedure was automated to detect the critical pressure while increasing shearrate quasi-steadily, so that less post-processing is required.The solution of the Navier-Stokes equations for the flow over the membrane surfacerequires specification of the appropriate boundary conditions. As shown in Fig. 1, thereare four types of boundary conditions used in the computational domain. The membranesurface is modeled as a no-slip boundary. A moving “wall” boundary condition is appliedat the top surface of the channel to induce shear flow between the moving top wall andthe stationary membrane surface. The bottom of the pore is also described by the “wall”boundary condition to prevent the mass flux and to keep the oil droplet pinned at thepore entrance. Periodic boundary conditions are imposed at the upstream and downstreamentries of the channel. On the lateral side of the channel in the ( Z +) direction, a pressure-inlet boundary condition is applied to allow mass transfer, and to ensure that the referencepressure is fixed. Finally, a “symmetry” condition is implemented and only half of thecomputational domain is simulated to reduce computational efforts. We performed testsimulations with an oil droplet r d = 2 µ m exposed to shear flow and found that the localvelocity profiles at the upstream, downstream, and the lateral sides remained linear when thewidth and length of the computational domain were fixed to 12 µ m and 36 µ m, respectively.These values were used throughout the study. The effect of confinement in the directionnormal to the membrane surface on the droplet deformation and breakup will be investigatedseparately in the subsection III B.The interface between two phases is described by a scalar variable, known as the volumefraction α , which is convected by the flow at every iteration via the solution of the transportequation as follows: ∂α∂t + ∇· ( α V ) = 0 , (1)where V is the three-dimensional velocity vector. The time dependence of the volumefraction is determined by the velocity field near the interface. Next, since the cells containingthe interface include both phases, the material properties are averaged in each cell; for6nstance, the volume-fraction-averaged density is computed as follows: ρ = α ρ + (1 − α ) ρ . (2)Using the averaged values of viscosity and density, the following momentum equation issolved: ∂∂t ( ρ V ) + ∇ · ( ρ VV ) = −∇ p + ∇ · [ µ ( ∇ V + ∇ V T )] + ρ g + F , (3)where V is the velocity vector shared between two phases, g is the gravitational acceleration,and F is the surface tension force per unit volume, which is given by F = σ ρ κ ∇ α ( ρ + ρ ) , (4)where σ is the surface tension coefficient and κ is the curvature of the oil-water interface,which in turn is defined as κ = 1 | n | h(cid:16) n | n | · ∇ (cid:17) | n | − ( ∇ · n ) i , (5)where n is the vector normal to the interface. The surface tension force given by Eq. (4) isnonzero only at the interface and it acts in the direction normal to the interface ( n = ∇ α ).Segments with higher interface curvature produce larger surface tension forces and tend tosmooth out the interface [50]. The orientation of the interface at the wall is specified by thecontact angle. The unit normal for a cell containing the interface at the wall is computedas follows: n i = n w cos θ + n t sin θ, (6)where n w and n t are the unit vectors normal to the wall and normal to the contact lineat the wall, respectively. The angle θ is the static contact angle measured in the dispersedphase [39].A SIMPLE method was utilized for the pressure-velocity decoupling. A second orderupwind scheme was used for discretization of the momentum equation and a staggered meshwith central differencing was used for the pressure equation. Piecewise Linear InterfaceReconstruction (PLIC) algorithm was employed to reconstruct the interface in each cell [51].The continuum surface force model of Brackbill et al. [39] was used to compute the surfacetension force.An accurate computation of the pressure and velocity fields for problems involving fluidinterfaces requires a precise estimate of the interfacial curvature. It is well known that7iscrete formulation of an interface produces a loss of accuracy in regions of high curvatureand, therefore, requires a sufficiently fine mesh. The numerical simulations were performedusing the mesh size of 0 . µ m, which corresponds to 32 mesh cells along the perimeter ofthe membrane pore. To ensure that the mesh resolution is sufficiently high, we performedsimulations at different shear rates using 2 and 4 times finer meshes and found that theresulting refinements in the final position of the droplet interface and the values of thecritical permeation pressure were negligible. The total volume of the oil phase inside thepore and above the membrane surface was used to calculate the droplet radius. Unlessotherwise specified, the following parameters were used throughout the study: the poreradius is r p = 0 . µ m, the droplet radius is r d = 2 µ m, the contact angle is θ = 135 ◦ , andthe surface tension coefficient is σ = 19 . III. RESULTSA. The critical pressure of permeation and the breakup capillary number
The pressure jump across a static interface between two immiscible fluids can be deter-mined from the Young–Laplace equation as a product of the interfacial tension coefficientand the mean curvature of the interface or ∆ P = 2 σ κ . For a pore of arbitrary cross-section,the mean curvature of the interface is given by κ = C p cos θ A p , (7)where C p and A p are the cross-sectional circumference and area of the pore, respectively [52].Therefore, the critical pressure of permeation of a liquid film into a pore of arbitrary cross-section is given by P cr = σ C p cos θA p . (8)In our recent study [47], the theoretical prediction for the critical permeation pressure,Eq. (8), was validated numerically for oil films on a membrane surface with rectangular,elliptical, and circular pores.In the case of a liquid droplet blocking a membrane pore, the critical pressure of per-meation, Eq. (8), has to be adjusted to account for the finite size of the droplet. It waspreviously shown [8, 9] that the critical pressure for an oil droplet of radius r d to enter a8ircular pore of radius r d is given by P cr = 2 σ cos θr p s − θ − cos θ r d /r p ) cos θ − (2 − θ + sin θ ) . (9)We showed earlier that the analytical prediction for the critical pressure given by Eq. (9)agrees well with the results of numerical simulations for an oil droplet at the pore entrancein the absence of crossflow [47]. In the presence of crossflow, however, Eq. (9) in not valid asthe shear flow deforms the droplet rendering its interface above the membrane surface non-spherical [47]. Furthermore, numerical simulations have shown that the critical pressure ofpermeation increases with increasing crossflow velocity up to a certain value, above which thedroplet breaks up [47]. Hence, the phase diagram was determined for the droplet rejection,permeation, and breakup depending on the transmembrane pressure and shear rate [47].In the present study, the critical permeation pressure is determined more accurately andits dependence on shear rate is studied numerically for a range of material properties andgeometrical parameters.In the presence of crossflow above the membrane surface, an oil droplet breaks up whenviscous stresses over the droplet surface exposed to the flow become larger than capillarystresses at the interface of the droplet near the membrane pore. Therefore, at the momentof breakup, the drag force in the flow direction is balanced by the capillary force at thedroplet interface around the pore D ≈ F σ . (10)Neglecting the contact angle dependence, F σ ∝ σ r p is the interfacial force acting in thedirection opposite to the flow at the droplet interface near the pore entrance. The dragforce generated by a linear shear flow on a spherical droplet attached to a solid surface isgiven by D ∝ f D ( λ ) µ ˙ γ r d , (11)where µ is the viscosity of the continuous phase, ˙ γ is the shear rate, and r d is the radius ofthe droplet [45, 53]. The coefficient f D ( λ ) is a function of the viscosity ratio λ = µ oil /µ water and it depends on the shape of the droplet above the surface. Sugiyama and Sbragaglia [45]have estimated this function analytically for a hemispherical droplet ( θ = 90 ◦ ) attached toa solid surface f D ( λ ) ≈ . λ . λ. (12)9y plugging Eq. (11) into Eq. (10) and introducing ¯ r = r d / r p , the critical capillary numberfor breakup of a droplet on a pore can be expressed as follows: Ca cr ∝ f D ( λ ) ¯ r , (13)where the capillary number is defined as Ca = µ w ˙ γr d /σ .The difference in pressure inside the pore in the presence of flow and at zero shear ratecan be estimated from the torque generated by the shear flow on the droplet surface. Thetorque around the center of the droplet projected on the membrane surface is given by T ∝ f T ( λ ) µ ˙ γ r d , (14)It was previously shown [45] that for a hemispherical droplet on a solid surface, f T ( λ ) is afunction of the viscosity ratio f T ( λ ) ≈ . λ . λ . (15)Hence, the balance of the torque due to shear flow above the membrane surface [givenby Eq. (14)] and the torque arising from the pressure difference, ( P cr − P cr ) A p r d , can bereformulated in terms of the capillary number and drop-to-pore size ratio as follows: P cr − P cr ∝ f T ( λ ) σ ¯ r Car p , (16)where P cr is the critical permeation pressure in the absence of crossflow.In what follows, we consider the effects of confinement, viscosity ratio, surface tension,contact angle, and droplet size on the critical pressure of permeation and breakup usingnumerical simulations and analytical predictions of Eq. (13) and Eq. (16). B. The effect of confinement on droplet deformation and breakup
In practical applications, the dimensions of a crossflow channel of a microfiltration systemare much larger than the typical size of emulsion droplets so that the velocity profile overthe distance of about r d from the membrane surface can be approximated as linear. Tomore closely simulate this condition in our computational setup, the shear flow above themembrane surface was induced by moving the upper wall of the crossflow channel (Fig. 1).To understand how the finite size of the channel affects droplet dynamics at the membrane10urface, we studied the influence of the channel height on the droplet behavior. The con-finement ratio is defined as the ratio of the height of the droplet residing on the pore atzero shear rate H d (i.e., the height of a spherical cap above the membrane surface) to thechannel height H ch . It is important to note that the degree of confinement is varied only inthe direction normal to the membrane surface and the computational domain is chosen tobe wide enough for the lateral confinement effects to be negligible (see Section II).We performed numerical simulations of an oil droplet with radius r d = 2 µ m in steady-state shear flow for the channel heights 3 . µ m H ch . µ m. Figure 3 illustrates theeffect of confinement on the shape of the droplet residing on a r p = 0 . µ m pore when thecapillary number is Ca = µ w ˙ γr d /σ = 0 . . µ m. It can be observed from Fig. 3 thathighly confined droplets become more elongated in the direction of flow than droplets withlower confinement ratios, which is in agreement with the results of previous simulations [54].When a droplet is highly confined, the distance between the upper moving wall and thetop of the droplet is relatively small. As a result, the effective shear rate at the surface ofthe droplet is higher and the droplet undergoes larger deformation. Furthermore, the cross-sectional profiles for the confinement ratios of 0 .
428 and 0 .
286 are nearly identical, indicatingthat the flow around the droplet is not affected by the upper wall when H d /H ch . . .
5, the breakup capillary number remains nearlyconstant. For the rest of the study, the channel height was fixed to 8 µ m, which correspondsto the confinement ratio of 0 .
428 for a droplet with radius r d = 2 µ m. For the resultspresented in the subsection III F, the channel height was scaled appropriately to retain thesame confinement ratio for larger droplets. C. The effect of viscosity ratio on the critical transmembrane pressure
The ratio of viscosities of the dispersed and continuous phases is an important factor thatdetermines the magnitude of viscous stresses at the interface between the two phases. For a11mall droplet at low Reynolds numbers, the viscous stresses are primarily counterbalancedby interfacial tension stresses. In a shear flow, viscous stresses tend to distort the surfaceof a droplet, while interfacial stresses assist in retaining its initial spherical shape. Thecompetition between the two stresses determines the breakup criterion, deformation, andorientation of the droplet [26, 55]. In this subsection, we investigate numerically the effectof viscosity ratio on the droplet deformation and breakup at the entrance of the membranepore.Figure 5 shows the effect of the viscosity ratio, λ = µ o /µ w , on the critical pressure ofpermeation and breakup of an oil droplet on a membrane pore as a function of the capillarynumber. The percent increase in critical pressure is defined with respect to the criticalpressure in the absence of crossflow P cr , i.e., ( P cr − P cr ) /P cr ×
100 %. Keeping in mindthat P cr does not depend on λ , the results shown in Fig. 5 demonstrate that at a fixed Ca , the critical pressure increases with increasing viscosity ratio, which implies that higherviscosity droplets penetrate into the pore at higher transmembrane pressures. Specifically,the maximum increase in critical pressure just before breakup is about 8 % for λ = 1 andabout 15 % for λ = 20. Furthermore, highly viscous droplets tend to break at lower shearrates because of the larger torque generated by the shear flow [see Eq. (14)]. As reported inFig. 5, the critical capillary number for breakup varies from about 0 .
018 for λ = 20 to 0 . λ = 1. The practical implication of these results is that in membrane emulsificationprocesses the use of liquids with lower viscosity ratios should be avoided as the dropletstend to break at higher shear rates.Examples of cross-sectional profiles of the oil droplet in steady shear flow are presented inFig. 6 for the viscosity ratio λ = 1. At small capillary numbers, no significant deformationoccurs and the droplet retains its spherical shape above the membrane surface. As Ca increases, a neck forms at the pore entrance while the rest of the droplet remains nearlyspherical. A closer look at the shapes of the droplet for Ca = 0 . . Ca .12e next estimate the breakup time and compare it with the typical deformation time ofthe droplet interface for different viscosity ratios. In our simulations, the upper wall velocityis increased quasi-steadily and the spontaneous initiation of the breakup process can beclearly detected by visual inspection of the droplet interface near the pore entrance. We thenidentify the moment when a droplet breaks into two segments and compute the breakup time.The deformation time scale, defined by µ w r d (1 + λ ) /σ , is a measure of the typical relaxationtime of the droplet interface with respect to its deformation at steady state [46, 56]. In Fig. 7,the breakup time is plotted against the deformation time scale for different viscosity ratios.Notice that the breakup time increases linearly with the deformation time scale, whichconfirms that highly viscous droplets break up more slowly. The inset in Fig. 7 displaysthe droplet cross-sectional profiles just before breakup for the same viscosity ratios. It canbe observed that the profiles nearly overlap with each other, indicating that droplets withdifferent viscosities are deformed identically just before breakup.According to Eq. (13), the breakup capillary number depends on the drop-to-pore sizeratio and the viscosity ratio via the function f D ( λ ). Therefore, it is expected that theproduct Ca cr f D ( λ ) will be independent of λ and the appropriate dimensionless number fora constrained viscous droplet in a shear flow is Ca f D ( λ ). Moreover, based on Eq. (16),the percent increase in the critical pressure is independent of the viscosity ratio when it isdivided by f T ( λ ). Figure 8 shows the same data as in Fig. 5 but replotted in terms of thenormalized critical pressure and the modified capillary number. As is evident from Fig. 8,the data for different viscosity ratios nearly collapse on the master curve. It is seen thatdroplets break at approximately the same value Ca f D ( λ ) ≈ .
09. In practice, the increasein critical pressure due to crossflow can be roughly estimated from the master curve in Fig. 8for any viscosity ratio in the range 1 λ
20. Also, if
Ca f D ( λ ) & .
09, the oil dropletswill break near the pore entrance for any viscosity ratio.
D. The effect of surface tension on the critical pressure of permeation
In this subsection, we investigate the influence of surface tension on the critical perme-ation pressure, deformation and breakup of an oil droplet residing at the pore entrance inthe presence of crossflow above the membrane surface. Figure 9 shows the critical pressureof permeation as a function of shear rate for five values of the surface tension coefficient.13s expected from Eq. (9), the critical pressure at zero shear rate increases linearly with in-creasing surface tension coefficient. Note that oil droplets with higher surface tension breakup at higher shear rates because larger stresses are required to deform the interface andcause breakup of the neck. Also, the difference between the critical pressure just beforebreakup and P cr is larger at a higher surface tension; for example, it is about 1.5 kPa for σ = 9 .
55 mN/m and 6 kPa for σ = 38 . γ = 1 . × s − . It can be observed that oil dropletswith lower surface tension become highly deformed along the flow direction. The elongationis especially pronounced when the surface tension coefficient is small; for σ = 9 .
55 mN/mthe droplet interface is deformed locally near the pore entrance and the neck is formed.To further investigate the effect of surface tension on the droplet breakup, we comparethe breakup time and the deformation time scale µ w r d (1 + λ ) /σ . The numerical resultsare summarized in Fig. 11 for the same values of the surface tension coefficient as in Fig. 9.Similar to the analysis in the previous subsection, the breakup time was estimated from thetime when a droplet becomes unstable under quasi-steady perturbation till the formation oftwo separate segments. It can be observed in Fig. 11 that the breakup time varies linearlywith increasing deformation time scale, which in turn indicates that the breakup time isinversely proportional to the surface tension coefficient. In addition, the inset in Fig. 11shows the cross-sectional profiles of the droplet just before breakup for the same surfacetension coefficients. Interestingly, the profiles nearly coincide with each other, indicatingthat the droplet shape at the moment of breakup is the same for any surface tension.In order to present our results in a more general form, we replotted the data from Fig. 9in terms of the percent increase in critical pressure, ( P cr − P cr ) /P cr ×
100 %, and thecapillary number in Fig. 12. Note that in all cases, the data collapse onto a master curveand breakup occurs at the same relative pressure ( P cr − P cr ) /P cr ≈ Ca cr ≈ . σ and Ca , and when it is divided by P cr , which itself is a linear function of σ [see Eq. (9)], the percent increase in critical pressure becomes proportional to the capillarynumber. In practice, the master curve reported in Fig. 12 can be used to predict the criticalpermeation pressure and breakup of emulsion droplets for specific operating conditions andsurface tension. E. The effect of contact angle on the droplet dynamics near the pore
Next, we focus on the effect of contact angle on the permeation pressure, deformation andbreakup of oil droplets on a membrane pore. The variation of the critical permeation pressureas a function of the capillary number is presented in Fig. 13 for nonwetting oil droplets withcontact angles 115 ◦ θ ◦ . The critical pressure at zero shear rate is higher for oildroplets with larger contact angles, which is in agreement with the analytical prediction ofEq. (9). As expected, with increasing shear rate, the critical pressure of permeation increasesfor all values of θ studied. We estimate the maximum change in the critical pressure to beabout 3 kPa and roughly independent of the contact angle. This corresponds to a relativeincrease of about 6% for the contact angle θ = 155 ◦ and 21% for θ = 115 ◦ . These resultssuggest that the relative efficiency of a microfiltration system due to crossflow is higher foremulsion droplets with lower contact angles. Interestingly, we find that the critical capillarynumber for breakup ( Ca cr ≈ . Ca can be used as a criterion for predicting breakup. Finally, the examples of the dropletcross-sectional profiles are shown in Fig. 14 for different contact angles when Ca = 0 . F. The effect of droplet size on the critical pressure of permeation
In the microfiltration process, the size of the membrane pore is one of the crucial param-eters that determine the permeate flux and membrane selectivity. Membranes with smallerpore sizes provide higher rejections but require higher transmembrane pressures to achievethe same permeate flux. In this subsection, we examine the influence of the drop-to-pore15ize ratio on the critical pressure of permeation and the breakup dynamics of oil droplets inthe presence of crossflow above the membrane surface.Figure 15 reports the critical permeation pressure as a function of shear rate for the dropletradii in the range from 1 . µ m to 2 . µ m, while the pore radius is fixed at r p = 0 . µ m. Inthe absence of crossflow, the critical pressure is higher for larger droplets because they havelower curvature of the interface above the membrane surface, which is in agreement with theanalytical prediction of Eq. (9). With increasing shear rate, the critical pressure increasesfor droplets of all sizes. Note also that the slope of the curves in Fig. 15 is steeper for largerdroplets because of the larger surface area exposed to shear flow, resulting in a higher dragtorque, and, consecutively, a higher transmembrane pressure needed to balance the torque.Furthermore, as shown in Fig. 15, smaller droplets break at higher shear rates, since highershear stress are required to produce sufficient deformation for the breakup to occur. Themaximum relative critical pressure is about 14% for r d /r p = 3 and 6% for r d /r p = 5.We next compute the difference in the critical permeation pressure with respect to thecritical pressure in the absence of flow, P cr − P cr , and define ¯ r = r d /r p . According toEq. (13), the product Ca cr × ¯ r is independent of the droplet radius. At the same time,Eq. (16) suggests that the increase in critical pressure depends on the droplet radius via theterm Ca × ¯ r . Figure 16 shows the critical pressure difference as a function of the modifiedcapillary number Ca × ¯ r for different droplet radii. It can be observed in Fig. 16 that allcurves nearly collapse on each other and the droplet breakup occurs at the same value Ca × ¯ r ≈ . r d . This approximation becomes more accurate for larger drop-to-pore size ratios, and, thus, the critical pressure difference in Fig. 16 is nearly the same forlarger droplets even at high shear rates.The inset of Fig. 16 shows the cross-sectional profiles of oil droplets just before breakupfor different drop-to-pore size ratios. Note that all droplets are pinned at the pore entranceand elongated in the direction of flow. It is seen that when ¯ r is small, the droplet shape issignificantly deformed from its original spherical shape. In contrast, larger droplets remainnearly spherical and only deform near the pore entrance. In general, the droplet-to-pore sizeratio should be large enough to make P cr sufficiently high for practicable separation. At thesame time, if the pore size is much smaller than the droplet size, the water flux through the16embrane decreases and the probability of breakup increases, which could result in lowerrejection rates and internal fouling of the membrane. Therefore, choosing a membrane withan appropriate pore size could greatly increase the efficiency of the microfiltration process. IV. CONCLUSIONS
In this paper, we performed numerical simulations to study the effect of material proper-ties on the deformation, breakup, and critical pressure of permeation of oil droplets pinnedat the membrane pore of circular cross-section. In our numerical setup, the oil droplet wasexposed to a linear shear flow induced by the moving upper wall. We used finite-volumenumerical simulations with the Volume of Fluids method to track the interface betweenwater and oil. The critical pressure of permeation was computed using a novel procedure inwhich the critical permeation pressure was found by adding pressure jumps across oil-waterinterfaces of the droplet inside the pore and above the membrane surface. First, the pres-sure jump across the static interface inside the pore was calculated using the Young-Laplaceequation. Then, the pressure jump across the dynamic interface above the membrane surfacewas computed numerically and added to the pressure jump inside the pore. This methodhas proven to be accurate, robust, and computationally efficient. To determine the dimen-sions of the computational domain, we also studied the effect of confinement on the dropletdeformation and breakup and concluded that in order to minimize finite size effects andcomputational costs, the distance between the membrane surface and the upper wall has tobe at least twice the droplet diameter. In particular, it was observed that highly confineddroplets become significantly deformed in a shear flow and break up more easily.In the absence of crossflow, we found that the analytical prediction for the critical perme-ation pressure derived by Nazzal and Wiesner [8] agrees well with the results of numericalsimulations for different oil-to-water viscosity ratios, surface tension, contact angles, anddroplet sizes. In general, with increasing crossflow shear rate, the critical permeation pres-sure increases with respect to its zero-shear-rate value and the droplet undergoes elongationin the flow direction followed by breakup into two segments. The results of numerical sim-ulations indicate that at a fixed shear rate, the critical permeation pressure increases as afunction of the viscosity ratio, which implies that more viscous droplets penetrate into thepore at higher transmembrane pressures. In agreement with a scaling relation for the critical17apillary number, we also found that droplets of higher viscosity tend to break at lower shearrates. Furthermore, with increasing surface tension coefficient, the maximum increase in thecritical permeation pressure due to crossflow becomes larger and the droplet breakup occursat higher shear rates. Interestingly, the percent increase in critical permeation pressure asa function of the capillary number was found to be independent of the surface tension co-efficient. Next, we showed that the breakup capillary number and the increase in criticalpressure of permeation are nearly independent of the contact angle. Last, it was demon-strated that smaller droplets penetrate into the pore at lower pressures and break up athigher shear rates because larger shear stresses are needed to deform the interface above themembrane surface.While most microfiltration membranes used in medium- to large-scale separation ap-plications have pores of complex morphologies and a distribution of nominal sizes, resultsobtained for the simple case of a pore of circular cross-section can be useful for identifyinggeneral trends. With the development of new methods of manufacturing micro-engineeredmembranes [57] and the rapid growth in the diversity and scale of applications of microflu-idic devices, conclusions obtained in this work can be of direct practical value for guidingmembrane design and optimizing process variables.
Acknowledgments
Financial support from the Michigan State University Foundation (Strategic Partner-ship Grant 71-1624) and the National Science Foundation (Grant No. CBET-1033662) isgratefully acknowledged. Computational work in support of this research was performed atMichigan State University’s High Performance Computing Facility. [1] J.T. Morgan, D.T. Gordon, J. Petrol. Technol. 22 (1970) 1199.[2] A. Erdemir, Tribology Int. 38 (2005) 249.[3] C. Lee, J. Lee, J. Cheon, K. Lee, J. Environ. Eng. 127 (2001) 639.[4] H. Kaiser, N. Legner, Plant Physiology, 143 (2007) 1068.[5] G. Brans, C.G.P.H. Schroen, R.G.M. van der Sman, R.M. Boom, J. Membr. Sci. 243 (2004)263.
6] J.A. Veil, Produced Water Management Options and Technologies. In: Produced Water,Springer, 2011, pp. 537-571.[7] R. Baker, Membrane Technology and Applications. John Wiley and Sons, 2013.[8] F.F. Nazzal, M.R. Wiesner, Water Environ. Res. 68 (1996) 1187.[9] I.W. Cumming, R.G. Holdich, I.D. Smith, J. Membr. Sci. 169 (2000) 147.[10] J. Hermia, Trans. Inst. Chem. Eng. 60 (1982) 183.[11] A.B. Koltuniewicz, R.W. Field, T.C. Arnot, J. Membr. Sci. 102 (1995) 193.[12] J. Mueller, Y. Cen, R.H. Davis, J. Membr. Sci. 129 (1997) 221.[13] A.J. Gijsbertsen-Abrahamse, A. van der Padt, R.M. Boom, J. Membr. Sci. 230 (2004) 149.[14] P. Walstra, Chem. Eng. Sci. 48 (1993) 333.[15] C.F. Christopher, S.L. Anna, J. Phys. D: Appl. Phys. 40 (2007) R319.[16] G.T. Vladisavljevi`c, S. Tesch, H. Schubert, Chem. Eng. Process. 41 (2002) 231.[17] B. Abisma¨ıl, J.P. Canselier, A.M. Wilhelm, H. Delmas, C. Gourdon, Ultrason. Sonochem. 6(1999) 75.[18] H. Karbstein, H. Schubert, Chem. Eng. Process. 34 (1995) 205.[19] N. Bremond, J. Bibette, Soft Matter 8 (2012) 10549.[20] L.L. Schramm, Adv. Chem. Series 231 (1992).[21] S.M. Joscelyne, G. Tr¨ag˚ardh, J. Membr. Sci. 169 (2000) 107.[22] S. Lee, Y. Aurelle, H. Roques, J. Membr. Sci. 19 (1984) 23.[23] A. Hong, A.G. Fane, R. Burford, J. Membr. Sci. 222 (2003) 19.[24] J. Atencia, D.J. Beebe, Nature 437 (2005) 648.[25] P.V. Puyvelde, A. Vananroye, R. Cardinaels, P. Moldenaers, Polymer 49 (2008) 5363.[26] H.A. Stone, Annu. Rev. Fluid Mech. 26 (1994) 65.[27] G.I. Taylor, Proc. R. Soc. Lond. A 138 (1932) 41.[28] G.I. Taylor, Proc. R. Soc. Lond. A 146 (1934) 501.[29] R.G. Cox, J. Fluid Mech. 37 (1969) 601.[30] E.J. Hinch, A. Acrivos, J. Fluid Mech. 98 (1980) 305.[31] F.D. Rumscheidt, S.G. Mason, J. Coll. Sci. 16 (1961) 238.[32] H.P. Grace, Chem. Eng. Commun. 14 (1982) 225.[33] B.J. Bentley, L.G. Leal, J. Fluid Mech. 167 (1986) 241.[34] J. Husny, J.J. Cooper-White, J. Non-Newton. Fluid Mech. 137 (2006) 121.
35] J.H. Xu, G.S. Luo, G.G. Chen, J.D. Wang, J. Membr. Sci. 266 (2005) 121.[36] J.M. Rallison, J. Fluid Mech. 109 (1981) 465.[37] T. Inamuro, T. Ogata, S. Tajima, N. Konishi, J. Comp. Phys. 198 (2004) 628.[38] G. Tryggvason, B. Bunner, A. Esmaeeli, D. Juric, N. Al-Rawahi, W. Tauber, J. Han, S. Nas,Y.-J. Jan, J. Comp. Phys. 169 (2001) 708.[39] J.U. Brackbill, D.B. Kothe, C. Zemach, J. Comp. Phys. 100 (1992) 335.[40] D. Gueyffier, J. Li, A. Nadim, R. Scardovelli, S. Zaleski, J. Comput. Phys. 152 (1999) 423.[41] R. Scardovelli, S. Zaleski, Ann. Rev. Fluid Mech. 31 (1999) 567.[42] M.E. O’Neill, Chem. Eng. Sci. 23 (1968) 1293.[43] T.C. Price, Q. J. Mech. Appl. Math. 38 (1985) 93.[44] C. Pozrikidis, J. Eng. Math. 31 (1997) 29.[45] K. Sugiyama, M. Sbragaglia, J. Eng. Math. 62 (2008) 35.[46] P. Dimitrakopoulos, J. Fluid Mech. 580 (2007) 451.[47] T. Darvishzadeh, N.V. Priezjev, J. Membr. Sci. 423-424 (2012) 468.[48] Fluent, Inc., 2003. FLUENT 6.1 Users Guide.[49] C.W. Hirt, B.D. Nichols, J. Comput. Phys. 39 (1981) 201.[50] D. Gerlach, G. Tomar, G. Biswas, F. Durst, Int. J. Heat Mass Transfer 49 (2006) 740.[51] W.J. Rider, D.B. Kothe, J. Comp. Phys. 141 (1998) 112.[52] P. Concus, R. Finn, Microgravity Sci. Technol. 3 (1990) 87.[53] Z. Zapryanov, S. Tabakova, Springer, Vol. 50 (1998).[54] Y. Renardy, Rheol. Acta. 46 (2007) 521.[55] J.M. Rallison, Annu. Rev. Fluid Mech. 16 (1984) 45.[56] H.A. Stone, PhD Thesis, Caltech (1988).[57] C.J.M. van Rijn, Micro-Engineered Membranes. In: Encyclopedia of Membrane Science andTechnology, Eds: E.M.V. Hoek, V.V. Tarabara, John Wiley and Sons, 2013. igures ressure InletSymmetry Periodic Boundary Condition X Wall YZ Moving Wall
FIG. 1: Schematic representation of the oil droplet residing at the pore entrance in a rectangularchannel with the corresponding boundary conditions. The width and length of the computationaldomain are fixed to 12 µ m and 36 µ m, respectively. Symmetry boundary conditions are used in theˆ z direction. The system dimensions are not drawn to scale. + P - P P P - P = = P - P P P P P P P FIG. 2: Schematic of the droplet cross-sectional profile at the membrane pore. The critical pressureof permeation ( P − P ) is calculated in three steps: (1) the pressure jump across the static interface( P − P ) is calculated from the Young–Laplace equation, (2) the pressure jump across the dynamicinterface ( P − P ) is computed numerically, and (3) the pressure jumps from steps 1 and 2 areadded. d /H ch = 0.286H d /H ch = 0.428H d /H ch = 0.686H d /H ch = 0.902Flow Direction µ m FIG. 3: The cross-sectional profiles of oil droplets in steady shear flow for the indicated confinementratios when the capillary number is Ca = 0 . r d = 2 µ m, the pore radiusis r p = 0 . µ m, the contact angle is θ = 135 ◦ , the surface tension coefficient is σ = 19 . λ = 1. onfinement Ratio (H d /H ch ) C a cr FIG. 4: The critical (breakup) capillary number as a function of the confinement ratio H d /H ch .Other parameters are the same as in Fig. 3. a% I n cre a s e i n C r i t i c a l P re ss u re λ = 1 λ = 2 λ = 3 λ = 5 λ = 10 λ = 20 FIG. 5: The percent increase in critical pressure of permeation as a function of the capillarynumber Ca = µ w ˙ γr d /σ for the indicated viscosity ratios λ = µ o /µ w . Typical error bars are shownon selected data points. For each value of λ , the data are reported up to the critical capillarynumber above which droplets break into two segments. The droplet and pore radii are r d = 2 µ mand r p = 0 . µ m, respectively. The contact angle is θ = 135 ◦ and the surface tension coefficient is σ = 19 . a = 0.0063Ca = 0.0126Ca = 0.0188Ca = 0.0251Ca = 0.0283Ca = 0.0314 Flow Direction µ m FIG. 6: The cross-sectional profiles of the oil droplet residing on the circular pore with r p = 0 . µ mfor the indicated capillary numbers. The viscosity ratio is λ = 1. Other parameters are the sameas in Fig. 5. eformation Time Scale, µ w r d (1 + λ )/ σ ( µ s) B re a kup T i m e ( µ s ) λ = 20 λ = 1 λ = 2 λ = 3 λ = 5 λ = 10 Flow Direction µ m FIG. 7: The breakup time versus deformation time scale µ w r d (1 + λ ) /σ for the tabulated valuesof the viscosity ratio λ = µ o /µ w . Other system parameters are the same as in Fig. 5. The straightline is the best fit to the data. The error bars for the breakup time are about the symbol size. Theinset shows the droplet profiles just before breakup for the same viscosity ratios. a f D ( λ ) ( % I n cre a s e i n C r i t i c a l P re ss u re ) / f T ( λ ) λ = 20 λ = 1 λ = 2 λ = 3 λ = 5 λ = 10 FIG. 8: The normalized percent increase in critical pressure of permeation versus the modifiedcapillary number
Ca f D ( λ ) for the selected values of the viscosity ratio λ = µ o /µ w . The functions f D ( λ ) and f T ( λ ) are given by Eq. (12) and Eq. (15), respectively. ×××× Shear Rate (10 C r i t i c a l P re ss u re o f P er m e a t i o n ( k P a ) σ = . m N / m σ = . m N / m σ = . m N / m σ = . m N / m σ = . m N / m FIG. 9: The critical pressure of permeation as a function of shear rate for the indicated surfacetension coefficients. The symbols ( × ) denote the analytical predictions of Eq. (9). The droplet andpore radii are r d = 2 µ m and r p = 0 . µ m, respectively. The viscosity ratio is λ = 1 and the contactangle is θ = 135 ◦ . = 0.0382 N/m σ = 0.0286 N/m σ = 0.0191 N/m σ = 0.0143 N/m σ = 0.00955 N/m Flow Direction µ m FIG. 10: The cross-sectional profiles of the oil droplet above the circular pore for the listed valuesof the surface tension coefficient. In all cases, the shear rate is ˙ γ = 1 . × s − . Other parametersare the same as in Fig. 9. eformation Time Scale, µ w r d (1 + λ )/ σ ( µ s) B re a kup t i m e ( µ s ) σ = 9.55 mN/m σ = 38.2 mN/m σ = 28.6 mN/m σ = 19.1 mN/m σ = 14.3 mN/m µ mFlow Direction FIG. 11: The breakup time versus deformation time scale µ w r d (1 + λ ) /σ for the surface tensioncoefficients in the range from 9 .
55 mN/m to 38 . a% I n cre a s e i n C r i t i c a l P re ss u re σ = 38.2 mN/m σ = 9.55 mN/m σ = 14.3 mN/m σ = 19.1 mN/m σ = 28.6 mN/m FIG. 12: The percent increase in critical pressure of permeation as a function of the capillarynumber Ca = µ w ˙ γr d /σ for the selected values of the surface tension coefficient. The rest of thematerial parameters are the same as in Fig. 9. ×××× Shear Rate (10 C r i t i c a l P re ss u re o f P er m e a t i o n ( k P a ) θ = o θ = o θ = o θ = o θ = o FIG. 13: The critical pressure of permeation as a function of the capillary number for the indicatedcontact angles. The critical pressure at zero shear rate, given by Eq. (9), is denoted by the symbols( × ). The droplet radius, pore radius, surface tension coefficient, and viscosity ratio are r d = 2 µ m, r p = 0 . µ m, σ = 19 . λ = 1, respectively. = 115 o θ = 125 o θ = 135 o θ = 145 o θ = 155 o Flow Direction µ m FIG. 14: The cross-sectional profiles of the oil droplet above the circular pore for the listed valuesof the contact angle when Ca = 0 . ×××× Shear Rate (10 C r i t i c a l P re ss u re o f P er m e a t i o n ( k P a ) r d / r p = r d / r p = r d / r p = . r d / r p = r d / r p = . FIG. 15: The critical pressure of permeation as a function of shear rate for the selected drop-to-pore size ratios. The symbols ( × ) indicate the critical pressure in the absence of flow calculatedfrom Eq. (9). The pore radius, surface tension coefficient, contact angle, and viscosity ratio are r p = 0 . µ m, σ = 19 . θ = 135 ◦ and λ = 1, respectively. a × r P cr - P cr ( k P a ) d /r p = 5.0r d /r p = 3.0r d /r p = 3.5r d /r p = 4.0 r d /r p = 4.5 Flow Direction µ m FIG. 16: The difference in the critical pressure, P cr − P cr , versus the modified capillary numberfor five drop-to-pore size ratios r = r d /r p . Other parameters are the same as in Fig. 15. Thecross-sectional profiles of the droplet just before breakup are shown in the inset for same r ..