Oil droplet behavior at a pore entrance in the presence of crossflow: Implications for microfiltration of oil–water dispersions
Graphical abstract
Introduction
Understanding the dynamics of an oil droplet at a pore entrance is a fascinating problem at the intersection of fluid mechanics and interface science that is of importance in such natural and engineering processes as extraction of oil from bedrock, lubrication, aquifer smearing by non-aqueous phase liquids, and sealing of plant leaf stomata [1], [2], [3], [4]. Membrane-based separation of liquid–liquid dispersions and emulsions is a salient example of a technology where the knowledge of liquid droplet behavior in the vicinity of a surface pore is critical for the success of practical applications. Milk fractionation, produced water treatment, and recovery of electrodeposition paint are examples of specific processes used in food, petroleum, and automotive industries where porous membranes are relied on to separate emulsions [5], [6], [7].
The membrane separation technique can be particularly useful when small droplets need to be removed from liquid–liquid dispersions or emulsions because other commonly used technologies, such as hydrocyclones and centrifugation-based systems, are either incapable of removing droplets smaller than a certain critical size (e.g., for hydrocyclones) or are expensive and have insufficient throughput (e.g., centrifuges). The early work by the Wiesner group [8] as well as other studies [9], [10] on oil droplet entry into a pore provided an estimate of the critical pressure of permeation; however, the understanding of the entire process of the droplet dynamics at a micropore entrance is still lacking, especially with regard to the practically relevant case of crossflow systems where blocking filtration laws [11] are, strictly speaking, not applicable. Crossflow membrane microfiltration is used to separate emulsions by shearing droplets of the dispersed phase away from the membrane surface and letting the continuous phase pass through [12]. In contrast to the normal, or dead-end, mode of filtration, crossflow microfiltration allows for higher permeate fluxes due to better fouling control [13], [14]. However, the accumulation of the dispersed phase on the surface of the membrane and inside the pores, i.e., fouling of the membrane, can eventually reduce efficiency of the process to an unacceptably low level even in the presence of crossflow.
Another important application that entails interaction of liquid droplets with porous media is membrane emulsification, where micron-sized droplets are produced by forcing a liquid stream through membrane pores into a channel where another liquid is flowing [15]. The emerging droplets break when the viscous forces exerted by crossflow above the membrane surface are larger than surface tension forces [16]. Membrane emulsification requires less energy and produces a more narrow droplet size distribution [17], [18] than conventional methods such as ultrasound emulsification [19] and stirring vessels [20].
In general, the studies of petroleum emulsions have been performed at two different scales, namely, macroscopic or bulk scales and mesoscopic or droplet scales [21], [22]. Early research on membrane emulsification and microfiltration involved bulk experiments aimed at determining averaged quantities and formulating empirical relations [23]. These studies considered macroscopic parameters such as droplet size distribution, dispersed phase concentration, and bulk properties such as permeate flux [24], [25]. These empirical approaches were adopted due to inherent complexity of two-phase systems produced by bulk emulsification, where shear stresses are spatially inhomogeneous and the size distribution of droplets is typically very broad [21], [26]. However, with the development of imaging techniques and numerical methods, the shape of individual droplets during deformation and breakup could be more precisely quantified for various flow types and material properties [27], [28].
First studies of the droplet dynamics date back to 1930s, when G.I. Taylor systematically investigated the deformation and breakup of a single droplet in a shear flow [29], [30]. Since then, many groups have examined this problem theoretically [31], [32] and in experiments [33], [34], [35]. A number of research groups have studied experimentally how a droplet pinned at the entrance of an unconfined pore deforms when it is exposed to a shear flow [17]. Experiments have also been performed to measure the size of a droplet after breakup as a function of shear rate and viscosity ratio [36], [37]. Numerical simulations of the droplet deformation and breakup have been carried out using various methods including boundary integral [38], Lattice Boltzmann [39], and Finite Volume [40] methods. These multiphase flow simulations generally use an interface-capturing method to track the fluid interfaces. Among other front-tracking methods, the Volume of Fluid method simply defines the fluid–fluid interface through a volume fraction function, which is updated based on the velocity field obtained through the solution of the Navier–Stokes equation [41], [42]. The Volume of Fluid method is mass-preserving, it is easily extendable to three dimensions, and it does not require special treatment to capture topological changes [43].
The drag force and torque on droplets or particles attached to a solid substrate and subject to flow-induced shear stress depend on their shape and the shear rate. Originally, O'Neill derived an exact solution for the Stokes flow over a spherical particle on a solid surface [44]. Later, Price computed the drag force on a hemispherical bump on a solid surface under linear shear flow [45]. Subsequently, Pozrikidis extended Price's work to study the case of a spherical bump with an arbitrary angle using the boundary integral method [46]. More recently, Sugiyama and Sbragaglia [47] varied the viscosity ratio to include values other than infinity (the only value considered by Price [45]) and found an exact solution for the flow over a hemispherical droplet attached to a solid surface. Assuming that the droplet is pinned to the surface, estimates for the drag force, torque, and the deformation angle as functions of the viscosity ratio were obtained analytically [47]. Also, Dimitrakopoulos showed that the deformation and orientation of droplets attached to solid surfaces under linear shear flow depend on the contact angle, viscosity ratio, and contact angle hysteresis [48].
More recently, Darvishzadeh and Priezjev [49] studied numerically the entry dynamics of nonwetting oil droplets into circular pores as a function of the transmembrane pressure and crossflow velocity. It was demonstrated that in the presence of crossflow above the membrane surface, the oil droplets can be either rejected by the membrane, permeate into a pore, or break up at the pore entrance. In particular, it was found that the critical pressure of permeation increases monotonically with increasing shear rate, indicating optimal operating conditions for the enhanced microfiltration process. However, the numerical simulations were performed only for one specific set of parameters, namely, viscosity ratio, contact angle, surface tension coefficient, and droplet-to-pore size ratio. One of the goals of the present study is to investigate the droplet dynamics in a wide range of material parameters and shear rates.
In this paper, we examine the influence of physicochemical parameters such as surface tension, oil-to-water viscosity ratio, droplet size, and contact angle on the critical pressure of permeation of an oil droplet into a membrane pore. In the absence of crossflow, our numerical simulations confirm analytical predictions for the critical pressure of permeation based on the Young–Laplace equation. We find that when the crossflow is present above the membrane surface, the critical pressure increases, and the droplet deforms and eventually breaks up when the shear rate is sufficiently high. Analytical predictions for the breakup capillary number and the increase in critical permeation pressure due to crossflow are compared with the 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 numerical simulations and a novel procedure for computing the critical pressure of permeation are described. In Section 3, the summary of analytical predictions for the critical pressure based on the Young–Laplace equation is presented, and the effects of confinement, viscosity ratio, surface tension, contact angle, and droplet size on the critical transmembrane pressure and breakup are studied. Conclusions are provided in the last section.
Section snippets
Details of numerical simulations
Three-dimensional numerical simulations were carried out using the commercial software ANSYS FLUENT [50]. The FLUENT flow solver utilizes a control volume approach, while the Volume of Fluid (VOF) method is implemented for the interface tracking in multiphase flows. In the VOF method, every computational cell contains a certain amount of each phase specified by the volume fraction. For two-phase flows, the volume fractions of 1 and 0 describe a computational cell occupied entirely by one of the
The critical pressure of permeation and the breakup capillary number
The pressure jump across a static interface between two immiscible fluids can be determined from the Young–Laplace equation as a product of the interfacial tension coefficient and the mean curvature of the interface or . For a pore of arbitrary cross-section, the mean curvature of the interface is given bywhere Cp and Ap are the cross-sectional circumference and area of the pore, respectively [54]. Therefore, the critical pressure of permeation of a liquid film into a pore of
Conclusions
In this paper, we performed numerical simulations to study the effect of material properties on the deformation, breakup, and critical pressure of permeation of oil droplets pinned at the membrane pore of circular cross-section. In our numerical setup, the oil droplet was exposed to a linear shear flow induced by the moving upper wall. We used finite-volume numerical simulations with the Volume of Fluid method to track the interface between water and oil. The critical pressure of permeation was
Acknowledgments
Financial support from the Michigan State University Foundation (Strategic Partnership Grant 71-1624) and the National Science Foundation (Grant No. CBET-1033662) is gratefully acknowledged. Computational work in support of this research was performed at Michigan State University's High Performance Computing Facility.
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