A multicontinuum approach for the problem of filtration of oily water systems across thin flat membranes: I. The framework

Investigation of the Different Regimes Associated with the Growth of an Interface at the Exit of a Capillary Tube into a Reservoir: Analytical Solutions and CFD Validation

The emergence of a droplet from a capillary tube opening into a reservoir is an important phenomenon in several applications. In this work, we are particularly interested in this phenomenon in an attempt to highlight the physics behind droplet appearance. The emergence of a droplet from a tube opening into a reservoir under quasi-static conditions passes through three stages. The first stage starts when the meniscus in the tube reaches the exit. At this moment, the meniscus intersects the wall of the tube at the equilibrium contact angle. The interface then develops until its radius of curvature becomes equal to the tube radius. During this stage, the capillary pressure increases. In the second stage, the interface continues to evolve with its radius of curvature increasing until the static contact angle with respect to the surface of the reservoir is achieved. This marks the end of the second stage and the start of the third in which the contact line (CL) starts to depart the tube opening along the reservoir surface and the contact angle remains constant. Analytical models for the three stages have been derived based on the law of conservation of linear momentum. The models account for pressure, gravitational, capillary, and viscous forces, while inertia force is ignored. The model can predict the profiles of the mean velocity in the tube, the capillary pressure, and the evolution of the contact angle. In addition, a computational fluid dynamics (CFD) simulation has been conducted to provide a framework for validation and verification of the developed model. The CFD simulation shows qualitative behavior in terms of snapshots of the emerging droplet with time similar to that speculated by the analytical model. In addition, quantitative comparisons with respect to velocity, pressure, and volume profiles of the droplet show very good agreement, which builds confidence in the modeling approach.

On the estimation of the size of a droplet emerging from a pore opening into a crossflow field

The problem of terminating a droplet at the surface of a membrane in a crossflow field is an important topic in the context of controlled emulsification of fluids for use in pharmaceutical and other industries. Some of these industries struggle to produce emulsions of uniform sizes for their products requiring higher levels of precision. In this work, we comprehensively investigated one such technique in which droplets were produced via membrane openings and were terminated via a crossflow field. Conditions of permeation and termination were identified. A model was developed to estimate the size of the emerging droplets from information about the interfacial properties, geometry, and operating conditions (i.e., pressure and crossflow velocity). Three forces, including capillary pressure, interfacial tension, and drag forces, were identified that account for a developed torque balance, which was then used to determine the onset of breakup of an emerging droplet. A comprehensive computational fluid dynamics (CFD) analysis has been conducted to highlight the physics involved in the process and also to provide scenarios for comparison exercises. The effects of crossflow velocity, applied pressure, and viscosity contrasts have been studied. It has been determined that the emerging droplet experiences deformation along the crossflow field because of the hydrodynamic drag. The receding portion of the contact line at the surface of the membrane wraps around the pore opening, generating an interfacial tension force that produces an opposing torque due to the crossflow drag and capillary pressure. Using this phenomenon, a framework for estimating the size of the droplet upon breakup is established. Comparisons with the results obtained from the CFD analysis under different conditions show very good agreement, which builds confidence in the modeling approach.

Velocity Profile Representation for Fully Developed Turbulent Flows in Pipes: A Modified Power Law

In the design practices of many engineering applications, gross information about the flow field may suffice to provide magnitudes of the parameters that are essential to complete the design with reasonable accuracy. If such design parameters can be estimated following simpler steps, it may be possible to abandon the need to conduct expensive numerical and/or experimental works to produce them. In this work, we are interested in providing a generalized power law that depicts the velocity profile for fully developed turbulent flows. This law incorporates two fitting parameters m and n that represent the exponents of (1) a nondimensional length scale and (2) an overall exponent, respectively. These two parameters may be determined by fitting the experimental and/or computational data. In this work, fitting benchmark experimental and computational fluid dynamics (CFD) data found in the literature reveals that the parameter m changes over a relatively smaller range (between 1 and 2), while the parameter n changes over a wider range (between 1 and 12 for the range of Reynolds number considered). These two parameters (m and n) are, generally, not universal, and they depend on the Reynolds number (Re). A correlation was also developed to correlate n and Re in the turbulent flow region. In order to preserve the continuity of the derivative of the velocity profile at the centerline, a value of m equals 2 over the whole range of Re is recommended. Apart from the near wall area, the new law fits the velocity profile reasonably well. This generalized law abides to a number of favorable stipulations for the velocity profile, namely the continuity of derivatives and reduction to the laminar flow velocity profile for lower values of Re.

The Effect of the Oleophobicity Deterioration of a Membrane Surface on Its Rejection Capacity: A Computational Fluid Dynamics Study

In this work, the effects of the deteriorating affinity-related properties of membranes due to leaching and erosion on their rejection capacity were studied via computational fluid dynamics (CFD). The function of affinity-enhancing agents is to modify the wettability state of the surface of a membrane for dispersed droplets. The wettability conditions can be identified by the contact angle a droplet makes with the surface of the membrane upon pinning. For the filtration of fluid emulsions, it is generally required that the surface of the membrane is nonwetting for the dispersed droplets such that the interfaces that are formed at the pore openings provide the membrane with a criterion for the rejection of dispersals. Since materials that make up the membrane do not necessarily possess the required affinity, it is customary to change it by adding affinity-enhancing agents to the base material forming the membrane. The bonding and stability of these materials can be compromised during the lifespan of a membrane due to leaching and erosion (in crossflow filtration), leading to a deterioration of the rejection capacity of the membrane. In order to investigate how a decrease in the contact angle can lead to the permeation of droplets that would otherwise get rejected, a CFD study was conducted. In the CFD study, a droplet was released in a crossflow field that involved a pore opening and the contact angle was considered to decrease with time as a consequence of the leaching of affinity-enhancing agents. The CFD analysis revealed that the decrease in the contact angle resulted in the droplet spreading over the surface more. Furthermore, the interface that was formed at the entrance of the pore opening flattened as the contact angle decreased, leading the interface to advance more inside the pore. The droplet continued to pass over the pore opening until the contact angle reached a certain value, at which point, the droplet became pinned at the pore opening.

Coalescence of an Oil Droplet with a Permeating One over a Membrane Surface: Conditions of Permeation, Recoil, and Pinning

When a droplet lands over a nonwetting surface it forms a convex interface that makes a contact angle larger than 90°. If the droplet lands over a pore opening, an interface is also formed at the pore opening that can prevent the droplet from permeating. The conditions for permeation and pinning are very much related to a threshold critical pressure that above which the droplet will permeate. This property defines a selectivity criterion for microfiltration processes of oily water systems using membrane technology. Such a feature of the membrane gets compromised, however, due to the permeation of droplets that are relatively smaller in size or whose critical entry pressure is smaller than the applied transmembrane pressure (TMP). In this work, we investigate what happens to a droplet when it coalesces with a droplet that undergoes permeation. Two scenarios are considered: namely, (1) a droplet coalesces with a permeating one whose interface inside the pore has not broken through the pore exit and (2) a droplet coalesces with a permeating one whose interface in the pore has broken through. We show that a larger droplet (that will essentially not permeate if pinned over a membrane opening) will now permeate when the pore is filled with oil from a preceding one or recoils when the interface inside the pore of a preceding droplet has not broken through the exit of the pore. This has interesting implications for the rejection capacity of the membrane, which decreases due to the permeation of droplets that would, otherwise, not permeate. A computational fluid dynamic (CFD) study has been conducted to confirm the conclusions obtained from the theoretical study and to reproduce the fates of the combined droplet after coalescence at the surface of the membrane. Furthermore, a simplified formula for estimating the critical entry pressure is developed.

A Unified, One Fluid Model for the Drag of Fluid and Solid Dispersals by Permeate Flux towards a Membrane Surface

The drag of dispersals towards a membrane surface is a consequence of the filtration process. It also represents the first step towards the development of the problem of fouling. In order to combat membrane fouling, it is important to understand such drag mechanisms and provide a modeling framework. In this work, a new modeling and numerical approach is introduced that is based on a one-domain model in which both the dispersals and the surrounding fluid are dealt with as a fluid with heterogeneous property fields. Furthermore, because of the fact that the geometry of the object assumes axial symmetry and the configuration remains fixed, the location of the interface may be calculated using geometrical relationships. This alleviates the need to define an indicator function and solve a hyperbolic equation to update the configuration. Furthermore, this approach simplifies the calculations and significantly reduces the computational burden required otherwise if one incorporates a hyperbolic equation to track the interface. To simplify the calculations, we consider the motion of an extended cylindrical object. This allows a reduction in the dimensions of the problem to two, thereby reducing the computational burden without a loss of generality. Furthermore, for this particular case there exists an approximate analytical solution that accounts for the effects of the confining boundaries that usually exist in real systems. We use such a setup to provide the benchmarking of the different averaging techniques for the calculations of properties at the cell faces and center, particularly in the cells involving the interface.

Simplified Formula for the Critical Entry Pressure and a Comprehensive Insight into the Critical Velocity of Dislodgment of a Droplet in Crossflow Filtration

Produced water treatment remains a challenging issue for the oil production industry. Finding ways to effectively treat oily water systems without incurring higher operational costs is the struggle and focus of recent research work. The success in establishing a modeling approach to study the filtration of oily water systems is dependent upon our understanding of the fate of oil droplets at the membrane surface. It has been determined that four fates confront oil droplets at the membrane surface, namely, permeation, breakup, pinning, and rejection. Conditions for manifestation of any of these four fates depend on two operating conditions (transmembrane pressure and crossflow velocity) in comparison with two critical conditions (entry pressure and critical velocity of dislodgment). In this work, a new simplified formula for the critical entry pressure is introduced. It compares very well with the formula already existing in the literature. Furthermore, the complete model for the critical velocity of dislodgment in crossflow filtration is presented and highlighted. More investigations on the physical processes that are involved during the pinning of a droplet at a pore opening are presented. In addition, a thorough analysis of the forces that are involved during the permeation of a droplet that could lead to its breakup is presented. It is found that, once the droplet reaches the pore opening, the interfacial tension force and the pressure force continue to increase. Following the critical configuration, these forces continuously decline and the drag force due to the crossflow field, therefore, becomes sufficient to break up the droplet.

On the design of sustainable antifouling system for the crossflow filtration of oily water systems: A multicontinuum and CFD investigation of the periodic feed pressure technique

The problem of fouling is considered a major reason for deteriorating the performance of porous membranes. Even though the accumulations of materials at the membrane surface are inevitable, efforts are continuously spent to minimize their drawbacks. Several techniques have been tested to minimize the problem of fouling. Some of these methods, however, confront some technical difficulties that make their use unfeasible. For example, in polymeric-type membranes, back flushing may result in the loss of bonding between the active and the support layers resulting thereby to the disintegration of the membrane. Recently, an interestingly new approach has been proposed that minimizes the problem of fouling and maintains the integrity of the membrane. The so-called periodic feed pressure technique, PFPT, cleans the surface of the membrane by reducing the adherence of the droplets to the membrane giving the chance to the crossflow field to sweep off pinned droplets. In this work, some of the features of the PFPT technique are highlighted using results from CFD simulation. Then we further investigate the PFPT technique in the realm of the multicontinuum modeling approach in which both the emulsion and the membrane are treated as overlapping continua. The behavior of the membrane is studied considering different transmembrane pressure values to highlight the fates of the different oil continua upon interacting with membrane continua. From the CFD highlights, it is found that during the half cycle when the TMP is set to zero, oil droplets at the surface of the membrane becomes unstable and it becomes easier for the crossflow field to dislodge them. The multicontinuum study, on the other hand, provides macroscopic analysis on the effects of different TMP cycles on important macroscopic parameters that influence the design, including the rejection capacity of membranes.

A novel antifouling technique for the crossflow filtration using porous membranes: Experimental and CFD investigations of the periodic feed pressure technique

Oily water production is one of the many drawbacks of petroleum and several other industries. Finding effective ways for the treatment of produced water remain one of the main areas of interest in membrane sciences. Albeit the many advantages of membrane technology, they suffer from the unavoidable problem of fouling, which results from the accumulation of dispersed materials at the surface of membranes. Membrane modification and operational optimization have been approached as a potential cure of the problem of fouling. In this work we introduce a new and novel method that minimizes the development of fouling and in the same time utilizes no chemicals (i.e., environmentally friendly). The core of this method is based on alternating the pressure in the feed channel in a periodic manner and is therefore named the periodic feed pressure technique, PFPT. The idea is to make pinned droplets at the surface of the membrane lose essential forces that keep them sticking to the surface. The drag force due to permeation flux and the capillary force due to interfacial tension represents the two forces that largely contribute to the pinning of oil droplets at the surface of the membrane. Other forces including buoyancy and lift forces are generally small to be of significant influence. The idea of the PFPT is, therefore, to eliminate the force due to permeation drag. This is done by setting the transmembrane pressure (TMP) to zero at fixed intervals allowing pinned oil droplets to dislodge the surface. When the TMP is set to zero, permeation flux stops and the force due to permeation drag vanishes. This significantly reduces the overall residence time of pinned oil droplets, minimizing the chance for other oil droplets to cluster and coalesce with pinned ones. The PFPT does not cause any damage to the support layer of the polymeric membrane, which is a drawback of back-flushing methodology. The novel PFPT displays minimal membrane fouling and very similar permeation recovery despite only half the cycle time is in filtration mode. In this work, we show how the permeation flux is recovered and provide comparisons between the PFPT and regular filtration methodology. Furthermore, we compare the overall amount of filtrate at the end of the experiments using both methods. It is interesting to note that, the amount of filtrate using the PFPT is very much comparable to that obtained using regular filtration methodology and even higher. By optimizing the frequency of the cycle and the amplitude of the pressure change, it is possible to customize the PFPT to various membrane technologies and to achieve the highest recovery of the flux. Visual inspections of the membranes post operation and post rinsing indicate that membranes undergoing filtration using the PFPT achieves a very clean surface compared with those undergoing regular filtration processes. This method is a promising solution to membrane fouling that is easy to implement without any additional use of chemicals or equipment. Computational fluid dynamics (CFD) investigation is also conducted on microfiltration processes to show why this technique works.