•The four fates confronting an oil droplet in crossflow filtration are highlighted.•Two critical conditions along with two operating conditions are used to construct fate maps.•An analysis of that ...portion of the volume of a droplet undergoing permeation is carried out.•Scaling analysis of the breakup time in conducted.•A macroscopic relationship for the estimation of the leaked volume during breakup is derived.
The fate of an oil droplet at the surface of a membrane in crossflow filtration has been identified as either of four destinies; namely, pinning, permeation, rejection or breakup. These fates are determined according to the operating conditions (transmembrane pressure, TMP, and crossflow velocity, CFV) in relation to the critical conditions (critical entry pressure,pcrit, and critical velocity of dislodgment, vcrit). These conditions have enabled the establishment of fate maps for every combination of pore and droplet sizes. Such maps have been the bases for the newly developed multicontinuum modeling approach, which describes the permeation process not only of the continuous phase but also of the inevitably associated dispersed phase. In the multicontinuum approach, both the droplet and pore size distributions are used to establish multitude of continua that interact according to the previously mentioned maps. Therefore, every oil continuum is checked against all membrane continua to determine how the droplets behave. According to this procedure, the permeation flux of the oil as well as the rejection capacity of the membrane can be estimated. At the moment of breakup, the portion of the droplet that has advanced inside the pore depends on the ratios TMP/pcrit andCFV/vcrit. In this work, we investigate this topic thoroughly by conducting computational fluid dynamics (CFD) analysis of various combination of operating and critical conditions. A macroscopic relationship is developed to allow for the estimation of the volume of the droplet inside the pore opening upon breakup under different conditions. This formula allows the accurate estimation of the permeation flux of those oil droplets that have experienced breakup conditions, which allow its use in the context of multicontinuum approach. From this study it is found that the volume of the droplet inside the pore is large the larger the ratio of the transmembrane pressure with respect to the critical entry pressure. It is also found that the larger the feed velocity compared with the critical velocity of dislodgment the smaller the size of that part of the droplet inside the pore. The latter is a manifestation of the increase in hydrodynamic drag as a result of the increased velocity. A scaling analysis has also been conducted to show how the breakup time is correlated with the ratio of the feed stream velocity and the critical velocity of dislodgment.
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.
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.
Display omitted
•A novel antifouling technique is introduced.•It is based on alternating the pressure of the feed side in cyclic manner.•During half the cycle when the pressure across the membrane is higher, permeation occurs.•During the other half the cycle when the pressure across the membrane is zero, no permeation occurs.•Crossflow field in the second half of the cycle cleans the surface of the membrane.
Display omitted
•A generalized model that considers the displacement of one fluid by another immiscible one is introduced.•Both wetting and nonwetting combinations are considered.•All combinations of ...pressure, gravity, capillarity, and friction forces are studied.•The model reduces to all special cases.•Comparisons with CFD simulations show very good agreement.
Displacing one fluid by another immiscible with it is a phenomenon of greatest importance and exists in various applications from large scale oil productions to small scale microfluidics. Modeling this phenomenon is crucial, particularly, with respect to estimating penetration rates. Although considerable amounts of research works have been conducted to understand the physics involved in this phenomenon, they remain adherent to particular applications. This includes, for example, imbibition scenarios in which a wetting fluid imbibes inside a capillary tube even without the need to initiate the penetration. On the other hand, drainage scenarios, in which a nonwetting fluid displaces a wetting one, requires such initiation by providing external boosting. There are several external factors that influence the rate at which the meniscus advances inside a capillary tube. The most common such forces include pressure force, capillary force, and gravity force. Several of the modeling approaches have considered the case in which the displaced fluids are gases (e.g., air), particularly in applications related to infiltration studies in porous media. This allowed researchers to ignore the frictional resistance of the displaced fluid and only considers the other denser fluid. However, there exists other applications in which this is not correct and it may not be appropriate to ignore the displaced fluid. In this work, a new generalized model is introduced that accounts for all the physics involved in this process and also does not ignore the displaced fluid. For validation, it is shown that the derived model reduces to all the special cases for which analytical solutions exist, (including that of a single-phase flows). Furthermore, comparisons with computational fluid dynamics simulation (CFD) of drainage/imbibition scenarios show very good match with the results of the derived model, which builds confidence in the modeling approach.
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.
In this work, we investigate the problem of imbibition/drainage of a fluid in capillaries of arbitrary axisymmetric cross-sections filled initially with another immiscible one. The model predicts the ...location of the meniscus and its speed along the tube length with time. The two immiscible fluids may assume any density and viscosity contrasts. In addition, the axisymmetric profile of the tube maintains a relatively small angle of tangency to warrant that the axial velocity distribution assumes, approximately, a parabolic profile. The driving forces that may be encountered in this system include the capillary force, pressure force, gravitational force and an opposing viscous force. The orientation of the capillary force can be in the direction of the flow (e.g. during imbibition) or opposite to the flow (e.g. during drainage). Likewise, the gravitational force can be in the direction of the flow or opposite to it. In this work we account for all these possibilities. A differential equation is developed that defines the location of the meniscus with time. A fourth-order-accurate Runge–Kutta scheme has been developed to provide solutions for the different scenarios associated with this system. It is shown that the developed model reduces to those appropriate for straight tubes, which builds confidence in the modelling approach. The effects of changing the tangent along the profile of the tube, which influences the calculation of the radius of curvature of the meniscus, is also considered. Unlike the cases of straight capillary tubes, in tubes with arbitrary symmetric profiles, the friction force depends on the variations of the tube profile. Examples of converging/diverging capillary tubes that follow straight and power law profiles are investigated. In addition, the case of sinusoidal profiles has also been considered.
In hydrodynamic modeling of flow through porous structures, the solution domain might encounter discontinuities. These include, for example, porous structure-open channel interface, porous dam-break, ...and heterogeneous porous structures. The treatment of discontinuity is challenging within a numerical scheme as it can be a source of instabilities. This study proposes a finite-volume method to solve coupled Saint-Venant and Darcy–Forchheimer equations for simulating free-surface flow through porous structures. For capturing shocks arising at discontinuous regions, an upwind scheme is utilized to maintain the solution monotone. Fully implicit methods can allow the choice of longer time steps. Since the current problem involves two nonlinear systems, namely the open-channel and seepage flow equations, the Picard method is adopted to linearize the system of equations. Unlike typical implicit schemes of seepage flows, herein, both flow depth and velocity matrices appear within the iterative process, threatening the convergence criterion. To converge iteration, the continuity equation's flux term is treated using the dynamic wave equation under the relaxation method. The present model is applicable to simulate gradually and rapidly unsteady flow through homogeneous and heterogeneous porous media under laminar, transitional, and fully developed turbulent flow regimes within various closed and/or open boundary conditions.
Fouling represents a bottleneck problem for promoting the use of membranes in filtration and separation applications. It becomes even more persistent when it comes to the filtration of fluid ...emulsions. In this case, a gel-like layer that combines droplets, impurities, salts, and other materials form at the membrane’s surface, blocking its pores. It is, therefore, a privilege to combat fouling by minimizing the accumulation of these droplets that work as seeds for other incoming droplets to cluster and coalesce with. In this work, we explore the use of the newly developed and novel periodic feed pressure technique (PFPT) in combating the fouling of ceramic membranes upon the filtration of oily water systems. The PFPT is based on alternating the applied transmembrane pressure (TMP) between the operating one and zero. A PFPT cycle is composed of a filtration half-cycle and a cleaning half-cycle. Permeation occurs when the TMP is set at its working value, while the cleaning occurs when it is zero. Three PFPT patterns were examined over two feeds of oily water systems with oil contents of 100 and 200 ppm, respectively. The results show that the PFPT is very effective in minimizing the problem of fouling compared to a non-PFPT normal filtration. Furthermore, the overall drops in permeate flux during the cleaning half-cycles are compensated by appreciable enhancement due to the significant elimination of fouling development such that the overall production of filtered water is even increased. Inspection of the internal surface of the membrane post rinsing at the end of the experiment proves that all PFPT cycles maintained the ceramic membranes as clean after a 2-h operation. This can ensure a prolonged lifespan of the ceramic membrane use and a continuous greater permeate volume production. The advantage of the PFPT is that it can be implemented on existing units with minimal modification, ease of operation, and saving energy.
Shale gas plays an increasingly important role in the current energy industry. Modeling of gas flow in shale media has become a crucial and useful tool to estimate shale gas production accurately. ...The second law of thermodynamics provides a theoretical criterion to justify any promising model, but it has been never fully considered in the existing models of shale gas. In this paper, a new mathematical model of gas flow in shale formations is proposed, which uses gas density instead of pressure as the primary variable. A distinctive feature of the model is to employ chemical potential gradient rather than pressure gradient as the primary driving force. This allows to prove that the proposed model obeys an energy dissipation law, and thus, the second law of thermodynamics is satisfied. Moreover, on the basis of energy factorization approach for the Helmholtz free energy density, an efficient, linear, energy stable semi-implicit numerical scheme is proposed for the proposed model. Numerical experiments are also performed to validate the model and numerical method.
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.
PDF
