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Microfiltration (MF) and ultrafiltration (UF), in which a semipermeable porous membrane is used to clean a slurry from the suspended particles, macromolecules, fine solids or viruses, are often accompanied by the formation of a cake on the membrane surface [1][2][3]. The hydraulic resistance of the cake, which may dramatically reduce the process driving force, is considered to be the factor limiting or narrowing the industrial and municipal application of MF and UF, for example, to wastewater and surface water treatment. A large number of methods to prevent or reduce the cake formation was proposed: high shear-inducing velocity of a liquid over the membrane surface, turbulence promoters, air sparging, vibratory membranes, a charged membrane surface, and the like [4].These methods often result in a more complicated design of membrane plants and always lead to higher operating and/or capital costs. In spite of all these efforts, almost all MF and UF plants require periodic membrane surface cleaning by flushing or backflushing. As a rule, it is especially difficult to "oppose" the cake formation in membrane separation processes where the size of colloidal particles to be removed is a couple of dozens of nanometers, that is, where the UF and MF particle-size ranges are overlapped.Many experimental and theoretical studies, for example [2,[5][6][7], suggest that the surface interactions and aggregation of colloidal particles at a membrane surface is the main factor accounting for the cake formation in membrane processes. As a result, all the efforts were directed to building mathematical models taking into account the surface interaction forces of colloidal particles, which are a necessary tool in predicting the performance of membrane plants. At the same time, no research was carried out to study microfiltration and ultrafiltration as separation processes in which the separation can be achieved not only by the rejection of suspended particles (SP) at the membrane surface but also by their adsorption on the same surface. It should be noted that the second mechanism, which is widely used in conventional depth filtration, adsorption, and chromatography, has never been considered as an engine that could produce an additional amount of clarified water in parallel to the production of permeate, the liquid that passes through the membrane.The membrane device most suitable to implement the idea of combining the rejection and adsorption of suspended particles at the membrane surface in one apparatus in order to increase the yield of clarified water is a hollow fiber filter in which the feed stream would flow around the shells of hollow fibers perpendicular to their axes. The permeate would be withdrawn from the HF lumens while the other product stream, the filtrate, which would be produced by collection of suspended particles on adsorbing or particle-collecting shells of hollow fibers, similar to conventional depth filtration [8], would be discharged from the other outlet (Fig. 1). The two clarified streams, the permeate and f...

Depth membrane filtration (DMF) with reversible adsorption is a novel pressure-driven membrane separation process in which the feed suspension is treated in a hollow fiber (HF) filter so that the clarified liquid produced by the semipermeable membranes is made up of a mixture of permeate and filtrate [1]. The permeate is a liquid that passed through the semipermeable membranes, which reject almost all suspended particles. The filtrate is a liquid in which the concentration of suspended particles was considerably reduced due to their adsorption on the membrane surface as the suspension flowed around the hollow fibers. DMF can be implemented in rectangular or radial hollow fiber devices. Generally speaking, it can be implemented in any membrane device with a high membrane packing density in which the suspension moves around the external surface of tubular, capillary, or hollow fiber membranes in a direction normal to their axes.DMF differs from crossflow membrane filtration in that it produces permeate and filtrate instead of permeate and retentate, which are the conventional outlet streams in crossflow filtration [2,3]. As compared to dead-end membrane filtration, DMF produces one more outlet stream, filtrate. By contrast with conventional membrane separation processes, DMF plants are designed to operate in a single-pass treatment (continuous-flow or batch) mode, rather than in a feed-andbleed or a multistage recycle operation, because these plants do not produce any retentate stream [2].In [1], we developed a mathematical model describing the process of depth membrane filtration with reversible adsorption in a rectangular hollow-fiber filter. The differential equations that account for the adsorption and peptization mass fluxes of particles in the boundary layer of surface interaction forces at the membrane surface and for the local and overall material and volume balance in the filter were solved using an approximate method based on the averaging of permeate flow rate. Example calculations with a latex suspension demonstrated the feasibility of DMF with reversible adsorption and its advantages over conventional ultrafiltration (UF) and microfiltration (MF) operations. The reason for the latter is that DMF beneficially uses the adsorption of suspended particles on the membrane surface instead of wasting power or other expensive resources on reducing the rate of particle deposition, which is now typical for UF and MF plants [2,3]. The purpose of this paper is to derive analytical expressions for evaluating the performance of a radial DMF filter and study the effect of transmembrane pressure, feed flow rate, and the geometry of DMF filters on their performance.As in [1], we assume that the suspension under treatment in a radial DMF filter (Fig. 1) is a dilute incompressible liquid with constant viscosity, which contains suspended quasi-lyophilic (quasi-stable) particles. The feed concentration, along with the process temperature, remains constant. The porous HF membranes completely reject the suspended particles. T...

We study experimentally and theoretically convective flows, which are induced in a horizontal liquid layer by a concentrated heat source: the split coherent beam of laser radiation. The layer surface is deformable. Depending on controlling factors, laboratory experiments demonstrate a variety of flow structures and surface configurations. The flow primarily has a single vortex pattern, but in a certain range of governing parameters, secondary nonstationary vortices are superimposed. Among the surface configurations there are the concave meniscus, the convex one, and the concavo-convex one. The numerical simulation is performed on a mathematical model, which involves nonlinear partial di¤erential equations for two-dimensional amplitude functions associated with distributions of temperature, vorticity, and surface deformation. In the long-wave approximation, the model describes the contribution of thermocapillary convection to the heat transfer as well as the degree of the interface deformation. The proposed model generalizes the existing one by taking into account the heating inhomogeneity.

The microstructure and dynamics of two-phase gasliquid flows in packings and granular beds remain a challenging problem in several fields of advanced engineering, such as chemical technology, the power industry, and petroleum and gas production. The creation of physical and computational models and methods is limited by the unobtainability of visual observations of phase interactions in such flows. Immersion tomography of a gas-liquid medium in a granular bed, a recently developed technique, provides a new means for investigating the hydrodynamics in this type of object [1]. The essence of the immersion visualization technique, in this case, consists of choosing a liquid with a refractive index practically coinciding with the refractive index of the transparent particles of the packing: as a result, the flooded packing becomes an optically homogeneous transparent object in which bubble movement can be observed in any projection. This movement, which depends on the interaction of a bubble with packing elements, is three-dimensional and very complex. For three-dimensional reconstruction of the bubble trajectory, the familiar methods of optimal tomography were used [2]: an object is illuminated in various directions, and optical-parameter distribution in the bulk of the probed object is reconstructed.The granular bed was modeled by a packing of K8 glass beads with a refractive index of 1.51. The liquid used was a solution of α -monobromonaphthaline with methylene iodide in n -decane. The immersion liquid was supplied by the All-Russian Research Institute of Physical Technical and Radio Engineering Measurements of the State Standard Office of the Russian Federation (Gosstandart). Required for the calculations were the following properties of the mixture: density ρ = 990 kg/m 3 , dynamic viscosity µ = 3.5 × 10 -3 Pa s, and surface tension coefficient σ = 3.5 × 10 -2 N/m. Beads with a diameter of d p = 6 × 10 -3 m in the cell were packed in ten layers in the vertical direction and four and eight layers in the transverse directions. However, in most of the experiments, the packing consisted of beads with a diameter of d p = 3 × 10 -3 m with an approximately doubled number of layers in all directions. A capillary used to feed gas microvolumes in order to create bubbles could be positioned either inside the packing or below it in the free liquid layer. The scheme and description of the experimental setup are found elsewhere [1]. The basic change in the measuring scheme was the use of a Lepton digital computer system (Zelenograd, Russia) as the image recorder. This avoided the use of a stroboscopic system and improved image quality. THEORETICAL ANALYSISPreliminary experiments, which are described elsewhere [1], showed that bubbles rose in a flooded granular bed at velocities 1.5-4 times slower than in a free liquid volume. Experiments in the granular beds also indicated the existence of significant transverse velocity components in bubbles. Therefore, roughly, we may regard a flooded granular bed as a highly viscous disper...