A filtration model for the flow of dilute,stable emulsions in porous media—II. Parameter evaluation and estimation

Part I of this work outlined a new theory, based on deep-bed filtration concepts, to describe the flow of dilute, stable emulsions in underground porous media. Here, in Part II, we quantitatively test the proposed theory against experimental data and we indicate how the filtration model parameters can be estimated from first principles.Comparison is made between the theory and data on transient permeability and effluent concentration for dilute, oil-in-water emulsions of mean drop-size diameters ranging from 1 to 10 μm and volume concentrations of 0.5–2.5% flowing in quartz sandpacks of 0.57–2.0 μm² permeability. The pH of the continuous aqueous phase is kept constant at 10. Filtration theory successfully represents the data, permitting unambiguous evaluation of the theoretical parameters.Procedures are described for a priori calculation of the filtration parameters from knowledge of the drop size and the pore-size and grain-size distributions of the porous medium. Good agreement is achieved between the experimentally determined parameters and their estimated values. Thus, the proposed filtration model provides a reliable tool for predicting emulsion flow behaviour in porous media.     

Dispersion in flow through porous media—II. Two-phase flow

In this paper we extend our previous study (Sahimi et al., 1986, Chem. Engng Sci.41, 2103–2122) of dispersion processes in porous media occupied by two fluid phases. We report results of Monte Carlo investigations of dispersion in two-phase flow through disordered porous media represented by square and simple cubic networks of pores of random radii. The percolation theory of Heiba et al. (1982, SPE 11015, 57th Annual Fall Meeting of the Soc. Petrol. Engrs) is used to determine the statistical distribution of phases in the porespace. One of the phases is assumed to be strongly wetting on the porewall in the presence of the other phase. A pore size distribution is chosen which yields through the percolation theory of Heiba et al. network relative permeabilities that are in agreement with the available experimental data.As in one-phase flow dispersion is diffusive in the cases simulated, i.e. it can be described by the convective-diffusion equation. Longitudinal dispersivity in a given phase rises greatly as the saturation of that phase approaches residual (i.e. its percolation threshold); transverse dispersivity also increases, but more slowly. As residual saturation of a phase is neared, the backbone of the subnetwork occupied by the phase becomes increasingly tortuous, with local mazes spotted along it that are highly effective dispersers. Dispersivities are found to be phase, saturation and saturation history dependent.Some limited Monte Carlo experiments with a residence time representation of the effects of deadend paths within a phase or reversible adsorption on the pore walls demonstrate that the approach developed can be extended to study the influence of such delay mechanisms on the dispersion process.     

Multiplicity features of porous catalytic pellets—III. Uniqueness criteria for the lumped thermal model

Exact uniqueness criteria are presented for a porous catalytic pellet in which a p-th order exothermic reaction occurs using a lumped thermal model, which accounts for intraparticle concentration gradients and assumes that the pellet temperature is uniform but different from that of the surrounding fluid. In addition, we determine the boundaries of the parameter regions in which five solutions exist in either cylindrical or spherical pellets.     

A CFD model for gas uniform distribution in turbulent flow for the production of titanium pigment in chloride process☆

The fluid dynamic behavior of feeding gas (TiCl4) in an annular channel affects the combination of O2 and TiCl4 in an oxidation reactor, a key piece of equipment in titanium pigment production. The numerical procedure was validated by a 3-dimensional gas flow in the annular channel. Applying the validated model, the flow character-istics of TiCl4 in the oxidation reactor with a tangential inlet were simulated and characterized. The flow distribu-tion with five rectifying rings of different structure was simulated and analyzed. The results showed that the rectifying ring improved the distribution uniformity of the pressure and outlet velocity. Compared to the original case without a rectifying ring, the non-uniformity of the pressure and outlet velocity could be reduced by up to 91%and 69%respectively. The rectifying ring #5, which can be instal ed and adjusted easily, is more effective in realizing even distribution. In addition, instal ation of the rectifying ring effectively reduced the circulating flow in an annular channel as well as the total energy loss.     

In situ measurement of conductivity and active surface area of porous electrodes by the current step method—I. Theory

If the electrical conductivity of a plate-shaped porous electrode immersed in an electrolyte is measured by the four-point d.c. method using a current step, the potential drop, Δφ₁, is a function of the time, t. The course of this function can be derived theoretically by solving two simultaneous partial differential equations describing the potential distribution in both the solid and liquid phases. It is shown that the Δφ₁ − t dependence permits not only to determine the conductivity of the electrode but also to estimate its true surface area and the conductivity of the pore electrolyte.     

Importance of emulsions in crystallization—applications for fat crystallization

Emulsions and crystallization are two independent research topics which normally do not overlap although a combination of the two could be applicable to many areas. Here, the importance of emulsions in the field of fat crystallization is described. Three applications with industrial relevance were chosen for investigation: fat fractionation, the solidification of phase change materials and solid lipid nanoparticles. For fat fractionation and phase change materials, emulsification can be applied as a tool to improve the fat crystallization process, and thus the product quality of the crystallized fat. Furthermore, the use of emulsification creates new application fields such as solid lipid nanoparticles in the area of fat crystallization.     

Numerical simulation of fine particle liquid–solid flow in porous media based on LBM-IBM-DEM

Fine particle liquid–solid flow in porous media is involved in many industrial processes such as oil exploitation, geothermal reinjection, and filtration systems. It is of great significance to master the behaviours of the fine particle liquid–solid flow in porous media. At present, there are few studies on the influences of the migration of fine particles on the flow field in porous media, and the effects of the porosity of porous media and inlet fluid velocity on the migration behaviours of fine particles in porous media. In this paper, a liquid–solid flow model was established based on the lattice Boltzmann method (LBM)-immersed boundary method (IBM)-distinct element method (DEM) and verified by the classical Drag Kiss Tumble (DKT) phenomena and flow around a cylinder successfully. In this model, the interaction between solid particles is analyzed using the distinct element method, and the interaction between fine particles and flow field is handled by IBM. Then, the migration and blockage of fine particles in porous media was studied using this model. It is found that, in addition to the blockage, a large amount of blocked-surface sliding-separation occur in fine particles. At the same time, the decrease in porosity increases the damage degree of fine particles on the permeability. The porosity exerts great influence on the penetration rate and dispersion behaviour of fine particles. The inlet fluid velocity mainly affects the residence time of fine particles and the average velocity of motion in the direction perpendicular to the main flow direction.     

May the Knudsen equation be legitimately,or at least usefully,applied to dilute adsorbable gas flow in mesoporous media?

The question of applicability of Knudsen's equation to dilute adsorbable gas flow in mesoporous media was the subject of a recent debate between Ruthven et al. (2009), Ruthven (2010) and Bhatia and Nicholson, 2010, Bhatia and Nicholson, 2011. Here, we present a critique of the said debate and introduce new information of critical importance, which has nevertheless been disregarded by both sides thereof. It is pointed out that von Smolukowski's derivation leaves no doubt that Knudsen's equation is meant to apply to the limiting case of zero gas adsorption. In practice, this limit may be expected to be approached at high temperature/low gas adsorbability but the crux of the matter is (i) choice of the right transport parameter (in this case the “reduced permeability”) to provide a useful criterion of approach to the Knudsen limit and (ii) thorough study of the course followed by the said parameter in its approach to the said limit. In this light, we begin our review of the aforesaid new information with conventional surface flow theory, which can deal satisfactorily with higher and moderate, but not with the case of weak, adsorption (because the reduced permeability does not converge smoothly to the Knudsen limit). Even so, we show that it can provide a good answer to a significant question raised in the aforesaid debate. We continue with the more advanced (but still analytical) surface flow approach of Nicholson and Petropoulos (1973), which has provided a plausible physical mechanism for the experimentally observed passage of the reduced permeability through a minimum with rising temperature, in the weak adsorption region. It is shown here that the said experimental data constitute the first observations (including the “fine structure”) of what we have called “apparent or quasiKnudsen” (to avoid confusion with proper Knudsen) flow behavior, which is the central theme of the debate in question. We then pass onto a discussion on the confirmation of the above interpretation of quasiKnudsen-flow “fine structure”, as well as the identification of conditions favorable to the appearance of such flow in practice, afforded by the results of ab initio evaluation of dilute adsorbable gas flow, under suitable adsorption potentials, in model pores (and model pore networks) by and Petropoulos (1981, 1985) and Petropoulos and Petrou (1991).     

A continuum model for gas–liquid flow in packed towers

Gas–liquid flow in packed towers is commonly encountered in the chemical and processing industry. A continuum model is developed based on the volume-and-time averaging of multiphase flows in isotropic rigid porous media/packed columns. Closures are presented for the evaluations of the extra surface/intrinsic phase integral terms. Both inertia and inter-phase interactions are retained in the volume averaged (Navier–Stokes) equations. These governing equations are solved for fully-developed axi-symmetric single and gas–liquid two phase flows in highly porous packed towers. It is found that the dispersion term is present in the continuity equation as well as the momentum equations. Numerical simulations with the models show that the volume-and-time averaged equations can predict the velocity, phase hold-up and pressure drop quite well for up to the loading point for gas–liquid counter-current flows.     

A model for a countercurrent gas—solid—solid trickle flow reactor for equilibrium reactions. The methanol synthesis

The theoretical background for a novel, countercurrent gas—solid—solid trickle flow reactor for equilibrium gas reactions is presented. A one-dimensional, steady-state reactor model is developed. The influence of the various process parameters on the reactor performance is discussed. The physical and chemical data used apply to the case of low-pressure methanol synthesis from CO and H₂ with an amorphous silica—alumina as the product adsorbent. Complete reactant conversion is attainable in a single-pass operation, so that a recycle loop for the non-converted reactants is superfluous.In the following article the installation and experiments for which this theory was developed will be described.     

Mathematical modeling of the simultaneous convection–anomalous diffusion processes in porous media

ABSTRACT

After a brief overview of most important general features of the nonclassical diffusion on the base of the extended irreversible thermodynamics, the relevant mathematical formulae are incorporated into general formalism of the simultaneous convection–diffusion processes taking place in porous media. Then, using the simplest variant of the convection–diffusion equation, novel-type analytic solutions are derived for transport processes with both subdiffusion and superdiffusion characters in Lagrangian representation.     

Measurement and analysis of particle—particle interaction in a cocurrent flow of particles in a dilute gas—solid system

The experimental apparatus of Arastoopour et al.[3] was modified to measure pressure drop and solid velocities for cocurrent flow of particles in a pneumatic conveying line. The data were translated into particle—particle interaction expression using a force balance over the particles. The particle interaction is a combination of collision and drag force in a particles low relative velocity region. A correlation for particle—particle interaction with relative velocity between the particles of 0.3–4.6 m/s has been developed. The correlation describes our experimental data within the 10% deviation.     

Microconvection in porous media during pervaporation of a liquid mixture—an experimental study

The objective of this investigation was to study experimentally the behaviour of a binary liquid mixture which evaporates from the surface of a porous plate. While the liquid flows towards the surface, an exponential concentration profile establishes in the pores. The plate acts as a mass transfer resistance which hinders the more volatile component from evaporating preferentially. It can be shown that under proper conditions evaporation is non-selective, that is, both components are removed equally.Experiments with the mixture 2-propanol-water and plates of different thickness and pore diameter proved the theoretical predictions to be valid, with exceptions: for low alcohol content evaporation is highly selective, the reason being intense microconvection in the porous plate. The intensity of this microconvection strongly depends on the mean pore size of the material.The mechanisms inducing the convection were attributed to Bénard and Marangoni instabilities. By using two different experimental set-ups, it could be shown that the Marangoni effect is dominant in thin plates, whereas Bénard convection is only of importance in thicker plates.A flow model is proposed to describe qualitatively how the Marangoni convection in a porous medium may take place.     

High-pressure filtration—a chance for the downstream processing of biological dispersions

High-pressure filtration is a common practice in solid-liquid separation. A pressure up to 150 bar is used first for short-time drainage and then to press the filter cake with the help of an elastomer diaphragm.After the principal investigation of this concept and measurements to separate biological dispersions, its use in several processes is proposed. For suspensions of cells a filter cake with a high proportion of dry residue can be obtained within a very short time. The optimal regulation of this process on an industrial scale can be obtained by evaluation of laboratory experiments. For separation procedures with biomass and fermentation liquid the high pressure can be used to shorten the filtration time. Bacterial dispersions may be separated effectively with a cross-flow filtration apparatus which is to be developed for work at high pressure.The filtration experiments were performed on bakers' yeast (leaven) and Escherichia coli (E. coli) bacteria as model substances with a laboratory filter press apparatus. The filtration velocity and the specific filtration resistance are reported. Furthermore, the processing costs in relation to the proportion of dry residue were calculated as a function of pressure. The minimum cost corresponds to the economical operation range. Though this process is unjustifiably considered as relatively expensive, it is possible that, worked at its best, it will be able to compete with other mechanical separation processes.     

An inequality for the empirical D.F. in the non–I.I.D. case

A generalization to the non-i.i.d. case of an inequality for the empirical df. due to Mason (1981) is proved     

Unified model for the prediction of consecutive solid–liquid filtration and expression at the constant pressure

Unified nonlinear model is proposed for the prediction of consecutive solid–liquid filtration and expression at the constant pressure. This model is based on the Darcy–Terzaghi filtration-consolidation equations modified to consider power-law pressure dependence of the specific cake resistance, and transforming Darcy law to the linear form. The model considers nonuniform structure of compressible filter cakes obtained by filtration and following expression. The profiles of local compressive pressure and local cake characteristics are simulated and compared for different moderately and highly compressible filter cakes (H.K. kaolin, CaCO₃, silica, activated sludge) based on the analytical and numerical solutions of the model. It is shown that the behavior of solid–liquid expression depends from the initial structure of compressed materials. Consolidation ratio U of the filter cakes with initially nonuniform structure formed by filtration differs from that of semi-solid materials with initially uniform structure. Different methods of determination of consolidation coefficient are analyzed and compared for nonuniformly structured filter cakes.     

Spout voidage distribution,stability and particle circulation rates in spouted beds of coarse particles—I. Theory

Mass and momentum balance equations are used with a derived expression for the interaction force between the fluid and particle phases to calculate the axial spout voidage distribution and particle circulation rate in a spouted bed of coarse particles. The voidage profiles above minimum spouting are found to follow a similarity relationship and an explicit equation for the circulation rate is developed using that relationship and the calculated voidage at the top of the spout. Criteria for stability and spouting regime identification are discussed.     

Multiplicity features of porous catalytic pellets—I. Single zeroth-order reaction case

The steady state multiplicity features of porous catalytic pellets, the temperature of which is uniform but different from that of the surrounding, are determined for a zeroth-order reaction. Simple algebraic expressions are derived which divide the global parameter space into regions with different number of solutions as well as with different types of bifurcation diagrams of surface temperature vs ambient Thiele modulus. The model predicts that up to five steady state solutions exist. The region of parameters with five solutions shrinks with increasing external mass transfer resistance and as the geometry of the pellet changes from a slab to a sphere.