Experimental evidence for the stochastic theory of particle flow under gravity

Experimental evidence is presented for a recent stochastic theory of flow which represents the convergent flow of cohesionless particles under gravity toward an open orifice as equivalent to a counterflow of voids from the orifice upward through the bed by biased random flight; the theory is summarized in a new phenomenological form. Data, taken from the literature, were obtained from cells initially loaded with alternate layers of differently colored granular material. The theory predicts that after a fixed flow, each layer develops a depression such that if z_o is the original height of a given layer above the orifice and z_m is the corresponding height of the depression minimum, then for three-dimensional flow a plot of z_m² vs. z_o² for different layers will yield a straight line of slope one; the intercept gives statistical information concerning the equivalent void jumps. For two-dimensional flow, the corresponding theoretical plot is z_m^{$\text{3}\text{2}$} vs. z_osu$\text{3}\text{2}$Data plotted from several sources conform closely to the above predictions, provided z_o is not too large for a given flow. The latter discrepancy is qualitatively explained by a transient effect requiring the density to fall to a certain level before steady-state flow can occur.     

Linear displacement of a wetting fluid by an immiscible non-wetting fluid in a porous medium: A predictive algorithm

The linear displacement of a wetting fluid by an immiscible non-wetting fluid in a two-dimensional porous medium composed of a network of sites multi-connected by bonds has been simulated mathematically. The algorithm involves Monte Carlo decision making, random walks and principles of the percolation theory. The algorithm described in the present work successfully predicts the three distinct behaviours of immiscible displacement in porous media. This algorithm is tested against experiments available in the literature for two-dimensional porous media. The agreement between the numerical results and the experiments is very good.     

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.     

Slow immiscible displacement flow in disordered porous media

The immiscible displacement of a wetting fluid by a non-wetting fluid in a disordered porous medium is studied in the capillary region, i.e. when capillary forces are dominant, by using the invasion percolation model to describe the displacement mechanism. The porous medium is represented by a two-dimensional network of interconnected capillaries whose radii follow a uniform size distribution. Disorder is assigned to the medium by considering the probabilities of occurrence of inaccessible pores, p_s, and non-conductive capillaries, p_b. It is found that the dynamic behaviour of the displacing fluid and the fraction of invaded pores depend on the degree of disorder of the medium. The results can provide an interpretation of the effects of the dead-end pore volume on the oil recovery and the displacement behaviour.     

Liquid distribution and redistribution in packed columns—II. Experimental

The theory of liquid redistribution in packed columns under the influence of a potential when the boundary conditions have been set by differences in permeability between the wall and bulk regions, has been tested in experiment. Liquid distribution in a packed column 0.3 m in diameter and 1.5 m long has been measured for $\text{1}\text{2}$, 1 and 1$\text{1}\text{2}$ in. Raschig rings both with and without gas flow. The trend of experimental points was found to be in general agreement with the theoretical relationship when there was no gas flow. For gas flow conditions in the loading region there was close agreement between experimental and theory.The experimental estimates of the permeability ratio are analysed for the range of experimental conditions.     

New improvements in ultraviolet sterilization of desalted and reuse water

Ultraviolet water sterilization inherently has the advantage that it adds nothing and does not change the chemical composition Of the fluid. However, the process has not been widely used because of operation and maintenance problems linked to quartz tubes which separate the ultraviolet source from the water. The Ultraviolet Technology, Inc. (UTI) process overcomes these problems by using ultraviolet transmitting fluorocarbon tubes to contain the water during sterilization. The fluorocarbon tubes are chemically inert, non-wetting, and resistant to fouling, USFDA approved for drinking water, and stable to ultraviolet as demonstrated in a 15 year exposure test. Commercial installations have been in continuous operations on turbid seawater for over 3 years without any need for cleaning. The largest plant in operation has a capacity of 1.3 MGD. Capital costs with the UTI process are less than $\text{1}\text{2}$ those of quartz tubes ultraviolet sterilizers. Operating costs are typically less than 1 cent per 1,000 gallons.     

Some aspects of wetting/dewetting of a porous medium

Various features of wetting/dewetting of porous media are examined. The phenomenon of capillary hysteresis is illustrated by a vertical capillary tube which consists of an alternating sequence of convergent—divergent conical sections. A study of the kinetics of wetting of this tube by a liquid shows that when the velocity of the liquid/vapour meniscus is plotted against the height of penetration, it oscillates about the Washburn velocity—distance curve and performs Haines jumps. A general macroscopic equation is derived for the rate of wetting/dewetting of a porous medium having randomly distributed, finely divided particles or pores. Use is made of the Forchheimer equation, which is an extension of Darcy's equation to higher Reynolds numbers. Dissipative energy terms due to internal fluid calculaton and to irreversible movements of the meniscus strongly affect the initial rate of imbibition, but as the wetting progresses the Reynolds number decreases and Washburn's equation prevails.The application of percolation theory to wetting/dewetting phenomena in porous media is studied. The use of percolation theory by Kirkpatrick and Stinchcombe to find the electrical conductivity of inhomogeneous solid mixtures is adapted to determining the permeability of a porous medium to fluid flow. It is also shown how the relation between the “precolation probability” and the concentration of “unblocked” channels or pores can be applied in calculating the capillary pressure—desaturation curve in drainage. In particular, percolation theory predicts that a threshold pressure or break-through pressure is required before a non-wetting fluid can displace a wetting fluid in a porous medium. It is often convenient to use tree-like or branching lattice networks as models of a porous medium, because these are amenable to exact solutions in regard to percolation probability and permeability. The percolation properties of porous medium models which consist of lattice networks of cylindrical channels with a distribution of cross-sections and also of randomly packed rotund particles are examined and their relevance to wetting/dewetting phenomena discussed.     

THE EFFECTS OF MEDIUM THICKNESS ON THE ISLAND SIZE DISTRIBUTION IN IMMISCIBLE DISPLACEMENT FLOW IN POROUS MEDIA

The invasion percolation algorithm is used to simulate two-fluid immiscible displacement of a wetting fluid by a non-wetting fluid in various porous media represented by two-dimensional and three-dimensional networks of interconnected capillaries. Trapping of the displaced fluid occurs, thereby creating isolated islands. The effects of the thickness of the porous medium on the island size distribution are studied for capillary displacements for the case in which buoyancy effects are negligible. It was found in a previous study that the number of islands of size s scales approximately as s~" in two-dimensional porous media, where a is a function of the fluid viscosity ratio. The present work reveals that there is a cross-over behavior between the two-dimensional and the three-dimensional problems.     

Effects of reaction order and convection around gas-bubbles in a gas-liquid reacting system

The mass transfer of gaseous reactant into liquid for chemical reaction is significantly affected by relative flow at the interface of gas and liquid. Two extreme cases are for a bubble behaving like a solid particle due to absorbed surfactant impurities and for a freely-internally circulating bubble with a relative interfacial velocity; the present calculations indicate a ratio of mass-transfer rates of a factor up to 1·4–2·45. The factor decreases with increasing reaction rate, becoming negligible for values of K > 2000 sec^?1. It is larger for a $\text{3}\text{2}$ order reaction than for a 1st order reaction.If there is internal circulation, the relative flow changes depending on whether the bubble is alone or in a rising bubble swarm. For small reaction rates the effect of this change in the mass transfer rate has been calculated to be 7–9% at typical bubble sizes of 0·1–0·2 cm radius. The mass transfer rate for a freely-circulating bubble is about 15% larger for a $\text{1}\text{2}$ order reaction than for a 1st order reaction at steady state. Transient and time averaged values of the Sherwood number were obtained. Shrinkage of bubbles from loss of reactant was also considered.     

Monte-Carlo simulations of surface and gas phase diffusion in complex porous structures

A new procedure for estimating surface diffusivities and tortuosities within realistic models of complex porous structures is reported. Our approach uses Monte-Carlo tracer methods to monitor mean-square displacements for molecules restricted to wander on pore walls within model random mesoporous solids typical of those used as adsorbents, heterogeneous catalysts, and porous membranes. We consider model porous solids formed from initial packings of spheres with unimodal, Gaussian, or bimodal distributions of size; changes in pellet porosity are achieved by increasing microsphere radii and by randomly removing spheres from highly densified packings in order to simulate densification and coarsening, respectively. Geometric tortuosities for the surface phase reached large values at void fractions near 0.04 and 0.42 for densified solids; the surface tortuosity gave a minimum value of 1.9 at a void fraction of ∼0.26. These high tortuosities correspond to percolation thresholds for the void and solid phases, which in turn reflect packing densities at which each phase becomes discontinuous. Surface tortuosities for coarsened solids at low void fractions were similar to those in densified solids; however, at void fractions above ∼0.3, surface tortuosities of coarsened solids increased only gradually with void fraction, because coarsening retains significant overlap among spheres at void fractions above those giving disconnected solids in densified structures. Simulations of bulk diffusion within voids were used to compare the transport properties and connectivity of the void space with those of surfaces that define this void space. Surface and void tortuosities were similar, except for void fractions near the solid percolation threshold, because unconnected solid particles interrupt surface connectivity but not gas phase diffusion paths. Surface and void tortuosities were also similar for channels within linear chains of overlapping hollow spheres as both tortuosities increased with decreasing extent of sphere overlap. These simulations provide a basis for estimates of surface and void tortuosities, which are essential in the interpretation and extrapolation of diffusion rates in complex porous media. Surface and void diffusivity estimates differed significantly from those obtained from lattice and capillary models of complex porous structures.     

NETWORK SIMULATION OF RELATIVE PERMEABILITY CURVES USING A BOND CORRELATED-SITE PERCOLATION MODEL OF PORE STRUCTURE

This paper deals with the conductivity and relative conductivity properties of irregular 3-D networks of pores that represent the continua of the oil phase and the aqueous phase respectively, during steady slate two phase flow in porous media. The relative conductivity properties presented, correspond to the saturation history defined by the drainage, imbibition and secondary drainage capillary pressure curves respectively. Use has been made of the pore accessibility history of a 20 × 20 × 20 network and a 10 × 10 × 10 nodes core portion of the network is used to write the flow equations. A set of 1001 linear equations is solved using the Preconditioned Conjugate Gradients Method for the conductivities of the wetting phase and the non-wetting phase respectively, as a function of network saturation and saturation history. The effects of pore throat size distribution and pore body size distribution on relative permeability behaviour has been investigated. Furthermore, the effect of conductivity function q(D) proportional to Dⁿ (n = 0, 1, 2, 3, 4) on relative permeability behaviour was investigated, where D stands for pore throat diameter and n is an exponent depending on pore geometry.

The results of this work are very significant in elucidating the following points that are not clearly stated in the literature: 1) using the bypassing as the only trapping mechanism, the primary drainage and secondary drainage relative permeability curves are in agreement with experimental findings; 2) more realistic displacement mechanisms in secondary imbibition are required to have better agreement with experimental findings; 3) the correlated network models after the site type problem of percolation theory are realistic models of pore structure; 4) the conductivity function q(D) proportional to D³ is the most appropriate pore throat conductivity function because of lamelar like pore geometries; and 5) accurate prediction of the effective permeability requires knowledge of the porosity and the detailed pore geometry in the pore network, in addition to pore size distributions used in the network simulation.     

Creeping flow mass transfer in a cloud of size distributed round drops at high Schmidt numbers

For large Schmidt numbers, analytical expressions of the continuous phase mass transfer coefficient for an active test drop or bubble located in a random fixed array of inactive falling drops or bubbles having an arbitrary size distribution have been obtained for the two limiting cases of rapidly circulating drops and almost solid-like drops in slow creeping motion. The theoretical basis for using a statistically expected velocity field around the test drop in the random cloud to calculate the statistically expected concentration field has been examined. Replacing the size distributed cloud of inactive drops by a uniform cloud of drops of size equal to the Sauter mean radius $\text{b}$₃₂ of the size distributed system causes a minor increase in the predicted mass transfer coefficient for the active test drop unless it is much larger than $\text{b}$₃₂. Theoretical calculations show a similar behavior if one deals with an active cloud and utilizes mass transfer coefficient expressions from an inactive cloud. The conclusion of Tan, Prasher and Guin that mass transfer coefficient in non-uniform active packings obtained experimentally is adequately described by correlations for uniform packings if $\text{b}$₃₂ is used is supported by the theoretical calculations for the size distributions used in the experiments of Tan et al. The predicted mass transfer coefficient in an inactive uniform cloud are somewhat lower than those predicted by the Waslo—Gal-Or model. There is reasonable agreement between the predicted values for a uniform inactive cloud of drops and available experimental data on active clouds of drops and bubbles.     

A new theory of aerosol deposition from turbulent fluids

The author's previously published theory[2] of mass (or heat) transfer to very rough surfaces predicts a power dependence on Sc of ?$\text{1}\text{2}$ (compared with ?$\text{2}\text{3}$ for smooth surfaces). This expected change of $\text{1}\text{6}$ in the power has previously proved difficult to test precisely, but it is shown here that available data on the deposition of small aerosol particles (Sc ～ 10⁶) confirm that the deposition is indeed 10 times faster on very rough surfaces.For larger aerosol particles, however, deposition rates depend strongly on roughness characteristics, as expected from the author's interception and impaction theory[13]. This assumes an x-directional deposition on to the roughness elements, the latter being very important to the process. Comparison of published data with the new theory, and with typical surface roughnesses, supports the proposed theory (eqn 4).     

Mass transfer in packed liquid—liquid extraction columns

The mass transfer characteristics of 3·5, 7·3, 10·16 and 15·6 cm i.d. packed liquid—liquid extraction columns were studied with a variety of packings such as, $\text{3}\text{8}$, $\text{1}\text{2}$ and 1 in. ceramic Raschig rings, $\text{5}\text{8}$ in. stainless steel Raschig rings, $\text{1}\text{2}$ and 1 in. ceramic Intalox saddles, $\text{5}\text{8}$ and 1 in. stainless steel Pall rings, and 1 in. polypropylene Intalox saddles and Pall rings. Some data were also obtained for the co-current mode of operation (up-flow) for packed columns, and without packings. In addition the mass transfer charactersitics of a packed extraction column with film flow were studied.The theory of extraction accompanied by a fast pseudo-first order reaction was employed to measure the values of effective interfacial area. The values of overall (continuous and/or dispersed phase) mass transfer coefficient were measured by the Colburn—Welsh technique. A fairly wide range of physical properties of the two phases was covered.The values of overall (continuous phase) mass transfer coefficient and effective interfacial area for new packings such as Pall rings and Intalox saddles, under otherwise similar conditions, are only about 10 and 35 per cent higher, respectively, than those provided by the conventional packings of the same nominal size. However, the flooding velocities for the newer packings are as much as 80 per cent higher than those for the conventional packings of the same nominal size.     

Temporary and spatial deposition of aerosol particles in the upper human airways during breathing cycle

The k–ε model was used to describe of airflow structure in the sequence of the two first bifurcations of the human respiratory system. Local and temporary distributions of the linear air velocity were calculated for a cyclic breathing pattern corresponding to symmetrical hyperventilation with spontaneous deep breathing. For such a flow, deposition efficiency of particles with diameters of 0.01, 0.1, 1 and $\text{10}\text{μm}$ was calculated using particle trajectories including random displacement of particles due to the Brownian motion. For each type of particles, the “hot spots” of deposition were identified. A specific temporary deposition pattern during breathing cycle was found. The enhancement of deposition was observed at the moment of transition between inspiratory and expiratory parts of the breathing curve.     

Laser-Doppler measurements of the turbulent flow in stirred vessels to establish scaling rules

A laser-Doppler velocimeter, equipped with a frequency shift so as to eliminate directional ambiguity, has been used to measure the turbulent flow in stirred vessels with diameters of 0.12, 0.29 and 0.90 m of the same geometry. The vessels contained water and measurements were done in the impeller stream region. Scaling rules have been derived for average velocity, the periodic component, turbulent intensities and turbulence power spectra.It appears that close to the impeller the flow is dominated by the periodically fluctuating flow of the trailing vortices behind the impeller blades. The normalized mean velocity in the trailing vortices, and therefore the turbulence intensity close to the impeller, is very sensitive to impeller geometry and shows a slight increase with size of the vessel. In the greater part of the impeller stream region the power spectra have a section with a ?$\text{5}\text{2}$ slope on a log-log scale and consequently the energy of the small eddies decreases with increasing scale. At the vessel wall the vortices have decayed completely to random turbulence and the spectrum shows a ? $\text{5}\text{3}$ slope.     

Gel formation in ionomers

A theoretical study of the phase diagram of ionomer solutions in water is presented. These systems show two phase transitions: a demixing transition and a gelation. The demixing transition is studied using Flory's theory of polymer solutions. The gelation concentration always scales as the overlap concentration $\text{c1}$; the dependence of the phase boundaries on temperature and on the fraction f of metallic groups along the chain is studied. At high temperatures, in a good solvent regime, c_gel is proportional to $\text{f}{}^{\mathrm{<mtext>-8</mtext><mtext>5</mtext>}}$; at lower temperatures c_gel is proportional to f^?1.