Copyright (c) 2013 Syracuse University All rights reserved. http://surface.syr.edu/bce Recent documents in Biomedical and Chemical Engineering en-us Sun, 27 Jan 2013 23:59:24 PST 3600 http://surface.syr.edu/bce/38 http://surface.syr.edu/bce/38 Fri, 06 Jul 2012 11:10:34 PDT Satvik N. Wani et al. http://surface.syr.edu/bce/37 http://surface.syr.edu/bce/37 Tue, 27 Mar 2012 11:20:26 PDT A problem of determining the macroscopic or effective thermal conductivity of an N-dimensional composite medium containing N-dimensional nonoverlapping hyperspherical inclusions is considered. Since the macroscopic conductivity is expected to become less sensitive to the detailed spatial distribution of the inclusions for N ≥ 4, only the special case of periodic arrangement of the inclusions is considered. An expression for the macroscopic conductivity correct to O(χ3N + 8), χ being the ratio of "diameter" of the inclusions to the spacing between them, is derived and the numerical results for the conductivity are presented as a function of χ and N for the two special cases of perfectly conducting and insulating inclusions. The effective conductivity of the composite is found to approach that of the continuous matrix in higher dimensions.

]]> Ashok S. Sangani http://surface.syr.edu/bce/36 http://surface.syr.edu/bce/36 Tue, 27 Mar 2012 11:20:25 PDT A numerical method is developed that takes into account the many-particle interactions in a rigorous manner to determine the effective thermal conductivity Km of a composite medium consisting of parallel circular cylinders of thermal conductivity ak suspended in a matrix of conductivity k. Numerical results for Km are presented for a wide range of a and o, the area fraction of the cylinders, after averaging over several computer-generated random arrays of cylinders. The results obtained via this exact method are compared with those of various approximate analytical methods to assess their utility in predicting Km.

]]> Ashok S. Sangani et al. http://surface.syr.edu/bce/35 http://surface.syr.edu/bce/35 Tue, 27 Mar 2012 11:20:23 PDT A numerical method is developed that takes into account the many particle interactions in a rigorous manner to determine the effective thermal conductivity of Km of a composite medium consisting of parallel circular cylinders of thermal conductivity ak suspended in a matrix of conductivity k. Numerical results for Km are presented for a wide rane of a and o, the area fraction of the cylinders, after averaging over several computer-generated random arrays of cylinders. The results obtained via this exact method are compared with those of various approximate analystical methods to assess their utility in predicting Km.

]]> Ashok S. Sangani et al. http://surface.syr.edu/bce/34 http://surface.syr.edu/bce/34 Tue, 27 Mar 2012 11:20:22 PDT The problem of determining the Nusselt number N, the nondimensional rate of heat or mass transfer, from an array of cylindrical particles to the surrounding fluid is examined in the limit of small Reynolds number Re and large Peclet number Pe. N in this limit can be determined from the details of flow in the immediate vicinity of the particles. These are determined accurately using a method of multipole expansions for both ordered and random arrays of cylinders. The results for N/Pe^1/3 are presented for the complete range of the area fraction of cylinders. The results of numerical simulations for random arrays are compared with those predicted using effective-medium approximations, and a good agreement between the two is found. A simple formula is given for relating the Nusselt number and the Darcy permeability of the arrays. Although the formula is obtained by fitting the results of numerical simulations for arrays of cylindrical particles, it is shown to yield a surprisingly accurate relationship between the two even for the arrays of spherical particles for which several known results exist in the literature suggesting thereby that this relationship may be relatively insensitive to the shape of the particles.

]]> Ashok S. Sangani et al. http://surface.syr.edu/bce/33 http://surface.syr.edu/bce/33 Tue, 27 Mar 2012 11:20:20 PDT The motion of bubbles dispersed in a liquid when a small-amplitude oscillatory motion is imposed on the mixture is examined in the limit of small frequency and viscosity. Under these conditions, for bubbles with a stress-free surface, the motion can be described in terms of added mass and viscous force coefficients. For bubbles contaminated with surface-active impurities, the introduction of a further coeflicient to parametrize the Basset force is necessary. These coefficients are calculated numerically for random configurations of bubbles by solving the appropriate multibubble interaction problem exactly using a method of multipole expansion. Results obtained by averaging over several configurations are presented. Comparison of the results with those for periodic arrays of bubbles shows that these coefficients are, in general, relatively insensitive to the detailed spatial arrangement of the bubbles. On the basis of this observation, it is possible to estimate them via simple formulas derived analytically for dilute periodic arrays. The effect of surface tension and density of bubbles (or rigid particles in the case where the no-slip boundary condition is applicable) is also examined and found to be rather small.

]]> Ashok S. Sangani et al. http://surface.syr.edu/bce/32 http://surface.syr.edu/bce/32 Tue, 27 Mar 2012 11:20:19 PDT A method for determining Stokes flow around particles near a wall or in a thin film bounded by a wall on one side and a nondeformable gas-liquid interface on the other side is developed. The no-slip boundary conditions at the wall are satisfied by constructing an image system based on Lamb’s multipoles. Earlier results for the image systems for the flow due to a point force or a force dipole are extended to image systems for force or source multipoles of arbitrary orders. For the case of a film, the image system consists of an infinite series of multipoles on both sides of the film. Accurate evaluation of the flow due to these images is discussed, including the use of Shanks transforms. The method is applied to several problems including chains of particles, radially expanding particles, drops, and porous particles.

]]> Ashok S. Sangani et al. http://surface.syr.edu/bce/31 http://surface.syr.edu/bce/31 Tue, 27 Mar 2012 11:20:18 PDT The wave equations for the interior and exterior of the particles are ensemble averaged and combined with an analysis by Allegra and Hawley @J. Acoust. Soc. Am. 51, 1545 ~1972!# for the interaction of a single particle with the incident wave to determine the phase speed and attenuation of sound waves propagating through dilute slurries. The theory is shown to compare very well with the measured attenuation. The inverse problem, i.e., the problem of determining the particle size distribution given the attenuation as a function of frequency, is examined using regularization techniques that have been successful for bubbly liquids. It is shown that, unlike the bubbly liquids, the success of solving the inverse problem is limited since it depends strongly on the nature of particles and the frequency range used in inverse calculations.

]]> Ashok S. Sangani et al. http://surface.syr.edu/bce/30 http://surface.syr.edu/bce/30 Tue, 27 Mar 2012 11:20:16 PDT A new method is described for incorporating close-field, lubrication forces between pairs of particles into the multiparticle Stokes flow calculations. The method is applied to the suspensions of both spherical as well as cyliridrical particles, and results computed by the method are shown to be in excellent agreement with the exact known results available in the literature.

]]> Ashok S. Sangani et al. http://surface.syr.edu/bce/29 http://surface.syr.edu/bce/29 Tue, 27 Mar 2012 11:20:15 PDT A method for computing Laplace and Stokes interactions among N spherical particles arbitrarily placed in a unit cell of a periodic array is described. The method is based on an algorithm by Greengard and Rokhlin [J. Comput. Phys. 73, 325 (1987)] for rapidly summing the Laplace interactions among particles by organizing the particles into a number of different groups of varying sizes. The far-field induced by each group of particles is expressed by a multipole expansion technique into an equivalent field with its singularities at the center of the group. The resulting computational effort increases only linearly with N. The method is applied to a number of problems in suspension mechanics with the goal of assessing the efficiency and the potential usefulness of the method in studying dynamics of large systems. It is shown that reasonably accurate results for the interaction forces are obtained in most cases even with relatively low-order multipole expansions.

]]> Ashok S. Sangani et al. http://surface.syr.edu/bce/28 http://surface.syr.edu/bce/28 Tue, 27 Mar 2012 11:20:14 PDT A method for computing Stokes flow interactions in suspensions of spherical objects is described in detail and applied to the suspensions of porous particles, drops, and bubbles to determine their hydrodynamic transport coefficients.

]]> Ashok S. Sangani et al. http://surface.syr.edu/bce/27 http://surface.syr.edu/bce/27 Tue, 27 Mar 2012 11:20:13 PDT A countercurrent gas–liquid flow through a fixed bed of spherical particles is examined numerically by solving the particle-scale equations governing the gas and liquid flows. The liquid is assumed to flow along the surface of the particles forming a thin film. The case of small gas flow rates is examined in detail first. In this limit the presence of the liquid film increases the gas pressure drop over its value for a dry bed by three mechanisms: The liquid film makes the apparent size of the particles larger, decreases the pore space for the gas flow, and, with its velocity pointing opposite to the mean gas flow, increases the apparent velocity of the gas compared with the particle surface. The excess pressure drop is determined for both periodic and random arrangements of particles. Next, the case of high gas flow rates where the traction exerted by the gas at the gas–liquid interface is comparable to the weight of the liquid film is examined. In this regime the liquid holdup increases with the gas flow rate and the pressure drop-gas velocity relation is nonlinear. The results of numerical simulations are compared with approximate models and it is shown that a simple capillary model yields reasonably accurate predictions for the liquid holdup and gas pressure drop.

]]> Ashok S. Sangani et al. http://surface.syr.edu/bce/26 http://surface.syr.edu/bce/26 Tue, 27 Mar 2012 11:20:12 PDT We consider the problem of predicting the rate of mass transfer to a fluid flowing parallel to the axes of randomly placed aligned tubes, a model of hollow-fibre contactors. The analysis is carried out for the limiting cases of short contactors, for which the concentration boundary layers remain thin compared with the radius of the tubes, and for the fully developed case corresponding to very long tubes. Numerical simulations for random arrays are carried out for N randomly placed tubes within a unit cell of a periodic array. It is shown that the mass transfer coefficient for the fully developed case is vanishingly small in the limit N to infinity. This suggests that the mass transfer coefficient for a random array of tubes of radius a enclosed in a shell of radius S will vanish logarithmically as the ratio S/a is increased. This behaviour arises due to the logarithmically divergent nature of concentration disturbances caused by each tube in the plane normal to its axis. A theory is developed for determining conditionally averaged velocity and concentration fields and its predictions are shown to compare very well with the results of rigorous numerical computations. The predictions of the theory are also shown to compare well with the measurements of the mass transfer coefficients in hollow-fibre contactors reported in the literature.

]]> Ashok S. Sangani et al. http://surface.syr.edu/bce/25 http://surface.syr.edu/bce/25 Tue, 27 Mar 2012 11:20:11 PDT Effective-medium theories for predicting conditionally averaged velocity field and hydrodynamic transport coefficients of monodisperse suspensions are extended to bidisperse suspensions. The predictions of the theory are shown to agree very well with the results of direct numerical simulations of bidisperse suspensions with hard-sphere configurations up to volume fractions at which phase separation in bidisperse hard-sphere systems are observed.

]]> Ashok S. Sangani et al. http://surface.syr.edu/bce/24 http://surface.syr.edu/bce/24 Tue, 27 Mar 2012 11:20:09 PDT A linear stability analysis is performed for the homogeneous state of a monodisperse gas-fluidized bed of spherical particles undergoing hydrodynamic interactions and solid-body collisions at small particle Reynolds number and finite Stokes number. A prerequisite for the stability analysis is the determination of the particle velocity variance which controls the particle-phase pressure. In the absence of an imposed shear, this velocity variance arises solely due to the hydrodynamic interactions among the particles. Since the uniform state of these suspensions is unstable over a wide range of values of particle volume fraction &phgr; and Stokes number St, full dynamic simulations cannot be used in general to characterize the properties of the homogeneous state. Instead, we use an asymptotic analysis for large Stokes numbers together with numerical simulations of the hydrodynamic interactions among particles with specified velocities to determine the hydrodynamic sources and sinks of particle-phase energy. In this limit, the velocity distribution to leading order is Maxwellian and therefore standard kinetic theories for granular\textfractionsolidus{}hard-sphere molecular systems can be used to predict the particle-phase pressure and rheology of the bed once the velocity variance of the particles is determined. The analysis is then extended to moderately large Stokes numbers for which the anisotropy of the velocity distribution is considerable by using a kinetic theory which combines the theoretical analysis of Koch (1990) for dilute suspensions (&phgr; ≪ 1) with numerical simulation results for non-dilute suspensions at large Stokes numbers. A linear stability analysis of the resulting equations of motion provides the first a priori predictions of the marginal stability limits for the homogeneous state of a gas-fluidized bed. Dynamical simulations following the detailed motions of the particles in small periodic unit cells confirm the theoretical predictions for the particle velocity variance. Simulations using larger unit cells exhibit an inhomogeneous structure consistent with the predicted instability of the homogeneous gas–solid suspension.

]]> Ashok S. Sangani et al. http://surface.syr.edu/bce/23 http://surface.syr.edu/bce/23 Tue, 27 Mar 2012 11:20:08 PDT Experiments were performed in a vertical channel to study the behaviour of a monodisperse bubble suspension for which the dual limit of large Reynolds number and small Weber number was satisfied. Measurements of the liquid-phase velocity fluctuations were obtained with a hot-wire anemometer. The gas volume fraction, bubble velocity, bubble velocity fluctuations and bubble collision rate were measured using a dual impedance probe. Digital image analysis was performed to quantify the small polydispersity of the bubbles as well as the bubble shape.A rapid decrease in bubble velocity with bubble concentration in very dilute suspensions is attributed to the effects of bubble–wall collisions. The more gradual subsequent hindering of bubble motion is in qualitative agreement with the predictions of Spelt & Sangani (1998) for the effects of potential-flow bubble–bubble interactions on the mean velocity. The ratio of the bubble velocity variance to the square of the mean is O(0.1). For these conditions Spelt & Sangani predict that the homogeneous suspension will be unstable and clustering into horizontal rafts will take place. Evidence for bubble clustering is obtained by analysis of video images. The fluid velocity variance is larger than would be expected for a homogeneous suspension and the fluid velocity frequency spectrum indicates the presence of velocity fluctuations that are slow compared with the time for the passage of an individual bubble. These observations provide further evidence for bubble clustering.

]]> Ashok S. Sangani et al. http://surface.syr.edu/bce/22 http://surface.syr.edu/bce/22 Tue, 27 Mar 2012 11:20:07 PDT A method is described for computing the motion of bubbles through a liquid under conditions of large Reynolds and finite Weber numbers. Ellipsoidal harmonics are used to approximate the shapes of the bubbles and the flow induced by the bubbles, and a method of summing flows induced by groups of bubbles, using a fast multipole expansion technique is employed so that the computational cost increases only linearly with the number of bubbles. Several problems involving one, two and many bubbles are examined using the method. In particular, it is shown that two bubbles moving towards each other in an impurity-free, inviscid liquid touch each other in a finite time. Conditions for the bubbles to bounce in the presence of non-hydrodynamic forces and the time for bounce when these conditions are satisfied are determined. The added mass and viscous drag coefficients and aspect ratio of bubbles are determined as a function of bubble volume fraction and Weber number.

]]> Ashok S. Sangani et al. http://surface.syr.edu/bce/21 http://surface.syr.edu/bce/21 Tue, 27 Mar 2012 11:20:06 PDT Observations of bubbles rising near a wall under conditions of large Reynolds and small Weber numbers have indicated that the velocity component of the bubbles parallel to the wall is significantly reduced upon collision with a wall. To understand the effect of such bubble-wall collisions on the flow of bubbly liquids bounded by walls, a model is developed and examined in detail by numerical simulations and theory. The model is a system of bubbles in which the velocity of the bubbles parallel to the wall is significantly reduced upon collision with the channel wall while the bubbles in the bulk are acted upon by gravity and linear drag forces. The inertial forces are accounted for by modeling the bubbles as rigid particles with mass equal to the virtual mass of the bubbles. The standard kinetic theory for granular materials modified to account for the viscous and gravity forces and supplemented with boundary conditions derived assuming an isotropic Maxwellian velocity distribution is inadequate for describing the behavior of the bubble-phase continuum near the walls since the velocity distribution of the bubbles near the walls is significantly bimodal and anisotropic. A kinetic theory that accounts for such a velocity distribution is described. The bimodal nature is captured by treating the system as consisting of two species with the bubbles (modeled as particles) whose most recent collision was with a channel wall treated as one species and those whose last collision was with another bubble as the other species. The theory is shown to be in very good agreement with the results of numerical simulations and provides closure relations that may be used in the analysis of bidisperse particulate systems as well as bounded bubbly flows.

]]> Ashok S. Sangani et al. http://surface.syr.edu/bce/20 http://surface.syr.edu/bce/20 Tue, 27 Mar 2012 11:20:04 PDT We have developed a dual impedance-based probe that can simultaneously measure the bubble velocity and the gas volume fraction in length scales comparable to the bubble diameter. The accurate determination of the profiles is very important for comparisons with existing theories that describe the rheological behavior of bubbly liquids. The gas volume fraction is determined by the residence time of bubble within the measuring volume of the probe. We have found that the details of the bubble-probe interactions must be taken into account to obtain an accurate measure of the gas volume fraction at a point. We are able to predict the apparent nonlinear behavior of the gas volume fraction measurement at large concentrations. The bubble velocity is obtained from the cross correlation of the signals of two closely spaced identical probes. Performance tests and results are shown for bubble velocity and bubble concentration profiles in a gravity driven shear flow of a bubbly liquid.

]]> Ashok S. Sangani et al. http://surface.syr.edu/bce/19 http://surface.syr.edu/bce/19 Tue, 27 Mar 2012 11:20:02 PDT The rheological behavior of rapidly sheared bubble suspensions is examined through numerical simulations and kinetic theory. The limiting case of spherical bubbles at large Reynolds number Re and small Weber number We is examined in detail. Here, Re = p y a^2 / u and We = p y^2 a^3 / s, a being the bubble radius, g the imposed shear, s the interfacial tension, and m and r , respectively, the viscosity and density of the liquid. The bubbles are assumed to undergo elastic bounces when they come into contact; coalescence can be prevented in practice by addition of salt or surface-active impurities. The numerical simulations account for the interactions among bubbles which are assumed to be dominated by the potential flow of the liquid caused by the motion of the bubbles and the shear-induced collision of the bubbles. A kinetic theory based on Grad’s moment method is used to predict the distribution function for the bubble velocities and the stress in the suspension. The hydrodynamic interactions are incorporated in this theory only through their influence on the virtual mass and viscous dissipation in the suspension. It is shown that this theory provides reasonable predictions for the bubble-phase pressure and viscosity determined from simulations including the detailed potential flow interactions. A striking result of this study is that the variance of the bubble velocity can become large compared with (y a)^2 in the limit of large Reynolds number. This implies that the disperse-phase pressure and viscosity associated with the fluctuating motion of the bubbles is quite significant. To determine whether this prediction is reasonable even in the presence of nonlinear drag forces induced by bubble deformation, we perform simulations in which the bubbles are subject to an empirical drag law and show that the bubble velocity variance can be as large as 15 y^2 a^2.

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