Deposition of spherical particles onto cylindrical solid surfaces. I. Numerical simulations

The permeability of fractal porous aggregates with realistic three-dimensional structure is investigated theoretically using model aggregates composed of identical spherical primary particles. Synthetic aggregates are generated by several techniques, including a lattice-based method, simulation of aggregation by differential settling and turbulent shear, and the specification of simple cubic structures, resulting in aggregates characterized by the number of primary particles, solid fraction, characteristic radius, and fractal dimension. Stokesian dynamics is used to determine the total hydrodynamic force on and the distribution of velocity within an aggregate exposed to a uniform flow. The aggregate permeability is calculated by comparing these values with the total force and velocity distribution calculated from the Brinkman equation applied locally and to the entire aggregate using permeability expressions from the literature. The relationship between the aggregate permeability and solid fraction is found to be best predicted by permeability expressions based on cylindrical rather than spherical geometrical elements, the latter tending to underestimate the aggregate permeability significantly. The permeability expressions of Jackson and James or Davies provide good estimates of the force on and flow through porous aggregates of known structure. These relationships are used to identify a number of general characteristics of fractal aggregates.     

Drag and Torque on Clusters of N Arbitrary Spheres at Low Reynolds Number

Hydrodynamics of particle clusters suspended in viscous fluids is a subject of considerable theoretical and practical importance. Using a multipole expansion of the flow velocity in a series of spherical harmonics, Lamb's fundamental solution of the Stokes flow outside a single sphere is generalized in this work to the case of N nonoverlapping spheres of arbitrary size with slip boundary conditions. The expansion coefficients are found by transforming the boundary conditions to the Lamb form and by transforming the spherical coordinates and solid spherical harmonics centered at different spheres. The problem is reduced to the solution of the linear system of equations for the expansion coefficients, which is carried out numerically. Based on the developed theory, the relation between the hydrodynamic and gyration radius of fractal-like aggregates with different structure is established. In another application, an asymptotic slip-regime dependence of the aggregate hydrodynamic radius on the Knudsen number and the number of particles is found by performing calculations of drag forces acting on the gas-borne fractal-like and straight chain aggregates. A good agreement is shown in comparing predictions of the described theory with available experimental and theoretical results on motion of various small sphere clusters in viscous fluid. Copyright 2000 Academic Press.     

Effect of adsorbed polymers on the slow motion of an assemblage of spherical particles relative to a fluid

 The effects of adsorbed polymers on the sedimentation of a homogeneous distribution of colloidal spheres and on the fluid flow through a bed of particles are investigated theoretically. The Reynolds number is assumed to be small, and the surface polymer layers are assumed to be thin with respect to the radius of particles and to the surface-to-surface spacing between neighboring particles. The effects of interaction of the individual particles are taken into explicit account by employing a fundamental cell-model representation which is known to provide good predictions for the motion of a swarm of spheres within a fluid in the absence of adsorbed polymers. To solve the Stokes flow equations within and outside the polymer layer a method of matched asymptotic expansions in a small parameter λ is used, where λ is the ratio of the length scale of the polymer layer to the particle radius. The results for the sedimentation rate and the pressure drop are expressed in terms of an effective hydrodynamic thickness (L) of the polymer layer, which are accurate to O(λ²). When the concentration of particles in a suspension or a bed is increased, L becomes larger, meaning the settling velocity decreases or the pressure drop increases. The O(λ) term for L normalized by its value in the limit λ→0 is found to be independent of the polymer segment distribution, the hydrodynamic inter-actions among the segments, and the volume fraction of the segments. The O(λ²) term for L, however, is a sensitive function of the polymer segment distribution and the volume fraction of the segments. In general, the particle-interaction effects on the motion of polymer-coated particles relative to a fluid can be quite significant. Received: 28 August 1996 Accepted: 23 January 1997     

Stable colloidal dispersions of a lipase-perfluoropolyether complex in liquid and supercritical carbon dioxide

The technique of hydrophobic ion pairing was used to solubilize the lipase from Candida rugosa in a fluorinated solvent, perfluoromethylcyclohexane (PFMC), in complex with a perfluoropolyether (PFPE) surfactant, KDP 4606. The enzyme-surfactant complex was determined to have a hydrodynamic diameter of 6.5 nm at atmospheric pressure by dynamic light scattering (DLS), indicating that a single lipase molecule is stabilized by surrounding surfactant molecules. The complex formed a highly stable colloidal dispersion in both liquid and supercritical carbon dioxide at high CO2 densities (>0.92 and 0.847 g/mL, respectively), with 4% by volume PFMC as a cosolvent, yielding a fluid that was orange, optically translucent, and very nearly transparent. DLS demonstrated aggregation of the enzyme-surfactant complexes in CO2 at 25 and 40 degrees C and various pressures (2000-5000 psia) with hydrodynamic diameters ranging from 50 to 200 nm. The mechanism by which the enzyme-surfactant particles aggregate was shown to be via condensation due to very low polydispersities as characterized by the size distribution moments. Interparticle interactions were investigated with respect to density and temperature, and it was shown that on decreasing the CO2 density, the particle size increased, and the stability against settling decreased. Particle size also decreased as the temperature was increased to 40 degrees C, at constant CO2 density. Nanoparticle aggregates of an enzyme-surfactant complex in CO2, which are nearly optically transparent and stable to settling, are a promising new alternative to previous types of dispersions of proteins in CO2 that either required water/CO2 microemulsions or were composed of large particles unstable to settling.     

Expansion and hydrodynamic properties of β-cyclodextrin polymer/tungsten carbide composite matrix in an expanded bed

The expansion and hydrodynamic properties of matrix are significant for expanded bed adsorption (EBA) processes. A series of new composite matrices CroCD-TuC are studied and estimated in an expanded bed. It is found that the heavier matrix is better suited for high operation fluid velocity than the lighters. Although the Richardson–Zaki equation can well correlate the bed voidage with fluid velocity for all CroCD-TuC matrices tested, the modifications are proposed to improve the accuracy of theoretical predictions of correlation parameters, including terminal settling velocity (U_t) and expansion index (n). Residence time distributions (RTDs) are determined, and the Bodenstein number (Bo) and axial dispersion coefficient (D_ax) are employed to analyze the liquid mixing in the expanded bed. It is found for CroCD-TuC matrices, both parameters notably changed with the variation of fluid velocity and viscosity. Furthermore, D_ax is an intuitive parameter estimating the bed stability on various operating conditions, and also a restriction on developing the matrix for high operation fluid velocity. The comparison of the hydrodynamic properties on different matrices reveals that CroCD-TuC 3 and CroCD-TuC 4 seem superior to other matrices in hydrodynamic properties, making them promising matrices for further use. The correlations as the functions of fluid velocity and viscosity have been established which may provide beneficial information for practical applications of CroCD-TuC matrices in EBA processes.     

Faxen's Laws of a Composite Sphere under Creeping Flow Conditions

Under creeping flow conditions, Faxen's laws are derived for a composite sphere comprising a solid core covered by a permeable layer of arbitrary thickness. The derivations are carried out by applying reciprocal theorem in combination with fluid velocity and pressure distributions in certain simple flow as a comparison field. In this regard, the fluid velocity disturbances caused by a composite sphere subject to a simple shear flow and a rotational flow are solved individually. In the limiting case where the solid core vanishes, the resulting Faxen expressions for the drag force, torque, and stresslet compare very well with the existing Faxen's law for a porous sphere. It is found that when the porous layer is thick enough and its permeability is sufficiently low, the hydrodynamic behavior of a composite sphere can be approximated by that of a porous particle with equal permeability. This can be explained by the fact that the fluid cannot penetrate deeply into a porous layer of low permeability to flow through the pores near the core surface, and thereby the fluid can hardly feel the resistance from the core surface. Copyright 2000 Academic Press.     

Transport studies of water and aqueous—dioxane through a pyrex sintered disc impregnated with cellulose acetate

Experimental results for the measurements of hydrodynamic permeability, electroosmotic velocity and streaming potential of water and dioxane—water (DH₂O) mixtures (10, 20, 30 and 40% by mass of dioxane) using a pyrex sintered disc (G₂) impregnated with cellulose acetate at 30°C and at voltages up to 40 V are reported. The data have been analysed in the light of non-equilibrium thermodynamics. Onsager's reciprocity relation for all compositions of aqueous—dioxane has been found to hold true. It has been found that the concentration dependence of the phenomenological coefficients conform to the Spiegler's frictional model. Efficiencies of electrokinetic energy conversion, i.e. electroosmosis and streaming potential, have been determined and the results are in accordance with the theory of non-equilibrium thermodynamics. The maximum value of the efficiency of energy conversion for both the modes has been found to be independent of the input force. Membrane characteristics such as pore radius and number of pores have been determined, whereas the membrane—permeant interface is characterised in term of the zeta potential.     

Expansion and hydrodynamic properties of cellulose-stainless steel powder composite matrix for expanded bed adsorption

For better understanding the influences of solid phase properties on the performance of the expanded bed, the expansion and hydrodynamic properties of cellulose-stainless steel powder composite matrix with a series of densities was investigated and analyzed in an expanded bed. Two kinds of matrix particle diameter fractions, the small one (60-125 microm) and the large (125-300 microm), were used in the present work. In general, the expansion factors decreased obviously with the increase of matrix density. A linear relation between the mean density of matrix and superficial velocity at expansion factor of 2.5 was found for same series of matrices. The Richardson-Zaki equation could correlate the bed expansion and operation fluid velocity for all matrices tested. The theoretical prediction of correlation parameters (the terminal settling velocity U(t) and expansion index n) was improved with the modification of equations in the literature. The residence time distributions were investigated to characterize the hydrodynamic property in expanded bed. Compared with three evaluation factors (the height equivalent of theoretical plate, Bo number and axial distribution coefficient D(ax)), the results indicated that D(ax) is the best parameter to analyze the bed stability of expanded bed under various operation conditions and matrix properties. In addition, it was found that fluid velocity is the most essential factor to influence the hydrodynamic properties in the bed. A linear relation between the D(ax) and superficial fluid velocity for all matrices tested was established.     

Hydrodynamic permeability of a membrane composed of porous spherical particles in the presence of uniform magnetic field

This work concerns the flow of an incompressible viscous fluid past a porous sphere in presence of transverse applied uniform magnetic field, using particle-in-cell method. The Brinkman equations are used in porous region and the Stokes equations for non-porous region. At the fluid-porous interface, the stress jump boundary condition for tangential stresses along with continuity of normal stress and velocity components are used. Four known boundary conditions on the hypothetical surface are considered and compared: Happel’s, Kuwabara’s, Kvashnin’s and Cunningham’s (Mehta-Morse’s condition). The hydrodynamic drag force experienced by a porous spherical particle in a cell and hydrodynamic permeability of membrane built up by porous spherical particles are evaluated. The patterns of streamlines are also obtained and discussed. The effect of stress jump coefficient, Hartmann number, dimensionless specific permeability of the porous particle and particle volume fraction on the hydrodynamic permeability and streamlines are discussed. Some previous results for hydrodynamic drag force and dimensionless hydrodynamic permeability have been verified.     

Effects of surfactants on wave-induced drainage of foam and emulsion films

It is well established that the plane-parallel models of foam and emulsion films underestimate the velocity of film thinning by up to several orders of magnitude and show an incorrect dependence of thinning velocity on film radius. A new theory of film thinning has been previously formulated for tangentially immobile films [12, 13], and shows that the reason for this discrepancy is the neglect of experimentally observed finite amplitude surface waves. For thin films of relatively large radii (> 1o^–2 cm), the pumping of the fluid generated by oscillations of the surface waves, provides the dominant contribution to film thinning velocity. The present hydrodynamic model includes the effects of surfactants (Marangoni-Gibbs-effect, surface viscosity and surface diffusion) and surface waves on thinning velocity. As in the case of a tangentially immobile film, it is concluded that the thinning velocity varies inversely with less than the first power of the film radius, and not with the square of the film radius, as predicted by the plane-parallel models of thin film. Also, the velocity of thinning is found to be up to several orders of magnitudes larger than that evaluated from the plane-parallel models. The influence of waves in enhancing the thinning velocity is found to be most significant for a tangentially immobile film and this effect decreases by a factor of up to 3, with a decrease in surface elasticity and surface viscosity.     

Sedimentation velocity and potential in concentrated suspensions of charged porous spheres

The body-force-driven migration in a homogeneous suspension of polyelectrolyte molecules or charged flocs in an electrolyte solution is analyzed. The model used for the particle is a porous sphere in which the density of the hydrodynamic frictional segments, and therefore also that of the fixed charges, is constant. The effects of particle interactions are taken into account by employing a unit cell model. The overlap of the electric double layers of adjacent particles is allowed and the relaxation effect in the double layer surrounding each particle is considered. The electrokinetic equations which govern the electrostatic potential profile, the ionic concentration (or electrochemical potential energy) distributions, and the fluid velocity field inside and outside the porous particle in a unit cell are linearized by assuming that the system is only slightly distorted from equilibrium. Using a regular perturbation method, these linearized equations are solved for a symmetrically charged electrolyte with the density of the fixed charges as the small perturbation parameter. An analytical expression for the settling velocity of the charged porous sphere is obtained from a balance among its gravitational, electrostatic, and hydrodynamic forces. A closed-form formula for the sedimentation potential in a suspension of identical charged porous spheres is also derived by using the requirement of zero net electric current. The dependence of the sedimentation velocity and potential of the suspension on the particle volume fraction and other properties of the particle-solution system is found to be quite complicated.     

Transportation and Settling Distribution of Microparticles in Low-Reynolds-Number Poiseuille Flow in Microchannel

In this article, velocity field and settling distribution of microparticles in a dilute suspension in low-Reynolds-number Poiseuille flow in a microchannel is experimentally investigated using microscopic image analysis. An effective technique is applied to manipulate single-particle tracking in order to determine the controlling parameters on transportation and settling of microparticles in microchannels. The results show that the velocities of dispersed phase are affected by the hydrodynamic properties, and this velocity deviation can be significant when the hydrodynamic coupling between particles and channel walls is considerable. Increasing the Reynolds number would result in decrease in total number of particles settled on the bottom wall of channel.     

Studies on the electrochemical and permeation characteristics of asymmetric charged porous membranes

Asymmetric charged porous membranes were prepared by imbedding 10% (W/W) ion-exchange resin in cellulose acetate binder. Membrane potential and conductance measurements have been carried out in sodium chloride solutions at different concentrations to investigate the relationship between concentration of fixed charges and electrochemical properties of developed nonselective cation- and anion-exchange membranes. Counterion transport number and permselectivity of these membranes were found to vary due to the presence of ion-exchange resin. The hydrodynamic and electroosmotic permeability of sodium chloride solutions has been studied in order to compute equivalent pore radius. For cation- and anion-exchange membranes good agreement was observed between pore radius values estimated from hydrodynamic and electroosmotic permeability coefficient separately, while for nonselective membranes no correlation was found. Membrane conductance data, along with values of concentration of fixed charges, were used for the estimation of the tortuosity factor, salt permeability coefficient, and frictional coefficient between solute and membrane matrix employing an interpretation by nonequilibrium thermodynamic principles based on frictional forces. Moreover, surface morphological studies of these membranes also have been carried out and the membranes were found to be reasonably homogeneous.     

Sedimentation of a charged colloidal sphere in a charged cavity

An analytical study is presented for the quasisteady sedimentation of a charged spherical particle located at the center of a charged spherical cavity. The overlap of the electric double layers is allowed, and the polarization (relaxation) effect in the double layers is considered. The electrokinetic equations that govern the ionic concentration distributions, electric potential profile, and fluid flow field in the electrolyte solution are linearized assuming that the system is only slightly distorted from equilibrium. Using a perturbation method, these linearized equations are solved for a symmetric electrolyte with the surface charge densities of the particle and cavity as the small perturbation parameters. An analytical expression for the settling velocity of the charged sphere is obtained from a balance among the gravitational, electrostatic, and hydrodynamic forces acting on it. Our results indicate that the presence of the particle charge reduces the magnitude of the sedimentation velocity of the particle in an uncharged cavity and the presence of the fixed charge at the cavity surface increases the magnitude of the sedimentation velocity of an uncharged particle in a charged cavity. For the case of a charged sphere settling in a charged cavity with equivalent surface charge densities, the net effect of the fixed charges will increase the sedimentation velocity of the particle. For the case of a charged sphere settling in a charged cavity with their surface charge densities in opposite signs, the net effect of the fixed charges in general reduces/increases the sedimentation velocity of the particle if the surface charge density of the particle has a greater/smaller magnitude than that of the cavity. The effect of the surface charge at the cavity wall on the sedimentation of a colloidal particle is found to increase with a decrease in the particle-to-cavity size ratio and can be significant in appropriate situations.     

Experimental and modeling study of breakage and restructuring of open and dense colloidal aggregates

In this work we present experimental and simulation analysis of the breakage and restructuring of colloidal aggregates in dilute conditions under shear. In order to cover a broad range of hydrodynamic and interparticle forces, aggregates composed of primary particles with two sizes, d(p) = 90 and 810 nm, were generated. Moreover, to understand the dependence of breakage and restructuring on the cluster structure, aggregates grown under stagnant and turbulent conditions, having substantially different initial internal structures with fractal dimension d(f) equal to 1.7 and 2.7, respectively, were used. The aggregates were broken by exposing them to a well-defined elongational flow produced in a nozzle positioned between two syringes. To investigate the evolution of aggregate size and morphology, respectively, the mean radius of gyration, , and d(f) were monitored during the breakup process using light scattering and confocal laser scanning microscopy. It was found that the evolution of aggregates' fractal dimension during breakage is solely controlled by their initial structure and is independent of the primary particles size. Similarly, the scaling of the steady-state vs the applied hydrodynamic stress is independent of primary particle size, however, depends on the history of aggregate structure. To quantitatively explain these observations, the breakage process was modeled using stokesian dynamics simulations incorporating DLVO and contact interactions among particles. The required flow-field for these simulations was obtained from computational fluid dynamics. The complex flow pattern was simplified by considering a characteristic stream line passing through the zone with the highest hydrodynamic stress inside the nozzle, this being the most critical flow condition experienced by the clusters. As the flow-field along this streamline was found to be neither pure simple shear nor pure extensional flow, the real flow was approximated as an elongational flow followed by a simple shear flow, with a stepwise transition between them. Using this approach, very good agreement between the measured and simulated aggregate size values and structure evolution was obtained. The results of this study show that the process of cluster breakup is very complex and strongly depends on the initial aggregate structure and flow-field conditions.     

On hydrodynamic permeability of a membrane built up by porous deformed spheroidal particles

This paper concerns the slow viscous flow of an incompressible fluid past a swarm of identically oriented porous deformed spheroidal particles, using particle-in-cell method. The Brinkman’s equation in the porous region and the Stokes equation for clear fluid region in their stream function formulations are used. Explicit expressions are investigated for both the inside and outside flow fields to the first order in a small parameter characterizing the deformation. The flow through the porous oblate spheroid is considered as the particular case of the porous deformed spheroid. The hydrodynamic drag force experienced by a porous oblate spheroid and permeability of a membrane built up by porous oblate spheroids having parallel axis are evaluated. The dependence of the hydrodynamic drag force and the hydrodynamic permeability on particle volume fraction, deformation parameter and viscosity of porous fluid are also discussed. Four known boundary conditions on the hypothetical surface are considered and compared: Happel’s, Kuwabara’s, Kvashnin’s and Cunningham’s (Mehta-Morse’s condition). Some previous results for hydrodynamic drag force and hydrodynamic permeability have been verified. The model suggested can be used for evaluation of changing hydrodynamic permeability of a membrane under applying unidirectional loading in pressure-driven processes (reverse osmosis, nano-, ultra- and microfiltration).     

Experimental study of small aggregate settling

The drag coefficient and hydrodynamic radius of particles are important parameters needed in crystallization science. Small aggregates of micrometric primary particles are mainly produced in stirred crystallizers. We present experimental results on the drag coefficient of macroscopic aggregates consisting of glass beads in the number range [2,100]. The drag coefficient is calculated from settling measurements in glycerol in order to preserve the Stokesian nature of typical flow around particles in a crystallizer. We show that the hydrodynamic radius of these aggregates is almost the radius based on the average projected area over all orientations. This result is extended to larger and more porous aggregates.     

Forces on a porous particle in an oscillating flow

We investigate theoretically forces acting on a porous particle in an oscillating viscous incompressible flow. We use the unsteady equations describing the creeping flow, namely the Stokes equations exterior to the particle and the Darcy or Brinkman equations inside it. The effect of particle permeability and oscillation frequency on the flow and forces is expressed via the Brinkman parameter beta = a/square root(k) and the frequency parameter Y = square root(a(2)omega/2nu) = a/delta, respectively. Here a is particle radius, k is its permeability, omega is the angular frequency, delta is the thickness of Stokes layer (penetration depth) and nu is the fluid kinematic viscosity. It is shown that the oscillations interact with permeable structure of the particle and affect both the Stokes-like viscous drag and the added mass force components.     

Size segregation in a fluid-like or gel-like suspension settling under gravity or in a centrifuge

We investigate size segregation effects in a bidisperse concentrated suspension when slowly settling under gravity or when submitted to a centrifugal field. Experiments are carried out with PMMA spheres of two different mean diameters (190 and 25 microm) suspended in a hydrophobic index-matched fluid. Spatial repartitions of both small and large spheres and velocity fluctuations of particles are measured using fluorescently labeled PMMA spheres and a particle-image-velocimetry method. Large particles behave as hard spheres in purely hydrodynamic interactions, while small spheres interact through weakly attractive forces. For a small amount of small spheres among large ones, the suspension remains fluid during settling and the organization of the velocity field of particles into finite-sized structures also called "blobs" promotes size segregation. A larger proportion of weakly attractive small spheres in the bidisperse suspension causes a considerable slowdown of the settling process under gravity and the occurrence of a large-scale collective behavior together with a loss of size segregation. When centrifuging the gel-like bidisperse suspension, a shear-induced melting of the particle network induces a spectacular segregation of species. As a consequence, aging tests of soft yielding materials using centrifugation methods are not representative of the shelf-life stability of the products. A tentative model based on the competition between viscous stresses acting upon particles and adhesive stresses gives a correct estimate of the critical stationary acceleration for the destabilization of the particle network and the onset of size segregation in a gel-like suspension.     

Structure-transport analysis for particulate packings in trapezoidal microchip separation channels

This article investigates the efficiency of particulate beds confined in quadrilateral microchannels by analyzing the three-dimensional fluid flow velocity field and accompanying hydrodynamic dispersion with quantitative numerical simulation methods. Random-close packings of uniform, solid (impermeable), spherical particles of diameter d(p) were generated by a modified Jodrey-Tory algorithm in eighteen different conduits with quadratic, rectangular, or trapezoidal cross-section at an average bed porosity (interparticle void fraction) of epsilon = 0.48. Velocity fields were calculated by the lattice Boltzmann method, and axial hydrodynamic dispersion of an inert tracer was simulated at Péclet numbers Pe = u(av)d(p)/D(m) (where u(av) is the average fluid flow velocity through a packing and D(m) the bulk molecular diffusion coefficient) from Pe = 5 to Pe = 30 by a Lagrangian particle-tracking method. All conduits had a cross-sectional area of 100d(p)(2) and a length of 1200d(p), translating to around 10(5) particles per packing. We present lateral porosity distribution functions and analyze fluid flow profiles and velocity distribution functions with respect to the base angle and the aspect ratio of the lateral dimensions of the different conduits. We demonstrate significant differences between the top and bottom parts of trapezoidal packings in their lateral porosity and velocity distribution functions, and show that these differences increase with decreasing base angle and increasing base-aspect ratio of a trapezoidal conduit, i.e., with increasing deviation from regular rectangular geometry. Efficiencies are investigated in terms of the axial hydrodynamic dispersion coefficients as a function of the base angle and base-aspect ratio of the conduits. The presented data support the conclusion that the efficiency of particulate beds in trapezoidal microchannels strongly depends on the lateral dimensions of the conduit and that cross-sectional designs based on large side-aspect-ratio rectangles with limited deviations from orthogonality are favorable.