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 force acting on an evaporating particle immersed in a rarefied plasma flow

Analytical expressions are presented for the drag force acting on an evaporating or nonevaporating particle immersed in a plasma flow for the extreme case of free-molecule flow regime and thin plasma .sheath. It is shown that the drag force on a spherical particle is proportional to the square of the particle radius and to the relative velocity between the particle and the bulk plasma at low speed ratios. The existence of a relative velocity between the particle and the plasma results in a nonuniform heat flux distribution with its rnaximum value at the frontal stagnation point of tire sphere. This nonuniform distribution of the local heat fux density causes a nonuniforrn distribution of the local evaporated-mass flux and vapor reaction force around the surface of an evaporating particle, and thus induces an additional force on the particle. Consequently, the drag force acting on art evaporating particle is always greater than that on a nonevaporating one. This additional drag force due to particle evaporation is more significant for nonmetallic particles and for particle materials with lower latent heat of evaporation and lower vapor molecular mass. It increases with increasing plasma temperature and with decreasing gas pressure at the high plasma temperatures associated with appreciable gas ionization. The drag ratio increases with increasing electron/heavy-particle temperature ratio at high electron temperatures for a two-temperature plasma.     

Restructuring and break-up of two-dimensional aggregates in shear flow

We consider single two-dimensional aggregates, containing glass particles, placed at a water/air interface. We have investigated the critical shear rate for break-up of aggregates with different sizes in a simple shear flow. All aggregates break-up nearly at the same shear rate (1.8 +/- 0.2 s(-)(1)) independent of their size. The evolution of the aggregate structure before break-up was also investigated. With increasing shear rate, the aggregates adopt a more circular shape, and the particles order in a more dense, hexagonal structure. A simple theoretical model was developed to explain the experimentally observed break-up. In the model, the aggregate is considered as a solid circular disk that will break near its diameter. The capillary and drag force on the two parts of the aggregate were calculated, and from this force balance, the critical shear rate was found. The model shows a weak size dependence of the critical shear rate for the considered aggregates. This is consistent with the experimental observations.     

Characterization of fractal particles using acoustics, electroacoustics, light scattering, image analysis, and conductivity

Fractals are aggregates of primary particles organized with a certain symmetry defined essentially by one parameter-a fractal dimension. We have developed a model for the interpretation of acoustic data with respect to particle structure in aggregated fractal particles. We apply this model to the characterization of various properties of a fumed silica, being but one example of a fractal structure. Importantly, our model assumes that there is no liquid flow within the aggregates (no advection). For fractal dimensions of less than 2.5, we find that the size and density of aggregates, computed from the measured acoustic attenuation spectra, are quite independent of the assumed fractal dimension. This aggregate size agrees well with light-scattering measurements. We applied this model to the interpretation of electroacoustic data as well. A combination of electroacoustic and conductivity measurements yields sufficient data for comparing the fractal model of the particle organization with a simple model of the separate primary particles. Conductivity measurements provide information on particle surface conductivity reflected in terms of the Dukhin number (Du). Supporting information for the zeta potential and Du can also be provided by electroacoustic measurements assuming thin double-layer theory. In comparing values of Du from these two measurements, we find that the model of separate solid particles provides much more consistent results than a fractal model with zero advection. To explain this, we first need to explain an apparent contradiction in the acoustic and electroacoustic data for porous particles. Although not important for interpreting acoustic data, advection within the aggregate does turn out to be essential for interpreting electrokinetic and electroacoustic phenomena in dispersions of porous particles.     

The distribution of stresses in rigid fractal-like aggregates in a uniform flow field

The distribution of stresses in rigid fractal-like aggregates moving in a uniform flow field was investigated for particle-cluster and cluster-cluster aggregates with fractal dimensions ranging from 1.7 to 2.3. The method of reflections was used to calculate the drag force on each monomer, while the internal inter-monomer interactions were calculated by applying force and torque balances on each primary particle. The stress distribution was found to be very dissimilar from that of the applied external forces. Although the highest external forces act on the monomers located at the periphery of the aggregate where the drag is more intense, the most stressed inter-monomer bonds are always located in the internal part of the aggregate. This phenomenon is a consequence of the structure of the studied fractal aggregates, which are made mainly of filaments of monomers: the stress generated by the external forces is propagated and progressively accumulated by such filaments up to their roots, which are situated in the inner part of the cluster. Such a behaviour is different from that exhibited by highly connected structures, in which the loads are absorbed locally by the structure and the largest stresses are normally found in the proximity of the highest applied external forces.     

Dependence of fragmentation behavior of colloidal aggregates on their fractal structure

The fragmentation dynamics of aggregate of non-Brownian particles in shear flow is investigated numerically. The breakup behaviors of aggregates having the same connectivity but the different space-filling properties are examined. The Lagrangian particle simulation in a linear flow field is performed. The effect of surrounding fluid on the motion of multiple particles is estimated by Stokesian dynamics approach. The inter-particle force is calculated from the retarded van der Waals potential based on the Lifshitz theory. The results obtained in this work indicate that the fragmentation behavior of colloidal aggregates depends on their fractal structure. However, if the resultant aggregate size is smaller than the critical one, the fragmentation behavior shows the universality regardless of their original structure. Furthermore, the restructuring of aggregate in shear flow and its effect on the fragmentation process are also discussed.     

The size of particle aggregates produced by flocculation with PNIPAM, as a function of temperature

This work investigates the effect of temperature on the size of alumina aggregates formed by flocculation with temperature responsive Poly(N-Isopropylacrylamide)(PNIPAM). The results are discussed in terms of the effects of temperature on particle collision, particle adhesion and aggregate breakage. It was found that the size of alumina aggregates increases with increasing solution temperature. Particle/particle collision and aggregate breakage are largely unaffected by increasing solution temperature and therefore could not account for the change in aggregate size. The dominant factor in aggregate growth with increasing temperature was found to be the increase in the force of adhesion between alumina particles. The appearance of the adhesive force is triggered by the increase in temperature above the lower critical solution temperature of PNIPAM.     

Stoichiometry and physical chemistry of promiscuous aggregate-based inhibitors

Many false positives in early drug discovery owe to nonspecific inhibition by colloid-like aggregates of organic molecules. Despite their prevalence, little is known about aggregate concentration, structure, or dynamic equilibrium; the binding mechanism, stoichiometry with, and affinity for enzymes remain uncertain. To investigate the elementary question of concentration, we counted aggregate particles using flow cytometry. For seven aggregate-forming molecules, aggregates were not observed until the concentration of monomer crossed a threshold, indicating a "critical aggregation concentration" (CAC). Above the CAC, aggregate count increased linearly with added organic material, while the particles dispersed when diluted below the CAC. The concentration of monomeric organic molecule is constant above the CAC, as is the size of the aggregate particles. For two compounds that form large aggregates, nicardipine and miconazole, we measured particle numbers directly by flow cytometry, determining that the aggregate concentration just above the CAC ranged from 5 to 30 fM. By correlating inhibition of an enzyme with aggregate count for these two drugs, we determined that the stoichiometry of binding is about 10,000 enzyme molecules per aggregate particle. Using measured volumes for nicardipine and miconazole aggregate particles (2.1 x 10(11) and 4.7 x 10(10) A(3), respectively), computed monomer volumes, and the observation that past the CAC all additional monomer forms aggregate particles, we find that aggregates are densely packed particles. Finally, given their size and enzyme stoichiometry, all sequestered enzyme can be comfortably accommodated on the surface of the aggregate.     

The drag of the row of parallel chains of spherical particles in the Stokes flow

The drag of thin-layered porous deposit consisting of dendrites of identical spherical particles with respect to the flow of viscous incompressible liquid is calculated. The deposit is approximated by a model system, a row of parallel chains of particles oriented perpendicular to a flow direction. The expression is derived for the dimensionless drag force acting on the unit chain length as a function of the ratio of a particle radius to a half-distance between chain axes, a/h. It is shown that, at a/h < 0.5, the hydrodynamic equivalent of the chains is the smooth cylinder whose radius is 1.16 times smaller than the particle radius that agrees with the experiment. It is also shown that, at a/h = 1, the drag force of a particle contacting with four adjacent particles in the layer with square packing is equal to F = 44F _St, where F _St is the Stokes drag force of a spherical particle. The pressure drop in this single layer is by 3.5% higher than in the layer of spherical particles with cubic packing. At a/h = 2/√3, drag force F of the particle contacting with six particles in a single layer with hexagonal packing is equal to 340F _St.     

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 electrostatic, hydrodynamic, and Brownian forces on particle trajectories and sieving in normal flow filtration

Particle deposition and fouling are critical factors governing the performance of microfiltration and ultrafiltration systems. Particle trajectories were evaluated by numerical integration of the Langevin equation, accounting for the combined effects of electrostatic repulsion, enhanced hydrodynamic drag, and Brownian diffusion. In the absence of Brownian forces, particles are unable to enter the membrane pores unless the drag associated with the filtration velocity can overcome the electrostatic repulsion. Brownian forces significantly alter this behavior, allowing some particles to enter the pore even at low filtration velocities. The average particle transmission, evaluated from the probability of having a particle enter the pore, increases with increasing filtration velocity due to the greater hydrodynamic drag force on the particle. These results provide important insights into particle behavior in membrane systems.     

Electrohydrodynamic Velocity and Pumping Measurements in Water and Alcohols

Bubble and particle velocities in water and alcohols, under the influence of an electric field, were investigated in this work. Air bubbles were injected into the liquids through an electrified metal capillary insulated by glass with its tip left exposed. The end of the capillary from which the bubbles were released was conical in shape. Due to an electric field formed between the noninsulated capillary tip and a ground electrode immersed in the solvent, small bubbles were formed and used as tracers for the electrohydrodynamic (EHD) flow field. The pressure inside the capillary was measured for all liquids used in this study. For water, ethanol, and n-propanol, it was found that, at relatively low applied voltage, the pressure increases with voltage, reaches a maximum (pressure breakpoint), and then sharply decreases. This behavior is a result of the competition between the electric force appearing at the interface and the force due to the EHD flow near the capillary tip. The electric force tends to increase the pressure inside the capillary, while the EHD flow tends to decrease this pressure. For isopropanol and butanol, the pressure breakpoint was not observed in the range of voltage applied in the experiments. The EHD flow velocity was measured by using microbubbles and particles as flow tracers. An adaptive phase-Doppler velocimeter was employed to measure the velocity of bubbles, while the velocity of particles was measured by trajectory visualization of fluorescent particles. A discrepancy was observed between the two methods because of the location at which the measurements were made. It was found that average velocities of both bubbles and particles increase linearly with applied voltage. Experiments were also conducted to investigate pumping of water, which is a result of the EHD velocity near the capillary tip. The pumping flow rate was linearly related to the applied voltage and agreed well with EHD velocity measurements obtained from particle trajectories. Copyright 2000 Academic Press.     

Dynamic drag force based on iterative density mapping: A new numerical tool for three‐dimensional analysis of particle trajectories in a dielectrophoretic system

Dielectrophoresis is a widely used means of manipulating suspended particles within microfluidic systems. In order to efficiently design such systems for a desired application, various numerical methods exist that enable particle trajectory plotting in two or three dimensions based on the interplay of hydrodynamic and dielectrophoretic forces. While various models are described in the literature, few are capable of modeling interactions between particles as well as their surrounding environment as these interactions are complex, multifaceted, and computationally expensive to the point of being prohibitive when considering a large number of particles. In this paper, we present a numerical model designed to enable spatial analysis of the physical effects exerted upon particles within microfluidic systems employing dielectrophoresis. The model presents a means of approximating the effects of the presence of large numbers of particles through dynamically adjusting hydrodynamic drag force based on particle density, thereby introducing a measure of emulated particle–particle and particle–liquid interactions. This model is referred to as “dynamic drag force based on iterative density mapping.” The resultant numerical model is used to simulate and predict particle trajectory and velocity profiles within a microfluidic system incorporating curved dielectrophoretic microelectrodes. The simulated data are compared favorably with experimental data gathered using microparticle image velocimetry, and is contrasted against simulated data generated using traditional “effective moment Stokes‐drag method,” showing more accurate particle velocity profiles for areas of high particle density.     

Plasma-particle momentum transfer under conditions of great knudsen numbers and thin plasma sheath

Particle drag force and thermophoresis results previous obtained are revised by including modified expressions for ion and electron components of the surface pressure. The present analysis shows that there is almost no difference between non-evaporating metallic and nonmetallic particles in their drags and there is only a little difference between those particles in their thermophoretic forces. The effect of evaporation on thermophoretic and drag forces is still marked, but the drag or thermophoretic force ratio with to without accounting for evaporation assumes somewhat different values from those obtained previously and depends notably on whether the particle is metallic or nonmetallic at high plasma temperatures.     

压力下流化床流动特性的实验研究

利用摄像技术在φ60mm×800mm压力为0.1MPa～1.5MPa的冷模加压流化床实验装置上, 对两种不同粒径颗粒的最小流化速度和床层膨胀高度进行了研究。研究结果表明,大粒径聚苯乙烯颗粒的Umf与p-0.3成比例,小粒径石英砂的Umf与p-0.21成比例,并根据实验值拟合出压力下最小流化速度公式为: Umf=μdp ρg｛［(34.15)2+0.05916×dp3 ρg( ρs- ρg)gμ2］12-34.15｝ 床层膨胀高度随气速的增大而不断增高,在相同U Umf下床层膨胀高度随压力的增大而增高,在大于0.7MPa时,压力对膨胀高度的影响减弱。对于聚苯乙烯颗粒,相同的H/Hmf下,U Umf随压力的增大而逐渐减小,当H/Hmf＝1.4～1.6时,U Umf与p-0.52～p-0.58成比例。     

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.     

Surface Potential of Coal Particles in Coal-Oil Mixture and Control of Settling Stability Under Gravity

This work deals with the problem of settling under gravity for coal-oil mixtures when the concentration of particles is large. The repulsive force necessary to ensure stability of coal particle is vital. The net forces acting on the particle include gravity, buoyancy, viscous drag force, and electrostatic repulsive force. Accordingly, the equation at the terminal velocity at settling is obtained along with a critical surface potential to prevent settling under gravity.     

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.     

Parameters of Equilibrium Spectrum of Particles in Coagulating System with Aggregate Disintegration

The formation of equilibrium spectrum of particles in a disperse system with the coagulation–fragmentation of aggregates at a steady-state shear flow was analyzed in terms of two-fraction model. It was suggested that an initial dispersed phase contains only small particles coagulating by the Brownian mechanism; the growth of larger aggregates proceeds by the gradient mechanism and is accompanied by the detachment of fragments. Parameters of equilibrium spectrum characterizing average masses and the number of particles in fine and coarse fractions were determined as functions of a flow shear rate, aggregate fractal dimension, parameters of particle interaction in aggregates, and the properties of the initial dispersed phase.     

Colloidal Fouling of Ultrafiltration Membranes: Impact of Aggregate Structure and Size

A close coupling between the structure and size of hematite flocs formed in suspension and the permeability of the cake that accumulates on ultrafiltration membranes is observed. Specific resistances of cakes formed from flocs generated under diffusion-limited aggregation conditions are at least an order of magnitude lower than those of cakes formed from flocs generated under reaction-limited aggregation conditions. Similar effects are observed whether the aggregation regime is controlled by salt concentration, pH, or added organic anions. This dramatic difference in cake resistance is considered to arise from the size and fractal properties of the hematite assemblages. The ease of fluid flow through these assemblages will be influenced both by the fractal dimension of the aggregates and by their size relative to primary particle size (since, for fractal aggregates, porosity increases as the size of the aggregate increases). The size and strength of aggregates are also important determinants of the relative effects of permeation drag, shear-induced diffusion, and inertial lift and result, in the studies reported here, in relatively similar rates of particle deposition for both rapidly and slowly formed aggregates. The results presented here suggest that control of cake permeability (and mass) via control of aggregate size and structure is an area with scope for further development though the nature and extent of compaction effects in modifying the fractal properties of aggregates generated in suspension requires attention. Copyright 1999 Academic Press.