A parametric study of cohesive particle agglomeration in a shear flow—numerical simulations by the discrete element method

In this study, we investigate numerically the shear-induced agglomeration of cohesive inertial particles in a simple shear flow. We conduct a series of numerical simulations by implementing a soft-sphere collision model. Post-processing tools are developed in-house to analyze the results of the simulations in terms of transient and terminal numbers of primary particles and agglomerates, magnitudes and distributions of agglomerate size, and fractal dimension as functions of the salient physical parameters. The obtained numerical results compared with existing transient models suggest that, as the system transitions from formation of duplets to larger agglomerates, in most cases the number of agglomerates in the system reveals clear maxima. The results also show evidence of agglomerate densification, as weaker agglomerates with lower fractal dimensions are broken up and stronger ones with higher fractal dimensions are formed. Furthermore, we found that a simple theoretical model proposed by, among others, Chimmili et al. is able to predict the temporal development of the agglomerate size properly.     

The self-preserving size distribution theory. I. Effects of the Knudsen number on aerosol agglomerate growth

Gas-phase synthesis of fine solid particles leads to fractal-like structures whose transport and light scattering properties differ from those of their spherical counterparts. Self-preserving size distribution theory provides a useful methodology for analyzing the asymptotic behavior of such systems. Apparent inconsistencies in previous treatments of the self-preserving size distributions in the free molecule regime are resolved. Integro-differential equations for fractal-like particles in the continuum and near continuum regimes are derived and used to calculate the self-preserving and quasi-self-preserving size distributions for agglomerates formed by Brownian coagulation. The results for the limiting case (the continuum regime) were compared with the results of other authors. For these cases the finite difference method was in good in agreement with previous calculations in the continuum regime. A new analysis of aerosol agglomeration for the entire Knudsen number range was developed and compared with a monodisperse model; Higher agglomeration rates were found for lower fractal dimensions, as expected from previous studies. Effects of fractal dimension, pressure, volume loading and temperature on agglomerate growth were investigated. The agglomeration rate can be reduced by decreasing volumetric loading or by increasing the pressure. In laminar flow, an increase in pressure can be used to control particle growth and polydispersity. For D(f)=2, an increase in pressure from 1 to 4 bar reduces the collision radius by about 30%. Varying the temperature has a much smaller effect on agglomerate coagulation.     

Pore-scale modeling of rarefied gas flow in fractal micro-porous media,using lattice Boltzmann method (LBM)

Due to the widespread use of rarefied gas flow in micro-porous media in industrial and engineering problems, a pore-scale modeling of rarefied gas flow through two micro-porous media with fractal geometries is presented, using lattice Boltzmann method. For this purpose, square- and circular-based Sierpinski carpets with fractal geometries are selected due to their inherent behavior for real porous media. Diffusive reflection slip model is used and developed for these porous media through this study. With this respect, the planar Poiseuille flow is selected as a benchmark and validated with the literature. The effect of Knudsen number (Kn) on the permeability is investigated and compared in each geometry. It is shown that as Knudsen number increases, the permeability will increase due to the gas slippage effect on the solid blocks. In addition, it is observed that the permeability is more sensitive to the gaseous flow behavior at the slip and beginning of transition flow regimes. At last, the permeability relationship with Knudsen number is presented with a higher coefficient of determination for both fractal geometries, showing that this relation is logarithmic.

     

Formation and Aggregation Kinetics of Mixed Metal Hydroxides Agglomerates

The aggregation and fractal structure of mixed metal hydroxides ( MMH ) agglomerates with increasing ionic strength have been studied by dynamic light scattering ( DLS ) and SEM techniques. The experiments indicate that the MMH agglomerates have two different structures in RLA regime and DLA regime, and also give the proof that the transition region between RLA and DLA may occur.     

Change in particle size distribution of fractal agglomerates during Brownian coagulation in the free-molecule regime

Time evolution of particle size distribution of fractal agglomerates undergoing Brownian coagulation in the free-molecule regime was investigated. A simple analytical solution for the size distribution change was obtained by using the assumption that the size distribution during the coagulation process can be represented by a time-dependent log-normal function. The derived solution consists of three parameters of the log-normal distribution function. This study is believed to provide the first analytical solution for all the parameters of the log-normal distribution of fractal agglomerates undergoing coagulation in the free-molecule regime. To validate the derived solution, numerical computations were performed. The results were compared with the analytical solution and good agreement was obtained.     

Numerical Analysis of 3‐D Flow Past a Stationary Sphere with Slip Condition at Low and Moderate Reynolds Numbers

Computations are performed to determine the steady 3‐D viscous fluid flow forces acting on the stationary spherical suspended particle at low and moderate Reynolds numbers in the range of 0.1≤Re≤200. A slip is supposed on the boundary so that the slip velocity becomes proportional to the shear stress. This model possesses a single parameter to account for the slip coefficient λ (Pa.s/m), which is made dimensionless and is called Trostel number (Tr=λ a/μ). Decreasing slip, increases drag in all Reynolds limits, but slip has smaller effects on drag coefficient at lower Reynolds number regimes. Increasing slip at known Reynolds number causes to delay of flow separation and inflect point creation in velocity profiles. At full slip conditions, shear drag coefficient will be zero and radial drag coefficient reaches to its maximum values. Flow around of sphere at full‐slip condition is not equal to potential flow around a sphere. Present numerical results corresponding to full slip (Tr→0) are in complete accord with certain results of flow around of inviscid bubbles, and the results corresponding to no‐slip (Tr→∞) have excellent agreement with the results predicted by the no‐slip boundary condition.     

Brownian Coagulation of Fractal Agglomerates: Analytical Solution Using the Log-Normal Size Distribution Assumption

An analytical solution to Brownian coagulation of fractal agglomerates in the continuum regime that provides time evolution of the particle size distribution is presented. The theoretical analysis is based on representation of the size distribution of coagulating agglomerates with a time-dependent log-normal size distribution function and employs the method of moments together with suitable simplifications. The results are found in the form that extends the spherical particle solution previously obtained by K. W. Lee (J. Colloid Interface Sci. 92, 315-325 (1983)). The results show that the mass fractal dimension has a significant effect on the size distribution evolution during coagulation. When the obtained solution was compared with numerical results, good agreement was found. The self-preserving size distribution of nonspherical agglomerates is discussed. Copyright 2000 Academic Press.     

The effect of gas slip on pressure drop and deposition of submicron particles in fibrous filters

The effect of gas slip at fibers on the drag to a flow and the deposition of submicron particles in model filters with a tree-dimensional flow field has been considered. The average values of the drag force and the efficiency of diffusion collection of particles with finite sizes in a double hexagonal three-dimensional model filter taken as a standard uniform filter have been calculated as depending on the packing density of fibers and the Knudsen number. It has been shown that, in the region of the sizes of the most penetrating particles, under preset conditions, and at specified filter parameters, the obtained collection efficiency values agree with the results of calculations performed by empirical formulas for a model fan filter. Moreover, formulas derived for a planar flow taking into account the slip effect are applicable to highly porous filters.     

Simulation of the hydrodynamic drag of aggregated particles

The drag force on aggregates and partially sintered agglomerates is assessed using the lattice Boltzmann method (LBM) and accelerated Stokesian dynamics (ASD). Both methods have been compared in terms of accuracy and computational effort. It is shown that they give comparable results if all numerical parameters are controlled carefully. LBM requires a much higher computational effort, however, in contrast to ASD it is able to simulate partially sintered agglomerates as well. The results show that even a very small amount of sintering leads to a significant reduction in the drag force. The analysis of the drag force on agglomerates as well as on aggregates shows that there is no simple geometric quantity which is uniquely related to the drag force. Moreover, there is a significant variation in drag force for single aggregates at different orientations or for the orientation averaged drag force of different aggregates of the same size. This is explained by the structural effects which may lead to a variation in the drag force up to +/-20%.     

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.     

Effects of molar mass, concentration and thermodynamic conditions on polymer-induced flow drag reduction

Drag reduction in Taylor flow of polystyrene solutions is investigated using a commercial rheometer equipped with a standard double-gap sample holder with axial symmetry. The dependence of drag reduction on various factors, including polymer molar mass, polymer concentration, and thermodynamic conditions is studied. Drag reduction induced by polystyrene in toluene is found to increase with increasing polymer concentration in the dilute concentration regime. It is also seen that molecules with high molar mass of the polymer promote drag reduction. In terms of hydrodynamic volume fraction normalisation, it is found that most of the drag reduction effect occurs at volume fractions below 0.2. It is observed that drag reduction is favoured by good thermodynamic conditions of the polymer-solvent system. Both the flow induced extension of the polymer chains and the hydrodynamic volume fraction occupied by the polymer molecules seem to play an important role for the drag reduction effect.     

Reliable measurements of interfacial slip by colloid probe atomic force microscopy. III. Shear-rate-dependent slip

We present experimental evidence and theoretical models that demonstrate that the slip length, which is the departure from the hydrodynamic no-slip boundary condition, cannot be constant as commonly assumed, but must decrease with increasing shear rate to avoid an unphysical divergence in the velocity of the fluid adjacent to the surface at small separations. The molecular origin of the shear rate dependence of the slip length is discussed. A new theoretical model for slip (the saturation model) is obtained, and it is shown to describe accurately colloid probe atomic force microscopy force measurements for all separations down to a few nanometers in two partially wetting situations (di-n-octyl phthalate on silanized silicon and bare silicon). Previous observations of slip length increasing with shear rate are explained as due to an imprecise calculation of the drag force on the cantilever. A new way of plotting experimental data is also presented, which provides a useful way to illustrate the slip length dependence on the shear rate.     

Boundary slip and wetting properties of interfaces: correlation of the contact angle with the slip length

Correlations between contact angle, a measure of the wetting of surfaces, and slip length are developed using nonequilibrium molecular dynamics for a Lennard-Jones fluid in Couette flow between graphitelike hexagonal-lattice walls. The fluid-wall interaction is varied by modulating the interfacial energy parameter epsilonr=epsilonsfepsilonff and the size parameter sigmar=sigmasfsigmaff, (s=solid, f=fluid) to achieve hydrophobicity (solvophobicity) or hydrophilicity (solvophilicity). The effects of surface chemistry, as well as the effects of temperature and shear rate on the slip length are determined. The contact angle increases from 25 degrees to 147 degrees on highly hydrophobic surfaces (as epsilonr decreases from 0.5 to 0.1), as expected. The slip length is functionally dependent on the affinity strength parameters epsilonr and sigmar: increasing logarithmically with decreasing surface energy epsilonr (i.e., more hydrophobic), while decreasing with power law with decreasing size sigmar. The mechanism for the latter is different from the energetic case. While weak wall forces (small epsilonr) produce hydrophobicity, larger sigmar smoothes out the surface roughness. Both tend to increase the slip. The slip length grows rapidly with a high shear rate, as wall velocity increases three decades from 100 to 10(5) ms. We demonstrate that fluid-solid interfaces with low epsilonr and high sigmar should be chosen to increase slip and are prime candidates for drag reduction.     

Hydrothermal stagnation point flow of Carreau nanofluid over a moving thin needle with non-linear Navier's slip and cubic autocatalytic chemical reactions in Darcy-Forchheimer medium

The present analysis deals with the hydrothermal and stagnation point flow of non-Newtonian fluid characterized by Carreau model over a moving thin needle. Darcy-Forchheimer medium is introduced. Homogeneous and heterogeneous chemical reactions are involved in the flow model. Non-linear Navier's slip condition is taken into account. Buongiorno model is adopted. The present hydrothermal flow model is useful in many complex industrial processes. The required numerical solution is obtained by 4th order Runge Kutta method along with shooting technique. The some remarkable outcomes of the current study are that axial velocity whittles down with rise in Weissenberg number and exhibits reverse trend in response to increase in non-linear slip parameter. Amplification of homogeneous and heterogeneous reaction parameters leads to diminution of nanoparticles concentration. Surface viscous drag intensify due to increment in Forchheimer number and heat transfer rate ameliorates with hike in Weissenberg number for both power law behavior (shear thinning and shear thickening) of Carreau nanofluid.     

Modeling molecular transport in slit pores

We examine the transport of methane in microporous carbon by performing equilibrium and nonequilibrium molecular dynamics simulations over a range of pore sizes, densities, and temperatures. We interpret these simulation results using two models of the transport process. At low densities, we consider a molecular flow model, in which intermolecular interactions are neglected, and find excellent agreement between transport diffusion coefficients determined from simulation, and those predicted by the model. Simulation results indicate that the model can be applied up to fluid densities of the order to 0.1-1 nm(-3). Above these densities, we consider a slip flow model, combining hydrodynamic theory with a slip condition at the solid-fluid interface. As the diffusion coefficient at low densities can be accurately determined by the molecular flow model, we also consider a model where the slip condition is supplied by the molecular flow model. We find that both density-dependent models provide a useful means of estimating the transport coefficient that compares well with simulation.     

Reliable measurements of interfacial slip by colloid probe atomic force microscopy. I. Mathematical modeling

We developed a stable spread-sheet algorithm for the calculation of the hydrodynamic forces measured by colloid probe atomic force microscopy to be used in investigations of interfacial slip. The algorithm quantifies the effect on the slip hydrodynamic force for factors commonly encountered in experimental measurements such as nanoparticle contamination, nonconstant drag force due to cantilever bending that varies with different cantilevers, flattening of the microsphere, and calibration at large separations. We found that all of these experimental factors significantly affect the fitted slip length, approximately in the order listed. Our modeling is applied to fit new experimental data reproducibly. Using this new algorithm, it is shown that the fitting of hydrodynamic theories to experimental data is reliable and the fitted slip length is accurate. A "blind test" protocol was developed that produces a reliable estimate of the fitting error in the determination of both the slip length and spring constant. By this blind test, we estimate that our modeling determines the fitted slip length with an average systematic error of 2 nm and the fitted spring constant with a 3% error. Our exact calculation of the drag force may explain previous reports that the fitted slip length depends upon the shape and spring constant of the cantilever used to perform the measurements.     

A capsule with a fractal shell in a uniform liquid flow

The problem of a liquid flow, which is uniform at infinity, around a capsule comprising a fractal shell filled with a liquid identical to the ambient liquid has been solved. The flow in the fractal layer is described by the Brinkman equation, assuming that the viscosity of the effective medium differs from the viscosity of the pure liquid. Velocity and pressure distributions have been found, and the viscous drag force applied to the capsule has been calculated.     

Gas flow in a layer of chains formed from permeable clusters of nanoparticles

The Stokes–Brinkman flow field has been calculated in a model deposit, i.e., a row of parallel chains formed from porous spherical clusters of nanoparticles and oriented perpendicularly to the gas flow direction. The force of drag to an air flow has been calculated for the row of chains taking into account their permeability and the distance to neighboring chains. The drag forces have been found for nanodendrites composing clusters with allowance for the gas slip effect. Corresponding approximation formulas have been derived. A method has been proposed for calculating the pressure drop across a highly porous deposit of clusters of aerosol nanoparticles deposited onto a filter.     

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.