Incipient motion of a small particle in the viscous boundary layer at a pipe wall

A force balance is derived for a hemispherical particle in the viscous boundary layer at the wall of a horizontal pipe conveying Newtonian fluid; the hemisphere, of radius much less than that of the pipe, rests on the bottom with its flat face against the wall. The drag on the hemisphere is calculated from the creeping flow field of Price (Q. J. Mech. Appl. Math. Pt. 1 (1993)). This yields a prediction of the maximum velocity gradient at the wall for equilibrium, with limiting friction between the hemisphere and the wall. It is shown that the flow field of Price predicts a zero lift force but the validity of this, for actual flows, is questioned. Use of a hemisphere formulates a relevant well-posed problem, capable of mathematical solution. However, the flow field around real particles, e.g. sand, is complex, because of their irregular shapes, but the hemisphere work gives a qualitative indication of the behaviour of irregular particles. For turbulent flow in a pipe it is pertinent to consider a particle wholly within the viscous sub-layer, because it is isolated from significant turbulence and therefore hard to move; for such flow, the theory gives Eq. (21) to predict the critical pipe velocity, v_C, for incipient motion of the hemisphere. For laminar flow, the wall shear rate is readily obtained from the parabolic velocity profile leading to Eq. (26) for v_C. The flow field of Price (and therefore the force acting on the hemisphere) is valid only for creeping flow (i.e. very low particle Reynolds number). Modifications to the force balance are tentatively suggested to account for inertial components to the drag force. The predictions of critical velocity are tested against our data for the incipient motion of small hemispheres at pipe walls in hydraulic conveying as well as new and previously published data for both hydraulic and pneumatic conveying. The new method of predicting incipient motion works well for both the pneumatic and hydraulic conveying of hemispheres and sand shaped particles but it overpredicts the critical velocity for more rounded particles. The dependency of critical velocity on particle shape is under-researched.     

Creeping motions of a composite sphere in a concentric spherical cavity

An analytical study is presented for the quasisteady translation and steady rotation of a spherically symmetric composite particle composed of a solid core and a surrounding porous shell located at the center of a spherical cavity filled with an incompressible Newtonian fluid. In the fluid-permeable porous shell, idealized hydrodynamic frictional segments are assumed to distribute uniformly. In the limit of small Reynolds number, the Stokes and Brinkman equations are solved for the flow field of the system, and the hydrodynamic drag force and torque exerted by the fluid on the particle which is proportional to the translational and angular velocities, respectively, are obtained in closed forms. For a given geometry, the normalized wall-corrected translational and rotational mobilities of the particle decrease monotonically with a decrease in the permeability of its porous shell. The boundary effects of the cavity wall on the creeping motions of a composite sphere can be quite significant in appropriate situations. In the limiting cases, the analytical solutions describing the drag force and torque or mobilities for a composite sphere in the cavity reduce to those for a solid sphere and for a porous sphere.     

Particle dispersion in a turbulent natural convection channel flow

Direct numerical simulations of particle dispersion in the turbulent natural convection flow between two vertical walls kept at constant but different temperatures are reported. It is assumed that the particles do not affect the flow (i.e. the dilute phase approximation is adopted). Particles with different levels of inertia, or Stokes numbers (0.843≤St≤17.45), are tracked according to the drag force imposed by the fluid. The gravity force is included for two cases, St=0.843 and St=17.45. The different levels of turbulence near the wall and near the center of the channel produce, as in isothermal turbulent channel or pipe flow, a larger concentration of particles near the wall. This effect becomes more important, and the deposition velocity of particles on the wall increases, as the particle inertia is increased. The simulations at St=8.38 and St=17.45 predict similar concentration profiles and deposition velocities according to the large inertia of these particles. The deposition velocities, obtained when the gravity force is ignored in the particle equations, follow the trend observed and measured for isothermal turbulent channel flows in the diffusion impaction regime. For the conditions considered, the gravity vector imposes a strong descending motion on the particles and this produces the increase of the particle concentration near the wall and a reduction of the deposition velocities in comparison with the results without the gravity force.     

Experimental study on collision rates of inertial particles in particulate flows under the effect of gravity

To investigate the inter-particle collision frequency, the present experiment conducts two streams of particulate flow converging into a main flow in a junction. Each stream of particulate flow consists of inertial steel particles of 1 and 2 mm diameter, respectively. Particles glide down inside two branches of slanting glass rectangular ducting. We use a high-speed camera to record the trajectory of particle and a PTV technique for data processing. The correlation between the inter-particle collision frequency, particulate concentration, effective diameter and mean fluctuation velocity of inertial particles is explored experimentally. The effect of inter-particle collision frequency under the effect of gravity force is considered and analyzed. Due to the high inertia of steel particles, the turbulent effect of gas-phase is eliminated. A significant difference between the characteristics of the gravitational particulate flow and the homogeneous isotropic flow is found. The physical interpretation for it and the statistical correlation with inter-particle collision frequency are discussed. The results support the analogy of kinetic theory for inertial particles in gas-solid flow provided the gravity effect is excluded.     

附加惯性力作用下竖直矩形流道内过冷流动沸腾的数值模拟

摇摆条件下附加惯性力的作用会对两相流动的压降及汽泡受力产生影响。考虑相变能量和质量输运,采用流体体积(VOF)多相流模型对附加惯性力条件下竖直矩形流道内过冷流动沸腾进行了数值模拟。汽液界面位置通过分段线性插值(PLIC)的方法获得。模拟结果获得了孤立汽泡周围压力、速度、温度分布以及二次流动现象,分析了汽泡聚合过程汽泡形态及内部速度矢量的演变过程,模拟结果与文献中结论吻合良好。附加惯性力作用使得流动压降比静止条件下要大,过冷流动沸腾压降由于汽相产生会在单相流动的基础上产生波动,且热通量越大,压降波动幅度越大。摇摆产生的附加惯性力相对汽泡所受的其他力而言可以忽略不计,而摇摆导致的流量波动会改变汽泡受力大小,进而影响沸腾换热。     

Effect of particle orientation on the drag force in random arrays of prolate ellipsoids in low-Reynolds-number flows

Direct numerical simulations are performed to study the effect of particle orientation on flows through fixed random arrays of prolate ellipsoids at low Reynolds numbers. The Hermans orientation factor and Beta distribution are introduced to quantify the mean orientation and orientation deviation of the particles. The simulation results show that the effect of particle orientation is profound especially when the solid volume fraction and the aspect ratio are large. With the increase of Hermans orientation factors, the drag force decreases when the flow follows a reference direction defined by the average direction of all particles' semi-major axes, while increases when the flow is perpendicular to the reference direction. Comparisons show that the traditional drag force correlations for ellipsoidal particles significantly under-predict the drag force. Based on current simulation results, new drag relations are proposed for prolate ellipsoidal particles at arbitrary aspect ratios, Hermans orientation factors and solid volume fractions.     

Relative importance of the lift force on heavy particles due to turbulence driven secondary flows

Saffman lift forces on dense particles due to gradients in both streamwise and cross-stream velocities in a downward, fully developed turbulent square duct flow at Re_τ = 360 are studied using large eddy simulations. Volume fraction of the dispersed phase is low enough (≤ 10^− 5) that the one-way coupling approach is reasonable, i.e., two-way coupling and particle-particle collisions are not considered. Eulerian and Lagrangian approaches are used to treat the continuous and dispersed phases, respectively. Subgrid stresses are modeled with the dynamic subgrid kinetic energy model of Kim and Menon [W.W. Kim and S. Menon. Application of the localized dynamic subgrid-scale model to turbulent wall-bounded flows, AIAA 97-0210, 1997.]. The particle equation of motion includes drag, lift forces due to both the streamwise and cross-stream velocity gradients, gravity, and is integrated using the fourth-order accurate Runge-Kutta scheme. Dependence of particle drag and lift forces on duct cross-sectional location and particle response time is demonstrated using the mean value contours and probability density functions (PDFs) of particle forces. It is shown that the streamwise component of the mean drag force experienced by particles of all response times is a deceleration force, i.e. on average, fluid streamwise velocity lags the particle streamwise velocity. Secondly, the two wall-normal (or lateral) components of the mean drag force are oriented such that the particles experience a net mean force toward the duct corners. PDFs of particle drag force components show that smaller response time particles experience a wider range of drag force about the mean value, as compared to the more inertial particles. Contours of mean wall-normal lift forces due to streamwise velocity gradients show that this force predominantly acts toward the duct walls and that the maximum lift force occurs close to the walls. PDFs of lift force due to streamwise velocity gradients show that the range of fluctuations increases with particle response time, but the dependence on particle response time is weaker compared to drag force. Lift forces due to cross-stream velocity gradients are at least an order of magnitude smaller than lift forces due to streamwise velocity gradients and are found to decrease in their range of fluctuations with particle response time. It is demonstrated that lift forces due to secondary flow velocity gradients are not as important as those due to streamwise velocity gradients in a square duct flow.     

Modification of Ergun equation for application in trickle bed reactors randomly packed with trilobe particles using computational fluid dynamics technique

Based on a slit model, a pellet scale model has been developed for calculation of drag force imposed on trilobe catalyst particles in a packed bed reactor. The drag coefficient for single gas phase flow in a porous media has been calculated by CFD simulation and the results compared to the Ergun equation. The results show that the drag coefficient predicted by Ergun equation should be modified for various bed porosities, particle aspect ratio and gas densities. Therefore, a correction factor has been proposed to correct the Ergun equation constants in various conditions for trilobe particles. Comparison between the proposed corrected Ergun equation results and experimental data indicates considerable agreement.     

Numerical study of cluster formation in a gas–particle circulating fluidized bed

Simulations with two-way coupling are performed for two-dimensional gas–solid flow in a circulating fluidized bed with a total solids concentration of 3% in the riser. The motion of particles is treated by a Lagrangian approach, and particles are assumed to interact through binary, instantaneous, non-frontal, and inelastic collisions with friction. The model for the interstitial gas phase is based on the Navier–Stokes equations for two-phase flow with fluid turbulence calculated by using LES. Several porosity functions exist in the literature relating the drag force for a particle in a cloud to the drag force on an isolated particle. We have studied the influences of this porosity function, observing large differences in the local flow structure. The fluctuating gas–solid motion has been investigated showing a strong anisotropic flow behaviour, which is similar to experimental findings. The instabilities in these flows are strongly linked to the non-linear drag function due to the group effect of particles in a cloud. The collision parameters have been found to have an important influence on the cluster structures.     

Initial Deposition of Colloidal Particles on a Rough Nanofiltration Membrane

The initial rate of colloid deposition onto semi‐permeable membranes is largely controlled by the coupled influence of permeation drag and particle‐membrane colloidal interactions. Recent studies show that the particle‐membrane interactions are subject to immense local variations due to the inherent morphological heterogeneity (roughness) of reverse osmosis (RO) and nanofiltration (NF) membranes. This experimental investigation reports the effect of membrane roughness on the initial deposition of polystyrene latex particles on a rough NF membrane during cross flow membrane filtration under different operating pressures and solution chemistries. Atomic force microscopy was used to characterize the roughness of the membrane and observe the structure of particle deposits. At the initial stages of fouling, the AFM images show that more particles preferentially accumulate near the “peaks” than in the “valleys” of the rough NF membrane surface.     

Gas‐Solid Drag Coefficient for Ordered Arrays of Monodisperse Microspheres in Slip Flow Regime

Numerous analytical and numerical correlations for the drag force of particles in packed arrays are not applicable to microspheres because of the invalidity of the no‐slip assumption at a solid wall. The slip flow through assemblages of spheres is investigated by the lattice Boltzmann method (LBM). Three periodic arrays of static and monodisperse particles, i.e., a simple cubic, a body‐centered cubic, and a face‐centered cubic array, each with a relatively wide range of solid volume fraction, are considered. The LBM is validated for the slip flow over a single unbounded sphere and the continuum flow through spheres in a simple cubic array. The LBM results agree well with the experimental and numerical data in the literature. Simulations of slip flow through the three ordered arrays of spheres are performed. The effects of solid volume fraction and slip are both quantified within the developed drag laws.     

地下水人工补给过程中流速对多孔介质胶体堵塞的影响机理

通过一系列室内砂柱模拟实验,研究了流速对胶体在饱和多孔介质中滞留-迁移行为的影响;运用COMSOL软件模拟,拟合实验数据后得到表征胶体沉积的关键参数。结果表明：流速增大缩短了胶体在多孔介质中的滞留时间,并增强水动力拖拽力,导致介质对胶体的吸附量减少,有利于胶体的迁移;回灌时间的延续造成的多孔介质渗透系数降低,可通过瞬间增大流速使渗透系数在较短时间内恢复,然而随后形成新的吸附渗透性仍会降低。水源离子强度、介质粗糙度等因素会影响胶体迁移的流速效应。在相同条件下,吸附系数随着离子强度的增大而增加,随着流速的增大而增加。综合来看,离子强度的增加可抵消一部分水动力拖拽力的影响,提高胶体在多孔介质中滞留的概率;介质表面粗糙度的增加,可削弱水动力拖拽力作用,同时增加胶体与介质的吸附、沉积点位和接触面积,导致胶体易于在多孔介质中发生滞留并可能进一步导致介质堵塞。     

Fluidization of elongated particles—Effect of multi-particle correlations for drag,lift, and torque in CFD-DEM

Having proper correlations for hydrodynamic forces is essential for successful CFD-DEM simulations of a fluidized bed. For spherical particles in a fluidized bed, efficient correlations for predicting the drag force, including the crowding effect caused by surrounding particles, are already available and well tested. However, for elongated particles, next to the drag force, the lift force, and hydrodynamic torque also gain importance. In this work, we apply recently developed multi-particle correlations for drag, lift and torque in CFD-DEM simulations of a fluidized bed with spherocylindrical particles of aspect ratio 4 and compare them to simulations with widely used single-particle correlations for elongated particles. Simulation results are compared with previous magnetic particle tracking experimental results. We show that multi-particle correlations improve the prediction of particle orientation and vertical velocity. We also show the importance of including hydrodynamic torque.     

Finite element simulation of deep bed filtration

In this paper, the numerical model for separation efficiency and transport in periodic porous media is studied. Finite element method was used to simulate the development of a predictive model of behavior of porous media during injection of particles. This paper describes the effects of injected particle size, Reynolds number and particle drag coefficient. The numerical results show that the separation efficiency increased with injected particle size increase. The separation efficiency is found to increase with increasing Reynolds number. For the effect of drag force, C_D, in porous media, numerical results show that for C_D<10 and C_D>100, the separation efficiency is not affected by drag coefficient in the range of drag coefficient from 10 to 100, and the separation efficiency significantly depends on the Reynolds number.     

Discrete particle simulation of gas fluidization of ellipsoidal particles

Fluidization is widely used in industries and has been extensively studied, either experimentally or theoretically, in the past decades. In recent years, a coupled simulation approach of discrete element method (DEM) and computational fluid dynamics (CFD) has been successfully developed to study the gas–solid flow and heat transfer in fluidization at a particle scale. However, to date, such studies mainly deal with spherical particles. The effect of particle shape on fluidization is recognized but not properly quantified. In this paper, the CFD–DEM approach is extended to consider the fluidization of ellipsoidal particles. In the simulation, particles used are either oblate or prolate, with aspect ratios varying from very flat (aspect ratio=0.25) to elongated (aspect ratio=3.5), representing cylinder-type and disk-type shaped particles, respectively. The commonly used correlations to determine the fluid drag force acting on a non-spherical particle are compared first. Then the model is verified in terms of solid flow patterns. The effect of aspect ratio on the flow pattern, the relationship between pressure drop and gas superficial velocity, and microscopic parameters such as coordination number, particle orientation and force structure are investigated. It is shown that particle shape affects bed permeability and the minimum fluidization velocity significantly. The coordination number generally increases with aspect ratio deviating from 1.0. The analysis of particle orientations shows that the bed structures for ellipsoids are not random as that for spheres. Oblate particles prefer facing upward or downward while prolate particles prefer horizontal orientation. Spheres have the largest particle–particle contact force and fluid drag force under the comparable conditions. With aspect ratio deviating from 1.0, particle–particle interaction and fluid drag become relatively weak. The proposed model shows a promising method in examining the effect of particle shape on different flow behaviour in gas fluidization.     

Influence of slip on the flow of a yield stress fluid around a flat plate

This experimental study deals with the influence of slip on the non‐inertial flow of a viscoplastic fluid around a flat plate moving at a constant velocity. The bulk and interfacial properties of the fluid have been finely characterized. The drag force has been analyzed with regards to the flow velocity and for two tribological conditions: adherence and slip. This force decreases with the velocity and is reduced in the presence of slip. Kinematic fields have also been measured by Particle Image Velocimetry (PIV), to determine the influence of both the velocity and the tribological conditions on the liquid and solid regions of the flow. The results highlight no significant influence of the flow velocity on the thickness of the boundary layer and rigid zones. The wall shear stresses along the plate obtained from force measurements and slip velocities are then compared to rheometrical measurements. © 2015 American Institute of Chemical Engineers AIChE J, 62: 1356–1363, 2016     

Shear-induced transport of a particle layer along a porous wall

A theory is presented which describes the transport of a concentrated layer of particles along a porous wall under laminar flow conditions. A shear-induced hydrodynamic diffusion mechanism is proposed to describe the lateral migration of particles away from the porous wall as the layer is sheared. At steady state, the particle diffusion within the layer is balanced by the convective flux of particles toward the porous wall due to the fluid flow into the wall. This model predicts the nonlinear velocity and concentration profiles within the sheared particle layer, as well as the layer thickness and the wall concentration. In addition, a criterion is found for predicting whether or not a stagnant particle layer will form on the porous wall. The application of this theory to crossflow microfiltration is discussed.     

Study on the Dynamics of Tribocharged Coal and Mineral Particles in Free-Fall Triboelectric Separator

The dynamics of 2.0?0.8 mm or 0.8?0.5 mm size fraction of tribocharged organic coal, pyrite, and calcite particles were studied under the electric field using the high-speed dynamic camera combined with high-speed motion analysis system. Motion images of these particles were obtained and used to analyze their dynamic parameters. Organic coal particles tribocharged positively move to the negative plate, while pyrite and calcite particles reach the positive plate under the influence of electric force. These results indicate that the trajectories of all 2.0?0.8 mm particles are similar to parabolic curves. For 0.8?0.5 mm particles, the trajectories are approximate straight line, except for the calcite. The vertical velocities of all particles increase with a fluctuant acceleration as a result of gravity and drag force. The horizontal velocities of all particles vary slightly. The dynamics of 0.8?0.5 mm particles prove that size is very important for the triboelectric separation. The actions of electric force and drag force are increased with the decrease of particle size.     

Aerosol Deposition in Sampling Probes

Deposition of aerosol particles on the inner walls of sampling probes is of concern in many aerosol sampling applications. Only inertial and gravitational effects have been considered in previous studies of the aerosol deposition; however, the lift force on particles is also of significance. In this investigation, experiments have been conducted to construct a database for aerosol deposition in Willeke-type sampling probes. An empirical correlation has been made between wall losses and the depositional forces of drag, gravity, inertia, and particle lift through the use of dimensionless parameters. Inclusion of the lift effect in this correlation not only helps to better predict particle behavior in the sampling inlet, but it also provides a basis for understanding of the intrinsic deposition phenomenon. The correlation has a geometric standard deviation of 1.13 and a 0.93 correlation coefficient relative to the experimental data.     

Numerical Simulation of Particle Segregation in Bubbling Gas‐Fluidized Beds

A multi‐fluid Eulerian model incorporating the kinetic theory of granular flow is used for the simulation of bubbling fluidized beds containing a binary mixture of Geldart B particles at low gas velocities. The cases of density, size and combined density/size segregation are investigated using computational fluid dynamic simulations. Various expressions for the drag force are evaluated for predicting different segregations. The simulation results show that summation of the particle‐particle drag force, i.e., the “hindrance effect” term, and the Stokes drag of particles, which is modified based on the Wen‐Yu drag model can be used for accurate simulation of a binary mixture of particles differing in size, density, or both. Bed expansion and dimensionless axial segregation profiles of CFD results are compared with the experimental data and good agreement is found.