Segregation kinetics in the rapid gravity flow of granular materials

An experimental analytical method for determining the kinetic segregation coefficient in the rapid gravity flow of a granular material down a rough incline is proposed. It is found that it is possible to predict the velocity of transverse displacement of individual large and small particles in the rapid gravity flow using a single kinetic segregation coefficient for different particle sizes and structural and kinematic characteristics of the flow. A previously proposed equation of segregation kinetics is refined, and its predictive power is analyzed.     

Phenomenological analysis of the interaction of nonelastic incoherent particles in a rapid gravity flow

Interaction of nonelastic incoherent spherical particles in a rapid gravity flow is analyzed. The analysis provides a refinement for the relationship between the pressure, dilatancy, and temperature of the granular medium (energy of mutual displacements of the particles). The equation of state obtained for the granular medium establishes the determined interrelation of the parameters of a rapid gravity flow of cohesionless nonelastic spherical particles and for the first time makes it possible to evaluate the effect of various forms of mutual displacement of particles on the dilatancy of the granular material during a rapid shift. The adequacy of the equation of state of the granular medium is confirmed with the use of the X-ray diffraction method.     

Dynamic flow rates of granular particles

Flow rates (W) of powder and granules are usually measured by letting the particulate solid pass through a stationary orifice. This is not a physical duplication of the event in, for instance, a rotary tablet machine, where the dies move in relation to the powder bed. An apparatus is described by which dynamic flow rates (i.e. flow into a moving orifice) can be measured. These dynamic flow rates (W₂ g/sec) differ from static flow rates. They are subject to the same dependence on particle diameter, d, as static flow rate, i.e.: W₂ ? W_m = ? k(d ? d_m)^s, where “m” refers to maximum and k and s are constants. They also exhibit the same type dependence on orifice opening (P cm) as static flow rates, i.e.W₂ = qPⁿ, where q and n are constants. The values for _n are 3 – 3.25, which is somewhat higher than for static flow where n = 2.3 – 2.6.     

Towards a turbulent modeling of rapid flow of granular materials

From the global equations of balance for particulate media and via a special ensemble averaging technique, the local equations governing the mean motion and the random fluctuations in a rapid flow of granular materials are derived. The resulting equations resemble those of a compressible turbulent flow. Approximate expressions for the components of the fluctuation stress tensor are obtained which are completely computable and in this sense are rather new. The order of magnitude analysis of these stresses indicate that they are in complete agreement with available experimental results. Furthermore, an explicit equation is derived for the evolution and transport of random fluctuation energy which is coupled with the energy dissipation due to fluctuating stress tensor and the mean motion.     

Segregation of granular particles by mass,radius, and density in a horizontal rotating drum

The impact of particle properties on segregation and mixing of bidisperse granular beds in a rotating horizontal drum have been studied by discrete element method (DEM) simulations. Bidispersities in radius, density, and mass have pronounced influences on the stationary mixing pattern, although they hardly affect the granules' flow regime. At 50% fill level, all beds mix well for a Froude number of ～0.56, corresponding to a flow regime intermediate to cascading and cataracting, while segregation occurs both at lower (rolling and cascading regime) and higher (cataracting/centrifuging regime) Froude numbers. These observations are explained qualitatively by noticing that the angular drum velocity dictates the flow regime, which in turn determines the effectiveness and direction of four competing (de)mixing mechanisms: random collisions, buoyancy, percolation, and inertia. A further dozen particle properties have been varied, including the friction coefficients and elastic modulus, but these proved inconsequential to the steady‐state degree of mixing. © 2013 American Institute of Chemical Engineers AIChE J, 60: 50–59, 2014     

Segregation of cohesive powders in a vibrated granular bed

The effects of small amounts of added liquid on the segregation behavior of a granular system under vertical vibration by DEM simulation are investigated in this study. The cohesive forces of grains are incorporated into DEM simulations via a simplified dynamic liquid bridge force model. The simulation results show that capillary forces in addition to viscous forces have an important effect on the segregation phenomenon. The segregation rate of larger intruder rises to the top of the bed is found to depend on the liquid content. The segregation rate is sharply increased when a small amount of liquid is added to granular system. A transition to the reduction of segregation rate occurs at a critical liquid content. It has shown that this transition can be interpreted as the increase of attractive force between grains due to viscous force. The viscous forces make the particles stick more tightly to each other and retard the movement of particles, thus reducing the segregation rate. The segregation rate is also related to the convection motion of the granular system. The presence of convection enhances the segregation rate of wet granular materials.     

Three-dimensional, rapid shear flow of particles with continuous size distributions

Rapid granular flows occur in nature and industry and often contain particles of many sizes. Over the last two decades, significant theoretical and experimental effort has been directed at rapid granular flows with monodisperse or binary particle-size distributions. In contrast, the behavior of rapid granular flows with more than two particle sizes has received only limited attention. The particle-size distributions in many natural and industrial granular flows may be represented as continuous distributions (e.g., Gaussian or lognormal distributions), providing incentive for the investigation of rapid granular flows with these particle-size distributions. As an extension of previous work for two-dimensional simulations of rapid shear flows with Gaussian and lognormal particle-size distributions, this work is directed at three-dimensional flows with continuous size distributions. Event-driven, discrete-particle (“molecular dynamic”) simulations are employed for the three-dimensional simple shear flow of smooth, spherical particles with Gaussian and lognormal particle-size distributions. The results parallel those found previously in two dimensions and demonstrate the effect of distribution width on the stress tensor and granular energy.     

Dispersion kinetics of nano-sized particles for different dispersing machines

For many applications nanoparticles have to be suspended in a fluid phase and dispersed into primary particles or to a definite agglomerate size. Thereby, the prediction of the dispersion kinetic is important among others for comparing the energy efficiency of different dispersion machines. The kinetic models existing today are not able to describe the kinetics over the entire process time correctly. Moreover, a prediction of the dispersion kinetics for new process parameters is not possible. For characterizing the dispersing process and deriving an enhanced model for the dispersion kinetic the stress intensity and the number of stress events in different dispersing machines were investigated. The dispersion kinetic was investigated by dispersing Alumina Alu C (Aeroxide^® Alu C, evonik) in distilled water and glycerol using a dissolver, kneader, 3-roller-mill and stirred media mill. Based on these investigations a new model was developed which is able to describe the dispersion process for different dispersing machines and operating parameters with high accuracy. The new model allows the prediction of the minimum reachable end-particle-size in the studied dispersion process and at varying process parameter based on only a few data points. Within the new model the dispersion kinetics depends on two fit parameters, which are only a function of stress intensity and stress frequency alone. Moreover, using the characteristic parameters stress intensity and stress number the dispersion kinetics for new process parameters can be predicted.     

Simulation of the granular flow of cylindrical particles in the rotating drum

This study aims at unveiling the effect of particle shape on granular flow behavior. Discrete element method is used to simulate cylindrical particles with different aspect ratios in the rotating drum operating in the rolling regime. The results demonstrate that the cylindrical particles exhibit similar general flow patterns as the spherical particles. As the aspect ratio of the cylindrical particles increases, the active‐passive interfaces become steeper, and the number fraction, solid residence time, and collision force in the active region decreases. The mechanism underlying the difference is the preferential orientation, with particles of greater aspect ratios increasingly orientating their longitudinal axes perpendicular to the drum length. Also, particle alignment in the active region is more uniform than that in the passive region. The results obtained in this work provide new insights regarding the impact of particle shape on granular flow in the rotating drum. © 2018 American Institute of Chemical Engineers AIChE J, 64: 3835–3848, 2018     

Segregation under flow of objects suspended in a yield stress fluid and NMR imaging visualisation

This study demonstrates the segregation of spheres suspended in a gelled fluid, in laminar flow, in a sudden expansion. The flow conditions are such that gravity effects are negligible. The spheres are stable in the gel at rest. The upstream pipe diameter is in a ratio of 4:1 to the diameter of the spheres. In these circumstances, the medium is discrete, not continuous. It is shown that the initial conditions and flow kinematics lead to segregation of the matter in the downstream pipe. Nuclear magnetic resonance imaging reveals an organised distribution of the solid matter downstream of the expansion. The effects of volume concentration, geometry and flow velocity were assessed. A model of the phenomenon is proposed.     

Dependence of initial cluster aggregation kinetics on shear rate for particles of different sizes under turbulence

Initial aggregation kinetics for three particle sizes and broad range of Péclet numbers were investigated under turbulent conditions in stirred tank. This allowed us to observe the transition from diffusion‐controlled to purely shear‐induced aggregation. The evolution of the root‐mean‐square radius of gyration, zero‐angle intensity of scattered light, and obscuration was obtained by small‐angle static light scattering. For a given particle size the measured evolution of all integral quantities obtained for various volume average shear rates 〈G〉, scales with a dimensionless time, τ_exp = α_exp × 〈G〉 × ? × t. The experimentally obtained aggregation efficiency α_exp, follows the power law α_exp = Pe^?n, where Pe is the primary particle Péclet number. With increasing particle size a decrease in n is observed in accordance with theory and literature data. As previously predicted by population balance equation simulations three aggregation regimes were observed experimentally. © 2009 American Institute of Chemical Engineers AIChE J, 2009     

Numerical simulation of flow behavior of particles and clusters in riser using two granular temperatures

Cluster in CFB riser significantly affects performance of circulating fluidized beds. To model hydrodynamic behavior in CFB risers, three phase flows were assumed in the riser, the gas phase, the dispersed particle phase, and the clusters phase. The gas-solid multi-fluid model is extended to give the macroscopic averaged equations with constitutive equations for both particle phases from kinetic theory of granular flow. The clusters and the dispersed particles have their own fluctuating energy or two individual granular temperatures. Interactions between the cluster and its surrounding dispersed particles were obtained from kinetic theory of granular flow. Drag force for gas to dispersed particles and the clusters are empirically determined. The momentum exchange between dispersed particles and clusters is modeled using the concept of molecular dynamics. Cluster properties are predicted with the cluster-based approach. The distributions of volume fractions and velocities of gas, dispersed particles and clusters are predicted. Computed solid mass fluxes and volume fractions agree with Manyele et al. [S.V. Manyele, J.H. Parssinen, J.X. Zhu, Characterizing particle aggregates in a high-density and high-flux CFB riser, Chemical Engineering Journal, 88 (2002) 151-161.] and Knowlton [T.M. Knowlton, Modelling benchmark exercise. Workshop at the Eighth Engineering Foundation Conference on Fluidization, Tours, France, 1995.] experimental data.     

Simulations of dense granular gases without gravity with impact-velocity-dependent restitution coefficient

We report two-dimensional simulations of strongly vibrated granular materials without gravity. The coefficient of restitution depends on the impact velocity between particles by taking into account both the viscoelastic and plastic deformations of particles, occurring at low and high velocities respectively. Use of this model of restitution coefficient leads to new unexpected behaviors. When the number of particles N is large, a loose cluster appears near the fixed wall, opposite the vibrating wall. The pressure exerted on the walls becomes independent of N, and linear in the vibration velocity V, quite as the granular temperature. The collision frequency at the vibrating wall becomes independent of both N and V, whereas at the fixed wall, it is linear in both N and V. These behaviors arise because the velocity-dependent restitution coefficient introduces a new time scale related to the collision velocity near the cross over from viscoelastic to plastic deformation.     

Experimental evidence for the stochastic theory of particle flow under gravity

Experimental evidence is presented for a recent stochastic theory of flow which represents the convergent flow of cohesionless particles under gravity toward an open orifice as equivalent to a counterflow of voids from the orifice upward through the bed by biased random flight; the theory is summarized in a new phenomenological form. Data, taken from the literature, were obtained from cells initially loaded with alternate layers of differently colored granular material. The theory predicts that after a fixed flow, each layer develops a depression such that if z_o is the original height of a given layer above the orifice and z_m is the corresponding height of the depression minimum, then for three-dimensional flow a plot of z_m² vs. z_o² for different layers will yield a straight line of slope one; the intercept gives statistical information concerning the equivalent void jumps. For two-dimensional flow, the corresponding theoretical plot is z_m^{$\text{3}\text{2}$} vs. z_osu$\text{3}\text{2}$Data plotted from several sources conform closely to the above predictions, provided z_o is not too large for a given flow. The latter discrepancy is qualitatively explained by a transient effect requiring the density to fall to a certain level before steady-state flow can occur.     

Stability of membrane fouling particles under perikinetic and orthokinetic flow conditions

Stability of fouling particles against Brownian motion (perikinetic) and simple shear flow (orthokinetic) was investigated. The effect of the various parameters such as particle radius, Hamaker constant, salt (1-1 type) concentration, and surface potential on the perikinetic stability using DLVO theory was studied. Conditions at which particles are stable at salt concentration = seawater salt concentration (∼0.6 M) were identified. The calculations showed that typical fouling particles of the Hamaker constant of 1 × 10⁻²⁰ J and surface potential of −25 mV are not stable against Brownian motion at concentrations larger than 0.1 M. Under these conditions the particle size plays a minor role on the stability. Particles of surface potential <−40 mV are stable at salt concentration = seawater concentration. The type of fouling particles expressed by the Hamaker constant strongly affects its stability. Particles of 100 nm in radius, −25 mV in surface potential and a Hamaker constant of <6 × 10⁻²¹ J are stable at salt concentration = seawater concentration. Above this value of the Hamaker constant, the fouling particles coagulate. The effect of particle radius, Hamaker constant, surface potential and shear rate represented by the two dimensionless parameters, C_A and C_R, on orthokinetic stability was also investigated. The investigation was made by solving the trajectory equations for the simple shear flow conditions. The results show that the coagulation efficiency goes through maxima as the ratio of the repulsive forces to the attraction forces is increased. Estimating the value of the Peclet number, the importance of the perikinetic to the orthokinetic coagulation can be determined.     

Experimental measurements and modelling of rapid granular flows

We present a review of the development of techniques available for the measurement of granular temperature in three dimensional dry granular flows, principally discussing positron emission particle tracking and magnetic resonance imaging. We will show how these techniques have given unique insight into granular behaviour, demonstrating phenomena such as buoyancy driven flows, non-equipartition and segregation, whilst also providing us with vital benchmark data for validating kinetic theory based models of rapid granular flow.     

Electrophoresis of colloidal particles in a salt-free medium

A general theory for the electrophoretic mobility of dilute spherical colloidal particles in a salt-free medium containing counterions only is given. This theory assumes that each particle is surrounded by a spherical free volume, within which electroneutrality as a whole holds. It is shown that the mobility can be obtained from the balance between the Stokes drag for the case where the particle is uncharged and the pressure due to the excess counterion charge on the outer boundary of the free volume. The mobility μ is thus expressed as μ=Qexp(y(R))/D, where Q is the particle surface charge, D is the drag coefficient of the particle when uncharged, and y(R) is the scaled potential value at the outer boundary of the free volume.     

不同氛围条件下PBT固相缩聚工艺与反应动力学研究

在真空和通氮气两种氛围下研究了聚对苯二甲酸丁二醇酯(PBT)固相缩聚的反应机理和反应动力学。考察了缩聚温度(185～205℃)、缩聚时间(3～12 h)、真空度(50～600 Pa)以及氮气流速(0.2～1 L/min)等因素对反应产物的特性黏数([η])和端羧基浓度的影响。结果表明:随着反应时间的延长,PBT端羧基浓度逐渐下降,PBT固相缩聚产物的[η]不断提高,并且随着反应温度的增加,端羧基浓度的下降速率和[η]的提高速率均加快;在反应中提高氮气速率(v)和降低压力(p)明显增加了PBT固相缩聚产物的[η],但p和v达到一定值后对[η]的影响逐渐变小,且真空氛围下[η]的增长速率高于氮气氛围;在高温及高真空下,PBT固相缩聚产物的[η]的增长趋势随着反应时间的增加而逐渐减慢并趋于平缓。PBT固相缩聚符合二阶反应动力学模型,两种氛围下反应速率常数(k)均随着温度的增加而升高,50 Pa真空下,k(185～205℃)为0.7～11.24 kg/(mol·h),反应活化能(E_a)为169.50 kJ/mol、指前因子(A)为1.42×10~(13) kg/(mol·h);1 L/min氮气氛围下,k(185～205℃)为0.52～2.81 kg/(mol·h),E_a为102.36 kJ/mol,A为2.12×10~5 kg/(mol·h)。