Calculation of the Velocity of Free Settling of Solid Particles in a Newtonian Liquid

A cubic equation for the velocity of free settling of solid particles in a Newtonian liquid is derived on the basis of a simple approximate expression for the drag coefficient of spherical particles as a function of the Reynolds number using the concept of the effective diameter of a particle of arbitrary shape, which characterizes the surface, volume, and midsection of the particle. The exact and approximate solutions are compared, and some experimental data are analyzed.     

Hindered settling velocity of a bimodal mixture of spherical solid particles in a Newtonian fluid

Works describing methods for calculating the settling velocity of a biomodal mixture of solid particles are analyzed. A method for calculating the settling velocity of a bimodal mixture of spherical particles is proposed that takes into account the contact interaction between particles of different sizes due to their collisions. The calculation results are in satisfactory agreement with experimental data on settling of a bimodal mixture of spherical solid particles.     

Respiratory Deposition of Fibers in the Non-Inertial Regime—Development and Application of a Semi-Analytical Model

A semi-analytical model describing the motion of fibrous particles ranging from nano- to micro scale was developed, and some important differences in respiratory tract transport and deposition between fibrous particles of various sizes and shapes were elucidated. The aim of this work was to gain information regarding health risks associated with inhalation exposure to small fibers such as carbon nanotubes. The model, however, is general in the sense that it can be applied to arbitrary flows and geometries at small fiber Stokes and Reynolds numbers. Deposition due to gravitational settling, Brownian motion and interception was considered, and results were presented for steady, laminar, fully developed parabolic flow in straight airways. Regarding particle size, our model shows that decrease in particle size leads to reduced efficiency of sedimentation but increased intensity of Brownian diffusion, as expected. We studied the effects due to particle shape alone by varying the aspect ratios and diameters of the microfibers simultaneously, such that the effect of particle mass does not come into play. Our model suggests that deposition both due to gravitational settling and Brownian diffusion decreases with increased fiber aspect ratio. Regarding the combined effect of fiber size and shape, our results suggest that for particles with elongated shape the probability of reaching the vulnerable gas-exchange region in the deep lung is highest for particles with diameters in the size range 10–100 nm and lengths of several micrometers. Note that the popular multi-walled carbon nanotubes fall into this size-range.     

Effect of particle shape on hindered settling in creeping flow

Solid particles of uniform size and shape were used to study the effect of particle shape on hindered settling in creeping flow (Re_o ? 0.2), where fluid flow patterns are independent of Reynolds number and the effect of shape is most prominent. The particles of different shape studied were spherical glass beads, cubical sodium chloride crystals and ABS plastic pellets, brick-like sugar crystals, and angular (imperfect octahedral) mineral silicate crystals. The liquids used were aqueous polyethylene glycol solutions and various blends of hydrocarbon oils. Two particle sizes on the average were investigated for each particle shape, and five settling column diameters were employed, so that the overall range of column-to-particle diameter ratio covered was 22 – 226.A Richardson—Zaki type equation of the form u = u_i?ⁿ was found to correlate the constant settling rate data for each particle size and shape over the voidage range ? = 0.65 – 0.9. However, the wall effect on hindered settling rate was found in most cases to be considerably smaller than that predicted by Richardson and Zaki. The term u_i, obtained by linearly extrapolating the settling velocity u (below ? = 0.9) to ? = 1 on a log—log plot of u versus ?, was found to be measurably lower than the corresponding free settling velocity. The index n varied from an average value of 4.8 for the smooth spheres to 5.4 for the cubes to 5.8 for both the brick-like and the angular particles. These values graphically display a definite trend with settled bed voidage, ?_b, which is shape-dependent and easily measured, and may therefore be a convenient parameter for taking account of shape variation generally.The method proposed by Beranek and Klumpar for correlating fluidization data on different shaped particles, which depends on ?_b, was found to be moderately successful in correlating the present settling data for different shapes.     

Determination of non-spherical particle terminal velocity using particulate expansion data

A new method is proposed for the determination of the terminal velocity of non-spherical particles and compared with experimental data. The method is based on particulate expansion data of fluidized bed and variational model for calculating fluid–particle interphase drag coefficient. Other methods require knowledge of the particle shape, a parameter that is not easy to obtain for real materials. We use pressure drops data in packed bed for indirect determination of particle shape factor which depends on the reliability of coefficients in the Ergun equation. Our data, however, show that these coefficients are system-specific. The proposed method for the determination of non-spherical particle settling velocity in liquid as well as extrapolation to system gas-particles gives results which are in good agreement with experimental data. The method is restricted to particles which can be fluidized particulately by liquid.     

Numerical calculation of parameters of an ensemble of particles in a perfectly mixed fluidized-bed reactor

The problem of numerical calculation of parameters (mass; fractional makeup; entrainment, settling, and discharge flow rates) of the solid phase in a perfectly mixed fluidized bed of reacting particles in flow and batch reactors and in a system of two flow-coupled reactors is considered. A system of equations is suggested for solving this problem for an arbitrary particle size distribution in the feed stream and for an arbitrary dependence of the shrinkage or growth of the particles on their diameter. Expressions for calculating the mass and fractional makeup of the dispersed material at the induction stage have been obtained. Calculation of parameters of an ensemble of particles in a recycle reactor is described. The particular cases of flow and batch reactors and a system of two flow-coupled reactors have been considered for different dependences of the transformation rate of the particles on their size at an arbitrary fractional makeup of the feed stream.     

Settling stability of coal slurries prepared by wet grinding in the szego mill

Static sedimentation experiments on coal-#2 oil and coal-water slurries, wet ground in a Szego mill, were conducted. These slurries contain more flaky particles as compared to those conventionally ground. Concentration and size distribution of the coal, diameter and height of the settling columns and settling time were the independent variables. Coal concentration and particle size distribution along the columns, and settling rates were measured. Despite their particle shape differences, these slurries were similar to others in terms of settling stability due to particle flocculation.     

Optimization of parameters for the synthesis of bimodal Ag nanoparticles by Taguchi method

Bimodal Ag nanoparticles were synthesized by modified precipitation process. Taguchi analysis was used in this study and the reaction time and temperature were the most influencing parameter on the particle size and the size distribution, respectively. The optimal conditions were determined by using Taguchi method and nano-sized Ag particles with bimodal shape were obtained.     

THE STOKES HYDRODYNAMIC RESISTANCE OF NONSPHERICAL PARTICLES

A review is presented of the motion of an isolated, nonspherical particle of general shape settling at small Reynolds numbers through an unbounded quiescent fluid—with a view towards establishing whether or not all particles ultimately attain a unique, time-independent terminal state, independently of their initial orientation and state of motion. Effects of inhomogeneities in internal mass distribution are incorporated into the analysis. Differences are pointed out between gravity and centrifugal settling rates for nonspherical particles. These arise from the tendency of such particles to adopt preferential orientations in a centrifugal field of force owing to variations in field strength over the length of the particle, ft is pointed out that Coriolis forces acting on both the fluid and particle in a centrifuge cause the particles to settle more slowly. Moreover, in the case of spherical particles, the particle path deviates from a purely radial trajectory. Effects of both translational and rotational Brownian motions on the mean settling velocities of submicron particles is discussed, again for generally-shaped particle. A detailed summary of the contents of this paper is provided at its conclusion.     

Measurement of coal particle size and shape with concentrated coal-water mixtures

The behavior of concentrated coal-water mixtures having narrow particle size ssfractions of coal was investigated. The pulverized coal was fractionated into six distinct particle size ranges, i.e. -70+80, -80+120, -120+140,-140+200, -200+400 and -400 mesh sizes by using a series of sieves. Settling rates were determined as functions of solids concentration for suspensions in water of coal particles to establish the measurement of particle size and shape factor and to assess concentration effect upon the observed hindered settling rates. The settling rates were modelled using the Richardson-Zaki model with the exponent n variable to account for the nonspherical shape of the coal particles. The data was also correlated with the Michaels-Bolger model which explicitly account for the excess water which is dragged down along with the particles undergoing sedimentation. In addition, coal particles and suspensions were characterized by coal analysis, heating value, solid heat capacity and thermal conductivity, densities, maximum packing concentrations and pore size distributions.     

In‐Line Monitoring of Crystallization Processes Using a Laser Reflection Sensor and a Neural Network Model

Laboratory‐scale experiments were carried out for measuring the chord length distribution of different particle systems using a laser reflection sensor. Samples consisted of monodisperse, polydisperse and bimodal FCC catalyst and PVC particles of different sizes, ranging from about 20 to 500 μm. The particles were dispersed in water, forming suspensions with solid‐phase mass fractions ranging from ca. 0.2 % until ca. 30 %. The experimental results, consisting of the particle number counting per chord length class, were used in fitting a neural network model for estimating the mass concentration of particles in the suspension and the volume‐based size distribution, eliminating the effects of suspension concentration and particle shape. The results indicate the feasibility of using such a model as a software sensor in crystallization processes monitoring.     

Emulsion copolymerization in a tubular reactor

A study was conducted on the emulsion copolymerization of vinyl acetate and butyl acrylate in a tubular reactor. It was performed at a constant temperature of 60°C and at different fluid velocities and feed compositions. Conversion, particle size distribution, and copolymer composition were measured, respectively, with gravimetric method, laser light scattering, and nuclear magnetic resonance. Maximum conversions were found for each of the monomer compositions; this maximum conversion varied, however, with the recipe used. The amount of butyl acrylate has a direct effect on the number of particles and on the final conversion. In lower levels of butyl acrylate particle size distribution is wide and bimodal. High levels of butyl acrylate leads to narrow and monomodal particle size distribution. Therefore the level of butyl acrylate and the velocity of fluid flowing inside the tube have strong effects on the shape (monomodal‐bimodal) and the width of particle size distributions. This effect may vary at different levels of butyl acrylate and flow rate. The results obtained from copolymer composition show that an alternating block copolymer is made during the reaction. © 2000 John Wiley & Sons, Inc. J Appl Polym Sci 75: 833–842, 2000     

柱状颗粒沉降过程的CFD-DEM联合仿真和实验研究

针对柱状催化剂颗粒相对于球形颗粒的不同运动特性,选择不同长度直径为2 mm的5种柱状颗粒,采用CFD-DEM数值模拟仿真,研究柱状颗粒在管状容器中沉降的运动行为,并建立柱状颗粒沉降试验台,采用高速摄像拍摄的方法进行实验研究。结果表明,在不同位置释放相同直径和长度的柱状颗粒时,靠近壁面释放的颗粒会在沉降过程中向中心漂移,且比中心释放的颗粒沉降更慢,时间更长;改变柱状颗粒与水平面的夹角,在圆管中心释放颗粒,最终颗粒都会旋转至水平状态,与水平面夹角越大,底部所受阻力越大,转动持续时间随之增加;推导柱状颗粒沉降斯托克斯方程,并通过实验数据对方程中的阻力系数进行修正,将修正后的阻力系数导入用户自定义函数(UDF)计算颗粒沉降末速度,相对误差从原来使用球形颗粒阻力系数的50%下降到17%以内,模拟较为可靠。     

Flow of suspensions in pipelines(Part 2: Two Mechanisms of Particle Suspension)

The distribution of sand and nickel particles with height in a horizontal channel of rectangular cross section has been measured and eddy diffusivities have been computed from these measurements. Turbulent suspension of particles appeared to occur where the concentration of solid particles was low, the particle diameter was small (< 0.2 mm.) and the ratio of the settling velocity of the particles to the friction velocity of the flow was less than 0.2. Where these criteria were not satisfied, concentration profiles were observed to deviate significantly from the shape associated with turbulent suspension. These deviations are attributed to the effect of particle interactions investigated by Bagnold.     

Primary Study of a Settling Fiber Particle in a Particle Cloud

Fiber particles have some unique behaviors due to their special shapes, which are important to those related industries. When a single fiber is in a particle cloud, its behavior will be influenced by others around it. Hence, the behavior of an isolated fiber particle will be different from that in a particle cloud, such as aggregation, orientation, and drag coefficient. However, little information is available on these phenomena, especially drag coefficient. Therefore, this article focuses on the settling process of a fiber particle in a particle cloud. Experiments were conducted to observe the settling behavior of fiber particles and to determine the drag coefficients of an isolated fiber particle in a particle cloud. The relationship between drag coefficient, orientation, and Re for different fiber particles is obtained, which is independent of volume concentration. It is further observed that the aspect ratio has little influence on the drag coefficients of fiber particles. By comparison, it is noticed that these relationships are similar to those found for an isolated single-fiber particle. Furthermore, the orientation of a fiber particle in a particle cloud fluctuates around the stable horizontal orientation in the same way as a single-fiber particle, but returns to the steady state more quickly.     

On the Settling Velocity in a Nonstationary Atmosphere

A general drag coefficient has been used in the equation of motion for solid spherical particles. The time constants, stopping times, and settling velocities in a still atmosphere are computed for a wide range of Reynolds numbers. The settling times are compared with the times calculated when a particle is falling in a fluctuating atmosphere. It is found that such particles will get significantly longer settling times owing to an enhancement in the drag coefficient caused by an increase of the relative velocity between the particle and the fluid. Surprisingly, this enhancement is present for a horizontal wind field due to a coupling between particle motion in different directions, but it is also present for a vertical field. The effect is most pronounced in the intermediate Reynolds number region, slightly above the Stokes range, where the increase in settling time can be more than 10% for certain fluctuation frequencies and amplitudes. This indicates that such particles must be carefully treated when they are falling in a nonstationary medium     

Attrition behavior of fine particles in a fluidized bed with bimodal particles: Influence of particle density and size ratio

To process the solid particulates in fluidized bed and slurry phase reactors, attrition is an inevitable consequence and is therefore one of the preliminary parameters for the catalyst design. In this paper, the mechanical degradation propensity of the zeolite catalysts (particles) was investigated in a bimodal distribution environment using a Gas Jet Attrition — ASTM standard fluidized bed test (D-5757). The experimentation was conducted in order to explore parameters affecting attrition phenomena in a bimodal fluidization. In a bimodal fluidization system, two different types of particles are co-fluidized isothermally. The air jet attrition index (AJI) showed distinct increases in the attrition rate of small particles in a bimodal fluidization environment under standard operating conditions, in comparison with single particle. A series of experiments were conducted using particles of various sizes, with large particles of different densities and sizes. Experimental results suggest that the relative density and particle size ratio have a significant influence on attrition behavior during co-fluidization. Therefore a generalized relationship has been drawn using Gwyn constants; those defined material properties of small particles. Moreover, distinct attrition incremental phenomenon was observed during co-fluidization owing to the change in collision pattern and impact, which was associated with relative particle density and size ratios.     

Drag coefficient and settling velocity for particles of cylindrical shape

Solid particles of cylindrical shape play a significant role in many separations processes. Explicit equations for the drag coefficient and the terminal velocity of free-falling cylindrical particles have been developed in this work. The developed equations are based on available experimental data for falling cylindrical particles in all flow regimes. The aspect ratio (i.e., length-over-diameter ratio) has been used to account for the particle shape. Comparisons with correlations proposed by other researchers using different parameters to account for the geometry are presented. Good agreement is found for small aspect ratios, and increasing differences appear when the aspect ratio increases. The aspect ratio of cylindrical particles satisfactorily accounts for the geometrical influence on fluid flow of settling particles.     

Determination of the characteristics of agglomerates in aqueous suspensions using nonlinear optimization

Measurement of characteristics of particles in suspensions without dilution has a practical interest in formulation, mineral processing, material sciences and environmental technologies.These characteristics are the size, shape, and surface properties of the primary particles, and also the size, structure and the number of primary particles in the agglomerates.In this work, the multiple light-scattering model through the optical analyzer, Turbiscan MA 2000 is used to determine the mean settling velocities of monodisperse glass beads and two polydisperse samples of powders, kaolin D and alumina, differing by their particle size distribution, their shape and their surface properties.Beyond the experimental validation of theoretical and empirical predictions, the nonlinear adjustment of experimental settling data gives the number of primary particles per agglomerate and the agglomerate size. These two characteristics lead to the determination of the fractal dimension of the agglomerates. The latter was found in the range of 2.5-2.7 for all suspensions examined. The calculation of permeability and spherical factor reveals the nonspherical impermeable agglomerates.     

Motion of a magnetic flow follower in two‐phase flow — application to the study of airlift reactor hydrodynamics

A low‐cost and simple magnetic particle tracer method was adapted to characterize the hydrodynamic behavior of an internal‐ and an external‐loop airlift reactor (ALR). The residence time distribution of three magnetic particles differing in diameter (5.5, 11.0 and 21.2 mm) and with a density very close to that of water was measured in individual reactor sections. The measured data were analyzed and used to determine the velocity of the liquid phase. Validation of the experimental results for liquid velocity was done by means of the data obtained by an independent reference method. Furthermore, analysis of the differences found in the settling velocity of the particle in single‐liquid and gas‐liquid phases was carried out, using a simplified 3D momentum transfer model. The model considering particle‐bubble interaction forces resulting from changes in the liquid velocity field due to bubble motion was able to predict satisfactorily the increase in the particle settling velocity in the homogeneous bubbly regime. The effective drag coefficient in two‐phase flow was found to be directly dependent on particle Reynolds number to the power of ? 2 but independent of gas flow‐rate for all particle diameters studied. Based on the experimental and theoretical investigations, the valid exact formulation of the effective buoyancy force necessary for the calculation of the correct particle settling velocity in two‐phase flow was done. In addition, recommendations concerning the use of flow‐following particles in internal‐loop ALRs for liquid velocity measurements are presented. Copyright © 2006 Society of Chemical Industry