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.     

An experimental study of non-newtonian fluid flow through fixed and fluidized beds of non-spherical particles

Extensive measurements of pressure drop in fixed beds, minimum fluidization velocity and expansion characteristics for beds of non-spherical particles are reported in the following ranges of conditions: 10^-3 ≤ Re ≤ 20; 0.66 ≤ n ≤ 1 and 0.41 ≤ ? ≤ 0.75. Based on an analysis of these results, it is illustrated that the existing frameworks originally developed for Newtonian fluid flow through beds of spherical particles are also satisfactory for power law fluid flow through beds of non-spherical particles, provided a volume equivalent diameter modified by a sphericity factor and a modified Reynolds number are used instead of their usual definitions.     

A numerical study of fluidization behavior of Geldart A particles using a discrete particle model

This paper reports on a numerical study of fluidization behavior of Geldart A particles by use of a 2D soft-sphere discrete particle model (DPM). Some typical features, including the homogeneous expansion, gross particle circulation in the absence of bubbles, and fast bubbles, can be clearly displayed if the interparticle van der Waals forces are relatively weak. An anisotropy of the velocity fluctuation of particles is found in both the homogeneous fluidization regime and the bubbling regime. The homogeneous fluidization is shown to represent a transition phase resulting from the competition of three kinds of basic interactions: the fluid-particle interaction, the particle-particle collisions (and particle-wall collisions) and the interparticle van der Waals forces. In the bubbling regime, however, the effect of the interparticle van der Waals forces vanishes and the fluid-particle interaction becomes the dominant factor determining the fluidization behavior of Geldart A particles. This is also evidenced by the comparisons of the particulate pressure with other theoretical and experimental results.     

Terminal settling velocity and drag coefficient of biofilm‐coated particles at high Reynolds numbers

The drag force (F_d) on bio‐coated particles taken from two laboratory‐scale liquid–solid circulating fluidized bed bioreactors (LSCFBBR) was studied. The terminal velocities (u_t) and Reynolds numbers (Re_t) of particles observed were higher than reported in the literature. Literature equations for determining u_t were found inadequate to predict drag coefficient (C_d) in Re_t > 130. A new equation for determining F_d as an explicit function of terminal settling velocity was generated based on Archimedes numbers (Ar) of the biofilm‐coated particle. The proposed equation adequately predicted the terminal settling velocity of other literature data at lower Re_t of less than 130, with an accuracy >85%. © 2010 American Institute of Chemical Engineers AIChE J, 2010     

Flow through beds of perfusive particles: effective medium model for velocity prediction within the perfusive media

The effective medium model is a modified model for flow through beds of perfusive particles. This model accounts for the advantages in the existing models and eliminate most of their disadvantages. This model consists of a perfusive particle surrounded by a hypothetical envelope that has fluid only. This envelope is surrounded by a swarm of perfusive particles with permeability equal to the bed permeability. The analytical solution has been obtained by solving Brinkman’s equation in the swarm and the perfusive particle and Navier–Stokes equation for creeping flow in the envelope. This model has proven to converge to the existing models. Where, the proposed model is more general. The validity of this model has been tested through the relation between its parameters, where a relation has been derived through the macroscopic and microscopic force balances on a perfusive bed and a single perfusive particle, respectively. The analysis shows that the result from the effective medium model is in a good agreement with the available experimental data for overall bed permeability.     

On the drag coefficient of flat and non-flat solid particles of irregular shapes: An experimental validation study

The very wide varieties of geometries of irregular particles makes it difficult to study their motion also propose a simple and general equation to estimate their drag coefficient (C_D). The goal of this research is to (1) experimentally validate the general C_D correlation previously proposed by the authors of this article as a simple tool to estimate the C_D of spherical to highly irregular particles over a wide ranges of Re from 0.001 to 300,000, and (2) examine and expand the reference model particles proposed by the authors to represent a wide range of irregular shapes (60 non-flat and 36 flat). A comparison between the C_D experimentally measured using reference model particles over a range of Re from 9 to 2481 and the C_D estimated using general C_D correlation, also comparing with C_D correlations by others, demonstrated the accuracy and simplicity of the proposed general C_D correlation.     

Critical gas velocity in a three‐phase internal loop airlift reactor with low‐density particles

In the present investigation the effects of the addition of organic additives (propanol, benzoic acid, iso‐amyl alcohol and carboxymethyl cellulose) on the critical gas velocity, (U_sg)_c, in an internal airlift loop reactor with low‐density particles (Nylon‐6 and polystyrene) were reported. Whereas the (U_sg)_c was reduced by adding the above additives, it increased with solids loading and density of the particles. The draft tube‐to‐reactor diameter ratio (D_E/D) in the range of 0.5–0.6 gave minimum (U_sg)_c values. The proposed dimensionless correlation predicted the experimental data well. Copyright © 2004 Society of Chemical Industry     

A prediction model for the critical gas velocity and pressure gradient of continuous liquid-carrying in inclined pipes

A generic model for the prediction of critical gas velocity and pressure gradient in slightly inclined pipes (β ≤ 6°) is presented in this article. The gas–liquid configuration was determined based on the minimum energy principle and consideration of wettability and surface tension. A visualization experiment was conducted to obtain the critical gas velocity and the critical pressure gradient of a gas–liquid flow through the 40 and 60-mm pipe diameter. The theoretical study shows that the configuration is close to a convex interface shape at the critical conditions, which is in accord with the experimental phenomenon. Experimental study shows interfacial waves are the main cause of increased interfacial friction factor and a linear functional relationship between the inclination angle and the flow correction factor f(β). The results demonstrate that the new model is capable of providing satisfactory prediction results for the critical gas velocity, pressure drop, and liquid holdup.     

高长径比三相内环流反应器中细颗粒相含率和循环液速的分布研究

在高径比为22的三相内环流反应器中,在常温常压下以空气-水-石英砂为物系,研究了在不同粒径下上升区固含率、下降区固含率和上升区循环液速随表观气速的变化规律和不同粒径下轴向固含率的分布情况,以及在固体体积分数不同的条件下,平均气含率和上升区气含率随表观气速的变化情况。结果表明:当粒径(ds)≤0.3 mm时,上升区固含率随表观气速的增加呈平缓变化趋势,下降区固含率随表观气速的增加而增加;当0.3 mm相似文献   

Experimental estimation of the settling velocity and drag coefficient of the hollow cylindrical particles settling in non-Newtonian fluids in an annular channel

The drag coefficient data of particles settling in an annular channel is very much essential for designing different solid–fluid handling equipment, such as the fluidized bed. Experimental settling velocity, wall factor, and drag coefficient data of the hollow-cylinder particle are presented. Carboxymethyl cellulose solution has been used as the working fluid with a flow index of 0.64 ≤ n ≤ 0.91 and a consistency index of 0.31 ≤ K ≤ 1.81. The experimental results covered a wide diameter ratio range (0.14 ≤ d_eq/L ≤ 0.46), hollow cylinder inner to outer diameter ratio (0.2 ≤ d_i/d_o ≤0.8), and Reynolds number (0.05 ≤ Re ≤ 51 and 0.09 ≤ Re_∞ ≤ 55). d_eq, d_i, and d_o are the equivalent inner and outer diameters of the particle, L is the annular gap, and Re and Re_∞ are the Reynolds numbers in the presence and absence of the wall effect, respectively. The wall factor decreased, and the drag coefficient increased with d_eq/L and d_i/d_o ratios. The above parameters declined with the Reynolds number. The hollow cylinder experienced a lesser wall effect than the spherical particles settling in a non-annular channel. In some cases, the wall factor of the hollow cylinder is found to be equal to the spherical particles settling in an annular channel. The developed correlations have successfully predicted the drag coefficients of the hollow cylinder.     

Rising velocity of a swarm of spherical bubbles in a power law non-newtonian liquid

Previous work on the rise of a swarm of bubbles in non-Newtonian media has been reviewed. Variational principles have been combined with the Happel free surface cell model to obtain upper and lower bounds on the swarm velocities of bubbles rising slowly through power law liquids. The predictions presented herein encompass wide ranges of gas holdup and shear-thinning behaviour.     

液速陡变时液固流化床中颗粒浓度的变化模型

在液速陡变时,分别考察了铅直管液固流化床内粒径为225 μm和511 μm的玻璃微珠的体积分数分布随时间变化的规律,发现大小颗粒在不同的液速变化幅度下都呈现出同样的体积分数变化趋势.联立颗粒速度和颗粒的连续方程式模拟颗粒的体积分数变化过程,建立了一个相对简单的颗粒体积分数变化的数学模型.在模型中,用定常状态的空隙率方程...     

The mechanism model of gas–liquid mass transfer enhancement by fine catalyst particles

In order to present the enhancement of gas–liquid mass transfer by heterogeneous chemical reaction near interface, the mechanism model has been proposed to describe the mass transfer rate for a gas–liquid–solid system containing fine catalyst particles. The composite grid technique has been used to solve the model equations. With this model the effect of particle size, first-order reaction rate constant, distance of particle to gas–liquid interface and residence time of particle near gas–liquid interface on the mass transfer enhancement have been discussed. The particle–particle interaction and slurry apparent viscosity can be considered in the model. The experimental data have been used to verify the model, and the agreement has been found to be satisfied.     

颗粒形状对包衣设备内药片颗粒运动特性影响的数值模拟

为研究颗粒形状对包衣设备内药片颗粒运动特性的影响,基于离散单元法及自行编写的喷雾区颗粒检测算法,采用数值模拟的方法对五种不同形状(棒状、长椭球、扁椭球、双凸形和球形)的药片颗粒在包衣设备内的运动行为规律进行了研究。分析了颗粒形状对颗粒系统的能量、床面颗粒平动速度、颗粒温度及颗粒流在喷雾区域停留时间的分布及其相对标准差的影响。结果表明,颗粒形状对颗粒的平均动能、颗粒床面速度、颗粒温度、喷雾区域停留时间分布及颗粒间包衣均匀性有重要影响。除双凸形颗粒系统外,对于其他四种形状的颗粒系统,随着颗粒球形度增大,颗粒系统具有的动能、床面速度和颗粒温度均呈减小趋势。除棒状颗粒系统外,对于其余四种形状的颗粒系统,随着颗粒球形度增大,颗粒系统在包衣喷雾区域内平均停留时间减小,平均停留时间的相对标准差增大,包衣均匀性变差。与球形颗粒系统相比,非球形颗粒系统的包衣均匀性更好;药片颗粒形状对包衣设备内颗粒运动特性及颗粒之间的包衣均匀性有重要影响。     

Theoretical model for multiple breakup of fluid particles in turbulent flow field

Generalized phenomenological model, based on the theories of probability and isotropic turbulence, is developed for multiple breakup of fluid particles in turbulent flow field. The approach uses a series of successive binary breakup events occur at a time scale comparable to the colliding eddy turnover time. It was found that the use of energy density, instead of energy, will increase the predicted binary breakup rate which is usually underestimated by the existing models in the literature. Generalization of the binary breakup model for multiple fragmentations is performed by defining a “remaining energy function” for the colliding eddy which means the contribution of original eddy to the later breakup events. For ternary breakage, the model shows a reasonably good agreement with the experimental data. The quaternary fragmentation frequency, however, is of negligible importance at lower energy dissipation rates but its contribution to breakage fraction at higher energy dissipation rates becomes considerable. The results also show that ternary and quaternary breakups have a considerable 90% contribution to the overall fragmentation, while pentenary and further fragmentations are of lower importance at low energy dissipation rates. At higher levels of energy dissipation rate, fragmentations up to six daughter particles contribute to more than 95% of the overall fragmentations. © 2016 American Institute of Chemical Engineers AIChE J, 62: 4508–4525, 2016     

A population model of quartzite particles in a lab-scale fluidized bed due to the thermal fragmentation

A population model has been developed to simulate the size distribution of quartzite particles in a lab-scale fluidized bed. While quartzite particles as a kind of bed materials in the differential density circulating fluidized bed were loaded into the high temperature bed, the size distribution of quartzite particles would be changed due to the thermal fragmentation, which could significantly influence fluidization characteristics and heat transfer.With the purpose of comprehending the population of quartzite particles after thermal fragmentation, a fragmentation experiment of quartzite particles has been carried out in a lab-scale circulating fluidized bed. The results show that the fragmentation of particles mainly occurred on particles surface. Based on this experimental phenomenon, a mathematical population model was established to estimate the particle size distribution. The predicted value from the population model is close to the experimental value.     

Development of a porosity prediction model based on shrinkage velocity and glass transition temperature

Abstract

Pore formation and evolution is a common physical phenomenon observed in food materials during different dehydration processes. This change affects heat and mass transfer process and many quality attributes of dried product. Many mathematical models ranging from emperical to classical models proposed in the literature for predicting porosity during drying of food materials. Classical model is in its infancy as the required materials properties during drying are not avaiable for the material charecterisation. Empirical and semi-empirical models are reasonably well developed in establishing relationships between pore evolution and moisture content and determining experimental based coefficients. However, there are no simplistic models that considered process conditions and material properties together to predict the porosity. The purpose of this work is to develop a simplistic theoretical model for pore formation taking both process parameters and changing material properties during drying into consideration. A new “shrinkage velocity” approach has been introduced and the model has been developed based on this shrinkage velocity taking into account the main factors that influence the porosity including the glass transition temperature. Experimental results show good agreement with simulated results and thus validated the model. This study is expected to enhance the current understanding of pore formation of deformable materials during drying.     

Discrete-continual model of flame propagation in a gas suspension of metal particles. I. One-dimensional approximation

Experimental data on combustion of magnesium particles in oxygen-containing gas mixtures are analyzed and generalized, which allowed us to develop a semi-empirical mathematical model that takes into account the integral effect of the initial particle size, pressure, and velocity of the oxidizing flow on combustion. Flame propagation in an air suspension of fine magnesium particles is considered in a one-dimensional approximation. For this purpose, a discrete-continual model of flame propagation in a gas suspension of metal particles is developed. The flammability limits in a vessel, caused by heat losses into the ambient medium and by spatial nonuniformity of the distribution of disperse-phase particles, are numerically determined, and the influence of nonuniformity and bidispersion of a one-dimensional ensemble of particles on flame characteristics is studied.__________Translated from Fizika Goreniya i Vzryva, Vol. 41, No. 2, pp. 81–93, March–April, 2005.     

Discrete-continual model of flame propagation in a gas suspension of metal particles. II. Allowance for the pre-flame oxidation reaction

The previously developed discrete-continual mathematical model of propagation of combustion waves in air suspensions of reactive particles is supplemented by the pre-flame oxidation process. This allows us to describe some quantitative parameters of the flame in mixtures of air and magnesium particles.__________Translated from Fizika Goreniya i Vzryva, Vol. 41, No. 2, pp. 94–97, March–April, 2005.