Drag of non-spherical solid particles of regular and irregular shape

The drag of a non-spherical particle was reviewed and investigated for a variety of shapes (regular and irregular) and particle Reynolds numbers (Re_p). Point-force models for the trajectory-averaged drag were discussed for both the Stokes regime (Re_p ? 1) and Newton regime (Re_p ? 1 and sub-critical with approximately constant drag coefficient) for a particular particle shape. While exact solutions were often available for the Stokes regime, the Newton regime depended on: aspect ratio for spheroidal particles, surface area ratio for other regularly-shaped particles, and min-med-max area for irregularly shaped particles. The combination of the Stokes and Newton regimes were well integrated using a general method by Ganser (developed for isometric shapes and disks). In particular, a modified Clift-Gauvin expression was developed for particles with approximately cylindrical cross-sections relative to the flow, e.g. rods, prolate spheroids, and oblate spheroids with near-unity aspect ratios. However, particles with non-circular cross-sections exhibited a weaker dependence on Reynolds number, which is attributed to the more rapid transition to flow separation and turbulent boundary layer conditions. Their drag coefficient behavior was better represented by a modified Dallavalle drag model, by again integrating the Stokes and Newton regimes. This paper first discusses spherical particle drag and classification of particle shapes, followed by the main body which discusses drag in Stokes and Newton regimes and then combines these results for the intermediate regimes.     

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     

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.     

Wall effects on the velocities of a single sphere settling in a stagnant and counter-current fluid and rising in a co-current fluid

Experimental results were obtained on the steady settling of spheres in quiescent media in a range of cylindrical tubes to ascertain the wall effects over a relatively wide range of Reynolds number values. For practical considerations, the retardation effect is important when the ratio of the particle diameter to the tube diameter (λ) is higher than about 0.05. A new empirical correlation is presented which covers a Reynolds number range Re = 53-15,100 and a particle to tube diameter ratio λ < 0.88. The absolute mean deviation between the experimental data and the presented correlation was 1.9%. The well-known correlations of Newton, Munroe and Di Felice agree with the presented data reasonably well. For steady settling of spheres in a counter-current water flow, the slip velocity remains practically the same as in quiescent media. However, for rising spheres in a co-current water flow, the slip velocity decreases with increasing co-current water velocity, i.e., the wall factor decreases with increasing co-current water velocity. Consequently, the drag coefficient for rising particles in co-current water flow increases with increasing water velocity.     

Segregation behavior of binary mixtures of cylindrical particles with different length ratios in the rotating drum

Particle shape impacts the flow behavior of granular material but this effect is still far from being fully understood. Using discrete element method, the current work explores the segregation phenomena of the binary mixtures of cylindrical particles (differing in length but with the same diameter) in the three-dimensional rotating drum operating in the rolling regime, with each cylindrical particle fully represented by the superquadric equation. The important characteristics and the effect of length ratio on the flow dynamics of the binary mixtures are discussed. Some trends are in sync with those of binary mixtures of spherical particles. Unique to nonspherical particles is the orientation of particles, with results indicating that the cylindrical particles align their major axes perpendicular to the drum axis and this behavior becomes more significant for large particles when the length ratio increases. The length-induced radial segregation causes the orientation of large cylindrical particles to be less uniform.     

Steady flow of power-law fluids across an unconfined elliptical cylinder

The momentum transfer characteristics of the power-law fluid flow past an unconfined elliptic cylinder is investigated numerically by solving continuity and momentum equations using FLUENT (version 6.2) in the two-dimensional steady cross-flow regime. The influence of the power-law index (0.2?n?1.8), Reynolds number (0.01?Re?40) and the aspect ratio of the elliptic cylinder (0.2?E?5) on the local and global flow characteristics has been studied. In addition, flow patterns showing streamline and vorticity profiles, and the pressure distribution on the surface of the cylinder have also been presented to provide further physical insights into the detailed flow kinematics. For shear-thinning (n<1) behaviour and the aspect ratio E>1, flow separation is somewhat delayed and the resulting wake is also shorter; on the other hand, for shear-thickening (n>1) fluid behaviour and for E<1, the opposite behaviour is obtained. The pressure coefficient and drag coefficient show a complex dependence on the Reynolds number and power-law index. The decrease in the degree of shear-thinning behaviour increases the drag coefficient, especially at low Reynolds numbers. While the aspect ratio of the cylinder exerts significant influence on the detailed flow characteristics, the total drag coefficient is only weakly dependent on the aspect ratio in shear-thickening fluids. The effect of the flow behaviour index, however, diminishes gradually with the increasing Reynolds number. The numerical results have also been presented in terms of closure relations for easy use in a new application.     

Membrane rejection of nonspherical particles: Modeling and experiment

The rejection coefficient of nonspherical particles from ultrafiltration and microfiltration membranes has been examined from both theoretical and experimental perspectives. Modeling efforts focused on incorporating the convective hindrance factor for a capsule shaped particle in a cylindrical pore into predictions of the rejection coefficient. First, the convective hindrance factor was approximated using previously reported results for the hydrodynamic resistances experienced by a sphere in a pore. Second, computational fluid dynamics calculations predicted the convective hindrance factor for a capsule in a cylindrical pore. Results from both approaches indicate that including hydrodynamic interactions in predictions of the rejection coefficient has a greater effect for smaller particles and particles with smaller aspect ratio (i.e., close to spherical shape). Rejections of several rod‐shaped Gram negative bacteria with aspect ratio from 2 to 5 by clean track‐etched membranes were in general agreement with theoretical predictions. © 2013 American Institute of Chemical Engineers AIChE J, 59: 3863–3873, 2013     

Liquid fluidization with cylindrical particles: Highly resolved simulations

We perform three-dimensional, time-dependent simulations of dense, fluidized suspensions of solid cylindrical particles in a Newtonian liquid in fully periodic domains. The resolution of the flow field is an order of magnitude finer than the diameter of the cylindrical particles. At their surfaces no-slip conditions are applied through an immersed boundary method (IBM), coupled to the lattice-Boltzmann method that is used as the fluid flow solver. The marker points of the IBM are also used to detect and perform collisions between the cylinders. With these particle-resolved simulations, we study the effects of the aspect ratio of the cylinders and the solids volume fraction on the superficial slip velocity between fluid and solids, on the solids velocity fluctuations, as well as on the orientation of the cylinders. The aspect ratio (length over diameter of the cylinders) ranges from 0.5 to 4, the solids volume fraction goes up to 0.48. Reynolds numbers based on average settling velocity are of the order of 1–10. At constant Archimedes number, we observe only minor sensitivities of the settling Reynolds number on the aspect ratio.     

The Filtration of Fibrous Aerosols

Fibrous particles constitute an important class of aerosols that are potential human health hazards. Filters can remove aerosols from the air. The capture of spherical and fibrous aerosols by fibrous filters was investigated in this study. The governing equations of motions for translation and rotation of fibrous particles are derived for airflow over a cylindrical object. Only impaction and interception losses were considered in this study. Transport and deposition of fibrous particles were found to depend on Stokes number, fibrous particle aspect ratio, and ratio of the fibrous particle diameter to the diameter of fibrous filters. Using the Kuwabara flow field, transport and single-fibrous filter capturing efficiency of spherical and fibrous particles were calculated numerically, and these calculations were compared with available data in the literature. The calculated results compared favorably with the results of Yeh and Liu (1974) for spherical particles. Good agreement for losses by interception for both spherical and fibrous particles was observed between our results and those of Lee and Liu (1982). Further experimental data are needed to verify the predicted losses of fibrous aerosols by impaction.     

Numerical study of cluster and particle rebound effects in a circulating fluidised bed

Gas-particle flows in a vertical two-dimensional configuration appropriate for circulating fluidised bed applications were investigated numerically. In the computational study presented herein the motion of particles was calculated based on a Lagrangian approach and particles were assumed to interact through binary, instantaneous, non-frontal, inelastic collisions including friction. The model for the interstitial gas phase is based on the Navier-Stokes equations for two-phase flows. The numerical study of cluster structures has been validated with experimental results from literature in a previous investigation. Numerical experiments were performed in order to study the effects of different cluster and particle rebound characteristics on the gas-particle flow behaviour.Firstly, we investigated the hard sphere collision model and its effect on gas-particle flow behaviour. The coefficient of restitution in an impact depends not only on the material properties of the colliding objects, but also on their relative impact velocity. We compared the effect of a variable restitution coefficient, dependent on the relative impact velocity, with the classical approach, which supposes the coefficient of restitution to be constant and independent of the relative impact velocity.Secondly, we studied the effects of different cluster properties on the gas-particle flow behaviour. Opposing clustering effects have been observed for different particle concentrations: within a range of low concentrations, groups of particles fall faster than individual particles due to cluster formation, and within a well-defined higher concentration range, return flow predominates and hindered settling characterises the suspension. We propose herein a drag law, which takes into account both opposing effects and have compared the resulting flow behaviour with that predicted by a classical drag law, which takes into account only the hindered settling effect.     

Effects of the variation of mass loading and particle density in gas-solid particle flow in pipes

The method of two dimensional Reynolds Averaged Navier-Stokes (RANS) equations has been employed for the simulation of turbulent particulate flow. This approach was fitted with appropriate closure equations that take into account all the pertinent forces and effects on the solid particles, such as: particle-turbulence interactions; turbulence modulation; particle-particle interactions; particle-wall interactions; gravitation, viscous drag and lift forces. The flow domain in all cases was a cylindrical pipe and the computations were carried for upward pipe flow. The finite volume technique was used for the numerical solution of the governing and closure equations. The results show the effect of loading and particle density on the profiles of the velocity, the turbulence intensity and the solids concentration.     

A new fluid model for particles settling in a viscoplastic fluid

The study of the settling behaviour of particles in viscoplastic fluids is closely related to the study of rheology. In this paper, a thorough examination of the flow behaviour of viscoplastic fluids, in the form of aqueous polyacrylamide solutions, has been presented. The results of this study suggest that the experimental fluids exhibit time-dependent flow characteristics, where the apparent viscosity of the solutions depends highly on their shear history. This time dependency has been attributed towards the processes of destruction and rejuvenation in the ‘structural network’ of the fluids (due to the presence of hydrogen bonding between polyacrylamide and water molecules), as they are subjected to changing rates of shear. A new fluid model was thus developed to capture this flow behaviour. This model, termed as ‘semi-viscoplastic’, features temporary yield stress characteristics that tend to dissipate once the structural network of the fluid is destroyed due to the application of shear. The time dependency of the fluid viscous parameters becomes apparent in the settling sphere experiment, where it has been demonstrated that a sphere that is following the flow path of another sphere tends to attain a fall velocity that is significantly higher than the preceding sphere. Based on this finding, a new generalised correlation has been developed, through which predictions of the fall velocity of spherical particles settling through viscoplastic fluids, of various shear history, can be made.     

Prediction of minimum bubbling velocity, fluidization index and range of particulate fluidization for gas-solid fluidization in cylindrical and non-cylindrical beds

A uniform fluidization exists between minimum fluidization velocity and minimum bubbling velocity. Experimental investigations have been carried out for determination of minimum bubbling velocity and fluidization index for non-spherical particles in cylindrical and non-cylindrical beds. In the present paper equations have been developed for the prediction of minimum bubbling velocity for gas-solid fluidization in cylindrical and non-cylindrical (viz. semi-cylindrical, hexagonal and square) beds for non-spherical particles fluidized by air at ambient conditions. A fairly good agreement has been obtained between calculated and experimental values. Based on the experimental data it is concluded that under similar operating conditions minimum bubbling velocity and the fluidization index are maximum in case of either semi-cylindrical conduit or hexagonal conduit for most of the operating conditions and minimum in case of square one. It is further observed that the range of uniform (particulate) fluidization is maximum in case of semi-cylindrical bed for identical operating conditions.     

沿同心环面层流流动的气体中颗粒的传热沉积物的预测:发展中的流动和已充分发展的流动的比较

Thermophoresis is an important mechanism of micro-particle transport due to temperature gradients in the surrounding medium. It has numerous applications, especially in the field of aerosol technology. This study has numerically investigated the thermophoretic deposition efficiency of particles in a laminar gas flow in a concentric annulus using the critical trajectory method. The governing equations are the momentum and energy equations for the gas and the particle equations of motion. The effects of the annulus size, particle diameter, the ratio of inner to outer radius of tube and wall temperature on the deposition efficiency were studied for both developing and fully-developed flows. Simulation results suggest that thermophoretic deposition increases by increasing thermal gradient, deposition distance, and the ratio of inner to outer radius, but decreases with increasing particle size. It has been found that by taking into account the effect of developing flow at the entrance region, higher deposition efficiency was obtained, than fully developed flow.     

DEVELOPMENT OF NEW EMPIRICAL EQUATIONS FOR ESTIMATION OF DRAG COEFFICIENT,SHAPE DEFORMATION,AND RISING VELOCITY OF GAS BUBBLES OR LIQUID DROPS

Some new correlations are derived to estimate the drag coefficient, the shape deformation, and the rising velocity of particles moving in an infinite liquid medium. The correlations are derived in terms of the dimensionless groups such as Reynolds number (Re), Morton number (Mo), and Weber number (We). The derivations are based on the experimental data or some other correlations given in the literature. A single statement is proposed to estimate the drag coefficient for the spherical solid particles that may be applicable in the range of 0.5 < Re < 10⁵. Similarly, some other equations are also derived to estimate the drag coefficient, the shape deformation, or the rising velocity for gas bubbles or liquid drops. The drag equation is applicable in the range of 0.5 < Re < 100 and 9 × 10^?7 ≤ Mo ≤ 7; the shape deformation equation is applicable in the range of 0.5 < Re < 100 and 1.1 × 10^?5 ≤ Mo ≤ 7; and the rising velocity equation is applicable in the range of 0.1 < Re < 100 and 9 × 10^?7 ≤ Mo ≤ 80. The model predictions are compared with the experimental data and with the other correlations given in the literature. The results indicated that the model predictions are in a good agreement with the literature data.     

Experiments and simulations of settling cylinders over a wide range of Archimedes numbers

Quantitative visualization experiments and particle-resolved simulations of rigid cylindrical particles settling in a Newtonian liquid have been conducted. By varying the viscosity of the liquid—a glycerol-water mixture—as well as the density of the cylinders, we were able to cover an Archimedes number range that spans almost six orders of magnitude in the experiments. The length over diameter aspect ratio of the cylinders ranged from 2.5 to 20. Cylinders were released vertically and rotated to a stable horizontal orientation in most of the lower viscosity solutions. The time required for reaching a horizontal orientation, as well as the Reynolds number at that stage, scale with the Archimedes number and only weakly depend on the aspect ratio. Particle-resolved numerical simulations based on the lattice-Boltzmann method complement the experimental study and illustrate the relevance of the experimental data as a benchmark for numerical approaches to solid–liquid flow with non-spherical particles.     

Thermophoretic deposition of aerosol particles in laminar tube flow with mixed convection

A model is developed to investigate the thermophoretic deposition of aerosol particles on the wall of a relatively cool cylindrical tube. A key aspect of the flow is the presence of mixed convection. Continuity, momentum, and energy equations are solved to determine the velocity and temperature profiles of the system. Additionally, an aerosol population balance that incorporates thermophoresis and Brownian diffusion is solved. The results indicate that more particles deposit at shorter axial distances in downward flow through a vertical pipe than in upward flow. This behavior is traced to the larger residence time associated with downward flow, despite the presence of a reduced radial temperature gradient relative to the upward configuration. Model predictions also indicate that a bulk Richardson number (Ri) of at least unity is necessary, but not sufficient for free convection effects to be important. The additional evaluation of a local Ri provides a more reliable indicator.     

Numerical simulation of horizontal jet penetration in a three-dimensional fluidized bed

The introduction of reactant gas as a jet into a fluidized bed chemical reactor is often encountered in various industrial applications. Understanding the hydrodynamics of the gas and solid flow resulting from the gas jet can have considerable significance in improving the reactor design and process optimization. In this work, a three-dimensional numerical simulation of a single horizontal gas jet into a cylindrical gas-solid fluidized bed of laboratory scale is conducted. A scaled drag model is proposed and implemented into the simulation of a fluidized bed of FCC particles. The gas and particles flow in the fluidized bed is investigated by analyzing the transient simulation results. The jet penetration lengths of different jet velocities have been obtained and compared with published experimental data as well as with predictions of empirical correlations. The predictions by several empirical correlations are discussed. A good agreement between the numerical simulation and experimental results has been achieved.     

MODELLING AND CORRELATING THE PERMEABILITY OF PULP AND PAPER

A sheet of paper is modelled as a network of cellulose fibres, either cylindrical or band-shaped. The equations for creeping flow through such structures are solved, and The calculated permeabilities are compared with measured values. Flow through some paper structures such as pulp sheets and handsheets of unbeaten sulphate pulp is adequately described by the structural model involving band-shaped fibres when a fibre aspect ratio of 3.5 is used. For newsprint sheets the measured permeability is lower than that predicted by the models when physically realistic values of the aspect ratio are taken. A total of 19 different paper grades have been characterised by measurement of the total specific surface area and The fibre orientation ratio in addition to the measurements of effective diffusivity, permeability and porosity. Permeability and effective diffusivity correlate with each other and permeability correlates with fibre orientation, so that at constant porosity, permeability decreases with increasing fibre orientation.     

EQUATIONS FOR CALCULATION OF THE TERMINAL VELOCITY AND DRAG COEFFICIENT OF SOLID SPHERES AND GAS BUBBLES

An analysis of the correlations proposed in the literature for calculation of the drag coefficient (C_D) and the terminal velocity of a falling rigid sphere has been made. Among the correlations describing C_D vs. Re, that of Turton and Levenspiel fits the experimental data almost perfectly. However, it is not explicit in the terminal velocity. The available explicit correlations do not fit the experimental data well. The present paper shows that a simple and precise explicit correlation can be developed if C_D is related to the Archimedes instead of the Reynolds number. The precision of the correlation proposed is similar to that of the Turton and Levenspiel (1986), while it is explicit in the terminal velocity. On the basis of this correlation, a model is proposed to calculate the drag coefficients and the terminal velocities of free falling or rising spherical particles in an infinite fluid as well as gas bubbles with any volume and shape rising in a contaminated liquid.