Numerical study of particle motion pattern in a rotating wavy tumbler based on angular momentum analysis

This work uses the hard-sphere model to simulate the particle motion pattern in a two-dimensional rotating tumbler with a sine-wave boundary. Based on the angular momentum analysis, the present work investigates the intrinsic effects of boundary configuration of tumblers, e.g. the wave number and the boundary amplitude, on the motion pattern of the particles separately. It is found that the particle motion pattern depends on the configuration of the tumbler significantly. There are two collision patterns, namely the primary and secondary patterns. The former pattern dominates when the boundary is composed of low wave numbers or small boundary amplitudes, whereas the latter pattern dominates when the boundary is composed of high wave numbers or large boundary amplitudes. The primary collision pattern is beneficial for the bulk motion of tumbling of the particle clusters, whereas the secondary pattern suppresses it. The interpretation for the different mechanisms of collision pattern indicates the existence of different regions, which is either positive or negative to the tumbling motion. It provides some guiding information for engineering application, such as the design of tumblers for efficient mixing.     

A CFD-DEM study of the cluster behavior in riser and downer reactors

This paper presents a numerical study of the particle cluster behavior in a riser/downer reactor by means of combined computational fluid dynamics (CFD) and discrete element method (DEM), in which the motion of discrete particles is obtained by solving Newton's equations of motion and the flow of continuum gas by the Navier-Stokes equations. It is shown that the existence of particle clusters, unique to the solid flow behavior in such a reactor, can be predicted from this first principle approach. The results demonstrate that there are two types of clusters in a riser and downer: one is in the near wall region where the velocities of particles are low; the other is in the center region where the velocities of particles are high. While the extent of particle aggregation appears to be similar, the duration time for the first type in a downer is shorter than in a riser. Furthermore, it is demonstrated that the formation of clusters is affected by a range of variables related to operational conditions, particle properties, and bed properties and geometry. The increase of solid volume fraction, sliding and rolling friction between particles or between particles and wall, or damping coefficient can enhance the formation of clusters. The use of multi-sized particles can also promote the formation of clusters. But the increase of gas velocity or use of a wider bed can suppress the formation of clusters. The van der Waals force may enhance the formation of clusters when solid concentration is high but suppress the formation of clusters near the wall region when solid concentration is low.     

液–固脉冲流化床中浓度波传播与衰减

采用数值计算和实验验证相结合的方法研究液–固脉冲流化床中浓度波的传播和衰减. 当脉冲开半周期T2和闭半周期T1都远大于颗粒弛豫时间tp时，两相的惯性力之差在一个周期的绝大部分时间中相对于重力很小，可忽略，这时由双流体模型的动量方程可推导出推广的Richardson–Zaki公式，双流体模型简化为局部平衡模型. 采用五阶精度WENO格式求解浓度波传播方程，得到了脉冲流化过程中浓度波传播与衰减的规律，与实验结果符合良好.     

Numerical simulation of pulsed fluidized bed with immersed tubes using DEM-LES coupling method

The present work is a 2-D numerical simulation of pulsed fluidized bed with immersed tubes using DEM-LES coupling method. The pulsed inflow of gas phase is modeled as U₀(1+sin(2πft)), in which four pulsating frequencies of f=5, 10, 15 and 20 of velocity inflow are used. The discrete element method (DEM) simulation for particle motion coupled with the large eddy simulation (LES) for gas phase is used. The fluidized bed with five immersed tubes of staggered arrangement and six immersed tubes of in-line arrangement is simulated, respectively. It is found that the pressure drop, the mean drag force and the mean pressure gradient force experienced by particles are forced oscillated. The different effects of pulsed fluidization on the circumferential distribution of particle-tube collision on the outer surface of tubes at different pulse frequencies and modes of arrangement of immersed tubes are numerically analyzed. Finally, it is found that the pulsed motion of fluid with high frequency leads to suppression of particle fluctuating motion.     

Thermophoretic motion of slightly deformed aerosol spheres

The thermophoretic motion of a slightly deformed aerosol sphere in a uniformly prescribed but arbitrarily oriented temperature gradient is analyzed in the steady limit of negligible Peclet and Reynolds numbers. The Knudsen number is assumed to be small so that the fluid flow is described by a continuum model with a temperature jump, a thermal slip, and a frictional slip at the surface of the particle. The energy and momentum equations governing the system are solved asymptotically using a method of perturbed expansions. To the second order in the small parameter characterizing the deformation of the aerosol particle from the spherical shape, the thermal and hydrodynamic problems are formulated for the general case, and explicit expressions for the thermophoretic velocity of the particle are obtained for the special cases of prolate and oblate spheroids. The agreement between our asymptotic results for a thermophoretic spheroid and the relevant exact or numerical solutions in the literature is quite good, even if the particle deformation from the spherical shape is not very small. Depending on the values of the relative thermal and surface properties of the aerosol spheroid, its thermophoretic mobility normalized by the corresponding value for a spherical particle with equal equatorial radius is not necessarily a monotonic function of the aspect ratio of the spheroid.     

On the unsteady forces during the motion of a sedimenting particle

We consider the unsteady motion of a sedimenting rigid spherical particle in order to examine the relative strength of the hydrodynamical forces acting on particles in fluid flows. The relative strength of the forces on all stages of the particle motion is a major concern for closing constitutive equations describing the more complex motion of particulate flows such as fluidised beds. The formulation results in a first order nonlinear integro-differential equation in terms of the instantaneous velocity of the sphere. This equation is made dimensionless and the particle Reynolds number and the fluid-particle density ratio are identified as the relevant physical parameters describing the particle motion. We obtain analytical solutions for the limits of small density ratios and small Reynolds number. In addition, a numerical solution is used for arbitrary values of the density ratio. The results show that the motion of spherical particles is significantly affected by the unsteady drag dominated by the memory Basset force on the early stages of the motion and on the approach to the steady state (terminal velocity). The present calculations indicate that the unsteady hydrodynamic drags might become of the same order of magnitude of the dominant viscous drag for flows with moderate particle-fluid density ratio. Therefore, unsteady drags should be taken into account on modelling multiphase particulate flows with moderate density ratio.     

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.     

Simulations of scalar dispersion in fluidized solid–liquid suspensions

Direct, particle‐resolved simulations of solid–liquid fluidization with the aim of quantifying dispersion have been performed. In addition to simulating the multiphase flow dynamics (that is dealt with by a lattice‐Boltzmann method coupled to an event‐driven hard‐sphere algorithm), a transport equation of a passive scalar in the liquid phase has been solved by means of a finite‐volume approach. The spreading of the scalar—as a consequence of the motion of the fluidized, monosized spherical particles that agitate the liquid—is quantified through dispersion coefficients. Particle self‐diffusivities have also been determined. Solids volume fractions were in the range 0.2–0.5, whereas single‐sphere settling Reynolds numbers varied between approximately 3 and 20. The dispersion processes are highly anisotropic with lateral spreading much slower (by one order of magnitude) than vertical spreading. Scalar dispersion coefficients are of the same order of magnitude as particle self‐diffusivities. © 2014 American Institute of Chemical Engineers AIChE J, 60: 1880–1890, 2014     

THE TRANSIENT MOTION OF A SPHERICAL FLUID DROPLET

A numerical method is developed for investigation of the unsteady motion of a spherical fluid droplet under the influence of gravity. This study extends previous work valid for creeping flow to moderate Reynolds number. The unsteady flow fields inside and outside of the fluid sphere are described by the two-dimensional, axisymmetric Navier-Stokes equations in the form of vorticity and stream function, along with the equation of motion of the droplet. The governing equations are approximated by a central difference and a second-order upwind difference, and are solved iteratively using the Gauss-Siedel and secant methods. Numerical results of the time-dependent vorticity, stream function and drop velocity are presented for a water droplet moving through air and for an air bubble rising in water. The steady state drop velocity and the drag coefficient at various Reynolds numbers are examined, and they are shown to agree very well with previous results.     

DEM-LES study of 3-D bubbling fluidized bed with immersed tubes

Based on Euler-Lagrange frame, a true three-dimensional numerical simulation of bubbling fluidized bed embedded with two immersed tubes is presented. The solid phase is composed of 178,200 particles of diameter and simulated by discrete element method (DEM, a soft-sphere approach). The gas phase is computed through solving the volume-averaged four-way coupling Navier-Stokes equations in which the Smagorinsky SGS tensor model is used in large eddy simulation (LES). Particle-tube collision is particularly treated as a transformation of DEM. The volume segmentation of a particle sphere for void fraction calculation is solved via a numerical sub-division approach. The numerical results are compared with the experimental results for validation. The results obtained with and without the LES model are also compared. The numerical results show a strong correlation between gas-particle interaction, particle-particle interaction, pressure drop, particle back mixing motion and bubble motion, and all of them follow a similar pattern of synchronous periodic variation though the periodicity may vary depending on different flow conditions. The effects of SGS tensor on evolution of fluidized bed are found in various aspects. Finally, the distribution of particle-tube impact frequency is given.     

Validation of a Discrete Particle Model in a 2D Spout‐Fluid Bed Using Non‐Intrusive Optical Measuring Techniques

To gain insight into the hydrodynamics of spout‐fluid beds, an experimental and numerical study was carried out. Particle image velocimetry was successfully developed and applied to determine particle velocity profiles, whereas voidage profiles were determined by digital image analysis. A 3D hard‐sphere discrete particle model was used to simulate the flow in a spout‐fluid bed. The simulations and experiments showed a similar influence of the background fluidization velocity on the spout behaviour.     

Numerical study on steady and transient mass/heat transfer involving a liquid sphere in simple shear creeping flow

A numerical method is utilized to examine the steady and transient mass/heat transfer processes that involve a neutrally buoyant liquid sphere suspended in simple shear flow at low Reynolds numbers is described. By making use of the known Stokes velocity field, the convection‐diffusion equations are solved in the three‐dimensional spherical coordinates system. For the mass transfer either outside or inside a liquid sphere, Sherwood number Sh approaches an asymptotic value for a given viscosity ratio at sufficiently high Peclet number Pe. In terms of the numerical results obtained in this work, two new correlations are derived to predict Sh at finite Pe for various viscosity ratios. © 2013 American Institute of Chemical Engineers AIChE J, 60: 343–352, 2014     

循环硫化床上升管中动态行为的拟流体模拟

The kinetic theory of granular flow (KTGF) is modified to fit the Einstein′s equation for effective viscosity of dilute flow. A pseudo-fluid approach based on this modified KTGF is used to simulate the dynamic formation and dissipation of clusters in a circulating fluidized bed riser. The agglomeration of particles reduces slip velocity within particle clusters, and hence results in two reverse trends: discrete particles are lifted by air while particle clusters fall down along the wall. The dynamic equilibrium of these two types of motion leads to the characteristic sigmoid profile of solid concentration along the longitudinal direction. The predicted solid velocity, lateral and longitudinal profiles of solid volume fraction and annulus thickness are in reasonable agreement with experimental results.     

Magnetically driven hydrodynamic interactions of magnetic and non-magnetic particles

The influence of magnetic forces on (i) the motion of magnetisable particles in a non-magnetisable fluid and (ii) the motion of non-magnetisable particles in a magnetisable fluid was investigated. A non-uniform magnetic field was used to induce a strong magnetic field gradient. The study was conducted at low particle Reynolds numbers allowing independent evaluation of the hydrodynamic forces along the tangential and normal directions, which in turn were used to deduce the strength and variation of the magnetic forces. Two iron spheres placed in a non-magnetisable fluid showed strong mutual attraction in the normal direction, consistent with an inverse distance to the power four law. The velocity of each sphere in the tangential direction prior to aggregation was constant. Upon aggregation, the velocity of the dumbbell in the tangential direction was approximately two times higher. This sudden increase in velocity was attributed to an increase in the magnetisation following aggregation of the two spheres. It was concluded that a dumbbell has a larger concentration of matter along the direction of the magnetic field and hence, with respect to motion in the tangential direction, a higher magnetisation. For the two acrylic spheres in a magnetised fluid the attractive force was found to be negligible, most probably because of the low magnetisation of the paramagnetic salt solution used. The two spheres did migrate towards each other because of the local field gradients that develop in the normal direction. Interestingly, the spheres developed a constant velocity prior to aggregation, which was also maintained after aggregation. Two acrylic spheres glued together also travelled in the tangential direction at the velocity observed for the individual spheres. It was concluded that there was no change in the magnetisation of the fluid following the aggregation of the spheres.     

Boundary effects on thermophoresis of aerosol cylinders

An analytical study is presented for the thermophoretic motion of a circular cylindrical particle in a gaseous medium with a transversely imposed temperature gradient near a large plane wall parallel to its axis in the quasisteady limit of negligible Peclet and Reynolds numbers. The Knudsen number is assumed to be small so that the fluid flow is described by a continuum model with a temperature jump, a thermal slip, and a frictional slip at the particle surface. The presence of the confining wall causes two basic effects on the particle velocity: first, the local temperature gradient on the particle surface is altered by the wall, thereby speeding up or slowing down the particle; secondly, the wall enhance the viscous retardation of the moving particle. Through the use of cylindrical bipolar coordinates, the transport equations governing this problem are solved and the wall effects on the thermophoresis of the aerosol cylinder are computed for various cases. The presence of the plane wall can reduce or enhance the particle velocity, depending upon the relative thermal conductivity and surface properties of the particle, the relative particle-wall separation distance, and the direction of the applied temperature gradient. The direction of the thermophoretic motion of a cylindrical particle near a plane wall is different from that of the prescribed thermal gradient, except when it is oriented parallel or perpendicular to the wall. The effects of the plane wall on the thermophoresis of a cylinder are found to be much more significant than those for a sphere at the same separation.     

Drag of nonspherical particles in a flow behind a shock wave

The early stage of velocity relaxation of nonspherical particles in a flow behind an incident shock wave is considered by the method of multiframe shadowgraphy. A procedure of processing the data on the motion of a free body for determining its acceleration is proposed; in combination with the diagnostic method used, the procedure forms something like a noncontact aerodynamic balance. Novel data on the drag of bodies of irregular shape in a flow behind a shock wave with Mach numbers of 0.5–1.5 and Reynolds numbers of 10⁵ typical of dust explosions are obtained. It is found that the values of drag of a nonspherical bluff body and a sphere under these conditions are similar and exceed the drag of a sphere in a steady flow by a factor of 2 to 3.Translated from Fizika Goreniya i Vzryva, Vol. 41, No. 1, pp. 81–88, January–February, 2005.     

Low-Reynolds-number motion of a heavy sphere between two parallel plane walls

A novel boundary-integral algorithm [Staben, M.E., Zinchenko, A.Z., Davis, R.H., 2003. Motion of a particle between two parallel plane walls in low-Reynolds-number Poiseuille flow. Physics of Fluids 15, 1711-1733; Erratum: Phys. Fluids 16, 4206] is used to obtain O(1)-nonsingular terms that are combined with two-wall lubrication asymptotic terms to give resistance coefficients for near-contact or contact motion of a heavy sphere translating and rotating between two parallel plane walls in a Poiseuille flow. These resistance coefficients are used to describe the sphere's motion for two cases: a heavy sphere driven by a Poiseuille flow in a horizontal channel and a heavy sphere settling due to gravity through a quiescent fluid in an inclined channel. When the heavy sphere contacts a wall in either system, which occurs when the gap between the sphere and the wall becomes equal to the surface roughness of the sphere (or plane), a contact-force model using the two-wall resistance coefficients is employed. For a heavy sphere in a Poiseuille flow, experiments were performed using polystyrene particles with diameters 10%-60% of the channel depth, driven through a glass microchannel using a syringe pump. The measured translational velocities for these particles show good agreement with theoretical results. The predicted translational velocity increases for increasing particle diameter, as the spheres extend further into the Poiseuille flow, except for particles that are so large (diameters of 80%-85% of the channel depth) that the upper wall has a dominant influence on the particle velocity. For a heavy sphere settling in a quiescent fluid in an inclined channel, the transition from the no-slip regime to slipping motion occurs for a larger inclination angle of the channel with respect to the horizontal for an increase in particle diameter, since the larger particles are more slowed by the second wall. Limited experiments were performed for Teflon spheres with diameters 64%-95% of the channel depth settling in a very viscous fluid along the lower wall of an inclined acrylic channel. The measured translational velocities, which are only about 15%-25% of the tangential component of the undisturbed Stokes settling velocity, are in close agreement with theory using physical parameters obtained from similar experiments with a single wall [Galvin, K.P., Zhao, Y., Davis, R.H., 2001. Time-averaged hydrodynamic roughness of a noncolloidal sphere in low Reynolds number motion down an inclined plane. Physics of Fluids 13, 3108-3119].     

Motion of a sphere parallel to plane walls in a Poiseuille flow. Application to field-flow fractionation and hydrodynamic chromatography

The motion of a solid spherical particle entrained in a Poiseuille flow between parallel plane walls has various applications to separation methods, like field-flow fractionation and hydrodynamic chromatography. Various handy formulae are presented here to describe the particle motion, with these applications in mind. Based on the assumption of a low Reynolds number, the multipole expansion method coupled to a Cartesian representation is applied to provide accurate results for various friction factors in the motion of a solid spherical particle embedded in a viscous fluid between parallel planes. Accurate results for the velocity of a freely moving solid spherical particle are then obtained. These data are fitted so as to obtain handy formulae, providing e.g. the velocity of the freely moving sphere with a 1% error. For cases where the interaction with a single wall is sufficient, simpler fitting formulae are proposed, based on earlier results using the bispherical coordinates method. It appears that the formulae considering only the interaction with a nearest wall are applicable for a surprisingly wide range of particle positions and channel widths. As an example of application, it is shown how in hydrodynamic chromatography earlier models ignoring the particle-wall hydrodynamic interactions fail to predict the proper choice of channel width for a selective separation. The presented formulae may also be used for modeling the transport of macromolecular or colloidal objects in microfluidic systems.     

Low peclet number particle-to-fluid heat and mass transfer in packed beds

For low Peclet numbers most of the experimentally obtained particle-to-fluid heat and mass transfer coefficients in packed beds were found to be some orders of magnitude below the values predicted for a single sphere in cross flow.From theoretical considerations one should expect the transfer coefficients in packed beds to exceed the single sphere predictions as they actually do for higher Peclet numbers.The obvious discrepancy between theory and experiment can be cxplained by a simple model accounting for a nonuniform distribution of the void fraction. The model consists of a packed bed of uniformly sized particles with an average void fraction ψ, where a small part ? of the total cross-sectional area f is assumed to have a larger void fraction ψ₂. Since the same pressure drop applies to both parts of the bed, the superficial velocity will be much larger in the section with the larger void fraction, especially in the range of low Reynolds numbers.Even though in both parts of the packed bed the individual transfer coefficients are taken from a correlation which is based on the single sphere predictions (as it is valid in the range of high Peclet numbers), the apparent overall transfer coefficients for the nonuniform system become much lower, and show the same characteristic variation with Peclet number and the ratio of particle diameter to bed height as the majority of the experimental data.     

硬球链流体在平板和硬球表面分布的密度泛函理论

采用Yethiraj和Woodward的密度泛函理论方法,结合胡英和刘洪来等发展的硬球链流体状态方程,得到了自由连接硬球链流体在平板狭缝中和球形固体颗粒表面附近的密度分布表达式,并计算了在两平行壁所组成的狭缝中和直径大小不同的球形固体颗粒周围硬球链分子的链节密度分布.理论计算结果与作者采用Dickman 和Hall 的方法进行Monte Carlo计算机模拟结果非常吻合.颗粒直径对链状分子的密度分布有一定的影响,随着固体颗粒直径的增加,靠近颗粒表面附近的链节密度降低.