Cross-sectional and axial flow characteristics of dry granular material in rotating drums

Cross-sectional and axial flow behaviors of dry granular material in rotating drums are closely related to the dynamic characteristics and velocity distributions between the surface layer and bed material. In this study, both 2D and 3D dry granular flow patterns in horizontal rotating drums are experimentally investigated with flow imaging analysis. A dimensionless flow parameter combining the effects of Froude number, relative particle size and volume filling is proposed in this study, which controls the flow characteristics in a rational drum such as dynamic angle of repose, thickness of the flowing layer, relative free surface velocity, and the shear rates in the flowing layer. The dimensionless granular temperature exhibits linear distribution in the flowing layer, being maximum at the free surface and being negligible at the interface in the rolling regime. The measured shear rate of the plug flow departs from drum angular velocity under the wall slip conditions when the drum surface is smooth. Due to the existence of axial convection and lateral surface profile, the mass flux in the flowing layer is always less than that of the plug flow in the 3D granular flows based on sidewall particle images. One the other hand, the mass flux in the flowing layer is always equal or greater than that of the plug flow in the 2D granular flows. 2D granular flows exhibit higher angles of repose and surface velocities than those of the 3D granular flows at the same volume fillings.     

Stability of ferrofluid flow between concentric rotating cylinders with an axial magnetic field

A linear stability analysis for the ferrofluid flow between two concentric rotating cylinders in the presence of an axial magnetic field is implemented in this study. Both of the wide-gap and small-gap cases are considered and the governing equations with respect to three-dimensional disturbances including axisymmetric and non-axisymmetric modes are derived and solved by a direct numerical procedure. A parametric study covering wide ranges of ?, the volume fraction of colloidal particles; ξ, the strength of axial magnetic field; μ, the ratio of angular velocity of the outer cylinder to that of the inner cylinder; and ε, the ratio of radius of the inner cylinder to that of the outer cylinder, is conducted. Results show that the stability characteristics depend heavily on these factors. It is found that the increases of ? and ξ, and decrease of ε tend to stabilize the basic flow for an assigned value of μ. The variations of the onset mode with these parameters are discussed in detail. An example for the practical application of present results is given to help the understanding of stability behaviour of this flow.     

Tangential velocity profiles of granular material within horizontal rotating cylinders modelled using the DEM

Flow regimes of granular materials in horizontal rotating cylinders are industrially important since they have a strong influence on the rates of heat and mass transfer within these systems. The tangential velocity profile, which describes how the average particle velocity in the direction parallel to the surface of the bed varies along a radius perpendicular to the surface of the bed, has been examined for many experimental and simulated systems. This paper is concerned with tangential velocity profiles within rotating cylinders simulated using the discrete element method. For high fill levels good agreement is found between the simulated velocity profiles and the equation proposed by Nakagawa et al. (Exp Fluids 16:54–60, 1993) based on magnetic resonance measurements. At lower fill levels slip is observed between the cylinder wall and the particles in contact with it and also between the outer layer of particles and the bulk of the bed. It is demonstrated that this slip occurs when the particles in contact with the wall are able to rotate and that it may be prevented either by using non-spherical particles or by attaching “lifters” to the cylinder wall.     

Radial segregation of ternary granular mixtures in rotating cylinders

We carry out an experimental study of the equilibrium segregation of ternary granular mixtures in a rotating cylinder. In all the experiments, 50% of the volume of the cylinder is filled with the granular mixture and the rotational speed used ensures operation in the rolling regime of flow. Mixtures of spherical particles differing only in size and of spherical particles differing in size and density are considered, using steel balls and glass beads of different sizes. Volume fractions of the components (f{\phi}) are measured by sampling at different radial positions (r) to yield the radial volume fraction profiles (f(r){\phi(r)}). Results for mixtures differing only in size of the components indicate that the segregation process is nearly independent of the sizes of the large and middle size particles for the same size of small particles. In the case of mixtures with different size and density components, the segregation patterns depend on the direction of the resultant driving force. In many of the mixtures considered, the pattern of segregation can be qualitatively predicted by considering binary interactions between the components. However, in some mixtures, ternary interactions are found to determine the pattern obtained.     

Modelling material separation processes in bulk recycling

Increasing environmental concerns about the disposal of mass produced products have resulted in efforts to recover value from components and materials before discarding such products. Methods include incineration, disassembly or de-manufacturing and bulk recycling. This paper investigates several cases of the problem of selecting the best sequence for recovery of materials by bulk recycling. A solution procedure for determining the optimal sequence for isolating all target materials present in a given batch based on dynamic programming is presented. A modification for targeting specific materials in a batch is discussed. Additional constraints arising from the operation of such facilities are also discussed. The general problem of target material identification is presented and a procedure for selecting a profitable target material mix, along with a numerical illustration, is documented. Finally, the integrated consideration of bulk recycling and disassembly for evaluating product disposal costs is presented.     

Hydrodynamics and external heat transfer in granular beds with rotating cylinders placed in them

The results are presented from an experimental investigation of the effect of rotation of a cylinder placed horizontally in fixed and fluidized granular media on the structure of the layer adjacent to the wall and on the intensity of external heat transfer.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 52, No. 1, pp. 5–9, January, 1987.     

Experimental investigation of axial dispersion in a horizontal rotating cylinder

This work reports experimental measurements of the dispersion of particles during rotation in a horizontal cylinder. The axial dispersion of a pulse of approximately monodisperse black glass ballotini into a bed of clear glass ballotini of the same size is analysed. This is done using a sectioning technique, where the concentration is determined throughout the cylinder for a given rotation time and speed. The concentration profile is fitted to an appropriate solution of Fick’s second law to determine the dispersion coefficient. The dispersion coefficient is compared for various drum rotation rates and glass ballotini sizes. The cylinder was filled to 35 % by volume and rotated at a range of speeds between 5 and 20 rpm. The particle sizes vary from 1.14 to 3.15 mm. The dispersion coefficient was found to be dependent on both particle size and rotation speed. As the rotation speed, \(\omega \), was increased the dispersion coefficient increased proportionally to \(\omega ^{0.8}\). As the particle diameter, \(d_p\), was increased the dispersion coefficient increased proportionally to \(d_p^{1.84}\). These results are compared with previous experimental and simulation data, in particular the simulations of Third et al. (Powder Technol 203:510, 2010). Strong agreement was found between the simulations of Third et al. and the experimental results.

     

Stability of axial Poiseuille-Couette flow between concentric cylinders

Summary The linear stability of axial parallel Poiseuille-Couette flow in an annulus between concentric circular cylinders is considered. Using a long-wave version of the axisymmetric Orr-Sommerfeld equation the stability chart of this flow in the velocity ratio-radius ratio plane is derived. It is shown that pure sliding Couette flow can become unstable if the radius ratio is below a specific threshold value. Finally, applying the results to other flow geometries, it is shown that the boundary layer along a slender cylinder can become unstable in a confined region downstream the leading edge only.     

The parameters governing the coefficient of dispersion of cubes in rotating cylinders

Axial dispersion of cubic particles in horizontal, rotating cylinders was investigated using discrete element modelling simulations. We found that, similar to the behavior of spheres, the axial dispersion coefficient of cubes depends on (1) the rotational speed of the cylinder \({\omega }\), (2) the acceleration due to gravity g and (3) the particle size d, satisfying the relationship \({D}_\mathrm {ax}\propto {\omega }^{1-2{\lambda }}{g}^{{\lambda }}{d}^{2-{\lambda }}\) with \({\lambda }\approx 0.15\) (\({\lambda }\approx 0.1\) for beds of spheres) (Third et al. in Powder Technol 203:510–517, 2010). This observation suggested that, although particle shape influences significantly the rate of axial dispersion (cubes disperse almost twice as fast as spheres of equal volume), the parameters controlling the coefficient of dispersion are independent of particle shape.     

Axial drive to nonlinear flow between rotating cylinders

Stability of pseudoplastic rotational flow between cylinders in presence of an independent axial component is investigated. The fluid is assumed to follow the Carreau model and mixed boundary conditions are imposed. The conservation of mass and momentum equations give rise to a four-dimensional low-order dynamical system, including additional nonlinear terms in the velocity components originated from the shear-dependent viscosity. In absence of the axial flow, as the pseudoplasticity effects increases, the purely-azimuthal base flow loses its stability to the vortex structure at a lower critical Taylor number. Emergence of the vortices corresponds to the onset of a supercritical bifurcation also present in the flow of a linear fluid. However, unlike the Newtonian case, pseudoplastic Taylor vortices lose their stability as the Taylor number reaches a second critical number corresponding to the onset of a Hopf bifurcation. Existence of an axial flow induced by a pressure gradient appears to further advance each critical point on the bifurcation diagram. In continuation, complete flow field together with viscosity maps is analyzed for different flow scenarios. Through evaluation of the Lyapunov exponent, flow stability and temporal behavior of the system for cases with and without axial flow are brought to attention.     

Stability of power law fluid flow between rotating cylinders

The centrifugal instability of power law fluid flow when the fluid fills the gap between two rotating concentric cylinders of infinite length is addressed. The instability of the circular Couette flow is determined via linear stability analysis. In the narrow gap case, the critical Taylor number is determined analytically, while in the general gap case, a finite-difference numerical method is employed. The results are shown to be in agreement with existing theoretical and experimental findings. The super-critical flow is investigated by means of a weakly non-linear analysis method. The Taylor-vortex flow (the secondary stable flow) is obtained. The instability of this flow is determined as we present the critical Taylor number when the Taylor vortices begin to exhibit a waviness in the azimuth     

Insights from simulations into mechanisms for density segregation of granular mixtures in rotating cylinders

The Discrete Element Method (DEM) is used to study the segregation of a binary mixture of differing density (but same size) granular material in an axially rotating cylinder. The rotation rates used produce a flow that is on the borderline between the avalanching and rolling regimes. The simulations replicate the experimental data well at both qualitative and quantitative levels. Both wall-induced and radial segregation are observed. The simulations show segregation is delineated into two main time regimes. At early times segregation is rapid (when the dense core is being established) and slows down appreciably thereafter. The final asymptotic state is found to be independent of the initial segregation state of the particles. We compare these results with previous theoretical models and relate these two distinct time regimes to the underlying segregation mechanisms. These comparisons suggest segregation varies as a function of two fundamental quantities (a) density ratio of particles and (b) angular speed of the rotating cylinder. It is shown that maximal segregation occurs for specific ranges of these quantities.     

Gravity flow of granular materials in contracting cylinders and tapered tubes

This paper presents two applications of the double-shearing theory for flow of granular materials under gravitational forces for axially symmetric flow. In the first the material is contained in a vertical circular cylinder which compresses the material by contracting radially. Exact solutions for the stress and velocity fields are derived, under the boundary conditions that the cylinder wall is either `perfectly rough' or subject to Coulomb friction. In the second problem the material flows under gravity through a tube with circular cross-section of radius that decreases slowly with depth. For this problem an approximate solution is derived that is accurate to first order in the slope of the tube wall relative to the vertical. It is also shown that an alternative theory, in which it is postulated that the principal axes of the stress and rate-of-strain tensors are coincident, leads to the prediction of unacceptable singularities in the flow field in the interior of the material.     

Hydromagnetic flow between rotating eccentric cylinders with suction at the porous walls

The effect of a radial magnetic field on the flow of a viscous, incompressible fluid between two eccentric rotating cylinders with suction at both the cylinders, for very small clearance ratio is investigated. The expressions for the stream function and pressure are obtained using perturbation analysis. Conclusions are drawn from the streamlines and pressure plots which are shown graphically for various values of flow parameters.     

Effects of end wall friction in rotating cylinder granular flow experiments

Dry granular flows in half-filled horizontal rotating cylinders were studied by NMR to ascertain end wall effects. We measured the velocity and density profiles in planes perpendicular to the axis of the drums for three cases: (1) Near the end wall of a long ``3D' cylinder, (2) Near the middle of the long cylinder and, (3) For the whole of a short ``2D' cylinder. The velocity near the free surface is fastest for the short cylinder and slowest near the end wall of the long cylinder. We believe that these differences are due to friction between the beads and the end walls, speeding up the flow for the 2D drum and reducing the near-wall speed for the 3D drum.     

Fully developed flow of granular materials down a heated inclined plane

Summary The mechanics of flowing granular materials such as coal, sand, metal ores, etc., and their flow characteristics have received considerable attention in recent years as it has relevance to several important technological problems. In a number of instances, these materials are also heated prior to processing, or cooled after processing. The governing equations for the flow of granular materials, taking into account the heat transfer mechanism by conduction, are derived using a continuum model (cf. Goodman and Cowin [1], [2], Rajagopal and Massoudi [3]). For a fully developed flow of these materials down an inclined plane, the equations reduce to a system of coupled non-linear ordinary differential equations. The resulting boundary value problem is solved numerically and the results are presented for cases where the viscosity and thermal conductivity are assumed to be functions of the volume fraction. It is shown that the equations admit multiple solutions for certain values of the parameters.List of symbols D Symmetric part of the velocity gradient - K thermal conductivity - L velocity gradient - T Cauchy stress tensor - b body force - h characteristic height - q heat flux - r radiating heat - u velocity vector - angle of inclination of the inclined plane with the horizontal - specific internal energy - distributed mass density - temperature - volume fraction - bulk mass density     

On the stability of secondary Taylor flow between rotating cylinders with a wide gap

In a general formulation, the problem of the stability of small perturbations in a homogeneous viscous fluid between rotating cylinders with a wide gap is investigated numerically.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 25, No. 1, pp. 99–106, July, 1973.     

Behaviour of spiral vortices on a rotating cone in axial flow

Summary The purpose of the present paper is to investigate experimentally in detail the boundary layer transition process and the behaviour of spiral vortices appearing in the transition range of the boundary layer on a 30°-cone, rotating in axial flow. Counterrotating spiral vortices in the transition range are visualized with a white smoke method, and observed the time dependent behaviour of them using a drum camera and a light sheet illumination method with a stroboscope flash light. The light passes a slit in order to illuminate only a thin sheet in the flow. With this method, the time dependent growing up and breaking down process of these spiral vortices is greatly clarified. A hot wire anemometer is also used for measuring in the flow field quantitatively. The results show that the spiral vortices are generated in the thin region of the steep shear velocity gradients near the wall. As the vortices grow up in z-direction, they are strongly distorted by the mean velocity field there, and finally they are teared off.With 7 Figures     

Effect of periodic boundary conditions on granular motion in horizontal rotating cylinders modelled using the DEM

Applying periodic boundary conditions to DEM simulations of granular motion in horizontal, rotating cylinders can result in unexpected bulk axial flow. This axial flow is shown to contribute significantly to the mean square deviation in particle position and consequently must be considered in the analysis of processes governed by slow axial motion such as axial dispersion.     

Plane steady shear flow of a cohesionless granular material down an inclined plane: A model for flow avalanches Part I: Theory

Summary A continuum mechanical model describing rapid shear flow of granular materials as deduced by Jenkins and Savage (1983) [11] from considerations of statistical mechanics is applied to steady plane shear flows down an inclined chute. Depending on the type and form of the physically suggested boundary conditions that are imposed at the base and the free surface, respectively, the emerging boundary value problems permit or prohibit existence of mathematical solutions. For instance, the model does not permit incorporation of an aerodynamic drag and requires special sliding boundary conditions at the base. Cause for the singular behavior is the fact that granular pressure and fluctuation energy vanish simultaneously. Rectification is e.g. possible by including particle density gradients in the constitutive relation of granular stress, but this requires postulation of additional boundary conditions.We present the differential equations and boundary conditions and suggest a procedure of non-dimensionalization which yields the dimensionless parameters governing the problem. Construction of local solutions close to the boundaries by means of Frobenius expansions discloses the singular behavior and yields the basis for the non-existence proof under limiting conditions. Adding to the particle stress a Newtonian viscous contribution is not sufficient to regularize the problem and neither is the form of the stress tensor resulting from Lun et al.'s statistical model that incorporates kinetic terms. The stress tensor must have a term proportional to the dyadic product of the particle concentration gradient with itself. Numerical solution techniques and computational results are given in a companion paper (Hutter, Szidarovszky, Yakowitz, 1986 [9]).With 3 Figures