Rough Spheres Research Articles

CFD simulations of bubbling beds of rough spheres

Hydrodynamics of three-dimensional gas–solid bubbling fluidized beds are numerically analyzed. The particle–particle interactions are simulated from the kinetic theory for flow of dense, slightly inelastic, slightly rough sphere proposed by Lun [1991. Kinetic theory for granular flow of dense, slightly inelastic, slightly rough sphere. Journal of Fluid Mechanics 233, 539–559] to account for rough sphere binary collisions and the frictional stress model proposed by Johnson et al. [1990. Frictional–collisional equations of motion for particulate flows and their application to chutes. Journal of Fluid Mechanics 210, 501–535] to consider the frictional contact forces between particles. The present model is evaluated by measured particle distributions and velocities of Yuu et al. [2001. Numerical simulation of air and particle motions in group-B particle turbulent fluidized bed. Powder Technology 118, 32–44] and experimental bed expansion of Taghipour et al. [2005. Experimental and computational study of gas–solid fluidized bed hydrodynamics. Chemical Engineering Science 60, 6857–6867]. Our computed results indicated that the present model gives better agreement with experimental data than the results from original kinetic theory for frictionless slightly inelastic sphere of Ding and Gidaspow [1990. A bubbling fluidization model using kinetic theory of granular flow. A.I.Ch.E. Journal 36, 523–538] with and without solid friction stress model.

The Effects of Interfacial Particles on the Contact of an Elastic Sphere With a Rigid Flat Surface

In chemical mechanical polishing (CMP), a rigid wafer is forced on a rough elastomeric polishing pad, while a slurry containing abrasive particles flows through the interface. One of the important factors that influence the material removal rate in CMP is the magnitude of contact force transmitted to the abrasive particles trapped at the contact interface. The total push-down force is distributed to the direct contact between the wafer and the pad, and to the three-body contact between the wafer, the pad, and the abrasive particles. The presence of the abrasive particles alters the asperity contact, which otherwise can be described by Hertz contact relationships. In this study, the effect of the interfacial particles on the single asperity contact is investigated. An approach used by Greenwood and Tripp (1967, “The Elastic Contact of Rough Spheres,” ASME J. Appl. Mech., 34, pp. 153–160) to study the contact of rough spheres is utilized since the presence of the particles provides a rough character to the contact. The results show that the contact behavior becomes non-Hertzian with decreasing contact force and increasing elastic modulus, particle size, and particle concentration. The role of the interfacial particles is to spread the contact over a larger area while lowering the maximum contact pressure at the center of contact predicted by Hertz contact. The conditions required to transfer the contact force on the particles effectively are also described.

Pairwise interaction of capsules in simple shear flow: Three-dimensional effects

We use a boundary integral formulation to investigate the collision of two identical capsules in simple shear flow. Each capsule consists of a viscous liquid drop enclosed by an elastic membrane. The hydrodynamic interaction is characterized by an irreversible cross-flow displacement after the capsules have crossed each other, as also observed for rough spheres or drops. This deflection would cause a shear-induced dispersion in a dilute suspension of capsules. After a previous work devoted to the interaction of two capsules located in the same shear plane, here, we focus on the more general case of capsules also shifted in the vorticity direction. For sufficiently spaced trajectories, the capsules exhibit negative deflections which displace them to closer streamlines after they have crossed. Furthermore, the cross-flow displacement occurs mainly in the vorticity direction. Both phenomena are in contradiction with the conclusions drawn for a pair of drops, and are, therefore, attributed to the elastic nature of the capsule interface.

Light harvesting enhancement for dye-sensitized solar cells by novel anode containing cauliflower-like TiO 2 spheres

Cauliflower-like TiO 2 rough spheres, which are about 200 nm large, have greatly enhanced light harvesting efficiency and energy conversion efficiency of dye-sensitized solar cells (DSC), due to their high light scattering effect and large BET surface area (80.7 m 2 g −1) even after calcinations at 450 °C for 30 min. The large size TiO 2 rough and smooth spheres, produced at different initial temperatures by hydrolysis of Ti(OBu) 4 with P105 (EO 37PO 56EO 37) or F68 (EO 78PO 30EO 78) tri-block copolymer as structural agents, have nearly the same diameter of ∼275 nm and strong light scattering effects in the wavelength of 400–750 nm. However, rough spheres have even higher light scattering effect and larger BET surface area than smooth spheres for the roughness of the surface. By adding 25 wt.% large TiO 2 spheres into the over-layer of TiO 2 film composed of ∼20 nm TiO 2 particles as light scattering centers, the energy conversion efficiency of the film containing rough spheres reaches 7.36%, much larger than that of smooth spheres (6.25%). From another point of view, the TiO 2 rough spheres may have the satisfying ability in other fields of application such as photo-catalysis, drug carriers and so on.

The analysis of self-diffusion and migration of rough spheres in nonlinear shear flow using a traction-corrected boundary element method

The phenomena of self-diffusion and migration of rough spheres in nonlinear shear flows are investigated using a new traction-corrected boundary element method (TC-BEM) in which the near-field asymptotics for the traction solution in the interstitial region between two nearly touching spheres is seamlessly coupled with a traditional direct boundary element method. The TC-BEM is extremely accurate in predicting particle trajectories, and hence can be used to calculate both the particle self-diffusivity and a newly defined migration diffusivity for dilute suspensions. The migration diffusivity is a function of a nonlinearity parameter characterizing the shear flow and arises from the net displacement of the centre of gravity of particle pairs. This net displacement of the centre of gravity of particle pairs does not occur for smooth particles, nor for rough particles in a linear shear flow. An explanation is provided for why two-particle interactions of rough spheres in a nonlinear shear flow result in particle migration.

Yttrium-86-labelled human serum albumin microspheres: relation of surface structure with in vivo stability

Radiolabelled particles are an attractive tool in the therapy of malignancies of the liver. We consider particles manufactured from denatured human serum albumin (HSA) as useful carriers of therapeutic radionuclides. Covalent attachment of suitable chelators onto the surface of the spheres promises an easy access to radiolabelled HSA microspheres. We synthesized 1,4,7,10-tetraazacyclododecane-N,N',N'',N'''-tetraacetic acid (DOTA) bearing smooth, medium-rough and rough surfaced HSA microspheres (mean diameter: 25 microm). In vitro stability of 86 Y-labelled particles was determined after incubation in human plasma and in a DTPA challenge experiment. In vivo stability of 86 Y DOTA-HSA microspheres was determined after single intravenous application in rats. Subsequently, the particles were completely trapped in the lung microvasculature. Thus, the lung serves in our experiments as target organ. DOTA-HSA microspheres were 86 Y labelled in reproducible high yields (>95%). No differences between smooth and rough surfaced spheres were found for both DOTA coupling and 86 Y labelling. Labelled microspheres showed high in vitro stability in human plasma and in DTPA solution with only 8+/-1% and 2+/-0% loss of radioactivity from the surface, respectively, 48 h postinjection (pi). The three batches (smooth, medium-rough and rough surfaced microspheres) differed considerably in their radioactivity recovery in the lungs of rats 48 h pi. Smooth particles showed the highest in vivo stability of the radiolabel on the surface of the spheres, presumably because of slower proteolytic degradation. We found that for the preparation of HSA-derived microspheres for radiotherapeutic application, smooth surfaced spheres are superior to rough spheres due to their higher in vivo stability of the radionuclide fixation.

Cauliflower-like TiO 2 rough spheres: Synthesis and applications in dye sensitized solar cells

A series of mesoporous TiO 2 rough spheres of alterable surface morphologies and pore structures were produced by altering initial temperatures and structural directing agents, Pluronic P105 (EO 37PO 56EO 37) or F68 (EO 78PO 30EO 78) tri-block copolymers. The average pore size increased with increasing initial temperature and in direct proportion to the polymer chain length. Various formation mechanisms were investigated including nucleation, growth and conglomeration of the initial TiO 2 particles. The morphologies of mesoporous TiO 2 rough spheres were completely maintained after calcination at 450 °C for 30 min. For dye sensitized solar cells (DSSCs) applications, efficient light scattering was obtained with the double layer film composed of a transparent layer containing 20 nm particles and a scattering layer containing ∼275 nm TiO 2 spheres. The rough TiO 2 spheres increase the overall conversion efficiency of DSSCs by 20.8% compared to TiO 2 smooth spheres, and reach 6.56%.

Contact Model for a Pad Asperity and a Wafer Surface in the Presence of Abrasive Particles for Chemical Mechanical Polishing

ABSTRACTA contact model for a single spherical (pad) asperity and a flat (wafer) surface, with an interface filled with spherical (abrasive) particles is introduced. Finite element method is used to determine the deformation characteristics of a single spherical abrasive indenting a hyper-elastic pad material. The asperities in Greenwood and Tripp (GT) model, which is widely used for the contact of rough spheres, are replaced with the spherical particles. Relations are developed to predict pad-to-particle-to-wafer contact in two distinct regimes, where a) pad and the wafer are separated by the particle; and b) all three bodies come in contact simultaneously. The results of the model show that the abrasive particles in the (pad) asperity-wafer interface causes the contact to be distributed over an area larger than that predicted by the Hertz model, while the maximum contact pressure becomes relatively lower. The fraction of the load carried by pad-to-wafer and pad-to-abrasive-to-wafer varies along the contact interface of the pad asperity and wafer.

The constitutive relation for the granular flow of rough particles, and its application to the flow down an inclined plane

A perturbation expansion of the Boltzmann equation is used to derive constitutive relations for the granular flow of rough spheres in the limit where the energy dissipation in a collision is small compared to the energy of a particle. In the collision model, the post-collisional relative normal velocity at the point of contact is in the present calculation), and is close to 1 if the angle between the relative velocity vector and the line joining the centres is less than the roughness angle. The conserved variables in this case are mass and momentum; energy is not a conserved variable in the ‘adiabatic limit’ considered here, when the length scale is large compared to the ‘conduction length’. The results for the constitutive relations show that in the Navier–Stokes approximation, the form of the constitutive relation is identical to that for smooth particles, but the coefficient of shear viscosity for rough particles is 10%–50% higher than that for smooth particles. The coefficient of bulk viscosity, which is zero in the dilute limit for smooth particles, is found to be non-zero for rough and partially rough particles, owing to the transport of energy between the translational and rotational modes. In the Burnett approximation, there is an antisymmetric component in the stress tensor for rough and partially rough particles, which is not present for smooth particles.The constitutive relations are used to analyse the ‘core region’ of a steady granular flow down an inclined plane, where there is a local balance between the production of energy due to the mean shear and the dissipation due to inelastic collisions. It is found that realistic results, such as the decrease in density upon increase in the angle of inclination near close packing, are obtained for the rough and partially rough particle models when the Burnett coefficients are included in the stress tensor, but realistic results are not obtained using the constitutive relations for smooth particles. This shows that the flow dynamics is sensitive to the numerical values of the viscometric coefficients, and provides an indication of the minimal model required to capture the flow dynamics.

The dynamic theory for the inelastically colliding particles

In the present work the dynamic system of the particles with model mechanism for the momentum and angular momentum exchange is studied. For hard rough spheres bounded by a wall the evolution equation for the delta functional microscopic density is constructed. In the case of chaotic initial state the kinetic equation for the one-particle distribution is obtained.

Tracking the flight of a spinning football in three dimensions

A football can be swerved considerably in flight by the application of spin. The paper considers the movement and spin of a football in three dimensions and seeks to quantify how much lateral deflection and spin can be achieved by a player in a realistic situation. The football is tracked in three dimensions by means of four markers on its surface, with tetrahedral symmetry, so that spin and deflection can be monitored under field conditions. The lift coefficients for the ball are compared with those for smooth and rough spheres.

An analytic form for the pair distribution function and rheology of a dilute suspension of rough spheres in plane strain flow

The effect of particle–particle contact on the stress of a suspension of small spheres in plane strain flow is investigated. We provide an analytic form for the particle pair distribution function in the case of no Brownian motion, and calculate the viscosity and normal stress difference based on this. We show that the viscosity is reduced by contact, and a normal stress difference induced, both at order boundary layer and demonstrating that diffusion reduces the magnitude of the contact effect but does not qualitatively alter it.

A Compact Model for Spherical Rough Contacts

A new model is developed that considers the effect of roughness on the elastic contact of spherical bodies. A general pressure distribution is proposed that encompasses the contact of rough spheres and yields the Hertzian theory for ideally smooth surfaces. A new parameter, nondimensional maximum contact pressure, is introduced and it is shown that this is the key parameter that controls the contact. The results of the present study are presented in the form of compact relationships. These relationships are compared against the experimental data collected by others and good agreement is observed.

Hydrodynamic interaction of rough spheres

The approach of two spheres along their line of centres is analyzed assuming that each sphere is covered by a porous layer. The slip of the fluid at the surface of the porous layers then permits the spheres to touch without a singularity in the associated force. In a corresponding analysis for flat surfaces, an analytical formula for the force is obtained. This force is also finite for vanishing gap width. It is shown how the properties of the porous layers can be related to the statistics of spatial distributions of surface asperities. Finally, it is emphasised that a finite force is associated with a finite asperity height.

The modelling of particle systems with real shapes.

The numerical modelling of particulate processes in environmental science increasingly requires an ability to represent the properties of individual natural particles. Considerable advances have been made in discontinuum modelling using spheres to represent particles. In this paper, we discuss recent developments that illustrate a way forward for tackling the complexity of realistically shaped bodies such as those exhibited by rock fragments. To address the validation of such approaches, we present a comparison of cube-packing experiments and their equivalent numerical simulation. Sensitivity to initial conditions, highlighted for non-spherical bodies, enters the discussion of problems with validation of numerical simulation. The algorithmic details behind these advances in modelling large systems of realistically shaped particles are summarized in our companion paper in this volume.

Experimental and Theoretical Modeling of the Effective Thermal Conductivity of Rough Steel Spheroid Packed Beds

The upper and lower bounds of the effective thermal conductivity of packed beds of rough spheres are evaluated using the theoretical approach of the elementary cell for two-phase systems. The solid mechanics and thermal problems are solved and the effects of roughness and packed bed structures are also examined. The numerical solution of the thermal conduction problem through the periodic regular arrangement of steel spheroids in air is determined using the Finite Element Method. The numerical results are compared with those obtained from an experimental apparatus designed and built for this purpose.

Bimodal distributions with the spiral-type Maxwellians for the Bryan–Pidduck model

The approximate explicit solution of the Boltzmann equation for the model of rough spheres is constructed in a form of bimodal distribution with stationary, inhomogeneous Maxwellians. It describes the interaction between two rotating flows of a gas. Sufficient conditions for the infinitesimality of the integral discrepancy are obtained.

Solid-solid contacts due to surface roughness and their effects on suspension behaviour.

Solid-solid contacts due to microscopic surface roughness in viscous fluids were examined by observing the translational and rotational behaviours of a suspended sphere falling past a lighter sphere or down an inclined surface. In both cases, a roll-slip behaviour was observed, with the gravitational forces balanced by not only hydrodynamic forces but also normal and tangential solid-solid contact forces. Moreover, the nominal separation between the surfaces due to microscopic surface roughness elements is not constant but instead varies due to multiple roughness scales. By inverting the system, so that the heavy sphere fell away from the lighter sphere or the plane, it was found that the average nominal separation increases with increasing angle of inclination of the plane or the surface of the lighter sphere from horizontal; the larger asperities lift the sphere up from the opposing surface and then gravity at large angles of inclination is too weak to pull the sphere back down to the opposing surface before another large asperity is encountered. The existence of microscopic surface roughness and solid-solid contacts is shown to modify the rheological properties of suspensions. For example, the presence of compressive, but not tensile, contact forces removes the reversibility of sphere-sphere interactions and breaks the symmetry of the particle trajectories. As a result, suspensions of rough spheres exhibit normal stress differences that are absent for smooth spheres. For the conditions studied, surface roughness reduces the effective viscosity of a suspension by limiting the lubrication resistance during near-contact motion, and it also modifies the suspension microstructure and hydrodynamic diffusivity.

Interactions between two rough spheres, water bridge and water vapour

Micromechanical considerations of the interactions between the granular skeleton of an unsaturated granular soil and the contained moisture involve the classical interaction model consisting of a water bridge in contact with two rigid smooth spheres of equal radius, at a separation determined by their actual surface roughness, and surrounded by a gas with a vapour pressure at equilibrium with the water. In this paper the water volume and the interparticle contact force are related to the water suction, the water–solid surface contact angle, and the size and roughness of the spheres in dimensionless terms. The dimensionless water suction is also related to the corresponding relative humidity of the water vapour. The calculated equilibrium relations are shown to possess non-uniqueness, which is interpreted in terms of soil mechanical properties while recognising that for low saturations in natural processes the humidity of the pore air is the independent parameter. The developed analytical solutions show the classical toroidal approximation to be reasonable. These solutions are applied in a companion paper to clarify the concept of an intergranular stress induced by both pore suction and related surface tension.

Interactions between two rough spheres, water bridge and water vapour

Micromechanical considerations of the interactions between the granular skeleton of an unsaturated granular soil and the contained moisture involve the classical interaction model consisting of a water bridge in contact with two rigid smooth spheres of equal radius, at a separation determined by their actual surface roughness, and surrounded by a gas with a vapour pressure at equilibrium with the water. In this paper the water volume and the interparticle contact force are related to the water suction, the water–solid surface contact angle, and the size and roughness of the spheres in dimensionless terms. The dimensionless water suction is also related to the corresponding relative humidity of the water vapour. The calculated equilibrium relations are shown to possess non-uniqueness, which is interpreted in terms of soil mechanical properties while recognising that for low saturations in natural processes the humidity of the pore air is the independent parameter. The developed analytical solutions show the classical toroidal approximation to be reasonable. These solutions are applied in a companion paper to clarify the concept of an intergranular stress induced by both pore suction and related surface tension.