Packing properties of binary mixtures in disordered sphere systems

Mixtures of binary spheres are numerically simulated using a relaxation algorithm to investigate the effects of volume fraction and size ratio, A complete profile of the packing properties of binary spheres is given. The density curve with respect to the volume fraction has a triangular shape with a peak at 70% large spheres. The density of the mixture increases with the size ratio, but the growth becomes slow in the case of a large size disparity, The volume fraction and size ratio effects are reflected in the height and movement, respectively, of specific peaks in the radial distribution functions. The structure of the mixture is further analyzed in terms of contact types, and the mean coordination number is demonstrated to be primarily affected by ＂large-small＂ contacts. A novel method for estimating the average relative excluded volume for binary spheres by weighting the percentages of contact types is proposed and extended to polydisperse packings of certain size distributions. The method can be applied to explain the density trends of polydisperse mixtures in disordered sphere systems,     

Rheology of concentrated suspensions containing mixtures of spheres and fibres

Optimising flow properties of concentrated suspensions is an important issue common for many industries. The rheology of concentrated suspensions has therefore been studied intensively both experimentally and theoretically. Most studies have focused on monodisperse and polydisperse suspensions of either spheres or fibres. In practice, most suspensions contain particles that are polydisperse both in size and shape. A mixing rule for such systems is expected to be a powerful tool for engineers and product designers. Therefore in this work, suspensions of spheres, fibres and mixtures thereof were characterised using rotational shear rheometry and in-line image analyses. Thereby, total solids volume concentration and fibre fraction was varied. Results from transient and steady-state shear rheometry are discussed with respect to concentration, fibre fraction, and shear induced microstructure. Experimentally obtained viscosity data were accurately fitted using the model proposed by Farris (T Soc Rheol 12:281, 1968) for mixtures of monodisperse non-interacting spheres of different sizes.Originally presented at the Annual European Rheology Conference 2003, AERC 2003     

Representative elementary volume analysis of polydisperse granular packings using discrete element method

The representative elementary volume (REV) for three-dimensional polydisperse granular packings was determined using discrete element method simulations. Granular mixtures of various sizes and particle size distributions were poured into a cuboid chamber and subjected to uniaxial compression. Findings showed that the minimum REV for porosity was larger compared with the REV for parameters such as coordination number, effective elastic modulus, and pressure ratio. The minimum REV for porosity and other parameters was found to equal 15, 10, and 5 times the average grain diameter, respectively. A study of the influence of sample size on energy dissipation in random packing of spheres has also confirmed that the REV size is about 15 times the average grain diameter. The heterogeneity of systems was found to have no effect on the REV for the parameters of interest for the narrow range of coefficient of uniformity analyzed in this paper. As the REV approach is commonly applied in both experimental and numerical studies, determining minimum REV size for polydisperse granular packings remains a crucial issue.     

Constitutive relations for the isotropic deformation of frictionless packings of polydisperse spheres

The isotropic compression of polydisperse packings of frictionless spheres is modeled with the Discrete Element Method (DEM). The evolution of coordination number, fraction of rattlers, isotropic fabric, and pressure (isotropic stress) is reported as function of volume fraction for different system parameters. The power law relationship, with power $\approx 1/2$, between coordination number and volume fraction is confirmed in the jammed state for a broad range of volume fractions and for different (moderate) polydispersities. The polydispersity in the packing causes a shift of the critical volume fraction, i.e., more heterogeneous packings jam at higher volume fractions. Close to jamming, the coordination number and the jamming volume fraction itself depend on both history and rate. At larger densities, neither the deformation history nor the loading rate have a significant effect on the evolution of the coordination number.Concerning the fabric tensor, comparing our DEM results to theoretical predictions, good agreement for different polydispersities is observed. An analytical expression for the pressure as function of isotropic (volumetric) strain is proposed for polydisperse packings, based on the assumption of uniform deformation. We note that, besides the implicit proportionality to contact number density (or fabric), no single power-law is evidenced in the relation between pressure and isotropic strain. However, starting from zero pressure at the jamming point, a linear term with a quadratic correction describes the stress evolution rather well for a broad range of densities and for various polydispersities. Finally, an incremental evolution equation is proposed for both fabric and stress, as function of isotropic strain, and involving the coordination number and the fraction of rattlers, as starting point for further studies involving anisotropic deformations.     

A micromechanical model for effective conductivity in granular electrode structures

Optimization of composition and microstructure is important to enhance performance of solid oxide fuel cells （SOFC） and lithium-ion batteries （LIB）. For this, the porous electrode structures of both SOFC and LIB are modeled as a binary mixture of electronic and ionic conducting particles to estimate effective transport properties. Particle packings of 10000 spherical, binary sized and randomly positioned particles are created numerically and densified considering the different manufacturing processes in SOFC and LIB： the sintering of SOFC electrodes is approximated geometrically, whereas the calendering process and volume change due to intercalation in LIB are modeled physically by a discrete el- ement approach. A combination of a tracking algorithm and a resistor network approach is developed to predict the con- nectivity and effective conductivity for the various densified structures. For SOFC, a systematic study of the influence of morphology on connectivity and conductivity is performed on a large number of assemblies with different compositions and particle size ratios between 1 and 10. In comparison to percolation theory, an enlarged percolation area is found, es- pecially for large size ratios. It is shown that in contrast to former studies the percolation threshold correlates to varying coordination numbers. The effective conductivity shows not only an increase with volume fraction as expected but also with size ratio. For LIB, a general increase of conductivity during the intercalation process was observed in correlation with increasing contact forces. The positive influence of cal- endering on the percolation threshold and the effective conductivity of carbon black is shown. The anisotropy caused by the calendering process does not influence the carbon black phase.     

Effect of particle polydispersity on micromechanical properties and energy dissipation in granular mixtures

A series of numerical tests was conducted to study the micromechanical properties and energy dissipation in polydisperse assemblies of spherical particles subjected to uniaxial compression. In general, distributed particle size assemblies with standard deviations ranging from 0% to 80% of the particle mean diameter were examined. The microscale analyses included the trace of the fabric tensor, magnitude and orien- tation of the contact forces, trace of stress, number of contacts and degree of mobilization of friction in contacts between particles. In polydisperse samples, the average coordination numbers were lower than in monodisperse assemblies, and the mobilization of friction was higher than in monodisperse assemblies due to the non-uniform spatial rearrangement of spheres in the samples and the smaller displacements of the particles. The effect of particle size heterogeneity on both the energy density and energy dissipation in systems was also investigated.     

Effect of particle polydispersity on micromechanical properties and energy dissipation in granular mixtures

A series of numerical tests was conducted to study the micromechanical properties and energy dissipation in polydisperse assemblies of spherical particles subjected to uniaxial compression. In general, distributed particle size assemblies with standard deviations ranging from 0% to 80% of the particle mean diameter were examined. The microscale analyses included the trace of the fabric tensor, magnitude and orientation of the contact forces, trace of stress, number of contacts and degree of mobilization of friction in contacts between particles. In polydisperse samples, the average coordination numbers were lower than in monodisperse assemblies, and the mobilization of friction was higher than in monodisperse assemblies due to the non-uniform spatial rearrangement of spheres in the samples and the smaller displacements of the particles. The effect of particle size heterogeneity on both the energy density and energy dissipation in systems was also investigated.     

Particle size distribution in CPFD modeling of gas–solid flows in a CFB riser

A computational particle fluid dynamics (CPFD) numerical method to model gas–solid flows in a circulating fluidized bed (CFB) riser was used to assess the effects of particle size distribution (PSD) on solids distribution and flow. We investigated a binary PSD and a polydisperse PSD case. Our simulations were compared with measured solids concentrations and velocity profiles from experiments, as well as with a published Eulerian-Eulerian simulation. Overall flow patterns were similar for both simulation cases, as confirmed by experimental measurements. However, our fine-mesh CPFD simulations failed to predict a dense bottom region in the riser, as seen in other numerical studies. Above this bottom region, distributions of particle volume fraction and particle vertical velocity were consistent with our experiments, and the simulated average particle diameter decreased as a power function with riser height. Interactions between particles and walls also were successfully modeled, with accurate predictions for the lateral profiles of particle vertical velocity. It was easy to implement PSD into the CPFD numerical model, and it required fewer computational resources compared with other models, especially when particles with a polydisperse PSD were present in the heterogeneous flow.     

Spectrum of density fluctuations in a particlefluid system—II. Polydisperse spheres

This paper presents an extension of the analysis shown in Part I to a polydisperse particle-fluid system. The density autocorrelation is shown to be a function of two quantities, a generalized Overlap function for which an analytical expression is derived, and the radial distribution function (RDF). In Fourier transform space, the density spectrum again appears to be a strong function of the mean particle size, and secondarily the mean particle separation distance. One unusual result is previously observed oscillations in the density spectrum of a monodisperse system of particles are severely dampened or even eliminated in the polydisperse case, depending on the width of the particle size distribution. Apparently contributions from different particle correlations interfere with each other, thereby reducing the coherent oscillations seen in the monodisperse particle-fluid system. Furthermore at large wavenumbers, the spectrum decays with a −2 power-law, independent of the shape of the particle size distribution. This behavior can be traced to the Overlap function which controls the behavior of the spectrum beyond the first peak. Remarkably the −2 power-law spectrum is determined by the shape of the particles (i.e. spheres) rather than their spatial distribution (RDF).

The effect of an asymptotically large pressure gradient on the correlation of several important higher-order moments is revisited for the polydisperse system. The relatively simple relationships developed for the monodisperse system are lost in the polydisperse case because particles of different sizes will be influenced differently by an applied pressure gradient. The result is moments that are of different order in velocity can no longer be related to each other (as they were in the monodisperse system), even in this idealized flow. A more comprehensive understanding of this phenomenon can only be achieved through direct numerical simulation or experiment.     

Particle size distribution in CPFD modeling of gas-solid flows in a CFB riser

A computational particle fluid dynamics(CPFD) numerical method to model gas-solid flows in a circulating fluidized bed(CFB) riser was used to assess the effects of particle size distribution(PSD) on solids distribution and flow.We investigated a binary PSD and a polydisperse PSD case.Our simulations were compared with measured solids concentrations and velocity profiles from experiments,as well as with a published Eulerian-Eulerian simulation.Overall flow patterns were similar for both simulation cases,as confirmed by experimental measurements.However,our fine-mesh CPFD simulations failed to predict a dense bottom region in the riser,as seen in other numerical studies.Above this bottom region,distributions of particle volume fraction and particle vertical velocity were consistent with our experiments,and the simulated average particle diameter decreased as a power function with riser height.Interactions between particles and walls also were successfully modeled,with accurate predictions for the lateral profiles of particle vertical velocity.It was easy to implement PSD into the CPFD numerical model,and it required fewer computational resources compared with other models,especially when particles with a polydisperse PSD were present in the heterogeneous flow.     

Study of fluid cell coarsening for CFD-DEM simulations of polydisperse gas–solid flows

Particle polydispersity is ubiquitous in industrial fluidized beds, which possesses a significant impact on hydrodynamics of gas–solid flow. Computational fluid dynamics-discrete element method (CFD-DEM) is promising to adequately simulate gas–solid flows with continuous particle size distribution (PSD) while it still suffers from high computational cost. Corresponding coarsening models are thereby desired. This work extends the coarse-grid model to polydisperse systems. Well-resolved simulations with different PSDs are processed through a filtering procedure to modify the gas–particle drag force in coarse-grid simulations. We reveal that the drag correction of individual particle exhibits a dependence on filtered solid volume fraction and filtered slip velocity for both monodisperse and polydisperse systems. Subsequently, the effect of particle size and surrounding PSD is quantified by the ratio of particle size to Sauter mean diameter. Drag correction models for systems with monodisperse and continuous PSD are developed. A priori analysis demonstrates that the developed models exhibit reliable prediction accuracy.     

Modeling of the random packing of a loose layer of polydisperse spherical particles

A method for calculating the loose packing structure of polydisperse spherical particles with a predetermined size distribution function is proposed. The coordinates of the particle centers in the loose layer are determined as the result of random fall of single spheres on a substrate under the action of gravity, assuming the inelastic collision of the spheres and considering the force of their adhesive interaction, and also assuming that the motion of one sphere on the surface of the other is pure slip. Numerical simulation is used to obtain the pattern of arrangement of polydisperse spherical particles in the loose powder layer, whose porosity depends on the particle size distribution function. The results are compared with experimental data.     

A simple semiempirical model for the effective viscosity of multicomponent suspensions

We propose a methodology to approximate the viscosity of multicomponent suspensions. The procedure consists of successive applications of expressions for the viscosity of binary mixtures, originally written as the product of monomodal stiffening functions. First, the viscosity of a binary mixture made of the two smallest components is calculated. This allows to extract a volume fraction that will be used, together with the volume fraction of the third component, to feed the next iteration of the procedure to calculate the viscosity of a trimodal mixture and so on. The application of this approach to arbitrary mixtures requires the detailed knowledge of the geometry of the system in the form of size ratios and compositions. When this information is unknown, an approximation of the model can still be used as a fitting tool. With that purpose, the final expression for the viscosity is written in terms of an effective volume fraction that is further approximated by the use of a (1,2) Padé approximant. This approximation allows to incorporate the crowding effects due to different species in a volume fraction-dependent crowding factor that can be used as a fitting parameter to match experimental or simulation data. We have applied the model to mixtures of particles with different sizes and tested its accuracy comparing with experimental results obtaining very good agreement.     

Pure liquids described as concentrated cluster dispersions

The Fulcher-Tammann-Hesse-Vogel equation, ln = A + B/(T – T ₀ ), is shown to be equivalent to the general viscosity-composition relationship, ln _r =k f /(1 – f ), for binary mixtures. The Cailletet-Mathias law of the Rectilinear Diameter is rearranged to represent a density mixture formula for two components. Temperature-independent viscosities and densities can then be calculated for dense, solid cluster fractions, dispersed in a low-density, low-viscosity non-clustered continuous phase. The cluster fraction decreases with temperature. The value ofT ₀ is shown to be related to the liquid- or solid-like behavior of the clusters. For liquids with a vapor pressure < 1 mm Hg at the melting point, the calculated cluster volume fraction suggests close packing of clusters, ranging in shape from monodisperse spheres to polydisperse non-spherical particles. Examples are given for molecular liquids, molten metals, and molten salts. The size of the clusters is estimated from the heat of evaporation.     

A scaling analysis of added-mass and history forces and their coupling in dispersed multiphase flows

Accurate momentum coupling model is vital to simulation of dispersed multiphase flows. The overall force exerted on a particle is divided into four physically meaningful contributions, i.e., quasi-steady, stress-gradient, added-mass, and viscous-unsteady (history) forces. Time scale analysis on the turbulent multiphase flow and the viscous-unsteady kernel shows that the integral representation of the viscous-unsteady force is required except for a narrow range of particle size around the Kolmogorov length scale when particle-to-fluid density ratio is large. Conventionally, the particle-to-fluid density ratio is used to evaluate the relative importance of the unsteady forces (stress-gradient, added-mass, and history forces) in the momentum coupling. However, it is shown from our analysis that when particle-to-fluid density ratio is large, the importance of the unsteady forces depends on the particle-to-fluid length scale ratio and not on the density ratio. Provided the particle size is comparable to the smallest fluid length scale (i.e., Kolmogorov length scale for turbulence or shock thickness for shock-particle interaction) or larger, unsteady forces are important in evaluating the particle motion. Furthermore, the particle mass loading is often used to estimate the importance of the back effect of particles on the fluid. An improved estimate of backward coupling for each force contribution is established through a scaling argument. The back effects of stress-gradient and added-mass forces depend on particle volume fraction. For large particle-to-fluid density ratio, the importance of the quasi-steady force in backward coupling depends on the particle mass fraction; while that of the viscous-unsteady force is related to both particle mass and volume fractions.     

Relating cellular structure of open solid food foams to their Young’s modulus: Finite element calculation

This paper investigates the role of structure on Young’s modulus of open cell materials of relative densities between 0.1 and 0.3. The cellular solid is obtained by generating mixture size of spherical voids using the Random Sequential Addition – RSA algorithm. The relative density of the material is controlled by increasing void number and overlap. Structural effects consider mainly a Gaussian distribution of spherical void size of varying width, distribution centre and void overlap distance. Finite element method is used to calculate effective Young’s modulus using a regular meshing scheme of 3D typical cellular solids and Conjugate Gradient solver. It is found that sphere overlap has the largest effect compared to sphere distribution width for a given density. A large scatter in the wall thickness distribution is predicted when overlapping is increased or when the width of sphere size distribution is decreased. Increased rigidity is found to be correlated to particular arrangement of mixture size spheres which is pointed out using the Pair Correlation Function. Experimental evidence of the role of void overlapping is treated in the case of bread crumbs structures determined using X-ray tomography. The scatter of effective Young’s modulus for a given relative density is sensitive to void overlapping.     

Multiscale modeling of heterogeneous propellants from particle packing to grain failure using a surface-based cohesive approach

In the present work,a computational framework is established for multiscale modeling and analysis ofsolid propellants.A packing algorithm,considering the ammonium perchlorate(AP) and aluminum(Al) particles asspheres or discs is developed to match the size distributionand volume fraction of solid propellants.A homogenizationtheory is employed to compute the mean stress and strainof a representative volume element(RVE).Using the meanresults,a suitable size of RVE is decided.Without considering the interfaces between particles and matrix,several numerical simulations of the relaxation of propellants are performed.The relaxation effect and the nonlinear mechanicalbehavior of propellants which are dependent on the appliedloads are discussed.A new technology named surface-basedcohesive behavior is proposed to describe the phenomenonof particle dewetting consisting of two ingredients:a damageinitiation criterion and a damage evolution law.Several examples considering contact damage behavior are computedand also nonlinear behavior caused by damaged interfaces isdiscussed in this paper.Furthermore the effects of the critical contact stress,initial contact stiffness and contact failuredistance on the damaged interface model have been studied.     

DEM simulation of polydisperse systems of particles in a fluidized bed

Numerical simulations based on three-dimensional discrete element model (DEM) are conducted for mono-disperse, binary and ternary systems of particles in a fluidized bed. Fluid drag force acting on each particle depending on its size and relative velocity is assigned. The drag coefficient corresponding to Ergun’s correlation is applied to the system of fluidized bed with particle size ratios of 1:1 for the mono-disperse system, 1:1.2, 1:1.4 and 1:2 for the binary system and 1:1.33:2 for the ternary system b...     

Multi-fluid modeling of density segregation in a dense binary fluidized bed

This paper presents simulation results of the density segregation in a dense binary gas fluidized bed using a multi-fluid model from Chao et al. (2011). The segregation behavior of two types of particles with approximately same particle diameters and different particle densities was studied and validated using the experimental data from Formisani et al. (2008). Some detailed information regarding the gas, particle velocity profiles, the distributions of the particle volume fractions and the flotsam-to-total particle volume fraction ratios is presented. The simulation results show that the simulated axial average flotsam-to-total particle volume fraction ratio distribution agrees reasonably with the experimental data of Formisani et al. (2008). The binary particle velocities are closely coupled though the segregation exists. The segregation behavior and the particle velocity profiles are superficial gas velocity dependent. The number and distribution of particle velocity vortices change dramatically with superficial gas velocity: at a comparatively low superficial gas velocity, the particles mainly segregate axially, and at a comparatively high superficial gas velocity, the particles segregate both axially and radially.     

Ethylene–air detonation in water spray

Detonation experiments are conducted in a 52 \(\hbox {mm}\) square channel with an ethylene–air gaseous mixture with dispersed liquid water droplets. The tests were conducted with a fuel–air equivalence ratio ranging from 0.9 to 1.1 at atmospheric pressure. An ultrasonic atomizer generates a polydisperse liquid water spray with droplet diameters of 8.5–12 \(\upmu \hbox {m}\), yielding an effective density of 100–120 \(\hbox {g}/\hbox {m}^{3}\). Pressure signals from seven transducers and cellular structure are recorded for each test. The detonation structure in the two-phase mixture exhibits a gaseous-like behaviour. The pressure profile in the expansion fan is not affected by the addition of water. A small detonation velocity deficit of up to 5 % was measured. However, the investigation highlights a dramatic increase in the cell size (\(\lambda \)) associated with the increase in the liquid water mass fraction in the two-phase mixture. The detonation structure evolves from a multi-cell to a half-cell mode. The analysis of the decay of the post-shock pressure fluctuations reveals that the ratio of the hydrodynamic thickness over the cell size (\(x_{{\mathrm {HT}}}/{\lambda }\)) remains quite constant, between 5 and 7. A slight decrease of this ratio is observed as the liquid water mass fraction is increased, or the ethylene–air mixture is made leaner.