Packing Of Monosized Spheres Research Articles

Benchmark polynomials for the porosity dependence of elastic moduli and conductivity of partially sintered ceramics

It is well known that no useful analytical models are available for predicting the porosity dependence of effective properties for partially sintered ceramics with an open (interconnected) pore space where concave pore surfaces are dominating. This does not mean, however, that that the problem of determining the effective properties of partially sintered ceramics does not have a unique solution for a given microstructure. It just means that this solution can be obtained only numerically. In this contribution we report the polynomial fits of our numerical results for the effective elastic moduli (Young’s modulus, shear modulus, bulk modulus) and conductivity (thermal or electrical) for model materials based on the random close packing of monosized spheres. These relations are recommended as benchmark relations for the porosity dependence of the effective properties of partially sintered ceramics.

Relationship between bed heterogeneity, chord length distribution, and longitudinal dispersion in particulate beds

We analyse a relationship between the bulk microstructure of randomly packed beds, which we quantify through chord length distribution (CLD) analysis of the interparticle void space, and the associated flow heterogeneity, as expressed by the longitudinal dispersion coefficient at a Péclet number of Pe = 10. A random collection of physically reconstructed packings is complemented with a systematic set of computer-generated packings of monosized spheres, for which the packing-generation algorithm has been carefully adjusted to realize a monotonic variation of the bed porosity and microstructural heterogeneity. The most relevant difference in the morphology between these computer-generated and the physically reconstructed packings are structural defects present in the real packings, such as particle oligomers and larger voids as well as contaminations and particle debris. These defects influence the pore space morphology and introduce additional structural heterogeneity. Hydrodynamic dispersion coefficients for all packings are derived by implementing the lattice-Boltzmann method to simulate fluid flow and a random-walk particle tracking technique to record the transport of passive, point-like tracers in the flow fields. We propose a morphological descriptor, σ/μ, based on statistical parameters of a CLD (standard deviation σ and mean chord length μ) that can be used to predict the dispersion coefficient in packed beds, independent from the underlying particle size distribution, packing-generation protocol, bed porosity, and the occurrence of structural defects.

X-ray tomography investigations of mono-sized sphere packing structures in cylindrical containers

The structure of mono-sized sphere packings (diameter d) in cylindrical containers (diameter D and height H) both with and without inner cylinders (diameter Di) has been investigated in detail by means of advanced X-ray computed tomography. The geometrical parameters were varied in a wide range; in all experiments 1d vertical vibration was applied. Five experiments were selected with characteristically differing local packing structures. The influence of container geometry, filling and vibration procedures on the formation of regular packings is discussed and a simple correlation is presented to assess whether structured packings occupy a significant fraction of the total packed volume.For a packing with moderate densification, the regular structures are restricted to small wall zones and a random packing exists in the largest part of the packing volume. By selecting appropriate vibration parameters, the zones with regular structures can increase considerably and can persist in the total packed volume. The increasing crystallisation causes an increase of the container packing fraction. For cylinders with H/D ≫1 and moderate D/d, regular structures develop preferentially in radial direction from a hexagonal layer at the concave wall. For H/D<1 and D/d≫1, hexagonal dense structures grow preferentially above the flat bottom plate and can occupy a great portion of the total volume. The role of granular convection on these crystallisation processes has been addressed. Previous statements that the thickness of wall zones is ≈(4–5)d are not generally valid for mono-sized sphere packings; the development of a comprehensive correlation is the task of a future work.Structural details of the packings close to concave, plane and convex walls are analysed via void fraction distributions, sphere centre positions, contact angle distributions, coordination numbers, radial distribution function and Voronoi tessellation. The combination of these methods provides a comprehensive understanding of structural details. Only a few characteristic results are presented; special topics will be the subject of forthcoming publications.

A mono-sized sphere packing algorithm in cylindrical container with two-phase gravity-based method

A mono-sized sphere packing algorithm in cylindrical container with two-phase gravity-based method

Structural and mechanical features of the order-disorder transition in experimental hard-sphere packings.

Here we present an experimental and numerical investigation on the grain-scale geometrical and mechanical properties of partially crystallized structures made of macroscopic frictional grains. Crystallization is inevitable in arrangements of monosized hard spheres with packing densities exceeding Bernal's limiting density ϕ(Bernal)≈0.64. We study packings of monosized hard spheres whose density spans over a wide range (0.59<ϕ<0.72). These experiments harness x-ray computed tomography, three-dimensional image analysis, and numerical simulations to access precisely the geometry and the 3D structure of internal forces within the sphere packings. We show that clear geometrical transitions coincide with modifications of the mechanical backbone of the packing both at the grain and global scale. Notably, two transitions are identified at ϕ(Bernal)≈0.64 and ϕ(c)≈0.68. These results provide insights on how geometrical and mechanical features at the grain scale conspire to yield partially crystallized structures that are mechanically stable.

Improved compaction of ZnO nano-powder triggered by the presence of acetate and its effect on sintering

The retention of nanocrystallinity in dense ceramic materials is still a challenge, even with the application of external pressure during sintering. The compaction behavior of high purity and acetate enriched zinc oxide (ZnO) nano-powders was investigated. It was found that acetate in combination with water plays a key role during the compaction into green bodies at moderate temperatures. Application of constant pressure resulted in a homogeneous green body with superior packing density (86% of theoretical value) at moderate temperature (85 °C) in the presence of water. In contrast, no improvement in density could be achieved if pure ZnO powder was used. This compaction behavior offers superior packing of the particles, resulting in a high relative density of the consolidated compact with negligible coarsening. Dissolution accompanying creep diffusion based matter transport is suggested to strongly support reorientation of ZnO particles towards densities beyond the theoretical limit for packing of ideal monosized spheres. Finally, the sintering trajectory reveals that grain growth is retarded compared to conventional processing up to 90% of theoretical density. Moreover, nearly no radial shrinkage was observed after sinter-forging for bodies performed with this advanced processing method.

Calculations for Density of Quark Core Consisting of Mono Flavored Closely Packed Quarks inside Neutron Star

The attempt has been taken to calculate the density of stars possessing quark matter core using sphere packing concept of crystallography. The quark matter has been taken as solid in nature as predicted in references 36 and 37, and due to immense gravitational pressure at the core of the star the densest packing of quarks as spheres has been assumed to calculate the packing fraction Φ, thus the density ρ of the matter. Three possible types of pickings—mono-sized sphere packing, binary sphere packing and ternary sphere packing, have been worked out using three possible types of quark matter. It has been concluded that no value about the ρ of quark matter can be calculated using binary and ternary packing conditions and for mono-sized packing condition different flavor quark matters of different values in the density have been calculated using results from the experiments done by HI, ZEUS, L3 and CDF Collaborations about the radius limit of quark. For example, for u quark matter ρ ranges from 4.0587 × 1048 - 7.40038 × 1048 MeV/c2 cm3 using results of L3 Collaboration, for s quark matter 15.91794 × 1048 - 17.6866 × 1048 MeV/c2 cm3, etc.

Impact of microstructure on the effective diffusivity in random packings of hard spheres

We present results of computer simulations of the effective diffusion coefficient in bulk random packings of hard monosized spheres with solid volume fraction between 0.54 (random-loose packing) and 0.634 (maximally random jammed). Six types of sphere packings were generated with different protocols and parameters resulting in a systematically varied degree of microstructural heterogeneity. The packing morphology is qualitatively characterized by statistical analyses of Voronoi cells obtained from spatial tessellation of the packing space. Diffusive transport of point-like tracers in the pore space of the packings was simulated with a random-walking particle-tracking technique. Our results indicate that the effective transport characteristics of the random sphere packings are not fully defined from the solid volume fraction but also depend on the packing microstructure. For the first time, we compared (i) the values of the effective diffusion coefficient Deff simulated in packings with different morphologies, and (ii) the corresponding values of Deff obtained from an approximate analytical formula involving the three-point microstructural parameter ζ2. This analysis reveals that this approximation involving ζ2 clearly reflects key morphological specificity of individual sphere packings and provides a sufficiently accurate estimate of the effective diffusion coefficient.

Flow-Induced Deformations Within Random Packed Beds of Spheres

Low Reynolds number flow-induced alterations of permeability of random packing of mono-sized spheres is studied. The number of spheres is several thousands and the porosities ranges between 0.4 and 0.6. The change of permeability is obtained for elastic deformations of the positions of the spheres using either of two methods. Each sphere is elastically attached to single points or the spheres that are connected via an elastic porous network. The system of spheres is divided into smaller volumes with Voronoi diagrams and the flow is derived by usage of a dual stream function. The local saturated flow fields are approximated as for close packed spheres and the overall flow pattern is obtained by minimising the dissipation rate of energy. The results show that the permeability for large random systems increases as a function of velocity and thus the deformation. The alteration is, however, much less than for two-dimensional cases like parallel cylinders. The relative increase in permeability becomes larger as the porosity increases from 0.4 to 0.6.

Constant gradient PFG sequence and automated cumulant analysis for quantifying dispersion in flow through porous media

This paper describes a new variant of established stimulated echo pulse sequences, and an analytical method for determining diffusion or dispersion coefficients for Gaussian or non-Gaussian displacement distributions. The unipolar displacement encoding PFGSTE sequence uses trapezoidal gradient pulses of equal amplitude g and equal ramp rates throughout while sampling positive and negative halves of q-space. Usefully, the equal gradient amplitudes and gradient ramp rates help to reduce the impact of experimental artefacts caused by residual amplifier transients, eddy currents, or ferromagnetic hysteresis in components of the NMR magnet. The pulse sequence was validated with measurements of diffusion in water and of dispersion in flow through a packing of spheres. The analytical method introduced here permits the robust determination of the variance of non-Gaussian, dispersive displacement distributions. The noise sensitivity of the analytical method is shown to be negligible, using a demonstration experiment with a non-Gaussian longitudinal displacement distribution, measured on flow through a packing of mono-sized spheres.

Mono-sized sphere packing algorithm development using optimized Monte Carlo technique

In this paper, fuel cell catalyst layer was developed using the optimized sphere packing algorithm. An optimization technique named adaptive random search technique (ARSET) was employed in this packing algorithm. The ARSET algorithm will generate the initial location of spheres and allow them to move in the random direction with the variable moving distance, randomly selected from the sampling range (α), based on the Lennard–Jones potential and Morse potential of the current and new configuration. The solid fraction values obtained from this developed algorithm are in the range of 0.610–0.624 while the actual processing time can significantly be reduced by 5.58–34% based on the number of spheres. The initial random number sampling range (α) was investigated and the appropriate α value is equal to 0.5.

The effect of different random number distributions on the porosity of spherical particles

A geometrical loose sphere packing model is initiated by randomly generating initial (x-y) coordinates of every sphere based on a specific distribution. In general, a normal distribution is applied in such models. However, other distributions, such as exponential, log-normal, and arbitrary distributions, may give better packing of particles, resulting in lower porosity values. In this study, the log-normal distribution provides the lowest porosity (40.72%) for mono-sized sphere packing. The same behavior is also shown in multi-sized sphere packing using a log-normal distribution.

DEM simulation of small particles clogging in the packing of large beads

The percolation of small particles through a periodic random loose packing of large beads is studied numerically with the Distinct Element Method. The representativity of periodic mono-sized sphere packing of varying system size was first studied by comparing their pore size distributions and tortuosities with those of a larger system, considered as an infinite medium. The results show that a periodic packing of size as low as 4-grain diameters gives a reasonable representation of the porous medium and allows reducing considerably the number of particles that has to be used in the simulations. The flow and clogging of small particles of varying concentrations and friction coefficients flowing through the former packing are then studied numerically. Results show that a steady state is rapidly reached where the mean velocity and mean vertical velocity of small particles are both constant. These mean velocities decrease with an increase in friction coefficient and in small particle concentration. The influence of the friction coefficient μ is much less marked for values of μ larger than or equal to 0.5. The distribution of small particles throughout the crossed packing becomes rapidly heterogeneous. Small particles concentrate in some pores where their velocity vanishes and where the density can reach values larger than the density of the random loose packing. The proportion of particles blocked in these pores varies linearly with concentration. Finally, the narrow throats of the porous medium responsible for blocking are identified and characterized for different values of the friction coefficient.

Experimental study of the packing of mono-sized spheres subjected to one-dimensional vibration

The packing of mono-sized spheres under one-dimensional (1D) vibration is studied experimentally. The effects of operational conditions, such as vibration amplitude A and vibration frequency ω, and feeding method on packing density have been analyzed. The results indicate that there exist optimum values for A and ω to achieve the maximum packing density. The effects of A and ω cannot be represented by a single parameter (i.e. vibration intensity Γ = Aω 2), but should be considered separately. The number of particles fed per batch affects the packing density significantly within a range of one to four layers per batch, but otherwise has no visible effect. Through the extrapolation on packing density using different sized containers, packing density can reach 0.636 in the total feeding method and 0.663 using the batch-wise feeding method. The values, however, are affected by material properties. The experimental results have therefore testified our previous numerical work on the transition from random loose packing to random close packing [An et al., Phys. Rev. Lett. 95, 205502 (2005)].

Near-wall porosity characteristics of fixed beds packed with wood chips

Packed beds of fuel wood chips are commonly found in thermal conversion processes such as combustion or gasification. Wood chips in particular are mostly used as fuel for small-scale domestic heating boilers but also for commercial-scale combustion units. The characterization of spatial voidage distribution inside the wood chip beds is of great importance for flow and reactor modelling. This study focuses on the radial porosity variations of cylindrical beds of three different types of commercially available wood chips including chips classified as G30 size class. The conventional technique of consolidating packed beds with a resin was chosen as the experimental procedure. The radial voidage distribution in different cylindrical beds is determined by image analysis of sections of the solidified packings. Additionally, a packing of monosized spheres was investigated in order to assess the selected procedure in comparison with widely available literature data for spheres. The results are discussed and summarized in a mathematical expression correlating the radial voidage distribution depending on average wood chip size, packing core porosity and dimensionless distance from the tube wall.

Geometrical structure of disordered sphere packings

The three-dimensional structure of large packings of monosized spheres with volume fractions ranging between 0.58 and 0.64 has been studied with x-ray computed tomography. We search for signatures of organization, classifying local arrangements and exploring the effects of local geometrical constrains on the global packing. This study is the largest and the most accurate empirical analysis of disordered packings at the grain-scale to date, mapping over 380,000 sphere coordinates with precision within 0.1% of the sphere diameters. We discuss topological and geometrical methods to characterize and classify these systems emphasizing the implications that local geometry can have on the mechanisms of formation of these amorphous structures. Some of the main results are (1) the observation that the average number of contacts increases with the volume fraction; (2) the discovery that these systems have a very compact contact network; (3) the finding that disordered packing can be locally more efficient than crystalline packings; (4) the observation that the peaks of the radial distribution function follow power law divergences; (5) the discovery that geometrical frustration plays no role in the formation of such amorphous packings.

Numerical study of the packing of wet coarse uniform spheres

AbstractThe discrete‐element method was extended to investigate the effect of liquid addition on the packing of monosized spheres. By explicitly taking into account the interaction forces between particles, such as contact forces and capillary force, the packing of wet spheres can be simulated by the method. Simulations are performed under the so‐called poured packing, where a different amount of water is added to spheres of different sizes. Comparison with data in the literature shows that the simulations yield a satisfactory agreement in the relationship between porosity and moisture content, macroscopically verifying the proposed model. The packing structure is then analyzed in terms of microscopic properties such as the radial distribution function and coordination number and are shown to vary with liquid content for a given particle size. The relationship between porosity and capillary force is finally quantified, which confirms those derived based on macroscopic assumptions.

Quantification of the Relationship between Porosity and Interparticle Forces for the Packing of Wet Uniform Spheres

This paper presents an attempt to quantify the relationship between porosity and interparticle forces. In particular, it concerns the packing of wet mono-sized spheres where capillary force is dominant. The interrelationships between capillary force, porosity, and liquid content are first analyzed based on well-established theories and experimental observations. The resultant data are then used to correlate porosity with the force ratio defined as the average capillary force to the gravity of a particle. Application of this correlation is finally demonstrated by examples of porosity prediction.

Dynamic simulation of the centripetal packing of mono-sized spheres

This paper presents a study of the centripetal packing of mono-sized spherical particles simulated by means of the granular dynamic or discrete element method. A packing is formed by imposing an assumed centripetal force on particles randomly generated in a spherical space. Different from the conventional simulation techniques, dynamic information of individual particles including transient forces and trajectory is traced in the present simulation. Structural properties, such as packing density, radial distribution function, coordination number distribution and homogeneity, are analyzed, with particular reference to the effects of the magnitude of the centripetal force and the number of particles. Comparison with the literature results suggests that such a dynamic model can satisfactorily simulate the dynamics of forming a packing and produce more realistic structural information. In particular, it is confirmed that a centripetal packing is not homogeneous in structure, becoming looser as its size or the number of particles increases. The packing has a limit packing density of 0.637–0.645, an overall mean coordination number of around 6.0 and a radial distribution function of clear split second peak. The centripetal force affects the rate of densification and the mean coordination number but not packing density and radial distribution function.

Effect of liquid addition on the packing of mono-sized coarse spheres

This paper reports an experimental study of the packing of spherical particles with special reference to the effect of liquid addition. It shows that the properties of both particle and liquid affect the packing behaviour significantly. Under given packing conditions, dry based porosity increases to a maximum and then keeps constant with the increase of liquid content, giving two regions: (a) a wetting region in which the porosity increases with liquid content because of an increased effect of liquid-related force; and (b) a filling region in which liquid increases bulk density by filling the voids among particles. Particle size and surface tension are the main factors in the quantification of this porosity-liquid content relation. Empirical equations have been formulated in terms of dimensionless groups for the purpose of prediction.