Packing fraction of trimodal spheres with small size ratio: An analytical expression

2013 ◽  
Vol 88 (3) ◽  
Author(s):  
H. J. H. Brouwers
Keyword(s):  
Packing Fraction ◽  
Size Ratio ◽  
Analytical Expression

scholarly journals How size ratio and segregation affect the packing of binary granular mixtures

Soft Matter ◽  
2020 ◽  
Vol 16 (39) ◽  
pp. 9094-9100
Author(s):  
Salvatore Pillitteri ◽  
Eric Opsomer ◽  
Geoffroy Lumay ◽  
Nicolas Vandewalle
Keyword(s):  
Packing Fraction ◽  
Size Ratio ◽  
Granular Mixtures ◽  
Granular Segregation ◽  
Small Grains

For reaching high packing fractions, grains of various sizes are often mixed together allowing the small grains to fill the voids created by the large ones. However, in most cases, granular segregation occurs leading to lower packing fractions. We show how a layered packing or a gradient segregation affects the global packing fraction.

Optimum formulation derivation for the ultimate packing fraction using monodispersed particle sizes when optimizing suspension viscosities

2019 ◽  
Vol 48 (1) ◽  
pp. 45-56
Author(s):  
Richard D. Sudduth
Keyword(s):  
Particle Size ◽  
Average Particle Size ◽  
Packing Fraction ◽  
Size Ratio ◽  
Average Particle ◽  
Particle Sizes ◽  
Monodisperse Particle ◽  
Content Type ◽  
Particle Size Ratio ◽  
Minimum Number

Purpose The importance of maximizing the particle packing fraction in a suspension by maximizing average particle size ratio of D5/D1 has been adequately shown to be important as previously reported in the literature. This study aims to extend that analysis to include the best formulation approach to maximize the packing fraction with a minimum number of monodisperse particle sizes. Design/methodology/approach An existing model previously developed by this author was modified theoretically to optimize the ratio used between consecutive monodisperse particle sizes. This process was found to apply to a broad range of particle configurations and applications. In addition, five different approaches for maximizing average particle size ratio D̅5/D̅1 were addressed for blending several different particle size distributions. Maximizing average particle size ratio D̅5/D̅1 has been found to result in an optimization of the packing fraction. Several new concepts were also introduced in the process of maximizing the packing fraction for these different approaches. Findings The critical part of the analysis to maximize the packing fraction with a minimum number of particles was the theoretical optimization of the ratio used between consecutive monodisperse particle sizes. This analysis was also found to be effectively independent of the maximum starting particle size. This study also clarified the recent incorrect claim in the literature that Furnas in 1931 was the first to generate the maximum theoretical packing fraction possible for n different particles that was actually originally developed in conjunction with the Sudduth generalized viscosity equation. In addition, the Furnas generated equation was also shown to give significantly different results from the Sudduth generated equation. Research limitations/implications Experimental data involving monodisperse particles of different blends with a minimum number of particle sizes that are truly monodisperse are often extremely difficult to obtain. However, the theoretical general concepts can still be applicable. Practical implications The expanded model presented in this article provides practical guidelines for blending pigments using a minimum number of monodisperse particle sizes that can yield much higher ratios of the particle size averages D̅5/D̅1 and thus potentially achieve significantly improved properties such as viscosity. Originality/value The model presented in this article provides the first apparent guidelines to control the blending of pigments in coatings by the optimization of the ratio used between consecutive monodisperse particle sizes. This analysis was also found to be effectively independent of the maximum starting particle size.

Influence of particle size on vertical plate penetration into dense cohesionless granular materials (large-scale DEM simulation using real particle size)

2019 ◽  
Vol 21 (4) ◽  
Author(s):  
Shinichiro Miyai ◽  
Murino Kobayakawa ◽  
Takuya Tsuji ◽  
Toshitsugu Tanaka
Keyword(s):  
Particle Size ◽  
Granular Materials ◽  
Vertical Plate ◽  
Plate Thickness ◽  
Shear Bands ◽  
Penetration Resistance ◽  
Packing Fraction ◽  
Size Ratio ◽  
Two Dimensional ◽  
The Mean

Abstract The influence of the particle size on the vertical plate penetration into dense cohesionless granular materials was numerically investigated. Simulations were performed in quasi-two-dimensional conditions by changing the mean particle diameters d50 but maintaining the plate thickness B from B/d50 = 63–2.6. The initial bulk packing fraction was kept high, irrespective of the particle size. In the smallest particle size case (B/d50 = 63), the size ratio reached almost the same level as that in the laboratory experiments using natural sand particles. The results demonstrated that the mean penetration resistance force acting on the plate tip surface increases with a decrease of B/d50, while the tangential force acting on the side surfaces does not change with B/d50. Tip resistances increase linearly with the penetration depth, while the tangential resistances increase with the square of the depth regardless of B/d50. The behavior of the resistance fluctuations changes qualitatively between B/d50 = 31 and 21. For all cases, we confirmed the formation of a wedge-shaped flow with a high forward velocity in front of the plate tip. The wedge flow width was larger than the plate thickness by almost a mean particle diameter, and was responsible for the increase in the mean resistance depending on the particle size. For the large B/d50 cases only, the resistance exhibited quasi-periodic fluctuations, which was attributable to the intermittent nucleation and disappearance of the shear bands. Moreover, we investigated the dependence of B/d50 on the band evolutions by analyzing the band thickness. Graphic abstract The influence of the particle size on the vertical plate penetration into dense cohesionless granular materials was numerically investigated using DEM. Simulations were performed in quasi-two-dimensional conditions by changing the median particle diameters d50 but maintaining the plate thickness B. The initial bulk packing fraction was kept high, irrespective of the particle size. Upper and lower figures show the result of small (B/d50 = 63) and large particle size case (B/d50 = 21), respectively. In the small particle size case (B/d50 = 63), the size ratio reached almost the same level as that in the laboratory and the dynamics of 35.5 million particles was considered. Right and left figures illustrate instantaneous shear strain rate and local packing fraction distributions, respectively. Large qualitative change in the granular behaviors as well as penetration resistance was observed between B/d50 = 31 and 21. The intermittent nucleation and disappearance of the shear bands were clearly observed only for large B/d50 cases.

On the influence of specimen thickness in TEM images of super-conducting vortices

1996 ◽  
Vol 54 ◽  
pp. 724-725
Author(s):  
J. Bonevich ◽  
D. Capacci ◽  
G. Pozzi ◽  
K. Harada ◽  
H. Kasai ◽  
...  
Keyword(s):  
Flux Line ◽  
Analytical Form ◽  
Realistic Model ◽  
Electron Holography ◽  
Specimen Thickness ◽  
Flux Tubes ◽  
Analytical Expression ◽  
Flux Lines ◽  
Normal Core ◽  
Two Parameters

The successful observation of superconducting flux lines (fluxons) in thin specimens both in conventional and high Tc superconductors by means of Lorentz and electron holography methods has presented several problems concerning the interpretation of the experimental results. The first approach has been to model the fluxon as a bundle of flux tubes perpendicular to the specimen surface (for which the electron optical phase shift has been found in analytical form) with a magnetic flux distribution given by the London model, which corresponds to a flux line having an infinitely small normal core. In addition to being described by an analytical expression, this model has the advantage that a single parameter, the London penetration depth, completely characterizes the superconducting fluxon. The obtained results have shown that the most relevant features of the experimental data are well interpreted by this model. However, Clem has proposed another more realistic model for the fluxon core that removes the unphysical limitation of the infinitely small normal core and has the advantage of being described by an analytical expression depending on two parameters (the coherence length and the London depth).

Contrast From Point Defects in The Infinitesimal Approximation

1980 ◽  
Vol 38 ◽  
pp. 350-351
Author(s):  
L. J. Sykes ◽  
J. J. Hren
Keyword(s):  
Electron Microscope ◽  
Point Defects ◽  
Broad Class ◽  
Stacking Fault ◽  
Crystalline Solids ◽  
Dislocation Loops ◽  
Analytical Expression ◽  
Stacking Fault Tetrahedra

In electron microscope studies of crystalline solids there is a broad class of very small objects which are imaged primarily by strain contrast. Typical examples include: dislocation loops, precipitates, stacking fault tetrahedra and voids. Such objects are very difficult to identify and measure because of the sensitivity of their image to a host of variables and a similarity in their images. A number of attempts have been made to publish contrast rules to help the microscopist sort out certain subclasses of such defects. For example, Ashby and Brown (1963) described semi-quantitative rules to understand small precipitates. Eyre et al. (1979) published a catalog of images for BCC dislocation loops. Katerbau (1976) described an analytical expression to help understand contrast from small defects. There are other publications as well.

FACTOR OF QUALITY FOR PASSIVE NON-TUNABLE BAND-STOP FILTERS

2019 ◽  
pp. 71-74
Author(s):  
N. I. Unru ◽  
E. I. Ashcherbagin
Keyword(s):  
Quality Criterion ◽  
Physical Structure ◽  
Electrical Characteristics ◽  
Central Frequency ◽  
Analytical Expression ◽  
Notch Band ◽  
Attenuation Level ◽  
Calculation Results ◽  
Design And Manufacture

The notion of a quality criterion for non-tunable band-stop filters is introduced, and on the basis of it a comparison of filters with different designs is performed. The quality criterion takes into account the electrical characteristics of the filter and its dimensions, including the volume, the central frequency of the notch band, the level of total losses in the passbands, the width of the notch band by the level of total losses, the width of the notch band by attenuation level. Thus, it allows you to compare the quality of design and manufacture of passive notch filters of various types. The necessary analytical expression is presented, and for a number of variants of filter execution, the corresponding calculation results are given. The stated materials allow us to estimate and optimize the system of interrelated parameters of filters of an arbitrary physical structure.

Simulation on Extrusion Comminution Model of Roller Press

2010 ◽  
Vol 156-157 ◽  
pp. 1702-1707
Author(s):  
Xiang Wen Cheng ◽  
Jinchao Liu ◽  
Qi Zhi Ding ◽  
Li Ming Song ◽  
Zhan Lin Wang
Keyword(s):  
Experimental Data ◽  
Particle Size ◽  
Field Experiments ◽  
Size Ratio ◽  
Optimum Operating ◽  
Extrusion Force ◽  
Particle Size Ratio ◽  
Small Constant ◽  
Working Parameters ◽  
The Relationship

How to predict the relationship among particle size and among product size, to establish the relationship between the granularity and working parameters in the process of grinding and to determine the optimum operating parameters. With proposing BS squeeze crush model by L. Bass and the idea of roll surface division as the material uneven extrusion force are adopted. Based on field experiments the experimental data is analyzed, the select function and the breakage functions are fitted with MATLAB software, and obtaining their model. The comminution model is determined by the roller division. We obtain the model parameter through the experimental data. Through model analysis shows: the relationship between particle breakage and energy absorption, namely the smaller size of the same power, the lower broken; the breakage diminishes with the decrease of particle size ratio and it will be tending to a small constant when the smaller particle size ratio. The breakage functions rapidly decrease within ratio of between 0.2-0.7. This shows: the energy consumption will rapidly increase when the particle size of less than 0.2 in broken; the selection diminish with the decrease of particle size. Pressure (8-9MPa) should be the most appropriate value.

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