scholarly journals Particle Size Distribution of Smoluchowski Coagulation Equation for Brownian Motion at Equilibrium

2021 ◽  
Author(s):  
Mingliang Xie
Keyword(s):  
Particle Size ◽  
Brownian Motion ◽  
Particle Size Distribution ◽  
Size Distribution ◽  
Information Entropy ◽  
Geometric Mean ◽  
Dissipative System ◽  
Smoluchowski Coagulation Equation ◽  
Coagulation Equation ◽  
Definition Of Information

The information entropy for Smoluchowski coagulation equation is proposed based on statistical mechanics. And the normalized particle size distribution is a lognormal function at equilibrium from the principle of maximum entropy and moment constraint. The geometric mean volume and standard deviation in the distribution function are determined as simple constant. The results reveal that the assumption that algebraic mean volume be unit in self-preserving hypothesis is reasonable in some sense. Based on the present definition of information entropy, the Cercignani’s conjecture holds naturally for Smoluchowski coagulation equation. Together with the proof that the conjecture is also true for Boltzmann equation, Cercignani’s conjecture will holds for any two-body collision systems, which will benefit the understanding of Brownian motion and molecule kinematic theory, such as the stability of the dissipative system, and the mathematical theory of convergence to thermodynamic equilibrium.

Determination of Particle-Size Distribution and Concentration of Cigarette Smoke by a Light-Scattering Method

1977 ◽  
Vol 9 (3) ◽  
Author(s):  
T. Okada ◽  
Y. Ishizu ◽  
K. Matsunuma
Keyword(s):  
Particle Size ◽  
Particle Size Distribution ◽  
Size Distribution ◽  
Cigarette Smoke ◽  
Scattered Light ◽  
Geometric Mean ◽  
Number Concentration ◽  
Scattering Method ◽  
Working Standard

AbstractA new method for determining particle-size distribution of cigarette smoke particles was developed by simultaneous measurement of scattered light at three angles for a fixed wavelength. A theoretical chart useful for this purpose, which was made of the relative intensities of scattered light at the angles 45° and 135° to that at the angle 90°, was calculated on the basis of the Mie theory. The number concentration was determined from the Rayleigh ratio using the working standard method. The measurements were rapidly performed, without change of particle size during measuring time, with a device for dilution. The geometric mean diameter, the logarithmic standard deviation and the number concentration of mainstream smoke were found to be about 0.18 um, 0.4 and 3 X 10

Size Distributions of Particles Extracted from Different Materials Compared with the MIL-STD-1246 Particle Size Distribution

2000 ◽  
Vol 43 (4) ◽  
pp. 25-29
Author(s):  
Roger Welker
Keyword(s):  
Particle Size ◽  
Particle Size Distribution ◽  
Size Distribution ◽  
Geometric Mean ◽  
Industrial Applications ◽  
Distribution Analysis ◽  
Size Distributions ◽  
Naturally Occurring ◽  
History Of ◽  
Log Normal

MIL-STD-1246 particle-size distribution is the basis for specifying the particle cleanliness of surfaces for many governmental and industrial applications. MIL-STD-1246 states that naturally occurring particle contamination on surfaces follows a log-normal particle-size distribution, with a geometric mean of 1 μm, following a very precise size specification. However, the naturally occurring particle-size distribution may be a function of the material under examination or the prior cleaning or surface treatment history of the material. This paper explores the relation between the MIL-STD-1246 particle-size distribution and particle-size distributions measured after extraction followed by liquidborne particle-size distribution analysis.

scholarly journals Particle-Size Distribution Models for the Conversion of Chinese Data to FAO/USDA System

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Wei Shangguan ◽  
YongJiu Dai ◽  
Carlos García-Gutiérrez ◽  
Hua Yuan
Keyword(s):  
Particle Size ◽  
Particle Size Distribution ◽  
Size Distribution ◽  
Soil Texture ◽  
Texture Classification ◽  
Geometric Mean ◽  
Coefficient Of Determination ◽  
Logistic Growth ◽  
United States Department ◽  
Data Points

We investigated eleven particle-size distribution (PSD) models to determine the appropriate models for describing the PSDs of 16349 Chinese soil samples. These data are based on three soil texture classification schemes, including one ISSS (International Society of Soil Science) scheme with four data points and two Katschinski’s schemes with five and six data points, respectively. The adjusted coefficient of determinationr2, Akaike’s information criterion (AIC), and geometric mean error ratio (GMER) were used to evaluate the model performance. The soil data were converted to the USDA (United States Department of Agriculture) standard using PSD models and the fractal concept. The performance of PSD models was affected by soil texture and classification of fraction schemes. The performance of PSD models also varied with clay content of soils. The Anderson, Fredlund, modified logistic growth, Skaggs, and Weilbull models were the best.

scholarly journals Particle size distributions from laboratory-scale biomass fires using fast response instruments

2010 ◽  
Vol 10 (16) ◽  
pp. 8065-8076 ◽  
Author(s):  
S. Hosseini ◽  
Q. Li ◽  
D. Cocker ◽  
D. Weise ◽  
A. Miller ◽  
...  
Keyword(s):  
United States ◽  
Particle Size ◽  
Particle Size Distribution ◽  
Size Distribution ◽  
Geometric Mean ◽  
The United States ◽  
Biomass Combustion ◽  
Size Distributions ◽  
Particle Size Distributions ◽  
Mean Diameter

Abstract. Particle size distribution from biomass combustion is an important parameter as it affects air quality, climate modelling and health effects. To date, particle size distributions reported from prior studies vary not only due to difference in fuels but also difference in experimental conditions. This study aims to report characteristics of particle size distributions in well controlled repeatable lab scale biomass fires for southwestern United States fuels with focus on chaparral. The combustion laboratory at the United States Department of Agriculture-Forest Service's Fire Science Laboratory (USDA-FSL), Missoula, MT provided a repeatable combustion and dilution environment ideal for measurements. For a variety of fuels tested the major mode of particle size distribution was in the range of 29 to 52 nm, which is attributable to dilution of the fresh smoke. Comparing mass size distribution from FMPS and APS measurement 51–68% of particle mass was attributable to the particles ranging from 0.5 to 10 μm for PM10. Geometric mean diameter rapidly increased during flaming and gradually decreased during mixed and smoldering phase combustion. Most fuels produced a unimodal distribution during flaming phase and strong biomodal distribution during smoldering phase. The mode of combustion (flaming, mixed and smoldering) could be better distinguished using the slopes in MCE (Modified Combustion Efficiency) vs. geometric mean diameter than only using MCE values.

scholarly journals Evolution of Aerosol Particles in the Rainfall Process via Method of Moments

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Fangyang Yuan ◽  
Fujun Gan
Keyword(s):  
Particle Size ◽  
Particle Size Distribution ◽  
Size Distribution ◽  
Method Of Moments ◽  
Particle Number ◽  
Aerosol Particles ◽  
Geometric Mean ◽  
Number Concentration ◽  
Brownian Diffusion ◽  
Particle Number Concentration

The method of moments is employed to predict the evolution of aerosol particles in the rainfall process. To describe the dynamic properties of particle size distribution, the population balance equation is converted to moment equations by the method of moments and the converted equations are solved numerically. The variations of particle number concentration, geometric mean diameter, and geometric standard deviation are given in the cases that the Brownian diffusion and inertial impaction of particles dominate, respectively. The effects of raindrop size distribution on particle size distribution are analyzed in nine cases. The results show that the particle number concentration decreases as time goes by, and particles dominated by Brownian diffusion are removed more significantly. The particle number concentration decreases much more rapidly when particle geometric mean diameter is smaller, and the particle size distribution tends to be monodisperse. For the same water content, the raindrops with small geometric mean diameters can remove particles with much higher efficiency than those with large geometric mean diameters. Particles in the “Greenfield gap” are relatively difficult to scavenge, and a new method is needed to remove it from the air.

scholarly journals Particle size distributions from laboratory-scale biomass fires using fast response instruments

2010 ◽  
Vol 10 (4) ◽  
pp. 8595-8621 ◽  
Author(s):  
S. Hosseini ◽  
L. Qi ◽  
D. Cocker ◽  
D. Weise ◽  
A. Miller ◽  
...  
Keyword(s):  
Particle Size ◽  
Particle Size Distribution ◽  
Size Distribution ◽  
Geometric Mean ◽  
Biomass Combustion ◽  
Climate Modelling ◽  
Size Distributions ◽  
Particle Size Distributions ◽  
Mean Diameter ◽  
Particle Sizer

Abstract. Particle size distribution from biomass combustion is an important parameter as it affects air quality, climate modelling and health effects. To date particle size distributions reported from prior studies vary not only due to difference in fuels but also difference in experimental conditions. This study aims to report characteristics of particle size distribution in a well controlled repeatable lab scale biomass fires for southwestern US fuels. The combustion facility at the USDA Forest Service's Fire Science Laboratory (FSL), Missoula, MT provided repeatable combustion and dilution environment ideal for particle size distribution study. For a variety of fuels tested the major mode of particle size distribution was in the range of 29 to 52 nm, which was attributable to dilution of the fresh smoke. Comparing volume size distribution from Fast Mobility Particle Sizer (FMPS) and Aerodynamic Particle Sizer (APS) measurements, ~30% of particle volume was attributable to the particles ranging from 0.5 to 10 μm for PM10. Geometric mean diameter rapidly increased during flaming and gradually decreased during mixed and smoldering phase combustion. Most of fuels gave unimodal distribution during flaming phase and strong biomodal distribution during smoldering phase. The mode of combustion (flaming, mixed and smoldering) could be better distinguished using slopes in Modified Combustion Efficiency (MCE) vs. geometric mean diameter from each mode of combustion than only using MCE values.

The Determination of Size Distribution of Particles in Suspension by a Diffusion Technique

1964 ◽  
Vol 3 (4) ◽  
Author(s):  
H. Reinhardt ◽  
J. O Liljenzin ◽  
R. Lindner
Keyword(s):  
Particle Size ◽  
Particle Size Distribution ◽  
Size Distribution ◽  
Geometric Mean ◽  
Diffusion Technique ◽  
Size Distribution Of Particles

SummaryUnder the assumption of a logarithmic particle-size distribution, the geometric mean radius

scholarly journals Saturated Hydraulic Conductivity and Textural Heterogeneity of Soils

2017 ◽  
Author(s):  
Carlos García-Gutiérrez ◽  
Yakov Pachepsky ◽  
Miguel Ángel Martín
Keyword(s):  
Particle Size ◽  
Hydraulic Conductivity ◽  
Particle Size Distribution ◽  
Size Distribution ◽  
Information Entropy ◽  
Fine Sand ◽  
Saturated Hydraulic Conductivity ◽  
Coarse Particles ◽  
Heterogeneity Parameter

Abstract. Saturated hydraulic conductivity Ksat is an important soil parameter that highly depends on soil's particle size distribution (PSD). The nature of this dependency is explored in this work in two ways, (1) by using the Information Entropy as a heterogeneity parameter of the PSD and (2) using descriptions of PSD in forms of textural triplets, different than the usual description in terms of the triplet of sand, silt and clay contents. The power of this parameter, as a descriptor of Ksat and log(Ksat) , was tested on a database of > 19 K soils. We found coefficients of determination of up to 0.977 for log(Ksat) using a triplet that combines very coarse, coarse, medium and fine sand as coarse particles, very fine sand as intermediate particles, and silt and clay as fines. The power of the correlation is analysed for different textural classes and different triplets. Overall, the use of textural triplets different than traditional, combined with IE, may provide a useful tool for predicting Ksat values.

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