Article

# Derivation of the Weibull Distribution Based on Physical Principles and its Connection to the Rosin-Rammler and Lognormal Distributions

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## Abstract

We describe a physically based derivation of the Weibull distribution with respect to fragmentation processes. In this approach we consider the result of a single‐event fragmentation leading to a branching tree of cracks that show geometric scale invariance (fractal behavior). With this approach, because the Rosin–Rammler type distribution is just the integral form of the Weibull distribution, it, too, has a physical basis. In further consideration of mass distributions developed by fragmentation processes, we show that one particular mass distribution closely resembles the empirical lognormal distribution. This result suggests that the successful use of the lognormal distribution to describe fragmentation distributions may have been simply fortuitous. © 1995 American Institute of Physics.

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... The mass on the right hand side of Eq. 6 is m and not m . Brown and Wohletz (1995) showed that a power law follows naturally from a single-event fragmentation that leads to a branching tree of cracks that have a fractal character. The spacing of the cracks is described by the fractal dimension D f = −3γ. ...
... The idea that extensive regrinding experienced in a hypervelocity impact leads to a power law SFD with a large exponent (Hartmann, 1969) is not supported by grinding experiments, which typically result in a Weibull distribution (Rosin & Rammler, 1933;Martin & Mills, 1977;Deb & Sen, 2013). Furthermore, Brown (1989) and Brown and Wohletz (1995) theorized that a power law distribution results from a single fragmentation event, whereas sequential fragmentation (i.e., regrinding) results in a Weibull distribution. We found that the power law is not a good model for the Vesta and Ceres boulder SFDs. ...
... A statistical test reveals that the power law is actually not a good model for the Vesta SFD, but the Weibull distribution fits the data very well. The Weibull distribution is commonly applied to describe SFDs resulting from rock grinding experiments, and results from the fractal nature of the cracks propagating in the rock interior (Brown & Wohletz, 1995). The Weibull distribution may provide a better description of the SFD of boulders on small bodies than the power law, and would naturally result in a steeper SFD for the relatively large boulders of Vesta. ...
Preprint
Dawn's framing camera observed boulders on the surface of Vesta when the spacecraft was in its lowest orbit (LAMO). We identified, measured, and mapped boulders in LAMO images, which have a scale of 20 m per pixel. We estimate that our sample is virtually complete down to a boulder size of 4 pixels (80 m). The largest boulder is a 400 m-sized block on the Marcia crater floor. Relatively few boulders reside in a large area of relatively low albedo, surmised to be the carbon-rich ejecta of the Veneneia basin, either because boulders form less easily here or live shorter. By comparing the density of boulders around craters with a known age, we find that the maximum boulder lifetime is about 300 Ma. The boulder size-frequency distribution (SFD) is generally assumed to follow a power law. We fit power laws to the Vesta SFD by means of the maximum likelihood method, but they do not fit well. Our analysis of power law exponents for boulders on other small Solar System bodies suggests that the derived exponent is primarily a function of boulder size range. The Weibull distribution mimics this behavior and fits the Vesta boulder SFD well. The Weibull distribution is often encountered in rock grinding experiments, and may result from the fractal nature of cracks propagating in the rock interior. We propose that, in general, the SFD of particles (including boulders) on the surface of small bodies follows a Weibull distribution rather than a power law.
... The mass on the right-hand side of Equation 6 is m and not m ′ . Brown and Wohletz (1995) showed that a power law follows naturally from a single-event fragmentation that leads to a branching tree of cracks that have a fractal character. The spacing of the cracks is described by the fractal dimension D f = −3 . ...
... The idea that extensive regrinding experienced in a hypervelocity impact leads to a power law SFD with a large exponent (Hartmann, 1969) is not supported by grinding experiments, which typically result in a Weibull distribution (Deb & Sen, 2013;Martin & Mills, 1977;Rosin & Rammler, 1933). Furthermore, Brown (1989) and Brown and Wohletz (1995) theorized that a power law distribution results from a single fragmentation event, whereas sequential fragmentation (i.e., regrinding) results in a Weibull distribution. We found that the power law is not a good model for the Vesta and Ceres boulder SFDs. ...
... A statistical test reveals that the power law is actually not a good model for the Vesta SFD but the Weibull distribution fits the data very well. The Weibull distribution is commonly applied to describe SFDs resulting from rock grinding experiments and results from the fractal nature of the cracks propagating in the rock interior (Brown & Wohletz, 1995). The Weibull distribution may provide a better description of the SFD of boulders on small bodies than the power law and would naturally result in a steeper SFD for the relatively large boulders of Vesta. ...
Article
Full-text available
Dawn's framing camera observed boulders on the surface of Vesta when the spacecraft was in its lowest orbit (LAMO). We identified, measured, and mapped boulders in LAMO images, which have a scale of 20 m per pixel. We estimate that our sample is virtually complete down to a boulder size of 4 pixels (80 m). The largest boulder is a 400 m-sized block on the Marcia crater floor. Relatively few boulders reside in a large area of relatively low albedo, surmised to be the carbon-rich ejecta of the Veneneia basin, either because boulders form less easily here or live shorter. By comparing the density of boulders around craters with a known age, we find that the maximum boulder lifetime is about 300 Ma. The boulder size-frequency distribution (SFD) is generally assumed to follow a power law. We fit power laws to the Vesta SFD by means of the maximum likelihood method, but they do not fit well. Our analysis of power law exponents for boulders on other small Solar System bodies suggests that the derived exponent is primarily a function of boulder size range. The Weibull distribution mimics this behavior and fits the Vesta boulder SFD well. The Weibull distribution is often encountered in rock grinding experiments, and may result from the fractal nature of cracks propagating in the rock interior. We propose that, in general, the SFD of particles (including boulders) on the surface of small bodies follows a Weibull distribution rather than a power law.
... By the way, this approach is not fully new. In literature of particle size analysis the idea of point processes has appeared already implicitly in the papers Brown [3], Brown and Wohletz [4] and Bernhardt [5], as explained in the Discussion section. ...
... The Introduction already mentioned that the use of point process ideas in particle size statistics, in particular the use of the intensity function, is not entirely new in the context of particle statistics. Indeed, it appears in hidden form and with another notation in the papers [3,4], known for a physicallybased derivation of the Weibull and RRSB distribution in the context of fragmentation. ...
... Brown and Wohletz [4] consider fragment or particle masses m (instead of x in the present paper) and use a function n(m) which "is the number distribution in units of ...
Article
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This paper re-considers the foundations of particle size statistics. While traditional particle size statistics consider their data as samples of random variables and use methods of classical mathematical statistics, here a particle sample is treated as a point process sample, and a suitable form of statistics is recommended. The whole sequence of ordered particle sizes is considered as a random variable in a suitable sample space. Instead of distribution functions, point process intensity functions are used. The application of point process data analysis is demonstrated for samples of fragments from single-particle crushing of glass balls. Three cases of data handling with point processes are presented: statistics for oversize particles, pooling of independent particle samples and pooling of piecewise particle data. Finally, the problem of goodness-of-fit testing for particle samples is briefly discussed. The point process approach turns out to be an extension of the classical approach, is simpler and more elegant, but retains all valuable traditional ideas. It is particularly strong in the analysis of oversize particles. Graphical abstract
... One such distribution which is very popular in the analysis of survival data is the Weibull distribution introduced Waloddi Weibull in 1951 (for more details, see [66]). The mathematical properties and its applicability and generalizations were studied by [15], [62], [45], [46], [67], [58], [10], [48], [38], [30], [11], [44], [49], among many others. The probability density function (pdf) of a continuous random variable X with a Weibull distribution with three parameters is given by, ...
... • Step 1. Generate a random sample from u 1 ∼ U (0, 1) and put this value as u 1 = F W (x | η 1 , β 1 , α 1 ), given in (10), that is, x = β 1 (− log(1 − u 1 )) 1/α1 + η 1 to generate a random observation on X; • Step 2. Generate a random sample from w ∼ U (0, 1) and put this value ...
... (this expression is the derivative of (1) with respect to u 1 , when F x (x) = u 1 and F y (y) = u 2 ); then solve this equation in relation to u 1 , generating a random observation u 2 , from u 2 ∼ U (0, 1), where u 1 is obtained from step 1; • Step 3. Considering u 2 obtained from step 2, put u 2 = F W (y | η 2 , β 2 , α 2 ), given in (10), that is, y = β 2 (− log(1 − u 2 )) 1/α2 + η 2 to generate a random observation on Y; ...
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Bivariate lifetime distributions are of great importance in studies related to interdependent components, especially in engineering applications. In this paper, we introduce two bivariate lifetime assuming three-parameter Weibull marginal distributions. Some characteristics of the proposed distributions as the joint survival function, hazard rate function, cross factorial moment and stress-strength parameter are also derived. The inferences for the parameters or even functions of the parameters of the models are obtained under a Bayesian approach. An extensive numerical application using simulated data is carried out to evaluate the accuracy of the obtained estimators to illustrate the usefulness of the proposed methodology. To illustrate the usefulness of the proposed model, we also include an example with real data from which it is possible to see that the proposed model leads to good fits to the data.
... The lognormal distribution and Weibull distribution are commonly used to model skewed distributions; however, the physical origin of the distribution is not well understood. Brown and Wohletz (1995) derived the Weibull distribution with respect to the fragmentation process, in which a power law was used to describe the breakup of a single particle into smaller particles. The Weibull distribution has been widely used as particle size distribution for coarse grains (Fang et al., 1993;Kondolf and Adhikari, 2000). ...
... Lognormal distribution has been observed in particle growth or coagulation processes (Smoluchowski, 1918;Friedlander and Wang, 1966), in which aggregation process dominates the dynamics. On the other hand, Weibull distribution has been commonly observed in the fragmentation process of large particles (Brown and Wohletz, 1995). In the flocculation process, both the aggregation and fragmentation processes play an important role. ...
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Floc size distribution is one of the key parameters to characterize flocculating cohesive sediment. An Eulerian–Lagrangian framework has been implemented to study the flocculation dynamics of cohesive sediments in homogeneous isotropic turbulent flows. Fine cohesive sediment particles are modeled as the dispersed phase by the discrete element method, which tracks the motion of individual particles. An adhesive contact model with rolling friction is applied to simulate the particle–particle interactions. By varying the physicochemical properties (i.e., stickiness and stiffness) of the primary particles, the dependence of the mathematical form of the floc size distribution on sediment properties is investigated. At the equilibrium state, the aggregation and breakup processes reach a dynamic equilibrium, in which construction by aggregation is balanced with destruction by breakup, and construction by breakup is balanced with destruction by aggregation. When the primary particles are less sticky, floc size distribution fits better with the lognormal distribution. When the primary particles are very sticky, both the aggregation of smaller flocs and breakup from larger flocs play an equally important role in the construction of the intermediate-sized flocs, and the equilibrium floc size distribution can be better fitted by the Weibull distribution. When the Weibull distribution develops, a shape parameter around 2.5 has been observed, suggesting a statistically self-similar floc size distribution at the equilibrium state.
... The Rosin-Rammler particle size distribution is assumed based on the proppant size. The Rosin-Rammler particle size distribution is a continuous probability distribution function to describe particle size distribution (Brown and Wohletz 1995). The top, bottom walls and fracture tip were specified as no-slip stationary walls for the liquid phase. ...
... Rosin-Rammler particle size distribution is assumed based on the 20/40 size sand. The Rosin-Rammler particle size distribution is a continuous probability distribution function to describe particle size distribution (Brown and Wohletz 1995). The top, bottom walls and fracture tip were specified as no-slip stationary walls for the liquid phase, as shown in Figure 5.1. ...
Thesis
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The distribution of proppant injected in hydraulic fractures significantly affects fracture-conductivity and well-performance. The proppant transport and suspension in thin fracturing fluid used in unconventional reservoirs are considerably different from those of fracturing fluids in conventional reservoirs, due to the very low viscosity of fracturing fluids used in the unconventional reservoirs, poor ability to suspend proppants and hence quick deposition of the proppants. This research presents the development of a three-dimensional computational fluid dynamics (CFD) modelling technique for the prediction of proppant-fluid multiphase flow in hydraulic fractures for unconventional reservoirs. The Eulerian-Lagrangian multiphase modelling approach has been applied to model the fluid flow and proppant transport, and the kinetic theory of granular flow is used to model the inter-proppant, fluid-proppant and proppantwall interactions. The existing proppant transport models ignore the fluid leak-off effect from the fracture side wall and the effect of fracture roughness. Thus, at the interface between the fracture and surrounding porous medium, the mass flow rate from the fracture to porous rock is calculated based on the permeability and porosity of the rock. The leakage mass flow rate is then used to define the mass and momentum source term at the fracture wall as a user-defined function, to investigate the proppant transport in hydraulic fractures with fluid leak-off effect. Furthermore, the hydrodynamic and mechanical behaviour of proppant transport on fracture roughness was studied in detail using different rough fracture profiles, and a relationship between the fracture roughness and proppant transport velocity is proposed. Lastly, an integrated model is developed, which simulates the proppant transport in dynamically propagating hydraulic fractures. The existing models either model the proppant transport physics in static predefined fracture geometry or account for the analytical models for defining the fracture propagation using linear elastic fracture mechanics. This limits the fracture propagation model to brittle rocks and neglect plastic deformations. Thus, in the present study, the fracture propagation was modelled using the extended finite element method (XFEM) and cohesive zone model (CZM), which can model the plastic deformations in the ductile rock. The fracture propagation was coupled with the CFD based proppant transport model, to model the fluid flow and proppant transport. The parametric study was then performed to investigate the effect of variation in proppant properties, fracturing fluid properties and geomechanical properties on the proppant transport. This study has enhanced the understanding of the flow and interaction phenomenon between proppant and fracturing fluid, and provides a technique with potential application in fracturing design for increasing well-productivity. The model can accurately simulate the proppant transport dynamics in hydraulic fracture and the present study proposes a solution to a frequent fracture tip screen out challenge faced in the petroleum industry. Thus, the developed modelling techniques provide petroleum engineers with a more suitable option for designing hydraulic fracturing operations, simultaneously modelling fracture propagation and fluid flow with proppant transport, and improves confidence by accurately tracking the distribution of proppants inside the fracture.
... The log-normal distribution has a peak skewed toward smaller sizes and is similar to observed size distributions of fragments at smaller sizes but tends to deviate from observed distribution at larger sizes, where the power law tends to fit better [20]. To the best of our knowledge, only the Weibull distribution [37,38] is similar to observed size distributions across the microplastics and mesoplastics ranges (fragments larger than 5 mm), at least in Cózar et al.'s study [20]. The Weibull distribution as applied to the size distributions of various types of particles is largely empirical but with an interpretation as resulting from the branching tree of cracks [38]. ...
... To the best of our knowledge, only the Weibull distribution [37,38] is similar to observed size distributions across the microplastics and mesoplastics ranges (fragments larger than 5 mm), at least in Cózar et al.'s study [20]. The Weibull distribution as applied to the size distributions of various types of particles is largely empirical but with an interpretation as resulting from the branching tree of cracks [38]. ...
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The size distribution of marine microplastics provides a fundamental data source for understanding the dispersal, break down, and biotic impacts of the microplastics in the ocean. The observed size distribution at the sea surface generally shows, from large to small sizes, a gradual increase followed by a rapid decrease. This decrease has led to the hypothesis that the smallest fragments are selectively removed by sinking or biological uptake. Here we propose a new model of size distribution, focusing on the fragmentation of marine plastics. The model is inspired by ideas from statistical mechanics. In this model, the original large plastic piece is broken into smaller pieces once by the application of “energy” or work by waves or other processes, under two assumptions, one that fragmentation into smaller pieces requires larger energy and the other that the occurrence probability of the “energy” exponentially decreases toward larger energy values. Our formula well reproduces observed size distributions over wide size ranges from micro- to mesoplastics. According to this model, the smallest fragments are fewer because large “energy” required to produce such small fragments occurs more rarely.
... The experimental measurements are taken from a Cavitron ultrasonic scaling device at a flow rate of 16.2 ml/min, a typical setting used in dental practices. A Rosin-Rammler fit, which is commonly used to describe particle distributions, is applied to the experimental data (Brown andWohletz, 1995 andRosin andRammler, 1933). The fit is shown to obey ...
... The experimental measurements are taken from a Cavitron ultrasonic scaling device at a flow rate of 16.2 ml/min, a typical setting used in dental practices. A Rosin-Rammler fit, which is commonly used to describe particle distributions, is applied to the experimental data (Brown andWohletz, 1995 andRosin andRammler, 1933). The fit is shown to obey ...
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COVID-19, caused by the SARS-CoV-2 (severe acute respiratory syndrome coronavirus 2) virus, has been rapidly spreading worldwide since December 2019, causing a public health crisis. Recent studies showed SARS-CoV-2's ability to infect humans via airborne routes. These motivated the study of aerosol and airborne droplet transmission in a variety of settings. This study performs a large-scale numerical simulation of a real-world dentistry clinic that contains aerosol-generating procedures. The simulation tracks the dispersion of evaporating droplets emitted during ultrasonic dental scaling procedures. The simulation considers 25 patient treatment cubicles in an open plan dentistry clinic. The droplets are modeled as having a volatile (evaporating) and nonvolatile fraction composed of virions, saliva, and impurities from the irrigant water supply. The simulated clinic's boundary and flow conditions are validated against experimental measurements of the real clinic. The results evaluate the behavior of large droplets and aerosols. We investigate droplet residence time and travel distance for different droplet diameters, surface contamination due to droplet settling and deposition, airborne aerosol mass concentration, and the quantity of droplets that escape through ventilation. The simulation results raise concerns due to the aerosols' long residence times (averaging up to 7.31 min) and travel distances (averaging up to 24.45 m) that exceed social distancing guidelines. Finally, the results show that contamination extends beyond the immediate patient treatment areas, requiring additional surface disinfection in the clinic. The results presented in this research may be used to establish safer dental clinic operating procedures, especially if paired with future supplementary material concerning the aerosol viral load generated by ultrasonic scaling and the viral load thresholds required to infect humans.
... From a formative perspective, the power-law SFD would indicate a single-event fragmentation 443 (for example during impact cratering) that leads to a branching tree of cracks that have a fractal 444 character (Turcotte et al., 1997, Schröder et al., 2021b. Whereas, the Weibull distribution is 445 thought to result from sequential fragmentation (Brown & Wohletz 1995) and it is largely used 446 in fracture and fragmentation theory (Grady and Kipp, 1987;Brown and Wohletz, 1995;447 Turcotte, 1997;McSaveney, 2002). In addition, the Weibull distribution (Weibull, 1951) is 448 often used to describe the particle distribution that is derived from grinding experiments (Rosin 449 and Rammler, 1933). ...
... From a formative perspective, the power-law SFD would indicate a single-event fragmentation 443 (for example during impact cratering) that leads to a branching tree of cracks that have a fractal 444 character (Turcotte et al., 1997, Schröder et al., 2021b. Whereas, the Weibull distribution is 445 thought to result from sequential fragmentation (Brown & Wohletz 1995) and it is largely used 446 in fracture and fragmentation theory (Grady and Kipp, 1987;Brown and Wohletz, 1995;447 Turcotte, 1997;McSaveney, 2002). In addition, the Weibull distribution (Weibull, 1951) is 448 often used to describe the particle distribution that is derived from grinding experiments (Rosin 449 and Rammler, 1933). ...
... The log-normal distribution has a peak skewed toward smaller sizes and is similar to observed size distributions of fragments at smaller sizes but tends to deviate from observed distribution at larger sizes, where the power law tends to fit better [20]. To the best of our knowledge, only the Weibull distribution [35,36] is similar to observed size distributions across the microplastics and mesoplastics ranges (fragments larger than 5 mm), at least in Cózar et al.'s study [20]. The Weibull distribution as applied to the size distributions of various types of particles is largely empirical but with an interpretation as resulting from the branching tree of cracks [36]. ...
... To the best of our knowledge, only the Weibull distribution [35,36] is similar to observed size distributions across the microplastics and mesoplastics ranges (fragments larger than 5 mm), at least in Cózar et al.'s study [20]. The Weibull distribution as applied to the size distributions of various types of particles is largely empirical but with an interpretation as resulting from the branching tree of cracks [36]. ...
Preprint
Full-text available
The size distribution of marine microplastics provides a fundamental data source for understanding the dispersal, break down, and biotic impacts of the microplastics in the ocean. The observed size distribution at the sea surface generally shows, from large to small sizes, a gradual increase followed by a rapid decrease. This decrease has led to the hypothesis that the smallest fragments are selectively removed by sinking or biological uptake. Here we propose a new model of size distribution without any removal of material from the system. The model uses an analogy with black-body radiation and the resultant size distribution is analogous to Planck's law. In this model, the original large plastic piece is broken into smaller pieces once by the application of "energy" or work by waves or other processes, under two assumptions, one that fragmentation into smaller pieces requires larger energy and the other that the probability distribution of the "energy" follows the Boltzmann distribution. Our formula well reproduces observed size distributions over wide size ranges from micro- to mesoplastics. According to this model, the smallest fragments are fewer because large "energy" required to produce such small fragments occurs more rarely.
... The Weibull distribution used here was first described as a family of curves [16] which has found applicability to describe the distribution of particle sizes following fragmentation or fractionation [17]. Brown and Wohletz provide a mechanistic derivation for the Weibull distribution which follows from repeated fragmentation of a larger structure, with each step resulting in a fractal fragmentation pattern (thus following a power law). ...
... In a scenario where all enhancers are equally active, a particular gene will be most strongly influenced by the closest enhancer (E2 in this figure). A Weibull model, as observed empirically in this analysis, can result from such a "superposition" of power-law distributions [17] should look carefully at whether there are multiple plausible causal genes, such as from paralogs, which exist within the 944 kb distance cutoff recommended here. ...
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Background A genome-wide association study (GWAS) correlates variation in the genotype with variation in the phenotype across a cohort, but the causal gene mediating that impact is often unclear. When the phenotype is protein abundance, a reasonable hypothesis is that the gene encoding that protein is the causal gene. However, as variants impacting protein levels can occur thousands or even millions of base pairs from the gene encoding the protein, it is unclear at what distance this simple hypothesis breaks down. Results By making the simple assumption that cis-pQTLs should be distance dependent while trans-pQTLs are distance independent, we arrive at a simple and empirical distance cutoff separating cis- and trans-pQTLs. Analyzing a recent large-scale pQTL study (Pietzner in Science 374:eabj1541, 2021) we arrive at an estimated distance cutoff of 944 kilobasepairs (95% confidence interval: 767–1,161) separating the cis and trans regimes. Conclusions We demonstrate that this simple model can be applied to other molecular GWAS traits. Since much of biology is built on molecular traits like protein, transcript and metabolite abundance, we posit that the mathematical models for cis and trans distance distributions derived here will also apply to more complex phenotypes and traits.
... In the blasting sciences and particularly in the metallurgy industry, particle size distributions are normally compared with empirical distributions such as the Rosin-Rammler (Cunningham, 1983) and Weibull distributions. Brown and Wohletz (1995) have shown that the Weibull distribution can be calculated from physical principals, and that the Rosin-Rammler is a derivative of the Weibull function. They also establish mathematically the association of the Weibull distribution to the more fundamental fractal dimension (D) that governs the scale-independent growth of cracks during fragmentation. ...
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This thesis explores the mechanics of kimberlite volcanic system processes across the scale range from local pipe brecciation to continental-scale control on the distribution of magma sources. It is not an exhaustive discussion of all details of kimberlite emplacement, but introduces new ideas, tools and understanding that ultimately provide practical benefits to kimberlite geology and exploration.
... The size distribution of the particles at the inlet to the domain is specified using Rosin-Rammler particle distribution [57][58][59] . According to this distribution, the mass fraction, ψ, of particles with a diameter greater than d p is given by, ψ = e −(d p /d 0 ) n , where n is the size distribution parameter 195 that controls the spread of the distribution, d p is the particle diameter, and d 0 represents the mean particle size. ...
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In the present work, particle-laden coaxial turbulent jet flow is studied using large-eddy simulation (LES). An Eulerian-Lagrangian framework is used to study the interaction between the continuous phase (air) and the discrete phase (glass bead particles). The solver is validated, using single-phase and particle-laden simulations, with reference data from experiments. A good match is observed between the present results and the reference data, for centerline velocity decay and radial profiles of axial velocity. Simulations are performed for three co-flow velocity ratios of 0, 1 and 1.5. The results pertaining to particle characteristics are presented for three different particle size classes. The effect of co-flow velocity ratio on particle size-velocity correlation and velocity statistics of both phases are studied with an emphasis on understanding the differences in the particle dispersion due to co-flow around the central jet. It is observed that the particle size-velocity correlation is negative in the potential core region, and it becomes positive as one moves downstream. For heavy particles, the axial distance required to attain the same velocity as that of air increases with an increase in the co-flow velocity ratio. The size-conditioned particle number density profiles along the axial and radial directions of coaxial jets showed some interesting trends that could be explained based on particle Stokes number effect. Significant radial dispersion of particles are realized when the corresponding Stokes number (St L), defined based on large-scale turbulent eddies, is of the order of one. The axial evolution of characteristic particle size exhibited non-monotonic trends for all co-flow ratios. Overall, the present work demonstrates potential application of LES for an in-depth study of dispersion of poly-disperse particles in turbulent coaxial jets.
... Originally Weibull's statistics was developed to describe the fracture of brittle materials [80], [81] and to calculate the probability of the damage-free survival of the given material. It can be derived from geometric scale invariance (fractal organized structures) by physical principles, [82] in mechanical mills. It is frequently applied in the study of mechanical fatigue and failure [83]. ...
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Lifetime analyses frequently apply a parametric functional description from measured data of the Kaplan-Meier non-parametric estimate (KM) of the survival probability. The cumulative Weibull distribution function (WF) is the primary choice to parametrize the KM. but some others (e.g. Gompertz, logistic functions) are also widely applied. We show that the cumulative two-parametric Weibull function meets all requirements. The Weibull function is the consequence of the general selforganizing behavior of the survival, and consequently shows self-similar death-rate as a function of the time. The ontogenic universality as well as the universality of tumor-growth fits to WF. WF parametrization needs two independent parameters, which could be obtained from the median and mean values of KM estimate, which makes an easy parametric approximation of the KM plot. The entropy of the distribution and the other entropy descriptions are supporting the parametrization validity well. The goal is to find the most appropriate mining of the inherent information in KM-plots. We show clinical examples of oncological hyperthermia treatment, evaluated by single-arm study. The method which we chosen is the modulated electrohyperthermia (mEHT, oncothermia ®) which is applied in the final stages of the cancer, and no conventional treatment as control is available in many studies. We show the two-parameter WF fits to the non-parametric KM survival curve in a real study of 1180 cancer patients offering satisfactory description of the clinical results. Two of the 3 characteristic parameters of the KM plot (namely the points of median, mean or inflection) are enough to reconstruct the parametric fit, which gives support of the comparison of survival curves of different patient’s groups. Objective in this chapter is to find a parametric description of overall survival, which fits the selforganized processes and is able to show the inherent information of survival measurements of cancer patients in advanced cases, when the curative conventional and evidence-based proven therapies fail. Keywords: Self-organizing; self-similarity; Avrami-function; Weibull-distribution; survival-time; allometry; entropy; bioscaling, mEHT.
... For the analytic scalings, several models to predict the resulting DSD were proposed. Among these, the most commonly adopted distributions are: normal [275,276], log-normal [277][278][279], Rosin-Rammler [280], Weibull [281], upper limit equation [282] and power law [271,274,[283][284][285][286]. perimental [271,272,287] and numerical [152,153,274,284,285,288] works, in which turbulent flows laden with drops are considered, report a good agreement with a power law scaling. ...
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Turbulent flows laden with large, deformable drops or bubbles are ubiquitous in nature and in a number of industrial processes. These flows are characterized by a physics acting at many different scales: from the macroscopic length scale of the problem down to the microscopic molecular scale of the interface. Naturally, the numerical resolution of all the scales of the problem, which span about eight to nine orders of magnitude, is not possible, with the consequence that numerical simulations of turbulent multiphase flows impose challenges and require methods able to capture the multi-scale nature of the flow. In this review, we start by describing the numerical methods commonly employed and discussing their advantages and limitations, and then we focus on the issues arising from the limited range of scales that can be possibly solved. Ultimately, the droplet size distribution, a key result of interest for turbulent multiphase flows, is used as a benchmark to compare the capabilities of the different methods and to discuss the main insights that can be drawn from these simulations. Based on this, we define a series of guidelines and best practices that we believe important in the simulation analysis and in the development of new numerical methods.
... To the best of our knowledge, only the Weibull distribution 33,34 is similar to observed size distributions across the microplastics and mesoplastics ranges, at least in Cózar et al.'s study 20 . The Weibull distribution as applied to the size distributions of various types of particles is largely empirical but with an interpretation as resulting from the branching tree of cracks 34 . ...
Preprint
Full-text available
The size distribution of marine microplastics (< 5 mm) provides a fundamental data source for understanding the dispersal, break down, and biotic impacts of the microplastics in the ocean. The observed size distribution generally shows, from large to small sizes, a gradual increase followed by a rapid decrease. This decrease has led to the hypothesis that the smallest fragments are selectively removed by sinking or biological uptake. Here we propose a new model of size distribution without any removal of material from the system. The model uses an analogy with black-body radiation and the resultant size distribution is analogous to Planck's law. In this model, the original large plastic piece is broken into smaller pieces once by the application of “energy” or work by waves or other processes, under two assumptions, one that fragmentation into smaller pieces requires larger energy and the other that the probability distribution of the “energy” follows the Boltzmann distribution. Our formula well reproduces observed size distributions over wide size ranges from micro- (< 5 mm) to mesoplastics ( > 5 mm). According to this model, the smallest fragments are fewer because large “energy” required to produce such small fragments occurs more rarely.
... where λ is the scale parameter and k is the shape parameter of distribution, with k & λ > 0. Weibull distribution finds its application in the system where dynamical evolution is driven by fragmentation and sequential branching [27,28]. Since the evolution of the system in hadrons and heavy-ion collision is dominated by a perturbative QCD based parton cascade model, we can apply q-Weibull distribution to study particle spectra. ...
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Transverse Momentum, $p_T$, spectra is of prime importance in order to extract crucial information about the evolution dynamics of the system of particles produced in the collider experiments. In this work, the transverse momentum spectra of charged hadrons produced in $PbPb$ collision at $5.02$ TeV has been analyzed using different distribution functions in order to gain strong insight into the information that can be extracted from the spectra. We have also discussed the applicability of Pearson distribution on the spectra of charged hadron at $5.02$ TeV.
... Considering the univariate situation, a distribution which is widely considered in the lifetime data analysis is the Weibull distribution (Weibull, 1951) given the flexibility of fit for the data. The mathematical properties and its applicability and generalizations have been studied by many authors (see for example, Cohen, 1965;Philip, 1974;Lai et al., 2003;Thoman et al., 1969;Stevens and Smulders, 1979;Rinne, 2008;Mudholkar et al., 1996;Brown and Wohletz, 1995;Pinder III et al., 1978;Cao, 2004;Pham and Lai, 2007;Saraiva and Suzuki, 2017; among many others). In this study, we explore a multivariate exponential distribution introduced by Marshall and Olkin (1967b) given as an extension of the fatal shock model to a multi-component system to build a new trivariate lifetime distribution denoted as the trivariate Marshall-Olkin-Weibull (TMOW) distribution. ...
Article
Multivariate lifetime data are common in many applications, especially in medical and engineering studies. In this paper, we consider a trivariate Marshall-Olkin Weibull distribution denoted by MOMW3 to model trivariate data in presence of right censored data.Maximum likelihood and Bayesian methods are used to get the parameter estimators of interest. An extensive simulation study was performed to verify the effectiveness of the maximum likelihood estimators. Reliability datasets related to fiber failure strengths were considered to illustrate the performance of the proposed model under the Classical and Bayesian approaches. As a result, it is observed that the MOMW3 model could be considered as a good alternative to model trivariate lifetime data, especially under a Bayesian approach which could be of interest for the reliability analysis, as observed with the real data application in industrial engineering presented in the study or any other area of interest.
... This method, however, assumes that the GSD follows a lognormal distribution, in other words the GSD is normally distributed on the φ-scale. Alternatively, the GSD can be described using a Weibull or Rosin-Rammler distribution (Rosin and Rammler, 1933;Weibull, 1951;Brown and Wohletz, 1995) from which shape and scale parameters can be described (Appendix B). Log-normal and Weibull distributions can be fit as mixture models to account for the multimodal form of many volcanic GSDs (Appendix B; Eychenne et al., , 2015Costa et al., 2016;Pioli et al., 2019;Mele et al., 2020). ...
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To quantify the size of tephra, two practical challenges must be addressed: the wide range of particle sizes (10 −8-10 1 m) and the diversity of particle shape, density and optical properties. Here we use dynamic image analysis (DIA) to simultaneously characterise the size and shape of tephra samples from Mount Mazama, Krafla, Mount St. Helens and Campi Flegrei. The Camsizer X2 instrument used in this study, which has a measurement range of 0.8 μm-8 mm, avoids the need to overlap different measurement methods and principles for fine (<125 μm) and coarse (>125 μm) particle sizes. Importantly, DIA does not require an assumption of particle properties. DIA also allows the measurement of grain size distributions (GSDs) using multiple size definitions. Quantification by particle long axis and the area equivalent sphere diameter, for example, make DIA GSDs compatible with the outputs of other methods such as laser diffraction and sieving. Parallel mass-based (sieving) and volume-based (DIA) GSDs highlight the effects of particle density variations on methods of size analysis; concentrations of dense crystals within a narrow size range, in particular, can affect mass-based GSDs and their interpretations. We also show that particle shape has an important effect on the apparent grain size of distal tephra. Extreme particle shapes, such as the platy glass shards typical of the distal Campanian Ignimbrite deposits, can appear coarser than other distal tephras if size is quantified according to the particle long axis. These results have important implications for ash dispersion models, where input GSDs assume that reported measurements are for volume-equivalent sphere diameters. We conclude that DIA methods are not only suitable for characterising, simultaneously , the size and shape of ash particles but also provide new insights into particle properties that are useful for both ash dispersion modelling and studies of explosive volcanism.
... The Weibull distribution was initially derived empirically, and is often used to describe the particle distribution resulting from grinding experiments (Rosin & Rammler 1933). Where the power law follows naturally from a single-event fragmentation that leads to a branching tree of cracks that have a fractal character, the Weibull distribution results from sequential fragmentation (Brown & Wohletz 1995). Because we only include boulders larger than a certain size in the fit, we employ a left-truncated Weibull distribution with the cumulative form (Wingo 1989): ...
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We mapped all boulders larger than 105 m on the surface of dwarf planet Ceres using images of the Dawn framing camera acquired in the Low Altitude Mapping Orbit (LAMO). We find that boulders on Ceres are more numerous towards high latitudes and have a maximum lifetime of $150 \pm 50$ Ma, based on crater counts. These characteristics are distinctly different from those of boulders on asteroid (4) Vesta, an earlier target of Dawn, which implies that Ceres boulders are mechanically weaker. Clues to their properties can be found in the composition of Ceres' complex crust, which is rich in phyllosilicates and salts. As water ice is though to be present only meters below the surface, we suggest that boulders also harbor ice. Furthermore, the boulder size-frequency distribution is best fit by a Weibull distribution rather than the customary power law, just like for Vesta boulders. This finding is robust in light of possible types of size measurement error.
... Furthermore, particles were generally found to be elongated, and axis ratios for boulders >2 m are close to 0.7 on average. The SFD of boulders on small bodies may better be described by a Weibull distribution than a power law (Schröder et al., 2020), and we have used a cumulative Weibull (Rosin-Rammler) distribution (Brown & Wohletz, 1995;Rosin, 1933;Weibull, 1951;Wingo, 1989) to represent the data provided by Michikami et al. (2019). The cumulative SFD N(D) is then given by ...
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The carbonaceous asteroid (162173) Ryugu formed from fragments which re-accreted after its parent body was disrupted by a catastrophic collision. Asteroids of this type are also known as rubble piles and the re-accumulation process is thought to be one of the causes for their large bulk porosity. We have applied mixing models to determine the amount of inter-boulder porosity taking the observed abundance of large and small boulders on the surface into account. We find that the relative abundances of differently sized boulders allow for a very efficient packing, such that inter-boulder porosity in Ryugu is rather small and only $16 \pm 3$~\%. This implies that a large part of { Ryugu's} total porosity must reside inside the boulders themselves. Using estimates of boulder intrinsic porosity, we furthermore constrain the average density of the boulder's constituent minerals to { $2848 \pm 152$ kg m$^{-3}$}, which is consistent with values measured for carbonaceous meteorites as collected on Earth. Thus, inter-boulder porosity of rubble pile asteroids may have been systematically overestimated in the past.
... The Weibull distribution was initially derived empirically and is often used to describe the particle distribution resulting from grinding experiments (Rosin & Rammler 1933). Where the power law follows naturally from a single-event fragmentation that leads to a branching tree of cracks that have a fractal character, the Weibull distribution results from sequential fragmentation (Brown & Wohletz 1995). Because we only include boulders larger than a certain size in the fit, we employ a left-truncated Weibull distribution with the cumulative form (Wingo 1989 ...
Article
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We mapped all boulders larger than 105 m on the surface of dwarf planet Ceres using images of the Dawn framing camera acquired in the Low Altitude Mapping Orbit. We find that boulders on Ceres are more numerous toward high latitudes and have a maximum lifetime of 150 ± 50 Ma, based on crater counts. These characteristics are distinctly different from those of boulders on asteroid (4) Vesta, an earlier target of Dawn, which implies that Ceres’ boulders are mechanically weaker. Clues to their properties can be found in the composition of Ceres’ complex crust, which is rich in phyllosilicates and salts. As water ice is thought to be present only meters below the surface, we suggest that boulders also harbor ice. Furthermore, the boulder size–frequency distribution is best fit by a Weibull distribution rather than the customary power law, just like for Vesta boulders. This finding is robust in light of possible types of size measurement error.
... Instead, the granules with specific distribution had to be utilized. Han et al. [20] modeled the combustion of ZPP in a closed vessel by applying the Rosin-Rammler distribution [28], considering the cumulative mass of all the granules. Based on this size distribution, Hwang et al. [29] proposed a precise analytical model for pyrotechnically actuated devices by investigating the combustion characteristics of ZPP charged in a small volume chamber. ...
Article
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A numerical model describing the ballistic behavior of a commercially used initiator is presented in this article. This model was built on the principle of conservation of mass and energy in the multi-phase framework incorporated with multi-loaded conditions. After obtaining the information about the grain size distribution in each composite, a fixing factor was proposed based on the surface area ratio of the composites. Thus, the solid propellant burning process based on distributed grain size was described, and the burn rate parameters of the applied pyrotechnic compositions were re-evaluated for different preconditioned temperature levels according to Vieille’s law. The influence of bridge wire and initiator metal cap was further modeled concerning their characteristic properties according to the observed measurements. The validation of the entire initiator model in the closed bomb test showed quantitative agreement with the measured pressure evolution, while the parameter study for evaluating the ballistic sensitivity of each component delivered some insights into the product development process. Furthermore, the configuration of a cold gas inflator was utilized to evaluate the initiator impact for a realistic application, where the shock wave intensity during deployment serves the main function in the inflator design. Incorporated with CFD simulations to capture the shock wave propagation, 0D-3D coupling strategy for initiator ballistics to inflator configuration was realized. Besides, the simulation results reflected the physical conditions in a proper manner. In particular, the parameter study led to a better understanding of interactions between inflator components, which were barely possible to be quantified through the measurements. The proposed initiator model could also be used in combination with other mechanical principles as a component of pyrotechnic devices such as pin-puller, electric line cutters, or airbag inflators. The detailed information gained in describing the physical properties enabled us to assess the existing design quantitatively and to have better control of the product quality.
... As previously invoked to explain the observed size distributions of plastics in aquatic environments (Brown and Wohletz, 1995;Timár et al., 2010), the sigmoid Weibull and lognormal distributions were applied to empirically model the plastic release, fragmentation, and agglomeration processes that may have occurred during the shear tests. ...
Article
Wearing face masks is a fundamental prevention and control measure to limit the spread of COVID-19. The universal use and improper disposal of single-use face masks are raising serious concerns for their environmental impact, owing to the foregone contribution to plastic water pollution during and beyond the pandemic. This study aims to uncover the release of micro/nanoplastics generated from face mask nonwoven textiles once discarded in the aquatic environment. As assessed by microscopy and flow cytometry, the exposure to different levels of mechanical stress forces (from low to high shear stress intensities) was proved effective in breaking and fragmenting face mask fabrics into smaller debris, including macro-, micro-, and nano-plastics. Even at the low level of fabric deterioration following the first second of treatment, a single mask could release in water thousands of microplastic fibers and up to 108 submicrometric particles, mostly comprised in the nano-sized domain. By contributing to the current lack of knowledge regarding the potential environmental hazards posed by universal face masking, we provided novel quantitative data, through a suitable technological approach, on the release of micro/nanoplastics from single-use face masks that can threaten the aquatic ecosystems to which they finally end-up.
... In this work, we take a novel approach to characterize crumpling and offer explanation for the logarithmic model by drawing a correspondence between crumpling and fragmentation processes. Fragmentation models have a rich history of theoretical development [22][23][24] as well as industrial applications 22,25 and use in modeling collision and fracture phenomena 26 . Here we concentrate on a theoretical, physically based rate equation for modeling time-dependent fragmentation detailed by Cheng and Redner 27 , which provides a general framework for processes that may be treated as successive, homogeneous breakups instigated by non-local stresses. ...
Article
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As a confined thin sheet crumples, it spontaneously segments into flat facets delimited by a network of ridges. Despite the apparent disorder of this process, statistical properties of crumpled sheets exhibit striking reproducibility. Experiments have shown that the total crease length accrues logarithmically when repeatedly compacting and unfolding a sheet of paper. Here, we offer insight to this unexpected result by exploring the correspondence between crumpling and fragmentation processes. We identify a physical model for the evolution of facet area and ridge length distributions of crumpled sheets, and propose a mechanism for re-fragmentation driven by geometric frustration. This mechanism establishes a feedback loop in which the facet size distribution informs the subsequent rate of fragmentation under repeated confinement, thereby producing a new size distribution. We then demonstrate the capacity of this model to reproduce the characteristic logarithmic scaling of total crease length, thereby supplying a missing physical basis for the observed phenomenon.
... Since then, dynamic fragment characterisation has been a subject of considerable research interest, and researchers have used a variety of statistical distributions in evaluating average fragment size. Some of the common statistical distributions used are: exponential (Grady and Kipp 1985), log-normal (Ishii and Matsushita 1992), power-law (Oddershede et al. 1993), Weibull (Brown and Wohletz 1995), Swebrec (Ouchterlony 2005) and Gilvarry (Sil'vestrov 2004). Another group of researchers have developed models based on principles of energy balance (Glenn and Chudnovsky 1986;Grady 1982;Yew and Taylor 1994). ...
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The aim of this study is to understand the strength behaviour and fragment size of rocks during indirect, quasi-static and dynamic tensile tests. Four rocks with different lithological characteristics, namely: basalt, granite, sandstone, and marble were selected for this study. Brazilian disc experiments were performed over a range of strain rates from ~ 10–5 /s to 2.7 × 10¹ /s using a hydraulic loading frame and a split Hopkinson bar. Over the range of strain rates, our measurements of dynamic strength increase are in good agreement with the universal theoretical scaling relationship of (Kimberley et al., Acta Mater 61:3509–3521, 2013). Dynamic fragmentation during split tension mode failure has received little attention, and in the present study, we determine the fragment size distribution based on the experimentally fragmented specimens. The fragments fall into two distinct groups based on the nature of failure: coarser primary fragments, and finer secondary fragments. The degree of fragmentation is assessed in terms of characteristic strain rate and is compared with existing theoretical tensile fragmentation models. The average size of the secondary fragments has a strong strain rate dependency over the entire testing range, while the primary fragment size is less sensitive at lower strain rates. Marble and sandstone are found to generate more pulverised secondary debris when compared to basalt and granite. Furthermore, the mean fragment sizes of primary and secondary fragments are well described by a power-law function of strain rate.
... The Weibull distribution typically describes a random fragmentation process where the probability of splitting a solid particle into fragments depends on the size of the particle. Recently, Brown and Wohletz demonstrated that the Weibull distribution arises naturally as a consequence of the fractal nature of the fragmentation process, [6]. According to this scenario, the fragmentation of a solid particle is initiated by generation of a fractal crack tree. ...
Article
An in-situ visualisation of the microstructure of a Carbopol gel flowing in a micro-channel is presented. By means of a novel chemical protocol able to stain the solid material units of the gel with a fluorescent molecule we are able to accurately identify the solid/fluid material units advected by the confined microscopic flow. Based on this novel visualisation technique a full statistical description of the distributions of solid-fluid material units in the flow in terms of space and time averages, probability density functions and space–time correlations is reported for the first time.
... This is surprising as it is likely that the fragmentation was more efficient for bigger aggregates than for smaller ones. However, it should be noticed that the PDF might not be exactly exponential, but was (at least for conditions 1 and 2) better fitted by a Weibull distribution, which can account for a size-dependent fragmentation [18]. When considering volume (or mass, not shown) averaged PDFs, they were will-fitted by a log-normal distribution. ...
Thesis
Whey protein isolates have received increasing interest due to their high nutritional value and their growing availability on the market as a co-product of cheese production. Whey protein isolates (WPI) can be aggregated upon heating to create new functional properties which depend on aggregate size and structural properties. Based on the fractal properties of these aggregates, one major application is to texture food products by two different ways: by forming a stable and thick suspension of aggregates or by forming a space filling network, through a gelation process. Fractal aggregate size generally ranges from a few hundred nanometres to a few microns at most. However, it would be interesting if their size could reach at least 30 microns (the limit of consumer perception) to increase their thickening power. Up to now, technological routes to create thickening particles were based mainly on the physico-chemical conditions of aggregation.The objective of the PhD is to study new aggregation and gelation processes with the aim to produce novel aggregate structures to enhance their texturising ability.Firstly, a good understanding of molecular self-assembly was required, taking into account the industrial process parameters. Indeed, in industrial conditions, aggregates are obtained under flow conditions at high temperature (≥75◦C) in few minutes. We developed a down scaling approach to study both the kinetics of aggregation after few seconds and its dependence with the mean shear rate in which heat transfer does not limit aggregation. The size and mass of aggregates and protein conformation were characterized by small-angle X-ray scattering and resonant mass measurement. We showed that for fractal aggregates formed at low protein and salt concentration, WPI aggregation at 92°C was limited by a step of nucleation, shear rate had no significant effect on the size of the aggregates, or on the aggregation kinetics and slower thermalization lead smaller aggregates size.Secondly, standard characterization methods of aggregates structure being limited to sub-micrometric aggregates, we developed a new method for aggregates of several tens of microns based on a covalent labelling of WPI and fluorescent microscopy. We have shown that, depending on the physico-chemical and heating conditions, the range of size where fractal aggregates exhibits a fractal dimension equal to 2 can be extended from 10 to 60 µm.Thirdly, we developed a new structuration process to spin gels of fractal aggregates by a combination of microflows and Ca2+-induced gelation. The WPI fibers presented a core-shell structure. Moreover, the size and the stability of the fiber was due to a complex interplay between different phenomena: hydrodynamics stresses, gelation kinetics, local pH changes during gelation and osmotic stresses.Finally, characterization of gelation of fractal aggregates by Ca2+ was studied in a bi-dimensional geometry in order to gain insights on the different phenomena that govern the formation and structure of the fiber. Especially, we showed the presence of osmotic flux which concentrates the aggregates in a front and form the shell of the fiber.
... As previously invoked to explain the observed size distributions of plastics in aquatic environments (Brown and Wohletz, 1995;Timár et al., 2010), the sigmoid Weibull and lognormal distributions were applied to empirically model the plastic release, fragmentation, and agglomeration processes that may have occurred during the shear tests. ...
Preprint
p>This study aims to assess the environmental impact of discarded face masks, that are a source of emerging concern as indicated by most recent literature, although still little investigated. Herein we evaluated micro- and nanoplastic particles that can be released from face mask once subject to environmental conditions. Exposure to simulated-low shear forces demonstrated to be effective in breaking and fragmenting face mask tissue into smaller debris. Even at low shear energy densities, a single mask could release in water thousands of microplastic fibers and up to 10^11 submicrometric particles. The latter were quantified using flow cytometry that was proven to be a promising technique for nanoplastic counting, thus improving our understanding on distribution and fate of NPs still representing a great analytical challenge in plastic pollution research. </p
... Other similar models are by Steacy and Sammis (1991) and Palmer and Sanderson (1991). A different model is by Brown and Wohletz (1995) who propose a self-similar model of breakage (Fig. 1e) and arrive at a Weibull distribution; this approach is a sub-set of the Filippov approach. ...
Article
The production of breccias and cataclasites is commonly proposed to result in power-law or log-normal probability distributions for fragment (grain) size. We show that in both natural and experimental examples, the common best fit probability distributions for the complete distributions are members of the Generalised Gamma (GG), Extreme Value (GEV) and Pareto (GP) families; power-law and log-normal distributions are commonly, but not always, poor fits to the data. An hierarchical sequence, GG → GEV → GP, emerges as the sample mean of the fragment size decreases. The physical foundations (self-similar fragmentation, collisional fragmentation, shattering) for these distributions are discussed. Particularly important is the shattering continuous phase transition that results in the simultaneous development of both coarse fragments and ultra-fine particles (dust). This phase transition leads to Generalised Pareto fragment size distributions for the coarse fragments. Also included is a discussion of the relations between fragment size distribution, processes and deformation history in the context of monomineralic rocks. The overall reported size distributions are compatible with theoretical developments but the topic would benefit from observations and experiments conducted with the theories in mind.
... Weibull distribution finds its application in the system where dynamical evolution is driven by fragmentation and sequential branching [28,29]. Since the evolution of the system in hadrons and heavy-ion collision is dominated by a perturbative QCD-based parton cascade model, we can apply q-Weibull distribution to study particle spectra. ...
Article
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Transverse momentum, p T , spectra are of prime importance in order to extract crucial information about the evolution dynamics of the system of particles produced in the collider experiments. In this work, the transverse momentum spectra of charged hadrons produced in P b P b collision at 5.02 TeV have been analyzed using different distribution functions in order to gain strong insight into the information that can be extracted from the spectra. We have also discussed the applicability of the unified statistical framework on the spectra of charged hadron at 5.02 TeV
... We consider a particle bed consisting of spherical rigid particles. The particle-size distribution is assumed to follow a Weibull distribution, as frequently found in several powder-based industrial applications [6]. The radii r = d/2 distribution is characterized by a shape parameter a and scale parameter b. ...
Article
Caking in amorphous powders compromises their quality during storage. Individual particles absorb water vapor, which changes their viscosity and promotes the formation of sinter bridges. Lumps of particles grow and eventually span the whole powder, affecting the mechanical properties and quality of the powder. Previous studies of the caking dynamics largely neglect the role of spatial heterogeneities in the particle-size distribution. We perform particle-based simulations and show that, if caking is mapped into a percolation transition, the role of spatial heterogeneities is well captured by the corresponding percolation threshold. Since this threshold only depends on the geometry of the granular assembly, we can separate the contribution of the spatial heterogeneities and of the individual particle properties. This enables a rational approach for interpreting and mitigating caking propensity of commercial products consisting of particle species with different particle size distributions. We corroborate the numerical and analytical predictions with experiments.
... This method, however, assumes that the GSD follows a log-normal distribution, in other words the GSD is normally distributed on the φ-scale. Alternatively, the GSD can be described using a Weibull or Rosin-Rammler distribution (Rosin and Rammler, 1933;Weibull, 1951;Brown and Wohletz, 1995) from which shape and scale parameters can be described ...
Thesis
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Around once every millenium, a large magnitude explosive eruption occurs on Earth dispersing volcanic ash across millions of square kilometers. Volcanic ash uniquely poses a wide-range of hazards to human health, infrastructure and the environment, the impacts of which are felt close to source to >1000’s of km from the volcano. Importantly for large eruptions, the removal of the ash is almost impossible, which means the material remains and is remobilised in the environment, posing secondary hazards for 100’s of years after the initial eruption. Here I present a study of the processes the produce, transport and deposit ash from large eruptions. I use the ~7.7 ka climactic eruption of Mount Mazama as a case study because the tephra was predominantly deposited on-land facilitating widespread data collection. I collate locations where the Mazama tephra has been recorded to produce a new isopach map and estimate of the total erupted volume (176 km^3 bulk or 61 km^3 Dense-Rock-Equivalent). The compilation of tephra thickness data also showed how the Mazama tephra deposit has been remobilised through time, exemplifying the uncertainties associated with field data. Remobilised deposits also provide insight into the types of secondary ash hazards that persist following large magnitude eruptions. I also investigate the physical and chemical properties of Mazama ash to provide insight into eruptive processes such as co-PDC plumes and distal ash transport. I determine from the composition of Fe-Ti oxides, that the distal ash can be attributed to the later stages of the climactic Mazama eruption. I also observe that the Grain Size Distribution (GSD) of the distal Mazama tephra is remarkably stable, a trend that is observed for other large distal deposits. This study also investigates the methods we use to analyse grain size in volcanology and outlines a new protocol for measuring the size and shape of volcanic ash using Dynamic Image Analysis (DIA). The benefits of DIA include the capacity for simultaneous particle size and shape characterisation, and the insight into particle density if used in parallel with sieve analysis. The new estimate of the total erupted volume and distal GSD of the Mazama were integrated with Ash3D, a numerical model of volcanic ash transport and deposition, to simulate the eruption and test the sensitivity of Ash3D to uncertainty in the eruption source parameters. The results stress the need to integrate radial spreading in the umbrella cloud region with advection-diffusion models when simulating the ash transport during large magnitude eruptions. Furthermore, it highlights significant knowledge gaps regarding the deposition of very fine-ash during any scale of eruption. This underscores the benefits of studying fine-ash deposition using the deposits from large eruptions where significant depositional areas and ash volumes facilitate extensive data collection and model testing.
... FSD in case G 1 S 2 C 1 also fits well with Weibull distribution with a shape factor of 2.5. Brown and Wohletz (1995) showed that at late stages, when the breakup and flocculation processes balance each other, the Weibull distribution is the outcome for a power-law breakup of a large floc into smaller particles. Our results suggest that fragmentation and reflocculation occur constantly at the equilibrium stage. ...
Article
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A two‐phase Euler‐Lagrangian framework was implemented to investigate the flocculation dynamics of cohesive sediment in isotropic turbulence. The primary particles are modeled as sticky soft spheres using the discrete element method (DEM). The attractive van der Waals forces are modeled by the DLVO theory and the JKR adhesive contact theory. The near steady state equilibrium floc size distribution (FSD) strongly depends on the ratio of the turbulent shear to the floc strength (or particle stickiness). When turbulence is strong, a single peak around the Kolmogorov length scale appears in the FSD, and the distribution fits the Weibull distribution well. A power‐law floc size distribution develops when the floc strength is greater than the destabilizing effect of turbulent shear. Sediment concentration does not significantly affect the shape of FSD or the average floc size. The average apparent floc settling velocity Ws increases with the average floc size. Fractal dimension of flocs decreases with the floc size following a power‐law relation for large flocs. Settling velocity of flocs as a function of floc size also follows a power‐law relation. Deviation from the power‐law relationship is found for large flocs because of their porous nature. At equilibrium stage, the construction by aggregation is balanced with the destruction by breakup, and the construction by breakup is balanced with the destruction by aggregation. The aggregation kernel by turbulent shear and power‐law breakup kernel can describe the dynamics reasonably well for the flocculation of cohesive sediment in homogeneous isotropic turbulence.
... It is worth noting that this distribution rightly bears Weibull's name, but it was independently described by Frechet, and Rosin, Rammler and Sperling (Brown and Wohletz 1995;Cook and DelRio 2019). The same probability distribution can also be derived from models other than the weakest link theory, such as the shot-noise model, the hazard rate approach, and the broken-stick model (Rinne 2008). ...
Article
Because slime water grain size variations cannot be accurately assessed in coal preparation plants during production, a method for determining the particle size distribution (PSD) based on Rosin – Rammler–Sperling – Bennett (RRSB) characteristic parameters is proposed. This study performed an industrial real-time sampling of coal slime water at four test points in a coal preparation plant, using an online sampler and a mechanical-pressing particle size measuring device. The measurement accuracy test was confirmed through testing such that production requirements were met. Taking the feeding of the concentration tank as an example, three distribution laws were verified, and the results showed that the PSD fit well with the RRSB distribution function, where the characteristic parameter n was 0.074 and the critical particle size Dewas 0.047 mm. In addition, after changing the parameters through gradation, n reflected the change of the dominant particle size range, while De indicated the direction of the particle size change. Through this successful application, show that employing a set of characteristic parameters based on RRSB PSD – obtained via an online granularity analysis system analyzing slurry particle size trends – has good potential for addressing the problem at hand.
Article
Blast block prediction is a complex non-stationary, nonlinear problem, the contribution of factors affecting results varies for different external conditions. Studies in a single environment are not universally applicable, the establishment of a blasted block size prediction model with fusion of multiple algorithms and reliable prediction results is the most urgent problem to be solved. In this study, a method is proposed that applies to different regions and rock conditions. To achieve the grouping prediction of blasted block size, this study firstly used hierarchical clustering to cluster the data in different areas, then used the random forest to establish the data grouping discriminant model based on the grouping results, and used back propagation neural network with genetic algorithm (GA-BP) to establish the blasted block size prediction model for each group of the blasted block size data separately. The results of the study show that (1) the block size data with different properties can be grouped according to the elastic modulus; (2) the grouping discriminant model established can correctly group the data; (3) compared with the GA-BP neural network prediction model without grouping, this model has a higher coefficient of determination (R2=0.982 ) and smaller root mean square error (RMSE=0.2) and mean relative error (MRE=4.857 ), which verifies the correctness of grouping prediction according to the elastic modulus; (4) by comparing with multiple regression, least squares support vector machines (LSSVM), and BP neural networks, the model outperforms the other models in R2RMSE, and MRE, and the prediction results are more accurate.
Thesis
In de huidige samenleving is duurzame energie één van de meest besproken onderwerpen. Om dit te bereiken is energie-efficiëntie van enorm groot belang. Zonnepanelen, elektrische voertuigen en stroomadapters zijn slechts enkele voorbeelden van toepassingen waarbij vermogen wordt getransformeerd en onvermijdelijk verloren gaat door middel van schakeling of geleiding. In de voorbije decennia hebben brede bandkloofmaterialen een grote belangstelling gewekt voor de vermogenshalfgeleiderindustrie. In het bijzonder toont gallium nitride (GaN) hogere efficiënties in toepassingen met hoge vermogens, vergeleken met traditionele Si tegenhangers. In recente literatuur is bezorgdheid geuit over de betrouwbaarheid en de geschatte levensduur van enerzijds de p-GaN/AlGaN/GaN schakelingssamenstelling en anderzijds de AlGaN/GaN-bufferlaag. In de literatuur worden extrapolatiemodellen voor levensduur van GaN transistoren gebruikt die specifiek voor Si transistoren enkele decennia geleden zijn ontwikkeld. In dit proefschrift wordt een stappenplan gepresenteerd met als doel een levensduurmodel op te bouwen voor de p-GaN schakelingssamenstelling en de AlGaN/GaN bufferlaag op basis van externe parameters zoals spanning, temperatuur en transistoroppervlakte. Dit statistisch-fysisch model zal van de grond op worden opgebouwd, door gebruik te maken van een combinatie van statistische faaldistributies en analyse van het fysische geleidingsmechanisme.
Article
The Orbiter High Resolution Camera (OHRC) is a very high spatial resolution panchromatic camera (0.45–0.70 μm) on-board Chandrayaan-2 orbiter. Its spatial resolution of 0.25 m from 100 km altitude is highest among all lunar orbiter missions. A simple crater with substantial boulder population was observed in an OHRC image of a region near Boguslawsky E crater. Boulders are distinctly seen in this image because of high spatial resolution of 0.28 m and low sun elevation angle (6°) which enhanced the boulders' shadows. We have identified and mapped >2000 boulders around this young un-named simple crater (74.9216° S, 54.5148° E). It is observed that the OHRC is capable of extending the lower limit of size for identifiable boulders below 1 m. The distributions of mapped boulders are studied and compared with previous studies. It was found that the coefficient values estimated by fitting power laws to various distributions, such as size-frequency, size-range, etc., are well within the ranges reported in literature for craters distributed on lunar surface around the landing sites. Weibull distribution was also fit to the data, and the fitting coefficients were compared with the values obtained in similar studies. The crater age was estimated to be in the range of 50–90 Ma using empirical relations, and comparison with areal density of other craters near lunar landing sites. This study also provides a glimpse of the low-light imaging capability of the OHRC showing inside the shadow regions, which were illuminated by reflected light from adjoining areas.
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To study the distribution characteristics and similarity laws of nuclei under different pressures, based on the self-designed decompression chamber and the acoustic measuring system, the size distributions of nuclei in the degassed tap water under negative ambient pressures were measured. A number density distribution function of nuclei based on the modified Weibull distribution function was proposed and verified by the experimental measurement results and some published data of nuclei size distribution. Based on this nuclei number density distribution function, the similarity law of the nuclei size distribution was analyzed: in the scale experiment, the value of exponential in the similarity law of the nuclei number density should be determined by the nuclei size distribution of the water in the prototype experiment and the actual nuclei size distribution of the water in the model experiment. And a precondition is that the nuclei size distributions are similar.
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A 100-mm-diameter split Hopkinson pressure bar was used to obtain the dynamic compressive properties and strain-rate sensitivity of rubber concrete, as well as to analyse the reason for the difference in the strain-rate sensitivities of rubber concrete and ordinary concrete. Rubber contents of 0%, 10%, 20%, 30%, 40%, and 50% of the fine aggregate volume were used. The test results showed that the quasi-static strength of rubber concrete decreased primarily because of the weak interfacial bond of the rubber–cement matrix. The dynamic increase factor (DIF) of the rubber concrete for different rubber contents increased with the strain rate. In addition, the rubber particles could slow down the accumulation of damage and increase the deformation lag effect, which increased the DIF of the rubber concrete compared to ordinary concrete. The toughness of the rubber concrete was also greater than that of ordinary concrete under impact loads because the stress decreased slowly after the peak stress value for the rubber concrete was reached. The fragment size of the post-test specimens gradually decreased with an increase in the strain rate, and the ordinary concrete had a larger number of cracks compared to rubber concrete, under a similar strain rate. The fragment size of the specimens followed a Weibull distribution, and the fragment size distribution model could correlate the strain rate and rubber content for failure mode prediction.
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The paper describes an experimental study of the influence of the composition and mechanical properties (ultimate and yield strength, reduction of area, impact toughness, and the fraction of fibrous fracture) of brittle, quasi-brittle, and ductile steels on the parameters of dynamic fragmentation and fracture mechanisms. The experiment showed that simple exponential functions could be used to describe cumulative mass distributions of shell fragments from the materials under study. The exponent of these functions is the reciprocal characteristic mass (1/μ) and it decreases as the fraction of fibrous fracture of shell fragments increases. It serves as a dynamic fracture criterion that is sensitive to changes in the composition and mechanical properties of the material. The relations between the fragmentation parameter 1/μ and mechanical properties and composition of shell materials were obtained. Reaching the critical value of the fragmentation criterion (1/μc) leads to changing the fracture mechanism from “ductile” to “brittle” mode which arises in a material with ∼40% of fibrous fracture at the fracture surface of impact specimens and ∼50% of shear fracture of fragments. The authors studied the structure and features of ductile and brittle fracture surfaces of shell fragments, compared the statistical distributions widely used to describe the fragmentation process (Mott, Weibull, and Grady-Odintsov) and the dependencies of their parameters on material properties. The study did not reveal any advantages of the parameters of these distributions over the 1/μ parameter of the exponential function. The analysis of the literature data on the fragmentation of shells of different steels showed that they did not contradict the obtained results, and the usage of simple exponential distributions makes it possible to reveal the physical meaning of parameters of statistical distributions. Several common patterns of fragmentation of solids are discussed.
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A study of the distribution of the value of traded goods under the Harmonized System is presented. The ramifications of this classification system are found to exhibit an approximate power law decay, indicating complexity and self-organization in the nomenclature of traded merchandises. For almost all countries with available data, log-values of annually imported and exported goods are well described by three-parameter Weibull distributions. This distribution commonly appears in particles size distributions, suggesting a connection between random fragmentation processes and the mechanisms behind the international trade of merchandises. Analysis of the resulting values for the fitting parameters from 1995 to 2018 shows a nearly constant linear relationship between the parameters of the Weibull distributions, so that, for each country, the distribution of log-values can be approximately characterized by a single shape parameter [Formula: see text]. The empirical findings of this paper suggest that specialization on trading a constant set of goods prevents the values of all traded merchandises from growing/decreasing simultaneously.
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We focus on a young (~ 4.5 Ma), 3.4 km long landslide located in the floor of Simud Vallis, Oxia Palus Quadrangle of Mars. By making use of a 2 m-scale HiRISE DEM we reconstruct the terrain surface before the landslide and in doing so we estimate the release and deposition heights and volumes related to the different stages of the landslide. Using the r.avaflow software we simulate the mass movement as a multi-stage event, and obtain simulated deposits that are both spatially and longitudinally comparable to the current landslide deposits. Through two 0.25 m-scale HiRISE images we identify and manually count >130,000 boulders that are located along the landslide, deriving their size-frequency distribution and spatial density per unit area for boulders with an equivalent diameter ≥1.75 m. Our analyses reveal that the distribution is of a Weibull-type, suggesting that the rocky constituents fractured and fragmented progressively during the course of the mass movement, consistent with our proposed two-stage model of landslide motion.
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Rocks around the InSight lander were measured in lander orthoimages of the near field (<10 m), in panoramas of the far field (<40 m), and in a high-resolution orbital image around the lander (1 km²). The cumulative fractional area versus diameter size-frequency distributions for four areas in the near field fall on exponential model curves used for estimating hazards for landing spacecraft. The rock abundance varies in the near field from 0.6% for the sand and pebble rich area to the east within Homestead hollow, to ~3-5% for the progressively rockier areas to the south, north and west. The rock abundance of the entire near field is just over 3%, which falls between that at the Phoenix (2%) and Spirit (5%) landing sites. Rocks in the far field (<40 m) that could be identified in both the surface panorama and a high-resolution orbital image fall on the same exponential model curve as the average near field rocks. Rocks measured in a high-resolution orbital image (27.5 cm/pixel) within ~500 m of the lander that includes several rocky ejecta craters fall on 4-5% exponential model curves, similar to the northern and western near field areas. As a result, the rock abundances observed from orbit falls on the same exponential model rock abundance curves as those viewed from the surface. These rock abundance measurements around the lander are consistent with thermal imaging estimates over larger pixel areas as well as expectations from fragmentation theory of an impacted Amazonian/Hesperian lava flow.
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Random breakage can be defined as the breakage patterns independent from the stressing environment and the nature of the broken particle. However, the relevant literature studies give contrary evidence against the random breakage of particles. A simple way to detect random breakage is to evaluate the fragment (progeny) size distributions. Such distributions are estimated analytically or through numerical models. The latter models generally treat random breakage as a geometric statistical problem with prior assumptions on particle/flaw geometry and external stressing environment, which may violate the randomness of the breakage process. This study presents a random-breakage algorithm that does not require such assumptions. The simulated progeny size distributions were compared with the experimental size distributions by impact loading (drop-weight) tests. Random breakage events should yield number-weighted size distributions that is fitted well to the lognormal distribution function. Also, a mass-weighted (sieve) size distribution function is presented for random breakage. Nevertheless, the results refute the random breakage of clinker and other brittle particles after impact loading. Instead, the sieve size distribution of fragments may evolve due to crack branching/merging and Poissonian crack nucleation processes.
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The assumption that distributions of mass versus size interval for fragmented materials fit the log normal distribution is empirically based and has historical roots in the late 19th century. Other often used distributions (e.g., Rosin-Rammler, Weibull) are also empirical and have the general form for mass per size interval: {ital n}({ital l})={ital kl}{sup Î±} exp(-{ital l}Î²), where {ital n}({ital l}) represents the number of particles of diameter {ital l}, {ital l} is the normalized particle diameter, and {ital k}, Î±, and Î² are constants. We describe and extend the sequential fragmentation distribution to include transport effects upon observed volcanic ash size distributions. The sequential fragmentation/transport (SFT) distribution is also of the above mathematical form, but it has a physical basis rather than empirical. The SFT model applies to a particle-mass distribution formed by a sequence of fragmentation (comminution) and transport (size sorting) events acting upon an initial mass {ital m}â²: {ital n}({ital x}, {ital m})={ital C} â«â« {ital n}({ital x}â², {ital m}â²){ital p}(Î¾) {ital dx}â² {ital dm}â², where {ital x}â² denotes spatial location along a linear axis, {ital C} is a constant, and integration is performed over distance from an origin to the sample location and mass limits from 0 to {ital m}.
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An analysis of fragmentation due to dynamic stress loading is presented which provides analytic functions for the distributions in fragment sizes. The analysis is restricted to one-dimensional bodies under uniform tensile loading. Concepts of survival statistics are used to account for spatially random fracture nucleation. Fragment size distribution curves for both brittle and ductile fracture are derived, and the curve for the latter is compared with experimental data. Fragment distribution curves are shown to depend on both material deformation properties and loading conditions.
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Suppose we have before us a large number of measurements. They may either be all approximations to the true value of a single unknown quantity, or may refer to the several members of a large class. The measurements will disagree among themselves, but on arranging them in order of size they show a tendency to cluster round some medium value. We are naturally inclined to infer that the true value of the unknown, or typical member of the class, is not far from this value. How to define and determine the appropriate medium in various classes of measurement becomes thus a natural object of inquiry. On examination we find that there is no strict and final criterion applicable to all cases.
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Carbon black exists as primary aggregates composed of primary particles fused together. We have measured the size, anisometry and bulkincss of these primary aggregates by electron microscopy. These parameters, which define ‘structure’, show a wide spread within a given sample. Weight-average parameters are calculated with the aid of a relation between projected area and number of particles per aggregate, based on computer-simulated flocs. As expected, the average values for all three parameters are high for acetylene black and low for thermal black. Furnace blacks give intermediate values, with aggregate size being the predominant parameter related to ‘structure’, at comparable levels of particle size.
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The structure of soot agglomerates formed by the combustion of acetylene in a coannular diffusion burner is studied. Structural data from electron micrographs were obtained by two methods, particle counting with the aid of stereopairs for small clusters and electronic digitization with high-resolution image processing, used for the larger agglomerates. Langevin dynamics computer simulations based on free molecular motion were performed as an aid to interpreting the experimental results.
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This paper discusses the applicability of statistics to a wide field of problems. Examples of simple and complex distributions are given.
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Zerkleinern ist ein statistischer Vorgang; die Kornzusammensetzung des Mahlgutes ist daher mit den Verfahrensweisen der mathematischen Statistik beschreibbar. Nachdem die Übertragung der Begriffe und Abbildungsweisen der Kollektivlehre auf das Kollektiv-„Mahlgut“ erörtert worden ist, werden die typischen Kornverteilungskurven des Mahlgutes von grober Zerkleinerung in Backenbrechern und Walzwerken über die mittelfeine Zerkleinerung in Schleudermühlen bis zur Feinstmahlung in Kugelmühlen behandelt, und es werden die analytischen Gleichungen für die einzelnen Kurventypen angegeben.
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A theory of sequential fragmentation is presented that describes a cascade of fragmentation and refragmentation,i.e., continued comminution. It is shown that the theory reproduces one of the two major empirical descriptors that have traditionally been used to describe the mass distributions from fragmentation experiments. Additional experimental evidence is presented to further validate the theory, and includes explosive aerosolization, grinding in a ball mill, and simulated volcanic action. Also presented are some astronomical applications of the theory including infalling extraterrestrial material, siderophile concentrations in black magnetic spherules of possible meteoritic origin, the asteroids, the distribution of galactic masses, and the initial mass function of stars
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The theoretical results of Gilvarry for the size distribution of the fragments in single fracture have been verified experimentally by fracturing spherical glass specimens under compression. The fragments were contained by a gelatin matrix to inhibit secondary fracture and thus make conditions conform as closely as possible to single fracture. Experimental values of the probability of fracture as obtained by sieve analysis show the predicted linear variation with the mean dimension x of the particles, over reasonably large intermediate ranges of the variables. It is shown that a logarithmic‐normal distribution does not represent the experimental results. The over‐all data exhibit three local maxima in the differential probability of fracture as a function of x, whereas the theory permits only two. Agreement in the number of peaks is obtained by subtracting the contribution to the over‐all probability of those fragments containing original surface of the specimen, which yields the true probability considered in the theory. In this manner, reasonably complete agreement between theory and experiment for single fracture is obtained. For plural fracture (carried out without use of gelatin), two additional peaks exist in the curve of the over‐all differential probability vs x, as compared to the case for single fracture. The theory of Gilvarry is confirmed down to a fragment dimension of at least 1 μ by means of an electrical counting instrument, and checked by direct microscopic sizing to 5 μ. The results yield numerical values of internal flaw densities, and thus provide a tool to study the distribution of Griffith flaws existing internally in a solid.
Article
Analytical solutions to the fragmentation equation are presented for specific rates of breakage and primary breakage distribution functions often used to correlate comminution data. These solutions, obtained by similarity arguments, compare favorably with the experimental data of Austin et al., and suggest that correlation of data via similarity techniques may facilitate determination of function parameters.