Probability Density Function Of Size Research Articles

Liquid sheet formation and spray characterization of N-heptane spray jet from a swirl atomizer: Numerical analysis and validation

This paper analyzes the liquid atomization mechanisms of an N-heptane spray jet emerging from a swirl simplex atomizer using numerical and experimental techniques. In particular, a direct seamless coupled Volume of Fluid and Lagrangian Particle Tracking approach together with adaptive mesh refinement within the Large Eddy Simulation framework offers a suitable way to accurately simulate the complex behavior of spray atomization, spray evolution, and droplet dispersion as a whole while using manageable computational cost. The achieved simulation results are first presented in terms of qualitative properties, characteristics of liquid sheet, air core generation, flow recirculation zones, and vortex patterns. For validation purposes, the numerical results are then compared with detailed experimental data obtained by a two-component Phase Doppler Anemometry technique. The assessment includes especially droplet statistics which strongly determine subsequent possible spray combustion process and related product and species emissions. The overall reported agreement demonstrates the capability of the adopted methodology in predicting and comprehensively investigating the complex phenomena associated with a pressure swirl fuel atomizer. In particular, it is found out that the normalized number-based probability density function of droplet size fits well with a lognormal distribution.

Characterizing inertial particle clustering with 1D signals

Preferential concentration in turbulent flows laden with inertial particles still presents many open fundamental physics questions. These gaps hinder the reconciliation of different experimental and numerical studies into a coherent quantitative view, which is needed to enable accurate high resolution modeling. We examine the influence of the dimensionality of the measurement technique (1D line, 2D planes or 3D volumes) on the characterization of cluster properties/preferential concentration, and proposes an approach to disentangle the cluster-characterizing results from random contributions that could contaminate the cluster statistical analysis. We studied this effect by projecting 2D and 3D data snapshots containing inertial particle clusters onto a 1D axis. The objective was to simulate 1D sensors (widely used experimentally: hot wire, fiber optics probes, LDV/PDA, etc) with different sensing lengths. These projected records were analysed by means of one-dimensional Voronoï tessellations. For the experimental parameters explored in this study, our results retrieved average clustering properties, when the measuring window is equal or larger than the Kolmogorov length scale (η), and smaller than of the integral length scale of the turbulence . These observations are consistent with previous research using 2D and 3D datasets taken under similar experimental conditions. The raw probability density function (PDF) of the Voronoï length in 1D does not provide error-free information on the clusters size or local concentration. This measurement bias is corrected based on the histograms of the number of particles within each cluster. Importantly, the analysis of these histograms explains the well-known clusters size PDFs power law reported in previous numerical and experimental research using 2D and 3D datasets. This power-law behavior, however, spanning less than a decade, is not robust in 1D. To address this, we use conditioned statistics (on the number of particles inside each cluster) to discern between turbulence-driven clustering and random particle concentration fluctuations. This secondary analysis complements standard cluster identification algorithms that rely on Voronoï tessellations.

Estimation of Fracture Size and Probability Density Function by Setting Scanlines in Rectangular Sampling Window

Abstract Rock masses are very important materials in geotechnical engineering. In engineering rock mass, fracture is the relatively weak part of mechanical strength in rock mass and is the most important factor controlling the deformation, damage, and permeability of rock mass. Therefore, investigating fractures is very important for characterizing rock mass. This paper proposed a new approach by using uniformly equidistant orthogonal scanlines. Within the study context, the solution formula of fracture size is derived by establishing the space intersection model of arbitrary fracture and scanline, rectangular window, and a rectangular box with a rectangular window. Then, fractures were randomly generated in a certain size cube and compared with the traditional Kulatilake trace length integral evaluation method. The study results have shown that the proposed method is more reasonable and accurate. Then, this method was applied to an adit of Songta Hydropower Station. Finally, a new fracture diameter probability density estimation method was proposed, the fracture diameter of the normal distribution was verified, and the parameters of the probability density function obtained by the scanlines method were in agreement with the initial set parameters. In summary, the proposed scanlines method can well estimate the mean value of the fracture diameter and the probability density function of the fracture size.

Shape- and polymer-considered simulation to unravel the estuarine microplastics fate

Environmental microplastics (MPs) constitute various sizes, polymers, and shape components. In estuaries, such differences are related to the reliability of assessing the seaward fate of MPs, aggregation hotspots, and ecological risks. This study sets the MP particle mass gradient using the shape factor and size probability density function to categorically estimate the MP load in the surface layer of the Yangtze River Estuary (YRE), which is the largest contributor of plastics to the sea. During the high plastic input period in July, the optimized estimated MP load through the surface layer of the YRE was 9766 kg/month, which was overestimated by 821 kg/month based on the empirical average particle mass. While tracking MP transport classified by shape and polymer type, the resuspension of MPs that accumulate in the intertidal zone cannot be neglected. The average relative error of the simulation was as low as 19.6% after including the abovementioned factors. Finally, the simulation results of the sensitive regions were extracted to assess the new MP risk index, which considers shape, abundance, and polymer type. By introducing these essential tools, this study helps to understand the fate of riverine MPs entering estuaries, where valuable opportunities for removing MPs exist before they spread to the oceans.

Time-dependent deep learning predictions of 3D electrode particle-resolved microstructure effect on voltage discharge curves

Battery electrode particle-resolved microstructure featured with a 3D active material (AM) particle network bonded by carbon and binder (CBD) phase plays a significant role in determining battery performances. Tremendous physical computational models were developed to investigate these microstructure effects on voltage discharge curves, which however are computationally expensive. Machine learning (ML) could be the solution to address the high computational cost, but they still suffer from two challenges. (1) 3D particle-resolved microstructure that cannot be modelled in traditional ML that only correlate the scalar inputs and outs, (2) the outputs (voltage discharge curves), are time dependent. In this work, a deep learning (DL) approach was developed to tackle these two challenges. Using NMC 111 as one example, the results show a great match between the predictions and ground truth with a 99.98% validation accuracy. The superb extrapolation capabilities were demonstrated through the accurate prediction of microstructures fabricated with distinct process conditions and graded microstructures that exhibit completely different probability density function (PDF) of particle size. The trained DL model can be employed to boost the optimization of fabrication process parameters and microstructure design of electrodes. Moreover, the proposed framework can be extended to predict other dynamic behaviors of particle-resolved electrode microstructures.

Fokker–Planck equation for the particle size distribution function in KJMA transformations

The Fokker–Planck (FP) equation has been derived for describing the temporal evolution of the particle size probability density function (PDF) for KJMA (Kolmogorov–Johnson–Mehl–Avrami) transformations. The classical case of transformations with constant rates of both nucleation and growth, in 3D space, has been treated. Integration of the equation shows that the PDF is given by the superposition of one-parameter gamma distributions with time-dependent mean size given by the KJMA theory. The asymptotic behavior of the FP solution offers a demonstration of the conjecture, previously proposed by Pineda et al. (2004), according to which the set of crystals formed at the same time are gamma-distributed, with the parameter depending on crystal birth time. Computer simulations of the transformation with constant nucleation and growth rates show that the temporal evolution of the PDF, in volume domain, is in good agreement with the Johnson–Mehl PDF. The approach based on the FP equation provides a particle size PDF that exhibits such behavior.

Investigation on start-up performance of gas foil bearing considering wear topography evolution of non-Gaussian surface

To reveal the influence of topography evolution caused by abrasive wear of the top foil on the start-up performance of gas foil bearing (GFB), the transient gas mixed lubrication-wear model for non-Gaussian surface is developed by deriving the expression of wear particle size probability density function and establishing the relationship between wear particle size and surface height distribution. The effects of wear topography evolution and initial surface parameters on the start-up performance of GFB are studied. The results show that compared with the results ignoring wear topography evolution, the standard deviation, skewness and the minimum lift-off speed are smaller when the wear topography evolution is considered. The minimum lift-off speed and the wear volume diminish with the decrease of the initial standard deviation and skewness. This study can predict the wear topography evolution of top foil surface from start-up to lift-off.

On the grain size distribution function in KJMA compliant growth

The temporal evolution of the probability density function (PDF) of grain size in phase transformations by nucleation and growth has been computed by means of Kolmogorov’s second equation. The transformation is modeled in accord with the classical KJMA (Kolmogorov-Johnson-Mehl-Avrami) theory for linear growth of nuclei and either simultaneous or progressive nucleation. For 3D growth, it is shown that for progressive nucleation, second order term can be neglected in the differential equation for the PDF of grain volume. For site-saturated nucleation the variance of the distribution increases with time to attain the value of the Gamma distribution at the end of the transformation. It is demonstrated that in this case the PDF is given by convolution of Gaussian-like solutions of Kolmogorov’s equation with Gamma distribution. The validity of the model is checked by computer simulations available in literature.

Exploration of the Underlying Space in Microscopic Images via Deep Learning for Additively Manufactured Piezoceramics.

There has been a surge of interest in applying deep learning (DL) to microstructure generation and materials design. However, existing DL-based methods are generally limited in generating (1) microstructures with high resolution, (2) microstructures with high variability, (3) microstructures with guaranteed periodicity, and (4) highly controllable microstructures. In this study, a DL approach based on a stacked generative adversarial network (StackGAN-v2) is proposed to overcome these shortcomings. The presented modeling approach can reconstruct high-fidelity microstructures of additively manufactured piezoceramics with different resolutions, which are statistically equivalent to original microstructures either experimentally observed or numerically predicted. Advantages of the proposed modeling approach are also illustrated in terms of its capability in controlling the probability density function (PDF) of grain size, grain orientation, and micropore in a large space, which would have significant benefits in exploring the effects of these microstructure features on the piezoelectricity of piezoceramics. In the meantime, periodicity of the microstructures has been successfully introduced in the developed model, which can critically reduce the simulation volume to be considered as a representative volume element (RVE) during computational calculation of piezoelectric properties. Therefore, this DL approach can significantly accelerate the process of designing optimal microstructures when integrating with computational methods (e.g., fast Fourier spectral iterative perturbation (FSIPM)) to achieve desired piezoelectric properties. The proposed DL-based method is generally applicable to optimal design of a variety of periodic microstructures, allowing for maximum explorations of design spaces and fine manipulations of microstructural features.

MESH RESOLUTION EFFECTS ON PRIMARY ATOMIZATION SIMULATIONS

In this work, we use direct numerical simulation through a volume of fluid solver with adaptivemesh refinement to analyze the atomization of a pulsating round liquid jet with a narrow length-scale range configuration. We propose three grid sizes based on characteristic scales we estimate from the deformation and fragmentation processes of this problem. We compute drop statistics and general spray features for each case. We found that mesh resolution affects the atomization rate and the probability density function of droplet size and position, not only for the underresolved drops but for all liquid structures. The two simulations with higher grid refinement presented volume-weighted distribution with minor differences. Therefore, we propose assessing the accuracy of atomization simulations based on the volume fraction of underresolved structures.

Wet and dry grinding of coal in a laboratory-scale ball mill: Particle-size distributions

In this study, experimental data for wet- and dry-ground coal samples under wet and dry grinding are characterized by commonly used distribution functions. First, both the R-R and Swrebec functions have superior fitting performances for cumulative particle size curves compared to the other studied functions. On this basis, a time-dependent expression is drawn to describe the cumulative particle size distribution. Second, the R-R function produces a significantly superior fit to the relative mass distributions of the ground products compared to those of the others at a short grinding time. The goodness of fit for all distribution functions studied performs marginally worse at approximately 3min, which can be associated with a change in the dominant breakage mechanisms from impact to abrasion-chipping. With an increase in the grinding time, the G-G-S function is the optimal function for characterizing the particle size probability mass distributions of wet grinding, whereas the G-M function provides the best fitting performance when applied to the experimental dry-grinding data. Further, the optimal particle size probability density functions are associated with the difference in breakage mechanisms between wet and dry grinding.

Experimental study on spray characteristics of pressure-swirl nozzle in China advanced PWR containment

Pressurized Water Reactor (PWR) is the most mature technology for nuclear power plants and nuclear power devices. Spray systems play an essential role in the safety of a PWR. The spray characteristics of the nozzle are of crucial importance for the performance of the spray system. In the present study, a test loop with four rotatable branches was established to obtain spray characteristics data, including flow characteristic, spray cone angle (SCA), spray flux distribution and droplet size spectrum. The effect of inclined angles was taken into consideration when designing and conducting the experiment. A specific type of pressure-swirl nozzle used in China advanced PWR was employed for the study. According to experimental data, variations of pressure drop and the spray angles versus volume flow rate were obtained, and typical curves of spray flux distribution were plotted, and the change in the flux distribution was evident with the change in inclined angle. The flux distribution results were used to build 3D spray flux distribution models together with the spray cone angles tested. The droplet size spectrum was measured by laser diffraction analysis. Typical droplet size distributions were obtained by the relevant software, and the experimental Probability Density Function (PDF) of droplet size was derived and compared with several empirical distributions.

Reagent Dosage Control for the Antimony Flotation Process Based on Froth Size Pdf Tracking and an Index Predictive Model

A reagent dosage hybrid control strategy for the antimony flotation process is proposed in this work. This strategy consists of two parts: reagent dosage tracking control based on a froth size probability density function (PDF) and reagent dosage compensation control based on a distributed-machine vision predictive model. The proposed method was tested on a gold-antimony flotation process, and it improved tailings qualification rate, and reduced the tailings standard deviation. This method also efficiently accounts for the influence of disturbances on the flotation system and improves the stability and effectiveness of the flotation system.

An integrated model of the water transport in nonuniform compressed gas diffusion layers for PEMFC

Abstract Water transport in gas diffusion layer (GDL) is a very important issue for high power density Proton Exchange Membrane Fuel Cell (PEMFC). During the GDL and bipolar plate (BPP) assembly process, the water transport behavior is greatly influenced by the nonuniform compression on the GDL, which leads to uneven distribution of the internal mass transport pores. In this study, an integrated model is developed to predict the water transport in nonuniform compressed GDL. Firstly, a GDL compression deformation model is built to obtain the relationship between the GDL deformation and assembly clamping force based on energy method. Then, a water transport model is established by considering the probability density function (PDF) of the pore size for the compressed GDL. The accuracy of the integrated model has been verified by comparing with the finite element method (FEM) and the computational fluid dynamics (CFD) simulation results. The influence of assembly clamping force, GDL thickness and channel geometry are analyzed based on the integrated model. Drainage pressure increases monotonically with the assembly clamping force and is divided into three stages. For the baseline case, 0.2 mm of GDL thickness and small rib-channel ratio is conducive to improving drainage capacity. It provides the guidance for matching of GDL/BPP assembly condition and performance prediction of PEMFC.

Effects of the wake flow on bubble patterns downstream of a cylindrical nozzle

The aim of the presented study is to reveal patterns of the bubbles traveling in liquid flow. Air was injected into flowing water via a cylindrical nozzle that protrudes in the flow. Bubble images were recorded using high-speed photography. Water flows under no-bubble condition were measured using particle image velocimetry (PIV). Effects of the mean water velocity and the air flow rate were considered. Three bubble patterns with distinct features were unveiled. They were purely rising bubbles, peeling-off bubbles from large air pocket, wake-dominated bubbles. As the mean water velocity increases, the probability density function of bubble size varies from unimodal to multi-mode distribution and eventually recovers to unimodal distribution. During this process, overall bubble size decreases continuously. The carrier flow imposes a significant effect on bubble patterns. For low mean water velocities, the driving force for bubble motion in streamwise direction is weak and bubbles rise in a well-organized manner. At the nozzle outlet, the sharp turning of streamlines results in the formation of large air pocket. Vorticity elements downstream of the nozzle are related to bubble patterns. As the mean water velocity increases, collapse of large bubbles is intensified and resultant small bubbles manifest wake flow features.

Extension of the Cylindrical Root Model for Water Uptake to Non‐Regular Root Distributions

Core Ideas An equivalent cylinder radius PDF can be derived from a soil to root distance PDF. Simple analytical solutions for random and clustered root systems were obtained. The obtained equations were validated with 2D simulations of soil water flow. The water flow toward a single root is efficiently solved in cylindrical coordinates using the matrix flux potential. The upscaling of single root fluxes to whole root systems is commonly based on the assumption of regularly distributed and parallel aligned roots. To provide more accurate solutions for randomly distributed roots, the point to root distance probability density function (PDF) is transformed to an equivalent cylinder size PDF (fC), which is used for upscaling steady‐state and steady‐rate cylindrical root models while simple analytical solutions are derived. The regular root distribution assumption leads to large overestimations (∼50%) of root water uptake for randomly distributed roots. Several numerical comparisons with simulations of the two‐dimensional (2D) Richards equation for five different soil textures and five root densities show the good performance of the derived solutions under both limiting and non‐limiting soil hydraulic conditions. Moreover, an explicit analytical solution is given for negative‐binomial clustered roots. Under this specific assumption, the solution can be transformed to an equivalent random solution with a modified root length density. Simulation of root water uptake of different 2D root maps generated by a Poisson cluster process shows the effectiveness of the derived approximation for clustered roots.

Field-theoretic approach to the universality of branching processes

Branching processes are widely used to model phenomena from networks to neuronal avalanching. In a large class of continuous-time branching processes, we study the temporal scaling of the moments of the instant population size, the survival probability, expected avalanche duration, the so-called avalanche shape, the $n$-point correlation function and the probability density function of the total avalanche size. Previous studies have shown universality in certain observables of branching processes using probabilistic arguments, however, a comprehensive description is lacking. We derive the field theory that describes the process and demonstrate how to use it to calculate the relevant observables and their scaling to leading order in time, revealing the universality of the moments of the population size. Our results explain why the first and second moment of the offspring distribution are sufficient to fully characterise the process in the vicinity of criticality, regardless of the underlying offspring distribution. This finding implies that branching processes are universal. We illustrate our analytical results with computer simulations.

Improvement of fractal model for porosity and permeability in porous materials

The investigation of porosity and permeability of porous materials is a challenge due to its complex pore microstructures. In this paper, an improved model of porosity is proposed based on the pore filling method, in which the facts of microscopic observation are considered that the smaller pores will be found with the decreasing observation scale or increasing magnification. The model is verified by two-dimensional and three-dimensional experiments, and the results show that the improved model can accurately describe the porosity and the pore volume distribution of fractal porous materials. Besides, a probability density function of pore size without conventional restrictions is also proposed based on the improved model of porosity, which makes the scope of application of the model of tortuosity fractal dimension much wider. Moreover, an improved model of permeability is finally proposed based on the improved model of porosity and the probability density function. By comparing the predicted values of the new permeability model with the experimental data, it is confirmed that the permeability model proposed in this paper could accurately predict the permeability of fractal porous materials.

Flow pattern and holdup phenomena of low velocity oil-water flows in a vertical upward small diameter pipe

This paper is dedicated to the investigation of flow pattern and holdup phenomena in low velocity oil-water flows through a vertical upward pipe with 20 mm inner diameter (ID). Four typical flow patterns, namely dispersed oil-in-water slug flow (D OS/W), dispersed oil-in-water flow (D O/W), very fine dispersed oil-in-water flow (VFD O/W) and transition flow (TF) were detected using a radial dual-sensors conductance probe array. We compared our experimental flow pattern boundaries with those from published physical models. To further explore the local flow structures of different flow patterns, we extracted probability density function (PDF) of oil droplet size across pipe section. In addition, slippage effect was evaluated by water holdup measurement using quick-closing valve (QCV). It was found that slippage effect generally weakens with increasing water-cut and mixture velocity. Finally, some typical drift-flux models were used to calculate water holdup, which was further compared with the measured data from published literatures. The results show that flow pattern-based modified drift-flux models proposed in the present study perform well on water holdup prediction for small ID pipes, while drift-flux models proposed by Flores et al. present superiorities on predicting water holdup in large ID pipes.

Prognostic Algorithms for Flaw Growth Prediction in an Aircraft Wing

Prediction of future health state of an aerospace structure allows airlines to plan repair or replacement activities well in time. Correct planning ensures low probability of failure as well as low aircraft's downtime. Accurate prediction is possible through usage of the effective prognostic algorithms, based on historical trends and underlying suitable degradation models. A particle filter (PF)-based prognostic algorithm is proposed to predict posterior probability density function of flaw size in future. PF allows usage of nonlinear state-transition and measurement functions, along with non-Gaussian/multimodal noise distributions. The proposed algorithm is further complemented by a learning-based error model to estimate associated noises and increase prediction accuracy once measurements are not available. The proposed algorithm is applied on actual aerospace historical data of an in-service Airbus A310 aircraft. The historical data is divided into training phase and validation phase. A common aerospace structural flaw due to mechanical fatigue is the damage around the edges of countersunk (CSK) holes of an aircraft wing. This study is based on three-dimensional flaw growth around the CSK rivet holes. Accurate prediction of flaw propagation will enable the maintenance managers to plan repair or replacement action well in time, thus reducing the downtime of the aircraft.