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NASA NTRS1999

Generalized Functions for the Fractional Calculus

Previous papers have used two important functions for the solution of fractional order differential equations, the Mittag-Leffler functionE(sub q)[at(exp q)](1903a, 1903b, 1905), and the F-function F(sub q)[a,t] of Hartley & Lorenzo (1998). These functions provided direct solution and important understanding for the fundamental linear fractional order differential equation and for the related initial value problem (Hartley and Lorenzo, 1999). This paper examines related functions and their Laplace transforms. Presented for consideration are two generalized functions, the R-function and the G-function, useful in analysis and as a basis for computation in the fractional calculus. The R-function is unique in that it contains all of the derivatives and integrals of the F-function. The R-function also returns itself on qth order differ-integration. An example application of the R-function is provided. A further generalization of the R-function, called the G-function brings in the effects of repeated and partially repeated fractional poles.

Lorenzo, Carl F.↗
NASA NTRS2000

Initialized Fractional Calculus

This paper demonstrates the need for a nonconstant initialization for the fractional calculus and establishes a basic definition set for the initialized fractional differintegral. This definition set allows the formalization of an initialized fractional calculus. Two basis calculi are considered; the Riemann-Liouville and the Grunwald fractional calculi. Two forms of initialization, terminal and side are developed.

Lorenzo, Carl F.↗
NASA NTRS1999

Trapped Melt in IIIAB Irons: Solid/Liquid Elemental Partitioning During the Fractionation of the IIIAB Magma

Group IIIAB, the largest iron-meteorite group, shows compositional trends (including a three-order-of-magnitude It concentration range) indicating that it formed by fractional crystallization of a metallic magma. Because about 200 irons are available, and all degrees of crystallization are well represented, IIIAB offers an excellent set of samples for the study of crystallization at all depths of the asteroidal core. On log-log Ir-Au, and Ir-As diagrams IIIAB forms a broad band; the breadth represents real meteorite-to-meteorite variations, far outside experimental or sampling uncertainties. A successful model must explain the width of this band; I suggest that it mainly resulted from the trapping of parental magma within the crystallizing solid. Because S is essentially insoluble in metal, the abundance of FeS is a measure of the fraction of trapped liquid. The trapped-melt model is supported by the observation that irons having higher S contents plot closer to the inferred composition of the magmatic parental liquid. The lowest S values are found in the irons occupying the left envelope of the IIIAB Ir-Au or Ir-As compositional fields, thus it is this set of irons that should be interpreted as the solid products of a fractionating magma. This simplifies the modeling of the crystallization process and allows inferences regarding the distribution ratios for other elements in the evolved IIIAB system. The large (multiton) Cape York irons show wide variations in their trapped-melt fractions; their compositions seem best understood in terms of a low initial S content of the IIIAB magma, about 20 mg/g. The inferred initial IIIAB distribution coefficient for Ir, 4.6, is much higher than published values based on laboratory studies of low-S systems; I suggest that low-S (and low-P) partition-ratio measurements tend to err in the direction of unity. In IIIAB distribution coefficients for Au, As, and Ni were still < 1 when the most evolved IIIAB irons formed, another indication of a low initial S content.

Wasson, John T.↗
NASA NTRS2002

Finite-Strain Fractional-Order Viscoelastic (FOV) Material Models and Numerical Methods for Solving Them

Fraction-order viscoelastic (FOV) material models have been proposed and studied in 1D since the 1930's, and were extended into three dimensions in the 1970's under the assumption of infinitesimal straining. It was not until 1997 that Drozdov introduced the first finite-strain FOV constitutive equations. In our presentation, we shall continue in this tradition by extending the standard, FOV, fluid and solid, material models introduced in 1971 by Caputo and Mainardi into 3D constitutive formula applicable for finite-strain analyses. To achieve this, we generalize both the convected and co-rotational derivatives of tensor fields to fractional order. This is accomplished by defining them first as body tensor fields and then mapping them into space as objective Cartesian tensor fields. Constitutive equations are constructed using both variants for fractional rate, and their responses are contrasted in simple shear. After five years of research and development, we now possess a basic suite of numerical tools necessary to study finite-strain FOV constitutive equations and their iterative refinement into a mature collection of material models. Numerical methods still need to be developed for efficiently solving fraction al-order integrals, derivatives, and differential equations in a finite element setting where such constitutive formulae would need to be solved at each Gauss point in each element of a finite model, which can number into the millions in today's analysis.

Freed, Alan D.↗
NASA NTRS2003

Ion Species Fractions in the Far-Field Plume of a High-Specific Impulse Hall Thruster

An ExB probe was used to measure the ion species fractions of Xe(+), Xe(2+), and Xe(3+) in the far-field plume of the NASA-173Mv2 laboratory-model Hall thruster. The thruster was operated at a constant xenon flow rate of 10 milligrams per second and discharge voltages of 300 to 900 V. The ExB probe was placed two meters downstream of the thruster exit plane on the thruster centerline. At a discharge voltage of 300 V, the species fractions of Xe(2+) and Xe(3+) were lower, but still consistent with, previous Hall thruster studies using other mass analyzers. Over discharge voltages of 300 to 900 V, the Xe(2+) species fractions increased from 0.04 to 0.12 and the Xe(3+) species fraction increased from 0.01 to 0.02.

Hofer, Richard R.↗
NASA NTRS2006

A Climatology of Midlatitude Continental Clouds from the ARM SGP Central Facility: Cloud Fraction and Radiative Forcing - Part II

Data collected at the Department of Energy Atmospheric Radiation Measurement (ARM) Southern Great Plains (SGP) central facility are analyzed for determining the variability of cloud fraction and radiative forcing at several temporal scales between January 1997 and December 2002. Cloud fractions are estimated for total cloud cover and for single-layer low (0-3 km), middle (3-6 km), and high clouds (greater than 6 km) using ARM SGP ground-based paired lidar-radar measurements. Shortwave (SW), longwave (LW), and net cloud radiative forcings (CRF) are derived from up- and down-looking standard precision spectral pyranometers and precision infrared radiometer measurements. The annual averages of total, and single-layer, nonoverlapped low, middle and high cloud fractions are 0.49, 0.11, 0.03, and 0.17, respectively. Total and low cloud amounts were greatest from December through March and least during July and August. The monthly variation of high cloud amount is relatively small with a broad maximum from May to August. During winter, total cloud cover varies diurnally with a small amplitude, mid-morning maximum and early evening minimum, and during summer it changes by more than 0.14 over the daily cycle with a pronounced early evening minimum. The diurnal variations of mean single-layer cloud cover change with season and cloud height. Annual averages of all-sky, total, and single-layer high, middle, and low LW CRFs are 21.4, 40.2, 16.7, 27.2, and 55.0 Wm(sup -2), respectively; and their SW CRFs are -41.5, -77.2, -37.0, -47.0, and -90.5 Wm(sup -2). Their net CRFs range from -20 to -37 Wm(sup -2). For all-sky, total, and low clouds, the maximum negative net CRFs of -40.1, -70, and -69.5 Wm(sup -2), occur during April; while the respective minimum values of -3.9, -5.7, and -4.6 Wm(sup -2), are found during December. July is the month having maximum negative net CRF of -46.2 Wm(sup -2) for middle clouds, and May has the maximum value of -45.9 Wm(sup -2) for high clouds. An uncertainty analysis demonstrates that the calculated CRFs are not significantly affected by the difference between clear-sky and cloudy conditions. A more comprehensive cloud fraction study from both surface and satellite observations will follow.

Dong, Xiquan↗
NASA NTRS2006

A Climatology of Midlatitude Continental Clouds from the ARM SGP Site: Cloud Fraction and Surface Radiative Forcing - Part II

Data collected at the Department of Energy Atmospheric Radiation Measurement (ARM) Southern Great Plains (SGP) Central Facility (SCF) are analyzed to determine the monthly and hourly variations of cloud fraction and radiative forcing between January 1997 and December 2002. Cloud fractions are estimated for total cloud cover and for single-layered low (0-3 km), middle (3-6 km), and high clouds (more than 6 km) using ARM SCG ground-based paired lidar-radar measurements. Shortwave (SW) and longwave (LW) fluxes are derived from up- and down-looking standard precision spectral pyranometers and precision infrared radiometer measurements with uncertainties of approximately 10 Wm(exp -2). The annual averages of total, and single-layered low, middle and high cloud fractions are 0.49, 0.11, 0.03, and 0.17, respectively. Both total and low cloud amounts peak during January and February and reach a minimum during July and August, high clouds occur more frequently than other types of clouds with a peak in summer. The average annual downwelling surface SW fluxes for total and low clouds (151 and 138 Wm(exp-2), respectively) are less than those under middle and high clouds (188 and 201 Wm(exp -2), respectively), but the downwelling LW fluxes (349 and 356 Wm(exp -2)) underneath total and low clouds are greater than those from middle and high clouds (337 and 333 Wm(exp -2)). Low clouds produce the largest LW warming (55 Wm(exp -2) and SW cooling (-91 Wm(exp -2)) effects with maximum and minimum absolute values in spring and summer, respectively. High clouds have the smallest LW warming (17 Wm(exp -2)) and SW cooling (-37 Wm(exp -2)) effects at the surface. All-sky SW CRF decreases and LW CRF increases with increasing cloud fraction with mean slopes of -0.984 and 0.616 Wm(exp -2)%(exp -1), respectively. Over the entire diurnal cycle, clouds deplete the amount of surface insolation more than they add to the downwelling LW flux. The calculated CRFs do not appear to be significantly affected by uncertainties in data sampling and clear-sky screening. Traditionally, cloud radiative forcing includes, not only the radiative impact of the hydrometeors, but also the changes in the environment. Taken together over the ARM SCF, changes in humidity and surface albedo between clear and cloudy conditions offset approximately 20% of the NET radiative forcing caused by the cloud hydrometeors alone. Variations in water vapor, on average, account for 10% and 83% of the SW and LW CRFs, respectively, in total cloud cover conditions. The error analysis further reveals that the cloud hydrometeors dominate the SW CRF, while water vapor changes are most important for LW flux changes in cloudy skies. Similar studies over other locales are encouraged where water and surface albedo changes from clear to cloudy conditions may be much different than observed over the ARM SCF.

Xi, B.↗
NASA NTRS2011

Obtaining the Grobner Initialization for the Ground Flash Fraction Retrieval Algorithm

At optical wavelengths and from the vantage point of space, the multiple scattering cloud medium obscures one's view and prevents one from easily determining what flashes strike the ground. However, recent investigations have made some progress examining the (easier, but still difficult) problem of estimating the ground flash fraction in a set of N flashes observed from space In the study by Koshak, a Bayesian inversion method was introduced for retrieving the fraction of ground flashes in a set of flashes observed from a (low earth orbiting or geostationary) satellite lightning imager. The method employed a constrained mixed exponential distribution model to describe the lightning optical measurements. To obtain the optimum model parameters, a scalar function of three variables (one of which is the ground flash fraction) was minimized by a numerical method. This method has formed the basis of a Ground Flash Fraction Retrieval Algorithm (GoFFRA) that is being tested as part of GOES-R GLM risk reduction.

Solakiewicz, R.↗
NASA NTRS2012

The Ground Flash Fraction Retrieval Algorithm Employing Differential Evolution: Simulations and Applications

The ability to estimate the fraction of ground flashes in a set of flashes observed by a satellite lightning imager, such as the future GOES-R Geostationary Lightning Mapper (GLM), would likely improve operational and scientific applications (e.g., severe weather warnings, lightning nitrogen oxides studies, and global electric circuit analyses). A Bayesian inversion method, called the Ground Flash Fraction Retrieval Algorithm (GoFFRA), was recently developed for estimating the ground flash fraction. The method uses a constrained mixed exponential distribution model to describe a particular lightning optical measurement called the Maximum Group Area (MGA). To obtain the optimum model parameters (one of which is the desired ground flash fraction), a scalar function must be minimized. This minimization is difficult because of two problems: (1) Label Switching (LS), and (2) Parameter Identity Theft (PIT). The LS problem is well known in the literature on mixed exponential distributions, and the PIT problem was discovered in this study. Each problem occurs when one allows the numerical minimizer to freely roam through the parameter search space; this allows certain solution parameters to interchange roles which leads to fundamental ambiguities, and solution error. A major accomplishment of this study is that we have employed a state-of-the-art genetic-based global optimization algorithm called Differential Evolution (DE) that constrains the parameter search in such a way as to remove both the LS and PIT problems. To test the performance of the GoFFRA when DE is employed, we applied it to analyze simulated MGA datasets that we generated from known mixed exponential distributions. Moreover, we evaluated the GoFFRA/DE method by applying it to analyze actual MGAs derived from low-Earth orbiting lightning imaging sensor data; the actual MGA data were classified as either ground or cloud flash MGAs using National Lightning Detection Network[TM] (NLDN) data. Solution error plots are provided for both the simulations and actual data analyses.

Koshak, William↗
NASA NTRS2012

Can Lightning Produce Significant Levels of Mass-Independent Oxygen Isotopic Fractionation in Nebular Dust?

Based on recent evidence that oxide grains condensed from a plasma will contain oxygen that is mass independently fractionated compared to the initial composition of the vapor, we present a first attempt to evaluate the potential magnitude of this effect on dust in the primitive solar nebula. This assessment relies on previous studies of nebular lightning to provide reasonable ranges of physical parameters to form a very simple model to evaluate the plausibility that lightning could affect a significant fraction of nebular dust and that such effects could cause a significant change in the oxygen isotopic composition of solids in the solar nebula over time. If only a small fraction of the accretion energy is dissipated as lightning over the volume of the inner solar nebula, then a large fraction of nebular dust will be exposed to lightning. If the temperature of such bolts is a few percent of the temperatures measured in terrestrial discharges, then dust will vaporize and recondense in an ionized environment. Finally, if only a small average decrease is assumed in the O-16 content of freshly condensed dust, then over the last 5 million years of nebular accretion the average delta O-17 of the dust could increase by more than 30 per mil. We conclude that it is possible that the measured " slope 1" oxygen isotope line measured in meteorites and their components represents a time-evolution sequence of nebular dust over the last several million years of nebular evolution O-16-rich materials formed first, then escaped further processing as the average isotopic composition of the dust graduaUy became increasingly depleted in O-16 .

Nuth, Joseph A.↗
NASA NTRS2014

A Statistical Treatment of Bioassay Pour Fractions

The binomial probability distribution is used to treat the statistics of a microbiological sample that is split into two parts, with only one part evaluated for spore count. One wishes to estimate the total number of spores in the sample based on the counts obtained from the part that is evaluated (pour fraction). Formally, the binomial distribution is recharacterized as a function of the observed counts (successes), with the total number (trials) an unknown. The pour fraction is the probability of success per spore (trial). This distribution must be renormalized in terms of the total number. Finally, the new renormalized distribution is integrated and mathematically inverted to yield the maximum estimate of the total number as a function of a desired level of confidence ( P(<n)=LOC ). Selected results of the indicated numerical calculations are presented. For LOC=0.5, or the likely value, the estimates differ little from the usual calculation: the number of spores counted divided by the pour fraction. The extension to recovery efficiency corrections is also presented. Now the product of recovery efficiency and pour fraction may be small enough that the likely value may be much larger than the usual calculation: the number of spores divided by that product. The use of this analysis would not be limited to microbiological data.

000000↗
NASA NTRS2014

Investigation of the Effects of Cathode Flow Fraction and Position on the Performance and Operation of the High Voltage Hall Accelerator

The National Aeronautics and Space Administration (NASA) Science Mission Directorate In- Space Propulsion Technology office is sponsoring NASA Glenn Research Center (GRC) to develop a 4 kW-class Hall thruster propulsion system for implementation in NASA science missions. Tests were performed within NASA GRC Vacuum Facility 5 at background pressure levels that were six times lower than what has previously been attained in other vacuum facilities. A study was conducted to assess the impact of varying the cathode-to-anode flow fraction and cathode position on the performance and operational characteristics of the High Voltage Hall Accelerator (HiVHAc) thruster. In addition, the impact of injecting additional xenon propellant in the vicinity of the cathode was also assessed. Cathode-to-anode flow fraction sensitivity tests were performed for power levels between 1.0 and 3.9 kW. It was found that varying the cathode flow fraction from 5 to approximately 10% of the anode flow resulted in the cathode-to-ground voltage becoming more positive. For an operating condition of 3.8 kW and 500 V, varying the cathode position from a distance of closest approach to 600 mm away did not result in any substantial variation in thrust but resulted in the cathode-to-ground changing from -17 to -4 V. The change in the cathode-to-ground voltage along with visual observations indicated a change in how the cathode plume was coupling to the thruster discharge. Finally, the injection of secondary xenon flow in the vicinity of the cathode had an impact similar to increasing the cathode-to-anode flow fraction, where the cathode-to-ground voltage became more positive and discharge current and thrust increased slightly. Future tests of the HiVHAc thruster are planned with a centrally mounted cathode in order to further assess the impact of cathode position on thruster performance.

Cathode Coupling↗
NASA NTRS2015

Effect of Cloud Fraction on Near-Cloud Aerosol Behavior in the MODIS Atmospheric Correction Ocean Color Product

Characterizing the way satellite-based aerosol statistics change near clouds is important for better understanding both aerosol-cloud interactions and aerosol direct radiative forcing. This study focuses on the question of whether the observed near-cloud increases in aerosol optical thickness and particle size may be explained by a combination of two factors: (i) Near-cloud data coming from areas with higher cloud fractions than far-from-cloud data and (ii) Cloud fraction being correlated with aerosol optical thickness and particle size. This question is addressed through a statistical analysis of aerosol parameters included in the MODIS (MODerate resolution Imaging Spectroradiometer) ocean color product. Results from ten Septembers (2002-2011) over part of the northeast Atlantic Ocean confirm that the combination of these two factors working together explains a significant but not dominant part (in our case, 15%-30%) of mean optical thickness changes near clouds. Overall, the findings show that cloud fraction plays a large role in shaping the way aerosol statistics change with distance to clouds. This implies that both cloud fraction and distance to clouds are important to consider when aerosol-cloud interactions or aerosol direct radiative effects are examined in satellite or modeling studies.

Remote sensing↗
NASA NTRS2020

A 3D Model to Predict Explicit Morphologies and Volume Fraction of Lack-of-Fusion Pores Generated in Selective Laser Melting Processes

The performance of an additively manufactured (AM) component is dependent on the distribution of process-induced defects in addition to the complex microstructure of the material, surface roughness of the component and the process-induced residual stresses. For instance, it has been well demonstrated that lack-of-fusion (LoF) pores produced in the selective laser melting (SLM) AM process can significantly limit the fatigue performance of the material. Although two-dimensional (2D) models exist to predict the 2D profiles of LoF pores, the 2D pore profiles cannot be directly inserted into a three-dimensional (3D) microstructure domain that is output from several prevailing process simulation packages. A few commercial packages that simulate the SLM process can predict LoF pores in a 3D domain, but the morphologies of LoF pores are voxelated and hence do not capture sharp corners of the pores, thereby obviating their use in fatigue crack initiation studies. In order to address the aforementioned gaps, a high-fidelity model that predicts not only the volume fraction, but also the explicit 3D morphologies and spatial distributions of LoF pores has been developed using a computer aided design-based environment. The model has been partially validated for Ti-6Al-4V alloy by comparing the predictions of the volume fraction of LoF pores predicted by the model with experimental data obtained from the literature. Absolute error in predicted volume fraction of LoF pores varied between 5.16% and 1.87% for energy density values between 13 J/mm3 and 45 J/mm3 where a significant amount (over 3% volume fraction) of LoF porosity was measured. The absolute error was within 1.87% for energy density values greater than 45 J/mm3.

Saikumar R. Yeratapally↗
NASA NTRS2021

Evaluation of VIIRS and MODIS snow cover fraction in High-Mountain Asia using Landsat 8 OLI

We present thefirst application of the Snow Covered Area and Grain size model (SCAG) tothe Visible Infrared imaging Radiometer Suite (VIIRS) and assess these retrievals withfiner-resolution fractional snow cover maps from Landsat 8 Operational Land Imager (OLI).Because Landsat 8 OLI avoids saturation issues common to Landsat 1–7 in the visiblewavelengths, we re-assess the accuracy of the SCAG fractional snow cover maps fromModerate Resolution Imaging Spectroradiometer (MODIS) that were previously evaluatedusing data from earlier Landsat sensors. Use of the fractional snow cover maps fromLandsat 8 OLI shows a negative bias of−0.5% for MODSCAG and−1.3% for VIIRSCAG,whereas previous MODSCAG evaluations found a bias of−7.6% in the Himalaya. Wefindsimilar root mean squared error (RMSE) values of 0.133 and 0.125 for MODIS and VIIRS,respectively. The Recall statistic (probability of detection) for cells with more than 15%snow cover in this challenging steep topography was found to be 0.90 for both MODSCAGand VIIRSCAG, significantly higher than previous evaluations based on Landsat 5Thematic Mapper (TM) and 7 Enhanced Thematic Mapper Plus (ETM+). In addition,daily retrievals from MODIS and VIIRS are consistent across gradients of elevation, slope,and aspect. Different native resolutions of the gridded products at 1 km and 500 m forVIIRS and MODIS, respectively, result in snow cover maps showing a slightly differentdistribution of values with VIIRS having more mixed pixels and MODIS having 7% morepure snow pixels. Despite the resolution differences, the snow maps from both sensorsproduce similar total snow-covered areas and snow-line elevations in this region, withR2values of 0.98 and 0.88, respectively. Wefind that the SCAG algorithm performsconsistently across various spatial resolutions and that fractional snow cover mapsfrom the VIIRS instruments aboard Suomi NPP, JPPS–1, and JPPS–2 can be asuitable replacement as MODIS sensors reach their ends of life.

Karl Rittger↗
NASA NTRS2020

Designing Trajectories Resilient to Missed Thrust Events Using Expected Thrust Fraction

With the adoption of efficient low-thrust propulsion methods, the probability of a missed thrust event occurring has become a significant concern for short and long- duration missions. If the missed thrust events take place during a critical portion of the trajectory, the mission can be compromised. Therefore, it is essential to de- velop trajectories that are resilient to missed thrust events. This paper investigates the use of expected thrust fraction, which embeds the stochastic nature of missed thrust events into a deterministic optimal control problem. The performance of trajectories designed using expected thrust fraction is compared with traditionally designed trajectories to measure changes in resiliency to missed thrust events. In this investigation, trajectories designed using expected thrust fraction arrive with a median lateness half that of traditionally designed trajectories. Using expected thrust fraction can help astrodynamicists mitigate risks posed by the use of low- thrust propulsion.

Laipert, Frank E.↗
NASA NTRSDate not supplied

Toward Consistent Long-term Records of Cloud Fraction from MODIS and VIIRS for CERES

In order to produce a long-term and stable climate record of Earth’s energy budget for NASA’s Clouds and Earth’s Radiant Energy System (CERES) project, a consistent cloud fraction record across different measurement platforms­­ is a crucial first step. As Aqua satellite is approaching the end of its operational lifetime, NOAA-20 VIIRS observations on the JPSS-2 spacecraft will be used for cloud detection to continue the long-term Earth energy budget record. Two steps are designed to achieve this goal. The first step is a quick approach that revises the CERES NOAA-20 VIIRS Edition 1 A (CV Ed1A) cloud mask to incorporate Cross-track Infrared Sounder (CrIS) water vapor and CO_2 channels into the VIIRS cloud mask (CV Ed1B) to produce a consistent cloud fraction with CERES MODIS Edition 4 (CM Ed4) to avoid potential discontinuity when Aqua orbit drifts beyond tolerance before the next Edition is completed. The second step is to develop, as part of CERES next edition (Ed5), a unified MODIS and VIIRS cloud mask using common channels to produce consistent cloud products with improved radiative transfer models, refined clear sky predictions, new GMAO reanalysis products, and the latest radiance collections of MODIS and VIIRS data. This poster will present cloud fraction comparison between CV Ed1B and CM Ed4, as well as the preliminary results of ongoing Ed5 including pixel level cloud mask results, monthly global cloud fraction differences between MODIS and VIIRS (consistency), and initial validation using CALIPSO data (accuracy).

CERES↗
DOE OSTI2025

Gaussian integral method for void fraction

Here, a novel method, the Gaussian Integral Method (GIM), is presented for calculating void fractions in Computational Fluid Dynamics–Discrete Element Method (CFD-DEM) simulations. GIM is versatile and applicable to various grid types, including structured and unstructured polyhedral meshes, without requiring special boundary treatments. An optimization technique is introduced to make GIM independent of grid resolution and type. The method is validated against experimental data from a fluidized bed, demonstrating that GIM produces realistic simulations closely resembling experimental observations. Additionally, unstructured polyhedral grids using GIM outperform structured grids of equivalent resolution, yielding results more aligned with experimental data. The gradient of the void fraction is computed in the CFD solver and utilized in the DEM solver for precise estimation at particle locations. Overall, GIM provides an effective solution for void fraction calculations in particulate media simulations with complex geometries, enhancing the accuracy and applicability of CFD-DEM simulations for industrial processes.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗