Monte Carlo Ray-tracing Simulations Research Articles

A novel solar tower simulator for hydrogen regeneration by Thermo-Catalytic cracking of Ammonia: Energy, Exergy, and CFD investigations

This paper presents a novel concept of a central solar tower-based hydrogen regeneration system by thermo-catalytic cracking of ammonia. This includes a closed volumetric receiver for ammonia heating and cracking, a 2.45 kWe high-concentration radiation source, and an air-cooling system for the radiation source. The thermodynamic analysis shows that the net solar-to-hydrogen generation efficiency, first law efficiency, and exergy efficiency are 31 %, 82 %, and 79.8 %, respectively, for the optical efficiency of 35 %, the conversion efficiency of 99 %, and the waste heat recovery fraction of 20 %. The simulations with an ammonia flow rate of 1 kg/hr and flux concentration of 1200 Suns show the regeneration of 1.27 kg H2/day at Jodhpur and 1.57 kg H2/day at Ladakh. Monte Carlo ray tracing simulations revealed that the developed radiation source provides an average flux of ∼ 2 MW/m2 onto a spot size of 20 mm with an efficiency of about 31 %. Reynolds-averaged Navier Stokes-based analyses are used to design a novel air-cooling system for the radiation source. The computations show that (a) an unequal distribution of air mass flow rates at the front mḟ and back mḃ, should be preferred with mḟ>mḃ, and (b) for a total mass flowrate of 0.1 kg/s with mḟmḃ = 4, the maximum surface temperature of reflectors and Xe-arc lamps is less than 315 K and 610 K, respectively. The investigations revealed the feasibility of the concept and provided valuable insight for the planned experimental assessment and scale-up of the solar tower simulator.

General guide concepts for compact, high-brilliance neutron moderators

The trend in neutron sciences is toward integrating compact, high-brightness moderators into new or upgraded facilities. Transporting neutrons from the source to the sample position with a phase-space distribution tailored to specific requirements is crucial to leverage high source brilliance. We have investigated four guide concepts using Monte Carlo ray tracing simulations: Montel beamline with nested Kirkpatrick–Baez mirrors, curved-tapered beamline with a bender and straight sections, straight-elliptical beamline, and curved-elliptical beamline. The straight-elliptical (curved-elliptical) beamline features two half-ellipse guides connected by a straight (non-straight) guide section. The neutron transport efficiency and phase space homogeneity have been quantitatively compared. Our results show that the straight-elliptical beamline performs best because of few neutron bounces on the guide surface with small reflection angles, minimizing flux loss. The Montel beamline provides the best spatial confinement of neutrons within the desired region; however, there is a high thermal-neutron loss due to large reflection angles. The curved-tapered beamline suffers from significant flux loss due to high bounces, and it shows a non-uniform angular distribution related to broad ranges of bounces and reflection angles. The non-straight guide section of the curved-elliptical beamline increases the phase space inhomogeneity, leading to a spatially non-uniform beam profile. The results apply to general neutron instruments that require transporting thermal and cold neutrons from a compact, high-brilliance moderator to the sample location with a moderate phase-space volume.

Enhancing optical and thermohydraulic performance of linear Fresnel collectors with receiver tube enhancers: A numerical study

Linear Fresnel collector system is a type of solar thermal utilization installation. It is mainly used to focus solar radiation on the receiver tube to generate high temperature thermal energy which can be applied in many aspects. This study investigates five different types of receiver tube models using water as the flowing medium, which leads to improving heat transfer performance by means of being equipped with enhancers. The optical results and thermohydraulic performance of the Linear Fresnel collector system are evaluated through Monte Carlo Ray Tracing and Computational Fluid Dynamics (CFD) simulation methods. The obtained optical results reveal that the distribution of energy flux density concentrating on the receiver tube surpasses that of the parabolic trough collector system. In order to maintain an energy reception rate of 90.0% or higher, the installation height range of the receiver tube should be between 1468–1532 mm. When the sun-tracking error increases from 0.0° to 1.0°, the overall optical efficiency of the Linear Fresnel collector system decreases from 98.9% to 79.7%. The thermohydraulic performance shows that the enhancers can generate a swirling flow as its main feature, which effectively promotes the homogenization of the temperature field and destroys the boundary layer near the receiver tube wall, thus greatly improving the convective heat transfer in the receiver tube. However, the introduction of the enhancers also means that the pressure drop loss of the receiver tube increases. Furthermore, the impact of the Kenics mixer's enhancer on heat transfer and flow characteristics is further amplified as the rotation rate decreases, owing to its alternating and complex structure. Through a comparative analysis of the influence of different receiver tube models on Nusselt number (Nu), Frictional resistance coefficient (f) and comprehensive evaluation criterion (PEC) under four rotation rate conditions (y = 6.3, 8.1, 11.34, 18.9), the conclusions present that the receiver tube with an enhancer at a low rotation rate has a larger Nu than that at a high rotation rate, while the Frictional resistance coefficient changes little at different rotation rates. Therefore, the PEC is higher when the receiver tube is equipped with a low-rotation-rate enhancer. Moreover, the receiver tube equipped with a Kenics mixer at a rotation rate of 6.3 was found to be the overall optimum case, with the PEC value varying from 1.47 to 1.73.

Monte–Carlo ray-tracing studies of multiplexed prismatic graphite analyzers for the cold-neutron triple-axis spectrometer at the High Flux Isotope Reactor

A modern cold triple-axis spectrometer to study quantum condensed matter systems is planned for the High Flux Isotope Reactor (HFIR) at Oak Ridge National Laboratory. Here, we describe the conceptual principles and design of a secondary spectrometer using a multiplexed, prismatic analyzer system relying on graphite crystals and inspired by the successful implementation of the Continuous Angle Multiple Energy Analysis (CAMEA) spectrometers at the Paul Scherrer Institute. This project is currently known as MANTA for Multi-Analyzer Neutron Triple-Axis. We report Monte-Carlo ray-tracing simulations on a simple but realistic sample scattering kernel to further illustrate the prismatic analyzer concept’s workings, calibration, and performance. Then, we introduce a new statistical analysis approach based on the prismatic analyzer concept to improve the number of final energies measured on the spectrometer. We also study possible evolutions in the CAMEA design relevant for MANTA.

Optimizing Horticulture Luminescent Solar Concentrators via Enhanced Diffuse Emission Enabled by Micro-Cone Arrays.

Optimizing the photon spectrum for photosynthesis concurrently with improving crop yields presents an efficient and sustainable pathway to alleviate global food shortages. Luminescent solar concentrators (LSCs), consisting of transparent host matrices doped with fluorophores, show excellent promise to achieve the desired spectral tailoring. However, conventional LSCs are predominantly engineered for photon concentration, which results in a limited outcoupling efficiency of converted photons. Here, we introduce a scheme to implement LSCs into horticulture (HLSC) by enhancing light extraction. The symmetry of the device is disrupted by incorporating microcone arrays on the bottom surface to mitigate total internal reflection. Both Monte Carlo ray tracing simulations and experimental results have verified that the greatest enhancements in converted light extraction, relative to planar LSCs, are achieved using microcone arrays (base width 50 μm, aspect ratio 1.2) with extruded and protruded profiles (85.15 and 66.55% improvement, respectively). Angularly resolved transmission measurements show that the HLSC device exhibits a broad angular radiation distribution. This characteristic indicates that the HLSC device emits diffuse light, which is conducive to optimal plant growth.

Optical characterization of a solar concentrating dish system

Solar dishes concentrate the quasi-collimated sun light towards a focal point. Typically, the optical characterization consists of experimental flux map measurements and Monte-Carlo ray-tracing (MCRT) simulations, which are fitted based on experimental data and serve for the prediction of the distribution of the irradiation in receivers or reactors. However, the MCRT simulations usually rely on idealized mirror geometries (i.e. parabolic mirrors), neglecting the actual dish geometry resulting from its construction method (such as petals, facets or segments). Thus, the parameter fitting might yield unrealistic parameters and local radiative flux peaks (“hot spots”), which are often experimentally detected behind the focal plane, cannot be predicted. Here, we characterized the 7m diameter solar dish at EPFL comprising 27 petals with a nominal focal length of 3.8 m and a rim angle of 50.3°. The measured peak concentration was 1781 suns and the received integrated solar power was 20.0 kW over an 18 cm diameter spot. We proposed an advanced geometry approach for the MCRT model considering the curvature of the dish, i.e. geometries mathematically described by a parabola with an exponent not equal, but close to 2. The advanced geometry approach was then applied for predicting experimentally measured flux distribution in focal and off-focus planes. The fitted reflectivities were 70% and 86% for the idealized and advanced geometry approach, respectively. The idealized geometry approach found an unrealistically low value whereas the advanced geometry approach’s value was within 1% of the manufacturer’s specification. The advanced geometry approach predicted a parabola exponent of 1.94, which emphasizes that the best geometry to describe the solar dish deviates from a perfect paraboloid. We conclude that MCRT models should consider non-perfect parabolic geometries for solar dishes in order to obtain more realistic values for the fitted parameters.

Interpreting Heat Flux Measurements in a Vitiated Backward-Facing Step Flow

The investigation of the relative impact of convective and radiative heat flux from vitiated gases to a combustor wall was performed in a backward-facing step combustor. Backward-facing step flows have many flow features of gas turbine combustors (recirculation, impingement, and boundary-layer development), but in a two-dimensional, nonproprietary geometry. Reynolds number, gas temperature, and plate temperature were varied as heat flux was measured in different regions of the flow. Measurements were made using a heat flux sensor and radiometer, which measure total and radiative heat flux, respectively. The highest total heat flux was measured at the impingement location and the lowest in the recirculation zone. Separating the total heat flux into the radiative and convective components is not straightforward and requires simulations of the experiment. Spectrally resolved Monte Carlo ray-tracing simulations are used to scale the radiative heat flux measurements for the separation of convective and radiative heat flux from the total heat flux measurements. These calculations show that only about 30% of incident radiation is captured by the radiometer due to spectral differences between the two sensors. Once the radiation measurement is corrected, experimental results show higher radiative than convective heat transfer at low Reynolds numbers (Reh=2000), highlighting the impact of radiative heat transfer in hot gas environments.

Verification of the McStas code using two double axis neutron diffractometers

The Monte Carlo ray-tracing simulation package McStas is verified by modeling two double axis neutron diffractometers, KARL and KANDI-II, located at the Israel Research reactors IRR-1 and IRR-2, respectively. The advantage of simulating these instruments is their simple design and small number of components. We focus on the simulation of a Soller slit collimator as well as Pyrolytic Graphite and Copper single crystal monochromators. The simulation is compared to neutron flux measurements, performed via activation analysis along the neutron beam as well as observed diffraction patterns from different powder samples. Calculated diffraction patterns are found to agree with observations to within 20%–50% in reflection peak width, while calculated attenuation of the neutron beam agrees to within 20% when the entire instrument is considered, and can show a discrepancy of up to 60% for individual components. Finally, a possible optimization of the neutron flux in the KARL diffractometer is presented.

Numerical MCRT Simulation and Inter-Symbol Interference Analysis of Laser Multipath Transmission in Airborne Air-to-Air Channels

To address the issues of signal waveform distortion and inter-symbol interference, both of which lead to performance degradation in airborne laser communication due to multipath transmission in airborne air-to-air channels, theoretical analysis and Monte Carlo ray tracing (MCRT) simulation methods were employed. Based on airborne application conditions, this research conducted numerical simulations of the Gaussian beam multipath transmission in an air-to-air channel, with a focus on analyzing the impact of meteorological conditions, communication distance, transmitter power, and initial pulse width on the pulse time spreading characteristics of received optical signals. In addition, an analysis of these parameters’ impact on the inter-symbol interference (ISI) and bit error rate (BER) was conducted. The research findings can serve as a reference for the design of anti-interference techniques in airborne laser communication links.

Multi-Objective Bayesian Optimization Design of Elliptical Double Serpentine Nozzle

The use of traditional optimization methods in engineering design problems, specifically in aerodynamic and infrared stealth optimization for engine nozzles, requires a large number of objective function evaluations, therefore introducing a considerable challenge in terms of time constraints. In this paper, this limitation is addressed by using a sample-efficient multi-objective Bayesian optimization that takes Kriging as a surrogate model and Expected Hypervolume Improvement as the infill criterion. Using this approach, the probabilistic model is continuously established and updated, and the approximate Pareto front is obtained at a relatively small computational budget. The objective of this work is to evaluate the applicability of employing a multi-objective Bayesian optimization framework for the aerodynamic-infrared shape optimization of an elliptical double serpentine nozzle at 6 km flight condition, where the objective functions are evaluated by means of high-fidelity computational fluid dynamics and reversed Monte Carlo ray tracing simulations. We achieve good results in both infrared radiation signature reduction and aerodynamic performance improvement with a reasonable number of evaluations, indicating that the proposed method is effective and efficient for tackling the computationally intensive optimization challenges in the aircraft design.

2D Helium Atom Diffraction from a Microscopic Spot.

A method for measuring helium atom diffraction with micron-scale spatial resolution is demonstrated in a scanning helium microscope (SHeM) and applied to study a micron-scale spot on the (100) plane of a lithium fluoride (LiF) crystal. The positions of the observed diffraction peaks provide an accurate measurement of the local lattice spacing, while a combination of close-coupled scattering calculations and MonteCarlo ray-tracing simulations reproduce the main variations in diffracted intensity. Subsequently, the diffraction results are used to enhance image contrast by measuring at different points in reciprocal space. The results open up the possibility for using helium microdiffraction to characterize the morphology of delicate or electron-sensitive materials on small scales. These include many fundamentally and technologically important samples which cannot be studied in conventional atom scattering instruments, such as small grain size exfoliated 2D materials, polycrystalline samples, and other surfaces that do not exhibit long-range order.

Optical measurements and flow modeling to characterize solid volume fractions in gravity-driven particle curtains

Solid volume fraction is a defining characteristic of multiphase gas–solid flows. In the case of gravity-driven particle curtains, it changes significantly along the flow direction. This has implications for the performance of falling-particle solar receivers, which use particle curtains to capture and store heat from concentrated sunlight. This study reports on the characterization of solid volume fraction as a function of height in gravity-driven particle curtains using optical techniques and simulations performed with a discrete element method (DEM) model. The experimental domain is a hopper-fed vertical channel, and the modeling domain is designed to match the experimental setup. Solid volume fraction was obtained from experimentally measured transmittance using two techniques: (1) a thermopile detector paired with a visible light source and (2) a high-speed camera. The DEM simulations were calibrated by identifying the coefficient of static friction between particles that resulted in equivalent mass flow rates to the experiments. All techniques, including an analytical model derived from first principles and empirical relationships, showed very similar trends of solid volume fraction decreasing along the flow direction, where it drops off rapidly initially before leveling off in a power law relationship. Both experimental and computational tests were performed with 1 mm and 2 mm particles, and solid volume fraction is found to decrease with increasing particle size at any location in the channel. Furthermore, the solid volume fractions calculated from transmittance measurements were found to be sensitive to the optical properties of the particles through Monte Carlo ray tracing simulations. To account for this effect, a new scale factor dependent on particle reflectivity is derived, which leads to significantly improved agreement between experimental and simulated results. A simple, closed-form equation is proposed to transform measured transmittance to a solid volume fraction while accounting for particle reflectivities. Even though solid volume fractions above 0.1 were not measured in this study, the technique developed is suitable to measure even larger solid volume fractions, especially for more reflective particles.

Engineered Features and Artificial Neural Networks for the Identification of Temperature-Dependent Radiative Characteristics in Porous Media

The accurate and efficient prediction of radiative properties of materials plays a crucial role in numerous industrial and engineering applications. However, experimental measurement can be cumbersome, particularly for complex materials. As a result, numerical techniques such as Monte Carlo ray tracing (MCRT) have been developed to assess radiative properties. MCRT is a well-known tool for solving radiation heat transport (RHT). Despite its advantages, MCRT requires tracing a large bundle of light beams for convergency and veracity of outcomes, which necessitates further advancements in its employment. To this end, supervised learning algorithms can be used in conjunction with MCRT to provide a quick and accurate substitution for typical MCRT simulations. In this study, machine learning algorithms employing expertly designed features are used to evaluate the radiative properties of heterogeneous porous media, while considering the effects of conductive heat transfer. Directional features have been introduced into the sets of the features implemented in our previous study for the enhancement of predictions. The results show that the applied method can improve prediction accuracy compared to prior studies. Moreover, this approach is generalizable to other radiative heat transfer problems by extracting relevant characteristics in accordance with the involved physics.

Predicting light-matter interaction in semi-transparent elliptical packed beds using hybrid deep learning (HDL) approach

In this study, we develop and train a domain-expert supervised learning framework for reliable prediction of the radiative properties of heterogeneous porous media. Our chosen model is based on a hybrid deep learning (HDL) approach employing convolutional neural networks (CNN), thereby extracting in-vitro geometric features, and an artificial neural network (ANN) to concatenate in-situ features as well as physical features while training. Our work is based on a framework of ground truth data generation where an in-house, fine-grained pixel-based Monte Carlo ray tracing (MCRT) simulation algorithm is used for data annotation, and an efficient custom packing algorithm is used for polydisperse and heterogeneous random input configuration generation of elliptical particles. Two distinct CNN architectures, namely LeNet and VGG-16 have been used as hybrid deep learning model cores, are trained to estimate macro radiative properties (e.g., reflectivity, transmissivity, absorptivity) and evaluated. The HDL model with a LeNet core achieves remarkable accuracy, predicting absorptivity, reflectivity, and transmissivity with R2 values exceeding 0.99, 0.93 and 0.63 respectively. Meanwhile, the HDL with a VGG16 core can predict absorptivity, reflectivity, and transmissivity with the accuracy of R2>0.94, R2>0.86, and R2>0.62 respectively. Our findings suggest that while the necessary directional geometric features can contribute to higher accuracy in predictions using the HDL model, achieving greater predictability in transmission requires incorporating more information related to particle materials and molecular structure during training. The results demonstrate that the trained model can reliably be applied to estimate radiation in densely packed porous beds with arbitrarily shaped particles, polydispersity, and varied eccentricity.

Precise derivations of radiative properties of porous media using renewal theory

This work uses the mathematical machinery of Renewal/Ruin (surplus risk) theory to derive preliminary explicit estimations for the radiative properties of dilute and disperse porous media otherwise only computable accurately with Monte Carlo Ray Tracing (MCRT) simulations. Although random walk and Lévy processes have been extensively used for modeling diffuse processes in various transport problems and porous media modeling, relevance to radiation heat transfer is scarce, as opposed to other problems such as probe diffusion and permeability modeling. Furthermore, closed-form derivations that lead to tangible variance reduction in MCRT are widely missing. The particular angle of surplus risk theory provides a richer apparatus to derive directly related quantities. To the best of the authors’ knowledge, the current work is the only work relating the surplus risk theory derivations to explicit computations of ray tracing results in porous media. The paper contains mathematical derivations of the radiation heat transfer estimates using the extracted machinery along with proofs and numerical validation using MCRT.

Radiative view factor correlations in particulate media from ray tracing simulations and data-driven modeling

Thermal radiation has been extensively modeled in static particulate media with effective radiative properties or with statistical ray tracing techniques. However, these techniques are not directly compatible to evaluate radiative fluxes on discrete particles, which is more suitable for particle flow systems. This study focuses on such a discrete approach to compute radiative fluxes by developing view factors correlations for particle-particle and particle-wall. Training data is generated from physics-based Monte Carlo ray tracing simulations on a monodisperse, packed bed with solid volume fractions ranging from 0.016 to 0.45. This data was used to develop reduced-order correlations to determine particle-particle and particle-wall radiative view factors as a function of particle-particle and particle-wall separation distance, viewing angle, and the number of shading particles. Uniquely, we determine best-fit functions that are physically interpretable to account for shading effects by particles. A sigmoid function with a non-linear dependence on viewing angle governs the extent of shading by an intermediate particle. A correction factor with the particle-wall normal separation distance as the feature variable is introduced to account for shading effects between a particle and a planar wall surface. View factor correlations result in reliable and reasonably accurate predictions. For a solid volume fraction of 0.45, the root mean squared errors of particle-particle and particle-wall view factors are 2.7e-4 and 0.021 with corresponding training data in the ranges of 0–0.08 and 0–0.5 respectively. To scale these correlations for large number of particles, restricting shading detection up to 5 nearest neighbors is demonstrated to be an effective strategy to balance prediction accuracy with computational efficiency. With thousands of particles, the computational cost of proposed view factor correlations with thresholding of 5 shading particles is about 100 times faster than serial Monte Carlo ray tracing simulations for a solid volume fraction of 0.45.

Low-cost photoreactors for highly photon/energy-efficient solar-driven synthesis

Low-cost photoreactors for highly photon/energy-efficient solar-driven synthesis

In-depth insights into polarization-dependent light extraction mechanisms of AlGaN-based deep ultraviolet light-emitting diodes.

The AlGaN-based deep ultraviolet light-emitting diodes (DUV LEDs) dominated by transverse-magnetic (TM) polarized emission suffer from extremely poor light extraction efficiency (LEE) from their top surface, which severely limits the device performance. In this study, the underlying physics of polarization-dependent light extraction mechanisms of AlGaN-based DUV LEDs has been explored in depth via simple Monte Carlo ray-tracing simulations with Snell's law. It is especially worth noting that the structures of the p-type electron blocking layer (p-EBL) and multi-quantum wells (MQWs) have a significant impact on light extraction behavior, especially for TM-polarized emission. Thus, an artificial vertical escape channel (named GLRV) has been constructed to efficiently extract the TM-polarized light through the top surface, by adjusting the structures of the p-EBL, MQWs, sidewalls, and using the adverse total internal reflection in a positive manner. The results show that the enhancement times of the top-surface LEE is up to 18 for TM-polarized emission in the 300 × 300 µm2 chip comprising a single GLRV structure, and further increases to 25 by dividing this single GLRV structure into a 4 × 4 micro-GLRV array structure. This study provides a new perspective for understanding and modulating the extraction mechanisms of polarized light to overcome the inherently poor LEE for the TM-polarized light.

Effects of complex terrain on the shortwave radiative balance: a sub-grid-scale parameterization for the GFDL Earth System Model version 4.1

Abstract. Parameterizing incident solar radiation over complex topography regions in Earth system models (ESMs) remains a challenging task. In ESMs, downward solar radiative fluxes at the surface are typically computed using plane-parallel radiative transfer schemes, which do not explicitly account for the effects of a three-dimensional topography, such as shading and reflections. To improve the representation of these processes, we introduce and test a parameterization of radiation–topography interactions tailored to the Geophysical Fluid Dynamics Laboratory (GFDL) ESM land model. The approach presented here builds on an existing correction scheme for direct, diffuse, and reflected solar irradiance terms over three-dimensional terrain. Here we combine this correction with a novel hierarchical multivariate clustering algorithm that explicitly describes the spatially varying downward irradiance over mountainous terrain. Based on a high-resolution digital elevation model, this combined method first defines a set of sub-grid land units (“tiles”) by clustering together sites characterized by similar terrain–radiation interactions (e.g., areas with similar slope orientation, terrain, and sky view factors). Then, based on terrain parameters characteristic for each tile, correction terms are computed to account for the effects of local 3D topography on shortwave radiation over each land unit. We develop and test this procedure based on a set of Monte Carlo ray-tracing simulations approximating the true radiative transfer process over three-dimensional topography. Domains located in three distinct geographic regions (Alps, Andes, and Himalaya) are included in this study to allow for independent testing of the methodology over surfaces with differing topographic features. We find that accounting for the sub-grid spatial variability of solar irradiance originating from interactions with complex topography is important as these effects led to significant local differences with respect to the plane-parallel case, as well as with respect to grid-cell-scale average topographic corrections. We further quantify the importance of the topographic correction for a varying number of terrain clusters and for different radiation terms (direct, diffuse, and reflected radiative fluxes) in order to inform the application of this methodology in different ESMs with varying sub-grid tile structure. We find that even a limited number of sub-grid units such as 10 can lead to recovering more than 60 % of the spatial variability of solar irradiance over a mountainous area.

A numerical study of the spectral radiative properties of packed bed with mixed bauxite and silica spheres

Bauxite and silica particles have gained increasing attention for applications in the field of concentrated solar power. In this work, a Monte Carlo ray-tracing simulation is performed to predict the radiative properties (absorptance, reflectance, and transmittance) of packed beds with mixed bauxite and silica spherical particles at wavelengths of 0.5, 2.6, 9.0, and 9.35 µm. These wavelengths are representative for the visible, near-infrared, and the mid-infrared regions that are important for solar and thermal radiation, respectively. A repeating unit-column approach is used to mathematically represent the particle bed. The effects of particle mixing ratio, volume fraction, and wavelength on the predicted radiative properties are examined. The obtained radiative properties are inputted to an inverse method to retrieve the effective absorption and scattering coefficients as well as the scattering albedo, which may be later used in a continuous-scale radiative heat transfer analysis. Furthermore, the independent scattering model is used to obtain the absorption coefficient and scattering albedo based on the absorption and scattering cross sections predicted by a Monte Carlo algorithm for a single particle. It is shown that the independent scattering model underpredicts the scattering coefficient for opaque particles but overpredicts the scattering coefficient for semitransparent particles for sufficiently high particle volume fractions. The radiative properties calculated from the independent scattering model are compared to the full Monte Carlo simulation of the particle bed to examine the influence of particle mixing on dependent scattering.