Radiation Distribution Factor Research Articles

Determination of interparticle radiative heat transfer and radiation absorption for ellipsoidal particle beds using Monte Carlo ray tracing

The calculation of radiative heat transfer between particles in particle-based solar systems presents significant challenges in terms of computational cost and accuracy. This study addresses these challenges by employing the discrete element method to generate random particle packings and the Monte Carlo ray tracing method to investigate the influence of particle shape and volume fraction (ϕ) on interparticle radiation transfer and the absorption of incident radiation. The results reveal that, at low volume fractions, the shape and orientation of ellipsoidal particles have a pronounced impact on direct radiative transfer. Oblate particles exhibit the highest average radiation distribution factor (RDF), and the dispersion of RDF data is more prominent for oblate and elongated prolate particles compared to spheres. As ϕ increases to 0.35, the contribution of indirect transfer from neighboring particles becomes significant for oblate particles, resulting in a reduction in the dispersion of RDF data. Furthermore, when ϕ reaches 0.55, the decisive factor in determining RDF becomes the shielding effect of particles positioned between the emitting and receiving particles. The average RDF tends to converge among particles of different shapes, although oblate and elongated prolate particles still exhibit considerably larger RDF dispersion compared to spherical and near-spherical particles. Additionally, the effective absorptivity of the particle bed for incident radiation decreases as the aspect ratio of particles increases at all volume fractions. Moreover, the transmitted fraction demonstrates an exponential relationship with the depth, and beds composed of spherical particles exhibit better radiation transmittance compared to beds containing non-spherical particles.

Lp-norm relaxation approach for 3D measurement via instantaneous light field of ethylene premixed flames

Optical imaging technology based on flame spontaneous radiation is a promising tool for combustion diagnosis. However, the existing radiation thermometry technology faces the problems of low angular resolution and low efficiency in solving radiative transfer. Therefore, a novel optical imaging model is proposed, which combines light field imaging theory with equation-solving integral equation method based on radiation distribution factor to effectively decouple the radiative properties related to transfer process and the temperature-related source terms. Aiming at the problem that traditional method may cause over-smooth effect, Lp-norm regularization based on non-negative constraint is established. Results show that Lp-norm can improve reconstruction quality and robustness, but p value needs to be reasonably selected according to noise level and flame inhomogeneity. Experimental study of ethylene/air-premixed flames shows that the methodology can realize 3D measurement (900 ∼ 1550 K). Compared with laser absorption spectroscopy and thermocouple, error is better than 3 % at height above burner 15 mm.

An efficient reconfigurable optimal source detection and beam allocation algorithm for signal subspace factorization

<span lang="EN-US">Now a <span>days, huge amount of data is communicated through channels in wireless network. It requires an efficient parallel operation for the optimal utilization of frequency, time allocation and coding model for signal subspace factorization in smart antenna. In view of this requirement, an efficient reconfigurable optimal source detection and beam allocation algorithm (RoSDBA) is proposed. The proposed algorithm is able to allocate desired signal to the user space to reduce the noise and also for efficient allocation of subspace to remove disturbance in all directions. The proposed method efficiently utilizes the antenna array elements by accurate identification and allocation of antenna array elements such as individual radiators, radiation beam, signal strength, and disturbance factor. With respect to simulation analysis, the proposed method shows better performance for the resolution, radiation beam allocations, identification bias, distribution factor and time taken for the detection of various</span> array arrangements and source numbers.</span>

Three-dimensional inhomogeneous temperature tomography of confined-space flame coupled with wall radiation effect by instantaneous light field

Optical tomography has been demonstrated to be a powerful tool for the three-dimensional (3D) measurement of the dynamic characteristics of turbulent flame. Many practical visualizations of reaction flows require imaging through optical windows on the combustor, which have complex interactions with wall temperature, reflection characteristics, and limited optical access, which may seriously influence the imaging quality. Previous work focused on unconfined flame imaging, and lacked the study of the multiple reflection effects of the wall and the tomographic reconstruction within a wide temperature range. To solve these problems, the effect of chamber wall radiation on the photothermal information is analyzed in the forward problem, and the flame temperature with wall radiation is reconstructed in the inverse problem. In the forward problem, the light field acquisition model of spontaneous emission of the confined flame is developed through the backward Monte Carlo method based on the radiation distribution factor, which can comprehensively consider the essential effects such as radiative attenuation, wall reflection, medium scattering interference, and limited optical access. The recorded light field signals indicate that the reflection pattern and transfer mechanism of the wall material could affect the contribution of radiative source terms in different discrete domains to the detected energy. A novel tomography algorithm, namely, the adaptive threshold segmentation iterative regularization (ATSIR) method, has been proposed to extend the temperature range of tomography. The key of this method is to carry out a two-step reconstruction of the high and low-temperature regions through the temperature threshold, and introduce the prior smoothing information to alleviate the ill-posedness of tomography. The visualization results show that the well-established algorithms (e.g., least-square QR-factorization and regularization) have lower reconstruction quality for the combustion field within a wide temperature range, even if the regularization term is added. However, the proposed ATSIR method has the advantages of better suppression of fluctuations, smaller reconstruction error, and stronger anti-noise ability. The detailed analysis of the radiative transfer mode and 3D temperature visualization of the combustion field in a confined space can provide a valuable guideline for the safety design of the engine combustion chamber and fuel utilization.

Particle non-isothermality effect on the radiative thermal conductivity in dense particulate systems

Emerging high-temperature applications require a deeper understanding of radiative heat transfer in particulate systems, while existing theoretical tools are still limited in estimating the influence of the highly variable particle parameters (morphology, size, and optical properties) on heat transfer characteristics. In this work, a novel method is developed to determine radiative heat transfer in optically thick particulate media based on the radiation distribution factor. The important parameters of characterizing radiative heat transfer in particulate systems, i.e., transport coefficient and radiative thermal conductivity, are directly calculated by the Monte Carlo ray tracing method at particle scale. Two ideal cases of high and low thermal conductivity of solid particles, characterized by radiative Biot number Bi, are considered to investigate the effect of particle non-isothermality on the radiative thermal conductivity, which correspond to the cases of isothermal (Bi<<1) and non-isothermal particles (Bi>>1), respectively. The results show that radiative transfer is more intensive in isothermal particles due to the additional contribution of internal heat transfer in highly conductive particles. The difference between the limits of isothermal particles and non-isothermal particles is significant at high values of solid fraction and surface emissivity. In addition, the results are used to estimate the applicability domain of the analytic multiphase approach. This work brings new insights to the understanding of radiative heat transfer in particulate media.

Thermal performance of a directional heat radiation device utilizing the Monte Carlo ray tracing method

Conventional air conditioning systems heat and cool an entire space for the total volume but are deficient in terms of energy efficiency. Partial and personalized devices rarely take into account moving occupants and are deficient in terms of flexibility. To address these problems, a directional heat radiation (DHR) device was introduced. The effectiveness of the DHR device under heating conditions was verified by experiments and the thermal performance of the DHR device under three different arrangements was analyzed. Based on energy balance, the radiation distribution factor calculation model and the heat transfer model of the DHR device were established by combining the Monte Carlo ray tracing method and the thermal network method. The results showed that the thermal performance of the DHR device arranged level downward was best and twice that of the level upward arrangement. The mathematical model of the heat transfer process was verified by experimental data, and the relative error was within 2 %. The simulation results showed that the radiation temperature at different positions θdprt in the center of the room increased with increasing tube diameter and reflecting cover opening width and decreased with increasing tube spacing. Under simulated heating conditions, the optimal structure of the DHR device has a tube diameter of 16 mm, reflecting cover opening width of 60 mm, and tube spacing of 80 mm. The heat flux per unit floor area of the DHR device was approximately 56 W/m2, which reduced the heating capacity by 20 % compared to that without the reflector cover, while improving the local thermal environment of the space and increasing energy efficiency.

An equation-solving method based on radiation distribution factor for radiative transfer in participating media with diffuse boundaries

An efficient and accurate method to solve the directional radiation intensity in the participating medium has always been a pursuit in radiative transfer fields, especially in radiation detection and inverse radiative transfer problems. As one of the few algorithms with both high accuracy and strong applicability, the reverse Monte Carlo (RMC) method suffers from the disadvantages of unavoidable random statistical errors and time-consuming ray tracing. Recently, a so-called equation-solving integral equation method based on the radiation distribution factor (ES-RDFIEM) can effectively overcome these shortcomings of RMC, which has comparable accuracy as RMC while taking only a few hundredths or thousandths of the calculation time of RMC. In this paper, ES-RDFIEM was extended to a radiation system with diffuse surfaces by constructing the radiative transfer equation (RTE) about the radiation distribution factor (RDF) of the wall and internal medium, respectively. The mathematical principle and formula were introduced in detail, and the computational performance was examined by several samples with different radiative parameters. The results showed that the RDF and radiation intensity solved by ES-RDFIEM are in good agreement with RMC, and ES-RDFIEM is more stable and reliable. More importantly, the computation time of ES-RDFIEM is significantly shorter than that of RMC, and it is not affected by wall emissivity, internal medium optical thickness and scattering albedo.

Efficient equation-solving integral equation method based on the radiation distribution factor for calculating radiative transfer in 3D anisotropic scattering medium

High-directional resolution radiation intensity (RI) can provide substantial measurement information inside the participating medium, such as the distribution of temperature and physical properties. Therefore, efficiently and accurately solving the radiative transfer equation (RTE) to obtain RI in any direction is the key and challenge of target-detection and inverse-radiation problems. In our previous works [1,2], the integral equation method based on the radiation distribution factor (RDFIEM) was proposed to accurately obtain an arbitrary directional RI. To overcome the inefficiency of the RDFIEM in building a radiation distribution factor (RDF) database from the time-consuming reverse Monte Carlo (RMC) method, a method of equation-solving RDFIEM (ES-RDFIEM) was improved and developed to obtain the solution of RTE in a three-dimensional (3D) anisotropic scattering medium in this work, which can effectively avoid the stochastic ray-tracing process of the traditional RMC method and obtain the value of RDF by solving linear equations directly. The mathematical principles and formulae of ES-RDFIEM are introduced and deduced in detail, whose core idea is to suppose a specified element with a unit blackbody emission, whereas the remaining elements have no energy emission. Subsequently, the linear equations for the RDF, which are only concerned with the physical properties and geometric factors of the radiative system, can be constructed and solved. The computational accuracy and efficiency of ES-RDFIEM and RMC in several cases with different parameters were comprehensively compared. The results showed an excellent agreement between the RDF values and the RI calculated by the two methods. The computational efficiency of ES-RDFIEM was significantly improved compared to RMC, which was almost unaffected by radiation properties.

An efficient equation-solving method for calculating radiative transfer in isotropic scattering medium

An efficient equation-solving integral equation method (ES-RDFIEM) is proposed to calculate the radiation distribution factor (RDF), which was required to be solved by the time-consuming ray tracing with reverse Monte Carlo method (RMC) in the past. Since the RDF is only related to the radiative properties and geometry of the medium, it can be assumed that the blackbody radiation intensity of a certain element n is 1, while there is no emission from the remaining elements of the radiation system. In this way, the RD(j, n) (j = 1, 2,…, N) can be obtained by solving N × N equations based on the equivalent relationship of incident radiation. As a result, the stochastic statistical process of the RMC can be effectively avoided. The numerical results show that the RDF and radiation intensity calculated by ES-RDFIEM and RMC are highly consistent for one-dimension and two-dimension emitting, absorbing, isotropic scattering medium with transparent boundary. The CPU time for ES-RDFIEM to calculate the RDFs of the whole radiation system is much shorter than RMC under the same condition, which, in addition, does not increase with the increase of scattering albedo or optical thickness. With the same calculation accuracy as RMC, ES-RDFIEM has the significant advantage of high efficiency, which shows great potential in solving inverse radiation problems.

Semi-analytical equation-solving RDFIEM method for radiative transfer in a plane-parallel anisotropic scattering medium

Once the geometry and physical properties of the radiation system are determined, the integral equation method based on radiation distribution factor (RDFIEM) can be effectively used to calculate the directional radiation intensity with high resolution. However, there exists an inherent shortcoming of low efficiency for statistical ray sampling since the radiation distribution factors (RDFs) are obtained by reverse Monte Carlo (RMC) method. For overcoming this deficiency, an efficient semi-analytical equation-solving RDFIEM (ES-RDFIEM) method is proposed to solve the RDFs directly. In this paper, the principles and formulas of ES-RDFIEM in 1D radiation system with anisotropic scattering medium are developed in detail. Benefit from that the RDF is only related to the geometry and radiation properties of the medium, it can be assumed that only a certain element n has unit blackbody radiation intensity, while the remaining elements do not emit energy. In this way, a semi-analytical set of linear equations can be constructed to solve the RDF matrix directly. Compared with RMC, multiple cases with different radiative parameters are adopted to examine the performance of ES-RDFIEM. Numerical results show that ES-RDFIEM is in good agreement with RMC in most conditions, especially in isotropic scattering or weak scattering of anisotropic scattering. In addition, ES-RDFIEM can effectively avoid the random errors produced by RMC, making the result of ES-RDFIEM smoother than RMC. More importantly, it only takes a few CPU times for computing all the RDF values by ES-RDFIEM, which is one or two orders of magnitude less than that of RMC, and its computational efficiency is almost independent of the radiation properties, showing an excellent efficiency.

Radiative heat transfer in the discrete element method using distance based approximations

A model to estimate radiative heat transfer in particle beds is developed for use in the Discrete Element Method (DEM). Monte Carlo ray tracing simulations are run to find the Radiation Distribution Factor (RDF) between pairs of particles and between particles and a wall, in particle beds with random packing. Curves are found to express the average RDF as a function of distance, and within DEM these curves are used to estimate particle-particle and particle-wall radiative transfer. The resulting Distance Based Approximation model is computationally efficient and simple to implement. RDF-distance curves are given as a set of tables covering two particle emissivities (0.65, 0.86), four wall emissivities (0.4, 0.6, 0.8, 1.0), and five solid fractions (0.25, 0.35, 0.45, 0.55, 0.64). The accuracy of the model is investigated, with accuracy sufficient for many engineering applications shown. An initial implementation is demonstrated for a heat exchanger with a dense granular flow.

McrtFOAM: A mesh-agglomeration Monte Carlo ray-tracing solver for radiative transfer in gray semitransparent solids

Monte Carlo ray-tracing method has become a promising technique for solving radiative transfer. However, large memory footprint or long CPU time hampers its further utilization. In this work, a Monte Carlo ray-tracing method radiation solver called mcrtFOAM is developed in the framework of open source CFD toolbox OpenFOAM. The radiative source term is calculated based on the radiation distribution factor to avoid repeated ray tracing. In addition, a mesh-agglomeration technique is used to overcome the problem of memory catastrophe for storing a huge radiation distribution factor matrix caused by a large number of mesh elements. The key principle is to agglomerate the refined mesh elements to the coarse ones, and the calculation of radiation distribution factor as well as radiative source term is carried out in the coarse mesh, while the procedure of ray tracing as well as solving the heat conduction is performed in the refined mesh. After validation of the algorithm code within a cubic enclosure, a coupled conductive–radiative heat transfer problem within a cylindrical enclosure filled with gray semitransparent solids is also tested. To test the present solver in realistic industrial systems, a coupled conductive–radiative heat transfer within a tomographic porous structure with gray surfaces is further investigated. All results are compared with previous studies and the solutions by the Finite Volume Discrete Ordinate Method (FVDOM) in OpenFOAM, and they all show great satisfaction. When the mcrtFOAM solver is used for solving the radiative transfer equation, it is found that the CPU time and the memory footprint consumed by the present mcrtFOAM solver with mesh-agglomeration technique are far less than that without this technique. Convincing first implementation exercises indicate that both significant memory and CPU benefits can be expected for configurations of industrial interest These findings will contribute greatly to achieve the calculation of coupled heat transfer involving radiation within large number of mesh elements.

Artificial Neural Networks in Radiation Heat Transfer Analysis

Abstract In the Monte Carlo ray-trace (MCRT) method, millions of rays are emitted and traced throughout an enclosure following the laws of geometrical optics. Each ray represents the path of a discrete quantum of energy emitted from surface element i and eventually absorbed by surface element j. The distribution of rays absorbed by the n surface elements making up the enclosure is interpreted in terms of a radiation distribution factor matrix whose elements represent the probability that energy emitted by element i will be absorbed by element j. Once obtained, the distribution factor matrix may be used to compute the net heat flux distribution on the walls of an enclosure corresponding to a specified surface temperature distribution. It is computationally very expensive to obtain high accuracy in the heat transfer calculation when high spatial resolution is required. This is especially true if a manifold of emissivities is to be considered in a parametric study in which each value of surface emissivity requires a new ray-trace to determine the corresponding distribution factor matrix. Artificial neural networks (ANNs) offer an alternative approach whose computational cost is greatly inferior to that of the traditional MCRT method. Significant computational efficiency is realized by eliminating the need to perform a new ray trace for each value of emissivity. The current contribution introduces and demonstrates through case studies estimation of radiation distribution factor matrices using ANNs and their subsequent use in radiation heat transfer calculations.

Continuum approach based on radiation distribution function for radiative heat transfer in densely packed particulate system

Radiative transfer equation (RTE) is the commonly accepted continuum scale governing equation for radiative heat transfer in particulate system. However, its applicability is questionable for non-random, densely and regularly packed particulate systems, due to dependent scattering and strong correlation of particle locations that may largely violate the basic assumptions of the RTE. In this work, a new continuum approach is proposed for radiative heat transfer in densely packed particulate system composed of optically large particles, which does not rely explicitly on the RTE. The new approach is based on an introduced concept of radiation distribution function (RDF), which is an extension of the concept of radiation distribution factor (RD) at discrete scale (particle scale) to continuum scale. The governing equations are in integral form of spatial coordinates, in which the RDF is the key continuum scale physical parameter that characterizes the radiative heat transfer in the system. The continuum approach is verified through heat transfer simulation at particle scale, and demonstrated to have excellent performance in predicting the temperature distribution in dense particulate system. This work provides an alternative continuum theory for the analysis and understanding of radiative heat transfer in densely packed particulate and porous materials.

Effective solution of three-dimensional inverse radiation problem in participating medium based on RDFIEM

A novel integral equation method based on radiation distribution factor (RDFIEM) is developed for solving radiation transfer equation (RTE) to obtain radiation intensity signal in a small solid angle on a small point, which is effectively applied to reconstruct the three-dimensional (3D) temperature distribution in an absorbing, emitting and scattering medium. The spatial distribution of radiation intensity inside the medium can be derived by the reverse Monte Carlo method based on the radiation distribution factor (RDF), and source term distribution can be obtained by integration of RDFs. Consequently, according to the idea of integral equation method (IEM), the boundary exit radiation intensity with respect to the RDF and temperature can be deduced. It is worth noting that the RDF is only related to the geometry and radiation properties of the medium, which can be calculated only once to make RDFIEM be more efficient. Meanwhile, the error caused by the random number of RMC can be reduced by integrating the RDFs. As a consequence, the RDFIEM is particularly suitable for temperature field reconstruction. The directional radiation intensity signals obtained by charge-coupled device (CCD) camera are used as the known information to reconstruct the temperature distribution, the effects of CCD combinations, detection directions, optical thickness, boundary conditions, scattering type and scattering albedo, measurement and coefficient errors on the implementation of RDFIEM have been examined. The least square QR decomposition (LSQR) algorithm is introduced to solve the 3D temperature reconstruction problem. The results show that the temperature field can be reconstructed accurately and efficiently.

Fast reconstructing two-dimensional temperature distribution in participating media with different surfaces conditions

The reverse Monte Carlo ray tracing technique (RMCRT) based on radiation distribution factor is applied to obtain high resolution directional radiative signals for solving the radiation transfer process and reconstructing the temperature distribution. The rigorous formula derivation shows that the radiation intensity in any direction at any position is only related to the radiation distribution factors (RDFs) if the temperature profiles is known, while the RDFs is only related to the physical properties and geometric parameters of the medium. The exact solution of one-dimensional parallel-plane gray medium is employed to validate the correctness of the RMCRT. Furthermore, it is successfully employed to reconstruct the two-dimensional temperature distribution of participating media with transparent, specular, diffuse and BRDF surfaces. Since the RDFs database is only related to the physical model and needs to be calculated once in advance, the temperature reconstruction is particularly fast and efficient. All the results show that the accuracy and efficiency of the single-peak temperature reconstruction on transparent surface is better than that of other combination of temperature field and surfaces.

Estimation and Use of the Radiation Distribution Factor Median for Predicting Uncertainty in the Monte Carlo Ray-Trace Method

In a recent contribution, the authors show that the uncertainty in heat transfer results obtained using the Monte Carlo ray-trace (MCRT) method is related to the median of the radiation distribution factor probability density function (PDF). The value of this discovery would be significantly enhanced if the median could be known a priori without first computing the distribution factors. This would allow the user to determine the number of rays required to achieve the desired accuracy of a subsequent heat transfer analysis. The current contribution presents a correlation for the median of the distribution factor PDF as a function of emissivity and the number of surface elements defining an enclosure. The correlation involves a single parameter whose value is unique for a given enclosure geometry. We find that the radiation behavior of a given enclosure can be classified on a scale ranging from reflection-dominated to geometry-dominated. The correlation is shown to work well for reflection-dominated enclosures but less well for geometry-dominated enclosures.

Uncertainty Analysis and Experimental Design in the Monte Carlo Ray-Trace Environment

Despite the dominant role of the Monte Carlo ray-trace (MCRT) method in modern radiation heat transfer analysis, the contemporary literature remains surprisingly reticent on the uncertainty of results obtained using it. After first identifying the radiation distribution factor as a population proportion, standard statistical procedures are used to estimate its mean uncertainty, to a stated level of confidence, as a function of the number of surface elements making up the enclosure and the number of rays traced per surface element. This a priori statistical uncertainty is then shown to compare favorably with the observed variability in the distribution factors obtained in an actual MCRT-based analysis. Finally, a formal approach is demonstrated for estimating, to a prescribed level of confidence, the uncertainty in predicted heat transfer. This approach provides a basis for determining the minimum number of rays per surface element required to obtain the desired accuracy.

An unstructured Monte Carlo ray-tracing method for solving radiative heat transfer in 3D gray semitransparent medium

Monte Carlo ray-tracing method (MCRTM) is widely used for solving the Radiative Transfer equation (RTE) due to its high accuracy and flexibility. In this paper, a MCRTM is developed for solving the radiative heat transfer in gray semitransparent media with complex geometries which are discretized by 3D unstructured grids. A novel algorithm associated to unstructured four-node tetrahedron element is proposed for generating random emitting positions of rays, and the ray tracing process is performed based on Beer's law. Meanwhile, radiation distribution factors are used for calculating radiative source terms which are loaded to the grid. The present method is first validated by comparing the predictive results with those from previous studies on a cubic enclosure, a 3D L-shaped enclosure, and a 3D elliptical enclosure, respectively. Furthermore, a radiation-conduction heat transfer problem is examined in a cylindrical enclosure. All the comparisons show that the present method is in good agreement with these previous cases. This method can be well adapted to various complex geometries. In addition, since the source term is calculated by the radiation distribution factor, which can be reused for different conditions as long as the volumetric radiative properties remain unchanged, this will bring great advantage when coupled with other heat transfer models such as conduction, convection, combustion, etc.

Numerical Modeling of Combined Radiation and Conduction Heat Transfer in Mineral Wool Insulations

This article addresses numerical modeling of coupled heat conduction and radiation in mineral wools under steady-state condition for prediction of its effective thermal conductivity. The radiative heat transfer is modeled using the Monte Carlo Ray-Trace Method. The radiation model is based on a random distribution of fibers in the media. The radiation distribution factor is employed in order to compute the fraction of the total radiation emitted from one fiber that is absorbed by another, due to both direct radiation and to all possible reflections within the enclosure. The radiation model is coupled with the nonlinear heat conduction equation. The results obtained by the proposed model compare well with experimental measurements of the heat flow meter apparatus. The method is easy to code, and the number of calculations during each iteration is considerably reduced.