Radiative Transfer Equation Solver Research Articles

Iterative space-angle discontinuous Galerkin method for radiative transfer equation

The radiative transfer equation (RTE) is an integro-differential equation that describes the radiation energy absorbing, emitting, and scattering in both space and angle, which can be up to five-dimensional problems. It is difficult for a RTE solver to satisfy both accuracy and efficiency (less computational resources) for such high dimensional problems. In this paper, an iterative solver for one-dimensional cylindrical radiative transfer problems using the space-angle discontinuous Galerkin (DG) method is developed to achieve both accuracy and efficiency. The iterative solver is based on the angular decomposition (AD) scheme, which slices the spatial-angular domain into slabs and decouples the angular integration between slabs. Both Jacobi and successive over-relaxation (SOR) iterative schemes are investigated by numerical analysis and examples. The comparison of the two iterative schemes suggests that an appropriate relaxation factor for the SOR method may accelerate the convergence. Finally, the iterative scheme is more efficient than the direct solution of the system both in terms of memory usage and computational time, especially for finer meshes.

Effects of non-collimated radiation during the decay of laser-induced spark

A fast-computing radiative heat transfer on-the-fly interpolation database was developed for high temperature radiative-gas dynamic simulations under local thermodynamic equilibrium (LTE). The absorption coefficient includes Atomic (Bound-Bound, Bound-Free, Free-Free) transitions and Molecular band systems. A 2D finite-volume radiative transfer equation (RTE) solver was developed and coupled with a compressible reacting flow solver to simulate a laser-induced decaying spark (LIDS) in air. Part spectrum correlated k-distribution with Gauss-Chebyshev quadrature and mixing model were applied to reduce the number of RTE runs. Non-collimated radiative heat loss was calculated in LIDS and the effects of radiation loss on the shock wave and kernel cooling were investigated. Total radiation loss was accounted to be approximately 2.3% of the absorbed energy which happens in a micro-second timescale and most of the kernel energy is taken away by the shock wave. Two simulations with and without RTE were compared to analyse the effects of radiative transfer. The radiation loss results in attenuation of shock wave velocity and delaying the emanation of blast wave. The cooling of the plasma kernel is accelerated due to energy loss by radiative transfer. Radiation increases the cooling rates in the primary cooling regime and decreases during the secondary regime due to an energy deficit in the kernel. The transition from primary to the secondary cooling regime is not uniform throughout the kernel.

Open-source non-gray gas radiation heat transfer solvers for combustion applications based on OpenFOAM

Numerical simulation of non-gray radiation heat transfer is essential for many thermal engineering processes, especially for combustion applications. This work develops open-source non-gray gas radiation solvers based on the OpenFOAM framework. The discrete ordinates method (DOM) and first order spherical harmonics (P1) method are implemented by the advanced numerical discretization utilities provided by the OpenFOAM toolbox. The weighted-sum-of-gray-gases (WSGG) model for the carbon dioxide and water vapor are coupled with the two radiative transfer equation (RTE) solvers. Validations with finite volume method (FVM) are performed to test the accuracy of the developed models. Comparisons show that both solvers can produce very close results to the FVM. The relative error of the DOM and P1 is about 3% and 10% respectively for radiative heat flux calculations.

Deterministic radiative transfer equation solver on unstructured tetrahedral meshes: Efficient assembly and preconditioning

Due to its integro-differential nature, deriving schemes for numerically solving the radiative transfer equation (RTE) is challenging. Most solvers are efficient in specific scenarios: structured grids, simulations with low-scattering materials... In this paper, a full solver, from the discretization of the steady-state monochromatic RTE to the solution of the resulting system, is derived.Using a mixed matrix-ready/matrix-free approach, our solver is able to discretize and solve a 45.7 billion unknown problem on 27 thousand processes in three minutes for a full physics involving scattering, absorption, and reflection. Because of the high dimensionality of the continuous equation, the linear system would have had more than 6×1015 nonzero entries if assembled explicitly. Our approach allows for large memory gains by only storing lower dimension reference matrices.The finite element-based solver is wrapped around open-source software, FreeFEM for discretization, PETSc for linear algebra, and hypre for the algebraic multigrid infrastructure. Overall, deterministic results are presented on arbitrarily-decomposed unstructured grids for radiative transfer problems with scattering, absorbing, and reflecting heterogeneities on up to 27 thousand processes.

Large Eddy Simulation of medium-scale methanol pool fires - effects of pool boundary conditions

This article reports Large Eddy Simulation (LES) of 30 cm diameter methanol pool fires in order to investigate the effects of the burner boundary conditions on the pool dynamics. The numerical model involves state-of-the-art subgrid scale (SGS) sub-models for mixing, combustion and turbulence-radiation interaction with all the model constants computed dynamically. The non-adiabatic steady laminar flamelet (SLF)/presumed filtered density function (FDF) combustion model is used whereas the radiation model combines the Rank Correlated Full Spectrum k-distribution (RCFSK) for spectral radiation with the finite volume method (FVM) as radiative transfer equation (RTE) solver. The baseline case considers a burner located one-pool diameter above the floor and a fuel lip height of 1 cm as specified in the experiments used for model validation. Model predictions for puffing frequency, mean and rms temperature and velocity, mean molar fractions of major species, radiative loss to the surrounding and radiative and total heat feedback to the fuel surface are in good agreement with the available experimental data. Two other burner boundary conditions are designed. The first disregards the fuel lip height while keeping the burner located one-pool diameter above the floor. The second modifies the first configuration by assuming that the burner rim is floor flush. Model results show that the burner boundary conditions affect significantly the flow structure and the pool fire dynamics by altering the flame base instability near the edge of the pan. A non-zero fuel lip height produces significantly wider and shorter flames which affect the heat feedback toward the fuel surface whereas altering the air entrainment at the pool basis enhances substantially the puffing frequency. This shows that the experimental burner boundary conditions have to be reproduced scrupulously for relevant model validations.

The impact of radiative heat transfer in combustion processes and its modeling – with a focus on turbulent flames

Radiative heat transfer (RHT) is often the dominant mode of heat transfer in flames, fires, and combustion systems and affects significantly temperature distributions directly and kinetically controlled chemical processes indirectly. Modeling RHT accurately in multidimensional flames and combustion systems is challenging mainly due to the highly spectrally dependent radiative properties of combustion products, the high computational cost of solving the radiative transfer equation (RTE), and the strong turbulence and radiation interactions (TRI). Significant progress has been made in all three aspects in the last three decades and the state-of-the-art models and methods have been incorporated into CFD practice for modeling fires, turbulent jet flames, and turbulent combustion in combustion systems. In this article, we first discuss the coupling between RHT and combustion and the important role played by RHT in some fundamental flame phenomena. Then we discuss the state-of-the-art radiation models with a focus on RTE solvers, radiative property models and TRI. Next, we review the recent simulations of turbulent combustion systems involving these state-of-the-art radiation models. Finally, we provide concluding remarks and some potential research areas to advance RHT modeling in multiphase reacting flows.

Underwater Optical Path Loss after Passage of a Tropical Storm

Underwater wireless optical communications (UWOCs) have attracted considerable attention in recent years as an alternative means for acoustic communication. However, optical path loss of light propagation from attenuation is in large part due to absorption and scattering in various water conditions. Identification of environmental effects, especially tropical storms on underwater optical path loss, is key to the success of using optics for UWOCs. Underwater inherent optical properties (IOPs), such as the beam attenuation coefficient for 470 nm light in the western North Pacific Ocean, were measured from U.S. Naval Oceanographic Office Seagliders deployed after Super Typhoon Guchol’s (June 7–20, 2012) passage from June 25 to June 30, 2012 and without any typhoon passage from January 9 to February 28, 2014. The two observed sets (with and without the super typhoon) of IOPs are taken as input for a recently developed radiative transfer equation solver. The simulated normalized received powers for the two durations show a large impact of typhoon passage on UWOCs.

Simulations of a turbulent line fire with a steady flamelet combustion model coupled with models for non-local and local gas radiation effects

The general objective of this study is to develop accurate combustion and radiation models for large eddy simulations (LES) of well-controlled laboratory-scale turbulent fires. The combustion model features a library of flamelet solutions corresponding to steady, laminar, strained diffusion flames. The radiation model features a classical description of non-local radiation phenomena through the Radiative Transfer Equation (RTE). Different treatments of coupled combustion and radiation effects at flamelet scale are considered: a treatment in which local radiation phenomena are neglected inside the flamelet solver; and a treatment in which these phenomena are included inside the flamelet solver and also subgrid-scale turbulence-radiation interactions are included in the RTE solver. This modeling framework is incorporated into the LES solver FireFOAM, developed by FM Global. It is evaluated in simulations of a buoyancy-driven, methane-air, turbulent diffusion flame experimentally characterized by measurements of local temperatures and global flame emissive power. Comparisons between simulated and measured temperatures show significant discrepancies that may be explained by overestimated values of the width of the presumed probability density function representing subgrid-scale variations of mixture fraction. Similar comparisons for the global radiative loss fraction show good agreement provided that local radiation phenomena and turbulence-radiation interactions are included in the model.

Diffusion Equation-Assisted Markov Chain Monte Carlo Methods for the Inverse Radiative Transfer Equation

Optical tomography is the process of reconstructing the optical properties of biological tissue using measurements of incoming and outgoing light intensity at the tissue boundary. Mathematically, light propagation is modeled by the radiative transfer equation (RTE), and optical tomography amounts to reconstructing the scattering coefficient in the RTE using the boundary measurements. In the strong scattering regime, the RTE is asymptotically equivalent to the diffusion equation (DE), and the inverse problem becomes reconstructing the diffusion coefficient using Dirichlet and Neumann data on the boundary. We study this problem in the Bayesian framework, meaning that we examine the posterior distribution of the scattering coefficient after the measurements have been taken. However, sampling from this distribution is computationally expensive, since to evaluate each Markov Chain Monte Carlo (MCMC) sample, one needs to run the RTE solvers multiple times. We therefore propose the DE-assisted two-level MCMC technique, in which bad samples are filtered out using DE solvers that are significantly cheaper than RTE solvers. This allows us to make sampling from the RTE posterior distribution computationally feasible.

Development of RTE solver for radiative transfer in absorbing-emitting medium using finite volume based CFD library OpenFOAM

In this study, the radiative transfer equation (RTE) solver radiationFoam within the Open Source Field Operation and Manipulation (OpenFOAM) C++ object-oriented framework is developed. As the RTE solver radiationFoam is recompiled based on the OpenFOAM C++ open source libraries, it inherits the main advantages of OpenFOAM software platform, such as the convenience of pre-processing for unstructured mesh generation by snappyHexMesh, MPI parallelization for domain-decomposition during calculation process, and post-processing with paraview. Particularly, the parameters required to solve RTE, such as the absorption coefficient, azimuthal and polar angles, and boundary conditions, could be easily set in the dictionary files. To verify the performance of radiationFoam, a preliminary test for classical benchmark problems of radiative transfer in absorbing-emitting medium is conducted. The RTE is discretized in solid angular space with the discrete ordinates method (DOM) and then integrated over both the control volume and solid angle with the finite volume method (FVM). The results obtained are compared with those previously reported in the literature to illustrate the performance and flexibility of the proposed method.

An Improved Accurate Solver for the Time-Dependent RTE in Underwater Optical Wireless Communications

Underwater optical wireless communication (UOWC) has been widely advocated as a viable way to satisfy these high-speed links constraints in the marine medium through the use of the visible spectrum. Nevertheless, UOWC faces several limitations, such as the path-loss due to the absorption and scattering phenomena, caused by underwater particles. Thus, quantifying this path-loss is of paramount importance in the design of futuristic UOWC systems. To this end, several approaches have been used in this regard, namely the Beer–Lambert’s law, Monte Carlo simulation, as well as radiative transfer equation (RTE). This last mentioned evaluates the optical path-loss of the light wave in an underwater channel in terms of the absorption and scattering coefficients as well as the scattering phase function (SPF). In this paper, an improved numerical solver to evaluate the time-dependent RTE for UOWC is proposed. The proposed numerical algorithm was improved based on the previously proposed ones, by making use of an improved finite difference scheme, a modified scattering angular discretization, as well as an enhancement of the quadrature method by involving a more accurate seven-point quadrature scheme in order to calculate the weight coefficients corresponding to the RTE integral term. Importantly, we applied the RTE solver to three different volume scattering functions, namely the single-term Henyey–Greenstein (HG) phase function, the two-term HG phase function, and the Fournier–Forand phase function, over both Harbor-I and Harbor-II water types. Based on the normalized received power evaluated through the proposed algorithm, the bit error rate performance of the UOWC system is investigated in terms of system and channel parameters. The enhanced algorithm gives a tightly close performance to its Monte Carlo counterpart by adjusting the numerical cumulative distribution function computation method as well as optimizing the number of scattering angles.

Inverse estimation of number and location of discrete heaters in radiant furnaces using artificial neural networks and genetic algorithm

The inverse problem of the determination of optimum number and location of discrete heaters in a three dimensional radiant heating furnace is solved by developing an artificial neural network based model to replace the radiative transfer equation solver and coupling it with genetic algorithm. The furnace medium is considered to be participating with a mixture of H2O and CO2. For the purpose of estimation of the best configuration of the heaters to achieve a desired uniform heat flux at the design surface, the inverse problem has been treated as an optimization problem. A generalized radiative transfer code based on Finite Volume Method and spectral line based weighted sum of gray gases method has been developed and validated with the available benchmarks. This code serves as the basis to develop the neural network based model for the flux estimation at design surface. The ANN function has been tested upon 5 random test cases. The ANN based replacement of RTE for this problem gives an error less than 4% for the local heat flux values at the order of 103 times less computation time. A design case is then considered where the desired flux levels should be uniform on the design surface in a very tight tolerance. Results obtained show that this novel method of ANN coupled with GA proves to be a fast, robust and efficient tool for solving such types of inverse problems.

MCmatlab: an open-source, user-friendly, MATLAB-integrated three-dimensional Monte Carlo light transport solver with heat diffusion and tissue damage.

While there exist many Monte Carlo (MC) programs for solving the radiative transfer equation (RTE) in biological tissues, we have identified a need for an open-source MC program that is sufficiently user-friendly for use in an education environment, in which detailed knowledge of compiling or UNIX command-line cannot be assumed. Therefore, we introduce MCmatlab, an open-source codebase thus far consisting of (a)a fast three-dimensional MC RTE solver and (b)a finite-element heat diffusion and Arrhenius-based thermal tissue damage simulator, both run in MATLAB. The kernel for both of these solvers is written in parallelized C and implemented as MATLAB MEX functions, combining the speed of C with the familiarity and versatility of MATLAB. We compare the RTE solver to Steven Jacques' mcxyz, which it is inspired by, and present example results generated by the thermal model. MCmatlab is easy to install and use and can be used by students and experienced researchers alike for simulating tissue light propagation and, optionally, thermal damage.

A detailed modeling study of radiative heat transfer in a heavy-duty diesel engine

In recent years, the importance of radiative heat transfer in combustion has been increasingly recognized. Detailed models have become available that accurately represent the complex spectral radiative properties of reacting gas mixtures and soot particles, and new methods have been developed to solve the radiative transfer equation (RTE). At the same time, the trends toward higher operating pressures and higher levels of exhaust-gas recirculation in compression-ignition engines, together with the demand for higher quantitative accuracy from in-cylinder CFD models, has led to renewed interest in radiative transfer in engines. Here an in-depth investigation of radiative heat transfer is performed for a heavy-duty diesel truck engine over a range of operating conditions. Results from 10 different combinations of turbulent combustion models, spectral radiation property models, and RTE solvers are compared to provide insight into the global influences of radiation on energy redistribution in the combustion chamber, heat losses, and engine-out pollutant emissions (NO and soot). Also, the relative importance of the individual contributions of molecular gas versus soot radiation, the spectral model, the RTE solver, and unresolved turbulent fluctuations in composition and temperature (turbulence–radiation interactions – TRI) are investigated. Local instantaneous temperatures change by as much as 100 K with consideration of radiation, but the global influences of radiation on heat losses and engine-out emissions are relatively small (in the 5–10% range). Molecular gas radiation dominates over soot radiation, consideration of spectral properties is essential for accurate predictions of reabsorption, a simple RTE solver (a first-order spherical harmonics – P1 – method) is sufficient for the conditions investigated, and TRI effects are small (less than 10%). While the global influences of radiation are relatively small, it is nevertheless desirable to explicitly account for radiation in in-cylinder CFD. To that end, a simplified CFD radiation model has been proposed, based on the findings reported here.

Full-spectrum correlated-k-distribution look-up table for use with radiative Monte Carlo solvers

The full-spectrum correlated-k-distribution (FSCK) look-up table previously developed by the authors Wang et al. (2018) provides an efficient means for accurate calculations of radiative properties in nonhomogeneous media in conjunction with conventional radiative transfer equation (RTE) solvers. However, the current FSCK look-up table cannot be combined with statistical RTE solvers. A random-number relation was derived using the new implementation proposed in Wang et al. (2018) for the use of FSCK tables together with photon Monte Carlo (PMC) RTE solvers, and a corresponding random-number database including CO2, H2O, CO and soot has been constructed. Radiative heat transfer for two realistic scaled flames is studied to test the performance of the FSCK/PMC method using both FSCK look-up table and random-number database. Results show that the FSCK/PMC method can achieve almost the same accuracy but at cheaper computational cost and much lower memory requirement compared to the benchmark line-by-line solutions.

A simplified CFD model for spectral radiative heat transfer in high-pressure hydrocarbon–air combustion systems

Detailed radiation modeling in piston engines has received relatively little attention to date. Recently, it is being revisited in light of current trends towards higher operating pressures and higher levels of exhaust-gas recirculation (EGR), both of which enhance molecular gas radiation. Advanced high-efficiency engines also are expected to function closer to the limits of stable operation, where even small perturbations to the energy balance can have a large influence on system behavior. Detailed radiation modeling using sophisticated tools like photon Monte Carlo/line-by-line (PMC/LBL) is computationally expensive. Here, guided by results from PMC/LBL, a simplified stepwise-gray spectral model in combination with a first-order spherical harmonics (P1 method) radiative transfer equation (RTE) solver is proposed and tested for engine-relevant conditions. Radiative emission, reabsorption and radiation reaching the walls are computed for a heavy-duty compression-ignition engine at part-load and full-load operating conditions with different levels of EGR and soot. The results are compared with those from PMC/LBL, P1/FSK (P1 with a full-spectrum k-distribution spectral model) and P1/Gray radiation models to assess the proposed model’s accuracy and computational cost. The results show that the proposed P1/StepwiseGray model can calculate reabsorption locally and globally with less than 10% error (with respect to PMC/LBL) at a small fraction of the computational cost of PMC/LBL (a factor of 30) and P1/FSK (a factor of 15). In contrast, error in computed reabsorption by the P1/Gray model is as high as 60%. It is expected that the simplified model should be broadly applicable to high-pressure hydrocarbon–air combustion systems, with or without soot.

The effects of flue-wall design modifications on combustion and flow characteristics of an aluminum anode baking furnace-CFD modeling

A modern aluminum smelter has a capacity of 1–2 million tons of aluminum per annum which requires more than 0.5–1.0 million tons of heat treated (baked) carbon anodes per year. The anode baking process is very energy intensive, approximately requires 2 GJ of energy per ton of carbon anodes which will be approximately 1–2 million GJ of energy per year. Since the plant testing is very expensive, anode baking furnace modeling is imperative to investigate the effects of different operational and geometrical parameters on the furnace energy consumption.In the numerical modeling of turbulent reactive flows, the accuracy of the model highly depends on the description of turbulence-chemistry interaction, and the robustness of the radiative transfer equation solver. Hence, the present study tested different turbulence-chemistry interaction frameworks and radiative transfer equation solvers for confined turbulent diffusion flames and results are compared with the experimental data. It was observed that the k-ε realizable turbulence model combined with the presumed probability density function method as turbulence-chemistry interaction, and discrete ordinates method as radiative transfer equation solver illustrate excellent agreement with the experimental data.Anode baking homogeneity is an important consideration in the design of the anode baking furnace, which requires appropriate flow distribution in the flue-wall cavity. Hence, the developed numerical framework was employed to investigate the effects of flue-wall design modifications on combustion and heat transfer characteristics of anode baking furnace energy consumption. It was observed that the modified design results in a higher thermal efficiency (lower energy consumption), and more homogenous temperature and flow fields which result in the baking of anodes more uniformly which consequently result in the evolution of more homogenous material properties. Consequently, the energy consumption in the aluminum reduction cell will also be significantly reduced. The present study also investigated the effect of flue-wall flow blockage due to packing coke infiltration on baking energy consumption for the aged anode baking furnaces and introduced three modified flue-wall designs.

A narrow-band k-distribution model with single mixture gas assumption for radiative flows

In the present study, the narrow-band k-distribution (NBK) model parameters for mixtures of H2O, CO2, and CO are proposed by utilizing the line-by-line (LBL) calculations with a single mixture gas assumption. For the application of the NBK model to radiative flows, a radiative transfer equation (RTE) solver based on a finite-volume method on unstructured meshes was developed. The NBK model and the RTE solver were verified by solving two benchmark problems including the spectral radiance distribution emitted from one-dimensional slabs and the radiative heat transfer in a truncated conical enclosure. It was shown that the results are accurate and physically reliable by comparing with available data. To examine the applicability of the methods to realistic multi-dimensional problems in non-isothermal and non-homogeneous conditions, radiation in an axisymmetric combustion chamber was analyzed, and then the infrared signature emitted from an aircraft exhaust plume was predicted. For modeling the plume flow involving radiative cooling, a flow-radiation coupled procedure was devised in a loosely coupled manner by adopting a Navier-Stokes flow solver based on unstructured meshes. It was shown that the predicted radiative cooling for the combustion chamber is physically more accurate than other predictions, and is as accurate as that by the LBL calculations. It was found that the infrared signature of aircraft exhaust plume can also be obtained accurately, equivalent to the LBL calculations, by using the present narrow-band approach with a much improved numerical efficiency.

High performance computation of radiative transfer equation using the finite element method

This article deals with an efficient strategy for numerically simulating radiative transfer phenomena using distributed computing. The finite element method alongside the discrete ordinate method is used for spatio-angular discretization of the monochromatic steady-state radiative transfer equation in an anisotropically scattering media. Two very different methods of parallelization, angular and spatial decomposition methods, are presented. To do so, the finite element method is used in a vectorial way. A detailed comparison of scalability, performance, and efficiency on thousands of processors is established for two- and three-dimensional heterogeneous test cases. Timings show that both algorithms scale well when using proper preconditioners. It is also observed that our angular decomposition scheme outperforms our domain decomposition method. Overall, we perform numerical simulations at scales that were previously unattainable by standard radiative transfer equation solvers.

Effect of aggregated black carbon aging on infrared absorption and longwave radiative forcing

Black carbon aerosols affect the shortwave and longwave radiation in climate in a strong yet uncertain way. In aging process, black carbon particles coated by co-emitted aerosols tend to reduce the shortwave radiative forcing of freshly emitted black carbon at the top of atmosphere (TOA), however, this effect is still unclear in the longwave range. Here in this work, we investigate the effect of black carbon aging on longwave radiative forcing. The freshly emitted black carbon aerosols are simulated to be fractal aggregates consisting of hundreds of small spherical primary particles, and these aggregated black carbon aerosols tend to be fully coated by the large sulfate particles after aging. The optical properties of these freshly emitted and internally mixed black carbon aerosols are simulated using the numerically exact superposition T-matrix method, and their longwave radiative forcings are calculated by the radiative transfer equation solver. The results indicate that the black carbon longwave radiative forcing at TOA is remarkably amplified (up to 3) by coating the large sulfate particles, while the black carbon shortwave radiative forcings decrease during their aging. Moreover, the thicker sulfate coatings tend to increase the longwave radiative forcings of black carbon aerosols at TOA. These findings should improve our understanding of the effect of black carbon aging on their longwave radiative forcings and provide guidance for assessing the climate change.