Solution Of Radiative Transfer Equation Research Articles

Two-dimensional P1 approximation (P1-2D) for the evaluation of the radiant field in annular and tubular photocatalytic reactors

The local volumetric rate of photon absorption (LVRPA) was formulated by solving the radiative transfer equation (RTE) in two dimensions using the P1 approximation approach (P1-2D) to describe the radiant field in annular and tubular photocatalytic reactors. The radiant sources employed were assumed to be periodic when applying the boundary conditions to facilitate the determination of the integration constants in the RTE solution. Three different systems were considered: S1 and S2 consisted of annular reactors, while S3 consisted of a tubular reactor. Simulations were conducted using commercial TiO2 P25 and anatase TiO2 as photocatalyst models, with their optical properties sourced from the literature. The linear source spherical emission (LSSE), Gaussian emission (GE), and uniform emission (UE) models were used as radiant sources. The LVRPA was found to decrease from the inner to the outer wall of the reactor for S1 and S2, and from the wall to the center of the reactor for S3; it was close to the values obtained using the Monte Carlo (MC) procedure in S2. The overall volumetric rate of photon absorption (OVRPA) increased exponentially with catalyst loading until a point where no significant increase was observed for each reactor. The apparent optical thickness τ App 1 ​formulated was a more reliable optimization parameter than the optical thickness.

Infinite series solution of the time-dependent radiative transfer equation in anisotropically scattering media

We solve the radiative transfer equation (RTE) in anisotropically scattering media as an infinite series. Each series term represents a distinct number of scattering events, with analytical solutions derived for zero and single scattering. Higher-order corrections are addressed through numerical calculations or approximations.The RTE solution corresponds to Monte Carlo sampling of photon trajectories with fixed start and end points. Validated against traditional Monte Carlo simulations, featuring random end points, our solution demonstrates enhanced efficiency for both anisotropic and isotropic scattering functions, significantly reducing computational time and resources. The advantage of our method over Monte Carlo simulations varies with the position of interest and the asymmetry of light scattering, but it is typically orders of magnitude faster while achieving the same level of accuracy. The exploitation of hidden symmetries further accelerates our numerical calculations, enhancing the method’s overall efficiency.In addition, we extend our analysis to the first and second moments of the photon’s flux, elucidating the transition between transport and diffusive regimes.

Radiative Transfer Equation Solution for Many Scattering Types

The radiative transfer equation is a mathematical equation that describes the changes in the number of photons within a specified volume of a medium over time, taking into account phenomena such as scattering, absorption, and re-emission resulting from photon interactions with the medium. In this study, the radiative transfer equation is considered for a finite slab which anisotropic scattering in a homogeneous medium. The equation solution is done by Legendre polynomials for linear anisotropic, pure quadratic and Rayleigh scattering types. The numerical results are displayed in the tables up to the 13th iteration of the Legendre polynomials. Tables are obtained using different scattering coefficients and single scattering albedo values. The results contain a wide range of data obtained from the method of solving the Legendre polynomial of the radiative transfer equation. Thus, with this study, the effect of different scattering types on the solution of the radiative transfer equation has been demonstrated.

Unified gas-kinetic particle method for frequency-dependent radiation transport

This paper proposes a unified gas-kinetic particle (UGKP) method for the frequency-dependent photon transport process. The photon transport is a typical multiscale process governed by the nonlinear radiative transfer equations (RTE). The flow regime of photon transport varies from the ballistic regime to the diffusive regime with respect to optical depth and photon frequency. The UGKP method is an asymptotic preserving (AP) scheme and a regime adaptive scheme. The teleportation error is significantly reduced, and the computational efficiency is remarkably improved in the diffusive regime. Distinguished from the standard multigroup treatment, the proposed UGKP method solves the frequency space in a non-discretized way. Therefore the Rosseland diffusion system can be precisely preserved in the optically thick regime. Taking advantage of the local integral solution of RTE, the distribution of the emitted photon can be constructed from its macroscopic moments. The Monte Carlo particles in the UGKP method need only be tracked before their first collision events, and a re-sampling process is performed to close the photon distribution for each time step. The high computational cost of calculating excessive scattering events can be saved, especially in the optically thick regime. The proposed UGKP method is implicit and removes the light speed constraint on the time step. The particle tracking approach combining the implicit formulation makes the proposed UGKP method an efficient solution algorithm for frequency-dependent radiative transfer problems. We demonstrate with numerical examples the capability of the proposed multi-frequency UGKP method.

An implicit unified gas-kinetic wave–particle method for radiative transport process

The unified gas-kinetic wave–particle method (UGKWP) has been developed for the multiscale gas, plasma, and multiphase flow transport processes for the past years. In this work, we propose an implicit UGKWP (IUGKWP) method to remove the Courant–Friedrichs–Lewy time step constraint. Based on the local integral solution of the radiative transfer equation (RTE), the particle transport processes are categorized into the long-λ streaming process and the short-λ streaming process compared to a local physical characteristic time tp. In the construction of the IUGKWP method, the long-λ streaming process is tracked by the implicit Monte Carlo method; the short-λ streaming process is evolved by solving the implicit moment equations; and the photon distribution is closed by a local integral solution of RTE. In the IUGKWP method, the multiscale flux of radiation energy and the multiscale closure of photon distribution are constructed based on the local integral solution. The IUGKWP method preserves the second-order asymptotic expansion of RTE in the optically thick regime and adapts its computational complexity to the flow regime. The numerical dissipation is well controlled, and the teleportation error is significantly reduced in the optically thick regime. The computational complexity of the IUGKWP method decreases exponentially as the Knudsen number approaches zero, and the computational efficiency is remarkably improved in the optically thick regime. The IUGKWP is formulated on a generalized unstructured mesh, and multidimensional 2D and 3D algorithms are developed. Numerical tests are presented to validate the capability of IUGKWP in capturing the multiscale photon transport process. The algorithm and code will apply in the engineering applications of inertial confinement fusion.

Modeling Thermal Radiation in Combustion Environments: Progress and Challenges

Modeling thermal radiation in combustion environments can be extremely challenging for two main reasons. First, the radiative transfer equation (RTE), which is the cornerstone of modeling radiation in such environments, is a five-dimensional integro-differential equation. Second, the absorption and scattering coefficients of molecular gases and particulates prevalent in combustion environments oscillate strongly with the wavenumber (or wavelength), i.e., the medium is strongly nongray, requiring the solution of the RTE for a large number of wavenumbers. This article reviews the progress that has been made in this area to date with an emphasis on the work performed over the past three decades. Progress in both deterministic and stochastic (Monte Carlo) solutions of the RTE is reviewed, in addition to the review of the treatment of the spectral properties of gases, soot, and fuel droplets that dominate combustion environments, i.e., spectral or nongray models. The application of the various state-of-the-art nongray models and RTE solution methods to flames (particularly turbulent), fires, combustors, and other combustion systems are summarized along with a critical discussion of the pros and cons of the models and methods. Finally, the challenges that remain in modeling thermal radiation in combustion systems are highlighted and future outlooks are shared.

Moment models for an axisymmetric inertial confinement experiment and one dimensional numerical study

In Inertial Confinement Fusion (ICF) experiments, radiation is well described by a kinetic model (radiative transfer equation). This model is usually too expensive to be used in numerical simulations of such phenomena. Hence, approximations are used. A common one is to use a moment model, in which the radiative transfer equation is replaced by its first and second order (in velocity) moments, together with a closure assumption. In this article, we propose a closure for 2D and 3D geometries, which are extensions of a one-dimensional spherically symmetric model called P1′. This model has proved to be very accurate in the study of ICF, which makes the models we propose promising in this respect. The closure is based on the fact that the model should, if possible, reproduce the exact solutions of radiative transfer equation in vacuum. We also design a numerical scheme for the one-dimensional spherically symmetric case only. This scheme is well-balanced and satisfies the diffusion limit of the model. This scheme is validated by various numerical tests. We plan to extend this method to higher dimensions in a future work.

A novel approach to solve forward/inverse problems in remote sensing applications

Inversion of electromagnetic (EM) signals reflected from or transmitted through a medium, or emitted by it due to internal sources can be used to investigate the optical and physical properties of a variety of scattering/absorbing/emitting materials. Such media encompass planetary atmospheres and surfaces (including water/snow/ice), and plant canopies. In many situations the signals emerging from such media can be described by a linear transport equation which in the case of EM radiation is the radiative transfer equation (RTE). Solutions of the RTE can be used as a forward model to solve the inverse problem to determine the medium state parameters giving rise to the emergent (reflected/transmitted/emitted) EM signals. A novel method is developed to determine layer-by-layer contributions to the emergent signals from such stratified, multilayered media based on the solution of the pertinent RTE. As a specific example of how this approach may be applied, the radiation reflected from a multilayered atmosphere is used to solve the problem relevant for EM probing by a space-based lidar system. The solutions agree with those obtained using the standard lidar approach for situations in which single scattering prevails, but this novel approach also yields reliable results for optically thick, multiple scattering aerosol and cloud layers that cannot be provided by the traditional lidar approach.

Effective parameterization of absorption by gaseous species and unknown UV absorber in 125–400 nm region of Venus atmosphere

We present an effective parameterization of molecular absorption of shortwave solar radiation in Venus atmosphere. It is addressed to general circulation modeling to accelerate radiative transfer calculations in the spectral interval 125 – 400 nm (25000 – 80000 cm−1). In F-UV and M-UV regions strong absorption of CO2 and SO2 enables to parameterize gaseous absorption with only three effective cross-sections. In N-UV region absorption of SO2 and unknown UV absorber are parameterized with six effective cross-sections for each species. For treatment of Rayleigh scattering and optical properties of Venus clouds eight effective spectral points are recommended. The parameterizations were validated by the original reference line-by-line Monte-Carlo radiative transfer model. The outcome of the validation shows discrepancy in fluxes less than 3%. Thus, it takes only eight-fold solution of radiative transfer equations to correctly describe solar fluxes and heating rates in the whole ultraviolet region.

Performance and validation of a radiation model coupled with a transient bubbling fluidized bed combustion model

None of the transient bubbling fluidized bed combustor (BFBC) models couples the solution of radiative transfer equation with the solution of conservation equations of other transport processes although thermal radiation is of considerable importance in such systems, as such an objective requires both accurate and computationally efficient methods for not only the solution of radiative transfer but also the estimation of radiative properties of particle-laden combustion gases. Therefore, in this study, a 1-D transient comprehensive system model for combustion of lignite is coupled with a 3-D radiation model based on method of line solution of discrete ordinates method utilizing one gas spectral line-based weighted sum of gray gases for spectral gas properties, Mie theory for gray particle properties, and gray wall properties. Predictive accuracy of the model is validated against (i) predictions of a computationally inefficient spectral radiation model and (ii) measurements of transient temperature and concentrations of the species O2 and CO obtained from METU 0.3 MWt atmospheric bubbling fluidized bed combustion test rig firing lignite. The outcome of this study provides a comprehensive, accurate, and computationally efficient tool for simulation of transient BFBCs and highlights the significant impact of radiative heat transfer in modeling of industrial boilers.

High-order iterative method for vector radiative transfer equation of particle layer on rough surface

In order to investigate the multiple scattering characteristics of energy transmission in a random particle layer on a typical rough surface, a high-order iterative method based on radiative transfer equation is proposed in this work. The particles are placed on the rough surface and the height dimension (<i>Z</i>-axis) is divided into several layers. Then the high-order iterative solution can be obtained by using the first-order iteration solution of radiative transfer equation and the rough surface theory. Moreover, the multiple scattering characteristics between particles, as well as between particles and rough surface can be obtained. Numerical results show that the multiple scattering characteristics can be better explored by the proposed high-order iterative method than by the first-order iteration method. All these interactions can be explained by the energy transfer theory. The proposed high-order iterative method can be used as an efficient tool to explain the multiple scattering characteristics and predict the change of the scattering coefficients.

Solution of radiative-convective heat transfer in irregular geometries using hybrid lattice Boltzmann-finite volume and immersed boundary method

Immersed boundary method (IBM) is introduced for solving the radiative transfer equation (RTE) to simulate the radiation heat transfer with natural convection in irregular geometries. IBM is organized based on the direct forcing method and sharp interface scheme. The Lattice Boltzmann method (LBM) with double multiple-relaxation-time (MRT)-D2Q9 is adopted in computing the momentum and energy equations, and the finite volume method (FVM) is applied in the numerical solution of RTE with a pseudo-time-stepping approach. The FVM-LBM code is tested in benchmark geometries in two cases. The ability and performance of IBM are examined in an irregular geometry for Rayleigh number, Planck number, and optical thickness on thermal and flow fields. Static and Hopf bifurcation as an attractive effect of radiation heat transfer is noted. The significant benefit of IBM is using a unique mesh in simulating these problems.

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.

High Improvement in Solar Heater Performance by Exploitation of Radiating Gas and Airfoil Shaped Ribs

​A novel design of solar air heater (SAH) is proposed in this paper based on the simultaneous exploitation of radiating gas and also airfoil shaped ribs. Using participating gases with high radiative characteristics respect to the usual working-gas, namely air, could show a more promising improvement in thermal performance, especially while this technique is combined with the method of extending surface area with low pressure drop. This new concept is demonstrated by simultaneous solution of radiative transfer equation (RTE) considering both diffuse and collimated beams coupled with momentum, energy and continuity equations. The set of governing equations are solved using the finite element method (FEM) in a steady-state condition by the Comsol Multi physics. The well known model is used in calculations of turbulent stress and heat flux. Through the presented results, radiative gas clearly proved well its potent potentials to serve as future working-gas in solar gas heaters (SGHs) and its combination with the airfoil shaped ribs shows magnificent 88% thermal efficiency and gas outlet temperature up to 85 C in the test cases. The contribution of airfoil shaped ribs and gas radiation effect in increasing the thermal performance are computed equal to 10% and 67%, respectively.

Solutions of Chandrasekhar’s basic problem in radiative transfer via theory of functional connections

We present a novel approach to solving Chandrasekhar’s problem in radiative transfer using the recently developed Theory of Functional Connections. The method is designed to elegantly and accurately solve the Linear Boundary Value Problem from the angular discretization of the integrodifferential Boltzmann equation for Radiative Transfer. The proposed algorithm falls under the category of numerical methods for the solution of radiative transfer equations. This new method’s accuracy is tested via comparison with the published benchmarks for Mie and Haze L scattering laws.

Solution of radiative transfer equation in finite medium involving Fresnel reflection

Radiative transfer problem in an absorbing, scattering and homogeneous finite medium with Rayleigh scattering subjected to externally incident radiation is considered. The solution is evaluated when the case of Fresnel and diffuse reflections is present at the boundary. Galerkin approximation is introduced to calculate some functionals arising in radiative transfer problems such as reflection and transmission coefficients. Numerical results are reported and compared with the available published data which show very good agreement.

Predicting radiative heat transfer in axisymmetric cylindrical enclosures using the collocation spectral method

In this paper, the collocation spectral method (CSM) is extended to solve the radiative transfer equation (RTE) in two-dimensional axisymmetric cylindrical medium, and an iterative solver based on Schur decomposition is developed for the spectral discretized equations. Then the performance of CSM is assessed by comparing with the semi-analytic solutions and the discrete ordinates method (DOM). The results show that the layout of angular grid points has great influence on the accuracy of CSM, and rearranging the angular grid can even reduce the ray effect. But a more reliable method to alleviate the ray effect is to decompose the radiative intensity into two components with separate treatments as the modified DOM proposed by Ramankutty and Crosbie (JQSRT, 57(1997) 107–140). Besides, the CSM can offer more accurate results than the DOM with acceptable computational cost and memory consumption, thus can be an alternative for the solution of RTE in cylindrical coordinates. However, the CSM loses its high order convergence accuracy for the two-dimensional problems due to the unavoidable non-smooth distribution of radiative intensity in both spatial and angular domains. Further treatments should be applied to improve the performance of CSM.

Analytical solutions of radiative transfer equations in accretion discs with finite optical depth

ABSTRACT The main purpose of this paper is to obtain analytical solutions for radiative transfer equations related to the vertical structure of accretion discs with finite optical depth. In the non-grey atmosphere, we employ the optical-depth dependent Eddington factor to define the relationship between the mean intensity and radiation stress tensor. Analytical solutions are achieved for two cases: (i) radiative equilibrium, and (ii) a disc with uniform internal heating and both cases are assumed to be in local thermodynamical equilibrium (LTE), too. These solutions enable us to study probable role of scattering and disc optical depth on the emergent intensity and other radiative quantities. Our results show that for the first case, the surface value of mean intensity with constant Eddington factor is three times larger than that with variable factor. Moreover, scattering has no role in the vertical radiative structure of discs with the assumptions of the first case. On the other hand, for the second case, we encounter reductions in all radiative quantities as the photon destruction probability decreases (which is equivalent to increasing scattering). Furthermore, for both cases with total optical depth less than unity, the outward intensity towards the polar direction becomes less than that from the edges of disc which is contrary to limb-darkening. At the end, we apply our results to find the spectrum from accretion systems, based on two dynamical models. Consequently, we can see that how the total optical depth varies with frequency and causes remarkable changes on the emergent spectra.

Acceleration of the numerical solution for the radiative transfer equation using a modified relaxation factor

PurposeThe purpose of this study is to propose a novel relaxation modified factor to accelerate the numerical solution of the radiative transfer equation (RTE) with several high-resolution total variation diminishing schemes. The methodology proposed is denoted as the X-factor method.Design/methodology/approachThe X-factor method was compared with the technique deferred-correction (DC) for the calculations of a two-dimensional cavity with absorting-emiting-scatteting gray media using the discrete ordinates method. Four parameters were considered to evaluate: the absorption coefficient, the emissivity of boundary surface, the scattering albedo and under-relaxation factor.FindingsThe results showed the central processing unit (CPU) time of X-factor method was lower than DC. The reductions of CPU time with the X-factor method were observed from 0.6 to 75.4%.Originality/valueThe superiority of the X-factor method over DC was showed with the reduction of CPU time of the numerical solution of RTE for evaluated cases.

An adaptive nested source term iteration for radiative transfer equations

We propose a new approach to the numerical solution of radiative transfer equations with certified a posteriori error bounds. A key role is played by stable Petrov--Galerkin type variational formulations of parametric transport equations and corresponding radiative transfer equations. This allows us to formulate an iteration in a suitable, infinite dimensional function space that is guaranteed to converge with a fixed error reduction per step. The numerical scheme is then based on approximately realizing this iteration within dynamically updated accuracy tolerances that still ensure convergence to the exact solution. To advance this iteration two operations need to be performed within suitably tightened accuracy tolerances. First, the global scattering operator needs to be approximately applied to the current iterate within a tolerance comparable to the current accuracy level. Second, parameter dependent linear transport equations need to be solved, again at the required accuracy of the iteration. To ensure that the stage dependent error tolerances are met, one has to employ rigorous a posteriori error bounds which, in our case, rest on a Discontinuous Petrov--Galerkin (DPG) scheme. These a posteriori bounds are not only crucial for guaranteeing the convergence of the perturbed iteration but are also used to generate adapted parameter dependent spatial meshes. This turns out to significantly reduce overall computational complexity. Since the global operator is only applied, we avoid the need to solve linear systems with densely populated matrices. Moreover, the approximate application of the global scatterer accelerated through low-rank approximation and matrix compression techniques. The theoretical findings are illustrated and complemented by numerical experiments with non-trivial scattering kernels.