Development and comparison of the DTM,the DOM and the FVM formulations for the short-pulse laser transport through a participating medium

The present article deals with the analysis of transient radiative transfer caused by a short-pulse laser irradiation on a participating medium. A general formulation of the governing transient radiative transfer equation applicable to a 3-D Cartesian enclosure has been presented. To solve the transient radiative transfer equation, formulations have been presented for the three commonly used methods in the study of radiative heat transfer, viz., the discrete transfer method, the discrete ordinate method and the finite volume method. To show the uniformity in the formulations in the three methods, the intensity directions and the angular quadrature schemes for computing the incident radiation and heat flux have been taken the same. To validate the formulations and to compare the performance of the three methods, effect of a square short-pulse laser having pulse-width of the order of a femtosecond on transmittance and reflectance signals in case of an absorbing and scattering planar layer has been studied. Effects of the medium properties such as the extinction coefficient, the scattering albedo and the anisotropy factor and the laser properties such as the pulse-width and the angle of incidence on the transmittance and the reflectance signals have been compared. In all the cases, results of the three methods were found to compare very well with each other. Computationally, the discrete ordinate method was found to be the most efficient.     

Participating media exposed to collimated short-pulse irradiation – A Laguerre–Galerkin solution

A new method is developed for the solution of radiative transfer in a one-dimensional absorbing and isotropically scattering medium with short-pulse irradiation on one of its boundaries. The time-dependent radiative intensity is expanded in a series of Laguerre polynomials with time as the argument. Moments of the radiative transfer equation, as well as of the boundary conditions, then yield a set of coupled time-independent radiative transfer problems. This set, in turn, is reduced to a set of algebraic equations by the application of the Galerkin method. The transient transmittance and reflectance of the medium are evaluated for various values of the optical thickness, scattering albedo and pulse duration. It is demonstrated that the Laguerre–Galerkin method is not only easier to implement and more efficient but also yields more accurate results compared to the direct application of the Galerkin method. The results are in very good agreement with those available in the literature.     

Analysis of Radiative Transport in a 2-D Cylindrical Participating Medium Subjected to Collimated Radiation

This article deals with the numerical analysis of radiative transport in a 2-D axisymmetric cylindrical enclosure containing absorbing, emitting, and scattering medium. The participating medium receives collimated radiation from the top boundary of the enclosure. Attenuation of the collimated radiation in the medium gives rise to the diffuse radiation. Thus, the governing radiative transfer equation accounts for both collimated and diffuse radiation. The radiative transfer equation is solved using the modified discrete ordinate method. Effects of extinction coefficient, scattering albedo, and aspect ratio on radial and axial distributions of heat flux and incident radiation are studied. In all cases, results are validated against those available in the literature. Modified discrete ordinate method has been found to provide accurate results.     

Improved social spider optimization algorithms for solving inverse radiation and coupled radiation–conduction heat transfer problems

A novel bio-inspired swarm algorithm, social spider optimization (SSO), is introduced to solve the inverse transient radiation and coupled radiation–conduction problems for the first time. Based on the original model, five improved SSO (ISSO) algorithms are developed to enhance search ability and convergence velocity. The sensitivity analysis of measured signals with respect to the physical parameters of the medium are described. After which, the SSO and ISSO algorithms are applied to solve the inverse estimation problems in a one-dimensional participating medium. Two cases concerns radiative transfer problems are investigated, in which the radiative source term, extinction coefficient, scattering albedo, and scattering symmetry factor are reconstructed. Furthermore, the coupled radiation–conduction heat transfer model is considered and the main parameters such as the conduction–radiation parameter, boundary emissivity, and scattering albedo are retrieved. All retrieval results show that SSO-based algorithms are robust and effective in solving inverse estimation problems even with measurement errors. Findings also show that the proposed ISSO algorithms are superior to the original SSO model in terms of computational accuracy and convergence velocity.     

Second-Order Radiative Transfer Equation and Its Properties of Numerical Solution Using the Finite-Element Method

The original radiative transfer equation is a first-order integrodifferential equation, which can be taken as a convection-dominated equation. The presence of the convection term may cause nonphysical oscillation of solutions. This type of instability can occur in many numerical methods, including the finite-difference method and the finite-element method, if no special stability treatment is used. To overcome this problem, a second-order radiative transfer equation is derived, which is a diffusion-type equation similar to the heat conduction equation for an anisotropic medium. The consistency of the second-order radiative transfer equation with the original radiative transfer equation is demonstrated. The perturbation characteristics of error are analyzed and compared for both the first- and second-order equations. Good numerical properties are found for the second-order radiative transfer equation. To show the properties of the numerical solution, the standard Galerkin finite-element method is employed to solve the second-order radiative transfer equation. Four test problems are taken as examples to check the numerical properties of the second-order radiative transfer equation. The results show that the standard Galerkin finite-element solution of the second-order radiative transfer equation is numerically stable, efficient, and accurate.     

Analysis of Conduction and Radiation Heat Transfer in a 2-D Cylindrical Medium Using the Modified Discrete Ordinate Method and the Lattice Boltzmann Method

This article deals with the analysis of radiative transport with and without conduction in a finite concentric cylindrical enclosure containing absorbing, emitting, and scattering medium. Isothermal medium as the radiation source confined between the cold cylinders and a nonisothermal medium with the inner cylinder as the radiation source are the two nonradiative and radiative equilibrium problems. They involve only calculation of radiative information. In the third problem, a conducting-radiating medium is thermally perturbed by raising the temperature of the inner cylinder. In all problems, radiative information is computed using the modified discrete ordinate method (MDOM), and in the third problem, the lattice Boltzmann method (LBM) is used to formulate and solve the energy equation. Depending on the problems, effects of various parameters such as the extinction coefficient, the scattering albedo, the boundary emissivity, the conduction-radiation parameter, and the radius ratio are studied on temperature and heat flux distributions. The MDOM and the LBM-MDOM results are compared with those available in the literature. To further establish the accuracy of the MDOM and the LBM-MDOM results, in all problems, comparisons are made with the results obtained from the finite volume method (FVM) and the finite difference method-FVM approach, in which FVM provides the radiative information. The selection of the FDM-FVM for the third problem is also with the objective that for this problem, not much work is reported in which the FVM is used to calculate the radiative information. MDOM and LBM-MDOM results are found to compare well with those available in the literature, and in all cases they are in excellent agreement with FVM and FDM-FVM approaches.     

Transient Radiative Transfer in a Graded Index Medium with Specularly Reflecting Surfaces

A modified Monte Carlo (MC) method has been developed for solving transient radiative transfer in a one-dimensional scattering medium with a graded refractive index. The accuracy and computational efficiency of the algorithm are validated initially. With the introduction of time shift and superposition principle into the MC model, the computational efficiency is greatly improved. We make a comparative analysis of the time-resolved incident radiation (and radiative heat flux) distributions in the media with diffuse and specular reflection boundaries. Results show that the temporal and spatial radiative signals of the medium with specular reflection boundaries greatly differ from those having diffuse reflection boundaries.     

Lattice Boltzmann Method and Modified Discrete Ordinate Method Applied to Radiative Transport in a Spherical Medium with and without Conduction

This article deals with the application of the modified discrete ordinate method (MDOM) to calculate volumetric radiative information with and without conduction in a concentric spherical enclosure containing a participating medium. With radiative information known from the MDOM, the energy equation of the combined mode transient conduction and radiation heat transfer is formulated and solved using the lattice Boltzmann method (LBM). Without conduction, for pure radiation case, two benchmark problems, representing nonradiative and radiative equilibrium situations are taken up. In the case of non-radiative equilibrium, an isothermal medium is bounded by cold walls and medium is the source of radiation, while in the case of radiative equilibrium, nonisothermal medium is confined between a hot and a cold wall, and the hot (inner sphere) wall is the radiation source. Depending upon the problem, heat flux, energy flow rate, emissive power, and temperature distributions in the medium are calculated for different values of parameters such as the extinction coefficient, the scattering albedo, the conduction-radiation parameter, the boundary emissivity, and the radius ratio. To validate the MDOM and the LBM-MDOM formulations, problems are also solved using the finite volume method (FVM) and the finite-difference method (FDM)–FVM approach, in which the FVM is used to calculate the volumetric radiation and the energy equation is also solved using the FDM. Results of the MDOM, LBM–MDOM, FVM and FDM–FVM are also benchmarked against those available in the literature. MDOM and LBM–MDOM have been found to provide accurate results.     

Lattice Boltzmann simulations of a radiatively participating fluid in Rayleigh–Benard convection

Thermal radiation is an integral part of the heat transfer process but it is often neglected due to the complexity involved in the analysis of radiative transfer. We use the lattice Boltzmann method as a common computational tool to solve all three modes of heat transfer: conduction, convection, and radiation. This tool is then used to analyze the effect of radiatively participating medium on Rayleigh–Benard convection. We find that increasing the effects of radiation (i) increases the critical Rayleigh number required for the onset of Rayleigh–Benard convection and (ii) affects the temperature and flow patterns of convection rolls significantly changing the net heat transfer between the hot and cold plates. Both these effects are due to the presence of radiation available as an additional mode of heat transfer. Thus, we establish that the unified lattice Boltzmann framework is an effective computational tool for heat transfer and propose to use this method for a large range of problems in science and engineering involving radiative heat transfer.     

Chebyshev collocation spectral method for one-dimensional radiative heat transfer in graded index media

Chebyshev spectral collocation method based on discrete ordinates equation is developed to solve radiative transfer problems in a one-dimensional absorbing, emitting and scattering semitransparent slab with spatially variable refractive index. For radiative transfer equation, the angular domain is discretized by discrete ordinates method, and the spatial domain is discretized by Chebyshev collocation spectral method. Due to the exponential convergence of spectral methods, a very high accuracy can be obtained even using few nodes for present problems. Numerical results by the Chebyshev collocation spectral-discrete ordinates method (SP-DOM) are compared with those available data in references. Effects of refractive index gradient on radiative intensity are studied for space dependent scattering media. The results show that SP-DOM has a good accuracy and efficiency for solving radiative heat transfer problems in even spatially varying absorbing, emitting, scattering, and graded index media.     

Radiative heat transfer in semitransparent solidifying slab considering space–time dependent refractive index

Radiative heat transfer in semitransparent phase-change media is of great interest in many engineering fields. Its essence is the transient coupled heat transfer of radiation and conduction along with liquid–solid phase change. The difficulty is to solve radiative heat transfer with the consideration of time–space dependent radiative properties. Especially when the refractive index is considered to vary with space and time in phase change, the problem becomes more complicated. This paper investigates the problem of the variable radiative properties with space and time during phase change in semitransparent media. The phase-change medium is assumed to have solid, mushy and liquid zones, and the solid/mushy and liquid/mushy interfaces are considered to be semitransparent and diffuse reflecting. In different zones, there are different physical property parameters. Phase interfaces are always moving in phase change, while the interfaces of control volumes are fixed. Therefore, the interfaces of control volumes and phase interfaces are not always coincided, which will bring errors into the simulation of radiative transfer in phase-change media. However, the errors can be reduced by dividing the medium into enough sub-layers. As long as the number of sub-layers is big enough, the errors can be limited in a very small range. Then using the multilayer radiative transfer model, we can solve the radiative transfer problem in the semitransparent phase-change medium. Considering time–space dependent refractive index, this paper analyzes coupled radiative and conductive heat transfer in semitransparent solidifying media. The results show that the effects of variable refractive index with time and space on transient coupled heat transfer are significant and could not be neglected inside the semitransparent phase-change medium under some conditions.     

The finite volume method approach to the collapsed dimension method in analyzing steady/transient radiative transfer problems in participating media

With the finite volume formulation (FVM) approach applied to the collapsed dimension method (CDM), this article deals with the application of the CDM to analyze radiative heat transfer problems in a participating medium subjected to a continuous diffuse or a continuous/short-pulse collimated boundary radiative loading. The planar medium contained between diffuse gray boundaries is absorbing, emitting and anisotropically scattering. With three categories of thermal boundary radiative loadings, for the four types of problems considered, the CDM results are compared for a wide range of radiative parameters with that of the FVM.     

Prediction of radiative heat transfer between two concentric spherical enclosures with the finite volume method

The radiative heat transfer between two concentric spheres separated by an absorbing, emitting, and isotropically scattering gray medium is investigated by using the finite volume method (FVM). Especially, a mapping that simplifies the solution of spherically symmetric radiative heat transfer problems is introduced, thereby, the intensity depending on spatial one-dimension and angular one-dimension is transformed into spatial two-dimensional one. By adopting this mapping process, angular redistribution, which appears in such curvilinear coordinates as cylindrical or spherical ones, is treated efficiently without any artifice usually introduced in the conventional discrete ordinates method (DOM). After a mathematical formulation and corresponding discretization equation for the radiative transfer equation (RTE) are derived, final discretization equation is introduced by using the directional weight, which is the key parameter in the FVM since it represents the inflow or outflow of radiant energy across the control volume faces depending on its sign. The present approach is then validated by comparing the present results with those of previous works by changing such various parameters as temperature ratio between inner and outer spherical enclosure, wall emissivity, and optical thickness of the participating medium. All the results presented in this work show that the present method is accurate and valuable for the analysis of spherically symmetric radiative heat transfer problems between two concentric spheres.     

Comparison of the Discrete-Ordinates Method and the Finite-Volume Method for Steady-State and Ultrafast Radiative Transfer Analysis in Cylindrical Coordinates

The time-dependent equation of radiative transfer is solved for an axisymmetric cylindrical medium using both the discrete-ordinates method and the finite-volume method. Steady and transient flux profiles are determined for absorbing and scattering media. Results for each solution method are compared and shown for various grid numbers, scattering albedos, and optical thicknesses. A comparison of computational time and memory usage between the methods is presented. It is found that the finite-volume method uses more memory and has a longer convergence time than the discrete-ordinates method for all cases, due to the difference in angular treatment.     

Development of a finite element radiation model applied to two‐dimensional participating media

A finite element method (FEM) for radiative heat transfer has been developed and it is applied to 2D problems with unstructured meshes. The present work provides a solution for temperature distribution in a rectangular enclosure with black or gray walls containing an absorbing, emitting, isotropically scattering medium. Compared with the results available from Monte Carlo simulation and finite volume method (FVM), the present FEM can predict the radiative heat transfer accurately. © 2005 Wiley Periodicals, Inc. Heat Trans Asian Res, 34(6): 386–395, 2005; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.20076     

Enhancement of Heat Transfer with Porous/Solid Insert for Laminar Flow of a Participating Gas in a 3-D Square Duct

In recent years, porous or solid insert has been used in a duct for enhancing heat transfer in high temperature thermal equipment, where both convective and radiative heat transfer play a major role. In the present work, the study of heat transfer enhancement is carried out for flow through a square duct with a porous or a solid insert. Most of the analyses are carried out for a porous insert. The hydrodynamically developing flow field is solved using the Navier–Stokes equation and the Darcy–Brinkman model is considered for solving the flow in the porous region. The radiative heat transfer is included in the analysis by coupling the radiative transfer equation to the energy equation. The fluid considered is CO₂ with temperature dependent thermophysical properties. Both the fluid and the porous medium are considered as gray participating medium. The increase in heat transfer is analyzed by comparing the bulk mean temperature, Nusselt number, and radiative heat flux for different porous size and orientation, Reyonlds number, and Darcy number.     

RADIATION ELEMENT METHOD FOR TRANSIENT HYPERBOLIC RADIATIVE TRANSFER IN PLANE-PARALLEL INHOMOGENEOUS MEDIA

In this study the radiation element method is formulated to solve transient radiative transfer with light radiation propagation effect in scattering, absorbing, and emitting media with inhomogeneous property. The accuracy of the method is verified by good agreement between the present calculations and Monte Carlo simulations. The sensitivity of the method against element size, ray emission number, and time increment size is examined. The transient effect of radiation propagation is essential in short-pulse laser radiation transport when the input pulse width is not considerably larger than the system radiation propagation time. The transient characteristics of radiative transfer are investigated in the media subject to collimated laser irradiation and/or diffuse irradiation withtemporal Gaussian and/or square profiles. The inhomogeneous profile of extinction coefficient of the medium affects strongly the transient radiative flux divergence inside the medium.     

A Lattice Boltzmann Formulation for the Analysis of Radiative Heat Transfer Problems in a Participating Medium

Use of the lattice Boltzmann method (LBM) has been extended to analyze radiative transport problems in an absorbing, emitting, and scattering medium. In terms of collision and streaming, the present approach of the LBM for radiative heat transfer is similar to those being used in fluid dynamics and heat transfer for the analyses of conduction and convection problems. However, to mitigate the effect of the isotropy in the polar direction, in the present LBM approach, lattices with more number of directions than those being used for the 2-D system have been employed. The LBM formulation has been validated by solving benchmark radiative equilibrium problems in 1-D and 2-D Cartesian geometry. Temperature and heat flux distributions have been obtained for a wide range of extinction coefficients. The LBM results have been compared against the results obtained from the finite-volume method (FVM). Good comparison has been obtained. The numbers of iterations and CPU times for the LBM and the FVM have also been compared. The number of iterations in the LBM has been found to be much more than the FVM. However, computationally, the LBM has been found to be much faster than the FVM.     

Radiative heat transfer in a participating medium with specular–diffuse surfaces

The results obtained by ray-tracing method can be regarded as benchmarks for its good accuracy. However, up to now, this method can be only used to solve radiative transfer within medium confined between two specular surfaces or two diffuse surfaces. This article proposes a hybrid ray-tracing method to solve the radiative transfer inside a plane-parallel absorbing–emitting–scattering medium with one specular surface and another diffuse surface (S–D surfaces). By the hybrid ray-tracing method, radiative transfer coefficients (RTCs) for S–D surfaces are deduced. Both surfaces of the medium under consideration are considered to be semitransparent or opaque. This paper examines the effects of scattering albedo, opaque surface emissivity and anisotropically scattering on steady-state heat flux and transient temperature fields. From the results it is found that the effects of anisotropic scattering is more for a bigger optical thickness medium; and keeping other optical parameters unchanged, anisotropic scattering affects transient temperature distributions so much in a small refractive index medium.     

Hybrid Laplace transform technique for Stefan problems with radiation-convection boundary condition

The hybrid application of the Laplace transform technique and the finite difference method (FDM) to one-dimensional Stefan problems involving the radiative and convective boundary condition is studied. The radiative term is linearized by Taylor's series approximation, and then the above hybrid method is used. This scheme is obtained by the use of the Laplace transform technique for the time-dependent terms and the fixed-grid FDM for space domain. It can be found from various illustrated examples that excellent agreement is obtained between the present results and those of early works. For the phase-change problem subjected to the nonlinear boundary condition, three or four iterations are required to obtain a convergent result at a specific time. The present analysis also demonstrates that the application of the Laplace transform technique is no longer limited to phase-change problems with the linear boundary condition.