Lattice Boltzmann Method Applied to Radiative Transport Analysis in a Planar Participating Medium

This article deals with the extension of the usage of the lattice Boltzmann method (LBM) to the analysis of radiative heat transfer with and without conduction in a one-dimensional (1-D) planar participating medium. A novel lattice needed for the calculation of the volumetric radiation spanned over the 4π spherical space has been introduced. The LBM formulation is tested for three benchmark problems, namely, radiative equilibrium, nonradiative equilibrium, and a combined mode conduction–radiation problem in a planar geometry. In the combined mode problem, with radiative information known from the proposed lattice structure, the energy equation is also formulated and solved using the LBM. The D1Q2 lattice is used in the energy equation. For validation, in problems 1 and 2, the LBM results are compared with the finite-volume method (FVM), while in problem 3, the LBM-LBM results are compared with the LBM-FVM in which FVM is used for the computation of radiative information. Comparisons are made for the effects of the governing parameters such as the extinction coefficient, the scattering albedo, and so on, on heat flux and emissive power (temperature) distributions. LBM results are found to be in excellent agreement with the benchmark results.     

Performance evaluation of four radiative transfer methods in solving multi-dimensional radiation and/or conduction heat transfer problems

This article reports results of the four popular and widely used numerical methods, viz., the Monte Carlo method (MCM), the discrete transfer method (DTM), the discrete ordinates method (DOM) and the finite volume method (FVM) used to calculate radiative information in any thermal problem. Different classes of problems dealing with radiation and/or conduction heat transfer problems in a 2-D rectangular absorbing, emitting and scattering participating medium have been considered. In radiative equilibrium and non-radiative equilibrium cases, the MCM results have been used as the benchmark data for comparing the performances of the DTM, the DOM and the FVM. In the combined radiation and conduction mode problem, the energy equation has been formulated using the lattice Boltzmann method (LBM). To compare the performance of the DTM, the DOM and the FVM, the required radiative field data computed using these methods have been provided to the LBM formulation. Temperature distributions obtained using the four methods and those obtained from the LBM in conjunction with the DTM, the DOM and the FVM have been compared for different parameters such as the extinction coefficient, the scattering albedo, the conduction-radiation parameter, the wall emissivity, the aspect ratio and heat generation rate. In all the cases, results of these methods have been found in good agreements. Computationally, the DTM was found the most time consuming, and the DOM was computationally the most efficient.     

Unstructured Polygonal Finite-Volume Solutions of Radiative Heat Transfer in a Complex Axisymmetric Enclosure

Radiative heat transfer in a complex axisymmetric enclosure with participating medium is investigated by using the finite-volume method (FVM). In particular, an implementation of the unstructured polygonal meshes is adopted by connecting each center point of the unstructured triangular meshes rather than joining the centroids of the triangular elements to the midpoints of the corresponding sides to form a polygonal element. Also, typical considerations regarding application of the present polygonal mesh system to axisymmetric radiation are discussed. After a mathematical formulation and corresponding discretization equation for the radiative transfer equation (RTE) are derived, the final discretization equation is introduced with the conventional finite-volume approach by using the directional weights. For validation and comparison, three test examples with complex axisymmetric geometries have been accomplished. The present study shows that not only is the method flexible in treating radiative problems with complex geometries, it is also accurate and efficient for the analysis of axisymmetric radiative heat transfer.     

Application of the lattice Boltzmann method for solving the energy equation of a 2-D transient conduction-radiation problem

A lattice Boltzmann method (LBM) is used to solve the energy equation in a test problem involving thermal radiation and to thus investigate the suitability of scalar diffusion LBM for a new class of problems. The problem chosen is transient conductive and radiative heat transfer in a 2-D rectangular enclosure filled with an optically absorbing, emitting and scattering medium. The energy equation of the problem is solved alternatively with a previously used finite volume method (FVM) and with the LBM, while the radiative transfer equation is solved in both cases using the collapsed dimension method. In a parametric study on the effects of the conduction-radiation parameter, extinction coefficient, scattering albedo, and enclosure aspect ratio, FVM and LBM are compared in each case. It is found that, for given level of accuracy, LBM converges in fewer iterations to the steady-state solution, independent of the influence of radiation. On the other hand, the computational cost per iteration is higher for LBM than for the FVM for a simple grid. For coupled radiation-diffusion, the LBM is faster than the FVM because the radiative transfer computation is more time-consuming than that of diffusion.     

THE EFFECTS OF RADIATION ON NATURAL CONVECTION IN A RECTANGULAR ENCLOSURE DIVIDED BY TWO PARTITIONS

The phenomena of radiation-affected steady-laminar natural convection in a rectangular enclosure with two incomplete partitions are numerically examined under a large temperature difference. Pure convection, convection with surface radiation, and convection with surface gas radiation are considered and compared. To examine the effects of two incomplete partitions on thermofluid dynamics behavior, they are assumed to be very thin and adiabatic. The finite-volume method (FVM) is used for solving the radiative transport equation, assuming that partitions are radiatively opaque. After validating the numerical procedures, the detailed radiation effects were sought. Based on the results of this study, it was found that the radiation played a significant role in developing the fluid dynamic and thermal distributions compared with cases without radiation. Once radiation was involved, the surface radiation was dominant over the gas radiation. The baffle configuration was also found to affect the results of radiation.     

Lattice Boltzmann Method Applied to the Solution of Energy Equation of a Radiation and Non-Fourier Heat Conduction Problem

This article concerns the application of the lattice Boltzmann method (LBM) to solve the energy equation of a combined radiation and non-Fourier conduction heat transfer problem. The finite propagation speed of the thermal wave front is accounted by non-Fourier heat conduction equation. The governing energy equation is solved using the LBM. The finite-volume method (FVM) is used to compute the radiative information. The formulation is validated by taking test cases in 1-D planar absorbing, emitting, and scattering medium whose west boundary experiences a sudden rise in temperature, or, with adiabatic boundaries, the medium is subjected to a sudden localized energy source. Results are analyzed for the various values of parameters like the extinction coefficient, the scattering albedo, the conduction-radiation parameter, etc., on temperature distributions in the medium. Radiation has been found to help in facilitating faster distribution of energy in the medium. Unlike Fourier conduction, wave fronts have been found to reflect from the boundaries. The LBM-FVM combination has been found to provide accurate results.     

Effect of Axial Radiation on Heat Transfer in a Thermally and Hydrodynamically Developing Flow between Parallel Plates

The present work investigates coupled convection-radiation heat transfer through parallel plates with both heated and cooled walls. The flow is both hydrodynamically and thermally developing. The momentum and energy equations are solved with the computational fluid dynamics (CFD) code FASTEST3D, based on the finite-volume method (FVM). The medium is considered to be radiatively participating and the radiative heat transfer is also treated by the finite-volume method. Both axial and transverse heat transfer are considered. The effect of axial radiation is shown for different optical parameters by comparing results with and without axial radiation. The effect of hydrodynamically developing flows is also considered.     

Analysis of Conduction-Radiation Heat Transfer in a 2D Enclosure Using the Lattice Boltzmann Method

Application of the lattice Boltzmann method (LBM) recently extended by Pietro et al. [P. Asinari, S. C. Mishra, and R. Borchiellini, A Lattice Boltzmann Formulation to the Analysis of Radiative Heat Transfer Problems in a Participating Medium, Numer. Heat Transfer B, 57(2), 126–146, 2010] for calculation of volumetric radiative information is extended for the analysis of a combined mode transient conduction and radiation heat transfer in a 2D rectangular enclosure containing an absorbing, emitting and scattering medium. Unlike all previous studies, with volumetric radiative information computed using the proposed LBM, the energy equation is formulated and solved using the LBM. In the combined mode conduction–radiation problem, to assess the computational advantage of computing the radiative information too using the LBM, the same problem is also solved using the LBM–finite volume method (FVM) formulation. In this LBM–FVM formulation, the FVM is used to calculate the volumetric radiative information needed for the energy equation, and the energy equation is solved using the LBM. Comparisons are made for the effects of the extinction coefficient, the scattering albedo and the conduction–radiation parameter on the temperature distributions in the medium. Although the number of iterations for the converged solution in LBM–LBM is much more than that of the LBM–FVM, computationally, the LBM–LBM is faster than the LBM–FVM.     

An Inverse Analysis for Parameter Estimation Applied to a Non-Fourier Conduction–Radiation Problem

Retrieval of parameters in a non-Fourier conduction and radiation heat transfer problem is reported. The direct problem is formulated using the lattice Boltzmann method (LBM) and the finite-volume method (FVM). The divergence of radiative heat flux is computed using the FVM, and the LBM formulation is employed to obtain the temperature field. In the inverse method, this temperature field is taken as exact. Simultaneous estimation of parameters, namely, the extinction coefficient and the conduction–radiation parameter, is done by minimizing the objective function. The genetic algorithm (GA) is used for this purpose. The accuracies of the estimated parameters are studied for the effects of measurement errors and genetic parameters such as the crossover and mutation probabilities, the population size, and the number of generations. The LBM-FVM in combination with GA has been found to provide a correct estimate of parameters.     

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.     

UNSTRUCTURED FINITE-VOLUME METHOD FOR RADIATIVE HEAT TRANSFER IN A COMPLEX TWO-DIMENSIONAL GEOMETRY WITH OBSTACLES

The radiative heat transfer in a complex two-dimensional enclosure with obstacles with participating medium is very important in practical engineering applications. In order to deal with this problem, in this study the finite-volume method (FVM) for radiation has been derived using the unstructured grid system. A general discretization equation was formulated by introducing the directional weight and the step scheme for spatial differencing. For its comparison and validation, two test cases, an equilateral triangular enclosure and a square enclosure with baffle, were chosen. Then, more complex and practical cases, such as a semicircular enclosure with cylinder hole, a square enclosure with finned internal cylinder, and a furnace with embedded cooling pipes, were investigated. All the results obtained by the unstructured FVM agreed very well with the exact solutions as well as the results obtained by the zone method. Furthermore, the wiggling behavior occurring in the blocked-off FVM was not produced by the unstructured FVM. Three types of manipulation of control angle overlap were also examined here. It was found that the solutions depended on the type of manipulation of control angle overlap, especially when the number of control angles was small. Usually, both the pixelation method and exact treatment introduced here yielded better solutions than the bold approximation.     

Improved Treatment of Anisotropic Scattering in Radiation Transfer Analysis Using the Finite Volume Method

Discretization of the integral anisotropic-scattering term in the equation of radiative transfer will result in two kinds of numerical errors: alterations in scattered energy and asymmetry factor. Though quadrature flexibility with large angular directions and further solid-angle splitting in the finite volume method (FVM) allow for reduction/minimization of these errors, computational efficiency is adversely impacted. A phase-function normalization technique to get rid of these errors is simpler and is applied to the three-dimensional (3-D) FVM for the first time to improve anisotropic radiation transfer computation accuracy and efficiency. FVM results are compared to Monte Carlo and discrete-ordinates method predictions of radiative heat transfer in a cubic enclosure housing a highly anisotropic participating medium. It is found that the FVM results generated using the normalization technique conform accurately to the results of the other two methods with little impact on computational efficiency.     

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.     

The Effect of Radiation on Flow in a Conical Diffuser

The effect of thermal radiation on the flow in a conical diffuser of half-cone angle 2.5° is studied numerically for constant thermophysical properties. The finite-volume method (FVM) is employed to solve both the Navier-Stokes equation (NSE) and the integro-differential radiative transfer equation (RTE). The effect of radiative parameters, e.g., optical diameter, scattering albedo and wall emissivity at two Reynolds numbers (Re = 500 and 1000) and Pr = 1 has been investigated for constant wall temperature of 1000 K and inlet temperature of 2000 K. The results show that radiation in a transparent or in a purely scattering medium does not affect the flow temperature profile and bulk mean temperature variation, whereas it increases the total Nusselt number; however, radiation in a absorbing-emitting participating medium significantly affects the temperature profile as well as Nusselt number, when compared with the pure convection case.     

Analysis of Conduction and Radiation Heat Transfer in a Differentially Heated 2‐D Square Enclosure

Radiative heat transfer with and without conduction in a differentially heated 2‐D square enclosure is analyzed. The enclosure with diffuse gray boundaries contains radiating and/or conducting gray homogeneous medium. Radiatively, the medium is absorbing, emitting and scattering. On the south boundary, four types of discrete heated regions, viz., the full boundary, the left one‐third, left two third and middle one third, are considered. In the absence of conduction, distributions of heat flux along the south boundary are studied for the effect of extinction coefficient. In the presence of conduction, distributions of radiation, conduction and total heat fluxes along the south boundary are analyzed for the effects of extinction coefficient, scattering albedo, conduction–radiation parameter, and south boundary emissivity. Effects of these parameters on centerline temperature distribution are also studied. To assess the performance of three commonly used radiative transfer methods, in all cases, the radiative transfer equation is solved using the discrete ordinate method (DOM), the conventional discrete ordinate method (CDOM) and the finite volume method (FVM). In the combined mode problem, with volumetric radiative information known from one of the three methods, viz., DOM, CDOM, and FVM, the energy equation is solved using the finite difference method (FDM). In all cases, the results from FDM‐DOM, FDM‐CDOM, and FDM‐FVM are in good agreement. Computationally, all three sets of methods are equally efficient.     

Analysis of combined mode heat transfer in a porous medium using the lattice Boltzmann method

ABSTRACT

Application of the lattice Boltzmann method (LBM) in solving a combined mode conduction, convection, and radiation heat transfer problem in a porous medium is extended. Consideration is given to a 1-D planar porous medium with a localized volumetric heat generation zone. Three particle distribution functions, one each for the solid temperature, the gas temperature, and the intensity of radiation, are simultaneously used to solve the gas- and the solid-phase energy equations. The volumetric radiation source term appears in the solid-phase energy equation, and it is also computed using the LBM. To check the accuracy of the LBM results, the same problem is also solved using the finite volume method (FVM). Effects of convective coupling, flow enthalpy, solid-phase conductivity, scattering albedo porosity, and emissivity on axial temperature distribution are studied and compared with the FVM results. Effects of flow enthalpy, solid-phase conductivity, and emissivity are also studied on radiative output. LBM results are in excellent agreement with those of the FVM.     

Numerical Study of the Swirl Effect on a Coaxial Jet Combustor Flame Including Radiative Heat Transfer

A numerical study of the swirl effect on a coaxial jet combustor flame including radiative heat transfer is presented. In this work, the standard k-ε model is applied to investigate the turbulence effect, and the eddy dissipation model (EDM) is used to model combustion. The radiative heat transfer and the properties of gases and soot are considered using a coupled of the finite-volume method (FVM), and the narrow-band based weighted-sum-of-gray gases (WSGG-SNB) model. The results of this work are validated by experiment data. The results clearly show that radiation must be taken into account to obtain good accuracy for turbulent diffusion flame in combustor chamber. Flame is very influenced by the radiation of gases, soot, and combustor wall. However, swirl is an important controlling variable on the combustion characteristics and pollutant formation.     

TREATMENT OF IRREGULAR GEOMETRIES USING A CARTESIAN COORDINATES FINITE-VOLUME RADIATION HEAT TRANSFER PROCEDURE

Abstract

This article presents a blocked-off-region procedure to model radiative transfer in irregular geometries using a Cartesian coordinates finite-volume method (FVM). Straight-edged, inclined and curved boundaries can be treated. It is capable of handling participating or transparent media enclosed by black or reflecting walls. With this procedure, irregular geometries can be specified through the problem specification portion of the program. Four test problems are used to show that the procedure is capable of reproducing available results for inclined and curved walls, transparent, nonscattering, and anisotropically scattering media.     

Inverse radiation–conduction design problem in a participating concentric cylindrical medium

Inverse conduction–radiation problem for design analysis in a two-dimensional concentric cylindrical absorbing, emitting and isotropically scattering medium has been solved, when the desired boundary conditions are available on the design surface. The finite-volume method was adopted to deal with energy conservation equation including conduction and radiation. The radiative transfer equation was also taken into consideration in direct problem, whereas the Levenberg–Marquardt method was used to solve a set of equations in inverse problem, which are expressed by errors between estimated and desired total heat fluxes on the design surface. The automatic differentiation as well as the Broyden combined update was utilized to reduce computational time in calculating the sensitivity matrix. The results have shown that the desired total heat flux distribution on design surface could be successfully estimated with less computational time using the present inverse procedure developed here.     

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.