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 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.     

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.     

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.     

Numerical Simulation of Radiative-Conductive Heat Transfer in an Enclosure with an Isotherm Obstacle

In this paper, the lattice Boltzmann method (LBM) and discrete ordinates method (DOM) were applied to investigate the heat transfer in a square radiative-conductive media with heat flux and temperature boundary conditions. Furthermore, an isothermal rectangular obstacle is located in the middle of participating media. The energy equation is solved using the LBM; while the radiative transfer equation is solved using DOM. The effects of various parameters such as the extinction coefficient, scattering albedo, and the conduction–radiation parameter in the presence of an obstacle are studied on temperature and heat flux distributions. It was shown that, decrease in scattering albedo value leads to decrease of the temperature field in participating media. In addition, with increase in scattering albedo value, conductive heat flux increases and radiative heat flux decreases. It was shown that increase in extinction coefficient and decrease in conduction–radiation parameter have some significant effects on increasing the temperature profile, especially in the region with longer distance from obstacle.     

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.     

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.     

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.     

Analysis of hyperbolic heat conduction in 1-D planar,cylindrical, and spherical geometry using the lattice Boltzmann method

Analyses of hyperbolic heat conduction in an 1-D planar, cylindrical, and spherical geometry are analyzed using the lattice Boltzamnn method (LBM). Finite time lag between the imposition of temperature gradient and manifestation of heat flow causes the governing energy equation to be hyperbolic one. Temporal temperature distributions are analyzed for thermal perturbation of a boundary by suddenly raising its temperature and also by imposing a constant heat flux to it. Wave-like temperature distributions in the medium are obtained when constant temperature boundary condition is used. However, when constant heat flux boundary condition is used, temperature distribution fluctuates before it becomes stable. To check the accuracy of the LBM results, the problems are also solved using the finite difference method (FDM). LBM and FDM results compare exceedingly well. LBM has computational advantage over the FDM.     

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.     

Thermal analysis of a 2-D heat recovery system using porous media including lattice Boltzmann simulation of fluid flow

The present work deals with the fluid flow simulation and thermal analysis of a two-dimensional heat recovery system using porous media. A basic high-temperature flow system is considered in which a high-temperature non-radiating gas flows through a random porous matrix. The porous medium, in addition to its convective heat exchange with the gas, may absorb, emit and scatter thermal radiation. It is desirable to have large amount of radiative heat flux from the porous segment in the upstream direction (towards the thermal system). The lattice Boltzmann method (LBM) is used to simulate fluid flow in the porous medium. The gas and solid phases are considered in non-local thermal equilibrium, and separate energy equations are applied to these phases. Convection, conduction and radiation heat transfers take place simultaneously in solid phase, but in the gas flow, heat transfer occurs by conduction and convection. In order to analyze the thermal characteristics of the heat recovery system, volume-averaged velocities through the porous matrix obtained by LBM are used in the gas energy equation and then the coupled energy equations for gas and porous medium are numerically solved using finite difference method. For computing of radiative heat flux in the porous medium, discrete ordinates method is used to solve the radiative transfer equation. Finally the effect of various parameters on the performance of porous heat recovery system is studied.     

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.     

Combined mode conduction and radiation heat transfer in a spherical geometry with non-Fourier effect

This article deals with the analysis of combined mode non-Fourier conduction and radiation heat transfer in a concentric spherical enclosure containing a conducting–radiating medium. The finite volume method (FVM) has been employed to calculate the volumetric radiative information and also to solve the governing energy equation, which is of hyperbolic nature. The non-Fourier effect which manifests in the form of a sharp discontinuity in the temporal temperature distribution and propagates with a finite speed has been investigated. As time progress, the discontinuity in the temperature distribution decays and in the steady-state, results with and without non-Fourier effect are the same. Detailed study of the effect of various parameters such as the extinction coefficient, the scattering albedo, the conduction radiation parameter, the emissivity and the anisotropy factor has been carried out. Results of the present work have been compared with the steady-state response of the combined mode Fourier conduction–radiation problems available in literature. Results have been found to agree well.     

Lattice Boltzmann method applied to the solution of the energy equations of the transient conduction and radiation problems on non-uniform lattices

Application of the lattice Boltzmann method (LBM) to solve the energy equations of conduction–radiation problems is extended on non-uniform lattices. In the LBM on non-uniform lattices, the single relaxation time based on the minimum velocity is used. This minimum velocity corresponds to the smallest size lattice. Because information propagates with the same minimum velocity in the prescribed directions from all the lattice centers, in a given time step, they are not equidistant from the neighboring lattices. Collisions in the LBM take place at the same instant. Therefore, in the LBM on non-uniform lattices, in every time step, interpolation is required to carry the information to the neighboring lattice centers. To validate this very concept in heat transfer problems involving thermal radiation, transient conduction and radiation heat transfer problems in a 1-D planar and a 2-D rectangular geometries containing absorbing, emitting and scattering medium are considered. The finite volume method (FVM) is used to compute the radiative information. In both the geometries, results for the effects of various parameters are compared for LBM–FVM on uniform and non-uniform lattices. To establish the LBM–FVM on non-uniform lattices for the combined conduction and radiation heat transfer problems, numerical experiments were performed with different cluster values. The accurate results were found in all the cases.     

Solidification of a 2-D semitransparent medium using the lattice Boltzmann method and the finite volume method

This article deals with the analysis of solidification of a 2-D semitransparent absorbing, emitting and scattering medium. An enthalpy based formulation was employed to simulate the phase-change process. Solidification was assumed to occur over a range of temperatures, and accordingly distinct liquid-, mushy- and solid-zones were considered. The problem was solved using the lattice Boltzmann method. The finite volume method was used to compute the radiative information required in the LBM formulation. Effects of various parameters such as the extinction coefficient, the scattering albedo, the conduction–radiation parameter and the latent heat were studied on temperature and liquid fraction distributions in the medium. These parameters were found to have significant effect on results.     

TRANSIENT CONDUCTION AND RADIATION HEAT TRANSFER WITH VARIABLE THERMAL CONDUCTIVITY

The effect of variable thermal conductivity on transient conduction and radiation heat transfer in a planar medium is investigated. Thermal conductivity of the medium is assumed to vary linearly with temperature, while the other thermophysical properties and the optical properties are assumed constant. The radiative transfer equation is solved using the discrete transfer method, (DTM) and the nonlinear energy equation is solved using an implicit scheme. Transient as well as steady state results are found for an absorbing, emitting, and anisotropically scattering gray medium. Thermal conductivity has been found to have significant effects on both transient as well as steady state temperature and heat flux distributions. Some steady state results are compared with the results reported in the literature.     

Effect of volumetric radiation on natural convection in a square cavity using lattice Boltzmann method with non-uniform lattices

This study deals with the effect of volumetric radiation on the natural convection in a square cavity containing an absorbing, emitting and scattering medium. Numerical simulation has been carried out using lattice Boltzmann method (LBM) with non-uniform lattices. Non-uniform lattices/control volumes have been implemented to deal with the sharp gradients and achieve reasonably accurate solutions. Separate equations dealing with different particle distribution functions in the LBM are used to calculate the density and velocity fields and the thermal fields. The finite volume method (FVM) is used to compute the radiative term of the energy equation. The results obtained in the present study is compared and validated against available results in literature. The centerline temperature across the cavity, the isotherms, the vertical velocity in the horizontal mid-plane, the horizontal velocity in the vertical mid-plane and the streamlines are studied for different parameters such as Rayleigh number, conduction–radiation parameter, extinction coefficient and scattering albedo. The results obtained by using the non-uniform lattices-based LBM are compared with the results for uniform lattice based-LBM. It is found that the non-uniform lattice-based LBM provides accurate results and it is computationally more efficient.     

Two-dimensional transient conduction and radiation heat transfer with temperature dependent thermal conductivity

This paper deals with the effect of the temperature dependent thermal conductivity on transient conduction and radiation heat transfer in a 2-D rectangular enclosure containing an absorbing, emitting and scattering medium. The thermal conductivity of the medium is assumed to vary linearly with temperature. The radiative part of the energy equation was solved using the collapsed dimension method. To facilitate solution of the energy equation, which is a highly nonlinear one, time linearization was done first and then the equation was solved using the alternating direction implicit scheme. Results for the effects of the variable thermal conductivity were found for temperature and heat flux distributions.     

Simultaneous Estimation of Properties in a Combined Mode Conduction–Radiation Heat Transfer in a Porous Medium

Simultaneous estimation of thermophysical and optical properties such as the thermal conductivity, the scattering albedo, and the emissivity of a 1‐D planar porous matrix involving combined mode conduction and radiation heat transfer with heat generation is reported. Coupled energy equations for the gas and solid phase account for the nonlocal thermal equilibrium between the two phases. Performances of the genetic algorithm (GA) and the global search algorithm (GSA) in simultaneous estimation of three properties are analyzed. Both the GA and the GSA utilize a priori knowledge of the axial gas temperature distribution, and the magnitudes of the convective and the radiative heat fluxes at the outer surface of the porous matrix. With volumetric radiative information needed in the solid‐phase energy equation computed using the discrete transfer method, the two energy equations are simultaneously solved using the finite volume method. GSA provides better estimation, and computationally, it is much faster than the GA.     

Analysis of Non-Fourier Conduction-Radiation Heat Transfer in a Cylindrical Enclosure

This article deals with the analysis of non-Fourier conduction and radiation heat transfer in a participating medium contained between 1-D concentric cylinders. The conducting-radiating medium is radiatively absorbing, emitting, and scattering. The non-Fourier effect is analyzed by suddenly perturbing the temperatures of the concentric cylinders. With radiative information computed using the finite volume method, the finite difference method is used to solve the hyperbolic energy equation. Effects of various parameters such as the extinction coefficient, the scattering albedo, the conduction-radiation parameter, the emissivity and the radius ratio are studied on the temporal evolution of temperature field in the medium. These parameters have been found to significantly influence the temporal temperature field, and non-Fourier effects are captured well. For non-Fourier conduction and Fourier conduction–radiation cases, results have been benchmarked against those available in the literature. A good comparison has been observed. In case of non-Fourier conduction-radiation, for some sample cases, the steady-state temperature distributions have been compared against those available in the literature. Results have been found to agree well.