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.     

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.     

Radiation Element Method Coupled with the Lattice Boltzmann Method Applied to the Analysis of Transient Conduction and Radiation Heat Transfer Problem with Heat Generation in a Participating Medium

This article deals with the implementation of the radiation element method (REM) with the lattice Boltzmann method (LBM) to solve a combined mode transient conduction-radiation problem. Radiative information computed using the REM is provided to the LBM solver. The planar conducting-radiating participating medium is contained between diffuse gray boundaries, and the system may contain a volumetric heat generation source. Temperature and heat flux distributions in the medium are studied for different values of parameters such as the extinction coefficient, the scattering albedo, the conduction-radiation parameter, the emissivity of the boundaries, and the heat generation rate. To check the accuracy of the results, the problem is also solved using the finite-volume method (FVM) in conjunction with the LBM. In this case, the data for radiation field are calculated using the FVM. The REM has been found to be compatible with the LBM, and in all the cases, results of the LBM-REM and the LBM-FVM have been found to provide an excellent comparison.     

Numerical analysis of combined-mode dual-phase-lag heat conduction and radiation in an absorbing,emitting, and scattering cylindrical medium

Combined-mode dual-phase-lag (DPL) heat conduction and radiation heat transfer is analyzed in a concentric cylindrical enclosure filled with a radiatively absorbing, emitting, and scattering medium. The governing energy equation is incorporated with volumetric radiation as a source term, essentially to take the effect of radiative heat flux into account. While the energy equation is solved using the lattice Boltzmann method (LBM), the finite volume method (FVM) is used to calculate the radiative information. To establish the accuracy of the proposed LBM formulation, the governing energy equation is also solved with the finite difference method (FDM). Thermal perturbation is caused by suddenly changing the temperature at the boundaries. Radial temperature distributions during transience as well as steady state (SS) are presented for a wide range of parameters such as lag ratio, extinction coefficient, scattering albedo, conduction–radiation (C-R) parameter, boundary emissivity, and radius ratio. Sample results are benchmarked with those available in the literature, and a good agreement between the present and reported results is found.     

Anisotropy Effects on Heat and Fluid Flow by Unsteady Natural Convection and Radiation in Saturated Porous Media

ABSTRACT

This article deals with a numerical study of fluid flow and heat transfer by unsteady natural convection and thermal radiation in a vertical channel opened at both ends and filled with anisotropic, in both thermal conductivity and permeability, fluid-saturated porous medium. The bounding walls of the channel are gray and kept at a constant hot temperature.

In the present study we suppose the validity of the Darcy law for motion and of the local thermal equilibrium assumption. The radiative transfer equation (RTE) is solved by the finite-volume method (FVM). The numerical results allow us to represent the time–space variations of the different state variables. The sensitivity of the fluid flow and the heat transfer to different controlling parameters, namely, the single scattering albedo ω, the temperature ratio R, the anisotropic thermal conductivity ratio R_c, and the anisotropic permeability ratio R_k, are addressed. Numerical results indicate that the controlling parameters of the problem, namely, ω, R, R_c, and R_k, have significant effects on the flow and thermal field behavior and also on the transient process of heating or cooling of the medium. Effects of such parameters on time variations of the volumetric flow rate q_v and the convected heat flux Q at the channel's outlet are also studied.     

Combined Mode Conduction and Radiation Heat Transfer in a Porous Medium and Estimation of the Optical Properties of the Porous Matrix

This article deals with the analysis of combined mode conduction and radiation heat transfer in a porous medium, and simultaneous estimation of the optical properties of the porous matrix. Simultaneous solution of the gas- and solid-phase energy equations encompasses local thermal nonequilibrium, while the convective heat exchange term couples the gas- and the solid-phase energy equations. A localized uniform volumetric heat generation zone is the source of heat transfer in the porous matrix. With volumetric radiative information needed in the solid-phase energy equation computed using the discrete transfer method, the solid- and gas-phase energy equations are simultaneously solved using the finite difference method. For a given set of boundary conditions and operating parameters, the computed temperature distribution serves as the exact temperature profile necessary in the estimation of parameters. In the estimation of parameters using inverse analysis, the objective function is minimized using the genetic algorithm. Effects of measurement error, number of generations, population size, crossover probability, and mutation probability are studied in regard to the accuracy of results and the computational time required. Reasonably accurate estimations of extinction coefficient, scattering albedo, and emissivity of the porous matrix are obtained.     

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.     

Numerical prediction of effective thermal conductivity of ceramic fiber board using lattice Boltzmann method

Abstract

Application of the lattice Boltzmann method has been extended for the analysis of combined transient conduction and radiation heat transfer through highly porous fibrous insulation media. Firstly, LBM has been employed for the analysis of combined mode of transient conduction radiation heat transfer in a 2?D rectangular enclosure containing an absorbing, emitting and scattering medium and results are compared with already published ones. The results have been found in good accord for different values of radiation-conduction parameter, scattering albedo and south (hot) wall emissivity. Furthermore, the proposed LBM for the calculation of effective thermal conductivity of ceramic fiber board has been employed. A random-generation growth method for generating micro morphology of natural ceramic fiber board has been selected. The conductive, radiative and effective thermal conductivity has been numerically estimated using the present LBM. It is found that the predicted effective thermal conductivity for different values of fibrous bulk density is in good agreement with the experimental data.     

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.     

Study on fluid–solid coupling heat transfer in fractal porous medium by lattice Boltzmann method

A lattice Boltzmann model is applied to simulate fluid–solid coupling heat transfer in fractal porous medium. The numerical simulation is conducted to investigate the influences of pressure drop and porosity on fluid flows and the effect of thermal conductivity ratio of solid matrix to fluid on heat transfer. The simulation results indicate that fluid flows still obey Darcy’s Law in the range of flow and pressure level in this paper, and that both velocity field and temperature evolution conform to the local structural characteristics of porous medium. The comparison of temperature results from lattice Boltzmann model against those from the finite-volume method (FVM, one of the conventional CFD methods) is also presented to demonstrate the reliability of LBM. The present results agree well with those from FVM, All these indicate the feasibility and the reliability for the lattice Boltzmann model to be used to reveal the phenomenon and rules of fluid–solid coupling heat transfer in complex porous structures.     

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.     

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.     

Simultaneous estimation of four parameters in a combined-mode heat transfer in a 2D porous matrix with heat generation

This article reports a study on simultaneous estimation of four parameters for combined-mode conduction and radiation heat transfer in a 2D rectangular porous matrix with a localized volumetric heat generation source. Air flows at uniform velocity through the conducting and radiating porous matrix. In the heat generation zone, and its downstream, the gas temperature is higher than that of the solid, and in the upstream the reverse situation occurs. This temperature difference between gas and the solid results in heat transfer by convection between the two phases, and the analysis thus requires consideration of separate energy equations for the two phases. The solid being involved radiatively, the volumetric radiative source term, in the form of the divergence of radiative heat flux, appears only in the solid-phase energy equation. The two equations are coupled through the convective heat transfer term. Four parameters—scattering albedo, emissivity, solid conductivity, and heat transfer coefficient—are simultaneously estimated based on the solid and gas temperature distributions, and convective and radiative heat fluxes at the outer surface of the porous matrix. In both direct and inverse approaches, the energy equations are solved using the finite volume method. For a test case, determining the genetic algorithm is much more time-consuming than the global search algorithm; in other cases, parameter estimations are done using the global search algorithm. Parameters are found to be estimated accurately.     

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.     

Simulation Studies on Multimode Heat Transfer from a Square-Shaped Electronic Device with Multiple Discrete Heat Sources

ABSTRACT

The results of a numerical study of the problem of multimode heat transfer from a square-shaped electronic device provided with three identical flush-mounted discrete heat sources are presented here. Air, a radiatively nonparticipating fluid, is taken to be the cooling medium. The heat generated in the discrete heat sources is first conducted through the device, before ultimately being dissipated by convection and surface radiation. The governing partial differential equations for temperature distribution are converted into algebraic form using a finite-volume based finite difference method, and the resulting algebraic equations are subsequently solved using Gauss-Seidel iterative procedure. A grid size of 151 × 91 is used for discretizing the computational domain. The effects of all relevant parameters, including volumetric heat generation, thermal conductivity, convection heat transfer coefficient, and surface emissivity, on various important results, such as the local temperature distribution, the peak temperature of the device, and the relative contributions of convection and surface radiation to heat dissipation from the device, are studied in sufficient detail. The exclusive effect of surface radiation on pertinent results of the present problem is also brought out.     

Modeling of trickle flow liquid fuel combustion in inert porous medium

Present work is a numerical analysis of fuel oil combustion inside an inert porous medium where fuel oil flows through the porous medium under gravity wetting its solid wall with concurrent movement of liquid fuel and air under steady state conditions. A one-dimensional heat transfer model has been developed under steady state conditions using a single step global reaction mechanism. The effects of optical thickness, emissivity of medium, flame position and reaction enthalpy flux on radiation energy output efficiency as well as the temperature, position and thickness of vaporization zone have been presented using kerosene as fuel. Low values of optical thickness and emissivity of porous medium will ensure efficient combustion, maximize downstream radiative output with minimum upstream radiative loss.     

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.     

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.     

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.     

Heat and Moisture Transfer in a Rectangular Cavity Partially Filled with Hygroscopic Porous Media

Abstract

This work deals with turbulent natural convection heat and moisture transfer with thermal radiation in a rectangular cavity partially filled with hygroscopic porous medium. The governing equations for the momentum and heat transfer in both free fluid and hygroscopic porous media and moisture content transfer in hygroscopic porous medium were solved by the finite element method. Comparisons with experimental and numerical results in the literature have been carried out. Effects of thermal radiation, Rayleigh number on natural convection and heat transfer in both free fluid and porous medium and moisture content transfer in porous medium were analyzed. It was found that surface thermal radiation can significantly change the temperature and moisture content fields in the regions of free flow and hygroscopic porous medium. With the increase in Rayleigh number, the temperature of porous medium at the interface increased slightly, and the magnitude of moisture change becomes smaller.