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.     

An axisymmetric bioheat transfer analysis through a unified model

A unified model is developed for the analysis of heat transfer (radiation and non-Fourier conduction) in an axisymmetric participating medium. The proposed model includes three different variants of hyperbolic–parabolic heat conduction models, that is, the single phase lag model, dual phase lag model, and the Fourier (no phase lag) model. The radiating-conducting medium is radiatively absorbing, emitting, and isotropically scattering. Significance of all the above mentioned models on the heat transfer characteristics is investigated in a two-dimensional axisymmetric geometry. The equation of transfer and the coupled non-Fourier conduction-radiation equation are solved via finite volume method. A fully implicit scheme is used to resolve the transient terms in the energy equation. For spatial resolution of radiation information, the STEP scheme is applied. Tri-diagonal-matrix-algorithm is used to solve the resulting set of linear discrete equations. Effects of two important influencing parameters: the scattering albedo and the radiation- conduction parameter are studied on the temporal evolution of temperature field in the radiatively participating medium. The non-Fourier effect of heat transport captured well with the proposed unified model. A good agreement can be found between the proposed model predictions and those available in the literature. It is also found that when the phase lag of the temperature gradient and the heat flux are the same, it reduces to conventional Fourier conduction-radiation and the wave behavior diminishes. However, the reduction to this Fourier model fails in the presence of constant blood perfusion and metabolic heat generation.     

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.     

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 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.     

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.     

Computational efficiency improvements of the radiative transfer problems with or without conduction--a comparison of the collapsed dimension method and the discrete transfer method

This paper deals with the performance evaluation of the collapsed dimension method (CDM) and the discrete transfer method (DTM) in terms of computational time and their abilities to provide accurate results in solving radiation and/or conduction mode problems in a 2-D rectangular enclosure containing an absorbing, emitting and scattering medium. For some pure radiation cases, studies were made for two representative benchmark problems dealing with radiative equilibrium and non-radiative equilibrium. For the combined mode, the transient conduction and radiation problem was solved. The alternating direction implicit scheme was used for the solution of the finite difference part of the energy equation. For the three types of problems considered, tests were performed for a wide range of aspect ratio, extinction coefficient, scattering albedo, conduction-radiation parameter and boundary emissivity. For pure radiation problems, results from the two methods were validated against the results from the Monte Carlo method. For the combined mode, some steady-state results were compared with results available in the literature. For the transient situations, results from the two methods were validated against each other. While both the methods were found to give the same results, the CDM was found to be much more economical than the DTM.     

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.     

Phase-change heat transfer during cyclic heating and cooling with internal radiation and temperature-dependent properties

This paper contains the results of a numerical investigation of the effects of internal radiation on heat transfer during cyclic heating and cooling of a phase-change energy–storage system. The effects of internal radiation, temperature distribution and energy stored and extracted were investigated. An absorbing, emitting, and isotropically scattering, finite, and semi-transparent, gray, phase-change medium bounded between two concentric cylinders was studied. The phase-change medium was a fluoride salt with LiF (41.27%), MgF₂ (48.76%) and KF (8.95% by weight). Radiative heat transfer affects the dynamics of phase changes significantly for low values of the conduction–radiation interaction parameter.     

Analyses of non-Fourier heat conduction in 1-D cylindrical and spherical geometry – An application of the lattice Boltzmann method

This article deals with the implementation of the lattice Boltzmann method (LBM) for the analyses of non-Fourier heat conduction in 1-D cylindrical and spherical geometries. Evolution of the wave like temperature distributions in the medium is obtained, and analysed for the effects of different sets of thermal perturbations at the inner and the outer boundaries of the geometry. The LBM results are validated against those available in the literature, and those obtained by solving the same problems using the finite volume method (FVM). Results of the LBM are in excellent agreement with those reported in the literature, and with the results from the FVM. Computationally, the LBM has an advantage over the FVM.     

A Conduction–Radiation Mixture Model for Laser-Assisted Phase Change of Semitransparent Material

A one-dimensional transient coupled conduction-radiation numerical model is developed to investigate the laser melting of semitransparent material under a continuous collimated laser pulse in a convective cooling environment. The medium is considered absorbing, emitting, and scattering. The thermophysical properties are taken to be different for different phase fields. Volumetric radiation is incorporated in the proposed model. The radiation information is obtained by solving the equation of transfer. The temperature field is obtained by solving the energy equation with internal radiation source. The finite-volume method is used to discretize both the equation of transfer and the energy equation. The enthalpy formulation is adopted to capture the continuously evolving solid–liquid interface during the phase change. The laser source is approximated with the collimated radiation source. Collimated intensity is captured directly (without splitting the total intensity into two parts: diffuse and collimated) by adjusting the control angles. The present model is first validated with the existing phase-change model in the literature. Then the effects of different parameters such as optical thickness, scattering albedo, and the conduction–radiation parameter on the liquid fractions and temperature distribution in the medium are studied. It is observed that when the radiation is dominant, the temperature in the medium is high and hence the liquid fraction is more, in contrast to conduction-dominated phase change.     

Combined conduction and radiation heat transfer with variable thermal conductivity and variable refractive index

This article deals with the solution of conduction–radiation heat transfer problem involving variable thermal conductivity and variable refractive index. The discrete transfer method has been used for the determination of radiative information for the energy equation that has been solved using the lattice Boltzmann method. Radiatively, medium is absorbing, emitting and scattering. To validate the formulation, transient conduction and radiation heat transfer in a planar participating medium has been considered. For constant thermal conductivity and constant and variable refractive indices, results have been compared with those available in the literature. Effects of conduction–radiation parameter and scattering albedo on temperature have been studied for variable thermal conductivity and constant and/or variable refractive index. Lattice Boltzmann method and the discrete transfer method have been found to successfully deal with the complexities introduced due to variable thermal conductivity and variable refractive index.     

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.     

Non-Fourier heat conduction in a finite medium subjected to arbitrary periodic surface disturbance

The non-Fourier transient heat conduction in a finite medium under arbitrary periodic surface thermal disturbance is investigated analytically. In order to obtain the desired temperature field from the known solution for non-Fourier heat conduction under a harmonic disturbance, the principle of superposition along with the Fourier series representation of an arbitrary periodic function is employed. The developed method can be applied for more realistic periodic boundary conditions occurred in nature and technology.     

Thermal radiation effects in phase-change energy-storage systems

A numerical model was developed to determine the transient temperature distribution, solid/liquid interface location, and energy-storage capacity of a semi-transparent phase-change medium. The medium is bounded between two concentric cylinders and internal energy transfer occurs simultaneously by conduction and thermal radiation. The radiation transport equation was coupled with the energy equation; both enthalpy and temperature were employed as dependent variables. The spherical harmonic approximation (P-N approximation) was used to obtain solutions for the radiative heat flux. The coupled conservation of energy and moment differential equations were solved by using iterative numerical finite-difference schemes with appropriate thermal and radiant boundary and interface conditions. The numerical model was used to study the effects of radiation on solidification (melting), transient temperature distribution and energy-storage capacity of an absorbing, emitting, and isotropically scattering, semi-transparent, gray medium contained in a cylindrical annulus. The results increase our understanding of internal energy transfer and show the effects of optical properties, conduction/radiation parameter, and geometric dimensions and should lead to better designs and optimization of phase-change energy-storage systems.     

Combined radiation and convection heat transfer in a porous channel bounded by isothermal parallel plates

Combined radiation and convection heat transfer in a porous medium confined between gray isothermal parallel plates is investigated. The medium is absorbing, emitting and scattering. Cases of boundaries at temperatures higher or lower than the medium are considered. In the porous medium, the boundary effect on the fully developed laminar velocity field as proposed by Kaviany is accounted for. For various values of the extinction coefficient, the scattering albedo, the conduction-radiation parameter and the boundary emissivity, Nusselt number, temperature and heat flux distributions are found for the range of values including the extreme limits of the porous medium shape parameter (PMSP), γ=(W²φ/K)^1/2, where W is the channel width, φ the porosity and K the permeability. For the lower limiting value of the PMSP γ, the effect of the porous medium is negligible and the situation approaches that of Poiseuille flow. For this limiting case, results from the present work are compared with those available in the literature. For medium to high values of the PMSP γ, for the purpose of comparison, some results are presented in tabular form. Radiation is found to have a significant effect on various parameters studied. The discrete transfer method was used for the solution of the radiative part of the energy equation. An iterative finite difference scheme was used to solve the energy equation.     

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.     

Simultaneous Retrieval of Parameters in a Transient Conduction-Radiation Problem Using a Differential Evolution Algorithm

This article deals with the application of the differential evolution (DE) algorithm for the inverse analysis of a transient conduction-radiation heat transfer problem. Thermophysical properties and/or optical properties of the medium are simultaneously retrieved with a known temperature field. The conducting-radiating planar enclosed medium bounded by diffuse-gray boundaries is absorbing, emitting, and scattering. In both the direct and inverse methods, the energy equations are solved using the lattice Boltzmann method (LBM), and the finite volume method (FVM) is used to compute the radiative information. In the inverse method, the objective function is minimized using the DE algorithm. Any two sets of parameters, viz., the extinction coefficient, the scattering albedo, emissivity and conduction-radiation parameter, are simultaneously retrieved. The effects of key DE algorithm parameters, such as the weighting factor and the crossover constant on the quality of solutions, are studied. Measurement errors are accounted. The accuracy of the DE algorithm is compared with the genetic algorithm. The DE algorithm is significantly faster, and it yields the global optimum for a wide range of parameters.     

Treatment of the interfacial temperature jump condition with non-Fourier heat conduction effects

Transient energy transport with non-Fourier heat conduction effects in a two-layer structure of dissimilar materials subject to ultra-fast laser heating is studied using the proper interfacial temperature jump condition. The solution obtained is compared with solutions available in literature that use diffusion-type interfacial conditions in conjunction with non-Fourier heat conduction effects. The dual-phase lag heat conduction model is used in this work as it includes both the temporal and spatial non-Fourier effects. It is found that the diffusion-type interfacial temperature jump condition with non-Fourier heat conduction models can lead to discrepancies and erroneous trends in theoretical predictions. Moreover, a comparison between the functional forms of the two solutions obtained utilizing both interfacial conditions shows that implementing the proper interfacial temperature jump condition does not add any complexity to the solution obtained. This study – implementing the proper interfacial temperature jump condition – is further extended to show the strong effects of the thermal contact conductance and the surface layer thickness on the transient thermal response of a two-layer material in a semi-infinite domain subject to ultra-fast laser heating processes in terms of the reflectivity change of the surface layer, the temperature distribution in the two-layer structure as well as the temporal variation of the interfacial temperature difference.