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.     

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.     

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.     

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.     

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.     

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.     

陶瓷基点阵夹芯结构等效导热系数的数值模拟分析

为了评价陶瓷基点阵夹芯结构的热防护效能，选取陶瓷点阵夹芯结构的胞元作为研究对象，给出了表征其传热效能的等效导热系数计算方法。建立了点阵夹芯结构胞元的导热-辐射稳态耦合传热的有限元模型，通过对模型施加不同的边界条件，利用数值模拟的方法求出了点阵夹芯结构稳态温度场分布和等效导热系数，并对其换热机理和影响因素进行了分析。研究表明：点阵夹芯结构的辐射换热是结构热传导的主要方式，且随温度的升高其作用效应就越强；相对较小的材料发射率和较大的芯杆高度能够降低结构的传热效能；减小芯杆直径会降低结构热防护能力；芯杆倾角对结构的热防护性能影响较小。     

Radiative properties and heat transfer characteristics of fiber-loaded silica aerogel composites for thermal insulation

The radiative properties and heat transfer in fiber-loaded silica aerogel composites were investigated using modified anomalous diffraction theory in a combined heat conduction and radiation model. The randomly parameterized 2-D fiber distribution was generated to simulate a very realistic material structure. The finite volume method was then used to solve a two flux radiation model and the steady-state energy equation to calculate the effective thermal conductivity of the composite. The numerical results provide theoretic guidelines for material designs with optimum parameters, such as the inclination angle, diameter and length-to-diameter ratio of the fibers. The results show that the fiber extinction coefficient increases as the fiber length-to-diameter ratio is reduced or the fiber inclination angle is increased. The effective thermal conductivity of the fiber-loaded aerogel can be reduced by reducing the fiber length-to-diameter ratio and the inclination angle and by moderately increasing the fiber volume fraction. The 4–6 μm diameter silicon fibers are optimum for high-temperature thermal insulation.     

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.     

Analytic determination of the effective thermal conductivity of PEM fuel cell gas diffusion layers

Accurate information on the temperature field and associated heat transfer rates are particularly important in devising appropriate heat and water management strategies in proton exchange membrane (PEM) fuel cells. An important parameter in fuel cell performance analysis is the effective thermal conductivity of the gas diffusion layer (GDL). Estimation of the effective thermal conductivity is complicated because of the random nature of the GDL micro structure. In the present study, a compact analytical model for evaluating the effective thermal conductivity of fibrous GDLs is developed. The model accounts for conduction in both the solid fibrous matrix and in the gas phase; the spreading resistance associated with the contact area between overlapping fibers; gas rarefaction effects in microgaps; and salient geometric and mechanical features including fiber orientation and compressive forces due to cell/stack clamping. The model predictions are in good agreement with existing experimental data over a wide range of porosities. Parametric studies are performed using the proposed model to investigate the effect of bipolar plate pressure, aspect ratio, fiber diameter, fiber angle, and operating temperature.     

Numerical Modeling of Combined Radiation and Conduction Heat Transfer in Mineral Wool Insulations

This article addresses numerical modeling of coupled heat conduction and radiation in mineral wools under steady-state condition for prediction of its effective thermal conductivity. The radiative heat transfer is modeled using the Monte Carlo Ray-Trace Method. The radiation model is based on a random distribution of fibers in the media. The radiation distribution factor is employed in order to compute the fraction of the total radiation emitted from one fiber that is absorbed by another, due to both direct radiation and to all possible reflections within the enclosure. The radiation model is coupled with the nonlinear heat conduction equation. The results obtained by the proposed model compare well with experimental measurements of the heat flow meter apparatus. The method is easy to code, and the number of calculations during each iteration is considerably reduced.     

Heat transfer through woven textiles

A mathematical model based on the principles of heat transfer to predict the thermal resistance of fabrics has been presented in this paper. The woven fabric is considered as a system of porous yarns, interlacements between warp and weft yarns and air pores and all the basic weaves can be depicted by this system. The conduction and radiation heat transfer together, was calculated based on the construction parameters of the fabric. The thermal insulation, which is equivalent to the thermal resistance, was predicted with the help of these parameters. The total heat transfer by conduction through each part was calculated using Fourier’s equation. Radiation heat transfer through the air pore was calculated with the help of net radiation method. Linear anisotropic scattering was used to model the radiation heat transfer through fibrous media. The total thermal resistance obtained was validated with actual values obtained from a standard thermal resistance measuring instrument.     

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.     

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.     

A lattice Boltzmann simulation of enhanced heat transfer of nanofluids

Due to its distinctive characteristics nanofluid has drawn much attention from academic communities since the last decade. Compared with conventional fluids, nanofluid has higher thermal conductivity and surface to volume ratio, which enables it to be an effective working fluid in terms of heat transfer enhancement. Recent experimental works have shown that with low nanoparticle concentrations (1–5 vol.%), the effective thermal conductivity of the suspensions can increase by more than 20% for various mixtures. Although many outstanding experimental works have been carried out, the fundamental understanding of nanofluid characteristics and performance is still not sufficient. Much more theoretical and numerical studies are required. Over the past two decades, the lattice Boltzmann method (LBM) has experienced a rapid development and well accepted as a useful method to simulate various fluid behaviours. In the present study, the LBM is employed to investigate the characteristics of nanofluid flow and heat transfer. By coupling the density and temperature distribution functions, the hydrodynamics and thermal features of nanofluids are properly simulated. The effects of the parameters including Rayleigh number and volume fraction of nanoparticles on hydrodynamic and thermal performances are investigated. The results show that both Rayleigh number and solid volume fraction of nanoparticles have influences on heat transfer enhancement of nanofluids; and there is a critical value of Rayleigh number on the performance of heat transfer enhancement.     

Lattice Boltzmann modeling of the effective thermal conductivity for fibrous materials

This paper presents a full set of numerical methods for predicting the effective thermal conductivity of natural fibrous materials accurately, which includes a random generation-growth method for generating micro morphology of natural fibrous materials based on existing statistical macroscopic geometrical characteristics and a highly efficient lattice Boltzmann algorithm for solving the energy transport equations through the fibrous material with the multiphase conjugate heat transfer effect considered. Using the present method, the effective thermal conductivity of random fibrous materials is analyzed for different parameters. The simulation results indicate that the fiber orientation angle limit will cause the material effective thermal conductivity to be anisotropic and a smaller orientation angle leads to a stronger anisotropy. The effective thermal conductivity of fibrous material increases with the fiber length and approach a stable value when the fiber tends to be infinite long. The effective thermal conductivity increases with the porosity of material at a super-linear rate and differs for different fiber location distribution functions.     

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.     

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.     

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.     

Experimental investigations of the coupled conductive and radiative heat transfer in metallic/ceramic foams

Due to their interesting thermal, mechanical and exchange properties, solid metal or ceramic foams have shown a strong development in numerous technological fields for which the knowledge of their thermal properties is of primary importance. In order to investigate the coupled conductive/radiative heat transfer in this kind of materials, we propose an identification method using thermograms obtained from laser-FLASH measurements. This permits us to evaluate, at ambient and high temperatures, the effective thermal conductivity and two global radiative properties of various metal or ceramic foams, describing the thermal behavior of their equivalent homogeneous semi-transparent materials. This new method of characterization of solid foams is promising since conduction and radiation contributions to heat transfer can be evaluated from a unique experiment.