ANALYSIS OF TWO-PHASE RADIATION IN THERMALLY DEVELOPING POISEUILLE FLOW

Combined conductive and radiative heat transfer in a thermally developing two-phase Poiseuille flow in a cylindrical duct is studied here. A two-phase radiative transfer equation (RTE) considering radiation by both gas and particles is taken into account. A complexform of nonlinear integrodifferential RTE is solved by the discrete ordinates method (DOM, or so called SN method) in axisymmetric geometry. After such validation, namely, the solution in a two-dimensional channel flow between two flat plates is compared with that solved by the zone method, the program is then applied to fully developed gas-particle two-phase flow in a cylindrical duct. A parametric study is performed for gas and particle absorption coefficients, particle number density, particle emissivity, and wall emissivity. The results show a significant effect of two-phase radiation on the thermal characteristics. However, in all cases, it was found that conduction is predominant near the wall.     

Numerical investigation of combustion with non-gray thermal radiation and soot formation effect in a liquid rocket engine

A numerical analysis was carried out in order to investigate the combustion and heat transfer characteristics in a liquid rocket engine in terms of non-gray thermal radiation and soot formation. Governing gas and droplet phase equations with PSIC model, turbulent combustion model with liquid kerosene fuel, soot formation, and non-gray thermal radiative equations are introduced. A radiation model was implemented in a compressible flow solver in order to investigate the effects of thermal radiation. The finite-volume method (FVM) was employed to solve the radiative transfer equation, and the weighted-sum-of-gray-gases model (WSGGM) was applied to model the radiation effect by a mixture of non-gray gases and gray soot particulates. After confirming the two-phase combustion behavior with soot distribution, the effects of the O/F ratio, wall temperature, and wall emissivity on the wall heat flux were investigated. It was found that the effects of soot formation and radiation are significant; as the O/F ratio increases, the wall temperature decreases. In addition, as the wall emissivity increases, the radiative heat flux on the wall increases.     

Prediction of radiative heat transfer between two concentric spherical enclosures with the finite volume method

The radiative heat transfer between two concentric spheres separated by an absorbing, emitting, and isotropically scattering gray medium is investigated by using the finite volume method (FVM). Especially, a mapping that simplifies the solution of spherically symmetric radiative heat transfer problems is introduced, thereby, the intensity depending on spatial one-dimension and angular one-dimension is transformed into spatial two-dimensional one. By adopting this mapping process, angular redistribution, which appears in such curvilinear coordinates as cylindrical or spherical ones, is treated efficiently without any artifice usually introduced in the conventional discrete ordinates method (DOM). After a mathematical formulation and corresponding discretization equation for the radiative transfer equation (RTE) are derived, final discretization equation is introduced by using the directional weight, which is the key parameter in the FVM since it represents the inflow or outflow of radiant energy across the control volume faces depending on its sign. The present approach is then validated by comparing the present results with those of previous works by changing such various parameters as temperature ratio between inner and outer spherical enclosure, wall emissivity, and optical thickness of the participating medium. All the results presented in this work show that the present method is accurate and valuable for the analysis of spherically symmetric radiative heat transfer problems between two concentric spheres.     

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.     

Three-dimensional numerical simulation of convection and radiation in a differentially heated cavity using the discrete ordinates method

The main purpose of this paper is to study, in a three-dimensional, differentially heated cavity, the phenomenon of radiation and natural convection in both transparent and participating media. The discrete ordinates method (DOM) is used to solve the radiative transfer equation. The Navier-Stokes equations (NSE), describing natural convection, are solved with a segregated SIMPLE-like algorithm. For non-participating media, the coupling between the radiative transfer and NSE is done via the radiative heat exchange between surfaces. For participating media, a source term is added in the energy equation. The local and mean heat flux as a function of the Rayleigh number is studied, for both transparent and participating media with different optical thicknesses. The effect of the Planck number on the heat flux is also analyzed for different values of the Rayleigh number. Also, a comparison between a purely two-dimensional case and the results obtained in the mid-plane of a long rectangular enclosure is presented.     

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.     

Second law analysis of coupled conduction–radiation heat transfer with phase change

This work considers an exergy-based analysis of two-dimensional solid-liquid phase change processes in a square cavity enclosure. The phase change material (PCM) concerns a semi-transparent absorbing, emitting and anisotropically scattering medium with constant thermodynamic properties. The enthalpy-based energy equation is solved numerically using computational fluid dynamics. Once the energy equation is solved, local exergy loss due to heat conduction and radiative heat transfer during the phase change process is calculated by post processing procedures. In this work, the radiation exergy loss in the medium and at the enclosure boundary is taken into consideration. It is found that radiation exergy loss is significant in the high-temperature phase change process. Parametric investigation is also carried out to study the effects of Stefan number, Biot number, Planck number, single scattering albedo and wall emissivity on exergy loss. The results show that the total exergy loss increases with Biot number, single scattering albedo and wall emissivity. The second law effects of the conduction–radiation coupling in the energy equation are also shown in this work.     

A formal averaging procedure for radiation heat transfer in particulate media

Radiative heat transfer in participating particulate media is modeled using a formal volume averaging procedure. The multiphase medium is composed of emitting-absorbing-scattering phases, i.e., a gas phase and several particle phases. Each particle phase contains large, opaque, gray, diffuse, and spherical particles having locally the same geometrical, thermophysical, and radiative properties. The resulting multiphase radiative transfer equation (MRTE) is solved using the discrete ordinates method. The present computed results are found to be in good agreement with those obtained using the Monte-Carlo theory and with the available experimental results. The coupling effect of the MRTE with the averaged energy equations in a three-dimensional cavity which is differentially heated or which contains a volumetric heat source is studied. A parametric study is performed for particle-phase and gas properties, and wall emissivity.     

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.     

Conjugate turbulent natural convection with surface radiation in air filled rectangular enclosures

Conjugate turbulent natural convection and surface radiation in rectangular enclosures heated from below and cooled from other walls, typically encountered in Liquid Metal Fast Breeder Reactor (LMFBR) subsystems, have been investigated by a finite volume method for various aspect ratios. The formulation comprises the standard two equation k–ε turbulence model with physical boundary conditions (no wall functions), along with the Boussinesq approximation, for the flow and heat transfer. As far as radiation is concerned, the radiosity – irradiation formulation for a transparent fluid of Prandtl number 0.7 has been employed. The conjugate coupling on the walls has been handled by using a fin type formulation. The Rayleigh number based on the width of the enclosure is varied from 10⁸ to 10¹² and the aspect ratio is varied from 0.5 to 2.0. Detailed results including stream lines, temperature profiles, and convective, radiative and overall Nusselt numbers are presented. A correlation for the mean convection Nusselt number in terms of Rayleigh number and aspect ratio is proposed for design purposes. The influence of the wall emissivity and the external heat transfer coefficient on the heat transfer from the enclosure has also been investigated.     

Coupled radiation and solidification heat transfer inside a graded index medium by finite element method

Effects of thermal radiation on solidification heat transfer must be considered inside semitransparent media. This paper investigates coupled heat transfer of solidification and radiation within a two-dimensional rectangular semitransparent medium having gradient index. Solidification process is supposed to happen at some temperature range, and accordingly three zones including liquid-, solid- and mushy-zones exist in phase-change media. In different phase field, parameters of thermophysical property are assumed different and those of radiative property are assumed same. Governing equation includes conduction, radiation and phase-change terms, and radiation and phase-change are treated as source terms in the equation, respectively. A Galerkin finite element method is used to solve energy equation of coupled radiation and phase-change heat transfer. This paper analyzes effect of thermal radiation on phase-change heat transfer and those of refractive index distributions on temperature fields and liquid fraction distributions during radiation–solidification coupled heat transfer. From the results, we can find that refractive index gradient has a major influence on phase-change process and compared with the case of smaller index gradient, bigger gradient can speed up phase-change heat transfer in semitransparent media.     

EFFECT OF RADIATIVE TRANSFER ON THE THERMOPHORETIC PARTICLE DEPOSITION AROUND A CIRCULAR CYLINDER IN A UNIFORM CROSS FLOW

We investigated the effects of radiative transfer on the particle deposition around a circular cylinder in a cross flow. The fluid is transparent to radiation, and radiating particles are suspended in flow. The finite volume method was applied to analyze radiative transfer in the flow utilizing the nonorthogonal coordinate system. The radiative transfer coupled with convection and mass transfer is reasonably predicted by the method. The results are in good agreement with those of boundary layer analysis and available experimental data. We discuss the effects of the conduction-to-radiation parameter, optical thickness, scattering albedo, and wall emissivity on the heat transfer and particle deposition.     

The Effect of Radiation on Flow in a Conical Diffuser

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

Modeling the 3-D radiation of anisotropically scattering media by two different numerical methods

An original model and code for 3-D radiation of anisotropically scattering gray media is developed where radiative transfer equation (RTE) is solved by finite volume method (FVM) and scattering phase function (SPF) is defined by Mie Equations (ME). To the authors’ best knowledge this methodology was not developed before. Missing the benchmark, another new 3-D model and code, which solve the same problems, based on a combination of zone method (ZM) and Monte Carlo method (MC), as a solution of RTE, is developed. Here SPF is also calculated by Mie Equations. The conception ZM + MC is numerically expensive and is used and recommended only as a benchmark. The 3-D rectangular enclosure and the spherical geometry of particles are considered. The both models are applied: (i) to an isotropic and to four anisotropic scattering cases previously used in literature for 2-D cases and (ii) to solid particles of several various coals and of a fly ash. The agreement between the predictions obtained by these two different numerical methods for coals and ash is very good. The effects of scattering albedo and of wall reflectivity on the radiative heat flux are presented. It was found that the developed 3-D model, where FVM was coupled with ME, is reliable and accurate. The methodology is also suitable for extension towards: (i) mixture of non-gray gases with particles and (ii) incorporation in computational fluid dynamics.     

LAMINAR AND TURBULENT NATURAL CONVECTION-RADIATION INTERACTIONS IN A SQUARE ENCLOSURE FILLED WITH A NONGRAY GAS

A numerical investigation of interactions oflaminar and turbulent natural convection and radiation in a differentially heated square enclosure was performed. Songray gas radiation was analyzed with the P^-approximation method for the radiative transfer equation and the weighted sum of gray gases model, A desirable compatibility was achieved in the resultant formulation for radiative transfer with the governing equations for natural convection flows. The Favre-averaged formulation was employed for analyzing turbulent flows together with a k-e model. In the analysis, a two-dimensional enclosure filled with carbon dioxide gas was considered. Solutions were obtained for a range of Grashof numbers and Prandtl numbers varying from 10⁴ to 10¹⁰ Characteristics of flow and temperature fields were compared with predictions in the literature and were found to be in good agreement in general.     

Measurements of Some Characteristics of Thermal Radiation in a 400-kW Grate-Fired Furnace Combusting Biomass

A comprehensive experimental investigation relevant to thermal radiation has been performed in a grate-fired test furnace. Thermal radiation is the dominating mode of heat transfer in the grate-fired furnace and yet only a few studies have focused on thermal radiation. No previous works, to the authors' knowledge, have been carried out concerning measurements on radiative heat transfer in grate-fired furnaces. In this work measurements of temperature have been carried out for all boundaries and the flue gases at a large number of locations. The gas species volume fraction, particle mass–size distributions, and wall irradiation have also been measured at a number of spatial locations. These data are useful in a computational framework to describe the radiative heat transfer reaching the boundaries. Comparing modeled wall irradiation to the measured one makes it possible to obtain a deeper insight into the thermal radiative transport inside the grate-fired furnace.     

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.     

Inverse Geometry Design of Radiating Enclosure Filled with Participating Media Using Meshless Method

A new inverse geometry design methodology is presented in this work for designing a two-dimensional radiating enclosure filled with participating media to meet the pre-specified radiative heat flux distribution on a designed boundary wall. Akima cubic interpolation is employed to approximate the shape of the unknown design surface and transform the continuous geometry shape design to the discrete points' position design. To avoid the tedious remeshing of the variable computational domain in the inverse geometry design processes, the direct collocation meshless method is adopted to solve the radiative transfer problem in the enclosure. The geometry shape of the design surface is optimized using the conjugate gradient method, and the zeroth order regularization method is chosen to stabilize the inverse solutions. A test example is taken to verify the new method presented in this work. The inverse design results show that pre-specified design requirement on the boundary wall can be successfully obtained using the new methodology.     

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 Study of the Radiant Heat Exchange within a Controlled-Atmosphere Furnace

Transient three-dimensional heat transfer between a traversing, structured, and rectangular object and an enclosure is studied. This study investigates the heat transfer process that occurs in brazing an aluminum heat exchanger in a controlled-atmosphere furnace. A model's development is discussed with prescribed enclosure temperature boundary conditions. The program determines the radiant heat exchange between gray diffuse surfaces, and solves the three-dimensional conduction equation for a solid with a radiant heat flux boundary condition using an implicit finite-difference method. The structured object's conduction and radiant thermal properties are described by effective values. It was shown that radiative thermal properties of the traversing object and the enclosure's temperature have a strong impact on the object's temperature history. The effective thermal emissivity was found to influence the object's rate of temperature change. The enclosure's temperatures influenced the object's equilibrium temperature. Also, it was shown that the object's position and rotation can alter its temperature distribution, but not as strong as the effect of boundary conditions and thermal properties. In addition to numerical methods, experiments were performed to further understand the process.