Combined non-gray radiative and conductive heat transfer in solar collector glass cover

A rigorous approach for the radiative heat transfer analysis in solar collector glazing is developed. The model allows a more accurate prediction of thermal performance of a solar collector system. The glass material is analysed as a non-gray plane-parallel medium subjected to solar and thermal irradiations in the one-dimensional case using the Radiation Element Method by Ray Emission Model (REM by REM).This method is used to analyse the combined non-gray convective, conductive and radiative heat transfer in glass medium. The boundary surfaces of the glass are specular. The spectral dependence of the relevant radiation properties of glass (i.e. specular reflectivity, refraction angle and absorption coefficient) are taken into consideration. Both collimated and diffuse incident irradiation are applied at the boundary surfaces using the spectral solar model proposed by Bird and Riordan. The optical constants of a commercial ordinary clear glass material have been used. These optical constants (100 values) of real and imaginary parts of the complex refractive index of the glass material cover the range of interest for calculating the solar and thermal radiative heat transfer through the solar collector glass cover. The model allows the calculation of the steady-state heat flux and temperature distribution within the glass layer. The effect of both conduction and radiation in the heat transfer process is examined. It has been shown that the real and imaginary parts of the complex refractive index have a substantial effect on the layer temperature distribution. The computational time for predicting the combined heat transfer in such a system is very long for the non-gray case with 100 values of n and k. Therefore, a simplified non-gray model with 10 values of n and k and two semi-gray models have been proposed for rapid computations. A comparison of the proposed models with the reference non-gray case is presented. The result shows that 10 bandwidths could be used for rapid computation with a very high level of accuracy.     

Solution of the radiative integral transfer equations in rectangular participating and isotropically scattering inhomogeneous medium

Radiative integral transfer equations for a rectangular participating and isotropically scattering inhomogeneous medium are solved numerically for the incident energy and the net partial heat fluxes using the method of “subtraction of singularity”. All the relevant single (surface integrals) and double integrals (volume integrals) are carried out analytically to reduce the computation time and numerical integration errors. The resulting system of linear equations are solved iteratively. A benchmark problem is chosen as a rectangular inhomogeneous cold participating medium which is subject to externally uniform diffuse radiation on the bottom surface. Solutions for linearly and quadratically varying scattering albedos are provided in tabular form.     

Moving boundary flow and heat transfer in an anisotropic porous medium

The axisymmetrical flow of a fluid injected through a circular opening into an anisotropic porous medium confined between two isothermal surfaces is studied. The governing equations are solved numerically according to the quasi-static approach, using a Landau type transformation to immobilize the interface in the new coordinate system. It is found that the anisotropy of the medium, as well as the inlet pressure, may significantly influence the shape and propagation speed of the moving front.     

Scaling anisotropic scattering in radiative transfer in three-dimensional nonhomogeneous media

Radiative heat transfer in three-dimensional nonhomogeneous participating medium was investigated by using REM² method. The anisotropic scattering phase function was dealt with the scaling technique based on delta function approximation. The three-dimensional scaled isotropic results were compared with the published anisotropic scattering computations. A good agreement between the scaled isotropic approaches and the anisotropic solutions was found. The effects of scattering albedo, forward fraction of phase function, and wall emissivity were discussed. It was found that, with the increase of the scattering albedo, the radiative heat flux increases for forward scattering media, but decreases for backward scattering media. The radiative heat flux is increased with the increases of forward fraction of phase function and wall emissivity. The emissive power at the center of a cubical nonhomogeneous medium in radiative equilibrium with gray diffuse walls equals to the averaged blackbody emissive power of the six walls.     

Combined nongray radiative and conductive heat transfer in multiple glazing taking into account specular reflection and absorption

The problem of combined nongray radiative and conductive heat transfer in multiple glazing subjected to solar irradiation is analyzed. A spectral solar model proposed by Bird and Riordan is used to calculate direct and diffuse solar irradiance. The radiation element method by ray emission model, REM², is used to analyze the spectral dependence of radiative heat transfer. Specular reflection at boundary surfaces is taken into account. The spectral dependence of radiation properties of glass such as specular reflectivity, refraction angle, and absorption coefficient is taken into account. The steady‐state temperature and heat flux distributions in the glass layer are obtained and the insulating efficiency of multiple glazing is examined. The overall heat transfer coefficients predicted by the present method are compared with those based on the JIS method. The values obtained by the present method are slightly lower than those obtained by the JIS method. To investigate the spectral variation of radiative heat flux attenuated in the glass layer, the spectral heat flux at the room‐side surface and incident radiation are compared. © 2003 Wiley Periodicals, Inc. Heat Trans Asian Res, 32(8): 712–726, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.10125     

Local RBF meshless scheme for coupled radiative and conductive heat transfer

ABSTRACT

A local radial basis function meshless (LRBFM) method is developed to solve coupled radiative and conductive heat transfer problems in multidimensional participating media, in which compact support radial basis functions (RBFs) augmented on a polynomial basis are employed to construct the trial function, and the radiative transfer equation (RTE) and energy conservation equation are discretized directly at nodes by the collocation method. LRBFM belongs to a class of truly meshless methods which require no mesh or grid, and can be readily implemented in a set of uniform or irregular node distributions with no node connectivity. Performances of the LRBFM is compared to numerical results reported in the literature via a variety of coupled radiative and conductive heat transfer problems in 1D and 2D geometries. It is demonstrated that the local radial basis function meshless method provides high accuracy and great efficiency to solve coupled radiative and conductive heat transfer problems in multidimensional participating media with uniform and irregular node distribution, especially for coupled heat transfer problems in irregular geometry with Cartesian coordinates. In addition, it is extremely simple to implement.     

Combined conduction and radiation heat transfer in a gray anisotropically scattering planar medium with diffuse-specular boundaries

Coupled conduction and radiation heat transfer in a gray planar nonlinearly anisotropic scattering medium bounded between two plane parallel surfaces reflecting both diffusely and specularly is analyzed. The governing integrodifferential equations are solved by a numerical iterative method consisting of Numerov's method to solve the energy equation and Chandarsekhar's discrete ordinates method in conjunction with the Crank-Nicolson method to solve the radiative transfer equation. Convergence of the solution is enhanced by Ng-acceleration. The numerical algorithm described is found to be fast and reliable. Numerical results based on S₃₂ method indicate that anisotropy plays an important role, and difference between the diffuse and specular reflections is found to be insignificant.     

Analysis of two-dimensional radiative heat transfer in a gray medium with internal heat generation

Radiative heat transfer in a two-dimensional rectangular enclosure with gray medium and internal heat generation is considered. Solutions are generated by a point allocation technique in which unknown temperature profiles are expressed as polynomials. Based on a recently developed generalized exponential integral function, the present solution technique is demonstrated to be computationally more efficient than most of the conventional solution methods. For the case with constant internal heat generation, numerical solutions for temperature and heat flux distributions for enclosures of different optical thicknesses and aspect ratios are obtained. Analytical solutions are developed in the optically thin limit. Both the optical thickness and the enclosure geometry are demonstrated to have strong effects on the temperature distribution within the medium. The average heat transfer to the different boundaries, on the other hand, appears to depend mainly on the enclosure geometry.     

Transient coupled radiative and conductive heat transfer in a composite of semitransparent media

INTRODUCTIONIhsemitransparentmaterial(STM),energyisusuallytransferredbyradiationinaddihontoheatconduchon.WhenthesemitransparentmaterialisexPOsedtohightemperatUresurroundingsorwhenanintensiveincidentradiationexists,theeffectoftheradiationonthetransienttempef~fieldsismoreimpobotthanthat'oftheconduchon.msfeatUreplaysanimpoftalltroleinmanufactUreandengineeringaPPlicationsofalotofsemitransparentnderialsll],suchasglassindustry,ceramicandfiber~rials,mulh-layersemiconductors,moltensaltmedia,se…     

Radiation heat transfer in a participating medium bounded by specular reflectors

This paper analyzes the problem of one dimensional plane-parallel absorbing emitting and isotropically scattering gray slab with gray, diffusely emitting and specularly reflecting boundaries, under the condition of radiative equilibrium. The problem is exactly formulated. The governing energy equation which is a Fredholm integral equation of the second kind involving the integral functions C_n(t) is solved by an accurate modified quadrature method based on the Gauss-Legendre rule. The temperature distribution and heat flux results are presented. Results of particular cases are compared to that published earlier. A three term exponential curve fit is presented to represent the heat transfer result of diffusely reflecting boundaries case.     

Radiative transfer in one-dimensional hollow cylindrical geometry with anisotropic scattering and variable medium properties

The effects of spatially varying absorption and scattering coefficients on radiation transfer in absorbing, emitting, anisotropically scattering hollow cylinders with reflecting boundaries were investigated using the discrete ordinates method (DOM) by Tsai et al. [Int. J. Heat Mass Transfer 33 (1990) 2651]. Their problem solutions for hollow cylinder cases are incorrect. The cause of this inaccuracies are identified and the correct solutions obtained using DOM S₆ are provided.     

Meshless local Petrov–Galerkin approach for coupled radiative and conductive heat transfer

A meshless local Petrov–Galerkin approach is employed for solving the coupled radiative and conductive heat transfer in absorbing, emitting and scattering media. The meshless local Petrov–Galerkin approach with upwind scheme for radiative transfer is based on the discrete ordinate equations. The moving least square approximation is used to construct the shape function. Three particular test cases for coupled radiative and conductive heat transfer are examined to verify this new approximate method. The dimensionless temperatures and the dimensionless heat fluxes are obtained. The results are compared with the other benchmark approximate solutions. By comparison, the results show that the meshless local Petrov–Galerkin approach has a good accuracy in solving the coupled radiative and conductive heat transfer in absorbing, emitting and scattering media.     

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.     

Solution of integral equations of intensity moments for radiative transfer in an anisotropically scattering medium with a linear refractive index

In this work, we derive the integral equations of radiative transfer in terms of intensity moments for radiative transfer in an anisotropically scattering slab with a spatially varying refractive index (VRI). The integral equations are solved by the Nyström method. We apply this method to study radiative heat transfer in a cold slab with higher-degree anisotropic scattering and linearly VRI. The slab lays on an opaque substratum. The refractive index may have a jump at the interface between the surroundings and the slab, while the interface between the slab and the substratum is assumed to be non-reflecting. To exemplify the application of the integral formulation, we consider the case with irradiation from external source in the surroundings and the case with an emitting substratum. We also solve the problems by the Monte Carlo method (MCM). The hemispherical reflectance and transmittance of the slabs obtained by solving integral equations are in excellent agreement with those obtained by the MCM. A positive gradient of refractive index (n′) enhances forward radiative transfer, and so the dimensionless radiative heat flux increases with the increase of n′ for the cases with irradiation from the surroundings. Effects of the optical thickness, the scattering albedo and the scattering phase function are also investigated.     

On the discrete ordinates method for radiative heat transfer in anisotropically scattering media

In the discrete ordinates method (DOM), the normalized condition for the numerical quadrature of some complex scattering phase functions may not be satisfied. In this paper, a revised discrete ordinates method (RDOM) is developed to overcome this problem, in which a renormalizing factor is added into the numerical quadrature of in-scattering term. The RDOM is used to solve the radiative transfer problem in one-dimensional anisotropically scattering media with complex phase function. The radiative heat fluxes obtained by the RDOM are compared with those obtained by the conventional discrete ordinates method (CDOM) and Monte Carlo method. The results show the RDOM can overcome the false scattering resulted from the numerical quadrature of in-scattering term and improve largely the accuracy of solution of the radiative transfer equation by comparison with the CDOM.     

Integral equation solutions based on exact ray paths for radiative transfer in a participating medium with formulated refractive index

The exact analytical path length of radiation traveling in a slab with formulated variable refractive index is derived. Based on the analytical path lengths, the integral equations in terms of intensity moments for radiative transfer in a participating slab with one of the family of spatially varying refractive indices are developed. We solve the integral equations for radiative transfer in a slab at radiative equilibrium or for radiative transfer in an isothermal slab. The boundaries are assumed to be black for the slab at radiative equilibrium and the index jumps at both boundaries for the isothermal slab are considered. For comparison purpose, we also solve the radiative equilibrium problems by the discrete ordinates method (DOM). The nondimensional emissive power and nondimensional radiative heat flux obtained by solving integral equations show an excellent agreement with those obtained by the DOM. For the slab at radiative equilibrium and with positive gradient of refractive index, the jump of the emissive power at bottom boundary decreases with the increase of optical thickness for the cases with slightly varying refractive index, but the trend may not hold for the cases with significantly varying refractive index. For the non-scattering slab with positive gradient of refractive index and fixed refractive indices at the boundaries, the directional emittances at both boundaries for the case with linear refractive index are smaller than those for the case with a refractive index of slope-increasing profile. Effects of the scattering albedo and the scattering phase function coefficient are investigated too.     

Conservation of asymmetry factor in phase function discretization for radiative transfer analysis in anisotropic scattering media

A new phase function normalization technique is developed for use with anisotropic scattering media and is applied to the conventional discrete-ordinates method. The new approach is shown to ensure conservation of both scattered energy and phase function asymmetry factor after directional discretization when considering the Henyey–Greenstein phase function approximation. Results show the necessity of conservation of the asymmetry factor as well as of the scattered energy. Lack of either conservation can lead to false results for radiation analysis in highly anisotropic media. Wall flux profiles predicted by the normalized DOM in a highly anisotropic scattering cylinder are compared with FVM and isotropic scaling profiles. The effect of scattering albedo and optical thickness is examined. For the tested benchmark problem, it is found that heat flux profiles generated with the present normalization approach conform more accurately to both FVM and isotropic scaling profiles than when the previous normalization techniques are implemented.     

Monte Carlo method for simulating the radiative characteristics of an anisotropic medium

This study introduces a Monte Carlo method for predicting the radiative characteristics of a semitransparent medium containing small particles and presents an analytical model of independent, multiple scattering. The anisotropic characteristics of the cloud of particles are considered directly using the Mie phase function. The bidirectional reflectivity and bidirectional transmittance of a flat plate containing the small particles are predicted. The calculated results are satisfactorily in agreement with the experimental data and the calculated results using the discrete-ordinate method. © 1999 Scripta Technica, Heat Trans Asian Res, 28(3): 201–210, 1999     

Radiative heat transfer in plane participating media

The problem of radiative heat transfer through a nonisothermal scattering, absorbing and emitting grey medium between reflecting, absorbing and emitting plates is analytically investigated. The solution technique is based on a projectional procedure, equivalent to a variational approach. The results concern the most meaningful physical quantities, and get an improved accuracy with respect to the data available in literature, with extremely low computational time.