Transient coupled radiative and conductive heat transfer in a semitransparent layer of ceramic

This work considers transient radiative and conductive heat transfer in a semitransparent layer of ceramic, submitted to several thermal and radiative boundary conditions. Each side of the layer is exposed to hot or cold radiative surroundings, while each boundary is heated or cooled by convection. The solution procedure must provide accurate temperature distribution in the layer, so a nodal analysis based on Hottel's zonal method extended by ray tracing method is carried out. A finite difference method with non-uniform space and time increments is used to solve the transient energy equation, including a radiative heat source, coupled to a equation of radiative transfer. Variable spacing was used to concentrate grid points in regions with large temperature gradients. The influence of refractive index, optical thicknesses and conduction-radiation parameters is investigated.     

Simultaneous radiative and conductive heat transfer in non-gray media

Steady-state energy transfer through non-gray radiating and conducting media enclosed by black walls of unequal temperature is studied. A rectangular Milne-Eddington type relation is used to describe the frequency dependence of the absorption coefficient. Temperature distributions and total heat transfer results are presented for materials which absorb radiation (a) of low frequency, (b) of high frequency, (c) within a finite band width, and (d) of all frequencies (gray). The influence of optical thickness (τ₀) and conduction to a radiation interaction parameter (N) are examined and the results for non-gray materials are compared with those for a gray analysis. Exact results are compared with those determined by using the optically-thin and the optically-thick approximations, as well as with those evaluated for purely conductive and purely radiative transfer.     

Discontinuous spectral element method for solving radiative heat transfer in multidimensional semitransparent media

A discontinuous spectral element method (DSEM) is presented to solve radiative heat transfer in multidimensional semitransparent media. This method is based on the general discontinuous Galerkin formulation. Chebyshev polynomial is used to build basis function on each element and both structured and unstructured elements are considered. The DSEM has properties such as hp-convergence, local conservation and its solutions are allowed to be discontinuous across interelement boundaries. The influences of different schemes for treatment of the interelement numerical flux on the performance of the DSEM are compared. The p-convergence characteristics of the DSEM are studied. Four various test problems are taken as examples to verify the performance of the DSEM, especially the performance to solve the problems with discontinuity in the angular distribution of radiative intensity. The predicted results by the DSEM agree well with the benchmark solutions. Numerical results show that the p-convergence rate of the DSEM follows exponential law, and the DSEM is stable, accurate and effective to solve multidimensional radiative transfer in semitransparent media.     

Least-squares collocation meshless approach for radiative heat transfer in absorbing and scattering media

A least-squares collocation meshless method is employed for solving the radiative heat transfer in absorbing, emitting and scattering media. The least-squares collocation meshless method for radiative transfer is based on the discrete ordinates equation. A moving least-squares approximation is applied to construct the trial functions. Except for the collocation points which are used to construct the trial functions, a number of auxiliary points are also adopted to form the total residuals of the problem. The least-squares technique is used to obtain the solution of the problem by minimizing the summation of residuals of all collocation and auxiliary points. Three numerical examples are studied to illustrate the performance of this new solution method. The numerical results are compared with the other benchmark approximate solutions. By comparison, the results show that the least-squares collocation meshless method is efficient, accurate and stable, and can be used for solving the radiative heat transfer in absorbing, emitting and scattering media.     

Numerical solution of radiative and conductive heat transfer in concentric spherical and cylindrical media

A new technique is presented to improve the performance of the discrete ordinates method when solving the coupled conduction-radiation problems in spherical and cylindrical media. In this approach the angular derivative term of the discretized one-dimensional radiative transfer equation is derived from an expansion of the radiative intensity on the basis of Chebyshev polynomials. The set of resulting differential equations, obtained by the application of the S_N method, is numerically solved using the boundary value problem with the finite difference algorithm. Results are presented for the different independent parameters. Numerical results obtained using the Chebyshev transform method compare well with the benchmark approximate solutions. Moreover, the new technique can easily be applied to higher-order S_N calculations.     

Development of a finite element model for coupled radiative and conductive heat transfer in participating media

Considering the geometrical applicability, a finite element model (FEM) for coupled radiative-conductive heat transfer has been developed which is applicable to enclosures of arbitrary geometry in present research. The present work provides a solution of coupled heat transfer in a rectangular, cylindrical or annulus enclosure with black or gray walls containing an absorbing-emitting-scattering medium. It is also applied to study the influence of conductive/radiation coefficient, albedo and wall emissivity on the temperature distribution in the medium. Compared with the results available in other references, the present FEM has no limitation with respect to geometry and can predict the coupled radiative-conductive heat transfer in participating media accurately.     

New method for solving radiation transfer problems in emitting, absorbing, and scattering media

The proposed method is based on a novel technique for approximating the angular dependence of the radiated intensity. The entire range of solid angles is divided into N cells, which are symmetric relative to the center of the sphere. In each of the cells the radiation is assigned in the form of the P ₁ approximation, and a system of differential equations is obtained to determine the set of local zeroth and first moments. In some special cases the proposed approach can be regarded as a generalization of the discrete-ordinates method, which makes it possible to solve the problem of selecting the weights in the quadrature formulas in a natural manner. The effectiveness of the method is demonstrated in two one-dimensional test cases. It is shown that in these cases fairly high accuracy is achieved in the solution of the problem already for N=2. Zh. Tekh. Fiz. 67, 1–7 (September 1997)     

Effects of specular reflection on radiative heat transfer in an absorbing,emitting, and anisotropically-scattering medium

The effect of specular reflection in a one-dimensional, absorbing, emitting, and anisotropically-scattering medium is analyzed. The mathematical formulation is shown to involve the function F_n(x)= ∝¹_o(e^-x/μ/1 - ?₁?^-2e_2L/μ)υ^n-2 dμ, which can be readily evaluated as a fast-coverging, infinite series of exponential integral functions. Numerical solutions to the resulting integral equations are generated by the method of point allocation.Physically, the radiative energy within a participating medium bounded by specularly reflective surfaces is observed to experience more multiple reflection than the corresponding diffuse reflection case. This leads to some differences between the two cases in the heat transfer and temperature profile results. These differences, however, are generally quite minor for the considered one-dimensional planar system.     

A multi-dimensional differential approximation for absorbing/emitting and anisotropically scattering media with collimated irradiation

By considering the intensity within a medium to consist of a collimated and a fairly diffuse part, the overall problem of radiative transfer is reduced to two simpler ones: first the collimated intensity is obtained (equivalent in complexity to a nonscattering medium); for the evaluation of the diffuse part of the radiation (due to emission and scattering), a new differential approximation has been developed. To demonstrate the accuracy and simplicity of the present method, two sample cases are presented for which some exact solutions can be found in the literature: results are presented (i) for cosine-varying irradiation incident upon a two-dimensional, isotropically scattering slab and (ii) for irradiation with a Gaussian intensity distribution of a two-dimensional, anisotropically scattering semi-infinite cylindrical medium.     

Discrete vs. continuum-scale simulation of radiative transfer in semitransparent two-phase media

The mathematical formulation of the continuum approach to radiative transfer modeling in two-phase semi-transparent media is numerically validated by comparing radiative fluxes computed by (i) direct, discrete-scale and (ii) continuum-scale approaches. The analysis is based on geometrical optics. The discrete-scale approach uses the Monte Carlo ray-tracing applied directly to real 3D geometry measured by computed tomography. The continuum-scale approach is based on a set of continuum-scale radiative transfer equations and associated radiative properties, and employs the Monte Carlo ray-tracing for computations of radiative fluxes and for computations of the radiative properties. The model two-phase media are reticulate porous ceramics and a particle packed bed, each composed of semitransparent solid and fluid phases. The results obtained by the two approaches are in good agreement within the limits of statistical uncertainty. The continuum-scale approach leads to a reduction in computational time by approximately one order of magnitude, and is therefore suited to treat radiative transfer problems in two-phase media in a wide range of engineering applications.     

Discrete ordinates solution of coupled conductive radiative heat transfer in a two-layer slab with Fresnel interfaces subject to diffuse and obliquely collimated irradiation

The coupled conductive radiative heat transfer in a two-layer slab with Fresnel interfaces subject to diffuse and obliquely collimated irradiation is solved. The collimated and diffuse components problems are treated separately. The solution for diffuse radiation is obtained by using a composite discrete ordinates method and includes the development of adaptive directional quadratures to overcome the difficulties usually encountered at the interfaces. The complete radiation numerical model is validated against the predictions obtained by using the Monte Carlo method.     

Transient conductive and radiative transfer in a planar layer with Arrhenius heat generation

Transient conductive and radiative energy transfer in a gray absorbing-emitting planar medium bounded by black walls is studied. The temperature distribution is uniform when the medium starts releasing energy according to the Arrhenius equation. The Crank-Nicolson method is used in the radiative-dominant case with good accuracy. Time-dependent temperature and heat flux distributions are calculated until steady-state is reached. Confirmation of the solution technique is made by comparison with previous investigations. The dimensionless activation energy required for ignition varies from 4.1 for pure conduction to 7.5 for pure radiation.     

Discontinuous finite element method for radiative heat transfer in semitransparent graded index medium

To avoid the complicated and time-consuming computation of curved ray trajectories, a discontinuous finite element method based on discrete ordinate equation is extended to solve the radiative transfer problem in a multi-dimensional semitransparent graded index medium. Two cases of radiative heat transfer in two-dimensional rectangular gray semitransparent graded index medium enclosed by opaque boundary are examined to verify this discontinuous finite element method. Special layered and radial graded index distributions are considered. The predicted dimensionless net radiative heat fluxes and dimensionless temperature distributions are determined by the discontinuous finite element method and compared with the results obtained by the curved Monte Carlo method in references. The results show that the discontinuous finite element method has a good accuracy in solving the multi-dimensional radiative transfer problem in a semitransparent graded index medium.     

Radiative and conductive heat transfer in a nongrey semitransparent medium. Application to fire protection curtains

This paper deals with heat transfer in nongrey media which scatter, absorb and emit radiation. Considering a two dimensional geometry, radiative and conductive phenomena through the medium have been taken into account. The radiative part of the problem was solved using the discrete ordinate method with classical S_n quadratures. The absorption and scattering coefficients involved in the radiative transfer equation (RTE) were obtained from the Mie theory. Conduction inside the medium was linked to the RTE through the energy conservation. Validation of the model has been achieved with several simulation of water spray curtains used as fire protection walls.     

The differential approximation for radiative transfer in an emitting,absorbing and anisotropically scattering medium

The validity of the well-established differential approximation in radiative transfer is extended to include linear-anisotropic scattering. Sample results demonstrate the accuracy of the method.     

Analysis of collimated radiation in participating media using the discrete transfer method

A general formulation of the discrete transfer method is provided to analyze radiative heat transfer problems in a participating medium subjected to collimated radiation. The formulation is validated by considering 1-D planar absorbing, emitting and anisotropically scattering gray medium in radiative equilibrium. Anisotropy of the medium is approximated by linear anisotropic phase function. For the purpose of comparison, the problem is also solved analytically. Results are obtained for different angles of incidence of the collimated radiation. At a given angle of incidence, results are obtained for forward, isotropic and backward scattering situations. Heat flux results are compared over a wide range of values of the extinction coefficient. Emissive power distributions in the medium are also obtained for some cases. The discrete transfer method results are found to compare very well with the analytic results.     

Multidimensional radiative transfer in absorbing,emitting, and linearly anisotropic scattering cylindrical medium with space-dependent properties

The integral form of three-dimensional radiative transfer equation for an absorbing, emitting, and linear-anisotropic scattering medium with space-dependent properties is formulated. A product-integration method is subsequently applied to develop a numerical scheme for solving the corresponding integral transfer equations in a two-dimensional, axisymmetric and nonhomogeneous medium subjected to externally incident radiation or bounded by emitting and diffusely-reflecting walls. The numerical solutions for cases of constant, continuous, and stepwise variations of scattering albedo are presented to illustrate its accuracy and flexibility, and validated by comparing with results available in the literature.     

Discrete transfer method applied to radiative transfer in a variable refractive index semitransparent medium

Application of the discrete transfer method (DTM) has been extended to the analysis of radiative heat transfer in a variable refractive index participating medium. To validate the DTM formulation, radiative heat transfer in an absorbing, emitting and isotropically scattering planar medium was considered. The participating medium was assumed to be in radiative equilibrium. For both constant and variable refractive indices of the medium, the DTM results were compared with those available in the literature. The DTM was found to provide accurate results.     

Measurement of thermal radiative and conductive properties of semitransparent materials using a photothermal crenel method

This paper deals with a theoretical and an experimental study allowing the measurement of the radiative and the conductive properties of semitransparent materials. The method consists of applying a crenel heat flux on the front face of a semitransparent sample and recording the temperature at the rear face using an open thermocouple junction.Parameter identification is performed by the minimization of the ordinary least-squares function comparing the measured and the calculated temperatures. This later is obtained from the thermal model describing the heat transfer by conduction and radiation in the medium. This model is built by the thermal quadrupole formalism.Measurements are reported on commercial glasses and plexiglass samples, and the used iterative algorithm is based on the Gauss-Newton method.