One-dimensional Radiative Heat Transfer Research Articles

Evaluation of spectral line mixing models and the effects on high pressure radiative heat transfer calculations

At higher pressures, the spectral lines in a gas spectrum can become more broadened and shifted due to collisional effects, leading to increased spectral line mixing. In the present study, we have reviewed different line mixing models for modeling the absorption spectra for three gases of CO2, CO, and H2O and applied these models to simulate radiative spectra for pressures within approximately 5–100 bar. The simulated data were compared with the yet-published high-pressure spectral experimental data as well as experimental data published in the literature. It has been found that the empirical “pseudo-Lorentz” line shape model considering the line mixing effects shows the best performance for modeling high-pressure gas spectra for the tested conditions, which is easy to be implemented and relatively accurate, thus is recommended for high-pressure radiative heat transfer calculations in engineering applications. Based on the selected “pseudo-Lorentz” line mixing model, we have constructed a high-pressure absorption coefficients database for CO2, CO, and H2O for pressure ranging from 1 bar to 80 bar. The newly generated database considering the spectral line mixing effects was tested for radiative heat transfer calculations, and the results were compared with the ones without considering the line mixing effects. The significance of ignoring the line mixing effects on radiative heat transfer calculations under different pressures was evaluated with one-dimensional radiative heat transfer benchmark cases. The high-pressure absorption coefficients database is available from the corresponding author upon request.

Incident energy transfer equation and its solution by collocation spectral method for one-dimensional radiative heat transfer

The ray-effect is a major discretization error in the approximate solution method for the radiative transfer equation (RTE). To overcome this problem, the incident energy transfer equation (IETE) is proposed. The incident energy, instead of radiation intensity, is obtained by directly solving this new equation. Good numerical properties are found for the incident energy transfer equation. To show the properties of numerical solution, the collocation spectral method (CSM) is employed to solve the incident energy transfer equation. Three test cases are taken into account to verify the performance of the incident energy transfer equation. The result shows that the radiative heat flux obtained based on IETE is much more accurate than that based on RTE, which means that the IETE is very effective in eliminating the impacts of ray-effect on the heat flux. However, on the contrary, the radiative intensity obtained based on IETE is less accurate than that based on RTE due to the ray-effect. So, this equation is more suitable for those radiative heat transfer problems, in which the radiation heat flux and incident energy are needed rather than the radiation intensity.

Chebyshev collocation spectral method for one-dimensional radiative heat transfer in linearly anisotropic-scattering cylindrical medium

In this study, the Chebyshev collocation spectral method (CCSM) is developed to solve the radiative integro-differential transfer equation (RIDTE) for one-dimensional absorbing, emitting and linearly anisotropic-scattering cylindrical medium. The general form of quadrature formulas for Chebyshev collocation points is deduced. These formulas are proved to have the same accuracy as the Gauss-Legendre quadrature formula (GLQF) for the F-function (geometric function) in the RIDTE. The explicit expressions of the Lagrange basis polynomials and the differentiation matrices for Chebyshev collocation points are also given. These expressions are necessary for solving an integro-differential equation by the CCSM. Since the integrand in the RIDTE is continuous but non-smooth, it is treated by the segments integration method (SIM). The derivative terms in the RIDTE are carried out to improve the accuracy near the origin. In this way, a fourth order accuracy is achieved by the CCSM for the RIDTE, whereas it's only a second order one by the finite difference method (FDM). Several benchmark problems (BPs) with various combinations of optical thickness, medium temperature distribution, degree of anisotropy, and scattering albedo are solved. The results show that present CCSM is efficient to obtain high accurate results, especially for the optically thin medium. The solutions rounded to seven significant digits are given in tabular form, and show excellent agreement with the published data. Finally, the solutions of RIDTE are used as benchmarks for the solution of radiative integral transfer equations (RITEs) presented by Sutton and Chen (JQSRT 84 (2004) 65–103). A non-uniform grid refined near the wall is advised to improve the accuracy of RITEs solutions.

A Discrete Transfer Method for Radiative Transfer through Anisotropically Scattering Media

A discrete transfer formulation for analyzing radiative heat transfer through participating medium with highly anisotropic scattering phase function is presented. This formulation used the finite volume method to evaluate integral of the radiative heat transfer problem. Hence, participating media anisotropic scattering phase function along a centre scattered sub-solid angle of the discrete transfer direction is defined by an average on the incident sub-solid angle. Numerical simulations on the efficiency of the proposed discrete transfer formulation for one-dimensional radiative heat transfer through participating, and anisotropic scattering medium under diffuse incidence is carried out. Results indicate that present discrete transfer radiative heat fluxes, transmittance and reflectance predictions have accuracy comparable to available analytical and numerical literature benchmark solutions. Excellent agreements are noted for different anisotropic scattering phase functions.

Chebyshev collocation spectral method for one-dimensional radiative heat transfer in graded index media

Chebyshev spectral collocation method based on discrete ordinates equation is developed to solve radiative transfer problems in a one-dimensional absorbing, emitting and scattering semitransparent slab with spatially variable refractive index. For radiative transfer equation, the angular domain is discretized by discrete ordinates method, and the spatial domain is discretized by Chebyshev collocation spectral method. Due to the exponential convergence of spectral methods, a very high accuracy can be obtained even using few nodes for present problems. Numerical results by the Chebyshev collocation spectral-discrete ordinates method (SP-DOM) are compared with those available data in references. Effects of refractive index gradient on radiative intensity are studied for space dependent scattering media. The results show that SP-DOM has a good accuracy and efficiency for solving radiative heat transfer problems in even spatially varying absorbing, emitting, scattering, and graded index media.

Iterative and direct Chebyshev collocation spectral methods for one-dimensional radiative heat transfer

In this paper, the Chebyshev collocation spectral method for one-dimensional radiative heat transfer equation with participating media is presented; and sequentially the iterative and direct solvers are developed. Implementation of the new method shows its flexibility to complex problems: highly anisotropic and space-dependent scattering. The new solvers can provide exponential convergence in space and can capture large oscillations. Numerical results verified the high accuracy of the new method, and its competitive ability compared with other newly appeared methods.

Isotropic scaling limits for one-dimensional radiative heat transfer with collimated incidence

As high-anisotropic scattering of materials induces difficulty in the resolution of radiative transfer equation, the isotropic assumption or approximate anisotropic phase functions are very useful. Isotropic scattering approximations need to be solved with a scaled optical depth and albedo. Isotropic scaling models involve the transformation of an anisotropic problem to an isotropic one. In this paper, the scaled optical depth and albedo are derived from the zero and first moment calculation of the scaled albedo and phase function product. Comparisons between the isotropic scaling, the anisotropic Henyey–Greenstein phase function approximations, and the exact solution are studied for one-dimensional steady state radiative heat transfer. The results show that both the isotropic scaling and the Henyey–Greenstein approximation are accurate for highly or weakly scattering media. For absorbing/scattering medium, the accuracy of the two approximations depends on the optical thickness and on the scattering function of the medium. The maximum errors between the two approximations and the benchmark solution are for albedo values around 0.5.

Discrete Ordinate Methods for Radiative Heat Transfer in Isotropically and Anisotropically Scattering Media

The objective of this note is to focus on the discrete ordinate method as a method for analyzing radiative heat transfer in isotropically and anisotropically scattering media. In particular, one dimension is chosen only for simplicity, since exact solutions are available for benchmarking the discrete ordinate method. One-dimensional radiative heat transfer has been analyzed in many previous investigations and has been extensively reviewed by Viskanta. The development of the discrete ordinate method has been described in detail by Chandrasekhar and Lathrop. The discrete ordinate method using Gauss quadrature has been compared with approximate solutions. In the cases previously analyzed solutions were obtained using ordinate sets that were larger than necessary. This note demonstrates that if ordinate sets are chosen to satisfy key moments of the radiative intensity, the resulting Nth order solution with the proposed quadrature is superior to the Nth order solution using Gauss quadrature.

Effects of Anisotropic Scattering on Steady One-Dimensional Radiative Heat Transfer through an Absorbing-Emitting Gray Medium

Effects of anisotropic scattering on temperature distribution of a steady one-dimensional radiative heat transfer through an absorbing, emitting gray medium in a semi-infinite space are analyzed by method of matched asymptotic expansions. An external radiation beam is imposed through a transparent boundary. Rayleigh scattering is considered as a typical model of an anisotropic scattering. The medium is assumed to be non-heat conducting, not to move and to have constant physical properties, The results show that no qualitative difference is shown between the temperature distribution in a scattering medium and that in a non-scattering medium. Only a little quantitative difference is shown for various values of the beam angle. The difference is in 10 percent of the non-scattering value at most.

Steady One-Dimensional Radiative Heat Transfer with Emission, Reflection and External Radiation

This paper deals theoretically with a radiative heat transfer through an absorbing-emitting medium which occupies a semi-infinite space bounded by a plane wall. A steady one-dimensional problem is considered. The medium is assumed to be grey and to be in a local thermodynamic equilibrium. Effects due to the emission from the wall, specular and diffuse reflection at the wall and external beam radiation through the wall are considered in contrast with the Milen problem, in which no radiation comes into the medium through a purely transparent boundary. The general solution is obtained. The solution is studied in detail in five special cases to show each effect of these boundary conditions. They are the cases of pure emission, pure beam radiation input, pure reflection, a uniform asymptotic field and emission accompanied by diffuse reflection. In the second case, a localized temperature change is found in the neighbourhood of the wall, even when the asymptotic field has a uniform temperature.

Generalizations in the theory of one-dimensional radiative heat transfer

Generalizations in the theory of one-dimensional radiative heat transfer