Comparison between radiative transfer theory and the simplified spherical harmonics approximation for a semi-infinite geometry

In this study, the third-order simplified spherical harmonics equations (SP3), an approximation of the radiative transfer equation, are solved for a semi-infinite geometry considering the exact simplified spherical harmonics boundary conditions. The obtained Green's function is compared to radiative transfer calculations and the diffusion theory. In general, it is shown that the SP3 equations provide better results than the diffusion approximation in media with high absorption coefficient values but no improvement is found for small distances to the source.     

An approximate mean absorption coefficient for H2O

An approximate mean absorption coefficient was obtained for the rotation spectrum of H₂O from the Fourier transform of the molecular dipole autocorrelation function. The correlation function was calculated using classical mechanics to describe the rotation of the molecule. The expression for the absorption coefficient was parameterized to facilitate the determination of the temperature dependence of the absorption coefficient. A comparison was made between the classical and the quantum-mechanical calculations of the mean absorption coefficient for the pure rotation band of H₂O at 250, 300 and 500°K. The agreement between the classical and quantum-mechanical calculations may be improved if corrections for quantum effects are introduced into the classical calculations. A method is given for generalizing the results for the pure rotation band to the vibration-rotation bands. The mean absorption coefficient of water vapor was used to compute the thermal radiative flux in a heavy pure water vapor atmosphere. The results of these calculations were compared with those of an existing calculation in which the absorption due to water vapor was given by a gray approximation.     

Radiative equilibrium in a rectangular enclosure bounded by gray walls

Two-dimensional temperature and heat flux distributions are calculated for an absorbing-emitting gray medium at radiative equilibrium in a rectangular enclosure. The bounding walls are gray and diffuse with arbitrary surface temperature distributions, and heat generation may take place inside the medium. As a first approximation, the problem is solved for optically thick systems (differential approximation). These results are subsequently improved by the introduction of a number of geometrical parameters to yield good accuracy for all optical thicknesses. As examples, two cases are discussed in detail: (1) uniform heat generation in a black enclosure and (2) an enclosure with one gray surface at constant temperature. Comparison with some numerical solutions generated by Hottel's zonal method shows excellent agreement.     

On the use of mean absorption coefficients in the presence of strong temperature gradients

We compare results obtained by using the Planck and Rosseland mean absorption efficientg with exact multifrequency transport results in a purely absorbing halfspace problem in the presence of strong temperature gradients. We give numerical results for both a prescribed temperature distribution in the presence of an Elsasser band absorption coefficient and a self-consistent radiative equilibrium distribution corresponding to a source deep within the halfspace transporting through a picket fence absorption coefficient. This latter problem (the Milne problem) is analyzed using the Wiener-Hopf factorization technique, and an explicit expression is given for the characteristic H-function. We conclude that the use of the Rosseland mean is generally more accurate than the use of the Planck mean, but that the Rosseland mean results can nevertheless be significantly in error.     

A full spectrum k-distribution based non-gray radiative property model for unburnt char

By using the concept of weighted sum of four gray particles and spectrum k-distribution (WSGP-SK), a non-gray radiative property model for unburnt char particles is developed. Based on the carbon burnout kinetic model for structure during oxidation, and the linear mixed approximation theory for complex index of refraction, spectral radiative properties of unburnt char particles are first calculated as function of the burnout ratio by Mie theory. Referring to the full spectrum k-distribution model, k-distribution is applied to reorder absorption and scattering efficiencies of particles. Then, weighting factors and efficiency factors of the non-gray radiative property model are directly obtained from Gaussian integral points of k-distribution. The model is validated against the benchmark solutions of line-by-line (LBL) model. Maximum relative errors of this model are 3% and 15% for radiative heat fluxes and source terms in non-isothermal inhomogeneous particulate media, respectively. The assumption of linearly varying radiative properties with burnout ratio (Lockwood et al. 1986) will result in a predicted deviation of 53% for radiative source terms. Results also show that this non-gray model is remarkably better than the Planck mean method. Moreover, a satisfactory comparison with LBL solutions is achieved in the gas and particle mixture by combining the non-gray WSGG-SK model (Guo et al. 2015). As a radiation sub-model, this non-gray radiative property model can significantly improve prediction accuracy of radiative heat transfer in oxy-fuel combustion.     

Comparison of exact and mean beam length results for a radiating hydrogen plasma

The purpose of this work is to investigate the accuracy of the mean beam length technique in high-temperature radiative heat transfer, combined with other modes of heat transfer. In order to study the validity of the mean beam length method, Nusselt numbers are presented for fully developed channel flow of a radiating nonisothermal hydrogen plasma. Black isothermal boundaries are considered. Numerical results obtained from the exact integrodifferential equation have been obtained previously for this problem. Linearized radiation and local thermodynamic equilibrium are assumed.The results show that the Nusselt numbers obtained by using the geometric mean beam length are in close agreement with results obtained by using the mean beam length. Therefore the complicated calculations needed to obtain the mean beam length are unnecessary.A comparison of the results obtained in the present work with previously reported work shows that the mean beam length technique is a better approximation to the exact solution than the optically-thick, or nongray differential approximation solutions.     

Radiative transfer in a nongray spherical layer: Simplified rectangular model

The problem of radiative transfer in a nongray, absorbing-emitting spherical layer is investigated. The absorption coefficient is assumed to be only a function of frequency, i.e. k_v = α(v)k, and the function α(v) is allowed only two values, zero or unity (simplified rectangular model). The nongray radiative transfer problem is reduced to a gray solution without the use of any approximation (such asthe Plank or Rosseland means) for an isothermal layer and for radiative equilibr     

Radiative heat transfer in axisymmetric domains of complex shape with Fresnel boundaries

A detailed study of the peculiarities of the radiative heat transfer (RHT) in axisymmetric domains bounded with Fresnel surfaces is undertaken. The analytical (exact) solutions of the RHT problem in conical and cylindrical domains with refractive index more than unity were obtained for a variety of absorption coefficient and geometrical parameters of the domains. It is shown that due to Fresnel reflections the net radiative flux strongly varies over the base of cone and cylinder. The difference in RHT processes for the cases of constant reflectivity of the boundaries and that calculated by Fresnel formula is demonstrated. The influence of specular reflection at the crystal side surface on the shape of the solid/liquid interface in growing bismuth germanate crystals is shown.     

Analysis of combined conduction and radiation heat transfer in presence of participating medium by the development of hybrid method

The current study addresses the mathematical modeling aspects of coupled conductive and radiative heat transfer in the presence of absorbing, emitting and isotropic scattering gray medium within two-dimensional square enclosure. A blended method where the concepts of modified differential approximation employed by combining discrete ordinate method and spherical harmonics method, has been developed for modeling the radiative transport equation. The gray participating medium is bounded by isothermal walls of two-dimensional enclosure which are considered to be opaque, diffuse and gray. The effect of various influencing parameters i.e., radiation-conduction parameter, surface emissivity, single scattering albedo and optical thickness has been illustrated. The adaptability of the present method has also been addressed.     

Near-infrared radiative properties of porous zirconia ceramics

Infrared radiative properties of zirconia ceramics of porosity about 16% are studied by means of the measurements of directional–hemispherical reflectance and transmittance in the wavelength range from 2.5 to 9 μm. The recently suggested modified two-flux approximation is examined as a simplified basis of the identification procedure. A comparison with the exact numerical solution confirms a good accuracy of this approach for identification of the absorption coefficient of ceramics. An analysis of the results for transport scattering coefficient showed that scattering is determined by isotropic pores with characteristic average radius about 1 μm. The corresponding approximate theoretical model of radiative properties of ceramics is suggested. The absorption coefficient of bulk zirconia in the semi-transparency range is obtained from the data for porous zirconia ceramics.     

Radiative decay times for thermal pertubations in a nongray medium

The radiative decay time of harmonic thermal perturbations in a nongray medium of infinite extent is obtained in closed form for two specific band absorption models. These models are the frequently used gray band and the exponential band, the latter being considered more realistic for molecular gases. It is found that the decay time at the boundary of a semi-infinite medium can be obtained in terms of that in an infinite medium. The decay time for combined thermal radiation and conduction is also discussed. The difference in radiative decay rates for a medium with gray bands and one with exponential-tailed bands is marked; in an infinite medium at large Bouguer number, the former falls to zero while the latter rises to a maximum.     

Spontaneous emission from a single two-level atom and the radiative correction to the final state

By using a modified Robertson projection technique exact equations of motion for expectation values of atomic population inversion and dipole moment operators are derived. A radiative correction to the unperturbed ground state expectation value of the population inversion operator is obtained in the Born and Markov approximation if no long-time limit approximation is used.     

An idealized radiative transfer scheme for use in a mechanistic general circulation model from the surface up to the mesopause region

A new and numerically efficient method to compute radiative flux densities and heating rates in a general atmospheric circulation model is presented. Our method accommodates the fundamental differences between the troposphere and middle atmosphere in the long-wave regime within a single parameterization that extends continuously from the surface up to the mesopause region and takes the deviations from the gray limit and from the local thermodynamic equilibrium into account. For this purpose, frequency-averaged Eddington-type transfer equations are derived for four broad absorber bands. The frequency variation inside each band is parameterized by application of the Elsasser band model extended by a slowly varying envelope function. This yields additional transfer equations for the perturbation amplitudes that are solved numerically along with the mean transfer equations. Deviations from local thermodynamic equilibrium are included in terms of isotropic scattering, calculating the single scattering albedo from the two-level model for each band. Solar radiative flux densities are computed for four energetically defined bands using the simple Beer-Bougert-Lambert relation for absorption within the atmosphere. The new scheme is implemented in a mechanistic general circulation model from the surface up to the mesopause region. A test simulation with prescribed concentrations of the radiatively active constituents shows quite reasonable results. In particular, since we take the full surface energy budget into account by means of a swamp ocean, and since the internal dynamics and turbulent diffusion of the model are formulated in accordance with the conservation laws, an equilibrated climatological radiation budget is obtained both at the top of the atmosphere and at the surface.     

Non-gray chemical composition based radiative property model of fly ash particles

Particle radiation has a spectral dependence and is closely related to the chemical composition of the material. Iron oxide, one of the main components of fly ash, observably affects the complex index of refraction of the particles. In this study, following the theory of the spectrum k-distribution based weighted sum of gray particles model (Guo et al. [4,13]), a non-gray fly ash radiative property model involving the chemical composition was developed. First, four typical fly ash particles with different iron oxide contents were selected, and the corresponding particle radiative parameters were obtained using the Mie theory. Then, the absorption efficiency and weighting factors of the non-gray model were directly obtained from the Gaussian integral points of the k-distribution. The scattering efficiency of the particles was obtained from the Planck mean. The accuracy of the newly developed model was evaluated in a one-dimensional plane-parallel slab system through comparison with the line-by-line (LBL) model and two commonly used gray radiative property models. The results show that the new non-gray model agrees well with the LBL solution and becomes more accurate as the iron oxide content increases. When the iron oxide content of the fly ash increased from 5.47% to 30.50%, the maximum relative error of the radiative heat flux and the radiative source term decreased from 12.50% to 5.68% and from 20.97% to 12.62%, respectively. The new model can improve the prediction accuracy of radiative heat transfer in pulverized coal-fired furnaces.     

Coaxial radiative and convective heat transfer in gray and nongray gases

Coupled radiative and convective heat transfer is investigated for an absorbing gas flowing in a finite length channel and heated by blackbody radiation directed along the flow axis. The problem is formulated in one dimension and numerical solutions are obtained for the temperature profile of the gas and for the radiation escaping the channel entrance, assuming both gray and nongray absorption spectra. Due to radiation trapping, the flowing gas is found to have substantially smaller radiation losses for a given peak gas temperature than a solid surface that is radiatively heated to this temperature. A greenhouse effect is also evident whereby radiation losses are minimized for a gas having stronger absorption at long wavelengths.     

Fast approximate technique for the cumulative wavenumber model to modeling radiative transfer in a mixture of real gas media

In the cumulative wavenumber (CW) model, the total range of the absorption cross-section Cη is subdivided into the supplementary absorption cross-section of gray gases C_j, j=1,…,n, where n is the number of gray gases; and the wavenumber region is subdivided into intervals Δ_i=[η_i−1, η_i], i=1, 2,…,p, where p is the number of intervals. The intersection of the two spectral subdivisions is used to define the modeling of the fractional gray gas D_ij. In the CW model, we solve the radiative transfer equation (RTE) in every subinterval D_ij; then it is necessary to solve n x p times the spectral form of the RTE for complete spectral integration. In this work, the CW model is used with a numerical approximation technique based on additive properties of radiative intensity to reduce the solution of RTE to n new fractional gray gas D_j for complete spectral integration. The CW model was first coupled with the discrete ordinates method and the accuracy of the simplified technique and the algorithm was first examined for one-dimensional homogeneous media; results are compared with line-by-line calculations and it is found that the CW model with the simplified technique is exact for the homogeneous media examined. Also, the fast approach is tested in the diffuse reflecting boundaries case. The CW model is implemented in a bi-dimensional enclosure containing real gases in isothermal cases. Afterwards, this approximate technique is extended to non-isothermal and non-homogeneous cases; the results are compared with line-by-line calculations taken from literature and good agreement was found. The results obtained using the acceleration technique for the CW model agree with the results of original CW model. With this acceleration technique the CPU time decreases p times. Spectral database HITRAN and HITEMP are used to obtain the molecular absorption spectrum of the gases.     

多维梯度折射率介质内辐射传递的扩散近似

推导多维梯度折射率介质内稳态辐射传递的扩散近似方程.使用有限元法对扩散近似进行离散和求解,利用两个二维半透明介质的稳态辐射传递问题验证该扩散近似的精度及适用性.算例考虑介质为均匀折射率及梯度折射率两种情况.利用扩散近似分别求解辐射平衡时的边界热流、介质内温度场分布,并与辐射传递方程的求解结果进行对比分析.结果表明:介质折射率变化、散射特性、光学厚度及散射反照率均直接影响扩散近似的精度;在光学厚及强散射条件下,该扩散近似可以作为一种快速算法应用于梯度折射率介质稳态辐射传递的求解.     

Sine-Gordon Theory for the Equation of State of Classical Hard-Core Coulomb Systems. III. Loopwise Expansion

We present an exact field theoretical representation of an ionic solution made of charged hard spheres. The action of the field theory is obtained by performing a Hubbard–Stratonovich transform of the configurational Boltzmann factor. It is shown that the Stillinger–Lovett sum rules are satisfied if and only if all the field correlation functions are short range functions. The mean field, Gaussian and two-loops approximations of the theory are derived and discussed. The mean field approximation for the free energy constitutes an exact lower bound for the exact free energy, while the mean field pressure is an exact upper bound. The one-loop order approximation is shown to be identical with the random phase approximation of the theory of liquids. Finally, at the two-loop order and in the pecular case of the restricted primitive model, one recovers results obtained in the framework of the mode expansion theory.     

Effect of band or line shape on radiative transfer in a nongray two-dimensional medium

An exact formulation is presented for a nongray two-dimensional, finite, planar, absorbing-emitting medium in radiative equilibrium. The absorption coefficient consists of an array of equal intensity, nonoverlapping bands or lines. Rectangular, triangular, exponential, Doppler and Lorentz shapes are specifically considered. Exact expressions are obtained for a medium subjected to collimated and diffuse radiation. The integral equations are linearized by the narrow-band approximation. The solution for the cosine-varying, collimated, monochromatic radiation model is used to construct the solutions for other boundary conditions. The two-dimensional equations are reduced to one-dimensional equations by the method of separation of variables. Results for the diffuse case are presented for several spatial variations.