Evaluation of solution methods for radiative heat transfer in gaseous oxy-fuel combustion environments

The exact solution to radiative heat transfer in combusting flows is not possible analytically due to the complex nature of the integro-differential radiative transfer equation (RTE). Many different approximate solution methods for the solution of the RTE in multi-dimensional problems are available. In this paper, two of the principal methods, the spherical harmonics (P₁) and the discrete ordinates method (DOM) are used to calculate radiation. The radiative properties of the gases are calculated using a non-gray gas full spectrum k-distribution method and a gray method. Analysis of the effects of numerical quadrature in the DOM and its effect on computation time is performed. Results of different radiative property methods are compared with benchmark statistical narrow band (SNB) data for both cases that simulate air combustion and oxy-fuel combustion. For both cases, results of the non-gray full spectrum k-distribution method are in good agreement with the SNB data. In the case of oxy-fuel simulations with high partial pressures of carbon dioxide, use of gray method for the radiative properties may cause errors and should be avoided.     

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.     

A theoretical study of radiation between two concentric spheres using a modified discrete ordinates method associated with Legendre transform

A modified discrete ordinates method (DOM) is used in spherical participating media. The radiative intensity is broken up into two components. One component is traced back to the enclosure's source. It is called direct intensity. The other component is rather traced back to the contribution of the medium itself. It is called diffuse intensity. Thus, the radiative transfer equation (RTE) is transformed into two simultaneous equations: a direct RTE and a diffuse RTE. The direct RTE is solved analytically. The diffuse RTE is solved numerically using the DOM. The streaming angular derivative term appearing in spherical geometry is modeled by making use of the Finite Legendre Transform. We study a pure radiation transfer problem between two concentric spheres. The medium is assumed to be gray and isotropically scattering. The limiting spheres are considered to be opaque, gray, diffusely emitting and diffusely reflecting with uniform emissivity over each surface. The obtained results are compared with available cases reported in the literature. In particular, relative importance of the direct radiation in optically thin media is studied.     

Coupled radiation and natural convection: Different approaches of the slw model for a non-gray gas mixture

The coupling between non-gray radiation heat transfer and convection-conduction heat transfer is studied. The spectral line weighted sum of gray gases model (slw) is used to account for non-gray radiation properties. The aim of this work is to analyze the influence of the different approaches used when calculating the parameters of the slw model. Such strategies include the use of optimized model coefficients to reduce the number of operations, and the interpolation of the distribution function instead of the use of mathematical correlations. Non-gray calculations are also compared to gray solutions using the Planck mean absorption coefficient, which can be also calculated with the slw model. The radiative transfer equation (rte) is solved by means of the discrete ordinates method (dom). A natural convection driven cavity is chosen to couple radiation and conduction-convection energy transfer. Several cases, with a significant variation of the ratio between radiation to convection heat transfer, as well as the ratio between radiation to conduction heat transfer, are discussed.     

Discrete ordinates interpolation method for radiative heat transfer problems in three-dimensional enclosures filled with non-gray or scattering medium

The discrete ordinates interpolation method (DOIM) is applied to three groups of problems of radiative heat transfer in three-dimensional rectangular enclosures containing non-gray or scattering medium. The original DOIM is first extended to a gray gas model using a new geometric interpolation scheme. It is applied to participating media for different scattering phase functions and optical thicknesses. For the non-gray gas model, the DOIM coupled with the narrow band-based weighted-sum-of-gray-gases (WSGG) model is developed. A few test problems with real gases such as pure H₂O and a mixture of CO₂, H₂O and N₂ are taken. The wall heat flux is calculated and compared with the exact solutions or reference values. All results of test problems are found to be reliable in this study. The DOIM closely reproduces the Monte Carlo reference solutions for different scattering phase functions and optical thicknesses. The non-gray gas results are compared with reference calculations based on the statistical narrow band model and they also show good agreements. The DOIM shows a remarkable merit in the computation time and the grid compatibility, to prove its usefulness for engineering applications.     

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.     

On the combined heat transfer in the multilayer non-gray porous fibrous insulation

A detailed numerical modeling is performed to investigate coupled heat transfer of natural convection, radiation and conduction in high-temperature multilayer thermal insulation (MTI), which consists of high-porous, non-gray semitransparent fibrous materials and reflective foils. Radiation within fibers, radiation between fibers and the reflective foils, conduction within fibers and convection from the fibers to the surrounding fluid are considered. Macroscopic (porous media) modeling is used to determine velocity, pressure and temperatures fields for fibrous insulation with a random packing geometry under natural convection, whereas the radiative transfer equation (RTE) is used to solve the radiative heat flux for non-gray materials. Key features of the macroscopic model include two-dimensional effects, non-gray radiative exchange, and the relaxation of the local thermodynamic non-equilibrium (LTNE). This model was validated by comparison with experimental data and it was used to investigate natural convection of coupled heat transfer in multilayer insulation, numerical results showed that natural convection is more likely to occur when the heated/cooled rate is low, while natural convection can be ignored in simulating steady-state coupled heat transfer in MTI.     

Numerical analyses of radiative heat transfer in any arbitrarily-shaped axisymmetric enclosures

A numerical approach for the treatment of radiative heat transfer in any irregularly-shaped axisymmetric enclosure filled with absorbing, emitting and scattering gray media is developed. Radiative transfer equation (RTE) is formulated for a general axisymmetric geometrical configurations, and the discretized equation is conducted using an unstructured meshes, generated by an appropriate computer algorithm, and the control volume finite element method which frequently adopted in CFD problems. A computer procedure has been done to solve the discretized RTE and to examine the accuracy and the computational efficiency of the proposed numerical approach. By using this computer algorithm, five test cases, a cylindrical enclosure with absorbing and emitting medium, a diffuser shaped axisymmetric enclosure, a finite axisymmetric cylindrical enclosure with a curved wall, a furnace with axially varying medium temperature and a rocket nozzle, are treated and the obtained results agree very well with other published works. Furthermore, the developed computer procedure has an accurate CPU time and it can be coupled easily with CFD codes.     

气粒混合物非灰辐射特性合并宽窄谱带K分布模型

气粒混合物辐射问题具有全场性、非灰性、耦合性等特点,准确预估高温燃气/粒子非灰辐射特性是非常重要的。本文将合并宽窄谱带K分布模础(CWNBCK)与离散坐标法(DOM)结合,开展了非灰气粒混合物辐射换热问题的模拟工作,分别验证了一维和三维情况下应用该模型的准确性,给出不同工况下的热流源项、壁面热流或辐射热流等。结果表明:该模型能够给出与SNB模型精度基本相同的结果,考虑其计算效率的提高,可以在工程实际中应用该模型计算非灰气粒混合物辐射换热。     

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.     

Modeling of radiative heat transfer in 3D complex boiler with non-gray sooting media

The radiative heat transfer problem is solved for 3D complex industrial boiler with five baffles containing a mixture of carbon dioxide and water vapor for non-uniform temperature fields. A numerical formulation using the FTn finite volume method coupled with the bounded high-order resolution CLAM scheme, the blocked-off-region procedure and the narrow-band based weighted-sum-of-gray-gases (WSGG) [Kim OJ, Song T-H. Data base of WSGGM-based spectral model for radiation properties of combustion products, JQSRT 2000; 64: 379-94] model is adapted. The effect of soot volumetric fraction, particle temperature and uniform particle concentration on the radiative heat flux and radiative heat source is investigated and discussed. Also the advantages, in non-gray media, of the FTnFVM compared to the classical FVM are highlighted.     

Radiative heat transfer in a plane-layer mixture of non-gray particulates and molecular gases

The interaction between particulate clouds and molecular gases with respect to radiative heat transfer in a one-dimensional, plane-parallel geometry is investigated. The particulates are assumed to have a non-gray absorption coefficient with a power-law dependence on wavenumber. The absorption coefficient of the molecular gases is described by the popular exponential wide-band model. Heat transfer rates are determined through the evaluation of internal and external slab emissivities of the particulate cloud and through evaluation of slab band absorptances for the molecular gases, modified to account for the interaction with the particles. Simple, closed-form expressions for the slab emissivities and slab absorptances are also introduced, resulting in good accuracy.     

Radiative heat transfer in a cylindrical mixture of non-gray particulates and molecular gases

The interaction between particulate clouds and molecular gases with respect to radiative heat transfer in a one-dimensional cylindrical medium with black walls is investigated. The particulates are assumed to have a non-gray absorption coefficient with a power-law dependence on wavenumber. The absorption coefficient of the molecular gases is described by the exponential wide-band model. Heat transfer rates are found through the evaluation of internal and external total hemispherical emissivities of the particulate cloud and through evaluation of total band absorptances for the molecular gases, modified to account for the interaction with the particles. Simple, closed-form expressions for the total hemispherical emissivities and total band absorptance are also presented, resulting in good accuracy.     

球形粒子的近场辐射换热研究

近场辐射引起的能量交换可以比远场辐射高若干个数量级.本文计算了SiC和Cu两种球形颗粒的近场辐射换热,发现当颗粒间距非常小时,辐射换热会大大强化.颗粒直径、颗粒间距以及颗粒的介电常数是决定近场辐射换热大小的重要因素,而颗粒温度对近场辐射换热的影响较小.     

Rigorous bounds on the radiative interaction between real gases and scattering particles

The radiative interaction due to the simultaneous presence of a real (non-gray) gas and isotropically scattering particles is examined rigorously for an isothermal plane layer, allowing completely general, frequency-dependent gas and particle radiative properties. A correction factor is defined to characterize the effects of interaction on the total hemispherical emittance of the layer, and rigorous bounds on the correction factor are determined, using the exact normal mode expansion technique of Case. It is shown that, under certain conditions for high albedo particles, interaction effects may lead to a slight enhancement of the total hemispherical emittance of the gas/particle mixture, compared to the sum of the emittances for the gas and particles considered separately, with the maximum enhancement possible being 3.2%. Under most conditions, however, the presence of the particles would be expected to shield the gas emittance. These rigorous bounds on the extent of the interaction effects are in contrast to the results of a previous numerical study, which predicted strong enhancement effects (nearly 100% enhancement for some cases) due to gas/particle radiative interaction.     

A concept of equivalent absorption and emission coefficients for gray analysis of radiative heat transfer

Due to the non-gray of gas radiation, the total emissivity differs from the total absorptivity. Therefore, in gray analysis, the equivalent absorption coefficient is not equal to the equivalent emission coefficient. Based on this idea, in this paper, a concept of equivalent absorption coefficient and equivalent emission coefficients is presented for gray analysis of gas radiative heat transfer. The equivalent emission coefficients are calculated by Leckner's formula and the equivalent absorption coefficients are estimated by inverse analysis. A one-dimensional gas radiation is taken as an example to show the efficiency of this concept. The results show that the concept of equivalent absorption and emission coefficients is feasible. It is necessary to use both the equivalent absorption coefficient and the equivalent emission coefficient for gray analysis.     

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.     

涂层的热红外发射率计算模型

根据Kubelka-Munk 模型和Mie散射理论,解出了涂覆在高发射率衬底上的红外涂层里的辐射传输方程（RTE）,建立了涂层在热红外（8～14 μm）波段的发射率的计算模型.在运用铝粉粒子作为颜料粒子的情况下,根据计算结果,得出了涂层发射率与粒子半径、涂层厚度的关系,以及最优的粒子半径范围.     

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.     

Numerical investigation of pulverized coal particle group combustion using tabulated chemistry

The ignition and combustion of coal particle groups are investigated numerically in a laminar flow reactor. The Flamelet Generated Manifold method is extended to account for the complex mixture of gases being released during devolatilization, which is calculated with a competing two-step model. A second mixture fraction is introduced to include the mixing with the second methane fuel stream. The interactions of the gas phase with particles are modeled within a fully coupled Euler-Lagrange framework. To investigate the influence of particle groups on ignition and combustion, successively increasing densities of particle streams have been analyzed. The ignition delay time is increased significantly by higher particle densities. This delay is validated successfully with the available measurements. Moreover, the shape of the volatile flame was found to be strongly influenced by the particle number density inside the flame. A transition from spherical flames around single particles to a conical flame around the particle cloud could be found in numerical results as well as in experiments. As the primary mechanism for the substantial ignition delay and the formation of the flame, the increased heat transfer from the gas-phase to the particle group, resulting in lower gas-phase temperatures, was identified.