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.     

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.     

The spectral-line moment-based (SLMB) modeling of the wide band and global blackbody-weighted transmission function and cumulative distribution function of the absorption coefficient in uniform gaseous media

The spectral-line moment-based (SLMB) modeling is proposed for the calculation of radiative properties of gases on any spectral width. The associated mathematical formulation is obtained by applying several concepts of the k-distribution methods such as the reordering of the wavenumber scale by monotonic variations of the absorption coefficient, together with the application of the k-moment method's principles. This approach gives both a general formula for the BTF and a simple and readily applicable approximation for the blackbody-weighted cumulated k-distribution function of the absorption coefficient. The model is applied for the computation of wide band BTFs and cumulative k-distributions for uniform columns of CO₂ and H₂O in the temperature range (300-2400 K) at atmospheric pressure. Model parameters are deduced from line-by-line (LBL) spectra calculated using the HITEMP database. Comparisons with LBL reference data as well as with contemporary modeling approaches (SLW, FSK, SNB) are performed and discussed.     

A simplified CFD model for spectral radiative heat transfer in high-pressure hydrocarbon–air combustion systems

Detailed radiation modeling in piston engines has received relatively little attention to date. Recently, it is being revisited in light of current trends towards higher operating pressures and higher levels of exhaust-gas recirculation (EGR), both of which enhance molecular gas radiation. Advanced high-efficiency engines also are expected to function closer to the limits of stable operation, where even small perturbations to the energy balance can have a large influence on system behavior. Detailed radiation modeling using sophisticated tools like photon Monte Carlo/line-by-line (PMC/LBL) is computationally expensive. Here, guided by results from PMC/LBL, a simplified stepwise-gray spectral model in combination with a first-order spherical harmonics (P1 method) radiative transfer equation (RTE) solver is proposed and tested for engine-relevant conditions. Radiative emission, reabsorption and radiation reaching the walls are computed for a heavy-duty compression-ignition engine at part-load and full-load operating conditions with different levels of EGR and soot. The results are compared with those from PMC/LBL, P1/FSK (P1 with a full-spectrum k-distribution spectral model) and P1/Gray radiation models to assess the proposed model’s accuracy and computational cost. The results show that the proposed P1/StepwiseGray model can calculate reabsorption locally and globally with less than 10% error (with respect to PMC/LBL) at a small fraction of the computational cost of PMC/LBL (a factor of 30) and P1/FSK (a factor of 15). In contrast, error in computed reabsorption by the P1/Gray model is as high as 60%. It is expected that the simplified model should be broadly applicable to high-pressure hydrocarbon–air combustion systems, with or without soot.     

Multi-scale k-distribution model for gas mixtures in hypersonic nonequilibrium flows

A k-distribution model is presented for gas mixtures in thermodynamic nonequilibrium, containing strongly radiating atomic species N and O together with molecular species of N₂, N₂⁺, NO and O₂. In the VUV range of the spectrum there is strong absorption of atomic radiation by bands of N₂. For this spectral range, a multi-scale model is presented, where RTEs are solved separately for each emitting species and overlap with other species is treated in an approximate way. Methodology for splitting the gas mixture into scales and evaluation of the overlap parameter between different scales is presented. The accuracy of the new model is demonstrated by solving the radiative transfer equation along the stagnation line flow field of the Crew Exploration Vehicle (CEV).     

Modeling nongray gas-phase and soot radiation in luminous turbulent nonpremixed jet flames

Much progress has been made in radiative heat transfer modeling with respect to treatment of nongray radiation from both gas-phase species and soot particles, while radiation modeling in turbulent flame simulations is still in its infancy. Aiming at reducing this gap, this paper introduces state-of-the-art models of gas-phase and soot radiation to turbulent flame simulations. The full-spectrum k-distribution method (Modest, M.F., 2003, Journal of Quantitative Spectroscopy & Radiative Transfer, 76, 69–83) is implemented into a three-dimensional unstructured CFD code for nongray radiation modeling. The mixture full-spectrum k-distributions including nongray absorbing soot particles are constructed from a narrow-band k-distribution database created for individual gas-phase species, and an efficient scheme is employed for their construction in CFD simulations. A detailed reaction mechanism including NO _x and soot kinetics is used to predict flame structure, and a detailed soot model using a method of moments is employed to determine soot particle size distributions. A spherical-harmonic P₁ approximation is invoked to solve the radiative transfer equation. An oxygen-enriched, turbulent, nonpremixed jet flame is simulated, which features large concentrations of gas-phase radiating species and soot particles. Nongray soot modeling is shown to be of greater importance than nongray gas modeling in sooty flame simulations, with gray soot models producing large errors. The nongray treatment of soot strongly influences flame temperatures in the upstream and the flame-tip region and is essential for accurate predictions of NO. The nongray treatment of gases, however, weakly influences upstream flame temperatures and, therefore, has only a small effect on NO predictions. The effect of nongray soot radiation on flame temperature is also substantial in downstream regions where the soot concentration is small. Limitations of the P₁ approximation are discussed for the jet flame configuration; the P₁ approximation yields large errors in the spatial distribution of the computed radiative heat flux for highly anisotropic radiation fields such as those in flames with localized, near-opaque soot regions.     

A Monte Carlo method for 3D thermal infrared radiative transfer

A 3D Monte Carlo model for specific application to the broadband thermal radiative transfer has been developed in which the emissivities for gases and cloud particles are parameterized by using a single cubic element as the building block in 3D space. For spectral integration in the thermal infrared, the correlated k-distribution method has been used for the sorting of gaseous absorption lines in multiple-scattering atmospheres involving 3D clouds. To check the Monte-Carlo simulation, we compare a variety of 1D broadband atmospheric fluxes and heating rates to those computed from the conventional plane-parallel (PP) model and demonstrate excellent agreement between the two. Comparisons of the Monte Carlo results for broadband thermal cooling rates in 3D clouds to those computed from the delta-diffusion approximation for 3D radiative transfer and the independent pixel-by-pixel approximation are subsequently carried out to understand the relative merits of these approaches.     

Modelling nongrey gas-phase and soot radiation in luminous turbulent nonpremixed jet flames

Much progress has been made in radiative heat transfer modelling with respect to the treatment of nongrey radiation from both gas-phase species and soot particles, while radiation modelling in turbulent flame simulations is still in its infancy. Aiming at reducing this gap, this paper introduces state-of-the-art models of gas-phase and soot radiation to turbulent flame simulations. The full-spectrum k-distribution method (M.F. Modest, 2003, Journal of Quantitative Spectroscopy & Radiative Transfer, 76, 69–83) is implemented into a three-dimensional unstructured computational fluid dynamics (CFD) code for nongrey radiation modelling. The mixture full-spectrum k-distributions including nongrey absorbing soot particles are constructed from a narrow-band k-distribution database created for individual gas-phase species, and an efficient scheme is employed for their construction in CFD simulations. A detailed reaction mechanism including NO _x and soot kinetics is used to predict flame structure, and a detailed soot model using a method of moments is employed to determine soot particle size distributions. A spherical harmonic P₁ approximation is invoked to solve the radiative transfer equation. An oxygen-enriched, turbulent, nonpremixed jet flame is simulated, which features large concentrations of gas-phase radiating species and soot particles. Nongrey soot modelling is shown to be of greater importance than nongrey gas modelling in sooty flame simulations, with grey soot models producing large errors. The nongrey treatment of soot strongly influences flame temperatures in the upstream and the flame-tip region and is essential for accurate predictions of NO. The nongrey treatment of gases, however, weakly influences upstream flame temperatures and, therefore, has only a small effect on NO predictions. The effect of nongrey soot radiation on flame temperature is also substantial in downstream regions where the soot concentration is small. Limitations of the P₁ approximation are discussed for the jet flame configuration; the P₁ approximation yields large errors in the spatial distribution of the computed radiative heat flux for highly anisotropic radiation fields such as those in flames with localized, near-opaque soot regions.     

Generalization of the k-moment method using the maximum entropy principle. Application to the NBKM and full spectrum SLMB gas radiation models

The k-moment method is generalized by applying the maximum entropy principle to get several estimates of the k-distribution function on any kind of spectral interval as a function of the first two moments of the absorption coefficient. Corresponding formulations of the blackbody weighted band averaged transmission function of a gaseous uniform path are obtained. Different constraints involving the first and second order positive, first order negative and logarithmic moments are introduced together with a physical meaning whenever it is possible. Different sets of these constraints are considered to get maximum entropy estimates of the distributions functions: the Dirac, exponential, Gamma, inverse Gaussian and reciprocal inverse Gaussian k-distribution functions. Analytical formulas are provided for each of these distributions and for their associated transmission function, as a function of the mean and variance of the absorption coefficient. The methodology can be applied considering any spectral interval: narrow, wide, the full spectrum, continuous or not. Thus the resulting associated transmission and cumulative k-distribution functions can be utilized in the frame of a large variety of gas radiation models. Hence the k-moment method using the maximum entropy principle is assessed in the frame of the NBKM and full spectrum SLMB gas radiation models. A series of test cases implying comparisons with reference Line-by-Line results exhibits which maximum entropy k-distributions are likely to give the best estimations of narrow band or total emitted intensities, curves-of-growth of the total emission function and full spectrum cumulative k-distribution functions. In particular, the inverse Gaussian and Gamma k-distributions seem most of the time to perform very well.     

Extension of the weak-line approximation and application to correlated-k methods

Global climate models require accurate and rapid computation of the radiative transfer through the atmosphere. Correlated-k methods are often used. One of the approximations used in correlated-k models is the weak-line approximation. We introduce an approximation T_γ which reduces to the weak-line limit when optical depths are small, and captures the deviation from the weak-line limit as the extinction deviates from the weak-line limit. This approximation is constructed to match the first two moments of the gamma distribution to the k-distribution of the transmission. We compare the errors of the weak-line approximation with T_γ in the context of a water vapor spectrum. The extension T_γ is more accurate and converges more rapidly than the weak-line approximation.     

The effects of the choice of the k-interval number on radiative calculations

The selection of the number of k-interval is a foundation to correlated k-distribution method and the problem of how to do it still remains unsettled. It is pointed out by numerical computation in this work that choosing the number of k-interval is a major factor affecting accuracy and speed in radiative calculation. To increase the number of k-interval is an efficient method to improve the accuracy. However, it is found by this study that there exists a saturation of the accuracy to an increase of the number. The optimal rules on the number of k-interval choosing are proposed in the paper. Then, five versions on atmospheric absorption by gases appropriate for GCMs are given according to them.     

Application of approximations for joint cumulative k-distributions for mixtures to FSK radiation heat transfer in multi-component high temperature non-LTE plasmas

Approximations for joint cumulative k-distribution for mixtures are efficient for full spectrum k-distribution (FSK) computations. These approximations provide reduction of the database that is necessary to perform FSK computation when compared to the direct approach, which uses cumulative k-distributions computed from the spectrum of the mixture, and also less computational expensive when compared to techniques in which RTE's are required to be solved for each component of the mixture. The aim of the present paper is to extend the approximations for joint cumulative k-distributions for non-LTE media. For doing that, a FSK to non-LTE media formulation well-suited to be applied along with approximations for joint cumulative k-distributions is presented. The application of the proposed methodology is demonstrated by solving the radiation heat transfer in non-LTE high temperature plasmas composed of N, O, N₂, NO, N₂⁺ and mixtures of these species. The two more efficient approximations, that is, the superposition and multiplication are employed and analyzed.     

A database for the SLMB modeling of the full spectrum radiative properties of CO2

Development and application of a database for the Spectral-Line Moment-Based (SLMB) modeling of the full spectrum radiative properties of mixtures of carbon dioxide and nitrogen is presented. The critical issue of the definition of a reference thermophysical condition is addressed together with the suggestion of a coherent and precise methodology to derive parameters of the model for any other configuration. The database is built accordingly from the CDSD-1000 high temperature spectroscopic databank for gas and blackbody-weighting temperatures in the range [300; 2700 K]. Accuracy of both the modeling and the database is assessed through comparisons with LBL results in terms of full spectrum k-distributions and emission functions. Results obtained from the application of FSK correlations and the Leckner's formula are also provided for extended analyses.     

Radiative heat transfer between two concentric spheres separated by a two-phase mixture of non-gray gas and particles using the modified discrete-ordinates method

The radiative heat transfer between two concentric spheres separated by a two-phase mixture of non-gray gas and a cloud of particles is investigated by using the combined finite-volume and discrete-ordinates method, named modified discrete-ordinates method (MDOM), which integrates the radiative transfer equation (RTE) over a control volume and a control angle simultaneously like in the finite-volume method (FVM) and treats the angular derivative terms due to spherical geometry as the conventional discrete-ordinates method (DOM). The radiative properties involving non-gray gas and particle behavior are modeled by using the extended weighted sum of gray gases model (WSGGM) with particles. Mathematical formulation and final discretization equations for the RTE are introduced by considering the behavior of a two-phase mixture of non-gray gas and particles in a spherically symmetric concentric enclosure. The present approach is validated by comparing with the results of previous works including gray and non-gray radiative heat transfer. Finally, a detailed investigation of the radiative heat transfer with non-gray gases and/or a two-phase mixture is conducted to examine the dependence of the radiative heat transfer upon temperature ratio between inner and outer spherical enclosure, particle concentration, and particle temperature.     

An efficient diffusion approximation for 3D radiative transfer parameterization: application to cloudy atmospheres

The three-dimensional (3D) diffusion radiative transfer equation, which utilizes a four-term spherical harmonics expansion for the scattering phase function and intensity, has been efficiently solved by using the full multigrid numerical method. This approach can simulate the transfer of solar and thermal infrared radiation in inhomogeneous cloudy conditions with different boundary conditions and sharp boundary discontinuity. The correlated k-distribution method is used in this model for incorporation of the gaseous absorption in multiple-scattering atmospheres for the calculation of broadband fluxes and heating rates in the solar and infrared spectra. Comparison of the results computed from this approach with those computed from plane-parallel and 3D Monte Carlo models shows excellent agreement. This 3D radiative transfer approach is well suited for radiation parameterization involving 3D and inhomogeneous clouds in climate models.     

A study of the radiative forcing and global warming potentials of hydrofluorocarbons

We developed a new radiation parameterization of hydrofluorocarbons (HFCs), using the correlated k-distribution method and the high-resolution transmission molecular absorption (HITRAN) 2004 database. We examined the instantaneous and stratospheric adjusted radiative efficiencies of HFCs for clear-sky and all-sky conditions. We also calculated the radiative forcing of HFCs from preindustrial times to the present and for future scenarios given by the Intergovernmental Panel on Climate Change Special Report on Emission Scenarios (SRES, in short). Global warming potential and global temperature potential were then examined and compared on the basis of the calculated radiative efficiencies. Finally, we discuss surface temperature changes due to various HFC emissions.     

Numerical explanation for accurate radiative cooling rates resulting from the correlated k-distribution hypothesis

Numerical simulations are conducted to demonstrate that (a) correlation between spectral absorption at two levels decreases with increasing level distance; and (b) the contribution to the flux gained by one layer from another decreases rapidly with the layer separation. The combination of the two facts explains why the existence of poor correlation between distant layers produces insignificant radiative cooling rate error. Therefore, the overall accuracy of cooling rate calculations under the correlated k-distribution hypothesis is high.     

A k-distribution-based radiation code and its computational optimization for an atmospheric general circulation model

The gas absorption process scheme in the broadband radiative transfer code “mstrn8”, which is used to calculate atmospheric radiative transfer efficiently in a general circulation model, is improved. Three major improvements are made. The first is an update of the database of line absorption parameters and the continuum absorption model. The second is a change to the definition of the selection rule for gas absorption used to choose which absorption bands to include. The last is an upgrade of the optimization method used to decrease the number of quadrature points used for numerical integration in the correlated k-distribution approach, thereby realizing higher computational efficiency without losing accuracy. The new radiation package termed “mstrnX” computes radiation fluxes and heating rates with errors less than 0.6 W/m² and 0.3 K/day, respectively, through the troposphere and the lower stratosphere for any standard AFGL atmospheres. A serious cold bias problem of an atmospheric general circulation model using the ancestor code “mstrn8” is almost solved by the upgrade to “mstrnX”.     

Random sets,q-distributions and quantum groups

We have shown that the non-extensivity of classical set theory is related to unitary quantum groups. Using this non-extensivity property, we define a q-distribution, a binomial q-distribution and a Poisson q-distribution.     

Narrow-band and full-spectrum k-distributions for radiative heat transfer—correlated-k vs. scaling approximation

The correlated-k and scaled-k distribution methods for radiative heat transfer in molecular gases are developed based on precise mathematical principles, for both narrow band and full spectrum models. Their differences and commonalities are high-lighted and discussed. Applications to narrow spectral bands of nonhomogeneous gases show both methods to be about equally accurate. For full-spectrum calculations, on the other hand, the scaled-k distribution consistently outperforms the correlated-k model.