Full-spectrum K-distributions Research Articles

Radiative Heat Transfer in Turbulent Combustion 2021 Max Jakob Memorial Award Paper

Abstract In many important combustion applications, heat transfer is dominated by thermal radiation from combustion gases and soot. Thermal radiation from combustion gases is extremely complicated, and accurate and efficient predictions are only now becoming possible with the use of accurate global methods, such as full-spectrum k-distributions, and with state-of-the-art line-by-line accurate Monte Carlo methods. The coupling between turbulence and radiation can more than double the radiative loss from a flame, while making theoretical predictions vastly more complicated. This paper is an embellished version of the 2021 Max Jakob Award lecture: Radiative properties and computational methods will be briefly discussed, and several examples of turbulent reacting flows, an oxy-fuel furnace, and high-pressure fuel sprays in combustion engines will be presented. Thermal radiation can also be used as an optical diagnostic tool to determine temperature and concentration distributions, which will be briefly discussed.

Comparison and refinement of the various full-spectrum k-distribution and spectral line weighted-sum-of-gray-gases models for nonhomogeneous media

Both the full-spectrum k-distribution (FSK) method and the spectral line weighted-sum-of-gray-gases (SLW) method are global spectral methods, with the former based on a spectral reordering concept, while the latter uses spectral binning. Both can provide excellent accuracy with outstanding numerical efficiency to model radiative heat transfer in combustion gases. In the present paper we aim to complete the development of both methods, and to tie them together, pointing out their commonalities and differences of their nonhomogeneous media extensions, employing either the correlated or scaled absorption coefficient assumption. Results from the complete set of plausible implementations are discussed and compared in detail for several radiative heat transfer calculations carried out in both 1D slabs and a real combustion field. The results show that emission is an important criterion to examine the accuracy of global methods applied to nonhomogeneous media, i.e., those that preserve emission generally give more accurate results than those that do not. It was also found that, while proper choice of the reference temperature required by all methods is important, the recommended methods appear to be only weakly dependent on that choice. Finally, equivalent SLW schemes are generally somewhat less accurate than their FSK counterparts due to their low-order spectral integration scheme; and this may be exacerbated if full-spectrum k-distributions are determined by mixing of values from individual species, as appears to be the preferred approach by SLW users to date.

Efficient full-spectrum correlated-k-distribution look-up table

To improve the efficiency of full-spectrum k-distribution look-up tables a new scheme has been devised to store correlated k-values. Unlike the previous look-up table developed by the authors [9,10], from which full-spectrum k-distributions must be determined with multi-step interpolations, the new table stores correlated values, which can be directly retrieved from the new table avoiding several calculations and interpolations, thus saving considerable CPU time during radiative calculations. The same species as in the previous table, i.e., CO2, H2O, CO and soot, are included in the new table, for which the construction details are illustrated. Previous and new tables as well as their implementations are compared in this work. Calculations of radiative heat sources for two scaled flames are employed to validate the new table. Results show that the new table gives results of similar accuracy but at considerably cheaper computational cost than the previous table for both gas mixtures and gas-soot mixtures.

Full-spectrum k-distribution look-up table for nonhomogeneous gas–soot mixtures

Full-spectrum k-distribution (FSK) look-up tables provide great accuracy combined with outstanding numerical efficiency for the evaluation of radiative transfer in nonhomogeneous gaseous media. However, previously published tables cannot be used for gas–soot mixtures that are found in most combustion scenarios since it is impossible to assemble k-distributions for a gas mixed with nongray absorbing particles from gas-only full-spectrum k-distributions. Consequently, a new FSK look-up table has been constructed by optimizing the previous table recently published by the authors and then adding one soot volume fraction to this optimized table. Two steps comprise the optimization scheme: (1) direct calculation of the nongray stretching factors (a-values) using the k-distributions (k-values) rather than tabulating them; (2) deletion of unnecessary mole fractions at many thermodynamic states. Results show that after optimization, the size of the new table is reduced from 5GB (including the k-values and the a-values for gases only) to 3.2GB (including the k-values for both gases and soot) while both accuracy and efficiency remain the same. Two scaled flames are used to validate the new table. It is shown that the new table gives results of excellent accuracy for those benchmark results together with cheap computational cost for both gas mixtures and gas–soot mixtures.

High fidelity radiative heat transfer models for high-pressure laminar hydrogen–air diffusion flames

Radiative heat transfer is studied numerically for high-pressure laminar H2–air jet diffusion flames, with pressure ranging from 1 to 30 bar. Water vapour is assumed to be the only radiatively participating species. Two different radiation models are employed, the first being the full spectrum k-distribution model together with conventional Radiative Transfer Equation (RTE) solvers. Narrowband k-distributions of water vapour are calculated and databased from the HITEMP 2010 database, which claims to retain accuracy up to 4000 K. The full-spectrum k-distributions are assembled from their narrowband counterparts to yield high accuracy with little additional computational cost. The RTE is solved using various spherical harmonics methods, such as P1, simplified P3 (SP3) and simplified P5 (SP5). The resulting partial differential equations as well as other transport equations in the laminar diffusion flames are discretized with the finite-volume method in OpenFOAM®. The second radiation model is a Photon Monte Carlo (PMC) method coupled with a line-by-line spectral model. The PMC absorption coefficient database is derived from the same spectroscopy database as the k-distribution methods. A time blending scheme is used to reduce PMC calculations at each time step. Differential diffusion effects, which are important in laminar hydrogen flames, are also included in the scalar transport equations. It was found that the optically thin approximation overpredicts radiative heat loss at elevated pressures. Peak flame temperature is less affected by radiation because of faster chemical reactions at high pressures. Significant cooling effects are observed at downstream locations. As pressure increases, the performance of RTE models starts to deviate due to increased optical thickness. SPN models perform only marginally better than P1 because P1 is adequate except at very high pressure.

Comparisons of Radiative Heat Transfer Calculations in a Jet Diffusion Flame Using Spherical Harmonics and k-Distributions

A new nongray radiation modeling library for combustion gases has been implemented in OpenFOAM. The spectral models for single species include gray, correlation tables and full spectrum k-distributions (FSK) assembled from a narrow-band database. Mixing models for k-distributions include the multiplication and uncorrelated mixture models. Radiative transfer equation solvers for the library include spherical harmonics such as P1, P3, SP3 and SP5 as well as the optically thin approximation. The performance of the different solution methods is compared for accuracy and speed as a tool for future model strategy selection.

On the modeling of radiative heat transfer in laboratory-scale pool fires

Two synthetic 30cm-diameter pool fires, generated from experimental data, are used to assess different radiative properties models. The first pool fire involves toluene while the second one is generated by reducing the soot volume fraction by one order of magnitude in order to study the effects of the optical thickness. Turbulence–Radiation Interactions (TRIs) are accounted for in a simplified manner, by considering the Optically Thin Fluctuation Approximation (OTFA) and by assuming that the emission soot volume fraction and the temperature are statistically independent. Decoupled radiative computations are carried out with the Finite Volume Method (FVM) by using the Statistical Narrow Band Correlated k method (SNBCK) and the Full Spectrum Correlated k model (FSCK). Full spectrum (FS) k-distributions are assembled from 43 and 367 narrow band (NB) databases. It is found that the FSCK can be used with confidence in such problems with a 5-point Gauss–Legendre quadrature scheme and k-distributions assembled from the 43NB database, allowing a reduction in Central Processing Unit (CPU) time by a factor of 14 as compared to the 43 band SNBCK. Finally the effects of the OTFA and of the correlation between the emission soot volume fraction and temperature are analyzed and quantified.

Multiscale Part-Spectrum k-Distribution Database for Atomic Radiation in Hypersonic Nonequilibrium Flows

An accurate and compact part-spectrum k-distribution database has been developed for the two most important radiating species N and O encountered in hypersonic nonequilibrium flows. The database allows users to calculate the desired full-spectrum k-distributions through look-up and interpolation, providing an efficient means to perform radiative transfer calculations. A detailed methodology of the k-distribution data generation is presented. An optimized Gauss quadrature scheme is implemented for reducing the size of the database. The accuracy of the database is determined by comparing part-spectrum emissivities with those obtained from line-by-line calculations. The application of the database to construct full-spectrum k-distributions at arbitrary gas states is discussed. Heat transfer results for the stagnation line of the Stardust vehicle are discussed and CPU-time studies are presented, demonstrating the accuracy and efficiency of the k-distribution database.

Assessment of radiative property models in non-gray sooting media

The radiative transfer equation (RTE) is solved by the Finite Volume Method (FVM) in 1D and 2D axisymmetric configurations containing mixtures of water vapor, carbon dioxide and soot. Several radiative property models, namely simple gray models, the Statistical Narrow Band Correlated- k (SNBCK) model, the Weighted-Sum-of-Gray-Gases (WSGG) model, the Full-Spectrum Correlated- k (FSCK) model, the Spectral-Line-Based Weighted-Sum-of-Gray-Gases (SLW) model, a Gray-Narrow-Band (GNB) model and a Gray-Wide-Band (GWB) model, are assessed by comparison with reference calculations based on the Ray Tracing (RT) method coupled with the Statistical Narrow Band (SNB) model. Different treatments of overlapping bands are implemented with the SNBCK model, namely the correlated approach, the uncorrelated approach, and the mixing schemes of Modest and Riazzi (J. Quant. Spectros. Rad. Trans. 90 (2005) 169–189) and of Solovjov and Webb (J. Quant. Spectros. Rad. Trans. 65 (2000) 655–672) (Superposition and Multiplication). The Full Spectrum (FS) k-distributions required to run both FSCK and SLW are assembled from the same narrow band (NB) database as for the SNB and the SNBCK models using either the mixing model of Modest and Riazzi or the Multiplication scheme. The correlated SNBCK is the most accurate model but is too time consuming for engineering applications. The computational efficiency of the SNBCK can be considerably improved without altering the quality of the solutions by using the gas mixing scheme of Modest and Riazzi. Numerical results show that, in this case, good accuracy for spectrally-integrated quantities is obtained by considering only a 2-point Gauss–Legendre quadrature scheme. The FSCK and the SLW, using FS k-distributions assembled with the Modest and Riazzi scheme, are found to be the best compromise in terms of accuracy and computational requirements. It is found that FSCK predictions remain satisfactory by reducing the number of quadrature points of the Gauss–Legendre scheme up to seven.

A database for the SLMB modeling of the full spectrum radiative properties of CO 2

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.

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.

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.

High-accuracy, compact database of narrow-band k-distributions for water vapor and carbon dioxide

An accurate and compact narrow-band k-distribution database has been constructed for water vapor and carbon dioxide from the HITEMP and CDSD-1000 spectroscopic databases. The systematic approach of k-distribution data generation and compaction, storage optimization and interpolation is discussed in this paper. The new database enables the user to obtain narrow-band, part-spectrum and full-spectrum k-distributions for inhomogeneous gas mixtures.

Assembly of full-spectrum k-distributions from a narrow-band database; effects of mixing gases, gases and nongray absorbing particles, and mixtures with nongray scatterers in nongray enclosures

Full-spectrum k-distributions provide great accuracy combined with outstanding numerical efficiency for the evaluation of radiative transfer in absorbing-emitting molecular gases, but they do have several shortcomings: (1) It is difficult to assemble k-distributions for gas mixtures from precalculated full-spectrum k-distributions of individual gas species (i.e., without calculating the mixture k-distribution directly from the HITRAN/HITEMP database), (2) it is impossible to assemble k-distributions for a gas mixed with nongray absorbing particles (such as soot) from gas-only full-spectrum k-distributions, and (3) like all global models, full-spectrum k-distributions cannot accommodate nongray scattering behavior and/or nongray wall reflectances. In the present paper we show how these restrictions can be relaxed by (1) assembling full-spectrum k-distributions for a gas mixture from a narrow-band k-distribution database created for individual gas species, (2) by assembling gas and nongray absorbing particle mixture full-spectrum k-distributions from the same narrow-band database, and finally (3) by showing how a group of part-spectrum k-distributions can be generated from the same database to accommodate nongray scattering and nongray walls.

Assembly of full-spectrum k-distributions from a narrow-band database; effects of mixing gases, gases and nongray absorbing particles, and mixtures with nongray scatterers in nongray enclosures

Full-spectrum k-distributions provide great accuracy combined with outstanding numerical efficiency for the evaluation of radiative transfer in absorbing-emitting molecular gases, but they do have several shortcomings: (1) It is difficult to assemble k-distributions for gas mixtures from precalculated full-spectrum k-distributions of individual gas species (i.e., without calculating the mixture k-distribution directly from the HITRAN/HITEMP database), (2) it is impossible to assemble k-distributions for a gas mixed with nongray absorbing particles (such as soot) from gas-only full-spectrum k-distributions, and (3) like all global models, full-spectrum k-distributions cannot accommodate nongray scattering behavior and/or nongray wall reflectances. In the present paper we show how these restrictions can be relaxed by (1) assembling full-spectrum k-distributions for a gas mixture from a narrow-band k-distribution database created for individual gas species, (2) by assembling gas and nongray absorbing particle mixture full-spectrum k-distributions from the same narrow-band database, and finally (3) by showing how a group of part-spectrum k-distributions can be generated from the same database to accommodate nongray scattering and nongray walls.

Full spectrum k-distribution correlations for CO 2 from the CDSD-1000 spectroscopic databank

Radiative heat transfer through molecular gases is a very important mode of energy transfer when dealing with combustion systems or modelling atmospheric processes. The most accurate radiative transfer results are obtained through line-by-line (LBL) calculations, which require, however, a prohibitively large computational effort. Consequently a number of global and band models have been proposed. Presently, the most accurate global models are the Full-Spectrum k-distributions (FSK) by Modest and coworkers [1, 2]. Earlier, somewhat less refined, related methods include the Spectral-Line-Based WeightedSum-of-Gray-Gases [3–5] or SLW and Absorption-Distribution-Function(ADF) [6, 7] models. All of these methods are orders-of-magnitude more efficient than LBL calculations, and they all use full-spectrum kdistributions, which, in general, need to be calculated from high-resolution databases, such as HITRAN [8], HITEMP [9] or —for CO2— the new CDSD-1000 [10, 11]. Both the SLW and ADF methods use simplified k-distributions reducing them to step-functions (gray gases), whereas the full-spectrum k-distribution method uses Gaussian quadrature to integrate over g-space resulting in better accuracy. Assembling such k-distributions is a rather tedious task and, therefore, to make simple engineering calculations feasible, Denison and Webb [12, 13] have proposed several simple full-spectrum k-distribution correlations for CO2 and H2O, based on the outdated HITRAN92 [8] database, combined with some extrapolations of their own for high temperatures. Recently, Zhang and Modest [14] provided an updated correlation for CO2 based on the newer HITEMP database which is purported to be accurate to at least 1000K. Unfortunately, it appears that the HITEMP database shows some erroneous behavior for temperatures beyond 1200K, as seen by comparison with experimental data [10,15]. The new CDSD-1000 databank [10,11], on the other hand, follows experimental data much more closely. Thus, it is the purpose of this note to provide a simple engineering correlation for full-spectrum k-distributions evaluated from the CDSD-1000 databank. ∗To whom all correspondence should be addressed. Fax: (814)863-8682; Email: mfm6@psu.edu

Scalable Multi-Group Full-Spectrum Correlated-k Distributions for Radiative Transfer Calculations

A new full-spectrum k-distribution method has been developed, in which spectral locations are sorted into M spectral groups, according to their absorption coefficient dependence on (partial) pressure and temperature. Calculating correlated-k full-spectrum k-distributions for each of the M groups, LBL accuracy can be obtained with M⩽32. Database values have been assembled for CO2 mixtures at atmospheric pressure. The method is fully scalable, i.e., spectral groups from the database can be combined to obtain coarser group models M=1,2,4,… for greater numerical efficiency (accompanied by slight loss in accuracy).

Full-Spectrum k-Distribution Correlations for Carbon Dioxide Mixtures

Full-spectrum k-distributions (or absorption-line blackbody distribution functions) have been calculated for CO 2 from the new HITEMP database for a large number of gas temperatures, and an approximate mathematical correlation is presented. This correlation supersedes the older one by Denison and Webb, which was based on the HITRAN92 database. Comparison between the correlations shows that, although giving excellent results for temperatures below 1000 K, the older correlation seriously underpredicts radiative losses at high temperatures

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.