Full-spectrum K-distribution Method Research Articles

Spectral effects of radiating gases and soot on an experimental aero-engine combustion chamber using full-spectrum k-distribution method

Radiative heat transfer is the main mode of heat transfer and has been proven to be important in the aero-engine combustion. Those radiating gases and soot resulted from combustion may display very strong spectral, or “nongray” behavior, which is difficult to both characterize and evaluate. In this work, the full-spectrum k-distribution (FSK) method is employed to resolve the spectral behavior of both radiating gases and soot during combustion in an experimental aero-engine. Different FSK-based spectral treatments are combined to illustrate how the spectral behavior affects the aero-engine combustion. Those weighted-sum-of-gray-gases (WSGG) results previously done by the authors (Wang, et al. 2024) are also partially compared to the FSK results in this work. Results show that different FSK-based spectral treatments result in small differences for temperature distribution of the flow field. However, since different spectral models are found to impose a considerable effect on radiative heat transfer rate, it is inferred that spectral behavior may be deeply coupled with the generation of pollutants. Results also show that the wall temperature of the combustor liner is highly affected by the spectral model: ignoration of soot radiation underestimates the wall temperature while assuming soot to be gray overestimates the wall temperature. Therefore, high-fidelity spectral models are recommended during the modelling of aero-engine combustion.

Evaluation of radiative absorption effect to estimate mean radiant temperature in environments with high water vapor concentration such as in a sauna

The importance of modeling the radiative absorption in an environment with a high concentration of water vapor, such as inside a sauna, was studied through numerical analyses and fieldwork. Radiative heat transfer analysis using the discrete ordinate method (DOM) was performed using OpenFOAM software to evaluate the influence of thermal radiation inside a sauna. In a high-temperature environment, such as a sauna, the effect of radiation heat transfer in the surrounding medium gradually increases. Therefore, the effects of the medium cannot be ignored. A full-spectrum k-distribution (FSK) was adopted to model the radiative absorption of the medium, which was composed of carbon dioxide and water vapor. The mean radiant temperature (MRT) was also determined using numerical simulations to compare the results with those measured directly using a globe thermometer. The measured MRT was lower than the air temperature because of radiative cooling by the surroundings. To evaluate the quality of the radiative absorption model, the MRT obtained using the transparent gas and FSK methods were compared. When the distance from the cooling front window was approximately 4 m, the FSK method improved the prediction of MRT by approximately 1 °C when compared with the transparent gas method, which is the conventional method. The FSK method improved the MRT prediction from the conventional method by approximately 27% in terms of the Root Mean Square Error. This result indicates the importance of modeling the MRT by considering radiative absorption in the presence of high water vapor pressure.

Research on the particle scale gas and solid non-gray thermal radiation transfer

Gas radiation emission is important for particle heat transfer in combustion scenarios where gas products are radiative participating media. However, it is usually neglected in particle scale simulations, and its magnitude and variation with void fraction are unclear. A particle scale thermal radiation model is developed to investigate the radiation heat transfer mechanism between the gas and solid phase. The radiative properties are modeled by the full spectrum k-distribution method, and the radiative transfer equation is solved by the discrete ordinates method. Results show that gas and solid radiative heat fluxes are about 1300 W/m2 and 4500 W/m2 for the dense and dilute conditions respectively, and decrease exponentially with the solid volume fraction. The local environmental temperature model is evaluated and shown to significantly over-estimate the gas radiation. Variations of the radiative exchange between the gas and solid with temperature and species fractions are also simulated and analyzed.

Flow and radiation modeling over a Martian entry vehicle

The high-temperature shock layer around re-entry vehicles strongly emits radiation through atomic lines and molecular bands. Radiative heat transfer strongly affects boundary layer flow and heat transfer. A novel spectral model based on k-distribution is implemented and coupled with a hypersonic flow solver in OpenFOAM to simulate heat transfer over a Martian entry vehicle. The Navier-Stokes equations with finite rate chemistry are used to model hypersonic chemically reacting flows. The convective terms in the governing equations are treated with various flux schemes (HLL, AUSM+, Kurganov, and Tadmor). The non-gray radiative properties of shock layer gases were modeled with the Emission-weighted Full Spectrum k-distribution (EFSK) method. The Radiative Transfer Equation (RTE) is solved with the P1 radiative solver to calculate the wall heat flux and radiative heat source term. At a high Mach number, the radiative heat flux is dominant compared to the convective heat flux. The numerical results for Mach number 26 are compared with published data and are found to be in good agreement. All the flux schemes provided results in close agreement, with minor differences, especially near the wall for the AUSM+ flux scheme.

Radiative heat transfer calculations using full spectrum k-distribution method for benchmark test cases

In the present work, the full spectrum k-distribution method (FSK) has been adopted to calculate the radiative transfer in the presence of participating gaseous medium within an enclosure. The radiative properties of the medium is obtained from the HITEMP-2010 database. Further, the radiative properties have been assembled into a monotonically increasing function using the FSK. Moreover, a look-up table has been developed for these properties at different thermodynamic states and a multi-dimensional linear interpolation technique for unavailable thermodynamic states of gases. Furthermore, the FSK method is extended for mixture of gases using different mixing models such as superposition, multiplication and hybrid models. The radiation transfer equation (RTE) is solved by finite angle method to calculate the wall heat fluxes and the divergence of radiative heat flux for various test cases in the categories of homogeneous and non-homogeneous medium having different conditions of temperature and mole-fraction. The multiplication mixing model produces most accurate results among the mixture models used here. The results obtained from FSK has been validated against benchmark solution of line by line method (LBL). The FSK method has been also successfully applied to non-homogeneous gaseous medium for single gas or mixture of gases with almost LBL accuracy at extremely less computational cost.

Radiative heat transfer calculation for mixture of gases using full spectrum k-distribution method

The full spectrum k-distribution method is used to obtain radiative heat flux and divergence of radiative heat flux for two test cases, containing mixture of CO 2 and H 2 O at different concentration and temperature keeping pressure constant. The k-distribution for mixture of gases is obtained from individual gas k-distribution using three different mixing models, viz., superposition, multiplication and hybrid model. Further, the radiative transfer equation (RTE) is solved by the finite volume discrete ordinate method (FVDOM) to obtain the radiative flux and the radiation source term. The results obtained were compared with the FSK from spectral addition and LBL method. The multiplication mixing model provides better accuracy compared to other mixing models considered in the present study.

Benchmark Test Cases for Non-Gray Radiative Heat Transfer Calculation using FSK Look-Up Table

Three test cases in the categories of homogeneous non-isothermal, non-homogeneous isothermal and non-homogeneous non-isothermal have been developed to validate the two-dimensional interpolation technique for calculation of non-gray radiative heat flux on the walls of the system. The participating gases H 2 O and CO 2 of different mole fractions and temperatures are considered in different zones of the test cases. HITEMP-2010 database has been used to calculate the absorption coefficients of H 2 O and CO 2 at different mole fractions and temperatures. Further, the random variation of absorption coefficients with spectrum has been reordered in smooth monotonically increasing smooth function using full spectrum k-distribution method (FSK). A look-up table is developed for different mole fractions and temperatures of gases H 2 O and CO 2. The calculation of absorption coefficients at thermodynamic states other than look up table has been performed using two dimensional interpolation techniques. The geometry of test cases have been divided into three zones whose conditions on the first and last zones are same as available in look-up table while interpolation is used for the middle zone. The radiative transfer equation is solved numerically by finite volume discrete ordinate method (FVDOM). The results have been compared with FSK method and have been found that interpolation techniques are giving satisfactory results with extremely less computational resource and time.

Gray and non-gray simulations of the combined conduction and radiation heat transfer in a complex enclosure utilizing FSK method considering the scattering influences

This research simulates the conduction-radiation problem in a complex enclosure for the gray/ non-gray assumptions with the presence of scattering influences to improve energy efficiency of thermal systems. The enclosure bottom wall includes an inclined recess with two diagonal walls. To convert these diagonal surfaces to the regular rectangular surfaces in Cartesian coordinates, the Blocked-off method is used. The air containing 10% CO2 and 20% H2O is regarded as participating medium in the enclosure. The absorption coefficient of this mixture is obtained across the spectrum using the information of HITRAN2008 database. Simulating the non-gray medium is done by employing the full-spectrum k-distribution (FSK) approach, whilst the gray medium simulation is performed utilizing the Planck mean absorption coefficient. The finite volume and discrete ordinates methods are respectively applied to calculate the energy and radiation transport equations. Impacts of surface emissivity on conductive, radiative and total heat fluxes are examined in details.

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.

Radiative heat transfer calculation of oxy-fuel flames using the multi-group full-spectrum K-distribution method

Oxy-fuel combustion has drawn much attention as a promising CO2 capture and storage/sequestration technology. Radiative heat transfer of oxy-fuel flames is more important than that of conventional flames owing to higher concentrations of radiating gases. The Full-Spectrum Correlated K-distribution (FSCK) methods with original correlated-K scheme (Modest and Zhang, 2002), the improved one (Cai and Modest, 2014), and the rank correlated one (Solovjov and Webb, 2018) provide obvious errors in the low-temperature regions of dry and wet oxy-fuel flames. To improve the accuracy of FSCK, the accuracy of the Multi-Group FSK (MGFSK) method with the three correlated-K schemes and two grouping strategies based on the scaling function and intervals of comonotonicity are investigated in two 1D, a 2D and a 3D oxy-fuel flames. The results show that MGFSK with different correlated-K schemes and grouping strategies demonstrate better accuracies than corresponding FSCK for both dry and wet oxy-fuel flames. MGFSK reduce the dependence of FSCK on the reference (or Planck) temperature and flame scale. The volume-averaged mole fractions of the maximum and minimum temperatures are recommend as the high- and low-temperature grouping states respectively. Dividing the spectral absorption coefficients into 2 to 4 groups is a good compromise between accuracy and efficiency.

Effects of the K-value solution schemes on radiation heat transfer modelling in oxy-fuel flames using the full-spectrum correlated K-distribution method

Radiation heat transfer in oxy-fuel flames is more important than in conventional fuel-air flames. The Full-Spectrum Correlated K-distribution methods (FSCK) with the original correlated-K solution scheme (Modest and Zhang, 2002) and a newly proposed one (Cai and Modest, 2014), and the Rank Correlated Full-Spectrum K-distribution method (RC-FSK) are used in radiative calculations of oxy-fuel flames. Twelve one-dimensional flames, including fuel-air, dry and wet oxy-fuel flames with various length scales, as well as a two-dimensional dry oxy-fuel flame are studied. The results show that the reference temperature has a non-negligible impact on the accuracy of original scheme and the emission-weighted temperature leads to a good accuracy. The accuracy of the new scheme is almost unaffected by the reference temperature, except for small-scale dry oxy-fuel flames. The error of the new scheme is mainly in the low-temperature region, and its accuracy depends on the length scale of computational domain. A hybrid correlated-K scheme using either the original or the new scheme according to local temperature is proposed. It combines the advantages of two schemes in different temperature regions, and is independent of the reference temperature. In addition, the RC-FSK demonstrates almost the same accuracy as FSCK with the hybrid scheme when the Planck temperature is set equal to the reference temperature.

A grouping strategy for the multi-group full-spectrum K-distribution method in gas mixtures

The Multi-Group Full-Spectrum K-distribution method (MGFSK) is a promising method for efficient and accurate thermal radiation calculations in participating media with strong temperature inhomogeneity. The current MGFSK method cannot be used for modeling radiative transfer in gas mixtures since it is applicable only to a single radiating gas. To make MGFSK applicable to mixtures without losing efficiency, a new grouping strategy for gas mixtures is proposed based on a scaling function of the mixture. Several one-dimensional two-layer systems with large temperature gradient and a two-dimensional oxy-fuel flame are simulated to demonstrate the accuracy of the modified MGFSK method (MGFSK-M). The simulation results are compared with those of the Line-by-Line model, the Full-Spectrum Correlated K-distribution method (FSCK) and the Rank Correlated Full-Spectrum K-distribution method (RC-FSK). MGFSK-M significantly outperforms both FSCK and RC-FSK in the two-layer systems. For the oxy-fuel flame, RC-FSK and MGFSK-M offer similar accuracy and both perform better than FSCK.

Investigation on the effect of fractal soot aggregation on radiative transfer in homogeneous gas-soot mixture using full-spectrum k-distribution method

Investigation on the effect of fractal soot aggregation on radiative transfer in homogeneous gas-soot mixture using full-spectrum k-distribution method

The rank correlated FSK model for prediction of gas radiation in non-uniform media, and its relationship to the rank correlated SLW model

Following previous theoretical development based on the assumption of a rank correlated spectrum, the Rank Correlated Full Spectrum k-distribution (RC-FSK) method is proposed. The method proves advantageous in modeling radiation transfer in high temperature gases in non-uniform media in two important ways. First, and perhaps most importantly, the method requires no specification of a reference gas thermodynamic state. Second, the spectral construction of the RC-FSK model is simpler than original correlated FSK models, requiring only two cumulative k-distributions. Further, although not exhaustive, example problems presented here suggest that the method may also yield improved accuracy relative to prior methods, and may exhibit less sensitivity to the blackbody source temperature used in the model predictions. This paper outlines the theoretical development of the RC-FSK method, comparing the spectral construction with prior correlated spectrum FSK method formulations. Further the RC-FSK model's relationship to the Rank Correlated Spectral Line Weighted-sum-of-gray-gases (RC-SLW) model is defined. The work presents predictions using the Rank Correlated FSK method and previous FSK methods in three different example problems. Line-by-line benchmark predictions are used to assess the accuracy.

Effects of concentration of participating media on turbulent natural convection in cubic cavity

To use and understand the large-scale natural convection in the building and systems, heat transfer analysis of the natural convection with radiation is important. The radiation effect is influenced by concentration of the participating media in the considered system. In this study, an effects of concentration of participating media on turbulent natural convection inside a cubic cavity were investigated. The turbulence effect was modeled by large eddy simulation. Further, the radiative heat transfer was calculated by the radiation element method using the ray emission model. In addition, the gas radiation was modeled by the full-spectrum k-distribution method. By varying the partial pressures of H2O and CO2, the effect of the participating media were evaluated. From the calculation, gas radiation effects affected the flow instability and restrained the temperature stratification when the concentration of the participating media was higher. However, the convective heat transfer have not been affected by the significant flow instability by the radiation effects.

Non-dimensional wavenumber in full-spectrum k-distribution computations with or without a reference state

In the full spectrum k-distribution method, the k-distributions are Planck function weighted. These k-distributions depend on temperature, which varies with location in nonisothermal media. Consequently, in order to apply the method to nonuniform media, k-distributions evaluated at different local temperatures are correlated by a k-distribution evaluated at an arbitrarily chosen reference blackbody temperature. On the other hand, in the original band method, the k-distributions are directly related to spectral locations. Such k-distributions can replace Planck function weighted k-distributions computed at a reference temperature. This approach was employed by Maurente et al. [1] in order to deal with gas mixtures in non-equilibrium media. In this paper the formulation is presented for the equilibrium case. The proposed formulation simplifies the application of the FSK method to nonuniform media, and brings new directions to k-distribution issues. It can be employed with a reference state, but not a reference temperature, with reference state and temperature, as in the traditional FSK, and without any reference state, being in this case the continuous limit of the recently developed Rank Correlated SLW [2]. Results were obtained for test cases considering one-dimensional, nonuniform media, constituted by CO2 and inert gases, and compared to line-by-line solutions.

Improvement of full-spectrum k-distribution method using quadrature transformation

The full-spectrum k-distribution (FSK) method can provide great accuracy combined with outstanding numerical efficiency for the evaluation of radiative transfer in nonhomogeneous nongray media. However, k-distributions cover several orders of magnitude, making their efficient numerical integration difficult. Different quadrature schemes give different levels of accuracy as compared to exact values unless a large number of quadrature points is employed. To improve efficient implementation of the FSK method, an integration method with quadrature transformation is investigated in this work. Several radiative heat transfer calculations were carried out to test the effects of quadrature transformation on accuracy at different thermodynamic states. Results show that for most thermodynamic states employing the transformation for the integration of k-values can greatly reduce the number of required quadrature points while maintaining good accuracy. Based on this, the FSK look-up table previously developed by the authors was updated to include k-values corresponding to the new quadrature points with transformation. Two scaled flames are used to verify both the accuracy and efficiency of the updated table.

Coupled Thermal Radiation and Mixed Convection Step Flow of Nongray Gas

The main goal of this paper is to analyze the thermal and hydrodynamic behaviors of laminar mixed convection flow of a nongray radiating gas over an inclined step in an inclined duct. The fluid is considered an air mixture with 10% CO2 and 20% H2O mole fractions, which is treated as homogeneous, absorbing, emitting, and nonscattering medium. The full-spectrum k-distribution (FSK) method is used to handle the nongray part of the problem, while the radiative transfer equation (RTE) is solved using the discrete ordinate method (DOM). In addition, the results are obtained for different medium assumptions such as pure mixed convection and gray medium to compare with the nongray calculations as a real case. The results show that in many cases, neglecting the radiation part in computations and also use of gray simulations are not acceptable and lead to considerable errors, especially at high values of the Grashof number in mixed convection flow.

A full-spectrum k-distribution look-up table for radiative transfer in nonhomogeneous gaseous media

A full-spectrum k-distribution (FSK) look-up table has been constructed for gas mixtures within a certain range of thermodynamic states for three species, i.e., CO2, H2O and CO. The k-distribution of a mixture is assembled directly from the summation of the linear absorption coefficients of three species. The systematic approach to generate the table, including the generation of the pressure-based absorption coefficient and the generation of the k-distribution, is discussed. To efficiently obtain accurate k-values for arbitrary thermodynamic states from tabulated values, a 6-D linear interpolation method is employed. A large number of radiative heat transfer calculations have been carried out to test the accuracy of the FSK look-up table. Results show that, using the FSK look-up table can provide excellent accuracy compared to the exact results. Without the time-consuming process of assembling k-distribution from individual species plus mixing, using the FSK look-up table can save considerable computational cost. To evaluate the accuracy as well as the efficiency of the FSK look-up table, radiative heat transfer via a scaled Sandia D Flame is calculated to compare the CPU execution time using the FSK method based on the narrow-band database, correlations, and the look-up table. Results show that the FSK look-up table can provide a computationally cheap alternative without much sacrifice in accuracy.

A Multi-Spectral Energy Bundle method for efficient Monte Carlo radiation heat transfer computations in participating media

The Monte Carlo method is a powerful technique for simulating radiation heat transfer processes. The method can easily deal with irregular three-dimensional geometries, anisotropic scattering of radiation and other complexities, which are usually difficult to be tackled with other numerical techniques. On the other hand, Monte Carlo computations are highly time consuming. In addition, accurate radiation heat transfer solutions require that spectral properties be taken into account in a detailed manner, being the line-by-line integration the more accurate but also more computationally expensive technique. Due to these reasons, Monte Carlo computations that accurately consider the spectral properties of radiation are impracticable for several cases of interest in engineering. Therefore reducing the computer time associated to Monte Carlo simulations is highly desirable. This paper presents the Multi-Spectral Energy Bundle method, which reduces the computational time of Monte Carlo simulations in which the spectral properties are accurately taken into account. The method can be applied for line-by-line computations as well as along with spectral models. Here, the proposed method is applied with the accurate full-spectrum k-distribution method, and the obtained results are analyzed.