K-distribution Method Research Articles

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.

An exact formulation of k-distribution methods in non-uniform gaseous media and its approximate treatment within the Multi-Spectral framework

The main restriction of k-distribution approaches for applications in radiative heat transfer in gaseous media arises from the use of a scaling or correlation assumption to treat non-uniform situations. It is shown that those cases can be handled exactly by using a multidimensional k-distribution that addresses the problem of spectral correlations without using any simplifying assumptions. Nevertheless, the approach cannot be suggested for engineering applications due to its computational cost. Accordingly, a more efficient method, based on the so-called Multi-Spectral Framework, is proposed to approximate the previous exact formulation. The model is assessed against reference LBL calculations and shown to outperform usual k-distribution approaches for radiative heat transfer in non-uniform media.

A database of narrow-band parameters for fuels commonly encountered in fire applications

The objective of this paper is to derive narrow band (NB) parameters for the Malkmus statistical narrow band model from the medium resolution transmission measurements carried out at NIST (Wakatsuki et al., Proc. Combust. Inst., vol. 31, 2007, pp. 2573–2580) up to about 1000K for nine fuels commonly encountered in fire applications, namely methane, methanol, ethane, ethylene, propane, propylene, heptane, methyl methacrylate and toluene. These NB parameters can be used to generate a database of NB k-distributions of these fuels in order to apply the k-distribution methods to model their contributions to radiative heat transfer. The accuracy of this NB database, the EM2C NB database for methane (Perrin and Soufiani, JQSRT, vol. 103, 2007, pp. 3–13), and HITRAN 2012 for methane, methanol, ethane, and ethylene is assessed by comparison with the measured medium resolution transmissivities and spectrally-integrated radiative properties. Results show that the EM2C database is the most accurate for methane over the entire range of the temperature and the fuel partial pressure path length considered. The line parameters in HITRAN 2012 are not available for some important bands of methanol, ethane and ethylene and this spectroscopic database cannot be used with confidence above 600K. The Malkmus narrow band database derived in the present study is found to reproduce spectrally-integrated radiative properties with error less than 20%, which suggests that it can be used for engineering applications. Finally, applications to radiation calculations along lines of sight with methane concentrations and temperatures representative of pool fires show that both the present Malkmus narrow band and the NIST databases can predict accurately the radiative intensity for small scale pool fires where as discrepancies of the order of 30% are expected in the case of methane pool fires with a heat release rate of 176kW.

HELIOS-K: AN ULTRAFAST, OPEN-SOURCE OPACITY CALCULATOR FOR RADIATIVE TRANSFER

We present an ultrafast opacity calculator that we name HELIOS-K. It takes a line list as an input, computes the shape of each spectral line and provides an option for grouping an enormous number of lines into a manageable number of bins. We implement a combination of Algorithm 916 and Gauss-Hermite quadrature to compute the Voigt profile, write the code in CUDA and optimise the computation for graphics processing units (GPUs). We restate the theory of the k-distribution method and use it to reduce $\sim 10^5$ to $10^8$ lines to $\sim 10$ to $10^4$ wavenumber bins, which may then be used for radiative transfer, atmospheric retrieval and general circulation models. The choice of line-wing cutoff for the Voigt profile is a significant source of error and affects the value of the computed flux by $\sim 10\%$. This is an outstanding physical (rather than computational) problem, due to our incomplete knowledge of pressure broadening of spectral lines in the far line wings. We emphasize that this problem remains regardless of whether one performs line-by-line calculations or uses the k-distribution method and affects all calculations of exoplanetary atmospheres requiring the use of wavelength-dependent opacities. We elucidate the correlated-k approximation and demonstrate that it applies equally to inhomogeneous atmospheres with a single atomic/molecular species or homogeneous atmospheres with multiple species. Using a NVIDIA K20 GPU, HELIOS-K is capable of computing an opacity function with $\sim 10^5$ spectral lines in $\sim 1$ second and is publicly available as part of the Exoclimes Simulation Platform (ESP; www.exoclime.org).

Technical Note: A novel parameterization of the transmissivity due to ozone absorption in the <i>k</i>-distribution method and correlated-<i>k</i> approximation of Kato et al. (1999) over the UV band

Abstract. The k-distribution method and the correlated-k approximation of Kato et al. (1999) is a computationally efficient approach originally designed for calculations of the broadband solar radiation at ground level by dividing the solar spectrum in 32 specific spectral bands from 240 to 4606 nm. Compared to a spectrally resolved computation, its performance in the UV band appears to be inaccurate, especially in the spectral intervals #3 [283, 307] nm and #4 [307, 328] nm because of inaccuracy in modeling the transmissivity due to ozone absorption. Numerical simulations presented in this paper indicate that a single effective ozone cross section is insufficient to accurately represent the transmissivity over each spectral interval. A novel parameterization of the transmissivity using more quadrature points yields maximum errors of respectively 0.0006 and 0.0143 for intervals #3 and #4. How to practically implement this new parameterization in a radiative transfer model is discussed for the case of libRadtran (library for radiative transfer). The new parameterization considerably improves the accuracy of the retrieval of irradiances in UV bands.

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.

Numerical analysis of interaction between non-gray radiation and forced convection flow over a recess using the full-spectrum k-distribution method

In the present work, the interaction between non-gray radiation and forced convection in a laminar radiating gas flow over a recess including two backward and forward facing steps in a duct is investigated numerically. Distributions of absorption coefficients across the spectrum (50 cm−1 < η < 20,000 cm−1) are obtained from the HITRAN2008 database. The full-spectrum k-distribution method is used to account for non-gray radiation properties, while the gray radiation calculations are carried out using the Planck mean absorption coefficient. To find the divergence of radiative heat flux distribution, the radiative transfer equation is solved by the discrete ordinates method. The effects of radiation–conduction parameter, wall emissivity, scattering coefficient and recess length on heat transfer behaviors of the convection–radiation system are investigated for both gray and non-gray mediums. In addition, the results of gray medium are compared with non-gray results in order to judge if the differences between these two approaches are significant enough to justify the usage of non-gray models. Results show that for air mixture with 10 % CO2 and 20 % H2O, use of gray model for the radiative properties may cause significant errors and should be avoided.

A new k-interval optimization technique for atmospheric upwelling radiance calculation in infrared absorption bands

A new optimization technique is proposed to generate the k-interval parameters including the quadrature weights and their corresponding equivalent absorption coefficients when using correlated k-distribution method for computations of spectrally integrated atmospheric upwelling radiances in infrared absorption bands. The main feature of this technique is the introduction of the upwelling contribution function (UCF) both in expressing the atmospheric upwelling radiance and in constructing the merit function for fitting, so that the optimization for the band radiance smoothly turns into the optimization for the band UCF of each atmospheric sublayer. The performance of the proposed technique is illustrated by comparing with the Δlog(k) technique and the Gauss–Legendre quadrature technique at three different band settings in both the US standard atmosphere and the tropical atmosphere, and the line-by-line results are treated as the reference. Results show that there are less relative errors by the proposed optimization technique than by the other two techniques under the same number of k-intervals. In most cases, the computational efficiency can be improved by more than 10% under the same precision requirement in upwelling radiance calculations, and even 80% in some preferable conditions.

Comparison of accuracy and computational expense of radiation models in simulation of non-premixed turbulent jet flames

The accuracy and computational expense of various radiation models in the simulation of turbulent jet flames are compared. Both nonluminous and luminous methane–air nonpremixed turbulent jet flames are simulated using a comprehensive combustion solver. The combustion solver consists of a finite-volume/probability density function-based flow–chemistry solver interfaced with a high-accuracy spectral radiation solver. Flame simulations were performed using various k-distribution-based spectral models and radiative transfer equation (RTE) solvers, such as P-1, P-3, finite volume/discrete ordinates method (FVM/DOM), and line-by-line (LBL) accurate Photon Monte Carlo (PMC) methods, with and without consideration of turbulence–radiation interaction (TRI). Various spectral models and RTE solvers are observed to have strong effects on peak flame temperature, total radiant heat source and NO emission. The P-1 method is found to be the computationally least expensive RTE solver and the FVM the most expensive for any spectral model. For optically thinner flames all radiation models yield excellent accuracy. For optically thicker flames the P-3 and the FVM with advanced k-distribution methods predict radiation more accurately than the P-1 method with any spectral model when compared to the benchmark LBL PMC. The LBL PMC yields exact results with sufficient number of samples and is found to be less expensive than the FVM (for all spectral models) and the P-3 (for some spectral models) in statistically stationary combustion simulations. TRI is found to drop the peak temperature by close to 150K for a luminous flame (optically thicker) and 25–100K for a nonluminous flame (optically thinner).

Estimating the photosynthetically active radiation under clear skies by means of a new approach

Abstract. The k-distribution method and the correlated-k approximation of Kato et al. (1999) is a computationally efficient approach originally designed for calculations of the broadband solar radiation by dividing the solar spectrum in 32 specific spectral bands from 240 to 4606 nm. This paper describes a technique for an accurate assessment of the photosynthetically active radiation (PAR) from 400 to 700 nm at ground level, under clear-sky conditions using twelve of these spectral bands. It is validated against detailed spectral calculations of the PAR made by the radiative transfer model libRadtran. For the direct and global PAR irradiance, the bias is −0.4 W m−2 (−0.2%) and −4 W m−2 (−1.3%) and the root mean square error is 1.8 W m−2 (0.7%) and 4.5 W m−2 (1.5%). For the direct and global Photosynthetic Photon Flux Density, the biases are of about +10.3 μmol m−2 s−1 (+0.8%) and 1.9 μmol m−2 s−1 (−0.1%) respectively, and the root mean square error is 11.4 μmol m−2 s−1 (0.9%) and 4.0 μmol m−2 s−1 (0.3%). The correlation coefficient is greater than 0.99. This technique provides much better results than two state-of-the-art empirical methods computing the daily mean of PAR from the daily mean of broadband irradiance.

Application of full-spectrum k-distribution method to combined non-gray radiation and forced convection flow in a duct with an expansion

This paper presents a numerical analysis of interaction between non-gray radiation and laminar forced convection flow in a duct with an expansion. Distributions of absorption coefficients across the spectrum (50 cm −1 < η < 20000 cm −1) are obtained from the HITRAN2008 database. The full spectrum k-distribution method (FSK) is used to deal with the non-gray part of the problem, while the gray radiation calculations are carried out using the Planck mean absorption coefficient. In addition, the results of non-gray medium are compared with the gray results in order to judge if the differences between these two approaches are significant enough to justify the usage of non-gray models. Results show that for air mixtures with different mole fractions of CO2 and H2O, use of gray model for the radiative properties may leads to considerable errors and should be eschewed.

How close to detailed spectral calculations is the k-distribution method and correlated-k approximation of Kato et al. (1999) in each spectral interval?

The k-distribution method and the correlated-k approximation of Kato et al. (1999) is a smart approach originally designed for broadband calculations of the solar radiation at ground level by dividing the solar spectrum in 32 spectral bands. The approach is a priori not suited for calculation of spectral irradiance. Nevertheless, this paper evaluates its performance when compared to more detailed spectral calculations serving as references for the spectral intervals no. 3 [283, 307] nm to 26 [1613, 1965] nm for clear and cloudy situations. The evaluation is based on numerical simulations. The clearer the sky, the greater the root mean square error (RMSE) in all bands. In the spectral intervals no. 3 and 4 [307, 328] nm, the irradiance is underestimated by large – approximately −90 % and −17 % in relative value -because the wavelength interval is large with respect to the absorption by ozone and a single value of ozone cross section is not enough for each interval. For each spectral interval from no. 5 [328, 363] nm to no. 18 [743, 791] nm, and for both global and direct radiation, the bias and the RMSE are less than 1.5 % of the irradiance in the corresponding interval under clear skies and may amount to 3 % in cloudy conditions. For greater wavelength intervals no. 19 to no. 26, the relative bias and RMSE show a tendency to increase with wavelength and may reach 8 % and 7 % for global and direct under clear skies respectively, and 11 % and 15 % under cloudy skies.

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.

Full-Spectrum k-Distribution Model for Radiation in Gas Based on Multi-Group Methods

The k-distribution method applied in narrow band and wide band is extended to the full spectrum based on spectroscopic datebase HITEMP, educing the full-spectrum k-distribution model. Absorption coefficents in this model are reordered into a smooth,monotonically increasing function such that the intensity calculations are performed only once for each absorption coefficent value and the resulting computations are immensely more efficent.Accuracy of this model is examined for cases ranging from homogeneous one-dimensional carbon dioxide to inhomogeneous ones with simultaneous variations in temperature. Comparision with line-by-line calculations (LBL) and narrow-band k-distribution (NBK) method as well as wide-band k-distribution (WBK) method shows that the full-spectrum k-distribution model is exact for homogeneous media, although the errors are greater than the other two models. After dividing the absorption coefficients into several groups according to their temperature dependence, the full-spectrum k-distribution model achieves line-by-line accuracy for gases inhomogeneous in temperature, accompanied by lower computational expense as compared to NBK model or WBK model. It is worth noting that a new grouping scheme is provided in this paper.

The multispectral gas radiation modeling: A new theoretical framework based on a multidimensional approach to k-distribution methods

A number of radiative heat transfer models dealing with uniform – homogeneous isothermal – or nonuniform gaseous media can be found in the literature. Usually, these models are considered to be distinct from each other by their developers, and hence the large number of different denominations. For a potential user, it makes difficult the choice of an optimal model for a given problem. The aim of the present work is to demonstrate that all k-distribution models, i.e. from the most accurate line-by-line up to the simplest correlated or Scaled-k ones, are based on a restricted set of concepts shared by all of them. By proposing the most general formulation of multidimensional k-distributions, we clearly define clustering techniques to build from reference line-by-line data approximate gas radiation models fully controllable in terms of accuracy. These techniques are presented in details and assessed in several test cases for demonstrating their relevance in radiative heat transfer.

A theoretical and experimental study of the time-dependent radiative properties of a solar bubbling fluidized bed receiver

In order to evaluate the potential of solar bubbling fluidized bed receivers compared to other methods for the solar heating of gases at high temperature, a thorough knowledge of the heat transfer of the receiver is necessary. Since the external energy source of the system is radiative and because of high working temperatures, it is particularly important to model the radiative heat transfer to later predict the temperature field in the solar receiver. The aim of this study is to model the radiative flux distribution in a fluidized bed by taking into account the time-dependent absorption and scattering of light in the particulate medium. For this purpose, we propose a model using the Monte Carlo Method as well as a time-dependent field of optical properties that was predicted using a Computational Fluid Dynamics tool implemented with an Eulerian model. A statistical treatment using the k-distribution method was later applied to the time-dependency of the radiative properties of the solar fluidized bed receiver. This method has proven to be useful to reduce computational time while keeping a good accuracy. An experimental set-up was designed to validate the numerical predictions of the particle volume fraction and the penetration of radiation into the fluidized bed. The good agreement of the current model with the experimental data confirms its suitability.

On Multilayer Modeling of Radiative Transfer for Use With the Multisource k-Distribution Method for Inhomogeneous Media

A nonisothermal medium is modeled using the multilayer approach in which the continuous temperature distribution in a one-dimensional system as modeled as being piecewise constant. This has been shown to provide accurate results for a surprisingly small number of layers. Analysis is performed on a nonisothermal gray medium to attempt to characterize the ways in which the errors introduced by the multilayer modeling change with various physical parameters namely, the optical thickness and the temperature or emissive power gradient. A demonstration is made of how the multisource k-distribution method is capable of evaluating the heat flux within a one-dimensional system with piecewise constant temperature distribution with line-by-line accuracy with a significant decrease in computational expense. The k-distribution method for treating the spectral properties of an absorbing–emitting medium represents a powerful alternative to line-by-line calculations by reducing the number of radiative transfer equation (RTE) evaluations from the order of a million to the order of 10 without any significant loss of accuracy. For problems where an appropriate reference temperature can be defined, the k-distribution method is formally exact. However, when no appropriate reference temperature can be defined, the method results in errors. The multisource k-distribution method extends the k-distribution method to problems with piecewise constant temperature and optical properties.

Improved full-spectrum k-distribution implementation for inhomogeneous media using a narrow-band database

A new implementation to determine the correlated k values as used in full-spectrum correlated k-distribution methods for inhomogeneous media is presented, which conserves radiative emission and significantly improves the accuracy of the full-spectrum correlated k-distribution method in strongly inhomogeneous media. In addition, an update of a previous high-accuracy, compact narrow-band k-distribution database is given. The new narrow-band database utilizes the latest spectroscopy databases to ensure accuracy at elevated temperatures and pressures. Differences between the two versions of narrow-band k-distribution databases as a result of the change of spectroscopy databases are illustrated.

K-bin and k-IR: k-distribution methods without correlation approximation for non-fixed instrument response function and extension to the thermal infrared—Applications to satellite remote sensing

k-distribution methods are efficient tools for large spectral band radiation transfer simulations in the atmosphere. A weakness of classical k-distribution method (correlated k-distribution method) is the simulation of layered atmosphere with variable gas absorption. This weakness is due to the use of the correlation approximation. We can avoid the use of the correlation approximation using a spectral mapping for the construction of the k-distribution intervals. This already-existing method is known as Spectral Mapping Transformation (SMT). Within SMT k-distribution, k-distribution is built to suit all the atmospheric layers. This difference of method, compared to the correlated k-distribution methods, leads to a loss of rapidity but to a gain of precision. Particularly, SMT k-distribution methods permit a balance between rapidity and precision by decreasing or increasing the number of bins (spectroscopically similar k-distribution intervals). Within this article, we recapitulate the main principles of SMT k-distribution methods and present two improvements: the k-IR method, which is an adaptation of an SMT k-distribution method to the thermal infrared, and k-bin, which is an adaptation of an SMT k-distribution method that allows the user to simulate narrow-band satellite channels with a non-fixed spectral response function. k-IR permits a modeling of the absorption and emission by gases, even if the spectral variability of the Planck function is completely different from the spectral variability of absorption/emission coefficients. This is achieved by associating to each bin (spectroscopically similar k-distribution interval) a precomputed value of the bin-averaged blackbody radiance. Within this paper, we outline k-IR and apply it for the simulation of Channel 3 (12.05μm) of the Imaging Infrared Radiometer (IIR) aboard the CALIPSO satellite. Results show that the accuracy steadily increases with an increasing number of bins. This will allow us to reach the instrument precision. k-bin, otherwise, is a more stringent approach of the k-distribution method. The major difference between a conventional k-distribution and k-bin is that, for a given spectral interval no assumption about the shape of the sensor weighting function has to be incorporated a priori. For a given spectral interval, any sensor response function can be constructed from a set of radiative transfer simulations. This requires somewhat different constraints in the way the subdivision in bins is performed, namely we must ensure that not only the band-averaged transmission is resembled to high accuracy, but also the transmission in each bin. Within this paper we outline k-bin method and apply it to high-resolution spectroscopic simulations in the oxygen A-band. k-bin allows for a representation of any channel (with a resolution of 0.04nm or lower) within the oxygen A-band with an accuracy of 0.2% or better with less than 600 simulations. This corresponds for the Orbiting Carbon Observatory-2 (OCO-2) to a computational cost of 0.6 simulations per channel.

A multi-source full spectrum k-distribution method for 1-D inhomogeneous media

The k-distribution method for treating the spectral properties of an absorbing–emitting medium represents an alternative to line-by-line calculations. It reduces the number of RTE evaluations from the order of a million to the order of ten without any significant loss of accuracy. For problems where an appropriate reference temperature can be defined, the k-distribution method is formally exact. However, when no appropriate reference temperature can be defined, the method results in errors. There have been several attempts to implement corrections to the k-distribution method to extend its application to inhomogeneous media by modeling the effects of temperature, pressure, and concentration gradient. Some of these techniques are effective at reducing these errors, but none successfully extend the k-distribution method to inhomogeneous media in a way that retains the analytical guarantee of line-by-line accuracy. The Multi-Source Full Spectrum K-Distribution Method (MSFSK) introduced here manages to attain this goal for 1-dimensional domains with piecewise constant temperature distributions by applying the superposition principle to the original RTE before applying the k-distribution transformation to decompose the problem into a set of sub-problems each of which is able to be solved exactly via the ordinary or modified full spectrum k-distribution method.