A k-DISTRIBUTION-BASED SPECTRAL MODULE FOR RADIATION CALCULATIONS IN MULTI-PHASE MIXTURES

k-distribution-based approaches are promising models for radiation calculations in strongly nongray participating media. Advanced k-distribution methods were found to achieve close-to benchmark line-by-line (LBL) accuracy for strongly inhomogeneous multi-phase media accompanied by several orders of magnitude smaller computational cost. In this paper, a k-distributionbased portable spectral module is developed, incorporating several state-of-the-art k-distribution methods along with compact and high-accuracy databases of k-distributions. The module construction is flexible – the user can choose among various kdistribution methods with their relevant k-distribution databases, to carry out accurate radiation calculations. The spectral module is portable, such that it can be coupled to any flow solver code with its own grid structure, discretization scheme, and solver libraries. This open source code module is made available for free for all noncommercial purposes. This article outlines in detail the design and the use of the spectral module. The k-distribution methods included in the module are briefly described with a discussion of their advantages, disadvantages and their domain of applicability. Examples are provided for various sample radiation calculations in multi-phase mixtures using the new spectral module and the results are compared with LBL calculations. INTRODUCTION Radiative heat transfer is often the dominant mode of heat transfer in commercial combustion systems and atmospheric processes. The magnitude of radiative heat fluxes can have profound effects on combustion performance and on environmental impact. In commercial combustion applications erroneous prediction of gas temperature by as little as 50oC may lead to vastly wrong prediction of its pollution characteristics. Therefore, accurate determination of radiation is necessary for correct prediction of overall heat transfer in combustion systems in order to achieve energy-optimized, economic and pollution-free performance. Traditionally, radiation in combustion systems has been treated using gray or even simpler models due to their simplicity ∗Address all correspondence to this author. and faster computation. Only during the past few years a number of investigators considered nongray radiation effects, using spectral models of varying levels of sophistication; all have shown a strong influence of spectral radiation in the combustion process [1–10]. Consideration of nongray thermal radiation introduces a new difficulty, as radiative properties exhibit strong and erratic variations with wavenumber for the participating species that are of interest in combustion [principally water vapor (H2O), carbon dioxide (CO2), and carbon monoxide (CO)]. Nongray radiation calculations in participating media can be most accurately performed using the line-by-line (LBL) approach. LBL accurately resolves the spectrum [11–15] requiring in excess of one million spectral solutions to the RTE, thus making spectral radiation calculations prohibitive (in both computational time and memory requirements) even with todays computer resources. For accurate and computationally efficient solutions of the radiative transfer equation (RTE), several models have been proposed, applying the concept of reordering the absorption coefficient across the entire spectrum. These include the spectral-line-based weighted-sum-of-gray-gases (SLW) model [16, 17], the absorption distribution function (ADF) method [18, 19], and the full-spectrum k-distribution (FSK) method [20]. The SLW and ADF methods are approximate schemes, in which the absorption coefficient is reduced to a few discrete values (chosen by the user), and the integration over the spectrum is achieved by adding contributions of the “gray gases” (effectively trapezoidal rule quadrature, which requires a large number of points for good accuracy). The FSK method, on the other hand, is an exact method for a correlated absorption coefficient using a continuous k-distribution over the entire spectrum. Spectral integration can be performed using high-accuracy Gaussian quadrature, which generally yields excellent accuracy for less than half the number of points required by the trapezoidal rule of integration. Although the FSK scheme is exact for radiative calculations in homogeneous media, its application to strongly inhomogeneous emitting–absorbing mixtures, containing both molecular gases and nongray soot particles, challenges its accuracy. Several advancements to the k-distribution method have been proposed to address the shortcomings of the basic FSK 1 Copyright c ⃝ 2009 by ASME scheme. To alleviate inaccuracies in inhomogeneous gas mixtures, two different approaches have been proposed, namely, the multi-scale (MS) [12] approach and the multi-group (MG) approach [21, 22]. In the multi-scale or fictitious gas approach the individual spectral lines comprising the absorption coefficient are placed into separate scales based on their temperature dependence. In the multi-group approach spectral positions, i.e., wavenumbers, are placed into several groups according to their dependence on temperature and partial pressure. The multigroup FSK (MGFSK) method has been shown to achieve great accuracy for a single gas species with inhomogeneity in temperature [12, 21, 22], whereas the multi-scale FSK (MSFSK) method can efficiently treat mixtures of absorbing gases with severe species inhomogeneity [23]. Combining the advantages of both methods, a hybrid multi-scale multi-group FSK (MSMGFSK) method has also been developed, which can accurately determine radiation from gas mixtures with extreme inhomogeneities in both temperature and concentration [24, 25]. Soot radiation constitutes an important part of radiation calculations in luminous flames. Because of the difficulties in soot modeling and because of its somewhat more benign spectral behavior, soot radiation in combustion has commonly been treated as gray [26]. Nongray soot has been investigated by Solovjov and Webb [27] using the SLWmethod, and byWang et al., who employed the single-scale FSK method [2]. Both these methods cannot address the species concentration inhomogeneity problem. The MSFSK method previously developed by Zhang and Modest [12] has been recently extended allowing for nongray soot in the gas mixtures with/without gray wall emission [28, 29]. However, the modified MSFSK method fails to produce accurate results in the presence of strong temperature inhomogeneities. Final advancement of the k-distribution concept was achieved by Pal and Modest by extending the full-spectrum-based hybrid MSMGFSK method to a narrow band-based hybrid MSMGFSK method to allow incorporation of nongray soot into the gas mixtures [30]. This narrow band-based MSMGFSK method was found to achieve close-to LBL accuracy in presence of both temperature and concentration inhomogeneities in a multiphase mixture. FSK calculations are very accurate and time efficient provided the required full-spectrum k-distributions are known, which are tedious to compile from spectroscopic databases, such as HITRAN [31], HITEMP [32] and CDSD-1000 [33]. To make accurate FSK calculations feasible for general engineering purposes, preassembled FSK distribution must be available in the form of accurate and compact databases. Wang and Modest [34] have compiled a high accuracy, compact database of narrow band k-distributions for the most important combustion gases, from which full-spectrum k-distributions can be obtained efficiently for arbitrary mixtures of combustion gases, including nongray absorbing and/or scattering particles. Full-spectrum multi-group databases have been constructed by Zhang and Modest for carbon dioxide and water vapor [21, 22]. It has been reported that close-to LBL accuracy can be achieved by considering only 4 such groups, within which the assumption of a correlated absorption coefficient holds. In the multi-group databases created by Zhang and Modest [21, 22] the absorption coefficients were obtained from the HITEMP spectroscopic database. Unfortunately, it has been found that the HITEMP database is not accurate for CO2 at temperatures higher than 1000 K [33, 35]. Recently Pal and Modest constructed more accurate and compact full-spectrum multi-group [24, 25] and narrow band multi-group databases [30] containing 4 groups for each species with spectral absorption coefficients for water vapor calculated from HITEMP 2000 [32], and for carbon dioxide from CDSD-1000 [33]. With accurate mixing models developed within the framework of advanced k-distribution methods in conjunction with the accurate and compact k-distribution databases, it is now possible to mix k-g distributions of multiphase species on the fly with efficient usage of memory and computational time. In this article, an open source code k-distribution methodbased portable spectral module, called “Spectral Radiation Calculation Software” (SRCS), has been developed. The SRCS includes all state of the art k-distribution methods. High-accuracy databases of k-distributions [25, 30, 34], together with our mixing models [12, 23, 25, 29, 30, 36], allow on-the-fly construction of FSK distributions. The module construction is flexible – the user can choose among various k-distribution methods with relevant k-distribution databases and perform accurate radiation calculations during the solution of combustion problems. The user has the choice to use the basic FSK method with a coarser grid during the initial stage of the combustion calculation and then move on to more sophisticated k-distribution models and finer grids during later stages of the computation. The spectral module is made portable, such that it can be coupled to any flow solver code with its own grid structure, discretization scheme, and solver libraries. Detailed module structure has been outlined in this paper including a discussion of various k-distribution methods with their applicability. Sample calculations were performed for a 1-D medium containing a gas–soot mixture with distributions in temperature and species concentrations using various k-distribution methods implemented within the SRCS. Computational time and accuracy of results obtained from different k-distribution methods are compared to LBL calculations.