The Pennsylvania State University The Graduate School College of Engineering SPECTRAL MODELING OF RADIATION IN COMBUSTION SYSTEMS

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1 The Pennsylvania State University The Graduate School College of Engineering SPECTRAL MODELING OF RADIATION IN COMBUSTION SYSTEMS A Dissertation in Mechanical Engineering by Gopalendu Pal c 21 Gopalendu Pal Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy May 21

2 The dissertation of Gopalendu Pal was reviewed and approved by the following: Michael F. Modest Distinguished Professor of Mechanical Engineering Dissertation Co-Adviser, Co-Chair of Committee Daniel C. Haworth Professor of Mechanical Engineering Dissertation Co-Adviser, Co-Chair of Committee Stephen R. Turns Professor of Mechanical Engineering Deborah A. Levin Professor of Aerospace Engineering Karen A. Thole Professor of Mechanical Engineering Department Head of Mechanical and Nuclear Engineering Signatures are on file in the Graduate School.

3 Abstract Radiation calculations are important in combustion due to the high temperatures encountered but has not been studied in sufficient detail in the case of turbulent flames. Radiation calculations for such problems require accurate, robust, and computationally efficient models for the solution of radiative transfer equation (RTE), and spectral properties of radiation. One more layer of complexity is added in predicting the overall heat transfer in turbulent combustion systems due to nonlinear interactions between turbulent fluctuations and radiation. The present work is aimed at the development of finite volume-based high-accuracy thermal radiation modeling, including spectral radiation properties in order to accurately capture turbulence radiation interactions (TRI) and predict heat transfer in turbulent combustion systems correctly and efficiently. The turbulent fluctuations of temperature and chemical species concentrations have strong effects on spectral radiative intensities, and TRI create a closure problem when the governing partial differential equations are averaged. Recently, several approaches have been proposed to take TRI into account. Among these attempts the most promising approaches are the probability density function (PDF) methods, which can treat nonlinear coupling between turbulence and radiative emission exactly, i.e., emission TRI. The basic idea of the PDF method is to treat physical variables iii

4 as random variables and to solve the PDF transport equation stochastically. The actual reacting flow field is represented by a large number of discrete stochastic particles each carrying their own random variable values and evolving with time. The mean value of any function of those random variables, such as the chemical source term, can be evaluated exactly by taking the ensemble average of particles. The local emission term belongs to this class and thus, can be evaluated directly and exactly from particle ensembles. However, the local absorption term involves interactions between the local particle and energy emitted by all other particles and, hence, cannot be obtained from particle ensembles directly. To close the nonlinear coupling between turbulence and absorption, i.e., absorption TRI, an optically thin fluctuation approximation can be applied to virtually all combustion problems and obtain acceptable accuracy. In the present study a composition-pdf method is applied, in which only the temperature and the species concentrations are treated as random variables. A closely coupled hybrid finite-volume/monte Carlo scheme is adopted, in which the Monte Carlo method is used to solve the composition-pdf for chemical reactions and the finite volume method is used to solve for the flow field and radiation. Spherical harmonics method-based finite volume solvers (P-1 and P-3 ) are developed using the data structures of the high fidelity open-source code flow software OpenFOAM. Spectral radiative properties of the participating medium are modeled using full-spectrum k-distribution methods. Advancements of basic k-distribution methods are performed for nongray nonhomogeneous gas- and particulate-phase (soot, fuel droplets, ash, etc.) participating media using multi-scale and multi-group based approaches. These methods achieve close-to benchmark line-by-line (LBL) accuracy in strongly inhomogeneous media at a tiny fraction of LBL s computational cost. A portable spectral iv

5 module is developed, which includes all the basic to advanced k-distribution methods along with the precompiled accurate and compact k-distribution databases. The P-1 /P-3 RTE solver coupled with the spectral module is used in conjunction with the combined Reynolds-averaged Navier-Stokes (RANS) and composition-pdf-based turbulence-chemistry solver to investigate TRI in multiphase turbulent combustion systems. The combustion solvers developed in this study is employed to simulate several turbulent jet flames, such as Sandia Flame D, and artificial nonsooting and sooting flames derived from Flame D. The effects of combustion chemistry, radiation and TRI on total heat transfer and pollutant (such as NO x ) generation are studied for the above flames. The accuracy of the overall combustion solver is assessed by comparing it with the experimental data for Flame D. Comparison of the accuracy and the computational cost among various spectral models and RTE solvers is extensively done on the artificial flames derived from Flame D to demonstrate the necessity of accurate modeling of radiation in combustion problems. v

6 Table of Contents List of Figures List of Tables List of Symbols Acknowledgments ix xiii xiv xviii Chapter 1 Introduction Motivation and Background Objectives Outline of the Chapters to Follow Chapter 2 Existing Methodologies Solution of the Radiation Transport Equation Photon Monte Carlo Method Discrete Ordinate Method Spherical Harmonics Method Spectral Modeling of Radiation LBL and Band Models Global Models k-distribution Method k-distribution Databases Radiation from Soot The Composition PDF Method The Transported Composition PDF Method The Composition PDF Formulation PDF Equation for General Scalar Modeled Terms in Composition PDF Solution Procedure for Composition PDF vi

7 2.4 Turbulence Radiation Interactions TRI Modeling Optically Thin Fluctuation Approximation Review of TRI Modeling Chapter 3 Full-Spectrum Multi-Scale Multi-Group k-distribution Method Introduction Hybrid MSMGFSK Approach Evaluation of Overlap Coefficient λ nm Multi-Group Database Construction Grouping of Wavenumbers: Construction of k g Distributions Interpolation of Database Combination of Groups Sample Calculations Summary Chapter 4 Modified Multi-Scale Full-Spectrum k-distribution Method Introduction The MSFSK Approach for Gas Soot Mixtures Evaluation of Overlap Parameter Evaluation of Modified Wall Stretch Factor Sample Calculations D Problem D Problem Summary Chapter 5 Narrow Band-Based Multi-Scale Multi-Group k-distribution Method The Narrow Band-Based MSMGFSK Approach Evaluation of Overlap Parameter Evaluation of Modified Wall Stretching Factor Narrow Band Multi-group Database Construction Sample Calculations D Problem D Problem Summary vii

8 Chapter 6 Portable Spectral Radiation Calculations Software (SRCS) Introduction Module Structure Methods Databases Spectral Input Main Operating Module Inter-Language Portability Module Sample Calculations Summary Chapter 7 Higher Order P-N Solver Introduction P-3 Approximation Solution Method Sample Calculations Summary Chapter 8 Flame Simulation Introduction Nonluminous Flames Problem Setup Results and Discussion Luminous Flame Summary Chapter 9 Conclusions 145 Bibliography 149 viii

9 List of Figures 3.1 Grouping of several spectral locations across 4.3 µm band of CO 2 in a CO 2 air mixture containing 1% CO Grouping of several spectral locations across 2.7 µm band of H 2 O in a H 2 O air mixture containing 1% H 2 O Original and smoothed weight function a and cumulative k-distributions of Group 2 for 1% H 2 O in H 2 O air mixture Schematic of the one dimensional problem setup Non-dimensional heat flux leaving an inhomogeneous slab of 1% CO 2 and 2% H 2 O at a total pressure of 1 bar with a step change in temperature: The hot left layer is at 15 K and the cold right layer is at 3 K Non-dimensional heat flux leaving an inhomogeneous slab at a total pressure of 1 bar with step changes in temperature and mole fraction: The hot left layer contains 2% CO 2 and 1% H 2 O at 15 K and the cold right layer contains 1% CO 2 and 2% H 2 O Non-dimensional heat flux leaving an inhomogeneous slab at a total pressure of 1 bar with step changes in temperature and mole fraction: The hot left layer contains 2% CO 2 and 1% H 2 O at 15 K and the cold right layer contains 1% CO 2 and 2% H 2 O Nondimensional heat flux leaving an inhomogeneous slab at a total pressure of 1 bar, with step changes in mole fraction of gas species: left layer contains 2% CO 2, 2% H 2 O and.1 ppm soot; right layer contains 2% CO 2, 2% H 2 O and.1 ppm soot Nondimensional heat flux leaving an inhomogeneous slab at a total pressure of 1 bar, with step changes in mole fraction: left layer contains 2% CO 2 and.1 ppm soot; right layer contains 2% H 2 O Nondimensional heat flux leaving an inhomogeneous slab at a total pressure of 1 bar, with step changes in mole fraction and temperature: left layer contains 2% CO 2, 2% H 2 O and.1 ppm soot at 15 K; right layer contains 2% CO 2, 2% H 2 O and no soot at 1 K ix

10 4.4 Nondimensional heat flux leaving an inhomogeneous slab at a total pressure of 1 bar, with wall emission and step changes in mole fraction: left layer contains 2% CO 2, 2% H 2 O, and.1 ppm soot; right layer contains 2% CO 2, 2% H 2 O and no soot; both Layers at 1 K Nondimensional heat flux leaving an inhomogeneous slab at a total pressure of 1 bar, with wall emission, step changes in mole fraction and temperature: left layer contains 2% CO 2, 2% H 2 O, and no soot at 15 K; right layer contains 2% CO 2, 2% H 2 O and.1 ppm soot at 1 K Distribution of soot volume fraction for 2-D test flame Relative error for radiative heat source calculations using single-scale FSCK compared to LBL in a gas (CO 2, H 2 O, CH 4 ) soot mixture in 2-D combustion chamber Relative error for radiative heat source calculations using MSFSK compared to LBL in a gas (CO 2, H 2 O, CH 4 ) soot mixture in 2-D combustion chamber Nondimensional heat flux leaving an inhomogeneous slab at a total pressure of 1 bar with step changes in mole fraction: The left layer contains 2% CO 2, 2% H 2 O and no soot and the right layer contains 2% CO 2, 2% H 2 O and.1 ppm soot, both layer at constant temperature 1 K Nondimensional heat flux leaving an inhomogeneous slab at a total pressure of 1 bar with step changes in temperature: Left layer at 15 K, and right Layer at 5 K, both layers contain 2% CO 2, 2% H 2 O,.1 ppm soot Nondimensional heat flux leaving an inhomogeneous slab at a total pressure of 1 bar with step changes in temperature and mole fraction: The hot left layer contains 2% CO 2, 2% H 2 O and no soot at 15 K, and the cold right layer contains 2% CO 2, 2% H 2 O and.1 ppm soot at 5 K Temperature and mole fraction distributions in numerically simulated KH87 flame, (a) temperature distribution; (b) mole fraction distribution of H 2 O and, approximately CO 2, wherever there is little CO; (c) mole fraction distribution of CO; (d) mole fraction distribution of C 2 H 4 ; (e) distribution of soot volume fraction (a) Local radiative heat source using LBL method and relative error (compared to LBL) for heat source calculations using: (b) the singlescale FSK method; (c) the modified MSFSK method; (d) the 2 group narrow band-based MSMGFSK method x

11 6.1 Software architecture diagram of SRCS Temperature and mole-fraction/volume-fraction distributions of CO 2, H 2 O and soot in a 1-D medium Local radiative heat source and relative error (compared to LBL) for heat source calculations using scalar distribution profile Local radiative heat source and relative error (compared to LBL) for heat source calculations using scalar distribution profile Local radiative heat source and relative error (compared to LBL) for heat source calculations using scalar distribution profile Incident radiation, G by P-3 approximation for optically thick case, C K = Incident radiation, G by photon Monte Carlo simulation for optically thick case, C K = Incident radiation, G by P-3 approximation for optically intermediate case, C K = Incident radiation, G by photon Monte Carlo simulation for optically intermediate case, C K = Incident radiation, G by P-3 approximation for optically thin case, C K = Incident radiation, G by photon Monte Carlo simulation for optically thin case, C K = Comparison of local radiative heat source ( Q) calculated using PMC, P-1 and P-3 RTE solvers Schematic of interfacing of the FV spectral radiation solver with the flow-chemistry solver Grid system used in flame simulations Comparison of centerline profiles of the computed mean temperature (P-1 +FSK+ARM2) with experimental data for Flame D Comparison of centerline profiles of the computed mean temperature in Flame D using various radiation models Radial profiles of mean temperature at various axial locations of Flame D Radial profiles of RMS temperature fluctuations at various axial locations (see legends in Fig. 8.5) of Flame D Radial profiles of mean CO 2 mass fraction at various axial locations (see legends in Fig. 8.5) of Flame D Radial profiles of RMS fluctuations of CO 2 mass fraction at various axial locations (see legends in Fig. 8.5) of Flame D xi

12 8.9 Radial profiles of mean H 2 O mass fraction at various axial locations (see legends in Fig. 8.5) of Flame D Radial profiles of RMS fluctuations of H 2 O mass fraction at various axial locations (see legends in Fig. 8.5) of Flame D Radial profiles of mean CH 4 mass fraction at various axial locations (see legends in Fig. 8.5) of Flame D Radial profiles of RMS fluctuations of CH 4 mass fraction at various axial locations (see legends in Fig. 8.5) of Flame D Radial profiles of mean NO mass fraction at various axial locations (see legends in Fig. 8.5) of Flame D Radial profiles of RMS fluctuations of NO mass fraction at various axial locations (see legends in Figs. 8.5 and 8.13) of Flame D Mean temperature contour of Flame D Mean NO mass fraction contour of Flame D Comparison of the centerline profiles of mean temperature using various spectral models with the P-1 RTE solver for Flame kl Comparison of the centerline profiles of mean NO mass fraction using various spectral models with the P-1 RTE solver for Flame kl Comparison of the contours of mean temperature using various spectral models with the P-1 RTE solver for Flame kl Comparison of the contours of mean of NO mass fraction using various spectral models with the P-1 RTE solver for Flame kl Comparison of the contours of mean local radiative heat source using various spectral models with the P-1 RTE solver for Flame kl Comparison of the centerline profiles of mean temperature using various RTE solvers for Flame kl Comparison of the centerline profiles of mean NO mass fraction using various RTE solvers for Flame kl Comparison of the contours of mean temperature using various RTE solvers for Flame kl Comparison of the contours of mean NO mass fraction using various RTE solvers for Flame kl Contour plots of mean temperature and soot volume fraction for Flame kl-3 with soot Contour plots of mean NO mass fraction and local radiative heat source for Flame kl-3 with soot xii

13 List of Tables 3.1 Precalculated gas states and reference temperatures Parameters for wavenumber grouping and multi-group databasing Advanced k-distribution methods Submodules within method module k-distribution databases Inlet boundary conditions of Sandia Flame D [1] Effect of radiation models and TRI in simulation of Flame D Comparison of CPU time and parallel efficiency Effect of radiation models and TRI in simulation of Flame kl Comparison of net heat loss in Flame kl-3 using P-1 RTE solver and various spectral models; frozen field study Comparison of net heat loss in Flame kl-3 using the LBL spectral model and various RTE solvers; frozen field study Effect of radiation models and TRI in simulation of Flame kl-3 with soot Comparison of net heat loss in Flame kl-3 with soot using P-1 RTE solver and various spectral models; frozen field study Comparison of net heat loss in Flame kl-3 with soot using the LBL spectral model and various RTE solvers; frozen field study xiii

14 List of Symbols A a b C φ C µ d j d p F(ψ; x, t) f(ψ; x, t) f v G g H I I b J α k anisotropic scattering coefficient nongray stretching factor for k-distribution method linear basis function model constant for mixing model constant in k-ɛ model jet diameter outer diameter of pilot flow composition mass density function composition probability density function, k-distribution soot volume fraction incident radiation cumulative k-distribution Heaviside step function radiative intensity blackbody intensity molecular diffusive flux for scalar φ α absorption coefficient variable in k-distribution, turbulent kinetic energy xiv

15 L M m N N nb n ˆn P q r S rad thickness of participating medium slab spectral group particle mass, index for spectral group species scale, order of spherical harmonics series total number of narrow bands index for species scale surface normal pressure, Legendre polynomials radiative heat flux position vector radiative heat source S α s ŝ T t ũ u W x x x creation rate of scalar φ α position variable direction unit vector temperature time Favre averaged velocity fluctuating component of velocity isotropic vector Wiener process species mole fraction species mole fraction vector position vector x, y, x coordinates xv

16 Y spherical harmonics, species mass fraction Greek symbols α parameter defined in Eq β Γ T extinction coefficient turbulent diffusivity γ parameter defined in Eq δ ɛ ε φ Φ φ φ ψ η η θ κ λ Kronecker s delta function parallel efficiency, gray wall emittance dissipation rate of turbulent kinetic energy physical scalar variable, azimuthal angle scattering phase function state variable vector reference state composition variable wavenumber narrow band width polar angle absorption coefficient overlap parameter µ T turbulent term in diffusivity Ω ω ρ solid angle scattering albedo mass density xvi

17 σ s σ φ τ scattering coefficient turbulent Schmidt or Prandtl number optical thickness Abbreviations ADA ADF ARM DOM FSK FVM IDA LES MDA MSFSK MSMGFSK OTFA PDE PDF PMC RANS RTE SHM SLW SRCS TRI WSGG advanced differential approximation absorption distribution function augmented reduced mechanism discrete ordinate method full-spectrum k-distribution finite volume method improved differential approximation large eddy simulation modified differential approximation mutli-scale FSK mutli-scale multi-group FSK optically thin fluctuation approximation partial differential equation probability density function photon Monte Carlo Reynolds averaged Navier-Stokes radiation transport equation spherical harmonics method spectral-line-based weighted-sum-of-gray-gases spectral radiation calculation software turbulence radiation interaction weighted-sum-of-gray-gases xvii

18 Acknowledgments I would like to thank my advisors, Dr. Michael Modest and Dr. Daniel Haworth, for their help, support and valuable guidance. Many thanks to Dr. Stephen Turns, who has provided a valuable critique of my work and to Dr. Deborah Levin, who has helped to improve this work with her insight. I am eternally grateful for the support and encouragement from my family. My friends and colleagues own a huge part in my success and I am thankful for their support and kindness. xviii

19 Chapter 1 Introduction 1.1 Motivation and Background Radiative heat transfer is often the dominant mode of heat transfer in 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 5 o C may cause serious errors in the estimation of pollution. Therefore, accurate determination of radiation is necessary for correct prediction of the overall heat transfer in combustion systems, in order to achieve optimized, economic and pollutionminimized performance. Unfortunately, to date it is not possible to make high-accuracy predictions of radiative heat transfer rates in high-temperature combustion applications. The reasons for this deficiency are: (i) lack of high accuracy and efficient radiative transport equation (RTE) solvers as the RTE is an integro-differential equation in five independent variables (three spatial coordinates and two directional coordinates), plus wavelength as one more variable for nongray media, and thus can only be solved numerically with approximate methods except for very few simple cases; (ii) lack of accurate modeling of radiative properties and (iii) lack of versatile, robust and computationally efficient models to predict radiation from nongray media. Consideration of nongray thermal

20 2 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 (H 2 O), and carbon dioxide (CO 2 )). In most practical combustion systems turbulence, radiation, and chemistry are coupled in interesting and highly nonlinear ways, leading to entirely new classes of interactions. In much the same way as convection is aided by turbulence, so is radiation, which in the presence of chemical reactions may increase several fold due to turbulence interactions. Traditional combustion calculations treat radiation and turbulence as uncoupled processes using mean temperatures and concentrations to evaluate radiative properties and intensities [2]. Turbulence-radiation interaction (TRI) has been largely ignored to date due to its extreme complexity, even though its importance has been widely recognized [3 7]. It has been shown that TRI always increases the heat loss from a flame, and that this additional heat loss can reach 6% of the total and more, leading to a reduction in the local gas temperature of 2 o C or more [8]. Erroneous prediction of TRI and thus overall heat transfer can cause severe errors in the estimation of pollution and drop in performance and efficiency of a combustion system. Therefore, in many turbulent combustion systems prediction of radiation and thus TRI is extremely important. Challenges in modeling turbulent combustion are 1) unavailability of accurate, robust, and computationally efficient turbulence models, 2) difficulty in prediction of nonlinear interaction between complex chemistry and turbulence, 3) lack of accurate and computationally efficient radiation models for multi-phase mixtures, and 4) difficulty of modeling TRI. 1.2 Objectives The objective of this research is to advance high-accuracy thermal radiation modeling in combustion system, including spectral radiation properties and deterministic RTE solvers, toward a level that is commensurate with its importance in chemically reacting turbulent flows. The advantages of a finite-volume based deterministic RTE

21 3 solver are: 1) the data structures of an existing flow solver can be used to solve the RTE and hence, the RTE solver matches seamlessly with the flow solver; 2) it is free from any statistical error; 3) it is computationally inexpensive, especially when a time accurate solution is needed; 4) spectral models of various levels of accuracy can be used, depending on the specific nature of the problem. Hence, a high-accuracy finitevolume-based radiation solver package is a necessary component of any comprehensive combustion solver. The objectives of the current research are the following: Development of full-spectrum k-distribution methods for spectral radiation calculations of nonhomogeneous gaseous and particulate-phase (soot, fuel droplets, ash, etc.) participating media, using multi-scale and multi-group based approaches to achieve close-to benchmark line-by-line (LBL) accuracy in strongly inhomogeneous media at a tiny fraction of LBL s computational cost. Compilation of full-spectrum and narrow band based compact k-distribution databases for rapid data retrieval and construction of full-spectrum k g distributions. Development of a k-distribution based portable spectral module, which includes all of the above k-distribution methods and databases. Development of a spherical harmonics based higher order accurate RTE solver (P-3 ) using the data structure of an existing flow solver Study of turbulence-radiation interactions in flames using this high accuracy and computationally efficient spectral radiation solver interfaced with the finite volume/monte Carlo-based turbulence chemistry solver. 1.3 Outline of the Chapters to Follow In addition to this chapter the rest of the thesis is divided into the following sections:

22 4 Chapter 2 will discuss existing methods for radiation calculations in combustion systems 1) RTE solution methods; 2) spectral modeling of radiation using the k-distribution method; 3) the composition-pdf method and how turbulenceradiation interactions are closed by this method. In Chapter 3 the full-spectrum, multi-scale, multi-group k-distribution method for nonhomogeneous gas mixtures will be developed. The accuracy of this spectral model will be assessed by sample radiation calculations for an artificial 1-D medium. Chapter 4 will discuss the mathematical development of the narrow band-based multi-scale k-distribution method for gas particle (soot) mixtures followed by sample calculations for an artificial 1-D medium and a more realistic 2-D combustion scenario. Chapter 5 will comprise the mathematical development of the narrow bandbased multi-scale multi-group k-distribution method for the most general case of combustion, i.e., a gas particle (soot) mixture containing both temperature and concentration inhomogeneities. Sample calculations will be performed to investigate the accuracy the this spectral model in an artificial 1-D medium and a 2-D combustion case involving a sooting flame. Chapter 6 will outline the development of a portable spectral module, called Spectral Radiation Calculations Software (SRCS). SRCS includes all k-distribution models, LBL and gray models for multiphase mixtures. Details of the software architecture, advantages and limitations of various spectral models and usage of the SRCS will be discussed in detail. In Chapter 7 the development of a higher order spherical harmonics-based RTE solver (P-3 ) will be discussed using the data structure of the open-source code flow software OpenFOAM [9]. The accuracy of the P-3 solver will be verified through sample calculations.

23 5 In Chapter 8 the newly developed finite volume/monte Carlo combustion solver (with finite volume-based high-accuracy spectral radiation solver) will be used to simulate jet flames. Comparison of the accuracy and computational time of various spectral models and RTE solvers will be investigated in the context of turbulence-radiation interactions. Chapter 9 will conclude this research by summarizing the important conclusions made in this work and proposing some research topics, which may be important in the future.

24 Chapter 2 Existing Methodologies 2.1 Solution of the Radiation Transport Equation The radiative transfer equation (RTE) for an emitting-absorbing-scattering medium on a spectral basis can be written as [1] di η ds = κ η(φ)i bη ( κ η (φ) + σ sη (φ) ) I η + σ sη(φ) 4π 4π I η (ŝ )Φ η (φ)(ŝ, ŝ )dω (2.1) Here I η is the spectral radiative intensity, κ η the absorption coefficient, I bη the spectral blackbody intensity (or Planck function), and wavenumber η is the spectral variable. The vector φ contains state variables that affect κ η and σ sη, which include temperature T, total pressure P, and gas mole fractions x: φ = (T, P, x). ŝ is a unit direction vector, Φ η is the spectral scattering phase function and Ω the solid angle. The governing equation, Eq. (2.1), is an integro-differential equation for the radiative intensity in five independent variables (three space coordinates and two direction coordinates). One more layer of complexity is added due to the nongray (wavelength dependent) nature of the participating medium. Similarly, the presence of other modes of heat transfer makes the problem more complex, because intensity is coupled to the source term of the overall energy conservation equation in a nonlinear manner.

25 7 Exact analytical solutions to the equation of radiative transfer are exceedingly difficult to obtain and explicit solutions are only possible for very simple situations, such as one-dimensional plane-parallel media without scattering. For more complicated scenarios, which may involve radiative equilibrium, multi-dimensional irregular geometry, anisotropic scattering or inhomogeneous media, etc, several approximate methods have been developed over time. Among these the most popular methods are: 1) the spherical harmonics method (SHM), 2) the discrete ordinate method (DOM) or finite volume method (FVM), 3) zonal method, and 4) Monte Carlo method. While the first three are deterministic in nature, the fourth one is a statistical method. Out of these three deterministic methods, SHM and DOM have received the most attention and have been developed over the years. In both these methods the integro-differential equation, Eq. (2.1), is simplified to spatial differential equations by approximating the direction dependence of intensity. The difference between SHM and DOM is how the directional dependence of the radiative intensity is approximated. Details of all these methods can be obtained from Modest [1] Photon Monte Carlo Method Among all RTE solvers, the photon Monte Carlo (PMC) can be applied to problems of arbitrary difficulty with relative ease [1]. The PMC method directly simulates the physical processes, i.e., emission, absorption, scattering and reflection from which the RTE is derived. In the standard Monte Carlo method, a ray carrying a fixed amount of energy is emitted and its progress is then traced until it is absorbed at a certain point in the participating medium or on the wall, or until it escapes from the enclosure. However, this method is inefficient when the medium is optically thin or walls are reflective [11]. Modest applied the concept of energy partitioning, in which the energy carried by a ray is no longer absorbed all at once, but rather attenuated gradually along its path until its depletion or until it leaves the enclosure [12, 13]. The PMC has been extended for use for continuous media represented by discrete

26 8 particles [14]. This scheme is capable of closing the absorption TRI term during turbulence-radiation interactions without any approximations. The main drawbacks of the PMC are: 1) it is subject to statistical errors and 2) it requires great computational power when very large number of photon bundles are needed to be sampled, e.g., for the case of time-accurate solutions Discrete Ordinate Method The DOM was first proposed by Chandrashekhar [15] with an application in stellar and atmospheric radiation. Although the DOM received little attention in the earlier stages, over the past few years it has seen considerable development. The DOM has been systematically optimized and applied to the solution of general radiation heat transfer problems [16 26]. The DOM is based on a discrete representation of the directional dependence of the radiative intensity with integrals over the total solid angle 4π obtained via numerical quadrature. Thus DOM is basically an application of the finite difference scheme over directions. If directional discretization is applied using solid angle volumes, it is commonly called the finite volume method (FVM) and has been gaining popularity as an RTE solver. Despite the simplicity and methodical development of the DOM, it suffers from several drawbacks. One of the most serious pitfall of DOM is false scattering. False scattering occurs due to spatial discretization error and numerical diffusion. DOM also suffers from the ray effect which is due to the discretization of the directional coordinates. To avoid ray effect the DOM needs equal order of discretization in space and directions, which makes it computationally very expensive when ray effect free solution is demanded. Finally in optically thick regimes the DOM loses its accuracy and is known to converge very slowly.

27 Spherical Harmonics Method The SHM was first proposed by Jeans [27]. The underlying principle of the SHM is similar to the moment method [1], and moments are taken in such a way so as to take advantage of the orthogonality of spherical harmonics. In the SHM (like DOM) the integro-differential equation, Eq. (2.1), is reduced to a relatively simple set of spatial partial differential equations while approximating the directional dependence of the radiative intensity. The directional dependence is separated from the spatial dependence by a generalized Fourier series expansion. Various orders of accuracy can be obtained by truncating the series after the desired number of terms [1]. The advantages of SHM are: The relatively simple partial differential equations reduced from the RTE, Eq. (2.1), can be solved using standard PDE solver packages, making it easy to implement using standard flow solver data structures. Since the SHM is a spectral method, it should require fewer terms than DOM for similar accuracy. The SHM is free from any ray effect due to the complete decoupling of space and direction. Unlike DOM/FVM, CPU time requirements for SHM do not increase for scattering media and reflecting walls, nor for optically thick media. The drawbacks of the SHM method are: The mathematical complexity increases rapidly if higher-order P-N -approximations for multi-dimensional geometry are desired, which is probably responsible for the fact that its development lags behind that for the DOM/FVM. SHM loses accuracy where strongly nonisotropic distributions of radiative intensity exist; such as intensity emanating from the wall, or an optically thin

28 1 medium acting as a radiation barrier between hot and cold surfaces, or in the presence of collimated irradiation [1]. The P-1 method has so far been the most popular RTE solver within the SHM framework because of its simplicity. But its accuracy is questionable as it is the lowest order SHM RTE solver. To alleviate the problems associated with the P-1 method, several strategies have been developed. Out of these, the modified differential approximation (MDA) [28 31] and improved differential approximation (IDA) [32 34] are widely recognized. Recently, Modest and Pal have developed the advanced differential approximations (ADA) in which the directional gradient of the intensity at the wall is minimized [35]. This makes the intensity distribution continuous for the P-1 method and mostly continuous for higher order P-N methods and, thus, reduces the loss of accuracy due to boundary emission. To achieve better accuracy, a number of higher-order P-N -approximations have been formulated for specific geometries by exploiting the symmetry of the media confined in either one-dimensional slabs, concentric cylinders or concentric spheres [36, 37]. The P-3 method was developed by Mengüç and Viskanta in a 3-D Cartesian coordinate system and for axisymmetric cylindrical geometry [38, 39]. Tong and Swathi [4] obtained higher-order solutions up to P 11 for an anisotropically scattering medium between one-dimensional concentric spheres. Ou and Liou [41] derived the three-dimensional P-N -approximation for arbitrary coordinate systems resulting in a set of (N + 1) 2 complex, coupled, first-order PDEs similar to DOM, without, however, discussing boundary conditions. Recently, Modest and Yang formulated a generic methodology that decomposes the RTE into N(N + 1)/2 coupled second-order elliptic partial differential equations for a given odd order N, allowing for variable properties and arbitrary three-dimensional geometries, including a set of generic boundary conditions [42, 43].

29 Spectral Modeling of Radiation Traditionally, radiation in combustion systems has been treated using gray models due to their simplicity 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 combustion calculations [44 5]. Spectral radiation models in combustion systems may be loosely grouped into the following methods: 1) line-by-line calculations (LBL); 2) band models; 3) global models; and 4) k-distribution methods (which can be formulated as narrow band and global models) LBL and Band Models LBL spectral calculations are performed using very detailed knowledge of every single spectral line, usually taken from high resolution spectroscopic databases. LBL calculations are the most accurate, but they require vast amounts of computer resources. Due to randomly varying spectral properties of media, LBL calculations [51 53] require in excess of one million spectral solutions to the radiative transfer equation (RTE). This is undesirable even with the availability of powerful supercomputers, since radiative calculations are only a small part of a sophisticated combustion code. Many studies have been devoted to narrow and wide band models. When calculating spectral radiative fluxes from a molecular gas one finds that the gas absorption coefficient (and with it, the radiative intensity) varies much more rapidly across the spectrum than other quantities, such as blackbody intensity, etc. It is, therefore, in principle possible to replace the actual absorption coefficient (and intensity) by smoothened values appropriately averaged over a narrow spectral range. A number of narrow band models have been developed some 4-5 years ago. On the other hand, wide band models make use of the fact that, even across an entire vibration-rotation band, blackbody intensity does not vary substantially. In principle, wide band calculations are found by integrating narrow band results across an entire band, resulting

30 12 in only slightly lesser accuracy. The limitations of these models are - 1) difficulty in applying them to nonhomogeneous gases and for enclosures that have nonblack walls and/or in the presence of scattering particles; 2) wide band correlations have a typical correlational accuracy of 3%, and in some cases may be in error by as much as 7%. A detailed description of band models can be obtained from the textbook by Modest [1] Global Models The most common global method is the so-called Weighted- Sum-of-Gray-Gases model. The concept of the WSGG approach was first presented by Hottel and Sarofim [54] within the framework of the zonal method. The method could be applied to arbitrary geometries with varying absorption coefficients, but was limited to nonscattering media confined within a black-walled enclosure. Modest [55] has shown that this model may be generalized for use with any arbitrary RTE solution method. In this method the nongray gas is replaced by a number of gray gases, for which the heat transfer rates are calculated independently by solving the RTE with weighted emissive powers for each of the gray gases. The total heat flux is then found by adding the fluxes of all gray gases. The different absorption coefficients κ i and emissive power weight factors for each gas were found from total emissivity data. Denison and Webb [56 59] improved on the WSGG model by developing the Spectral-Line- Based Weighted-Sum-of-Gray-Gases (SLW) model with absorption coefficients based on detailed spectral line data. They also extended the SLW model to nonisothermal and nonhomogeneous media by introducing a cumulative distribution function of the absorption coefficient, calculated over the whole spectrum and weighted by the Planck function. The absorption distribution function (ADF) approach developed by Pierrot et al [6, 61] is almost identical to the SLW model and differs from the SLW only in the calculation of the gray-gas weights. These weights are chosen in such a manner that emission by an isothermal gas is rigorously predicted for actual spectra. This

31 13 method has been further generalized [61] by introducing fictitious gases (ADFFG) employing a joint distribution function that separates the κ η into two or more fictitious gases, and is designed to be more suitable for the treatment of nonhomogeneous media k-distribution Method An alternative to the traditional spectral models discussed above is the so-called k- distribution method. It was observed that over a small spectral interval the Planck function and nongaseous radiation properties remain essentially constant, while the gas absorption coefficient varies wildly even across a very narrow spectrum, attaining the same value many times (at slightly different wavenumbers), each time resulting in identical intensity and radiative flux. It would, therefore, be advantageous to reorder the absorption coefficient field into a smooth, monotonically increasing function, assuring that each intensity field calculation is carried out only once. This is the basic underlying principle of the k-distributions. Today s k-distribution methods come in two different versions: 1) narrow band based k-distributions [62 68], and 2) the full-spectrum k-distribution method [53, 69]. The narrow band-based correlated k-distribution (CK) method uses the fact that inside a spectral band η, which is sufficiently narrow to assume a constant Planck function, the precise knowledge of each line position is not required for the computation. The full-spectrum k-distribution (FSK) approach has been developed based on Weighted-Sum-of-Gray-Gases arguments. Through these arguments the SLW and ADF models have been shown to be simply crude implementations of the FSK method [53]. Whereas the SLW and ADF methods are weighted-sum-of-gray-gases approaches, i.e., the assumed to be correlated absorption coefficient is reduced to a few discrete values, the FSK method distinguishes itself in that it is an exact method for a correlated absorption coefficient, utilizing a continuous k-distribution over the whole spectrum, and this allows a quadrature scheme of arbitrary order of accuracy to be

32 14 employed. The FSK method achieves LBL accuracy for homogeneous media with only a tiny fraction of LBL s computational cost. Since its introduction, the FSK method has undergone several major developments, including the formal mathematical development of full-spectrum correlated-k (FSCK) and scaled-k (FSSK) methods [69], development of the multi-scale FSK (MSFSK) method [7 72], and the multi-group FSK (MGFSK) method [73, 74]. FSCK, FSSK, MSFSK, and MGFSK methods are designed to apply the FSK scheme to radiative transfer problems in inhomogeneous media, which remains a challenging topic for reordering models and any other global methods. The challenge is that spatial inhomogeneities in total pressure, temperature, and component gas mole fraction (partial pressure) change the spectral distribution of the absorption coefficient, which is critical to the FSK reordering process. The effect of varying total pressure on the FSK reordering process is relatively small, as evidenced by the success of applying the correlated-k method in the field of meteorology, where strong total pressure variations occur while temperatures stay relatively uniform [62 64]. The effect of varying temperature can be substantial, as first recognized by Rivière et al [65 67]: Spectral lines that are negligible at room temperatures can become more and more important at elevated temperatures, giving rise to the so-called hot lines. The effect of varying gas concentrations can also be substantial, as first recognized by Modest and Zhang [53]: At one spatial location the absorption coefficient may be dominated by one species, but by another (with totally different spectral lines) at a different location. The FSCK and FSSK methods can produce accurate results for media that have large gradients in total pressure but small variations in temperature and partial pressure. For media that have large temperature and partial pressure variations, sophisticated MSFSK and MGFSK methods have been developed to improve the accuracy of the FSK method. The MSFSK method deals with the inhomogeneity problem by grouping individual spectral lines comprising the absorption coefficient into M separate scales according to their temperature dependence [7]. The overlap

33 15 in spectrum between different scales is treated in an approximate way so that a number of M independent radiative transfer equations (RTEs), instead of M 2 required for the fictitious gas approach [7], need to be solved. In contrast to the MSFSK method, which assigns spectral lines to scales, the MGFSK method places spectral positions into M separate groups according to their temperature and partial pressure dependencies [73, 74]. This avoids the problem of overlap among different groups and, therefore, also requires only M RTE evaluations but without any further approximation. However, with the MGFSK method it appears impossible to obtain k-distributions for arbitrary gas mixtures from mixing those of individual species. The need for mixing comes from practical considerations, since it would be impossible to precalculate and database the infinite number of possible k g distributions that are required in a practical heat transfer calculation. In practice, the k g distributions of individual gases are precalculated and then are mixed during the calculation to obtain local mixture k g distributions. Although the MGFSK method can achieve great accuracy for individual gases, groups from different gases are found to be incompatible, making its application to inhomogeneous mixtures problematic. The MSFSK method, on the other hand, can treat the absorption coefficient of an individual species in a mixture as one of its scales. The problem of mixing FSKs of individual species in a gas mixture is thus transformed into a problem of resolving the overlap among scales. To date the basic FSK method has been modified to allow accurate mixing of inhomogeneous gaseous media only. Hence, using recent advanced FSK methods accurate radiation calculations can be done only for very limited cases of combustion systems, e.g., nonluminous flames with gaseous fuels, such as methane or ethylene. Most combustion systems, in general, consist of multi-phase mixtures comprising gaseous and/or liquid fuel droplets, gaseous combustion products, soot, ashes, etc. Thus, accurate radiation calculations in general combustion systems, e.g., luminous flames, spray flames, etc, still remain challenged using the currently available advanced FSK

34 16 methods. Hence, further advancement of k-distribution methods is needed to accurately treat the most general cases of combustion problem k-distribution Databases FSK calculations are very accurate and time efficient provided the required fullspectrum k-distributions are known, which, however, are tedious to compile from spectroscopic databases, such as HITRAN [75], HITEMP [76] and CDSD-1 [77]. Several very approximate correlations have been generated by Denison and Webb [57, 59] and Modest et al. [53, 7, 73, 74]. Such correlations are of limited accuracy, and assembling mixture distributions from those for individual species has proven problematical [78]. However, 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 have compiled a highly accurate, compact database of narrow band k-distributions for the most important combustion gases, from which highaccuracy full-spectrum k-distributions can be obtained efficiently for arbitrary mixtures of combustion gases, including nongray absorbing and/or scattering particles [79]. Full-spectrum multi-group databases have been constructed by Zhang and Modest for carbon dioxide and water vapor [73, 74]. The spectral positions of each gas were placed into 32 exclusive spectral groups depending on their temperature and partial pressure dependence. The absorption coefficients at the 32 group level satisfy the scaling approximation. The spectral groups from the databases are combinable, i.e., for better numerical efficiency, the spectral groups can be combined to reduce the number of groups. It has been recently reported by Wang and Modest [8] 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 [73, 74] the absorption coefficients were obtained from the HITEMP spectroscopic database. Unfortunately, it has been found that

35 17 the HITEMP database is not accurate for CO 2 at temperatures higher than 1 K [77, 81] Radiation from Soot Soot radiation constitutes an important part of radiation calculations in luminous flames. Unfortunately, soot radiation in combustion flames has mostly been treated using the optically-thin approximation with the assumption of gray soot [44] for simplicity. Nongray soot has been investigated by Solovjov and Webb [82] using the SLW method and by Wang et al, who employed the single-scale FSK method [45]. Solovjov and Webb treated nongray soot as an additional nongray gas species and the multi-component gas mixture with soot was treated as a single gas within the SLW method [82]. To include nongray soot in the single-scale FSK method [45], the local mixture k g distributions, consisting of contributions from gas-phase species and soot particles, can be constructed in two ways: (1) by summing the gas-phase and soot absorption coefficients in an LBL manner from spectroscopic databases, and (2) by mixing precalculated single-gas k-distributions (SGKs) according to the local gas mixture composition; the contribution from soot particles is then incorporated into the local gas mixture k g distributions. While the first method is computationally expensive, the second method can produce local k g distributions on the fly. Local k g distributions from the SGKs can be constructed in two formats, viz., full-spectrum and narrow band k-distributions. In the full-spectrum format, soot must be treated as gray and its spectrally averaged mean property (a constant) can then be added directly to full-spectrum k-distributions of the gas mixture. In the narrow band format, the mixing of component SGKs is performed at the narrow band level and this allows consideration of nongray soot particles: absorption due to nongray soot can be added directly to the narrow band k-distributions of the gas mixture, since the soot absorption coefficient is essentially constant across each narrow band [83]. In realistic combustion problems, soot is found to be present only close to the flame (locally),

36 18 rather than everywhere in the domain, i.e., there are strong soot concentration gradients in the domain. For most combustion flames local peak soot volume fraction is in the order of parts per million (ppm), in which case the radiation from the gases is comparable to the radiation from the soot. For such cases the accuracy of radiation calculations using the single scale FSK method can be severely challenged, whereas the MSFSK method has already been shown to accurately accommodate mixing of gas species with concentration gradients. Although mixing of nongray soot with gases has been performed using the single-scale FSK and SLW methods, additional problems are encountered when the MSFSK method is employed (for inhomogeneous nongray gas soot mixtures). 2.3 The Composition PDF Method The concept of a probability density density function (PDF) has been applied in turbulent reacting flows for many years, motivated by the difficulties in the closure of the chemical source terms by moment methods [84], which demand a large number of partial differential equations to be solved simultaneously. The significant difficulties encountered in extracting the mean reaction rates are due to the fact that the terms are highly nonlinear and the fluctuations so large that the higher moments of the fluctuations cannot be neglected without introducing errors. An alternative to moment closures is the PDF closure, which was initiated by Dopazo and O Brien [85], and later developed by Pope [86], Kollmann [87] and Dopazo [88]. A comprehensive description of the PDF methodology has been given by Pope [86] and by Haworth [89]; PDF methods have established themselves as viable alternatives to conventional deterministic closure models and numerical techniques. Kollmann [87] has given an excellent review on functional formulations, PDF modeling and applications to both low and high Mach number combustion flows. Dopazo [88] has given a good review of developments in PDF methods. There are two different ways to obtain the desired probability density function.

37 19 The most common approach had been to assume the PDF to have a particular shape that is parameterized by its first and second moments. These first and second moments are then obtained by solving modeled transport equations. Measurements of the PDF s of scalar quantities have been reported for a wide range of flows including plug flow reactors, mixing layers, jets, wakes and boundary layers [9]. A more powerful technique is to obtain the joint PDF of the scalars directly from their transport equations. This eliminates the need to specify PDF shapes. This method is referred to as the transported composition joint PDF method or simply the composition PDF method [86]. The composition PDF method has been widely used in the study of the turbulent reacting flows. The compostion PDF is not a self-contained model. The mean momentum equations must be solved for the velocity field ũ, and a turbulence model (such as k-ε) is required to determine both turbulent diffusivity and the mixing rate used in the stochastic mixing model [91, 92]. More advanced PDF methods have been developed to include additional independent variables into the PDF evolution equation such as the velocity compostion PDF method and the velocity compostion dissipation joint PDF method [86, 93, 94]. However, these are still relatively premature and/or too mathematically complex The Transported Composition PDF Method In composition PDF methods all the scalars necessary to establish the thermodynamic state of the reacting mixture (i.e., species mass fractions and enthalpy or temperature) are treated as random variables. The composition PDF then quantifies the probability of having a certain gas composition (including temperature) at each location in the computational domain. The variances and higher moments of all orders can be calculated directly from the PDF, as well as the means, which are the zeroth moments. The most remarkable feature of the PDF method is that one-point nonlinear sources, such as chemical reactions, can be evaluated exactly no matter how complicated they are. As an extension, any term in the equations, as long as it is

38 2 a function of the local scalar variables only, can be evaluated exactly by composition PDF methods. This is in contrast to conventional moment methods, where the mean reaction rates can be determined only under special constraints of linearity or when the reactions are very fast or very slow compared to the turbulent time scales. Modest and coworkers [95 97] used this fact to accurately model turbulent radiative emission. This is possible because the nonlinearity of radiative emission (a function of the local temperature only) and absorption coefficient (a function of the local composition variables only) can be treated exactly. Mazumder and Modest [98] applied the velocity composition PDF, solved by the Monte Carlo method, to investigate the turbulence-radiation interactions in nonluminous flames, qualitatively demonstrating the importance of the absorption coefficient temperature correlation. Similarly, Modest, Haworth and coworkers [97, 99 12] used the composition PDF/Monte Carlo method to study the radiative heat transfer in reactive flow The Composition PDF Formulation In the PDF method any one-point statistic of any quantity Q, which is only a function of the scalar field φ, can be evaluated directly from the PDF. The set of scalars considered includes all those necessary to specify the local thermodynamic state of the reacting mixture; species mass fractions and enthalpy are often an appropriate set. The quantity Q then can include the absorption coefficient κ, specific heat c p, density ρ and any other quantities that depend only on φ. The following identities are the basis of evaluating mean quantities and fluctuationaround-mean quantities (both conventional and Favre averages) Q = f(ψ)q(ψ)dψ (2.2) ρ Q(φ) = Q = Q Q (2.3) f(ψ)ρ(ψ)q(ψ)dψ (2.4)

39 21 Q = Q Q (2.5) where ψ represents the composition space vector and f(ψ) is defined as the probability density of the joint event ψ = φ (i.e., ψ 1 = φ 1, ψ 2 = φ 2,, ψ n = φ n ). This also means that, f(ψ)dψ = Probability ( ψ φ < ψ + dψ ) (2.6) f(ψ) is known as the composition PDF and is the simplest form used in the PDF methods. It carries information only about the composition variables. It does not carry information about the velocity fields, or turbulence time and length scales. It is also a one-point PDF, i.e., it carries information about the probability distribution only of a single point. However, it contains complete information about the scalars at that particular point. The biggest advantage is that any quantity Q can be represented exactly, even if it is a highly nonlinear function of the composition vector φ PDF Equation for General Scalar In turbulent reacting flow the composition PDF is also a function of space x and time t. The transport equation for the PDF f(ψ; x, t) can be derived [93] from the conservation laws of scalars, which for a composition variable φ α (x, t) is given generically by ρ Dφ α Dt = J α i x i + ρs α (2.7) where D/Dt is the material derivative, J α isthe molecular diffusive flux of φ α, and S α is the rate of production of φ α. The transport equation for the mass density function F(ψ; x, t) = ρ(x, t) f(ψ; x, t) can be obtained from [86, 99]: F t + x i [ũf] + [S α (ψ)f] = [ ] u i ψ F + ψ α x i ψ α [ ] 1 Ji α ψ F ρ x i (2.8)

40 22 In Eq. (2.8), summation is implied over repeated indices i and α within the terms and i and α represent physical and composition space components, respectively. The notation A B denotes the conditional probability of event A given that B occurs. The first two terms in Eq. (2.8) are transport of the mass density function F when following the Favre averaged mean flow. The third term is the divergence of the flux of probability in composition space due to sources of the first kind, e.g., chemical reaction and radiative emission and absorption. This equation also shows that no matter how nonlinear these sources are, they can be represented exactly Modeled Terms in Composition PDF The terms on the right-hand side of Eq. (2.8) must be modeled. The first term on the right-hand side represents transport of the mass density function F(ψ; x, t) in physical space due to turbulent velocity fluctuations. Since the joint composition PDF contains no information about the velocity field, the conditional expectation of u i ψ needs to be modeled. This is generally done based on a gradient-diffusion model with information supplied for the turbulent flow field by a flow solver as [86] F u i ψ F Γ t (2.9) x i where the turbulent diffusivity Γ t is estimated by an ad-hoc relationship, which is derived from dimensional analysis and other considerations [86]. It is expressed as Γ t = C µ ρ σ φ k 2 /ɛ = µ T σ φ (2.1) where µ T and σ φ are, respectively, the modeled turbulent diffusivity in the standard two-equation k-ε model and a turbulent Schmidt or Prandtl number. The second term on the right-hand-side of Eq. (2.8) represents transport in scalar (composition) space due to molecular transport. This modeled term becomes very important when dealing with reactive flows, and has been one of the main limitations

41 23 of PDF methods for such flows. One of the simplest models proposed by Dopazo [85] is interaction by exchange with the mean, or the IEM model, also known as the LMSE (least mean square estimation) model. Pope [86] developed a particle-pairing model and Euclidean Minimum Spanning Tree (EMST) model [13, 14]. If a particlepairing model is used, then ψ α [ ] [ 1 Ji α ψ F 2C φ ω 2 ρ x i F ( ψ + ψ ) F ( ψ ψ ) dψ F ( ψ )] (2.11) The final modeled transport equation for the mass density function F can be written as F t + [ ] u i F + [ Sreaction (ψ)f ] + [S α,radiation F] = x i ψ s ψ α x i [ ] F Γ t + F mix x i (2.12) where F mix is the flux due to molecular mixing and is obtained from a mixing model. This equation can be solved using Monte Carlo methods as discussed in Section Solution Procedure for Composition PDF In principle, the PDF transport equations, Eqs. (2.8) and (2.12), can be solved by traditional finite-volume and finite-element methods. This approach was used by Janicka et al [15] for a jet diffusion flame. In general, however, the PDF is a function of a large number of independent variables, which makes it prohibitively expensive to use standard finite volume methods (FVM) and finite element methods (FEM) methods to solve Eq. (2.12). Monte Carlo approaches to the solution of the PDF equation have, therefore, been developed [86]. The basic idea is to represent the PDF by a sufficiently large number of notional fluid particles (point masses), each of which can be interpreted as a delta function discretization of the PDF, and these particles evolve with time according to a set of stochastic Lagrangian differential equations [86]. The mean quantities at any point in the domain are then calculated as an ensemble average of the particles in a sufficiently small neighborhood. By its nature,

42 24 the particle tracking method is grid-free; however, in general, a grid is required to extract the statistics of the scalars at discrete locations. The algorithm was developed by Pope and coworkers [86, 16, 17] and by Subramaniam and Haworth [18]. An algorithm was developed by Zhang and Haworth [19] to ensure consistency between the flowfield values obtained by a FV solution and the Monte Carlo solution. All use a coupled flow solver to provide information about the velocity field. Conventional particle tracing suffers from various weaknesses, which have been identified and described in detail by Li [99]. The imbalance in particle counts due to large variation in cell size has been targeted by Subramaniam and Haworth [18], who proposed a scheme to control the particle number density in each cell of the computational domain. Li and Modest [97, 99, 11] developed a scheme to handle the limitation of using constant time steps. Instead of using a single time step in integration of all particle equations for each particle, they used locally divided adaptive substeps governed by the local flowfield. To reduce statistical error and further improve the efficiency, a particle splitting and recombination procedure has also been used [99]. It can be shown that the equation for the mean of any quantity Q = Q(φ) can be obtained by multiplying the PDF transport Eq. (2.12) by the composition variable ψ α, followed by an integration over the entire ψ space. Thus, one obtains for a mean composition variable, [ ρ φ ] ] [ ] α [ ρ ũ i φα + = φ α Γ T + ρ t x i x i x S α,reaction δ α,s [ q R ] (2.13) i Instead of solving Eq. (2.13), as is done in moment methods, a stochastic approach is employed to solve for the PDF. For a modeled mass density PDF function, Eq. (2.12), the corresponding particle equations for location x and composition φ are governed by the following stochastic equations: dx (t) = [ũ + Γ T / ρ ] x (t) dt + [2Γ T / ρ ] 1/2 x (t) dw (2.14)

43 25 dφ α(t) = S α,reactiondt + F mix + δ αs S radiation ρ dt, α = 1,..., s (2.15) where F mix is the mixing model term, S radiation is the radiative heat source calculated for the particle. In these equations, the species are numbered from α = 1,..., s 1 and the s th scalar is the enthalpy. The radiation source term only affects the s th scalar and does not contribute to the transport of other scalars directly. Variables with an asterisk refer to values of a Lagrangian particle, W is an isotropic vector Wiener process and Γ T is the turbulent diffusivity given in Eq. (2.1), and can be obtained by using a different turbulence model as well. During solution a large number of particles are traced according to the set of stochastic equations, Eqs. (2.14) and (2.15). The PDF is then obtained approximately as the histogram of the particles properties in sufficiently small neighborhoods in physical space, and the means of the flow field are deduced statistically from the properties of these particles. The particle tracking scheme developed by Subramaniam and Haworth [111] is adopted here. It is suitable for three-dimensional unstructured stationary or deforming meshes with relatively large Courant numbers (> 1) and large variations in the finite-volume element size (up to factor of 1 6 variation in element volumes). Particles are tracked from element to element using trilinear basis functions. Details can be found in [19, 111]. To maintain an acceptable distribution of particles as they move in physical space, particle splitting and recombination is performed. If the number of particles in a particular cell falls below a preset N min, then a particle of mass m is split into two particles with mass m /2, each having the same properties as the original particle. The largest-mass particles are cloned in each control volume. This conserves the mean scalars at the element level and also the higher-order moments. If the number of particles in a cell exceeds a maximum N max, then particles are annihilated (maintaining global mass consistency) with preference given to low-mass particles. An important issue in PDF/Monte-Carlo methods is the estimation of means. Estimates of mean quantities from particle values are needed to evaluate Favre means,

44 26 which also appear as coefficients in the particle evolution equations. The straightforward method of taking ensemble averages over particles in a cell was found to be inaccurate [86]. Various approaches have been used to estimate the mean fields from a notional particle field [111]. Currently, field means are estimated by a particlecloud-in-cell method, in which a grid system sits on top of the computational domain and local means are approximated by weighted averages of the values of particles contained in a nearby region. Thus, the field means at a nodal point V can be calculated from: φ v = 1 b pv m m pφ ; p v m v = p C b pv m p (2.16) p C where b pv are linear basis functions that act as appropriate weight functions. 2.4 Turbulence Radiation Interactions The radiation source term in Eq. (2.13) can be expressed as [1] S radiation = q R = κ η ( 4π I η dω 4πI bη ) dη (2.17) where q R is the radiative flux. The radiative intensity I η in Eq. (2.17) is governed by the RTE, Eq. (2.1). Only low-mach-number flows are considered here so that radiative energy does not contribute to the mass and momentum equations as the fluid velocity is very small. For Reynolds average simulations (RAS), Eq. (2.17) is averaged to yield S radiation = ( 4π ) κ η I η dω 4π κ η I bη dη (2.18) The absorption coefficient κ η depends on both species concentrations and the temperature so that, in general, κ η I bη = κ η I bη + κ ηi bη κη ( x, h ) I bη, (2.19)

45 27 I bη = I bη ( T ), (2.2) and κ η I η = κ η I η + κ ηi η κη ( x, h ) I η. (2.21) The essence of TRI modeling is to accurately estimate the left-hand sides of Eqs. (2.19) and (2.21). In these equations, κ ηi η represents the correlation between the fluctuating absorption coefficient and spectral incident intensity, and κ ηi bη represents the correlation between the fluctuating absorption coefficient and the local blackbody intensity. Following Li [99], these two correlations are loosely defined as absorption coefficient incident intensity correlation and absorption coefficient Planck function correlation, respectively TRI Modeling Depending on which spectral model is used to solve the RTE and the averaging process that is adopted, there will be different unclosed terms evolving from Eqs. (2.19) and (2.21). However, as pointed out by Li and Modest [97, 99, 11], the unclosed terms can be grouped into two categories: (a) correlations that can be calculated directly or indirectly from local scalars φ; and (b) correlations that cannot be calculated from local scalars. Here, the local scalars include all composition variables like species concentrations and temperature, absorption coefficients, etc. Using the composition PDF method it is possible to accurately evaluate terms from group (a). However, there are major difficulties in modeling the terms belonging to group (b) [112]. Among current state-of-the-art methods, only the photon Monte Carlo method seems to have potential to completely account for absorption TRI. All other approaches use the so-called Optically Thin Fluctuation Assumption (OTFA), which neglects absorption TRI with some justification.

46 Optically Thin Fluctuation Approximation The absorption coefficient intensity correlation has been generally neglected by most of the research community, partly due to the difficulties in its modeling. The most commonly used justification in the literature was provided by Kabashnikov and Myasinkova [113]. They suggested that if the mean free path for radiation is much larger than the turbulent length scale then the local incident radiative intensity would only be weakly correlated with the local absorption coefficient. Thus one can then assume κ η I η (2.22) so that κ η I η κ η I η (2.23) where κ η is the absorption coefficient self-correlation. Kabashnikov and Myasinkova provided several conditions for the validity of this thin-eddy approximation [113]. In general, however, the thin-eddy approximation will depend on the assumption that the optical thickness of the turbulent eddies is small κ η l 1 (2.24) where l is an appropriate turbulent eddy length scale. The validity of this assumption will depend on the eddy-size distribution and radiative properties of the absorbing gases. In the numerical simulation of a combustion chamber Hartick [114] showed that the thin-eddy approximation may not be valid over some highly absorbing spectral regions. This observation suggests that the optically thin-eddy approximation may result in miscalculation of the re-absorption of radiative energy when the medium has fluctuating radiative properties. As mentioned by Li [99], the fact that the radiative intensity I η does not depend only on the local compostion makes this a very challenging and difficult problem. This is also the main reason why almost all researchers accept OTFA as reasonable and use it in their analysis. However, the degree of

47 29 approximation introduced is still unknown. Modest and Mehta [112] obtained expressions for the absorption TRI using a diffusion approximation, which is valid in optically very thick parts of the spectrum. Only a temperature dependent absorption coefficient was considered. It was shown that absorption TRI may not be negligible over certain parts of the spectrum, though the overall contribution to radiative transfer may not be significant [112] Review of TRI Modeling In recent years, much progress has been made in the modeling of TRI. An exhaustive review of the literature up to 21 has been given by Li [99]. Over the past few decades, experimentalists like Faeth and Gore and coworkers [ ] have conducted studies on both luminous and nonluminous flames with different fuels. Depending on the fuel and other flow conditions, they have shown that radiative emission from a turbulent flame may be as much as 5 to 3% higher than what is predicted from the mean values of temperature and absorption coefficient. Nelson [ ] studied an idealized, single-step, irreversible reaction of fuel and oxidizer to yield a radiating product. A mixture fraction approach was used and turbulent fluctuations were emulated by considering a two-parameter assumed PDF for the mixture fraction. Band radiation was studied, and it was concluded that the Planck function fluctuations dominated the radiative heat transfer rates entirely and the effects of fluctuations of the radiation properties of the medium (like species concentration, absorption coefficient etc) was much weaker. Thus, it was suggested that it could be possible to adequately predict the effect of fluctuations on radiative transfer by accounting for variations in the Planck function alone, and all other parameters can be based on the mean values of temperature and absorption coefficient. Soufiani and Taine [122] and Soufiani et al [123] investigated similar effects on radiative flux for a channel flow of radiating, but nonreacting, gas and found that temperature fluctuation effects on mean radiative fluxes were limited to about 1%

48 3 in such flows. A similar conclusion was reached by Mazumder and Modest [124], who used a joint velocity composition PDF method with no assumptions about the PDF of fluctuating scalars. Large Eddy Simulation (LES) of a similar channel flow was done by Singh [125] who also reported very small differences between radiative fluxes calculated using mean values and those calculated using full TRI correlations. Radiative calculations accounting for TRI and their effects feeding back to the flow field were first carried out by Song and Viskanta [126], who investigated combustion in a two-dimensional furnace. Hartick et al [114] extended this approach to an enclosed diffusion flame with assumed PDF shapes for the mixture fraction and chemical heat release rate. They concluded that turbulence radiation interactions had a small impact on the spatial distribution of temperature and other scalar fields. However, the drop in temperatures caused significant effects on local nitrogen oxide production. Mazumder and Modest [95, 96] first introduced transported PDF methods, which actually solve for the PDF instead of assuming one, to investigate TRI in methaneair diffusion flames, solving a modeled joint velocity composition PDF equation. A similar approach was taken by Li and Modest, except that they used the joint composition PDF method [97, 99, 11]. They also studied methane air diffusion flames and well documented experimental flames like the Sandia-D, Sydney-L flames, etc. Coelho et al [127] used a discrete ordinates method to analyze TRI for nonluminous flames. A second order closure of turbulence and a steady laminar flamelet model was used for combustion modeling of a methane air diffusion flame [128]. An assumed beta PDF was used to model mixture fraction fluctuations. A photon Monte Carlo method can potentially account for all TRI including absorption TRI. A first attempt in that direction was done by Tessé et al [129], who calculated radiative transfer for a sooty, turbulent ethylene air diffusion jet flame using a photon Monte Carlo method together with a correlated-k (CK) model for the gas properties. In their TRI model, the radiative properties of the assumed homogeneous turbulent structures are randomly obtained from a multidimensional PDF of the reaction progress variable,

49 31 mixture ratio and the soot volume fraction. They concluded that TRI yielded an increase of radiative heat loss of about 3%. The main limitation of their approach was that the turbulence structures were modeled using ad-hoc arguments. A new approach based on photon Monte Carlo method for media represented by particle fields has been developed by Wang and Modest [14]. This approach removes the limitation of assumed turbulent structures and models TRI in a most comprehensive manner.

50 Chapter 3 Full-Spectrum Multi-Scale Multi-Group k-distribution Method 3.1 Introduction The full-spectrum correlated k-distribution method is a high-accuracy spectral model. Although exact for homogeneous media, the accuracy of the full-spectrum correlated k-distribution method is challenged in the presence of severe temperature and concentration inhomogeneities in gas mixtures [7 73, 8]. To alleviate the mixing problem of gases in a combustion system containing strong concentration and temperature inhomogeneities, a new hybrid multi-scale multi-group FSK (MSMGFSK) method is developed [8, 13, 131]. The MSMGFSK method resolves the absorption coefficient of an individual species in a mixture as one of its scales. Within each scale, the wave numbers are placed into exclusive spectral groups according to their temperature dependence. Mixing of species is addressed by introducing an overlap parameter to approximate the effect of overlap among scales.

51 Hybrid MSMGFSK Approach Although the multi-scale multi-group full-spectrum correlated k-distribution (MSMGFSK) method can be easily extended to include gray absorbing and scattering particles, for brevity a medium consisting of a mixture of molecular gases is considered and the radiative transfer equation (RTE), Eq. (2.1), for this medium can be written as [1] di η ds = κ η(φ)(i bη I η ), (3.1) subject to the boundary condition at s = : I η = I wη. (3.2) The boundary intensity I wη may be due to emission and/or reflection from the enclosure wall [132]. The total absorption coefficient κ η is first separated into contributions from N component gases, e.g., CO 2 and H 2 O, and then the spectral locations of the n-th gas absorption coefficient are sorted into M exclusive spectral groups, i.e., κ η = N M n n=1 m=1 κ nmη, I η = N M n n=1 m=1 I nmη, (3.3) and the radiative intensity I η is broken up accordingly. Note that the spectral locations constituting the m-th group may not be consecutive. The RTE is then transformed into N M n component RTE s, one for each group of each gas or scale: n=1 di nmη ds = κ nmη (φ)i bη κ η (φ)i nmη, for n = 1,, N; m = 1,, M n. (3.4) It is observed, physically, that the intensity I nmη is due to emission from the m-th group of the n-th gas species (the nm-th group) but subject to absorption by all groups of other gases plus its own group. There is no overlap among groups of a single species

52 34 and, therefore, there is no emission over wavenumbers where κ nqη (q m) absorbs. Thus in Eq. (3.4) κ η = κ nmη + N M l κ lqη. (3.5) l n q=1 We now apply the FSK scheme [1] to the RTE of each group: first Eq. (3.4) is multiplied by Dirac s delta function δ(k nm κ nmη (φ )), followed by division with f nm (T, φ, k nm ) = 1 I bη (T ) I bη (T )δ(k nm κ nmη (φ ))dη. (3.6) where φ and T refer to a reference state. The resulting equation is then integrated over the entire spectrum, leading to di nmg ds = k nm a nm I b λ nm I nmg, for n = 1,, N; m = 1,, M n, (3.7) where I nmg = I nmη δ(k nm κ nmη (φ ))dη f nm (T, φ, k nm ), (3.8) g nm = k nm f nm (T, φ, k)dk, (3.9) a nm = f nm(t, φ, k nm ) f nm (T, φ, k nm ), (3.1) λ nm I nmg = k nm I nmg + ( l n. ) ( ) Ml q=1 κ lqη(φ) I nmη δ k nm κ nmη (φ ) f nm (T, φ, k nm ) dη. (3.11) Here the absorption coefficient within each group has been assumed to be correlated. This implies that k nm = k nm (T, φ, g nm ) is evaluated from the k-distribution of the

53 35 local absorption coefficient of the nm-th group weighted by the Planck function at the reference temperature [1]. The last term in Eq. (3.7) is due to the overlap of the absorption coefficient of the nm-th group, κ nmη, with groups of all other gases, and this overlap only occurs over part of the spectrum. Physically, the overlap coefficient λ nm is a reordered absorption coefficient of the nm-th group taking into account the overlap with groups of all other gases. Based on the MSFSK approach, the λ nm can be determined approximately since the overlap effects between groups are relatively small. There are many ways of approximating λ nm. Here the approach used in the original MSFSK development is followed, that is, the overlap coefficient λ nm is determined in such a way that the emitted intensity emanating from a homogeneous layer bounded by cold black walls is predicted exactly [7]. In Eq. (3.7) the reordering is performed in terms of κ nmη and the overlap between κ nmη and κ η during the reordering process is lumped into λ nm. In order to determine λ nm, the reordering can also be performed in terms of κ η, which, for a homogeneous layer at state φ = φ(p, T, x), leads to di nmg ds = k nmi b f(t, φ, k) ki nmg, for n = 1,..., N; m = 1,..., M, (3.12) where f(t, φ, k) = 1 I b (T ) I nmg = I bη (T )δ(k κ η (φ))dη, (3.13) I nmη δ(k κ η (φ))dη f(t, φ, k), (3.14) k nm = 1 I b (T ) I bη (T )κ nmη δ(k κ η (φ))dη. (3.15) In Eq. (3.12), the overlap between κ nmη and the remainder of κ η is lumped into k nm. The solutions to Eqs. (3.7) and (3.12) for a homogeneous layer at temperature T bounded by cold black walls can be obtained analytically, and the total exiting

54 36 intensity at s = L, obtained from Eq. (3.7) and (3.12), respectively, are and I nm = 1 I nmg dg = I nm = 1 k nm λ nm I b [1 exp( λ nm L)]f nm (T, φ, k nm )dk nm, (3.16) I nmgdg = k nm k I b[1 exp( kl)]dk. (3.17) The spectrally integrated intensity, I nm, must be equal to I nm, and this requirement leads to λ nm = k and k nm f nm (T, φ, k nm )dk nm = k nm(k)dk, (3.18) or k nm(λ nm )dλ nm = k nm f nm (T, φ, k nm )dk nm. (3.19) Equation (3.19) provides the relationship between λ nm and k nm that is required to solve Eq. (3.7). One convenient way of determining λ nm is using the relationship [7] k nm k nmf nm (T, φ, k nm)dk nm = k =λ nm k nm(k )dk. (3.2) In wavenumber space this can also be expressed as κ nmηi bη(t )dη = κ nmηi bη(t )dη. (3.21) η κ nmη k nm η κ η λ nm Equation (3.2) is an implicit relation for the λ nm k nm relationship. In practice, the left and right hand sides of Eq. (3.2) are evaluated for a set of predetermined k nm and λ nm values and the results are stored in two arrays. The corresponding λ nm values for the k nm values used in the RTE evaluations are determined by interpolation from the two arrays. The so-determined λ nm will be a function of the state variables (i.e., temperature and gas species concentration) as well as k nm (or g nm ).

55 Evaluation of Overlap Coefficient λ nm The left-hand side (LHS) of Eq. (3.2) can be readily evaluated since LHS = g nm (k nm ) k nmdg nm, (3.22) and the k nm g nm distribution of the nm-th group can be obtained from a multi-group database. The right hand side (RHS) of Eq. (3.2) contains the knm term, which may be calculated directly from Eq. (3.15) using high resolution spectroscopic databases, if the spectral locations of each spectral group are known. This direct calculation, however, is extremely tedious and impractical for the solution of general problems. It is desirable to evaluate the RHS of Eq. (3.15) using databased multi-group k g distributions for faster and efficient computation. The multi-group database construction for combustion gases CO 2 and H 2 O is discussed in detail in the following section. In order to do so, the quantity Q nm is considered for the nm-th group: Q nm = 1 I bη κ nmη exp( κ η L)dη. (3.23) I b Physically, Q nm is related to the emission from the nm-th group, attenuated over a path L by the groups of all other gases and itself. Q nm can be rewritten as Q nm = 1 I b κ nmη exp( kl)δ(k κ η )dkdη = knm exp( kl)dk = L(knm), (3.24) i.e., Q nm is the Laplace transform of k nm. Q nm can also be written as Q nm = 1 I bη κ nmη exp( κ nmη L) I b N exp( κ lη L)dη (3.25) l n

56 38 where κ lη is the total (of all groups) absorption coefficient of the l-th gas species. Assuming absorption coefficients of the nm-th group and all the other gases are statistically uncorrelated with each other [83], i.e., Planck function weighted transmissivities are multiplicative, Eq can be written as Q nm 1 I bη κ nmη exp( κ nmη L)dη I b N 1 I b l n I bη exp( κ lη L)dη, (3.26) Since the mixing in Eq. (3.26) is at the full-spectrum level, it is expected to be somewhat less accurate than on a narrow band level [83]. However, the uncorrelated assumption should be reasonable here since the nm-th group not only comes from a different gas but also from scattered parts of the spectrum. The k-distribution method then can be applied to Eq. (3.26), which is written as Q nm = 1 1 k nm exp( k nm L)dg nm 1 1 N l n 1 exp( k l L)dg l ( k nm (g nm ) exp ) k l L k nm L l n g nm = g 1 = g N 1 = dg N 1 dg 1 dg nm, (3.27) Here k l (g l ) is the k g distribution of the entire l-th gas, obtained by combining all groups into one [73], M l g l (T, φ, k) = g lq (T, φ, k) M l + 1, (3.28) q=1 evaluated at any given k (same for all groups).

57 39 Equating Eqs. (3.27) and (3.24) leads to L(k nm) = ( k nm exp ) k l L k nm L dg N 1 dg 1 dg nm, l n g nm = g 1 = g N 1 = (3.29) and the integral property of the Laplace transform gives L k=λ nm knm(k)dk = 1 1 g nm = g 1 = 1 g N 1 = ( exp ) k l L k nm L l n k nm L dg N 1 dg 1 dg nm. (3.3) Finally, taking the inverse of the Laplace transformation results in RHS = k=λ nm k nm(k)dk o ( k nm H k ) k l k nm dg N 1 dg 1 dg nm, (3.31) l n g nm= g 1 = g N 1 = where H is the Heaviside step function. By equating the LHS and RHS, a generic expression is obtained for the determination of the overlap coefficient λ nm based on group k-distributions constructed from a multi-group database, i.e., g nm (k nm ) k nmdg nm = k nm g nm = g 1 = g N 1 = ( H λ nm k l k nm )dg N 1 dg 1 dg nm, (3.32) l n The number of multiple integrals on the right hand side of Eq. (3.32) is N, the number of component gases in the mixture.

58 4 3.3 Multi-Group Database Construction Accurate and compact databases of multi-group full spectrum k-distributions are constructed as a part of this work. The spectral absorption coefficients for water vapor are calculated from HITEMP 2, and for carbon dioxide CDSD-1 is used, which is considered more reliable than HITEMP for temperatures higher than 1 K [77, 81], as assembled by Wang and Modest [79]. The resulting multi-group k g distributions of the combustion gases are stored for various values of total pressure, local gas temperature, species mole fraction and Planck function temperature and are summarized in Table Grouping of Wavenumbers: Step 1: The wavenumbers of each gas species in.1cm 1 intervals are grouped according to the temperature dependence of their absorption coefficient. Some typical results are shown in Figs. 3.1 and 3.2. Figure 3.1 corresponds to a few chosen spectral locations across the 4.3 µm band of CO 2 in a mixture with 1% CO 2 and 9% N 2, while Fig. 3.2 corresponds to a few chosen spectral locations across the 2.7 µm band of H 2 O in a H 2 O N 2 mixture with 1% H 2 O. It is observed that there are distinct behaviors of absorption coefficients with temperature, which are consistent for all spectral locations and gas species. This temperature dependence is classified into 4 categories and used for initial grouping of the spectral locations. As seen from Figs. 3.1 and 3.2 lines with decreasing κ η with temperature at room temperature are collected into Group 1; those which have increasing κ η at low temperature but then decrease later on are placed into Group 2; lines that first decrease but later on wake up are placed into Group 3; and finally the lines that wake up at low temperature and continue to rise are collected into Group 4. Step 2: After this initial grouping each group is assigned an average k-profile,

59 41 1% CO2-9% N2 Group Group 4 κ(η)/κ(η,3 Κ) Group 2 Group T (K) Figure 3.1. Grouping of several spectral locations across 4.3 µm band of CO 2 in a CO 2 air mixture containing 1% CO 2 which is expressed as: k ij = [κ η (T j )/κ η (T ref )]κ η (T ref )I bη (T ref ) η i κ η (T ref )I bη (T ref ) η = κ η (T j )I bη (T ref ) η i κ η (T ref )I bη (T ref ). (3.33) η where i is the counter indicating group and j temperature. Unlike the earlier database constructed by Zhang and Modest [73, 74], a weighted average of κ η (T )/κ η (T ref ) is employed, using spectral emission κ η (T ref )I bη (T ref ) as the weight factor. In typical combustion applications the maximum temperature is expected to vary between 5 K and approximately 16 K, and for such temperature range a value of T ref = 85 K was found to give the best grouping results. The following are the steps used to optimize the grouping of wavenumbers: Step 3: The group tag for each wavenumber is determined by minimizing the

60 % H2O - 9% N2 Group 3 Group 4 κ(η)/κ(η,3 Κ) Group 2 Group T (K) Figure 3.2. Grouping of several spectral locations across 2.7 µm band of H 2 O in a H 2 O air mixture containing 1% H 2 O normalized departure of its relative temperature dependence from the average k-profile of the groups: j (k ij C ηi κ ηj ) 2 [max(.1κ η,max, k ij )] 2 = ε ηi, (3.34) where C ηi = k ij κ ηj j. (3.35) j κ 2 ηj In Eq. (3.34), the departure of the absorption coefficients from the average k- shape of the group is normalized in such a way that very low values of the absorption coefficient are given less importance. The numerator of Eq. (3.34) is the absolute departure of the absorption coefficient from the average k-shape, whereas the denominator normalizes that departure with respect to the average k-shape of the group. C ηi is a constant optimized for each wavenumber, since

61 43 the departure from the average k-shape is to be minimized, not departure from k itself. After calculating the total departure (ε η ) from the average k-shape the spectral locations are assigned to that group for which the value of ε η is minimum. Step 4: After regrouping the average k-profile of each group is updated based on Planck function weighted absorption coefficients: k ij = w ηj κ ηj C ηi η, where w ηj = κ η (T ref )I bη (T ref ), (3.36) w ηj η and the process is repeated until less than 1% of the total number of wavenumbers change groups. Table 3.1. Precalculated gas states and reference temperatures Parameter Sampling Number of Samples Species CO 2 and H 2 O 2 Total Pressure.1-.5 bar every.1 bar;.7 bar; bar every 1. bar; bar every 5. bar Local Gas 3-25 K 23 Temperature every 1 K Mole-fraction every.25 Planck Function 3-25 K 23 Temperature every 1 K Construction of k g Distributions Once all spectral locations are grouped, the full-spectrum k g-distributions are calculated for each group and each gas species for the parameters presented in Table 3.1. When calculating FSK distributions for each group of a gas species, a set of nominal

62 44 Table 3.2. Parameters for wavenumber grouping and multi-group databasing Parameter values T ref 85 K k bins (initial) 2 k bins (final) 128 wavenumber step.1 cm 1 k-values between the group maximum and minimum k-values must be chosen at the local gas state. Here a power law distribution of k-values is considered [79] and a total number of k-bins of N k = 2. Parameters used for grouping of wavenumbers and databasing of the full-spectrum k g-distributions are summarized in Table 3.2. Details on the calculations of (multi-group) full-spectrum k g distributions, and the weight function a from the absorption coefficients can be obtained from Zhang and Modest [53, 73]. After calculation of initial k-distributions with 2 k-bins, data compaction is performed using a Gaussian quadrature scheme with a fixed-g concept as outlined by Wang and Modest [79]. The final database contains 128 k-distribution data points for each group. To eliminate the detrimental effect of noise in the stretch function a on quadrature efficiency, natural B-splines are used to smoothen the k g distributions by a small amount, resulting in better-behaved a-functions. Figure 3.3 shows the original and smoothened k g distributions and stretching functions a for Group 2 of a H 2 O N 2 mixture with 1% H 2 O at 15 K local gas temperature, and 1 K as the reference temperature. It is observed that a very small change (nearly indiscernible smoothening) in the k-distributions can result in a much smoother a- function Interpolation of Database The multi-group full-spectrum database is used to obtain the k g distributions for each group of each gas species. To obtain the k-distribution for an arbitrary state, an interpolation is needed between precalculated states stored in the database. For

63 % H2O - 9% N2 original after smoothening k 1-2 k(1k, 1K, g) 1 a 1-3 k(15k,1k,g) a(15k,1k) g Figure 3.3. Original and smoothed weight function a and cumulative k-distributions of Group 2 for 1% H 2 O in H 2 O air mixture a single gas species the k-distribution is specified by total pressure (P), local gas temperature (T ), mole fraction (x) and a Planck function temperature (T ). Hence, for full-spectrum cases four-dimensional interpolations in (P, T, T, x) are required. In order to achieve acceptable accuracy with small computational cost, a hybrid scheme is employed for database interpolation: 1-D spline interpolation is used for P, and trilinear interpolation for T T x. The multi-group database of important combustion gases CO 2 and H 2 O containing 4 groups for each gas species has a total size of 5 MB. For a given state, multi-group k g-distributions can be obtained from the database in 2 milli-seconds on a Pentium Zeon 2.4 GHz computer Combination of Groups The newly constructed multi-group database is combinable, i.e., for faster computation the groups can be combined to obtain coarser groups. The k g distributions of

64 46 the combined group can be calculated [73] from Eq. (3.28) as 1 g n (k) = m (1 g m (k)). (3.37) where g n and g m are the cumulative k-distributions for the same k-values of the combined groups and original groups, respectively. Details of group combination can also be obtained from Zhang and Modest [73]. When groups are combined to obtain a 2-group k g-distribution from 4 groups, the first two groups are combined into one group and the last two into second group, since the average characteristics of Group 1 and 2 are similar (both of them contain lines with decreasing κ η with increase in temperature at higher temperature). Group 3 and 4 also have similarity in the sense that they contain lines with increasing κ η at higher temperature. 3.4 Sample Calculations Figure 3.4. Schematic of the one dimensional problem setup To illustrate the performance of the newly created multi-group database in conjunction with the new hybrid MSMGFSK method in predicting radiative heat transfer in inhomogeneous gas mixtures, a few sample calculations are performed with extreme temperature and mole-fraction (partial pressure) inhomogeneities. In all cases a one dimensional medium containing CO 2 H 2 O N 2 gas mixtures confined between cold

65 47 black walls are considered. The mixture consists of two different homogeneous layers (denoted as left and right layers/column) adjacent to each other. Two cases of total pressure, i.e., 1 bar and 1 bar, are considered. The left layer is at 15 K and has a fixed width of 5 cm. The right layer is cold and at 3 K. The width of the cold layer is varied in the calculations. The radiative heat transfer leaving from the right layer is calculated using the LBL method, the single-scale FSK method, and the hybrid MSMGFSK method. A schematic of the problem setup is shown in Fig For consistency all three methods employ absorption coefficients as obtained from CDSD-1 and HITEMP for CO 2 and H 2 O, respectively. In the MSMGFSK calculations the required k g distributions are obtained by interpolating the database, while in the single-scale FSK calculations the required k g distributions are calculated directly from the absorption coefficients without mixing k-distribution of species, i.e., the absorption coefficients are calculated for the mixture before the k-distributions are formed. The LBL calculations serve as benchmark and the FSK calculations serve to demonstrate the improvement made by the hybrid MSMGFSK method. Figure 3.5 compares the nondimensional radiative heat fluxes for the case of 1 bar total pressure and a step change in temperature only, as calculated by several methods. The mole fractions of the component gases are kept uniform throughout the two-layer slab. The percentage errors of the MSMGFSK and the single scale FSK calculations compared to LBL results are also shown in the figure. In order to compare accuracy of the new database with respect to the previous one, results are obtained from both the previous multi-group database created by Zhang and Modest [73, 74] and the newly created multi-group database in this paper. For a temperature inhomogeneity of this magnitude (a drop from 15 K to 3 K), the FSK method gives more than 2% error as the cold layer length becomes larger than 8 cm, indicating failure of the correlated absorption coefficient assumption. The errors of the MSMGFSK method while using the new multi-group database, on the other hand, stay below 1.6% for any cold layer thickness. The number of the

66 48 groups indicated in the figure is the number of groups into which each gas scale is separated. As can be seen from Fig. 3.5, the accuracy of the hybrid MSMGFSK method is excellent, with the accuracy of the two group case actually being better than for the four group case, which apparently is due to the presence of compensating errors between grouping of absorption coefficients and mixing between two different absorbing gases. It has been shown by Modest and Riazzi [83] that for an isothermal layer of a gas mixture the error incurred during heat transfer calculations is about 2% when the k g distributions are mixed at the full-spectrum level. Hence, a multi-group database of combustion gases with four or even two groups is sufficient to optimize between accuracy and numerical efficiency. Results are also compared with the data from Wang and Modest [8] in which the LBL calculations (not shown in Fig. 3.5) were carried out using the absorption coefficients of CO 2 obtained from the HITEMP.2 Step Change in Temperature 25 2 Flux 15 Non-Dimensional Flux Error LBL FSCK 4 Groups - New Database 2 Groups - New Database 4 Groups - Old Database [1,16] 2 Groups -Old Database [1,16] Error Relative to LBL (%) Length of the Cold Layer (cm) Figure 3.5. Non-dimensional heat flux leaving an inhomogeneous slab of 1% CO 2 and 2% H 2 O at a total pressure of 1 bar with a step change in temperature: The hot left layer is at 15 K and the cold right layer is at 3 K

67 49 spectroscopic database and all the FSK calculations using the database created by Zhang and Modest [73, 74] where again the HITEMP spectroscopic database was used to obtain the absorption coefficients of CO 2. Calculations using the new multi-group database verifies the superior accuracy of the new database compared to the earlier one, especially for the two group case..2 Step Change in Temperature and Mole Fraction 4 Flux Non-Dimensional Flux Error LBL FSCK 4 Groups - New Database 2 Groups - New Database 4 Groups - Old Database [1,16] 2 Groups -Old Database [1,16] 2-2 Error Relative to LBL (%) Length of the Cold Layer (cm) Figure 3.6. Non-dimensional heat flux leaving an inhomogeneous slab at a total pressure of 1 bar with step changes in temperature and mole fraction: The hot left layer contains 2% CO 2 and 1% H 2 O at 15 K and the cold right layer contains 1% CO 2 and 2% H 2 O Figure 3.6 shows the results for a case including mole fraction step changes in both CO 2 and H 2 O in addition to a temperature step change as in Fig The left hot layer contains 2% CO 2 and 1% H 2 O, and this composition is switched in the right cold layer. Total pressure of the gas mixture is 1 bar. The error of the FSK method reaches more than 4% for this extremely inhomogeneous problem. On the other hand, the four- and two-group hybrid models provide excellent results with the maximum error remaining below 2%. Results from the new database are considerably more accurate than those of Wang and Modest [8] for the 4-group case, and even

68 Step Changes in Temperature and Mole-Fraction Non-dimensional Flux Error Flux LBL FSCK MSMGFSCK (4 Groups) MSMGFSCK (2 Groups) Error Relative to LBL (%) Length of Cold Layer (cm) 8 1 Figure 3.7. Non-dimensional heat flux leaving an inhomogeneous slab at a total pressure of 1 bar with step changes in temperature and mole fraction: The hot left layer contains 2% CO 2 and 1% H 2 O at 15 K and the cold right layer contains 1% CO 2 and 2% H 2 O more significantly for the 2-group case. Radiative calculations were also performed for a case of higher total gas pressure. Figure 3.7 shows the results for a case similar to the one in Fig. 3.6 except that the total gas pressure is raised to 1 bar. Again, it is observed that the MSMGFSK formulations in conjunction with the new multi-group database predicts heat transfer calculations very accurately. The errors for the four and two group models remain limited to within 2% whereas the FSK method produces more than 5% error. Hence, the results suggest that this new hybrid MSMGFSK model together with the multigroup databases can perform efficiently over a wide range of gas states.

69 Summary A new full-spectrum k-distribution method was developed for radiative transfer in strongly inhomogeneous gaseous media. The method combines the advantages of the multi-scale, full-spectrum k-distribution method to accommodate mixing and partial pressure inhomogeneities, and those of the multi-group, full-spectrum k-distribution method designed to deal with temperature inhomogeneities. In this method the absorption coefficient of the mixture is broken up into contributions from the gas components, and the absorption coefficient of each component gas is subsequently separated into exclusive correlated groups. The overlap between a group and all other gases is treated approximately. As the number of groups in each gas scale increases, the effect in the approximate treatment of the overlap becomes diminished. A new multi-group full-spectrum k-distribution database of improved accuracy has also been constructed for the most important combustion gases, carbon dioxide and water vapor. To optimize between computational efficiency and accuracy in radiative heat transfer calculations this multi-group database contains four groups for each gas species. The accuracy of the new method was established by performing sample calculations for radiative transfer in strongly inhomogeneous media using the newly constructed database. It was found that the new method successfully handles extreme inhomogeneous problems with only two or four groups for each gas component.

70 Chapter 4 Modified Multi-Scale Full-Spectrum k-distribution Method 4.1 Introduction Radiation calculations for the general cases of combustion involve nonhomogeneous multi-phase participating media, i.e., gas particulate (soot, ash, fuel droplets) mixtures. The k-distribution method developed so far can accommodate only nonhomogeneous gas mixtures or homogeneous gas-particle (soot) mixtures. Hence, the MSFSK method previously developed by Zhang and Modest [7] is extended to allow the incorporation of nongray soot in the inhomogeneous gas mixture [133, 134]. This is done by adding one more scale (soot scale) to the gas scales. For the overlap parameter calculation it is assumed that 1) on a narrow band basis, the spectral behavior of different species is essentially statistically uncorrelated, and 2) the soot absorption coefficient is essentially constant across each narrow band [71, 83]. This model is also designed to include both black and gray wall emission. In the previous model, wall emission was incorporated into the MSFSK method by distributing wall emission across all gas scales according to the absorption coefficient of each scale [72]. Since

71 53 wall emission is continuous in nature across the spectrum, similar to the radiation from soot, inclusion of wall emission in the soot scale is more appropriate and hence this is done in this modified MSFSK model. 4.2 The MSFSK Approach for Gas Soot Mixtures A brief mathematical derivation of the modified MSFSK method is presented here. A participating medium containing molecular gases and nongray soot is considered. Scattering from the medium is assumed to be gray. The radiative heat transfer equation (RTE), Eq. (2.1), for such a medium can be written as [1] di η ds = κ η(φ)i bη ( κ η (φ) + σ s (φ) ) I η + σ s(φ) 4π 4π I η (ŝ )Φ(φ)(ŝ, ŝ )dω (4.1) subject to the boundary condition at s = : I η = ɛi bηw + 1 ɛ π 2π I η ˆn ŝ dω (4.2) The boundary wall is assumed to be gray and diffuse with ɛ being the emittance, and ˆn the surface normal. If one separates the contributions to κ η from the M component gases and soot and breaks up the radiative intensity I η accordingly, i.e., M M κ η = κ mη, I η = I mη (4.3) m=1 m=1 then the RTE Eq. (4.1) is transformed into M component RTE s, one for each species or scale. For each scale this leads to di mη ds = κ mη(φ)i bη κ η (φ)i mη + σ s 4π 4π I mη (ŝ )Φ(ŝ, ŝ )dω, for m = 1,..., M (4.4) It is observed that, physically, the intensity I mη for the m-th scale is due to emission from the m-th species but subject to absorption by all species.

72 54 It is important to note that, if there is no soot or wall emission present in the medium, the spectral locations where κ η contributes to the absorption of I mη (i.e., absorption by all the gas scales) are only those wavenumbers for which κ mη is nonzero. Therefore, the overlap region is only a subset of those wavenumbers with κ mη, across which absorption from other gases occurs as well. The original MSFSK formulation takes advantage of the fact that the overlap regions for each scale are relatively small compared to the total emission/absorption spectrum of each scale [7, 71]. In the presence of soot and wall emission this assumption no longer holds. Hence, addition of soot radiation and wall emission into already existing gas scales is not possible and an additional scale for soot radiation and wall emission is needed. Since radiation from soot and from wall emission are both continuous in nature both are combined into a single scale. When all the wall emission is added to the soot scale, Eq. (4.2) can be written as: at s = : I mη = 1 ɛ π 2π I sη = ɛi bηw + 1 ɛ π I mη ˆn.ŝ dω for m = 1,..., M 1 (gas scales) 2π I sη ˆn.ŝ dω for m = s = M (soot scale). (4.5) where the subscript s denotes the soot scale. We now apply the FSK scheme [69] to the RTE of each scale: first Eq. (4.4) is multiplied by Dirac s delta function δ(k m κ mη (φ )), followed by division with f m (T, φ, k m ) = 1 I b (T ) I bη (T )δ(k m κ mη (φ ))dη (4.6) where, φ and T refer to a reference state. The resulting equation is then integrated over the entire spectrum, leading to di mg ds = k ma m I b λ m I mg + σ s 4π 4π I mg (ŝ )Φ(ŝ, ŝ )dω, for m = 1,..., M (4.7)

73 55 where I mg = / )) ( ) I mη δ (k m κ mη (φ dη f m T, φ, k m (4.8) λ m I mg = k m I mg + ( n m g m = km f m (T, φ, k)dk (4.9) a m = f m(t, φ, k m ) f m (T, φ, k m ) ) / )) ( ) κ nη I mη δ (k m κ mη (φ dη f m T, φ, k m (4.1) (4.11) Similarly, FSK reordering is performed on the boundary condition(s) with respect to κ mη (φ ), which results in at s = : I mg = 1 ɛ π 2π I sg = ɛa w I bw + 1 ɛ π I mg ˆn ŝ dω for m = 1,..., M 1 2π I sg ˆn ŝ dω for s = M. (4.12) where I sg = / )) ( ) I sη δ (k s κ sη (φ dη f s T, φ, k s (4.13) and a w is the wall stretch factor for soot defined as a w = f s(t w, φ, k s ) f s (T, φ, k s ) (4.14) T w is the wall temperature, which may be different from the medium temperature T. The second term on the RHS of Eq. (4.7) is due to the overlap of the absorption coefficient of the m-th scale, κ mη, with those of all other scales, which occurs over part of the spectrum. In the MSFSK method the overlap parameter λ m is evaluated in an approximate way, such that the emitted intensity emanating from a homogeneous nonscattering layer bounded by black walls is predicted exactly. The so-determined λ m is a function of the state variables as well as of k m (or g m ). Here we follow the same

74 56 approximate approach that was used in the original MSFSK development [7, 71]. In Eq. (4.7) the reordering is performed in terms of the scale absorption coefficients κ mη and the interaction between κ mη and κ η during the reordering process is lumped into the overlap parameter λ m. The reordering can also be performed in terms of κ η, which, for a homogeneous layer at temperature T, leads to di mg ds = k mi b f(t, φ, k) ki mg, for m = 1,..., M (4.15) where f(t, φ, k) = 1 I b (T ) I bη (T )δ ( k κ η (φ) ) dη (4.16) Img = I mη δ ( ( )) k κ η φ dη /f ( T, φ, k ) (4.17) km = 1 I bη (T )κ mη δ ( k κ η (φ) ) dη (4.18) I b Reordering the boundary condition(s) with respect to κ η (φ) leads to at s = : Img = 1 ɛ π 2π f ( T Isg w, φ, k ) = ɛi bw f ( T, φ, k ) + 1 ɛ π I mg ˆn.ŝ dω for m = 1,..., M 1 2π I sg ˆn.ŝ dω for m = s = M. (4.19) where I sg = I sη δ ( k κ η ( φ )) dη /f ( T, φ, k ) (4.2) In Eq. (4.15), the interaction between κ mη and κ η is lumped into km. The solutions to Eqs. (4.7), (4.12), (4.15), and (4.19) for a homogeneous layer at temperature T bounded by black walls can be obtained analytically, and the total exiting intensities from the gas scales at s = L are I m = 1 I mg dg = k m λ m I b [1 exp( λ m s)] f m (T, φ, k m )dk m for m = 1,..., M 1 (4.21)

75 57 and I m = 1 I mgdg = k m k I b [1 exp( ks)] dk, for m = 1,..., M 1 (4.22) respectively. Since wall emission is added to the soot scale, the total exiting intensity from the soot scale at s = L is I s = = 1 I sg dg = I s1 + I s2 a w I bw exp( λ s L)f s (T, φ, k s )dk s + k s λ s I b [1 exp( λ s L)] f s (T, φ, k s )dk s (4.23) where I s1 is short-hand for the first term (wall emission), and I s2 for the second term (medium emission) and I s = = 1 I sgdg I bw f(t w, φ, k) exp( kl)dk + k k I b [1 exp( kl)] dk = I s1 + I s2 (4.24) where again I s1 abbreviates the first term (wall emission), and I s2 the second term (medium emission). The spectrally integrated intensity, I m, should be equal to I m (for gas scales), and I s, should be equal to I s (for the soot scale). For gas scales this requirement leads to λ m = k and k m f m (T, k m )dk m = k m(k)dk (4.25) or k m(λ m )dλ m = k m f m (T, k m )dk m (4.26) Similar to the original MSFSK method Eq. (4.26) provides the relationship between

76 58 λ m and k m that is required to solve Eq. (4.7). One convenient way of determining λ m is using the relationship [7] km k m f m (T, k m )dk m = k=λm k m(k)dk (4.27) For the soot scale we use the strategy that the overlap parameter λ s is determined by equating medium emission I s2 and Is2, as was done in the original MSFSK formulation. This leads to the same equation as Eq. (4.27). To equate overall intensity from the soot scale, the wall emissions I s1 and Is1 must also be equal. The expression for Is1 is rearranged employing the approximation for λ s, Eq. (4.26): I s1 = = f(t w, φ, k) k s(t, φ, k) k s(t, φ, k)i bw exp( kl)dk f(t w, φ, λ s ) k s(t, φ, λ s ) k s(t, φ)i bw exp( λ s L)f s (T, φ, k s )dk s (4.28) By comparison with the expression for I s1 in Eq. (4.23), it is clear that if a w = f(t w, φ, λ s ) k s(t, φ, λ s ) k s(t, φ) (4.29) then I s1 equals I s Evaluation of Overlap Parameter The overlap parameter is determined efficiently and accurately from a database of narrow band (NB) k-distributions of individual species (scales). The advantages of using NB k-distributions are that assembling mixture FS k-distributions from NB k- distributions of individual gas species mixed at the narrow band level is more accurate than mixing entire FS k-distributions of individual species. In addition, the use of NB k-distributions of individual species allows the inclusion of nongray absorbing particles in the participating medium [83]. For the m-th scale, substituting Eq. (4.18), the right-hand side (RHS) of Eq. (4.27)

77 59 may be rewritten in terms of narrow band k m RHS = k=λm N nb i=1 I bi 1 N nb I k=λm bi κ mη δ(k κ η )dη dk = k I b η η I m,i(k)dk (4.3) i=1 b where k m,i is the narrow band counterpart of k m, N nb is the number of narrow-bands comprising the entire spectrum, and the NB Planck function I bi is defined as I bi = η I bη dη (4.31) As always in the NB k-distribution approach, we have assumed that I bη is constant over η and can be approximated by I bi / η. In order to evaluate the integrals involving k m,i in Eq. (4.3) in terms of NB k-distributions, we consider the quantity Q m Q m = 1 κ mη exp( κ η L)dη (4.32) η η for the i-th narrow band. Physically, Q m is related to narrow band emission from scale m, attenuated over path L by the entire gas mixture. Q m can be rewritten as Q m = 1 η η κ mη exp( kl)δ(k κ η )dk dη = i.e., Q m is the Laplace transform of k m,i. k m,i exp( kl)dk = L(k m,i) (4.33) Previously it has been shown that, on a narrow band basis, the spectral behavior of different species is essentially statistically uncorrelated, while the soot absorption coefficient is essentially constant across each narrow band [71, 83]. With these two assumptions Q m for the i-th narrow band of the soot scale can also be written as (using subscript m = s) Q s k s,i exp( k s,i L) n s ( ) 1 exp( κ nη L)dη, i = 1,, N nb (4.34) η η

78 6 where k s,i is the NB average value of the soot absorption coefficient. The k-distribution method can then be applied to Eq. (4.34) and we obtain Q s k s,i exp( k s,i L) ( 1 ) exp( k n,i L)dg n n s 1 ( 1 = k s,i exp k s,i L ) k n,i L dg M 1,i dg 1,i (4.35) g 1,i = g M 1,i = n s Equating Eqs. (4.33), and (4.35), we have L(k s,i) 1 g 1,i = ( 1 k s,i exp k s,i L ) k n,i L dg M 1,i dg 1,i (4.36) g M 1,i = n s and, using the integral property of the Laplace transform, ( k=λs ) L ks,i(k)dk 1 1 k exp( k s,i L n s k n,il) g 1,i = g M 1,i = s,i dg L M 1,i dg 1,i (4.37) Finally, taking the inverse Laplace transform, we obtain k=λs k s,i(k)dk k s,i 1 g 1,i = where H is the Heaviside step function. ( 1 H λ s k s,i ) k n,i dg M 1,i dg 1,i g M 1,i = n s (4.38) For the m-th gas scale, using the same approach, we obtain k=λm k m,i(k)dk 1,i g 1 = ( 1 k m,i H λ m k s,i ) k n,i dg M 1,i dg 1,i g M 1,i = n s (4.39) The LHS of Eq. (4.27) is also readily expressed in terms of NB k-distributions for the m-th scale as: LHS = km k m 1 I b I bη δ(k m κ mη )dη dk m

79 61 = = N nb i=1 N nb i=1 I bi I b I bi I b km gm,i (k m) 1 k m δ(k m κ mη )dη dk m η η k m,i dg m,i (4.4) Equating the LHS and RHS, we obtain a generic expression for the determination of the overlap parameter λ m of the m-th gas scale based on NB k-distributions of individual gas species as N nb i=1 = I bi I b gm,i (k m ) N nb i=1 k m,i dg m,i I 1 1 bi k m,i H(λ m k s,i k n,i )dg M 1,i dg 1,i, I b g 1,i = g M 1,i = n s for m = 1,..., M 1 (4.41) For the soot scale λ s is evaluated from N nb i=1 = I bi I b gs,i (k s ) N nb i=1 k s,i dg s,i I 1 1 bi k s,i H(λ s k s,i k n,i )dg M 1,i dg 1,i, I b g 1,i = g M 1,i = n s s = soot scale (4.42) The integrals in Eqs.(4.41), (4.42) can be evaluated efficiently based on the narrow band database compiled by Wang and Modest [79], as was shown by Wang and Modest [71] Evaluation of Modified Wall Stretch Factor Incorporation of wall emission into the soot scale, Eq. (4.12), introduces the stretch factor a w in the MSFSK formulation. The parameter a w can be evaluated in two ways: from the direct definition, Eq. (4.14), and from the modified definition, Eq. (4.29). MSFSK calculations using the directly calculated a w may not recover the LBL result

80 62 for a homogeneous medium bounded by a gray wall at a different temperature from that of the medium, due to the approximation made for λ s. On the other hand, MSFSK calculations using the modified a w from Eq. (4.29) recover the LBL result for homogeneous media with arbitrary boundary wall temperatures, since it is formulated to incorporate the approximation made for λ s. If the gas is nonisothermal, then the modified a w is evaluated at the reference temperature. The narrow band k-distributions constructed by Wang and Modest [79] are used to calculate the wall stretch factor for both cases direct and modified. Calculation of the modified a w, from Eq. (4.29), requires evaluation of ks and f(t w, φ, λ s ). ks can be determined by differentiating the RHS of Eq. (4.27). This approach is found to be accurate and robust and is implemented as follows: the RHS of Eq. (4.42) is calculated for a set of λ s values using the narrow band database; since it is a monotonically increasing function of λ s, a monotonic cubic spline (third order polynomial) can be readily constructed; then the polynomial coefficient corresponding to the first order term are the ks for the corresponding λ s values [72]. For the calculation of f(t w, φ, λ s ) the soot absorption coefficient must be mixed with gas absorption coefficients at the narrow band level. Details of this method can be found in Modest and Riazzi [83]. Then f(t w, φ, λ s ) is evaluated using the Planck function at the wall temperature. 4.3 Sample Calculations D Problem To demonstrate the performance of the new MSFSK model for gas soot mixtures first a one-dimensional medium containing CO 2 H 2 O N 2 with and without soot, confined between cold black walls, is considered (see schematic of problem configuration in Fig. 3.4 of Chapter 3). The mixture consists of two different homogeneous layers (denoted as left and right layer/column) adjacent to each other at a total pressure of 1 bar, with a step jump in species concentrations. The temperature of both layers

81 63 Step Change in Mole Fraction 5 K 1 K LBL FSCK MSFSCK Non-dimensional heat flux K 5 K Error (%) Length of the left layer (cm) Figure 4.1. Nondimensional heat flux leaving an inhomogeneous slab at a total pressure of 1 bar, with step changes in mole fraction of gas species: left layer contains 2% CO 2, 2% H 2 O and.1 ppm soot; right layer contains 2% CO 2, 2% H 2 O and.1 ppm soot are either the same or an additional step jump in temperature is introduced. Two different cases are considered for radiation calculations in this 1-D medium: 1) the right layer has a fixed width of 5 cm, while the width of the left layer is varied in the calculations; 2) the left layer has a fixed width of 5 cm, while the width of the right layer is varied in the calculations. Only emitting/absorbing (non-scattering) soot is considered in the sample calculations, as would be the case for nonagglomerated soot, which is assumed in most predictions of radiative transfer in gas/soot mixtures [82]. The radiative heat flux leaving from the right layer is calculated using the LBL method, the single-scale FSK method, and the new MSFSK method. The errors are calculated with respect to the benchmark LBL calculations. Such problems with steps in species concentration and/or temperature provide an acid test for these methods

82 64 because of their extreme inhomogeneity gradients. Step Change in Mole Fraction 5 K K LBL FSCK MSFSCK Non-dimensional heat flux 4 Error (%) K 5 K Length of the right layer (cm) Figure 4.2. Nondimensional heat flux leaving an inhomogeneous slab at a total pressure of 1 bar, with step changes in mole fraction: left layer contains 2% CO 2 and.1 ppm soot; right layer contains 2% H 2 O First we investigate the case where the right layer has a fixed width of 5 cm, while the width of the left layer is varied. Figure 4.1 shows the results for such a case of gas-soot mixture with a step change in concentration in the gas scales. The soot is uniform (.1 ppm) throughout. It is observed that for both temperatures MSFSK calculations predict heat flux more accurately than the FSK method. For the lower temperature (5 K) the maximum error in the MSFSK calculations is limited to 1%, whereas for 1 K the maximum error is 3.5%. On the other hand, the maximum error for the FSK method is 1% and 5.5% for 5 K and 1 K, respectively. Since soot is uniformly (.1 ppm) present in the medium, at higher temperature, as the case in 1 K, the radiation is mainly dominated by the soot scale and hence, the single-

83 65 Case 2 Step Change in Mole Fraction and Temperature Case 1 LBL FSCK MSFSCK Non-dimensional heat flux 12 8 Error (%) 4-4 Case 2 Case Length of the right layer (cm) Figure 4.3. Nondimensional heat flux leaving an inhomogeneous slab at a total pressure of 1 bar, with step changes in mole fraction and temperature: left layer contains 2% CO 2, 2% H 2 O and.1 ppm soot at 15 K; right layer contains 2% CO 2, 2% H 2 O and no soot at 1 K scale FSK calculations are fairly accurate with small improvement from single-scale FSK to MSFSK. Next we perform radiation calculations for the case where the left layer has a fixed width of 5 cm, while the width of the right layer is varied. Figure 4.2 shows the results for a case of gas soot mixture including mole fraction step changes in all the three scales (species). In this inhomogeneous problem the error of the single-scale FSK method reaches more than 26% for the 5 K case and 14% for the 15 K case. In comparison to that, if the gas soot mixture is broken into several scales, one for each species, the new MSFSK method produce more accurate solutions, with maximum errors limited to 3% for the 5 K case and 4% for the 15 K case.

84 66 After performing tests with step changes in concentration only for the 1-D problem, we now introduce step changes in temperature also. The left and right layers are now at 15 K and 1 K, respectively. Two different cases are considered Case 1: the left layer contains 2% CO 2, 2% H 2 O and.1 ppm soot and the right layer contains 2% CO 2, 2% H 2 O and no soot; Case 2: the concentrations are exactly reversed from the previous case, i.e., the left layer contains 2% CO 2, 2% H 2 O and no soot and the right layer contains 2% CO 2, 2% H 2 O and.1 ppm soot. Figure 4.3 shows that MSFSK calculations are more accurate than the single-scale FSK results, although the improvement in accuracy is small. For Case 1 the maximum error in the MSFSK calculations is limited to 4% while the FSK method incurs a maximum error of 1%. For Case 2 the maximum MSFSK error is 8% while the maximum FSK error is 16%. Although the MSFSK method has been developed to accommodate mixing and concentration inhomogeneity problems, it is observed that breaking up the mixture into scales (species) helps to reduce errors in temperature inhomogeneities as well; but since the MSFSK method does not address the temperature inhomogeneity problem directly as opposed to the MGFSK method, hence, improvement in accuracy is small from the single-scale FSK to the MSFSK in presence of temperature inhomogeneity in addition to concentration inhomogeneity. 1-D problems with wall emission are investigated in Figs. 4.4 and 4.5. Figure 4.4 shows results for the case where both layers are at 1 K; length of both the left and the right layers are kept at 5 cm while the wall temperature of the left layer is varied. Gray wall emission is considered and the wall emittance is taken as ɛ =.6. Step changes in concentration are introduced in all scales. Comparison is done between LBL, single-scale FSK, MSFSK [with direct calculation of a w from Eq. (4.14)] and MSFSK [with calculation using the modified a w from Eq. (4.29)]. It is observed that MSFSK calculations with modified a w are the most accurate (maximum error of 2%). MSFSK calculations with direct a w cannot recover the intensity emitted by a homogeneous medium surrounded by black wall and, hence, can become very

85 67 8 Step Change in Mole Fraction with Wall Emission LBL FSCK MSFSCK (direct aw) MSFSCK (modified aw) Non-dimensional heat flux Error(%) ε = Wall Temperature (K) Figure 4.4. Nondimensional heat flux leaving an inhomogeneous slab at a total pressure of 1 bar, with wall emission and step changes in mole fraction: left layer contains 2% CO 2, 2% H 2 O, and.1 ppm soot; right layer contains 2% CO 2, 2% H 2 O and no soot; both Layers at 1 K inaccurate for very disparate layers. Although the single-scale FSK method incurs larger errors at low temperature (maximum error of 6%), it performs much better at higher wall temperatures, i.e. when wall emission becomes dominant compared to radiation from the medium and, hence, single-scale FSK calculations provide closeto-lbl accuracy. Figure 4.5 shows results for a 1-D problem similar to Fig. 4.4 but with concentrations reversed from the previous case and also including a step change in temperature, with temperatures of the left and the right layers at 15 K and 1 K, respectively. It is again observed that MSFSK calculations with modified a w performs best with a maximum error of 3% at low wall temperatures where the FSK method has more

86 Step Change in Mole Fraction and Temperature with Wall Emission LBL FSCK MSFSCK (direct aw) MSFSCK (modified aw) Non-dimensional heat flux Error(%) ε = Wall Temperature (K) Figure 4.5. Nondimensional heat flux leaving an inhomogeneous slab at a total pressure of 1 bar, with wall emission, step changes in mole fraction and temperature: left layer contains 2% CO 2, 2% H 2 O, and no soot at 15 K; right layer contains 2% CO 2, 2% H 2 O and.1 ppm soot at 1 K than 1% error D Problem Next we consider a more realistic, but still severe, two-dimensional problem of the axisymmetric methane burner considered by Modest and Zhang during the development of the FSK method [53], with its sharp temperature and (independently varying) concentration gradients. In that work a pure gas mixture was considered, while here soot will be added to the gas mixture. The soot volume fraction is obtained from a state relationship for the fuel air equivalence ratio [135]. Temperature and concentration distributions for CO 2, H 2 O and CH 4 can be obtained from previous work by Modest

87 69 fv (soot) 1 3.8E-7 1.5E-6 2.7E-6 3.8E-6 5.E-6 6.2E-6 7.3E-6 radial distance axial distance Figure 4.6. Distribution of soot volume fraction for 2-D test flame error: radial distance %min 3%max axial distance Figure 4.7. Relative error for radiative heat source calculations using single-scale FSCK compared to LBL in a gas (CO 2, H 2 O, CH 4 ) soot mixture in 2-D combustion chamber and Zhang [53]. The distribution of the soot volume fraction is shown in Fig. 4.6, and shows some discontinuities caused by slight wiggles in concentration, which are greatly amplified by the state relationship. The local radiative heat source term is calculated using the LBL, FSK and MSFSK approaches, employing the P-1 method as the RTE solver, and relative errors are determined by comparison with LBL as error(%) =.q LBL.q FSCK/MSFSCK.q LBL,max 1 (4.43) Figure 4.7 shows that the single-scale FSK method generates large errors for gas soot mixtures with varying ratios of concentrations (the maximum error in the present problem reaches as much as 3% near the inlet). In the multi-scale approach, CO 2 and

88 7 error: radial distance %max -7%min axial distance Figure 4.8. Relative error for radiative heat source calculations using MSFSK compared to LBL in a gas (CO 2, H 2 O, CH 4 ) soot mixture in 2-D combustion chamber H 2 O are combined into a single scale since they have the same ratio of concentration throughout the combustion chamber, while CH 4 is treated as a second scale. The maximum error is now limited to 7% near the inlet (region of high errors) as seen in Fig This is a substantial improvement and the accuracy of the new MSFSK approach for gas soot mixtures is clearly demonstrated. 4.4 Summary The multi-scale full-spectrum k-distribution method was extended in this chapter to make it applicable to nongray gas soot mixtures with gray wall emission. Due to the continuous nature of, both, wall emission and radiation from soot, they are lumped into a single scale. Sample calculations were performed for both 1-D and 2-D media for gas soot mixtures with and without wall emission. The MSFSK method is more accurate than the single scale FSK method and is observed to efficiently mix gases with soot and accurately predicts radiative fluxes in the presence of concentration inhomogeneities. The MSFSK method with modified wall stretch factor produces close to line-by-line accuracy. It is seen that, at higher wall temperatures when wall emission dominates over radiation from the medium, use of single-scale FSK can produce sufficiently accurate results. In realistic combustion problems the multi-scale

89 71 method is able to provide very accurate results (an order of magnitude more accurate than the FSK).

90 Chapter 5 Narrow Band-Based Multi-Scale Multi-Group k-distribution Method This modified MSFSK (described in Chapter 4), however, is incapable of accurately dealing with temperature inhomogeneities in gas soot mixtures. A final advancement of the FSK method is performed by extending the hybrid full-spectrum-based MSMGFSK scheme to a narrow band based MSMGFSK scheme to address the most general case of combustion problem, i.e., a nongray, multi-phase mixture with temperature and concentration inhomogeneities [136]. In this method, wavenumbers within each narrow band are classified into M different groups according to their temperature dependence. Mixing of the species is performed at the narrow band group level where the two assumptions of the modified MSFSK, i.e., 1) absorption coefficients of different species are essentially statistically uncorrelated and 2) soot absorption coefficient is essentially constant across each narrow band, hold. This way both mixing in a multi-phase mixtures and temperature inhomogeneities are accounted for and, thus, the narrow band based MSMGFSK serves as the ultimate spectral model for radiation calculations in combustion problems.

91 5.1 The Narrow Band-Based MSMGFSK Approach 73 A brief mathematical derivation of the narrow band based MSMGFSK method is presented here. A participating medium containing molecular gases and nongray particles, such as soot, is considered. Scattering from the medium is assumed to be gray. The radiative heat transfer equation (RTE) for such a medium can be written as (using Eq. 4.1 from Chapter 4) di η ds = κ η(φ)i bη ( κ η (φ) + σ s (φ) ) I η + σ s 4π (φ) 4π I η (ŝ )Φ(φ)(ŝ, ŝ )dω, (5.1) subject to the boundary condition (using Eq. 4.2 from Chapter 4) at s = : I η = ɛi bηw + 1 ɛ π 2π I η ˆn ŝ dω. (5.2) The mixture s spectral absorption coefficient κ η is first separated into contributions from N -1 component gases and soot, and the radiative intensity I η is also broken up accordingly. N N κ η = κ nη, I η = I nη, (5.3) n=1 n=1 The RTE in Eq. (5.1) is transformed into N component RTE s, one for each species or scale. For each scale this leads to di nη ds = κ nη(φ)i bη ( κ η (φ) + σ s (φ) ) I nη + σ s 4π (φ) 4π I nη (ŝ )Φ(φ)(ŝ, ŝ )dω, for n = 1,..., N. (5.4) The intensity I nη is due to emission by the n-th scale but subject to absorption by all scales. Now the spectral locations of the n-th gas absorption coefficients κ nη along with the n-th gas scale s radiative intensity I nη, are sorted into M exclusive groups, i.e., κ nη = M n m=1 κ nmη, I nη = M n m=1 I nmη for n = 1,, N 1, (5.5)

92 74 Considering the soot scale (m = N) as a single group scale, the RTE for the m-th group of the n-th gas scale is transformed into di nmη ds = κ nmη (φ)i bη ( ) κ η (φ) + σ s Inmη + σ s 4π 4π I nmη (ŝ )Φ(ŝ, ŝ )dω, for n = 1,, N 1; m = 1,, M n (gas scales). (5.6) Note that the intensity I nmη is due to emission by the m-th group of the n-th gas species (the nm-th group) but subject to absorption by all groups of the other gases, soot (single group scale) and its own group. There is no overlap among groups of a single species and, therefore, there is no emission over wavenumbers where κ nqη (q m) absorbs. Thus, in Eq. (5.6) κ η = κ nmη + N M l κ lqη. (5.7) l=1 l n q=1 As was done in the modified MSFSK formulations [134], radiation from soot and from wall emission are combined into a single scale due to their continuous nature. When all wall emission is added to the soot scale, Eq. (5.2) can be written as: at s = : I nmη = 1 ɛ π 2π I sη = ɛi bηw + 1 ɛ π I nmη ˆn.ŝ dω 2π for n = 1,..., N 1; m = 1,..., M n (gas scales), I sη ˆn.ŝ dω for m = N(= s) (soot scale). (5.8) where the subscript s denotes the soot scale. We now apply the FSK scheme [69] to each RTE. This process is demonstrated for the RTEs of each group of the gas scales. For the soot scale, i.e., the N-th scale (n = s), the same procedure needs to be followed for a scale with a single group, M s = 1. First Eq. (5.6) is multiplied by Dirac s delta function δ(k nm κ nmη (φ )),

93 75 followed by division with f nm (T, φ, k nm ) = 1 I b (T ) I bη (T )δ(k nm κ nmη (φ ))dη, (5.9) where φ and T refer to a reference state and k nm is the reordered absorption coefficient variable of the nm-th group of a gas scale. The resulting equation is then integrated over the entire spectrum, leading to di nmg ds = k nm a nm I b λ nm I nmg + σ s I nmg (ŝ )Φ(ŝ, ŝ )dω, 4π 4π for n = 1,..., N 1; m = 1,..., M n for n = s; m = 1 (nm = s)., (5.1) where I nmg = / )) ( I nmη δ (k nm κ nmη (φ dη f nm T, φ, k nm ). (5.11) The cumulative k-distribution g is the nondimensional spectral variable of the reordered spectrum and for the m-th group of the n-th scale [131] g nm = knm f nm (T, φ, k)dk, (5.12) a m is the stretching factor for the m-th group of the n-th scale [131] and is calculated from a nm = f nm(t, φ, k nm ) f nm (T, φ, k nm ), (5.13) and, finally λ nm is the overlap parameter of the m-th group of the n-th scale [131] with all other scales and can be written as λ nm I nmg = k nm I nmg + ( ) M l ( ) κ lqη (φ) I nmη δ k nm κ nmη (φ ) dη l n q=1 f nm (T, φ, k nm ). (5.14)

94 76 Similarly, FSK reordering is performed on the boundary condition(s) with respect to κ nmη (φ ) for each group of the gas scales and κ sη (φ ) for the soot scale, which results in at s = : I nmg = 1 ɛ π 2π I sg = ɛa w I bw + 1 ɛ π I nmg ˆn.ŝ dω 2π I sg ˆn.ŝ dω for n = 1,..., N 1; m = 1,..., M n, for m = N(= s). (5.15) where a w is the stretching factor for wall emission defined as a w = f s(t w, φ, k s ) f s (T, φ, k s ). (5.16) T w is the wall temperature, which may be different from the medium temperature T. Finally, total radiative intensity is found by integrating each group over spectral space g, followed by summing over all groups and scales. I = N M n n=1 m=1 I nm = N M n n=1 m=1 1 g min I nmg dg nm. (5.17) The second term on the right hand side of Eq. (5.1) is due to the overlap of the absorption coefficient of the m-th group of the n-th scale, κ nmη, with those of all other scales, which occurs over part of the spectrum. The overlap parameter is a function of the state variables as well as of the k-g distributions. Here we follow the approximate approach for overlap parameter calculations as was done in the modified MSFSK method for gas soot mixtures, by demanding that the intensity emanating from a homogeneous nonscattering layer, bounded by black walls, is predicted exactly [134]. In Eq. (5.1) the reordering is performed in terms of absorption coefficients κ nmη and the interaction between κ nmη and κ η during the reordering process is lumped into the overlap parameter λ nm. The reordering can also be performed in terms of

95 77 κ η, which, for a nonscattering homogeneous layer at temperature T, bounded by a black wall at temperature T w, leads to di nmg ds = k nmi b f(t, φ, k) ki nmg for n = 1,..., N 1; m = 1,..., M n, for n = s; m = 1 (nm = s). (5.18) where f(t, φ, k) = 1 I b (T ) I bη (T )δ ( k κ η (φ) ) dη, (5.19) Inmg = I nmη δ ( ( )) k κ η φ dη /f ( T, φ, k ), (5.2) knm = 1 I bη (T )κ nmη δ ( k κ η (φ) ) dη. (5.21) I b Reordering the boundary condition(s) with respect to κ η (φ) leads to at s = : I nmg = 1 ɛ π 2π I nmg ˆn.ŝ dω Isg = ɛ f(t w, φ, k) f(t, φ, k) I bw + 1 ɛ π 2π I sg ˆn.ŝ dω for n = 1,..., N 1; m = 1,..., M n, for m = N(= s). (5.22) The solutions to Eqs. (5.1), (5.15), (5.18), and (5.22) for a homogeneous layer at temperature T bounded by black walls can be obtained analytically, and the total exiting intensities from each group of the gas scales from a layer of thickness L are I nm = 1 I nmg dg = k nm λ nm I b [1 exp( λ nm L)] f nm (T, φ, k nm )dk nm for n = 1,..., N 1; m = 1,..., M n, (5.23) and I nm = 1 I nmgdg = k nm k I b [1 exp( kl)] dk, for n = 1,..., N 1; m = 1,..., M n. (5.24)

96 78 respectively. Since wall emission is included in the soot scale, the total exiting intensity from the soot scale from a layer of thickness L is I s = = 1 I sg dg a w I bw exp( λ s L)f s (T, φ, k s )dk s + = I s1 + I s2, k s λ s I b [1 exp( λ s L)] f s (T, φ, k s )dk s (5.25) where I s1 is short-hand for the first term (wall emission), and I s2 for the second term (medium emission) and I s = = 1 I sgdg I bw f(t w, φ, k) exp( kl)dk + k k I b [1 exp( kl)] dk = I s1 + I s2, (5.26) where again Is1 abbreviates the first term (wall emission), and Is2 the second term (medium emission). The spectrally integrated intensity, I nm, should be equal to Inm (for each group of the gas scales), and I s should be equal to Is (for the soot scale). For the m-th group of the n-th gas scale this requirement leads to λ nm = k and k nm f nm (T, φ, k nm )dk nm = k nm(k)dk, (5.27) Eq. (5.27) provides the relationship between λ nm and k nm that is required to solve Eq. (5.1). One convenient way of determining λ nm is using the relationship [7] knm k nm f nm (T, k nm )dk nm = k=λnm k nm(k)dk. (5.28) For the soot scale we use the strategy that the overlap parameter λ s is determined by equating medium emission I s2 and I s2, as was done in the modified MSFSK formula-

97 79 tion [134]. To equate overall intensity for the soot scale, the wall emissions I s1 and I s1 must also be equal. The expression for I s1 is rearranged employing the approximation for λ s : I s1 = f(t w, φ, λ s ) k s(t, φ, λ s ) k si bw exp( λ s L)f s (T, φ, k s )dk s, (5.29) By comparison with the expression for I s1 in Eq. (5.24), it is clear that if a w (k s ) = f(t w, φ, λ s ) k s(t, φ, λ s ) k s λ s = λ s (k s ), (5.3) then I s1 equals I s Evaluation of Overlap Parameter For efficient calculations the overlap parameter needs to be available from a database of narrow band multi-group (NBMG) k-distributions for individual gas species. The advantages of using NBMG k-distributions are: 1) groups within each narrow band are scalable, and hence can be combined to obtain coarser k-g distributions, 2) the use of NBMG k-distributions of individual gas species allows the inclusion of nongray absorbing particles in the participating medium [83], 3) mixing of k-g distributions is more accurate when performed at the narrow band level as compared to the fullspectrum level, and 4) since the wavenumbers within a narrow band are grouped according to their temperature dependence, NBMG k-distributions can be used to construct full-spectrum multi-group k-g distributions, which are known to be more accurate for temperature inhomogeneities in multi-phase mixtures. For the m-th group of the n-th gas scale, substituting Eq. (5.21), the right-hand side (RHS) of Eq. (5.28) may be rewritten in terms of a narrow band-based k nm RHS = k=λnm N nb i=1 I bi I b k nm,i(k)dk = k=λnm N nb i=1 I bi I b 1 η η κ nmη δ(k κ η )dη dk, (5.31)

98 8 where k nm,i is the narrow band counterpart of k nm, N nb is the number of narrow-bands comprising the entire spectrum, and the narrow band integrated Planck function I bi is defined as I bi = I bη dη. (5.32) η As always in the narrow band-based k-distribution approach, we have assumed that I bη is constant over η and can be approximated by I bi / η. In order to evaluate the integrals involving k nm,i in Eq. (5.31) in terms of NBMG k-distributions, we consider the quantity Q nm Q nm = 1 κ nmη exp( κ η L)dη, (5.33) η η for the i-th narrow band. Physically, Q nm is related to emission from the m-th group of the n-th scale for the given narrow band i, attenuated over path L by the entire gas mixture. Q nm can be rewritten as Q nm = 1 η η κ nmη exp( kl)δ(k κ η )dkdη = i.e., Q nm is the Laplace transform of k nm,i. k nm,i exp( kl)dk = L(k nm,i), (5.34) Previously, in the modified MSFSK development [134] it was shown that, 1) on a narrow band basis, the spectral behavior of different species is essentially statistically uncorrelated, and 2) the soot absorption coefficient is approximately constant across each narrow band [71, 83, 134]. Since the wavenumbers within a narrow band are placed into exclusive spectral groups, the assumption of statistical uncorrelatedness in spectral behavior between a group (within a narrow band) of one gas species and the narrow band of another gas species still holds. With this assumption, Q nm can

99 81 be written as (after applying the k-distribution method [134]) Q nm 1 N g nm,i = l s,n ( 1 k nm,i g l,i = ( exp l s,n ) k l,i L k nm,i L k s,i L dg l,i )dg nm,i. (5.35) where k s,i is the narrow band average value of the soot absorption coefficient. Equating Eqs. (5.34) and (5.35), we have L(k nm,i) 1 g nm,i = 1 g 1,i = 1 ( k nm,i exp ) k l,i L k nm,i L k s,i L g N 2,i = l s,n dg N 2,i dg 1,i dg nm,i. (5.36) Using the integral property of the Laplace transform and then taking the inverse we obtain k=λnm knm,i(k)dk k nm,i g nm,i = g 1,i = g N 2,i = H (λ nm k l,i k nm,i k s,i )dg N 2,i dg 1,i dg nm,i. l s,n (5.37) where H is the Heaviside step function. The LHS of Eq. (5.28) is readily expressed in terms of narrow band k-distributions for the m-th group of n-th gas scale as: LHS = = = knm N nb i=1 N nb i=1 I bi I b I bi I b k nm 1 I b knm k nm 1 η gnm,i (k nm) I bη δ(k nm κ nmη )dη dk nm η δ(k nm κ nmη )dη dk nm k nm,i dg nm,i. (5.38) Equating the LHS and RHS, we obtain a generic expression for the determination of the overlap parameter λ nm of the m-th group of the n-th gas scale based on NB

100 82 k-distributions of individual gas species as N nb I bi I b i=1 N nb = i=1 gnm,i (k nm ) I bi I b 1 g nm,i = k nm,i dg nm,i 1 g 1,i = 1 k nm,i H (λ nm ) k l,i k nm,i k s,i g N 2,i = l s,n dg N 2,i dg 1,i dg nm,i, for n = 1,..., N 1, m = 1,..., M n (gas scales). (5.39) The integrals in Eq. (5.39) can be evaluated efficiently based on the NBMG database, as outlined by Wang and Modest [71]. Soot being treated as a single-group scale, its overlap parameter calculations can be obtained from the modified MSFSK formulation from Chapter Evaluation of Modified Wall Stretching Factor Incorporation of wall emission into the soot scale, Eq. (5.15), introduces the wall stretching factor a w [134]. It was demonstrated in Chapter 4 that MSFSK calculations using the modified a w from Eq. (5.3) are more accurate as compared to calculations using the direct a w from Eq. (5.16), because only the modified a w recovers the LBL results for homogeneous media with arbitrary boundary wall temperatures. For calculation of the modified a w in the present narrow band-based hybrid method, the same approach has been considered as outlined in the modified MSFSK formulations [134]. 5.2 Narrow Band Multi-group Database Construction Accurate and compact databases of narrow band multi-group (NBMG) k-distributions are constructed as part of this work. The spectral absorption coefficients for water vapor are calculated from HITEMP 2, and for carbon dioxide from CDSD-1.

101 83 The resulting NBMG k-g distributions of the combustion gases are stored for various values of total pressure, local gas temperature, and species mole fraction as described in Wang and Modest [79] but now for 4 groups. The wavenumbers within each narrow band of the gas species in.1 cm 1 intervals are placed into 4 exclusive spectral groups according to the temperature dependence of the absorption coefficients. Details of the grouping of wavenumbers can be obtained from Chapter 3. Once all spectral locations are grouped, the narrow band k-g-distributions are calculated for each group and each gas species. Details of the k-g distribution construction can be obtained from Wang and Modest [79]. After calculation of the initial k-distributions, data compaction is performed using a Gaussian quadrature scheme with fixed g-values as outlined by Wang and Modest [79]. To obtain the k-distribution for an arbitrary state, interpolation is needed between precalculated states stored in the database. For a single gas species, the k-distribution is specified by total pressure (P), local gas temperature (T ), and mole fraction (x). Hence, three-dimensional interpolation in (P,T,x) is required. In order to achieve acceptable accuracy with small computational cost, a 1-D spline interpolation is used for T and bilinear interpolation for P and x [79]. The newly constructed NBMG database is scalable, i.e., for faster computation the groups can be combined to obtain coarser groups both at the narrow band and full-spectrum level. The narrow band k-g distributions of the combined group n from finer groups m can be calculated [73, 131] as 1 g n,i (k) = m (1 g m,i (k)). (5.4) where g n,i and g m,i are the i-th narrow band s cumulative k-distributions for the same k-values of the combined groups and original groups, respectively [73].

102 Sample Calculations D Problem Sample calculations were performed for a 1-D medium, containing emitting absorbing CO 2 H 2 O N 2 gas mixtures as well as soot, confined between cold black walls. The mixture consists of two different homogeneous layers (denoted as left and right layers/column) adjacent to each other at a total pressure of 1 bar. The left layer has a fixed width of 5 cm. The width of the right layer was varied in the calculations. The radiative heat flux leaving from the right layer (i.e., radiative flux at the right wall) was calculated using the LBL method, the single-scale FSK method, the modified MSFSK method, and the present narrow band-based MSMGFSK method (using 2 and 4 groups). For all LBL calculations absorption coefficients of CO 2 and H 2 O were obtained from the CDSD-1 and the HITEMP spectroscopic databases, respectively, and for the k-distribution based calculations the k-g data for CO 2 and H 2 O from the new narrow band multi-group databases. Soot absorption coefficients were evaluated invoking the assumption of small particles (scattering from the agglomerated soot particles was ignored for all sample calculations) with the complex index of refraction given by Chang and Charalampopoulos [137]. Figure 5.1 shows the results for the case of a gas soot mixture with mole fraction step changes in all three scales: CO 2, H 2 O, and soot. The left layer contains 2% CO 2 and 2% H 2 O and no soot, while the right layer contains 2% CO 2 and 2% H 2 O and.1 ppm soot. Both layers are at a constant 1 K. In this inhomogeneous problem, the error of the basic single-scale FSK method reaches more than 2%. In comparison to that, if the gas soot mixture is broken up into several scales, one for each species, the modified MSFSK method produces considerably more accurate solutions, with a maximum error below 4%. The narrow band based MSMGFSK calculations were performed using 2 or 4 groups for each gas scale and soot was considered as a single

103 85.6 Step Changes in Mole-Fraction Non-dimensional Flux Error Flux LBL FSK MSMGFSK (4 Groups) MSMGFSK (2 Groups) MSFSK Error (%) Length of Right Layer (cm) Figure 5.1. Nondimensional heat flux leaving an inhomogeneous slab at a total pressure of 1 bar with step changes in mole fraction: The left layer contains 2% CO 2, 2% H 2 O and no soot and the right layer contains 2% CO 2, 2% H 2 O and.1 ppm soot, both layer at constant temperature 1 K group scale. Since this problem does not have any temperature inhomogeneity (only concentration inhomogeneities considered), hence, both the modified MSFSK and narrow band-based MSMGFSK perform equally well. Both the 2 and 4 groupsbased MSMGFSK calculations result in slightly better accuracy (maximum error limited to less than 3%) compared to the modified MSFSK method. The improvement of accuracy in the narrow band-based MSMGFSK is due to the enhancement of correlation by breaking up the spectrum into several exclusive spectral groups. Figure 5.2 shows the results for the case of a gas soot mixture with step changes in temperature. Both layers contain 2% CO 2 and 2% H 2 O and.1 ppm soot. The left layer is at 15 K while the right layer is at 5 K. In this case, the maximum error of the basic single-scale FSK method reaches 9%. The modified MSFSK method reduces the maximum error to below 5%. Both the 2 and 4 groups based MSMGFSK

104 86.3 Flux Step Changes in Temperature 2 Non-dimensional Flux Error LBL FSK MSMGFSK (4 Groups) MSMGFSK (2 Groups) -.3 MSFSK Length of Right Layer (cm) 1 Error (%) Figure 5.2. Nondimensional heat flux leaving an inhomogeneous slab at a total pressure of 1 bar with step changes in temperature: Left layer at 15 K, and right Layer at 5 K, both layers contain 2% CO 2, 2% H 2 O,.1 ppm soot calculations still yield better accuracy (maximum error limited to 2% for both). It is observed that the accuracy of the 2 and 4 group based calculations are close to each other, which apparently is due to the presence of compensating errors between grouping of absorption coefficients and mixing among different absorbing species. Radiative transfer calculations were also performed for the case of a gas soot mixture with mole fraction step changes in all the three scales (two gas species and soot) in addition to a step change in temperature, and results are shown in Fig The left hot layer contains 2% CO 2 and 2% H 2 O and no soot; the right cold layer has.1 ppm of soot with the gas compositions reversed. It is observed here that the 2 and 4 group based MSMGFSK method have a maximum error of only 5% for very high optical thickness, whereas the single scale FSK method incurs a maximum error close to 6%. The modified MSFSK method incurs a maximum error of 4%, demonstrating its inability to handle strong temperature inhomogeneities in multi-

105 87.15 Step Changes in Temperature and Mole-Fraction 6.1 Flux 4 Non-dimensional Flux Error LBL FSK MSMGFSK (4 Groups) MSMGFSK (2 Groups) MSFSK Error (%) Length of Right Layer (cm) Figure 5.3. Nondimensional heat flux leaving an inhomogeneous slab at a total pressure of 1 bar with step changes in temperature and mole fraction: The hot left layer contains 2% CO 2, 2% H 2 O and no soot at 15 K, and the cold right layer contains 2% CO 2, 2% H 2 O and.1 ppm soot at 5 K phase mixtures. In all cases it is seen that the 2-group based calculations result in excellent accuracy, and only 2N RTEs need to be solved (with N the number of species/scales) D Problem Next we consider a two-dimensional axisymmetric ethylene air jet flame numerically studied by Mehta [12]. This flame simulates the jet flame experimentally studied by Kent and Honnery [138]. The burner of this Kent and Honnery flame (KH87) consists of cylindrical nozzle of diameter d j = 3 mm. The Reynolds number varies from 75 to 15. A three-dimensional wedge-like (wedge-angle 1 o ) grid system was employed to simulate the axisymmetric flame by applying periodic boundary conditions on the sides. The dimensions in the x- and z- directions are 3d j and 25d j, respectively.

106 88 (a) T (b) xh2o ~ xco (c) xco (d) xc2h (e) fv 1E-7 3E-7 5E-7 7E-7 9E-7 Figure 5.4. Temperature and mole fraction distributions in numerically simulated KH87 flame, (a) temperature distribution; (b) mole fraction distribution of H 2 O and, approximately CO 2, wherever there is little CO; (c) mole fraction distribution of CO; (d) mole fraction distribution of C 2 H 4 ; (e) distribution of soot volume fraction The details of modeling the KH87 flame can be found elsewhere [12]. The converged results of that study were used as a frozen data field for radiation calculations. CO 2, H 2 O, CO and soot are the major products of combustion and hence are considered in radiation calculations in addition to ethylene (fuel). The concentrations of the major species and the temperature data are shown in Fig The pressure is uniform (equal to 1 bar). The local radiative heat source term is calculated using the LBL, the basic single-scale FSK, the modified MSFSK, and the 2 and 4 group narrow band-based MSMGFSK approaches, employing the P-1 method as the RTE solver.

107 89 (a) q(w/m 3 ) (1.5 million E+61.3E+6 RTE) (b) FSK Error (%) (1X1 RTE) -3% min 35% max (c) Modified MSFSK Error (%) (4X1 RTE) 7% max -6% min (d) 2 Group NB-based MSMGFSK Error (%) (7X1 RTE) 4% max -3% min Figure 5.5. (a) Local radiative heat source using LBL method and relative error (compared to LBL) for heat source calculations using: (b) the single-scale FSK method; (c) the modified MSFSK method; (d) the 2 group narrow band-based MSMGFSK method Relative errors are determined by comparison with LBL as error(%) =.q LBL.q FSK/MSFSK/MSMGFSK.q LBL,max 1 (5.41) For 2-D LBL calculations the absorption coefficients of C 2 H 4 and CO were obtained from the HITRAN-24 [139] and HITEMP [76] spectroscopic databases respectively. Narrow band single-group databases of k-g distributions were compiled for the additional gas species, C 2 H 4 and CO, as outlined by Wang and Modest [79] and were used for 2-D calculations. The total number of RTEs solved in each method for the 2-D problem was: 1.5 million for LBL, 1 for single-scale FSK, 4 1 for modified MSFSK (CO 2 and H 2 O as combined scale, each other species as one scale),

108 9 7 1 and 11 1 for narrow band based MSMGFSK (each species as one scale, CO 2 and H 2 O scales having 2 and 4 groups each, respectively), where 1 is the number of quadrature points. The local radiative heat source term calculated using the LBL method is shown in Fig. 5.5(a). Figure 5.5(b) shows that the single-scale FSK method generates large errors for gas soot mixtures with varying ratios of concentrations (the maximum error in the present problem reaches as much as 35% near the inlet). In the multi-scale approach, CO 2 and H 2 O are combined into a single scale since they have approximately the same ratio of concentration throughout the combustion chamber, while C 2 H 4, CO and soot are treated as single-group individual scales. Mixing of CO 2 and H 2 O is performed with their local concentrations using the narrow band based k-distribution mixing rule [83]. The maximum error is now limited to 7% near the inlet (region of high errors) as seen in Fig. 5.5(c). Figure 5.5(d) shows the errors incurred in the 2 group narrow band-based MSMGFSK calculations. In this approach the C 2 H 4, CO and soot are treated as single-group scales while CO 2 and H 2 O are treated as two separate scales each having two spectral groups. The maximum error for this case is limited to 4%. The results from the 4 group-based MSMGFSK method are approximately the same as the 2 group case and, hence, are not shown here. This is a substantial improvement and the accuracy of the new narrow band-based MSMGFSK approach for gas soot mixtures is clearly demonstrated. CPU time for the LBL calculations is approximately 56 hours on a 2.4 GHz AMD Opteron machine while the single scale FSK, the modified MSFSK, the 2 group-based MSMGFSK, and the 4 group-based MSMGFSK take only 7 sec, 41 sec, 78 sec, and 11 sec (i.e., typically times required for chemistry calculations in a combustion problem), respectively, for this calculation. This implies factors of 3 1 4, 5 1 3, , and CPU time improvement, respectively, over LBL cost.

109 Summary In this chapter, a new narrow band-based multi-scale multi-group full-spectrum k- distribution method has been developed for radiation calculations involving nongray gas soot mixtures with gray wall emission. This spectral method is capable of producing close-to-lbl accuracy for radiation calculations in general combustion problems with multi-phase mixtures and temperature and concentration inhomogeneities. Accurate and compact narrow band multi-group databases were constructed for the most important combustion gases, CO 2 and H 2 O. Sample calculations were performed for both 1-D media and for a 2-D ethylene-air jet flame with gas soot mixtures. The narrow band-based hybrid method is more accurate than the single-scale FSK method for all the cases, and more accurate than the modified MSFSK method for cases with temperature inhomogeneity. It is observed that the 2 group-based calculations produce similar accuracy as the 4 group-based calculations, both yielding closeto LBL accuracy but requiring less computational time. In realistic combustion problems the narrow band-based multi-scale multi-group method is able to provide very accurate results (an order of magnitude more accurate than the FSK, and with several orders of magnitude less computational cost than LBL).

110 Chapter 6 Portable Spectral Radiation Calculations Software (SRCS) 6.1 Introduction In this chapter, a k-distribution method-based 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 [79, 131, 136], together with our mixing models [7, 71, 83, 131, 134, 136], 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 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 chapter including a discussion of various k-distribution methods with their applicability and limitations.

111 93 Figure 6.1. Software architecture diagram of SRCS 6.2 Module Structure The Spectral Radiation Calculation Software (SRCS) has been developed in FOR- TRAN9. The schematic diagram in Fig. 6.1 shows the software architecture of SRCS. The various modular parts of SRCS are: 1) methods (k-distribution and LBL) 2) databases (k-distribution and LBL), 3) spectral inputs, 4) main operating module, and 5) inter-language portability module. The various modular parts of the SRCS are discussed in detail in the following sections Methods The SRCS includes all state-of-the-art k-distribution methods (both full-spectrum and narrow band based) and the LBL method for spectral radiation calculations. k- distribution methods implemented in the SRCS are summarized in tabular form with their advantages and shortcomings in Table 6.1. The narrow band (NB) module, full-spectrum (FS) routines, all data retrieval

112 94 Table 6.1. Advanced k-distribution methods Method Advantages Shortcomings Single scale FSK [53] (8 1) RTEs Narrow band based single-scale FSK [83] (8 1) RTEs Multi-scale FSK/Fictitious gas [61, 7] N (8 1) RTEs Multi-group FSK [73, 74] M (8 1) RTEs Narrow band-based multi-scale FSK [71] N (8 1) RTEs Multi-scale multi-group FSK [ref. Chapter 3] N M (8 1) RTEs Narrow band-based modified multi-scale FSK [ref. Chapter 4] N (8 1) RTEs Narrow band-based multi-scale multi-group FSK [ref. Chapter 5] N M (8 1) RTEs Most CPU efficient; accurate for moderately inhomogeneous media Most CPU efficient; mixing of multi-phase species; accurate for moderately inhomogeneous media Accurate for mixing and species (gas only) inhomogeneity Accurate for temperature inhomogeneity in a single gas Better accuracy for mixing of gases; potential for multi-phase mixing Accurate for general inhomogeneity problem for gas mixtures Accurate for multi-phase mixing and species inhomogeneity Accurate for multi-phase mixing and strong inhomogeneous media Inaccurate for strongly inhomogeneous media; problems in mixing of species Inaccurate for strongly inhomogeneous media Inaccurate for strong temperature inhomogeneity and multiphase mixing No mixing of species; inaccurate for concentration inhomogeneity Inaccurate for strong temperature inhomogeneity Inaccurate for multiphase mixing Inaccurate for strong temperature inhomogeneity CPU expensive relative to single-scale FSK modules, and line-by-line (LBL) routines are submodules of the method module. Functionality of each of the submodules is summarized in Table 6.2 and is described next in details. 1) LBL Submodule: Retrieves LBL absorption coefficient data from line-byline database (only for CO 2 and H 2 O) compiled by Wang and Modest [79]. This submodule is also capable to calculate absorption coefficient data from the spectroscopic

113 95 Table 6.2. Submodules within method module Module Submodule Functionality LBL Module Retrieves LBL data from LBL database; constructs LBL FS and NB k-g distributions FSMG Data Retrieval k-g data from FSMG database Data Retrieval Module by interpolation NBSG Data Retrieval k-g data from NBSG database by interpolation NBMG Data Retrieval k-g data from NBMG database by interpolation NB Module NBSG routines Narrow band based mixing and overlap parameter for single-group k-g data NBMG routines Narrow band multi-group based mixing and overlap parameter for multi-group k-g data; group combination to produce single-group k-g data FS Module FSMG routines Full-spectrum based mixing and overlap parameter for multi-group k-g data; group combination to produce single-group k-g data NB to FS conversion Conversion of narrow band data from NB module to full-spectrum databases on the fly. The LBL submodule contains routines to construct k-g distributions (both full-spectrum and narrow band based) directly from high-resolution databases. 2) Data Retrieval Submodule: This submodule retrieves k-g distribution data from various k-distribution databases (databases discussed under database section). Depending on the scalar data (P, T, x) (state variable) information and the user s choice of k-distribution database, the data retrieval module opens the corresponding database and obtains the k-g data by interpolation. Two different data retrieval schemes have been implemented (subject to user s choice): 1) loading of the

114 96 whole database into memory (computationally efficient), or 2) keeping the database files open and set a pointer to it so that information from given (P-T-x) grid space can be read when needed (memory efficient). Flexibility has been offered to the user in database interpolation also. The user can choose either linear or spline interpolation for any state variable (i.e., pressure, temperature, and mole-fraction) during database interpolation. The linear interpolation method is faster but can be less accurate, while spline interpolation can be more accurate, but is always more CPU intensive. A detailed discussion of database interpolation methods can be found in Wang and Modest [79]. The data retrieval module contains three submodules (as mentioned in Table 6.2): 1) FSMG data retrieval module to retrieve full-spectrum multi-group data from the full-spectrum multi-group database; 2) NBSG data retrieval module to retrieve narrow band single-group data from the narrow band single-group database; and 3) NBMG data retrieval module to retrieve narrow band multi-group data from the narrow band multi-group database. 3) Narrow band Submodule: The narrow band submodule is also divided into two submodules: 1) narrow band single-group and 2) narrow band multi-group submodules as shown in Table 6.2. The narrow band single group module contains routines for narrow band based mixing of species (outlined by Modest and Riazzi [83]) and narrow band based singlegroup overlap parameter calculations (outlined in Chapter 4). The narrow band single-group module receives k-g data via the NBSG data retrieval module; performs narrow band based mixing of k-g data and overlap parameter calculations and returns the narrow band data to the FS module (shown in Fig. 6.1). The narrow band multi-group module contains routines for narrow band based multi-group overlap parameter calculations and group combination at the narrow band level (outlined in Chapter 5). The narrow band multi-group module receives k-g data via the NBMG data retrieval module; performs narrow band based overlap parameter calculations and returns the narrow band data to the FS module (shown

115 97 in Fig. 6.1). 4) Full-spectrum Submodule: The full-spectrum submodule contains two submodules: 1) full-spectrum multi-group routines, and 2) narrow band to fullspectrum assembly routines. The full-spectrum multi-group module contains routines for full-spectrum based multi-group overlap parameter calculations and group combination at the full-spectrum level (outlined in Chapter 3). As shown in Fig. 6.1 the full-spectrum multi-group module receives k-g data via the FSMG data retrieval module; performs full-spectrum based overlap parameter calculations and returns the full-spectrum data to Spectral Main (the main operating module, discussed later). The narrow band to full-spectrum assembly module combines narrow band single/ multi-group data obtained from the NB module to full-spectrum single/ multi-group data (developed by Modest and Riazzi [83]) and returns data to the Spectral Main. This module is also capable of combining groups at the full-spectrum level [73, 74] to produce single-group full-spectrum k-g distributions Databases High-accuracy compact k-distribution databases are a part of SRCS. All available k-distribution databases are summarized in Table 6.3. In all these databases, the spectral absorption coefficients for water vapor and carbon monoxide were calculated from HITEMP 2 [76], for carbon dioxide from CDSD-1 [77], and all other species from HITRAN-24 [139]. The narrow band single group k-distribution databases constructed by Wang and Modest [79] have been extended to include more combustion species, such as CO, CH 4,and C 2 H 4. Various values of total pressure, local gas temperature, and species mole fraction and/or Planck function temperature, for which k-g distribution data are stored, can be obtained from Wang and Modest [79]. For any given arbitrary state (P, T, x), the k-g data can be obtained from the database by interpolation (as described in the Data Retrieval Module section). Typ-

116 98 ical data retrieval time for interpolated state conditions from the FSMG database is 1 ms (4 full-spectrum k-g distributions), from the NBSG database 3 ms (248 narrow band k-g distributions), and from the NBMG database 6 ms (4 248 narrow band k-g distributions) using a 3. GHz Intel Xeon machine. Table 6.3. k-distribution databases Databases Species Full-spectrum multi-group CO 2 and H 2 O Narrow band single group CO 2,H 2 O,CO,CH 4,C 2 H 4 Narrow band multi-group CO 2,H 2 O Spectral Input SRCS has been designed to offer user flexibility. Through the spectral input module the user interacts with the core software. The spectral input module has two parts: 1) spectral input file and 2) input check submodule. In the spectral input file the user can specify: 1) k-distribution method, 2) k- distribution database, 3) number of quadrature points for spectral integration, 4) species identifers, 5) if a multi-scale method is chosen, species identifiers for combination of species scales (optional) are chosen, otherwise each species is treated as one scale, 6) database interpolation method, and 7) whether database is to be loaded into memory (yes/no). The input check module performs compatibility checks of the spectral input parameters specified by the user. If the input parameters are incompatible, an error message is generated and an instruction is given to the user to choose compatible input parameters for running applications. As an example, if the user chooses the narrow band based modified-msfsk method [ref. Chapter 4], as listed in Table 6.1, with the full-spectrum multi-group database [ref. Chapter 3], an error message is generated as the method and database are incompatible with each other. A quick suggestion for the choice of database is given to the user based on the user s choice

117 99 of k-distribution method Main Operating Module The Spectral Main (shown in Fig. 6.1) or main operating module is the central processing and handling core of the SRCS software. This main module receives scalar (P, T, x) data from the RTE solver, and reads in the spectral input data specified by the user. If all inputs are correct, it passes the information to the corresponding data retrieval module. Spectral Main receives full-spectrum data from the FS module (the FS module obtains full-spectrum data directly from the FSMG database or narrow band from the NB Module and assembles them to full-spectrum data) and supplies the full-spectrum spectral property scalars to the RTE solver Inter-Language Portability Module Since SRCS is designed to supply spectral radiation properties to an RTE solver, it should have allowance to be interfaced with any existing RTE solver. For portability of this FORTRAN9 based software with C/C++ based RTE solvers, a module is provided at the top level for interfacing C/C++ based RTE solvers with SRCS. This interface is placed between the RTE solver and Spectral Main of the SRCS (since Spectral Main handles the in- and outflow of data to and from SRCS). This interfacing module is shown in Fig. 6.1 as wrapper routines. The module contains header files containing definitions of FORTRAN9 functions and subroutines, which are needed for the data flow, and C/C++ based routines to map C/C++ format data to FORTRAN9 format data. Currently, the FORTRAN9 based SRCS is coupled with the C++ based P-1 and P-3 RTE solvers implemented in the open source-code flow software OpenFOAM (Version 1.5) [9].

118 1 T (K) x/l.2 Profile 1 Profile 2 Profile 3 xco x/l xh2o x/l E-7 Soot fv 1.5E-7 1E-7 5E x/l Figure 6.2. Temperature and mole-fraction/volume-fraction distributions of CO 2, H 2 O and soot in a 1-D medium 6.3 Sample Calculations Sample calculations were performed for a 1-D medium containing CO 2 H 2 O N 2 gas mixtures and soot confined between cold black walls to demonstrate the functionality of the SRCS. Total pressure is kept constant at 1 bar. Gaussian distributions of temperature and concentration of each species within the medium are considered and are shown in Fig As can be seen, three different scalar distribution profiles were considered. From Profile 1 to Profile 3, the local gradient of scalar distributions (i.e., temperature and species concentrations) increases. Thus Profile 1 serves as a benign test case, and Profile 2 and 3 are progressively more severe test cases. For each of these cases the local radiative source term was calculated using the LBL method, the single-scale FSK method, the modified MSFSK method, and the narrow band-based MSMGFSK method (using 2 and 4 groups) as spectral models and a finite volume based P-1 as RTE solver. The 1-D medium is discretized into 4 finite volume cells. 1 Gaussian quadrature point integration over spectral space is

119 11 4 Calculation with scalar Profile Error (%) q (W/cm 3 ) LBL Single-scale FSK Modified MSFSK NB-MSMGFSK (2 Group) NB-MSMGFSK (4 Group) x/l.3 Figure 6.3. Local radiative heat source and relative error (compared to LBL) for heat source calculations using scalar distribution profile 1 used for all k-distribution methods. The total number of RTEs solved for each finite volume cell is: 1.5 million for LBL, 1 for single-scale FSK, 3 1 for modified MSFSK (each species as one scale), 5 1 and 9 1 for narrow band based MSMGFSK (each species as one scale, CO 2 and H 2 O scales having 2 and 4 groups each, respectively). Calculations were done on a 3. GHz Intel Xeon machine. CPU times for the various spectral models are typically as follows: 892 sec (8916 sec property acquisition and 4 sec RTE solution) for LBL, 23 sec (22 sec k-distribution assembly and 1 sec RTE solution) for the single-scale FSK, 122 sec (12.5 sec k-distribution assembly and 1.5 sec RTE solution) for the modified-msfsk, 216 sec (214 sec k-distribution assembly and 2 sec RTE solution) for the 2 group narrow band based MSMGFSK, and 41 sec (398.5 sec k-distribution assembly and 2.5 sec RTE solution) for the 4 group narrow band based MSMGFSK method. Obviously, CPU savings achieved by the k-distribution methods, while substantial, will be much greater for three-dimensional problems (when CPU time becomes dominated by the

120 12 3 Calculation with scalar Profile Error (%) q (W/cm 3 ) LBL Single-scale FSK Modified MSFSK NB-MSMGFSK (2 Group) NB-MSMGFSK (4 Group) x/l Figure 6.4. Local radiative heat source and relative error (compared to LBL) for heat source calculations using scalar distribution profile 2 number of RTEs to be solved). Relative errors are determined by comparison with LBL as error(%) = q LBL q FSK/MSFSK/MSMGFSK q LBL,max 1 (6.1) Results of radiation calculations with scalar Profile 1 are presented in Fig In this case, the maximum error the basic single-scale FSK method incurs is 3% very close to the boundaries, otherwise remains bounded within 1%. The modified MS- FSK method and both the 2 and 4 groups based MSMGFSK calculations yield better accuracy (maximum error limited to 1% for each case). For such benign scalar gradients all the k-distribution methods perform almost equally well and, hence, the basic single-scale FSK method is recommended to be used due to its lower computational cost. Results of radiation calculations with scalar Profile 2 are shown in Fig As seen from the figure, the maximum error of the basic single-scale FSK method again

121 Calculation with scalar Profile Error (%) q (W/cm 3 ) LBL Single-scale FSK Modified MSFSK NB-MSMGFSK (2 Group) NB-MSMGFSK (4 Group) x/l Figure 6.5. Local radiative heat source and relative error (compared to LBL) for heat source calculations using scalar distribution profile 3 is only 3%. The modified MSFSK method reduces the maximum error to below 2%. Both the 2 and 4 groups based MSMGFSK calculations yield better accuracy (maximum error limited to 1% for both). It is observed that the accuracy of the 2 and 4 group based calculations are close to each other, which apparently is due to the presence of compensating errors between grouping of absorption coefficients and mixing among different absorbing species as explained in Chapters 3 and 5. Finally, calculations were performed with scalar Profile 3 (most severe case) and results are shown in Fig It is observed here that the 2 and 4 group based MSMGFSK method have a maximum error of only 2%, whereas the single scale FSK method incurs a maximum error close to 1%, and the modified MSFSK method of 4%. For such a case the choice of the basic single-scale FSK method will lead to marginal accuracy predicting the radiative source term. If closeto LBL accuracy is desired rather than faster computation, the user may want to choose a more advanced method, such as the narrow band based MSMGFSK or the modified MSFSK method.

122 Summary In this chapter a portable spectral module, called Spectral Radiation Calculation Software has been developed, which includes all state-of-the art k-distribution methods and databases. The module construction is made flexible to offer the user choices of various k-distribution methods with compatible k-distribution databases. A discussion of the architecture (modules and submodules and their structures) of the SRCS has been given as well as a summary of k-distribution methods and databases implemented in SRCS. This software is designed in a modular structure for better maintenance, further development and portability (i.e., ease of interfacing with any existing flow solver). Example calculations for a 1-D medium have been provided to demonstrate the performance of SRCS, for both accuracy and performance. Guidelines have been given based on sample calculations to facilitate the choice of methods to achieve optimized computational performance and accuracy of spectral radiation calculations.

123 Chapter 7 Higher Order P-N Solver 7.1 Introduction In this chapter a higher order P-N (P-3 ) solver is developed using the data structures and solver libraries of OpenFOAM [9]. Based on the formulations of Modest and Yang [43], six coupled simultaneous PDEs are solved with corresponding boundary conditions. A brief discussion of the P-3 approximation is provided next, including sample results to check the accuracy of the P-3 solver. The RTE solver has been designed with user flexibility in terms of the choice of discretization schemes and solver libraries as offered by OpenFOAM. 7.2 P-3 Approximation The general form of the RTE (Eq. (2.1)) is given as [1] ŝ τ I + I = (1 ω) I b + ω 4π I(ŝ )Φ(ŝ, ŝ )dω (7.1) 4π where ω is the scattering albedo. Augmentation of intensity due to in-scattering is represented by the last term in Eq.(6.1), where Φ is the scattering phase function and describes the probability that a ray from direction ŝ will be scattered into a

124 16 certain direction, ŝ. The intensity gradient, τ, along direction ŝ is written in terms of nondimensional optical coordinates with the extinction coefficient β and dτ = βds. The radiative intensity field I(r, ŝ) at a point r can be thought of as a surface of a unit sphere surrounding the point r. Such a function may be expressed as a two-dimensional generalized Fourier series, I (r, ŝ) = n n= m= n I m n (r)y m n (ŝ) (7.2) where the I m n (r) are position-dependent coefficients and the Y m n (ŝ) are spherical harmonics, given by Yn m (ŝ) = cos(mφ)p m n (cos θ), for m, sin( m φ)p m n (cos θ), for m <. (7.3) Here θ and φ are the polar and azimuthal angles defining the direction of the unit vector ŝ, and P m n (cos θ) are associated Legendre polynomials, given by P m n (µ) = ( 1) m (1 µ2 ) m /2 2 n n! d n+ m ( µ 2 1 ) n d n+ m µ (7.4) The P-3 approximation is obtained by substituting the truncated Fourier series of Eq. (7.2) at n = 3 into Eq. (2.1). The highest value for n that is retained gives the method its order and name. After (tedious) algebra, the result is a set of (3+1) 2 = 16 first-order partial differential equations. Modest and Yang employed successive elimination of spherical harmonic tensors to reduce the set to 3(3 + 1)/2 = 6 second-order elliptic partial differential equations. Since OpenFOAM does not provide any builtin coupled PDE solver, the solution of the coupled PDEs are obtained by solving them in a segregated manner, i.e., in each PDE one different intensity variable is solved implicitly while the others are considered with explicit values from the previous iteration. The entire set of PDEs are iterated until convergence is achieved for each intensity. For this purpose the equations are rearranged in such a way that the

125 17 unknown intensity variable for a given equation is in conservative form and the intensities with cross derivatives are treated explicitly. Thus, the system of rearranged equations for a Cartesian co-ordinate system are given as follows [ ( ) τ 2γ8,3 τ I2 2 I + { 2 I2 2 γ 1,1 + 2γ 7, 3 τ x τ y + [ I2 I2 2 I2 1 γ 1,1 2γ 7, 3 + γ 2, ] I τ y τ x τ x τ z α 1 τ x + [( ) 1 I 2 2 2γ 8,3 5 I2 1 5 I2 1 τ z α 3 τ z α 3 τ y α 3 τ x I2 1 + γ 2, 3 τ y + 5 ] I τ z α 1 τ y ] } 2α 2 I2 2 =, for Y2 2 (7.5) ( ) τ γ6,1 τ I2 + { τ x + I2 [3γ 1 4,1 τ y + τ z [ 3γ 4,1 I 1 2 τ z 6γ 1,1 I 2 2 I2 2 6γ 1,1 τ y I γ 1,1 [ τ z (γ 9,4 γ 6,1 ) I 2 I γ 3, 2 τ z I2 2 6γ 1,1 τ x ] 5 I τ x α 1 τ y I γ 3, I τ x τ y α 1 τ z 5 ] I τ y α 1 τ x ] } α 2 I 2 =, for Y 2 (7.6) [ ( ) τ 2γ8,3 τ I2 2 I + { 2 I2 2 I2 1 γ 1,1 2γ 7, 3 + γ 2, ] I τ x τ x τ y τ z α 1 τ x + [ I2 I2 2 I2 1 γ 1,1 + 2γ 7, 3 γ 2, 3 5 ] I (7.7) τ y τ y τ x τ z α 1 τ y + [( ) 1 I 2 2γ 2 8,3 5 I I 1 ] 2 } 2α 2 I2 2 =, for Y2 2 τ z α 3 τ z α 3 τ x α 3 τ y ( ) 5 τ τ I + { [ 1 I2 + 6 I I2 2 3 ] I2 1 α 1 τ x α 1 τ x α 1 τ x α 1 τ y α 1 τ z + [ 1 I2 6 I I2 2 3 I 1 ] 2 (7.8) τ y α 1 τ y α 1 τ y α 1 τ x α 1 τ z + [ 2 I2 3 I2 1 3 I 1 ] 2 } 5α I = 5α 4πI b, for Y τ z α 1 τ z α 1 τ x α 1 τ y [( ) ( ) τ γ8,3 τ I2 1 5 I 1 + { 2 γ 8,3 1 I ] I2 1 τ x α 3 τ x α 3 τ z α 3 τ y + [ I2 I2 1 γ 3, 2 γ 2, I2 2 5 ] I τ y τ z τ x α 3 τ z α 1 τ z + [ I2 I2 2 I2 2 γ 4,1 2γ 2, 3 + 2γ 2, 3 5 ] I } α 2 I2 1 =, for Y2 1 τ z τ y τ y τ x α 1 τ y (7.9)

126 18 [ ( ) τ γ8,3 τ I2 1 I + { 2 I2 1 γ 3, 2 γ 2, 3 1 I2 2 5 ] I τ x τ z τ y α 3 τ z α 1 τ z + [( ) 5 I 1 γ 2 8,3 1 I I 1 ] 2 τ y α 3 τ y α 3 τ z α 3 τ x + [ I2 I2 2 I2 2 γ 4,1 2γ 2, 3 + 2γ 2, 3 5 ] I } α 2 I2 1 =, for Y2 1 τ z τ x τ x τ y α 1 τ x (7.1) where and ( i γ i,j = + j ) α 3 α 1 (7.11) α n = (2n + 1) ωa n (7.12) A n is a coefficient for higher-order approximation of anisotropic scattering. Because the intensity is expressed as a truncated series, the boundary conditions are satisfied approximately by minimizing the difference between the intensity predicted by P-3 approximation, I(r, ŝ) and the actual surface intensity, I(r w, ŝ) at the boundary. Marshaks boundary conditions, which minimize this difference in an integral sense, were chosen by Modest and Yang [43] as they appear to be flexible and accurate. Their general form can be given as ˆn ŝ> I (τ w, ŝ ) Ȳ m 2i 1dΩ = ˆn ŝ> I s (τ w, ŝ ) Ȳ 2i 1dΩ, m i = 1,, 1 (N + 1), all relevant m 2 (7.13) where the Ȳ denote spherical harmonics measured from a local spherical coordinate system, with polar angle θ measured from the surface normal (local z-axis), and φ in the plane of the surface (measured from a local x-axis). All relevant m implies choosing a set consistent with the P-N -approximation. For example, for the P- 3 approximation six boundary conditions are needed: i = 1 provides three (m = 2;, +2), and another three must come from i = 2 (here the chosen m-values are m = 1,, +1). The global spherical harmonics are then rotated into the local coordinate system to obtain the boundary conditions. The details of the methods and boundary conditions for P-3 can be obtained from Modest and Yang [43] and Deshmukh [14].

127 Solution Method The P-3 equations with the boundary conditions are implemented for any generalized grid (structured and unstructured) using the data structures of OpenFOAM [9]. As mentioned earlier, Eqs. (7.5) (7.1) are solved in a segregated manner, i.e., in each PDE one different intensity variable, which is given a conservative form, is solved implicitly while the others are considered with explicit values from the previous iteration. Eq. (7.5) solves for I 2 2, Eq. (7.6) for I 2, Eq. (7.7) for I 2 2, Eq. (7.8) for I, Eq. (7.9) for I 1 2 and Eq. (7.1) for I 1 2. OpenFOAM data structures provides operators, such as, 2, / x etc, as a part of the implemented discretization scheme. Details of the OpenFOAM mesh definition and discretization of equations can be obtained from the OpenFOAM programmer s manual [9]. The set of Eqs. (7.5) (7.1) are solved by iteration. In each iteration the most updated values of intensity is used. The entire set is iterated until convergence is achieved for each intensity. 7.4 Sample Calculations The P-3 implementation was verified by performing sample test calculations. First, a one-dimensional medium aligned with any of the three primary global coordinates is studied. Next, a two-dimensional nonscattering gray medium with specified I b and κ P profiles was considered: I b = 1 + 2r 2 ( 1 r 2) (7.14) κ P = C K [ ( 1 r 2) 2 ] (7.15) r 2 = 1 2 ( y 2 + z 2), 1 y, z 1 (7.16) where C K is a multiplier to change the optical thickness of the medium. C K = 1 corresponds to an optically thick medium, C K =.1 corresponds to an optically

128 11 Figure 7.1. Incident radiation, G by P- 3 approximation for optically thick case, C K = 1 Figure 7.2. Incident radiation, G by photon Monte Carlo simulation for optically thick case, C K = 1 Figure 7.3. Incident radiation, G by P-3 approximation for optically intermediate case, C K =.1 Figure 7.4. Incident radiation, G by photon Monte Carlo simulation for optically intermediate case, C K =.1 Figure 7.5. Incident radiation, G by P- 3 approximation for optically thin case, C K =.1 Figure 7.6. Incident radiation, G by photon Monte Carlo simulation for optically thin case, C K =.1

129 111 intermediate medium, and C K =.1 corresponds to an optically thin medium. The walls are cold and black. The P-3 solution is compared with a photon Monte Carlo (PMC) simulation, which uses 1 7 photon bundles, in Figs. 7.1 and 7.2 for the optically thick case, in Figs. 7.3 and 7.4 for the optically intermediate case, and in Figs. 7.5 and 7.6 for the optically thin case. It is observed that the P- 3 solution matches the photon Monte Carlo solution except at the boundaries, in particular the corners for the optically thick case. This is an improvement over the P-1 approximation, which is known to fail at the optically thin limit [1]. The P-3 solution does not match at the boundaries due to the approximate treatment of boundaries in the solution method and the discontinuity at the boundary. This discontinuity is severe at the corners of the domain boundary, where the P-3 solution over predicts the incident radiation. For the optically intermediate case, the solution agrees well with the photon Monte Carlo solution at the center of the domain but the agreement deteriorates as one moves away from the center and approaches the domain boundary. In this case, radiation travels farther than in the optically thick case and the local properties are not well correlated with the incident radiation. The disagreement between the photon Monte carlo solution and the P-3 solution is relatively largest for the optically thin case. The radiation travels farthest for this case of the three cases considered here and the incident radiation is not correlated with the local properties. The disagreement of the P-3 solution with the photon Monte Carlo solution is for the most part of the domain, except at the center. In general, the agreement of the P-3 solution with the photon Monte Carlo solution decreases with decreasing optical thickness. The local radiative heat source term along the diagonal of the same sample problem, calculated using various RTE solvers, are shown in Fig It is seen that for the optically thick case (C K = 1) P-1 method incurs a maximum error of 28% (compared to the PMC solution) whereas the maximum error for the P-3 case is limited to 8%. For the optically intermediate case (C K =.1) the maximum relative errors for the

130 PMC P-3 P-1 Radiative Heat Source, Q C k =.1 C k = C k = Distance Along Diagonal, S Figure 7.7. Comparison of local radiative heat source ( Q) calculated using PMC, P-1 and P-3 RTE solvers P-1 and P-3 method are 18% and 9%, respectively. Thus, a significant improvement in accuracy is noticed for the case of P-3 method over the P-1 method. Both these methods perform equally accurately for the optically thin case (C K =.1) as the local radiative heat source term is almost entirely contributed by emission which is correctly predicted by all these RTE solvers. Similar comparisons were made for x-y and z-x planes and the solutions were found to be identical to the y-z plane solution. 7.5 Summary In this chapter, a P-3 solver has been developed using the data structures and solver libraries of OpenFOAM [9]. The solution of the six coupled PDEs in P-3 are obtained by solving them in a segregated manner, i.e., in each PDE one different intensity variable is solved implicitly while the others are considered with explicit values from the

131 113 previous iteration. The entire set of PDEs are iterated until convergence is achieved for each intensity. Sample test calculations have been performed for one and twodimensional media with given Planck function and absorption coefficient profiles for optically thin, intermediate and thick cases. Results from the P-3 solver matches excellently with the PMC simulations.

132 Chapter 8 Flame Simulation 8.1 Introduction Advanced k-distribution-based spectral models and RTE solvers have been developed and tested so far in this thesis. In this chapter the spectral models will be incorporated into a hybrid finite-volume/monte Carlo (FV/MC) architecture to simulate jet flames. This work is a part of the development of a comprehensive combustion solver containing the newly developed high-accuracy deterministic spectral radiation solver. As discussed in Chapter 2, the composition PDF method is not self-contained, requiring the solution of the mean flow field to supply mean velocity and turbulence quantities to be used in the stochastic differential equations. In this study an open source-code flow calculation software OpenFOAM [9] is employed as a finite-volume (FV) solver to solve the mean flow field. A composition PDF module [19] using a particle Monte Carlo method to solve the PDF transport equation is connected to OpenFOAM and is employed to solve for scalars (enthalpy and species concentrations). Consistency is maintained between the mean particle mass and FV fluid mass at the FV element level [19]. The newly developed FV-based RTE solver is then interfaced with the hybrid FV/PDF flow chemistry solver. A schematic of the interface of various solver modules and the data flow are shown in Fig This interface requires data passing between the C++-based FV solvers (flow and RTE solvers in OpenFOAM) and the

133 115 Figure 8.1. Schematic of interfacing of the FV spectral radiation solver with the flowchemistry solver FORTRAN9-based PDF solver and spectral module, SRCS. The passing of data is done through a C++ to FORTRAN interface. This interface includes header files containing function and subroutine definitions and wrapping functions for data exchange. The interface has been designed to offer maximum user flexibility. The user can choose various RTE solvers in conjunction with various spectral models (from the SRCS) through a generic case set-up configuration. As mentioned in earlier sections, the P-1 and P-3 RTE solvers are developed using the data structures of the FV flow solver OpenFOAM. Hence these RTE solvers are a part of the FV solver package. The spectral module, i.e., SRCS, is coupled with the PDF module in order to allow the particle-based spectral radiative property calculations to correctly compute the turbulence radiation interaction (TRI) terms. Provisions are given to the user during the property calculations using the SRCS to bypass the PDF solver and directly supply the property values to the FV RTE solver