Modelling nongrey gas-phase and soot radiation in luminous turbulent nonpremixed jet flames

Much progress has been made in radiative heat transfer modelling with respect to the treatment of nongrey radiation from both gas-phase species and soot particles, while radiation modelling in turbulent flame simulations is still in its infancy. Aiming at reducing this gap, this paper introduces state-of-the-art models of gas-phase and soot radiation to turbulent flame simulations. The full-spectrum k-distribution method (M.F. Modest, 2003, Journal of Quantitative Spectroscopy & Radiative Transfer, 76, 69–83) is implemented into a three-dimensional unstructured computational fluid dynamics (CFD) code for nongrey radiation modelling. The mixture full-spectrum k-distributions including nongrey absorbing soot particles are constructed from a narrow-band k-distribution database created for individual gas-phase species, and an efficient scheme is employed for their construction in CFD simulations. A detailed reaction mechanism including NO _x and soot kinetics is used to predict flame structure, and a detailed soot model using a method of moments is employed to determine soot particle size distributions. A spherical harmonic P₁ approximation is invoked to solve the radiative transfer equation. An oxygen-enriched, turbulent, nonpremixed jet flame is simulated, which features large concentrations of gas-phase radiating species and soot particles. Nongrey soot modelling is shown to be of greater importance than nongrey gas modelling in sooty flame simulations, with grey soot models producing large errors. The nongrey treatment of soot strongly influences flame temperatures in the upstream and the flame-tip region and is essential for accurate predictions of NO. The nongrey treatment of gases, however, weakly influences upstream flame temperatures and, therefore, has only a small effect on NO predictions. The effect of nongrey soot radiation on flame temperature is also substantial in downstream regions where the soot concentration is small. Limitations of the P₁ approximation are discussed for the jet flame configuration; the P₁ approximation yields large errors in the spatial distribution of the computed radiative heat flux for highly anisotropic radiation fields such as those in flames with localized, near-opaque soot regions.     

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

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

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

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

A full spectrum k-distribution based non-gray radiative property model for unburnt char

By using the concept of weighted sum of four gray particles and spectrum k-distribution (WSGP-SK), a non-gray radiative property model for unburnt char particles is developed. Based on the carbon burnout kinetic model for structure during oxidation, and the linear mixed approximation theory for complex index of refraction, spectral radiative properties of unburnt char particles are first calculated as function of the burnout ratio by Mie theory. Referring to the full spectrum k-distribution model, k-distribution is applied to reorder absorption and scattering efficiencies of particles. Then, weighting factors and efficiency factors of the non-gray radiative property model are directly obtained from Gaussian integral points of k-distribution. The model is validated against the benchmark solutions of line-by-line (LBL) model. Maximum relative errors of this model are 3% and 15% for radiative heat fluxes and source terms in non-isothermal inhomogeneous particulate media, respectively. The assumption of linearly varying radiative properties with burnout ratio (Lockwood et al. 1986) will result in a predicted deviation of 53% for radiative source terms. Results also show that this non-gray model is remarkably better than the Planck mean method. Moreover, a satisfactory comparison with LBL solutions is achieved in the gas and particle mixture by combining the non-gray WSGG-SK model (Guo et al. 2015). As a radiation sub-model, this non-gray radiative property model can significantly improve prediction accuracy of radiative heat transfer in oxy-fuel combustion.     

Simulations of sooting turbulent jet flames using a hybrid flamelet/stochastic Eulerian field method

The stochastic Eulerian field method is applied to simulate 12 turbulent C₁?C₃ hydrocarbon jet diffusion flames covering a wide range of Reynolds numbers and fuel sooting propensities. The joint scalar probability density function (PDF) is a function of the mixture fraction, enthalpy defect, scalar dissipation rate and representative soot properties. Soot production is modelled by a semi-empirical acetylene/benzene-based soot model. Spectral gas and soot radiation is modelled using a wide-band correlated-k model. Emission turbulent radiation interactions (TRIs) are taken into account by means of the PDF method, whereas absorption TRIs are modelled using the optically thin fluctuation approximation. Model predictions are found to be in reasonable agreement with experimental data in terms of flame structure, soot quantities and radiative loss. Mean soot volume fractions are predicted within a factor of two of the experiments whereas radiant fractions and peaks of wall radiative fluxes are within 20%. The study also aims to assess approximate radiative models, namely the optically thin approximation (OTA) and grey medium approximation. These approximations affect significantly the radiative loss and should be avoided if accurate predictions of the radiative flux are desired. At atmospheric pressure, the relative errors that they produced on the peaks of temperature and soot volume fraction are within both experimental and model uncertainties. However, these discrepancies are found to increase with pressure, suggesting that spectral models describing properly the self-absorption should be considered at over-atmospheric pressure.     

Radiative transfer in a nongray spherical layer: Simplified rectangular model

The problem of radiative transfer in a nongray, absorbing-emitting spherical layer is investigated. The absorption coefficient is assumed to be only a function of frequency, i.e. k_v = α(v)k, and the function α(v) is allowed only two values, zero or unity (simplified rectangular model). The nongray radiative transfer problem is reduced to a gray solution without the use of any approximation (such asthe Plank or Rosseland means) for an isothermal layer and for radiative equilibr     

Numerical study of the effects of gravity on soot formation in laminar coflow methane/air diffusion flames under different air stream velocities

Numerical simulations of laminar coflow methane/air diffusion flames at atmospheric pressure and different gravity levels were conducted to gain a better understanding of the effects of gravity on soot formation by using relatively detailed gas-phase chemistry and complex thermal and transport properties coupled with a semi-empirical two-equation soot model. Thermal radiation was calculated using the discrete-ordinates method coupled with a non-grey model for the radiative properties of CO, CO₂, H₂O, and soot. Calculations were conducted for three coflow air velocities of 77.6, 30, and 5 cm/s to investigate how the coflowing air velocity affects the flame structure and soot formation at different levels of gravity. The coflow air velocity has a rather significant effect on the streamwise velocity and the fluid parcel residence time, especially at reduced gravity levels. The flame height and the visible flame height in general increase with decreasing the gravity level. The peak flame temperature decreases with decreasing either the coflow air stream velocity or the gravity level. The peak soot volume fraction of the flame at microgravity can either be greater or less than that of its normal gravity counterpart, depending on the coflow air velocity. At sufficiently high coflow air velocity, the peak soot volume fraction increases with decreasing the gravity level. When the coflow air velocity is low enough, soot formation is greatly suppressed at microgravity and extinguishment occurs in the upper portion of the flame with soot emission from the tip of the flame owing to incomplete oxidation. The numerical results provide further insights into the intimate coupling between flame size, residence time, thermal radiation, and soot formation at reduced gravity level. The importance of thermal radiation heat transfer and coflow air velocity to the flame structure and soot formation at microgravity is demonstrated for the first time.     

A computational framework for predicting laminar reactive flows with soot formation

Numerical modeling is an attractive option for cost-effective development of new high-efficiency, soot-free combustion devices. However, the inherent complexities of hydrocarbon combustion require that combustion models rely heavily on engineering approximations to remain computationally tractable. More efficient numerical algorithms for reacting flows are needed so that more realistic physics models can be used to provide quantitative soot predictions. A new, highly-scalable combustion modeling tool has been developed specifically for use on large multiprocessor computer architectures. The tool is capable of capturing complex processes such as detailed chemistry, molecular transport, radiation, and soot formation/destruction in laminar diffusion flames. The proposed algorithm represents the current state of the art in combustion modeling, making use of a second-order accurate finite-volume scheme and a parallel adaptive mesh refinement (AMR) algorithm on body-fitted, multiblock meshes. Radiation is modeled using the discrete ordinates method (DOM) to solve the radiative transfer equation and the statistical narrow-band correlated-k (SNBCK) method to quantify gas band absorption. At present, a semi-empirical model is used to predict the nucleation, growth, and oxidation of soot particles. The framework is applied to two laminar coflow diffusion flames which were previously studied numerically and experimentally. Both a weakly-sooting methane–air flame and a heavily-sooting ethylene–air flame are considered for validation purposes. Numerical predictions for these flames are verified with published experimental results and the parallel performance of the algorithm analyzed. The effects of grid resolution and gas-phase reaction mechanism on the overall flame solutions were also assessed. Reasonable agreement with experimental measurements was obtained for both flames for predictions of flame height, temperature and soot volume fraction. Overall, the algorithm displayed excellent strong scaling performance by achieving a parallel efficiency of 70% on 384 processors. The proposed algorithm proved to be a robust, highly-scalable solution method for sooting laminar flames.     

Evaluation of solution methods for radiative heat transfer in gaseous oxy-fuel combustion environments

The exact solution to radiative heat transfer in combusting flows is not possible analytically due to the complex nature of the integro-differential radiative transfer equation (RTE). Many different approximate solution methods for the solution of the RTE in multi-dimensional problems are available. In this paper, two of the principal methods, the spherical harmonics (P₁) and the discrete ordinates method (DOM) are used to calculate radiation. The radiative properties of the gases are calculated using a non-gray gas full spectrum k-distribution method and a gray method. Analysis of the effects of numerical quadrature in the DOM and its effect on computation time is performed. Results of different radiative property methods are compared with benchmark statistical narrow band (SNB) data for both cases that simulate air combustion and oxy-fuel combustion. For both cases, results of the non-gray full spectrum k-distribution method are in good agreement with the SNB data. In the case of oxy-fuel simulations with high partial pressures of carbon dioxide, use of gray method for the radiative properties may cause errors and should be avoided.     

The spectral-line moment-based (SLMB) modeling of the wide band and global blackbody-weighted transmission function and cumulative distribution function of the absorption coefficient in uniform gaseous media

The spectral-line moment-based (SLMB) modeling is proposed for the calculation of radiative properties of gases on any spectral width. The associated mathematical formulation is obtained by applying several concepts of the k-distribution methods such as the reordering of the wavenumber scale by monotonic variations of the absorption coefficient, together with the application of the k-moment method's principles. This approach gives both a general formula for the BTF and a simple and readily applicable approximation for the blackbody-weighted cumulated k-distribution function of the absorption coefficient. The model is applied for the computation of wide band BTFs and cumulative k-distributions for uniform columns of CO₂ and H₂O in the temperature range (300-2400 K) at atmospheric pressure. Model parameters are deduced from line-by-line (LBL) spectra calculated using the HITEMP database. Comparisons with LBL reference data as well as with contemporary modeling approaches (SLW, FSK, SNB) are performed and discussed.     

The importance of thermal radiation transfer in laminar diffusion flames at normal and microgravity

The importance of radiation heat loss in laminar and turbulent diffusion flames at normal gravity has been relatively well recognized in recent years. There is currently lack of quantitative understanding on the importance of radiation heat loss in relatively small scale laminar diffusion flames at microgravity. The effects of radiation heat transfer and radiation absorption on the structure and soot formation characteristics of a coflow laminar ethylene/air diffusion flame at normal- and microgravity were numerically investigated. Numerical calculations were conducted using GRI-Mech 3.0 combustion chemistry without the NO_x mechanism and complex thermal and transport properties, an acetylene based soot formation model, and a statistical narrow-band correlated-k non-grey gas radiation model. Radiation heat transfer and radiation absorption in the microgravity flame were found to be much more important than their counterparts at normal gravity. It is important to calculate thermal radiation transfer accurately in diffusion flame modelling under microgravity conditions.     

Investigation of conditional source-term estimation applied to a non-premixed turbulent flame

A turbulent combustion model, Conditional Source-term Estimation (CSE) is applied to a non-premixed turbulent jet methane flame. The conditional chemical source terms are determined on the basis of first order closure and the conditional averaged species concentrations are obtained by inverting an integral equation. The Tikhonov method is implemented for regularisation. Detailed chemistry is tabulated using the trajectory generated low-dimensional manifold method. Radiation due to the gaseous species is included. Reynolds Averaged Navier–Stokes calculations are performed using two different turbulence models. The objectives of the paper are (i) assessment of the impact of the main numerical parameters in CSE and (ii) comparison of the CSE numerical predictions with available experimental data and results from previous simulations for the selected flame. The number of CSE domains and the number of points in each CSE domain are shown to have a significant impact on the results if not selected appropriately. The present CSE calculations always converge to unique and stable predictions. The corrected k–ε model yields mixture fraction profiles in good agreement with the experimental data values for axial locations in the first half of the flame. Farther downstream, the RNG k–ε model performs better. Overall, the current predictions for the mixture fraction are in good agreement with the experimental data. The predicted temperatures using CSE and the k–ε turbulence model with a modified value of C_ε1 = 1.47 are found to be in very good agreement with the experimental data. Further, the current CSE results are of comparable quality with previous simulations using the flamelet model and conditional moment closure. Future work may include further investigation on optimal determination of the regularisation parameter and alternative regularisation techniques, soot modelling within the CSE formulation, and improved formulation of radiation.     

A simplified CFD model for spectral radiative heat transfer in high-pressure hydrocarbon–air combustion systems

Detailed radiation modeling in piston engines has received relatively little attention to date. Recently, it is being revisited in light of current trends towards higher operating pressures and higher levels of exhaust-gas recirculation (EGR), both of which enhance molecular gas radiation. Advanced high-efficiency engines also are expected to function closer to the limits of stable operation, where even small perturbations to the energy balance can have a large influence on system behavior. Detailed radiation modeling using sophisticated tools like photon Monte Carlo/line-by-line (PMC/LBL) is computationally expensive. Here, guided by results from PMC/LBL, a simplified stepwise-gray spectral model in combination with a first-order spherical harmonics (P1 method) radiative transfer equation (RTE) solver is proposed and tested for engine-relevant conditions. Radiative emission, reabsorption and radiation reaching the walls are computed for a heavy-duty compression-ignition engine at part-load and full-load operating conditions with different levels of EGR and soot. The results are compared with those from PMC/LBL, P1/FSK (P1 with a full-spectrum k-distribution spectral model) and P1/Gray radiation models to assess the proposed model’s accuracy and computational cost. The results show that the proposed P1/StepwiseGray model can calculate reabsorption locally and globally with less than 10% error (with respect to PMC/LBL) at a small fraction of the computational cost of PMC/LBL (a factor of 30) and P1/FSK (a factor of 15). In contrast, error in computed reabsorption by the P1/Gray model is as high as 60%. It is expected that the simplified model should be broadly applicable to high-pressure hydrocarbon–air combustion systems, with or without soot.     

Transported JPDF modelling and measurements of soot at elevated pressures

Accurate measurements and modelling of soot formation in turbulent flames at elevated pressures form a crucial step towards design methods that can support the development of practical combustion devices. A mass and number density preserving sectional model is here combined with a transported joint-scalar probability density function (JDPF) method that enables a fully coupled scalar space of soot, gas-phase species and enthalpy. The approach is extended to the KAUST turbulent non-premixed ethylene-nitrogen flames at pressures from 1 to 5 bar via an updated global bimolecular (second order) nucleation step from acetylene to pyrene. The latter accounts for pressure-induced density effects with the rate fitted using comparisons with full detailed chemistry up to 20 bar pressure and with experimental data from a WSR/PFR configuration and laminar premixed flames. Soot surface growth is treated via a PAH analogy and soot oxidation is considered via O, OH and O₂ using a Hertz-Knudsen approach. The impact of differential diffusion between soot and gas-phase particles is included by a gradual decline of diffusivity among soot sections. Comparisons with normalised experimental OH-PLIF and PAH-PLIF signals suggest good predictions of the evolution of the flame structure. Good agreement was also found for predicted soot volume statistics at all pressures. The importance of differential diffusion between soot and gas-phase species intensifies with pressure with the impact on PSDs more evident for larger particles which tend to be transported towards the fuel rich centreline leading to reduced soot oxidation.     

Level set simulations of turbulent thermonuclear deflagration in degenerate carbon and oxygen

We study the dynamics of thermonuclear flames propagating in fuel stirred by stochastic forcing. The fuel consists of carbon and oxygen in a state which is encountered in white dwarfs close to the Chandrasekhar limit. The level set method is applied to represent the flame fronts numerically. The computational domain for the numerical simulations is cubic, and periodic boundary conditions are imposed. The goal is the development of a suitable flame speed model for the small-scale dynamics of turbulent deflagration in thermonuclear supernovae. Because the burning process in a supernova explosion is transient and spatially inhomogeneous, the localized determination of subgrid scale closure parameters is essential. We formulate a semi-localized model based on the dynamical equation for the subgrid scale turbulence energy k _sgs. The turbulent flame speed s _t is of the order √2k _sgs. In particular, the subgrid scale model features a dynamic procedure for the calculation of the turbulent energy transfer from resolved toward subgrid scales, which has been successfully applied to combustion problems in engineering. The options of either including or suppressing inverse energy transfer in the turbulence production term are compared. In combination with the piece-wise parabolic method for the hydrodynamics, our results favour the latter option. Moreover, different choices for the constant of proportionality in the asymptotic flame speed relation, s _t∝√2k _sgs, are investigated.     

Investigation of mass and energy coupling between soot particles and gas species in modelling ethylene counterflow diffusion flames

A numerical model is developed aiming at investigating soot formation in ethylene counterflow diffusion flames. The mass and energy coupling between soot solid particles and gas-phase species is investigated in detail. A semi-empirical two-equation model is chosen for predicting soot mass fraction and number density. The model describes particle nucleation, surface growth, and oxidation. A detailed kinetic mechanism is considered for the gas phase and the effect of considering radiation heat losses is also evaluated. Simulations were done for a range of conditions that produce low-to-significant amounts of soot using three strategies: first by changing the strain rate imposed on the flow field, second, by changing the oxygen content in the oxidant stream, and third, by changing the pressure. Additionally, the effect of the transport model chosen was analysed. The results showed that, for the flames studied and within the limits of the present work, the soot and gas radiation terms are of primary importance for numerical simulations. Additionally, it was shown that the soot mass and thermodynamic properties coupling terms are, in general, a second-order effect, with an importance that increases as soot amount increases. As a general recommendation, the radiation terms have always to be considered, whereas full coupling has to be employed only when the soot mass fraction, Y_S, is equal to or larger than 0.008. If a higher precision is required, with errors less than 1%, full coupling should be taken into account for Y_S ≥ 0.002. For lower soot amounts, the coupling through soot mass and thermodynamic properties may be neglected as a first approximation, but an error on the total mass conservation will be present. Additionally, discrepancies from considering different transport models (detailed or simplified) are larger than those found from not fully coupling the phases.     

Sooting turbulent jet flame: characterization and quantitative soot measurements

Computational fluid dynamics (CFD) modelers require high-quality experimental data sets for validation of their numerical tools. Preferred features for numerical simulations of a sooting, turbulent test case flame are simplicity (no pilot flame), well-defined boundary conditions, and sufficient soot production. This paper proposes a non-premixed C₂H₄/air turbulent jet flame to fill this role and presents an extensive database for soot model validation.     

Extension of the weak-line approximation and application to correlated-k methods

Global climate models require accurate and rapid computation of the radiative transfer through the atmosphere. Correlated-k methods are often used. One of the approximations used in correlated-k models is the weak-line approximation. We introduce an approximation T_γ which reduces to the weak-line limit when optical depths are small, and captures the deviation from the weak-line limit as the extinction deviates from the weak-line limit. This approximation is constructed to match the first two moments of the gamma distribution to the k-distribution of the transmission. We compare the errors of the weak-line approximation with T_γ in the context of a water vapor spectrum. The extension T_γ is more accurate and converges more rapidly than the weak-line approximation.     

Effect of soot shape on soot radiation

It is increasingly recognized that soot particles play an important role in the radiative heat transfer from flames and smoke. After their formation, these minute particles usually conglomerate into different forms, with the limiting shapes being the spheres and long chains which can be modeled as infinite cylinders. The present work analyzes the effect of soot shape on soot radiation. The spectral extinction coefficient of spheres, being lower than that of the cylindrical particles, falls off rapidly in the near i.r. The shape effect on soot radiation is found to be more pronounced at low temperatures than at high temperatures. In flame radiation calculations the radiative contribution of the various conglomerated soot shapes can be properly accounted for by assuming spherical and polydisperse soot particles. Based on the extinction characteristics of the particles, an experimental method for determining the amount of spherical and cylindrical particles in a soot cloud is suggested.     

Non-gray chemical composition based radiative property model of fly ash particles

Particle radiation has a spectral dependence and is closely related to the chemical composition of the material. Iron oxide, one of the main components of fly ash, observably affects the complex index of refraction of the particles. In this study, following the theory of the spectrum k-distribution based weighted sum of gray particles model (Guo et al. [4,13]), a non-gray fly ash radiative property model involving the chemical composition was developed. First, four typical fly ash particles with different iron oxide contents were selected, and the corresponding particle radiative parameters were obtained using the Mie theory. Then, the absorption efficiency and weighting factors of the non-gray model were directly obtained from the Gaussian integral points of the k-distribution. The scattering efficiency of the particles was obtained from the Planck mean. The accuracy of the newly developed model was evaluated in a one-dimensional plane-parallel slab system through comparison with the line-by-line (LBL) model and two commonly used gray radiative property models. The results show that the new non-gray model agrees well with the LBL solution and becomes more accurate as the iron oxide content increases. When the iron oxide content of the fly ash increased from 5.47% to 30.50%, the maximum relative error of the radiative heat flux and the radiative source term decreased from 12.50% to 5.68% and from 20.97% to 12.62%, respectively. The new model can improve the prediction accuracy of radiative heat transfer in pulverized coal-fired furnaces.