Funding for this work was provided by the National Science Foundation under SBIR Phase II .... TNF website, http://www.ca.sandia.gov/TNF/abstract.html.
Advanced CFD Tools for Modeling Lean Premixed Combustion in Ultra-Low NOx Burners in Process Heaters AFRC-JFRC Joint International Symposium 2004
Advanced CFD Tools for Modeling Lean Premixed Combustion in Ultra-Low NOx Burners in Process Heaters Q. Tang, B. Adams, M. Bockelie, M. Cremer, M. Denison, C. Montgomery, A. Sarofim Reaction Engineering International http://www.reaction-eng.com D. J. Brown Stone & Webster, Inc. http://www.shawgrp.com/StoneWebster/index.cfm Abstract Air quality emissions regulations, NOx in particular, are impacting the chemical process industry and forcing adoption of expensive selective catalytic reduction (SCR) units or retrofit of less expensive but relatively unproven ultra-low NOx burners. CFD modeling provides a potentially cost-effective method for evaluating NOx emissions and furnace performance for new burner technologies, thus minimizing furnace start-ups times and unforeseen performance impacts. However, existing CFD models are incapable of accurately predicting NOx emissions due to the complex geometries, turbulent mixing, lean premixed combustion chemistry, heat transfer and low NOx levels in the new generation of burners. This paper reviews a new computational tool developed by REI specifically designed to model lean premixed combustion associated with low NOx burners in process heaters/furnaces. The new tool is based on a Hybrid Scalar Transport Probability Density Function Solver and includes adaptive mesh refinement to capture near-burner mixing, reduced chemical kinetic mechanisms with in-situ adaptive tabulation to compute finite rate chemistry, a combination of conventional Eulerian turbulence modeling with Monte-Carlo PDF methods to model turbulent reactions and mixing, and a matrix-free Newton-Krylov method to reduce solver computational time and improve robustness. Predicted flame characteristics and emission levels for low NOx burners in a test furnace will be discussed and compared with measured and observed data. 1. Introduction Air quality emissions regulations resulting from the National Ambient Air Quality Standards (NAAQS) are requiring increasingly stringent ozone emissions from numerous furnaces in the chemical process industry. NOx emissions have been identified as a major contributor to ground level ozone, and have been targeted for significant reduction. These reductions are impacting a large number of industrial boilers and heaters in non-attainment areas, approximately 1900 in the Houston-Galveston area alone, for example, and are forcing furnace operators to adopt expensive selective catalytic reduction (SCR) technology (an SCR retrofit can exceed $3M for a large ethylene cracking furnace) or retrofit less expensive but relatively unproven ultra-low NOx burners. These ultra-low NOx burners can provide a relatively inexpensive solution to NOx reduction requirements, but reductions must be achieved without compromising the heat release profiles or flame ‘quality’ of the burners: it is not uncommon for ultra-low NOx burners to have wider and longer flames than conventional burners or even, in the worst case, for flames to impinge directly on the heat absorbing process tubes. CFD modeling provides a potentially cost-
1
REACTION ENGINEERING INTERNATIONAL
Advanced CFD Tools for Modeling Lean Premixed Combustion in Ultra-Low NOx Burners in Process Heaters AFRC-JFRC Joint International Symposium 2004
effective method for evaluating NOx emissions and furnace performance for new burner technologies, thus minimizing furnace start-ups times and unforeseen performance impacts. However, existing CFD models have shown poor accuracy in predicting furnace NOx emissions due to the combination of complex geometries, turbulent mixing, lean premixed combustion, and low NOx levels in the new generation of low NOx burners. Existing CFD software has been used to study more general process furnace characteristics such as furnace exit temperatures, heat transfer to process tubes, and flame shapes, as well as specific combustion components such as single burners (Brown, et. al 2003, Brown, et. al 1997). However, current commercially available CFD tools are generally limited in one or more of the following capabilities necessary for accurate low NOx process heater modeling: •
sufficient computational points to resolve the detailed burner geometry, multiple fuel mixing zones, flame interactions between burners, and full furnace geometry;
•
combustion sub-models to predict fuel-lean, premixed, turbulent combustion;
•
radiative heat transfer sub-models to describe gas-wall-tube heat exchange;
•
flow and chemistry sub-models to account for turbulence-chemistry interactions; and
•
finite-rate chemical kinetics sub-models to account for NOx reactions.
Developing software that provides all of these capabilities in a single modeling tool has proven challenging due to limitations in computer capacity and descriptions of detailed turbulence and chemistry. This paper describes a new computational tool developed by Reaction Engineering International (REI) that is specifically designed to model lean premixed combustion associated with ultra-low NOx burners in process furnaces. The new tool is based on a Hybrid Scalar Transport Probability Density Function Solver and includes: adaptive mesh refinement to capture near-burner mixing; reduced chemical kinetic mechanisms with in-situ adaptive tabulation to compute finite rate chemistry; a combination of conventional Eulerian turbulence modeling with Monte-Carlo PDF methods to model turbulent reactions and mixing; and a matrix-free Newton-Krylov method to reduce solver computational time and improve robustness. The remainder of the paper is organized as follows: A brief review of relevant turbulent combustion models is provided first followed by details of the included sub-models and their implementation into the new tool. This is followed by sections that provide simulation results for a well characterized, partially premixed methane flame and an industrial scale furnace firing a low NOx burner, respectively. Last, conclusions and planned future work are summarized.
2
REACTION ENGINEERING INTERNATIONAL
Advanced CFD Tools for Modeling Lean Premixed Combustion in Ultra-Low NOx Burners in Process Heaters AFRC-JFRC Joint International Symposium 2004
2. Modeling Turbulent Combustion Most practical combustion systems involve turbulent flames, which are characterized by various complex and rate-controlling processes associated with turbulent transport, mixing, chemical kinetics, and convective and radiative heat transfer, among others. The success of any methodology used in modeling turbulent combustion depends a great deal on how well it can model these processes, which often strongly interact with each other at various disparate timescales and length scales. In combustion processes which are characterized by fast chemistry, use of the flamelet model (Peter, 1984; Pitsch and Steiner, 2000) or conditional moment closure (CMC) (Bilger, 1993) based on a conserved scalar is known to be quite accurate in making qualitative as well as quantitative predictions. However, these methods may not be viable in slow chemistry regimes such as the formation and destruction of NOx. The flamelet model requires the specification of a scalar dissipation rate and assumes the shape of the PDF at the sub-grid level. The first-order CMC model ignores any fluctuations about the conditional mean (Raman et al., 2003). Probability density function (PDF) methods have been demonstrated to be a successful modeling approach for turbulent combustion (Pope, 1985, Lindstedt et al., 2000, Tang et al, 2000, Xu and Pope, 2000). The approach offers an important advantage over other methods in that chemical reactions appear in closed form in the PDF equations. As a consequence, realistic combustion chemistry can be incorporated without the need for closure approximations pertaining to the reactions. Therefore, PDF methods are able to accurately describe finite-rate chemistry effects and turbulent-chemistry interactions in turbulent flames. There are basically two categories of PDF methods: the joint scalar PDF (or composition PDF) method and the joint velocity-turbulent frequency-composition PDF method. Both methods have been used for modeling turbulent combustion. The methods are general, apply equally well to premixed, partially premixed, and non-premixed combustors. In this study, the joint scalar PDF method is selected as the foundation of the advanced modeling tools for modeling practical lean premixed combustion problems due to its simpler implementation and ease of handling the complex 3-D geometry needed to model industrial combustion equipments. In this approach, the modeled transport equation for the joint scalar PDF equation is solved, and a coupled or uncoupled RANS solver is applied to solve modeled momentum equations. In addition to the unclosed terms in the traditional RANS equations, the molecular flux terms in the PDF equations also need to be modeled. It is straightforward to combine this method with well established RANS solvers. In addition, the statistical error due to randomness is reduced because the convection is not stochastic. The implementation of well-established physics models for other processes such as radiative heat transfer, and the treatment of complex boundary conditions are straightforward, which is important for comprehensive modeling of industrial combustion systems. New approaches such as large eddy simulation (LES) and direct numerical simulation (DNS) have been applied to combustion modeling (e.g., Pitsch and Steiner, 2000, Veynante and Vervisch, 2002) in recent years. These methods have the potential to resolve turbulent combustion phenomena in unprecedented detail. However, these methods have not reached the state of maturity required for implementation in a model intended for practical combustors with
3
REACTION ENGINEERING INTERNATIONAL
Advanced CFD Tools for Modeling Lean Premixed Combustion in Ultra-Low NOx Burners in Process Heaters AFRC-JFRC Joint International Symposium 2004
complex geometry and chemistry. For a more thorough review of turbulent combustion models and their use in modeling turbulent combustion processes, see the textbook by Peters (2000) or the recent review paper by Hilbert et al. (2004). 3. Description of New Modeling Tool This section describes the new modeling tool. Presented are: 1) An overview on the structure of the tool and the overall solution strategy; 2) The adaptive mesh refinement (AMR) turbulent flow solver; 3) The approach for problem initialization; 4) Sub-models and solution scheme used within the PDF solver. Described are the submodels used to treat turbulence diffusion, molecular mixing, and heat transfer processes, respectively; 5) A brief explanation on the reduced chemical mechanism; 6) The in-situ adaptive tabulation (ISAT) algorithm that incorporates realistic combustion chemistry into PDF calculations is discussed. 3.1. Overview The modeling tool solves the joint scalar PDF transport equations for statistically stationary turbulent reactive flows. In the PDF approach, the high dimensional joint probability density function is represented by a large ensemble of notional ‘fluid’ particles. The particle properties evolve according to particle sub-models such that the evolution of the statistics of Lagrange Particle Solver ~ Finite Volume Solver the particle ensemble corresponds to the U, k , ε modeled PDF evolution. Solve for: Particle properties:
Solve for Mean fields:
~ U, p , k , ε
ρ ,υ
m * , x* , φ * φ* , ∆t
CARM Work Bench Optimized Reduced Mechanism
∆φ * , ρ *
φ S(φ)
In-Situ Adaptive Tabulation
Figure 3.1 Modules in new CFD modeling tool.
The new modeling tool consists of four major components, namely an Eulerian finite-volume flow solver, a Lagrangian particle solver, an in-situ adaptive tabulation (ISAT) (Pope, 1997) module for reaction source term evaluation, and a chemical mechanism generator (CARM work bench) (Montgomery et al., 2002). The relationships among different components are illustrated in Figure 3.1.
During the simulation, the conservation equations for mean mass, momentum, pressure, and turbulent kinetic energy and dissipation rate are solved by the finite volume flow solver while the Lagrangian particle solver is employed to solve the joint composition PDF transport equation. A reduced chemical mechanism (i.e., a FORTRAN subroutine) is generated by CARM offline, and is linked to the other modules to form the executable. The highly efficient in-situ adaptive tabulation module is called by the particle solver to evaluate the reduced mechanism when solving the joint PDF equations.
4
REACTION ENGINEERING INTERNATIONAL
Advanced CFD Tools for Modeling Lean Premixed Combustion in Ultra-Low NOx Burners in Process Heaters AFRC-JFRC Joint International Symposium 2004
FV inner iteration
Figure 3.2 shows the solution scheme adopted in the modeling tool. The FV and particle methods are periodically used in the hybrid algorithm to solve their respective equations. The hybrid FV/particle algorithm can be designed to run in a tightly or loosely coupled manner. In a loosely coupled algorithm, the outer iteration START is completed by running the FV code for a specified number of iterations Initialize FV (steps) or until convergence and the and particle code particle code is run for a specified number of time steps. Note that tight coupling, in which both FV and Advance FV code particle codes are run for a single step to complete an outer iteration, is in fact No a special case of loose coupling. FV converged? Yes Favre averaged velocity field
Particle inner iteration
Advance particle code
Evaluate particle fields and time average Yes
More particle time steps?
1) The Eulerian finite volume solver solves for mean velocity, pressure, turbulent kinetic energy and turbulent dissipation. 2) Given these mean fields, the Lagrangian particle solver solves for the joint composition mass density function. 3) Effects of thermo-chemistry are solved in the Lagrangian algorithm.
No No
The overall solution sequence can be summarized as follows.
Global convergence Yes STOP
Figure 3.2 Solution scheme - hybrid FV/particle method.
4) Mean thermo-chemical properties that influence the flow field, like density and molecular viscosity fields, are then coupled back to the finite-volume method that solves for the new turbulent flow field. Subsequent sub-model calculations are
performed until a stable solution is reached. 3.2. Unstructured Grid Flow Solver with Adaptive Mesh Refinement An AMR flow solver developed by REI is used for the 3-D finite volume solver. The solver is an adaptive grid code that uses an orthogonal grid with local mesh refinement. A structured base grid is input and grid refinements are performed. Cells that are tagged for refinement are divided into eight smaller cells. Cells are tagged for refinement either manually via user input, automatically to resolve geometry, or based on gradients in field variables in the solution. A typical AMR grid is shown in Figure 3.3.
5
REACTION ENGINEERING INTERNATIONAL
Advanced CFD Tools for Modeling Lean Premixed Combustion in Ultra-Low NOx Burners in Process Heaters AFRC-JFRC Joint International Symposium 2004
The flow model of the AMR code uses a SIMPLEM algorithm (Acharya and Moukalled, 1989). This algorithm is similar to SIMPLER (Patankar, 1980) except that the mass-satisfying flow field defined on the cell faces is calculated from the pressure equation directly in lieu of a pressure correction equation (Acharya and Moukalled, 1989). The momentum equations are solved in a collocated manner so that the velocity field is stored at the cell centers and velocities from the discretized equations, without the pressure gradient source term, are interpolated to Figure 3.3. AMR solution for an injector study. the cell faces via a momentum interpolation. The pressure field, also stored at cell centers, is solved from the continuity equation discretized over the cells. The resulting pressure gradient sources are then added to the interpolated face velocities resulting in a mass satisfying flow field at the faces. The discretized partially differential equations are solved using an algebraic multi-grid (AMG) strategy. 3.3. Problem Initialization In the particle-based Monte Carlo method, the fluid is represented by a large number of “particles” having properties that evolve according to the model stochastic differential equations. For a typical 3-D turbulent combustion simulation, millions of particles are tracked in the simulation to achieve a quality solution. Even with the latest numerical techniques and computational power, the calculation can be very time consuming. Hence, a good problem initialization strategy using a simple, yet sufficiently accurate, turbulent combustion model is crucial for the success of the PDF method in practice. The AMR code includes an assumed PDF turbulent combustion model based on mixture fraction and its variance. For non-premixed, diffusion-limited combustion the approach works well. However, its extension and application to premixed combustion with multi-step chemistry is awkward because of the assumption of statistical independence for each additional reaction variable and the significant increase in the required computational effort. To avoid this difficulty, an eddy dissipation concept (EDC) model has been implemented to provide an initial estimate of the solution for pre-mixed combustion. The implemented EDC model is an extension to the EDC model of Gran and Magnussen (1996), and can account for multi-step chemistry. The EDC model is based on the assumption that chemical reactions occur in the regions where the dissipation of turbulence energy takes place. In flows from moderate to intense turbulence, these regions are concentrated in isolated regions, occupying only a small fraction of the flow. These regions consist of fine structures whose characteristic dimensions are of the order of the Kolmogorov length scale in one or two
6
REACTION ENGINEERING INTERNATIONAL
Advanced CFD Tools for Modeling Lean Premixed Combustion in Ultra-Low NOx Burners in Process Heaters AFRC-JFRC Joint International Symposium 2004
dimensions. Based on this assumption, the mean reaction rate is expressed as a function of the volume fraction of the fine structure regions, the lifetime of the fine structure, and the species mean mass fraction. In the implementation of the modified EDC model, a pseudo time splitting scheme is applied in the AMR code to solve the mean species and enthalpy transport equations, where a convection-diffusion step and a reaction step, which solves the EDC mean reaction rates, are performed to update the scalar fields. Note that the ISAT algorithm is applied in the reaction step, which speeds up the calculation significantly. The work of implementing another solution scheme based on a Newton-Krylov solution strategy for the EDC model is in-progress. Unlike the traditional “Picard” iteration scheme used in the time splitting scheme, the Newton-Krylov based solver solves the species transport equation simultaneously. Previous study has shown that the Newton-Krylov solver can cut the CPU time by a factor of two compared to the Picard iteration (Wang, et al., 2002) 3.4. Sub-models for Lean Premixed Combustion 3.4.1 Problem Formation In the transport PDF approach, species concentration and specific enthalpy are treated as random variables, and the transport of their joint PDF rather than their finite moments is considered. Once the PDF is determined, the mean of any quantity can be evaluated exactly from the PDF, if it is a function of the species concentration and/or specific enthalpy. The composition vector is defined as ψ ≡ (ψ 1 ,ψ 2 , K ,ψ ns ,ψ h ) T , where ψ 1 , ψ 2 , L, ψ ns are species mass fractions and ns is the number of species, and ψ h is the specific enthalpy of the mixture. The sub-model calculates the local one-point, one-time, joint composition PDF f ϕ . The transport equation for f ϕ can be derived from the general equations for conservation of species mass fraction and energy for low-Mach-number flow. Following the procedure outlined by Pope (1985), the transport equation for f ϕ can be expressed as
[
]
[
]
[
∂ ∂ ∂ ρ (ψ ) f φ + ρ (ψ )U i f φ + ρ (ψ )S α (ψ ) f φ ∂t ∂x i ∂ψ α
]
⎤ ∂ ∂ ∂ ⎡ ∂J αi ρ (ψ ) u i′′ ψ f ϕ + ψ fφ ⎥ − ∇ ⋅ q ψ fφ = ⎢ ∂ψ h ∂ψ α ⎣⎢ ∂x i ⎥⎦ ∂x i
[
]
[
]
(1)
where ρ , U , u ′′ , S, J and q denote density, mean velocity, turbulent fluctuation velocity, chemical reaction source term, diffusion flux and radiation heat flux, respectively. The notation of 〈 A | B〉 is the expectation of the conditional probability of event A, given that event B occurs. All terms on the left-hand-side of Equation (1) appear in closed forms. The first term on the right-hand-side represents transport in composition space due to molecular mixing. The second term represents transport in physical space due to turbulent velocity fluctuations. The third term represents transport in composition space due to radiation heat loss. These three terms need to be modeled. The approach for modeling these terms is provided below. 3.4.2 Turbulent Diffusion Model The transport term due to turbulent velocity fluctuations is modeled using the gradient-diffusion model. Detail about this model can be found in Pope (1985).
7
REACTION ENGINEERING INTERNATIONAL
Advanced CFD Tools for Modeling Lean Premixed Combustion in Ultra-Low NOx Burners in Process Heaters AFRC-JFRC Joint International Symposium 2004
3.4.3 Mixing Model Two mixing models are implemented to model the transport of probability density function in composition space by molecular fluxes, namely the interaction-by-exchange-with-the-mean (IEM) model (Dopazo, 1975) and the Euclidean Minimum Spanning Tree (EMST) model (Subramaniam, 1997, Ren et al., 2002). IEM: The IEM model has the effect of moving the value of the instantaneous composition towards the mean composition at a controlled rate and thus reduces the distribution in the composition values. The model is simple and easy to implement, but could be problematic when there are strong interactions between turbulent mixing and finite-rate chemical reaction (Subramaniam, 1997). EMST: The EMST model is a complicated particle-interaction model designed to overcome shortcomings of simpler models. A full description of the EMST model can be found in Subramaniam (1997) and Ren et al. (2003). The EMST model is a localized mixing model, and was successfully used in several joint PDF calculations of Piloted-Jet flames and Bluff-body flames to predict local extinction and re-ignition (Tang et al., 2000, Xu and Pope, 2000, Liu, 2004). The mixing frequency required by the IEM and the EMST mixing models is obtained from the κ − ε turbulent model (Jones and Launder, 1973) contained in the finite volume flow solver. 3.4.4 Wall Heat Transfer Model Radiation heat transfer and near-wall convection/conduction heat transfer must be accurately represented in modeling wall-bounded combustion flows that are typical in industrial combustion equipment. Hence, models to handle the heat transfer between the combustion gas and wall have been implemented into the hybrid PDF solver. The heat transfer between gas phase and wall due to convection and conduction is described by a linear model (Fox, 2004) of the form dT (n ) = C w (Tw − T (n ) ) dt
(2)
where T (n ) , Tw are particle temperature and wall temperature, respectively. C w is a coefficient chosen to yield the desired heat transfer rate, which will vary from cell to cell and change with time as the flow evolves. It should be noted that the maximum heat transfer will occur when C w = ∞ (i.e., T (n ) = Tw ), and that C w = 0 corresponds to a zero-flux wall. 3.4.5 Radiation Heat Transfer Model Radiation is typically the most significant mode of heat transfer in practical combustion equipment. Accurately simulating radiation heat transfer to specific regions in a system requires a model that can account for both absorbing-emitting radiation processes and complex system geometries, including arbitrary structures such as convective tube passes. REI’s model utilizes the discrete-ordinates method (DOM) that has proven to be a viable choice for modeling radiation in combustion systems, both in terms of computational efficiency and accuracy (Adams and Smith, 1995). The model has been implemented into the AMR flow solver and is modified to couple with the Monte Carlo PDF solver.
8
REACTION ENGINEERING INTERNATIONAL
Advanced CFD Tools for Modeling Lean Premixed Combustion in Ultra-Low NOx Burners in Process Heaters AFRC-JFRC Joint International Symposium 2004
The discrete-ordinates method solves the integral-differential radiation heat transfer equation (RTE) in a number of discrete angular directions spanning the total solid angle of 4π steradians. The discrete-ordinates representation of the mean radiation heat transfer equation can be written in Cartesian coordinates as µm
∂ Im ∂x
+ ηm
∂ Im ∂y
+ ξm
∂ Im
= − k ag I m + ebg
∂z
(3)
where µ m , η m , and ξ m are the direction cosines for the discrete angular direction Ωm, 〈 I m 〉 is the mean radiation intensity in the direction Ωm , 〈 k ag 〉 is the mean gas absorption coefficients, and 〈 ebg 〉 is the mean gas blackbody radiation intensity, which can be calculated as
σ 4 k ag Tg (4) π where σ = 5.669 × 10 −8 W / m 2 K 4 is the Stefan-Boltzmann constant and Tg is the gas temperature. The radiation heat loss rate per unit volume that forms the radiation source term in the energy transport equation can be expressed as ebg =
M
Qr = ∇ ⋅ q = 4πebg − k ag ∑ wm I m
(5)
m =1
where wm is the quadrature weight for the mth direction, and M is the total number of discrete directions. In traditional finite volume CFD codes, Equation (3) is solved together with the energy transport equation in an iterative manner where 〈 I m 〉 and 〈ebg 〉 are updated periodically to achieve a converged temperature field. To model the radiation term in the joint composition PDF equation, we substitute Equation (5) into the third term in the right-hand side of equation (1) and split it into two terms
[
]
M ⎤ ∂ ⎡ ⎛ ∂ ∂ ⎞ 4 4 4σ k ag Tg f φ − ⎢ ⎜ 4σkag Tg − kag ∑ wm I m ⎟ ψ f φ ⎥ = ∂ψ h ⎢⎣ ⎝ ∂ψ h m =1 ⎠ ⎥⎦ ∂ψ h
M ⎡ ⎛ ⎤ ⎞ ⎢ ⎜ kag ∑ wm I m ⎟ ψ f φ ⎥ (6) ⎠ ⎢⎣ ⎝ m=1 ⎥⎦
The first term on the right-hand side of equation (6) represents radiative emission and appears in closed form because the conditional mean can be calculated from local composition exactly. The second term on the right-hand side of equation (6) needs to be modeled because I m is not a part of the composition space, and the conditional mean can not be calculated by local composition. The model adopted here is the optically-thin eddy approximation (Kabashnikov and Myasnikova, 1985) that suggests that the local incident radiation intensity I m is only weakly correlated to the local radiative properties. This leads to M M ⎤ ∂ ⎡ ⎛ ∂ ⎡⎛ ⎞ ⎞ ⎤ ⎜ kag ∑ wm I m ⎟ f φ ⎥ ⎢ ⎜ kag ∑ wm I m ⎟ ψ f φ ⎥ ≈ ⎢ ∂ψ h ⎢⎣ ⎝ m=1 ⎠ ⎠ ⎦ ⎥⎦ ∂ψ h ⎣⎝ m=1
(7)
Equation (7) can be coupled with equation (3) to solve for the change in particle enthalpy due to radiation and hence the mean temperature field. 〈 I m 〉 in equation (7) is solved from equation (3) using a finite difference scheme, while 〈ebg 〉 in equation (3) is obtained from the ensemble mean of particle radiation intensity determined by particle composition via equation (4).
9
REACTION ENGINEERING INTERNATIONAL
Advanced CFD Tools for Modeling Lean Premixed Combustion in Ultra-Low NOx Burners in Process Heaters AFRC-JFRC Joint International Symposium 2004
3.4.5 Solution Strategy The joint composition PDF is solved using a modified Euler scheme (Chao and Pope, 2002) to sequentially calculate the change in particle properties due to the simultaneous processes of mixing, etc. 1) The particles are first convected for half a time step. 2) The changes in particle properties due to mixing, reaction, and radiation are then calculated for the full time step using the Strang’s operator splitting scheme. 3) Finally, the particles are convected to their final positions for the time step. A local time-stepping strategy (Muradoglu and Pope, 2001) is adopted in the modified Euler scheme where the time steps to advance particles are determined by local residence and mixing times. In the FV solver, convergence is measured by residuals and a FV solution is considered to be converged when the magnitude of residuals is smaller than a specified tolerance value. Here the residual is defined as the mean of absolute residuals of the continuity, momentum, turbulent kinetic energy, and turbulent kinetic energy dissipation equations averaged over the entire computation domain. The global convergence of the particle algorithm is monitored by the changes in the particle mean fields. These changes are typically measured at important physical locations and serve the same purpose as the residuals in the FV solver. In addition, overall mass and energy balances are also used to help determine convergence. 3.5. Reduced Mechanism Based on Quasi-Steady State Assumption The reduced mechanism is constructed using the quasi-steady-state assumption (QSSA) introduced by Bodenstein (1906), where given a detailed mechanism with ns chemical species, a small number of nc species are retained to represent the dynamics of the detailed system, while the other nss = ns - nc species that are associated with fast processes are assumed to be in steady state with their net chemical production rates being set to zero. Then, in the turbulent combustion calculation, the relevant equations are solved for the nc unsteady-state species instead of for the ns species. Because the computational cost increases at least linearly with the number of species represented, substantial gains can be achieved for nc
