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Modeling of lean premixed combustion in stationary gas turbines
PERGAMON
Progress in Energy and Combustion Science 25 (1999) 353–385 www.elsevier.com/locate/pecs
Modeling of lean premixed combustion in stationary gas turbines B. Scott Brewster*, Steven M. Cannon, James R. Farmer, Fanli Meng Advanced Combustion Engineering Research Center, Brigham Young University, Provo, UT 84602, USA
Abstract Lean premixed combustion (LPC) of natural gas is of considerable interest in land-based gas turbines for power generation. However, modeling such combustors and adequately addressing the concerns of LPC, which include emissions of nitrogen oxides, carbon monoxide and unburned hydrocarbons, remains a significant challenge. In this paper, characteristics of published simulations of gas turbine combustion are summarized and methods of modeling turbulent combustion are reviewed. The velocity–composition PDF method is selected for implementation in a new comprehensive model that uses an unstructured-grid flow solver. Reduced mechanisms for methane combustion are evaluated in a partially stirred reactor model. Comprehensive model predictions of swirl-stabilized LPC of natural gas are compared with detailed measurements obtained in a laboratory-scale combustor. The model is also applied to industrial combustor geometries. q 1999 Elsevier Science Ltd. All rights reserved.
Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Simulations of gas turbine combustion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Modeling turbulent premixed flames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Moment closure methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1. Eddy dissipation concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2. Assumed PDF method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3. Laminar flamelet models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4. Linear eddy model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.5. Conditional moment closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Monte Carlo PDF methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1. Node-based composition PDF method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2. Particle-tracking composition PDF method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3. Velocity–composition PDF method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4. Velocity–composition–dissipation PDF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Large eddy simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Direct numerical simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Combustion chemistry of natural gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Global reduced mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1. Two-step mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2. Four-step mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3. Five- and nine-step mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.4. Evaluation of reduced mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . * Tel/fax: 1 1-801-325-1633. E-mail address: [email protected] (B.S. Brewster) 0360-1285/99/$ - see front matter q 1999 Elsevier Science Ltd. All rights reserved. PII: S0360-128 5(98)00014-8
354 360 361 361 361 361 362 362 362 362 363 363 363 364 364 365 365 365 365 366 366 367
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4.2. ILDM method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. In situ tabulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Comprehensive model description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Unstructured grid flow solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Submodel for lean premixed combustion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Model evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4. Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1. Introduction Gas turbines are of considerable interest for power generation in the utility and industrial sectors for several reasons. First, their combined-cycle efficiencies can exceed 50% when coupled to a heat-recovery steam generator and steam turbine [1–3]. Second, their modularity means that the capacity to generate power can be purchased in smaller increments (less than 10 MWe [1]) than with conventional power plants, thus making it easier to plan for future needs and thereby reducing the risk associated with capital investment. Third, gas turbine power plants require minimal onsite construction [3] and can be brought on line relatively quickly [1]. Fourth, NOx emissions can be reduced to ultralow levels (,10 ppm at 15% O2) [4–13]. Gas turbines can also burn a variety of fuels and can be adapted to coal use when natural gas supplies dwindle [14,15]. These advantages and others have led to renewed interest in gas turbines for power generation in recent years. In order to promote the development of land-based gas turbines, the US Department of Energy (DOE) is sponsoring the Advanced Turbine Systems (ATS) program to demonstrate commercial prototypes of both utility and industrial turbines by the turn of the century. ATS program objectives include: system efficiency 60% (lower heating value [LHV]) or greater for utility-scale systems, and an equivalent 15 percent improvement for industrial systems; environmental superiority without the use of post combustion emissions controls; busbar energy costs 10 percent lower than current systems; adaptability to coal or biomass firing; and reliability, availability and maintainability equivalent to current advanced turbine systems [16]. Modeling can play a significant role in achieving these objectives. A recent survey [17] gives insight into the modeling capabilities that are needed. Representatives from twelve US gas turbine manufacturers, industrial research groups and users of gas turbines indicated their organization’s level of interest in various fuels, operational modes, and geometric configurations for land-based gas turbine
368 369 370 370 371 373 377 377 378 379
combustion. NOx emissions were of vital concern and lean premixed combustion (LPC) of natural gas was the chief operational mode of interest. Geometric configurations of interest included cannular, aeroderivative, and annular. Thus, a model capable of predicting NOx emissions for LPC in a wide variety of configurations is needed. Gas turbine combustors have traditionally been operated in the non-premixed mode for safety and stability [1,3,18]. Unfortunately, this mode leads to unacceptably high levels of thermal NOx, which is produced in the high-temperature, near-stoichiometric regions that occur in the combustor during the mixing of fuel and air. Traditional methods that have been used to moderate the temperature and control thermal NOx formation include steam or water injection and Rich/Quench/Lean (RQL) combustion [1,3]. However, injection of steam or water lowers the process efficiency and increases emissions of carbon monoxide and unburned hydrocarbons (UHC), and the effectiveness of RQL combustion is limited by the rate at which air can be mixed with hot gases in the lean zone. Although regulations limiting NOx emission are site-specific [19], the general objective of the gas turbine industry is to achieve levels of less than 10 ppm (at 15% O2) [20–22]. Achieving this “ultra-low” level of NOx is not feasible by traditional methods. In order to achieve single-digit NOx, manufacturers have turned to LPC [23–33]. With LPC, the amount of postcombustion dilution air is reduced, and the combustion zone is operated with excess air to reduce the flame temperature. A combustion zone equivalence ratio of 0.4–0.6 is typical [10,34]. In addition, the fuel and air are premixed to eliminate stoichiometric regions [35]. The flame is cooler (,1800 K) and thermal NOx is virtually eliminated. Prompt NOx is also negligible at these lean conditions [11] and the nitrous oxide mechanism accounts for essentially all of the NOx formed [10,32,36]. However, combustion efficiency is reduced at the lower temperatures of LPC, as it is in combustors using steam or water injection, and emissions of CO and UHC are of concern [4,37–39]. In addition, stability limits are significantly reduced by pre-mixing the fuel and air and by operating at fuel-lean conditions near the lean flammability limit [40,41]. Accurately predicting CO, UHC, and NOx in premixed combustors is difficult because of the complexity of the
Table 1 Gas turbine combustion simulations Authors (year)
Combustion model
Radiation
Pollutant chemistry
1 Anand et al. (1996) [148,149] Premixed natural gas/air
k– e
No information
CO
2 Mueller and Knill (1996) [150] Non-premixed propane
k– e
Scalar PDF with two-step methane chemistry Eddy dissipation concept (EDC) and laminar flamelet Laminar chemistry for propane combustion Scalar PDF Assumed PDF with fast chemistry; two-step EDC; laminar flamelet Four-step EDC
Adiabatic walls; radiation neglected No information
Thermal/prompt NO processor; CO; UHC Thermal NO and CO
Evaporating Jet-A liquid spray k– e
4 Leonard and Dai (1994) [152] No information 5 Bai and Fuchs (1994) [153] Non-premixed methane
No information k– e
6 Rizk and Mongia (1994) [154,155]
Evaporating liquid spray
k– e
7 Gran et al. (1994) [156]
Non-premixed methane
k– e
8 Micklow et al. (1994) [157], (1993) [158] 9 Di Martino et al. (1994) [159] 10 Baron et al. (1994) [160] 11 Gulati et al. (1994) [161]
Evaporating Jet-A liquid spray k– e Evaporating liquid spray Non-premixed methane Non-premixed, gaseous fuel
k– e k– e k– e
12 Chen and Shuen (1993) [162] Evaporating n-pentane liquid k– e spray 13 Gran et al. (1993) [163] Non-premixed propane k– e 14 Cline et al. (1993) [164,165] Evaporating Jet-A liquid spray k– e
One-step EDC and laminar flamelet with extinction Laminar/equilibrium chemistry for slow/fast reactions Assumed-shape PDF Assumed-shape PDF Assumed-shape PDF One-step laminar chemistry
15 Bond et al. (1992) [166] 16 Tolpaid (1992) [167] 17 Yang et al. (1992) [168], [169] 18 Yan et al. (1991) [170] 19 Melconian et al. (1990) [171] 20 Nguyen (1990) [172]
Non-premixed C2H2 Evaporating liquid spray Non-premixed propane
Modified k– e k– e k– e
21 Nguyen and Ying (1990) [173] 22 Talpallikar et al. (1990) [174] 23 Shyy et al. (1989) [175]
Evaporating liquid (Jet-A) spray No information Non-premixed CH2
k– e
EDC Laminar chemistry for propane Assumed-shape PDF Assumed-shape PDF Laminar and equilibrium chemistry Assumed-shape PDF One-step EDC One-step with mixing-limited rate Laminar chemistry
No information k– e
No information Assumed-shape PDF
Evaporating liquid spray k– e Evaporating liquid spray k– e Evaporating Jet-A liquid spray k– e
No information CO Absorbing/emitting gas CO with discrete transfer method Quasi-3D zone method to calculate liner temperature No information
Semi-analytic postprocessor for NOx, CO, UHC, and smoke None
Adiabatic walls, radiation neglected Neglected Neglected Adiabatic; internal heat transfer neglected Adiabatic wall; neglected
CO and thermal NO
None
Adiabatic wall; neglected No information
None Thermal NO and CO
No information Neglected No information
No information None Thermal NO and CO
Thermal NO post-processor None None
Six-flux model No information No information
None Thermal NO post-processor Thermal NO
No information
CO and NOx
No information No information
No information None
B.S. Brewster et al. / Progress in Energy and Combustion Science 25 (1999) 353–385
Turbulence model
3 Cline et al. (1995) [151]
Fuel injection
355
356
Authors (year)
Turbulence model
Combustion model
Radiation
Pollutant chemistry
24 Pridden and Coupland (1988) No information [176] 25 Shyy et al. (1988) [177] Non-premixed, gaseous fuel 26 Chleboun et al. (1987) [178] Non-premixed propane
k– e
Assumed-shape PDF
No information
Thermal NO post-processor
k– e
Assumed-shape PDF
Neglected
None
k– e
Assumed-shape PDF
No information
27 Wild et al. (1987) [179]
28 Shyy and Braaten (1987) [180,181] 29 Shyy et al. (1986) [182,183] 30 Coupland and Pridden (1985) [184]
Fuel injection
Evaporating liquid (kerosene) spray; nonpremixed propane Non-premixed, gaseous fuel
k– e and algebraic stress
Two-step EDC
No information
CO departure from equilibrium based on eddy break-up CO
k– e
Assumed-shape PDF
Neglected
None
Non-premixed, gaseous fuel Gaseous fuel (kerosene)
k– e k– e
Assumed-shape PDF Assumed-shape PDF
Neglected No information
None Thermal NO post-processor
B.S. Brewster et al. / Progress in Energy and Combustion Science 25 (1999) 353–385
Table 1 (continued)
Table 2 Gas turbine combustion simulations
1 2
Grid
Solution method
Data
Computer/Code
Sensitivity studies
Allison RSPN low emission combustor Lab-scale can type combustor
2D, staggered
Finite volume/Monte Carlo PDF; SIMPLE No information
Mean velocities, mix fraction and temperature Exit temp, equivalence ratio, CO2, O2, and NOx
IBM RS6000 workstation
None reported
TASCflow 3D commercial CFD package No information d100 cpu-hr on IBM RS6000 with 120 MBytes; CFD-ACE No information
B.C.s; comb model; temperature fluctuations in NO production Air mass flow split None reported
3D, non orthogonal BFC with grid embedding
3 4
Staged RQL combustor Pratt & Whitney annular combustor
3D, non-orthogonal BFC 3D, non-orthogonal BFC
SIMPLE-like Finite volume/Monte Carlo PDF; SIMPLE
None reported None reported
5
Model gas turbine combustor RQL; can and annular; premixed automotive combustor Axisymmetric can and annular Fuel nozzle/rich burn section of RQL combustor Two Alfa Romeo Avio reverse-flow annular combustors Reverse-flow annular combustor Full-scale, double annular research combustor GE annular combustor with dual can dome Annular combustor sector
3D cylindrical, staggered
Quasi-time marching with multi-grid solver SIMPLE
None reported
6
7 8 9
10 11 12 13 14 15 16 17 18 19
Sector of annular gas turbine combustor Vaporizer and flametube
Wall temperature, NOx, CO, UHC, smoke
No information
3D, non-orthogonal BFC; co-located 2D axi-symmetric, nonorthogonal BFC 3D non-orthogonal BFC; colocated
SIMPLE
None reported
No information
Time-dependent (transient)
None
KIVA-II [185] code
SIMPLE-like
Exit temperature at idle and full load power conditions
7–11 cpu-hr on VAX 9410 computer
Combustion model; extinction Air mass flow split; wall geometry; swirl direction Idle versus takeoff conditions
3D, Cartesian with stair-step boundaries 3D, non-orthogonal BFC; staggered 2D, non-orthogonal BFC; multiblock 3D, non-orthogonal BFC; co-located 3D, non-orthogonal BFC
Unsteady-state; fractional time steps SIMPLE-like; QUICK differencing Dual time-stepping integration method SIMPLE
Velocity, temperature, and species Mean and rms temperature and species None reported
Computer code ESTET
Inlet air temperature
CONCERT-3D computer code ALLSPD computer code
Dilution air
SIMPLE-like algorithm
3D BFC
No information
3D, non-orthogonal BFC; staggered 3D BFC 3D, staggered
Multi-grid; SIMPLE-like
Qualitative temperature comparisons Water analogy and full combustor Exit temperature
3D, Cartesian, stair-step boundaries
Langrangian–Eulerian SIMPLE SIMPLE
None reported
None reported Velocity and temperature profiles Water analog tests in the VRT combustor
None reported
37–47 CPU-min on Cray X-MP No information
Finite differencing scheme
No information
KIVA-II code VAX 780
Fuel injector location and geometry Vaporization; takeoff vs. idle Equivalence ratio None reported
FLUENT commercial code
VRT configuration
No information
Air mass flow split
357
GE single annular CFM56 turbofan engine Staged RQL combustor Axisymmetric swirl combustor Variable residence time (VRT) and annual combustors
3D, non-orthogonal BFC
Effect of combustion model on CO/temperature Fuel type; spray quality; air split
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Combustor
358
Combustor
Grid
Solution method
Data
Computer/Code
Sensitivity studies
20 21
Staged RQL combustor Model airblast fuel nozzle
2D, staggered 2D, non-orthogonal BFC
None None
RQL staged combustor
3D
None
No information KIVA-2 computer code; CRAY X-MP REFLEQS computer code
None reported Chemical mechanism
22
SIMPLE algorithm Implicit-continuous Eulerian (ICE) No information
23
Three annual GE combustors Annular Rolls-Royce RB211 combustor Annular GE turbofanengine combustor Can-type test combustor
3D, non-orthogonal BFC
SIMPLE-like; multi-grid
Exit temperature patterns
No information
2D rotated orthogonal, BFC
SIMPLE algorithm
PACE computer program
3D, non-orthogonal BFC
SIMPLE-like
3D cylindrical
SIMPLE
Exit plane temperature distribution Exit plane temperature and pattern factor Fuel and CO at exit plane Gas composition, temperature, exit velocity and residence time Exit temperature and pattern factor None reported
No information
Non-equilibrium CO model None reported
No information
Secondary dilution holes
No information
Jet-to-main flow velocity ratio Grid density
24 25 26 27
Cylindrical can and sector of annual combustor
3D, cylindrical; stair-step boundaries
SIMPLE
28
Annular GE turbofanengine combustor Annular combustor sector
3D, non orthogonal BFC
SIMPLE-like; multi-grid
3D, non-orthogonal BFC; staggered 2D, axi-symmetric orthogonal BFC
SIMPLE-like; multi-grid
29 30
Rolls-Royce RB211 annular combustor
SIMPLE
Flow patterns; exit temperature; NOx
No information PACE computer program
, 3 cpu-hr on IBM3081
Mixing geometry; jet momentum flux Dilution hole geometry Effect of combustor geometry on NOx Effect of secondary holes
B.S. Brewster et al. / Progress in Energy and Combustion Science 25 (1999) 353–385
Table 2 (continued)
B.S. Brewster et al. / Progress in Energy and Combustion Science 25 (1999) 353–385
359
Table 3 Modeling approaches for turbulent combustion with kinetics Method
Advantages
Disadvantages
EDC [186,187]
Complex chemistry Simple to add to CDF codes based on laminar chemistry
Assumed PDF [129,192–199]
Simpler than Monte Carlo PDF methods
Flamelet [202–204]
Complex chemistry
Conditional moment closure [188,207,208]
Allows use of full kinetic mechanisms [188] Inexpensive [209]
Empirically based [188–190] Limited to wrinked flames [191] Assumes gradient diffusion Can produce spurious peaks in mean temperature in reaction zone [217] Multiple solutions possible; application to premixed flames not straightforward [55] Limited to binary mixing [210] Limited to two or three reactive scalars [200] Usually assumes statistical independence Assumes gradient diffusion PDF shape must be assumed Complex phenomena such as extinction and differential diffusion precluded Small errors in PDF shape may result in large errors in mean reaction rate [201] Does not account rigorously for interactions between turbulence and chemistry [200]. Only valid in thin flamelet regime [188,189,205,206] Advanced versions require strain history of the flow [188] Limited to binary mixing [210] Has difficulty dealing with backmixing [210] Scalar fluctuations about conditional mean assumed small [136,210]. Results for multi-step chemistry and inhomogeneous flows have not been reported [210] Shape of the PDF must be assumed [209]
Monte Carlo PDF [211–216]
Eulerian formulation; can be readily added to existing CFD codes [210] PDF shape calculated [217] Reaction treated exactly [217] Amenable to parallel processing [217]
Linear eddy [218]
Detailed parametric studies are economically feasible [219] Local integral moment [220–223]
Distinguishes among effects of turbulent mixing, molecular diffusion, and chemical reaction at all scales of the flow [219] Practical application still in early stages of development [219] High computational requirements [210] Economical Complex flows; predicts large-scale structure Complex chemistry
chemistry and its interactions with the turbulent fluctuations. Turbulent flow is often modeled with time-averaged equations. However, time-averaged reaction rates cannot, in general, be calculated from time-averaged temperature and concentration. Also, fluctuating rates with their associated heat release cause density fluctuations that affect the turbulent velocity field. A number of modeling approaches have been devised to
Expensive Currently affordable only for reduced chemistry [106,188] Molecular transport and transport due to the fluctuating pressure gradient must be modeled [188,215] Requires mixtures fraction gradients; not yet applied to premixed combustion where reaction kinetics dominate
treat chemical reactions in turbulent flow. “Laminar chemistry” neglects the turbulent fluctuations. The “fast chemistry” or “mixed-is-burnt” approach neglects kinetic limitations and assumes instantaneous reaction, limited only by turbulent mixing. Fast chemistry has worked reasonably well for predicting heat release and temperature in non-premixed combustion. In such cases, thermal NOx has been predicted reasonably well by kinetically-based
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post-processors that neglect the effects of NOx species on the gross flame properties. However, for premixed systems and to more accurately predict kinetically limited phenomena in non-premixed flames (e.g. CO emission), finite-rate chemistry with proper accounting of the chemistry/turbulence interactions is needed. In addition to the challenges presented by chemistry/ turbulence interactions, the geometric complexity of gas turbine combustors presents its own challenge. Typical combustors often contain more than 10 000 holes, with swirlers, film-cooling slots, contracting ducts, etc. [42]. Generating a computational grid for such geometries can be extremely difficult, requiring man-months of effort [43]. In spite of these challenges, computer modeling has played a key role in the design of gas turbine combustors for several years [44,45]. Every major gas turbine manufacturer routinely uses CFD-based, turbulent combustion modeling as part of the design procedure [46]. Current modeling technology, however, does not adequately address the concerns of LPC in modern combustors. In addition, the need for modeling is greater because the stability range is narrower with LPC and the emissions of CO, UHC, and NOx are likely to be more sensitive to design modifications. This paper summarizes the characteristics of gas turbine combustion simulations published in the last decade. Approaches for modeling turbulent combustion are reviewed, and the Monte Carlo velocity–composition PDF method is selected for implementation in a new model. Techniques for implementing advanced chemistry schemes are discussed and several reduced mechanisms are evaluated. Predictions of LPC in a laboratory-scale combustor are compared with recent experimental data [47,48]. Finally, application of the model to two industrial premixers and an industrial combustor is demonstrated for non-reacting flow.
2. Simulations of gas turbine combustion Comprehensive modeling of gas turbine combustion has been an active area of research for the last decade and numerous models and simulations have been reported in the literature. Characteristics of simulations published during this period are summarized in Tables 1 and 2, with the most recent simulations listed first. Only CFD-based simulations of reacting flow in gas turbine geometries have been included. Although an attempt has been made to include all reported simulations that meet the above criteria, some omissions have undoubtedly occurred. As the tables indicate, at least 30 simulation studies have been reported, with several of the studies involving more than one combustor. The majority of simulations have been carried out for jet engine combustors as opposed to stationary, land-based gas turbines. In jet engines, the predominant fuel of interest is kerosene or Jet-A, which is injected as an evaporating, liquid spray. The dynamics of spray evaporation have often been taken into account in the simulations,
perhaps to the detriment of modeling turbulence/chemistry interactions, which have often been neglected in such simulations. The k– e model has been universally used for turbulence closure. Turbulent combustion modeling approaches that have been applied to gas turbines include laminar chemistry, the eddy dissipation concept (EDC) method, the assumedshape PDF method, and the scalar PDF method. The laminar chemistry approach ignores turbulent fluctuations. The empirical EDC model includes the fluctuations by relating the reaction rate to the gross turbulence properties. The assumed-shape PDF method incorporates the details of the turbulent fluctuations, but for an assumed shape of their PDF. It is usually implemented with the fast chemistry assumption, ignoring kinetic effects. Monte Carlo PDF methods, including the scalar PDF method, treat complex, finite-rate chemistry without assuming the shape of the PDF. Each of these methods (except laminar chemistry) is discussed in more detail in the next section. Only three of the simulations included radiation. The others either explicitly neglected it (presumably due to the difficulty of including it) or did not mention it at all and presumably neglected it. The fact that radiation was typically ignored does not mean that it is considered unimportant in gas turbine combustors, as several of the investigators specifically suggested the neglect of radiation as a primary reason for discrepancies between their predictions and the data [159,162,163,167,177,180,181]. Pollutants that have been considered in simulations of gas turbine combustion include NOx, CO, UHC, and smoke. Thermal NO is the NOx-forming mechanism most often included. Prompt NO was considered in only one case. NOx-forming reactions should be treated kinetically, even when fast chemistry is assumed for the combustion reactions. NOx calculations are, therefore, often performed in a post-processor, neglecting the effects of these reactions on the gross flame structure, since NOx species are present in only trace amounts. CO is more difficult to predict accurately, because CO formation is also kinetically limited and it is not a trace species. Several of the attempts to predict non-equilibrium CO used laminar chemistry. A variety of geometric configurations, including both can-type and annular, are represented. The most common type of grid was non-orthogonal, body-fitted coordinates (BFC). Pressure-correction methods, such as the SIMPLE algorithm, were used almost universally, and comparisons have been made with data for several of the combustors. Some of the simulations were performed with commercially available codes, such as FLUENT; others were performed with proprietary industry or university research codes. There seems to be a trend toward running the simulations on workstations rather than supercomputers, consistent with general trends in computations. The large number of sensitivity studies that have been conducted to investigate the effect of one or more design variables indicates the importance of simulation to the design process.
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It is obvious from this list of reported simulations that there is current interest in simulating premixed gas turbine combustors, but that the technology to do so is just developing and is not widely available. While the k– e model has limitations, it has been adequate for many combustor applications. However, improved methods of incorporating combustion and pollutant chemistry as well as including radiation are needed.
3. Modeling turbulent premixed flames Modeling of multi-dimensional, turbulent flames such as those found in modern gas turbine combustors has been carried out on several levels. At the lowest level, the timeaveraged Navier–Stokes equations are solved for the mean fields. At the next level, a transport equation derived from the Navier–Stokes equations is solved for the joint PDF of composition and/or velocity, providing a complete statistical description from which the mean as well as higher moments can be determined. The third level of modeling is large eddy simulation (LES), where the large scales of the turbulent flowfield are resolved explicitly while the small scales are modeled. The highest level of modeling is direct numerical simulations (DNS), where the Navier–Stokes equations are solved directly on a fine grid and the instantaneous, small-scale structure of the flame is resolved. Turbulent premixed flames have been modeled at all of these levels. 3.1. Moment closure methods Moment closure methods include the classical approach of time averaging as well as the recently proposed method of conditional averaging. Moment closure generally requires a turbulence model to provide closure for the Reynolds stresses in the momentum equations and a turbulent combustion model to provide closure for the time-averaged reaction-rate term in the species continuity and energy equations. The accuracy of moment closure is limited by both models. The focus here is on the turbulent combustion model, consistent with the need for a comprehensive model that will address the concerns of LPC. While its deficiencies are well-known, especially for swirling flows [49], and while some researchers have obtained improved predictions with the second-order Reynolds stress models [50], the k– e model continues to be widely used in gas turbine combustor modeling. Since the k– e model is adequate in many cases, the need for an improved turbulence model is secondary to the need for more detailed chemical kinetics with improved modeling of chemistry/turbulence interactions. Turbulent combustion models have been used to provide closure for the time-averaged reaction term in the energy and species continuity equations include the EDC model, the assumed PDF model, laminar flamelet models, and the linear eddy
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model (LEM). The advantages and disadvantages of each are summarized in Table 3. 3.1.1. Eddy dissipation concept The EDC model [187] is based on the assumption that combustion occurs at the small scales, where mixing occurs on a molecular level and the rate is assumed to be proportional to the inverse of the turbulence time scale k/e . It was developed from the original eddy break-up model proposed by Spalding [51,52], the most significant difference being that the EDC model accounts for the fact that reaction cannot occur unless both fuel and oxidizer mix on a molecular scale at a sufficiently high temperature [163]. This is accomplished by relating the reaction rate to the limiting species. The EDC model has been extended to multiple-step chemistry by treating the small-scale structures as perfectly stirred reactors (PSRs) [53], although some investigators have reported it to be difficult to use with realistic multistep kinetics [188]. One interesting feature of the extended EDC model is that it can give different results depending on the initial conditions that are assumed for the PSR calculations of the fine-scale structures. The reason for this is the well-known ignition–extinction hysteresis loop for PSRs [54]. Multiple steady-states are possible and the converged solution depends on the initial guess. Gran [55] used the extended EDC model to predict turbulent premixed methane–air flame and investigated the effect of initial conditions. He obtained good agreement with the measured data when the appropriate initial conditions were used and concluded that the only general method of achieving an accurate prediction in premixed flames, where equilibrium chemistry cannot be used to give reasonable initial conditions, is to integrate in time from some known initial condition. This requires detailed knowledge of the experimental procedure, and this level of detail is typically not available in papers that describe experimental work. He did not experience the same dilemma for non-premixed flames where equilibrium chemistry was successfully used to provide reasonable initial conditions for the PSR calculations in the fine-scale structures. 3.1.2. Assumed PDF method In the assumed PDF method, the chemistry is determined by one or more “progress variables” which are allowed to fluctuate. The usual approach is to assume “fast chemistry” where the instantaneous properties of the gas are in equilibrium and can be calculated from the mixture fraction. The effects of turbulent fluctuations are incorporated by assuming a shape for the PDF of the mixture fraction, solving transport equations for its first and second moments, and integrating all instantaneous properties over the PDF to obtain time-averaged values. This integration is efficiently performed by pre-calculating gas properties as functions of the mixture fraction and storing the values in a table for subsequent retrieval.
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Reaction kinetics can be included by defining additional “reaction progress variables”, assuming the shape of a joint PDF, solving equations for the first and second moments of all the progress variables, and integrating instantaneous properties over the joint PDF. The mixture fraction and reaction progress variables are usually assumed to be mutually independent. Thus, the joint PDF is separable and the integration for each variable can be performed independently. Multi-dimensional table look-up and interpolation procedures are employed for efficiency. The assumed PDF method has been applied to nonpremixed, diffusion-limited combustion, where it works reasonably well. Its extension and application to premixed combustion with multiple-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. 3.1.3. Laminar flamelet models Flamelet models are relatively simple and can be used with complex chemistry, but they assume that the reactions are all fast and that their length scales are smaller than the smallest (i.e. Kolmogorov) turbulence length scale. Since the important reaction time scales (and the corresponding length scales) in gas turbine combustion vary over as much as seven orders of magnitude [3], some reactions are clearly in the flamelet regime, some are in the distributed reaction regime, and some are in between [56]. Therefore, the flamelet approach is not sufficiently general to treat all of the reactions of interest in gas turbine combustion and their interactions with turbulence. 3.1.4. Linear eddy model The recently proposed LEM focuses on the small-scale mixing effects in high-Reynolds number flows and explicitly distinguishes between the processes of turbulent mixing, molecular diffusion, and chemical reaction [219]. It does this by simplifying the representation of the scalar field statistics to one dimension and resolving all length scales (from integral to Kolmogorov) [57]. The effects of molecular diffusion on the one-dimensional representation are incorporated by explicitly solving the diffusion equation. Turbulent mixing is incorporated by stochastic rearrangements of the one-dimensional representation. The rearrangement blocks correspond to eddies and their size distribution and the frequency of the rearrangements events are determined from known scaling laws of turbulent mixing. Details are given by McMurtry et al. [219]. Goldin and Menon [201] demonstrated the use of LEM in a conventional, steady-state, turbulent combustion model of a non-premixed jet flame. However, they were unable to demonstrate superiority over the conventional assumed PDF approach, and further work is needed to extend the method to premixed flames. LEM remains in the early stages of development, particularly in its implementation and application in CFD codes, and is not yet considered a
viable option for a comprehensive, three-dimensional combustion model. 3.1.5. Conditional moment closure In conditional moment closure (CMC), species mass fractions and enthalpy are conditionally averaged over a fluctuating progress variable to account for the effects of turbulent fluctuations on the mean reaction rate. In non-premixed combustion, the mixture fraction is used as the conditioning variable. For premixed combustion, conditioning on a reaction progress variable has been proposed [188]. The fluctuations about the conditional mean are assumed to be small, thus limiting the range of application to flames that are far from extinction. Also, the shape of the PDF of the conditioning variable must be assumed [209]. Conditioning on both the mixture fraction and a reaction progress variable has been proposed as a means of extending the range of application to near extinction [58], but the results of this approach are not yet available. Presumably, double conditioning would also be needed to treat practical premixed combustors, where premixing is imperfect and the effects of non-premixedness are important to consider. Thus, CMC is a promising approach, particularly when the effects of detailed chemistry (e.g. full mechanisms) must be considered. The computational workload associated with CMC is considerably less than PDF methods. However, it is not yet sufficiently generalized for application in practical, pre-mixed combustor simulations. 3.2. Monte Carlo PDF methods Monte Carlo PDF methods can efficiently treat the effects of complex chemistry with turbulence interactions. PDF methods are general, applying equally well to premixed, partially premixed, and non-premixed flames. They treat complex chemistry exactly, with no modeling of terms involving reaction rates. However, modeling is required for other processes, such as molecular diffusion, because the PDF that is considered gives the statistical description of the scalar field at only a single point and time. Joint probabilities between the scalar field at multiple locations or for multiple times are not included. Hence, conditional expectations involving simultaneous scalar values at multiple locations such as spacial derivatives are not exact and must be modeled. Monte Carlo PDF methods solve a PDF transport equation using a large number of particles to represent the statistics of the turbulent flowfield. PDF equations can also be solved by conventional finite-difference techniques but this is not practical for the multi-dimensional PDFs that are of interest in turbulent combustion. By appropriately modeling the molecular diffusion term in the PDF transport equation, PDF methods can be applied to both the laminar flamelet and distributed regimes of turbulent combustion. The approaches typically employed, e.g. Curl’s models [92,93] and the interaction-by-exchangewith-the-mean (IEM) model [94,95], are applicable to
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distributed combustion. They consider only the turbulence dissipation time scale, ignoring the detailed effects of flame structure. Pope and Anand [59,65] adapted the PDF approach to the flamelet regime. Sakak [60] developed a unified approach that provides a consistent description of both regimes and demonstrated its applicability to regimes in between these two extremes. They showed that the PDF transport equation resembles the field equation in conventional laminar flamelet modeling when the combined reaction and diffusion terms are expressed in terms of the laminar flame velocity. PDF methods are classified according to the variables that are treated as random variables and the type of Monte Carlo method that is used. Composition, velocity–composition, and velocity–composition–dissipation PDFs have all been used in turbulent combustion modeling. The latter two are also referred to as the velocity–scalar PDF method. Both Eulerian and Lagrangian Monte Carlo methods have been used to solve the composition PDF transport equation. Only Lagrangian methods have been used with PDFs that treat the components of velocity as random variables. PDF methods are compared with other methods in Table 3. Table 4 shows the relative advantages and disadvantages of the various Monte Carlo PDF methods that have been employed in combustion. They are listed in order of increasing level of statistical description, but not necessarily in terms of increasing computational cost for a solution of given accuracy. Most applications of PDF methods have been limited to incompressible flows, but PDF methods have been recently extended to compressible flows where the effects of pressure fluctuations on density are taken into account [61,226–228]. A brief description of each method is given below.
3.2.1. Node-based composition PDF method Composition PDF methods treat all of the scalars that describe enthalpy and abundance of chemical species as random variables. From the composition PDF, one can determine the probability of having a certain gas composition at each location in the reactor. Variances and higher moments can be calculated from the PDF as well as means, although the accuracy of the higher moments becomes increasingly dependent on the number of particles used to obtain the solution. The transport equation for the composition PDF can be solved by both node-based [211] and particle-tracking [213] methods. The former uses an Eulerian approach with a grid and the particles are assigned to cells. The particles do not individually represent physical mass and, therefore, do not have continuous physical location. They can be thought of as residing at the cell nodes and moving from cell to cell according to the effects of diffusion and convection. The number of particles in a cell is constant, and the particles in each cell collectively represent the statistics (i.e. the PDF) of the scalars in that cell. The node-based composition PDF
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method was the first PDF method to be applied to combustion [62]. The advantage of the node-based, composition PDF method is its relative simplicity. Its major disadvantage is that it assumes gradient diffusion, which detracts from its usefulness for premixed combustion where counter-gradient diffusion is known to occur [230]. It also neglects the effects of density fluctuations on the turbulence and introduces randomness in the implementation of both convection and turbulent diffusion, leading to statistical error [229]. 3.2.2. Particle-tracking composition PDF method A Lagrangian approach is used in the particle-tracking method. The particles represent physical mass and their physical locations are tracked continuously throughout the flowfield. While the particle-tracking algorithm is grid free, its implementation employs a grid to calculate the scalar statistics at discrete locations. Unlike the node-based method, the number of particles in a cell is not constant and fluctuates in time as the particles move throughout the flowfield. Because velocity is not treated as a random variable in the PDF transport equation, neither composition PDF method calculates the flowfield and a conventional CFD solver is required for that purpose. The CFD solver provides the velocity and turbulence fields to the PDF code and the PDF code provides the density field to the CFD solver. The particle-tracking composition PDF method is more complicated and difficult to implement than the node-based method. It retains many of the disadvantages of the nodebased method, namely the gradient diffusion assumption and the inability to include the effects of density fluctuations on the turbulence. However, the statistical error due to randomness is reduced because the convection is not stochastic [229]. There is still randomness in the implementation of turbulent diffusion, which is treated by the random walk method [127]. 3.2.3. Velocity–composition PDF method In this method, the components of velocity are treated as random variables along with the species mass fractions and enthalpy. This method has been implemented with the Lagrangian, particle-tracking approach. Even though the velocity field is calculated, the method is not stand-alone, since the one-point, one-time PDF contains no information on the time or length scales of the turbulent motions. The turbulence dissipation (or time scale) field needs to be supplied for closure [63]. A major advantage of this method is that turbulent diffusion is now treated exactly. In addition, the effects of variable density are included. Although the mean pressure field needed for the Monte Carlo calculations can be extracted from the velocity PDF solution, it is sometimes more convenient to supply the turbulence time scale as well as the mean pressure field from a concurrently running finite-volume-based conventional CFD solver [214–216]. In this case, the PDF code supplies the Reynolds stresses to the CFD solver, thus avoiding the
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need for a turbulence closure model in the CFD solver. As with the composition-only method, the PDF code also provides the density field to the CDF solver. The first three-dimensional calculations with the velocity– composition PDF method were performed by Haworth and El Tahry [215]. Their transient calculations were obtained for non-reacting, constant-density flow. Hulek and Lindstedt [64] recently performed both steady-state and fully transient calculations of premixed methane–air and propane–air combustion using the velocity–composition method. Their run times for transient one-dimensional computations were “of the order of weeks on a standard level UNIX workstation”, even though they used a simplified kinetic approach. However, their results demonstrate that transient calculations of combustion are possible with the velocity– composition PDF method. In the Lagrangian Monte Carlo algorithm, the simultaneous processes that affect the particle properties and position are calculated sequentially. This is known as the method of fractional steps [213] or operator-splitting [103]. The error introduced by this technique can be controlled by using small time steps at the expense of more computational time. Gran [55] modeled a turbulent premixed flame with both the EDC and velocity–composition PDF methods and found that the PDF method underpredicted the reaction rate compared with the EDC model. They attributed this discrepancy to the coupling between reaction and diffusion and the effects of operator-splitting. Several investigators have commented on the weakness of the gradient diffusion assumption and the importance of treating the components of velocity as random variables for reacting flow, especially for premixed flames where countergradient diffusion is known to occur [65,66]. Anand et al. [230] have shown that gradient diffusion can lead to erroneous results in the case of reacting flow. They stress that the full potential of the PDF method cannot be realized by a composition-only PDF approach. Haworth and El Tahry [215], who first extended the velocity–composition PDF method to time-dependent flows, concur. 3.2.4. Velocity–composition–dissipation PDF In this method, turbulence dissipation is added to the list of random variables. The method now contains time- and length-scale information, thus enabling it to stand alone without an external CFD solver or turbulence model. Sometimes the method is referred to as a velocity–scalar PDF method were dissipation is numbered with the random scalars, along with the chemical series and enthalpy. Anand et al. [230] extended the velocity–composition– dissipation method of Pope and Chen [235] to swirling flows and applied it to a non-reacting, coaxial swirling jet. In addition to the stand-alone PDF calculations, they obtained predictions using an Eulerian transport equation for the mean dissipation, which is equivalent to using an external CFD solver to obtain length-scale information, and found that the stochastic model (i.e. the stand-alone PDF method
with dissipation treated as a random variable) gave predictions that agreed more closely with measurements. The stochastic model required approximately one-third more CPU time. One of the challenges in applying the velocity–composition–dissipation PDF method to recirculating flows has been the determination of the pressure field. Recently, Anand et al. [231] applied a new elliptic flow algorithm developed by Pope [67] to several swirl combustor flows. The new flow algorithm utilizes variance reduction techniques to maintain mean momentum conservation when turbulent processes such as mixing and viscous dissipation are performed on the particles. It also corrects particle positions to maintain consistency between the total volume of the particles in a cell and the geometric volume of the cell. Hsu et al. [217] compared the predictions of Anand et al. [231] for the stand-alone method with results obtained using the composition-only approach. They used the nodebased method to solve the composition PDF transport equation. Their results show the stand-alone method to be more accurate and they estimated it to be only 20% more time-consuming than the composition-only method. Unfortunately, there are no published results comparing the stand-alone method with the composition-only method solved by the distributed-particle approach or with the velocity–composition method. While showing considerable promise, the stand-alone PDF method is not yet considered sufficiently robust to use in a practical gas turbine combustion model. 3.3. Large eddy simulation LES has been applied successfully to premixed combustion, but it has not yet reached the state of maturity required for implementation in a model intended for practical combustors with complex geometry and chemistry. LES resolves the large-scale motion of the fluid by solving the instantaneous Navier–Stokes equations directly on a coarse mesh, but requires subgrid turbulence and turbulent combustion models for closure. The same methods that have been developed to model turbulent combustion at all scales have been applied to the subgrid scales in LES. Ryde´n et al. [68] developed a subgrid combustion model based on the EDC method and obtained improved predictions, as compared with k– e , for a premixed flame behind a bluff-body flame holder. Givi [69] and Gao and O’Brien [70] have used subgrid models based on PDF methods. Calhoon and Menon [71] developed a subgrid model using LEM and applied it to non-premixed flames. Menon et al. [72] have demonstrated the approach of using a subgrid model based on LEM in an LES simulation of premixed combustion using a simplified approach that avoids the significant closure problems that arise from combustion on the subgrid scale, namely the evaluation of filtered reaction source terms and the temperature–species correlations in the state equations. In view of the ability of
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LES to resolve the large-scale structures of the flow and the ability of LEM to resolve the fine scales where mixing and reaction take place, LES with sub-grid closure based on LEM might be the preferred approach for modeling turbulent combustion in the future. The recently proposed local integral moment (LIM) model [220,221] is a type of LES model, in that the large scales of the turbulent flow are computed explicitly. However, it uses a Lagrangian approach where the Navier–Stokes equations are transformed to a set of ordinary differential equations through a local parabolization around a time-evolving material surface on which the scalar gradients are concentrated. The parabolization is based on the observation that molecular mixing processes in turbulent flows are concentrated on universal, self-similar structures [73]. The chemical species fields are obtained from the scalar field constructed from the integral moments on the time-evolving surface via a strained diffusion and reaction layer formulation. The method is economical and has produced accurate calculations of complex flows with complex chemistry, but it has not been applied to partially premixed flames where the reaction kinetics dominate. Also, the method is not applicable to premixed flames where the effects of mixing are neglected and there are no gradients in the mixture fraction [74]. 3.4. Direct numerical simulation DNS has been applied to turbulent premixed flames [75– 78] but it is unlikely to be practical for industrial design calculations in the foreseeable future. Its main role is as a research tool to provide fundamental insight into turbulent flows and to help develop improved statistical submodels, such as the LEM, that will be applied in engineering codes for practical combustors in the future.
4. Combustion chemistry of natural gas The combustion chemistry of natural gas is complex. Natural gas contains numerous species and the composition is variable. Methane is the dominant component, with the average volume fraction varying from 0.86 to 0.95 in 14 large cities in the US [79]. Other important species include heavier hydrocarbons (ethane, propane, butanes, etc.), CO2 and N2. Many reaction mechanisms have been developed for describing the combustion of hydrocarbons in natural gas. These mechanisms range from simple one- and two-step models [80] to detailed models with hundreds of elementary reactions [83]. The simpler mechanisms ignore hydrocarbons heavier than methane and treat only a few species for a limited range of conditions, while more detailed mechanisms may describe the chemical behavior of dozens of species for a broad range of conditions and may include
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C21 chemistry for describing fuel breakdown and ignition [81,82]. The most comprehensive chemical mechanism for natural gas combustion is GRI-Mech 2.11 [83], with 276 elementary reactions and 49 species, including C2 and nitrogen species. Its development is being sponsored by the Gas Research Institute. C3 and heavier hydrocarbons are not currently included. However, the development of GRIMech is ongoing, and additional species, reactions and features are expected in future versions. Although Monte Carlo PDF methods treat chemical reactions exactly and there is no theoretical limitation preventing the use of a detailed mechanism such as GRI-Mech 2.11 in a PDF calculation, there is a practical limit of three to five reactive scalars with present-day computers and standard table look-up procedures. It has been estimated that 6 years of CPU time would be required on a typical engineering workstation to perform a practical, full-scale PDF method calculation with a 16-species mechanism for methane combustion [106], using direct integration (i.e. without table look-up). Therefore, reduced chemistry with pre-calculated table look-up, efficient table storage methods or both must be used. Reduced chemistry can be implemented through the use of global, reduced mechanisms (see Section 4.1) or by the dynamical systems approach using intrinsic low-dimensional manifolds (ILDM) [99] (see Section 4.2). 4.1. Global reduced mechanisms Reduced mechanisms can be developed empirically or from skeletal mechanisms that typically contain dozens of elementary reactions. In the latter approach, judiciously selected species are assumed to be at steady-state and selected reactions are assumed to be in equilibrium. These assumptions provide algebraic relationships that reduce the number of independent variables (i.e. reaction scalars) needed to describe the chemistry. Global reduced mechanisms must be used with caution because the assumptions or empiricism on which they are based may not be valid at all conditions, such as at low temperature or high pressure. 4.1.1. Two-step mechanism A two-step mechanism for methane combustion was proposed by Dryer and Glassman [84]: CH4 1 CO 1
3 O $ CO 1 2H2 O 2 2 1 O $ CO2 2 2
1 2
This mechanism was based on the observation that CO is not appreciably consumed until all of the hydrocarbon species have disappeared. Carbon monoxide is primarily formed in hydrocarbon flames by fast, fuel-consuming reactions [80]. CHx radicals are also generated by these reactions. Since
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Table 4 Comparison of Monte Carlo PDF methods Method
Advantages
Disadvantages
Node-based composition PDF [224– 228]
Simplest PDF method
Same as distributed-particle composition PDF method plus: Randomness in the implementation of convection leading to statistical error [229] For non-orthogonal grids, may be difficult to compute effective cell flowrates [210] Difficult to choose proper time step [210] Same as velocity–composition PDF method plus: Convection by turbulent velocity modeled by gradient diffusion [230] Randomness in the implementation of turbulent diffusion (i.e. a random walk) [229] Neglects effects of density fluctuations on velocity field Contains no information on the time or length scales of the turbulent motions [230] Molecular mixing must be modeled [232] Particles must be sorted to determine cell location at each time step, a computationally intensive task No. of particles per cell varies randomly
No. of particles in each cell is constant No particle sorting to determine cell location
Distributed-particle composition PDF
Convection is deterministic resulting in decreased statistical error [229] Time step can be made arbitrarily small [210] Particle equation of motion is grid independent [210]
Velocity–composition PDF
Turbulent diffusion exact [191] Variable density effects (i.e. fluctuations) exact [231]
Velocity–composition–dissipation PDF [230–234]
Provides complete closure; stand-alone method [235,236] Allows for multiple time scales, accounts for internal intermittency, has potential to account for large scale structures, and better accounts for particle history effects on local turbulent structure [230]
Needs a robust algorithm for calculating the mean pressure field; recently developed pressure algorithm needs more development [231]
they are more reactive than CO, they consume the available oxygen. It is only after the CHx radicals have disappeared that CO can oxidize [37] at an appreciable rate. With optimized rate parameters, Westbrook and Dryer [85] used the two-step mechanism to suitably reproduce lean and rich flammability limits as well as the dependence of flame speed on pressure and equivalence ratio.
velocity in a premixed laminar flame between 1 and 80 atm [87]. The intermediate species OH, O, HO2, CH3, CH2O, CHO, CH3O and H2O2 were assumed to be in steady state. Some of the resulting algebraic relationships were truncated in order to simplify the solution procedure. Several of the simplifications are valid only at high temperature [89].
4.1.2. Four-step mechanism Several studies with reduced mechanisms have emphasized the role of hydrogen-containing free radicals such as H and OH in predicting CO [86,136]. The following mechanism incorporates H and was used to analyze the inner structure of a methane–air premixed flame [87]:
4.1.3. Five- and nine-step mechanisms Two reduced mechanisms were recently developed specifically for conditions of interest in lean premixed gas turbines (0.4–0.6 equivalence ratio and 30 atm) [90]. The first mechanism consists of four combustion steps and one NO-forming step:
CH4 1 2H 1 H2 O $ CO 1 4H2
3
3H2 1 O2 1 CO2 $ 3H2 O 1 CO
7
CO 1 H2 O $ CO2 1 H2
4
H2 1 2OH $ 2H2 O
8
2H 1 M $ H2 1 M
5
3H2 1 CO $ H2 O 1 CH4
9
O2 1 3H2 $ 2H 1 2H2 O
6
H2 1 CO2 $ H2 O 1 CO
10
3H2 1 CO2 1 2NO $ 3H2 O 1 CO 1 N2
11
This mechanism was derived from a skeletal mechanism [88] containing 25 elementary reactions and 16 species and was used to make reasonable predictions of flame
The rate of NO formation in Eq. (11) is a function of
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elementary reaction rates in the thermal, prompt, and N2Ointermediate pathways, and so all three NO-forming mechanisms are represented. The second mechanism consists of seven combustion steps and two steps for nitrogen chemistry: H2 1 O $ H 1 OH
12
4H 1 O2 $ H2 1 2OH
13
H 1 O $ OH
14
H2 1 2O 1 CH3 $ 4H 1 OH 1 CO
15
O 1 CH4 $ OH 1 CH3
16
H 1 OH 1 CO $ H2 1 CO2
17
H 1 OH $ H2 O
18
5H 1 2NO $ 2H2 1 O 1 OH 1 N2
19
O 1 N2 O $ O2 1 N2
20
As in the five-step mechanism, all three NO-forming mechanisms are represented in Eq. (19). A summary of all four reduced mechanisms is shown in Table 5. Both the five- and nine-step mechanisms were derived from the detailed, 276-step, 49-species, GRI-Mech 2.11 mechanisms [96,97]. Nitrogen chemistry was included. Steady-state assumptions were applied to 38 and 34 intermediate species, respectively, and these equations were not truncated as was the case with the four-step mechanism. 4.1.4. Evaluation of reduced mechanisms Predictions with the four-, five-, and nine-step mechanisms were compared with GRI-Mech 2.11 in a PSR model [90] at 1 and 30 atm, with the equivalence ratio varying from 0.4 to 1.0 at 1 atm and from 0.6 to 1.0 at 30 atm. The agreement for predicted temperature was within 3% of the full mechanism at all conditions for all three reduced mechanisms and was slightly better for the five- and ninestep mechanisms. For CO, all three reduced mechanisms did fairly well at low pressure (less than 7% discrepancy for the five- and nine-step mechanisms and 12% for the four-step mechanism). At the higher pressure, however, the five- and nine-step mechanisms considerably outperformed the fourstep mechanism (2% compared with 65%). For NO, the agreement was 5% at 30 atm for both the five- and ninestep mechanisms. At low pressure, however, the accuracy of the five-step mechanism deteriorated significantly at equivalence ratios greater than 0.7. At lean conditions, both mechanisms were within 2% of the full mechanism. Further evaluation was performed with a Partially Stirred Reactor (PaSR) model. In a PaSR model, the contents of the reactor are not completely mixed on a molecular level, even though the reactor is homogeneous [91,113]. The finite rate of molecular mixing is governed by an input parameter, the
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mixing frequency. In a premixed flame, the incoming fuel and air mixture cannot react until it is mixed with hot combustion products and brought to reaction temperature. Thus, the PaSR provides a means of evaluating chemical mechanisms int he context of finite-rate, turbulent mixing. The PaSR can be thought of as a zero-dimensional composition PDF model. Indeed, the governing equation for the PaSR can be obtained by integrating the composition PDF transport equation over three-dimensional space. It is conveniently solved with the Monte Carlo method. The reactor is assumed to be filled with a large number (e.g. 1000) of particles to represent the statistics. At each time step, the particles undergo mixing, chemical reaction, and throughput. Mixing is accomplished between randomly selected pairs of particles (Curl [92] or modified Curl [93] mixing model) or by relaxing the properties of all fluid particles deterministically toward the ensemble mean values. The deterministic approach is referred to as the IEM model [94,95]. Throughout is accomplished by adding fresh, unreacted particles and randomly removing an equal number of particles at each time step. When the IEM model is used, it is this random removal process that gives the PaSR model its stochastic nature. In the case of premixed combustion, the fresh particles consist of both fuel and air. PaSR predictions for atmospheric LPC were made with all four reduced mechanisms and with GRI-Mech 2.11 [96,97]. Typical results are shown in Fig. 1. In this case, the reactor statistics were represented by 650 particles. The equivalence ratio was 0.8, the mixing frequency was 5000 Hz, the inlet temperature was 300 K, and the reactor residence time was 2 ms. Sufficient mixing of hot, partially reacted particles with fresh, unreacted particles was required to avoid blowout at this low inlet temperature and residence time. Fig. 1 shows the evolution of mean temperature, CO mole fraction, and NO mole fraction as functions of non-dimensional time (actual time normalized by the residence time). Initially, all of the particles were assumed to be fully reacted (i.e. at equilibrium). The predictions evolve from the assumed initial condition to a stochastic steady-state. The mean temperature predicted by the detailed mechanism (Fig. 1a) is most closely followed by the nine-step mechanism. The five-, four-, and two-step mechanisms overpredicted the steady-state, mean temperature by approximately 20 K, 25 K, and 105 K, respectively. The overprediction was likely due to the greater degree of dissociation that was allowed in the full mechanism. The evolution of CO mole fraction is shown in Fig. 1b. The five- and nine-step mechanisms were about 5% higher in the mean, steady-state value than the detailed mechanism while the four-step mechanism was about 12% lower. The assumption of partial equilibrium for OH in the four-step mechanism might have contributed to this discrepancy. The significant underprediction by the two-step mechanisms shows the importance of hydrogen chemistry. The evolution of NO mole fraction for the reduced and
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Table 5 Characteristics of reduced mechanisms Reduced mechanism
Parent mechanism
Reduction conditions
Steady-state assumptions
Other
Two-step [85]
None
None
None
Empirically derived from experimental data Does not include H2 chemistry
Four-step [87]
25-Step [88]
Laminar premixed flame
Seven S–S species
Eight species
16 Species
f 1.0
OH, O, CH3, CH3O, CH2O, HCO, HO2, H2O2
Truncated and explicit steadystate species equations OH partial equilibrium assumption Does not include NO chemistry
Six species CH4, O2, N2, CO2, H2O, CO
P 1 atm
CH4, O2, N2, CO2, H2O, CO, H, H2 Five-step [90]
276-Step [83]
Perfectly stirred reactor
38 S–S species
Nine species
49 Species
H, O, C2H, H2CN, HCNN, C2H3, C2H5, C
P 30 atm
CH3O, CH, CN, N, C2H6, NH, HCCO, CH3, NNH, CH2OH, N2O, NH2, HCCOH, CH2S, NCO, C2H2, HOCN, C2H4, HNO, HCO, CH2CO, NO2, CH2, CH3OH, H2O2, HCNO, HO2, HNCO, HCN, CH2O
f 0.4–0.6
CH4, O2, N2, CO2, H2O, CO, OH, H2, NO
Nine-step [90]
276-Step [83]
Perfectly stirred reactor
34 S–S species
13 Species
49 Species
f 0.4–0.6
Same as five-step plus:
P 30 atm
H, O, CH3, N2O
CH4, O2, N2, CO2, H2O, CO, H, OH, H2, O, N2O, CH3, NO
detailed mechanisms is shown in Fig. 1c. A period of approximately five residence times (ten times longer than for temperature or CO) was required to reach steady-state for NO. The five- and nine-step mechanisms agreed with the detailed mechanism to within about 2 and 9 ppm, respectively. The higher discrepancy in the five-step mechanism was probably due to the additional steady-state assumptions for O, H, and N2O. A detailed PaSR study of premixed methane–air combustion was performed by Cannon et al. [98] with the mechanisms presented here (except the two-step mechanism). This study showed that the evolution of mean and rms temperature, CO, and NO was accurately described with the ninestep mechanism over a wide range in mixing frequency (2000–10 000 Hz) and equivalence ratio (0.65–1.0). The five-step mechanism performed less reliably than the ninestep mechanism at f 1.0 but performed similarly to the nine-step mechanism at f 0.65. The four-step mechanism underpredicted mean CO values and overpredicted instantaneous temperature reaction rates, most likely due to its inferior parent mechanism, partial equilibrium assumption for OH, and unallowed dissociation of neglected radical species.
Implicit steady-state species equations No partial equilibrium assumptions
Includes thermal, prompt, and N2O-intermediate NO chemistry
Implicit steady-state species equations No partial equilibrium assumptions Includes thermal, prompt, and N2O-intermediate NO chemistry
4.2. ILDM method The ILDM method [99] is a generalized approach to reduced chemistry that obviates the need for a global reduced mechanism that has narrow applicability and must be carefully validated for the conditions of interest. In the ILDM approach, a full mechanism is used and the chemistry is reduced locally to a user-specified number of degrees of freedom. The method is based on the observation that, in any region of composition space, all starting compositions are quickly attracted to low-dimensional manifolds where only a few reactions are controlling, and then move more slowly along the manifolds towards equilibrium. The ILDM method, which is based on the dynamical systems approach [100], can be thought of as a reduced mechanism approach where the mechanism is optimized locally rather than globally. The manifold is identified by eigenvector analysis of the governing chemical kinetic equations for the full mechanism. It is then parameterized and tabulated a priori. A three-dimensional manifold has been used successfully in combustion calculations of H2/O2 –air, CO/H2/N2 –air and natural gas–air flames [101,102,233]. Yang and Pope [103] have summarized the difficulties of
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Fig. 1. Evolution of mean (a) temperature, (b) CO mole fraction, and (c) NO mole fraction predicted by the PaSR model for the reduced and full mechanisms.
the ILDM approach. First, reduced manifolds are not straightforward to parameterize. Second, manifolds of different dimension are appropriate in different regions of composition space. Third, look-up table generation is not automatic and must be performed at each set of conditions. And finally, fourth, the major portion of the manifold is wastefully tabulated since it is not known a priori which regions will be needed. In an attempt to overcome these difficulties, Yang and Pope [103] have developed a new variable-dimensional manifold method which defines the manifold in terms of a time scale rather than a global dimension. Properties of this manifold can be efficiently calculated and stored using in situ tabulation. 4.3. In situ tabulation In the Monte Carlo PDF method, fluid particles evolve by simultaneous processes of convection, mixing and reaction. In order to calculate the changes in particle composition due
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to reaction at each time step, reaction rates must be calculated and used to calculate the new properties of the fluid particle. Due to the presence of a wide range of reaction time scales, implicit numerical methods are required to integrate the chemical kinetic equations. The solution to these systems can be obtained by the use of stiff solvers such as LSODE [104] or VODE [105]. These software packages use a variable step-size, backward differentiation formula method and require significant computational effort. The main advantage of the reduced mechanism or manifold methods is that only a few independent scalars are needed to define the thermo-chemical state of the system. With a three- or four-step mechanism, it is feasible to tabulate the change in particle properties as a function of the independent scalars for the allowable composition domain in a pre-processing phase. The expensive calculation of reaction rates and integration of kinetic equations in the simulation phase is then replaced by table look-up and interpolation. The look-up table quickly becomes very large as the number of reactive scalars increases, and table size is a limiting factor in the number of scalars that can be considered. Fig. 2 shows how the size of the look-up table varies with the number of reactive scalars. The four-step mechanism, for example, would require approximately 2.3 gigabytes for one reactor pressure and one time step, if 20 grid points were used to discretize the domain of each of the four reactive scalars, the mixture fraction, and enthalpy. Information stored at each grid point would include the change in each of the four scalars for the specified time step, temperature, density, and specific heat. Since chemical compositions evolve through composition space along low-dimensional manifolds, table values are needed for only a fraction of the allowed composition space. With an aim toward improving the efficiency of table generation and storage, in situ tabulation has been suggested by Pope [106]. In situ tabulation occurs during the simulation phase and stores the data in a binary-tree data structure. Only compositions that are actually needed during the simulation phase are stored. In situ tabulation therefore requires significantly less storage than a priori tabulation, because only a small fraction of the allowable domain is tabulated. Fig. 3 shows the evolution of the in situ table size for the PaSR calculations at a mixing frequency of 2500 Hz and with the four-step mechanism. The tabulation was performed with a maximum refinement level of 6 (normalized cell size 2 26), which was sufficient for these calculations. As shown, a steady-state condition was reached where the table size remained fairly constant and table look-up (rather than direct integration) provided most of the required information. Pope [106] has extended the in situ tabulation method to automatically adapt the local refinement level and effectively control the tabulation errors. The evolution of the mean methane mole fraction at the
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Fig. 2. Table size for a priori tabulation with 20 grid points per scalar (including mixture fraction and enthalpy).
conditions of Fig. 3 is shown in Fig. 4. Results are shown for three cases: in situ integration (i.e. without storage), in situ tabulation and a priori tabulation. In situ integration is the benchmark for comparing the two tabulation methods. As shown, the in situ tabulation method produced results that agree almost identically with the benchmark whereas the a priori method produced results that are about 8% lower. These calculations demonstrate the potential for achieving improved accuracy with the in situ tabulation method while at the same time reducing the size of the look-up table. Required CPU time and table size for the PaSR calculations using the in situ tabulation, a priori tabulation, and in situ integration methods are compared in Table 6. As expected, table look-up significantly decreased the required CPU time. The in situ tabulation method required less CPU time than a priori tabulation and the table was significantly smaller. Most of the time required by the a priori method was for table generation.
Fig. 4. Mean CH4 mole fraction predicted by the PaSR model with the in situ (direct) integration, in situ tabulation, and a priori tabulation methods.
As was shown earlier, the size of the a priori look-up table increases considerably with each additional scalar and the method is therefore limited to four or five reactive scalars. It has been shown that an in situ tabulation method, however, can be applied to mechanisms with 12 reactive scalars while still maintaining reasonable storage requirements [106]. Therefore, the in situ tabulation method shows good potential for including detailed kinetic mechanisms in turbulent combustion calculations.
5. Comprehensive model description A new combustion model is being developed to address the concerns in modern land-based gas turbines. The new model uses an unstructured grid flow solver to accommodate geometrical complexity. An important element of the new model is a new sub-model for lean premixed combustion of natural gas which is based on the velocity–composition PDF method. The model is being validated with detailed data obtained in a laboratory-scale gas turbine combustor and applied to practical gas turbine combustor geometries obtained from industry. 5.1. Unstructured grid flow solver
Fig. 3. Evolution of table size in the PaSR calculation using in situ tabulation with the four-step mechanism.
A control-volume finite element method [107] and a variant of the SIMPLE technique [108] are used to solve the Navier–Stokes equations on an unstructured, tetrahedral grid. A skewed, mass-weighted, upwind interpolation function [109] is used for the convective discretization, while linear interpolation functions are used for the diffusion and pressure gradient discretization. The standard k– e model is employed to compute the turbulent eddy viscosity. The fluid density field is provided by the PDF submodel.
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Table 6 CPU time and table size for PaSR calculations Method
Relative CPU time
Table size (MB)
In situ integration 38 – A priori tabulation 5.9 (table generation only) 5.8 6.5 (total including table generation) In situ tabulation 1 0.6
5.2. Submodel for lean premixed combustion The PDF submodel was developed from the foundation of PDF2DS, a two-dimensional code developed at Cornell University [110,214,216]. PDF2DS was extended to three dimensions, to non-isothermal, reacting flows and to unstructured, tetrahedral grids. The submodel calculates the local one-point, one-time, joint velocity–composition PDF. The PDF transport equation is as follows and can be derived from the general equations for conservation of mass, momentum, energy, and species for low-Mach-number flow [213]:
r C
2f 2f 2f 1 r CVj 1 r CSa Cf a±h 2t 2Xj 2Ca 1 r Cgj 2
2kpl 2xj
!
Fig. 5. Swirl-stabilized combustor (taken from Ref. [139]).
2f 2Vj
+ # 2tij 2p 0 2 1 jV; C f 2xi 2xj ai E i 2 2J 2 hD 1 jV; C f 2 7·qjV; C f 2Ca 2Ch 2xi 21
2 2Vj
"*
Here, f represents the velocity–composition PDF, r is the density, V is the velocity, Ca is the a -dimensional composition scalar, Sa is the reaction source term, gj is the body force, p is the pressure, t ij is the stress tensor, Ji a is the diffusive flux, C h is the specific enthalpy, and 7·q is the specific enthalpy source term (due to radiation). This equation illustrates the capability of the velocity–composition PDF method to completely describe the important chemical kinetic and convective transport effects in reacting, turbulent flows. These effects are described by the underlined terms in Eq. (21) and appear in closed form. The terms on the right-hand side must be modeled since the required information for computing the expected values (indicated by the angle brackets) is not available from the single-point, joint PDF. The first term on the right-hand side of Eq. (21) represents transport of the PDF in velocity space by the viscous stress and fluctuating pressure gradient. This term is described with a simplified Langevin model [111]. Some implicit assumptions of the simplified Langevin model include
Fig. 6. Bluff-body combustor (taken from Ref. [142]).
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[127]: (i) due to the Markov nature of the Weiner process and since the model terms are a function of local mean quantities, the turbulence structure must be expressed in terms of local mean quantities; (ii) local isotropy, i.e. flows that are so strongly distorted or have such a low Reynolds number that the small scales are not even approximately isotropic, are beyond the scope of this model. Further assumptions in the simplified Langevin model are a linear return-to-isotropy and negligible “rapid pressure” effects. For homogeneous turbulence, the simplified Langevin model is equivalent to Rotta’s [112] model [111]. The second term on the right-hand side of Eq. (21) represents transport in composition space by the diffusive flux and is described by the IEM model [46,113–116,213,214]. 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 [148]. In homogeneous isotropic turbulence, the IEM model predicts that the shape of the composition PDF does not always relax to a Gaussian distribution, but instead preserves the shape of the initial PDF [233]. Stochastic, pair-exchange models [92,117] can also be used in PDF methods but their computational requirements are higher and their superiority has not been shown for reacting flow simulations. For example, it has been demonstrated that the choice of mixing model is not critical in the distributed reaction regime of lean premixed combustion [118]. When turbulent mixing frequencies were above 1000 Hz, the differences between the IEM and pairexchange mixing models were overwhelmed and blurred by the finite rates of the chemical reactions. The k– e turbulence model [119] is used to provide the mixing frequency (e /k) to both the simplified Langevin and IEM models. In the k– e model, the turbulent viscosity, m t, has no directional dependence. This isotropic restriction can limit the model’s usefulness in high swirling or other complex flows with varying normal stresses [120]. However, the k– e model has been recommended for highswirl calculations due to the marginal improvements offered by other higher-order models [49]. The five constants in the standard k– e model have also been modified for modeling swirling flows [121]. The mean radiation flux divergence is provided by the discrete ordinates radiation model [122,123]. The discrete ordinates model is a multi-flux model that discretizes the intensity field into a finite number of streams or ordinates [124]. The axi-symmetric calculations presented here were performed for 12 discrete directions. This S4 approximation has been shown to be adequate for radiation calculations in both two- and three-dimensional furnace geometries [123,125]. A spectral-line weighted-sum-of-gray-gases model [126] provides gas emissivity. This spectral model provides an accurate way of obtaining radiative gas properties for use in the discrete ordinates model, as it does not require the often-made gray gas assumption. Because of the large dimensionality of the velocity–
composition PDF (7 1 a ), Eq. (21) is not amenable to conventional solution with finite differences. It is, however, efficiently solved indirectly with a Monte Carlo method [127–134,211,214]. A large number of Lagrangian fluid particles evolve in time and are used to represent the joint velocity–composition statistics. Each particle may be thought of as coming from a different realization of the flow [111]. The time step for the Lagrangian solution is determined from the minimum of all local residence and mixing times in the reactor. An improved time-splitting technique [229] is used to sequentially calculate the change in particle properties due to the simultaneous processes of mixing, etc. The particles are first convected for half of a time step. The changes in particle properties due to mixing, reaction, and radiation are then calculated for the full time step. Finally, the particles are convected to their final positions for the time step. Pope [229] found this method to be superior to the method of fractional steps which was originally implemented in PDF2DS, where the particle properties first evolved at the initial position for the entire time step before the particles were convected [213]. In addition to performing the reaction and mixing calculations at the half-step position, these calculations are performed sequentially for several reduced time steps in order to minimize the error due to the coupling between mixing and reaction. In his velocity–composition PDF predictions of a premixed flame, Gran [55] observed that the mean reaction rate was apparently underpredicted and inferred that this underprediction may be due to coupling between reaction and diffusion that is not accounted for in the time-splitting Lagrangian approach. The size of the reduced time step was determined from PaSR calculations where it was gradually reduced until it was not affecting the results. In the limit of an infinite number of particles and an infinitely small time step, the Monte Carlo method provides an exact solution to Eq. (21) [215]. Fluid particle velocities evolve in the PDF submodel according to the simplified Langevin model [135] and are also adjusted to maintain consistency with the mean velocity and kinetic energy fields provided by the Eulerian flow code [133]. This consistency ensures that the no-slip condition is satisfied at the walls [136,211]. Convective heat loss is calculated based on the law of the wall [137]. The unstructured grid is used to store mean values of particle properties. The mean fields of velocity, pressure gradients, and turbulence (e /k) are passed to the PDF submodel and fluid particles are then allowed to evolve for one reactor residence time or a specified fraction thereof. Overall convergence requires a statistical steady-state in the PDF calculation and simultaneous satisfaction of the Navier–Stokes equations. Overall mass and energy balances are used to help determine convergence. One macro-iteration for a three-dimensional case involving approximately 3500 nodes, 1.5 × 10 4 tetrahedral cells, and 1.5 × 10 6 particles requires approximately 2 hr of CPU time on an HP-
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735 workstation. A macro-iteration consists of converging the Navier–Stokes equations and then performing Lagrangian PDF calculations for the specified length of time, which in this case, was 20% of the reactor residence time. 5.3. Model evaluation Recent experimental data in swirl-stabilized [138] and bluff-body-stabilized [34] lean premixed combustors were used to evaluate the model. The swirl-stabilized combustion chamber, shown in Fig. 5, was configured such that the geometry around the injector was nearly axi-symmetric with an average diameter of 170 mm. Flat quartz windows, 60 mm wide, were located on each side of the chamber to allow the use of laser diagnostic equipment. Laser Doppler anemometry (LDA) measurements were made to obtain axial, radial, and tangential components of velocity at over 130 locations [48,139]. This swirl-stabilized combustor has been simulated for methane–air equivalence ratios of 0.8 and 0.65, a swirl number of 0.74, 1 atm, a 300 K inlet temperature, and assuming an axi-symmetric geometry [140]. An unstructured grid consisting of 4302 triangular elements and a total fluid particle number of 310 000 were used in the simulations. Both convective and radiative heat losses were considered. A uniform wall temperature of 700 K was assumed. The five-step chemical mechanism [90], with in situ tabulation [106], was used to describe the combustion chemistry. The total CPU time was about 210 hr on an HP-735 workstation. The bluff-body-stabilized combustion chamber, shown in Fig. 6, was configured such that a stainless steel, conical bluff body was mounted coaxially at the center of the combustor and served as a flame holder. A combination of Rayleigh scattering, spontaneous Raman scattering, and laser-induced fluorescence (LIF) measurements of chemical species (CH4, O2, H2O, CO2, NO, and CO) and temperature were obtained radially at several axial locations [34]. This bluff-body-stabilized combustor has been simulated at 0.586 methane–air equivalence ratio, 1 atm, 300 K inlet temperature, and assuming axi-symmetric geometry [140,142]. An unstructured grid consisting of 1596 triangular elements and a total fluid particle number of 116 000 were used in the simulations. Both convective and radiative heat losses were considered. A uniform wall temperature of 500 K was assumed. The five-step chemical mechanism [90], with in situ tabulation [106], was used to describe the combustion chemistry. The total CPU time was about 100 hr on an HP735 workstation. Fig. 7 shows the predicted mean axial velocity profiles compared with experimental data for equivalence ratios of 0.8 and 0.65 at z axial locations of 10, 20, 40, 100, and 175 mm. These results show that recirculation zones existed
Fig. 7. Measured and predicted mean axial velocity profiles in the swirl-stabilized combustor at equivalence ratios of 0.8 and 0.65.
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Fig. 8. Scatterplots of measured and predicted axial/tangential velocity at an axial location of 10 mm and radial locations of (a) 9 mm, (b) 12 mm, and (c) 21 mm in the swirl-stabilized combustor.
along the centerline (centerline recirculation zone (CRZ)) and out radially near the wall (outer recirculation zone (ORZ)). The swirling nature of this flow created a vortex funnel where downstream gases were drawn into the CRZ as a result of the incoming fluid being flung outward [48]. These recirculation zones allowed hot combustion products and radicals to mix and subsequently react with the incoming, cold CH4 –air mixture and stable combustion was obtained without the use of a pilot flame. The predictions compared well with the experimental data at the lowest axial location of 10 mm. The location and magnitude of the maximum mean velocity was well represented at z 20 mm. At locations higher than z 20 mm it is obvious that the model overpredicted the radial spread rate of the high velocity jet. The often, overpredicted turbulent viscosity with the k– e turbulence model [143,144] could be attributed to the overpredicted spread of the high velocity fluid. Also, the negative centerline velocities were overpredicted (by at most 10 m s 21), which is a known shortcoming of the standard k– e turbulence model for predicting confined, swirling flows [121,145]. The effect of varying the equivalence ratio from 0.8 to 0.65 on the mean axial velocity appeared to be small for both the predictions and the measurements. The predictions and measurements both suggested a slightly weaker CRZ for the leaner case at z 20 and 40 mm. Also, the predictions indicated a slightly lower peak velocity for the f 0.65 case (at z 10, 20, and 40 mm), as would be expected for the higher density fluid in this lower heat release case. Comparisons of fluid particle scatterplots in axial and
tangential velocity space were made at several radial locations and at an axial location of about 10 mm. Fig. 8 shows these measured and predicted axial and tangential fluid particle velocities. The experimental data and measurements in Fig. 8a, 8b, and 8c represent fluid particle velocities at radial locations of 9, 12, and 21 mm, respectively. These locations are marked as points A, B, and C in Fig. 7. At each location, the measurements were represented with 4096 data points, and the predictions were represented with 250–750 fluid particles. The fluid particles were sampled from a rectangular cell, centered at the location of interest, with an axial and radial length of 5 and 2 mm, respectively. The highest axial and tangential velocities were observed at r 9 mm and these velocities decreased with radial location and were near zero (for axial velocity) at r 21 mm. Overall, the predictions matched the mean, or central location of fluid particles, though the model did underpredict the spread of the fluid particle scatter plots. Overall, the velocity fields in this swirl-stabilized, lean premixed combustor were well represented near the combustor inlet and far downstream. Discrepancies were observed in the middle region of the combustor (z 20– 40 mm), mainly due to an overpredicted radial spread of the high-velocity fluid. Comparisons of the predicted and measured mean, rms, and the PDF of temperature and species mole fractions of CO2, CO, and NO will be presented at various locations within the bluff-body-stabilized combustor. Fig. 9 shows comparisons between the measurements and predictions for the mean temperature at x/d axial locations 0.1, 0.3,
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Fig. 9. Measurements and predictions of mean temperature profiles in the bluff-body-stabilized combustor (taken from Ref. [142]).
0.6, and 1.0. These comparisons show that the predicted temperatures were well represented. The mean temperatures in the recirculation zone region (r/d , 0.4) were at most only 6–7% (100 K) higher than the measurements. The locations of temperature gradients, where cold and hot gases mixed, were predicted accurately, especially at x/d locations of 0.1, 0.3, and 0.6. Only the mean temperature gradient in the shear layer region at x/d 1.0 was underpredicted, and thus mean temperatures at the edge of the recirculation zone and in
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the shear layer (0.3 , r/d , 0.05) were underpredicted by about 40%. The comparisons of predicted CO2 mole fractions showed excellent agreement with the measurements in the recirculation zone (Fig. 10). The widened shear layer that was predicted at x/d locations of 0.6 and 1.0 compared to the measurements confirm that the model underpredicts the amount of fully burned products at this location. This may be due to underpredictions in CO oxidation and to overpredictions in the mixing of unburned reactants into the shear layer with no subsequent CH4 oxidation. Overpredicted amounts of both available CO and available CH4 at this shear layer location were observed [142] and confirm the lower than expected CO2 values. The radial profiles of mean CO mole fraction reveal the location of the reaction zone or reacted fluid that has been quenched (high CO concentrations). Fig. 11 shows comparisons between the measured and predicted mean CO mole fractions at several axial locations in the bluff-body combustor. The equilibrium value of CO is 7 ppm, much lower than the concentrations that were predicted and measured in the recirculation zone and shear layer region. At the lowest axial location of x/d 0.1, the model showed underpredicted levels of CO in the recirculation zone and shear layer region. These results indicate that the model underpredicts the initial CO formation in the shear layer and then also overpredicts the CO oxidation in the recirculation zone. At x/d 0.3, the comparison between predictions and measurements was excellent in the shear layer region and recirculation zone CO levels were slightly underpredicted (500–700 ppm). Further downstream at x/d 0.6, the peak CO in the shear layer region was overpredicted by about 2400 ppm, though the recirculation zone CO showed excellent comparisons to data. Also, at x/d 1.0, the CO was overpredicted in the shear-layer region and compared well at the centerline along the recirculation zone. Despite the noticeable discrepancies between the measurements and predictions, the comparisons were reasonably good. The ability to accurately predict CO emissions in turbulent methane flames is an extremely difficult task and is strongly dependent upon the chemical model, as well as the other submodels in the comprehensive combustion model. Previous studies of turbulent methane–air non-premixed flames have shown CO discrepancies as high as orders of magnitude [136]. The results presented here have shown marked improvements in predicting CO in turbulent methane flames. The radial profiles of mean NO mole fraction reveal the location of burned fluid that has had sufficient time to form the slow-reacting NO species. Fig. 12 shows comparisons between the measured and predicted mean NO mole fractions at several axial locations in the bluff-body combustor. At the lowest axial location of x/d 0.1, the model showed underpredicted levels of NO in the recirculation zone. The measurements were as high as 11 ppm, while the predictions reached maximum levels of 5 ppm. Apparently the model
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Fig. 10. Measured and predicted profiles of mean CO2 mole fraction in the bluff-body-stabilized combustor (taken from Ref. [142]).
slightly underpredicted the overall formation of NO in this lean premixed system. At x/d 0.3 and 0.6 the comparison between predictions and measurements also showed the NO mole fractions to be underpredicted by about 5–7 ppm. The measurements indicated a trend of decreasing centerline NO from x/d 0.6 to x/d 1.0. The predictions, on the other hand, showed the centerline mean NO to remain at about 5 ppm even at x/d 1.0. Thus, at x/d 1.0, the measurements and predictions compared well at the centerline, where both showed values near 5 ppm. Nandula et al. [34]
Fig. 11. Measured and predicted profiles of mean CO mole fraction (ppm) in the bluff-body-stabilized combustor (taken from Ref. [142]).
report that the NO measurements could be 2–3 ppm higher than the actual values due to the unaccounted interference of O2 on the NO channel. This could partly explain the 5 ppm underpredictions observed in the model. These predictions represented the first NO predictions in premixed combustors using the velocity–composition PDF method. The accessed composition space in a turbulent combustion calculation with the PDF method represents the values of species mole fraction and enthalpy for each fluid particle
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an a priori table [140]. The in situ tabulation approach allowed for practical chemical kinetic calculations using the five-step mechanism in these multi-dimensional, lean premixed combustors. 5.4. Application
Fig. 12. Measured and predicted profiles of mean NO mole fraction (ppm) in the bluff-body-stabilized combustor (taken Ref. [142]).
at each time step during the simulation. The accessed compositions in the swirl-stabilized and bluff-bodystabilized combustor simulations were found to lie in small, well-defined regions that were defined by the mixing, reaction, and heat transfer in the combustor. These accessed composition regions were obtained in situ and required less than 35 MB of memory once a steady-state and converged solution was reached. This memory requirement was more than three orders of magnitude less than would be needed in
Application of the new model to gas turbine configuration from industry has also been demonstrated. Two industrial premixers have been simulated. The first premixer consists of an annulus with swirling flow. A computational grid was created for a 458 segment with a single fuel jet located on the inner wall of the annulus. Periodic boundary conditions were imposed on each face of the segment. The grid contains 9623 nodes and 43 029 elements, and was created using GeoMesh [146] and TGrid [147]. The two periodic planes contain approximately 2000 element faces each. The second premixer also consists of an annulus with radial fuel pegs attached to the inner and outer walls. Multiple fuel inlets are distributed radially along the fuel pegs. The unstructured grid for the smallest repeating segment of this premixer geometry was created by decomposing a structured hexahedral grid into tetrahedral elements. This grid contains 26 622 nodes and 142 800 elements. Both premixer simulations were carried out on an HP-735 workstation and required between 10 and 20 hr of CPU time. Predicted mixing for both premixers is shown in Fig. 13. In both cases, air enters at the bottom and the fuel and air mixture exits at the top. The mixture fraction is color-coded, with red indicating fuel and dark blue indicating air. In both premixers, there are small gradients in fuel concentration predicted at the outlet. These gradients could contribute to the formation of thermal NOx in the combustor and should be considered in the combustor simulation. Simply assuming a perfectly premixed fuel and air mixture at the combustor inlet could significantly underpredict the NOx. Non-reacting premixer simulations such as these will be used to provide realistic inlet conditions for reacting combustor simulations. Preliminary cold-flow simulation of the primary zone of an industrial combustor is shown in Fig. 14. The grid used for these predictions consisted of 16 120 nodes and 83 376 tetrahedral elements. Boundary conditions consisted of two upstream inlets and a single downstream outlet, inner and outer walls, and periodic side planes at both faces of a 908 segment of the combustor. The tetrahedral grid was created by decomposing a multi-block hexahedral grid. The predictions in Fig. 14 are for the 458 plane located halfway between the periodic planes.
6. Conclusions Based on a review of methods for modeling turbulent combustion, the Monte Carlo velocity–composition PDF
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Fig. 13. Predicted mixture fraction in industrial premixers.
method was selected as the best approach to use in a new comprehensive model for land-based gas turbines. This decision was based on the fact that PDF methods are general, applying equally well to premixed, partially premixed, and non-premixed combustors, and can incorporate arbitrarily complex chemistry. The velocity–composition PDF method was selected over the composition PDF method because it treats turbulent diffusion exactly instead of assuming gradient diffusion. Also, variable density effects due to fluctuating temperature are included. The velocity–composition–dissipation PDF method was judged to be potentially superior because an external turbulence model is not needed, but it is insufficiently developed to apply in comprehensive modeling at the present time. Likewise, large-eddy simulation and the linear eddy model have significant potential but require more development before practical application.
Fuel Research, Inc., East Hartford, CT, is gratefully acknowledged. The authors also wish to express appreciation to Dr. M. S. Anand and Profs. J. Y. Chen,
Acknowledgements Funding from the US Department of Energy, under Contract No. DE-FC21-92MC29061 administered through the South Carolina Energy Research and Development Center at Clemson University, and Contract No. DE-AC21-93MC30040 administered through Advanced
Fig. 14. Predicted normalized axial velocity (cold-flow) in an industrial combustor.
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Stephen B. Pope, and L. Douglas Smoot for their help and guidance. [15] [16]
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Progress in Energy and Combustion Science 25 (1999) 353–385 www.elsevier.com/locate/pecs
Modeling of lean premixed combustion in stationary gas turbines B. Scott Brewster*, Steven M. Cannon, James R. Farmer, Fanli Meng Advanced Combustion Engineering Research Center, Brigham Young University, Provo, UT 84602, USA
Abstract Lean premixed combustion (LPC) of natural gas is of considerable interest in land-based gas turbines for power generation. However, modeling such combustors and adequately addressing the concerns of LPC, which include emissions of nitrogen oxides, carbon monoxide and unburned hydrocarbons, remains a significant challenge. In this paper, characteristics of published simulations of gas turbine combustion are summarized and methods of modeling turbulent combustion are reviewed. The velocity–composition PDF method is selected for implementation in a new comprehensive model that uses an unstructured-grid flow solver. Reduced mechanisms for methane combustion are evaluated in a partially stirred reactor model. Comprehensive model predictions of swirl-stabilized LPC of natural gas are compared with detailed measurements obtained in a laboratory-scale combustor. The model is also applied to industrial combustor geometries. q 1999 Elsevier Science Ltd. All rights reserved.
Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Simulations of gas turbine combustion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Modeling turbulent premixed flames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Moment closure methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1. Eddy dissipation concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2. Assumed PDF method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3. Laminar flamelet models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4. Linear eddy model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.5. Conditional moment closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Monte Carlo PDF methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1. Node-based composition PDF method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2. Particle-tracking composition PDF method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3. Velocity–composition PDF method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4. Velocity–composition–dissipation PDF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Large eddy simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Direct numerical simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Combustion chemistry of natural gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Global reduced mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1. Two-step mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2. Four-step mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3. Five- and nine-step mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.4. Evaluation of reduced mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . * Tel/fax: 1 1-801-325-1633. E-mail address: [email protected] (B.S. Brewster) 0360-1285/99/$ - see front matter q 1999 Elsevier Science Ltd. All rights reserved. PII: S0360-128 5(98)00014-8
354 360 361 361 361 361 362 362 362 362 363 363 363 364 364 365 365 365 365 366 366 367
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4.2. ILDM method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. In situ tabulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Comprehensive model description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Unstructured grid flow solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Submodel for lean premixed combustion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Model evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4. Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1. Introduction Gas turbines are of considerable interest for power generation in the utility and industrial sectors for several reasons. First, their combined-cycle efficiencies can exceed 50% when coupled to a heat-recovery steam generator and steam turbine [1–3]. Second, their modularity means that the capacity to generate power can be purchased in smaller increments (less than 10 MWe [1]) than with conventional power plants, thus making it easier to plan for future needs and thereby reducing the risk associated with capital investment. Third, gas turbine power plants require minimal onsite construction [3] and can be brought on line relatively quickly [1]. Fourth, NOx emissions can be reduced to ultralow levels (,10 ppm at 15% O2) [4–13]. Gas turbines can also burn a variety of fuels and can be adapted to coal use when natural gas supplies dwindle [14,15]. These advantages and others have led to renewed interest in gas turbines for power generation in recent years. In order to promote the development of land-based gas turbines, the US Department of Energy (DOE) is sponsoring the Advanced Turbine Systems (ATS) program to demonstrate commercial prototypes of both utility and industrial turbines by the turn of the century. ATS program objectives include: system efficiency 60% (lower heating value [LHV]) or greater for utility-scale systems, and an equivalent 15 percent improvement for industrial systems; environmental superiority without the use of post combustion emissions controls; busbar energy costs 10 percent lower than current systems; adaptability to coal or biomass firing; and reliability, availability and maintainability equivalent to current advanced turbine systems [16]. Modeling can play a significant role in achieving these objectives. A recent survey [17] gives insight into the modeling capabilities that are needed. Representatives from twelve US gas turbine manufacturers, industrial research groups and users of gas turbines indicated their organization’s level of interest in various fuels, operational modes, and geometric configurations for land-based gas turbine
368 369 370 370 371 373 377 377 378 379
combustion. NOx emissions were of vital concern and lean premixed combustion (LPC) of natural gas was the chief operational mode of interest. Geometric configurations of interest included cannular, aeroderivative, and annular. Thus, a model capable of predicting NOx emissions for LPC in a wide variety of configurations is needed. Gas turbine combustors have traditionally been operated in the non-premixed mode for safety and stability [1,3,18]. Unfortunately, this mode leads to unacceptably high levels of thermal NOx, which is produced in the high-temperature, near-stoichiometric regions that occur in the combustor during the mixing of fuel and air. Traditional methods that have been used to moderate the temperature and control thermal NOx formation include steam or water injection and Rich/Quench/Lean (RQL) combustion [1,3]. However, injection of steam or water lowers the process efficiency and increases emissions of carbon monoxide and unburned hydrocarbons (UHC), and the effectiveness of RQL combustion is limited by the rate at which air can be mixed with hot gases in the lean zone. Although regulations limiting NOx emission are site-specific [19], the general objective of the gas turbine industry is to achieve levels of less than 10 ppm (at 15% O2) [20–22]. Achieving this “ultra-low” level of NOx is not feasible by traditional methods. In order to achieve single-digit NOx, manufacturers have turned to LPC [23–33]. With LPC, the amount of postcombustion dilution air is reduced, and the combustion zone is operated with excess air to reduce the flame temperature. A combustion zone equivalence ratio of 0.4–0.6 is typical [10,34]. In addition, the fuel and air are premixed to eliminate stoichiometric regions [35]. The flame is cooler (,1800 K) and thermal NOx is virtually eliminated. Prompt NOx is also negligible at these lean conditions [11] and the nitrous oxide mechanism accounts for essentially all of the NOx formed [10,32,36]. However, combustion efficiency is reduced at the lower temperatures of LPC, as it is in combustors using steam or water injection, and emissions of CO and UHC are of concern [4,37–39]. In addition, stability limits are significantly reduced by pre-mixing the fuel and air and by operating at fuel-lean conditions near the lean flammability limit [40,41]. Accurately predicting CO, UHC, and NOx in premixed combustors is difficult because of the complexity of the
Table 1 Gas turbine combustion simulations Authors (year)
Combustion model
Radiation
Pollutant chemistry
1 Anand et al. (1996) [148,149] Premixed natural gas/air
k– e
No information
CO
2 Mueller and Knill (1996) [150] Non-premixed propane
k– e
Scalar PDF with two-step methane chemistry Eddy dissipation concept (EDC) and laminar flamelet Laminar chemistry for propane combustion Scalar PDF Assumed PDF with fast chemistry; two-step EDC; laminar flamelet Four-step EDC
Adiabatic walls; radiation neglected No information
Thermal/prompt NO processor; CO; UHC Thermal NO and CO
Evaporating Jet-A liquid spray k– e
4 Leonard and Dai (1994) [152] No information 5 Bai and Fuchs (1994) [153] Non-premixed methane
No information k– e
6 Rizk and Mongia (1994) [154,155]
Evaporating liquid spray
k– e
7 Gran et al. (1994) [156]
Non-premixed methane
k– e
8 Micklow et al. (1994) [157], (1993) [158] 9 Di Martino et al. (1994) [159] 10 Baron et al. (1994) [160] 11 Gulati et al. (1994) [161]
Evaporating Jet-A liquid spray k– e Evaporating liquid spray Non-premixed methane Non-premixed, gaseous fuel
k– e k– e k– e
12 Chen and Shuen (1993) [162] Evaporating n-pentane liquid k– e spray 13 Gran et al. (1993) [163] Non-premixed propane k– e 14 Cline et al. (1993) [164,165] Evaporating Jet-A liquid spray k– e
One-step EDC and laminar flamelet with extinction Laminar/equilibrium chemistry for slow/fast reactions Assumed-shape PDF Assumed-shape PDF Assumed-shape PDF One-step laminar chemistry
15 Bond et al. (1992) [166] 16 Tolpaid (1992) [167] 17 Yang et al. (1992) [168], [169] 18 Yan et al. (1991) [170] 19 Melconian et al. (1990) [171] 20 Nguyen (1990) [172]
Non-premixed C2H2 Evaporating liquid spray Non-premixed propane
Modified k– e k– e k– e
21 Nguyen and Ying (1990) [173] 22 Talpallikar et al. (1990) [174] 23 Shyy et al. (1989) [175]
Evaporating liquid (Jet-A) spray No information Non-premixed CH2
k– e
EDC Laminar chemistry for propane Assumed-shape PDF Assumed-shape PDF Laminar and equilibrium chemistry Assumed-shape PDF One-step EDC One-step with mixing-limited rate Laminar chemistry
No information k– e
No information Assumed-shape PDF
Evaporating liquid spray k– e Evaporating liquid spray k– e Evaporating Jet-A liquid spray k– e
No information CO Absorbing/emitting gas CO with discrete transfer method Quasi-3D zone method to calculate liner temperature No information
Semi-analytic postprocessor for NOx, CO, UHC, and smoke None
Adiabatic walls, radiation neglected Neglected Neglected Adiabatic; internal heat transfer neglected Adiabatic wall; neglected
CO and thermal NO
None
Adiabatic wall; neglected No information
None Thermal NO and CO
No information Neglected No information
No information None Thermal NO and CO
Thermal NO post-processor None None
Six-flux model No information No information
None Thermal NO post-processor Thermal NO
No information
CO and NOx
No information No information
No information None
B.S. Brewster et al. / Progress in Energy and Combustion Science 25 (1999) 353–385
Turbulence model
3 Cline et al. (1995) [151]
Fuel injection
355
356
Authors (year)
Turbulence model
Combustion model
Radiation
Pollutant chemistry
24 Pridden and Coupland (1988) No information [176] 25 Shyy et al. (1988) [177] Non-premixed, gaseous fuel 26 Chleboun et al. (1987) [178] Non-premixed propane
k– e
Assumed-shape PDF
No information
Thermal NO post-processor
k– e
Assumed-shape PDF
Neglected
None
k– e
Assumed-shape PDF
No information
27 Wild et al. (1987) [179]
28 Shyy and Braaten (1987) [180,181] 29 Shyy et al. (1986) [182,183] 30 Coupland and Pridden (1985) [184]
Fuel injection
Evaporating liquid (kerosene) spray; nonpremixed propane Non-premixed, gaseous fuel
k– e and algebraic stress
Two-step EDC
No information
CO departure from equilibrium based on eddy break-up CO
k– e
Assumed-shape PDF
Neglected
None
Non-premixed, gaseous fuel Gaseous fuel (kerosene)
k– e k– e
Assumed-shape PDF Assumed-shape PDF
Neglected No information
None Thermal NO post-processor
B.S. Brewster et al. / Progress in Energy and Combustion Science 25 (1999) 353–385
Table 1 (continued)
Table 2 Gas turbine combustion simulations
1 2
Grid
Solution method
Data
Computer/Code
Sensitivity studies
Allison RSPN low emission combustor Lab-scale can type combustor
2D, staggered
Finite volume/Monte Carlo PDF; SIMPLE No information
Mean velocities, mix fraction and temperature Exit temp, equivalence ratio, CO2, O2, and NOx
IBM RS6000 workstation
None reported
TASCflow 3D commercial CFD package No information d100 cpu-hr on IBM RS6000 with 120 MBytes; CFD-ACE No information
B.C.s; comb model; temperature fluctuations in NO production Air mass flow split None reported
3D, non orthogonal BFC with grid embedding
3 4
Staged RQL combustor Pratt & Whitney annular combustor
3D, non-orthogonal BFC 3D, non-orthogonal BFC
SIMPLE-like Finite volume/Monte Carlo PDF; SIMPLE
None reported None reported
5
Model gas turbine combustor RQL; can and annular; premixed automotive combustor Axisymmetric can and annular Fuel nozzle/rich burn section of RQL combustor Two Alfa Romeo Avio reverse-flow annular combustors Reverse-flow annular combustor Full-scale, double annular research combustor GE annular combustor with dual can dome Annular combustor sector
3D cylindrical, staggered
Quasi-time marching with multi-grid solver SIMPLE
None reported
6
7 8 9
10 11 12 13 14 15 16 17 18 19
Sector of annular gas turbine combustor Vaporizer and flametube
Wall temperature, NOx, CO, UHC, smoke
No information
3D, non-orthogonal BFC; co-located 2D axi-symmetric, nonorthogonal BFC 3D non-orthogonal BFC; colocated
SIMPLE
None reported
No information
Time-dependent (transient)
None
KIVA-II [185] code
SIMPLE-like
Exit temperature at idle and full load power conditions
7–11 cpu-hr on VAX 9410 computer
Combustion model; extinction Air mass flow split; wall geometry; swirl direction Idle versus takeoff conditions
3D, Cartesian with stair-step boundaries 3D, non-orthogonal BFC; staggered 2D, non-orthogonal BFC; multiblock 3D, non-orthogonal BFC; co-located 3D, non-orthogonal BFC
Unsteady-state; fractional time steps SIMPLE-like; QUICK differencing Dual time-stepping integration method SIMPLE
Velocity, temperature, and species Mean and rms temperature and species None reported
Computer code ESTET
Inlet air temperature
CONCERT-3D computer code ALLSPD computer code
Dilution air
SIMPLE-like algorithm
3D BFC
No information
3D, non-orthogonal BFC; staggered 3D BFC 3D, staggered
Multi-grid; SIMPLE-like
Qualitative temperature comparisons Water analogy and full combustor Exit temperature
3D, Cartesian, stair-step boundaries
Langrangian–Eulerian SIMPLE SIMPLE
None reported
None reported Velocity and temperature profiles Water analog tests in the VRT combustor
None reported
37–47 CPU-min on Cray X-MP No information
Finite differencing scheme
No information
KIVA-II code VAX 780
Fuel injector location and geometry Vaporization; takeoff vs. idle Equivalence ratio None reported
FLUENT commercial code
VRT configuration
No information
Air mass flow split
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GE single annular CFM56 turbofan engine Staged RQL combustor Axisymmetric swirl combustor Variable residence time (VRT) and annual combustors
3D, non-orthogonal BFC
Effect of combustion model on CO/temperature Fuel type; spray quality; air split
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Combustor
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Combustor
Grid
Solution method
Data
Computer/Code
Sensitivity studies
20 21
Staged RQL combustor Model airblast fuel nozzle
2D, staggered 2D, non-orthogonal BFC
None None
RQL staged combustor
3D
None
No information KIVA-2 computer code; CRAY X-MP REFLEQS computer code
None reported Chemical mechanism
22
SIMPLE algorithm Implicit-continuous Eulerian (ICE) No information
23
Three annual GE combustors Annular Rolls-Royce RB211 combustor Annular GE turbofanengine combustor Can-type test combustor
3D, non-orthogonal BFC
SIMPLE-like; multi-grid
Exit temperature patterns
No information
2D rotated orthogonal, BFC
SIMPLE algorithm
PACE computer program
3D, non-orthogonal BFC
SIMPLE-like
3D cylindrical
SIMPLE
Exit plane temperature distribution Exit plane temperature and pattern factor Fuel and CO at exit plane Gas composition, temperature, exit velocity and residence time Exit temperature and pattern factor None reported
No information
Non-equilibrium CO model None reported
No information
Secondary dilution holes
No information
Jet-to-main flow velocity ratio Grid density
24 25 26 27
Cylindrical can and sector of annual combustor
3D, cylindrical; stair-step boundaries
SIMPLE
28
Annular GE turbofanengine combustor Annular combustor sector
3D, non orthogonal BFC
SIMPLE-like; multi-grid
3D, non-orthogonal BFC; staggered 2D, axi-symmetric orthogonal BFC
SIMPLE-like; multi-grid
29 30
Rolls-Royce RB211 annular combustor
SIMPLE
Flow patterns; exit temperature; NOx
No information PACE computer program
, 3 cpu-hr on IBM3081
Mixing geometry; jet momentum flux Dilution hole geometry Effect of combustor geometry on NOx Effect of secondary holes
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Table 2 (continued)
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Table 3 Modeling approaches for turbulent combustion with kinetics Method
Advantages
Disadvantages
EDC [186,187]
Complex chemistry Simple to add to CDF codes based on laminar chemistry
Assumed PDF [129,192–199]
Simpler than Monte Carlo PDF methods
Flamelet [202–204]
Complex chemistry
Conditional moment closure [188,207,208]
Allows use of full kinetic mechanisms [188] Inexpensive [209]
Empirically based [188–190] Limited to wrinked flames [191] Assumes gradient diffusion Can produce spurious peaks in mean temperature in reaction zone [217] Multiple solutions possible; application to premixed flames not straightforward [55] Limited to binary mixing [210] Limited to two or three reactive scalars [200] Usually assumes statistical independence Assumes gradient diffusion PDF shape must be assumed Complex phenomena such as extinction and differential diffusion precluded Small errors in PDF shape may result in large errors in mean reaction rate [201] Does not account rigorously for interactions between turbulence and chemistry [200]. Only valid in thin flamelet regime [188,189,205,206] Advanced versions require strain history of the flow [188] Limited to binary mixing [210] Has difficulty dealing with backmixing [210] Scalar fluctuations about conditional mean assumed small [136,210]. Results for multi-step chemistry and inhomogeneous flows have not been reported [210] Shape of the PDF must be assumed [209]
Monte Carlo PDF [211–216]
Eulerian formulation; can be readily added to existing CFD codes [210] PDF shape calculated [217] Reaction treated exactly [217] Amenable to parallel processing [217]
Linear eddy [218]
Detailed parametric studies are economically feasible [219] Local integral moment [220–223]
Distinguishes among effects of turbulent mixing, molecular diffusion, and chemical reaction at all scales of the flow [219] Practical application still in early stages of development [219] High computational requirements [210] Economical Complex flows; predicts large-scale structure Complex chemistry
chemistry and its interactions with the turbulent fluctuations. Turbulent flow is often modeled with time-averaged equations. However, time-averaged reaction rates cannot, in general, be calculated from time-averaged temperature and concentration. Also, fluctuating rates with their associated heat release cause density fluctuations that affect the turbulent velocity field. A number of modeling approaches have been devised to
Expensive Currently affordable only for reduced chemistry [106,188] Molecular transport and transport due to the fluctuating pressure gradient must be modeled [188,215] Requires mixtures fraction gradients; not yet applied to premixed combustion where reaction kinetics dominate
treat chemical reactions in turbulent flow. “Laminar chemistry” neglects the turbulent fluctuations. The “fast chemistry” or “mixed-is-burnt” approach neglects kinetic limitations and assumes instantaneous reaction, limited only by turbulent mixing. Fast chemistry has worked reasonably well for predicting heat release and temperature in non-premixed combustion. In such cases, thermal NOx has been predicted reasonably well by kinetically-based
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post-processors that neglect the effects of NOx species on the gross flame properties. However, for premixed systems and to more accurately predict kinetically limited phenomena in non-premixed flames (e.g. CO emission), finite-rate chemistry with proper accounting of the chemistry/turbulence interactions is needed. In addition to the challenges presented by chemistry/ turbulence interactions, the geometric complexity of gas turbine combustors presents its own challenge. Typical combustors often contain more than 10 000 holes, with swirlers, film-cooling slots, contracting ducts, etc. [42]. Generating a computational grid for such geometries can be extremely difficult, requiring man-months of effort [43]. In spite of these challenges, computer modeling has played a key role in the design of gas turbine combustors for several years [44,45]. Every major gas turbine manufacturer routinely uses CFD-based, turbulent combustion modeling as part of the design procedure [46]. Current modeling technology, however, does not adequately address the concerns of LPC in modern combustors. In addition, the need for modeling is greater because the stability range is narrower with LPC and the emissions of CO, UHC, and NOx are likely to be more sensitive to design modifications. This paper summarizes the characteristics of gas turbine combustion simulations published in the last decade. Approaches for modeling turbulent combustion are reviewed, and the Monte Carlo velocity–composition PDF method is selected for implementation in a new model. Techniques for implementing advanced chemistry schemes are discussed and several reduced mechanisms are evaluated. Predictions of LPC in a laboratory-scale combustor are compared with recent experimental data [47,48]. Finally, application of the model to two industrial premixers and an industrial combustor is demonstrated for non-reacting flow.
2. Simulations of gas turbine combustion Comprehensive modeling of gas turbine combustion has been an active area of research for the last decade and numerous models and simulations have been reported in the literature. Characteristics of simulations published during this period are summarized in Tables 1 and 2, with the most recent simulations listed first. Only CFD-based simulations of reacting flow in gas turbine geometries have been included. Although an attempt has been made to include all reported simulations that meet the above criteria, some omissions have undoubtedly occurred. As the tables indicate, at least 30 simulation studies have been reported, with several of the studies involving more than one combustor. The majority of simulations have been carried out for jet engine combustors as opposed to stationary, land-based gas turbines. In jet engines, the predominant fuel of interest is kerosene or Jet-A, which is injected as an evaporating, liquid spray. The dynamics of spray evaporation have often been taken into account in the simulations,
perhaps to the detriment of modeling turbulence/chemistry interactions, which have often been neglected in such simulations. The k– e model has been universally used for turbulence closure. Turbulent combustion modeling approaches that have been applied to gas turbines include laminar chemistry, the eddy dissipation concept (EDC) method, the assumedshape PDF method, and the scalar PDF method. The laminar chemistry approach ignores turbulent fluctuations. The empirical EDC model includes the fluctuations by relating the reaction rate to the gross turbulence properties. The assumed-shape PDF method incorporates the details of the turbulent fluctuations, but for an assumed shape of their PDF. It is usually implemented with the fast chemistry assumption, ignoring kinetic effects. Monte Carlo PDF methods, including the scalar PDF method, treat complex, finite-rate chemistry without assuming the shape of the PDF. Each of these methods (except laminar chemistry) is discussed in more detail in the next section. Only three of the simulations included radiation. The others either explicitly neglected it (presumably due to the difficulty of including it) or did not mention it at all and presumably neglected it. The fact that radiation was typically ignored does not mean that it is considered unimportant in gas turbine combustors, as several of the investigators specifically suggested the neglect of radiation as a primary reason for discrepancies between their predictions and the data [159,162,163,167,177,180,181]. Pollutants that have been considered in simulations of gas turbine combustion include NOx, CO, UHC, and smoke. Thermal NO is the NOx-forming mechanism most often included. Prompt NO was considered in only one case. NOx-forming reactions should be treated kinetically, even when fast chemistry is assumed for the combustion reactions. NOx calculations are, therefore, often performed in a post-processor, neglecting the effects of these reactions on the gross flame structure, since NOx species are present in only trace amounts. CO is more difficult to predict accurately, because CO formation is also kinetically limited and it is not a trace species. Several of the attempts to predict non-equilibrium CO used laminar chemistry. A variety of geometric configurations, including both can-type and annular, are represented. The most common type of grid was non-orthogonal, body-fitted coordinates (BFC). Pressure-correction methods, such as the SIMPLE algorithm, were used almost universally, and comparisons have been made with data for several of the combustors. Some of the simulations were performed with commercially available codes, such as FLUENT; others were performed with proprietary industry or university research codes. There seems to be a trend toward running the simulations on workstations rather than supercomputers, consistent with general trends in computations. The large number of sensitivity studies that have been conducted to investigate the effect of one or more design variables indicates the importance of simulation to the design process.
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It is obvious from this list of reported simulations that there is current interest in simulating premixed gas turbine combustors, but that the technology to do so is just developing and is not widely available. While the k– e model has limitations, it has been adequate for many combustor applications. However, improved methods of incorporating combustion and pollutant chemistry as well as including radiation are needed.
3. Modeling turbulent premixed flames Modeling of multi-dimensional, turbulent flames such as those found in modern gas turbine combustors has been carried out on several levels. At the lowest level, the timeaveraged Navier–Stokes equations are solved for the mean fields. At the next level, a transport equation derived from the Navier–Stokes equations is solved for the joint PDF of composition and/or velocity, providing a complete statistical description from which the mean as well as higher moments can be determined. The third level of modeling is large eddy simulation (LES), where the large scales of the turbulent flowfield are resolved explicitly while the small scales are modeled. The highest level of modeling is direct numerical simulations (DNS), where the Navier–Stokes equations are solved directly on a fine grid and the instantaneous, small-scale structure of the flame is resolved. Turbulent premixed flames have been modeled at all of these levels. 3.1. Moment closure methods Moment closure methods include the classical approach of time averaging as well as the recently proposed method of conditional averaging. Moment closure generally requires a turbulence model to provide closure for the Reynolds stresses in the momentum equations and a turbulent combustion model to provide closure for the time-averaged reaction-rate term in the species continuity and energy equations. The accuracy of moment closure is limited by both models. The focus here is on the turbulent combustion model, consistent with the need for a comprehensive model that will address the concerns of LPC. While its deficiencies are well-known, especially for swirling flows [49], and while some researchers have obtained improved predictions with the second-order Reynolds stress models [50], the k– e model continues to be widely used in gas turbine combustor modeling. Since the k– e model is adequate in many cases, the need for an improved turbulence model is secondary to the need for more detailed chemical kinetics with improved modeling of chemistry/turbulence interactions. Turbulent combustion models have been used to provide closure for the time-averaged reaction term in the energy and species continuity equations include the EDC model, the assumed PDF model, laminar flamelet models, and the linear eddy
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model (LEM). The advantages and disadvantages of each are summarized in Table 3. 3.1.1. Eddy dissipation concept The EDC model [187] is based on the assumption that combustion occurs at the small scales, where mixing occurs on a molecular level and the rate is assumed to be proportional to the inverse of the turbulence time scale k/e . It was developed from the original eddy break-up model proposed by Spalding [51,52], the most significant difference being that the EDC model accounts for the fact that reaction cannot occur unless both fuel and oxidizer mix on a molecular scale at a sufficiently high temperature [163]. This is accomplished by relating the reaction rate to the limiting species. The EDC model has been extended to multiple-step chemistry by treating the small-scale structures as perfectly stirred reactors (PSRs) [53], although some investigators have reported it to be difficult to use with realistic multistep kinetics [188]. One interesting feature of the extended EDC model is that it can give different results depending on the initial conditions that are assumed for the PSR calculations of the fine-scale structures. The reason for this is the well-known ignition–extinction hysteresis loop for PSRs [54]. Multiple steady-states are possible and the converged solution depends on the initial guess. Gran [55] used the extended EDC model to predict turbulent premixed methane–air flame and investigated the effect of initial conditions. He obtained good agreement with the measured data when the appropriate initial conditions were used and concluded that the only general method of achieving an accurate prediction in premixed flames, where equilibrium chemistry cannot be used to give reasonable initial conditions, is to integrate in time from some known initial condition. This requires detailed knowledge of the experimental procedure, and this level of detail is typically not available in papers that describe experimental work. He did not experience the same dilemma for non-premixed flames where equilibrium chemistry was successfully used to provide reasonable initial conditions for the PSR calculations in the fine-scale structures. 3.1.2. Assumed PDF method In the assumed PDF method, the chemistry is determined by one or more “progress variables” which are allowed to fluctuate. The usual approach is to assume “fast chemistry” where the instantaneous properties of the gas are in equilibrium and can be calculated from the mixture fraction. The effects of turbulent fluctuations are incorporated by assuming a shape for the PDF of the mixture fraction, solving transport equations for its first and second moments, and integrating all instantaneous properties over the PDF to obtain time-averaged values. This integration is efficiently performed by pre-calculating gas properties as functions of the mixture fraction and storing the values in a table for subsequent retrieval.
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Reaction kinetics can be included by defining additional “reaction progress variables”, assuming the shape of a joint PDF, solving equations for the first and second moments of all the progress variables, and integrating instantaneous properties over the joint PDF. The mixture fraction and reaction progress variables are usually assumed to be mutually independent. Thus, the joint PDF is separable and the integration for each variable can be performed independently. Multi-dimensional table look-up and interpolation procedures are employed for efficiency. The assumed PDF method has been applied to nonpremixed, diffusion-limited combustion, where it works reasonably well. Its extension and application to premixed combustion with multiple-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. 3.1.3. Laminar flamelet models Flamelet models are relatively simple and can be used with complex chemistry, but they assume that the reactions are all fast and that their length scales are smaller than the smallest (i.e. Kolmogorov) turbulence length scale. Since the important reaction time scales (and the corresponding length scales) in gas turbine combustion vary over as much as seven orders of magnitude [3], some reactions are clearly in the flamelet regime, some are in the distributed reaction regime, and some are in between [56]. Therefore, the flamelet approach is not sufficiently general to treat all of the reactions of interest in gas turbine combustion and their interactions with turbulence. 3.1.4. Linear eddy model The recently proposed LEM focuses on the small-scale mixing effects in high-Reynolds number flows and explicitly distinguishes between the processes of turbulent mixing, molecular diffusion, and chemical reaction [219]. It does this by simplifying the representation of the scalar field statistics to one dimension and resolving all length scales (from integral to Kolmogorov) [57]. The effects of molecular diffusion on the one-dimensional representation are incorporated by explicitly solving the diffusion equation. Turbulent mixing is incorporated by stochastic rearrangements of the one-dimensional representation. The rearrangement blocks correspond to eddies and their size distribution and the frequency of the rearrangements events are determined from known scaling laws of turbulent mixing. Details are given by McMurtry et al. [219]. Goldin and Menon [201] demonstrated the use of LEM in a conventional, steady-state, turbulent combustion model of a non-premixed jet flame. However, they were unable to demonstrate superiority over the conventional assumed PDF approach, and further work is needed to extend the method to premixed flames. LEM remains in the early stages of development, particularly in its implementation and application in CFD codes, and is not yet considered a
viable option for a comprehensive, three-dimensional combustion model. 3.1.5. Conditional moment closure In conditional moment closure (CMC), species mass fractions and enthalpy are conditionally averaged over a fluctuating progress variable to account for the effects of turbulent fluctuations on the mean reaction rate. In non-premixed combustion, the mixture fraction is used as the conditioning variable. For premixed combustion, conditioning on a reaction progress variable has been proposed [188]. The fluctuations about the conditional mean are assumed to be small, thus limiting the range of application to flames that are far from extinction. Also, the shape of the PDF of the conditioning variable must be assumed [209]. Conditioning on both the mixture fraction and a reaction progress variable has been proposed as a means of extending the range of application to near extinction [58], but the results of this approach are not yet available. Presumably, double conditioning would also be needed to treat practical premixed combustors, where premixing is imperfect and the effects of non-premixedness are important to consider. Thus, CMC is a promising approach, particularly when the effects of detailed chemistry (e.g. full mechanisms) must be considered. The computational workload associated with CMC is considerably less than PDF methods. However, it is not yet sufficiently generalized for application in practical, pre-mixed combustor simulations. 3.2. Monte Carlo PDF methods Monte Carlo PDF methods can efficiently treat the effects of complex chemistry with turbulence interactions. PDF methods are general, applying equally well to premixed, partially premixed, and non-premixed flames. They treat complex chemistry exactly, with no modeling of terms involving reaction rates. However, modeling is required for other processes, such as molecular diffusion, because the PDF that is considered gives the statistical description of the scalar field at only a single point and time. Joint probabilities between the scalar field at multiple locations or for multiple times are not included. Hence, conditional expectations involving simultaneous scalar values at multiple locations such as spacial derivatives are not exact and must be modeled. Monte Carlo PDF methods solve a PDF transport equation using a large number of particles to represent the statistics of the turbulent flowfield. PDF equations can also be solved by conventional finite-difference techniques but this is not practical for the multi-dimensional PDFs that are of interest in turbulent combustion. By appropriately modeling the molecular diffusion term in the PDF transport equation, PDF methods can be applied to both the laminar flamelet and distributed regimes of turbulent combustion. The approaches typically employed, e.g. Curl’s models [92,93] and the interaction-by-exchangewith-the-mean (IEM) model [94,95], are applicable to
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distributed combustion. They consider only the turbulence dissipation time scale, ignoring the detailed effects of flame structure. Pope and Anand [59,65] adapted the PDF approach to the flamelet regime. Sakak [60] developed a unified approach that provides a consistent description of both regimes and demonstrated its applicability to regimes in between these two extremes. They showed that the PDF transport equation resembles the field equation in conventional laminar flamelet modeling when the combined reaction and diffusion terms are expressed in terms of the laminar flame velocity. PDF methods are classified according to the variables that are treated as random variables and the type of Monte Carlo method that is used. Composition, velocity–composition, and velocity–composition–dissipation PDFs have all been used in turbulent combustion modeling. The latter two are also referred to as the velocity–scalar PDF method. Both Eulerian and Lagrangian Monte Carlo methods have been used to solve the composition PDF transport equation. Only Lagrangian methods have been used with PDFs that treat the components of velocity as random variables. PDF methods are compared with other methods in Table 3. Table 4 shows the relative advantages and disadvantages of the various Monte Carlo PDF methods that have been employed in combustion. They are listed in order of increasing level of statistical description, but not necessarily in terms of increasing computational cost for a solution of given accuracy. Most applications of PDF methods have been limited to incompressible flows, but PDF methods have been recently extended to compressible flows where the effects of pressure fluctuations on density are taken into account [61,226–228]. A brief description of each method is given below.
3.2.1. Node-based composition PDF method Composition PDF methods treat all of the scalars that describe enthalpy and abundance of chemical species as random variables. From the composition PDF, one can determine the probability of having a certain gas composition at each location in the reactor. Variances and higher moments can be calculated from the PDF as well as means, although the accuracy of the higher moments becomes increasingly dependent on the number of particles used to obtain the solution. The transport equation for the composition PDF can be solved by both node-based [211] and particle-tracking [213] methods. The former uses an Eulerian approach with a grid and the particles are assigned to cells. The particles do not individually represent physical mass and, therefore, do not have continuous physical location. They can be thought of as residing at the cell nodes and moving from cell to cell according to the effects of diffusion and convection. The number of particles in a cell is constant, and the particles in each cell collectively represent the statistics (i.e. the PDF) of the scalars in that cell. The node-based composition PDF
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method was the first PDF method to be applied to combustion [62]. The advantage of the node-based, composition PDF method is its relative simplicity. Its major disadvantage is that it assumes gradient diffusion, which detracts from its usefulness for premixed combustion where counter-gradient diffusion is known to occur [230]. It also neglects the effects of density fluctuations on the turbulence and introduces randomness in the implementation of both convection and turbulent diffusion, leading to statistical error [229]. 3.2.2. Particle-tracking composition PDF method A Lagrangian approach is used in the particle-tracking method. The particles represent physical mass and their physical locations are tracked continuously throughout the flowfield. While the particle-tracking algorithm is grid free, its implementation employs a grid to calculate the scalar statistics at discrete locations. Unlike the node-based method, the number of particles in a cell is not constant and fluctuates in time as the particles move throughout the flowfield. Because velocity is not treated as a random variable in the PDF transport equation, neither composition PDF method calculates the flowfield and a conventional CFD solver is required for that purpose. The CFD solver provides the velocity and turbulence fields to the PDF code and the PDF code provides the density field to the CFD solver. The particle-tracking composition PDF method is more complicated and difficult to implement than the node-based method. It retains many of the disadvantages of the nodebased method, namely the gradient diffusion assumption and the inability to include the effects of density fluctuations on the turbulence. However, the statistical error due to randomness is reduced because the convection is not stochastic [229]. There is still randomness in the implementation of turbulent diffusion, which is treated by the random walk method [127]. 3.2.3. Velocity–composition PDF method In this method, the components of velocity are treated as random variables along with the species mass fractions and enthalpy. This method has been implemented with the Lagrangian, particle-tracking approach. Even though the velocity field is calculated, the method is not stand-alone, since the one-point, one-time PDF contains no information on the time or length scales of the turbulent motions. The turbulence dissipation (or time scale) field needs to be supplied for closure [63]. A major advantage of this method is that turbulent diffusion is now treated exactly. In addition, the effects of variable density are included. Although the mean pressure field needed for the Monte Carlo calculations can be extracted from the velocity PDF solution, it is sometimes more convenient to supply the turbulence time scale as well as the mean pressure field from a concurrently running finite-volume-based conventional CFD solver [214–216]. In this case, the PDF code supplies the Reynolds stresses to the CFD solver, thus avoiding the
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need for a turbulence closure model in the CFD solver. As with the composition-only method, the PDF code also provides the density field to the CDF solver. The first three-dimensional calculations with the velocity– composition PDF method were performed by Haworth and El Tahry [215]. Their transient calculations were obtained for non-reacting, constant-density flow. Hulek and Lindstedt [64] recently performed both steady-state and fully transient calculations of premixed methane–air and propane–air combustion using the velocity–composition method. Their run times for transient one-dimensional computations were “of the order of weeks on a standard level UNIX workstation”, even though they used a simplified kinetic approach. However, their results demonstrate that transient calculations of combustion are possible with the velocity– composition PDF method. In the Lagrangian Monte Carlo algorithm, the simultaneous processes that affect the particle properties and position are calculated sequentially. This is known as the method of fractional steps [213] or operator-splitting [103]. The error introduced by this technique can be controlled by using small time steps at the expense of more computational time. Gran [55] modeled a turbulent premixed flame with both the EDC and velocity–composition PDF methods and found that the PDF method underpredicted the reaction rate compared with the EDC model. They attributed this discrepancy to the coupling between reaction and diffusion and the effects of operator-splitting. Several investigators have commented on the weakness of the gradient diffusion assumption and the importance of treating the components of velocity as random variables for reacting flow, especially for premixed flames where countergradient diffusion is known to occur [65,66]. Anand et al. [230] have shown that gradient diffusion can lead to erroneous results in the case of reacting flow. They stress that the full potential of the PDF method cannot be realized by a composition-only PDF approach. Haworth and El Tahry [215], who first extended the velocity–composition PDF method to time-dependent flows, concur. 3.2.4. Velocity–composition–dissipation PDF In this method, turbulence dissipation is added to the list of random variables. The method now contains time- and length-scale information, thus enabling it to stand alone without an external CFD solver or turbulence model. Sometimes the method is referred to as a velocity–scalar PDF method were dissipation is numbered with the random scalars, along with the chemical series and enthalpy. Anand et al. [230] extended the velocity–composition– dissipation method of Pope and Chen [235] to swirling flows and applied it to a non-reacting, coaxial swirling jet. In addition to the stand-alone PDF calculations, they obtained predictions using an Eulerian transport equation for the mean dissipation, which is equivalent to using an external CFD solver to obtain length-scale information, and found that the stochastic model (i.e. the stand-alone PDF method
with dissipation treated as a random variable) gave predictions that agreed more closely with measurements. The stochastic model required approximately one-third more CPU time. One of the challenges in applying the velocity–composition–dissipation PDF method to recirculating flows has been the determination of the pressure field. Recently, Anand et al. [231] applied a new elliptic flow algorithm developed by Pope [67] to several swirl combustor flows. The new flow algorithm utilizes variance reduction techniques to maintain mean momentum conservation when turbulent processes such as mixing and viscous dissipation are performed on the particles. It also corrects particle positions to maintain consistency between the total volume of the particles in a cell and the geometric volume of the cell. Hsu et al. [217] compared the predictions of Anand et al. [231] for the stand-alone method with results obtained using the composition-only approach. They used the nodebased method to solve the composition PDF transport equation. Their results show the stand-alone method to be more accurate and they estimated it to be only 20% more time-consuming than the composition-only method. Unfortunately, there are no published results comparing the stand-alone method with the composition-only method solved by the distributed-particle approach or with the velocity–composition method. While showing considerable promise, the stand-alone PDF method is not yet considered sufficiently robust to use in a practical gas turbine combustion model. 3.3. Large eddy simulation LES has been applied successfully to premixed combustion, but it has not yet reached the state of maturity required for implementation in a model intended for practical combustors with complex geometry and chemistry. LES resolves the large-scale motion of the fluid by solving the instantaneous Navier–Stokes equations directly on a coarse mesh, but requires subgrid turbulence and turbulent combustion models for closure. The same methods that have been developed to model turbulent combustion at all scales have been applied to the subgrid scales in LES. Ryde´n et al. [68] developed a subgrid combustion model based on the EDC method and obtained improved predictions, as compared with k– e , for a premixed flame behind a bluff-body flame holder. Givi [69] and Gao and O’Brien [70] have used subgrid models based on PDF methods. Calhoon and Menon [71] developed a subgrid model using LEM and applied it to non-premixed flames. Menon et al. [72] have demonstrated the approach of using a subgrid model based on LEM in an LES simulation of premixed combustion using a simplified approach that avoids the significant closure problems that arise from combustion on the subgrid scale, namely the evaluation of filtered reaction source terms and the temperature–species correlations in the state equations. In view of the ability of
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LES to resolve the large-scale structures of the flow and the ability of LEM to resolve the fine scales where mixing and reaction take place, LES with sub-grid closure based on LEM might be the preferred approach for modeling turbulent combustion in the future. The recently proposed local integral moment (LIM) model [220,221] is a type of LES model, in that the large scales of the turbulent flow are computed explicitly. However, it uses a Lagrangian approach where the Navier–Stokes equations are transformed to a set of ordinary differential equations through a local parabolization around a time-evolving material surface on which the scalar gradients are concentrated. The parabolization is based on the observation that molecular mixing processes in turbulent flows are concentrated on universal, self-similar structures [73]. The chemical species fields are obtained from the scalar field constructed from the integral moments on the time-evolving surface via a strained diffusion and reaction layer formulation. The method is economical and has produced accurate calculations of complex flows with complex chemistry, but it has not been applied to partially premixed flames where the reaction kinetics dominate. Also, the method is not applicable to premixed flames where the effects of mixing are neglected and there are no gradients in the mixture fraction [74]. 3.4. Direct numerical simulation DNS has been applied to turbulent premixed flames [75– 78] but it is unlikely to be practical for industrial design calculations in the foreseeable future. Its main role is as a research tool to provide fundamental insight into turbulent flows and to help develop improved statistical submodels, such as the LEM, that will be applied in engineering codes for practical combustors in the future.
4. Combustion chemistry of natural gas The combustion chemistry of natural gas is complex. Natural gas contains numerous species and the composition is variable. Methane is the dominant component, with the average volume fraction varying from 0.86 to 0.95 in 14 large cities in the US [79]. Other important species include heavier hydrocarbons (ethane, propane, butanes, etc.), CO2 and N2. Many reaction mechanisms have been developed for describing the combustion of hydrocarbons in natural gas. These mechanisms range from simple one- and two-step models [80] to detailed models with hundreds of elementary reactions [83]. The simpler mechanisms ignore hydrocarbons heavier than methane and treat only a few species for a limited range of conditions, while more detailed mechanisms may describe the chemical behavior of dozens of species for a broad range of conditions and may include
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C21 chemistry for describing fuel breakdown and ignition [81,82]. The most comprehensive chemical mechanism for natural gas combustion is GRI-Mech 2.11 [83], with 276 elementary reactions and 49 species, including C2 and nitrogen species. Its development is being sponsored by the Gas Research Institute. C3 and heavier hydrocarbons are not currently included. However, the development of GRIMech is ongoing, and additional species, reactions and features are expected in future versions. Although Monte Carlo PDF methods treat chemical reactions exactly and there is no theoretical limitation preventing the use of a detailed mechanism such as GRI-Mech 2.11 in a PDF calculation, there is a practical limit of three to five reactive scalars with present-day computers and standard table look-up procedures. It has been estimated that 6 years of CPU time would be required on a typical engineering workstation to perform a practical, full-scale PDF method calculation with a 16-species mechanism for methane combustion [106], using direct integration (i.e. without table look-up). Therefore, reduced chemistry with pre-calculated table look-up, efficient table storage methods or both must be used. Reduced chemistry can be implemented through the use of global, reduced mechanisms (see Section 4.1) or by the dynamical systems approach using intrinsic low-dimensional manifolds (ILDM) [99] (see Section 4.2). 4.1. Global reduced mechanisms Reduced mechanisms can be developed empirically or from skeletal mechanisms that typically contain dozens of elementary reactions. In the latter approach, judiciously selected species are assumed to be at steady-state and selected reactions are assumed to be in equilibrium. These assumptions provide algebraic relationships that reduce the number of independent variables (i.e. reaction scalars) needed to describe the chemistry. Global reduced mechanisms must be used with caution because the assumptions or empiricism on which they are based may not be valid at all conditions, such as at low temperature or high pressure. 4.1.1. Two-step mechanism A two-step mechanism for methane combustion was proposed by Dryer and Glassman [84]: CH4 1 CO 1
3 O $ CO 1 2H2 O 2 2 1 O $ CO2 2 2
1 2
This mechanism was based on the observation that CO is not appreciably consumed until all of the hydrocarbon species have disappeared. Carbon monoxide is primarily formed in hydrocarbon flames by fast, fuel-consuming reactions [80]. CHx radicals are also generated by these reactions. Since
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Table 4 Comparison of Monte Carlo PDF methods Method
Advantages
Disadvantages
Node-based composition PDF [224– 228]
Simplest PDF method
Same as distributed-particle composition PDF method plus: Randomness in the implementation of convection leading to statistical error [229] For non-orthogonal grids, may be difficult to compute effective cell flowrates [210] Difficult to choose proper time step [210] Same as velocity–composition PDF method plus: Convection by turbulent velocity modeled by gradient diffusion [230] Randomness in the implementation of turbulent diffusion (i.e. a random walk) [229] Neglects effects of density fluctuations on velocity field Contains no information on the time or length scales of the turbulent motions [230] Molecular mixing must be modeled [232] Particles must be sorted to determine cell location at each time step, a computationally intensive task No. of particles per cell varies randomly
No. of particles in each cell is constant No particle sorting to determine cell location
Distributed-particle composition PDF
Convection is deterministic resulting in decreased statistical error [229] Time step can be made arbitrarily small [210] Particle equation of motion is grid independent [210]
Velocity–composition PDF
Turbulent diffusion exact [191] Variable density effects (i.e. fluctuations) exact [231]
Velocity–composition–dissipation PDF [230–234]
Provides complete closure; stand-alone method [235,236] Allows for multiple time scales, accounts for internal intermittency, has potential to account for large scale structures, and better accounts for particle history effects on local turbulent structure [230]
Needs a robust algorithm for calculating the mean pressure field; recently developed pressure algorithm needs more development [231]
they are more reactive than CO, they consume the available oxygen. It is only after the CHx radicals have disappeared that CO can oxidize [37] at an appreciable rate. With optimized rate parameters, Westbrook and Dryer [85] used the two-step mechanism to suitably reproduce lean and rich flammability limits as well as the dependence of flame speed on pressure and equivalence ratio.
velocity in a premixed laminar flame between 1 and 80 atm [87]. The intermediate species OH, O, HO2, CH3, CH2O, CHO, CH3O and H2O2 were assumed to be in steady state. Some of the resulting algebraic relationships were truncated in order to simplify the solution procedure. Several of the simplifications are valid only at high temperature [89].
4.1.2. Four-step mechanism Several studies with reduced mechanisms have emphasized the role of hydrogen-containing free radicals such as H and OH in predicting CO [86,136]. The following mechanism incorporates H and was used to analyze the inner structure of a methane–air premixed flame [87]:
4.1.3. Five- and nine-step mechanisms Two reduced mechanisms were recently developed specifically for conditions of interest in lean premixed gas turbines (0.4–0.6 equivalence ratio and 30 atm) [90]. The first mechanism consists of four combustion steps and one NO-forming step:
CH4 1 2H 1 H2 O $ CO 1 4H2
3
3H2 1 O2 1 CO2 $ 3H2 O 1 CO
7
CO 1 H2 O $ CO2 1 H2
4
H2 1 2OH $ 2H2 O
8
2H 1 M $ H2 1 M
5
3H2 1 CO $ H2 O 1 CH4
9
O2 1 3H2 $ 2H 1 2H2 O
6
H2 1 CO2 $ H2 O 1 CO
10
3H2 1 CO2 1 2NO $ 3H2 O 1 CO 1 N2
11
This mechanism was derived from a skeletal mechanism [88] containing 25 elementary reactions and 16 species and was used to make reasonable predictions of flame
The rate of NO formation in Eq. (11) is a function of
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elementary reaction rates in the thermal, prompt, and N2Ointermediate pathways, and so all three NO-forming mechanisms are represented. The second mechanism consists of seven combustion steps and two steps for nitrogen chemistry: H2 1 O $ H 1 OH
12
4H 1 O2 $ H2 1 2OH
13
H 1 O $ OH
14
H2 1 2O 1 CH3 $ 4H 1 OH 1 CO
15
O 1 CH4 $ OH 1 CH3
16
H 1 OH 1 CO $ H2 1 CO2
17
H 1 OH $ H2 O
18
5H 1 2NO $ 2H2 1 O 1 OH 1 N2
19
O 1 N2 O $ O2 1 N2
20
As in the five-step mechanism, all three NO-forming mechanisms are represented in Eq. (19). A summary of all four reduced mechanisms is shown in Table 5. Both the five- and nine-step mechanisms were derived from the detailed, 276-step, 49-species, GRI-Mech 2.11 mechanisms [96,97]. Nitrogen chemistry was included. Steady-state assumptions were applied to 38 and 34 intermediate species, respectively, and these equations were not truncated as was the case with the four-step mechanism. 4.1.4. Evaluation of reduced mechanisms Predictions with the four-, five-, and nine-step mechanisms were compared with GRI-Mech 2.11 in a PSR model [90] at 1 and 30 atm, with the equivalence ratio varying from 0.4 to 1.0 at 1 atm and from 0.6 to 1.0 at 30 atm. The agreement for predicted temperature was within 3% of the full mechanism at all conditions for all three reduced mechanisms and was slightly better for the five- and ninestep mechanisms. For CO, all three reduced mechanisms did fairly well at low pressure (less than 7% discrepancy for the five- and nine-step mechanisms and 12% for the four-step mechanism). At the higher pressure, however, the five- and nine-step mechanisms considerably outperformed the fourstep mechanism (2% compared with 65%). For NO, the agreement was 5% at 30 atm for both the five- and ninestep mechanisms. At low pressure, however, the accuracy of the five-step mechanism deteriorated significantly at equivalence ratios greater than 0.7. At lean conditions, both mechanisms were within 2% of the full mechanism. Further evaluation was performed with a Partially Stirred Reactor (PaSR) model. In a PaSR model, the contents of the reactor are not completely mixed on a molecular level, even though the reactor is homogeneous [91,113]. The finite rate of molecular mixing is governed by an input parameter, the
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mixing frequency. In a premixed flame, the incoming fuel and air mixture cannot react until it is mixed with hot combustion products and brought to reaction temperature. Thus, the PaSR provides a means of evaluating chemical mechanisms int he context of finite-rate, turbulent mixing. The PaSR can be thought of as a zero-dimensional composition PDF model. Indeed, the governing equation for the PaSR can be obtained by integrating the composition PDF transport equation over three-dimensional space. It is conveniently solved with the Monte Carlo method. The reactor is assumed to be filled with a large number (e.g. 1000) of particles to represent the statistics. At each time step, the particles undergo mixing, chemical reaction, and throughput. Mixing is accomplished between randomly selected pairs of particles (Curl [92] or modified Curl [93] mixing model) or by relaxing the properties of all fluid particles deterministically toward the ensemble mean values. The deterministic approach is referred to as the IEM model [94,95]. Throughout is accomplished by adding fresh, unreacted particles and randomly removing an equal number of particles at each time step. When the IEM model is used, it is this random removal process that gives the PaSR model its stochastic nature. In the case of premixed combustion, the fresh particles consist of both fuel and air. PaSR predictions for atmospheric LPC were made with all four reduced mechanisms and with GRI-Mech 2.11 [96,97]. Typical results are shown in Fig. 1. In this case, the reactor statistics were represented by 650 particles. The equivalence ratio was 0.8, the mixing frequency was 5000 Hz, the inlet temperature was 300 K, and the reactor residence time was 2 ms. Sufficient mixing of hot, partially reacted particles with fresh, unreacted particles was required to avoid blowout at this low inlet temperature and residence time. Fig. 1 shows the evolution of mean temperature, CO mole fraction, and NO mole fraction as functions of non-dimensional time (actual time normalized by the residence time). Initially, all of the particles were assumed to be fully reacted (i.e. at equilibrium). The predictions evolve from the assumed initial condition to a stochastic steady-state. The mean temperature predicted by the detailed mechanism (Fig. 1a) is most closely followed by the nine-step mechanism. The five-, four-, and two-step mechanisms overpredicted the steady-state, mean temperature by approximately 20 K, 25 K, and 105 K, respectively. The overprediction was likely due to the greater degree of dissociation that was allowed in the full mechanism. The evolution of CO mole fraction is shown in Fig. 1b. The five- and nine-step mechanisms were about 5% higher in the mean, steady-state value than the detailed mechanism while the four-step mechanism was about 12% lower. The assumption of partial equilibrium for OH in the four-step mechanism might have contributed to this discrepancy. The significant underprediction by the two-step mechanisms shows the importance of hydrogen chemistry. The evolution of NO mole fraction for the reduced and
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Table 5 Characteristics of reduced mechanisms Reduced mechanism
Parent mechanism
Reduction conditions
Steady-state assumptions
Other
Two-step [85]
None
None
None
Empirically derived from experimental data Does not include H2 chemistry
Four-step [87]
25-Step [88]
Laminar premixed flame
Seven S–S species
Eight species
16 Species
f 1.0
OH, O, CH3, CH3O, CH2O, HCO, HO2, H2O2
Truncated and explicit steadystate species equations OH partial equilibrium assumption Does not include NO chemistry
Six species CH4, O2, N2, CO2, H2O, CO
P 1 atm
CH4, O2, N2, CO2, H2O, CO, H, H2 Five-step [90]
276-Step [83]
Perfectly stirred reactor
38 S–S species
Nine species
49 Species
H, O, C2H, H2CN, HCNN, C2H3, C2H5, C
P 30 atm
CH3O, CH, CN, N, C2H6, NH, HCCO, CH3, NNH, CH2OH, N2O, NH2, HCCOH, CH2S, NCO, C2H2, HOCN, C2H4, HNO, HCO, CH2CO, NO2, CH2, CH3OH, H2O2, HCNO, HO2, HNCO, HCN, CH2O
f 0.4–0.6
CH4, O2, N2, CO2, H2O, CO, OH, H2, NO
Nine-step [90]
276-Step [83]
Perfectly stirred reactor
34 S–S species
13 Species
49 Species
f 0.4–0.6
Same as five-step plus:
P 30 atm
H, O, CH3, N2O
CH4, O2, N2, CO2, H2O, CO, H, OH, H2, O, N2O, CH3, NO
detailed mechanisms is shown in Fig. 1c. A period of approximately five residence times (ten times longer than for temperature or CO) was required to reach steady-state for NO. The five- and nine-step mechanisms agreed with the detailed mechanism to within about 2 and 9 ppm, respectively. The higher discrepancy in the five-step mechanism was probably due to the additional steady-state assumptions for O, H, and N2O. A detailed PaSR study of premixed methane–air combustion was performed by Cannon et al. [98] with the mechanisms presented here (except the two-step mechanism). This study showed that the evolution of mean and rms temperature, CO, and NO was accurately described with the ninestep mechanism over a wide range in mixing frequency (2000–10 000 Hz) and equivalence ratio (0.65–1.0). The five-step mechanism performed less reliably than the ninestep mechanism at f 1.0 but performed similarly to the nine-step mechanism at f 0.65. The four-step mechanism underpredicted mean CO values and overpredicted instantaneous temperature reaction rates, most likely due to its inferior parent mechanism, partial equilibrium assumption for OH, and unallowed dissociation of neglected radical species.
Implicit steady-state species equations No partial equilibrium assumptions
Includes thermal, prompt, and N2O-intermediate NO chemistry
Implicit steady-state species equations No partial equilibrium assumptions Includes thermal, prompt, and N2O-intermediate NO chemistry
4.2. ILDM method The ILDM method [99] is a generalized approach to reduced chemistry that obviates the need for a global reduced mechanism that has narrow applicability and must be carefully validated for the conditions of interest. In the ILDM approach, a full mechanism is used and the chemistry is reduced locally to a user-specified number of degrees of freedom. The method is based on the observation that, in any region of composition space, all starting compositions are quickly attracted to low-dimensional manifolds where only a few reactions are controlling, and then move more slowly along the manifolds towards equilibrium. The ILDM method, which is based on the dynamical systems approach [100], can be thought of as a reduced mechanism approach where the mechanism is optimized locally rather than globally. The manifold is identified by eigenvector analysis of the governing chemical kinetic equations for the full mechanism. It is then parameterized and tabulated a priori. A three-dimensional manifold has been used successfully in combustion calculations of H2/O2 –air, CO/H2/N2 –air and natural gas–air flames [101,102,233]. Yang and Pope [103] have summarized the difficulties of
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Fig. 1. Evolution of mean (a) temperature, (b) CO mole fraction, and (c) NO mole fraction predicted by the PaSR model for the reduced and full mechanisms.
the ILDM approach. First, reduced manifolds are not straightforward to parameterize. Second, manifolds of different dimension are appropriate in different regions of composition space. Third, look-up table generation is not automatic and must be performed at each set of conditions. And finally, fourth, the major portion of the manifold is wastefully tabulated since it is not known a priori which regions will be needed. In an attempt to overcome these difficulties, Yang and Pope [103] have developed a new variable-dimensional manifold method which defines the manifold in terms of a time scale rather than a global dimension. Properties of this manifold can be efficiently calculated and stored using in situ tabulation. 4.3. In situ tabulation In the Monte Carlo PDF method, fluid particles evolve by simultaneous processes of convection, mixing and reaction. In order to calculate the changes in particle composition due
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to reaction at each time step, reaction rates must be calculated and used to calculate the new properties of the fluid particle. Due to the presence of a wide range of reaction time scales, implicit numerical methods are required to integrate the chemical kinetic equations. The solution to these systems can be obtained by the use of stiff solvers such as LSODE [104] or VODE [105]. These software packages use a variable step-size, backward differentiation formula method and require significant computational effort. The main advantage of the reduced mechanism or manifold methods is that only a few independent scalars are needed to define the thermo-chemical state of the system. With a three- or four-step mechanism, it is feasible to tabulate the change in particle properties as a function of the independent scalars for the allowable composition domain in a pre-processing phase. The expensive calculation of reaction rates and integration of kinetic equations in the simulation phase is then replaced by table look-up and interpolation. The look-up table quickly becomes very large as the number of reactive scalars increases, and table size is a limiting factor in the number of scalars that can be considered. Fig. 2 shows how the size of the look-up table varies with the number of reactive scalars. The four-step mechanism, for example, would require approximately 2.3 gigabytes for one reactor pressure and one time step, if 20 grid points were used to discretize the domain of each of the four reactive scalars, the mixture fraction, and enthalpy. Information stored at each grid point would include the change in each of the four scalars for the specified time step, temperature, density, and specific heat. Since chemical compositions evolve through composition space along low-dimensional manifolds, table values are needed for only a fraction of the allowed composition space. With an aim toward improving the efficiency of table generation and storage, in situ tabulation has been suggested by Pope [106]. In situ tabulation occurs during the simulation phase and stores the data in a binary-tree data structure. Only compositions that are actually needed during the simulation phase are stored. In situ tabulation therefore requires significantly less storage than a priori tabulation, because only a small fraction of the allowable domain is tabulated. Fig. 3 shows the evolution of the in situ table size for the PaSR calculations at a mixing frequency of 2500 Hz and with the four-step mechanism. The tabulation was performed with a maximum refinement level of 6 (normalized cell size 2 26), which was sufficient for these calculations. As shown, a steady-state condition was reached where the table size remained fairly constant and table look-up (rather than direct integration) provided most of the required information. Pope [106] has extended the in situ tabulation method to automatically adapt the local refinement level and effectively control the tabulation errors. The evolution of the mean methane mole fraction at the
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Fig. 2. Table size for a priori tabulation with 20 grid points per scalar (including mixture fraction and enthalpy).
conditions of Fig. 3 is shown in Fig. 4. Results are shown for three cases: in situ integration (i.e. without storage), in situ tabulation and a priori tabulation. In situ integration is the benchmark for comparing the two tabulation methods. As shown, the in situ tabulation method produced results that agree almost identically with the benchmark whereas the a priori method produced results that are about 8% lower. These calculations demonstrate the potential for achieving improved accuracy with the in situ tabulation method while at the same time reducing the size of the look-up table. Required CPU time and table size for the PaSR calculations using the in situ tabulation, a priori tabulation, and in situ integration methods are compared in Table 6. As expected, table look-up significantly decreased the required CPU time. The in situ tabulation method required less CPU time than a priori tabulation and the table was significantly smaller. Most of the time required by the a priori method was for table generation.
Fig. 4. Mean CH4 mole fraction predicted by the PaSR model with the in situ (direct) integration, in situ tabulation, and a priori tabulation methods.
As was shown earlier, the size of the a priori look-up table increases considerably with each additional scalar and the method is therefore limited to four or five reactive scalars. It has been shown that an in situ tabulation method, however, can be applied to mechanisms with 12 reactive scalars while still maintaining reasonable storage requirements [106]. Therefore, the in situ tabulation method shows good potential for including detailed kinetic mechanisms in turbulent combustion calculations.
5. Comprehensive model description A new combustion model is being developed to address the concerns in modern land-based gas turbines. The new model uses an unstructured grid flow solver to accommodate geometrical complexity. An important element of the new model is a new sub-model for lean premixed combustion of natural gas which is based on the velocity–composition PDF method. The model is being validated with detailed data obtained in a laboratory-scale gas turbine combustor and applied to practical gas turbine combustor geometries obtained from industry. 5.1. Unstructured grid flow solver
Fig. 3. Evolution of table size in the PaSR calculation using in situ tabulation with the four-step mechanism.
A control-volume finite element method [107] and a variant of the SIMPLE technique [108] are used to solve the Navier–Stokes equations on an unstructured, tetrahedral grid. A skewed, mass-weighted, upwind interpolation function [109] is used for the convective discretization, while linear interpolation functions are used for the diffusion and pressure gradient discretization. The standard k– e model is employed to compute the turbulent eddy viscosity. The fluid density field is provided by the PDF submodel.
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Table 6 CPU time and table size for PaSR calculations Method
Relative CPU time
Table size (MB)
In situ integration 38 – A priori tabulation 5.9 (table generation only) 5.8 6.5 (total including table generation) In situ tabulation 1 0.6
5.2. Submodel for lean premixed combustion The PDF submodel was developed from the foundation of PDF2DS, a two-dimensional code developed at Cornell University [110,214,216]. PDF2DS was extended to three dimensions, to non-isothermal, reacting flows and to unstructured, tetrahedral grids. The submodel calculates the local one-point, one-time, joint velocity–composition PDF. The PDF transport equation is as follows and can be derived from the general equations for conservation of mass, momentum, energy, and species for low-Mach-number flow [213]:
r C
2f 2f 2f 1 r CVj 1 r CSa Cf a±h 2t 2Xj 2Ca 1 r Cgj 2
2kpl 2xj
!
Fig. 5. Swirl-stabilized combustor (taken from Ref. [139]).
2f 2Vj
+ # 2tij 2p 0 2 1 jV; C f 2xi 2xj ai E i 2 2J 2 hD 1 jV; C f 2 7·qjV; C f 2Ca 2Ch 2xi 21
2 2Vj
"*
Here, f represents the velocity–composition PDF, r is the density, V is the velocity, Ca is the a -dimensional composition scalar, Sa is the reaction source term, gj is the body force, p is the pressure, t ij is the stress tensor, Ji a is the diffusive flux, C h is the specific enthalpy, and 7·q is the specific enthalpy source term (due to radiation). This equation illustrates the capability of the velocity–composition PDF method to completely describe the important chemical kinetic and convective transport effects in reacting, turbulent flows. These effects are described by the underlined terms in Eq. (21) and appear in closed form. The terms on the right-hand side must be modeled since the required information for computing the expected values (indicated by the angle brackets) is not available from the single-point, joint PDF. The first term on the right-hand side of Eq. (21) represents transport of the PDF in velocity space by the viscous stress and fluctuating pressure gradient. This term is described with a simplified Langevin model [111]. Some implicit assumptions of the simplified Langevin model include
Fig. 6. Bluff-body combustor (taken from Ref. [142]).
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[127]: (i) due to the Markov nature of the Weiner process and since the model terms are a function of local mean quantities, the turbulence structure must be expressed in terms of local mean quantities; (ii) local isotropy, i.e. flows that are so strongly distorted or have such a low Reynolds number that the small scales are not even approximately isotropic, are beyond the scope of this model. Further assumptions in the simplified Langevin model are a linear return-to-isotropy and negligible “rapid pressure” effects. For homogeneous turbulence, the simplified Langevin model is equivalent to Rotta’s [112] model [111]. The second term on the right-hand side of Eq. (21) represents transport in composition space by the diffusive flux and is described by the IEM model [46,113–116,213,214]. 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 [148]. In homogeneous isotropic turbulence, the IEM model predicts that the shape of the composition PDF does not always relax to a Gaussian distribution, but instead preserves the shape of the initial PDF [233]. Stochastic, pair-exchange models [92,117] can also be used in PDF methods but their computational requirements are higher and their superiority has not been shown for reacting flow simulations. For example, it has been demonstrated that the choice of mixing model is not critical in the distributed reaction regime of lean premixed combustion [118]. When turbulent mixing frequencies were above 1000 Hz, the differences between the IEM and pairexchange mixing models were overwhelmed and blurred by the finite rates of the chemical reactions. The k– e turbulence model [119] is used to provide the mixing frequency (e /k) to both the simplified Langevin and IEM models. In the k– e model, the turbulent viscosity, m t, has no directional dependence. This isotropic restriction can limit the model’s usefulness in high swirling or other complex flows with varying normal stresses [120]. However, the k– e model has been recommended for highswirl calculations due to the marginal improvements offered by other higher-order models [49]. The five constants in the standard k– e model have also been modified for modeling swirling flows [121]. The mean radiation flux divergence is provided by the discrete ordinates radiation model [122,123]. The discrete ordinates model is a multi-flux model that discretizes the intensity field into a finite number of streams or ordinates [124]. The axi-symmetric calculations presented here were performed for 12 discrete directions. This S4 approximation has been shown to be adequate for radiation calculations in both two- and three-dimensional furnace geometries [123,125]. A spectral-line weighted-sum-of-gray-gases model [126] provides gas emissivity. This spectral model provides an accurate way of obtaining radiative gas properties for use in the discrete ordinates model, as it does not require the often-made gray gas assumption. Because of the large dimensionality of the velocity–
composition PDF (7 1 a ), Eq. (21) is not amenable to conventional solution with finite differences. It is, however, efficiently solved indirectly with a Monte Carlo method [127–134,211,214]. A large number of Lagrangian fluid particles evolve in time and are used to represent the joint velocity–composition statistics. Each particle may be thought of as coming from a different realization of the flow [111]. The time step for the Lagrangian solution is determined from the minimum of all local residence and mixing times in the reactor. An improved time-splitting technique [229] is used to sequentially calculate the change in particle properties due to the simultaneous processes of mixing, etc. The particles are first convected for half of a time step. The changes in particle properties due to mixing, reaction, and radiation are then calculated for the full time step. Finally, the particles are convected to their final positions for the time step. Pope [229] found this method to be superior to the method of fractional steps which was originally implemented in PDF2DS, where the particle properties first evolved at the initial position for the entire time step before the particles were convected [213]. In addition to performing the reaction and mixing calculations at the half-step position, these calculations are performed sequentially for several reduced time steps in order to minimize the error due to the coupling between mixing and reaction. In his velocity–composition PDF predictions of a premixed flame, Gran [55] observed that the mean reaction rate was apparently underpredicted and inferred that this underprediction may be due to coupling between reaction and diffusion that is not accounted for in the time-splitting Lagrangian approach. The size of the reduced time step was determined from PaSR calculations where it was gradually reduced until it was not affecting the results. In the limit of an infinite number of particles and an infinitely small time step, the Monte Carlo method provides an exact solution to Eq. (21) [215]. Fluid particle velocities evolve in the PDF submodel according to the simplified Langevin model [135] and are also adjusted to maintain consistency with the mean velocity and kinetic energy fields provided by the Eulerian flow code [133]. This consistency ensures that the no-slip condition is satisfied at the walls [136,211]. Convective heat loss is calculated based on the law of the wall [137]. The unstructured grid is used to store mean values of particle properties. The mean fields of velocity, pressure gradients, and turbulence (e /k) are passed to the PDF submodel and fluid particles are then allowed to evolve for one reactor residence time or a specified fraction thereof. Overall convergence requires a statistical steady-state in the PDF calculation and simultaneous satisfaction of the Navier–Stokes equations. Overall mass and energy balances are used to help determine convergence. One macro-iteration for a three-dimensional case involving approximately 3500 nodes, 1.5 × 10 4 tetrahedral cells, and 1.5 × 10 6 particles requires approximately 2 hr of CPU time on an HP-
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735 workstation. A macro-iteration consists of converging the Navier–Stokes equations and then performing Lagrangian PDF calculations for the specified length of time, which in this case, was 20% of the reactor residence time. 5.3. Model evaluation Recent experimental data in swirl-stabilized [138] and bluff-body-stabilized [34] lean premixed combustors were used to evaluate the model. The swirl-stabilized combustion chamber, shown in Fig. 5, was configured such that the geometry around the injector was nearly axi-symmetric with an average diameter of 170 mm. Flat quartz windows, 60 mm wide, were located on each side of the chamber to allow the use of laser diagnostic equipment. Laser Doppler anemometry (LDA) measurements were made to obtain axial, radial, and tangential components of velocity at over 130 locations [48,139]. This swirl-stabilized combustor has been simulated for methane–air equivalence ratios of 0.8 and 0.65, a swirl number of 0.74, 1 atm, a 300 K inlet temperature, and assuming an axi-symmetric geometry [140]. An unstructured grid consisting of 4302 triangular elements and a total fluid particle number of 310 000 were used in the simulations. Both convective and radiative heat losses were considered. A uniform wall temperature of 700 K was assumed. The five-step chemical mechanism [90], with in situ tabulation [106], was used to describe the combustion chemistry. The total CPU time was about 210 hr on an HP-735 workstation. The bluff-body-stabilized combustion chamber, shown in Fig. 6, was configured such that a stainless steel, conical bluff body was mounted coaxially at the center of the combustor and served as a flame holder. A combination of Rayleigh scattering, spontaneous Raman scattering, and laser-induced fluorescence (LIF) measurements of chemical species (CH4, O2, H2O, CO2, NO, and CO) and temperature were obtained radially at several axial locations [34]. This bluff-body-stabilized combustor has been simulated at 0.586 methane–air equivalence ratio, 1 atm, 300 K inlet temperature, and assuming axi-symmetric geometry [140,142]. An unstructured grid consisting of 1596 triangular elements and a total fluid particle number of 116 000 were used in the simulations. Both convective and radiative heat losses were considered. A uniform wall temperature of 500 K was assumed. The five-step chemical mechanism [90], with in situ tabulation [106], was used to describe the combustion chemistry. The total CPU time was about 100 hr on an HP735 workstation. Fig. 7 shows the predicted mean axial velocity profiles compared with experimental data for equivalence ratios of 0.8 and 0.65 at z axial locations of 10, 20, 40, 100, and 175 mm. These results show that recirculation zones existed
Fig. 7. Measured and predicted mean axial velocity profiles in the swirl-stabilized combustor at equivalence ratios of 0.8 and 0.65.
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Fig. 8. Scatterplots of measured and predicted axial/tangential velocity at an axial location of 10 mm and radial locations of (a) 9 mm, (b) 12 mm, and (c) 21 mm in the swirl-stabilized combustor.
along the centerline (centerline recirculation zone (CRZ)) and out radially near the wall (outer recirculation zone (ORZ)). The swirling nature of this flow created a vortex funnel where downstream gases were drawn into the CRZ as a result of the incoming fluid being flung outward [48]. These recirculation zones allowed hot combustion products and radicals to mix and subsequently react with the incoming, cold CH4 –air mixture and stable combustion was obtained without the use of a pilot flame. The predictions compared well with the experimental data at the lowest axial location of 10 mm. The location and magnitude of the maximum mean velocity was well represented at z 20 mm. At locations higher than z 20 mm it is obvious that the model overpredicted the radial spread rate of the high velocity jet. The often, overpredicted turbulent viscosity with the k– e turbulence model [143,144] could be attributed to the overpredicted spread of the high velocity fluid. Also, the negative centerline velocities were overpredicted (by at most 10 m s 21), which is a known shortcoming of the standard k– e turbulence model for predicting confined, swirling flows [121,145]. The effect of varying the equivalence ratio from 0.8 to 0.65 on the mean axial velocity appeared to be small for both the predictions and the measurements. The predictions and measurements both suggested a slightly weaker CRZ for the leaner case at z 20 and 40 mm. Also, the predictions indicated a slightly lower peak velocity for the f 0.65 case (at z 10, 20, and 40 mm), as would be expected for the higher density fluid in this lower heat release case. Comparisons of fluid particle scatterplots in axial and
tangential velocity space were made at several radial locations and at an axial location of about 10 mm. Fig. 8 shows these measured and predicted axial and tangential fluid particle velocities. The experimental data and measurements in Fig. 8a, 8b, and 8c represent fluid particle velocities at radial locations of 9, 12, and 21 mm, respectively. These locations are marked as points A, B, and C in Fig. 7. At each location, the measurements were represented with 4096 data points, and the predictions were represented with 250–750 fluid particles. The fluid particles were sampled from a rectangular cell, centered at the location of interest, with an axial and radial length of 5 and 2 mm, respectively. The highest axial and tangential velocities were observed at r 9 mm and these velocities decreased with radial location and were near zero (for axial velocity) at r 21 mm. Overall, the predictions matched the mean, or central location of fluid particles, though the model did underpredict the spread of the fluid particle scatter plots. Overall, the velocity fields in this swirl-stabilized, lean premixed combustor were well represented near the combustor inlet and far downstream. Discrepancies were observed in the middle region of the combustor (z 20– 40 mm), mainly due to an overpredicted radial spread of the high-velocity fluid. Comparisons of the predicted and measured mean, rms, and the PDF of temperature and species mole fractions of CO2, CO, and NO will be presented at various locations within the bluff-body-stabilized combustor. Fig. 9 shows comparisons between the measurements and predictions for the mean temperature at x/d axial locations 0.1, 0.3,
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Fig. 9. Measurements and predictions of mean temperature profiles in the bluff-body-stabilized combustor (taken from Ref. [142]).
0.6, and 1.0. These comparisons show that the predicted temperatures were well represented. The mean temperatures in the recirculation zone region (r/d , 0.4) were at most only 6–7% (100 K) higher than the measurements. The locations of temperature gradients, where cold and hot gases mixed, were predicted accurately, especially at x/d locations of 0.1, 0.3, and 0.6. Only the mean temperature gradient in the shear layer region at x/d 1.0 was underpredicted, and thus mean temperatures at the edge of the recirculation zone and in
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the shear layer (0.3 , r/d , 0.05) were underpredicted by about 40%. The comparisons of predicted CO2 mole fractions showed excellent agreement with the measurements in the recirculation zone (Fig. 10). The widened shear layer that was predicted at x/d locations of 0.6 and 1.0 compared to the measurements confirm that the model underpredicts the amount of fully burned products at this location. This may be due to underpredictions in CO oxidation and to overpredictions in the mixing of unburned reactants into the shear layer with no subsequent CH4 oxidation. Overpredicted amounts of both available CO and available CH4 at this shear layer location were observed [142] and confirm the lower than expected CO2 values. The radial profiles of mean CO mole fraction reveal the location of the reaction zone or reacted fluid that has been quenched (high CO concentrations). Fig. 11 shows comparisons between the measured and predicted mean CO mole fractions at several axial locations in the bluff-body combustor. The equilibrium value of CO is 7 ppm, much lower than the concentrations that were predicted and measured in the recirculation zone and shear layer region. At the lowest axial location of x/d 0.1, the model showed underpredicted levels of CO in the recirculation zone and shear layer region. These results indicate that the model underpredicts the initial CO formation in the shear layer and then also overpredicts the CO oxidation in the recirculation zone. At x/d 0.3, the comparison between predictions and measurements was excellent in the shear layer region and recirculation zone CO levels were slightly underpredicted (500–700 ppm). Further downstream at x/d 0.6, the peak CO in the shear layer region was overpredicted by about 2400 ppm, though the recirculation zone CO showed excellent comparisons to data. Also, at x/d 1.0, the CO was overpredicted in the shear-layer region and compared well at the centerline along the recirculation zone. Despite the noticeable discrepancies between the measurements and predictions, the comparisons were reasonably good. The ability to accurately predict CO emissions in turbulent methane flames is an extremely difficult task and is strongly dependent upon the chemical model, as well as the other submodels in the comprehensive combustion model. Previous studies of turbulent methane–air non-premixed flames have shown CO discrepancies as high as orders of magnitude [136]. The results presented here have shown marked improvements in predicting CO in turbulent methane flames. The radial profiles of mean NO mole fraction reveal the location of burned fluid that has had sufficient time to form the slow-reacting NO species. Fig. 12 shows comparisons between the measured and predicted mean NO mole fractions at several axial locations in the bluff-body combustor. At the lowest axial location of x/d 0.1, the model showed underpredicted levels of NO in the recirculation zone. The measurements were as high as 11 ppm, while the predictions reached maximum levels of 5 ppm. Apparently the model
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Fig. 10. Measured and predicted profiles of mean CO2 mole fraction in the bluff-body-stabilized combustor (taken from Ref. [142]).
slightly underpredicted the overall formation of NO in this lean premixed system. At x/d 0.3 and 0.6 the comparison between predictions and measurements also showed the NO mole fractions to be underpredicted by about 5–7 ppm. The measurements indicated a trend of decreasing centerline NO from x/d 0.6 to x/d 1.0. The predictions, on the other hand, showed the centerline mean NO to remain at about 5 ppm even at x/d 1.0. Thus, at x/d 1.0, the measurements and predictions compared well at the centerline, where both showed values near 5 ppm. Nandula et al. [34]
Fig. 11. Measured and predicted profiles of mean CO mole fraction (ppm) in the bluff-body-stabilized combustor (taken from Ref. [142]).
report that the NO measurements could be 2–3 ppm higher than the actual values due to the unaccounted interference of O2 on the NO channel. This could partly explain the 5 ppm underpredictions observed in the model. These predictions represented the first NO predictions in premixed combustors using the velocity–composition PDF method. The accessed composition space in a turbulent combustion calculation with the PDF method represents the values of species mole fraction and enthalpy for each fluid particle
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an a priori table [140]. The in situ tabulation approach allowed for practical chemical kinetic calculations using the five-step mechanism in these multi-dimensional, lean premixed combustors. 5.4. Application
Fig. 12. Measured and predicted profiles of mean NO mole fraction (ppm) in the bluff-body-stabilized combustor (taken Ref. [142]).
at each time step during the simulation. The accessed compositions in the swirl-stabilized and bluff-bodystabilized combustor simulations were found to lie in small, well-defined regions that were defined by the mixing, reaction, and heat transfer in the combustor. These accessed composition regions were obtained in situ and required less than 35 MB of memory once a steady-state and converged solution was reached. This memory requirement was more than three orders of magnitude less than would be needed in
Application of the new model to gas turbine configuration from industry has also been demonstrated. Two industrial premixers have been simulated. The first premixer consists of an annulus with swirling flow. A computational grid was created for a 458 segment with a single fuel jet located on the inner wall of the annulus. Periodic boundary conditions were imposed on each face of the segment. The grid contains 9623 nodes and 43 029 elements, and was created using GeoMesh [146] and TGrid [147]. The two periodic planes contain approximately 2000 element faces each. The second premixer also consists of an annulus with radial fuel pegs attached to the inner and outer walls. Multiple fuel inlets are distributed radially along the fuel pegs. The unstructured grid for the smallest repeating segment of this premixer geometry was created by decomposing a structured hexahedral grid into tetrahedral elements. This grid contains 26 622 nodes and 142 800 elements. Both premixer simulations were carried out on an HP-735 workstation and required between 10 and 20 hr of CPU time. Predicted mixing for both premixers is shown in Fig. 13. In both cases, air enters at the bottom and the fuel and air mixture exits at the top. The mixture fraction is color-coded, with red indicating fuel and dark blue indicating air. In both premixers, there are small gradients in fuel concentration predicted at the outlet. These gradients could contribute to the formation of thermal NOx in the combustor and should be considered in the combustor simulation. Simply assuming a perfectly premixed fuel and air mixture at the combustor inlet could significantly underpredict the NOx. Non-reacting premixer simulations such as these will be used to provide realistic inlet conditions for reacting combustor simulations. Preliminary cold-flow simulation of the primary zone of an industrial combustor is shown in Fig. 14. The grid used for these predictions consisted of 16 120 nodes and 83 376 tetrahedral elements. Boundary conditions consisted of two upstream inlets and a single downstream outlet, inner and outer walls, and periodic side planes at both faces of a 908 segment of the combustor. The tetrahedral grid was created by decomposing a multi-block hexahedral grid. The predictions in Fig. 14 are for the 458 plane located halfway between the periodic planes.
6. Conclusions Based on a review of methods for modeling turbulent combustion, the Monte Carlo velocity–composition PDF
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Fig. 13. Predicted mixture fraction in industrial premixers.
method was selected as the best approach to use in a new comprehensive model for land-based gas turbines. This decision was based on the fact that PDF methods are general, applying equally well to premixed, partially premixed, and non-premixed combustors, and can incorporate arbitrarily complex chemistry. The velocity–composition PDF method was selected over the composition PDF method because it treats turbulent diffusion exactly instead of assuming gradient diffusion. Also, variable density effects due to fluctuating temperature are included. The velocity–composition–dissipation PDF method was judged to be potentially superior because an external turbulence model is not needed, but it is insufficiently developed to apply in comprehensive modeling at the present time. Likewise, large-eddy simulation and the linear eddy model have significant potential but require more development before practical application.
Fuel Research, Inc., East Hartford, CT, is gratefully acknowledged. The authors also wish to express appreciation to Dr. M. S. Anand and Profs. J. Y. Chen,
Acknowledgements Funding from the US Department of Energy, under Contract No. DE-FC21-92MC29061 administered through the South Carolina Energy Research and Development Center at Clemson University, and Contract No. DE-AC21-93MC30040 administered through Advanced
Fig. 14. Predicted normalized axial velocity (cold-flow) in an industrial combustor.
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Stephen B. Pope, and L. Douglas Smoot for their help and guidance. [15] [16]
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