A G-equation Combustion Model Incorporating Detailed Chemical Kinetics for PFI/DI SI Engine Simulations

Transcription

1 Sixteenth International Multidimensional Engine Modeling User s Group Meeting at the SAE Congress, April 2, 2006, Detroit, Michigan A G-equation Combustion Model Incorporating Detailed Chemical Kinetics for PFI/DI SI Engine Simulations Long Liang,RolfD.Reitz Engine Research Center, University of Wisconsin-Madison, Madison, WI 53706, USA Jianwen Yi, Claudia O. Iyer Ford Research and Advanced Engineering, Dearborn, MI 48124, USA A G-equation-based combustion model incorporating detailed chemical kinetics has been developed and implemented in KIVA-3V for Spark-Ignition (SI) engine simulations for better predictions of flame propagation and pollutant formation. A progress variable concept is introduced into the turbulent flame speed correlation to account for the laminar to turbulent evolution of the spark kernel flame. The flame front in the spark kernel stage is tracked using the Discrete Particle Ignition Kernel (DPIK) model. In the G-equation model, it is assumed that after the flame front has passed, the mixture within the mean flame brush tends to local equilibrium. The subgrid-scale burnt/unburnt volumes of the flame-containing cells are tracked for the primary heat release calculation. An iso-octane kinetic mechanism coupled with a reduced NO X mechanism is used to describe the chemical processes in the post-flame region and the potential heat release from the end gas. The integrated model was used to simulate the combustion process in a Ford four-valve single-cylinder SI engine, which is equipped with both Port-Fuel-Injection (PFI) and Direct-Injection (DI) fuel systems. For both PFI and DI operational modes, good agreement with experimental in-cylinder pressure, heat release rates and engine-out NO X was obtained for different spark timings and internal residual levels. 1 Introduction The in-cylinder turbulent combustion in SI engines is a complicated aero-thermo-chemical process especially due to the turbulence and chemistry interactions on tremendously different time-scale and length-scale levels. In this paper, we present a G-equation-based flamelet combustion model incorporating detailed chemical kinetics for both Port- Fuel-Injection (PFI) and Direct-Injection (DI) Spark- Ignition (SI) engine combustion simulations. The level set method is a powerful tool for describing in- Corresponding author. address: lliang@wisc.edu terface evolution. With its application to combustion, Williams [1] first suggested a transport equation of a non-reactive scalar, G, for laminar flame propagation. Peters [2] [3] subsequently extended this approach to the turbulent flame regime. The turbulent G-equation concept has been successfully applied to SI engine combustion simulations by Dekena et al. [4], Tan [5] and Ewald et al. [6]. In recent years, to better understand the fundamental engine combustion process and to further improve the versatility of multidimensional models, attention is being given to models incorporating comprehensive elementary chemical kinetic mechanisms. The objective of the current work is to incorporate detailed chemical kinetics into the G-equation-based turbulent combustion model which was implemented into the KIVA-3V code by Tan et al. [5] [7]. Specifically, detailed fuel oxidation mechanisms coupled with a reduced NO X mechanism are applied behind the mean flame front for modeling post flame combustion and NO X formation. The chemical kinetic mechanisms are also applied in front of the flame front for potential capability of predicting the compression autoignition of the end-gas. In the course of coupling detailed chemistry with the G-equation combustion model for the primary heat release calculation within the flame front, it was required to revisit and improve laminar and turbulent flame speed correlations for better description of the turbulent flame propagation process. 1

2 2 Model Formulation 2.1 G-equation description of turbulent flame propagation In the flamelet modeling theory of premixed turbulent combustion by Peters [3], two regimes of practical interest, i.e., the corrugated flamelet regime and the thin reaction zone regime, were described by the same group of level set equations, including the transport equations for the Favre averaged, G, and its variance, G 2, and a model equation for the turbulent/laminar flame surface area ratio, which in turn gives an algebraic solution for steady-state planar turbulent flame speed, ST 0. The set of equations used in the current implementation is [5] [7]: G t +(ṽ f v vertex ) G = ρ u ρ S0 T G D T κ G (1) G 2 +(ṽ f v vertex ) G t 2 = ( ρ u ρ D T G 2 )+2D T ( G) 2 ɛ c s G k 2 (2) ST 0 SL 0 =1+ 1 exp( C m2 t/τ) [ a 4b 2 3 2b 1 l + ( a 4b 2 3 l F 2b 1 ] l ) l 2 + a 4 b 2 u l 3 F SL 0 l F (3) where is the tangential gradient operator, ṽ f is the fluid velocity vector, v vertex is the vertex moving velocity. D T is the turbulent diffusivity, and a 4,b 1,b 3 and c s are constants from the turbulence model or experiment (cf. [3]). ɛ and k are the Favre mean turbulence kinetic energy and its dissipation rate from the RNG k-ɛ model [8]. u is the turbulence intensity. SL 0 is the unstretched laminar flame speed. l and l F are the turbulence integral length scale and laminar flame thickness, respectively. κ is the mean flame front curvature. These equations together with the Reynolds averaged Navier-Stokes equations and the turbulence modeling equations form a complete set to describe premixed turbulent flame front propagation. One significant advantage of the G-equation formulation of turbulent premixed flames is the absence of chemistry source terms in the transport equations. As a consequence, the turbulent flame speed ST 0 plays a crucial role as a predetermined input. Compared with the correlation for ST 0 derived by Peters [2], an exponentially increasing term is added in Eq. (3) in the present model to account for the laminar to turbulent evolution of the spark kernel flame. This term can be obtained by solving Eq. (2) assuming a uniform turbulence profile [3]. Physically, the additional exponential term can be interpreted as a progress variable which accounts for the increasing disturbing effect of the surrounding eddies on the flame front surface as the ignition kernel grows from the laminar flame stage into the fully developed turbulent stage. In the present study, C m2 in Eq. (3) is selected as a tunable model constant for different engines considering uncertainties due to other sub-models and/or mesh resolution. However, for the same Ford engine studied in this paper, C m2 is fixed over all operating conditions. Laminar flame speed SL 0 is one of the key scaling factors in Eq. (3). Metghalchi et al. [9] suggested a correlation for SL 0 as a function of equivalence ratio φ, temperature and pressure of the unburnt mixture given by: SL 0 = S0 L,ref ( T u ) α ( p ) β (1 2.1 Y dil ) (4) T u,ref p ref where the subscript ref means the reference condition of 298 K and 1 atm. Y dil is the mass fraction of diluent. The SL,ref 0 and fuel-type independent exponents α and β are correlated as functions of φ as: S 0 L,ref = B M + B 2 (φ φ M ) 2 (5) α = (φ 1) (6) β = (φ 1) (7) Since Eq. (5) is invalid for very lean and very rich mixtures (by predicting negative flame speed values), which is unacceptable in modeling DI operating conditions, a formula proposed by Gülder [10] was adopted in this study: S 0 L,ref = ωφη exp( ξ(φ σ) 2 ) (8) Due to the relatively coarse mesh resolution in engine simulations, the growth of the ignition kernel is tracked by using the DPIK model [5], where the flame front position is marked by Lagrangian particles. The kernel growth rate is: dr k dt = ρ u (S plasma + S T ) (9) ρ k where r k is the kernel radius, ρ u and ρ k are the local unburnt gas density and the gas density inside the kernel, respectively. The plasma velocity S plasma is given as [5]: S plasma = Q spk η eff 4πr 2 k [ρ u(u k h u )+p ρ u /ρ k ] (10) 2

3 where Q spk is the electrical energy discharge rate, η eff is the electrical energy transfer efficiency due to heat loss to the spark plug. h u is the enthalpy of unburnt mixture, u k is the internal energy of the mixture inside the kernel. To account for turbulent strain and curvature effects on the kernel flame, the unstretched laminar flame speed SL 0 in Eq. (4) was multiplied by a stretch factor I 0, which takes the following form according to Herweg et al. [11]: I 0 =1 (l F /15l) 1/2 (u /SL 0 )3/2 2 (l F /r k )(ρ u /ρ k ) (11) where the second and third terms on the right hand side represent the contributions due to turbulent strain and due to the geometrical curvature of the kernel, respectively. Note that the mean curvature effects are also considered in the G-equation combustion model by the last term of Eq. (1). The transition from the kernel model to the turbulent G-equation combustion model follows the same criterion as the one used in the previous work by Tan and Reitz [5], i.e., the transition is controlled by a comparison of the kernel radius with a critical size which is proportional to the locally averaged turbulence integral length scale,viz., r k C m1 l = C m1 0.16k 3/2 /ɛ (12) where C m1 is a model constant. Although elements of the above G-equation description were originally developed for premixed flames, it is also successfully applied to partially premixed flames in the DI operating mode in this study. 2.2 Primary heat release within the turbulent flame brush In the present implementation of the G-equation model, it is assumed that after the flame front has passed, the mixture within the turbulent flame brush tends to the local and instantaneous thermodynamic equilibrium. Based on this assumption, an updated method is suggested to calculate the species conversion rate and the associated primary heat release at the flame front, viz., dρ i /dt = ρ u (Y i,u Y i,b )S 0 T A f,i4 /V i4 (13) where i4 is the computational cell index in KIVA [12]. Y i,u and Y i,b are the mass fractions of species i with respect to the unburnt and burnt mixtures, respectively. A f is the mean flame front area and V is the cell volume. Figure 1: Numerical descriptions of the turbulent flame structure and the flame containing cells. As shown in Fig. 1, in this method, in order to predict Y i,u for all the species in the flame-containing cells, the sub-grid scale unburnt/burnt volumes partitioned by the mean flame front are tracked based on geometrical information. As the mean flame front sweeps forward, the mixture behind the sweeping volume tends to local equilibrium following a constant pressure, constant enthalpy process. Y i,u can be calculated as follows: (1) Determine the equilibrium species mass fractions Y i,b and the equilibrium flame temperature. In this study, an element potential method-based code by Pope [13] was used for the chemical equilibrium calculation. (2) Calculate the burnt gas density and the burnt species densities based on the equation of state and the Y i,b from step (1): ρ i,b = Y i,b (pmw mix,b )/(RT b ) (14) where MW mix,b is the average molecular mass of the burnt mixture, and R is the universal gas constant. (3) Calculate the unburnt species densities ρ i,u based on species mass conservation: ρ i,u =(ρ i V i4 ρ i,b V b )/V u (15) (4) Finally, determine the unburnt species mass fractions: Y i,u = ρ i,u / ρ i,u (16) i In KIVA, the heat release rate due to the chemistry source term is directly related to the species conversion rate [12]. It needs to be noted that the four species associated with the NO X formation mechanism, i.e., NO,NO 2,N,andN 2 O, are excluded from the equilibrium calculation due to their relatively short residence time within the flame front, and the relative slow rate of the NO X chemical reactions. 2.3 Post-flame heat release and pollutant formation In this study, the computational cells ahead and behind the propagating flame front are modeled as Well 3

4 Stirred Reactors (WSR). A detailed PRF chemical kinetic mechanism was applied to account for the further oxidation of CO and other intermediate species, such as small hydrocarbon molecules and the species in the H 2 -O 2 system. To consider the effects of turbulent mixing, the reaction rates could be adjusted by considering the eddy turnover time as an additional timescale, and by combining this timescale with the kinetic timescale. However, this was not done in the present study, i.e., only kinetic rates were used. A nine-reaction reduced NO X mechanism was coupled with the hydrocarbon oxidation mechanism for predicting the formation of NO and NO 2 [14]. constants C m1 =2.0 andc m2 =1.0 were held fixed (see Eqs. (12) and (3)). The predicted in-cylinder pressure traces match the measured data well in terms of peak pressure and combustion phasing in the PFI operating mode with spark timing sweeps, as shown in Fig. 2. Figure 3 shows the evolution of the mean turbulent flame front ( G = 0 iso-surface) in the PFI mode with spark timing = -40 ATDC and engine speed 1500 rev/min). As seen, in the PFI mode, the flame propagates throughout the whole cylinder reaching the wall at about 20 ATDC. 3 Results and Discussion The engine studied is a Ford four-valve single-cylinder SI gasoline engine which features a pentroof combustion chamber and a converging-shaped piston bowl. The engine is equipped with both PFI and DI fuel systems. The DI system employs the wide-spacing arrangement, with a centrally mounted spark plug and an intake-side-mounted swirl-type injector. The test data used in this work was reported by Muñoz et al. [15]. The specifications of the engine and the modeled operating conditions are listed in Table 1. Table 1: Ford engine specifications and operating conditions [15]. Bore / Stroke 89 mm / 79.5 mm Compression Ratio 12 Engine Speed 1500 rev/min PFI Mode Spark Timings ( ATDC) -44, -40, -36, -32 Internal Residual 28% MAP 65kPa DI Mode Spark Timings ( ATDC) -32, -28, -24, -20 Internal Residual 6% MAP 75kPa End-of-Injection ( ATDC) -72 The computational mesh contains around 170,000 cells, including the intake and exhaust manifolds and the cylinder. A reduced 25-species, 51-reaction iso-octane mechanism [16] including the NO X reactions was used to model the post-flame chemistry and the low temperature chemistry in the end gas. CHEMKIN II was used to solve the detailed chemical kinetic equations. In all simulated cases, the model Figure 2: Measured (EXPT) and predicted (SIMU) incylinder pressure in PFI mode. (c) Figure 3: Evolution of the mean turbulent flame front (shaded G = 0 iso-surface) in PFI mode (Spark timing = -40 ATDC, Engine speed=1500 rev/min). Compared to PFI cases, it is more challenging to accurately predict the pressure evolution and heat release in the DI mode. The equivalence ratio of the stratified charged mixture varies from very rich to very lean. Therefore the laminar flame speed correlation needs to be reliable over a wide range of equiv- (d) 4

5 alence ratios, and also as a function of temperature and pressure of the unburnt mixture. As seen in Fig. 4, the pressure and heat release rate predicted by the present model agree with the measured data reasonably well for all spark timings. the dark lines denote stoichiometric conditions, the bright lines represent the flame front surfaces. According to the simulation, most of the NO is formed around the stoichiometric lines, while CO is mainly generated within the fuel rich region, as expected. Figure 6 shows the comparison of normalized engine-out NO X in both the PFI and DI modes. Although there are discrepancies in absolute values, the general trends as functions of spark timing are well captured. Figure 6: Measured and predicted engine-out NO X. PFImode;DImode. (c) (d) Figure 4: Comparisons of in-cylinder pressure and heat release rate in the DI mode. Figure 5: In-cylinder NO and CO mass fraction contours at 20 ATDC (DI, spark timing = -32 ATDC). Figures 5 and show the spatial distribution of NO and CO species mass fractions at 20 ATDC for the DI case with spark timing = -32 ATDC, where Figure 7: Calculated evolution of kernel radius. PFI mode; DI mode. During the calibration of the present ignition and combustion models, it was found that the stretch effects on the kernel flame, as described by Eq. (11), played an important role in the prediction of combustion phasing. Referring to Eq. (11), the stretch effect due to flame curvature (the last term on the right hand side) decays very quickly with the growth of r k, while the strain effect due to strong turbulence and a 5

6 thick laminar flame structure (the second term on the right side) might significantly delay the kernel flame s fast propagation, as in the PFI case with spark timing =-44 ATDC shown in Fig. 7. (Here for the first 10 CA from spark timing, the kernel flame was sustained mainly by the plasma velocity). Comparing Figs. 7 and, the stretch effects in the DI cases are generally less significant compared to those in the PFI cases, mainly because of the lower internal residual fractions and correspondingly higher unstretched laminar flame speeds in the DI cases. 4 Conclusions A G-equation-based turbulent combustion model was coupled with detailed chemistry, so that the postflame heat release and the low temperature heat release in the end gas could be modeled based on detailed chemical kinetics. A sub-grid scale burnt/unburnt volume-based method was developed for calculating the primary heat release and species conversion within the turbulent flame brush. Both the laminar and turbulent flame speed correlations were updated for more accurate descriptions of kernel flame evolution and turbulent flame propagation in stratified mixtures. A Ford SI Gasoline engine was modeled in both PFI and DI operating modes with spark timing sweeps. Good agreement with the measured pressure traces and engine-out NO X was achieved in all cases with fixed values of the model constants. References [1] F. A. Williams. Turbulent Combustion. SIAM, [2] N. Peters. The turbulent burning velocity for large scale and small scale turbulence. J. Fluid Mech., 384: , [3] N. Peters. Turbulent Combustion. Cambridge University Press, [4] M. Dekena and N. Peters. Combustion modeling with the G-equation. Oil & Gas Science and Technology-Rev., IFP, 54(2): , [5] Z. Tan and R. D. Reitz. Modeling ignition and combustion in spark-ignition engines using a level set method. SAE Paper , [6] J. Ewald and N. Peters. A level set based flamelet model for the prediction of combustion in spark ignition engines. 15th International Multidimensional Engine Modeling User s Group Meeting, Detroit, MI, [7] Z. Tan, S.-C. Kong, and R. D. Reitz. Modeling premixed and direct injection si engine combustion using the G-equation model. JSAE Paper , [8] Z. Han and R. D. Reitz. Turbulence modeling of internal combustion engines using rng k-e models. Comb. Sci. Tech., 106: , [9] M. Metghalchi and J. C. Keck. Burning velocities of mixtures of air with methanol, isooctane, and indolene at high pressures and temperatures. Combustion and Flame, 48: , [10] O. L. Gülder. Correlations of laminar combustion data for alternative s.i. engine fuels. SAE Paper , [11] R. Herweg and R. R. Maly. A fundamental model for flame kernel formation in s.i. engines. SAE Paper , [12] A. A. Amsden. A block-structured kiva program for engines with vertical or canted valves. Technical Report LA MS, Los Alamos National Lab, [13] S. B. Pope. Ceq: A fortran library to compute equilibrium compositions using gibbs function continuation. pope/ceq, [14] S.-C. Kong, Y. Sun, and R. D. Reitz. Modeling diesel spray flame lift-off, sooting tendency and nox emissions using detailed chemistry with phenomenological soot model. Proceedings of ASME ICES2005, [15] R. H. Muñoz, Z. Han, B. A. VanDerWege, and J. Yi. Effect of compression ratio on stratified-charge direct-injection gasoline combustion. SAE Paper , [16] S. Tanaka, F. Ayala, and J. C. Keck. A reduced chemical kinetic model for hcci combustion of primary reference fuels in a rapid compression machine. Combustion and Flame, 133: ,