On the Influence of Composition on the Thermally-dominant Recompression Hcci Combustion Dynamics

A zero dimensional, mean-value, control-oriented model for recompression homogeneous charge compression ignition (HCCI) combustion with two discrete states representing temperature and composition dynamics is presented. This model captures steady state magnitudes and trends in combustion phasing, residual gas fraction, and mass flows caused by sweeps in valve timings, fueling rate, and fuel injection timing. It is shown that the coupling of the composition state with the mainly thermally-driven combustion dynamics causes competing slow and fast dynamics that shape the transient response of the phasing. A decoupled version of the model, where composition does not affect combustion phasing, is also developed in an effort to further simplify the model. This version matches the steady state fidelity of the coupled model, but has a qualitatively different dynamical behavior. Both models exhibit complex behaviors such as limit cycles at extremely late phasing. Both realizations are valid contenders as low order steady state representations of HCCI behavior. High-fidelity transient data will be necessary to further clarify the necessity of including composition effects on combustion phasing. NOMENCLATURE a/bTDC After / before top dead center CA## Crank angle at which ##% of fuel energy is released EVO/C Exhaust valve opening / closing HCCI Homogeneous charge compression ignition ibd Inert gas fraction of blowdown gases ic Inert gas fraction of charge before combustion IVO/C Intake valve opening / closing m f Mass of fuel NVO Negative valve overlap SOC Start of combustion ∗Address all correspondence to [email protected] SOI Start of injection pα In-cylinder pressure at event α Tα In-cylinder temperature at event α Tbd Temperature of blowdown gases xr Residual gas fraction ω Engine speed, in RPM INTRODUCTION A major challenge with homogeneous charge compression ignition (HCCI) engines is adequate combustion control. To enable model-based control approaches this contribution develops, validates, and analyzes control-oriented models. Previous low-order model structures [1–5] are extended to obtain satisfactory estimation results over a fairly extensive data set from a four cylinder 2.0 liter gasoline engine with several actuator sweeps at one engine speed. Specifically, the energy losses during combustion are modeled as a function of the states and a semiempirical model of the residual gas fraction is adapted from [2]. The influence of early fuel injection on combustion phasing is included through an Arrhenius expression. It is also shown that the charge composition influences the transient behavior due to the coupling between the composition and the thermal dynamics. This dynamic influence is investigated here, as earlier sensitivity analysis [6] showed that temperature at IVC is the dominant variable for capturing the steady state combustion phasing. Hence steady-state sensitivity is not sufficient to judge the necessity of including composition effects on combustion phasing. To this end, we compare here two models, one where both temperature and composition influence combustion and a second where only the charge temperature affects combustion. The latter involves the tuning of less parameters and captures steady state combustion phasing as accurately as the former. We also show that both models are capable of predicting limit cycle behavior at Proceedings of the ASME 2011 Dynamic Systems and Control Conference DSCC2011 October 31 November 2, 2011, Arlington, VA, USA 1 Copyright © 2011 by ASME Figure 1. INPUT / OUTPUT MODEL OVERVIEW conditions associated with late phasing. The dynamic response, however, of the two models is different. Namely, the temperature and composition influenced model exhibits overshooting response, whereas decoupling composition results in mostly damped behavior. Hence for effective controller design there is a need to understand and evaluate different model parameterizations that predict both steady state and transient behavior. MODELING PROCESS A control-oriented model that describes HCCI combustion must be sophisticated enough to capture all relevant physical trends, while being simple enough to enable effective controller design and analysis. This work deals with recompression HCCI [7], in which the exhaust valve is closed early and the intake valve is opened late, thus resulting in a negative valve overlap (NVO). The model developed is a discrete-time model with two states – (i) the temperature of the blowdown gases (Tbd), representing thermal dynamics and (ii) the inert gas fraction of the blowdown gases (ibd), representing composition dynamics. An input / output overview of the model is presented in Fig. 1. The actuator inputs considered are the negative valve overlap (NVO), the mass of fuel (m f ), the start of injection (SOI), and engine speed (ω). The valve timing is controlled by a cam phasing actuator with fixed cam profiles. The use of variable fuel injection timing shows promising results in HCCI combustion phasing control [5], both on a cycle-to-cycle and a cylinder-to-cylinder basis. Fuel injection in the recompression region causes thermal and chemical changes to the fuel due to the moderately high temperatures and pressures. The primary output of the model is combustion phasing. This is quantified by the location of CA10, CA50 and CA90, which are the crank angles at which 10%, 50% and 90% respectively of the total heat release occur. A number of other important outputs are calculated such as the residual gas fraction (xr), the in-cylinder Air-Fuel ratio (AFRc), and the in-cylinder pressure and temperature traces. Figure 2 shows a typical in-cylinder pressure trace for a recompression-based HCCI engine. Each engine cycle lasts for 720 ◦CA and is considered to end at EVO, that is after combustion finishes. In each cycle 0 ◦CA is considered to be at the combustion TDC. Figure 2 clearly shows the negative valve overlap resulting Figure 2. TYPICAL IN-CYLINDER PRESSURE TRACE in a recompression region of moderately high temperatures and pressures, into which the fuel is injected at the desired time. The parameters used in the model equations are fitted to experimental data. The nature of the dataset and the estimation results are presented in the section “Steady State Validation”. Residual Gas Fraction The mass of charge in the cylinder (mc) includes the mass of air and fuel inducted per cycle plus the residual mass (mres) trapped from the previous cycle. The residual gas fraction (xr) is defined as the ratio between mres and mc. A basis function for xr was developed for rebreathing-based HCCI in [2] by emulating the engine as a pumping device. This has been adapted to the residuals trapped in recompression HCCI with good results, and is given in Equation (1) xr(k)= ( α1 +α2 ( ω(k) 2000 )α3 NVO(k) )1+ α4 ( pem(k) pim(k) )α5 √ Tbd(k)  (1) where pim and pem are the pressures of the intake and exhaust manifold respectively. Composition in the cylinder The composition in the cylinder before and after combustion is shown in Fig. 3. At any point in the cycle, the charge in the cylinder is assumed to be a mixture of three components – air, fuel and inert gases. The chemical composition of these mixtures is determined from the stoichiometric combustion equation of the fuel considered. For example, from Eq. (17), which represents the stoichiometric combustion of one mole of iso-octane fuel, the constituents of the in-cylinder charge are considered to be: 1. Fuel (iso-octane) : C8H18 2. Air : O2 +3.773N2 3. Inert gases : 8CO2 +9H2O+47.16N2 2 Copyright © 2011 by ASME Figure 3. IN-CYLINDER COMPOSITION, BEFORE AND AFTER COMPLETE LEAN COMBUSTION The inert gas fraction (IGF), which is defined to be the ratio of the mass of inert gases to the total mass of charge, is used to account for composition effects. Similar to [2], the IGFs before combustion (ic) and after combustion (ibd) are related to each other by Eq. (2) and (3). ic(k) = xr(k)ibd(k) (2) ibd(k+1) = ( AFRs +1 AFRc(k)+1 ) (1− ic(k))+ ic(k) (3) where ibd(k) is the IGF of the blowdown gases of the previous cycle. The air-fuel ratio in the cylinder before combustion (AFRc) is described later by Eq. (5), while the stoichiometric AFR (AFRs) is assumed to be 14.7. The composition of the residual gases is assumed to be the same as that of the blowdown gases. The state update equation for ibd , presented in Eq. (4), can be derived from Eq. (2), (3), (5) and (7). ibd(k+1) = (AFRs +1)m f (k)R(xr(k)) pivc(k)Vivc(k) Tivc(k)+ xr(k)ibd(k) (4) where Eq. (10) and (9) give the dependence of Tivc(k) on Tbd(k). Relationship between ibd and χO2 Instead of ibd , the mole fraction of oxygen before combustion (χO2) can be equivalently chosen as the composition state. This is shown in Appendix A, where the relationship between ibd and χO2 is derived. AFR before combustion (AFRc) As external EGR is not used in this study, the fresh charge is assumed to consist purely of air. This considerably simplifies the comprehensive air charge model presented in [2]. The trapped residuals contain excess air that is left over after lean combustion. Both of these sources of air are considered while calculating AFRc in Eq. (5). Relative AFR in the cylinder before combustion (λc) is calculated in Eq. (6). AFRc(k) = (1− ibd(k)xr(k))mc(k) m f (k) −1 (5) λc(k) = AFRc(k) AFRs = (1− ibd(k)xr(k))mc(k)−m f (k) 14.7m f (k) (6) Mass of charge The mass of charge (mc) is estimated by the ideal gas law. Here the gas constant varies with xr. mc(k) = pivc(k)Vivc(k) R(xr(k))Tivc(k) (7) R(xr(k)) = 274(1− xr(k))+290xr(k) (8) State of Charge at IVC Estimating the state of charge at IVC is critical to the accuracy of combustion phasing prediction, as the actuators have no control authority after IVC. Pressure at IVC (pivc) can be approximated as a linear function of the intake manifold pressure. Temperature at IVC (Tivc) is calculated by a mass weighted average between temperatures of the fresh charge from the intake manifold (Tim) and the residual gases from the previous cycle’s combustion (Tr) as shown in Eq. (10). Thermal effects of early injection on the temperature of the residuals are captured in Eq. (9) based on fuel injection timing and mass. Here f1 (m f (k),SOI(k)) is a bilinear function that increases with larger and earlier fuel injection. This estimates the charge cooling effects of fuel vaporization, and possible heat release due to exothermic reactions. Tr = Tbd(k)− f1 (m f (k),SOI(k)) (9) Tivc(k) = xr(k)Tr +(1− xr(k))Tim(k) (10) Combustion Process The combustion is assumed to be complete if it is lean. For the effects of partial fuel burn at very late phasing or when close to stoichiometric levels, see [8, 9]. CA10 is considered to be the start of combustion (SOC) while CA90 is considered to be the end of combustion. The model is parameterized using experimental data to predict the temperature and pressure of the charge at important angles in the combustion process, such as CA10, CA50, CA90, and EVO. The intermediate regions between these points are assumed to be ideal polytropic or adiabatic processes to form complete pressure and temperature traces. The in-cylinder volume at any desired crank angle is given by standard cylinder geometry relations [10]. The combustion process starts at IVC and can be split into three phases – (1) a polytropic compression phase that leads to combustion, (2) a burn phase, and (3) a polytropic expansion phase followed by blowdown and exhaust. Phase 1: IVC to SOC The start of combustion in the HCCI combustion process is determined by an Arrhenius rate reaction, given by Eq. (11) and (12). This is motivated by work such as [11] and the expression developed for the ignition time 3 Copyright © 2011 by ASME delay for the autoignition of iso-octane in [12]. Sensitivity analysis of the autoignition integral in [6] suggests that temperature is the dominant factor in determining the start of combustion. Hence, Eq. (12) includes pressure and temperature effects only as a simplification in determining SOC. In [5], the effect of early fuel injection was captured by varying the Arrhenius threshold in the autoignition integral. A more physical strategy [13] considered in this study is to compute the autoignition integral given by Eq. (11) in the recompression region. This method is attractive as it provides a more physical basis function for parameterizing the phenomenon. Equation (11) is evaluated and the result is used as the initial value for the main combustion integral in Eq. (12), thus reducing the effective Arrhenius threshold. ∫ IVO SOI Arc p np c ( m f (k) mc(k) )n f exp ( −Ea,rc RTc ) dθ ω(k) = ξ (11) ∫ CA10 IVC A(pivc(k)υc ivc) np exp ( −Eaυc ivc