Energy Fuels 2009, 23, 5383–5393 Published on Web 09/10/2009
: DOI:10.1021/ef900559x
Predictive Modeling of a Homogeneous Charge Compression Ignition (HCCI) Engine with EGR Fueled with Diesel Miguel Torres Garcı´ a,* Fco Jimenez-Espadafor Aguilar, Elisa Carvajal Trujillo, and Jose Antonio Becerra Villanueva Escuela Superior de Ingenieros de Sevilla, Avda. Camino de los Descubrimientos, s/n 41092 Sevilla, Spain Received May 31, 2009. Revised Manuscript Received August 21, 2009
In this article, a predictive model of an engine running in homogeneous charge compression ignition (HCCI) combustion mode with diesel fuel is presented and validated with experimental results. The predictive model behaves as a transfer function with four independent input variables; engine speed, air mass flow, fuel consumption, and constant pressure specific heat. Combustion pressure and NOx emissions are given as results. The model is based on several submodels that include intake flow characterization, the start of combustion from chemical kinetics, and a new heat release law applied in a unique volume. This new heat release rate for HCCI combustion mode allows reproducing combustion chamber pressure for any load condition, including EGR. The predictive model has been developed in a Matlab environment from 269 observations that cover the full operation range of the engine in HCCI combustion mode. Research is underway to estimate other relevant engine performance parameters, such as brake mean effective pressure, fuel consumption, or engine torque. investigation of HCCI combustion diesel fuel where the difficulty in achieving HCCI combustion without accessory methods is highlighted.6 Several potential control systems have been proposed to control HCCI combustion without changing the cetane number of diesel fuel: an intake charge heating system,7 internal exhaust gas recirculation (EGR),8 variable compression ratio (VCR),9 and variable valve timing (VVT)10,11 have been used to change the effective compression ratio and/or the quantity of hot exhaust gases retained in the cylinder. HCCI combustion is initiated by spontaneous auto ignition of multiple sites under high-temperature and high-pressure conditions. Environmental conditions including pressure and temperature and, also, temperature fluctuations of the charge inside the cylinders may be large enough to initiate local combustion in small volumes, which plays an important role in ignition timing.12,13 Then, the fuel’s physical and chemical properties, mixture components, and engine operating conditions (such as
1. Introduction Homogeneous charge compression ignition (HCCI) combustion integrates features of both spark ignition (SI) and compression ignition (CI) engines, obtaining a promisingly high efficiency in a diesel engine with virtually almost no NOx and soot emissions.1-3 At the same time, the huge responsibility of the injection system in the conventional diesel engine, which requires a considerable investment in research and development, is not needed in HCCI combustion mode.4 However, there are several problems blocking the road to successful integration of the HCCI concept in automotive and other applications such as power generation. The most important problems are the control of ignition timing and combustion duration at overall operating ranges and the operation at high loads.5 Nevertheless, because of its many potential advantages, substantial efforts have been made to understand diesel-fueled HCCI and to advance some approaches to commercial application. In the past decade, a considerable amount of research has been dedicated to the
(7) Qian, Z.-q.; Lu, X.-c. Characteristics of HCCI engine operation for additive, EGR, and intake charge temperature while using iso-octane as a fuel. J. Zhejiang Univ. Sci., A 2006, 7 (Suppl. II), 252-258. (8) Epping, K.; Aceves, S.; Bechtold, R.; Dec, J. The Potential of HCCI Combustion for High Efficiency and Low Emissions. SAE Tech. Pap. 2002, Paper No. 2002-01-1923. (9) Kim, M.; Kim, J. W.; Lee, C. S.; Lee, J. H. Effect of Compression Ratio and Spray Injection Angle on HCCI Combustion in a Small DI Diesel Engine. Energy Fuels 2006, 20 (1), 69–76. (10) Liu, H.; Yao, M.; Zhang, B.; Zheng, Z. Effects of Inlet Pressure and Octane Numbers on Combustion and Emissions of a Homogeneous Charge Compression Ignition (HCCI) Engine. Energy Fuels 2008, 22 (4), 2207–2215 (DOI: 10.1021/ef800197b). (11) Hiraya K. et al. A study of gasoline fuelled compression ignition engine;A trail of operation region expansion. In Proceedings of the JSAE Convention (in Jpn.), 2001; pp 9-14. (Paper No. 98-01) (12) Hultqvist, A.; Christiensen, M.; Johanson, B.; Richter, M., Nygren, J.; Hult, J. The HCCI combustion process in a single cyclehigh-speed fuel tracer LIF and chemiluminescence imaging. SAE Tech. Pap. 2002, Paper No. 2002-01-0424. (13) Chen, J. H.; Hawkes, E. R.; Sankaran, R; Mason, S. D.; Im, H. G. Direct numerical simulation of ignition front propagation in a constant volume with temperature inhomogeneities: I. Fundamental analysis and diagnostics. Combust. Flame 2006, 145, 128–144.
*To whom correspondence should be addressed. Phone: 0034954486111. Fax: 0034954487243. E-mail:
[email protected]. (1) Thring, R. H. Homogeneous Charge Compression Ignition (HCCI) Engines. SAE Tech. Pap. 1989, Paper No. 892068. (2) Kelly-Zion, P. L. et al. A Computational Study of the Effect of Fuel Type on Ignition Time in HCCI Engines. Int. Combust. Symp., 2000. (3) Liu, H.; Yao, M.; Zhang, Bo.; Zheng, Z. Effects of Inlet Pressure and Octane Numbers on Combustion and Emissions of a Homogeneous Charge Compression Ignition (HCCI) Engine. Energy Fuels 2008, 22, 2207–2215. (4) Chen, H.; Lu, X.; Huang, Z. New Reduced Chemical Mechanism for Homogeneous Charge Combustion Ignition Combustion Investigation of Primary Reference Fuels. Energy Fuels 2008, 22 (2), 935–944. (5) Zheng, J.; Yang, W.; Miller, D. L. Cernansky, N. P. A Skeletal Chemical Kinetic Model for the HCCI Combustion Process. SAE Tech. Pap. 2002, Paper No. 2002-01-0423. (6) Lund, C. M. A General Computer Program for Calculating TimeDependent Phenomena Involving One-Dimensional Hydrodynamics, Transport, and Detailed Chemical Kinetics. LLNL Report No. UCRL52504, August 1987. r 2009 American Chemical Society
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have a tendency to focus on the degree-by-degree variation of engine variables, generally in detail, and are hence generally slow-running. In this category, we can consider, with or without detailed chemical kinetics, single zone models,18 multizone models,19,20 and a computational fluid dynamics (CFD) approach.21-23 Conversely, empirically based models take a wider view, predicting the mean values or trends of the major engine variables, which allows shorter run times. Inside empirical models applied to internal combustion engines can be considered, such as polynomial methods,24,25 modal analysis,26,27 neural networks,28,29 or predictive modeling from steady-state maps.30 This is the approach that has been developed in this paper. Predictive modeling is a process used in predictive analysis to create a statistical model that has been developed to try to best predict an outcome. A usual application is for modeling processes (input-output relationship) from which there are not a simple or analytical physical model. Predictive analysis is the area of data mining concerned with forecasting probabilities and trends. A predictive model is composed of several predictors and variable factors that are likely to influence future behavior or results. To create a predictive model, data are collected for the relevant predictors, a statistical model is formulated, predictions are made, and the model is validated (or revised) as additional data become available. In this work, a predictive model of an engine running in HCCI combustion mode has been built. The objective of the model is to simulate fundamental engine results such as combustion pressure, heat-release rate, indicated mean pressure (IMEP), gas temperature, and exhaust gas emission from easily measured engine parameters. Some of these parameters are angular engine speed, fuel consumption, or intake temperature. The predictive model provides (1) an increase of information from a given series of tests, including the relative contributions of known factors (measured engine parameters), the interactions between them, and other uncontrollable effects; (2) the prediction of optimum engine performance with associated confidence limits, even when the optimum condition is not part of the test; and (3) reduced testing time and cost through a reduction in the number of tests.
Figure 1. Schematic procedure of the predictive model.
engine speed, load, and swirl) could play an important role in combustion duration and heat release rate (HRR) of HCCI combustion (according to Christensen et al.14). However, the influence of combustion chamber geometry and turbulence on the HCCI process is rather unexplored experimentally, especially with diesel fuel. Christensen et al.15 experimentally investigated the influence of turbulence on HCCI combustion in isooctane/n-heptane fuel mixtures, concluding that there is a lower combustion rate and an increase in combustion duration when the turbulence was increased. Kong et al.16 developed a simulation model based on KIVA-3 V coupled with a CHEMKIN chemistry solver to more deeply analyze the effect of turbulence on HCCI combustion, comparing simulation results with the above experiments. The model fit experimental combustion pressure and NOx emissions quite well; the authors concluded that in-cylinder flow turbulence affects HCCI combustion mainly through its influence on wall heat transfer. Zhang et al.17 studied the influence of wall temperature, swirl level, mixture inhomogeneities, top-ring-land crevice, and other variables on ignition timing and emissions in a PDF model (transported probability density function method) to establish trends and sensitivities rather than a quantitative comparison with experimental measurements. A high number of HCCI engine models have been developed by researchers, most of them based on a combination of analytical and empirical methods. Analytical-based models
(18) Duffy, K.; Fluga, E.; Faulkner, S.; Heaton, D.; Schleyer, C.; Sobotowski, R. Latest Developments in Heavy Duty Diesel HCCI. Which Fuels for Low CO2 Engines? Duret, P., Montagne, X., Eds.; Editions Technip: Paris, 2004; pp 79-87. (19) Easley, W. L.; Agarwal, A.; Lavoie, G. A. SAE Tech. Pap. 2001, Paper No. 2001-01-1029. (20) Ogink, R.; Golovitechv, V. SAE Tech. Pap. 2002, Paper No. 200201-1745. (21) Kong, S. S.; Reitz, R. D.; Christensen, M.; Johansson, B. SAE Tech. Pap. 2003, Paper No. 2003-01-1088. (22) Zhang, Y. Z.; Kung, E. H.; Haworth, D. C. Proc. Combust. Inst. 2005, 30, 2763–2771. (23) Liu, C.; Karim, G. A. Int. J. Hydrogen Energy 2008, 33, 3863– 3875. (24) Jiang, Q.; Gerpen, V. SAE Tech. Pap. 1992, Paper No. 920466. (25) Stronach, A. F.; Smith, R. J. Electr. Power Energy Syst. 1988, 10 (2), 123-129. (26) Boam, D. J.; Finlay, I. C.; Ma, T. H.; Wallace, S.; Bloomfield, J. H.; Lee, R. ImechE paper, 1993; Paper No. C465/031. (27) Ma, T. H.; Finlay, I. C. ImechE paper, 1993; Paper No. C465/030. (28) Shayler, P. J.; Darnton, N. J.; Ma, T. Predicting the fuel consumption of vehicles for drive cycles starting from cold ambient conditions. Presented at the EAEC 5th International Congress, Strasbourg, June 21-23, 1995; Paper No. SIA9506A27. (29) O’Reilly, P.; Thompson, S. Eng. Syst. Des. Anal. 1994, 64 (6), 191-198. (30) Watson, H. C.; Alimoradian, B. ImechE paper, 1989; Paper No. C382/098.
(14) Christensen, M., Johanson, B., Ammeus, P., Mauss, F. Supercharged Homogeneous Charge Compression Ignition. SAE Tech. Pap. 1998, Paper No. 980787. (15) Christensen, M.; Johansson, B. The effect of combustion chamber geometry on HCCI operation. SAE Tech. Pap. 2002, Paper No. 200201-0425. (16) Kong, S. S.; Reitz, R. D.; Christensen, M.; Johansson, B. Modeling the effects of geometry generated turbulence on HCCI engine combustion. SAE Tech. Pap. 2003, Paper No. 2003-01-1088. (17) Zhang, Y. Z.; Kung, E. H.; Haworth, D. C. A PDF method for multidimensional modeling of HCCI engine combustion: Effects of turbulence/chemistry interactions on ignition timing and emissions. Proc. Combust. Inst. 2005, 30, 2763–2771.
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This paper is organized as follows. Section 2 gives a description of the developed predictive model, problems underlying the modeling process and the experimental installation to obtain test results. Section 3 develops the predictive model theory and validation of the HCCI combustion mode. In section 4, both experimental and simulation results are presented to validate the model, and section 5 is the conclusion. 2. Material and Methods The scheme of the proposed methodology is represented in Figure 1 and has been divided in two blocks: A and B. Block A is based on a direct type of integral engine model that includes three submodels: (i) Intake flow model developed from an intake engine geometry. (ii) Start of combustion model, based on chemical-kinetic simulation with Chemkin code.32 (iii) A zero-dimensional combustion model based on a dedicated heat-release rate law, whose parameters have been adjusted through the fitting between measured and simulated combustion chamber pressure. Block B is a predictive model built from the best fitting parameters of the dedicated heat-release rate law and nonintrusive engine parameters as engine speed (given in units of rpm), fuel flow, or ambient temperature. 2.1. Experimental Study. The work has been developed in a modified diesel engine running in HCCI combustion mode. The specifications of the modified engine are given as follows: cylinder bore, 95 mm; stroke, 100 mm; displacement, 708 cm3; and compression ratio, 16:1. The test rig allows the intake engine temperature and the EGR ratio to be changed. All the details of the experimental procedure are explained in the work by Gordon.31 2.2. Intake Flow Models. The knowledge of in-cylinder temperature and trapped mass at intake valve closing is very important for a successful HCCI combustion model. A onedimensional CFD model was developed, based on mass conservation, momentum, and energy equations applied to each of the volumes into which the intake system was divided. The model is described in detail in the work of Torres Garcia et al.32 2.3. Start of Combustion Model. In HCCI engines, combustion is dominated by chemical kinetics reactions, with a minor influence from transport phenomena or injection characteristics. Good accuracy is achieved in combustion simulation using a substitution fuel of equal cetane index13 and detailed kinetic oxidation models.14 In this work, a zerodimensional kinetic model has been used to model the start of the combustion angle. The combustion chamber is considered to be a perfect mixture reactor with variable volume, with even pressure distribution, temperature, and concentration of the chemical species. Just before the start of combustion in a HCCI engine, a homogeneous mixture is compressed and the start of combustion occurs in the central zone of the cylinder. This model considers that the thermodynamic variables (pressure P, temperature T, volume V) is
Figure 2. Measured and predicted cylinder pressure traces. (Conditions: 1.3 10-5 kg/cycle fuel consumption, engine speed = 1800 rpm.)
only dependent on the crank angle. Therefore, the model does not include diffusive mass or heat models, with the mixture being perfectly homogeneous in the entire volume. In this model, the autoignition of the mixture happens in the entire volume in a simultaneous way, and, therefore, it predicts an extremely fast combustion. The model is composed by 1064 species and 3491 kinetic reactions. The expression of the evolution for the concentration of the i component (Ci) is η R X β φ d½Ci X ½Ci dV ¼ Kη ðTÞ P½Cφη KR ðTÞ P½CβR 1 1 dt V dt 1 1 ð1Þ Here, η is the number of reactions, and Ci is a product; each one of these reactions has φ reactives. The number of reactions where Ci is a reactive is denoted as R; each one of these reactions has β reactives, and among them would be Ci. The stoichiometric coefficients have not been included here to simplify the expression. In this expression, the first term corresponds to the Ci generation rate, the second term to the Ci consumption rate, and the last term is due to the variance of the volume. In this work, the values of the coefficients have been obtained from the high-pressure kinetic detailed model for paraffin that was reported by Curran et al.,33,34 in which the oxidation of any long hydrocarbon chain has, in place, in a series of steps that can be classified according to the elementary reactions that intervene in each one of them. 2.4. Parametric Law HRR. HRR curves in HCCI combustion mode reveal a very high heat-release rate, which causes a rapid increase in pressure in the engine cylinder.35 From this, combustion is characterized by a sudden steep pressure increase in the cylinder pressure curve and the peak in the HRR. After all the already-formed flammable mixture is burned, the HRR decreases until the end of combustion. (33) Curran, H. J.; Gaffuri, P.; Pitz, W. J.; Westbrook, C. K. Combust. Flame 1998, 114, 149–177. (34) Curran, H. J.; Gaffuri, P.; Pitz, W. J.; Westbrook, C. K. Combust. Flame 2002, 129, 253–280. (35) Torres Garcia, M.; Jimenez-Espadafor Aguilar, F.; Sanchez Lencero, T. M. Experimental Study of the Performances of a Modified Diesel Engine Operating in Homogeneous Charge Compression Ignition (HCCI) Combustion Mode Versus the Original Diesel Combustion Mode. Energy 2009, 34, 159–171.
(31) Gordon, P. B. The Design and Simulation of a 4 Stroke Engine; Society of Automotive Engineers: Warrendale, PA, 1999. (32) Torres Garcia, M.; Chacartegui Ramirez, R.; Jimenez-Espadafor Aguilar, F; Sanchez, L. Analysis of the Start of Combustion of a Diesel Fuel in a HCCI Process through an Integral Chemical Kinetic Model and Experimentation. Energy Fuels 2008, 22, 987–995 (DOI: 10.1021/ ef700541z).
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Table 1. Parameters of Simulated HRR: 1800 rpm -5
injected mass fuel ( 10
kg/cycle)
1.3 1.3 1.3
Φ
%EGR rate (mass)
K1
K2
Mp
Mpp
0.11 0.21 0.25
8 20 23
0.70 0.57 0.51
52.53 52.10 53.14
1.70 1.62 1.52
0.99 0.54 0.46
For all the fuel consumption rates, engine speeds, and EGR rates tested in HCCI mode, the HRR has shown the same development. Therefore, combustion is completely controlled by chemical kinetics. Because gas cylinder temperature is not sufficiently high, the fuel does not reach pyrolysis conditions, almost avoiding soot formation and drastically diminishing NOx formation.17 A model for the HRR which can be efficiently adapted to a combustion process controlled only by chemical kinetics is that of Wiebe.31 In this work, it has been found that modifying the Wiebe function31 has allowed the establishment of a new HRR law that fits quite well with the HCCI combustion process; this new HRR law is expressed in eq 1. 2 ! !Mp !Mpp þ1 3 dQ Qp θ θ 5 ¼ a1 ðMp þ1Þ exp4 -a2 dθ θp θp θp
Figure 3. Measured and predicted cylinder pressure and HRR in HCCI combustion mode. (Conditions: engine speed = 2100 rpm; constant fuel consumption = 2.75 10-5 kg/cycle; initial Φ = 0.57; and different EGR rates.)
ð2Þ where a1, a2, Mp, and Mpp are shape factors, θp is the duration of the energy release, and Qp characterizes the heat release in HCCI combustion mode. To reproduce the HRR, see eq 1; parameter Mp must be considered differently from parameter Mpp (this is the opposite of the original Wiebe function).31 Defining the terms: ! Qp ðMp þ1Þ K1 ¼ a 1 ð3aÞ p þ1 θM p K2 ¼ a2
1
Figure 2 compares simulated cylinder pressures with measured values derived from the experimental study. The modeled cylinder pressures shown have been obtained from the new HRR law for different mass flow and EGR; the engine speed is 1800 rpm and fuel consumption is constant (1.3 10-5 kg/ cycle). In terms of the area under the pressure curve between IVC and EVO, the maximum error is ∼1%. An interesting aspect to highlight is the good reproduction of pressure during the combustion process when an abrupt increase of pressure occurs. Table 1 shows the parameters of simulated HRR. Another case is analyzed at constant engine speed and fuel consumption (2100 rpm and 2.75 10-5 kg/cycle, respectively). Figure 3 shows measured and simulated results where good concordance can be observed. As can be observed, the new HRR law adapts perfectly to any EGR rate. From the minimum EGR rate (5%), each increment of the EGR rate has three effects on the HRR: (a) a delay in start of combustion, (b) a diminution of the maximum heat-release rate, and (c) an increase in combustion duration. The increase of torque is due to the diminution of combustion pressure along the compression stroke, which is due to the delay in start of combustion (effect (a)). Because of this power increase at the same engine speed and fuel consumption, there is also an improvement in the engine’s specific fuel consumption to the weight of effect (c) which reduces thermodynamic efficiency.36 Effect (b) produces a diminution in the combustion chamber temperature, so there will be a diminution of NOx emissions.37 Table 2 shows the parameters of simulated HRR.
!
pp þ1 θM p
It follows the HRR as dQ ¼ K1 θMp expð -K2 θMpp þ1 Þ dθ
ð3bÞ
ð4Þ
2.4.1. Combustion Model. In this work, a thermodynamic zero-dimensional model has been used to conduct the analysis and evaluation of the parameters of an analytical HRR law, following an iterative optimization process. The combustion pressure (Psimulated) is evaluated through a zero-zone model, which includes the following submodels: combustion chamber volume, thermal state equation, heat-release rate law (see section 2.4), the first principle of thermodynamics, the start of the combustion model,32 and heat losses.35 The equations of the thermodynamic model presented are solved numerically step by step, using a simple time-marching technique, and they give the estimated combustion chamber pressure (Psimulated). The theory-experimental model is direct type; the start is the pressure chamber curve and the end is the HRR curve. The time of computation can be increased on occasion, because during the optimization process, a direct model must be resolved in each iteration, and the rate of convergence is dependent on the initial conditions used and the accuracy. Some of the most decisive factors in computation time are the initial conditions.
(36) Chang, J. et al. New heat Transfer Correlation for an HCCI engine derived from measurements of instantaneous surface heat flux. SAE Tech. Pap. 2004, Paper No. 2004-01-2996. (37) Torres Garcı´ a, M.; Jimenez-Espadafor Aguilar, F. J.; Sanchez Lencero, T. Combustion characteristics, emissions and heat release rate (HRR) analysis of a homogeneous charge compression ignition (HCCI) engine with EGR fuelled with diesel. Energy Fuels 2009, DOI: 10.1021/ ef801010m.
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Table 2. Parameters of Simulated HRR: 2100 rpm -5
injected mass fuel ( 10
kg/cycle)
2.7 2.7 2.7 2.7
Φ
%EGR rate (mass)
K1
K2
Mp
Mpp
0.34 0.35 0.38 0.42
8 10 13 22 28
0.35 0.28 0.25 0.18 0.14
50.26 52.33 51.62 50.26 49.24
2.64 2.10 2.60 2.35 2.30
1.36 1.36 1.22 1.28 1.20
Table 3. Matrix of Experimental Tests engine speed (rpm)
injected fuel mass ( 10-5kg/cycle)
mCp range (J/K)
Φ
9 10 10 8 7 6 8 13
1200 1200 1200 1200 1200 1200 1200 1200
1.16 1.30 1.45 1.75 2.04 2.19 2.34 2.71
0.3-1.3 0.4-1.4 0.6-1.69 0.35-1.7 0.4-1.6 0.55-1.46 0.6-1.35 1.37-1.79
0.24-0.16 0.27-0.18 0.33-0.20 0.36-0.24 0.44-0.28 0.45-0.30 0.48-0.32 0.57-0.37
7 11 7 8 7 9 7 11
1500 1500 1500 1500 1500 1500 1500 1500
1.16 1.30 1.45 1.75 2.04 2.19 2.34 2.71
0.33-0.59 0.41-0.91 0.40-0.99 0.45÷0.93 0.42÷0.78 0.6-0.96 0.64-0.68 0.71-1.36
0.21-0.17 0.28-0.19 0.3-0.22 0.38-0.26 0.44-0.31 0.47-0.33 0.49-0.35 0.55-0.41
9 7 8 8 7 9 7 9
1800 1800 1800 1800 1800 1800 1800 1800
1.16 1.30 1.45 1.75 2.04 2.19 2.34 2.71
0.39-0.86 0.48-0.9 0.48-0.90 0.48-0.8 0.48-0.8 0.48-0.8 0.48-0.8 0.71-0.86
0.21-0.17 0.27-0.20 0.30-0.22 0.37-0.26 0.43-0.3 0.45-0.33 0.47-0.35 0.52-0.40
8 9 9 8 7 8 9 9
2100 2100 2100 2100 2100 2100 2100 2100
1.16 1.30 1.45 1.75 2.04 2.19 2.34 2.71
0.1-0.3 0.1-0.4 0.2-0.5 0.36-0.64 0.39-0.72 0.6-0.89 0.6-1.05 0.55-1.14
0.20-0.17 0.25-0.21 0.26-0.23 0.34÷0.30 0.40÷0.33 0.42÷0.35 0.64÷0.40 0.73-0.47
number of tests
Table 4. Best-Fit Surface Parameters from eq 4 b0 b1 b2 b3 b4 b12 b13 b14 b23 b24 b34 b11 b22 b33 b44
K1
K2
Mp
Mpp
NOx
-4.0952 5.2313 0.8912 3.9754 0.0842 -0.5549 -0.2384 -0.2144 -0.7740 -0.0090 -0.3597 -0.8527 0.1695 -0.1720 0.0473
14.2359 -4.6824 -7.4076 -13.1110 2.4544 1.9398 6.7862 -0.9351 3.8742 0.8278 -1.2349 0.1234 -0.7311 0.6875 -0.1530
-22.0995 21.2952 -2.1442 13.9503 1.0859 1.1436 -5.3949 -0.2438 1.4255 -0.0146 -0.2002 -4.7466 -0.2339 -3.1413 -0.0409
10.0419 -0.4448 -0.9893 -3.8443 -0.7110 0.3785 -0.7384 0.2979 1.4078 -0.0644 0.3255 -0.4338 -0.0105 -0.2842 0.0082
286.7489 -39.6406 13.6279 -552.5610 -4.2689
many different variables, such as intake air mass, fuel injection, engine speed, injection angle, intake temperature, EGR, and compression ratio. The influence of the injection angle has not been included in the study because all tests were done with a fixed injection angle of 45 BTDC. The compression ratio was fixed to the optimum 16:1.31 The influence of intake temperature was not considered, because all experiments were done at atmospheric temperature. Main engine characteristics are described in the work by Torres Garcia et al.32 This is a direct injection (DI) engine, with no swirl inlet flow and with very low internal swirl. The inlet valve is poppet type, without any shroud. In addition, engine speed in HCCI mode is very low (1200-2100 rpm), and, therefore, internal swirl, which is accelerated with an almost cubic law with engine speed, is not important. Thus, neither has been considered.37 3.1. Response Surface Design. Response surface designs are widely used in research as well as in industrial settings, sometimes, however, for very different purposes.38 The primary goal in scientific research is usually to show the statistical significance of an effect that a particular factor exerts on the dependent variable of interest. The design
3. Description of the Predictive Model An important problem in regression analysis is the selection of the number of variables, which allows one to obtain good results. These variables must be independent because, if not, the regression shows anomalies and confidence intervals for the regression will be large. Performance and exhaust emissions of internal combustion engines running in HCCI combustion mode are dependent on
(38) Montgomery, D. C.; Peck, E. A. Introduction to Linear Regression Analysis, 2nd Edition; Wiley: New York, 1992.
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Figure 4. Residuals result for the five model results.
The model proposed for fitting the observed dependent variable y from the independent xi includes main effects for factors x1, ..., xk, their interactions (x1x2, x1x3, ..., xk-1xk), and their quadratic components (x12, ..., xk2). This equation is capable of defining a variety of different shapes of response surfaces, such as a hill, a saucer, a saddle, or a rising ridge.
adopted looks for the estimation (fitting) of response surface, following the general model: y ¼ b0 þ b1 x1 þ ::: þ bk xk þ b12 x1 x2 þ b13 x1 x3 þ ::: þ bk-1, k xk1 xk þ b11 x1 þ ::: þ bkk xk
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Figure 5. Response surface results of predictive model corresponding to K1, K2, Mp, Mpp, and NOx.
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: DOI:10.1021/ef900559x
Torres García et al. Table 5. Regression Errors K1 Mp K2 Mpp NOx
Multiple regression is a powerful technique because it allows one to consider as many predictor variables as can be thought of, although only a few of them will turn out to be significant. One of the limitations of this technique is related to the relationship between the number of observations and variables. Most authors recommend having at least 10-20 times as many observations (cases, respondents) as variables; otherwise, the estimates of the regression line are probably very unstable and unlikely to replicate. This work has taken 269 observations and considered four variables ( j; j=1,2,3,4) for each test condition i: • Engine speed (rpm), denoted as xi,1 • Injected fuel mass, denoted as xi,2 • mC p x, denoted as i,3 (where mCp (given in units of J/K) is the product of the intake mass by mean specific heat of the gases mixture, evaluated at the temperature of start of combustion provided by the combustion model) (see Figure 1) • Air mass intake, denoted as xi,4 Matrix X (see Table 3) is formed by variables xi,j;i=1,2,...,269; j=1,2,3,4 and is the same for the entire fitting process. On the other hand, there are five dependent variables, K1, K2, Mp, Mpp, and NOx, which form vector Yr, which is formed by yi,r;i=1,2,...,269;r=1,2,3,4,5, with yi,1 = K1i, yi,2 = K2i, yi,3 = Mpi, yi,4 = Mppi, yi,5 = NOxi. 2 3 x1, 1 ... x1, 4 5 3 : X ¼4 3 x269, 1 ... x269, 4
Yi ¼ 4
3 y1, r
“leave one out”
“10-fold cross validation”
0.4 0.13 0.11 0.08 0.23
0.27 0.12 0.10 0.27 5.49
0.31 0.14 0.12 0.31 6.56
10-3, 105, 1, and 103. This is, the physical quantities of the four independent variables, once multiplied, are transformed to variables in the following ranges: (i) engine speed = (1.2-2.1) 103; (ii) fuel consumption = (1.1-2.71) 10-5; (iii) heat capacity=0.1-1.8; and (iv) air mass=(3.67.2) 10-3. These transformed variables are the ones that feed the mathematical model described by eq 5. The results of regression are shown in Table 4. Extended residual analysis is used for inspecting different residual and predicted values; therefore, to examine the adequacy of the predictive model, there is a need for a transformation of the variables in the model and the existence of outliers in the data. Residuals are deviations of observed values of the dependent variable from predicted values, given the current model. The models used to analyze responses to the dependent variable make certain assumptions about the distributions of residual (but not predicted) values of the dependent variable. These assumptions can be summarized by saying that the model assumes normality, linearity, homogeneity of variances and covariances, and independence of residuals .39 All of these properties of residuals for a dependent variable can be inspected using residual analysis. Residual representation shown in Figure 4 (expressed as a percentage) clearly reveals the adequacy of the regression model to the information, mainly because residuals have a null mean and are evenly distributed for the five dependent variables considered. The representation of the response surface is not possible in three-dimensional graphs because it is a hyper plane of four variables; for this reason, it is represented in two dimensions. Figure 5 shows response surface results from the predictive model that correspond to HRR parameters and NOx emissions. This figure shows a confidence interval of 95% in prediction and stands out; these intervals are thin, because of the mathematical procedure and adequate treatment of variables. Furthermore, experimental test density is wellstructured; otherwise, the confidence intervals would be wide and the predictive model would not show good results. The quadratic behavior of K1, K2, Mp, and Mpp versus the linear behavior of NOx emissions can be observed. This response surface allows knowing the HRR law parameters between (intermediates) that are tested. The response surface for NOx has the highest sensibility with mCp. NOx emissions decrease when mCp grows, which is behavior that is typical of the HCCI combustion mode.35
Figure 6. Schematic representation of the procedure used for predictive model validation.
2
RMSPE
5
y269, r Table 3 shows the experimental test input. A least-squares algorithm is applied to obtain parameters bij from eq 5, which define the equations of regression, equations of interaction, or quadratic equations. The following step is to obtain the regression that best adjusts to the information. To improve the results of the least-squares problem, dependent variables have been scaled. The purpose of this scaling is to make all the variables of a similar order of magnitude in the region of interest. The aim is to cause each variable to be of similar numerical weight during the mathematical procedure, which improves the numerical results. This way, engine speed, fuel consumption, heat capacity, and air mass have been multiplied, respectively, by factors of
4. Validation of the Simulation Predictive Model and Accuracy The predictive model has been implemented and simulated in the Matlab environment. To assess its capability of predicting combustion chamber pressure and NOx emissions, (39) Milton, J. S.; Arnold, J. C. Introduction to Probability and Statistics: Principles and Applications for Engineering and the Computing Sciences; McGraw-Hill: New York, 2003. (ISBN: 007246836X)
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: DOI:10.1021/ef900559x
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Table 6. Examples of HRR Law Parameters Obtained from Predictive Model NOx (ppm) fuel consumption ( 10-5 kg/cycle)
%EGR rate (mass)
2.54 2.54 2.54
2 15 34
1.24 1.24 1.24 2.45 2.45 2.45
Mp
Mpp
test
model
Engine Speed = 1200 rpm 0.35 39.74 0.68 51.83 0.72 67.3
1.23 1.64 1.79
0.23 0.31 0.53
380 136 42
369 149 55
5 18 35
Engine Speed = 1350 rpm 0.87 43.76 0.73 48.65 0.64 51.94
1.64 1.46 1.14
0.34 0.39 0.48
198 105 24
185 120 37
2 13 20
Engine Speed = 2000 rpm 0.61 52.86 0.31 52.66 0.15 2.63
2.50 2.52 4.49
1.23 1.10 1.07
256 179 85
270 185 99
K1
K2
Figure 7. Measured and estimated combustion chamber pressure in HCCI combustion mode. (Conditions: engine speed = 1350 rpm, constant fuel consumption = 1.24 10-5 kg/cycle, initial Φ = 0.31, with 13% mass EGR rate.)
simulation results have been compared with both experimental data obtained in an experimental study (see Figure 2) and those provided by the simulation predictive model. The schematic validation process is shown in Figure 6. The predictive model allows one to know the combustion pressure curve in the HCCI engine without using experimental results. The validation process is as follows: first, an operating condition is defined by the independent variables (engine speed, injected fuel per cycle, air mass intake, and mCp). With the predictive model, the parameters of the HRR law can be obtained, and by introducing this HRR in an inverse combustion model, the results provide the estimated pressure curve.40 These results have been compared with the experimental pressure curve obtained from the operating conditions in the first step. One of the statistical methods used to validate the model is known as cross validation. Cross validation is a process
that involves assessment of the predictive accuracy of a model in a test sample (sometimes also called a crossvalidation sample), compared to its predictive accuracy in the learning sample from which the model was developed. Cross validation can be simply used to estimate the generalization error of a given model, or it can be used for model selection by choosing one of several models that has the smallest estimated generalization error. With a small training set, it is possible that a subset smaller than the “best” subset may provide better generalization error. In k-fold cross validation, you divide the data into k subsets of (approximately) equal size. You train the net k times, each time omitting one of the subsets from the training set, but using only the omitted subset to compute whatever error criterion interests you. If k equals the sample size, this is called “leave-one-out” cross validation. However, if k gets too small, the error estimate is pessimistically biased, because of the difference in training-set size between the full-sample analysis and the cross-validation analyses. A k value of 10 is popular for estimating the generalization error.
(40) Yasar, H.; Soyhan, H. S.; Walmsley, H.; Head, B.; Sorusbay, C. Doueble-Wiebe function: An approach for single-zone HCCI engine modeling. Appl. Thermal Eng. 2008, 28, 1284–1290.
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: DOI:10.1021/ef900559x
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Figure 8. Measured and estimated combustion chamber pressure in HCCI combustion mode. (Conditions: engine speed = 2000 rpm, constant fuel consumption = 2.52 10-5 kg/cycle, initial Φ = 0.31, with 2%, 13%, and 20% EGR rates (mass).)
the estimation of the maximum pressure is ∼0.5% for the three EGR rates. It can be concluded that the precision in the estimation of combustion chamber pressure and NOx emissions from the predictive model is acceptable for applications concerning engine diagnostics, power estimation, and NOx predictions.
In this work, there have been “RMSPE”,41 “Leave one out”,42 and “10 fold cross validation”.43 Table 5 shows the parameters error obtained by the three different methods tested. The widely used measure is the prediction error sum of squares corrected (RMSPE), which P is defined as RMSPE ¼ ½ð ni¼1 ðyei -ypi Þ2 Þ=n1=2 because it gives error on a “per compound” basis, where yei are experimental values of the property, ypi are predicted values for the external validation test, and n is the total number of tests. In the “leave one out” and “10-fold cross validation” methods, the prediction error is defined aspredicted error ¼ MAX½yi -ypi ni¼1 where yi is the “value left out” variable and ypi denotes the predicted values of the model without this data. Table 6 shows experimental results and output from the predictive model for tests not included in Table 3. The HRR obtained from the predictive model is introduced in the inverse combustion model (without using any optimization iterative process) and the combustion pressure is obtained. Figure 7 shows the simulated pressure curve and the experimental one. In this figure, for the engine running at 1350 rpm, very good agreement is observed between both curves, because of good accuracy of the predictive model (versus experimental tests). The error in the estimation of the maximum pressure is
