Energy Fuels 2010, 24, 954–964 Published on Web 12/04/2009
: DOI:10.1021/ef901137j
Influence of a Combustion Parametric Model on the Cyclic Angular Speed of Internal Combustion Engines. Part II: Statistical Sensitivity Assessment Results Fernando Cruz-Perag on,*,† Francisco J. Jimenez-Espadafor,‡ Jose M. Palomar,† and M. Pilar Dorado§ †
Department of Mechanics and Mining Engineering, EPS, Universidad de Jaen, Campus Las Lagunillas s/n, 23071 Jaen, Spain, ‡ Department of Energetic Engineering, ESI, Universidad de Sevilla, Camino de los Descubrimientos s/n, Isla de la Cartuja, 41092 Sevilla, Spain, and §Department Chemistry Physics and Applied Thermodynamics, EPS, Edif. Leonardo da Vinci, Campus de Rabanales, 14071 Cordoba, Spain Received October 7, 2009. Revised Manuscript Received November 12, 2009
The most extended parametric form that describes the combustion process in an engine is based on the fuel rate of heat release (ROHR), derived from a mass burned fraction function (MBF), such as Wiebe’s function. When a combustion model is included in either identification or optimization processes, the obtained cyclic pressure and speed profiles must be very close to real values, thus helping to estimate the ROHR accurately. However, in most cases deep knowledge about the input parameters associated to the combustion model is not available, thus indicating the same system response (engine instantaneous angular speed) can lead to multiple solutions. In this work, a procedure to establish a relationship between the combustion model input parameters and the engine dynamic response (cyclic angular speed), next to the interrelation between input factors, is presented. In a previous work, the necessity of a sensitivity assessment has been justified. (Cruz-Peragon, F.; Jimenez-Espadafor, F. J.; Palomar, J. M.; Dorado, M. P. Energy Fuels 2009, 23, 2921-2929.) According to this, a design of experiment (DoE) including data from a test set provided by a single-cylinder direct injection compression ignition engine (CIE) has been carried out. Later, a statistical assessment has been performed to evaluate both the influence of the main parameters of the combustion model over the system response and the interrelation between parameters. Finally, some improvements about the parameters estimation procedure are discussed. It can be concluded that the equivalence ratio, the ignition timing for spark ignition engines or SIE (injection timing for CIE), and the form factor are parameters with high influence over the engine response. The proposed methodology helps to analyze engine combustion but needs adjustments to extend its application to other engines. However, the procedure demonstrates the effectiveness of this methodology if both adequate experimental designs and system model are used.
combustion chamber,11-15 the Wiebe’s parametric model that provides the mass burned fraction (MBF) corresponds to one of the most internationally extended.16 In this sense, many studies predict the correlations between some of the previous parameters.17,18 For these reasons, the goal of the present work is to evaluate the influence of the combustion parameters of the Wiebe’s function on the engine response (instantaneous angular speed). This will help to improve the existing indirect modeling
Introduction To improve the exhaust emissions and performance of internal combustion engines, the influence of the design parameters over the engine behavior must be identified.2 Depending on the purpose (design, control, or diagnosis), pressure cycle measurements and modeling may be required.3-10 Among the different procedures to model the combustion process into a *To whom correspondence should be addressed. Phone: þ34 953 212367. Fax: þ34 953 212870. E-mail:
[email protected]. (1) Cruz-Perag on, F.; Jimenez-Espadafor, F. J.; Palomar, J. A.; Dorado, M. P. Energy Fuels 2009, 23, 2921–2929. (2) Palomar, J. M.; Cruz-Perag on, F.; Jimenez-Espadafor, F. J.; Dorado, M. P. Energy Fuels 2007, 21 (1), 110–120. (3) Chamaillard, Y.; Higelin, P.; Charlet, A. Control Eng. Pract. 2004, 12 (4), 417–429. (4) Cruz-Perag on, F.; Jimenez-Espadafor, F. J. Energy Fuels 2007, 21 (5), 2600–2607. (5) Cruz-Perag on, F.; Jimenez-Espadafor, F. J. Energy Fuels 2007, 21 (5), 2627–2636. (6) Cruz-Perag on, F.; Jimenez-Espadafor, F. J.; Palomar, J. A.; Dorado, M. P. Energy Fuels 2008, 22 (5), 2972–2980. (7) Gu, F.; Jacob, P. J.; Ball, A. D. Proc. Inst. Mech. Eng., Part D 1999, 213 (D2), 135–143. (8) Jacob, P. J.; Gu, F.; Ball, A. D. Proc. Inst. Mech. Eng., Part D 1999, 213 (D1), 73–81. (9) Kalogirou, S. A. Prog. Energy Combust. Sci. 2003, 29 (6), 515–566. (10) Kegl, B.; Pehan, S. Thermal Sci. 2008, 12 (2), 171–182. (11) Barba, C.; Burkhardt, C.; Boulouchos, K.; Bargende, M. MTZ Motortech. Z. 1999, 60. r 2009 American Chemical Society
(12) Bonnier, J.; Tromp, C.; Klein, J. In Decoding Torsional Vibration Recordings for Cylinder Process Monitoring, 22nd CIMAC Congress, 1998; 1998; pp 639-649. (13) Chmela, F. G.; Orthaber, G. C., Rate of heat release prediction for direct injection Diesel engines based on purely mixing controlled combustion. SAE Paper 1999-01-0186 1999. (14) Connolly, F. T.; Yagle, A. E. J. Eng. gas Turbines Power 1993, 115, 801–809. (15) Connolly, F. T.; Yagle, A. E. Mech. Syst. Signal Process. 1994, 8 (1), 1–19. (16) Miyamoto, N.; Chikahisa, T.; Murayama, T.; Sawyer, R. In Description and analysis of diesel engine rate of combustion and performance using Wiebe’s functions, SAE Paper No. 850107, SAE International Congress and Exposition, Detroit, MI, March, 8-11, 1985; Detroit, Michigan, 1985. (17) Cruz-Perag on, F.; Jimenez-Espadafor, F. J. An. Ing. Mec. 2004, 15 (3), 1891–1896. (18) Roberts, J. B.; Peyton Jones, J. C.; Lansboroguh, K. J. Mech. Systems Signal Process. 2001, 15 (2), 419–438.
954
pubs.acs.org/EF
Energy Fuels 2010, 24, 954–964
: DOI:10.1021/ef901137j
Cruz-Peragon et al.
procedures (IM), based on either pressure measurement (IMP) or instantaneous angular speed measurements (IMV). This work follows the next sequence. In a first step, the relation between the instantaneous rate of heat release (ROHR) profile and the instantaneous angular speed is demonstrated. This study was successfully performed in a previous work.1 Additionally, when indirect methods are applied, the necessity of evaluating the influence of combustion parameters on the engine response (angular speed) to achieve unique solutions is justified. In this sense, a sensitivity analysis based on statistical methods provides an effective procedure to assess those relationships into the system model. A detailed revision about the cyclic profiles provided by both pressure and angular speed modeling can be found in the literature1 and is summarized as follows: an exponential-based combustion parametric model provides MBF (as described by Wiebe’s function) and calculates ROHR in the combustion chamber by deriving the MBF profile. ROHR can be determined by the measured pressure profile and the First Law of Thermodynamics, considering the chamber as a single homogeneous zone (zero-order model). This profile, as well as the input parameters (IP), can be obtained by either a direct method (DM) or IMP and provides an input signal to the dynamic engine behavior. In any case, the accuracy is enough to allow the consideration of the combustion parameters as empirical data. On the other hand, the cyclic angular speed curve can be modeled with a high degree of precision using a torque balance equation, considering also the input pressure signal to the dynamic system. An engine system model, composed by both pressure and dynamic submodels, is then defined. It then becomes easy to establish the cyclic speed profile of the engine shaft from the parametric combustion model input parameters (direct problem). On the other hand, an inverse problem could be proposed to determine the input parameters from the output speed measurements by IMV. However, this method does not reach the level of accuracy provided by DM or IMP procedures, thus leading to multiple solutions. To increase the degree of precision of the model, the relationship between input and output parameters must be approached. In this sense, sensitivity analysis via statistical methods helps to elucidate the effect of the combination of parameters into the system model. In the second part of the present work, a design of experiments (DoE) is carried out. Experiments have been performed in a test bench with a Deutz Dieter LKRS-A single-cylinder direct-injection compression ignition engine (DI CIE), as mentioned by Cruz-Perag on et al.1 A DM procedure provides the input terms or predictors of the combustion model, while the crankshaft instantaneous angular speed measurements help to establish the output indicators. In the third part of this work, the sensitivity analysis is performed by stochastic assessment. Considering the input combustion terms, a regression fitting function19 associated to each output variable is established. Then, the most suitable predictor variables for these equations are selected. In this sense, a forward selection (FS) procedure is developed.20 It ranks by importance the input parameters in the output
Figure 1. Indicators for a sensitivity assessment in an engine system: (a) Inputs: parameters of the ROHR profile in a DI CIE (j, equivalence ratio; qp, fraction of the fuel energy released in the premixed phase; θp, premixed phase period; θd, diffusion phase period; θj, injection timing; θi, ignition timing; mp, form factor of the premixed phase; md, form factor of the diffusion phase); (b) Outputs: instantaneous angular speed profile (ω) indicators (Δω1, angular speed increment before TDC; Δω2, angular speed increment after TDC; dωd1/dt, derivative of the first deceleration section; dωd2/dt, derivative of the second deceleration section; dωu1/dt, derivative of the first acceleration section; dωu2/dt, derivative of the second acceleration section); (c) control factors (n, mean angular speed; TL, load).
function, keeping only the main terms in the regression functions. At this stage, analysis of variance tests (ANOVA) for multiple predictors is used.21 Once all the output variables have been evaluated, the most repeated main parameters are then considered to be the main input parameters of the engine system model. Results will help to improve the combustion model for indirect modeling applications (such as design, diagnosis, or control). Moreover, it will help significantly reduce the number of input parameters to be identified, in order to obtain a unique solution by IM procedures. In this sense, a particular example has been included. Sensitivity Analysis Methodology 1. Design of Experiments and Steady-State Test. Due to lack of relevant knowledge about the empirical inputoutput data of the engine system, a new simple model is proposed to evaluate the influence of the combustion terms on the variation of the angular speed profile. If the output signal is required along the whole cycle it could complicate the evaluation, being useless. In this sense, the parametric expression of this new model may explain the variations of the output parameter (indicator of the speed profile) by linking it to changes in specific noise factors, such as the Wiebe’s model terms besides the equivalence ratio. These noise factors are disturbing sources, which are technically or economically difficult to control and can disrupt the ideal function. Thus, certain indicators related to the measured instantaneous angular speed profile have been adopted to provide an output parametric expression related to the instantaneous angular speed profile. Figure 1 shows the input and output parameters that are associated to a CIE engine system (more complex than a spark ignition engine or SIE). Analysis of a multicylinder engine could be hampered because of overlapping between the effects of different pressure curves over the system response (angular velocity). Furthermore, as the combustion model in a CIE is more complex than in a SIE, a
(19) Kleinbaum, D. G.; Kupper, L. L.; Muller, K. E., Applied Regression Analysis and Other Multivariable Methods; Duxbury Press, cop.: California, 1988; p 718. (20) Mason, R. L.; Gunst, R. F.; Hess, J. L., Statistical Design and Analysis of Experiments, with Applications to Engineering and Science, 2nd ed.; Wiley-Interscience: Hoboken, New Jersey, US, 2003.
(21) Canavos, G. C., Applied Probability and Statistical Methods; McGraw-Hill: New York, 1984.
955
Energy Fuels 2010, 24, 954–964
: DOI:10.1021/ef901137j
Cruz-Peragon et al.
single-cylinder CIE constitutes the ideal system to perform an exhaustive sensitivity analysis of the combustion parameters. Results can be extrapolated to SIE and multicylinder engines models. In the proposed engine system model, mp and md denote form factors (dimensionless) related to premixed and diffusive phases (subscripts p and d), respectively. Their corresponding periods (measured in crank angle degrees, CAD) are: premixed phase period (θp) and diffusion phase period (θd), so the overall duration of combustion (θC) matches: θC =θp þ θd (other authors16 identify θC as θd). During the premixed phase, an amount of energy is released (qp) and in addition to the energy released in the diffusion phase (qd), it can be considered that qp þ qd=1. The inclusion of these two energy terms leads to the definition of two exponential laws for the MBF and, consequently, their derivatives. In CIE, it is very important to take into consideration the ignition timing (θi) as follows: θi =θj - θD, considering the ignition delay (θD) and the start of the injection (θj). In SIE, only θi is considered. Finally, the fuel consumption per cylinder and cycle is evaluated throughout the equivalence ratio (j). The ROHR is obtained from the MBF derivative multiplied by the fuel consumption and the low heating value of the fuel (LHV). Concerning modern engines, a split of the fuel injection into several stages makes the combustion process more complex.11 In this case, the Wiebe’s function will require more than two exponential laws to be applied. In the case of HCCI engines, where early injection occurs, the ignition delay determination is essential.22 In any case, the methodology does not differ, with the only exception of a possible increase in the number of input parameters to be evaluated. A robust engineering design and development needs a parametric model in order to desensitize the system from noise factors, hence reducing the output variation. The best way to assess the relation engine system input-output parameters is by a sensitivity analysis. This analysis provides information from linear models, whereas comparison between multiple outputs and evaluation of the interaction between parameters remain unsolved. Alternatively, statistical analysis approaches can be used to elucidate the effect of parameters combination into the system model.23 A statistically designed experiment (DoE) is a planned sequence of tests that must be completed before any data analysis is carried out. It depends on the type of data that will be derived from the experiment, the model needed to represent the data, the degree of prediction accuracy required, and the available testing resources.23 It provides the different operating conditions of the engine to be tested, reducing the number of tests and including a wide range of load and speeds values. Systematic sampling is needed to develop a strategy for predicting the distributional characteristic of parameter space, required for the purpose of this work, that includes a second-order DoE. The parameters of the combustion model (see Figure 1) that cannot be controlled comprise noise factors of the system model. Only j, θi, average angular speed (n), and torque (TL) could be considered as control factors, acting over the inner characteristics of the
Table 1. Test Array of the Designed Experiment mean speed load minimum intermediate centered intermediate maximum
minimum intermediate centered intermediate maximum X X X
X X
X
X X
X
fuel injection-ignition system (FIS). However, the difficulty controlling the injection system during the tests leads to consider only n and TL as control factors during the performance of the experiments. Thus, although both parameters are linked to the engine response, they will not be considered as output parameters in this study. The study of a process or system must be focused on the relationship between the response and the input factors. If the number of input factors is reduced and their nature is quantitative, a response surface methodology provides an effective tool to study this relationship.24 Particularly, the central composite designs are the most commonly used second-order designs in response surface studies because it assesses the minimum number among the possible control variable combinations. In the present study, the controllable parameters of each test are load and mean angular speed. Also, it is necessary to describe a number of conditions concerning the central composite DoE. This design is composed of three parts and the sample space is divided by k control factors as follows: (i) A central point, where both factors (load and speed) are fixed to the mean values into their domain. It points the model tendency for the mean conditions. (ii) Star points (or axial points). One of the input factors takes the two extreme values into its domain. So, the total number of points to be considered is equal to 2k (in this case, 4). These values determine the effect that only one factor variation produces in the model. (iii) Cube points or corner points, located in the center of each semidomain. They form the called factorial portion of the design, equal to 2k (in this work, this is equal to 4). They allow the study of the effect of parameters interrelations into the assessed model. The cube points and some central points constitute a firstorder DoE. The rest of the components expand the design to a second-order DoE. The experiments design is composed by combinations of n and TL values between their extreme values, as shown in Table 1.17 The engine test bench included an electric dynamometer attached to a Deutz Dieter LKRS-A single-cylinder DI CIE.1 Engine behavior makes difficult to perform the linear combinations required in Table 1. However, case studies considering very low speed are not representative of usual engine performance, although a very high speed could lead to engine malfunction that could end with engine failure. For these reasons, the engine testing conditions were in the range of 1800-2600 rpm (n) and 0-34 N m (TL). To perform the predefined 9 conditions, an increased number of tests up to 12 were carried out. Chamber pressure and angular speed were measured for each operative cycle and predefined conditions. Later, combustion parameters were identified by a DM. Both injection timing (θj) and ignition timing (θi) were identified, as well as
(22) Yamane, K.; Shimamoto, Y. J. Eng. Gas Turbines Power - Trans. ASME 2002, 124 (3), 660–667. (23) Edwards, S. P.; Grove, D. M.; Wynn, H. P., Statistics for Engine Optimization; John Wiley & Sons, Inc.: 2000; p 208.
(24) Wu, C. F. J.; Hamada, M., Experiments: Planning, Analysis, and Parameter Design Optimization; 2nd ed.; John Wiley & Sons: New York, 2000.
956
Energy Fuels 2010, 24, 954–964
: DOI:10.1021/ef901137j
Cruz-Peragon et al.
assessing the predictive ability of a reduced model representative to the complete model is the comparison of the coefficients of determination (R2). If the adjusted coefficients of determination are approximately equal for the null and a reduced model, the two fits are considered to exhibit equal predictive ability. 3. Introducing an Indirect Modeling Improvement. The evaluation of the interdependency between input parameters and the procedure to minimize their number using IM methods is extremely important. Statistical analysis from previous section considers output variables those input parameters of the combustion submodel with less importance into the engine system model. So, the main input components are now the only input parameters to be evaluated. The difference with the previous section is the higher degree of precision and the use of a more complex nonlinear approach to the model functions, compared to eq 1. Many studies concerning combustion processes can be used to identify some combustion parameters.26 Miyamoto16 establishes the relationship between these terms with θj, obtaining second-order fitting functions with smooth dependencies, that constitutes an adequate starting point. In CIE, Arregle et al.27,28 analyze how the diffusive phase is developed from injection parameters and running conditions. In the present work, it can be used to characterize θd, md, and (1 - qp). Another contribution evaluates the ignition phenomena in iso-octane/air mixtures.29 Results can help to both refine the ignition theory in modern engines (especially in HCCI engines and the associated ignition delay time prediction next to the propagation of reaction fronts) and to characterize the parameters of the proposed parametric combustion model. HCCI models including the evaluation of the ROHR profile, θi and θC, have been proposed by Torres-Garcia et al.30 To find a semiempirical function that describes the less important terms of the system model, the use of modern space-filling experimental designs, such as Latin Hypercube Sampling designs are recommended. For this purpose, factorial experiments with different engine operating conditions (considering the controllable parameters n, TL, j, and θi) are required. Three replications of each test must be performed to estimate the experimental measurement-error variability. In the experiments, abnormal conditions such a misfire must be induced. So, the proposed procedure is able to provide correct fitting functions, even for abnormal conditions. Provided that outputs depend on several random sources of variation, a component of variance model can be used, instead of a parametric model.23 The main disadvantage of this approach is the high amount of requested experiments. For this reason, in the present work results from the previously described response surface experimental design of a DI CIE have been used. Thus, conclusions may be taken as indicative of the procedure to follow in futures studies.5,6
the ignition delay (θD). In this study, the start of combustion or ignition timing is considered the input parameter to be evaluated. This proposal constitutes a good approach for both CIE and SIE: in CIE, because the start of ignition can be well approached considering θj and the fitting for θD; on the other hand, θi is selected for SIE. Additionally, the equivalence ratio j can be easily estimated by measuring the fuel consumption at each test, besides engine geometric data and volumetric efficiency. Thus, the combustion parameters that belong to the Wiebe’s function can be determined, once j and θj are known. Later, results of both parameters were check out by means of an IMP method. 2. Statistical Assessment. Once the tests have been run, it is important to rank the input parameters for each output variable using statistical methods. The targets were to perform a regression model (for each output variable) based on a least-squares method and to select the main input parameters from the regression models.19 Considering the different approaches, the most repeated parameters were then considered as the main system model input parameters. For these approaches, an analysis of variance (ANOVA) for multiple parameters was used.21 In many statistical procedures, it is assumed that model errors are normally distributed. Thus, previous to assessment, values must be standardized to a mean value (μ) of 0 and standard deviation (σ) of 1, thus avoiding scaling discrepancies between different parameters.25 In this sense, seven input parameters and six output parameters (Figure 1) were considered to perform the sensitivity analysis (see ^ (with seven normalized Table 1). Thus, the input vector W ^ (with six normainput parameters) and the output vector Y lized output variables) were obtained.17 Flexible models that are able to fit different shapes of surfaces are preferred. The simplest and most used type of response model for the output variable Yj assumes that every input factor Wk has an independent linear effect on the response, as eq 1 shows, where linear coefficients are denoted by akj . Other functions, such as exponential, potential, or logarithmic, can be conveniently transformed, adopting linear forms as in eq 1. X akj Wk ð1Þ Yj ¼ a0j þ k
The goal of this assessment is to provide a procedure to evaluate the input-output parameters relations of the system model. In this study, as outputs values correspond to angular speed profile indicators, a qualitative study is more important than a quantitative one. For this reason, although the degree of precision of the model is low, the proposed linear expression suffices to reach the goal. At this stage, a FS procedure for variable selection starting with no predictor variables in the model was used. Variables are added one at a time until a satisfactory fit is achieved or until all predictors have been added. For this procedure the use of F-statistics is preferred. As the selection criteria are based on the principles of reduction of the root mean square error (RMSE), it is possible to measure the incremental contribution of a predictor variable above that provided by the model variables. The process ends when the addition of a predictor does not result in a statistically significant F-statistic.20 An often used complementary criterion for
(26) Ramos, J. I., Internal Combustion Engine Modeling; Hemisphere Publishing Corporation: New York, 1989. (27) Arregle, J.; Lopez, J. J.; Garcia, J. M.; Fenollosa, C. Appl. Therm. Eng. 2003, 23, 1301–1317. (28) Arregle, J.; Lopez, J. J.; Garcia, J. M.; Fenollosa, C. Appl. Therm. Eng. 2003, 23, 1319–1331. (29) Walton, S. M.; He, X.; Zigler, B. T.; Wooldridge, M. S.; Atreya, A. Combust. Flame 2007, 150 (3), 246–262. (30) Torres Garcia, M.; Jimenez-Espadafor, F. J.; Sanchez Lencero, T.; Becerra Villanueva, J. A. Appl. Therm. Eng. 2009, 29 (17-18), 3654– 3662.
(25) Gill, P. E.; Murray, W.; Wright, M. H., Practical Optimization; 11th ed.; Academic Press: London, 1997.
957
Energy Fuels 2010, 24, 954–964
: DOI:10.1021/ef901137j
Cruz-Peragon et al.
indicated input components, neglecting the others. As the results are standardized, values of RMSE and R2 can be compared. The evaluated output variables have a qualitative importance in this study. Thus, the degree of prediction accuracy required does not constitute a key factor, although it is mandatory to reach a minimum R2 of 0.9 and a maximum RMSE of 0.1. Then, it guarantees that the combustion terms will not exhibit a significant influence over the indicator dωd1/dt. It can be initially inferred that the most frequent parameters from Table 2 constitute the main components of the combustion model. Thus, the strong influence of some parameters over the output indicators j, θj, and mp can be observed from Table 2. Figure 5 shows the best-fitting function type for each case study. A high accuracy for first- and second-order polynomials is not required in this assessment. The parameter θi affects the rest of the output parameters and has the most significant influence on the engine response. Additionally, the first term j shows a high influence in the overall response of the engine (Δω2 and TL). The higher the fuel injected, the higher the oscillation in the crankshaft. It corroborates that the fuel injection system in CIE (and the ignition system in SIE) exhibits a high influence in the overall engine response. Finally, it is worth mentioning the important role that mp plays in the combustion process. It affects the initial instants of the combustion associated to the premixed phase, besides the period associated to the diffusion phase (it is the main parameter associated to the regression model of dωu2/dt). Although these relations are obvious, this study proposes a scientific methodology to evaluate their interactions. In a previous study, it was demonstrated that both terms j and θi (or θj) could be fairly well estimated by an indirect identification procedure (IMV).4,31 Nevertheless, the third parameter (mp) is directly related to the rest of terms, thus providing a good agreement with the pressure profile, but showing higher differences in relation with DM identification. It means that the number of input parameters initially considered as independent terms in the Wiebe’s function must be reduced, in order to ensure an adequate identification of this parameter. For this purpose, the inclusion of the partial correlation matrix is crucial. 3. Toward the Minimization of the Number of Input Parameters to be identified by IMV Procedures. In a few cases, results from IMV identification for input parameters are very accurate (Figures 9 and 10, Tables 3 and 4 in CruzPerag on et al.1). Nevertheless, in most cases, a strategy to improve the method, such as minimizing the number of combustion parameters to be identified by the IMV procedure, is required. For this reason, it is highly important to evaluate the influence of each term and the correlation between them. In this sense, it has been found that the previously mentioned three main terms (reduced to only one in case the SIE feeding and ignition systems are wellknown) of the engine system model can be accurately estimated by IMV. Thus, the rest of the terms (qp, θp, θd, and md considering a CIE, or form factor A and θC
Results 1. Preliminary Results. Initially, the simplest relationship between input and output parameters was analyzed. The only controllable factors in this test set were n and TL, according to eq 2 and as shown in Figure 2. The input surfaces were estimated by second-order polynomials in n and TL. Yð jÞ ¼ Yj ðn, TL Þ; j ¼ 1:::6 WðkÞ ¼ Wk ðn, TL Þ; k ¼ 1:::7
ð2Þ
Usually, experimental and simulated data are very close, with the exception of the data associated to θi and the premixed form factor (mp) (Figures 2c and 2e). Later, it will be demonstrated that both terms play an important role in the statistical system modeling. It will also corroborate the direct relation between TL and j (Figure 2a), as expected due to the qualitative performance of the injection pump. Results concerning output variables are depicted in Figure 3. Second-derivative indicators (Figures 3c-f) were rescaled from time domain (s-2) to crank angle domain (s-1), in order to present results with more clarity. It can be seen that response surfaces fit to real data with an acceptable accuracy. Differences are not relevant at this stage, because output variables correspond to indicators of the system, exhibiting more qualitative importance than quantitative. When the IMP procedure was carried out, θj showed values from 9.8 to 13.5° BTDC (before top dead centre). Additionally, j reached a maximum estimation error of 10% in relation with the experimental measurements of fuel consumption. This means that combustion efficiency was always above 90%. These findings demonstrate the robustness of the IMP method as a procedure to evaluate the combustion process in an engine cylinder. The sensitivity analysis, defined as the variation of Y depending on one parameter, can be obtained directly through the combination of results from Figures 2 and 3, although there are several disadvantages, that is, the impossibility to evaluate input parameter interrelations. At first sight, a preliminary sensitivity analysis considering the control factors TL and n could be inferred from the analysis of Figures 2 and 3. This first assessment shows the variation of each parameter in relation to the other factors. Nevertheless, the aim of this work is to evaluate how the input terms affect the engine response throughout the output indicators. Thus, the influence of control factors TL and n is not considered during the analysis. Additionally, for multivariate analysis, there are many possible combinations between parameters, whereas figures can only depict relations between each output factor and two predictors. Then, it is important to establish a complementary method to evaluate the sensitivity analysis, that is, via statistical methods. 2. Statistical Assessment for Sensitivity Analysis. The previous approach does not exhibit enough accuracy to determine the correlations between parameters. So, sensitivity assessment is required. A linear regression function for each one of the output indicators has been initially established. An example of the correlation curves between input parameters (standardized) and the all the output indicators is depicted in Figure 4. Later, the FS procedure will provide the most relevant parameters considering the different regression functions, ranked by importance, as shown in Table 2. That means the initial regression functions only depend on the
(31) Cruz-Perag on, F. An alisis de metodologı´as de optimizaci on inteligentes para la determinaci on de la presi on en c amara de combusti on para motores alternativos de combusti on interna por m etodos no intrusivos; Ph.D. Thesis; University of Seville: Spain, Seville, 2005.
958
Energy Fuels 2010, 24, 954–964
: DOI:10.1021/ef901137j
Cruz-Peragon et al.
Figure 2. Combustion parameters results from the test matrix. Inputs: n, mean angular speed; TL, load. Outputs: j, equivalence ratio; qp, fraction of the fuel energy released in the premixed phase; θp, premixed phase period; θd, diffusion phase period; θi, ignition timing; θD, ignition delay; mp, form factor of the premixed phase; md, form factor of the diffusion phase (dots: real data; grid: input surface approach).
results do not follow the tendencies observed by Miyamoto,16 thus another approach is needed.
considering a SIE) must be previously determined before their inclusion into the combustion submodel. However, 959
Energy Fuels 2010, 24, 954–964
: DOI:10.1021/ef901137j
Cruz-Peragon et al.
Figure 3. Instantaneous angular speed indicators results from the test matrix. Inputs: n, mean angular speed; TL, load. Outputs: Δω1, angular speed increment before TDC; Δω2, angular speed increment after TDC; dωd1/dt, derivative of the first deceleration section; dωd2/dt, derivative of the second deceleration section; dωu1/dt, derivative of the first acceleration section; dωu2/dt, derivative of the second acceleration section (dots: real data; grid: response surface approach).
md exhibits a low correlation with θp and mp, thus it could be considered to be independent from these two predictors. (iv) The specific energy released in the premixed phase (qp) is poorly correlated with both mp and θd. Previous considerations constitute the starting point to evaluate new approaches toward the definition of a combustion submodel from parameters totally or almost independent. It has been demonstrated that the main input parameters of the model are j, θj (for CIE or θi for SIE) and mp, which are also considered input terms of the combustion submodel and the objective data to be identified in diagnosis applications. To remove the rest of the terms in the identification process, it is important to establish the relationship between them and the three main components, even removing from the fittings functions the influence of control factors n and TL. Thus, it can be found a high
At this stage, it is necessary to focus on the partial correlation matrix. This matrix shows the interrelation between each two parameters, removing the effect of the rest of terms over the evaluated output.21 A zero value indicates that parameters are completely independent. Per contra, a value equal to 1 indicates their total dependency. Associated sign indicates the direct or inverse relation between both evaluated terms. The values that the triangular matrix shows for a linear exponential regression function is shown in Table 3. Other linear models provide similar results. From this table, the following remarks could be highlighted: (i) The equivalence ratio (j) is strongly correlated with the rest of the parameters, including θi. (ii) The duration of the diffusion phase (θd) is strongly correlated with the rest of the parameters, especially with md. (iii) The factor 960
Energy Fuels 2010, 24, 954–964
: DOI:10.1021/ef901137j
Cruz-Peragon et al.
Figure 4. Correlation between modeled and empirical standardized data considering: (a) Y = Δω1 ; (b) Y = Δω2 ; (c) Y = dωd1/dt; (d) Y = dωd2/dt; (e) Y = dωu1/dt; (f) Y = dωu2/dt; (solid line: measured values; dots: predicted values). Table 2. Main Input Components (MIC) of the Regression Functions, Ranked by Importance and Associated to the Output Parameters, Considering Different Linear Models function
linear
exponential 2
logarithmic
2
potential
output Indicator
MIC
R /RMSE
MIC
R /RMSE
MIC
R /RMSE
MIC
R2/RMSE
Δω1 Δω2 dωd1/dt dωd2/dt dωu1/dt dωu2/dt
θi θi , q p mp θi θi mp
0.85/0.15 0.98/0.015 0.67/0.44 0.9/ 0.1 0.91/0.09 0.81/0.2
θi θi , j θi θi θi mp, θd
0.9/ 0.1 0.99/0.01 0.72/0.36 0.93/0.07 0.95/0.04 0.85/0.16
θi θi , qp, j θi θi θi mp
0.8/0.22 0.97/0.03 0.66/0.47 0.89/0.11 0.87/0.14 0.86/0.15
θi θi ,j θi θi θi θi , mp, θp
0.82/0.22 0.975/0.022 0.7/0.39 0.92/0.08 0.9/0.096 0.9/0.1
At this stage, the procedure remains the same, although new regression functions are required, following the structure of eq 3, where output functions Y0 correspond to the combustion terms with less significance in the system model: qp, θp, θd, and md. These input terms are dependent on the others, considering the exponential laws of the parametric Wiebe’s model.
Table 3. Partial Correlation Matrix for Input Parameters in an Exponential Regression Function, Corresponding to Table 2 parameter j mp md qp θi θp θd
j
mp
md
qp
θi
θp
1.000 0.3444 -0.6464 -0.6522 0.7194 0.5551 1.000 -0.0986 -0.0485 0.1718 0.5073 1.000 0.4100 -0.7457 -0.0161 1.000 -0.4532 -0.2314 1.000 0.3005 1.000
2
θd -0.3228 -0.2117 0.80888 0.0572 -0.4628 0.3613 1.000
Y 0 ðjÞ ¼ y0j ðj, θi , mp Þ;
j ¼ 1:::4
ð3Þ
If a linear model is evaluated, such as eq 1, results provide a R2 value of less than 0.8, wherezas the RMSE value is higher than 0.25. The approach cannot be considered to exhibit an
dependency between qp, md, and j, being independent from mp. 961
Energy Fuels 2010, 24, 954–964
: DOI:10.1021/ef901137j
Cruz-Peragon et al.
^ and the main parameters in W ^ from Table 3: (a) Y = Δω1; (b) Y = Δω2 ; (c) Y = dωd1/dt; (d) Y = dωd2/ Figure 5. Correspondence between Y ^ j, equivalence ratio; θi, ignition timing; mp, form factor of the premixed phase). dt; (e) Y = dωu1/dt; (f) Y = dωu2/dt (Inputs of W:
coefficients.
acceptable adjusting of the system model. Nevertheless, the focus should be placed on the new partial correlation matrix, where a high independence between j and mp on one side, and between mp and θi on the other (with corresponding partial correlation coefficient less than 0.2 in any case), is found. A certain correlation exists between j and θi, because ignition timing depends on both the start of the injection and in-cylinder thermodynamic conditions, derived from the ignition delay θD. In this case, the degree of accuracy of the regression functions to be evaluated must be high, because new parametric functions for the physical engine model must be found. In this sense, a way to add flexibility to the model is to introduce extra terms on the right side of eq 1. As an example, a system with three input parameters (W1, W2, and W3) and a second-order polynomial response Y is shown in eq 4, where bj corresponds to
Y ¼ b0 þ b1 W1 þ b2 W2 þ b3 W3 þ b4 W12 þ b5 W22 þ b6 W32 þ b7 W1 W2 þ b8 W1 W3 þ b9 W2 W3
ð4Þ
As a result of the FS procedure, the main components are shown in Table 4. Different second-order polynomials have been defined, predicting the parameters (standardized values) with a high degree of accuracy. Response surfaces considering original values are depicted in Figure 6. As an example, Figure 7 and Table 5 show new approaches for indirect evaluation of the pressure profile at 2200 rpm and 17 N m, with a specific fuel consumption (BSFC) of 343 g/kWh, based on angular speed measurements (IMV). DM and IMP procedures provide similar results. An IMV method applied to this example uses a Newton-based optimization procedure in three cases where the number of identified 962
Energy Fuels 2010, 24, 954–964
: DOI:10.1021/ef901137j
Cruz-Peragon et al.
Figure 6. Correspondence between dependent and independent input parameters of the combustion model. Inputs: j, equivalence ratio; mp, form factor of the premixed phase. Outputs: qp, fraction of the fuel energy released in the premixed phase; md, form factor of the diffusion phase; θp, premixed phase period; θd, diffusion phase period (dots: real data; grid and solid line: fitting functions).
independent and easily controllable by the electronic control unit (ECU) of the engine. Its behavior is opposite to CIE, where uncertainties related to θD estimation make the start of combustion dependent on the injection process and thermodynamic properties of gases inside the cylinder. Parameters such as j, θj in CIE, or θi in SIE are not easily measurable. Nevertheless, when the engine system that controls these terms is practicable, it is very useful to measure and manipulate their values, being unnecessary their identification. In this case, there is only one term of the combustion submodel that needs to be identified, that is mp in CIE or m in SIE. As previously mentioned, results drawn out from the current analysis are only of application to the tested engine, because they are useless for other engines unless adjustments are provided. Even considering the last assessment, data correspond to healthy engine conditions. When the proposed procedure is applied, there are several advantages once the independent parameters are identified, such as the drastic reduction of the computation time. For diagnosis and control procedures, such as artificial neural network designs (ANN), the number of variables to be treated is low and the designed and trained architectures may be more robust. If an IMP procedure is applied, provided that partial correlation between them is reduced, the identification method (i.e., Newton-based methods or genetic algorithms) reaches a unique solution quickly.
IP differs: (a) all the input terms have been identified; (b) the three main parameters have been identified (j, θj, mp); and (c) both j and θj are considered as known data, so only mp has been searched. According to Figures 6 and 7, pressure and ROHR profiles fit better under this new situation, although results still miss accuracy. This is due to the difficulty to fit the dynamic model response of the system (instantaneous angular speed) to real angular speed measurements. Discrepancies in the pressure curve appear in the proximities of TDC, where uncertainties associated to the dynamic model exists. In any case, Table 5 shows that combustion parameters exhibit values near to the original ones (from IMP). Thus, the ROHR shape is closer to the target curve (considering 1- or 3-parameter-IMV) than before (using 7-parameter-IMV). Considering either known or unknown values of j and θj, no influence of the approach fitting has been found. Thus, it is extremely important to keep the approach including all main input terms. An additional advantage is the capability to diagnose abnormal combustion processes that leads to a partly unburnt fuel, no matter whether the injected fuel mass is known or not. The comparison of the effective value of j with the measured one provides the value of the combustion efficiency under these running conditions. This parameter allows engine failure diagnosis derived from combustion deficiencies, that is, low values correspond to combustion faults. When these considerations are extrapolated to SIE, the number of parameters is reduced. Furthermore, θi is 963
Energy Fuels 2010, 24, 954–964
: DOI:10.1021/ef901137j
Cruz-Peragon et al.
Figure 7. Results and comparison of indirect modeling procedures for combustion model identification in a DI CIE at 2200 rpm and 17 N m: (a) Measured pressure profiles; (b) measured instantaneous angular speed profile; (c) pressure profiles comparison; (d) ROHR profiles comparison.
Conclusions
cylinder CIE, results can be extrapolated to a SIE (showing a simpler combustion model). In the same way, this methodology can be used to analyze combustion parameters in models applied to modern engines, with split fuel injection. Results show that there are three main terms of this model that influence the engine response: the equivalence ratio (j), the ignition timing (θi) for SIE or injection timing (θj) for CIE, and the form factor (m, particularly mp in premixed phase for CIE). The two first terms have a higher influence over responses, whereas the third one presents a high correlation with other combustion terms. A strong interrelation exists between the rests of input factors (noise factors), which explains why different combinations provide nearly same results (pressure and angular speed profiles into a cycle), consistent with literature. If an objective of the diagnosis consists of the identification of combustion problems, it is important to evaluate the combustion parameters, as well as the pressure profile. The robustness fault derived from the correlation matrix leads to the proposal of a particular procedure that makes possible a unique solution combustion model. The rest of the input parameters are considered dependent from the previous three ones, thus helping to obtain their corresponding fitting functions. Results from Figures 6 and 7 are only useful for the analyzed engine and are not general. Nevertheless, the procedure demonstrates the effectiveness of this methodology if adequate experimental designs and system model are used.
The introduced sensitivity analysis allows the assessment of the influence of the parameters that conform the combustion model (input parameters) over the engine response (angular speed), as well as the partial correlations between inputs. Although the proposed procedure has been applied to a single
Acknowledgment. Authors gratefully acknowledge support for this research from the Spanish Ministry of Education and Science (ENE2007-65490/ALT and Integrated Actions Program, HI2008-0229) and from CICE, “Junta de Andalucı´ a”, Spain (PAIDI TEP 169 BIOSAHE and TEP-4994).
Table 4. Main Independent Parameters in Normalized Fitting Functions, Considering Input Parameters with Low Significance in the Engine System Model input parameter qp θp θd md
components
RMSE
R2
j2, mp j2, j 3 mp, mp, j j2, j j2, j
0.04 0.013 0.0085 0.019
0.96 0.9855 0.99 0.98
Table 5. Comparison of Combustion Parameters Values at Each Evaluated Situation for DI CIE at 2200 rpm and 17 N ma method parameter
DM-IMP
IMV 7 IP
IMV 3 IP
IMV 1 IP
j (dimensionless) mp (dimensionless) md (dimensionless) qp (dimensionless) θj (CAD BTC) θi (CAD BTDC) θp (CAD) θC (CAD) θd (CAD) AROHR (J/cycle)
0.59 1.26 1.79 0.56 10.1 3.1 22.4 53.4 31 812
0.67 4.00 1.22 0.25 12.9 5.9 17.0 43.3 26.3 923
0.54 1.2 1.68 0.64 13.4 6.5 22.5 51.1 29 737
0.59 0.5 1.59 0.45 10.1 3.1 23.7 55.7 32 812
a Direct method: DM; indirect method comparing pressures: IMP; indirect method comparing velocities: IMV; number of input parameters: IP.
964
