Available online at www.sciencedirect.com
ScienceDirect Procedia Engineering 157 (2016) 451 – 456
IX International Conference on Computational Heat and Mass Transfer, ICCHMT2016
Determining selected diesel engine combustion descriptors using the bootstrap method Andrzej Bąkowskia*, Leszek Radziszewskia, Žmindak Milanb a
Kielce University of Technology, Al. Tysiąclecia Państwa Polskiego 7, 25314 Kielce Poland b University of Žilina, Univerzitná 8215/1, 010 26 Žilina, Slovakia
Abstract This paper presents the method that uses the coefficient of variation (COV) of combustion chamber pressure to determine the crank angle degree (CAD) at which the rate of heat release (HRR) reaches its maximum. The results were obtained for a diesel engine running at full load on diesel. The bootstrap statistical method was used to find the crank angle degree of the peak rate of heat release. The uncertainty of the COV for the combustion chamber pressure was calculated. A good agreement between the experimental results and the literature data shows the validity of the analysis. © 2016 2016The TheAuthors. Authors.Published Published Elsevier © by by Elsevier Ltd.Ltd. This is an open access article under the CC BY-NC-ND license Peer-review under responsibility of the organizing committee of ICCHMT2016. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of ICCHMT2016 Keywords: Engines, heat release rate, signal descriptors, bootstrap method
1. Introduction Optimisation of modern diesel engine operation as regards its performance, fuel consumption or compliance with emission standards is forcing researchers to look for new algorithms which will ensure effective and easy control of the engine in real time [1]. One of the signals used to control the fuel injection process in modern diesel engines is pressure in the combustion chamber, ୡ Ǥ The knowledge of ୡ values in consecutive operating cycles allows determining parameters [2] such as mean indicated pressure, crank angle degree of the peak pressure ୡ or CAD at which half of the dosage of fuel is burnt. To calculate these parameters, it may be necessary to use super-computing microcontrollers in engine control systems and the algorithms which may be complicated and thereby insufficiently
* Corresponding author. Tel.: +48-41-34-24-444; fax: +48-41-34-42-997. E-mail address:
[email protected]
1877-7058 © 2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of ICCHMT2016
doi:10.1016/j.proeng.2016.08.388
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effective. Since the distribution of ୡ values recorded in consecutive operating cycles for the same crank angles is unknown, the accuracy of determining the quantities used in the control process is difficult to attain. In [3], the authors proposed the algorithm for calculating the start of combustion angle, ןୗେ , and for the crank angle at which the heat release rate reaches its peak, ןୌୖୖ . The algorithm uses the COV of pressure ୡ and is easy to implement. In order to find the value of ןୌୖୖ , the authors of [3] performed a theoretical analysis based on the thermodynamic model of combustion. The theoretical equation for ןୌୖୖ has the form: ௗమ ௗఈಲವ మ
ௗ మ
ௗఈಲವ మ
ൌ Ͳ ൌെ
ௗ ௗఈಲವ
ଵ
ఒ
ଵ
ଶ
ଶ
ଶ
ఒ
ቂߢ ߙ݊݅ݏ ߢ ߙ݊݅ݏ ܿߙݏ ߙ݊݅ݏ ߙ݊݅ݏ ܿߙݏ ቃ ଶ
ଵ
ଵ
ఒ
ଵ
ఒ
ఒ
ଶ
ଶ
ସ
ଶ
ଶ
ଶ
ቂ ܿߙݏ െ ଶ ߙ ቃ െ ߢ ቂ ܿߙݏ
ଶ ߙ െ ଶ ߙ ቃ
(1) ோ
where ߢ is the isentropic exponent, ܴ is the crank radius, L is the connecting rod length, and ߣ ൌ . The results obtained from the solution of equation 1 show a satisfactory agreement with those obtained based on the COV of the combustion chamber pressureǤ All algorithms presented in this work were implemented using the generally available statistical package R [4]. 2. Test stand and sample results The results presented in this article were obtained from the Laboratory of Internal Combustion Engines at the Kielce University of Technology [5]. The tests were carried out on a Perkins AD3.152 UR compression ignition engine. The detailed data of the engine are given in [5]. The combustion chamber pressure was recorded using an AVL QC34D piezoelectric sensor. The pressure ୡ ǡ was measured as a function of the crank angle with resolution of 1.4o, which gave 512 measurement points per cycle. The values from 50 full cycles were recorded for each operating condition. The results of the tests are presented in [5]. The tests used in this study assessed the full-load operation of the engine running on diesel at speeds from 1000 rpm to 2000 rpm. Figure 1a shows an example of pressure ୡ recorded during the first measurement cycle for the engine running at n=1600 rpm. a
b
Figure 1 Example of pressure recorded in the combustion chamber as a function of CAD for: a) the first operating cycle, b) bootstrap-determined pressures – engine run on diesel, running at 1600 rpm
3. Analysis of COVs Analysis of the results from the pressure measurements in the combustion chamber indicates that the values recorded for the same crank angle positions vary from cycle to cycle. This non-uniformity can be evaluated by
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analysing the coefficient of variation for standard deviation [3]. The coefficient of variation is defined using the following equation: ܸܱܥ ሺן ሻ ൌ
ఙ ሺןಲವ ሻ
(2)
ሺןಲವ ሻ
Where: ɐୡ ሺןେୈ ሻ – is the standard deviation of the combustion chamber pressure determined for the crankshaft position defined by ןେୈ, ୡ ሺןେୈ ሻ- is the mean value of pressure in the combustion chamber determined for ןେୈ . The method proposed in [3] is based on the analysis of ୡ ሺןେୈ ሻǡ performed to find the peak HRR angleןୌୖୖ ǡ and the combustion start angle ןୗେ . Figure 2 shows one of the methods for determining angles ןୗେ andןୌୖୖ . The angle at which the value of ୡ starts rising rapidly leading to the local extreme corresponds to the combustion start point (hereןୗେ ൌ ͵ͷͷǤͺ୭ ሻ. The angle at which the coefficient reaches its local maximum (ןୌୖୖ ൌ ͵ͷǤͳͻ ) corresponds to the angle of peak heat release rate. The pressure values recorded for the crankshaft position defined by angle ן are the only implementation of the random variable ሺן ሻ known to us. The knowledge of ሺן ሻ values allows determining statistical characteristics of this variable such as its expected value, standard deviation or the coefficient of variation. Estimators of these parameters are based on only one implementation of the variable ሺן ሻ, which contributes to lack of information about parameter estimation errors relating to the probability distribution of the variable ሺן ሻ. The use of the bootstrap statistical method based on the resampling with replacement from the only known implementation of the random variable ሺן ሻ allows obtaining additional information about its distribution. The parameter ܸܱܥ ሺן ሻis assumed to have the value of the estimator of the expected value determined on the basis of coefficients ܸܱܥ ሺן ሻ generated by sampling N times [6]: ଵ
ܸܱܥ ሺן ሻ ൌ σே ୀଵ ܸܱܥ ሺן ሻ ே
(3)
Where: ܸܱܥ ሺן ሻ – is the value of the COV determined for the i-th bootstrap sample. The standard error of determining the estimator of the ܸܱܥ ሺן ሻ parameter can be expressed as [6] మ σಿ సభሺைು ሺןಲವ ሻିைು ሺןಲವ ሻ ሻ
ܵܧ൫ܸܱܥ ሺן ሻ൯ ൌ ට
ேିଵ
(4)
In this study the number of replications of the variableୡ ሺןେୈ ሻ was taken to be N=1000. Table 1 shows script which was designed to use the bootstrap method for determining coefficients of variation for the combustion chamber pressure, ୡ ሺןେୈ ሻ, at the crankshaft positions defined by ןେୈ. Table 1 Bootstrap-determination of the combustion chamber pressure coefficients of variationܸܱܥ ሺן ሻǤ pc
