A Bayesian approach to the determination of ignition delay
Introduction
Standard methods for determining the start of combustion have changed little in the last few decades. Most current studies [1], [2], [3], [4], [5], [6], [7] that examine the start of combustion, or ignition delay, use the net rate of heat release, with most citing the 1988 book Internal Combustion Engine Fundamentals written by John Heywood [8]. This method is commonly used because it is considered reliable and the net rate of heat release is simple to calculate. However, this paper will introduce the use of a statistical model in the Bayesian paradigm to accurately determine the start of combustion.
Calculation of the net rate of heat release comes from analysing the heat losses in an engine from a first law of thermodynamics perspective, in its most commonly used form [8]:where, dQn/dt is the net rate of heat release, γ is the ratio of specific heats, p is the in-cylinder pressure, V is the in-cylinder volume and t is time. More complicated versions of Equation (1) exist that take into account heat loss to the walls, effects of crevice regions and other possible sources for heat loss—which are mostly engine specific and not general. The start of combustion is defined as the point when the net rate of heat release begins increasing rapidly—some authors use the point that the net rate of heat release becomes positive [6].
From experimental in-cylinder pressure, another method for determining the start of combustion is from the rate of pressure rise [9]. This method locates the point at which the rate of pressure rise begins to increase rapidly, and can be done by analysing either the first or second derivative of the in-cylinder pressure signal. It has parallels with the net rate of heat release, which also requires the differentiation of the in-cylinder pressure data.
In a recent study by Rothamer and Murphy [10], six methods of determining the ignition delay were compared. The six methods used were:
- 1.
location of 50% of pressure rise due to premixed burn combustion;
- 2.
extrapoliation of the peak slope of pressure rise due to combustion to the zero crossing point;
- 3.
location of the first peak of the second derivative of the pressure trace;
- 4.
location of the first peak of the third derivative of the pressure trace;
- 5.
location of 10% of the maximum heat release rate in the premixed burn; and,
- 6.
a repeat of (5) using a low-pass (threshold 2000 Hz) filtered in-cylinder pressure trace.
Their study focused on jet fuels and diesel fuel in a heavy-duty direct-injection single-cylinder diesel and the data analysis was performed using 250 cycles of averaged data. A conclusion from their study found that the methods which required second or third derivatives were not optimal owing to the presence of noise and that the ignition delay determined by the heat release method using the low-pass filtered in-cylinder pressure signal gave a result 200–330 μs shorter than the other methods.
An early method for estimating ignition delay was proposed by Hardenberg and Hase [11]. They developed an empirical relationship between the parameters which they determined had the most impact on ignition delay: mean piston speed (m/s), MPS; in-cylinder temperature at the time of injection (K), T; the compression ratio, rc; the polytropic index of compression, n; the cetane number, CN; and, the absolute charge-air pressure at the time of injection (bar), P. In crank-angle degrees the empirical relationship as determined by Hardenberg and Hase is:where, R is the universal gas constant (8.31434 J/mole). The polytropic index of compression, n, and the compression ratio, rc, impact on the temperature and pressure of the charge-air. Estimates of T and P can be obtained from the inlet manifold conditions [8], [11].
Later work done by Prakash et al. [12] extended this model to incorporate dual-fuel operation of diesel engines.
Since the work done by Hardenberg and Hase, other estimators of ignition delay based on engine parameters have been developed. Assanis et al. [13] have extensively reviewed these and proposed their own method for estimating ignition delay. However, for experimentally validating their ignition delay estimator, Assanis et al. compared their estimator to measured values by taking the peak of the second derivative to be the start of combustion.
Flame luminosity is another method used by researchers for determining the start of combustion. Heywood [8] argues that the use of flame luminosity detectors as a means to determine the start of combustion increases the potential for error. This is because the first appearance of the flame occurs after the increase in pressure. However, a recent study has argued that the first appearance of the flame coincides well with results from analysing the net rate of heat release [14]. Perhaps, with improving technology this method is becoming more reliable. Flame luminosity sensors are, however, prohibitively expensive for wide-spread practical use.
All of these approaches have practical difficulties or offer little, or no, information regarding cycle-by-cycle changes and hence do not allow for inter-cycle variability studies. In this paper a methodology for determining the start of combustion is proposed that requires no knowledge of difficult to estimate parameters such as heat loss to the walls and is still accurate with noisy data. Using only the in-cylinder pressure signal a statistical modelling approach is used to determine the start of combustion. A Bayesian approach to statistical modelling is used because it estimates the plausible range of parameter values (which includes the start of combustion), given the data observed [15]. In contrast a classical statistical analysis would provide the logical reverse: being estimates of the plausibility of the data, under specific (null or alternative) hypothesis of the parameter values. The latter would be better suited to confirmatory analyses where experimentalists wished to confirm whether parameter values took on specific values in a new situation. In this paper a new Bayesian modelling framework which provides posterior estimates of the start of combustion is given and is implemented across 4000 consecutive cycles at various engine loads to demonstrate its utility.
Section snippets
Experimental configuration
Experiments were conducted at the QUT Biofuel Engine Research Facility (BERF) in June 2011. Table 1 contains the technical specifications of the engine and data acquisition equipment. The engine was run at 2000 rpm on neat automotive diesel at full load (760 Nm) and at three quarters (570 Nm) and half (380 Nm) of full load.
Experimental data
Band-pass filtering was applied directly to the in-cylinder pressure signal, Fig. 1 is an example of the in-cylinder pressure signal, prior to digitising. This was achieved with a two channel analogue Krohn-Hite model 3202 filter, by first passing the in-cylinder pressure signal through a high-pass filter with a threshold of 4 kHz and then through a low-pass filter with a threshold of 20 kHz. The high-pass threshold was set this close to the frequency of interest, approximately 5–7 kHz, to
Determination of ignition delay
Ignition delay will be defined as being the time period between the start of diesel injection and the start of combustion [16]. Therefore, in order to determine the ignition delay it is important to obtain accurate knowledge of the diesel injection timing, detailed in Section 4.1, and the start of combustion, detailed from Section 4.3. In the engine setup under investigation in this paper nominal diesel injection timing was able to be determined by directly interrogating the signal sent from
Inter-cycle variability
An important justification for using the analysis methodology described in the previous section was to allow for cycle-by-cycle analysis of the data. As the Bayesian method is reliable with noisy data the need to cycle average is removed and individual cycles can be examined independently. Hence, the data can be investigated across large numbers of consecutive cycles—the results shown in the section are from 4000 consecutive cycles (4 min of data). This investigation was performed on a standard
Comparison to net rate of heat release
Traditionally the start of combustion was determined by examining a net rate of heat release plot [8]—the start of combustion is usually characterised to be the point at which the net rate of heat release becomes positive. Fig. 13 shows a net rate of heat release plot generated from the average of 4000 engine cycles at 2000 rpm and full load using Equation (1). On the opposing axis of Fig. 13 is the start of combustion distribution shown in Fig. 11, generated from the same 4000 engine cycles as
Conclusion
This paper has introduced a powerful new method for determining ignition delay. In contrast to standard techniques this method uses combustion resonance, in the form of band-pass filtered in-cylinder pressure, and Bayesian statistical modelling to accurately resolve the start of combustion. It was also demonstrated that this method allows for true inter-cycle variability studies and does not suffer from some of the issues surrounding the use of in-cylinder pressure directly to determine the
Acknowledgements
The authors wish to thank Mr. Anthony Morris for assisting with the design of the experimental campaigns and his invaluable technical knowledge and Mr. Noel Hartnett for his work in maintaining the engine for research. Further thanks also to Technologist Mr. Ken McIvor for his assistance in setting up the data acquisition software. This work was undertaken under an Australian Research Council Linkage Grant (LP0775178) in association with Peak3 P/L.
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