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The torque ratio concept for combustion monitoring of internal combustion engines
Control Engineering Practice 20 (2012) 561–568
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Control Engineering Practice journal homepage: www.elsevier.com/locate/conengprac
The torque ratio concept for combustion monitoring of internal combustion engines Ingemar Andersson n, Mikael Thor, Tomas McKelvey Department of Signals and Systems, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden
a r t i c l e i n f o
abstract
Article history: Received 17 June 2011 Accepted 23 December 2011 Available online 2 March 2012
This work presents a method to analyze combustion events in an internal combustion engine, called the torque ratio concept. The method is based on crankshaft torque measurements, but an extension to angular speed measurements is possible. The torque ratio concept provides a parametrized model for the combustion progress from which, e.g. combustion phasing can be extracted. The torque ratio concept is derived mathematically and related theoretically to other combustion analysis methods, such as pressure ratio and net heat release. Finally, analysis on recorded data from a five cylinder spark ignited engine verifies the relationships between the three methods. For combustion phasing, the 50% torque ratio is an equivalent measure to 50% pressure ratio and can be transformed into the 50% net heat release position by using a derived volume ratio function. & 2012 Elsevier Ltd. All rights reserved.
Keywords: System identification Automotive application Mechanical system Engine management systems Inverse filtering
1. Introduction One key step in closed loop combustion control of internal combustion engines is to estimate a value of the combustion property one wishes to control. Several sensors are subject to research to allow for more detailed measurements of the combustion events, such as in-cylinder pressure, ion current, crankshaft torque and flywheel angular speed. While pressure sensors offer a measurement that is closely connected to combustion theory, the torque and angular sensors offer robustness and cost benefits but require more signal processing to analyze combustion performance. Several methods do exist to describe the combustion process, e.g. burned mass fraction and cumulative heat release (Heywood, 1988; Gatowski et al., 1984; Rassweiler & Withrow, 1938) or pressure ratio (Matekunas, 1986). These methods are based on measurements of the cylinder pressure and are not applicable when the sensor is located on the crankshaft or flywheel. Crankshaft based sensors like the flywheel angular speed or the crankshaft torque sensor measure the combined effect of all cylinders attached to the crankshaft. To utilize such a signal for individual cylinder control, the effect of each cylinder action on the measured signal needs to be separated into individual cylinder torque contributions. Still, the reverse translation of the cylinder torque to cylinder pressure through the crank slider mechanism poses a difficult problem.
n
Corresponding author. Tel.: þ46 31 7721784; fax: þ46 31 7721782. E-mail addresses: [email protected] (I. Andersson), [email protected] (M. Thor), [email protected] (T. McKelvey). 0967-0661/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.conengprac.2011.12.007
This paper presents the torque ratio concept, a method to estimate combustion properties from an estimated individual cylinder torque contribution. The torque ratio concept has previously been presented in Andersson and McKelvey (2004b), Larsson and Andersson (2008) and is given here with a more theoretical analysis of the relationship to other combustion analysis methods such as pressure ratio and net heat release. Torque sensor measurements are shown as an introduction to crankshaft based torque sensing and how the separation of cylinder torque contributions work. However, as system properties are easier to isolate when using ‘‘ideal’’ input signals, the cylinder torque contributions are calculated from cylinder pressure to benefit the evaluation of the torque ratio concept. 1.1. Experimental equipment The engine under study is a five cylinder four stroke spark ignited engine. A torque sensor is mounted between the last crank throw and the flywheel and provides a measurement of the torque with a bandwidth of 3 kHz. The engine testbed is, besides the standard engine management system, also instrumented with spark plug mounted pressure sensors in all five cylinders. A one crank angle degree (CAD) resolution angular decoder is mounted on the free end of the crankshaft to provide an angle base sampling clock to the data acquisition system. Two data sets were collected, named A and B. Data set A was recorded at 0.27 bar intake pressure and 2000 rpm with the ignition angle sweeping from 45 to 10 CAD before top dead center (BTDC). In data set B sharp load transients are present. The standard engine calibration was used for the ignition timing and
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the load changed from 20% to 60% in sharp ramps back and forth, twice. During the load transients, the engine speed also changed by 720% from the nominal 2000 rpm due to the dynamic features of the engine dyno. Data set B ends with a load and speed reduction down to idle conditions. Data set A generated a wider range of combustion phasing variation than data set B. 1.2. Simulations
Tfp,i Tp,i
Tg,1
+
Tg,2
+
T1
T2 Trest
As part of the investigations, simulations of combustion events were used to compare the properties of the torque ratio function to the pressure ratio and the net heat release. The net heat release is represented by a Vibe function (Vibe, 1956), since it is known to make a fair representation of the spark ignited combustion: ( 0, y o ySOC xb ¼ ð1Þ mþ1 1eaðyySOC Þ , y Z ySOC Here, xb denotes the released fraction of the total energy, SOC denotes the start of combustion and fa,mg are the shape parameters (Heywood, 1988). For the simulation, the shape parameters fa,mg were calculated based on an assumption of a burn profile defined in Fig. 1. The shape parameters are calculated as lnð1x1 Þ ln lnð1x2 Þ ð2Þ m¼ lnðyd Þlnðyd þ yb Þ
a¼
lnð1x1 Þ
ð3Þ
ydm þ 1
where the flame development angle yd and the fast burn angle yb are the crank angle durations for the combustion of 0–10% and 10–85% of the total energy respectively. 1.3. Crankshaft torque measurements The crankshaft torque sensor measures the instantaneous combined effect of the torque contributions from all cylinders and the torsional deflections of the crankshaft itself. The upper graph of Fig. 3 shows an example of the measured crankshaft torque. The measured torque has a few very clear peaks distanced by 1441 which corresponds to the combustion in a five cylinder engine. Definition of burn angles
1 0.9 0.8
xb
0.7
x2
+ Tg,4
+
T4
+
Tm
^T
g,5
^ −
T5
−
Tm,5
Fig. 2. A schematic of the torque separation algorithm. Tm is the measured torque, Tg is the cylinder indicated torque, Tfp is the piston friction, Tp is the mass torque created by the reciprocating piston and H is the crankshaft model.
As combustion analysis is performed for each cylinder, the first step is to separate the measured torque signal into individual cylinder torque contributions. When completed, combustion analysis is performed on the estimated separated cylinder torque. The torque separation algorithm uses a model for the crankshaft which transforms the cylinder torque into the measured torque (Andersson & McKelvey, 2004a). The model also considers the torsional deflection effects which are present in the measurements. The cylinder torque separation is performed in two steps for every window of a firing cylinder, see Fig. 2. First, the combined effect on the measured signal from the previous combustion on the other cylinders in the last cycle is calculated using the crankshaft model. The result is subtracted from the measured signal. The remaining torque trace is filtered though the crankshaft model to form the estimated indicated cylinder torque. The lower graph in Fig. 3 shows the result from the separation, an estimate of the cylinder torque, compared to the actual cylinder torque which was derived from the cylinder pressure. The estimated cylinder torque has some residual oscillations which is a result of crankshaft model properties in the separation algorithm. Crankshaft modeling has been presented in Schagerberg and McKelvey (2003), Andersson and McKelvey (2004a) and McKelvey and Andersson (2006), and the separation algorithm was developed in Andersson and McKelvey (2004a).
0.6
2. Pressure ratio vs. net heat release
0.5
The torque ratio is derived from the pressure ratio, and since the net heat release is an established tool for combustion analysis it is interesting to analyze how the pressure ratio relates to the net heat release. This analysis will later be extended to the torque ratio as well. The net heat release rate is calculated from the cylinder pressure as (Heywood, 1988)
0.4 x1
0.3 0.2 θd
0.1 0
θb
dQ n ¼ 0
10
20
30 CAD
40
50
60
Fig. 1. Definition of burn angles for simulation. The flame development angle yd and the fast burn angle yb are connected to certain burn ratios x1 and x2.
g 1 p dV þ V dp g1 g1
ð4Þ
and the accumulated net heat release Q n ðyÞ is found by integrating Eq. (4). A motored cycle occurs when the combustion is omitted and the cylinder pressure pmot can be approximated by a polytropic
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The derivative of the pressure ratio with respect to crank angle y is
1997 RPM, 91 Nm
400
D
200
g1 d pcyl ðyÞ VðyÞ dmb dqw dm 1 ¼ qLHV þC dy pmot V0 dt dy dy |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
Nm
V r ðyÞ
0 raw torque
−200 −400
−300
−200
−100
0
100
CAD 400 est cyl trq real cyl trq
300 200 Nm
563
ð8Þ
dQ n ðyÞ
The volume ratio function V r ðyÞ puts an angle based weight to the net heat release rate dQ n ðyÞ when forming the pressure ratio. Fig. 4 shows the volume ratio function with a datum V0 at TDC and g ¼ 1:3. When forming the pressure ratio, the Vr function alters the net heat release rate dQ n ðyÞ and puts higher weight on the parts of the dQ n furthest from TDC. This introduces a shift where the pressure ratio rate peak is shifted away from TDC, and consequently the PR50 occurs later than the 50% point of the net heat release (Q n,50 ) for the normal post-TDC combustion phasing. The size of the shift depends on the phasing itself and the combustion duration. Fig. 5 shows a range of simulated combustion scenarios where the phasing of the Vibe function was shifted from early to late combustion and the combustion duration was varied. The duration
100
Volume ratio function 2.4
0
2.2
−100 −400
−300
−200
−100
0
100
CAD
2 1.8 Vr
Fig. 3. An example of torque separation at 2000 rpm, 91 N m. The upper graph shows the measured torque and the lower shows the estimated and real cylinder torque.
1.6
process with a polytropic coefficient of g as pmot ðyÞ ¼ p0
V0 g VðyÞ
1.4
1.2
ð5Þ
1 −200
V is the cylinder volume and 0 denotes conditions at a datum point, e.g. when the intake valves are closing. The volume V is a known function of crank angle and p0 can be estimated in several ways (Eriksson & Andersson, 2002). The concept of pressure ratio (PR) (Matekunas, 1986) relates the cylinder pressure pcyl from combustion to its corresponding motored pressure pmot as pcyl ðyÞ PRðyÞ ¼ 1 pmot ðyÞ
−150
−100
−50
0
50
100
150
200
[CAD] Fig. 4. The volume ratio function, normalized with the TDC volume.
PR50 deviation vs. Qn,50
4
dur=40 dur=70 dur=100
3
ð6Þ
PR50−Qn,50 [CAD]
2 The pressure ratio is a function that starts at zero and during the combustion increases to a higher value where it stays constant when the combustion is finished. There is a point in crank angle where the PR has reached half its final value, which is referred to as PR 50% or PR50 . A theoretical formulation of the cylinder pressure is derived in Appendix A, which combined with (6) arrives at
1 0 −1 −2
pcyl ðyÞ 1 PRðyÞ ¼ 1 ¼ D pmot
Z
y
y0
V g1 dmb dqw dm qLHV þC dt V0 dt dt dt
−3 ð7Þ
where dmb =dt is the mass burn rate of fuel, dqw =dt is the wall heat transfer rate, dm=dt is the change in total mass, D is a constant and C is a mass specific energy measure which depends mainly on the gas temperature. See Appendix A for more details.
−4 −100
−50
0 50 Qn,50 pos [CAD]
100
150
Fig. 5. The difference in the 50% point between the pressure ratio (PR) and the net heat release (Qn).
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(dur) and burn angles fyd , yb g for the three test cases are defined in Table 1. As seen, a combustion event with a longer duration shows a larger difference between the PR50 and the Q n,50 values than for shorter durations. Also, the PR50 and the Q n,50 deviates the most for combustion phasing around 30 CAD ATDC. The net heat release can be calculated from the pressure ratio function in the following way: 1. Calculate the pressure ratio function PRðyÞ as in (6) 2. Calculate the net heat release rate as PR
dQ n ðyÞ dPRðyÞ D ¼ dy dy V r ðyÞ
ð9Þ
3. The cumulative net heat release is Z y PR dQ n ðtÞ Q PR ð y Þ ¼ dt n d t soc
gas torque Tg on the crankshaft, according to T g ðyÞ ¼ ðpcyl ðyÞpcc ÞLðyÞA
ð11Þ
where pcc is the crank case pressure and A is the piston area. The geometric transformation of gas pressure to crankshaft torque can be described with a crank angle dependent effective lever, LðyÞ see Fig. 7. One straightforward way to extract combustion information from the estimated cylinder torque would be to reverse Eq. (11) and estimate the cylinder pressure. However, this is difficult since at the top dead center Lð0Þ ¼ 0 and the slightest noise in the estimated torque would make the inverse impossible. Instead, a method called the torque ratio was developed to address this problem. The torque ratio concept is inspired by the pressure ratio concept, which was presented in Matekunas (1986) (Fig. 8).
ð10Þ
Fig. 6 shows the PR50 vs. the Q n,50 and the PR based net heat
Pcyl
release Q PR n,50 vs. the Q n,50 . calculated from the data set. Fig. 6 shows that PR50 and Q n,50 are similar but not the same and the volume ratio function Vr explains the difference between them.
F=Pcyl A 3. The torque ratio concept The combustion process creates a gas pressure pcyl inside each cylinder that via the piston and connecting rod is transformed to a
Pcc
Table 1 Burn rate parameters for the simulation cases.
yd
yb
40 70 100
10 15 20
20 40 60
45
45
40
40
35
35
30
30
PR
25 20
25 20
15
15
10
10
5
5
0
0
20 Qn,50 [CAD]
Pcc A
Fig. 7. The effective lever L, created by the geometry of piston, connecting rod and crankshaft. The force on the piston is the net pressure pcyl pcc times the piston area A.
50
0
T T
50
−5
L
r
Qn,50 [CAD]
PR50 [CAD]
dur
l
40
−5
0
20 Qn,50 [CAD]
40
Fig. 6. From engine data set A; a comparison of 50% net heat release and PR. Left graph: PR50 vs. Q n,50 . Right graph: The Q PR n,50 from a PR curve adjusted with the volume ratio function Vr, vs. Q n,50 .
I. Andersson et al. / Control Engineering Practice 20 (2012) 561–568
Let
400
[Nm]
565
motored torque fired torque
200
Hp ðyÞ ¼ p0
V0 g LðyÞA VðyÞ
xðyÞ ¼ T g ðyÞT mot ðyÞ
0
Eq. (14) is re-written as −200 −100
−50
0
50
100
Crank angle [CAD]
xðyÞ ¼ Hp ðyÞ f TR ðyÞ Let (
2
Hðc, yÞ ¼
TR
1 TR50
0 −100
−50
0
50
100
Crank angle [CAD]
PRðyÞ ¼
2 6 HðcÞ ¼ 4
Hðc, y1 Þ
ð12Þ
So in theory, the torque ratio and the pressure ratio are identical, but calculated from different sensor sources. However, the torque ratio has a singularity at TDC where the lever is zero, since the estimated torque T g ðyÞ will never be completely noise free. This property prevents an easy calculation of the cylinder pressure by using (11), and hence use any pressure based combustion analysis algorithm. To solve the singularity problem, the torque ratio function is modeled by a parametrized mathematical function which is selected with the combustion type in mind. For spark ignited engines the model becomes ( 0, y o ysoc mþ1 f TR ðyÞ ¼ ð13Þ bð1eaððyysoc Þ=cÞ Þ, ysoc o y where SOC denotes start of combustion. The fTR function is a variation of the Vibe function in (1) and has four parameters fa,b,c,mg. The parameters a and m are the shape parameters and in this case they are set to constant values. Parameters b and c are left for optimization in least squares sense. Parameter b is the amplitude of the TR and c corresponds to the burn duration of the combustion. These two parameters hold all the information about each combustion. For diesel combustion, this model can be selected as a double Vibe function, which then represents two injection pulses per combustion (Thor, Andersson, & McKelvey, 2009).
3.1. Torque ratio parameter estimation When the fb,cg parameters have been estimated from the measured data, other combustion information like the phasing (TR 50%) can be calculated. Combining Eqs. (5) and (12) gives V0 g T g ðyÞT mot ðyÞ ¼ p0 LðyÞA f TR ðyÞ ð14Þ VðyÞ
y Z ysoc y o ysoc
ð16Þ
^ Hðc, yN Þ
3 7 5
The separable least squares problem is formulated as x ¼ bHðcÞ
pcyl ðyÞpcc þ pcc ðpcyl ðyÞpcc þ pcc ÞLðyÞA 1 1 ¼ pmot ðyÞpcc þ pcc ðpmot ðyÞpcc þpcc ÞLðyÞA
T g ðyÞ þ pcc LðyÞA T g ðyÞT mot ðyÞ ¼ 1 ¼ 9TRðyÞ T mot ðyÞ þ pcc LðyÞA pmot LðyÞA
Þ,
Form a sequence of N sampled data into a column array and the corresponding geometry functions of cylinder volume and piston lever likewise: 2 3 xðy1 Þ 6 ^ 7 x¼4 5 xðyN Þ
Fig. 8. Torque ratio for a single cylinder case, simulation.
The torque ratio function TRðyÞ is formed in the following way. Starting with (6) and (11), the expression is evolved with the crank case pressure pcc and effective lever LðyÞ as
mþ1
Hp ðyÞð1eaððyysoc Þ=cÞ 0,
1.5
0.5
ð15Þ
ð17Þ
Parameter c and b are found by standard methods for separable LS problems (Kay, 1994): ^ ¼ xT ½IHðcÞðHT ðcÞHðcÞÞ1 HT ðcÞx Jðc, bÞ
ð18Þ
^ c^ ¼ minJðc, bÞ
ð19Þ
b^ ¼ ðHT ðcÞHðcÞÞ1 HT ðcÞx
ð20Þ
c
The next step is to find the position where f TR ðyÞ attains 50% of its final value b: b ð21Þ TR50 ¼ y : f TR ðyÞ ¼ 2 The general solution is 1=ðm þ 1Þ ln 2 TR50 ¼ ysoc þ c a
ð22Þ
If the shape parameters a and m are chosen such that fTR becomes close to a symmetric function, TR50 can be approximated by TR50 ysoc þ
c 2
ð23Þ
4. Validation of torque ratio To validate the TR50 measure it is compared to the cylinder pressure based PR50 and 50% net heat release Q n,50 measures. Cylinder pressure data from the test engine was analyzed by the pressure ratio and net heat release algorithms and the corresponding cylinder torque Tg was calculated and analyzed by the proposed torque ratio algorithm. The torque ratio shape parameters were selected to match the burn profile as good as possible, with a slight favor towards the early phasing data. The parameter values were a¼5 m¼4 First, the TR50 is compared to the PR50 , and finally the TR50 is compared to the Q n,50 . Also, a derivation is presented on how to calculate the net heat release Qn from TR.
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4.1. Torque ratio vs. pressure ratio
4.2. Torque ratio vs. net heat release
Fig. 9 shows the comparison between the TR50 and PR50 phasing measures. The measure TR50 corresponds well to the PR50 measure. A deviation is seen for the most late combustion, due to that the shape parameters fa,mg were kept constant over the whole data range. A control algorithm based on the TR50 measure should have approximately the same target value for the combustion phasing as for a PR50 based algorithm. The deviation for late combustion can be addressed using phasing dependent shape parameters in the torque ratio function f TR ðyÞ.
The torque ratio relates to the net heat release in the same way as the pressure ratio does, except for the misfit of the burn rate model. The net heat release is calculated from the estimated cylinder torque in the following way:
1. Calculate the torque ratio function TRðyÞ by estimating the parameters (b,c) through Eq. (19) 2. Use parameters (b,c) to calculate the derivative of the torque ratio dTRðyÞ abðm þ1ÞðyySOC Þm aððyysoc Þ=cÞm þ 1 ¼ e dy cm þ 1
Torque Ratio vs. Pressure Ratio
50
Cycle−to−cycle TR50 vs. PR50
45
3. Calculate the net heat release rate as
1−to−1
TR
40
dQ n ðyÞ dTRðyÞ D ¼ dy dy V r ðyÞ
35 TR50 ref [CAD]
ð24Þ
30
ð25Þ
4. The cumulative net heat release is
25
Q TR n ðyÞ ¼
20
Z
y
TR
dQ n ðtÞ dt dt soc
ð26Þ
15 10 5 0 −5
0
10
20 30 PR50 [CAD]
40
50
50
50
45
45
40
40
35
35
30
30 Qn,50 [CAD]
25
TR
TR50 [CAD]
Fig. 9. Comparison of TR50 and PR50 measures from cylinder pressure data. The dashed line marks the 1:1 relation. The deviation at late phasing is due to less accuracy in the burn rate profile parameters fa,mg. Data set A was used.
Fig. 10 shows the result. It turns out that the misfit of the combustion model in the torque ratio approach compensates somewhat the deviation between the pressure ratio and the net heat release, and the torque ratio is very similar to the net heat release already. The TR50 differs by an average of 1.3 CAD from the Q n,50 in the data sets, and the torque ratio based Q TR n,50 shows an average bias of 0.2 CAD and a standard deviation of 0.7 CAD compared to the cylinder pressure based Q n,50 . The results obtained here can be compared with results from earlier experiments, where the TR50 measure was estimated from crankshaft torque sensor measurements on an engine in a dyno
20
25 20
15
15
10
10
5
5
0
0
−5
0
20 Qn,50 [CAD]
40
−5
0
20 Qn,50 [CAD]
40
Fig. 10. From engine data sets A and B combined; a comparison of 50% net heat release and TR. Left graph: TR50 vs. Q n,50 . Right graph: The Q TR n,50 from a TR curve adjusted with the volume ratio function Vr, vs. Q n,50 .
I. Andersson et al. / Control Engineering Practice 20 (2012) 561–568
(McKelvey & Andersson, 2006) and on an engine in a vehicle (Andersson, McKelvey, & Thor, 2008). The torque sensor based TR50 estimation showed an average bias of 3–4 CAD compared to the cylinder pressure based TR50 estimation for moderate excitation of the crankshaft dynamics ( r 3000 rpm). For severe excitation of the crankshaft dynamics ( Z 4000 rpm) the bias rose to 9 CAD. Hence, the cylinder separation accuracy is an important factor for the overall performance of crankshaft torque based combustion phasing estimation.
where R is the mass specific gas constant. It is calculated as R¼
The torque ratio concept is presented as a method to estimate combustion properties in an engine where the combustion sensor is located on the crankshaft or the flywheel of the engine. Standard combustion analysis tools as the burned mass fraction, heat release or pressure ratio are difficult to use due to the problems of reversely estimate a cylinder pressure from an estimated cylinder torque. The torque ratio is estimated as a parametrized function which is intended to resemble the combustion profile as good as possible. For a gasoline engine the parametrized function was chosen to be based on the well known Vibe function where the parameters of combustion amplitude and duration are estimated. The 50% point of the torque ratio is used as a measure for combustion phasing, TR50 . The TR50 is a monotonous function of combustion phasing and therefore a useful measure of the same. Further, the TR50 is close to the 50% pressure ratio, and the difference appears due to the limited accuracy of the torque ratio combustion model. The relationship between the pressure ratio and the net heat release is established in theory. A volume ratio function is defined and identified that connects the derivative of the pressure ratio to the net heat release rate. The relationship is also illustrated by engine data for a wide range of combustion phasing. It is shown that the torque ratio function can be used as a base for net heat release calculations. The phasing measurement TR50 turns out to deviate from the 50% net heat release with an average bias of 1.3 CAD, and the torque ratio based 50% net heat release has a standard deviation of 0.7 CAD and a bias of 0.2 CAD compared to the cylinder pressure based ditto.
Appendix A. Combustion pressure for the SI engine This is a derivation of the combustion pressure for a spark ignited engine on a closed form. The derivation is based on the description in Matekunas (1986) and is given here to support the derivations in Section 2 and forward. Consider a closed system where a gas resides. It has some pressure p, temperature T, volume V and mass m. The first law of thermodynamics states that a change in internal energy for a gas, dU, is the result of heat added to the gas, dQ, and the work that the gas performs on its environment, dW, and mass flow across the system boundaries dm: 0
dU ¼ dQ dW þ h dm
ðA:1Þ
0
where h is the enthalpy of the mass element dm at its source. Furthermore, the following relations hold: dW ¼ p dV
ðA:2Þ
dU ¼ mcv dT þ u dm
ðA:3Þ
Here, p is the gas pressure, dV is the volume change of the gas, m is the trapped gas mass, cv is the heat capacity of the gas and dT is the temperature change. The ideal gas law states that pV ¼ mRT
ðA:4Þ
R~ M
ðA:5Þ
where R~ is the universal gas constant and M is the molar mass of the gas. R is related to the heat capacities cv and cp of the gas and the ratio of specific heats g as R ¼ cp cv
ðA:6Þ
cp cv
ðA:7Þ
g¼ 5. Conclusions
567
The net heat dQ added to the system is formed by the added heat from the combustion and the heat transfer to the cylinder walls dQ ¼ qLHV dmb dqw
ðA:8Þ
The sum of the unburned mass mu and the burned mass mb is constant since no mass enters the system. Differentiate Eq. (A.4): V dp þ p dV ¼ mR dT þRT dm
ðA:9Þ
and re-write dU dU ¼
mcv ðV dp þ p dVRT dmÞ þu dm mR
ðA:10Þ
Form dU þ dW ¼
cv ðV dp þp dVRT dmÞ þ u dm þp dV R 0
dU þ dW ¼ dQ þ h dm ) dp þ g
ðA:11Þ ðA:12Þ
p R 0 dV ¼ ðdQ þh dmu dm þ cv T dmÞ V cv V
ðA:13Þ
The differential equation is solved for the pressure pðyÞ. Multiply by V g : V g dpþ V g g
p R 0 dV ¼ V g ðdQ þðcv T þ h uÞ dmÞ V cv V
V g dpþ gpV g1 dV ¼ V g1
R 0 ðdQ þðcv T þ h uÞ dmÞ cv
ðA:14Þ
ðA:15Þ
Note that the independent variable for p, V and Q is the crank angle y. Replace the differentials with their derivatives with respect to y. The left side of Eq. (A.14) is the differential of pV g : d R dQ dm 0 ðpV g Þ ¼ V g1 þ ðcv T þh uÞ ðA:16Þ dy c v dy dy The solution to Eq. (A.16) is written as Z R dQ dm 0 þ ðcv T þh uÞ dy pV g ¼ C þ V g1 cv dy dy y
ðA:17Þ
When the intake valve closes at angle y0 , the trapped gas conditions are noted p0, V0, T0 and m0. The mass remains constant through the compression and expansion phases in a pre-mixed combustion system as the Otto-engine. The added heat to the system, dQ, is zero before y0 since the gas mixture is fresh by then. If there would be no heat added to the system through the whole cycle, Eq. (A.17) is simply pV g ¼ C
ðA:18Þ
which is the expression for an adiabatic process. The constant C is easily found as g
C ¼ p0 V 0 Eq. (A.17) then becomes Z y R dQ dm 0 g V g1 þ ðcv T þh uÞ dt pV g ¼ p0 V 0 þ c v dt dt y0
ðA:19Þ
ðA:20Þ
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I. Andersson et al. / Control Engineering Practice 20 (2012) 561–568
Isolate p on the left side of Eq. (A.20) g Z y V0 1 R dQ dm 0 þ g V g1 þ ðcv T þ h uÞ pðyÞ ¼ p0 dt cv dt dt V V y0
References ðA:21Þ
Use Eq. (A.4) to express R in p0, V0, T0 and m0 and insert in Eq. (A.21) pðyÞ ¼ p0
! Z y V0 g 1 dm 0 g1 1 dQ dt 1þ V þ ðc T þ h uÞ v g1 V cv dt dt y0 T 0 m0 V 0
ðA:22Þ Define the motored pressure pmot as gu V0 pmot ¼ p0 V
ðA:23Þ
and combine with Eqs. (A.22) and (A.8) to get pðyÞ ¼ pmot
ggu V0 V
1þ
1 m0 cv T 0
Z
y
y0
! V g1 dmb dqw dm 0 dt qLHV þ ðcv T þ h uÞ dt dt V0 dt
ðA:24Þ Approximate g gu and set C ¼ cv T þh u
0
ðA:25Þ
D ¼ m0 c v T 0
ðA:26Þ
then (A.24) becomes ! Z 1 y V g1 dmb dqw dm dt pðyÞ ¼ pmot 1 þ qLHV þC dt dt D y0 V 0 dt
ðA:27Þ
Andersson, I., & McKelvey, T. (2004a). A system inversion approach on a crankshaft of an internal combustion engine. In: Proceedings of the 43rd IEEE conference on decision and control (pp. 5449–5454). Andersson, I., & McKelvey, T. (2004b). Torque ratio concept for combustion phasing detection of a spark ignited engine. In: Proceedings of the 43rd IEEE conference on decision and control (pp. 1703–1708). Andersson, I., McKelvey, T., & Thor, M. (2008). Evaluation of a closed loop spark advance controller based on a torque sensor. SAE Technical Paper Series 200801-0987. Eriksson, L., & Andersson, I. (2002). An analytic model for cylinder pressure in a four stroke SI engine. SAE 2002 Transactions: Journal of Engines, 726–733. Gatowski, J. A., Balles, E. N., Chun, K. M., Nelson, F. E., Ekchian, J. A., & Heywood, J. B. (1984). Heat release analysis of engine pressure data. SAE Technical Paper 841359. Heywood, J. B. (1988). Internal combustion engine fundamentals. McGraw-Hill. Kay, S. M. (1994). Fundamentals of statistical signal processing – estimation theory. Prentice Hall. Larsson, S., & Andersson, I. (2008). Self-optimising control of an SI-engine using a torque sensor. Control Engineering Practice, 16, 505–514. Matekunas, F. A. (1986). Engine combustion control with ignition timing by pressure ratio management. US Patent 4,622,939. McKelvey, T., & Andersson, I. (2006). System identification of the crankshaft dynamics in a 5 cylinder internal combustion engine. In: Proceedings of the 14th IFAC symposium on system identification (pp. 440–445). Rassweiler, G. M., & Withrow, L. (1938). Motion pictures of engine flames correlated with pressure cards. SAE Transactions, 42. Schagerberg, S., & McKelvey, T. (2003). Instantaneous Crankshaft torque measurements – modeling and validation. SAE Technical Paper Series 2003-01-0713. Thor, M., Andersson, I., & McKelvey, T. (2009). Parameterized diesel engine heat release modeling for combustion phasing analysis. SAE Technical Paper Series 2009-01-0368. Vibe, I. I. (1956). Semi-empirical expression for combustion rate in engines. In: Proceedings of a conference on piston engines (pp. 185–191).
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Control Engineering Practice journal homepage: www.elsevier.com/locate/conengprac
The torque ratio concept for combustion monitoring of internal combustion engines Ingemar Andersson n, Mikael Thor, Tomas McKelvey Department of Signals and Systems, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden
a r t i c l e i n f o
abstract
Article history: Received 17 June 2011 Accepted 23 December 2011 Available online 2 March 2012
This work presents a method to analyze combustion events in an internal combustion engine, called the torque ratio concept. The method is based on crankshaft torque measurements, but an extension to angular speed measurements is possible. The torque ratio concept provides a parametrized model for the combustion progress from which, e.g. combustion phasing can be extracted. The torque ratio concept is derived mathematically and related theoretically to other combustion analysis methods, such as pressure ratio and net heat release. Finally, analysis on recorded data from a five cylinder spark ignited engine verifies the relationships between the three methods. For combustion phasing, the 50% torque ratio is an equivalent measure to 50% pressure ratio and can be transformed into the 50% net heat release position by using a derived volume ratio function. & 2012 Elsevier Ltd. All rights reserved.
Keywords: System identification Automotive application Mechanical system Engine management systems Inverse filtering
1. Introduction One key step in closed loop combustion control of internal combustion engines is to estimate a value of the combustion property one wishes to control. Several sensors are subject to research to allow for more detailed measurements of the combustion events, such as in-cylinder pressure, ion current, crankshaft torque and flywheel angular speed. While pressure sensors offer a measurement that is closely connected to combustion theory, the torque and angular sensors offer robustness and cost benefits but require more signal processing to analyze combustion performance. Several methods do exist to describe the combustion process, e.g. burned mass fraction and cumulative heat release (Heywood, 1988; Gatowski et al., 1984; Rassweiler & Withrow, 1938) or pressure ratio (Matekunas, 1986). These methods are based on measurements of the cylinder pressure and are not applicable when the sensor is located on the crankshaft or flywheel. Crankshaft based sensors like the flywheel angular speed or the crankshaft torque sensor measure the combined effect of all cylinders attached to the crankshaft. To utilize such a signal for individual cylinder control, the effect of each cylinder action on the measured signal needs to be separated into individual cylinder torque contributions. Still, the reverse translation of the cylinder torque to cylinder pressure through the crank slider mechanism poses a difficult problem.
n
Corresponding author. Tel.: þ46 31 7721784; fax: þ46 31 7721782. E-mail addresses: [email protected] (I. Andersson), [email protected] (M. Thor), [email protected] (T. McKelvey). 0967-0661/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.conengprac.2011.12.007
This paper presents the torque ratio concept, a method to estimate combustion properties from an estimated individual cylinder torque contribution. The torque ratio concept has previously been presented in Andersson and McKelvey (2004b), Larsson and Andersson (2008) and is given here with a more theoretical analysis of the relationship to other combustion analysis methods such as pressure ratio and net heat release. Torque sensor measurements are shown as an introduction to crankshaft based torque sensing and how the separation of cylinder torque contributions work. However, as system properties are easier to isolate when using ‘‘ideal’’ input signals, the cylinder torque contributions are calculated from cylinder pressure to benefit the evaluation of the torque ratio concept. 1.1. Experimental equipment The engine under study is a five cylinder four stroke spark ignited engine. A torque sensor is mounted between the last crank throw and the flywheel and provides a measurement of the torque with a bandwidth of 3 kHz. The engine testbed is, besides the standard engine management system, also instrumented with spark plug mounted pressure sensors in all five cylinders. A one crank angle degree (CAD) resolution angular decoder is mounted on the free end of the crankshaft to provide an angle base sampling clock to the data acquisition system. Two data sets were collected, named A and B. Data set A was recorded at 0.27 bar intake pressure and 2000 rpm with the ignition angle sweeping from 45 to 10 CAD before top dead center (BTDC). In data set B sharp load transients are present. The standard engine calibration was used for the ignition timing and
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the load changed from 20% to 60% in sharp ramps back and forth, twice. During the load transients, the engine speed also changed by 720% from the nominal 2000 rpm due to the dynamic features of the engine dyno. Data set B ends with a load and speed reduction down to idle conditions. Data set A generated a wider range of combustion phasing variation than data set B. 1.2. Simulations
Tfp,i Tp,i
Tg,1
+
Tg,2
+
T1
T2 Trest
As part of the investigations, simulations of combustion events were used to compare the properties of the torque ratio function to the pressure ratio and the net heat release. The net heat release is represented by a Vibe function (Vibe, 1956), since it is known to make a fair representation of the spark ignited combustion: ( 0, y o ySOC xb ¼ ð1Þ mþ1 1eaðyySOC Þ , y Z ySOC Here, xb denotes the released fraction of the total energy, SOC denotes the start of combustion and fa,mg are the shape parameters (Heywood, 1988). For the simulation, the shape parameters fa,mg were calculated based on an assumption of a burn profile defined in Fig. 1. The shape parameters are calculated as lnð1x1 Þ ln lnð1x2 Þ ð2Þ m¼ lnðyd Þlnðyd þ yb Þ
a¼
lnð1x1 Þ
ð3Þ
ydm þ 1
where the flame development angle yd and the fast burn angle yb are the crank angle durations for the combustion of 0–10% and 10–85% of the total energy respectively. 1.3. Crankshaft torque measurements The crankshaft torque sensor measures the instantaneous combined effect of the torque contributions from all cylinders and the torsional deflections of the crankshaft itself. The upper graph of Fig. 3 shows an example of the measured crankshaft torque. The measured torque has a few very clear peaks distanced by 1441 which corresponds to the combustion in a five cylinder engine. Definition of burn angles
1 0.9 0.8
xb
0.7
x2
+ Tg,4
+
T4
+
Tm
^T
g,5
^ −
T5
−
Tm,5
Fig. 2. A schematic of the torque separation algorithm. Tm is the measured torque, Tg is the cylinder indicated torque, Tfp is the piston friction, Tp is the mass torque created by the reciprocating piston and H is the crankshaft model.
As combustion analysis is performed for each cylinder, the first step is to separate the measured torque signal into individual cylinder torque contributions. When completed, combustion analysis is performed on the estimated separated cylinder torque. The torque separation algorithm uses a model for the crankshaft which transforms the cylinder torque into the measured torque (Andersson & McKelvey, 2004a). The model also considers the torsional deflection effects which are present in the measurements. The cylinder torque separation is performed in two steps for every window of a firing cylinder, see Fig. 2. First, the combined effect on the measured signal from the previous combustion on the other cylinders in the last cycle is calculated using the crankshaft model. The result is subtracted from the measured signal. The remaining torque trace is filtered though the crankshaft model to form the estimated indicated cylinder torque. The lower graph in Fig. 3 shows the result from the separation, an estimate of the cylinder torque, compared to the actual cylinder torque which was derived from the cylinder pressure. The estimated cylinder torque has some residual oscillations which is a result of crankshaft model properties in the separation algorithm. Crankshaft modeling has been presented in Schagerberg and McKelvey (2003), Andersson and McKelvey (2004a) and McKelvey and Andersson (2006), and the separation algorithm was developed in Andersson and McKelvey (2004a).
0.6
2. Pressure ratio vs. net heat release
0.5
The torque ratio is derived from the pressure ratio, and since the net heat release is an established tool for combustion analysis it is interesting to analyze how the pressure ratio relates to the net heat release. This analysis will later be extended to the torque ratio as well. The net heat release rate is calculated from the cylinder pressure as (Heywood, 1988)
0.4 x1
0.3 0.2 θd
0.1 0
θb
dQ n ¼ 0
10
20
30 CAD
40
50
60
Fig. 1. Definition of burn angles for simulation. The flame development angle yd and the fast burn angle yb are connected to certain burn ratios x1 and x2.
g 1 p dV þ V dp g1 g1
ð4Þ
and the accumulated net heat release Q n ðyÞ is found by integrating Eq. (4). A motored cycle occurs when the combustion is omitted and the cylinder pressure pmot can be approximated by a polytropic
I. Andersson et al. / Control Engineering Practice 20 (2012) 561–568
The derivative of the pressure ratio with respect to crank angle y is
1997 RPM, 91 Nm
400
D
200
g1 d pcyl ðyÞ VðyÞ dmb dqw dm 1 ¼ qLHV þC dy pmot V0 dt dy dy |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
Nm
V r ðyÞ
0 raw torque
−200 −400
−300
−200
−100
0
100
CAD 400 est cyl trq real cyl trq
300 200 Nm
563
ð8Þ
dQ n ðyÞ
The volume ratio function V r ðyÞ puts an angle based weight to the net heat release rate dQ n ðyÞ when forming the pressure ratio. Fig. 4 shows the volume ratio function with a datum V0 at TDC and g ¼ 1:3. When forming the pressure ratio, the Vr function alters the net heat release rate dQ n ðyÞ and puts higher weight on the parts of the dQ n furthest from TDC. This introduces a shift where the pressure ratio rate peak is shifted away from TDC, and consequently the PR50 occurs later than the 50% point of the net heat release (Q n,50 ) for the normal post-TDC combustion phasing. The size of the shift depends on the phasing itself and the combustion duration. Fig. 5 shows a range of simulated combustion scenarios where the phasing of the Vibe function was shifted from early to late combustion and the combustion duration was varied. The duration
100
Volume ratio function 2.4
0
2.2
−100 −400
−300
−200
−100
0
100
CAD
2 1.8 Vr
Fig. 3. An example of torque separation at 2000 rpm, 91 N m. The upper graph shows the measured torque and the lower shows the estimated and real cylinder torque.
1.6
process with a polytropic coefficient of g as pmot ðyÞ ¼ p0
V0 g VðyÞ
1.4
1.2
ð5Þ
1 −200
V is the cylinder volume and 0 denotes conditions at a datum point, e.g. when the intake valves are closing. The volume V is a known function of crank angle and p0 can be estimated in several ways (Eriksson & Andersson, 2002). The concept of pressure ratio (PR) (Matekunas, 1986) relates the cylinder pressure pcyl from combustion to its corresponding motored pressure pmot as pcyl ðyÞ PRðyÞ ¼ 1 pmot ðyÞ
−150
−100
−50
0
50
100
150
200
[CAD] Fig. 4. The volume ratio function, normalized with the TDC volume.
PR50 deviation vs. Qn,50
4
dur=40 dur=70 dur=100
3
ð6Þ
PR50−Qn,50 [CAD]
2 The pressure ratio is a function that starts at zero and during the combustion increases to a higher value where it stays constant when the combustion is finished. There is a point in crank angle where the PR has reached half its final value, which is referred to as PR 50% or PR50 . A theoretical formulation of the cylinder pressure is derived in Appendix A, which combined with (6) arrives at
1 0 −1 −2
pcyl ðyÞ 1 PRðyÞ ¼ 1 ¼ D pmot
Z
y
y0
V g1 dmb dqw dm qLHV þC dt V0 dt dt dt
−3 ð7Þ
where dmb =dt is the mass burn rate of fuel, dqw =dt is the wall heat transfer rate, dm=dt is the change in total mass, D is a constant and C is a mass specific energy measure which depends mainly on the gas temperature. See Appendix A for more details.
−4 −100
−50
0 50 Qn,50 pos [CAD]
100
150
Fig. 5. The difference in the 50% point between the pressure ratio (PR) and the net heat release (Qn).
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(dur) and burn angles fyd , yb g for the three test cases are defined in Table 1. As seen, a combustion event with a longer duration shows a larger difference between the PR50 and the Q n,50 values than for shorter durations. Also, the PR50 and the Q n,50 deviates the most for combustion phasing around 30 CAD ATDC. The net heat release can be calculated from the pressure ratio function in the following way: 1. Calculate the pressure ratio function PRðyÞ as in (6) 2. Calculate the net heat release rate as PR
dQ n ðyÞ dPRðyÞ D ¼ dy dy V r ðyÞ
ð9Þ
3. The cumulative net heat release is Z y PR dQ n ðtÞ Q PR ð y Þ ¼ dt n d t soc
gas torque Tg on the crankshaft, according to T g ðyÞ ¼ ðpcyl ðyÞpcc ÞLðyÞA
ð11Þ
where pcc is the crank case pressure and A is the piston area. The geometric transformation of gas pressure to crankshaft torque can be described with a crank angle dependent effective lever, LðyÞ see Fig. 7. One straightforward way to extract combustion information from the estimated cylinder torque would be to reverse Eq. (11) and estimate the cylinder pressure. However, this is difficult since at the top dead center Lð0Þ ¼ 0 and the slightest noise in the estimated torque would make the inverse impossible. Instead, a method called the torque ratio was developed to address this problem. The torque ratio concept is inspired by the pressure ratio concept, which was presented in Matekunas (1986) (Fig. 8).
ð10Þ
Fig. 6 shows the PR50 vs. the Q n,50 and the PR based net heat
Pcyl
release Q PR n,50 vs. the Q n,50 . calculated from the data set. Fig. 6 shows that PR50 and Q n,50 are similar but not the same and the volume ratio function Vr explains the difference between them.
F=Pcyl A 3. The torque ratio concept The combustion process creates a gas pressure pcyl inside each cylinder that via the piston and connecting rod is transformed to a
Pcc
Table 1 Burn rate parameters for the simulation cases.
yd
yb
40 70 100
10 15 20
20 40 60
45
45
40
40
35
35
30
30
PR
25 20
25 20
15
15
10
10
5
5
0
0
20 Qn,50 [CAD]
Pcc A
Fig. 7. The effective lever L, created by the geometry of piston, connecting rod and crankshaft. The force on the piston is the net pressure pcyl pcc times the piston area A.
50
0
T T
50
−5
L
r
Qn,50 [CAD]
PR50 [CAD]
dur
l
40
−5
0
20 Qn,50 [CAD]
40
Fig. 6. From engine data set A; a comparison of 50% net heat release and PR. Left graph: PR50 vs. Q n,50 . Right graph: The Q PR n,50 from a PR curve adjusted with the volume ratio function Vr, vs. Q n,50 .
I. Andersson et al. / Control Engineering Practice 20 (2012) 561–568
Let
400
[Nm]
565
motored torque fired torque
200
Hp ðyÞ ¼ p0
V0 g LðyÞA VðyÞ
xðyÞ ¼ T g ðyÞT mot ðyÞ
0
Eq. (14) is re-written as −200 −100
−50
0
50
100
Crank angle [CAD]
xðyÞ ¼ Hp ðyÞ f TR ðyÞ Let (
2
Hðc, yÞ ¼
TR
1 TR50
0 −100
−50
0
50
100
Crank angle [CAD]
PRðyÞ ¼
2 6 HðcÞ ¼ 4
Hðc, y1 Þ
ð12Þ
So in theory, the torque ratio and the pressure ratio are identical, but calculated from different sensor sources. However, the torque ratio has a singularity at TDC where the lever is zero, since the estimated torque T g ðyÞ will never be completely noise free. This property prevents an easy calculation of the cylinder pressure by using (11), and hence use any pressure based combustion analysis algorithm. To solve the singularity problem, the torque ratio function is modeled by a parametrized mathematical function which is selected with the combustion type in mind. For spark ignited engines the model becomes ( 0, y o ysoc mþ1 f TR ðyÞ ¼ ð13Þ bð1eaððyysoc Þ=cÞ Þ, ysoc o y where SOC denotes start of combustion. The fTR function is a variation of the Vibe function in (1) and has four parameters fa,b,c,mg. The parameters a and m are the shape parameters and in this case they are set to constant values. Parameters b and c are left for optimization in least squares sense. Parameter b is the amplitude of the TR and c corresponds to the burn duration of the combustion. These two parameters hold all the information about each combustion. For diesel combustion, this model can be selected as a double Vibe function, which then represents two injection pulses per combustion (Thor, Andersson, & McKelvey, 2009).
3.1. Torque ratio parameter estimation When the fb,cg parameters have been estimated from the measured data, other combustion information like the phasing (TR 50%) can be calculated. Combining Eqs. (5) and (12) gives V0 g T g ðyÞT mot ðyÞ ¼ p0 LðyÞA f TR ðyÞ ð14Þ VðyÞ
y Z ysoc y o ysoc
ð16Þ
^ Hðc, yN Þ
3 7 5
The separable least squares problem is formulated as x ¼ bHðcÞ
pcyl ðyÞpcc þ pcc ðpcyl ðyÞpcc þ pcc ÞLðyÞA 1 1 ¼ pmot ðyÞpcc þ pcc ðpmot ðyÞpcc þpcc ÞLðyÞA
T g ðyÞ þ pcc LðyÞA T g ðyÞT mot ðyÞ ¼ 1 ¼ 9TRðyÞ T mot ðyÞ þ pcc LðyÞA pmot LðyÞA
Þ,
Form a sequence of N sampled data into a column array and the corresponding geometry functions of cylinder volume and piston lever likewise: 2 3 xðy1 Þ 6 ^ 7 x¼4 5 xðyN Þ
Fig. 8. Torque ratio for a single cylinder case, simulation.
The torque ratio function TRðyÞ is formed in the following way. Starting with (6) and (11), the expression is evolved with the crank case pressure pcc and effective lever LðyÞ as
mþ1
Hp ðyÞð1eaððyysoc Þ=cÞ 0,
1.5
0.5
ð15Þ
ð17Þ
Parameter c and b are found by standard methods for separable LS problems (Kay, 1994): ^ ¼ xT ½IHðcÞðHT ðcÞHðcÞÞ1 HT ðcÞx Jðc, bÞ
ð18Þ
^ c^ ¼ minJðc, bÞ
ð19Þ
b^ ¼ ðHT ðcÞHðcÞÞ1 HT ðcÞx
ð20Þ
c
The next step is to find the position where f TR ðyÞ attains 50% of its final value b: b ð21Þ TR50 ¼ y : f TR ðyÞ ¼ 2 The general solution is 1=ðm þ 1Þ ln 2 TR50 ¼ ysoc þ c a
ð22Þ
If the shape parameters a and m are chosen such that fTR becomes close to a symmetric function, TR50 can be approximated by TR50 ysoc þ
c 2
ð23Þ
4. Validation of torque ratio To validate the TR50 measure it is compared to the cylinder pressure based PR50 and 50% net heat release Q n,50 measures. Cylinder pressure data from the test engine was analyzed by the pressure ratio and net heat release algorithms and the corresponding cylinder torque Tg was calculated and analyzed by the proposed torque ratio algorithm. The torque ratio shape parameters were selected to match the burn profile as good as possible, with a slight favor towards the early phasing data. The parameter values were a¼5 m¼4 First, the TR50 is compared to the PR50 , and finally the TR50 is compared to the Q n,50 . Also, a derivation is presented on how to calculate the net heat release Qn from TR.
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I. Andersson et al. / Control Engineering Practice 20 (2012) 561–568
4.1. Torque ratio vs. pressure ratio
4.2. Torque ratio vs. net heat release
Fig. 9 shows the comparison between the TR50 and PR50 phasing measures. The measure TR50 corresponds well to the PR50 measure. A deviation is seen for the most late combustion, due to that the shape parameters fa,mg were kept constant over the whole data range. A control algorithm based on the TR50 measure should have approximately the same target value for the combustion phasing as for a PR50 based algorithm. The deviation for late combustion can be addressed using phasing dependent shape parameters in the torque ratio function f TR ðyÞ.
The torque ratio relates to the net heat release in the same way as the pressure ratio does, except for the misfit of the burn rate model. The net heat release is calculated from the estimated cylinder torque in the following way:
1. Calculate the torque ratio function TRðyÞ by estimating the parameters (b,c) through Eq. (19) 2. Use parameters (b,c) to calculate the derivative of the torque ratio dTRðyÞ abðm þ1ÞðyySOC Þm aððyysoc Þ=cÞm þ 1 ¼ e dy cm þ 1
Torque Ratio vs. Pressure Ratio
50
Cycle−to−cycle TR50 vs. PR50
45
3. Calculate the net heat release rate as
1−to−1
TR
40
dQ n ðyÞ dTRðyÞ D ¼ dy dy V r ðyÞ
35 TR50 ref [CAD]
ð24Þ
30
ð25Þ
4. The cumulative net heat release is
25
Q TR n ðyÞ ¼
20
Z
y
TR
dQ n ðtÞ dt dt soc
ð26Þ
15 10 5 0 −5
0
10
20 30 PR50 [CAD]
40
50
50
50
45
45
40
40
35
35
30
30 Qn,50 [CAD]
25
TR
TR50 [CAD]
Fig. 9. Comparison of TR50 and PR50 measures from cylinder pressure data. The dashed line marks the 1:1 relation. The deviation at late phasing is due to less accuracy in the burn rate profile parameters fa,mg. Data set A was used.
Fig. 10 shows the result. It turns out that the misfit of the combustion model in the torque ratio approach compensates somewhat the deviation between the pressure ratio and the net heat release, and the torque ratio is very similar to the net heat release already. The TR50 differs by an average of 1.3 CAD from the Q n,50 in the data sets, and the torque ratio based Q TR n,50 shows an average bias of 0.2 CAD and a standard deviation of 0.7 CAD compared to the cylinder pressure based Q n,50 . The results obtained here can be compared with results from earlier experiments, where the TR50 measure was estimated from crankshaft torque sensor measurements on an engine in a dyno
20
25 20
15
15
10
10
5
5
0
0
−5
0
20 Qn,50 [CAD]
40
−5
0
20 Qn,50 [CAD]
40
Fig. 10. From engine data sets A and B combined; a comparison of 50% net heat release and TR. Left graph: TR50 vs. Q n,50 . Right graph: The Q TR n,50 from a TR curve adjusted with the volume ratio function Vr, vs. Q n,50 .
I. Andersson et al. / Control Engineering Practice 20 (2012) 561–568
(McKelvey & Andersson, 2006) and on an engine in a vehicle (Andersson, McKelvey, & Thor, 2008). The torque sensor based TR50 estimation showed an average bias of 3–4 CAD compared to the cylinder pressure based TR50 estimation for moderate excitation of the crankshaft dynamics ( r 3000 rpm). For severe excitation of the crankshaft dynamics ( Z 4000 rpm) the bias rose to 9 CAD. Hence, the cylinder separation accuracy is an important factor for the overall performance of crankshaft torque based combustion phasing estimation.
where R is the mass specific gas constant. It is calculated as R¼
The torque ratio concept is presented as a method to estimate combustion properties in an engine where the combustion sensor is located on the crankshaft or the flywheel of the engine. Standard combustion analysis tools as the burned mass fraction, heat release or pressure ratio are difficult to use due to the problems of reversely estimate a cylinder pressure from an estimated cylinder torque. The torque ratio is estimated as a parametrized function which is intended to resemble the combustion profile as good as possible. For a gasoline engine the parametrized function was chosen to be based on the well known Vibe function where the parameters of combustion amplitude and duration are estimated. The 50% point of the torque ratio is used as a measure for combustion phasing, TR50 . The TR50 is a monotonous function of combustion phasing and therefore a useful measure of the same. Further, the TR50 is close to the 50% pressure ratio, and the difference appears due to the limited accuracy of the torque ratio combustion model. The relationship between the pressure ratio and the net heat release is established in theory. A volume ratio function is defined and identified that connects the derivative of the pressure ratio to the net heat release rate. The relationship is also illustrated by engine data for a wide range of combustion phasing. It is shown that the torque ratio function can be used as a base for net heat release calculations. The phasing measurement TR50 turns out to deviate from the 50% net heat release with an average bias of 1.3 CAD, and the torque ratio based 50% net heat release has a standard deviation of 0.7 CAD and a bias of 0.2 CAD compared to the cylinder pressure based ditto.
Appendix A. Combustion pressure for the SI engine This is a derivation of the combustion pressure for a spark ignited engine on a closed form. The derivation is based on the description in Matekunas (1986) and is given here to support the derivations in Section 2 and forward. Consider a closed system where a gas resides. It has some pressure p, temperature T, volume V and mass m. The first law of thermodynamics states that a change in internal energy for a gas, dU, is the result of heat added to the gas, dQ, and the work that the gas performs on its environment, dW, and mass flow across the system boundaries dm: 0
dU ¼ dQ dW þ h dm
ðA:1Þ
0
where h is the enthalpy of the mass element dm at its source. Furthermore, the following relations hold: dW ¼ p dV
ðA:2Þ
dU ¼ mcv dT þ u dm
ðA:3Þ
Here, p is the gas pressure, dV is the volume change of the gas, m is the trapped gas mass, cv is the heat capacity of the gas and dT is the temperature change. The ideal gas law states that pV ¼ mRT
ðA:4Þ
R~ M
ðA:5Þ
where R~ is the universal gas constant and M is the molar mass of the gas. R is related to the heat capacities cv and cp of the gas and the ratio of specific heats g as R ¼ cp cv
ðA:6Þ
cp cv
ðA:7Þ
g¼ 5. Conclusions
567
The net heat dQ added to the system is formed by the added heat from the combustion and the heat transfer to the cylinder walls dQ ¼ qLHV dmb dqw
ðA:8Þ
The sum of the unburned mass mu and the burned mass mb is constant since no mass enters the system. Differentiate Eq. (A.4): V dp þ p dV ¼ mR dT þRT dm
ðA:9Þ
and re-write dU dU ¼
mcv ðV dp þ p dVRT dmÞ þu dm mR
ðA:10Þ
Form dU þ dW ¼
cv ðV dp þp dVRT dmÞ þ u dm þp dV R 0
dU þ dW ¼ dQ þ h dm ) dp þ g
ðA:11Þ ðA:12Þ
p R 0 dV ¼ ðdQ þh dmu dm þ cv T dmÞ V cv V
ðA:13Þ
The differential equation is solved for the pressure pðyÞ. Multiply by V g : V g dpþ V g g
p R 0 dV ¼ V g ðdQ þðcv T þ h uÞ dmÞ V cv V
V g dpþ gpV g1 dV ¼ V g1
R 0 ðdQ þðcv T þ h uÞ dmÞ cv
ðA:14Þ
ðA:15Þ
Note that the independent variable for p, V and Q is the crank angle y. Replace the differentials with their derivatives with respect to y. The left side of Eq. (A.14) is the differential of pV g : d R dQ dm 0 ðpV g Þ ¼ V g1 þ ðcv T þh uÞ ðA:16Þ dy c v dy dy The solution to Eq. (A.16) is written as Z R dQ dm 0 þ ðcv T þh uÞ dy pV g ¼ C þ V g1 cv dy dy y
ðA:17Þ
When the intake valve closes at angle y0 , the trapped gas conditions are noted p0, V0, T0 and m0. The mass remains constant through the compression and expansion phases in a pre-mixed combustion system as the Otto-engine. The added heat to the system, dQ, is zero before y0 since the gas mixture is fresh by then. If there would be no heat added to the system through the whole cycle, Eq. (A.17) is simply pV g ¼ C
ðA:18Þ
which is the expression for an adiabatic process. The constant C is easily found as g
C ¼ p0 V 0 Eq. (A.17) then becomes Z y R dQ dm 0 g V g1 þ ðcv T þh uÞ dt pV g ¼ p0 V 0 þ c v dt dt y0
ðA:19Þ
ðA:20Þ
568
I. Andersson et al. / Control Engineering Practice 20 (2012) 561–568
Isolate p on the left side of Eq. (A.20) g Z y V0 1 R dQ dm 0 þ g V g1 þ ðcv T þ h uÞ pðyÞ ¼ p0 dt cv dt dt V V y0
References ðA:21Þ
Use Eq. (A.4) to express R in p0, V0, T0 and m0 and insert in Eq. (A.21) pðyÞ ¼ p0
! Z y V0 g 1 dm 0 g1 1 dQ dt 1þ V þ ðc T þ h uÞ v g1 V cv dt dt y0 T 0 m0 V 0
ðA:22Þ Define the motored pressure pmot as gu V0 pmot ¼ p0 V
ðA:23Þ
and combine with Eqs. (A.22) and (A.8) to get pðyÞ ¼ pmot
ggu V0 V
1þ
1 m0 cv T 0
Z
y
y0
! V g1 dmb dqw dm 0 dt qLHV þ ðcv T þ h uÞ dt dt V0 dt
ðA:24Þ Approximate g gu and set C ¼ cv T þh u
0
ðA:25Þ
D ¼ m0 c v T 0
ðA:26Þ
then (A.24) becomes ! Z 1 y V g1 dmb dqw dm dt pðyÞ ¼ pmot 1 þ qLHV þC dt dt D y0 V 0 dt
ðA:27Þ
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