SAE TECHNICAL
PAPER SERIES
1999-01-1489
TDC Determination in IC Engines Based on
the Thermodynamic Analysis of the
Temperature-Entropy Diagram
M. Tazerout, O. Le Corre and S. Rousseau
DSEE-Ecole des Mines de Nantes
International Spring Fuels & Lubricants Meeting
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May 3-6, 1999
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Printed in USA
1999-01-1489
TDC Determination in IC Engines Based on the Thermodynamic
Analysis of the Temperature-Entropy Diagram
M. Tazerout, O. Le Corre and S. Rousseau
DSEE-Ecole des Mines de Nantes
Copyright © 1999 Society of Automotive Engineers, Inc.
ABSTRACT
A thermodynamic methodology of TDC determination in
IC engines based on a motoring pressure-time diagram
is presented. This method consists in entropy calculation
and temperature-entropy diagram analysis. When the
TDC position is well calibrated, compression and expansion strokes under motoring conditions are symmetrical
with respect to the peak temperature in the (T,S) diagram. Moreover, in case of error on the TDC position, a
loop appears, which has no thermodynamic significance.
Hence, an easy methodology has been conceived to
obtain the actual position of TDC. This methodology is
applied to motoring measurements in order to present its
performance, which are compared to usual methods.
INTRODUCTION
The recording of accurate indicator diagrams is very difficult but is of great importance. Among the many sources
of error, a wrong TDC position, leading to an incorrect
(P,V) diagram, has been recognized as the major source
of error on thermodynamic calculation results such as
IMEP (Indicated Mean Effective Pressure), mass fraction
burned or combustion duration. Hribernik [4] proposes a
formulation giving the ratio between the IMEP error (in %)
and the TDC position error (in °CA).
IMEPerror[%]
≈9
TDC error [°CA]
Figure 1.
efinition of the TDC phase lag error
In this paper, the TDC phase lag error ∆ϕ is defined as
the angle between the peak pressure with the actual TDC
and the peak pressure with a wrong TDC, as shown on
figure 1.
BACKGROUND
In 1967, Brown [1] identified the main sources of error in
pressure measurements. One of them was the shift
between pressure and crank angle. He proposed a correction using the polytropic exponent obtained from
experimental data:
(1)
The main difficulty in locating TDC is that the peak pressure under motoring conditions precedes the TDC position (corresponding to the minimum volume) because of
heat transfer and mass losses: it is the loss angle (figure
1). Thus, previous studies [1] [2] [4] [6] [7] [9] have
investigated several methods for determining the TDC
position.
n=
ln( Pi ) − ln( Pi +1 )
ln(Vi +1 ) − ln(Vi )
(2)
Stas [9] proposed another method based on polytropic
coefficients calculated at the inflexion point occurring during the compression and expansion strokes (figure 2).
1
Figure 2.
Figure 3.
Pressure vs. crank angle under motoring
conditions
In that context, the purpose of the method proposed in
this paper is to present a general method to determine
the TDC position using an experimental pressure-time
diagram under motoring conditions and the analysis of its
transformation into temperature-entropy diagram. Such a
method is fully independent of any heat transfer correlation or polytropic exponent calculation.
According to Stas, this methodology allows to locate TDC
with an accuracy of about ± 0.1°CA. Note that the calcu2
2
lation of polytropic coefficients uses d V and ( dV ) ,
which can cause numerical errors.
The location of TDC can be determined using the symmetry of the pressure-time diagram under motoring conditions. It is the method used by the MACAO software
OSIRIS [7]. It constitutes pairs of points (A, A’), (B, B’),
and so on. Points A, B are on the compression curve of a
motoring cycle, A’, B’ are on the expansion curve (figure
3). The pressure in A and A’ is the same. Then, it calculates by linear regression the straight line passing
through the center of segments [ A, A’], [B, B’], … The
intersection between this line and the crank angle axis
gives the thermodynamic TDC.
THERMODYNAMIC MODEL
A thermodynamic model for motoring simulations has
been developed and simulates the whole engine cycle,
from the inlet to the exhaust. It corresponds to a onezone thermodynamic model for performance predictions
but with no combustion, Heywood [3], and Ramos [8].
In cylinder gas are assumed to be ideal gas, and specific
heats are supposed to depend on the temperature only:
This angle must be corrected to give the actual TDC location. OSIRIS calculates this correction on the basis of the
engine characteristics, the peak pressure and the inlet
pressure.
2
Cp = 1403.06 − 360.72
IMEPu
Pmax
1000
 1000 
 1000 
+108.24
 -10.79

 T 
 T 
T
The ideal gas law and the first principle of thermodynamics applied to the chamber respectively give cylinder
pressure and temperature:
pV = m r T
(
0.8 − 0.53
1 A r Pmax T Pmax T Pmax − T0
K=
π Am c p P 0.8T − 0.53 (T − T0 )
3
(4)
Calibration results from most of methods depend on the
heat transfer coefficient. For instance, Pinchon [6] proposed a calibration based on the IMEP and the peak
pressure:
θ TDC = θ P max + K
Pressure vs. crank angle under motoring
conditions
)
m Cv
T&
=−
p V&
− Q&
(5)
w
+ u m& cyl
(6)
(3)
In equation (6), p V& is the work due to in-cylinder gas,
and Q& w is heat losses through the chamber walls.
This method uses Woschni's correlation with a corrective
factor of 1.7. Its accuracy is then very dependent on the
accuracy of the heat transfer correlation.
Assuming that leakage is negligible, the change of the
mass in the control volume is only due to flow-rates
through inlet and exhaust valves:
m&
2
= δ valve C d Avalve Pupst
2γ Rp 2 / γ − Rp γ +1 / γ
(γ − 1)RTupst
(7)
where Cd is a constant valve discharge coefficient,
Avalve (θ ) is the geometrical valve flow area taken as a
function of sine, and δ valve equals 1 during intake, -1 during exhaust and 0 during the other strokes. The pressure
ratio R p is defined according to δ valve :
(8)
Where the sonic pressure ratio is defined by:
Rplim upst

2 
=
 γ upst + 1 


γ upst γ upst −1
(9)
The subscript upst designates the manifold during intake
and the cylinder during exhaust.
The heat transfer to the walls under motoring conditions
is due to convection:
Q& w =h g S w (T −Tw )
(10)
The wall temperature is assumed constant and uniform.
The heat transfer coefficient is calculated with Hohenberg’s correlation:
h g ( θ ) = 130
[
p( θ ) 0.8 S p + 1.4
V (θ )
0.06
T (θ )
] 0. 8
Figure 4.
(11)
Tmax − T1 = Tmax − T2 = ∆T > 0
The instantaneous cylinder volume and heat exchange
surface between gas and chamber walls are known analytically with respect to the engine’s geometrical characteristics and to the crankshaft angle.
Note that the peak temperature is not necessary located
at TDC.
The specific entropy leads to:
This model allows simulating engine cycles under motoring conditions with actual and wrong TDC.
d S = Cp
TEMPERATURE-ENTROPY DIAGRAM
dT
dP
−R
T
P
(14)
Near TDC, the change of the volume can be neglected.
Under these conditions:
During compression and expansion strokes, the in cylinder mass is supposed constant, and the temperature is
obtained from the pressure diagram (either numerically
or experimentally) using a reformulation of eq (5):
pV
mr
(13)
Two points ( T1 , T2 ) are defined in part of other of the peak
temperature Tmax , such as (figure 5):
Input and output of the predictive model are summarised
on figure 4.
T=
Predictive model from Le Corre and al. [5]
0.4
d P d T
=
P
T
(15)
As ∆T is small, specific heat Cp and Cv can be
assumed constant. During the compression stroke, equation 14 becomes:
(12)
3
( ) ∫ dTT
∆T
= (C − R ) ln (1 −
)
T
∆S max→1 = C p − R
1
max
p
max
(16)
Using a first order Taylor's development, equation (16)
can be rewritten:
(
∆S max→1 = R − C p
) T∆ T
max
(17)
The same assumptions are applied to the expansion
stroke:
∆S max →2 = −( R − C p )
∆T
Tmax
(18)
Figure 6.
(19)
As shown on figure 7, the loop still exists until a TDC
phase lag of 0.45°CA is observed.
Equations 17 and 18 lead to:
∆S max →1 = − ∆S max → 2
Simulated (T,S) diagram with different TDC
phase lags
Equation 19 shows that the entropy varies symmetrically
around the peak temperature. In other words, the temperature-entropy diagram must be completely symmetrical
with respect to Tmax .
Figure 7.
Figure 5.
Simulated (T,S) diagram with different TDC
phase lags
The existence of this loop appears to be a new way to
locate TDC. But one must verify the robustness of this
way before conceiving a new methodology. In order to be
validated, the new method is then compared to other
existing methods and is applied to an experimental pressure-time diagram.
Simulated (T,S) diagram under motoring
conditions with the actual TDC calibration
When an error exists on the TDC position, simulations
show that a loop appears in the (T,S) diagram. This loop
has no thermodynamic significance. Figure 6 presents
four (T,S) diagrams, one with the actual TDC position,
three other curves with a wrong TDC position (error of
+1, +0.75 and +0.5°CA). In the three last cases, a loop
appears, which size increases with the error on the TDC
position.
TESTS OF ROBUSTNESS
Three variables are tested. The first one is the engine
throttle. The second one is the in cylinder mass since this
value is difficult to known accurately. The last one is the
volumetric compression ratio, in order to extend the
method to any kind of engines.
4
Even if a mass error is introduced, figure 9 shows that the
loop exists for a shift of +0.5°CA and is always vanished
for a shift of +0.45°CA (figure 10).
1. PARTIAL OPEN THROTTLE – Three open throttles
have been analyzed. The robustness of the method is
important at partial open throttle since it can be impossible or dangerous to obtain motoring cycles at wide open
throttle (WOT) for some kind of engines (gas engines
with carburetor for example).
Figure 10. Simulated (T,S) diagram for different incylinder masses
Figure 8.
Simulated (T,S) diagram for different open
throttles
This means that the error on the mass inside the cylinder
has no effect on the loop.
Since the loop exists for any throttle (figure 8), this
method can be applied on motoring cycles obtained at
partial open throttle.
3. COMPRESSION RATIO – In order to apply the
method to any kind of engines, the loop must be independent of the compression ratio. Simulated (T,S) diagrams
for three different compression ratios have shown that the
loop exists for all of them (figure 11).
2. IN-CYLINDER MASS – Errors can be done in the calculation of the mass contained in the cylinder. These are
mainly due to errors on the inlet flow-rate measurement
and to assumptions made to estimate the mass of residual gas. Moreover, the in-cylinder mass is not constant
during compression and expansion strokes because of
leakage.
The following sensitivity study uses the following definition:
Mass used = Mass True (1 − Mass Error )
Figure 11. Simulated (T,S) diagram for several
compression ratios
In any case, the loop is vanished for an error on the TDC
position of +0.45°CA.
The three tests of sensitivity allow concluding that this
new method can be applied for any kind of engine, at any
open throttle.
Figure 9.
Simulated (T,S) diagram for different incylinder masses
5
In the polytropic exponent diagram, the curve with no
error of phase could be interpreted as a negative error of
phase on TDC (figure 14). So, in order to apply its
method to real engines cycles, Hribernik [4] explained
how to transform the measured pressure-time history into
adiabatic pressure-time history.
COMPARISON WITH THE POLYTROPIC
EXPONENT METHOD
The (T,S) diagram can easily be compared to the polytropic exponent curvature. Figure 12 shows the evolution
of the polytropic exponent versus crankshaft angle proposed by Hribernik [4] for an ideal adiabatic cycle.
Figure 14. Simulated (n,θ) diagram for a real cycle with
heat transfer
Figure 12. Simulated (n,θ) diagram for an ideal cycle
This underlines the interest of the (T,S) diagram, which
avoids such confusion (figure 6), and shows the limits of
the polytropic exponent diagram.
He wrote that "the curves lie in the quadrants II and IV
when the phase lag error is positive and in the quadrants
I and III when the error is negative"
Note that, the hyperbolic aspect of the polytropic exponent is due to its formula (equation 2).
METHODOLOGY FOR THE CALIBRATION OF
TDC
The curve (n,θ) is a horizontal line for an actual TDC in
this case (adiabatic cycle).
A new methodology can be proposed if one remarks that:
( )
For an actual calibration and an ideal adiabatic cycle, the
(T,S) diagram becomes a vertical line, as shown on figure
13.
( )
max scomp = sTmax = min sexp
(20)
When the compression stroke is described, one retains
( )
the maximum of entropy max scomp . This value is compared to the value of the entropy sTmax at the peak tem-
( )
perature. If max scomp
is superior to sTmax (case a on
figure 15) then the TDC position must be decreased by a
constant step of 0.1°CA. This operation must be iterated
until the convergence. Note θlim this angle, (case B).
Finally, adding θlim and -0.45°CA gives the actual calibration (case C).
Figure 13. Simulated (T,S) diagram for an ideal cycle
6
occur during these strokes, as the cylinder gas temperature is higher than the wall temperature.
Being given that the criteria for a negative TDC phase lag
is difficult to obtain but not for a positive TDC phase lag,
the new methodology proposed in this paper is essential.
TEST RESULTS
The method has been tested on a SI engine. Pressuretime diagrams have been recorded with OSIRIS with an
acquisition every 1°CA.
Engine characteristics are the following:
Designation
Ignition
Admission
Displacement Volume
Number of cylinders
Bore B
Stroke S
Engine Speed
Figure 15. Schematic (T,S) diagrams according to the
correction on the TDC position
Lister-Petter
TS1
SI
Natural
Aspiration
633 cm3
1
95.3
88.9
1500 RPM
The data acquisition system for the cylinder pressure is
composed by:
• a Sensor AVL QH32D, gain 25.28pC/Bar Range 0200 BAR
The following point must be underlined: this methodology
assumes that initially, the peak pressure under motoring
conditions is before the actual TDC position.
• a Piezo Amplifier AVL 3066A01, gain 400pC/V with
no reference of pressure
This condition is imperative since the loop does not exist
for negative TDC phase lag (figure 16).
• a Druck type PTX 510 - Range 2.5 BARA inside inlet
manifold to give the reference of pressure
The first transformation in the (T,S) diagram can be done
(figure 17). A loop appears in this diagram: the TDC position is wrong.
Figure 16. Simulated (T,S) diagram with positive TDC
phase lag
In this case, an anomaly in a thermodynamic point of
view appears on the entropy variation during the compression stroke (positive instead of negative) and at the
expansion stroke (idem). This phenomenon can not
Figure 17. Initial experimental (T,S) diagram
7
A first correction of -0.1°CA is applied (figure 18).
Figure 20. Experimental (T,S) diagram for the actual
TDC
Figure 18. Experimental (T,S) diagram with a correction
of 0.1°CA
Figure 21 shows the actual (P, V) diagram.
After several corrections, the limit diagram is obtained
(figure 19).
Figure 21. Experimental (P,V) diagram with actual TDC
Figure 19. Experimental (T,S) diagram with a correction
of -1.3°CA
As simulations have shown that the loop disappears for a
TDC phase lag of 0.45°CA, the actual correction that
should be applied to the physical position of the crankshaft angle encoder is –1.75°CA.
Note that it is impossible to obtain the conditions required
by the automatic procedure of OSIRIS for locating TDC
(3000 rev/min, 25% WOT, no fuel injection), since the
engine speed is imposed by the generator at 1500 rev/
min. Despite this, the difference observed between
OSIRIS procedure and the new methodology is only 0.15°CA.
With this correction, the experimental temperatureentropy diagram is fairly symmetric with respect to the
peak temperature (figure 20).
Figure 22. Experimental (P, θ) diagram with actual TDC
8
REFERENCES
The usual method [7] is in accordance with the calibration since the pressure diagram versus crankshaft angle
is symmetric (figure 22).
1. W.L. Brown Methods for Evaluating Requirements
and Errors in Cylinder Pressure Measurement, SAE
Paper N°670008
2. M.F.J. Burnt and A. L. Emtage Evaluation of IMEP
Routines and Analysis Errors SAE Paper N°960609
3. J. Heywood Internal Combustion Engine Fundamentals, McGraw-Hill International Editions, 1988, ISBN
0-07-100499-8
4. A. Hribernik Statistical Determination of Correlation
Between Pressure and Crankshaft Angle During Indication of Combustion Engines, SAE Paper N°982541
5. O. Le Corre, S. Rousseau and C. Solliec One Zone
Thermodynamic Model Simulation of a Stationary
Spark Ignition Gas Engine : Static and Dynamic Performances, SAE Paper N°982694
6. P. Pinchon Calage Thermodynamique du Point Mort
Haut des Moteurs à Piston Revue de l'institut du
Pétrole, Vol 39, N°1, Janv-Fev 1984
7. OSIRIS Guide Version 2.0, Copyright MACAO 199497
8. J. Ramos, Internal Combustion Engine Modeling, Ed.
Hemisphere Publishing Corporation, 1989, ISBN 089116-157-0
9. M. J. Stas Thermodynamic Determination of T.D.C. in
Piston Combustion Engines SAE Paper N°960610
In addition, the polytropic exponent evolution is in accordance too, especially in the quadrants II and IV (figure
23).
Figure 23. Comparison between experimental and
simulation on (n,θ) diagram
ACCURACY OF THE NEW METHOD
NOMENCLATURE
The last point of this work concerns the accuracy of this
method. Stas's method [9] should have an accuracy of
±0.1°CA. Hribernik's method [4] should have a precision
of ±0.025°CA.
A correction step of 0.1°CA has been chosen. It determines the accuracy of the method, since CASE B – figure
15 will be reached with a maximum error equal to the
step, i.e. 0.1°CA. Thus, the authors evaluate the accuracy of the new method at ±0.1°CA.
CONCLUSION
The interpretation of the temperature-entropy diagram is
a right way to obtain the TDC position. In fact, a physical
error on the crankshaft encoder induces a loop, which
has no thermodynamic sense. On this basis, a simple
algorithm is proposed to obtain the actual TDC position.
Two advantages can be underlined in relation with previous papers. Firstly, this new methodology is robust and
independent of heat transfer coefficient and mass losses.
Secondly, the algorithm is easy to implement and does
not generate numerical difficulties or errors. It has been
applied with success on a real engine.
9
p
V
θ
T
n
r
cp
cv
s
m
TDC:
IMEP
WOT:
∆ϕ
CR:
Mpa:
m3 :
°CA:
K:
-:
J/kg/K:
J/kg/K:
J/kg/K:
J/kg/K:
kg:
Qw
man:
upst:
W:
BAR:
°CA:
Pressure
Volume
Crank Shaft Angle
Temperature
Polytropic exponent
Mayer Coefficient
Specific Heat at Constant Pressure
Specific Heat at Constant Volume
Entropy
Mass inside the cylinder
Top Dead Center
Indicated Mean Effective Pressure
Wide Open Throttle
Phase Lag Error
Compression Ratio
Heat losses
Manifold
Upstream