Determination of TDC in internal combustion engines by a newly developed thermodynamic approach. Part 1: base theory

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DETERMINATION OF TDC IN INTERNAL COMBUSTION ENGINES BY A NEWLY DEVELOPED THERMODYNAMIC APPROACH PART 1: BASE THEORY Emiliano Pipitone, Alberto Beccari Dipartimento di Meccanica – University of Palermo

ABSTRACT It is known to internal combustion engine researcher that in-cylinder pressure analysis represents an indispensable tool in the research and development process, and must be carried out with maximum accuracy. A key factor is the precise knowledge of the Top Dead Centre (TDC) position, since a phase error of just 1 degree between in-cylinder pressure and volume can cause large errors in the estimation of the most important performance related parameters. It is also known that the TDC position does not coincide with the location of the peak pressure (LPP) and their angular lag is known as the “loss angle”, being related to the two losses: heat transfer with wall and mass leakage. To gain an acceptable accuracy, the TDC position should be known within the precision of 0.1 degrees: this can be accomplished by means of an expensive dedicated sensor or employing a “thermodynamic method”, i.e. an algorithm whose input is the in-cylinder pressure measured under motored condition. In this paper the authors, starting from the study of the compression-expansion process in a motored cylinder, developed an original thermodynamic method, based on the evaluation of an expressly defined “loss function”, i.e. a function whose derivative although combines both losses, can be simply evaluated once the engine geometry and the motored incylinder pressure are known. The determination of the loss angle, and hence of the TDC position, can be performed through the evaluation of the loss function variation at the peak pressure point. The key point of the new method relies on its self-referencing procedure, which allows an adequate evaluation of the TDC position without a direct estimation of both mass leakage and heat transfer: this has been demonstrated, in the appendix, by means of models commonly used to describe the two losses. Keywords: Top dead centre determination, Spark ignition engine, Compression ignition engine

INTRODUCTION In-cylinder pressure analysis is nowadays an indispensable tool in internal combustion engine research and development. It allows the measure of some important performance related parameters, such as indicated mean effective pressure (IMEP), mean friction pressure, indicated fuel consumption, heat release rate, mass fraction burned, etc.. Moreover, future automotive engine will probably be equipped with in-cylinder pressure sensors for continuous combustion monitoring and control, in order to fulfil the more and more strict emission limits. For these reasons, in-cylinder pressure analysis must be carried out with maximum accuracy, in order to minimize the effects of its characteristic measurement errors. The exact determination of crank position when the piston is at top dead centre (TDC) is of vital importance, since a 1 degree error can cause up to a 10% evaluation error on IMEP and 25% error on the heat released by the combustion: the position of the crankshaft (and hence the volume inside the cylinder) should be known with the precision of at least 0.1 crank angle degrees, which is not an easy task, even if the engine dimensions are well known: it corresponds to a piston movement in the order of one tenth of micron, which is very difficult to estimate. A good determination of the TDC position can be pursued by means of a dedicated capacitive TDC sensor, which allows a dynamic measurement (i.e. while engine is running) within the required 0.1 degrees precision. Such a sensor has a substantial cost and its use is not really fast,

since it must be fitted in the spark plug or injector hole of the cylinder. A different approach can be followed using a thermodynamic method, whose input is in-cylinder pressure sampled during the compression and expansion strokes: some of these methods, more or less valid, can be found in literature [1,2,3,4,5,6,7]. This paper will discuss a new thermodynamic approach to the problem of the right determination of the TDC position.

BASE THEORY The compression and expansion processes in a motored (i.e. without combustion) engine can be described by observing the energy transformations regarding the unity mass which remains in the cylinder. The first law of thermodynamics states that:

δ q − pδ v = δ u

(1)

where δq represents the specific heat received by the gas from the cylinder walls during the crank rotation δϑ, p and v represent the gas pressure and specific volume, and δu the specific internal energy variation. The gas involved in the process can be assumed to be a perfect gas, thus the following equations are also valid:

dp dv dT ⎫ + = p v T ⎪ ⎪ ⎬ ⎪ δu = cV δT ⎪ ⎭

p v = R' T ⇒

(2)

being R’ and T the gas constant and temperature respectively, while cV and cP represent the constant volume and constant pressure specific heat, which are functions of the gas temperature. The influence of flow-friction during the compression-expansion process in a reciprocating machine is not significant, since the fluid is macroscopically “motionless”. Then heat transfers are largely more important than flow-friction losses, hence, according to the second law of thermodynamics, the specific entropy variation δS of the in-cylinder gas during the crank rotation δϑ is: δq δS = (3) T from equation (1) and (2) the specific entropy variation thus results:

δS =

δq T

= cP

δv v

+ cV

δp p

(4)

Due to mass leakage through valve seats and piston rings, the available volume V for the in-cylinder gas increases, hence its specific volume changes:

V = v⋅m ⇒

dv dV dm = − v V m

(5)

where m represents the in-cylinder mass. Hence, considering the finite increment “δ” due to a crank rotation δϑ, the specific entropy variation in equation (4) will now result:

δS = cP

δV V

+ cV

δp p

− cP

δm m

(6)

being δm the mass entering the cylinder (hence δm≤0) during the crank rotation δϑ; hence the incylinder pressure changes is:

δp =

1 [δQ(k − 1 ) − kpδV ] + k p δm V m

(7)

where δQ=m δq represents the heat received by the gas and k the isentropic coefficient =cP/cV. In an ideal adiabatic motored engine both δQ and δm would be zero, and pressure would reach its maximum (δp=0) when the volume is minimum (δV=0): the compression and the expansion strokes would cause in-cylinder pressure variations symmetric with respect to TDC and the position of the peak pressure (which can be easily determined with 0.1 precision by means of polynomial interpolation of the pressure curve sampled with 1 crank angle degree resolution) would coincide with the position of the TDC. Vmin pmax Tmax

Twall

Vmax

Fig. 1 Temperature-Entropy diagram of the compression-expansion process in a motored cylinder: ideal engine (segments AB and BA) and non-adiabatic engine (dashed curve)

Fig. 2 Temperature-Entropy diagram of the non-adiabatic compression-expansion process: peak pressure (point D) occurs before the TDC (point E)

In a Temperature-Entropy (T, S) diagram these two evolutions would be represented by two coincident segments (AB and BA in Fig. 1). In a real motored engine the pressure variation is caused both by volume changes and by two phenomena related to the real machine, i.e. the heat received by the gas from the cylinder walls δQ (which is negative when the gas temperature is higher than wall temperature, i.e. δQ∝(Twall–T)) and the mass leakage δm (which is negative whenever in-cylinder pressure is higher than outer pressure): equation (7) clearly shows that both of them cause pressure rise to be zero when the volume changes are still negative (i.e. during compression). Both heat transfer and gas leakage cause the pressure curve to be asymmetric with respect to the TDC, shifting the Location of the Peak Pressure (LPP) in advance with respect to the TDC position (see the non-adiabatic evolution in Fig. 2 or a real pressure curve in Fig. 3): the angular distance between LPP and the TDC position is called “loss angle”, being related to the energy and mass losses, and usually assumes values between 0.4 and 1 CA degree, depending on the entity of the heat transferred and the escaped mass:

ϑloss= LPP−LTDC

(8)

Equation (6) shows that two easily measurable quantities, the in-cylinder pressure and volume, allow the evaluation of the entropy variation (i.e. heat transfer) together with the mass leakage by means of the functions δV/V and δp/p, which are plotted as example in Fig. 5.; defining the “loss function” F so that:

8.5

65

Pressure [bar]

60 7.5 6.5

δF = cP

70 Pressure Volume

ϑLOSS

55 50 45

5.5

40 4.5

Volume [cm 3]

9.5

δV V

it will result:

δF = δS + c P

+ cV

δm m

δp p

(9)

(10)

35 [CAD ATDC]

The entity of the variation of the “loss function” δF, which gathers the -30 -20 -10 0 10 20 30 sum of the two “losses”, is then Fig. 3 In-cylinder pressure and volume near TDC determined by the capability of the (MAP = 0.4 bar, n=1000 rpm, ρ=10) cylinder walls to exchange heat with the gas and by the amount of gas 0.2 escaping from the cylinder. The 0.0 qualitative progress of the “loss -0.2 function” variation in a real cylinder during a compression-expansion -0.4 process, together with its two -0.6 dFF δ constitutive terms δS and cP δm/m, is dS = δdQ q/T/ mT -0.8 δS= shown for example in Fig. 4 for a cp cpdm/m δm/m -1.0 compression-expansion process in a ideal adiabatic real cylinder: the entropy variation -1.2 -180 -120 -60 0 60 120 180 starts with a positive value (being CAD ATDC T