in order to reach an acceptable level of accuracy. Also, the time spent on a complete 1D engine model is not negligible, not only in terms of simulation times for a good convergence, but also the time needed to set up experiments, measure data, do the posttreatment and model again. Another way to investigate wave action at the intake is to employfrequency basedanalysistechniquesthatmakeuse of reciprocating nature of the engine to access the frequency content and characterize a given system. Ohata and Ishida [4] and Matsumoto and Ohata [5] used analytical acoustic equations to characterize the behavior of the propagating waves at the intake written in terms of the Laplace variable. Using simple models for ducts, volumetric efficiency curves could be deduced as well as self-induction and interference effects of the different cylinders. This meant that such techniques can be used successfully to evaluate the unsteady behavior of the intake system. Harrison and Stanev [6] presented a technique based on wave decomposition methods using a pair of distant pressure sensors to evaluate forward and backward pressurecomponents.Thislead to the calculation of the reflectioncoefficient at the intake of a single cylinder engine and highlighted the fact that wave action at the intake, despite the large amplitudes of pressures, can be considered linear in nature. Concretely, this means that transfer function models which are linear parameterized models are quite capable of reproducing pressure traces. Desmet [7] derived an analytical relationship that links the air speed entering a combustion chamber to the pressure fluctuations knowing the geometry and speed of the piston. This equation was used to highlight the interest of delayed intake-valve closing, this isduetotheinertiaoftheairmassintheintake,knownastheinertial ram effect. Acoustic methodology was developed by Harrison et al. [8,9] where a linear acoustic model was presented for studying wave action at the intake. This differs from acoustics dedicated to radiated noise because the frequency range of interest is much lower and the wave propagation itself is coupled to inertial effects especially at the primary runners. The acoustic model considers the intake valve and throat area as the acoustic source and uses impedance characterization and mass flow through the intake valve to calculate acoustic pressure fluctuations. It was found that an inertial effect parameter [10] must be added to the modeling in
order to obtain correct amplitudes. The following work is based on identifying a transfer function that similarly uses mass flow through the intake valve to calculate pressure response; however it incorporates a ram parameter that takes into consideration inertial effects. This transfer function can be derived from a second order differential equation that characterizes pressure response as a function of the excitation mass flow. Fontana and Huurdeman [11] deduced such an equation of pressure and mass flow. They based their equation on an analogy between the column of air existing in the intake of an engine andelectricRLCcircuits.Chaletetal.[12]employedasimilarequation by making an analogy with a mechanical mass-spring-damper oscillatory system. This equation can be written in the form given by Eq. (1). This equation contains different parameters such as the relative pressure p, the transient mass flow excitation qm, the angular frequency x, the damping parameter e and the inertial parameter Xin.
1 x2
d2p dt2 þ2
e x
dp dt þp¼Xin
dqm dt ð1Þ The relative pressure response p is given by the following equation pabsolute ¼p0 þploss þp ð2Þ where p0 and ploss are respectively the initial pressure at equilibrium and pressure drop corresponding to a mean flow. Fontana and Huurdeman [11] proposed a new technique for engine simulation, it relies on a link between pressure and mass flow rate in the frequency domain. This was achieved by applying a Laplace transformation on Eq. (1) and arranging terms, Eq. (3) is thus obtained
TFðsÞ¼
PðsÞ QmðsÞ¼
Xins
s x 2
þ2e x sþ1 ð3Þ P(s) and Qm(s) are respectively the Laplace transform of the pressure and mass flow. The transfer function TF(s) in Eq. (3) is identified for a given geometry following an impulse excitation of mass flow. This is possible thanks to a unique test rig called the dynamic flow bench: the considered engine intake line is mounted on the
Nomenclature
BSFC brake specific fuel consumption c0 local speed of sound (m/s) C1, C2 propagation constants (mbar) F force of air mass (N) FFT fast Fourier transform fi frequency of the ith mode (Hz) Dfi resonant mode i bandwidth (Hz) G function found by curve fitting i harmonic frequency number j complex constant (j2 =1) L physical length of pipe (m) L corrected length of pipe (m) DL end correction (m) losscoef Pressure loss coefficient (mbar/kgh) p relative pressure (mbar) P(s) Laplace of pressure (mbar) p0 initial pressure at rest (mbar) pabsolute absolute pressure (mbar) ploss steady pressure losses (mbar) Pr Prandtl number Dp pressure rise hammer effect (Pa) qm mass flow (kg/h) Qm(s) Laplace of mass flow (kg/h)
s Laplace variable (s = jx) S cross section of the tube (m2) t time (s) TF transfer function (mbar/kgh) u velocity (m/s) Du velocity rise hammer effect (m/s) Xi inertial parameter for the ith mode (1/m) Xin total inertial parameter (1/m)
Greek letters x angular velocity (rad/s) e damping parameter q air density (kg m3) c ratio of specific heats (1.4) u momentum (kg m/s) a attenuation coefficient (Np/m)
Subscripts 0 Initial state absolute absolute pressure maximum maximum pressure loss constant pressure drop