Crank Angle Resolved White Box Modelling

A purely physics based model to represent the gas exchange process of the cylinder can be established by applying the modelling techniques for restrictions and volumes [57]. Figure 3.6 illustrates all components which are relevant for the cylinder gas exchange process.

Figure 3.6: Physics based cylinder model Adopted from [17]

The mass flows through the intake and exhaust valves are modelled by applying the equation set for compressible flow as defined by Equations 3.3 and 3.4. The Equation Set 3.33 gives the air mass flow across the intake valve [61]

() s ;) •BI) € ;!45 ;) • ~ } ˜_{P − 1 ™1 − €}2P ;_;!45 ) • d} ~e } š› ~ 3.33 () s ;!45 •BI) •P € 2 P + 1• }‚~ d} ~e

and the Equation Set 3.34 gives the mass flow from the cylinder into the exhaust system [61]. ( s ;!45 •BI!45 – ; ;!45— ~ } ˜_{P − 1 ™1 − –}2P _;; !45— d} ~e } š› ~ 3.34 ( s ;!45 •BI!45 •P € 2 P + 1• }‚~ d} ~e

Temperature inside the cylinder is derived from energy balance as described by Equation 3.35 [61]. bI!45 bF s y • () I) − ( I!45ž − I!45d () − ( e + C!− CD − ;!45bL_{bF | b}!45 _;BI!45 !45L!45 3.35

The mass inside the cylinder is solved from the mass balance, given by Equation 3.36 [17].

b !45

bF s () − ( 3.36

and pressure inside the cylinder is finally estimated from the ideal gas law as shown in Equation 3.37 [17].

;!45s !45_LBI!45

describe the kinematics of the piston, camshaft, and the valves. These equations deliver the required information of the actual cylinder volume _L!45, its derivative

²³´_{, as well as the physical open areas of the intake and exhaust valves}

) and

respectively. The required equations can be found in Heywood [59], Pezouvanis [62] and Ferguson [61].

Equations 3.33 to 3.37 describe the gas exchange process of the cylinder. The equation set indicates the complexity of the process and allows a technical analysis of all factors which have an influence. The equation set reveals that there are seven factors which have a direct influence on the final cylinder air charge at Intake Valve Closing (IVC). Those factors are discussed in the following:

• Flow conditions upstream from the intake valve: The pressure difference across the intake valve drives the air mass flow into the cylinder described by Equation 3.33. Consequently, pressure and temperature upstream from the intake valve have a very strong influence on the amount of fresh air inside the cylinder at IVC.

• Flow conditions downstream from the exhaust valve: The pressure difference across the exhaust valve drives the mass flow from the cylinder into the exhaust system as described by Equation 3.34. Consequently, the higher the back pressure in the exhaust system, the higher the pressure inside the cylinder at EVC. A higher in-cylinder pressure at IVO reduces the amount of fresh air that can be induced during the induction stroke. • Valve timing: The timing of the intake and exhaust valve actuate the mass

flows into and out of the cylinder. The profile of the camshaft and the timing with respect to the engine crankshaft determine the physical open area ₎ and in Equations 3.34 and 3.35 respectively. Valve lift, lift duration as well as the timing of EVO, EVC, IVO and IVC have a significant impact on the amount of fresh air inside the cylinder by the end of induction. The reader is referred to [59] for more detailed information about the impact of valve timing upon the gas exchange process.

o EVO: Early exhaust valve opening helps emptying the cylinder from exhaust gases due to the high in-cylinder pressure. This allows reducing the mass of the residual gases.

o EVC: The most important factor, which determines the amount of residual gases, is the Exhaust Valve Closing (EVC) event. A too

early EVC prohibits the exhaust gases from leaving the cylinder while a too late EVC causes exhaust gases to be sucked back into the cylinder as the piston moves from TDC to BDC.

o IVO: Intake valve opening has a very small impact on the air charge. However, early IVO causes overlap where intake and exhaust valves are open at the same time. This enables blow through of fresh air into the exhaust system or flow of residual gases into the intake manifold. Which phenomenon occurs depends on the current pressures _{; , ;!45} and _;> .

o IVC: Intake valve closing is the most important valve event for air charge. Late IVC can increase the cylinder filling at high engine speeds where _;!45 at BDC is still smaller than _{; due to the flow} restriction across the valves. However, at low engine speed late IVC causes backflow from the cylinder into the intake manifold as the piston moves from BDT to TDC.

• Valve design: The size and the geometry of the intake and exhaust valve affect the flow restriction of the valve. This determines the flow coefficient of each valve in Equations 3.34 and 3.35 respectively which directly affects the mass flows into and out of the cylinder.

• Intake and exhaust system geometry: The length of the intake and exhaust runners, as well as the size of the plenums in the intake and exhaust system, determine the propagation of the pressure waves through the air-path system. Tuning the length of the pipes in the exhaust system allows the reduction of pressure downstream from the exhaust valve to enhance emptying the cylinder. This is achieved if an expansion wave arrives at the exhaust valve just before EVC. Tuning of the intake system allows to increase the pressure upstream from the intake valve just before IVC which can significantly increase the cylinder air charge. This is achieved if a pressure wave arrives at the intake valve just before IVC. Tuning the length of the pipes is straightforward since the pressure waves always travel with the speed of sound.

• Charge heating: Heating of the air inside the cylinder during the intake stroke reduces the density of the air charge, which consequently reduces

o Heat transfer: Heat transfer from the cylinder walls and the piston into the air increase the in-cylinder temperature as described by Equation 3.35. This phenomenon is primarily dependent on cylinder wall temperature _ID and the temperature upstream the intake valve, I) . A higher wall temperature increases the in-cylinder heat

transfer due to the increased temperature difference between _I!45 and _ID. A higher temperature _I) reduces the heat transfer due to the reduced difference between _I!45 and _ID.

o Mixing with residual gas: The in-cylinder temperature at EVC defines the temperature of the residual gas which mixes with the fresh gas. The higher the temperature at EVC, the higher is the temperature of the mixed gas at IVC.

• Engine speed: The local velocities of the gases along the engine air-path are proportional to the mean piston speed [59]. Consequently, engine speed affects the cylinder air charge in a number of different ways.

o Charge heating: The duration of the induction stroke is proportional to the speed of the engine. For this reason, the impact of charge heating reduces with increasing engine speed, since less time is available to heat the induced air in the cylinder.

o Friction losses: The friction losses across valves increases as the square of the velocity. Specifically, at high engine speeds the flow at the end of the induction stroke becomes choked as described in Equations 3.33 and 3.34. Once this occurs, the air mass flow into the cylinder can only be increased by a higher pressure upstream the intake valve.

o Induction ram: Since the pressure waves always propagate with the speed of sound, tuning of the intake and exhaust system is limited to a specific engine speed range. The length of the runners determines at which engine speed the pressure wave arrives just before IVC which increase the cylinder air charge. At other engine speeds this effect, which is also known as induction ram [147]– [149], is not present or might even have a negative effect in case an expansion wave arrives just before IVC.

The physics based cylinder model (Equations 3.33 to 3.37) allows accurate cylinder air charge predictions since it includes most physical effects. However, accurate predictions require accurate models for the flow coefficients of the intake and exhaust valve, as well as an accurate representation of the pressure wave propagation in the intake and exhaust systems which requires gas dynamic models. As mention in Subsection 3.3.2, this involves time consuming solvers and a small simulation step size.