## 2 Models and Methods
### Internal Combustion Engine Model
#### 2.3.1 Fuel Flow  
The air flow rate is proportional to the speed and intake pressure. The fuel flow rate \(m_{fuel}\) is itself proportional to the air flow rate, with the exception that it depends to the chemical composition of the mixture.  
To increase the accuracy of the model, the volumetric efficiency \(\eta_{vol}\) is introduced. It expresses the real efficiency of the engine compared to a perfect volumetric pump and therefore its capacity to introduce fresh gases (Equation (5)).  
\[\dot{m}_{fuel}=\frac{\phi}{\phi+(m_{air}/m_{fuel})_{st}}\cdot\eta_{vol}\cdot P_ {intake}\times 10^{5}\cdot V_{d}\cdot\frac{N}{60R_{c}}\cdot\frac{M_{gaz}}{R_{gaz} \cdot T_{adm}} \tag{5}\]  
The fill rate is highly dependent on the speed that controls the gas flow rate at the inlet. The flow is limited by flow friction, especially around the valves, which increases with speed (and fluid velocity). Moreover, at very high speed, a critical flow rate is reached and limits the flow around the valve during at least part of the intake phase (chocking phenomenon) [29].  
Conversely, filling at high speed can be improved by delaying the closing of the intake valve to take advantage of the inertia of the air column. This type of adjustment can result in intake backflow that degrades efficiency at low speed [5]. Finally, filling can be maximized for certain engine speeds thanks to an acoustic tuning that increases intake pressure by wave propagation phenomena [29].  
The speed-dependent filling therefore has a maximum and decreases at both low and high speed. The position of the maximum depends on design choices and in particular on the geometry of the nozzles [27; 30]. According to the studies, the filling rate evolves in a range of about 0.1 to 0.3 points and the corresponding maximum speed varies from 1250 to 4000 rpm [5; 12; 29; 30].  
Figure 11: Emission model at (\(N=2000\) rpm, \(\phi=0.9\) Bar)—relative spark advance \(\Delta_{SA}\) varies from \(-20^{\circ}\) to \(20^{\circ}\) (see Equation (14)).  
We have chosen to model the effect of engine speed on fill rate by a quadratic function which has an amplitude of 0.2 points and a maximum for an engine speed of 2250 rpm (Equation (6) and Figure 12).  
The filling rate is also influenced by the equivalence ratio of the mixture. This is because the fuel is injected as a liquid and vaporizes in the intake pipes. The heat consumed lowers the temperature and increases the density of the gases and the filling rate of the cylinder [5].  
We chose to model this effect by varying the maximum value of the filling ratio (Equation (7) and Figure 12). This value is equal to 0.9 for a filling rate less than 1 (excess air) and increases for excess fuel (filling rate greater than 1).  
\[\eta_{vol}=\eta_{vol,\phi}-1.4\times 10^{-8}\cdot(N-2250)^{2} \tag{6}\]  
\[\eta_{vol,\phi}=\begin{cases}0.9&\phi\leq 1\\ 0.5\cdot\phi+0.4&1<\phi\leq 1.2\end{cases} \tag{7}\]  
Finally, the engine speed has an influence on the intake temperature [5]. Indeed, at low speed, the residence time of gases in the intake system is more important, which favors thermal exchanges with the hot walls and increases their temperature [31]. At high speed, the gases do not have time to warm up and remain at room temperature.  
We have modeled this influence by a linear function of the speed, which has the largest temperature increase observed during the measurement campaign (16 \({}^{\circ}\)C) for the idle speed and decreases to room temperature for the maximum speed of 6000 rpm (Equation (8) and Figure 12).  
\[T_{intake}=T_{amb}-0.003\cdot N+18 \tag{8}\]  
We can see in Figure 13 that the model has a good fit to the measurements, with a determination coefficient of 0.96 and a mean error of less than 10%. Parameters values are listed in Table A3.  
Figure 12: Influence of parameters on fuel flow model.