## 2 Models and Methods
### Emissions Model
#### 2.2.3 NO Model  
The oxides of nitrogen NO, N\({}_{2}\)O and NO\({}_{2}\) are all present in the exhaust of a spark ignition engine. However, nitrous oxide is not significant as it represents less than 1% of NO emitted. Nitrogen dioxide can be more present with the maximum occuring at  
Figure 6: Correlogram of HC emission with governing parameters. Figure 7: Comparison between \([HC]\) measurement and \([HC]\) prediction based on \([HC]\) model. wide open throttle and a rich mixture. Nevertheless, its concentration is less than a few percent of NO [25]. This is confirmed by [PERSON] et al. [21], which states that: \"NO is the predominant nitrogen oxide emitted by combustion devices\". In this context, the present work will focus on NO species. It can be produced by three distinct mechanisms:  
* Early NO are formed at the flame front and in fuel-rich areas. They result from the combination of nitrogen N\({}_{2}\) with hydrocarbon radicals (C, C2, CH\({}_{2}\) or CH\({}_{2}\)) to give the intermediate products HN, HCN, CN or CNH\({}_{2}\). These can recombine with oxygen to give nitrogen oxide NO [21]. * The NO-fuel are derived from the oxidation of the nitrogen atoms present in the fuel [26]. * Thermal NO is formed by the direct reaction of nitrogen and oxygen present in the air when the temperature and oxygen concentration are sufficient. This process is an important contributor to the total NO\({}_{\rm x}\) emissions and is known by the extended Zeldovich mechanism [5; 9; 27]. The rate of NO formation depends strongly on temperature as described by the Zeldovich mechanism. The temperatures necessary for NO formation are reached by the intense heat released during combustion (see Equation (3)). \[\frac{d[NO]}{dt}=\frac{6\cdot 10^{6}}{\sqrt{T}}\cdot\exp{(\frac{-69090}{T})} \cdot[O_{2}]^{1/2}\cdot[N_{2}] \tag{3}\]  
The heat release varies depending on the mass of fuel injected. In gasoline engines, this is mainly controlled by the intake pressure, which regulates the amount of mixture introduced into the cylinder. Figure 8 shows that NO emissions increase with intake pressure. This behavior will be modeled by a linear function. The spark advance also influences the maximum temperature reached during combustion. Indeed, the timing of the ignition with the compression phase of the gases by the piston influences the temperature and pressure at the beginning of combustion and during the propagation of the flame [28]. Higher pressure increases the volume concentration of the mixture and reduces the average free path of the molecules. Combustion is faster and the heat release is more intense, which increases the maximum combustion temperature and NO emissions. Figure 8 thus shows that NO emissions increase with spark advance. This influence will be expressed by a linear function. Equivalence ratio \(\phi\) also plays an important role in the formation of NO. Under slightly lean fuel conditions (0.9 \(\leq\phi\leq\) 0.95), the fuel consumed allows the temperature of the cylinder to increase sufficiently and since oxygen is present in significant quantities, this  
Figure 8: NO concentration as a function of control parameters (each curve represents a series of measurements where only the parameter in abscissa changes). favors the formation of nitrogen oxides. Leaner fuel conditions reduce the burned mass and therefore the maximum combustion temperature, which reduces NO emissions. Finally, a fuel-rich mixture reduces NO emissions because oxygen is consumed to produce CO or CO\({}_{2}\) via reactions that require less energy. The concentration of NO is therefore at a maximum and decreases in both rich and lean mixes. This corresponds well to the behavior observed in Figure 8 that we have modeled by a second degree function \((\phi-0.9)^{2}\). The correlogram (see Figure 9) also show a good correlation with \((\phi-0.9)^{2}\). The engine speed does not have a strong influence on NO emissions as shown in Figure 8. We have therefore chosen not to include it in our model.