## 2 Models and Methods
### Emissions Model
#### 2.2.2 HC Model  
Unburned hydrocarbons in gasoline engines mainly come from 4 mechanisms: flame quenching at the cylinder walls, entrapment of the unburned mixture in dead volumes (crevices), adsorption-desorption of the fuel in the oil film that covers the walls, and incomplete combustion or misfire [5].  
Equivalence ratio has an influence on the emissions of unburned hydrocarbons. Indeed, if the mixture is very low in fuel, the flame speed is reduced which can lead to misfire and higher cyclic variability leading to significant increase in HC concentration. With a rich mixture, some of the fuel remains after combustion when all the oxygen has been used to produce carbon monoxide or dioxide and water. Furthermore, increasing \(\phi\) results in the raise of fuel vapor concentration in the bulk gas which will increase the absorption and hence the desorption from the oil layer [22]. Therefore the concentration of HC increases in both rich and lean mixes. This corresponds well to the behavior observed in Figure 5. Thus, we decided to introduce an augmented variable which is modeled by a second degree function around a minimum \(\phi\) value of 0.9.  
The spark advance changes the HC emissions due to the storage of fresh gases in the crevices volumes and in the oil film on the walls [23]. This is because trapped hydrocarbons can be released and oxidized during the expansion phase if the temperature and amount of oxygen are sufficient. When the spark advance is high, combustion ends earlier, which does not allow efficient oxidation and increases HC concentrations. Figure 5 shows this behavior, which we have modeled by a linear function.  
The speed has an influence on the internal aerodynamics of the engine. Increasing the speed improves the homogeneity of the mixture and promotes turbulence during combustion. The flame speed increases which improves combustion efficiency and reduces HC emissions [24]. Figure 5 shows this effect, which we modeled using a linear function.  
When the mixture is lean, the fuel concentration is lower and the average free path of the molecules increases. On the other hand, when the intake pressure decreases, the fraction of residual gas in the cylinder increases which also contributes to dilute the fresh gas. Simultaneously, a decrease in the wall temperature is observed [22]. Low intake pressure can thus lead to incomplete combustion and increased HC emissions, especially in combination with lean burning. We have chosen to model this influence with a linear function.  
As shown in Figure 6, HC emissions are positively correlated with Spark Advance SA and the augmented variable \((\phi-0.9)^{2}\), and negatively correlated with intake pressure \(P_{intake}\) and engine rotation speed \(N\). The complete HC model expresses the causalities with the variables where \(hc1\) to \(hc4\) are the adjusting parameters. It is a linear model saturated  
Figure 5: HC concentration as a function of control parameters (each curve represents a series of measurements where only the parameter in abscissa changes).  
to avoid negative concentrations that are not physically correct. Its expression is given by the Equation (2).  
\[[HC]=\max(0,\;\;hc_{1}\cdot N+hc_{2}\cdot(\phi-0.9)^{2}+hc_{3}\cdot SA+hc_{4}\cdot P _{intake}+hc_{5}) \tag{2}\]  
The values for the different parameters are listed in Table 11. Figure 7 shows that the proposed adjustment correlates with the observations. The coefficient of determination is 0.74 and the mean error between the measurements and the estimate is 27%. About ten points have measured HC emissions higher than 840 ppm and are slightly off-centre with respect to the fit line between the model and the data. These points correspond to low equivalence ratio (\(\phi<0.8\)) and low or intermediate intake pressures (\(P_{intake}\leq 0.5\) bar) except for one point (\(\phi=0.65\), \(P_{intake}=1\) bar).