## 2 Models and Methods
### Emissions Model  
#### 2.2.1. CO Model  
Figure 2 is a graphical illustration of the measurements of CO concentration. Each graph represents the CO concentration as a function of one control parameter, i.e., (a) engine speed, (b) fuel/air equivalence ratio, (c) spark advance and (d) intake pressure. Each curve represents a series of measurements where only the parameter in abscissa changes.  
The fuel is oxidized during combustion producing mainly CO, particularly abundant around the flame front [21]. This species is then itself oxidized in a second step to give carbon dioxide if the temperature and the quantity of oxygen are sufficient. [PERSON] expresses CO concentration only as a function of the equivalence ratio that controls the amount of oxygen present in the cylinder [16]. This approach is consistent with the experimental results obtained as shown in Figure 2. Indeed, the curves representing the series where the speed, spark advance and pressure vary, are very close to horizontal straight lines. Whatever the value chosen for the variable on the abscissa, CO concentration is constant. Figure 3 is a correlogram between intake pressure, CO concentration, fuel/air ratio \(\phi\), engine speed N and Spark Advance SA. It allows analyzing the relationship between each pair of numeric variables in a dataset. We notice that neither spark advance, nor engine speed or intake pressure show high correlation factor with CO concentration. [PERSON] [16] proposes to use a polynomial of degree 5 to model the [PERSON] of the CO concentration on fuel/air ratio. [PERSON] and [PERSON] [13] use the root of a polynomial of degree 2, which gives a suitable shape by reducing the number of coefficients to be adjusted. We have therefore chosen to use the form proposed by [PERSON] and [PERSON] (Equation (1))  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Intake manifold pressure (bar)** & & 0.3 & 0.5 & 0.7 & [PERSON] & 1 & \\ \hline
**Engine speed (rpm)** & 800 & 1000 & 1500 & 2000 & 3000 & 4000 \\ \hline
**fuel/air eq. ratio (-)** & 0.7 & 0.8 & 0.85 & 0.9 & 0.95 & 1 & 1.1 & 1.2 \\ \hline
**Spark Advance (*BTDC)** & & & & \(-10:5:50\) & & & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Operating conditions.  
Figure 2: CO concentration as a function of control parameters: (**a**) engine speed, (**b**) fuel/air equivalence ratio, (**c**) spark advance and (**d**) intake pressure. Each curve represents a series of measurements where only the parameter in abscissa changes.  
and to adjust the coefficients with the least squares method. In Equation (1), \(\phi\) represents the fuel/air equivalence ratio, \(co_{1}\) and \(co_{2}\) are the adjusting parameters. Symbols used in the document are described in the nomenclature section and values of the coefficients are listed in Table 1.  
\[[CO]=co_{1}\bigg{(}\phi-1+\sqrt{(\phi-1)^{2}+co_{2}}\bigg{)} \tag{1}\]  
Figure 4 shows that the proposed fit is in good agreement with the measurements. A large number of observations are concentrated around the origin of the graph which corresponds to low values of CO concentration. Indeed, a major part of the data have a fuel/air ratio lower than 1 (lean fuel mixture), so there is enough oxygen to convert CO into CO\({}_{2}\). The coefficient of determination is 0.99 and the mean error between the data and the predictions is 31%.  
Figure 4: Comparison between \([CO]\) measurement and \([CO]\) prediction based on \([CO]\) model.  
Figure 3: Correlogram of CO emission with governing parameters.