## 2 Models and Methods
### Internal Combustion Engine Model
#### 2.3.3 Exhaust Temperature  
Last but not least exhaust temperature model has to be defined with regards to its action on catalytic converter behavior. Figure 16 represents the measurements. In the bottom left graphic, the exhaust temperature is described versus the relative spark advance. It should be noted that this was not the case for emissions plots presented above.  
The temperature rise at the engine outlet depends on the proportion of energy involved in combustion that remains in the exhaust and the thermal capacity of the gases. The proportion of energy still present in the exhaust depends on the engine's close control parameters. The exhaust temperature depends on the temperature rise and the intake temperature (Equation (15)). The latter is calculated using the same formula as for the flow model (Equation (8)).  
\[T_{exh}=T_{intake}+\frac{k_{\phi}\cdot k_{SA}\cdot k_{N}\cdot k_{Pintake}\cdot LHV }{C_{p,gaz}\cdot(1+\frac{(m_{air}/m_{fad})_{it}}{\phi})} \tag{15}\]  
Engine speed and intake pressure control the flow of gases into the engine. The higher the flow rate, the less time available for heat exchange with the walls relative to the quantities involved [5; 12]. Heat losses at the walls decrease and the exhaust temperature  
Figure 16: Exhaust temperature as a function of control parameters (each curve represents a series of measurements where only the parameter in abscissa changes).  
Figure 15: Comparison between \(\Gamma_{ICE}\) measurement and \(\Gamma_{ICE}\) prediction based on \(\Gamma_{ICE}\) model.  
increases. This behavior is apparent in Figure 16, and we have chosen to model the influence of these parameters with two linear functions (Equations (16) and (17)).  
\[k_{N}=a_{N}+b_{N}\cdot N \tag{16}\]  
\[k_{Pintake}=a_{Pintake}+b_{Pintake}\cdot P_{intake} \tag{17}\]  
The influence of equivalence ratio on exhaust temperature is similar to its effect on maximum combustion temperature (see Section 2.2.3) [5]. Both excess fuel and excess air form an inert mass that lowers the temperature in the cylinder and at the exhaust. The latter has therefore a maximum around the stoichiometric air/fuel ratio as shown in Figure 16. As the temperature model already depends on the equivalence ratio by the multiplicative factor in the denominator, we have added a linear function of the equivalence ratio to the numerator (Equation (18)) which allows obtainign a parabolic shape for the global influence of this parameter.  
\[k_{\phi}=a_{\phi}+b_{\phi}\cdot\phi \tag{18}\]  
Finally, the later the ignition occurs (negative relative spark advance, i.e., spark retard), the more the combustion shifts towards the opening of the exhaust valve and the more the temperature increases, as shown in Figure 16. We have chosen to model this effect with a parabola (Equation (19)).  
\[k_{SA}=a_{SA}+b_{SA}\cdot(\Delta_{SA}-\Delta_{SA,min})^{2} \tag{19}\]  
The 9 model fitting parameters were determined using the method of least squares (2 constants for speed, pressure and fuel/air ratio, and 3 for spark advance). The Figure 17 shows the shapes of the coefficients which express the portion of residual energy at the exhaust. The global contribution of the wealth (in orange) corresponds to the (scaled) ratio between \(k_{\phi}\) and the coefficient depending on \(\phi\) in the denominator. The model corresponds well to the measurements as can be seen in Figure 18, with a determination coefficient of 0.94 and a mean error of 5.6%.  
Figure 17: Influence of parameters on exhaust temperature model.