## 2 Materials and Methods
### Proposed Cabin Thermal Model
Equation (6) considers the heat transfer direction depending on the difference between the interior air temperature and one of the windows; if such a difference is positive (\(sign(T_{i1}-T_{w1})=+1\), i.e., \(T_{i1}-T_{w1}>0\)), the heat transfer occurs from the interior air to the windows and in the opposite direction if it is negative. \(\dot{Q}_{12}\) is the heat load due to the air temperature difference between Zones 1 and 2, which causes air circulation from one zone to the other due to the stack effect [31], and it is calculated from Equation (7). \[\dot{Q}_{12}=\dot{V}_{12}\cdot\rho_{i}\cdot c_{pi}\cdot(T_{i1}-T_{i2}) \tag{7}\] \begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Value** \\ \hline \(V_{i1}\) & 12 m\({}^{3}\) \\ \(V_{w1}\) & 0.0425 m\({}^{3}\) \\ \(\rho_{i}\) & 1.225 Kg\(\cdot\)m\({}^{-3}\) \\ \(\rho_{glass}\) & 2.5\(\cdot\)10\({}^{3}\) Kg\(\cdot\)m\({}^{-3}\) \\ \(c_{pi}\) & 1005 \(\cdot\)Kg\({}^{-1}\cdot\)K\({}^{-1}\) \\ \(c_{pglass}\) & 850 \(\cdot\)Kg\({}^{-1}\cdot\)K\({}^{-1}\) \\ \(\beta\) & 1 \\ \(A_{12}\) & 6.25 m\({}^{2}\) \\ \(H\) & 2.5 m\({}^{2}\) \\ \(A_{front1}\) & 7 m\({}^{2}\) \\ \(A_{left1}\) & 2 m\({}^{2}\) \\ \(A_{right1}\) & 2 m\({}^{2}\) \\ \hline \end{tabular}
\end{table} 
Table 2: Values of model parameters from equations regarding Zone 1. Volumetric air flow \(\dot{V}_{12}\) can be calculated from Equation (8) [31], which takes into account the air transfer from the warmer zone to the colder one,  
\[\dot{V}_{12}=sign(T_{i1}-T_{i2})\cdot 0.03\cdot A_{12}\cdot\sqrt{g\cdot H\cdot|T_{i1 }-T_{i2}|/T_{ref}} \tag{8}\]  
where \(A_{12}\) is the contact area between the two zones, \(H\) is the height of that area, \(g\) is the acceleration of gravity and \(T_{ref}=T_{i1}\) if \(T_{i1}-T_{i2}>0\) or \(T_{ref}=T_{i2}\) otherwise; the factor 0.03 has been determined using least-squares estimation from an actual coach's experimental data. Airflow direction is also considered the same way as in Equation (6), but in this case from the difference \(T_{i1}-T_{i2}\) between interior air temperatures of both zones. \(\dot{Q}_{owl}\) is the convective heat transfer between the outside air and the windows embedded in the coach body in Zone 1; it can be calculated from Equation (9),  
\[\dot{Q}_{owl}=sign(T_{o}-T_{w1})\cdot U_{ow}\cdot A_{w1}\cdot|T_{o}-T_{w1}| \tag{9}\]  
where \(T_{o}\) is the outside air temperature, which is another model input, \(U_{ow}\) is the overall heat transfer coefficient between the outside air and the vehicle windows in Zone 1 and \(A_{w1}\) is the surface of those windows. As in previous cases, Equation (9) also considers that heat transfer may occur from any side to the other. Finally, \(\dot{Q}_{sun1}\) is the heat transfer due to incident solar radiation over the windows in Zone 1 and is calculated using Equation (10),  
\[\dot{Q}_{sun1}=\alpha_{w}\cdot\sum I_{side}\cdot A_{side1}\quad,\ side=(front, right,left) \tag{10}\]  
where \(\alpha_{w}\) is the absorptivity of the windows, \(I_{side}\) is the incident solar irradiance on each side of Zone 1 (front, left and right) dependent on the environmental conditions and \(A_{side1}\) is the surface of the corresponding window section. Obviously, \(A_{w1}=A_{front1}+A_{left1}+A_{right1}\). Energy balance equations for Zone 2 (passengers) can be similarly stated as Equations (1) and (2). Specifically, Equations (11) and (12) describe the effect of heat loads and heat transfers on the interior air temperature of Zone 2 (\(T_{i2}\)) and the temperature of vehicle windows in that zone (\(T_{w2}\)). \(C_{2}\) and \(C_{w2}\) are the heat capacities of the air in Zone 2 and of the windowpanes of that zone and are calculated using Equations such as (3) and (4).