# A Hybrid Model for Freight Train Air Brake Condition Monitoring
## 2 Materials and Methods
### Main Components Implemented in the Model
#### 2.3.7 Brake Release Valve  
The release phase is modelled through the connection of the brake cylinder to the atmosphere by the brake release valve (identified with (7) in Figure 6). The release phase begins when the distributor pressure drops below the release activation threshold (identified as Sact in Table 1). This parameter has been empirically tuned to model the mechanical delay during the release process, causing the release activation time in the main brake pipe to differ from that in the brake cylinder. The orifice mass flow rate has been modelled using Equation (3) as a function of the brake cylinder pressure (\(\text{p}_{\text{bc}}\)), the brake release valve inlet temperature (identified as \(\text{T}_{\text{in}}\) in Table 1) and the atmospheric pressure (identified as \(\text{P}_{\text{atm}}\) in Table 1). While the brake cylinder pressure is iteratively calculated by the model at each simulated time, the inlet temperature is considered constant since the efflux is modelled as an isothermal transformation [35]. Additionally, atmospheric pressure is treated as a  
Figure 16: Alimentation valve opening model for braking phase.  
Figure 15: Simplified scheme of the relay valve.  
constant physical parameter. The orifice area (identified as A in Table 1 for freight brake mode) has been experimentally estimated based on the observed releasing time.  
\[\dot{\text{m}}=\text{AC}_{\text{q}}\text{C}_{\text{M}}\frac{\text{P}_{\text{R} _{\text{c}}}}{\sqrt{\text{T}_{\text{in}}}} \tag{3}\]  
where C\({}_{\text{M}}\) is a correction factor that is calculated from an isentropic approach to the problem [36], as shown in Equation (4).  
\[\text{C}_{\text{M}}=\begin{cases}\sqrt{\frac{2\gamma}{\text{R}(\gamma-1)}} \sqrt{\left(\frac{\text{P}_{\text{R}_{\text{c}}}}{\text{P}_{\text{atm}}} \right)^{\frac{2}{\gamma}}-\left(\frac{\text{P}_{\text{R}_{\text{c}}}}{\text{ P}_{\text{atm}}}\right)^{\frac{(\gamma+1)}{\gamma}}},\text{subsonic}\\ \sqrt{\frac{2\gamma}{\text{R}(\gamma-1)}}\left(\frac{2}{\gamma+1}\right)^{\frac {1}{(\gamma-1)}},\text{supersonic}\end{cases} \tag{4}\]  
Due to the irreversibility of the thermodynamic process, an empirical correction factor C\({}_{\text{q}}\) evaluated according to the Perry correlation (proposed in [23]) is considered, as shown in Equation (5).  
\[\begin{split}\text{C}_{\text{q}}&=0.8414-0.1002\left( \frac{\text{P}_{\text{R}_{\text{c}}}}{\text{P}_{\text{atm}}}\right)+0.8415 \left(\frac{\text{P}_{\text{R}_{\text{c}}}}{\text{P}_{\text{atm}}}\right)^{2} -3.9\left(\frac{\text{P}_{\text{R}_{\text{c}}}}{\text{P}_{\text{atm}}}\right) ^{3}\\ &\quad+4.6001\left(\frac{\text{P}_{\text{R}_{\text{c}}}}{\text{P}_{ \text{atm}}}\right)^{4}-1.6827\left(\frac{\text{P}_{\text{R}_{\text{c}}}}{ \text{P}_{\text{atm}}}\right)^{5}\end{split} \tag{5}\]  
The saturation of the efflux orifice area is simulated with a similar approach to the algorithm presented in Figure 16 to account for the command pressure during releasing. The cross-sectional area of the orifice can be changed to account for the P I G mode different releasing time of around 17.5 s for passengers and 52.5 s for freight [33]. Knowing the orifice mass flow rate, the continuity equation can be applied to evaluate the brake cylinder pressure at the brake release valve. This is achieved by specifying the brake cylinder volume, represented as Vmin or Vmax in Table 1, depending on whether the efflux occurs with the minimum or maximum brake cylinder volume. These parameters were obtained from the wagon's technical specification provided by Mercitalia Intermodal.