# Optimal Control for a Hydraulic Recuperation System Using Endoreversible Thermodynamics
## 4 Endoreversible Model  
Based on the above described modeling approach we develop an endoreversible model of the mechanical energy recuperation system. A central element of that model is the cardan shaft as the connecting link between recuperation system, combustion engine and brakes. The endoreversible model of the hybrid drive train is shown in Figure 1.  
On the left side of the figure we can see the subsystems 1 to 4, which represent the recuperation system. Here, reservoirs 1 and 2 are the gas and the hydraulic fluid within the bladder accumulator, the dissipative engine 3 represents the hydraulic unit, and reservoir 4 represents the hydraulic fluid tank. The hydraulic fluid flow is modeled by a coupled reversible particle and entropy flux (\(J^{n}\) and \(J^{S}\), respectively). Such coupled fluxes can be efficiently treated by a so-called multi-extensity flux [29]. The reversible volume fluxes (\(J^{V}\)) represent the volume exchange at equal pressure between the inside and outside of the accumulator bladder. In case of the hydraulic unit--the dissipative engine 3--we consider two extensity transfers namely a combined particle and entropy transfer, as well as an angular momentum transfer. Here, the angular momentum flux \(J^{L}_{\text{hyd}}\) connects the hydraulic unit with engine 5 which represents the cardan shaft of the vehicle with angular velocity \(\omega\). Since the latter is assumed to not store but only transfer energy in a reversible manner, the modeling as a simple endoreversible engine is appropriate. Further, it is connected to the dissipative engine 6 which represents the internal combustion engine. The irreversible angular momentum flux \(J^{L}_{\text{brake}}\) towards an infinite reservoir with intensity \(\omega_{0}=0\) represents the energy absorption and dissipation at the conventional disc brakes, and a reversible angular momentum flux \(J^{L}_{\text{truck}}\) is carrying the transferred energy to accelerate or decelerate the truck. The internal combustion engine is modeled as a dissipative engine with torque output represented by the flux \(J^{L}_{\text{comb}}\) and with a power input \(P_{\text{fuel}}\) without specified carrying extensive from the work reservoir 7. Note that there is actually no direct angular momentum transfer between the cardan shaft and the disc brakes. The cardan shaft transfers the angular momentum of the hydraulic unit and the combustion engine towards the wheels which then transfer angular momentum to the disc brakes when braking. However, the simplified representation here suffices for the modeling. Having an overview of the model, we begin again from the left side of Figure 1 describing the interactions as well as the gas and the hydraulic fluid within the system. The gas within the bladder accumulator is modeled as a van der Waals gas and there is equilibrated pressure \(p_{1}=p_{2}\) and a constant overall volume \(V_{\text{acc}}=V_{1}+V_{2}\) within the accumulator--implemented by means of the reversible volume flux \(J^{V}\). Hence we can write  
Figure 1: Endoreversible model used for fuel consumption minimization. The recuperation system is represented by the gas 1 and the hydraulic fluid 2 within the bladder accumulator, the hydraulic unit 3 and the hydraulic fluid tank 4. Engine 5 represents the cardan shaft and engine 6 is the combustion engine connected to a work reservoir 7 representing the diesel fuel tank. Energy transfers via the cardan shaft are carried by angular momentum fluxes. The angular momentum flux towards the chassis of the vehicle with angular velocity \(\omega_{0}=0\) represents energy dissipation at the conventional disc brakes. \[\dot{V}_{1}=J^{V}=-\dot{V}_{2} \tag{6}\]  
for the volume flux between the gas and the hydraulic fluid. The hydraulic fluid is modeled as an incompressible liquid, which is a suitable assumption for liquids in general.