## 2 Consideration of the Energy Losses for Converting Chemically Stored Energy into Electrical Energy  
To justify the pros and cons for different piston trajectories regarding the thermodynamic and electric efficiency of the free-piston engine, a loss analysis is used. Depending on the piston trajectory, different losses may result in terms of combustion characteristics, heat losses, gas exchange, compression and expansion characteristics. The chemical fuel energy \(E_{fuel}\), which enters the engine via the injector, represents the maximal supplied energy for the engine and is the basis of the loss analysis. This energy only depends on the injected mass \(m_{fuel}\) and the lower heat value \(LHV\) according to:  
\[E_{fuel}=m_{fuel}\cdot LHV. \tag{1}\]  
This chemical energy is transformed to thermal energy during the combustion process. Due to the fact that an ideal combustion is not realizable, there is always an amount of unburned fuel \(m_{unb}\), which leads to an energy loss during the transformation from chemical to thermal energy \(E_{th}\). Besides the imperfect combustion, incomplete combustion could occur when injected fuel is not completely thermally converted and oxidized. When fuel reaches the cylinder wall or the upper face of the piston during combustion, the flame extinguishes and is, thus, not available for further thermal conversion. Out of the difference of the incomplete converted fuel and the injected fuel, incomplete combustion efficiency can be derived. The parameter \(\eta_{comb}\) describes the efficiency of the combustion based on the converted fuel mass, whereby the thermal energy yields to:  
\[E_{th}=\left(m_{fuel}-m_{unb}\right)\cdot LHV=\eta_{comb}\cdot E_{fuel}. \tag{2}\]  
This thermal energy is the basis for the volume change work. Due to the fact that an energy loss analysis is used for the comparison, the volume change work is called here the volume change energy \(E_{cc}\). The combustion process creates high temperature gradients, which lead to an energy loss through the walls \(E_{val}\). This is described by the heat flux through the walls of the combustion chamber, which is defined by:  
\[E_{val}=\alpha\cdot A\cdot\Delta T\cdot\Delta t. \tag{3}\]where \(\alpha\) is the heat transfer coefficient (HTC), \(A\) is the area of the combustion chamber, \(\Delta T\) is the temperature gradient between the surrounding solid material and the gas inside the combustion chamber and \(\Delta t\) is the time in which the process is performed. All parameters are space- and time-dependent. The parameters in Equation (3) are mean values over one work cycle to simplify the formulas for a better understanding [32]. The time-dependent temperature at the wall, which is causing \(\Delta T\), is influenced by all three ways of heat transfer. Due to the very low thermal conductivity coefficient of air, the influence of the conduction within the combustion chamber is low as well. The influence of the radiation to the wall temperature depends on the used ignition strategy and the corresponding injection system. Diesel engines with direct injection are more influenced as gasoline engines in general and especially with port fuel injection. The last influence is the convection. Due to the fast motion of the gas mixture inside the combustion chamber, it has strongest influence. Not all of the thermal energy can be transformed into volume change energy within the given time for the expansion stroke. Therefore, thermal energy \(E_{exh}\) is lost within the exhaust gas when leaving the combustion chamber over the exhaust valve, as defined as:  
\[E_{exh}=E_{int\ out}-E_{int\ in}=(T_{out}\cdot c_{p_{out}}-T_{in}\cdot c_{p_{ in}})\cdot m_{gas}. \tag{4}\]  
where \(E_{int\ out}\) is the internal energy within the exhaust gas, \(E_{int\ in}\) is the internal energy within the supplied air-fuel-mixture, T is the mean temperature, \(c_{p}\) is the mean specific heat capacity and \(m_{gas}\) is the gas mass which is supplied and exhausted.