## 4 Thermodynamic Model of the Free-Piston Engine  
Based on the data of Table 1, a one-dimensional thermodynamic model for the free-piston engine is built up. The combustion engine is working with an external mixture formation via a gasoline injector with a constant injected fuel mass in the intake manifold. The fuel mass is chosen this way so that the equivalence ratio \(\phi=1\). This model of the free-piston engine is quite similar to an engine with a conventional crank train. To model the combustion process and the corresponding heat release, the predictive combustion model, the so-called Spark-Ignition Turbulent Flame Model (SITurb) from GT-Suite, was used. This model can be used for several operating points after it has been parameterized once. Additionally, it is normally used for gasoline engines, which is the test bench engine. The heat release is calculated by the burned fuel mass \(m_{b}\):  
\[\frac{dm_{b}}{dt}=\frac{m_{e}-m_{b}}{\tau} \tag{19}\]  
where \(m_{e}\) is the entrained mass and \(\tau\) is the time constant. The entrained mass is calculated by:  
\[\frac{dm_{e}}{dt}=\rho_{u}\cdot A_{e}\cdot(S_{T}+S_{L}) \tag{20}\]  
where \(\rho_{u}\) is the unburned density, \(A_{e}\) is the surface area at flame front, \(S_{T}\) is the turbulent flame speed and \(S_{L}\) is the laminar flame speed. The turbulent flame speed is calculated by:  
\[S_{T}=C_{TFS}\cdot u^{\prime}\cdot\left(1-\frac{1}{1+C_{FKG}\cdot\left(\frac{ R_{f}}{L_{i}}\right)^{2}}\right) \tag{21}\]  
where \(C_{TFS}\) is the turbulent flame speed multiplier, \(u^{\prime}\) is the turbulent intensity, \(C_{FKG}\) is the flame kernel growth multiplier, \(R_{f}\) is the flame radius and \(L_{i}\) is the integral length scale. The laminar flame speed is calculated by:  
\[S_{L}=\left(B_{m}+B_{\phi}\cdot\left[\phi-\phi_{m}\right]^{2}\right)\cdot \left(\frac{T_{u}}{T_{ref}}\right)^{\alpha_{e}}\cdot\left(\frac{p}{p_{ref}} \right)^{\beta}\cdot f(Dilution) \tag{22}\]  
Figure 5: Force \(F_{EM}\) depending on position \(x\) and current \(i_{q}\). where \(B_{m}\) is the maximum laminar speed, \(B_{\phi}\) is the laminar speed roll-off value, \(\phi\) is the equivalence ratio, \(\phi_{m}\) is the equivalence ratio at maximum speed, \(T_{u}\) is the unburned gas temperature, \(T_{ref}\) is defined as 298 K, \(\alpha_{e}\) is the temperature exponent, \(p\) is the pressure, \(p_{ref}\) is defined as 101,325 Pa and \(\beta\) is the pressure exponent. The heat transfer is defined by the WoschniClassic model from GT-Suite. This model is used for engines without swirl flows, which correspond to the test bench engine. First, a convective heat transfer is calculated; then, it is calibrated to the measurement data by a multiplier to take the influences of conduction and radiation into account. The heat transfer coefficient is calculated by:  
\[\alpha=\frac{K_{1}\cdot p^{0.8}\cdot w^{0.8}}{B^{0.2}\cdot T^{K_{2}}} \tag{23}\]  
where \(K_{1}\) and \(K_{2}\) are model constants, \(p\) is the cylinder pressure, \(w\) is the average cylinder gas velocity, \(B\) is the cylinder bore and \(T\) is the cylinder temperature. All multipliers are used to calibrate the model to the measurement data. This heat transfer coefficient only takes the convection into account. To consider the influence of the radiation and conduction, an additional multiplier is used to calculate \(\alpha\) for Equation (3). The major difference of the model and the conventional engine models is the completely free definition of the piston motion. Specifically, the determination of the piston movement in the model is done via a table with free choice of the piston position as a function of time. In contrast to that, the piston motion normally is defined by the geometrical parameters of the crank train and the corresponding relation between crank angle and piston position.