## 3 Modeling Methods and Results  
The mathematical model describing the expander working cycle was constructed based on Equations (1)-(15) and was implemented in computer software [22]. The flowchart of the computational procedure is presented in Figure 6.  
The goal of the analytical modeling was to examine the influence of the following expander characteristics on power output:  
* Eccentricity \(e\),
* Length of the cylinder \(L\),
* Expansion ratio \(\sigma=p_{s}/p_{d}\),
* Polytropic exponent \(m\). These parameters were varied according to arithmetic progression in the following ranges: \(e=0.008\)-\(0.2R\), \(L=1.5\)-\(3.0R\), \(\sigma=2\)-\(8\), \(m=1.333\)-\(1.99\). As the eccentricity and the length of the expander are functions of a cylinder radius \(R\), calculations were carried out for a fixed \(R=0.15\) m. Working fluid, treated as a polyatomic ideal gas, was natural gas. Due to methane accounts for 98% of the natural gas, its properties were used in the calculations (\(R_{CH4}=518.33\) J /(kg K)). Additionally, it  
Figure 6: Flowchart of the computational procedure. was assumed that clearance volume is equal to zero and pressure fluctuations at the inlet and outlet, caused by valves inertness, are negligible and can be omitted. In each case, the pressure at the outlet was held constant and set to \(p_{d}=500\) kPa, while rotational speed was equal to \(n=3000\) rpm. In Figure 7, the volume of a working chamber versus rotor rotation angle, for different eccentricities, is depicted. As a whole cycle of the expander requires two full revolutions, the volume varies from 0 to \(V_{max}\) twice in a one cycle. The volume increases with increasing rotor rotation angle according to an S-shaped curve. With increasing eccentricity, the volume greatly increases, and in the middle range of the half-cycle, linear growth is observed. In Figures 8 and 9, pressure and temperature distributions versus rotor rotation angle, for different polytropic exponents \(m\), are presented. The area under the \(p\)-curve was proportional to the indicated work. With increasing polytropic exponent this area also increased. The variation of \(m\) influenced the rotor rotation angle for which inlet valve opened. With increased \(m\), the inlet valve opened for greater values of rotor rotation angle, i.e., the filling cycle was lengthened. During the filling, both pressure and temperature stayed constant up to the opening of the inlet valve. Then, the expansion process started and pressure decreased. For a fixed \(m\), temperature decreased from inlet temperature \(T_{s}\) to a minimal temperature \(T_{min}\) after expansion. It can be seen that in Figure 9, \(T_{s}\) was very high and increased with increasing \(m\). This temperature resulted from heat input that had to be provided to the working fluid before the expansion started. Heating was necessary in order to heat up the working fluid to the temperature \(T_{s}\) for which the temperature of the working fluid after expansion would be higher than the limiting temperature \(T_{min}\). This was done in order to prevent hydrate formation in the gas [2]. During all calculations, \(T_{min}\) was set to 274 K. In \(T\)-chart heat is the area under the \(T\)-curve, and it was observed that with increasing \(m\), the heat necessary for heating the working fluid increased as well. Figure 7: Working chamber volume versus rotor rotation angle for different eccentricities (\(e\)). In order to assess the analytical model, the authors decided to compare the obtained results with experimental data presented in the literature. As the use of rolling piston expanders in power systems, and in particular for the expansion of natural gas, is a novel and constantly evolving topic, finding publications containing a sufficient range of experimental data and description of machine geometrical dimensions was difficult.