## 2 The Assembly and Mathematical Description of a Rolling Piston Expander
The volume of the working chamber, whose position is defined by the angle \(\varphi\) (see Figure 5), can be described by the equation:  
\[V(\varphi)=A(\varphi)L=A_{ABC}(\varphi)L \tag{2}\]  
where: \(A_{ABC}(\varphi)\) is a cross sectional area of the working chamber. The cross sectional area can be determined using the calculation procedure valid for multi-vane expanders. This procedure is comprehensively described in [14; 21]. Figure 4: The rolling piston expander thermodynamic cycle in the \(p\)-\(V\) plane. The angle between Point B and Point C (see Figure 5) can be described by the equation:  
\[\lambda_{\text{u}}=\pi-\varphi_{\text{u}} \tag{3}\]  
The volume of the working chamber, whose position is defined by the angle \(\varphi\), can be described by the equation:  
\[V(\varphi)=LR^{2}Z_{k}(\varphi_{\text{u}}) \tag{4}\]  
where \(Z_{k}(\varphi_{\text{u}})\) is a function dependent only on construction parameters and the chamber position:  
\[Z_{k}(\varphi_{\text{u}})=f\bigg{[}\varphi_{\text{u}}(\varphi);\,\frac{e}{R},\,\frac{b}{R},\,\frac{y}{R},\,\psi_{\text{u}}(\varphi),\,z_{\text{lu}}( \varphi)\bigg{]} \tag{5}\]  
The \(Z_{k}(\varphi_{\text{u}})\) function is comprehensively described in [21]. The additional parameters can be described by the Equations (6)-(8):  
Figure 2:0  
Figure 2:1  
Figure 2:2  
If \(\varphi>\pi\), \(A_{ABC}(\varphi)\) (see Figure 5a) the cross section area can be described by the equation:  
Figure 2:3  
Figure 2:4 The scheme used for determination of the working chamber volume. (**a**) \(\pi>\varphi>0\); (**b**) \(2\pi>\varphi>\pi\). The \(A_{ABD}(\varphi)\) cross sectional area can be calculated similarly to \(A_{ABC}(\varphi)\) if \(\varphi<\pi\) (see Figure 5b). The position of the working chamber is described by the angle \(\varphi_{\mathrm{u}}=\pi\), the angle of vane inclination to the piston radius is \(\psi_{\mathrm{u}}\), and the angle between point B and point C is \(\lambda_{\mathrm{u}}\) (see Figure 5b). The additional parameters can be described by Equations (10)-(12):  
Figure 2:5  
Figure 2:6  
Figure 2:7  
The ABD cross sectional area can be described by the equation:  
Figure 2:8  
If \(2\pi<\varphi<4\pi\), the volume of the working chamber can be described by the equation:  
Figure 2:9  
The theoretical power output of a rolling piston expander can be obtained with the equation:  
Figure 3:0  
It is also possible to calculate the theoretical power output using the equation:  
Figure 3:1  
where: \(\overline{N}_{a}\) is the reference expander power and depends on the geometric parameters and working fluid properties. The method of the reference expander power calculation is comprehensively described in [21]. The presented model is a novel and helpful tool that can be applied for the modeling and optimization of the geometry of a rolling piston expander, and the selection of optimal operating parameters for high efficiency. The model is based on the relationships between specific geometrical parameters (\(e,\,R,\,L\), etc.) rather than the absolute values. This feature allows for the straightforward design of a series of types of expanders that would fit in complex systems, with a variable demand for flow rates, pressure ratios or temperature requirements, which is crucial in the case of natural gas pressure reduction stations.