## 2 The Assembly and Mathematical Description of a Rolling Piston Expander
At this stage a portion of the working fluid is still enclosed within the working chamber. During further counter-clockwise piston movement, the working chamber increases its volume, and the gas is expanded. The expansion ends when the piston reaches the position defined by the \(\alpha_{3}\) angle (in Figure 2 it is represented with Point D, and in Figure 4 with Point 3). The working cycle proceeds, and when the piston reaches the position defined by the angle \(\varphi=2\pi+\alpha_{1}\), the inlet valve opens again. The second working chamber is filled with working fluid flowing through the inlet port. When the piston reaches the position defined by the angle \(\varphi=2\pi+\alpha_{2}\), the outlet valve opens and the evacuation of the gas enclosed in the first working chamber starts (in Figure 4 it is represented by Point 4). The evacuation proceeds simultaneously with the expansion of the working fluid in the second working chamber, and ends when the piston reaches the position defined by the angle \(\varphi=4\pi-\alpha_{4}\) (in Figure 4 it is represented by Point 5). The outlet valve closes at this point. During further counter-clockwise piston movement, the residual working fluid (remained in the first working chamber after the evacuation) is compressed (see process 5-1 in Figure 4). The expansion of the working fluid in the second working chamber ends when the piston reaches the position defined by the angle \(\varphi=4\pi\) (i.e., after two full revolutions). The expander working cycle is completed at this point. Figure 3: The process of gas expansion in the rolling piston expander. (**a**) Beginning of the working cycle; (**b**) end of the working chamber filling; (**c**) end of expansion; (**d**) the moment of the inlet valve re-opening; (**e**) the moment of the outlet valve opening (beginning of the evacuation); (**f**) end of the evacuation; (**g**) end of the working cycle. The ideal rolling piston expander working cycle (see Figure 4) is formed by four ideal thermodynamic processes: isobaric filling (1-2), adiabatic expansion (2-3), isobaric evacuation (3-4-5) and adiabatic compression (5-1). The real working cycle is different from the ideal cycle as it is affected by energy dissipation phenomena i.e., under and overexpression, friction, internal leakages, pressure losses, etc. These issues are comprehensively described in [19; 20]. The mathematical description of the rolling piston expander can be described on the basis of Figure 2. The rolling piston expander working parameters are related to its geometrical dimensions (i.e., the cylinder radius \(R\), the piston radius \(r\), the eccentricity \(e\), the vane thickness \(b\), the vane coordinate \(x(\varphi)\), the cylinder length \(L\), the radial gap width \(y\)) and the steering angles. The position of the piston is defined by the angle \(\varphi\). In the Cartesian coordinate system, the position of the contact point (W), between the piston (2) and the vane (3) (see Figure 2), is described by the variable angle \(\varphi_{\rm w}\) and the variable piston radius-vector \(\rho_{\rm w}\). While \(\psi\) is the angle between \(\rho_{\rm w}\) and the vane vertical axis. The length of \(\rho_{\rm w}\) changes with the changing piston position and can be described by the equation:  
\[\rho_{\rm w}(\varphi)=r\frac{\sin\bigl{(}\varphi+\arcsin\frac{\varepsilon}{ r}\sin\varphi\bigr{)}}{\sin\varphi} \tag{1}\]  
As observed in Figure 2, the vane divides the working volume into two working chambers. In the following description, it is assumed that the position of the working chamber is described by the position of the contact point (C) between the piston and the cylinder. As mentioned earlier, the expander working cycle is completed after two full revolutions of the piston (\(\Delta\varphi=4\pi\)). Figure 5 shows a scheme useful for the determination of a working chamber volume.