## 2 The Assembly and Mathematical Description of a Rolling Piston Expander  
The assembly of the rolling piston expander is described as in Figure 2. The main machine elements are: cylinder (1), rolling piston (2) and sliding vane (3). The vane (3) is placed in a slot (4) and is pressed to the piston surface with help of a spring (5). The vane remains in close contact with the piston. The piston is mounted eccentrically in the cylinder on rolling bearings. The radii of the pistons \(r\) and cylinders \(R\), the lengths of the cylinders \(L\) and the value of eccentricity \(e\) may vary, depending on the machine power and design. It is generally assumed that these values are related to each other as follows [18; 19; 20]: \(e=(0.008\)-\(0.2)R\), \(L=(1.5\)-\(3)R\). Between the surfaces of the cylinder and the piston, radial gap of width \(y\) is formed. The width of the gap ranges from 0.1 to 0.15 mm [19]. Small dimensions are characteristic of small power expanders, while larger dimensions are used in large power expanders. The cylinder is usually made of cast aluminum alloy, while the piston is usually made of rolled steel. The vane is made of different materials. The commonly used materials are graphite, or cured silicone doped with graphite and textolite. The inlet and outlet port edges (G, B, E and D in Figure 2) are referred to as the machine \"steering edges\" [21]. The timing of the machine operation may be performed using steering valves, or by the proper arrangement of the \"steering edges\", i.e., the edges of the inlet (points G and B in Figure 2) and the outlet port (points E and D in Figure 2). The position of the \"steering edges\" is defined by the angles \(\alpha_{1}\), \(\alpha_{2}\), \(\alpha_{3}\) and \(\alpha_{4}\) (see Figure 2 for details). \(\alpha_{1}\) and \(\alpha_{2}\) are the angles which define the inlet port edges. \(\alpha_{3}\) (also referred as the filling angle [21]), and \(\alpha_{4}\) (also referred as the evacuation angle [21]) are the angles which define the outlet port edges. The proper arrangement of the \"steering edges\" has a significant influence on the expander operation. This issue is comprehensively described in [21]. The working principle of a rolling piston expander can be described on the basis of the Figures 2-4. The expander working cycle begins when the rotating piston reaches the position defined by the angle \(\varphi=0\) (in Figure 4 it is represented by Point 1). In this position of the piston,  
Figure 2: The simplified scheme of the rolling piston expander. chamber is equal to \(V(\varphi)=0\) and both the inlet and the outlet valve are closed. The working chamber is limited by the surfaces of the cylinder (1), piston (2) and vane (3) (see Figure 2). In the rolling piston expander two working chambers coexist in a single moment of time and the gas is expanded concurrently in both chambers. This feature results in continuous machine operation. The working cycle continues, and when piston reaches the position defined by the \(\alpha_{1}\) angle (in Figure 2 it is represented by Point B), the volume of the working chamber slightly increases, the inlet valve opens, and the working fluid flows into the working chamber through the radial gap (\(y\)) formed between the surfaces of the cylinder and the piston. The rate of pressure increase depends on the width of the radial gap \(y\), the machine rotational speed, and the working fluid thermal properties, namely the pressure (\(p_{s}\)) and temperature (\(T_{s}\)) at the expander inlet. When piston reaches the position defined by the \(\alpha_{2}\) angle (in Figure 2 it is represented with Point G) the inlet port is fully opened to the working chamber. During further counter-clockwise piston movement, the filling of the working chamber continues. The filling of the working chamber ends when the inlet valve closes (typically when the piston reaches the position defined by the angle \(\varphi=\pi/3\)[20,21]). In Figure 4 it is represented by Point 2.