## 1 Introduction
The comparison of isenthalpic throttling and polytropic expansion of natural gas is depicted in Figure 1.  
Figure 1: Comparison of isenthalpic throttling and polytropic expansion of natural gas depicted in a Mollier diagram, (gas pressure before reduction: 6 MPa, gas pressure after reduction: 2.5 MPa), 0–1–2: isobaric heating, 1–3: isenthalpic throttling, 2–4: polytropic expansion. The applicability of turbines to natural gas expansion has been analyzed in a number of studies, see, e.g., Refs. [3; 4; 5; 6; 7; 8; 9]. In [3] it was indicated that the turbine may be an interesting alternative to the pressure reduction station. However, the turbine is mechanically complicated. In case of low power turbines, the very high rotational speeds is an important problem. This necessitates very precise turbine fitting, which in turn results in high manufacturing costs. Moreover, when the explosive natural gas is considered as the working fluid, special sealings and shaft bearings should be adopted [10]. An alternative to a complicated and expensive turbine can be a volumetric machine. Volumetric expanders are mechanically simple, and thus cheap. Examples of such machines are: rolling piston, screw, scroll, piston, and lobe expanders. Nowadays, volumetric expanders are present in almost all industry branches. These machines are applied, e.g., in small and medium power organic Rankine cycle systems (ORCs) [11; 12; 13; 14; 15] and as pneumatic engines in mining or automotive industry [16]. The general advantages of volumetric expanders compared to turbines are [15]:  
* Low cycle frequency,
* Low specific speed,
* High pressure drops in one stage,
* Low rotational speed (approx. 3000 rev/min),
* Ease of hermetic sealing. The disadvantages are [15]:  
* Internal friction, reducing the efficiency and reliability of the machine,
* Need for lubrication and frequent replacement of wearing parts,
* Large weight in relation to power and efficiency,
* Internal and external leakages reducing expander efficiency. Rolling piston expanders are especially promising for natural gas pressure reduction. Their most important advantages are: very simple design, suitability for wet gas conditions, negligible clearance volume, and ease of gas-tight sealing. The issues concerning the rolling piston expander design and its application in the industrial systems utilizing waste or renewable energies are innovative, and not fully described scientifically. The results of initial experiments on a small power ORC system adopting the rolling piston expander system, are reported in [17]. The results of this experiments confirmed that it is possible to feed the rolling piston expander with low-boiling working fluid instead of air. However, it was presented in this paper that rolling piston expanders are still at the level of lab prototypes, or under research. The analysis of the suitability of this type of expanders for natural gas pressure reduction has not been conducted so far, and therefore requires comprehensive study, i.e., analytical as well as experimental analysis. Encouraged by the advantages of rolling piston expanders described above, as well as by promising experimental results [17], the authors decided to comprehensively assess the suitability of rolling piston expanders for natural gas pressure reduction. In the consecutive sections we analyze the assembly and introduce a mathematical model of a rolling piston expander, and present the results of analytical modeling. Computational analysis of the operating parameters of the expander and its performance were carried out using a novel mathematical model, taking into account various geometrical dimensions and construction parameters. Moreover, the authors defined a novel assessment parameter, i.e., Power Recovery Factor (PRF), which can be used for assessing the efficiency of the energy regeneration in a rolling piston expander fed by natural gas. The PRF was used to assess the various construction solutions of the rolling piston expander.