# Heat-Only-Driven Vuilleumier Refrigeration
## 1 Introduction  
Villeumier refrigerators [1; 2] are special, heat-driven machines which provide cooling power by utilizing heat available at temperatures above environment conditions. They operate by simultaneously taking heat from the hot source and the cold source and delivering it to the environment based on a special combination of thermodynamic processes. One can think of the operation as a combination of two Stirling engines, one running as a heat engine utilizing the hot heat source to provide power, and one running as a Stirling refrigerator driven by that power to extract heat from the cold source. Thus, the Vuilleumier machines can be used as heat-driven heat pumps [3; 4; 5] or as heat-driven refrigerators [6; 7; 8; 9; 10], air conditioners [11], or waste-heat-driven cargo refrigerators for instance on fishing vessels [3]. These possible applications make them attractive, especially when the driving heat is inexpensive, and have led to increased interest in their thermodynamic performance [12; 13; 14]. As in Stirling engines, the working fluid--usually a gas--undergoes a thermodynamic cycle, in which heat is removed from and supplied to the working fluid in between the compression and expansion strokes. In order to increase the efficiency of the overall process, one uses regenerators to temporarily store the heat and supply it back to the process later. Such regenerators are here used on both the hot and the cold side of the Vuilleumier machine. The actual technical realization of the Vuilleumier machine can come in a variety of configurations, depending on how the fluid movement is controlled and where the regenerators are located. In this study, we use a configuration in which the regenerators are integrated into the displacer pistons which actuate the working fluid transport. The resulting schematic view of the Vuilleumier machine is shown in Figure 1.  
The three working spaces 1, 2, and 3 are displayed by colors indicating their general temperature levels. Red indicates the hot working space, blue the cold one, and green the intermediate one. They are connected to heat exchangers which facilitate the input and output of heat from and into the hot heat source, the cold heat source (the refrigerated space), and the environment. The two displacer pistons move the wthe working spaces, which change their volume periodically. The regenerators are thus subjected to alternating hot and cold fluxes of gas, which leads to the temporary storage of energy as described above. The piston motion in mechanically-driven Stirling and Vuilleumier machines is usually approximately harmonic due to the crankshaft and connecting rod mechanisms. Here, we will use a pure sinusoidal time dependence with a phase shift between the two pistons. In this short note, our aim is to study a special feature of Vuilleumier refrigerators that makes them interesting for a particular application area. Obviously, the cooling operation is economically attractive if the driving energy is available as waste heat from other processes. Then the efficiency of the refrigeration process is less important than the cooling power of a given machine. This cooling power depends, of course, on the internal losses occurring in the refrigerator's operation. Such losses are due to mechanical friction but also due to heat losses and irreversibilities caused by the gas transport through the regenerators. Especially the mechanical friction losses can either be compensated by an auxiliary drive (usually electric) or by designing the Vuilleumier refrigerator such that the pressure differences acting on the pistons lead to the desired forces. The latter solution is, in particular, attractive in situations where the energy input is from waste heat and is thus \"free\". In that case, no permanent auxiliary drive is needed.