# Heat-Only-Driven Vuilleumier Refrigeration
## 2 The Endoreversible Thermodynamics Model  
The thermodynamic modeling of Vuilleumier refrigerators needed here goes beyond classical equilibrium approaches. In order to include the main loss terms in a quantitative fashion we use Endoreversible Thermodynamics [15; 16; 17]. In this approach, the loss  
Figure 1: The longitudinal section of a Vuilleumier refrigerator (VR) displaying its general structure. Note the different cross-sectional areas on each of the two pistons’ sides. The piston positions are defined via the periodic functions \(x_{\text{II}}(t)\) and \(x_{\text{C}}(t)\). The VR is, by finite heat conduction, connected to three heat baths with the constant temperatures \(T_{\text{H}}>T_{\text{M}}>T_{\text{C}}\). These heat exchanges are characterized by the entropy fluxes \(J_{\text{1,h}}^{S}\), \(J_{\text{3,m}}^{S}\), and \(J_{\text{2,c}}^{S}\). The VR consists of three working spaces, \(i=1,2,3\), that can exchange working gas through two regenerators. In those working spaces the temperatures \(T_{i}\), pressures \(p_{i}\), and chemical potentials \(\mu_{i}\) vary over time and are here assumed to be homogeneously distributed. terms and the ensuing entropy production are captured quantitatively by viewing the Vuilleumier refrigerators as a network of reversible subsystems which can be described by equilibrium thermodynamics. The irreversibilities are mapped into the interactions between these subsystems, which describe the fluxes of thermodynamic extensities such as volume, entropy, and mol number. Such interactions are characterized by transport laws such as, for instance, Fourier or Newtonian heat conduction. Endoreversible modeling has shown to be an effective tool for describing nonequilibrium systems and has been applied to a variety of systems, among them solar-power-driven systems [18; 19; 20; 21], thermoelectric [22; 23] and chemical devices [24; 25; 26; 27; 28], as well as thermal engines in general [29; 30; 31; 32; 33]. The endoreversible approach is very useful in the context of optimization studies [34; 35; 36; 37; 38], as it allows to concentrate on the relevant loss mechanisms in thermal devices and can thus also be used in systems with fluctuating environments [39; 40; 41]. For this work, it is of particular interest that it has been successfully used in the context of waste heat recovery and recuperation [38; 42; 43; 44; 45; 46]. The setup of the endoreversible Vuilleumier refrigerator model used here is presented in Figure 2. It consists of nine endoreversible subsystems and seven additional reservoirs needed for bookkeeping purposes. The main external interactions are provided by three external (infinite capacity) heat baths H, M, and C with constant temperatures \(T_{\text{H}}\), \(T_{\text{M}}\), and \(T_{\text{C}}\), respectively. For simplicity, we avoid modeling the technical aspects of transporting heat into and out of the device. Instead, we consider the \"external\" heat sources to be the input sides of the Vuilleumier engine's heat exchangers. The output sides of these heat exchangers are connected to the respective working spaces--labeled 1, 2, and 3--containing the working gas. Due to the finite heat exchange, the working gas temperatures (\(T_{1}\), \(T_{2}\), and \(T_{3}\)) are not identical with the corresponding reservoir temperatures (\(T_{\text{H}}\), \(T_{\text{M}}\) and \(T_{\text{C}}\)). Nonetheless, the differences are rather small. The pressure differences between the working spaces can be used to gain mechanical power by the two engines TH and TC. This \(pV\)-power is needed to compensate the frictional losses, which are here accounted for by the bookkeeping reservoir WF. Depending on the engine speed, there may be surplus power or an additional power demand, which is balanced in the bookkeeping reservoirs WTH and WTC separately for each engine. Depending on the engine design, there is possibly a coupling between the two engines, which facilitates a power exchange \(P_{\text{PE}}\) between them. In a heat-only operation the net cycle average vanishes.