# Heat-Only-Driven Vuilleumier Refrigeration
## 2 The Endoreversible Thermodynamics Model
As the overall volume of the working spaces changes during a cycle, an infinite volume reservoir E with fixed pressure \(p_{\text{E}}\) provides the needed volume fluxes to TH and TC. The regenerators RH and RC between the working spaces are modeled by the two complex interactions rh and rc. These interactions are irreversible, and use two bookkeeping reservoirs each--WRH and SRH for the hot side as well as WRC and SRC for the cold side--which store those energy and entropy amounts during a cycle, which are regenerated. In our endoreversible model, we will use a linear transport law for the gas transport through the regenerators, i.e., the mol number flux is assumed to be proportional to the pressure difference, with mass transport coefficient \(\alpha\) being the proportionality constant. The heat fluxes are modeled as Newtonian heat transport with the heat conduction coefficient \(\kappa\), which applies to all heat fluxes. Finally, the friction coefficient \(\beta\) is used for the frictional losses being quadratic in the volumetric changes on the hot and cold side of each of the two displacer pistons in the refrigerator. These choices capture the main effects of these irreversibilities, but of course more complicated transport laws can be considered if desired. For the regenerators a variety of endoreversible models exist ranging from ideal regeneration [47] over the r-regenerator model [48,49] to the ES, EE, and EEn models [50,51]. Here, we used the r-regenerator model, which captures the regeneration extent by the regeneration coefficient \(r\). It describes which fraction of the enthalpy difference between the gas states in the two working spaces connected by the regenerator is temporarily stored. The best possible regeneration would take out the whole difference, the worst would take out nothing. As we have two regenerators--a hot and the cold one--and for each of them gas can come from the two connected working spaces, we have four regeneration coefficients \(r_{\rm h1}\) and \(r_{\rm h3}\) for the hot side, and \(r_{\rm c2}\) and \(r_{\rm c3}\) for the cold side. They describe which fraction of the enthalpy of the gas entering working space 1 and working space 3 for the hot side and which fraction of the enthalpy of the gas entering working space 2 and working space 3 for the cold side are taken out. For details see [49]. When operating the refrigerator, it turns out that the requirement of cyclicity for the regenerators enforces different values for \(r_{\rm h1}\) and \(r_{\rm h3}\) as well as for \(r_{\rm c2}\) and \(r_{\rm c3}\). As described in [49], the overall regeneration extent is then given by the respective mean value  
\[r_{\rm h} = (r_{\rm h1}+r_{\rm h3})/2, \tag{1}\] \[r_{\rm c} = (r_{\rm c2}+r_{\rm c3})/2. \tag{2}\]  
The thermal dynamics which unfolds in the [PERSON] refrigerator depends crucially on the piston motion which we use. Here we will consider the standard harmonic motion. The corresponding volume dynamics in working spaces 1, 2, and 3 is given by  
\[V_{1}(t) = V_{0}+D_{1}\sin(t/t_{0}), \tag{3}\] \[V_{2}(t) = V_{0}+D_{2}\sin(t/t_{0}+\Delta), \tag{4}\]  
where \(V_{0}\) is the dead volume, \(D_{1}\) and \(D_{2}\) are the respective displacements caused by the diameter of the piston in working spaces 1 and 2 (see Figure 1), \(t_{0}\) is the cycle time, and \(\Delta\) is the phase shift between the two displacers. Working space 3 is considered as a sum  
\[V_{3}(t)=V_{31}(t)+V_{32}(t). \tag{5}\]  
Figure 2: Endoreversible setup of the Vuilleumier refrigerator considered here. The external heat sources/sinks are modeled by the three reservoirs (\(T_{\rm H}\), \(T_{\rm M}\), \(T_{\rm C}\)). These are closely coupled by heat exchangers to the corresponding inner working spaces modeled as reservoirs with temperatures (\(T_{1}\), \(T_{2}\), \(T_{3}\)). The two engines TH and TC provide the mechanical power to compensate the frictional forces on the hot and cold side, respectively.