## 3 Results and Discussions
### Viscous Dissipation  
Typically, it is assumed that viscous dissipation effects play a small role for low speed flows. However, in the flow situations presented in this paper, the contribution of viscous dissipation due to the existence of internal structures within the investigated domain may be significant and hence it is considered here. Dimensional analysis presented in Appendix A reveals the dimensionless parameters that need to be considered. In particular, it is found that comparison between viscous dissipation in areas of empty resonator and heat exchanger plates should take into consideration porosity, \(\sigma\), and Eckert number, \(E\)c--cf., Equation (A7). In a viscous flow, the energy of the fluid motion (kinetic energy) is transformed into the internal energy of the fluid through the existence of viscous dissipation. Viscous dissipation for a two-dimensional model is defined as [28]:  
\[\Phi=2\Bigg{\{}\bigg{(}\frac{\partial u}{\partial x}\bigg{)}^{2}+\bigg{(}\frac {\partial v}{\partial y}\bigg{)}^{2}\Bigg{\}}+\bigg{(}\frac{\partial v}{ \partial x}+\frac{\partial u}{\partial y}\bigg{)}^{2}-\frac{2}{3}\bigg{(} \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}\bigg{)}^{2}. \tag{10}\]  
Figure 28: Temperature profiles for two selected phases, when there is no gravity (g = 0) and with gravity, g, at different tilt angles; \(0^{\circ}\), \(90^{\circ}\), \(270^{\circ}\), at a location 15 mm from the joint above the cold (CHX) and hot (HHX) plates. Figure 27: The effect of gravity and device orientation on the wall heat flux of the hot (HHX) and cold (CHX) plates. The computational domain is divided into four regions defined in Figure 29. The normalised viscous dissipation is averaged over the cycle and space, \(A\), corresponding to each region and calculated as:  
\[\langle\Phi^{*}\rangle=\frac{1}{2\pi A}\int_{0}^{A}\int_{0}^{2\pi}(\mu^{*}\Phi^{* })d\phi dA. \tag{11}\]  
The term in the bracket on the right-hand-side of Equation (11) is calculated as follows:  
\[\mu^{*}\Phi^{*}=\frac{1}{\mu_{c}}\bigg{(}\frac{d_{c}}{u_{c}\sigma}\bigg{)}^{2 }\mu\bigg{[}2\bigg{\{}\bigg{(}\frac{\partial u}{\partial x}\bigg{)}^{2}+\bigg{(} \frac{\partial v}{\partial y}\bigg{)}^{2}\bigg{\}}+\bigg{(}\frac{\partial v}{ \partial x}+\frac{\partial u}{\partial y}\bigg{)}^{2}-\frac{2}{3}\bigg{(} \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}\bigg{)}^{2}\bigg{]}. \tag{12}\]  
Following the dimensional analysis, the weighted-average value is further multiplied by the dimensionless parameters identified as porosity, \(\sigma\), and Eckert number, \(Ec\):  
\[\langle\Phi\rangle=\sigma^{2}Ec\langle\Phi^{*}\rangle. \tag{13}\]  
The reference velocity is taken at 300 mm from the joint of the heat exchangers (labelled as \(m\) in Figures 2 and 29) towards the right end of the figure within the region called the \"cold end\". The reference location is selected to avoid the influence of temperature on the open area next to the HHX, where a hot plume is observed. A value of porosity, \(\sigma\), is appropriately used when calculating dissipation for the open area and the area of the heat exchanger following the earlier discussion. For the open area, the porosity is set to one while an appropriate value of porosity (0.65 for the geometry investigated) is used for the heat exchangers. Figure 30 shows the effect of the heat exchanger wall temperature on the dimensionless viscous dissipation. The viscous dissipation at the open areas at both ends of the heat exchangers are one order of magnitude lower than the dissipation in the area containing the plates. This is consistent with the fact that the working medium interacts with more wall surfaces within the area of the heat exchanger where viscous resistance is significant, hence causing more energy dissipation. When the HHX becomes hotter, the dissipation becomes bigger. This is a consequence of gas viscosity that becomes bigger as the temperature increases. Figure 30b has an expanded \(y\)-axis to show dissipation levels at the ends of the heat exchanger assembly.