## 3 Results and Discussion
### Viscous Dissipation  
In a viscous flow, the energy of the fluid motion (kinetic energy) is turned into internal energy of the fluid through the existence of viscous dissipation. A dimensionless approach, introduced by [PERSON] [11] and [PERSON] [28], has been used with appropriate consideration of the porosity of the area within the heat exchanger and the open areas next to both ends of the heat exchanger (0.65 and one, respectively) [22]. The total dimensionless viscous dissipation is calculated as:  
\[\langle\Phi\rangle=\sigma^{2}Ec\left[\frac{1}{2\pi A}\int_{0}^{A}\int_{0}^{2 \pi}\left(\mu^{*}\Phi^{*}\right)d\phi dA\right], \tag{15}\]  
where:  
\[\mu^{*}\Phi^{*}=\frac{1}{\mu_{c}}\bigg{(}\frac{d_{c}}{u_{c}\sigma}\bigg{)}^{2 }\mu\left[2\left\{\left(\frac{\partial u}{\partial x}\right)^{2}+\left(\frac{ \partial v}{\partial y}\right)^{2}\right\}+\left(\frac{\partial v}{\partial x }+\frac{\partial u}{\partial y}\right)^{2}-\frac{2}{3}\bigg{(}\frac{\partial u }{\partial x}+\frac{\partial v}{\partial y}\bigg{)}^{2}\right]. \tag{16}\]  
The terms \(A\), \(\mu\) and \(\phi\) in Equation (15) are the area, viscosity and phase of a flow cycle, respectively; the term in square brackets is the area-weighted-average value of dimensionless dissipation, which is also averaged over one flow cycle. The dimensionless dissipation in Equation (16) is related to the mean flow velocity. The results in Figure 18 are shown as a product of porosity, \(\sigma\), Eckert number, \(Ec=\ \left(u_{c}^{2}\right)/c_{p,c}T_{c}\) (where \(u_{c}\), \(c_{p,c}\) and \(T_{c}\) are the reference values of velocity, specific heat and temperature, respectively) and the area-weighted-average value of dimensionless dissipation. The reference values are taken at the location 200 mm away from the end of CHX. More detailed explanations and dimensional analysis were already given by [PERSON] and [PERSON] [21]. The Eckert number, \(Ec\), expresses the relationship between the flow kinetic energy and enthalpy and is used to characterise dissipation.  
The viscous dissipation presented by Equation (16) is also known as \"direct dissipation\", not to be confused with turbulent dissipation [25]. The velocity components \(u\) and \(v\) in Equation (16) are the mean velocity components of the flow. As with the laminar model, the direct dissipation represents the transfer of mechanical energy to internal energy through viscosity. The turbulent dissipation (which is the transfer of energy from the mean motion into the turbulence fluctuation and then into the internal energy) is indirectly affecting the mean flow when the turbulence model is used. The effect of turbulence on the internal energy of the flow is reflected in the final velocity of the mean flow used in Equations (1), (2) and (16). The laminar model was used for the drive ratio of 0.30%, the transition-SST model for drive ratios of 0.45% and 0.65%, while the SST k-\(\omega\) model for the drive ratio of 0.83%.  
Figure 18 shows that the viscous dissipation increases as the drive ratio increases. The increment occurs in areas both within the channel (Figure 18a) and the area outside (Figure 18b). Hot areas have  
Figure 17: CFD predictions of vorticity contours at the end of the cold heat exchanger for phase \(\phi\)8 at drive ratios of (**a**) 0.3% using the laminar model and (**b**) 0.83% using the SST k-\(\omega\) model. Distances \(x\) and \(y\) measured from an arbitrary origin.  
a higher viscous dissipation compared to cold ones, due to the increase of viscosity with temperature. Furthermore, thermal expansion at the area bounded by the wall of the HHX and the buoyancy-driven flow as reported in [16], particularly prominent at the hot end, may well be responsible for the higher viscous dissipation within these areas. As the drive ratio increases, the amplitude of flow increases, and the viscous dissipation becomes larger.