## 5 Numerical Study  
### Conditions  
To investigate the influence of bath heating on the system performance, the following two systems were investigated: one is the multi-functional system with the functions of both hot water supply and bath heating (system M); the other is the uni-functional system with the function only of hot water supply (system U). Table 2 shows the specifications of the multi-functional CO\({}_{2}\) HP WHS investigated here. Their values are given based on an existing system. Figure 6 shows values measured using an existing system and approximate functions identified based on measured values for the power consumption, HP COP, and their product, or heat output of the CO\({}_{2}\) HP [34]. As an example, these are shown in relation only to the inlet water temperature for constant air and outlet water temperatures of 16 and 85 \({}^{\circ}\)C, respectively. Here, each value is relative to its rated one for the air and inlet/outlet water temperatures of 16, 17, and 65 \({}^{\circ}\)C, respectively. The reason why the performance of the CO\({}_{2}\) HP is expressed by relative values is as follows: the performance of the CO\({}_{2}\) HP was measured for an existing system; however, the performance has been enhanced significantly after that because components such as compressors, gas coolers, etc. have been improved technologically; thus, absolute values for the performance obtained by the measurement may mislead one to the performance of CO\({}_{2}\) HPs in the current market, and hence, they are not suitable. For the same reason, the performance of the WHS is also expressed by relative values. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Equipment** & **Specification** & **Value** \\ \hline \hline  & Rated heat output & 4.50 kW \\  & Parameters \(\alpha_{1}\) & 2.90 \\  & \(\beta_{1}\) & \(35.9\times 10^{-3}\) \\  & \(\gamma_{1}\) & \(23.8\times 10^{-6}\) \\  & \(\alpha_{2}\) & \(-0.386\)  
The air temperature was set at 16 \({}^{\circ}\)C as the ambient condition, while the feed water temperature was set at 17 \({}^{\circ}\)C as one of the boundary conditions. These are prescribed for mid-season by the Japanese Industrial Standards [39]. Hot water and bath heating demands shown in Figure 7 were used, which were also prescribed as standard on the representative day by the Japanese Industrial Standards. Figure 7a shows the hourly change in the volumetric flow rate of the demand for general and bath hot water supply. The temperature of hot water supplied to the tapping site was set at 40 \({}^{\circ}\)C. Figure 7b shows the hourly change in the heat of the demand for bath heating. Here, the height of each vertical line corresponds to the total heat. Since the temperature of hot water at the top of the storage tank changed with time, the temperature of hot water and duration could not be prescribed. Thus, when the total heat attained its prescribed value, the bath heating was considered to be finished. In addition, the volumetric flow rates of hot water with higher temperature on the storage tank side and lower temperature on the bathtub side were set at 0.06 and 0.15 L/s, respectively. On the other hand, in system U, the heat equivalent to the total heat was considered as an additional demand for bath hot water supply. Thus, the start time was common to both systems M and U, however, the stop time was independent in each system. At the initial time, the temperature of water in the storage tank was set at 17 \({}^{\circ}\)C. In addition, an infinitesimal positive value was given in place of zero to the mass of water in the bathtub, so that the set of equations could be solved; its temperature was set at 17 \({}^{\circ}\)C. The CO\({}_{2}\) HP started up at 2:00 on each day, and shut down when the inlet water temperature attained its prescribed value. Here, the outlet water temperature during HP operation was set at 85 \({}^{\circ}\)C, and the inlet water temperature for HP shutdown was set at 30 \({}^{\circ}\)C.  
Figure 6: Performance characteristics of CO\({}_{2}\) heat pump.