## 5 Numerical Study
### Results and Discussion  
First, system M was investigated for when hot water returned to the bottom of the storage tank after heat exchange for bath heating. The performance of system M was compared to that of system U. Figures 8 and 9 show the hourly changes in the temperature distribution in the storage tank for systems M and U, respectively. Figures 8a and 9a show the hourly changes during HP operation from 0:00 to 7:00, and Figures 8b and 9b show the hourly changes during hot water supply and bath heating, and hot water supply for systems M and U, respectively, from 7:00 to 24:00. In system U, since only the feed water flows into the bottom of the storage tank, the region with temperatures close to 17 \({}^{\circ}\)C expands. In system M, on the other hand, since only hot water demand occurs during 7:00 to 20:00, the temperature distribution at 20:00 was almost the same as that in system U. However, since bath heating demand occurred four times in addition to hot water demand during 20:00 to 22:00, the temperature from the bottom to the middle of the storage tank rose at 22:00. Afterwards, hot water demand occurred, and the temperature at the bottom of the storage tank  
Figure 10 shows the comparison in the hourly change in the HP COP between systems M and U. 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. It shows that the hourly change in the HP COP depends significantly on the temperature distribution from the bottom to the middle of the storage tank at 24:00 after daily hot water supply and bath heating, as shown in Figures 8b and 9b This is because water from the bottom to the middle of the storage tank enters the CO\({}_{2}\) HP during HP operation, and the performance of the CO\({}_{2}\) HP depends significantly on the inlet water temperature, as shown in Figure 6. Since system U had an almost uniform temperature distribution at the lowest quarter part of the storage tank, it had almost constant HP COP during the first half of HP operation. On the other hand, since system M had an increase in the temperature at the middle of the storage tank, it had an almost constant (although still lower) HP COP during the middle of HP operation. This difference resulted in a lower daily HP COP in system M as compared toto that in system U. Table 3 shows the comparison in the following daily system performance values between systems M and U: HP COP, storage and system efficiencies, and volumes of stored and unused hot water. Here, the HP COP is defined as the ratio of the daily heat output to daily power consumption, the storage efficiency is defined as the ratio of daily heat supply to daily heat output, and the system efficiency is defined as the ratio of daily heat supply to daily power consumption, which is equal to the product of the HP COP and storage efficiency. The HP COP, and the storage and system efficiencies in system M are relative to those in system U. The volumes of stored and unused hot water are defined as the ones at 7:00 after HP operation and 24:00 after hot water supply and bath heating, respectively. They were evaluated as the volumes of hot water with a temperature of 40 \({}^{\circ}\)C obtained by mixing the hot water with temperatures higher than 40 \({}^{\circ}\)C and the feed water of 17 \({}^{\circ}\)C. Since the temperature at the middle of the storage tank rose in system M, the HP COP in system M decreases by 4.7% as compared toto that in system U. In addition, since the average temperature in the storage tank in system M becomes higher than that in system U, the storage efficiency in system M also decreased as compared toto that in system U, but its decrease rate remains only 0.2%. As a whole, the system efficiency in system M decreases by 4.9% as compared to that in system U.