## 1 Introduction
In addition, an exergy analysis has been conducted: [PERSON] et al. have made an exergy analysis of the HP systems including heat transfer and fluid flow effects [25]; [PERSON] and [PERSON] have made an exergoeconomic analysis of the HP refrigeration machines [26]. In recent years, attention has also been paid to analyzing and enhancing the performance of a WHS in case that it is operated under daily and hourly changes in hot water demand. This is because the performance of the system depends significantly on the temperature distribution in the storage tank, which changes daily and hourly by air and feed water temperatures, hot water demand, and HP operation. Many studies have been conducted to analyze and enhance the performance of CO\({}_{2}\) HP WHSs theoretically and experimentally as follows. Some studies have been conducted to analyze the performance under steady states: [PERSON] et al. have analyzed the performance of an HW by a simulation model by taking account of the effect of inlet water temperature and assuming perfect stratification and mixing in a storage tank [27]; [PERSON] has investigated the performance of an HP for combined space and water heating in consideration of the effect of inlet water temperature due to heat conduction in a storage tank theoretically and experimentally [28]; [PERSON] has analyzed the performance of a WH by taking account of the effect of inlet water temperature theoretically and experimentally [29]. Other studies have been conducted to analyze the performance under unsteady states including charging and tapping modes: [PERSON] et al. have analyzed the performance of a WHS composed of an HP and a storage tank in the charging mode [30]; [PERSON] et al. have examined the influence of the ambient and operating conditions on the performance of such a system under a daily cyclic change in hot water demand [31; 32]; [PERSON] has made an exergy analysis of the system [33]; additionally, [PERSON] et al. have analyzed the system performance under daily changes in standardized and simulated hot water demands [34; 35]; [PERSON] et al. have conducted a study to enhance the performance by extracting tepid water from a storage tank [36]. In addition, operation and design optimizations have been tried: [PERSON] et al. have estimated system performance values and have clarified the optimal operating conditions under daily change in simulated hot water demand [37]; [PERSON] et al. have analyzed and optimized the performance of a combination of a solar heater and an HP WHS [38]. These results have been obtained for uni-functional systems with the function only of hot water supply. This is because hot water is retrieved from the top of the storage tank, and water is fed to its bottom, which keeps the temperature distribution in the storage tank stratified because of the dependence of water density on temperature, and thus, the stratified temperature distribution is expressed accurately by a one-dimensional model. On the other hand, multi-functional CO\({}_{2}\) HP WHSs with the functions not only of hot water supply but also of bath heating have also been developed. In addition, the Japanese Industrial Standard \"Residential Heat Pump Water Heaters\" has been established, and the method of testing the performance of systems with the functions of hot water supply and bath heating has been prescribed [39]. It is necessary to distinguish these two functions for the performance analysis of a multi-functional system. This is because hot water retrieved from the top of the storage tank is returned to its bottom or side after heat exchange for bath heating, which destroys a stratified temperature distribution in the storage tank and causes three-dimensional convectional water flow, and the temperature distribution in the storage tank is essentially different from that of a uni-functional system. As a result, system performance values of the multi-functional system differ from those of the uni-functional system.