## 3 System Modeling  
### Component Models  
To conduct a numerical simulation, models of components which constitute the WHS were created. Mathematical equations of the component models are given in Appendix A. Only brief descriptions of the component models are given here.  
The component models for the CO\({}_{2}\) HP and mixing valve were conventional ones which have been proposed previously [34; 40]. A simplified static component model was adopted and expressed by algebraic equations for the CO\({}_{2}\) HP. Mass and energy balance relationships, energy input-output relationship, and approximate relationships for the power consumption and HP COP were basic equations to be satisfied. A static component model was also adopted and expressed by algebraic equations for the mixing valve. Mass and energy balance relationships were basic equations to be satisfied.  
The component model for the storage tank was revised from the one which has been proposed previously [31; 34; 40]. A one-dimensional dynamic component model was adopted and expressed by differential algebraic equations for the storage tank. It was revised partly for the control volume where hot water is returned by bath heating and the region where a stratified temperature distribution is destroyed by hot water return. Mass and energy balance relationships for the control volumes, which were created by dividing the storage tank vertically, were basic equations to be satisfied. Heat transfer by water flow was related with mass flow rates and temperatures. Outlet water temperatures were equalized with the temperatures at the corresponding positions in the storage tank. On the other hand, when hot water retrieved from the top of the storage tank is returned to its bottom or side after heat exchange for bath heating, a stratified temperature distribution is destroyed, and three-dimensional convectional water flow arises. To consider such a phenomenon, it was necessary to conduct a three-dimensional thermo-fluid numerical analysis. However, this takes a long computation time, and it does not meet the purpose of this paper. Here, a partly perfect mixing model was adopted for the region where a stratified temperature distribution was destroyed and three-dimensional convectional water flow arises. Namely, the region where the temperatures in lower control volumes become higher than those in higher ones is identified; it is assumed that water in the region was perfectly mixed, and that the temperature of water in the region became uniform instantaneously.  
Figure 1: Configuration of multi-functional CO\({}_{2}\) heat pump water heating system.  
The component models for the heat exchanger and bathtub were created newly to consider the function of bath heating. The counter flow type was selected for the heat exchanger. Since the capacity of the heat exchanger was much smaller than that of the storage tank, a static component model was adopted and expressed by algebraic equations. Mass and energy balance relationships for the flows with higher and lower temperatures, which correspond to the storage tank and bathtub sides, respectively, were basic equations to be satisfied. The exchanged heat was expressed as the product of overall heat transfer coefficient, heat transfer area, and log mean temperature difference. Since the capacity of the bathtub was as large as that of the storage tank, a dynamic component model was adopted and expressed by differential algebraic equations for it. Here, perfect mixing was assumed in the bathtub. Mass and energy balance relationships were basic equations to be satisfied. Outlet water temperatures were equalized with the temperature in the bathtub.