## 2 Materials and Methods
### Tank Capacity and Water Balance Model  
Dimensioning rainwater tanks in Polish hydrological conditions is relatively simple. It is usually assumed that for every 25 m\({}^{2}\) of the area from which rainwater is collected, the tank should be able to store 1 m\({}^{3}\) of water. Additionally, the time of storing water in the tank should not be longer than 2-3 weeks [24]. In the study analyzed in this manuscript, the area of the roof from which rainwater could be collected was 300 m\({}^{2}\). In accordance with the above, a tank with a capacity of 12 m\({}^{3}\) was selected, which, at a consumption of 1030 dm\({}^{3}\) per day, will guarantee a water retention time in the tank not longer than 11 days. In order to verify the selection of the tank size, the research was extended to include a comparison of different tank storage capacities, and their influence on cost-benefit analysis. Four different sizes were selected for further examination. Both, smaller and larger tank volumes were chosen (6, 9, 15, and 18 m\({}^{3}\)). The water balance model was run to evaluate the hypothetical rainwater harvesting (RWH) system, taking into account the 20-year daily rainfall data, runoff capture, rainwater storage, and water demand. The water balance model was operated in a day-step mode, as described in the diagram in Figure 2.  
The other water balance model was run to evaluate the hypothetical combined RWH system with water recovery from the exhaust air system presented in Figure 3. A cross-flow exchanger operating in air-conditioning systems was proposed as an element for heat recovery, which would enable additional water recovery from the exhaust air. The contact of humid exhaust air with the cold plate (lower than the dew point of the exhaust air) of the exchanger caused water vapor to accumulate in the air condensation in the exhaust air channels. The low plate temperature was the result of its contact with the ambient air. In calculations, the 8-year period of the actual parameters of the ambient air in Wroclaw was taken into account. The rainwater inflows were calculated based on the rainfall amounts, the rooftop area, and the surface runoff coefficient (0.95). The rainwater storage was determined as the in-, out-, and overflows balance in the tank volume. The RWH system performance was defined as a share of rainwater outflow in water demand, calculated based on [PERSON] et al. [23] and shown in Part 1 of this study [23]. In case of insufficiency of the rainwater outflow to meet the water demand, tap water flows were automatically supplied by a switch-over controller. The simulation took into consideration the rainwater losses and rainfall overflow. In practice, two algorithms were often used to describe the behavior of the collecting system during a given time interval: the yield after a spillage (YAS) and the yield before a  
Figure 3: The diagram of the water flow in the rainwater harvesting system cooperating with water recovery from the exhaust air. spillage (YBS). In this investigation, each simulation was tested by the YAS algorithm, in which the withdrawal occurred before the rainfall:  
\[Q_{D_{t}}=\max\left\{ \begin{array}{c}V_{t-1}+A\cdot R_{t}-S\\ 0\end{array} \right. \tag{1}\]  
\[V_{t}=\min\left\{ \begin{array}{c}D_{t}\\ V_{t-1}\end{array} \right. \tag{2}\]  
\[V_{t}=\min\left\{ \begin{array}{c}V_{t-1}+A\cdot R_{t}-Y_{t}\\ S-Y_{t}\end{array} \right. \tag{3}\]  
where, \(Q_{Dt}\) (m\({}^{3}\)) is the volume discharged as overflow from the storage tank at time step \(t\), \(V_{t}\) (m\({}^{3}\)) is the volume stored at time step \(t\), \(R_{t}\) (m) is the rainfall at time \(t\), \(A\) (m\({}^{2}\)) is impervious surface area (m\({}^{2}\)), \(Y_{t}\) (m\({}^{3}\)) is the yield of rainwater from the storage tank at time step \(t\), \(D_{t}\) (m\({}^{3}\)) is the water demand at time step \(t\), and \(S\) (m\({}^{3}\)) is the tank storage capacity.