## 2 Material and Methods
### The F-Chart Method  
As it was previously said, the f-chart method [26; 27; 28] is a well-validated and acceptable method for the estimation of the solar fraction on the solar thermal units for DHW and space heating. This work regards a system that produces useful heat only for covering the DHW needs and also it includes an electrical heater as the auxiliary heat source. In this method, there are two logistic parameters (X) and (Y) which lead to the calculation of the monthly solar fraction (f). These parameters are defined as below: \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Collector Type** & \({\rm F_{R}(\tau\alpha)}\) & \({\rm F_{R}U_{L}}\) & \({\rm R^{2}}\) \\  & **(-)** & **(W/m\({}^{2}\)K)** & **(\%)** \\ \hline Non-selective simple FPC & 0.73 & 5.85 & 99.93 \\ Selective advanced FPC & 0.77 & 4.59 & 99.34 \\ Collector with evacuated tubes (ETC) & 0.70 & 2.92 & 97.38 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Thermal efficiency parameters of the collectors after the approximation. Figure 2: Depiction of the collector efficiency and approximation lines (Analysis for \({\rm T_{am}}=25\)\({}^{\circ}\)C and \({\rm G_{T}}=1000\) W/m\({}^{2}\)). \[X=F_{R}U_{L}\cdot\frac{F_{R}^{\prime}}{F_{R}}\cdot(T_{ref}-T_{am,m})\cdot\Delta t \cdot\frac{A_{col}}{L} \tag{3}\]  
\[Y=F_{R}(\tau\alpha)_{n}\cdot\frac{F_{R}^{\prime}}{F_{R}}\cdot\frac{(\tau\alpha) }{(\tau\alpha)_{n}}\cdot H_{T}\cdot N\cdot\frac{A_{col}}{L} \tag{4}\]  
where (\(A_{col}\)) is the collecting area, (\(\tau\alpha\))\({}_{n}\) the thermal transmittance product at normal irradiation, (\(F_{R}^{\prime}/F_{R}\)) the ratio of the ret removal factor of the system to the collector, (\(H_{T}\)) the daily solar energy on the solar field area, (\(\Delta t\)) the total seconds of the month, (N) the number of the days in the month, (T\({}_{am,m}\)) the mean ambient temperature of the month, (T\({}_{ref}\)) is the maximum system temperature chosen at 100 \({}^{\circ}\)C and (L) is the thermal load of the month for the DHW. In this work, the ratio (\(F_{R}^{\prime}/F_{R}\)) is selected to be 0.95 as a typical value, while the ratio [(\(\tau\alpha\))/(\(\tau\alpha\))\({}_{n}\)] has a small monthly variation between 0.94 for winter and 0.90 for summer [26; 27; 28]. The use of the parameters (X) and (Y) makes possible the calculation of the monthly solar fraction (f) for the liquid systems by using the following formula [26; 27; 28]:  
\[f=1.029\cdot Y-0.065\cdot X-0.245\cdot Y^{2}+0.0018\cdot X^{2}+0.02\cdot Y^{3} \tag{5}\]  
It is critical to state that for the cases that f \(>\) 100% then it is selected at f = 100% in order to have reasonable results. Moreover, the storage tank of this work is selected to have a storage volume that is dependent on the collecting area according to the following expression [28]:  
\[V\left[m^{3}\right]=0.075\cdot A_{col}[m^{2}\right] \tag{6}\]  
The monthly solar fraction is practically the ratio of the energy that is exploited by the sun to the total load. Usually, this parameter is expressed in an indirect way by introducing the consumption from the auxiliary source (\(L_{aux}\)) as below:  
\[f=1-\frac{L_{aux}}{L} \tag{7}\]  
The yearly solar coverage (F) is found by utilizing the monthly solar coverage (f\({}_{i}\)) and the monthly load (L\({}_{i}\)):  
\[F=\frac{\sum_{i=1}^{12}f_{i}\cdot L_{i}}{\sum_{i=1}^{12}L_{i}} \tag{8}\]  
In the present work, the thermal load is calculated as below:  
\[L=\rho\cdot V_{w}\cdot c_{p}\cdot(T_{hot}-T_{cold})\cdot N \tag{9}\]  
where (\(V_{w}\)) is the daily water need expressed in [\(m^{3}\)], (N) are the days of the month, (T\({}_{hot}\)) is the desired temperature at 45 \({}^{\circ}\)C, (T\({}_{cold}\)) is the temperature from the cold grid water equal to (T\({}_{cold}=T_{w,m}\)), (\(\rho\)) is the water density and (c\({}_{p}\)) is the water-specific heat capacity. The cold temperature is calculated according to the Greek Technical Chamber data for every location separately [29].