## 2 Materials and Methods
### Thermodynamic Analysis  
One of the critical tools in defining the performance of energy transformations and energy storage systems is thermodynamic analysis. Therefore, the first and second laws of thermodynamics have been used in this study. In this condition, energy and exergy  
\begin{table}
\begin{tabular}{c c c c c c} \hline  & **V\({}_{\text{mantle}}\) m/s** & **T\({}_{\text{mantle}}\) (K)** & **V\({}_{\text{main}}\) m/s** & **T\({}_{\text{main}}\) (K)** & **Model** \\ \hline Case 13 & 0.147 & 353 & 0.036 & 290 & a.60- \\ \hline Case 14 & 0.147 & 353 & 0.073 & 290 & m.100- \\ \hline Case 15 & 0.147 & 353 & 0.11 & 290 & l.200 P \\ \hline Case 16 & 0.147 & 353 & 0.147 & 290 & \\ \hline Case 17 & 0.147 & 353 & 0.036 & 290 & \\ \hline Case 18 & 0.147 & 353 & 0.073 & 290 & m.150- \\ \hline Case 19 & 0.147 & 353 & 0.11 & 290 & l.100 P \\ \hline Case 20 & 0.147 & 353 & 0.147 & 290 & \\ \hline Case 21 & 0.147 & 353 & 0.036 & 290 & a.60- \\ \hline Case 22 & 0.147 & 353 & 0.073 & 290 & m.150- \\ \hline Case 23 & 0.147 & 353 & 0.11 & 290 & l.200 P \\ \hline Case 24 & 0.147 & 353 & 0.147 & 290 & \\ \hline \end{tabular}
\end{table}
Table 2: _Cont._balances are written for each case. Some guesses were made to decrease the complication of analyzes during thermodynamic analyzes. These guesses are as follows.  
* It was assumed that the physical properties of the water did not change with temperature.
* The flow inside the tank was considered to be frictionless.
* It was accepted that the pump power was zero.
* Heat losses from the tank to the environment were disregarded.
* The ambient temperature was 24 \({}^{\circ}\)C.  
These assumptions raise energy and exergy efficiency but do not significantly affect the relative values between their states. With these assumptions, critical parameters on the performance of horizontally heated hot water tanks are investigated.  
Figure 4: Experimental and numerical analysis results for the mantle outlet and main outlet temperature for the obstacle placed at \(\mathrm{a}=60^{\circ}\), \(\mathrm{m}=100\) mm, \(\mathrm{l}=100\) mm tank.  
Figure 3: (**a**) The experimental system of horizontal hot water tank, (**b**) Schematic view of the experimental system.  
Energy Analysis:  
Energy balance of the horizontal mantled hot water tank can be written as:  
\[Energy\ input\ -\left(Energy\ recovered\ -\ Heat\ loss\right)=Energy\ Accumulation \tag{1}\]  
This expression may be written as follows:  
\[\left(\dot{E}_{A}-\dot{E}_{B}\right)-\left[\left(\dot{E}_{D}-\dot{E}_{C} \right)-\dot{E}_{loss}\right]=\Delta\text{E}, \tag{2}\]  
The expression \((\dot{E}_{A},\dot{E}_{B})\) is the amount of exergy entering the inner tank, while the expression \(\left(\dot{E}_{D}-\dot{E}_{C}\right)\) is the amount of heat exergy taken from the tank.  
Energy Analysis:  
Energy balance for the horizontal mantled hot water tank can be expressed as:  
\[Energy\ input\ -\left[Energy\ recovered\ +\ Exergy\ loss\right]-\ Exergy\ destruction\ =\ Exergy\ accumulation \tag{3}\]  
\[\left(\dot{E}x_{A}-\dot{E}x_{B}\right)-\left[\left(\dot{E}x_{D}-\dot{E}x_{C} \right)-\dot{E}x_{dest}\right]-I=\Delta\text{E}x, \tag{4}\]  
This represents the amount of exergy entering the \((Ex_{A}-Ex_{B})\) volume in this statement. The expression \((Ex_{D}-Ex_{C})\) shows the net exergy amount from the storage volume. (I) represents the amount of exergy caused by irreversibility within the horizontal mantle hot water tank.  
Exergy efficiency statement for horizontally heated hot water tank can be written as follows.  
\[\psi=\frac{\left(\dot{E}x_{D}-\dot{E}x_{C}\right)-\dot{E}x_{dest}}{\left(\dot{ E}x_{A}-Ex_{B}\right)}, \tag{5}\]  
Exergy values can be calculated by the following expressions in the inlet and outlet of the tank.  
\[\dot{E}x_{i}=\dot{m}\ c\ \bigg{[}(T_{i}-T_{\infty})-T_{\infty}\ ln\frac{T_{i}}{T_ {\infty}}\bigg{]}. \tag{6}\]  
The \"i\" in this expression represents any entrance or exit of the tank.