# Numerical Investigation of Thermal Efficiency of a Solar Cell
## 2 Materials and Methods
### Experimental Setup
#### 2.1.2 Heat Balance of the Experimental Setup  
In order to balance the heat input against the heat lost, and the heat gained by the water, uncertain heat transfer was calculated for different values of net heat input. Net  
Figure 2: Top view of PV panel, back view of PV panel, and thermal resistance for individual component of PV cells.  
heat input is the total solar radiation incident minus the electrical power output, as shown in Equation (1), which was further used to calculate the absorbed heat (Equation (2)). The electrical power output is measured with the help of a digital multi-meter (Figure 2). Experiments were carried out for various heat inputs to the PV cell ranging from 2 W to 30 W and the mass flow rate of water being fixed to 0.025 kg/s. Heat loss because of conduction and radiation from the cross-section surface of PV cells were individually accounted for every material (Figure 2) using the commercial equation of conduction and radiation Equations (4) and (5). The average temperature in line with the thickness of the PV cell at the individual surface was calculated using Equation (3).  
\[Q_{net,\ input}=Q_{input}-Q_{Electrical} \tag{1}\]  
\[Q_{absorbed}=0.93\times Q_{net,\ input} \tag{2}\]  
\[T_{j}=T_{i}-\frac{L_{ij}}{K_{mat}A_{s\ mat}}\times Q_{absorbed} \tag{3}\]  
\[Q_{Cond}=K_{Mat}\times A_{s\ Mat}\times\left(\frac{T_{j}+T_{i}}{2}-T_{a}\right) \tag{4}\]  
\[Q_{Rad}=\sigma eA_{s\ Mat}\times\left[\left(\frac{T_{i}+T_{j}}{2}\right)^{4}-T_ {a}{}^{4}\right] \tag{5}\]  
\[Q_{Cow,\ PV\ Cell\ Bottom}=h_{c,\ Air}\times A_{PV\ cell}\times(T_{6}-T_{a}) \tag{6}\]  
\[Q_{Rad,\ PV\ Cell\ Bottom}=\sigma eA_{PV\ cell}\times\left(T_{6}{}^{4}-T_{a}{}^{4}\right) \tag{7}\]  
\[Q_{water}=m^{\prime}\times C_{p,water}\ \times(T_{water,\ out}-T_{water,\ in}) \tag{8}\]  
The convection and radiation heat loss from the back surface of the PV cell was accounted for by using Equations (6) and (7) (Figure 2). A similar method was adopted for calculating the heat transfer because of convection and radiation from the pipe surface exposed to the atmosphere (Figure 2). Figure 3 shows the actual surface responsible for heat transfer because of convection and radiation from the pipe surface. Last, the temperature difference between the inlet and outlet of the heat exchanger was measured, which enables the magnitude of heat gained by the water (Equation (8)). The heat transfer through physical modes of heat transfer highly depended on human skills, thus increasing the probability of calculation errors in model validation.  
Figure 3: Components of heat lost from various surfaces of the pipe.  
In order to validate the results of thermal efficiency computed numerically, a graph of theoretical efficiency versus numerical efficiency is plotted later in the Results and Discussion section. The numerical efficiency was calculated using numerically computed temperature of water at the outlet, while the theoretical efficiency is defined using Equation (9). In Equation (9), the suffix \(T\) and \(N\) stand for theoretical and numerical, respectively.  
\[\eta_{thermal\ T,\ N}=\frac{m^{\prime}\times\mathrm{C}_{p,water}\ \times(T_{water,\ out}-T_{water,\ in})}{Q_{net,\ input}} \tag{9}\]