# Sustainable Approach of Generating Water and Energy:
## 3 Mathematical Model
### Hybrid Photovoltaic Thermal System  
Desalination of seawater or brackish water is mostly powered by fossil fuels, which raises questions regarding greenhouse gas emissions, especially in the dry Middle East. There have been several initiatives to use solar resources to solve this problem, but all of them have focused on turning solar energy into electricity. To overcome these difficulties, previous research proposed a small-scale hybrid photovoltaic/thermal (PVT) solar collector-powered reverse osmosis (RO) desalination system suitable for a rural village in the Kingdom of Saudi Arabia (KSA) was constructed, and its power needs were estimated [28]. The PVT model developed previously in prior research was utilized in this work.  
The type of PVT considered here is a flat-plate collector. A cross-section area of a typical flat-plate collector is shown in Figure 3, where I indicates the solar radiation falling on the collector's surface.  
Figure 2: Artificial Neural Network.  
The governing equations of the PVT model that are used to predict electric and thermal energy generation were adopted from the literature [29; 30]. The estimated array is made up of collectors wired in parallel \(N_{p}\) and \(N_{s}\) collector in series, totaling \(N_{p}*N_{s}\) panels. Equation (1) is used to estimate the total thermal energy \((Q_{h})\) transfer to the fluid from a connected series panel [30].  
\[\dot{Q}_{h_{\text{ }total}}=N_{p}*\dot{Q}_{h*N_{s}} \tag{1}\]  
The thermal energy transmitted to the working fluid through \(N_{s}\) panels connected in series is estimated by Equation (2), as follows [29]:  
\[\dot{Q}_{h*N_{s}}=N_{s}A_{c}F_{R}\Bigg{(}S\Bigg{[}\frac{1-(1-K_{K})^{N_{S}}}{N_ {S}K_{K}}\Bigg{]}-U_{L}\Bigg{[}\frac{1-(1-K_{K})^{N_{S}}}{N_{S}K_{K}}\Bigg{]} \Big{(}T_{fi}-T_{a}\Big{)}\Bigg{)} \tag{2}\]  
The expression \(K_{k}\) is given by [29]:  
\[K_{k}=\frac{A_{C}F_{R}U_{L}}{mC_{p}} \tag{3}\]  
In Equation (3), the fluid mass flow rate is denoted by m and the fluid specific heat is \(C_{p}\). Equation (4) calculates the amount of solar radiation absorbed per unit area of a single PVT panel.  
\[S=I(\alpha\tau)\Big{(}1-\frac{\eta_{c}}{\alpha}\Big{)} \tag{4}\]  
where \(I\) refers to solar irradiation intensity, \(\tau\) is the collector glazing transmittance. \(\eta_{c}\) and \(\alpha\) are the electric efficiency and the collector absorptance. The outlet temperature of the fluid is given by Equation (5) [29].  
\[T_{fo}=\bigg{[}\frac{S}{U_{L}}+T_{a}\bigg{]}\bigg{[}1-e^{(-\frac{-N_{s}A_{c}F_ {B}U_{L}}{mC_{p}})}\bigg{]}+T_{fi}e^{(-\frac{-N_{s}A_{c}F_{B}U_{L}}{mC_{p}})} \tag{5}\]  
The following equation gives the collector heat removal factor [29]:  
\[F_{R}=\frac{mc_{p}}{U_{L}}\bigg{[}1-e^{-\frac{U_{L}f}{mC_{p}}}\bigg{]} \tag{6}\]  
Electric equations can be expressed as the following Equations (7) and (8) [28].  
\[\dot{Q}_{e\text{ }total}=Q_{e}N_{s}N_{p} \tag{7}\]  
\[Q_{e}=\frac{A_{c}S\eta_{a}}{\alpha}\bigg{[}1-\frac{\eta_{p}B_{r}}{\eta_{c}} \bigg{[}F_{R}\Big{(}T_{fi}-T_{a}\Big{)}+\frac{S}{U_{L}}(1-F_{R})\bigg{]}\bigg{]} \tag{8}\]  
Figure 3: Block diagram for the PVT [28].