## 3 Results and Discussion
### Efficiency Correlation with Artificial Neural Networks  
Artificial neural networks (ANN) or other practical prediction methods can be used to obtain the correlations for efficiencies of PV-thermal module or thermal engineering systems [31; 32; 33; 34; 35; 36; 37; 38; 39; 40; 41]. Radial basis function networks consist of three-layer network structures that include input, hidden and output  
Figure 8: Efficiency versus external heat transfer coefficient for two values of nanoparticle volume concentration (q = 1000 W/m\({}^{2}\), \(T_{in}\) = 30 \({}^{\circ}\)C, cylindrical shape).  
layers. The hidden layer nodes are radial basis functions. The outputs are calculated by a weighted average sum of the radial basis functions, which can be given as [37]:  
\[y(x_{i})=\sum_{k=1}^{N}w_{k}\Psi\left(||x_{i}-d_{k}||\right)+b. \tag{20}\]  
Radial basis function response decreases monotonically from a center point with distance. Gaussian function is a radial basis function which has central point c and smoothness parameter \(\sigma\) which controls the shape of the function. It is given in the following form:  
\[f(x)=e^{\left(-(x-c)^{2}/\sigma^{2}\right)}. \tag{21}\]  
A schematic representation of network topology is given in Figure 9 with three inputs: (solid particle volume fraction (\(\phi\)), convective heat transfer coefficient (h) and solar irradiation (q)) and two outputs (thermal and PV-efficiency).  
In order to evaluate the performance of trained networks, different error measures can be used. Mean square error (MSE) and coefficient of determination (\(R^{2}\)) performance parameters can be given as:  
\[\text{MSE}=\frac{1}{N}\sum_{k=1}^{N}(y^{CFD}-y_{k}^{ANN})^{2}, \tag{22}\]  
\[R^{2}=1-\frac{\sum_{k=1}^{N}(y_{k}^{CFD}-y_{k}^{ANN})^{2}}{\sum_{k=1}^{N}(y_{k }^{CFD}-\bar{y})^{2}}, \tag{23}\]  
where \(y_{k}^{ANN},y_{k}^{CFD}\), \(N\) and \(\bar{y}\) represent the predicted value form ANN, CFD value, sample number and the mean value of CFD values, respectively.  
The MATLAB Neural Network Toolbox (Version 2010, The Mathworks, Natick, USA) was used to select the number of hidden layers, number of neurons in each layer, training algorithm [42]. Feed-forward network structure with one hidden layer and a linear output layer was selected. The number of the neurons of ANN model was taken as 10. The feed-forward network structure with hidden layers and the linear output layer was selected and Levenberg-Marquardt back-propagation was used as the training algorithm. The random data division property of MATLAB is used and 70% of the data was used for estimation while 15% was used for validation and 15% of the data was used for testing purposes. Table 5 shows the number of samples for training, validation and testing, mean squared error (MSE) values and regression \(R\) values. A higher \(R\) value denotes a higher correlation between the outputs and target values.  
Table 6 represents the comparison results of CFD data between the predicted data by artificial neural networks for various values of pertinent parameters. The difference between the actual CFD data and established artificial neural network model is very small. This modeling strategy with ANN  
Figure 9: Schematic representation of the layers in the PV-thermal module.