# Modeling Liquid Thermal Conductivity of Low-GWP Refrigerants Using Neural Networks
## 4 Experimental Settings and Results  
A simple ANN with five input parameters and only one hidden layer was chosen to calculate the thermal conductivity of liquid low-GWP refrigerants. It has the same input parameters of Equation (1), developed to describe \(\lambda_{\mathrm{L}}\) of alternative refrigerants: reduced temperature (\(T_{r}\)), critical pressure (\(P_{c}\) (\(bar\))), acentric factor (\(\omega\)), molecular weight (\(M\) (\(\frac{kg}{mol}\))) and reduced pressure (\(P_{r}\)). Since all the selected \(\lambda_{L}\) data are positive, and to account for the nonlinearity of the model, a sigmoid was chosen as the activation function. In this study, to avoid overfitting the proposed network, we chose to split the entire dataset as follows: the training set comprised 80% of the data, the validation set contained 10%, and the remaining 10% was used for the test set. Table 1 details the number of points. Another way to avoid overfitting is by using a network with few neurons with a lean and non-rigid architecture. For this reason, it was chosen to design a network with only 1 hidden layer and 19 neurons. As shown in Figure 2, the ANN with 19 neurons ensured excellent results, which are summarized in Table 3. The same table reports additional statistical parameters that characterize the various datasets. The average absolute relative deviation (AARD%), maximum absolute relative deviation (MARD%), and root mean square error (RMSE) were compared:  
\[AARD\%=\frac{100}{N}\sum_{i=1}^{N}\left|\frac{\lambda_{i}^{\text{exp}}-\lambda_ {i}^{\text{calc}}}{\lambda_{i}^{\text{exp}}}\right| \tag{11}\]  
\[MARD\%=Max\left(\frac{100}{N}\sum_{i=1}^{N}\left|\frac{\lambda_{i}^{\text{ exp}}-\lambda_{i}^{\text{calc}}}{\lambda_{i}^{\text{exp}}}\right|\right) \tag{12}\]  
\[RMSE=\sqrt{\frac{1}{N}\sum_{i=1}^{N}\left(\lambda^{Calc}-\lambda^{Exp}\right) ^{2}} \tag{13}\]  
As evident in Table 3, all statistical values are aligned, confirming the goodness of the proposed network, which manages to model the thermal conductivity of alternative liquid refrigerants accurately. In particular, the ANN provided very low deviations over the entire database, reaching an AARD% of 0.389%. It is also important to note that the deviations of the test set are aligned with those of the other sets, confirming the prediction capability of the network and the low overfitting tendency. In fact, the test set has an AARD% of 0.375% with a MARD% of 4.598%, in line with the deviations of all other datasets. Moreover, the K-fold cross-validation technique was used after dividing the dataset into 4 subsets. The model used the first subset in the first iteration to test the model, while the remaining dataset was used to train the model. The second subset helped to test the model, and the other subsets supported the training process. The same procedure was repeated until the test set used every subset. The results are shown in Table 4, in which all the deviations are in line, and there are no particular critical issues in either the test or validation sets. Figure 2: Absolute average percent deviation between data collected and the ANN results vs. the number of neurons in the hidden layer. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Data Set** & **Point N.** & **AARD\%** & **MARD\%** & **RMSE** \\ \hline Training set & 2723 & 0.390 & 6.074 & 0.0005 \\ Validation Set & 340 & 0.396 & 4.945 & 0.0005 \\ Test Set & 341 & 0.375 & 4.598 & 0.0005 \\ Overall & 3404 & 0.389 & 2.070 & 0.0003 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Statistical parameters for the individual datasets including the training set, validation set, test set, and the entire dataset. Table 5 shows the AARD% per fluid. Again, homogeneity in the results can be observed. The fluid with the largest deviation is R1336 mzz(E), with an AARD% of 1.223% and a MARD% of 5.499%. Figure 3 shows the experimental and calculated conductivity of R1336 mzz(E) fluid and the relative deviations.