# Thermal Conductivity of Low-GWP Refrigerants Modeling with Multi-Object Optimization
## 3 Results and Discussion  
In this section, the final result of the MOO process is described. The algorithm started with an initial guess for the solution and gradually improved it until a satisfactory result was obtained. It incorporated techniques from the NSGAII genetic algorithm to minimize the AARD% and maximize the R\({}^{2}\). After all the steps described above, the following optimal result was obtained: AARD% = 1.40660 and R\({}^{2}\) = 0.991193. Figure 3 shows the result of the choice process of the optimal result from the Pareto frontier made with the abovementioned tool [31; 32]. It shows a graphical representation of the position of the optimal solution compared to the rest of the solutions analyzed. In this figure, the best solution is indicated by the red circle located at the bottom-left corner. This solution was selected as the optimal one of the problem because it is the only one that provides the maximum of the R\({}^{2}\) and the minimum of the AARD% with respect to the weighted objective functions. Therefore, the chosen point is the best compromise between the two weighted objective functions. The parameters found at the end of the minimization procedure are: \(a=-0.43751\), \(b=0.0055\), \(c=0.3000\), \(d=0.25228\), \(g=0.76287\), \(f=0.0805\), \(f_{0}=-0.0023\), and \(\lambda_{0}=0.2885\). Table 2 shows a comparison between the old and new parameters. It is possible to see that there are a few significant differences between the old and new parameters. Nevertheless, the main differences are between the very low parameters, where the new and old parameters differ by a relatively small amount. On the other hand, we can notice that the magnitude and the sign are the same between the two parameter sets. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & **Original Parameters** & **New Parameters** & **Difference** & **Rate of Change** \\ \hline \(\lambda_{0}\) & 0.43693 & 0.2885 & 0.14843 & 34\% \\ \(a\) & \(-0.28725\) & \(-0.43751\) & 0.15026 & \(-52\%\) \\ \(b\) & 0.00372 & 0.0055 & \(-0.00178\) & \(-48\%\) \\ \(c\) & 0.26967 & 0.3000 & \(-0.03033\) & \(-11\%\) \\ \(d\) & 0.36436 & 0.25228 & 0.11208 & 31\% \\ \(f_{0}\) & \(-0.00135\) & \(-0.0023\) & 0.00095 & \(-70\%\) \\ \(f\) & 0.05484 & 0.0805 & \(-0.02566\) & \(-47\%\) \\ \(g\) & 0.88049 & 0.76287 & 0.11762 & 13\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison between the parameters obtained with the method used in [19] and the new method (NSGAII). Figure 3: Pareto frontier for the chosen configuration. In red is the point chosen as optimal. As shown in Table 3, the new method for computing the coefficients provided better values of AARD% and RMSE (Equation (4)) with respect to the original coefficients. However, it did not ensure the best solution for R\({}^{2}\) due to the weighing in the optimal point selection on the Pareto frontiers. In fact, as we stated in the previous section, the objective functions were weighted to obtain better results of AARD%. However, the R\({}^{2}\) is very close to the previous one, with a change on the third decimal digit. \[\text{RMSE}=\frac{100}{N}\sum_{i=1}^{n}\sqrt{\frac{\left(\lambda_{L,i}^{exp} -\lambda_{L,i}^{calc}\right)^{2}}{\lambda_{L,i}^{exp}}} \tag{4}\]  
Table 4 shows the deviations between the experimental thermal conductivity data of the studied liquid refrigerants and the calculations provided by Equation (1) using the coefficients corresponding to the chosen point in the Pareto frontier. The results given by Equation (1) using the original coefficients (20) and by REFPROP 10.0 are also reported in Table 4. REFPROP 10.0 provides the thermal conductivity values using either fluid-specific correlations or a modification of the extended CSP method. In particular, these models describe the thermal conductivity dependence on temperature and pressure at different thermodynamic conditions. As expected, REFPROP 10.0 ensured the lowest deviations since it uses fluid-specific correlations or extended corresponding states models.