# Enhanced Heat Transfer Study of Spherical Heat Storage Based on Response Surface Methodology
## 2 Experimental Procedure
### Design of RSM  
Response surface methodology (RSM) has been widely used in recent years for experimental design and optimization. RSM is an optimization method for experimental design that combines mathematics and statistics to enable correlation between the independent and dependent variables. The optimal working conditions are determined by analyzing the mathematical model established using RSM. Central composite design (CCD) is one of the most widely used experimental design methods in RSM that helps reduce the number of experiments used to achieve optimal conditions and analyze the interactions between parameters. In this study, CCD was applied to optimize the design of the complete melting time of phase change materials. The number of pin fins (N), water bath temperature (T), and the content of CuO in the nanocomposite phase change materials (M) were used as the factors for the study, and the effect of the three parameters on the complete melting time  
Figure 4: Temperature change at measurement point 3 in the heat storage unit: (**a**) various numbers of fins; (**b**) different mass fractions of CuO.  
Figure 3: Diagram of melting experiment system.  
was analyzed using the response surface methodology. According to the central composite design principle, the values of three parameters were designed: \(1\leq N\leq 7\), \(62\ ^{\circ}\text{C}\leq T\leq 78\ ^{\circ}\text{C}\), and \(0\) wt% \(\leq M\leq 6\) wt%. The real values of each parameter were code-transformed to facilitate response surface analysis. After the coding conversion to determine the range of values, specific experimental conditions were designed, and the experiments were arranged on the melting test system according to the designed experimental conditions. The experimental data under each experimental condition were recorded. The coding levels of the influence factors of the operational parameters are shown in Table 1, and Table 2 shows the design scheme and results. For the measurement error of the measurement time, the experiment was conducted three times for each group under the same experimental conditions, and the melting time in Table 2 is the mean value of the three experiments. Table 3 shows the uncertainty analysis that existed during this experimentation  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Variable**} & \multirow{2}{*}{**Symbol**} & \multicolumn{6}{c}{**Real Values of Code Levels**} \\ \cline{3-6}  & & \(-\alpha\) & \(-\mathbf{1}\) & \(\mathbf{0}\) & \(\mathbf{1}\) & \(-\alpha\) \\ \hline \(N\) & \(X_{1}\) & \(1\) & \(2\) & \(4\) & \(6\) & \(7\) \\ \(T\ (^{\circ}\text{C})\) & \(X_{2}\) & \(62\) & \(65\) & \(70\) & \(75\) & \(78\) \\ \(M\) (wt\%) & \(X_{3}\) & \(0\) & \(1\) & \(3\) & \(5\) & \(6\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Actual and coded values of the independent variables used for the experimental design.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Run**} & \multicolumn{6}{c}{**Design Factors**} & \multicolumn{3}{c}{**Responses**} \\ \cline{2-7}  & \(\mathbf{X_{1}}\) & \(\mathbf{X_{2}}\) & \(\mathbf{X_{3}}\) & \\ \cline{2-7}  & \(\mathbf{N}\) & \multicolumn{3}{c}{\(T\ (^{\circ}\text{C})\)} & \multicolumn{3}{c}{\(M\) (wt%)} & \multirow{2}{*}{\(t\) (min)} \\ \hline No. & Actual & Code & Actual & Code & Actual & Code & \\ \hline \(1\) & \(6\) & \(+1\) & \(65\) & \(-1\) & \(1\) & \(-1\) & \(101.0\) \\