# Enhanced Heat Transfer Study of Spherical Heat Storage Based on Response Surface Methodology
## 3 Results and Discussion  
The experimental CCD included 20 groups. Groups 1-8 were orthogonal designs, and groups 9-14 formed a central composite design with \(\alpha=\pm 1.68\). Groups 15-20 were centroid designs used to estimate the experimental error. The experimental data obtained from Table 2 were analyzed using the second-order polynomial model obtained from Equation (1). \[y=\beta_{0}+\sum_{i=1}^{3}\beta_{i}X_{i}+\sum_{i=1}^{3}\beta_{ii}X_{i}^{2}+\sum _{\text{j}}\sum_{\text{i=1}}^{3}\beta_{ij}X_{i}X_{j}+\varepsilon \tag{1}\]  
where \(y\) is the predicted response value; \(X_{i}\) and \(X_{j}\) are coded independent variables; \(\beta_{0}\), \(\beta_{1}\),..., \(\beta_{k}\), \(\beta_{ij}\) are regression coefficients; and \(\varepsilon\) is the statistical error. The least-squares method was used to determine the regression coefficients \(\beta\). For each response, regression coefficients were obtained using coded variables, and then regression coefficients were calculated for the actual variables. Analysis of variance (ANOVA) was used to determine the valid parameters of the model and to interpret its significance. After the regression fitting of each operational parameter, the regression model for the coding level of the complete melting time \(t\) is obtained as  
\[ \begin{array}{l}t=94.04-5.32X_{1}-6.52X_{2}-2.53X_{3}-1.01X_{12}-0.2125X_{13 }-0.8125X_{23}\\ +0.2274X_{1}{}^{2}+1.54X_{2}{}^{2}+0.3385X_{3}{}^{2}\end{array}  \tag{2}\]  
The regression model for the actual level of complete melting time is  
\[ \begin{array}{l}t=458.09+4.13N-9.30T+4.13M-0.101 NT-0.053 NM-0.081 TM\\ +0.057N^{2}+0.062T^{2}+0.085M^{2}\end{array}  \tag{3}\]  
From Equation (2), it can be seen that the water bath temperature has a greater effect on the melting time because the absolute value coefficient of the water bath temperature is greater than the absolute value coefficient of the number of pin fins and the mass fraction of CuO nanoparticles. Although the three primary term coefficients are negative, the required melting time is as short as possible, so all three factors have a coordinating effect on the complete melting time. In the actual level of the quadratic regression mathematical model, comparing the magnitude of each interaction term coefficient, the order of interaction between interaction term factors can be derived from Equation (3), which shows that the order of interaction between interaction factors is \(M^{2}>T^{2}>N^{2}>NT>NM>TM\). Table 4 shows the ANOVA data for the complete melting time. The F-value was used to determine the statistical significance of the second-order regression model. When \(p<0.05\), the corresponding parameters were all significant, and when \(p<0.0001\) and F = 34.37 in the complete melting time model, the model was apparently significant. The coefficient of determination R\({}^{2}\) is the ratio of the response contribution of the regression model, and the closer it is to one, the higher the model explanation. R\({}^{2}\) = 0.9687. This indicates that more than 96.87% of the data deviations can be explained. The adequacy of the model can be tested by the correction decision coefficient R\({}^{2}\)adj, and R\({}^{2}\)adj = 0.9405, indicating a strong interaction between the predicted response values and the experimental results. AP is defined as a measure of the ratio of signal to noise, and AP > 4 is desirable. The model has AP = 22.2023, indicating that the regression equation is well-adapted. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Source** & **Sum of Square** & **Degree of Freedom** & **Mean Square** & **F-Value** & **\(p\)-Value** & **R\({}^{2}\)** & **Adjusted R\({}^{2}\)** & **Adequate Precision** \\ \hline Model & 1036.96 & 9 & 115.22 & 34.37 & \textless{}0.0001 & 0.9687 & 0.9405 & 22.2023 \\ Error & 4.37 & 5 & 0.8737 & - & - & - & - & - \\ Total & 1070.48 & 19 & - & - & - & - & - & - \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Results of ANOVA.