# Enhanced Heat Transfer Study of Spherical Heat Storage Based on Response Surface Methodology
## 3 Results and Discussion
The residuals are the values fitted by the regression model subtracted from the experimental measurements. The smaller the residuals, the more accurately the regression model describes the experimental results. Figure 5a shows the relationship between the experimental values and the response prediction values. From the ideal linear distribution curve, we can see that the experimental and simulated values are basically distributed along a straight line, indicating that the experimental and simulated values are highly fitted. A normal plot of the experimental residuals can be used to determine the fit of the model, as shown in Figure 5b, which shows the fit of the residuals to the standard deviation. If the distribution of points on the plot is a straight line, the residuals are normally distributed. As can be seen from the figure, most of the experimental values fall on the predicted values in a linear distribution, indicating that the model fits the actual results well. Figure 5c,d shows the relationship between the residuals and the predicted values. The points in the diagnostic plot that are beyond the upper and lower boundary lines are anomalies, and these anomalies can indicate whether the model has large deviations. It can be seen from the figure that no points are located outside the boundary line, indicating that the model and the data match. Moreover, there are no large interference points in the model points, and the distribution of model points is also concentrated, so the model diagnostic plot results match the ANOVA data. Response surfaces are used to analyze the interaction between the factors and the optimal response values at the optimal factor conditions. Using Design-Expert 13.0, 3D surface plots of the response values were obtained, and the contour lines were circular for insignificant interaction of the two factors and elliptical for significant interaction. The melting time of PCM in a spherical thermal storage unit is affected by the number of pin  
Figure 5: The diagnostic plots of RSM models: (**a**) actual vs. predicted values for complete melting time; (**b**) normal probability plots; (**c**) residuals vs. experimental runs; (**d**) residuals vs. predicted values. fins, the water bath temperature, and the content of CuO nanoparticles, all three of which have a synergistic effect on the complete melting time. Figure 6 shows the effect of the number of pin fins and the water bath temperature on the melting time. The complete melting time decreases with increasing the number of pin fins and increases with decreasing the water bath temperature, and the water bath temperature has a greater effect on the melting time than the number of pin fins. This is because the melting of PCM is mainly influenced by temperature; the higher the external ambient temperature, the faster the PCM reaches the phase change point. Therefore, if the water bath temperature is high, the heat that can be absorbed by the PCM in the spherical unit increases; the melting rate is faster, and the melting time decreases. During the melting of the phase change material, the melting of the material in the units is faster because of the large temperature difference between the inner wall of the sphere and the solid phase change material, and the heat transfer is mainly heat conduction. As melting proceeds, the thickness of the liquid PCM layer increases, leading to an increase in thermal resistance and resulting in a gradual decrease in the melting rate. At this point, the addition of internal pin ribs is equivalent to increasing the heat transfer area in contact with the inside of the sphere, which speeds up the melting process to a certain extent and shortens the melting time. Figure 7 shows the interaction effect of the number of pin fins and the mass fraction of CuO nanoparticles, which is not significant, as shown by the circular shape of the contour plot. From Equation (2), it is clear that the number of pin fins has a greater effect on the melting time of PCM than the CuO nanoparticles content.