# Numerical Study on Cooling Performance of a Steam-Cooled Blade Based on Response Surface Method
## 4 Analysis of the Results of the Response Surface Model Fitting
### Correlation Fitting Based on Response Surface Model  
The correlation equations of the average cooling efficiency, temperature non-uniformity and average dimensionless temperature of the steam-cooled blade can obtain the characteristics of the heat transfer of the blade in the range of the working parameters, which is important for the safe operation and performance prediction of the blade. The response surface model (RSM) can obtain an explicit correlation between the operating parameters and the characteristics of the heat transfer of the blade with high accuracy, which is given by the following equations [30; 31]:  
\[y=B_{0}+\sum_{i=1}^{k}B_{i}x_{i}+\sum_{i=1}^{k}B_{ii}x_{i}^{2}+\sum_{i<j}^{k}B _{ii}x_{i}x_{j}+\omega \tag{13}\]  
where \(B_{0}\), \(B_{i}\), \(B_{ii}\) and \(B_{ij}\) are regression coefficients; \(x_{i}\) and \(x_{j}\) are design variables; \(y\) is the response; and \(\omega\) is the prediction error. The second-order polynomial response surface model was obtained based on CCF fitting. The specific test design table is shown in Table 3. The design parameters are \(T_{g}\), \(p_{\rm go}\), \(p_{\rm r}\), \(T_{\rm r}\) and \(M_{\rm r}\) and the responses are \(\varepsilon_{\rm ave}\), \(\zeta\) and \(\eta_{\rm ave}\). The coefficients of the response surface model are given in Table 4. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Coefficients** & \(\varepsilon_{\rm ave}\) & \(\zeta\) & \(\eta_{\rm ave}\) \\ \hline \(B_{0}\) & 0.285 & \(-\)0.008 & 0.999 \\ \(B_{1}\) & \(-\)0.00048 & 0.000216 & 0.000188 \\ \(B_{2}\) & 0.000036 & 0.000135 & 0.00018 \\ \(B_{3}\) & \(-\)1.229 & \(-\)1.023 & 0.0413 \\ \(B_{4}\) & 0.317 & \(-\)0.0224 & \(-\)0.2213 \\ \(B_{5}\) & 0.06165 & 0.018726 & \(-\)0.034086 \\ \(B_{11}\) & 0 & 0 & 0 \\ \(B_{22}\) & 0.000001 & 0 & 0 \\ \(B_{33}\) & 0.694 & 0.0187 & 0.1034 \\ \(B_{44}\) & \(-\)0.0983 & 0.01049 & 0.0342 \\ \(B_{55}\) & \(-\)0.001819 & \(-\)0.000304 & 0.000636 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Coefficients of response surface model. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Order** & \multicolumn{4}{c}{**Design Variables**} & \multicolumn{4}{c}{**Responses**} \\ \hline  & \(T_{\rm gj}\)/K & \(p_{\rm go}\)/kPa & \(p_{\rm r}\) & \(T_{\rm r}\) & \(M_{\rm r}\) & \(\varepsilon_{\rm ave}\) & \(\zeta\) & \(\eta_{\rm ave}\) \\ \hline  
\hline \hline \end{tabular}
\end{table} 
Table 3: _Cont._Figure 18 presents a comparison of the numerical simulation value and RSM predicted value for the sample points of the experimental design of the steam-cooled blade. Figure 18a-c shows the comparison of the two values when the response is average cooling efficiency \(\varepsilon_{\text{ave}}\), temperature non-uniformity \(\zeta\) and average dimensionless temperature \(\eta_{\text{ave}}\) of the steam-cooled blade, respectively. In the figure, the solid black line indicates the numerically simulation values of the sample points, the red scatter indicates the RSM predicted values, and the blue line indicates the \(\pm 1\%\) deviation of the numerical simulation values. From Figure 18, it can be seen that the RSM predicted values of the average cooling efficiency \(\varepsilon_{\text{ave}}\), temperature nonuniformity \(\zeta\) and average dimensionless temperature \(\eta_{\text{ave}}\) of the steam-cooled blade are all distributed within \(\pm 1\%\) of the numerical simulation values of the sample points, which indicates that the fitted response surface model can well predict the numerical simulated values.