## 4 Analysis and Discussion of Results
### Full 2\({}^{3}\) Factorial Analysis  
The calculation of the main effects and interactions involves comparing the average responses at the high and low levels of each factor. The main effect of a factor is the difference between the average response at the level (+) and the average response at the level (\(-\)) of that factor. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Entry** & **A** & **B** & **C** & \
\begin{tabular}{c} **\(\mathbf{G_{TR}}\)** \\ **(\(\times\mathbf{10^{-3}}\)\({}^{\circ}\)C-W\({}^{-1}\))** \\ \end{tabular} &
\begin{tabular}{c} **Average** \\ **(\(\times\mathbf{10^{-3}}\)\({}^{\circ}\)C-W\({}^{-1}\))** \\ \end{tabular} \\ \hline  
\) & \(+\) & \(+\) & \(+\) & \(540.450\) (1) \(|\) \(518.984\) (3) & 529.717 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Design matrix and results of the full 2\({}^{3}\). \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Entry** & **A** & **B** & **C** & **AB** & **AC** & **BC** & **ABC** & \
\begin{tabular}{c} \(\mathbf{G_{TR}}\)** \\ **(\(\times\mathbf{10^{-3}}\)\({}^{\circ}\)C-W\({}^{-1}\))** \\ \end{tabular} &
\begin{tabular}{c} \(\mathbf{h}\)** \\ **(\(\times\mathbf{10^{-3}}\)\({}^{\circ}\)C-W\({}^{-1}\))** \\ \end{tabular} \\ \hline
1 & \(-\) & \(-\) & \(-\) & \(+\) & \(+\) & \(-\) & 606.629 & 452.894 \\ \hline
2 & \(+\) & \(-\) & \(-\) & \(-\) & \(+\) & \(+\) & 632.359 & 467.149 \\ \hline
3 & \(-\) & \(+\) & \(-\) & \(-\) & \(+\) & \(-\) & \(+\) & 582.834 & 486.639 \\ \hline
4 & \(+\) & \(+\) & \(-\) & \(+\) & \(-\) & \(-\) & \(-\) & 634.962 & 516.025 \\ \hline
5 & \(-\) & \(-\) & \(+\) & \(+\) & \(-\) & \(-\) & \(+\) & 620.334 & 469.824 \\ \hline
6 & \(+\) & \(-\) & \(+\) & \(-\) & \(+\) & \(-\) & \(-\) & 645.460 & 498.295 \\ \hline
7 & \(-\) & \(+\) & \(+\) & \(-\) & \(-\) & \(+\) & \(-\) & 621.791 & 489.627 \\ \hline
8 & \(+\) & \(+\) & \(+\) & \(+\) & \(+\) & \(+\) & \(+\) & 618.622 & 529.717 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Contrast matrix for the 2\({}^{3}\) factorial design. Thus, to quantify the main effect of the power supply (factor A) on the \(G_{TR}\), for instance, Equation 10 should be used:  
\[E_{A}=\frac{1}{4}[(y_{2}+y_{4}+y_{6}+y_{8})-(y_{1}+y_{3}+y_{5}+y_{7})] \tag{11}\]  
In Table 5, \(y_{i}\) represents the response for the ith entry, resulting in \(E_{A}=24.954\). In a similar way, it is possible to calculate the main effects of factors B (temperature) and C (flow rate) for both responses (\(G_{TR}\) and \(h\)). The interaction effects between the factors can also be determined by examining their combined effects. To calculate the interaction effect between factors \(A\) and \(B\), we analyzed the difference between the average response values (using \(h\) as an example) at the high level (entries 1, 4, 5, and 8 in Table 5) and at the low level (entries 2, 3, 6, and 7). By applying this reasoning, all the effects of two-factor and three-factor interactions were determined. The values are presented in Table 6. The values in red indicate statistical significance at the 95% confidence level. In statistical analysis, it is necessary to decide which effects are significant and different from zero, and therefore worthy of interpretation. First, the effect whose absolute value is greater than \(t_{8}\times\)\(s_{\mathit{effect}}\) was considered statistically significant. The parameter \(t_{8}\), used to determine the significance threshold for the effects, originates from the t-distribution, which is commonly employed in statistical hypothesis testing. The value \(t_{8}\) = 2.306 corresponds to the critical t-value for a two-tailed test with eight degrees of freedom (from the design matrix, \(d_{f}\) = \(N-k\), where \(N\) is the number of experimental runs and \(k\) is the number of model parameters), at a 95% confidence level (\(\alpha\) = 0.05). This critical value is then multiplied by the standard error of the effect (\(s_{\mathit{effects}}\)) to establish the minimum magnitude that an effect must be considered as significantly different from zero.