## 2 Methods
### Transfer Function of the \(Cop\) Response  
After performing the ANOVA, the transfer function was determined, using the estimated uncoded coefficients derived from MINITAB 19 for the \(COP\) responses. The uncoded coefficients shown in Table 6 were based on the ANOVA analysis, including up to four-way interactions. It is important to emphasize the importance of including the appropriate number of interactions, to minimize error in the transfer function. The estimated uncoded coefficients, which considered up to four-way interactions, helped in reducing the error. As indicated in Table 6, the R-Sq, which measure the goodness of fit, decreased to 55.69% when only the main effects were considered. However, it improved to 88.99% when two-way interactions were included. The R-Sq value was calculated by dividing the Seq-SS of the main effects by the total sum of squares.  
The regression equation in uncoded units with all terms is observed in Equation (14):  
\[COP=5.5922+0.00656D-0.41781x_{n}-0.33544\Delta T_{mn}-5.5487B\\ -0.25144 Ls-0.000438D\cdot x_{n}-0.000563D\cdot\Delta T_{mn}-0.016 25D\cdot B\\ -0.000813D\cdot L_{sn}+0.024912x_{n}\cdot\Delta T_{mn}+0.69325X \cdot B+0.021062x_{n}\cdot L_{sn}\\ +0.44975\Delta T_{mn}\cdot B+0.018988\Delta T_{mn}\cdot L_{sn}+0. 22775B\cdot L_{sn}\\ +0.000038D\cdot x_{n}\cdot\Delta T_{mn}+0.000750D\cdot x_{n}\cdot B +0.000038D\cdot x_{n}\cdot L_{sn}\\ +0.001250D\cdot\Delta T_{mn}\cdot B+0.000062D\cdot\Delta T_{mn} \cdot L_{sn}+0.002250D\cdot B\cdot L_{sn}\\ -0.047450x_{n}\cdot\Delta T_{mn}\cdot B-0.001553x_{n}\cdot\Delta T _{mn}\cdot L_{sn}-0.032450x_{n}\cdot B\cdot L_{sn}\\ -0.024550\Delta T_{mn}\cdot B\cdot L_{sn}-0.000050D\cdot x_{n} \cdot\Delta T_{mn}\cdot B\\ -0.00002D\cdot x_{n}\cdot\Delta T_{mn}\cdot L_{sn}-0.000050D\cdot x _{n}\cdot B\cdot L_{sn}\\ -0.000150D\cdot\Delta T_{mn}\cdot B\cdot L_{sn}+0.002790x_{n} \cdot\Delta T_{mn}\cdot B\cdot L_{sn}+0.14844 Ct\cdot Pt \tag{14}\]  
The regression equation in uncoded units with \(p\)-values \(\leq 0.05\) is defined in Equation (15):  
\[COP=5.5922-0.41781x_{n}-0.33544\Delta T_{mn}-5.5487B\\ -0.25144L_{sn}+0.024912x_{n}\cdot\Delta T_{mn}+0.69325x_{n}\cdot B +0.021062x_{n}\cdot L_{sn}\\ +0.44975\Delta T_{mn}\cdot B+0.018988\Delta T_{mn}\cdot L_{sn}+0. 22775B\cdot L_{sn}\\ -0.047450x_{n}\cdot\Delta T_{mn}\cdot B-0.001553x_{n}\cdot\Delta T _{mn}\cdot L_{sn}-0.032450x_{n}\cdot B\cdot L_{sn}\\ -0.024550\Delta T_{mn}\cdot B\cdot L_{sn}+0.002790x_{n}\cdot \Delta T_{mn}\cdot B\cdot L_{sn}=1.759313 \tag{15}\]  
The transfer function in Equation (14) includes all coefficients, while Equation (15) features only those coefficients with a \(p\)-Value \(\leq 0.05\). Both functions were constructed using these uncoded units and highlight the most significant coefficients affecting the \(COP\) response. Specifically, Equation (15) contains only the significant coefficients, including the four-way interactions identified through the ANOVA of the \(COP\) results. Given the ANOVA results, the R-sq value from the response surface methodology (RSM) was not included, as it indicated a minimal contribution from the curvature in the full factorial (less than 1%); see Table 6. Consequently, the optimal \(COP\) value for the analyzed space could be calculated and validated using the \"Solver\" function in 'Excel' or 'Libreoffice calc', based on the transfer function derived from the \(COP\) parameters \(x_{n}\), \(\Delta T_{mn}\), \(B\), and \(L_{sn}\) and their interaction with the uncoded coefficients; see Equation (15). The residual error related in the ANOVA for this case was zero (0.00%). Additionally, the transfer function indicated that the parameter \(D\) and its interactions were not significant, based on the \(p\)-values presented in Table 6.