## 4 Numerical Example: Transient Fin Problem
### Sensitivity Analysis of the Heat Flux History: Defining the Fin's Performance  
Using the heat flux output, two measurements of performance were calculated for the fin: heat transfer effectiveness, \(\varepsilon\), calculated using Equation (26), and the thermal efficiency, \(\eta\), defined in Equation (27). The heat transfer effectiveness corresponds to the ratio of the heat flux (\(\overset{\rightarrow}{q}\)) from a wall with a fin to the heat flux of the same wall without a fin, and the efficiency is the relationship of effective heat transfer from the fin to the room regarding the heat transfer of an ideal fin kept at the initial temperature \(\theta_{0}\)[24]. The sensitivity analysis of these two performance indices represents useful information for improving the design of fins since it can identify which variables positively affect their performance. In this regard, the main idea is to reduce the values of parameters with negative sensitivities and increase the importance of parameters with positive sensitivities. \[\varepsilon(t)=\frac{\overset{\rightarrow}{q}(\theta(X=1,t))}{\overset{ \rightarrow}{q}(X=1)} \tag{26}\]  
\[\eta(t)=\frac{\dot{Q}(\theta(X,t))}{\dot{Q}(\theta(X)=\theta_{0})}=\varepsilon (t)\frac{\delta}{2b} \tag{27}\]  
To calculate the performance parameters using Transient Thermal ZFEM and built-in Abaqus functions, the heat flux was obtained at the wall-fin interface (\(x\approx b\)). The evolution of \(\varepsilon\) and \(\eta\) are presented in Figure 9a,b, respectively, showing good agreement between the Transient Thermal ZFEM and the Abaqus built-in element response and with NRMSE values of 0.052 and 0.052, respectively, when compared with the analytical equations. \(\eta\) and \(\varepsilon\) quickly reach the steady-state value with an asymptotic behavior tending to positive infinity in \(t=0\) s. This behavior is explained by the discontinuity of the temperature profile at the wall-fin interface at that time. In addition, this is similar to the behavior of the transient heat transfer coefficient studied in similar problems [30,31]. A comprehensive sensitivity analysis of the fin performance parameters involving all of the parameters affecting the model was performed. The time history evolutions for all sensitivities are shown in the Supplementary Material, and only the results for \(k\), \(T_{\infty}\), and \(b\), are included in Figure 10 for \(\varepsilon\) and Figure 11 for \(\eta\). The maximum error for the sensitivities corresponded to 0.048 for \(S^{\varepsilon}\) and 0.048 for \(S^{\eta}\) when compared to the analytical solution. As observed in the time history for the sensitivities, the values quickly stabilized to the steady-state. Therefore, pie charts were developed showing the values of the contribution  
Figure 9: (**a**) Time history of fin \(\varepsilon\), (**b**) time history of fin \(\eta\). Black line, Equation (13), yellow star, Transient Thermal ZFEM solution, blue square, Abaqus built-in element solution, black thin dashed line, black dashed line, the steady-state solution. of the individual sensitivities to the total sum of the sensitivities for all variables in the model after steady-state conditions have been reached. Figures 10d and 11d show the sensitivities for \(\varepsilon\) and \(\eta\), respectively. In the pie charts, the \((+)\) symbol inside each of the colored sections implies a direct relationship between the output and the input variable while the \((-)\) implies an inverse relationship. From Figure 10d, it is possible to identify that \(\delta\), \(h_{U}\), and \(k\) are the most influential parameters to modify \(\varepsilon\). Each of these variables accounts for nearly a third of the total sensitivity of the effectiveness. Based on Equation (14), if \(\delta\), \(h_{U}\), or \(k\) are duplicated, an equivalent thermal effectiveness increment (or reduction) in the fin at steady-state will occur. In this regard, the \((-)\) inside the pie sections for \(\delta\) and \(h_{U}\) means that they should be reduced to increase the effectiveness.