# Definition and Determination of Fin Substitution Factors Accelerating Thermal Simulations
## 4 Discussion and Conclusions  
The overarching goal of the authors' research is to accelerate heat transfer simulations of finned arrays. For this purpose, the authors have defined the \(FSF\) and developed a process whose output is the most accurate analytical \(FSF\) calculation path for a given case. The performance of the approach was demonstrated using the example of a vertical finned array under natural convection with an isothermal temperature at the fin base. Based on \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Unit** & \multicolumn{3}{c}{**Examples from Literature**} \\ \hline
**Source** & **-** & **[**25**]** & **[**26**]** & **[**27**]** \\ \hline \(s\) & mm & 5.00 & 25.00 & 4.00 \\ \(W\) & mm & 180.00 & 150.00 & 250.00 \\ \(B\) & mm & 3.00 & 2.00 & 3.00 \\ \(H\) & mm & 25.00 & 50.00 & 15.00 \\ \(L\) & mm & 340.00 & 200.00 & 100.00 \\ \(b\) & mm & 14.70 & 16.50 & 16.00 \\ \(n\) & - & 11.00 & 9.00 & 14.00 \\ \(k_{f}\) & W/m K & 130.00 & 200.00 & 200.00 \\ \(T_{\infty}\) & K & 293.00 & 301.30 & 293.00 \\ \(T_{\beta b}\) & K & 314.5 & 339.0 & 360.0 & 314.3 & 333.8 \\ \(h_{\rm BP}\) & W/m\({}^{2}\) K & 4.37 & 5.34 & 5.86 & 4.01 \\ \(FSF_{A,b,Ai}\) & - & 3.66 & 3.64 & 3.62 & 7.01 \\ \(Q_{CFD}\) & W & 18.48 & 50.20 & 82.26 & 10.99 & 35.89 & 63.54 \\ \(\bar{Q}_{FEA+FSF}\) & W & 21.03 & 54.74 & 87.01 & 10.93 & 34.50 & 59.43 \\ \(D_{EN.}\) & \% & 13.77 & 9.04 & 5.77 & \(-0.57\) & \(-3.88\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Dimensions, boundary conditions, output and relative deviation of the finned arrays. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**FSF**} & \multirow{2}{*}{**Overall**} & \multicolumn{3}{c}{\(T_{fb}\) (K)} & \multicolumn{3}{c}{\(H\) (mm)} & \multicolumn{3}{c}{\(L\) (mm)} & \multicolumn{3}{c}{\(b\) (mm)} \\ \cline{3-14}  & & **303** & **323** & **373** & **423** & **473** & **13** & **25** & **50** & **100** & **125** & **250** & **500** & **8.8** & **14.7** & **32.4** & **56.0** \\ \hline \(FSF_{A}\) & 1.40 & 2.06 & 1.43 & 1.07 & 0.91 & 0.80 & 0.44 & 0.70 & 1.22 & 2.13 & 0.77 & 1.20 & 1.78 & 2.19 & 0.82 & 0.44 & 0.33 \\ \(FSF_{A,\gamma,EL}\) & 1.18 & 1.75 & 0.91 & 0.56 & 0.63 & 0.77 & 0.42 & 0.66 & 1.13 & 1.86 & 0.81 & 1.07 & 1.46 & 1.95 & 0.81 & 0.42 & 0.35 \\ \(FSF_{A,\gamma,Ai}\) & 1.17 & 1.75 & 0.91 & 0.55 & 0.60 & 0.74 & 0.42 & 0.66 & 1.13 & 1.85 & 0.80 & 1.06 & 1.45 & 1.93 & 0.78 & 0.40 & 0.35 \\ \(FSF_{A,\gamma,Bu}\) & 1.19 & 1.77 & 0.91 & 0.56 & 0.63 & 0.77 & 0.42 & 0.66 & 1.13 & 1.87 & 0.81 & 1.08 & 1.47 & 1.97 & 0.80 & 0.41 & 0.35 \\ \(FSF_{A,\gamma,R01}\) & 1.16 & 1.77 & 0.93 & 0.53 & 0.56 & 0.69 & 0.42 & 0.66 & 1.13 & 1.84 & 0.78 & 1.05 & 1.45 & 1.93 & 0.74 & 0.38 & 0.33 \\ \(FSF_{A,\gamma,R01}\)-32 & 1 this case study and the results presented in Tables 4 and 5, the following conclusions can be drawn:  
1. _FSFs_ were determined based solely on geometric ratios, and the heat conduction showed the largest deviations both overall and concerning low-level parameters. Thus, these calculation paths were unsuitable. 2. The same applies to the calculation paths, which consider all three components. Looking at all, the results _FSF\({}_{A,h,B,EL}\)_ were the exception, as this calculation path was among the five most accurate ones. 3. The paths considering the geometric and heat transfer components gave the best results. Four of the five most accurate calculation paths were among these concerning the overall mean deviation and the overall standard deviation. The _FSF\({}_{A,h,R01}\)_ path had the best results for low fin base temperatures. There was no single calculation path for higher temperatures and different fin heights that turned out to be particularly accurate. The performance of the _FSF\({}_{A,h,El}\)_, _FSF\({}_{A,h,Ai}\)_, _FSF\({}_{A,h,Ra}\)_ and _FSF\({}_{A,h,R01}\)_ paths were very similar for these parameters. For longer fin lengths and almost all examined fin spacing, the _FSF\({}_{A,h,Ai}\)_ path was the best one. 4.