## 2 Materials and Methods
### Validation of Computed Results  
A self-developed computer program based on the \"Partial Differential Equations Toolbox of MATLAB\({}^{\otimes\prime\prime}\)\" was employed in the present investigation. The program is already validated in [15] for the cases of the annulus of double pipe without fins, the annular sector, and very thin finned annulus. Moreover, in the present problem, if both crown height and angle of the diamond fin are assumed to be zero, then the present problem corresponds to triangular-finned annulus [15], and when crown height is considered as a maximum of fin-height, then it presents rectangular-finned annulus [11]. Table 2 shows the comparison of the above-stated cases for specified parameters. The maximum error for the case of the triangular-finned annulus for \(fRe\) and \(Nu\) are 0.48% and 1.12%, respectively. For the case of the rectangular-finned annulus, \(fRe\) and \(Nu\) are 0.99% and 0.83%, respectively.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(M\) & \(l^{\star}\) & \(D_{h}\) & \(fRe\) & \(\overline{Nu}\) & \(j\)/\(f\) & **Triangles** \\ \hline \multirow{6}{*}{12} & 0.2 & 1.0371 & 16.1870 & 2.5926 & 0.1821 & 1662 \\ \cline{2-7}  & 0.4 & 0.7714 & 15.2731 & 1.8899 & 0.1407 & 1573 \\ \cline{2-7}  & 0.6 & 0.6069 & 17.1333 & 2.1223 & 0.1409 & 1461 \\ \cline{2-7}  & 0.8 & 0.4942 & 18.3535 & **3.3330** & 0.2065 & 1294 \\ \cline{2-7}  & 1.0 & 0.4116 & 14.8160 & 3.1601 & 0.2426 & 1174 \\ \hline \multirow{6}{*}{16} & 0.2 & 0.9404 & 13.8735 & 1.9727 & 0.1617 & 1323 \\ \cline{2-7}  & 0.4 & 0.6617 & 12.3026 & 1.3041 & 0.1205 & 1256 \\ \cline{2-7}  & 0.6 & 0.5022 & 14.3244 & 1.3904 & 0.1104 & 1147 \\ \cline{2-7}  & 0.8 & 0.3981 & 17.4476 & 2.6178 & 0.1706 & 951 \\ \cline{2-7}  & 1.0 & 0.3240 & 14.6330 & **2.9529** & 0.2295 & 866 \\ \hline \multirow{6}{*}{20} & 0.2 & 0.8596 & 11.8639 & 1.5470 & 0.1483 & 1026 \\ \cline{2-7}  & 0.4 & 0.5780 & 9.9060 & 0.9504 & 0.1091 & 1000 \\ \cline{2-7}  & 0.6 & 0.4264 & 11.7185 & 0.9574 & 0.0929 & 884 \\ \cline{2-7}  & 0.8 & 0.3308 & 16.0240 & 1.9603 & 0.1391 & 764 \\ \cline{2-7}  & 1.0 & 0.2641 & 14.2170 & **2.7342** & 0.2187 & 644 \\ \hline \multirow{6}{*}{24} & 0.2 & 0.7911 & 10.2341 & 1.2498 & 0.1389 & 802 \\ \cline{2-7}  & 0.4 & 0.5120 & 8.0371 & 0.7221 & 0.1022 & 780 \\ \cline{2-7}  & 0.6 & 0.3690 & 9.5238 & 0.6912 & 0.0825 & 739 \\ \cline{2-7}  & 0.8 & 0.2810 & 14.3850 & 1.4505 & 0.1147 & 611 \\ \cline{2-7}  & 1.0 & 0.2205 & 13.7050 & **2.5339** & 0.2103 & 509 \\ \hline \multirow{6}{*}{28} & 0.2 & 0.7323 & 8.8284 & 1.0255 & 0.1321 & 753 \\ \cline{2-7}  & 0.4 & 0.4588 & 6.5799 & 0.5655 & 0.0977 & 723 \\ \cline{2-7}  & 0.6 & 0.3240 & 7.7713 & 0.5203 & 0.0761 & 624 \\ \cline{2-7}  & 0.8 & 0.2427 & 12.7328 & 1.0745 & 0.0960 & 535 \\ \cline{2-7}  & 1.0 & 0.1874 & 13.0630 & **2.3334** & 0.2031 & 426 \\ \hline \multirow{6}{*}{32} & 0.2 & 0.6812 & 7.6880 & 0.8581 & 0.1269 & 707 \\ \cline{2-7}  & 0.4 & 0.4148 & 5.4547 & 0.4549 & 0.0948 & 647 \\ \cline{1-1} \cline{2-7}  & 0.6 & 0.2878 & 6.3751 & 0.4042 & 0.0721 & 572 \\ \cline{1-1} \cline{2-7}  & 0.8 & 0.2123 & 11.1190 & 0.7998 & 0.0818 & 476 \\ \cline{1-1} \cline{2-7}  & 1.0 & 0.1614 & 12.3670 & **2.1563** & 0.1983 & 358 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Cont.