## 3 Results and Discussion
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Triangular Fin Case**} & \multicolumn{3}{c}{**Rectangular Fin Case**} \\ \hline \(\boldsymbol{l}^{*}\) & \(\boldsymbol{N}\) & \multicolumn{2}{c}{**Present Results**} & \multicolumn{3}{c}{**Literature [15]**} & \multicolumn{3}{c}{**Present Results**} & \multicolumn{3}{c}{**Literature [11]**} \\  & & & & **Results** & & & **Results** & & **Results** \\ \hline  & & _fRe_ & \(\boldsymbol{Nu}\) & _fRe_ & \(\boldsymbol{Nu}\) & _fRe_ & \(\boldsymbol{Nu}\) & _fRe_ & \(\boldsymbol{Nu}\) \\ \hline  & 12 & 20.071 & 3.9125 & 20.0691 & 3.9118 & 19.153 & 3.4864 & 19.125 & 3.4833 \\ \cline{2-10}
0.2 & 24 & 16.563 & 2.6075 & 16.5504 & 2.606 & 14.319 & 2.0474 & 14.302 & 2.0466 \\ \hline  & 12 & 19.608 & 3.3675 & 19.5545 & 3.3543 & 19.391 & 3.1677 & 19.358 & 3.1644 \\ \cline{2-10}
0.4 & 24 & 15.067 & 1.7428 & 15.0379 & 1.7399 & 13.032 & 1.358 & 12.998 & 1.3572 \\ \hline  & 12 & 19.937 & 3.8454 & 19.9076 & 3.8319 & 20.083 & 3.9379 & 20.027 & 3.9256 \\ \cline{2-10}
0.6 & 24 & 17.251 & 1.9279 & 17.1831 & 1.9172 & 15.874 & 1.4214 & 15.815 & 1.4213 \\ \hline  & 12 & 18.282 & 4.3454 & 18.251 & 4.3235 & 17.197 & 4.2436 & 17.145 & 4.2311 \\ \cline{2-10}
0.8 & 24 & 19.59 & 3.6711 & 19.4964 & 3.6305 & 21.024 & 3.7524 & 20.818 & 3.7216 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of the triangular- and rectangular-finned annulus results for \(\hat{R}=0.5\) and \(\beta=3^{\circ}\). (* in the superscript represents dimensionless quantity). Figure 3: Dimensionless velocity contours for diamond-finned annulus with number of fins varying from 4 to 32, having 20% (**a–e**) and 40% (**f–j**) height of the annulus and R = 0.5 and \(\beta=3^{\circ}\) [Colored Figure]. For \(\hat{R}=0.5\), \(\beta=3^{\circ}\), \(N=4\),12,20,28,32, and \(0.2\leq l^{*}\leq 1.0\), the isotherms are shown in Figures 5 and 6. As the definition of temperature \(l^{*}\) shows the difference of fluid and temperature of the wall, the lower values of isotherms in Figures 5 and 6 show high temperature, and the higher values of isotherms indicate low temperature. Figure 5 shows isotherms for \(\hat{R}=0.5\), \(\beta=3^{\circ}\), \(N=4\),12,20,28,32, and \(l^{*}=0.2\) and \(0.4\). In Figure 5a, high temperature gradients are observed at the tip, the whole pavilion surface of diamond fins, and the surface of the inner pipe away from the base of diamond fins, showing that there is high heat convection. The negligible heat convection is observed due to very hot fluid in the corner of the crown surface of the fin and the surface of the inner pipe. This supports the stagnant fluid flow in this region. The low temperature isotherms are noted on the outer wall, which are in the form of circular arcs between two consecutive diamond fins. It is reflecting that diamond fins with 20% of annulus width have no significant impact on the temperature distribution near the outer wall. With an increase in \(N\), the high temperature gradients gradually shift from the inner pipe surface to the tip and pavilion, as depicted in Figure 5b-d. In other words, the extent of the stagnant flow area has increased due to the reduction of the gap between two consecutive diamond fins. Moreover, the hotter region near the outer wall increases with increments in \(N\). Figure 4: _Cont._  
Figure 4: Dimensionless velocity contours for diamond-finned annulus with number of fins varying from 4 to 32, having 60% (**a**–**e**), 80% (**f**–**j**), and 100% (**k**-**o**) height of the annulus and \(\bar{R}=0.5\) and \(\beta=3^{\circ}\). Figure 5: Dimensionless temperature contours for diamond-finned annulus with number of fins varying from 4 to 32, having 20% (**a–e**) and 40% (**f–j**) height of the annulus and \(\hat{R}=0.5\) and \(\beta=3^{\circ}\). The effects of fin-height are depicted in Figures 5 and 6. The height of the diamond fin is increased from 40% to 100% of the annulus width.