## 3 Results and Discussion
With an increase in the fin-height, the scorching region near the inner pipe and crown surface of the fin first increases then decreases due to a decrease in the gap between the fin tip and the outer wall. For \(N\geq 12\), the temperature rises between any two fins near the crown surface, showing negligible convection there. For \(N\geq 12\), high temperature gradients can be observed at the surface of the inner pipe. Conclusively, the convection rate is high at the fin surface and inner pipe where high velocity gradients exist. Both height and number of diamond fins are found to be influential for heat convection. Inspection of the present heat transfer system based on the diamond-finned annulus is presented in terms of _fRe_, \(Nu\), and _jf_. _fRe_ is an important parameter that is used for the analysis of pressure drop in a heat exchanger. We computed _fRe_ and _fRe_ on the basis of the diameters \(D_{H}^{s}\) and \(D_{e}^{s}\) where \(D_{H}^{s}=4\frac{A_{e}^{s}}{w_{p}}\) and \(D_{e}^{s}=2\big{(}1-\hat{R}\big{)}\) denote the hydraulic and equivalent diameters, respectively, in a dimensionless form as follows:  
\[fRe=\frac{2D_{H}^{s\,2}}{C\overline{U}^{s}} \tag{14}\]  
\[\left(fRe\right)_{e}=\frac{2D_{e}^{s\,2}}{C\overline{U}^{s}} \tag{15}\]  
Figure 7a,b shows the curves of _fRe_ against \(N\) for the specific parameters. For an increase in \(N\), _fRe_ strictly decreases for \(0.2\leq l^{s}\leq\)0.6, and for \(l^{s}>0.6\) it shows non-monotonic behavior. With an increase in \(l^{s}\), _fRe_ shows a decreasing trend for \(N=4\), _fRe_ shows a wavy trend for \(8\leq N\leq 24\), and _fRe_ first decreases to attain a respective minimum value and then shows strictly increasing behavior or \(N>24\). The minimum values of _fRe_ for any choice of fin-height exist for 32 fins. Likewise, \(l^{s}=1.0\) gives the minimum value of _fRe_ when \(N\) varies from 4 to 12. For \(N>12\), \(l^{s}=0.4\) depicts the lowest values of _fRe_. With an increase in the thickness of the fin \(\beta\) from \(3^{\circ}\) to \(5^{\circ}\), _fRe_ increases for small values of \(N\), and _fRe_ decreases for large values of \(N\). The effect of increasing the radii ratio \(\hat{R}\) are shown in Figure 7b. _fRe_ is higher for higher values of \(\hat{R}\). The wavy trend of the curves _fRe_ is due to its dependence on the hydraulic diameter. Figure 6: _Cont._  
Figure 6: \(Cont.\)  
Figure 6: Dimensionless temperature 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 \(\hat{R}=0.5\) and \(\beta=3^{\circ}\). Figure 3:0 _Cont._  
Figure 7c-d presents the graphs of the normalized \(fRe_{e}\) against \(N\). The normalized \(fRe_{e}\) corresponds to a finless annulus. It is useful in understanding the effect of the finned annulus on pressure drop. With an increase in \(N\), the curves of the normalized \(fRe_{e}\) show a monotonically increasing trend. For \(l^{*}=0.4\), the normalized \(fRe_{e}\) shows a steady rise for \(l^{*}=0.4\) and a rapid rise for other values of \(l^{*}\). The normalized \(fRe_{e}\) increases with an increase in \(l^{*}\). The impact of \(\beta\) on the normalized \(fRe_{e}\) is significant for higher values of \(l^{*}\) as well as \(N\). Further, the normalized \(fRe_{e}\) decreases with an increase in \(\hat{R}\). It is concluded that the normalized \(fRe_{e}\) rises with an increase in \(\beta\), \(l^{*}\), and \(N\), and it decreases with an increase in \(\hat{R}\). We presented the thermal performance of the proposed DPHE in terms of the average Nusselt number (\(Nu\)) and Colburn \(j\)-factor. \(Nu\) is defined based on \(D_{H}\) as \(Nu=\frac{D_{H}\overline{h}}{\lambda_{f}}\). Here, \(\overline{h}=\frac{\hat{Q}}{p_{h}(T_{w}-T_{b})}\) and \(\lambda_{f}\) are the average heat transfer coefficient and the fluid thermal conductivity, respectively.