## 3 Results and Discussion
\(\hat{Q}\), \(p_{h}\), \(T_{b}\), and (\(T_{w}-T_{b}\)) and are the rate of heat transfer per unit length, the heated perimeter, bulk fluid mean temperature, and driving temperature  
Figure 3:1 \(fRe\) (**a**–**b**) and normalized \(fRe_{e}\) (**c**–**d**), \(Nu\) (**e**–**f**), normalized \(Nu_{e}\) (g-h) and normalized \(jf_{e}\) (**i**–**j**) plotted against the number of fins at \(\hat{R}=0.25\), \(0.5\) and \(\beta=3^{\circ}\), \(5^{\circ}\). difference, respectively. The dimensionless forms of \(Nu\) for both characteristic diameters are given by [14]:  
Figure 3:2  
Figure 3:3  
In the above equations \(D_{H}^{*}\), \(D_{e}^{*}P_{h}^{*}\) and and \(\tau_{b}\) are dimensionless forms of hydraulic diameter, equivalent diameter, heated perimeter, and bulk fluid mean temperature (\(\tau_{b}=\frac{\int\left[U^{*}T^{*}\right]}{\int U^{*}}\)), respectively. Figure 7e-f presents the curves of \(Nu\) versus \(N\). The plots of \(Nu\) for \(\beta=3^{\circ}\)and \(5^{\circ}\), \(\hat{R}=0.25\), \(0.2\leq l^{*}\leq 1.0\) and \(4\leq N\leq 32\) are shown in Figure 7e. With an increase in \(N\), the \(Nu\) shows a decrease for \(l^{*}\geq\)0.8, and an eventual decrease for \(l^{*}\geq 0.8\). This trend may also be noted in Figure 3:4 The peak values of \(Nu\) are observed at \(l^{*}=0.2\) for \(N=4\), \(l^{*}=0.8\) for \(8\leq N\leq 12\), and \(l^{*}=1.0\) for \(16\leq N\leq 32\). The peak values of \(Nu\) are made bold in Figure 3:5  
With an increase in \(\beta\), the \(Nu\) shows an increase for \(0.2\leq l^{*}\leq 0.6\), irrespective of \(N\). However, for the smaller \(\beta\), the \(Nu\) increases for \(l^{*}\geq\)0.8, and attains its maximum at \(N=8\). The non-monotonic dependence of \(Nu\) on \(l^{*}\) and \(N\) may be credited to its dependence on \(D_{H}^{*}\). For a large number of fins, \(\beta\) should be reduced to have enough free flow area. Figure 7f shows that the peak value of \(Nu\) exists for \(4\leq N\leq 8\) at \(l^{*}=0.2\), for \(12\leq N\leq 20\) at \(l^{*}=0.8\), and for \(N>20\) at \(l^{*}=1.0\). The behavior of normalized \(Nu_{e}\) versus \(N\) is shown in Figure 7g-h. The trend of normalized \(Nu_{e}\) is increasing with the N for \(l^{*}\geq 0.8\). The difference between the values of the normalized \(Nu_{e}\) for \(\beta=3^{\circ}\) and \(5^{\circ}\) becomes prominent with an increase in \(N\). The normalized \(Nu_{e}\) decreases as with the increase in \(\hat{R}\), and the extent of radii increases. From these figures, it is concluded that the normalized \(Nu_{e}\) larger than unity shows that the augmentation of diamond fins increases the rate of coefficient of heat transfer as compared to finless DPHE. For \(l^{*}\leq 0.6\), there is no significant influence of the increasing number of diamond fins on the normalized \(Nu_{e}\). However, for \(l^{*}>0.6\), the significant change in the normalized \(Nu_{e}\) is observed for any value of \(\beta\) and \(\hat{R}\). The Colburn \(j\)-factor is generally represented in the ratio form as below:  
Figure 3:6  
where \(Pr\) is the Prandtl number selected to be 0.68 (for the air). Figure 7i-j represents the curves of the normalized \(jf_{e}\) versus \(N\). Figure 7i shows the graphs of the normalized \(jf_{e}\) for \(\beta=3^{\circ}\) and \(5^{\circ}\), \(\hat{R}=0.25\), \(0.2\leq l^{*}\leq 1.0\) and \(4\leq N\leq 32\). The behavior of graphs is somewhat similar to those of \(Nu\). Further, the normalized \(jf_{e}\) shows its peak values at \(l^{*}=1.0\) for the given ranges of the other parameters. From these figures, it is concluded that the thermal performance of diamond-fin DPHE is significantly better in terms of the corresponding rise in the friction factor as compared to the finless double pipe. In Figure 8a-b, \(Nu/\!fRe\) is plotted against \(N\) at \(\beta=3^{\circ}\) and \(5^{\circ}\) for \(\hat{R}=0.25\) and \(0.5\), respectively. \(Nu/\!fRe\) shows an enhancement of heat transfer per unit rise in the frictional loss. The maximum value of \(Nu/\!fRe\) is attained at \(N=4\) for \(0.2\leq l^{*}\leq 0.8\), and at \(N=8\) for \(l^{*}=1.0\).