## 4 Conclusions  
An innovative design of diamond-shaped augmented fins for DPHEs was proposed for thermal efficiency and energy saving. The triangular-shaped and rectangular-shaped fin designs of DPHE, available in the literature, can be recovered as special cases of the proposed design. An \(h\)-adaptive finite element method was employed for the solution of the governing equations. The results were computed for various performance measures against the emerging parameters. The significant findings of the investigation are listed below.  
* Flow analysis suggests that the height and the number of diamond fins play a significant role in the velocity profile.
* The reduction of the pressure drop in the proposed DPHE design depends on the choice of fin thickness, fin-height, radii ratio, and the number of diamond fins.
* Enhanced heat transfer is observed in a DPHE with proposed diamond-shaped fins, at a radii ratio of 0.25, for the following cases: (i) 4 fins for a fin-height of 20% to 80% of the annulus, (ii) 8-12 fins for a fin-height of 80% of the annulus, and (iii) 16-32 fins for a fin-height of 100% of the annulus.
* The normalized heat transfer coefficient is greater than unity, showing that the augmentation of the fins increases the coefficient of heat transfer as compared to finless DPHE.  
**Conflicts of Interest:** The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: [PERSON], [PERSON], [PERSON], [PERSON], and [PERSON] have a patent pending at the Intellectual Property Organization (IPO) Pakistan.  
**Patents:** [PERSON], [PERSON], [PERSON], [PERSON], and [PERSON] have a patent pending at the Intellectual Property Organization (IPO) Pakistan.  
**Nomenclature**  
\begin{tabular}{l l} \(A_{c}\) & flow cross-sectional area, m\({}^{2}\) \\ \(D_{H}\) & hydraulic diameter of finned geometry, m \\ \(f\) & Fanning friction factor, dimensionless \\ \(j\) & Colburn j-factor, dimensionless \\ \(N\) & number of fins \\ \(Nu\) & average Nusselt number, dimensionless \\ \(\frac{dp}{dz}\) & pressure gradient in finned geometry, pa/m \\ \(\mathbf{Pr}\) & Prandtl number, dimensionless \\ \(\mathbf{Pn}\) & heated perimeter, m \\ \(\mathbf{R}\) & dimensionless radial coordinate, dimensionless \\ \(\mathbf{Re}\) & Reynolds number, dimensionless \\ \(\mathbf{r}\), \(\varnothing\), \(\mathbf{z}\) & cylindrical coordinates \\ \(r_{i}\) & radius of inner pipe, m \\ \(r_{o}\) & radius of outer pipe, m \\ \(\mathbf{T}\) & temperature, \({}^{o}\)C \\ \(T_{b}\) & bulk mean fluid temperature, \({}^{o}\)C \\ \(T_{w}\) & fluid temperature at the solid-fluid interface, \({}^{o}\)C \\ \(U^{\prime}\) & dimensionless axial velocity component, dimensionless \\ \(U_{\text{max}}\) & maximum axial fluid speed at a cross-section, m/s \\ \(\theta_{1}\) & crown angle (6% of the angle \(\alpha\)),rad \\ \(\alpha\) & half-angle between successive fins, rad \\ \(\beta\) & fin half angle, rad \\ \(k\) & thermal diffusivity, m\({}^{2}\)/s \\ \(\lambda_{f}\) & thermal conductivity of the fluid, W/m K \\ \(\tau^{*}\) & dimensionless temperature \\ \(\overline{h}\) & average heat transfer coefficient, Wm\({}^{-2}\)K\({}^{-1}\) \\ \(Q\) & heat transfer per unit length, W/m \\ \end{tabular}  
**Subscripts**  
\begin{tabular}{l l} \(b\) & bulk \\ \(e\) & equivalent diameter \\ \(h\) & heated parameter \\ \(H\) & hydraulic diameter \\ \(w\) & solid wall \\ \end{tabular}  
**Superscripts**  
\begin{tabular}{l l} \(\ast\) & dimensionless quantity \\  & overbar ( \({}^{-}\) ) & average value \\ \end{tabular}