## 2 Materials and Methods  
### Problem Description  
A DPHE with four diamond-shaped fins is shown in Figure 1a. These are augmented longitudinally and distributed uniformly at the inner pipe surface. These fins are non-porous, non-empty, and straight. The material of the pipes and fins is considered the same and assumed to be highly conductive. The thickness of the pipes is assumed to be negligible. The cross-section of the proposed DPHE with four diamond fins is illustrated in Figure 1b. We highlighted two surfaces relevant to a diamond-shaped fin: the crown surface and the pavilion surface. The angles \(\alpha\) and \(\beta\) represent the angle between two consecutive fins and the fin half-angle, respectively. In the present work, the crown height was set as 30% of the overall fin-height, and the crown angle was set as 6% of the angle \(\alpha\). The computational domain, chosen due to symmetry, is presented in Figure 1c. The fluid motion was considered as steady, laminar, and fully developed. The length of DPHE was sufficient to guarantee that the flow becomes fully developed. Viscous dissipation and body forces were ignored. The density and viscosity of the fluid were considered constant. The thermal boundary condition imposed at the inner pipe was constant heat input per unit axial length with the circumferentially constant temperature at any cross-section. Due to the assumption of highly conductive material of the diamond fins and inner pipe, this thermal condition may be assumed to be imposed at the interface of the inner pipe-diamond fin assembly and the fluid. At the outer pipe, an adiabatic thermal boundary condition was considered. The edges at which the boundary conditions were imposed are mentioned \(i\) to \(vi\) in Figure 1c. The governing equations are described below with the corresponding boundary conditions. The momentum equation:  
\[\frac{\partial^{2}U}{\partial r^{2}}+\frac{1}{r}\frac{\partial U}{\partial r}+ \frac{1}{r^{2}}\frac{\partial^{2}U}{\partial\varnothing^{2}}=\frac{1}{\mu} \frac{dp}{dz} \tag{1}\]  
The energy equation:  
\[\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial T}{\partial r} \right)+\frac{1}{r^{2}}\frac{\partial^{2}T}{\partial\varnothing^{2}}+\frac{ \partial^{2}T}{\partial z^{2}}=\frac{1}{k}U\frac{\partial T}{\partial z} \tag{2}\]  
Figure 1: (**a**) The double-pipe heat exchanger with extended diamond-shaped fins; (**b**) Cross-section of DPHE with diamond-shaped fins; (**c**) numerical domain due to the symmetry. The corresponding boundary conditions are given by:  
\[ \begin{array}{ll}\text{(i)}&U=0\text{ and }T=T_{w}(z)\text{ at }r=r_{i}\text{ and }0\leq\varphi\leq\alpha\\ \text{(ii)}&\frac{\partial U}{\partial\varnothing}=0\text{ and }\frac{\partial T}{ \partial\varnothing}=0\text{ at }r_{i}\leq r\leq r_{o}\text{ and }\varphi=0\\ \text{(iii)}&U=0\text{ and }\frac{\partial T}{\partial r}=0\text{ at }r=r_{o}\text{ at }0\leq \varphi\leq\alpha+\beta\\ \text{(iv)}&\frac{\partial U}{\partial\varnothing}=0\text{ and }\frac{\partial T}{ \partial\varnothing}=0\text{ at }r_{i}\leq r\leq r_{o},\ \varphi=\alpha+\beta\\ \text{(v)}&U=0\text{ and }T=T_{w}(z)\text{ at }r=\frac{r_{o}r_{i}\text{ sin}(-\theta_{1})}{r_{i}\text{ sin}(\varphi-\alpha)+r_{o}\text{ sin}(\alpha-\theta_{1}-\varphi)}\text{ and }\alpha-\theta_{1}\leq\varphi\leq\alpha\\ \text{(vi)}&U=0\text{ and }T=T_{w}(z)\text{ at }r=\frac{r_{o}r_{i}\text{ sin}(\theta_{1}+\beta)}{r_{1}\text{ sin}(\alpha+\beta-\varphi)+r_{o}\text{ sin}(\varphi-\alpha+\theta_{1})}\text{ and }\\ &\alpha-\theta_{1}\leq\varphi\leq\alpha+\beta\end{array}