## 1 Introduction
The longitudinal fins augmented internally in the tube, in the form of a wave structure, were investigated experimentally by [PERSON] et al. [12]. They studied the characteristics of pressure and heat in both regions: fully developed and entry. They considered Re in the range of 900 to 3500 and reported that the fins' wave structure raises the heat transfer. [PERSON] and [PERSON] [13] proposed the idea of using baffle to increase the rate of heat exchange in the tube. They considered three arrangements of the baffles. They concluded that baffles arranged in an inclined downstream manner exhibit a significant role in heat enhancement. [PERSON] et al. [14] numerically examined heat transfer characteristics at the entrance region of a finned DPHE. A triangular-shaped finned DPHE was studied by [PERSON] et al. [15] for laminar convection. They numerically simulated the fully developed, steady, and laminar flow in the DPHE with constant heat flux at the boundary. [PERSON] et al. [16] carried out a simulation of laminar convection in an annulus of DPHE with triangular fins of different heights. [PERSON] et al. [17] presented various optimal designs of finned DPHEs. They considered various configurations of fins in the DPHE. They used the finite element method for numerical solutions. [PERSON] et al. [18] numerically studied the conjugate heat analysis in the triangular-finned DPHE. They reported that the thermal performance of the proposed DPHE is dependent on the ratio of thermal conductivities. [PERSON] et al. [19] numerically performed the experiments to study the pressure drop and heat characteristics in DPHE using nanofluid, vortex generators, and twisted tape. They reported that heat transfer characters showed a significant increase due to variation in the vortex generation. [PERSON] et al. [20] investigated the thermal performance of DPHE using interruptions. These are generated by splitting the longitudinal fins (SLF). They showed that a significant enhancement in the rate of heat transfer due to SLF as compared to simple longitudinal fins. [PERSON] and [PERSON] [21] investigated the flow and the characteristics of heat transfer in DPHE having a swirl generator. They used 12 blades with holes in the turbulator for swirl flow. They concluded that swirl flow due to the turbulator had shown significant effectiveness in the enhancement of heat. [PERSON]\(\mathrm{\SIUnitSymbolMicro}\) et al. [22] studied the geometry, number, and dimensions of fins in the DPHE using various nanoparticles for promotion of heat transfer. [PERSON] and [PERSON] [23] proposed and investigated the heat transfer problem in the oval annulus of a double-tube heat exchanger. They reported their results in terms of aspect and twist ratios of the annuli. [PERSON] and [PERSON] [24] performed pressure drop and heat analysis in a DPHE having shot peening. They reported an up to 1.19 times enhancement in the thermal performance of the proposed design. The diamond fin shape in the present work is considered the first time in the literature for the flow regime in double-pipe heat exchangers. The objective of the present work is to study the characteristics of convective heat transfer in the annular region of a finned DPHE with an innovative diamond-shaped fin design. The diamond-shaped fins are longitudinally augmented on the outer surface of the inner pipe of the DPHE. The configuration of the diamond-finned annulus is determined by the various values of the geometrical parameters, such as the radii ratio (\(\hat{R}\)), number of fins (\(N\)), fin-height (\(l^{*}\)), and fin thickness (\(\beta\)). The effects of these parameters on various performance measures, such as the product of the Reynolds number and friction factor, Nusselt number, and \(j\)-factor, are computed. The model equations are solved using an \(h\)-adaptive finite element method. The rest of the article is structured as follows. The problem geometry, the governing partial differential equation with the corresponding boundary conditions, and finite element scheme are described in Section 2.