# Heat Transfer Correlations for Star-Shaped Fins
## 1 Introduction
### Influence of Outside Tube Diameter  
[PERSON] [8] tested three tubes of different diameters with spiral fins. [PERSON] [22] investigated the heat transfer coefficient and pressure drop in a tube bundle with eight staggered rows of tubes, using tube diameters of 38.1 mm and 50.8 mm while pitch and fin height were kept constant. As the tube diameter increases, the inefficient rear part of the tube surface is raised. [PERSON] concluded that the pressure drop and the Nusselt number increase with increasing tube diameter. [PERSON] and [PERSON] came to a slightly different conclusion about the influence of diameter on heat transfer [23]. They numerically studied a finned tube bundle with two rows of tubes. They assumed a transient flow. They concluded that increasing the tube diameter barely increases the heat transfer coefficient but increases the pressure drop. There are several correlations in the literature for determining the properties of \(Nu\) and \(Eu\). Some correlations use the above studies, whereas others are based on laboratory experiments. [PERSON] et al. [24] summarized and analyzed the results of a study of the airside performance of spiral fin heat exchangers and provided an overview of the available correlations for heat exchange. [PERSON] et al. [25] experimentally tested the heat transfer and flow resistance for air-flow spiral fin-tube heat exchangers in dry conditions and provided a correlation. [PERSON] et al. [26] laboratory-tested tubes with three different fin designs (circular plain fin, circular integrated pin fin, and serrated integrated pin fin) fabricated by additive technology using selective laser melting. The experiment was conducted in the range of Reynolds numbers between 1800 and 7800. They found that the highest heat transfer per volume heat exchanger was for the circular integrated pin. They derived heat transfer correlations from the experimental data. [PERSON] [27] experimentally tested a finned heat exchanger with oval tubes under wet conditions. He changed the dimensions and the pitch of the tubes. The largest Colburn \(j\)-factor was obtained for the two-row tube configuration. He developed correlations for heat transfer and pressure drop for the tested configurations. [PERSON] et al. [28] experimentally and numerically analyzed the heat transfer from straight rectangular fins on a horizontal surface in the range of Reynolds numbers from 2600 to 6800. Their experimental results showed that Nusselt and Reynolds number have an approximately linear relationship. Common correlations for \(Nu\) and \(Eu\) for annular fins according to [PERSON] [11], [PERSON], and [PERSON] [20] are shown in Tables 1 and 2. Correlations for serrated fins can also be found in the literature. For example, [PERSON] et al. [29] proposed a correlation to determine the Nusselt number over a serrated fin tube for Reynolds numbers between 1300 and 8000. In order to obtain correlations for heat transfer and pressure drop for star-shaped fins, taking into account all influencing factors and based on experimental results, several heat exchangers with different geometries should be made, where the thickness and height of the fins, the number of fin vortices, the pitch of the fins and tubes, the tube diameter, etc. would change. As it requires larger material costs, a reasonable solution is a correlation based on the results of test models of heat exchangers and additional information obtained by numerical analysis. For this purpose, several heat exchanger model variants were selected, varying the fin thickness and pitch. Tables 3 and 4 show geometry and boundary conditions for the heat transfer calculations. An overview of the tested variants is shown in Table 5.  
A numerical analysis was carried out for all the above variants using the ANSYS Fluent software. For this purpose, a corresponding geometric model (Figure 2) and a geometric mesh (Figure 3) were created.