# Heat Transfer Correlations for Star-Shaped Fins
## 1 Introduction
### Influence of Fin Spacing  
The proper choice of fin spacing has a significant effect on heat exchange on finned surfaces. The experimental results of a larger number of researchers [1, 2, 3, 4, 5, 6] show that the heat transfer along the base surface of the fins is lower for a smaller spacing between fins because of the thicker boundary layer than for a larger spacing when the boundary layeris thinner. In general, a smaller gap between fins creates a thicker boundary layer. The stagnation zone along the fin root and the tube surface forms a laminar flow, and heat transfer is much lower in this area. [PERSON] [7] concluded that the heat transfer increases when the fin pitch (\(s_{\text{f}}\)) is increased to 6 mm (the gap between fins is 4.5 mm), and further increases in fin pitch do not increase the heat transfer for the range 48,000 \(<Re<\) 760,000. It is expected that the influence of fin pitch on the heat transfer coefficient decreases with an increasing Reynolds number. However, [PERSON] et al. [4] found that the heat transfer does not depend on the Reynolds number in the case of high fin density (0.98 fins/mm). A larger pressure drop and unfavorable conditions in terms of surface fouling occur when the distance between fins is smaller. [PERSON] [8] found that increasing the number of fins above 275.6 fins/m increases the pressure drop without increasing the heat transfer. [PERSON] et al. [9; 10] defined the influence of the dimensionless tube diameter ratio (do) and fin pitch (\(s_{\text{f}}\)) on heat transfer by the coefficient \"m\" in the correlation \(Nu=\text{C }Re^{\text{m}}\) (C is constant). The exponent \"m\" decreased from 0.73 to 0.59 when the ratio do/sf increased from 0.034 to 0.103. When the do/sf ratio was greater than 0.241, the value of \"m\" was 0.55. The influence of the do/sf ratio on the heat transfer was more significant for small Reynolds numbers. [PERSON] and [PERSON] [11] and [PERSON] [12] found in the analysis that the ratio of fin height (\(h_{\text{t}}\)) and fin pitch (\(s_{\text{f}}\)) was the most influential variable affecting both heat transfer coefficient and pressure drop. [PERSON] and [PERSON] provided heat transfer correlations that accounted for fin height, thickness and spacing, and fin root diameter. [PERSON] and [PERSON] [13] modified the correlations of [PERSON] and [PERSON] and [PERSON] [14] by adding a term that accounted for the number of rows of the tube bundle. They tested the heat exchanger with finned tubes in the staggered arrangement by changing the number of tube rows from one to six. [PERSON], [PERSON], and [PERSON] [15] studied the heat transfer coefficient on the airside in a heat exchanger with spiral fins. The pitch of the fins ranged from 5.0 mm to 12.5 mm, and the number of rows of tubes also varied. When the tubes were staggered (two rows of tubes were tested), the heat transfer coefficient increased by approximately 27.3% as the fin pitch increased from 5.0 to 12.5 mm. They also indicated that the Colburn factor for the staggered tube arrangement is on average 11% higher and for the parallel tube arrangement 7.3% higher than for the parallel tube layout for the interval 400 \(<Re<\) 850, respectively. The aim of [PERSON] and [PERSON] [16] investigation was to determine the pitch of L-shaped fins spirally attached to a tube with an outer diameter of 16.35 mm to achieve the maximum performance of the heat exchanger. In the experiment, the fin pitches (2.4 mm, 3.2 mm, and 4.2 mm) were varied in the range of Reynolds numbers from 5000 to 15,000 (the characteristic length for calculating \(Re\) is the tube outer diameter). They concluded that the optimum fin pitch is 3.2 mm. [PERSON] and [PERSON] [17] examined the influence of fin spacing in a tube bundle with four rows of tubes with circular fins. The experiment was carried out on a staggered and parallel tube arrangement using numerical analysis. The development of the boundary layer on fins and tubes depends mainly on the ratio of fin pitch and fin height (\(s_{\text{f}}/h_{\text{f}}\)).