# Heat Transfer Correlations for Star-Shaped Fins
## 3 Results and Discussion
### Determination of Correlation for Nu Number  
To determine the correlations for the \(Nu\) and \(Eu\) numbers and to produce the corresponding graphs, numerical and graphical methods and analysis and synthesis methods were used. The general form of the correlations was adopted based on the review of numerous literature sources for annular and serrated fins and the assumption that a similar form of correlations applies to novel star-shaped fins. A very similar procedure for determining correlations for \(Nu\) and \(Eu\) was performed by [PERSON] [34]. He determined new correlations for the heat exchanger with annular fins. He considered the influence of fin height, spacing, pitch, tube and fin outer diameter, transversal and longitudinal tube pitch, and number of tube rows. In this study, the influence of the fin thickness on the heat transfer, i.e., on the \(Nu\) number, was first analyzed. In this sense, variants 1 to 12 were analyzed. The calculation results are shown in Figure 4. It can be seen that the \(Nu\) number increased with increasing fin thickness, but this increase was not large. The influence of the fin pitch is shown in Figure 5. By increasing the fin pitch to 4.5 mm, the \(Nu\) number increased significantly; in the range of the fin pitch above 4.5 mm, this increase was much less pronounced. Figure 4: Influence of fin thickness on heat transfer. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**Variant** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **10** & **11** \\ \hline \multirow{2}{*}{\(Pr\)} & 0.7028 & 0.7038 & 0.7047 & 0.7031 & 0.7041 & 0.7050 & 0.7034 & 0.7044 & 0.7053 & 0.7039 & 0.7050 \\ \cline{2-11}  & 2437 & 5918 & 12,470 & 2323 & 5644 & 11,892 & 2256 & 5484 & 11,555 & 2296 & 5349 \\ \cline{2-11} \multirow{2}{*}{\(Nu\)} & 29.07 & 47.25 & 76.45 & 28.34 & 46.0 & 72.7 & 27.47 & 44.81 & 70.64 & 26.65 & 43.59 \\ \cline{2-11}  & 0.5321 & 0.4620 & 0.4226 & 0.5215 & 0.4501 & 0.4103 & 0.5150 & 0.4443 & 0.4051 & 0.5091 & 0.4380 \\ \hline
**Variant** & **12** & **13** & **14** & **15** & **16** & **17** & **18** & **19** & **20** & **21** & \\ \hline \multirow{2}{*}{\(Pr\)} & 0.7059 & 0.7030 & 0.7042 & 0.7052 & 0.704 & 0.705 & 0.705 & 0.704 & 0.705 & 0.705 & \\ \cline{2-11}  & 11,267 & 2340 & 5694 & 12,016 & 2217 & 5384 & 11,331 & 2189 & 5595 & 11,162 & \\ \cline{2-11} \multirow{2}{*}{\(Nu\)} & 69.41 & 26.20 & 42.68 & 69.54 & 27.63 & 44.48 & 70.18 & 27.40 & 44.45 & 69.80 & \\ \cline{2-11}  & 0.3987 & 0.6106 & 0.5021 & 0.4468 & 0.4685 & 0.4167 & 0.3840 & 0.4390 & 03938 & 0.3666 & \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Calculation results of heat transfer and pressure drop for particular variants of the analyzed geometry. To numerically determine the influence of the fin thickness and pitch on the \(Nu\) number, the general form of the relationship was assumed:  
\[Nu=C_{1}\cdot Re^{C_{2}}\cdot Pr^{1/3}\left(\frac{t_{\text{f}}}{s_{\text{f}}} \right)^{C_{3}}\left(\frac{s_{\text{f}}}{d_{\text{o}}}\right)^{C_{4}} \tag{5}\]  
The initial values of parameters C\({}_{1}\) to C\({}_{4}\) were supposed. The value of \(Nu\) number for the variants 1 to 21 was obtained by substituting the values of \(Re\), \(t_{\text{f}}\), \(s_{\text{f}}\), and \(d_{\text{o}}\) into Equation (5). By analyzing the influence of each parameter C\({}_{1}\), C\({}_{2}\), C\({}_{3}\), and C\({}_{4}\), on the \(Nu\) function and applying the least-squares method, the value of the parameters can be iteratively determined so that the deviation of Equation (5) from the curves shown in Figures 4 and 5 is minimal (the obtained average deviation of the \(Nu\) feature was 1.9006):  
\[C_{1}=0.38\]  
\[C_{2}=0.587\]  
\[C_{3}=0.0254\]  
\[C_{4}=0.07\]  
The next step is to analyze and present the results of the laboratory tests and numerical analysis. For this purpose, the results of the laboratory tests [28] and the corresponding results of the numerical analysis of the heat exchanger model with a fin pitch of 4.5 mm and a fin thickness of 0.5 mm were used [33].