# Heat Transfer Correlations for Star-Shaped Fins
## 3 Results and Discussion
### Determination of Correlation for the Eu Number  
First, the analysis of the influence of the fin thickness on the pressure drop, i.e., on the _Eu_ number, was carried out. In this sense, variants 1 to 12 were analyzed. The calculation results are shown in Figure 9. From the figure, it can be seen that the value of _Eu_ number decreased with increasing _Re_ number, and the value of _Eu_ number increased with increasing fin thickness. Figure 8: Influence of fin pitch in correlation on the _Nu_ number. Figure 7: Influence of fin thickness in correlation on the _Nu_ number. The results of the numerical analysis of variants 13 to 21 show the influence of fin pitch on the \(Eu\) number, which is shown in Figure 10. As the fin pitch decreases, the \(Eu\) number increases. It is also evident from Figures 9 and 10 that the influence of fin pitch on the pressure drop, and hence on the value of \(Eu\), was greater than the influence of fin thickness. To numerically determine the influence of fin thickness and pitch on the \(Eu\) number, the general form of correlation for the \(Eu\) number was assumed to be of the form:  
\[Eu=\mathrm{C}_{5}\cdot Re^{\mathrm{C}_{6}}\cdot\left(\frac{t_{\mathrm{f}}}{s_{ \mathrm{f}}}\right)^{\mathrm{C}_{7}}\left(\frac{s_{\mathrm{f}}}{d_{\mathrm{o} }}\right)^{\mathrm{C}_{8}} \tag{7}\]  
Assuming the initial values of the parameters \(\mathrm{C}_{5}\) to \(\mathrm{C}_{8}\) and substituting the value of the \(Re\) number, \(t_{\mathrm{f}}\), \(s_{\mathrm{f}}\), and \(d_{\mathrm{o}}\) for individual variants from 1 to 21 into Equation (7), the values of \(Eu\) number were obtained. By analyzing the influence of the individual parameters \(\mathrm{C}_{5}\),  
Figure 10: Influence of fin pitch on pressure drop. Figure 9: Influence of fin thickness on pressure drop. \(\mathrm{C_{6}}\), \(\mathrm{C_{7}}\), and \(\mathrm{C_{8}}\) on the \(Eu\) function and applying the least-squares method, its value can be determined iteratively:  
\[\mathrm{C_{5}}=1.438\]  
\[\mathrm{C_{6}}=-0.145\]  
\[\mathrm{C_{7}}=0.071\]  
Figure 8:0  
so that the deviation of Equation (7) from the curves shown in Figures 9 and 10 is minimal. The next step was to analyze and present the results of the laboratory tests and the numerical analysis. Figure 11 shows a certain deviation of these two curves. This means that a new expression for the \(Eu\) function needed to be derived for the actual heat exchanger. Since we do not have experimental results for the actual heat exchanger for different fin thickness and pitch, it was assumed that the influence of fin pitch and thickness was the same as in the numerical analysis, i.e., \(\mathrm{C_{7}}=0.071\) and \(\mathrm{C_{8}}=-0.273\) were assumed. It was then necessary to determine the value of parameters \(\mathrm{C_{5}}\) and \(\mathrm{C_{6}}\) to match the \(Eu\) function as closely as possible with the experimental results. Analogous to the previous observation, the value of parameters \(\mathrm{C_{5}}=2.6\) and \(\mathrm{C_{6}}=-0.2\) were determined. The correlation for the \(Eu\) then looks like:  
Figure 8:1  
This correlation is shown in Figure 11 and can be assumed to be valid for the interval:  
Figure 8:2  
Figure 8:3  
Figure 8:4  
The deviation of the \(Eu\) number obtained based on experimental testing and the basis of numerical analysis was relatively large. It was highest in the range of \(Re<4000\) where it was up to \(29\%\). In the range of \(Re>6000\), the deviation was less than \(12\%\). In [31; 32; 33] it was shown that the correlations available in the literature also provide very different results, differing even by more than \(50\%\). It was also noted that the numerical analysis always provides slightly lower values of \(Eu\) numbers compared to the values obtained from the experimental tests. In the range of lower values of \(Re\), the measurement error in the  
Figure 8:5 \(Eu\) number obtained by laboratory testing, numerical analysis, and correlation. experimental test is often somewhat higher because of the characteristics of the measuring devices [32].