# Numerical Analysis of Tube Heat Exchanger with Trimmed Star-Shaped Fins
## 3 Results and Discussion
### Data Reduction and Interpretation  
The calculation of the dimensionless numbers \(Nu\), \(Pr\) and \(Eu\) is based on the physical properties of the working fluid, which depend on the temperature and pressure. For an accurate calculation of the dimensionless characteristics, it is necessary to determine the exact temperature at which the physical properties of the working medium are measured. This temperature is the boundary layer temperature, which is defined as follows:  
\[T_{\text{bl}}=(T_{\text{w}}+T_{\text{av}})/2 \tag{4}\]  
The average temperature of the free air flow is considered at  
\[T_{\text{av}}=(T_{\text{in}}+T_{\text{out}})/2 \tag{5}\]  
For the calculation of the \(Re\) number, the density and the dynamic viscosity are applied to the average air temperature \(T_{\text{av}}\). The outer diameter of the tube is taken as the characteristic dimension (\(L\)). The characteristic air velocity (\(u_{\text{ff}}\)) is the air velocity through the narrowest flow section within the tube bundle.  
\[Re=\frac{\rho_{\text{av}}\cdot u_{\text{ff}}\cdot L}{\mu_{\text{av}}} \tag{6}\]  
The narrowest area of free flow is between the tubes in the transverse direction and can be expressed as follows:  
\[A_{\text{ff}}=s_{\text{f}}\cdot(s_{\text{I}}-d_{0})-2\cdot h_{\text{f}}\cdot t _{\text{f}} \tag{7}\]  
The dimensionless Nusselt number is defined as follows:  
\[Nu=\frac{\alpha_{0}\cdot d_{0}}{\lambda_{\text{bl}}} \tag{8}\]  
where \(\alpha_{0}\) is the actual average heat transfer coefficient on the air-side. The expression for the dimensionless \(Eu\) number is:  
\[Eu=\frac{\Delta p}{N_{\text{I}}\cdot\rho_{\text{av}}\cdot u_{\text{ff}}^{2}} \tag{9}\]  
Thermal performance is calculated based on mass air flow and total temperature change.  
\[\dot{m}=\rho_{\text{in}}\cdot A_{\text{tot}}\cdot u_{\text{in}} \tag{10}\]  
\[Q_{\text{air}}=\dot{m}\cdot c_{\text{p,av}}\cdot(T_{\text{out}}-T_{\text{iu}}) \tag{11}\]  
The total heat transfer coefficient (\(L\)) is calculated using the log average method (LMTD).  
\[\Delta T_{\text{in}}=\frac{T_{\text{in}}-T_{\text{out}}}{\ln\frac{T_{\text{ in}}-T_{\text{w}}}{T_{\text{out}}-T_{\text{w}}}} \tag{12}\]  
\(T_{\text{w}}\) is the temperature of the inner wall of the tube, i.e., the water in the tube. The total heat transfer coefficient can be calculated as follows:  
\[U=\frac{Q_{\text{air}}}{A\cdot\Delta T_{\text{ln}}} \tag{13}\]  
The heat transfer coefficient on air-side can be calculated as follows:  
\[\alpha_{e}=\frac{1}{\left(\frac{1}{d}-A_{\text{tot}}\frac{\ln\frac{\alpha_{0} }{2}}{\pi T_{\text{d}}\cdot\lambda}-\frac{A_{\text{tot}}}{A_{\text{i}}\,a_{ \text{i}}}\right)} \tag{14}\]The effective heat transfer coefficient in Equation (14) is the apparent heat transfer coefficient and includes the fin efficiency. To calculate the actual average heat transfer coefficient (\(\alpha_{0}\)), the efficiency of the fin \(\eta_{\text{f}}\) must first be calculated.  
It was assumed that the heat transfer coefficient on the tube-side is large enough (several thousand W/(m\({}^{2}\)K)) so that the last term in the denominator is approximately zero.  
\[\alpha_{0}=\frac{\alpha_{\text{e}}\cdot A_{\text{tot}}}{(A_{\text{t}}+\eta_{ \text{f}}\cdot A_{\text{f}})} \tag{15}\]  
Fin efficiency is calculated using the average fin surface temperature from the numerical analysis (see Tables 3-6).  
\[\eta_{\text{f}}=1-\frac{(T_{\text{W}}-T_{\text{f}\text{av}})\cdot\alpha_{\text {e}}}{U\cdot(T_{\text{W}}-T_{\text{av}})} \tag{16}\]  
The pressure drop in the tube bundle is calculated using the following formula:  
\[\Delta p=p_{\text{in}}-p_{\text{out}} \tag{17}\]  
where \(p_{\text{in}}\) and \(p_{\text{out}}\) are the mass-weighted average pressures at the inlet and outlet of the tube bundle, respectively, obtained from the numerical analysis.