# Heat and Flow Study of the Internally Finned Tubes with Different Fin Geometries
## 2 Material and Methods
### Model Validation  
The created model was properly validated by comparing the obtained results with the theoretical expressions for the Nusselt number and the friction factor for the reference case. More specifically, the validation process uses the equations of Gnielinski [18] which have been described in Section 2.2. More specifically, the equations of Gnielinski [18] have been used in order to determine properly the pressure drop (\(\Delta\)P) and the heat transfer coefficient (h) following the described mathematical formulas in Section 2.2. These results are compared with the results of the simulation tool in order to determine the deviations and evaluate them.  
Figure 3 shows the heat transfer coefficient for different values of the Reynolds number. These points correspond to the cases with different mass flow rates (from 0.20 kg/s up to 0.35 kg/s) and to different inlet temperatures (from 290 K up to 360 K). The results are given for the theoretical case and the simulation analysis. It is obvious that the respective points are close to each other for the examined range of the Reynolds number. Figure 4 shows that the deviations are relatively low and are inside the error bar of 5%. Therefore, it is obvious that the maximum error is up to 4.95%, while the mean deviation is found at 1.2% which is a low value.  
The next step is to check the results of the pressure drop. Figure 5 shows the pressure drop for the different investigated cases (combinations of different mass flow rate values and inlet temperature levels). It is clear that the found points between the theory and the simulation are close to each other. Figure 6 exhibits the deviations between the theory and the simulation, and it is obvious that the found deviations are lower than 5%. The mean deviation of the pressure drop is found at 2.54% which is a relatively small and acceptable value.  
To conclude, the aforementioned comparison analysis makes clear that the present created model in SolidWorks Flow Simulation is able to predict with high accuracy the heat transfer coefficient and the pressure drop of the examined problem. Therefore, the results of this study can be assumed to be reliable.  
\begin{table}
\begin{tabular}{c c c c c} \hline  & **Mesh 1** & **Mesh 2** & **Mesh 3** & **Mesh 4** \\ \hline
**Total cells** & 12,200 & 72,664 & 715,641 & 1,359,609 \\
**Fluid cells** & 4864 & 36,120 & 333,921 & 743,585 \\
**Solid cells** & 7336 & 36,544 & 381,720 & 616,024 \\ \hline
**u (m/s)** & 0.5054 & 0.4889 & 0.4819 & 0.4827 \\
**T\({}_{\text{out}}\) (K)** & 309.62 & 309.95 & 309.96 & 309.98 \\
**T\({}_{\text{wall}}\) (K)** & 325.02 & 322.11 & 321.71 & 321.68 \\
**\(\Delta\)P (Pa)** & 351.4 & 321.0 & 323.3 & 323.6 \\ \hline \end{tabular}
\end{table}
Table 1: The examined meshes for the reference case.  
Figure 4: Deviations in the heat transfer coefficient between simulation studies and theoretical calculations.  
Figure 3: Heat transfer coefficient from simulation studies and theoretical calculations.  
Figure 5: The pressure drop from simulation studies and theoretical calculations.  
Figure 6: Deviations in the pressure drop between simulation studies and theoretical calculations.