# Heat and Flow Study of the Internally Finned Tubes with Different Fin Geometries
## 3 Results and Discussion
### Final Evaluation of the Different Cases  
The last step in the present investigation concerns the evaluation of the Nusselt number increase with the simultaneous friction factor increase. Figure 23 shows that clearly the Nusselt number ratio (Nu/Nu\({}_{0}\)) is enhanced with the fins. The rectangular fins lead to 36% enhancement, circular fins to 25% enhancement, and triangular fins to 23% enhancement, approximately, compared to the reference tube. It is useful to state that in low Reynolds numbers, the circular tube leads to lower enhancements, which is considered an interesting result. Figure 24 shows the variation of the friction factor ratio (f/f\({}_{0}\)) for the finned tubes. The rectangular and the circular fins lead to a higher friction factor increase of around 65% and 63%, respectively. The triangular fins lead to a 43% friction factor increase. Moreover, it is useful to state that in low Reynolds numbers, the circular fins lead to a lower increase in the friction factor. The last step is the proper evaluation of the aforementioned ratios of the (Nu) of the (f). The performance evaluation criterion (PEC) is used as a proper criterion for this purpose which has been defined in Equation (1). The results of Figure 25 show that the PEC is over 1 for all the cases, so there is flow enhancement in all the examined cases with all the finned tubes. This is an important result that indicates that the idea of using internal fins works for enhancing the flow totally. More specifically, the PEC is around 1.16 for the rectangular finned tube, 1.09 for the triangular finned tube, and 1.06 for the circular finned tube. Thus, the most appropriate design is the one with rectangular fins. Figure 23: Nusselt number ratio in the different examined operating conditions for the different finned tubes. Figure 24: Friction factor ratio in the different examined operating conditions for the different finned tubes. Figure 25: Performance evaluation criterion ratio in the different examined operating conditions for the different finned tubes. Moreover, the calculated results for the (Nu) and the (f) are presented in the following equations. These equations have been found with the proper approximations, and the mean error (\(\rm{Er_{m}}\)) is relatively low; thus, these equations are accurate. These equations can be used for water and Reynolds number in the range of 10,000 \(<\) Re \(<\) 70,000. The format of these equations follows typical formats of similar equations in the literature and so a very accurate approximation was achieved. **Reference tube:**  
\[\rm{Nu_{ref}}=0.02405\cdot Re^{0.8033}\cdot Pr^{0.4450}\,\ \rm{Er_{m}}=0.72\% \tag{13}\]  
\[\rm{f_{ref}}=0.2762\cdot Re^{-0.2417},\ \rm{Er_{m}}=0.27\% \tag{14}\]  
**Circular finned tube:**  
\[\rm{Nu_{circ}}=0.02446\cdot Re^{0.8194}\cdot Pr^{0.4712}\,\ \rm{Er_{m}}=1.54\% \tag{15}\]  
\[\rm{f_{circ}}=0.4772\cdot Re^{-0.2468},\ \rm{Er_{m}}=0.52\% \tag{16}\]  
**Rectangular finned tube:**  
\[\rm{Nu_{rect}}=0.02537\cdot Re^{0.8239}\cdot Pr^{0.4804}\,\ \rm{Er_{m}}=1.18\% \tag{17}\]  
\[\rm{f_{rect}}=0.4246\cdot Re^{-0.2351},\ \rm{Er_{m}}=0.13\% \tag{18}\]  
**Triangular finned tube:**  
\[\rm{Nu_{tri}}=0.02445\cdot Re^{0.8167}\cdot Pr^{0.4710}\,\ \rm{Er_{m}}=1.38\% \tag{19}\]  
Figure 24:0  
Furthermore, it is useful to define the mean error (\(\rm{Er_{m}}\)) according to Equation (21). More specifically, when the approximated value is (X) and the real calculated values are (X\({}_{0}\)), while there are (N) points of approximation, the error is defined as:  
Figure 24:1  
In the end, it is useful to discuss the present results with other studies from the literature. The present work uses 8 internal fins, and this number is correct according to the literature. [PERSON] and [PERSON] [16] found that the internal fins are the optimum choice.