# Heat and Flow Study of the Internally Finned Tubes with Different Fin Geometries
## 2 Material and Methods
### The Investigated Cases  
Figure 1 depicts the examined tube which has a length of L = 2 m, an internal diameter D\({}_{\text{in}}\) = 20 mm, and an outer diameter D\({}_{\text{out}}\) = 22 mm. This tube is made of copper, and it is heated with a total heat input of 6281 W, which is given uniformly. Practically, this heat input corresponds to a specific heat flux of q = 50,000 W/m\({}^{2}\) in the internal tube surface of the reference case. The working medium is selected to be water at a pressure of 4 bar in order to operate always on the liquid phase during the simulation studies. The inlet temperature was examined from 290 K up to 360 K with a step of 10 K, while the inlet mass flow rate was from 0.2 kg/s up to 0.35 kg/s with a step of 0.05 kg/s. The examined conditions correspond to turbulent flow, and so the conclusions of this work are valid only for the turbulent flow cases.  
Figure 2 shows the different examined cross-sections of the examined tube. Figure 2a shows the reference case, Figure 2b the internally fined tube with circular fines, Figure 2c the case with rectangular fins, and Figure 2d the case with triangular fins. The height of every fin was selected at 2 mm, which is a reasonable value according to the selected diameter. Totally, 8 fins were selected for every case in order to use a relatively high number of fins in order to take a suitable percentage of thermal enhancement.  
Figure 1: The examined tube for the performed simulations.  
The mean fluid's velocity (u) can be written as below:  
\[\mathrm{u}=\frac{\dot{\mathrm{m}}}{\rho\cdot\left(\frac{\pi\cdot\mathrm{D}_{ \mathrm{in}}^{2}}{4}\right)} \tag{7}\]  
The pumping work (W\({}_{\mathrm{P}}\)) is defined according to the next expression:  
\[W_{\mathrm{P}}=\frac{\dot{\mathrm{m}}\cdot\Delta\mathrm{P}}{\rho} \tag{8}\]  
The Bejan criterion (Be) is a proper criterion that evaluates the entropy generation type. More specifically, it evaluates the relationship between entropy generation due to the fluid's temperature increase (\(\Delta\mathrm{S_{T}}\)) and the entropy generation due to the fluid's pressure drop (\(\Delta\mathrm{S_{P}}\)). More specifically, it is defined as below [17]:  
\[\mathrm{Be}=\frac{\Delta\mathrm{S_{T}}}{\Delta\mathrm{S_{T}}+\Delta\mathrm{S_ {P}}} \tag{9}\]  
For the empty tube, accurate equations about the (Nu\({}_{\mathrm{th}}\)) and the (f\({}_{\mathrm{th}}\)) have been suggested by [PERSON] [18]:  
\[\mathrm{Nu_{th}}=\frac{\left(\frac{f}{8}\right)\cdot\left(\mathrm{Re}-1000 \right)\cdot\mathrm{Pr_{fluid}}}{1+12.7\cdot\sqrt{\frac{\mathrm{f_{eff}}}{8} \cdot\left(\mathrm{Pr_{fluid}^{\frac{2}{3}}}-1\right)}}\cdot\left(\frac{ \mathrm{Pr_{fluid}}}{\mathrm{Pr_{wall}}}\right)^{0.11} \tag{10}\]  
\[\mathrm{f_{th}}=\left[1.82\cdot\log_{10}(\mathrm{Re})-1.64\right]\ ^{-2} \tag{11}\]  
where the water Prandtl number has to be used twice, one calculated for the mean fluid temperature (Pr\({}_{\mathrm{fluid}}\)) and one for the mean wall temperature (Pr\({}_{\mathrm{wall}}\)). The Reynolds number (Re) is defined as below:  
\[\mathrm{Re}=\frac{4\cdot\dot{\mathrm{m}}}{\pi\cdot\mathrm{D}_{\mathrm{in}} \cdot\mathrm{\mu}} \tag{12}\]