# Heat Transfer Performance of Fruit Juice in a Heat Exchanger Tube Using Numerical Simulations
## 3 Results and Discussion
### Model Validation with Liquid Water
Velocity magnitude obtained with Grid 1 differs with the ones obtained with Grids 2 and 3, which are very similar. The highest differences are shown in the region close to the wall at the three locations, where Grid 1 overestimates velocity magnitude, which is approximately 15% higher than with the other grids. It indicates that Grid 1 also overestimates the hydrodynamic entry length in the tube. Hence, the velocity contour (Figure 7) using Grid 1 shows that the flow is fully developed at a distance of approximately 0.18 m from the inlet. Considering Grids 2 and 3, the flow development is reached earlier (at approximately of 0.11 m distance from the inlet). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Variable** & **Grid 1** & **Grid 2** & **Grid 3** & **Smooth Pipe** \\ \hline Fanning factor (\(C_{f}\)) & 0.0049 & 0.0055 & 0.0056 & 0.0054 \\ \hline Nusselt number (\(Nu\)) & 318.4 & 304.8 & 306.1 & 275.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Theoretical and numerical Fanning friction factor and Nusselt number obtained at Grids 1–3 using water. Figure 6: Velocity magnitude distribution across centerline for Grids 1–3 using water: at a distance of 0.15 m from the inlet (**a**); at 90\({}^{\circ}\) of section bend (**b**); and at 0.3 m from the outlet (**c**). According to the results obtained above, the lowest differences between numerical and theoretical results were shown using Grids 2 and 3. Moreover, in both cases, low \(y^{+}\) contributed to the accuracy of the wall treatment of the turbulence model. Grid 2 was considered the grid size with accurate results in turbulent regime, representing approximately 28% fewer cells than Grid 3. Grid 2 was selected because the \(C_{f}\) and Nu numerically obtained showed low differences in comparison to the smooth pipe values. Accordingly, the selected grid was also used to simulate the case under laminar regime. Figure 7: Velocity magnitude contour distribution along the inlet section using water at: Grid 1 (a); Grid 2 (b); and Grid 3 (c). Considering Grid 2, the pressure drop along the curved tube was calculated in the three analysed sections. The total pressure drop considering the entire computational domain was approximately 5560 Pa, as a difference between total pressure at Planes 2 and 18. Analysing the three sections of the computational domain, it can be highlighted that the curved section (between Planes 8 and 12) represented approximately 19% of the total pressure drop in the pipe. The straight inlet (between Planes 2 and 8) and outlet (between Planes 12 and 18) sections represented 41% and 40%, respectively, of the total pressure drop. With regard to the temperature, Figure 8 shows the temperature contour distribution along the computational domain using the grid selected. The fluid temperature is more uniform along the inlet (Planes 2, 4 and 6) and outlet section (Planes 14, 16 and 18). For both sections, the average temperature at the centre of the pipe is lower than the region close to the wall. For the inlet section, the temperature at the wall reached maximum values of approximately 305.2 K, slightly reducing at the centre (303.3 K). In the curved section (Planes 9-11), temperature variability is high, and smaller temperature differences can be seen between the temperature at the wall and the fluid temperature at the centre, related to the effect of the elbow on the flow behaviour, which promotes secondary and mixing flow. Figure 9 shows the average inner wall temperature and bulk fluid temperature across the plane locations. The wall temperature slightly increases in the upstream and downstream of the curved pipe, while it decreases in the curved pipe section, reaching the minimum values near the 90\({}^{\circ}\) location (Planes 10 and 11). Regarding the bulk fluid temperature, it increases linearly along the computational domain. The temperature distribution help explain local Nu values across the plane locations (Figure 10), where the maximum values were reached at the curved section.