# Heat Transfer Performance of Fruit Juice in a Heat Exchanger Tube Using Numerical Simulations
## 3 Results and Discussion
### Model Validation with Liquid Water  
Regarding the turbulent numerical simulation using water, with a Reynolds number (Re) = 45 \(\times\) 10\({}^{3}\) at the inlet, and considering the entire computational domain, the \(y^{+}\) distribution along the wall is shown in Figure 5. Hence, it is possible to compare these values according to the proposed grids. For the coarser grid (Grid 1), approximately 87% of data reached \(y^{+}\) values ranging from 3.5 to 4.5, only representing 3.3% those data with \(y^{+}\) values ranged from 2.5 to 3.5, slightly reducing the accuracy of the turbulence model used in the region close to the wall. It can be seen that the \(y^{+}\) obtained for Grid 2 and 3 showed the refined mesh generated using both grids. For Grid 2, approximately 99% of data showed \(y^{+}\) values ranging from 0.5 to 1.5. For Grid 3, approximately 33% of data reached \(y^{+}\) values below 0.5, and the remainder ranged from 0.5 to 1.5. To validate the proposed numerical simulations, we compared the numerical results using water in turbulent regime and those obtained with theoretical equations for a smooth tube. This comparison was restricted to the simulation results obtained in the straight inlet section of the tube. With this aim, the Fanning friction factor (\(C_{f}\)) and Nusselt number (Nu) along the straight inlet section (from Plane 2 to 8) of the computational domain were determined. Regarding \(C_{f}\), the most significant differences appear when comparing \(C_{f}\) values at Grid 1 with respect to theoretical \(C_{f}\) for a smooth tube (Table 5), where, at Grid 1, it reached a value approximately 10.5% lower than a smooth pipe. Similar \(C_{f}\) values were obtained for Grids 2 and 3, where these discrepancies were greatly reduced when comparing both grids with the theoretical values, with a numerically obtained \(C_{f}\) around 0.5% higher than the smooth pipe in both grids. With regard to the Nu, the average value obtained at the planes located along the computational domain are shown in Table 5. Although no high differences were obtained for the three grids analysed, the Nu obtained at Grids 2 and 3 showed a value approximately 10.6% and 11% higher than the theoretical Nu, respectively, with this difference being slightly higher (15.6%) when using Grid 1. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Activation Energy (kJ/mol)** & **A (Pa-s)** & \(r^{2}\) \\ \hline
8.203 & 0.001047647 & 0.983 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Arrhenius equation fitting parameters. Figure 5: \(y^{+}\) distribution along the computational domain for the turbulent numerical simulation using water (Re = 45 \(\times\) 10\({}^{3}\)) with: Grid 1 (**a**); Grid 2 (**b**); and Grid 3 (**c**). In the present study, the comparison of the numerical and theoretical Nu shows that the numerical results overestimate the theoretical data slightly but accurately predict the average Nu. These results are in good agreement with those obtained by [PERSON] et al. (2013), who carried out a numerical simulation of the heat transfer process in a simple smooth tube in turbulent regime using water. In that study, Nu numerically obtained was higher than the experimental Nu, obtaining maximum discrepancies (for \(\text{Re}=40\times 10^{3}\)) ranging from 12% to 15%, for Prandtl numbers of 2.9 and 4.3, respectively. Regarding the \(C_{f}\), [PERSON] et al. (2013) obtained maximum differences between experimental and numerical results around 5.5%. Other researchers (e.g., ([PERSON] et al., 2018)), reported that the comparison between the experimental and numerical overall heat transfer coefficients showed discrepancies within 5%, in a study related to experimental and CFD estimation of the heat transfer in helically coiled heat exchangers. Figure 6 shows velocity magnitude distribution across centerline in three locations of the computational domain, at a distance of 0.15 m from the inlet, at 90\({}^{\circ}\) of section bend (Plane 10) and at 0.3 m from the outlet (Plane 17).