# Heat Transfer Performance of Fruit Juice in a Heat Exchanger Tube Using Numerical Simulations
## 3 Results and Discussion
### Model Validation with Liquid Water
Figure 8: Temperature contour distribution at each plane location using water at Grid 2. To complement the numerical results validation previously carried out for a straight smooth tube section, experimental and theoretical results for a curved pipe from other studies are included. Therefore, to validate the current numerical study in turbulent regime, Nu, \(C_{f}\) and loss coefficient (\(K_{1}\)) were compared to theoretical values and experimental data proposed by other researchers focused on the curved section. Regarding Nu, [PERSON] [32] carried out an investigation of heat transfer characteristics of swirling flow in a 180\({}^{\circ}\) circular section bend with uniform heat flux (\(q=5100\,\mathrm{W/m^{2}}\)). Figure 10: Local Nusselt number across the plane locations along the computational domain using fruit juice and water. Figure 9: Wall and bulk fluid temperature at each plane location using water at Grid 2. In that study, a bend mean radius of 255 mm with an inner diameter of 54.5 mm was used. Although Nu values (151 and 181 for Re of \(60\times 10^{3}\) and \(80\times 10^{3}\), respectively) were slightly lower than the ones obtained in the current study, it can be highlighted that the maximum local Nu was reached along the near 90\({}^{\circ}\) location, similar to the numerical Nu of the present study. Moreover, [PERSON] [32] obtained a similar wall and bulk fluid temperature distribution as shown in Figure 9. In a previous study, [PERSON] and [PERSON] [33] analysed turbulent flow (Re ranging from \(12.5\times 10^{3}\) to \(50\times 10^{3}\)) behaviour around a 180\({}^{\circ}\) bend comparing with heat transfer measurements. In that study, experiments using a transient heat transfer technique with liquid crystal thermography were conducted, using a curved pipe with 29.7 mm radius and 25.1 mm inner diameter. Considering Re = \(50\times 10^{3}\), the authors reported that the highest Nu (approximately 250) was reached near the 90\({}^{\circ}\) location, similar to the results of the present study. Some studies performed heat transfer analysis on the average heat transfer rate in curved and helical pipes [10; 34; 35], but most of them used different geometry shape to the proposed in the present study or used correlations only useful for a limited range of parameters. To validate the Nu numerically obtained for a curved pipe, the theoretical [PERSON]'s correlation [36] (Equation (21)) was used, where the ratio between Nu for a curved coil (\(Nu_{c}\)) and for the straight tube are computed (\(Nu_{s}\)). According to [PERSON] et al. [37], although this correlation was proposed for a curved coil, it can also be used in 90\({}^{\circ}\) bends for practical applications, applicable for Re ranging from \(20\times 10^{3}\) to \(15\times 10^{4}\) and ratio between radius of curvature (R) and tube inside radius (a) ranging from 5 to 84. In the current study, the \(Nu_{c}/Nu_{s}\) computed theoretically and numerically were 1.77 and 1.41, respectively. \[\frac{Nu_{c}}{Nu_{s}}=1.0+3.6\left[1-\left(\frac{a}{R}\right)\right]\left( \frac{a}{R}\right)^{0.8} \tag{21}\]  
With regard to the \(C_{f}\) and \(K_{1}\), the numerical results have been compared to the theoretical equations proposed by other authors. According to [PERSON] [38], the pressure drop in a bend for Re ranging from \(20\times 10^{3}\) to \(40\times 10^{4}\) is computed as follows. \[\Delta p_{b}=K_{1}\frac{\rho\,u^{2}}{2} \tag{22}\]  
for \(Re(R/a)^{-2}\,>\,91\)  
\[K_{1}=0.00241B\,\phi\,Re^{-0.17}(R/a)^{0.84} \tag{23}\]  
for \(\phi=180^{\circ}\)  
Figure 6:0  
[PERSON] [38] proposed the following correlation (Equation (25)) to compute \(C_{f_{c}}\) in a curved pipe in turbulent flow, which has a high degree of approximation according to other studies [39; 40]. Figure 6:1  
According to Equation (22), the numerical \(K_{1}\) reached a value of 0.327, obtained as a difference between total pressure at Planes 8 and 12 (approximately 1022 Pa).