# Heat Transfer Performance of Fruit Juice in a Heat Exchanger Tube Using Numerical Simulations
## 3 Results and Discussion
### Simulation Results with a Non-Newtonian Fruit juice in Laminar Regime  
With regard to the non-Newtonian simulations, the results obtained for the \(C_{f}\) and Nu were compared to the theoretical values obtained for a smooth tube in laminar regime. Figure 10 shows the local Nu corresponding to several plane locations along the computational domain. According to these values, the Nu presented similar values along the inlet and outlet section, increasing the values in the planes located in the curved section, which showed a similar tendency to the results obtained under water. It should be noted that the Nu values obtained did not correspond to a fully developed thermal flow in laminar regime. The numerically obtained Nu was compared to the theoretical correlation defined for a fluid with thermally profile in development [41] (Equation (26)). Considering a length of the straight tube (\(L_{\text{s}}\)) of 0.75 m, the Nu obtained was approximately 37.5, slightly higher (31.4) than the average Nu numerically obtained in the current study. These discrepancies are related to the fact that the fruit juice properties are not constant along the computational domain, and this equation might not be useful for non-Newtonian fluids. \[Nu_{sl}=4.36+\frac{\left[0.1156+\frac{0.08569}{Pr^{0.4}}\right]Gz}{\left[1+0. 1158\,Gz^{0.6}\right]} \tag{26}\]  
where the Graetz number (Gz) is given by:  
\[Gz=\frac{D\,Re_{g}\,Pr}{L_{\text{s}}} \tag{27}\]  
With regard to the \(C_{f}\), as well as the \(C_{f}\) obtained at the straight inlet section with the empirical Equation (18), the proposed correlation by [PERSON] [38] for \(C_{f_{c}}\) in a curved pipe in laminar flow (Equation (28)) has been used. \[C_{f_{c}}=\frac{16}{Re_{g}}\frac{21.5 De}{\left(1.56+\log De\right)^{5.73}} \tag{28}\]  
where the Dean number (De) is given by:  
\[De=Re_{g}\left(\frac{a}{R}\right)^{1/2} \tag{29}\]  
According to these results, good agreement was found between the numerically obtained \(C_{f}\) at the straight inlet section and the empirical Equation (18). The numerical \(C_{f}\) obtained (0.0356) was approximately 5% higher than that theoretically computed (0.0336). Regarding the \(C_{f_{c}}\), this parameter showed a value of 0.064 using the empirical Equation (28), approximately 3% higher than the \(C_{f_{c}}\) numerically obtained (0.062). Regarding the fruit juice simulation, the fluid velocity along the computational domain reached maximum values of 0.75 m/s (Figures 11 and 12). The velocity behaviour is similar at the straight inlet and outlet section, with notable differences between the fluid velocity at the wall and the centre of the pipe. Slight variation in the fluid velocity can be seen in the planes located in the elbow. In this case, the fluid in the centre of the pipe tends to move to the outer side due to centrifugal forces, while the slow parts close to the wall are forced inward, to a zone with low pressure (Figure 13), making the flow move inward along the wall. Due to the curvature effect, fruit juice flows faster on the outer side (O) than on the inner side (I) curvature. In this case, this difference between inner and outer velocity is due to centrifugal force moving the fluid. Similarly to the turbulent regime, the values for the pressure drop along the computational domain were calculated in three sections of the curved tube. The total pressure drop considering the entire computational domain was found to be 64% lower than the case of water under turbulent regime. The straight inlet and outlet section represented 42% and 41%, and the elbow section around 17%. With regard to the temperature distribution (Figures 14 and 15), significant differences between the temperature at the wall and the centre of the pipe can be observed in comparison to the simulations carried out with water. At the straight inlet and outlet section (Figure 14), the temperature at the centre of the pipe is more homogenous in comparison to the temperature distribution along the elbow, as can be also seen in Planes 9-11 (Figure 15).