3.2. Model Validation with Liquid Water
Regarding the turbulent numerical simulation using water, with a Reynolds number (Re) =
at the inlet, and considering the entire computational domain, the
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
values ranging from 3.5 to 4.5, only representing 3.3% those data with
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
obtained for Grids 2 and 3 showed the refined mesh generated using both grids. For Grid 2, approximately 99% of data showed
values ranging from 0.5 to 1.5. For Grid 3, approximately 33% of data reached
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 () and Nusselt number (Nu) along the straight inlet section (from Plane 2 to 8) of the computational domain were determined.
Regarding
, the most significant differences appear when comparing
values at Grid 1 with respect to theoretical
for a smooth tube (
Table 5), where, at Grid 1, it reached a value approximately 10.5% lower than a smooth pipe. Similar
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
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.
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 Córcoles-Tendero et al. [
23], 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 Re =
) ranging from 12% to 15%, for Prandtl numbers of 2.9 and 4.3, respectively. Regarding the
, Córcoles-Tendero et al. [
23] obtained maximum differences between experimental and numerical results around 5.5%. Other researchers (e.g., [
31]), 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° of section bend (Plane 10) and at 0.3 m from the outlet (Plane 17). 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).
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 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 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.
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° 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.
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,
and loss coefficient (
) were compared to theoretical values and experimental data proposed by other researchers focused on the curved section. Regarding Nu, Chang [
32] carried out an investigation of heat transfer characteristics of swirling flow in a 180° circular section bend with uniform heat flux (
). 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
and
, 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° location, similar to the numerical Nu of the present study. Moreover, Chang [
32] obtained a similar wall and bulk fluid temperature distribution as shown in
Figure 9. In a previous study, Besserman and Tanrikut [
33] analysed turbulent flow (Re ranging from
to
) behaviour around a 180° 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 =
, the authors reported that the highest Nu (approximately 250) was reached near the 90° 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 Schmidt’s correlation [
36] (Equation (
21)) was used, where the ratio between Nu for a curved coil (
) and for the straight tube are computed (
). According to Kakaç et al. [
37], although this correlation was proposed for a curved coil, it can also be used in 90° bends for practical applications, applicable for Re ranging from
to
and ratio between radius of curvature (R) and tube inside radius (a) ranging from 5 to 84. In the current study, the
computed theoretically and numerically were 1.77 and 1.41, respectively.
With regard to the
and
, the numerical results have been compared to the theoretical equations proposed by other authors. According to Itō [
38], the pressure drop in a bend for Re ranging from
to
is computed as follows.
for
for
= 180°
Itō [
38] proposed the following correlation (Equation (
25)) to compute
in a curved pipe in turbulent flow, which has a high degree of approximation according to other studies [
39,
40].
According to Equation (
22), the numerical
reached a value of 0.327, obtained as a difference between total pressure at Planes 8 and 12 (approximately 1022 Pa). Considering Equations (
23) and (
24), the theoretical
(0.299) was approximately 8.5% lower than the numerical
. Comparing the
, the theoretical (Equation (
25)) and numerical values obtained were 0.0084 and 0.0091, respectively.
3.3. Simulation Results with a Non-Newtonian Fruit Juice in Laminar Regime
With regard to the non-Newtonian simulations, the results obtained for the
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 (
) 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.
where the Graetz number (Gz) is given by:
With regard to the
, as well as the
obtained at the straight inlet section with the empirical Equation (
18), the proposed correlation by Itō [
38] for
in a curved pipe in laminar flow (Equation (
28)) has been used.
where the Dean number (De) is given by:
According to these results, good agreement was found between the numerically obtained
at the straight inlet section and the empirical Equation (
18). The numerical
obtained (0.0356) was approximately 5% higher than that theoretically computed (0.0336). Regarding the
, this parameter showed a value of 0.064 using the empirical Equation (
28), approximately 3% higher than the
numerically obtained (0.062).
Regarding the fruit juice simulation, the fluid velocity along the computational domain reached maximum values of 0.75 m/s (
Figure 11 and
Figure 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 (
Figure 14 and
Figure 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). At the curvature, the maximum temperature reached a value close to 336 K, in the region located at the inner side, reaching the minimum values (303 K) at the outer side. Regarding the inner wall temperature and bulk fluid temperature,
Figure 16 shows a similar tendency to the results obtained under water, where the fluid temperature increases along the computational domain, while the wall temperature decreases in the curved pipe. Velocity changes indicated in
Figure 12 can help explain temperature variability in the elbow, with moderately high temperatures being reached at the centre of the elbow, compared to the centre of the straight inlet and outlet section.
Figure 17 and
Figure 18 show the strain rate and dynamic viscosity distribution along the straight inlet and the elbow section. The viscosity distribution for the straight outlet section was very similar to the inlet. The strain rate showed the highest values at the wall, reaching maximum values ranging from 300 to 800 s
(
Figure 17). At the centre of the pipe, the strain rate decreases, reaching values between 0 to 50 s
. This is related to the velocity variation between the fluid layers being lower at the centre of the pipe compared to the wall. The highest dynamic viscosity values were found at the centre of the pipe (between 0.05 and 0.09 Pa·s), with the lowest values at the wall (0.0076 Pa·s) (
Figure 18). This behaviour is explained by the pseudoplastic condition of the fruit juice, where viscosity increases as the shear rate decreases.