Using nite volume method for investigating the turbulent ow eld and heat transfer of a non-Newtonian nanouid in a channel with triangular vortex generators

This study examines the turbulent ow eld and heat transfer rate (HTR) of the non-Newtonian H2O-Al2O3-carboxymethyl (CMC) in a channel with vortex generators. The nite volume method and SIMPLE algorithm were employed for solving the partial differential equations. The mean Nusselt numbers (Num) and pressure drops were studied at angles of 30-60°, vortex generator depths of 1-3 mm, Reynolds numbers (Re) of 3000-30000, and nanoparticles volume fractions (φ) of 0.5% and 1.5%. According to the numerical results, the use of triangular vortex generators signicantly incremented the Nusselt number (Nu) of the non-Newtonian nanouid (NF), while it had a lower effect on the enhancement of pressure drop (DP). It was also indicated that a change in the vortex generator depth in the range of a few millimeters had no signicant effects on the Nu and pressure drop. Moreover, a rise in the Re (i.e., more turbulent ow) signicantly incremented HTR. An increase in the Re raised pressure drop; however, the Num incremented more than the pressure drop. Also, the variations of the local Nu indicated that the local Nu signicantly incremented around vortex generators due to the formation of vortex ows. An enhancement in the volume fraction of the base uid’s nanoparticles (NPs) from 0.5% to 1.5% signicantly incremented HTR and the Nu.


Introduction
Because of the importance of HTR in many industrial phenomena such as energy consumption management [1][2][3][4][5][6][7][8], heat storage [9,10], cooling [11][12][13][14], and desalination [15,16], numerous scientists have focused on HTR enhancement. The distribution of vortex generations is widely employed to enhance HTR in heat exchangers and the cooling of turbine blades and electronic pieces. Considering the importance of vortex generators in improving HTR, they have been used to break thermal layers and increment turbulence in many cases. The use of vortex generators on surface walls has been considered as a passive HTR method. When a uid ows through a channel with vortex generators, the ow becomes turbulent due to the growth of rotational regions near the vortex generators, leading to the NF mixture and HTR improvement. Parsaei et al. [17] simulated the turbulent ow and HTR of the Al 2 O 3 -H 2 O NF in a rectangular vortex generator channel. They examined the impacts of the vortex generator angle of attack in a rectangular channel, Re, and Nu of the NPs on HTR enhancement. The results showed that the Nu incremented by 2.37, 1.96, and 2 times at the Re ranging from 15000 to 30000 and an attack angle of 60°c ompared to an attack angle of 0°, respectively. Toghraei et al. [18] numerically studied a ow and HTR in at, sine-shaped, and angled microchannels with and without a NF. They employed the amplitude and microchannel length, the sine shape of waves and vortex generators, φ, and Re on HTR as the performance evaluation criteria of the channel. The results demonstrated that the use of sin-shaped microchannels without a NF is more effective than the use of NPs in at microchannels in cases where only HTR is considered. By analyzing the effects of the wavelength and amplitude on the Nu, it was found that a decline in the sine wavelength and rib length would raise the Nu. It was also indicated that the use of zigzag vortex generators in the channel produced better performance than the use of sine waves. Toghraei et al. [19] studied the effects of employing the CuO-H2O NF and L-shaped porous vortex generators on microchannel performance evaluation criteria. In addition to the pure H2O, the effects of using the CuO-H2O NF on the microchannel were examined. Flows were simulated at four Re and two φ.
The results indicated that the NF had no signi cant effects on the enhancement of HTR, despite porous ribs. The use of porous vortex generators improved the HTR to 42% and 25% at the Re of 1200 and 100, respectively. Hamdi et al. [20] improved the HTR of NF ow with a turbulent ow regime in a channel with triangular vortex generators on the wall. They included parameters such as NP type, NP content, NP diameter, and Re to indicate their effects on the HTR coe cient and friction coe cient. The NP contents of 1-6% and the Re of 4000-32000 were included. The results demonstrated that the Nu m was 84.45% higher at a Re of 32000 than at a Re= 4000 and φ=6% for the ethylene glycol-SiO 2 NF. The results also showed that a rise in the NP concentration exerted a slight rise in the friction coe cient. Ahmad et al. [21] numerically studied the ow features and HTR of the H2O-CuO NF within direct channels with sine, triangular, and trapezoidal vortex generators. The effects of the φ and Re on the ow velocity, temperature contour, Nu m , dimensionless DP, and thermal-hydraulic performance were examined. The results demonstrated that the Nu m and thermal-hydraulic performance rose as the φ and Re increment for all the channel shapes. Also, they reported the trapezoidal channel to have the largest Nu, followed by sine, triangular, and direct channels, respectively. Raghoub and Mansouri [22] analytically investigated the forced convection ow of a non-Newtonian uid in a channel with an oval cross-section. They considered the input uid temperature as an alternating function of the temperature. They also employed a power model to describe non-Newtonian uid's behavior. The results revealed the signi cant effects of the shape factor and the power equation's powers on the reduction of the temperature variation range.
Tehrani et al. [23] evaluated the convective HTR of the non-Newtonian CuO-carboxymethyl (CMC) NF in a tube equipped with a spiral band. They showed that the NF's behavior was similar to that of pseudoplastic uids. According to their results, the use of the spiral band improved the HTR and DP; however, HTR improvement dominated DP rise. Ghorbani and Javaherdeh [24] studied improving the thermal e ciency of non-Newtonian NF ow in a ba ed porous environment. They selected CMC cellulose as the base uid, solving the problem for the φ=0.05%, 0.1%, and 0.2%. Their variables included the Nu, friction coe cient, and thermal e ciency. The results revealed that adding 0.2 wt.% of carbon nanotubes to the base uid improved the mean conductivity coe cient and heat transfer coe cient (HTC) by 12.4% and 39.4%, respectively. The results also indicated that the use of a non-Newtonian NF ow along with a porous environment could improve the thermal e ciency by up to 29%. Lamaroei et al. [25] numerically examined the thermal and dynamic behavior of the non-Newtonian H2O-Al2O3 NF. They solved the problem for the volumes fractions of 0-5% and the Re of 25-300. Also, they employed Stowald's model to model the NF's ow behavior. The variables included the HTC, Nu m , streamlines, and isothermal contours at different Re. They demonstrated that an enhancement in the Re and φ in the base uid incremented HTR. They also showed that the required pumping power of a non-Newtonian uid was signi cantly larger than that of a Newtonian uid in the same conditions. Rashed et al. [26] numerically studied the non-Newtonian CuO-CMC NF in a ned microchannel. They examined the hydrodynamic behavior and entropy generation of the non-Newtonian NF. The results indicated the optimal ratio of HTR to DP to be nearly 2.29. Also, NF's entropy generation was 2.7% lower than that of the base uid in the optimal conditions. Lee et al. [27] numerically evaluated the HTR e ciency of a non-Newtonian uid in a microchannel. They simulated the problem in a three-dimensional setting and employed the power law to simulate the non-Newtonian uid's behavior.
Sajjadifar et al. [28] investigated the ow eld and HTR of a non-Newtonian NF in micro-tubes with the velocity sliding condition and temperature jump. They evaluated the effects of the φ. They demonstrated that a rise in the Re, sliding coe cient and φ raised the dimensionless Nu. Aliakbari et al. [29] studied the effects of NP velocity and size on the HTR of non-Newtonian NF ow. They limited their work to laminar ows, a Re range of 10-100, and φ=0.5% and 1.5%. Geravandian et al. [30] investigated the effects of the vortex generator shape factor on the HTR of the non-Newtonian H2O-TiO2 NF ow in a two-dimensional rectangular microchannel. They limited their work to laminar ows, Re of 10, 50, 100, and 300, and φ=0%, 2%, and 4% for shape factors of 10, 15, 20, and 25. The results revealed that a rise in the φ in the base uid enhanced HTR, friction coe cient, e ciency evaluation index, and DP. They also indicated that the friction coe cient was almost independent of the shape factor but was a function of the φ. Shamsaei et al. [31] evaluated the HTR of a non-Newtonian NF's ow in a rectangular microchannel with triangular vortex generators. They employed the power-law model to model the non-Newtonian uid's ow. The results showed that the use of triangular ribs with an angle of 30° lead to the maximum Nu rise and minimum DP. Rahmati et al. [32] simulated the ow of a non-Newtonian NF with different φ in the base uid and sliding and non-sliding boundary conditions. They applied the power-law model for simulating the non-Newtonian uid's behavior. The results suggested that an enhancement in the φ and velocity sliding factor reduced the temperature gradient in the layers near the uid surface. Nike and Windod [33] studied the forced convective HTR enhancement of a non-Newtonian NF within a shell and tube heat exchanger with a spiral coil. They investigated FeO, Al2O3, and CuO NPs in a CMC base uid. The results demonstrated that a rise in the φ incremented the shell=side temperature and Nu m . Also, the use of CuO as for NPs yielded the highest performance. Sun et al. [34] studied the convective HTR of the non-Newtonian H 2 O-Al 2 O 3 NF ow at φ=0 to 3%. They used the power model to simulate the non-Newtonian uid's behavior. Rashed et al. [35] numerically studied the ow of the non-Newtonian CuO-CMC in a nned microchannel. They evaluated the effects of the φ in the base uid, Re, and n shape on the thermal performance from the rst and second laws of thermodynamics. Javadpour et al. [36] experimentally investigated the effects of a magnetic eld on the forced convective HTR of a non-Newtonian NF within a channel. They evaluated laminar and steady ows subjected to a xed magnetic eld and constant heat ux. They employed the CMC-H2O combination as the base uid, reporting that an enhancement in the φ in the base uid enhanced local HTR. They also found that the presence of a xed magnetic eld incremented the mean HTC by 13% for the NF with φ=1%.
The previous studies indicate that no study investigated the turbulent ow eld and HTR of the non-Newtonian H 2 O-Al 2 O 3 -CMC NF in a channel with triangular vortex generators. The present study investigates the effects of angles of 30°-60°, vortex generator depths of 1-3 mm, Re of 3000-30000, and φ=0.5 to 1.5% on the Nu m and ow eld. Fig. 1 illustrates the channel with triangular vortex generators. Also, Table 1 represents the included geometry.

Geometry And Nf Properties
The ow was considered as a two-dimensional, incompressible, non-Newtonian, turbulent, and singlephase ow. The properties of non-Newtonian NF were used at the xed temperature. The ow was assumed to enter uniformly at a xed velocity, and the NPs were treated as completely spherical. The ow eld and HTR were numerically solved for all Re and hydrodynamic development conditions. The power model was employed to simulate the non-Newtonian NF. The constants of the power model were extracted from the data provided by [25]. Also, non-Newtonian uid coe cients (i.e., n and k) were treated to remain unchanged for each nanoparticles concentration. The non-Newtonian H 2 O-Al 2 O 3 -CMC NF was included. Table 2

Governing Equations
The continuity and momentum conservation equations are written as where C, D, E, F, G, and H are de ned as [24] The Reynolds, Peclet, and Prandtl number are de ned as [24] in which v is the kinematic viscosity. Also, the local convective HTC is de ned as

Numerical Simulation
The governing equations have been solved by the nite volume method and the SIMPLE procedure. A uniform grid was applied to the solving eld. Then, control volumes were created around the nodes. The governing equations were integrated, separating the equations and obtaining a system of algebraic equations. The second-order central difference was used for distribution terms, while the hybrid method was applied to displacement terms. The central difference method was employed for Peclet numbers below 0.2, while the upstream method was used for Peclet numbers above 2. The under-relaxation factors of 0.5 and 0.7 were employed for the velocity components and temperature to obtain convergence.

Validation
The numerical results provided by Akbari et al. [29] were used to validate the proposed numerical model. Akbari et al. [29] numerically studied the ow of a non-Newtonian NF composed of H2O and 0.5 wt.% CMC at φ=0.5% and 1.5% in a rectangular channel. They evaluated laminar ows. Fig. 4 compares the numerical results provided by Akbari et al. [29] to those provided by the present work. This study's results are provided for the Re of 100 and 500 (laminar ows) and a φ=1.5%. As can be seen, the numerical results are in good agreement, suggesting the accuracy of the proposed numerical model.

Grid-independence of the results
This section investigates the grid-independence of the solution. Generally, grid-independence indicates that the reduction of meshing elements (i.e., increasing the number of elements) does not signi cantly change the numerical results, i.e., result changes are ignorable in comparison to the reduction of computational efforts. The present study employed 5000-12000 structured rectangular meshes to investigate grid-independence. Table 3 provides the friction coe cient and Nu for the lower wall in the rst sample at φ=1.5% for different meshes.
According to Table 3, the solution becomes independent from the grid for a grid number of 600000, with an error of smaller than 5%.

Flow eld
Figs. 5 and 6 illustrate the NF's velocity distributions at fraction volumes of 0.5% and 1.5% for a Re of 5000 and three different triangular vortex generator angles, respectively. As can be seen, the maximum velocity occurs on the right side of the vortex generators' tips at the attack angles of 30° and 45°. However, a rise in the attack angle shifts the maximum velocity location backwards. Also, very few abrupt changes are observed in the velocity contour at the attack angles of 30° and 45°. As can be seen, at the attack angle of 60°, some vortexes form behind the vortex generators, leading to abrupt velocity changes on the contact surface of the vortex generators and the uid on them. Considering the abrupt thinning of the channel due to the vortex generators in the ow direction (particularly at the attack angle of 60°), the channel's effective cross-section reduces, increasing the maximum uid velocity in vortex generator locations. Considering the gradual change in the slopes of the vortex generators at the attack angles of 30° and 45°, the velocity contour is symmetric, as shown in Figs. 5 and 6. For the attack angle of 60°, however, the velocity contour is observed to be asymmetric due to dramatic and abrupt slope changes. Changes in the velocity can in uence the growth of the boundary layer formed next to the vortex generator due to the attack angle change. This negative effect is higher for the attack angle of 60°. Fig. 7 demonstrates the static temperature distribution contour at φ=1.5% and Re= 5000 for three attack angles. The temperature change contour is indicated for different layers of the uid, from locations near the warm wall to the surrounding of the channel's upper wall, for vortex generator attack angles. For the attack angles of 30° and 45°, the static temperature change (thermal boundary layer growth) gradually reduces as the uid moves toward the vortex generator tip in the upper locations of the warm wall, but it begins to rise after the vortex generator. In fact, it can be said that the mentioned locations represent warm locations or locations with low HTR, and the growth of such locations should reduce to some extent, so HTR would increment. However, the growth of the thermal boundary layer is inevitable due to the effect of HTR from the warm wall to the cold wall in the middle part of the channel. Unlike the attack angles of 40° and 60°, the thermal boundary layer's growth before and after the vortex generator is considerably smaller for the attack angle of 30°, which increments HTR at this angle. However, the temperature gradient behind the vortex generator dramatically incremented for the attack angle of 60°, while the static temperature does not show a symmetric rise before and after the vortex generator. This created some warm locations with low HTCs. This is a drawback of channels with vortex generators, which is resolved by selecting suitable attack angles for vortex generators. vortex generator heights. As can be seen in Fig. 8, the maximum Nu occurs in the channel's inlet for all the cases because no thermal boundary layer has formed in the inlet yet, and the highest temperature difference exists between the channel surface and NF, which rationally enhances HTR and the Nu. As the uid ow passes through the channel, a thermal boundary layer forms and grows, reducing the temperature gradient slope. As can be seen, the reduction trend begins after the channel's inlet. It can also be understood that the local Nu takes a larger value in the vortex generator tip for all the cases. The crosssection decreases as the vortex generators' tips are approached, enhancing the ow velocity. The ow velocity continues to enhance up to the vortex generator tip, where HTR reduces due to the inverse vortex generator slope and reduced velocity. Concerning the local Nu graphs, it should be noted that the Nu of the vortex generator tip reduces by moving forward through the channel.

Local Nu variations
Because the uid exchanges heat with the wall while passing through the channel, the uid temperature decreases in the channel direction, with the temperature gradient between the channel surface and uid reducing. This reduces local Nu. Also, Fig. 8 implies that a rise in the Re raises the Nu in the channel direction, leading to improved HTR. This is more obvious around the vortex generators because the velocity rises as the uid ows on the vortex generator due to the cross-section reduction, enhancing the Re, and HTR rises as the ow becomes more turbulent, vortexes form, and the thermal boundary layer disappears.
The same case holds for the Nu in all the situations. According to the local Nu graphs, a change in the attack angle and vortex generator shape considerably in uences HTR improvement. The variations of the local Nu near the vortex generators are observed by comparing Figs. 4-12 and 4-14. Fig. 11 shows Nu m variations at the Re of 5000, 15000, and 30000 for different attack angles, vortex generation depths, and volume fractions. As can be seen, a rise in the φ raises the Nu for all the vortex generator depths. In fact, as the φ increment, the NF's thermal conductivity coe cient increments, enhancing HTR. Also, it can be found that the Nu m decreases as the attack angle increments. The presence of obstacles helps the uid to be mixed more effectively near the warm zone, producing vortexes. Such vortexes separate the warm uid near the wall and replace it with cold uid, improving HTR and enhancing the Nu. However, the incremented vortex generator attack angle brings abrupt changes onto the channel surface. Therefore, more powerful vortexes form, and HTR improves. For a xed vortex generator height, however, a rise in the triangular vortex generator's attack angle decreases the Nu m because the heat exchange area reduces for a larger attack angle. Fig. 12 represents DP variation for different Re, different triangular vortex generator heights, and two φ. At the attack angles, a rise in the φ often raises DP due to the incremented number of NPs and high power coe cient k for a non-Newtonian uid with a higher φ. Moreover, the incremented k helps the adhesive force overcome the inertia force, leading to the uid being more in uenced by the adhesive force. Therefore, the DP of the NF with φ=1.5% is larger than that of the NF with φ=0.5%. According to Fig. 12, the vortex generator pro le has no signi cant effect on DP for low Re. However, this effect becomes larger for high Re (e.g., 15000 and 30000). A larger DP occurs for the attack angle of 60° within the channel. An abrupt change in the area forms powerful vortexes and increments DP. The triangular vortex generator's DP variation is larger at φ=0.5% than at φ=1.5% because of the apparent viscosity of the non-Newtonian NFs. In fact, the apparent viscosity of a non-Newtonian uid is not only a function of k and n in the power model but is also a function of the uid's shear stress. For xed k and n values, a rise in the uid's stress enhances the apparent viscosity. As the vortex generator attack angle rises, the uid ow abruptly changes in the vortex generator zone, leading to a larger DP, particularly at an attack angle of 60°. In this case, the apparent viscosity can be calculated using the power model. According to Fig. 12, a change in the vortex generator depth slightly changes the DP due to the incremented shear stress and more turbulent ow. However, such changes are small. A change in the φ in the base uid has a more signi cant effect on the DP because a rise in the φ raises the apparent viscosity, leading to a larger DP.

Conclusion
The present study investigated the turbulent ow and HTR of a non-Newtonian uid in a channel with triangular vortex generators. The Nu m and DP were studied for the attack angles of 30°, 45°, and 60°, vortex generator depths of 1-3 mm, Re of 3000-30000, and φ=0.5% and 1.5%. The most important numerical results included 1. The use of triangular vortex generators signi cantly enhanced the Nu, while it had a smaller effect on the DP rise.
2. An enhancement in the Re (i.e., a more turbulent ow) signi cantly enhanced HTR (mean dimensionless Nu). The incremented Re also incremented the DP. However, the Nu underwent a more signi cant rise than the DP.
3. The variations of the local Nu indicated that the local Nu signi cantly incremented near the vortex generators due to the formation of vortex ows. 4. For low Re, the vortex generator pro le had a small effect on the DP; however, the effect became larger for higher Re (e.g., 15000 and 30000).
5. The DP of the triangular vortex generators was larger at φ=0.5% than at φ=1.5% due to the apparent viscosity of non-Newtonian NFs.
6. A larger DP occurred for an attack angle of 60° within the channel.
7. As the vortex generator attack angle incremented, the uid ow abruptly changed in the vortex generator zone, leading to a larger DP, particularly for an attack angle of 60°.
8. For an attack angle of 60°, some vortexes formed behind the vortex generators, leading to abrupt changes in the velocity on the vortex generator contact surface.