Simulation approach on turbulent thermal performance factor of Al2O3-Cu/water hybrid nanofluid in circular and non-circular ducts

This paper presents the numerical study of thermal performance factor of Al2O3-Cu/water hybrid nanofluid in circular and non-circular ducts (square and rectangular). Turbulent regime is studied with the Reynolds number ranges from 10000 to 100000. The heat transfer performance and flow behaviour of hybrid nanofluid are investigated, considering the nanofluid volume concentration between 0.1 and 2%. The thermal performance factor of hybrid nanofluid is evaluated in terms of performance evaluation criteria (PEC). This present numerical results are successfully validated with the data from the literature. The results indicate that the heat transfer coefficient and Nusselt number of Al2O3-Cu/water hybrid nanofluid are higher than those of Al2O3/water nanofluid and pure water. However, this heat transfer enhancement is achieved at the expense of an increased pressure drop. The heat transfer coefficient of 2% hybrid nanofluid is approximately 58.6% larger than the value of pure water at the Reynolds number of 10000. For the same concentration and Reynolds number, the pressure drop of hybrid nanofluid is 4.79 times higher than the pressure drop of water. The heat transfer performance is the best in the circular pipe compared to the non-circular ducts, but its pressure drop increment is also the largest. The hybrid nanofluid helps to improve the problem of low heat transfer characteristic in the non-circular ducts. In overall, the hybrid nanofluid flow in circular and non-circular ducts are reported to possess better thermal performance factor than that of water. The maximum attainable PEC is obtained by 2% hybrid nanofluid in the square duct at the Reynolds Number of 60000. This study can help to determine which geometry is efficient for the heat transfer application of hybrid nanofluid.


Introduction
The development of industrial activities in the past few decades has created a substantial demand of energy sources. Heat transfer technologies are constantly improving through progressive research. Extensive efforts are made for heat transfer enhancement purposes, but the performance is limited by the poor thermal conductivity of conventional heat transfer fluids, such as water, oil and ethylene glycol [1]. The advancement in material engineering has made the creation of nano-sized particles (average particle size less than 100 nm) possible, leading to the development of a new class of heat transfer fluid. The term "nanofluid" was introduced by Choi and Eastman [2] to represent this heat transfer fluid which contains nano-sized metallic or non-metallic particles. The most commonly used nanoparticle includes oxide ceramic (Al 2 O 3 , CuO), metal carbides (SiC), nitrides (AlN, SiN), metal (Al, Cu), carbon in different form (diamond, fullerene), and non-metal (graphite) [3]. Nanofluid is found to enhance the poor thermal properties of conventional heat transfer fluids. Dispersing nanoparticle into fluid can greatly augment the thermal conductivity of the base fluid, which subsequently increases the convective heat transfer rate [4,5]. This heat transfer ability is also found to improve with the increase of nanoparticle loading. It is postulated that the increment of particle loading limits the mean free path between particles within the fluid. This causes the nanoparticles to collide more aggressively, forming compactly packed thermal interfaces and chain between particles in fluid, thus improving the thermal conductivity. Moreover, with the decrement of the mean free path, the Brownian motion of particles is amplified, and hence the heat transfer rate is augmented [6,7].
Although the introduction of nanofluid is proven to increase the heat transfer rate of the base fluid by raising its thermal properties and delay the development of thermal boundary, the heat transfer augmentation is still considered low. Therefore, new studies carried out are focusing on hybrid nanofluid. Hybrid nanofluid is synthesized by dispersing two or more different nanoparticles into the base fluid. Most of the research claims that hybrid nanofluid possesses better thermal properties compared to nanofluid with single nanoparticles [8,9]. Hameed et al. [8] had compared the laminar heat transfer and flow characteristics of three different nanofluids, which one of it was Al 2 O 3 /water nanofluid, while the other two were hybrids, namely Al 2 O 3 -Cu/water and Al 2 O 3 -CNT/water nanofluids. From their experiments, it was revealed that the heat transfer enhancement of Al 2 O 3 -CNT hybrid nanofluid was the greatest, followed by Al 2 O 3 -Cu/water and Al 2 O 3 /water nanofluids. They also remarked an average increment of Nusselt number from 13.66 to 26.62% when the nanoparticle loading of Al 2 O 3 -CNT/water nanofluid was increased from 0.1 to 0.3 vol.%.
Furthermore, Takabi and Shokouhmand [9] investigated the heat transfer characteristic of Al 2 O 3 /water nanofluid and Al 2 O 3 -CuO/water hybrid nanofluid with the Reynolds number ranged from 10000 to 100000. 1% Al 2 O 3 -CuO/ water hybrid nanofluid was reported to increase the Nusselt number by 32.07% when compared to that of water while Al 2 O 3 /water mono nanofluid with the same volume concentration increased the Nusselt number by merely 13.68%. The heat transfer potential of hybrid nanofluid was higher than water and mono nanofluid due to the attribution of hybrid nano-composite, which possessed high crystalline nature and surface functionalized, thus allowing the development of effective thermal interfaces within the fluid medium [6].
Moreover, in terms of flow behaviour, the mono nanofluid and hybrid nanofluid could increase the friction factor and pressure drop of the host fluid. These friction factor and pressure drop increment of nanofluid were attributed to the increased effective viscosity of the fluid [10,11]. When the volume concentration of nanofluid was raised, the viscosity increased and caused the pressure drop to increase, which unfortunately increased the penalty of pumping power [10,12,13]. Moghadassi et al. [10] studied the laminar flow behavior of 0.1% Al 2 O 3 /water mono nanofluid and Al 2 O 3 -Cu/water hybrid nanofluid. The friction factor of 0.1% Al 2 O 3 -Cu/water hybrid nanofluid was discovered to be 6.93% and 15.53% higher than those of Al 2 O 3 /water mono nanofluid and water, respectively.
Research of hybrid nanofluid has been carried out extensively in circular duct [8][9][10][12][13][14][15][16][17][18][19]. Labib et al. [12] performed a numerical study on the effect of nanoparticle loading and Reynolds number on hybrid nanofluid consisting of Al 2 O 3 nanoparticles and carbon nanotubes (CNTs). 22.8% augmentation in heat transfer coefficient was reported for (0.05 vol% CNTs + 1.6 vol% Al 2 O 3 )-water hybrid nanofluid in comparison to 0.05 vol% CNTs-water nanofluid. Suresh et al. [13] examined experimentally the characteristic of Al 2 O 3 -Cu/water hybrid nanofluid in laminar circular flow. An average Nusselt number increment of 10.94% was observed for 0.1% Al 2 O 3 -Cu/water hybrid nanofluid when comparison was made with the pure water. This indicated that significant heat transfer enhancement could be achieved by incorporating a small amount of copper nanoparticles into the alumina matrix. On the other hand, Suresh et al. [14] also analyzed the turbulent behaviour of Al 2 O 3 -Cu/water hybrid nanofluid in a uniformly heated circular tube. The average heat transfer enhancement of 0.1% Al 2 O 3 -Cu/water hybrid nanofluid was claimed to be approximately 8% in comparison to water. Besides, the friction factor of 0.1% hybrid nanofluid was slightly higher than that of 0.1% Al 2 O 3 /water nanofluid. Gupta et al. [15] scrutinized the heat transfer and flow behaviour of Zn-Fe 2 O 4 /water hybrid nanofluid in a circular duct. From their results, 0.5 wt% of Zn-Fe 2 O 4 / water hybrid nanofluid could enhance the heat transfer coefficient by 43% at the Reynolds number of 2200. At the same Reynolds number and nanofluid concentration, the pressure drop increment was recorded to be approximately 72%. Rahman et al. [16] stated that the dominance nanoparticles of Al 2 O 3 -Cu/water hybrid nanofluid had a stronger effect on the thermal characteristic of hybrid nanofluid. Moreover, Ramadhan et al. [19] studied experimentally and numerically the convective heat transfer of TiO 2 -SiO 2 / (ethylene-glycol/water) hybrid nanofluid. At the Reynolds number of 11000 and volume concentration of 3%, a heat transfer coefficient increment of 36% was obtained in comparison to the base fluid. The friction factor of the hybrid nanofluid was also stated to be larger than the base fluid.
If compared to the work done on the circular duct, less studies were conducted on hybrid nanofluid flowing in plain ducts with non-circular cross-section [20][21][22]. Besides circular pipe, non-circular ducts such as square and rectangular ducts are worth to be scrutinized as they have gained popularity in thermal application in industries like compact heat exchangers, electronics and aerospace due to their lower pressure drop when compared to the circular duct [23,24]. Tekir et al. [20] investigated numerically the forced convection of Al 2 O 3 -Cu/water hybrid nanofluid in a square duct under the turbulent regime. They noticed an increment of 53% in heat transfer coefficient for 2% hybrid nanofluid. Concentration of hybrid nanofluid was also discovered to have insignificant effect on the friction factor. Benkhedda et al. [21] examined the laminar mixed convection of Ag-TiO 2 /water hybrid nanofluid in a horizontal annulus. They declared that the Ag-TiO 2 / water hybrid nanofluid contributed to better thermal and dynamic performance compared to pure mono nanofluid and water. Huminic and Huminic [22] conducted numerical study on the laminar heat transfer and flow behaviour of graphene oxide (GO)-cobalt oxide (Co 3 O 4 )/water hybrid nanofluid in ducts with different cross-section (circular, flat and elliptical ducts). The range of volume concentration they considered was between 0.1 and 0.2%. They reported that the type of geometry had a more significant impact on flow performance and heat transfer if compared to the volume concentration of hybrid nanofluid. The thermal performance factor of hybrid nanofluid were lower than that of water for all volume concentration and geometries studied.
According to the review above, there is limited information on the comparison of the thermal performance factor of hybrid nanofluid in plain ducts with different crosssection (circular, square and rectangular ducts). Therefore, it is the aim of this study to address this research gap. It is strongly motivated to study and compare the thermal performance factor of hybrid nanofluid in different geometries as the thermal performance factor is a way to assess the effectiveness of hybrid nanofluid in terms of its heat transfer and flow performance. The pressure drop enhancement of hybrid nanofluid was claimed to be insignificant to their augmented convective heat transfer ability [14,20]. However, some researchers also pointed out that the friction factor enhancement might exceed the thermal enhancement [7,11,15,22,25,26]. Hence, it is vital and necessary to evaluate the thermal performance factor of the usage of hybrid nanofluid. The analysis from this study can contribute in determining the type of geometry which is more efficient for the utilization of hybrid nanofluid for heat transfer application. This manuscript is arranged into different sections. Section 2 provides the methodology employed for this numerical study, including the computation domain of the geometries, the solved governing equations, the thermo-physical properties of hybrid nanofluid, the boundary conditions and numerical solution strategy applied, and lastly the equations adopted for the analysis. The grid independence study is presented in Sect. 3. Section 3 is followed by the results and discussion in Sect. 4. The main findings of the study are then concluded in Sect. 5.

Geometry and computational domain
In this study, the geometries investigated consisted of circular and non-circular (rectangular and square) ducts. Fig. 1 shows a quarter of the computational grids for the geometries. All these geometries shared the same hydraulic diameter and length. The dimension and sizing of all the ducts studied are listed in Table 1. The length of the tubes was set to be 3 m to ensure the flow was hydrodynamically and thermally fully developed.

Governing equations
The numerical study of nanofluid was carried out using the single-phase model. The single-phase model considers the mixture of base fluid and nanoparticles to be a single homogeneous fluid. The nanoparticles are assumed to be well dispersed within the base fluid. The singlephase approach was used for this present study due to its simplicity and some researchers [9,[20][21][22] claimed that the single-phase approach was able to model the hybrid nanofluid. The present study focused only on the turbulent flow regime. The range of Reynolds number studied was from 10000 to100000. The assumptions incorporated for the simulations were: 1. The flow was in three-dimensional steady state condition. 2. Hybrid nanofluid was incompressible. 3. The nanoparticles and the base fluid were in thermal equilibrium and they had the same velocity. 4. Heat losses to the surrounding were negligible. 5. The pipes were smooth with zero roughness. 6. Natural convection was negligible. 7. The thermo-physical properties of hybrid nanofluid were constant and temperature independent.
Based on the assumptions made, the governing equations of mass, momentum and energy are expressed as follows [27]: where is the fluid density, and u are the velocity component, P is the static pressure, μ is the viscosity, ij is the Kronecker delta, h is the sensible enthalpy, k + k t is the effective thermal conductivity, k t is the turbulent thermal conductivity, and T is the temperature. Standard ƙ-ɛ turbulence model with enhanced wall treatment was selected to solve the turbulent kinetic energy, ƙ and dissipation rate, ɛ. The term of x j − u � i u � j in Eq. (2), and k t in Eq. (3) were also estimated with the turbulence model. Enhanced wall treatment was adopted to obtain accurate result regards the temperature and pressure difference, especially near the wall of ducts. The wall y+ value was checked to be smaller than or close to one. The transport equations of ƙ-ɛ turbulence model are shown in the following equations [27]. The Eddy viscosity, t and turbulence intensity, I of fluid were calculated using Eq. (6) and Eq. (7) [27].  In Eq. (4) and Eq. (5), t is the time, u i is the velocity component, is the viscosity, and are the turbulent Prandtl Number for turbulent kinetic energy (ƙ) and dissipation rate (ɛ), respectively, G is the turbulent kinetic energy generation due to mean velocity gradient, and Re D is the Reynolds number. The constants, C used for the transport equations [Eq. (5) and Eq. (6)] are 1.44 for C 1 , 1.92 for C 2 and 0.09 for C [27].
These governing and transport equations were solved using CFD software ANSYS Fluent 19.1 to obtain the fluid temperature and pressure drop along the ducts. These data were then used to study the heat transfer, pressure drop and thermal performance factor of hybrid nanofluid.

Thermo-physical properties
The hybrid nanofluid considered in this study was alumina oxide-copper/water (Al 2 O 3 -Cu/water) hybrid nanofluid. Al 2 O 3 -Cu/water was selected as it was widely studied and proven to increase the heat transfer rate with the increase of particle loading. The heat transfer rate of some other hybrid nanofluids would reduce at a high volume concentration due to the agglomeration of particles [20]. This hybrid nanofluid was also chosen due to non-reactivity of Al 2 O 3 and high thermal conductivity of Cu. In addition, the dispersion of nanoparticles in Al 2 O 3 -Cu/water hybrid nanofluid and the stability of the hybrid nanofluid were found to be good. The volume fraction of hybrid nanoparticles was specified within the range of 0.1-2%. The weightpercentage ratio of hybrid nanoparticle, Al 2 O 3 -Cu used for the simulations was 9:1.
The density ( ) and heat capacity ( C p ) of hybrid nanofluid were estimated using Eqs. (8) to (11) as listed [9,20]. The base fluid (water), hybrid nanofluid and singleparticle nanofluid are denoted by the subscript bf, hnf and nf, respectively. In Eq. (12), is the hybrid nanofluid volume fraction. In addition, the thermal conductivity (k) and dynamic viscosity ( ) for various hybrid nanofluid concentrion were determined using the experimental data of (7) I = 0.16Re − 1 8 D Al 2 O 3 -Cu/water hybrid nanofluid obtained by Suresh et al. [28], which was applicable for the concentration range from 0-2%.
The properties of water, aluminium oxide and copper nanoparticles, as well as hybrid nanofluid with different volume concentration are shown in Table 2 below [20].

Boundary conditions
The governing equations were solved based on the defined boundary condition. The fluid entered the domain boundary with a uniform velocity profile with the flow normal to the boundary. The axial velocity of the nanofluid was calculated based on the Reynolds number used. An inlet temperature of 300 K was imposed. The outflow boundary condition was applied at the outlet boundary. No slip boundary condition was enforced at the tube walls. A constant heat flux of 20 kW/m 2 along the tube wall was considered.

Numerical solution strategy
The Computational Fluid Dynamic (CFD) solver ANSYS FLUENT version 19.1 was utilized in this present study. The governing equations with the boundary conditions were solved iteratively using a segregated, pressurebased solver. The double precision solver was turned on to adjust the heat transfer rate in the computational domain. Besides, the Semi Implicit Method for Pressure Linked Equations (SIMPLE) algorithm was employed to resolve the problem of velocity and pressure coupling. For the spatial discretization, the least squares cell-based gradient was adopted. The standard scheme was implemented for the pressure term while the second order upwind scheme was applied for the discretization of momentum and energy terms. Moreover, the under relaxation factors for the converged solutions were fixed at their default values. The convergence criterion was based on the residual values of calculated variables of mass, velocity and energy components. The residual values were set to 10 −6 for all the variables in the calculations for this present numerical computation. The simulation was monitored to converge when the sum of normalized residuals of each equation and variable was less than the set residual values. Furthermore, the convergence was also judged by the mass flux report, where the net mass imbalance must be smaller than 1% of the inlet mass flux.
where q ′′ is the heat flux, T w is the wall temperature, T f is the film temperature, T in is the inlet temperature, T out is the outlet temperature, A is the cross-sectional area and P is the perimeter. For flow behaviour of hybrid nanofluid, the friction factor (f) and pressure drop ( ΔP ) were calculated using Eq. (18) and Eq. (19), respectively [20]. Moreover, the thermal performance factor of hybrid nanofluid was evaluated in terms of performance evaluation criteria (PEC) as in Eq. (20) [20].
For validation purposes, the Dittus-Boelter correlation [Eq. (21)] was used to compute the Nusselt number of water while Eq. (23) was applied for friction factor of water [20]. The validity ranges of the Dittus-Boelter correlation are: 0.6 ≤ Pr ≤ 160, Re ≥ 10000 and L

Grid independence study
Prior to performing the simulations, the grid independence studies were conducted to find the optimum mesh size. The mesh needs to be grid independent to ensure the accuracy of the simulations. The effect of mesh size on the Nusselt number and pressure drop was examined. The error rate decreases as the mesh number increases. The computational time will only increase after this certain mesh number. From Fig. 2, it can be noticed that the Nusselt number and pressure drop for water flow in the square duct remain nearly unchanged after a certain number of mesh nodes. Hence, it is concluded that the mesh has reached its grid independence. These are also true for circular and rectangular ducts. Therefore, mesh elements of 501832, 507828, and 502272 have been used as a base for other simulations for square, rectangular, and circular ducts, respectively. The maximum skewness of the mesh chosen for all the ducts is smaller than 0.8, which indicates a good mesh quality.

Result validation
Before the simulations for hybrid nanofluid, the numerical heat transfer and pressure drop results of water in the circular and non-circular ducts were validated against the values from Eq. (21) and Eq. (23), respectively. This was to ensure the reliability and accuracy of the present numerical model and method. The deviation between the Nusselt number and pressure drop from simulation and those of correlations for the square duct is shown in Table 3. From the table, the numerical results compare well with the values from correlations, such that the deviation is lower than 10%. A remarkable agreement between the Nusselt number and pressure drop is also observed for the circular and rectangular ducts. The deviation is all below 15%. This indicates that the simulation results are reliable such that the simulation technique and model selected as well as the geometries and mesh generation are suitable for simulations that are going to be performed.
To ensure the validity of the simulation for hybrid nanofluid, the simulated results of 1% and 2% hybrid nanofluid (Al 2 O 3 -Cu/H 2 O) in the square duct were compared with numerical results obtained by Tekir et al. [20]. The square duct considered in this present study has the same dimension as the one used by Tekir et al. [20]. Besides, the amount of heat flux applied for constant heat flux condition was also made the same as that of Tekir et al. [20] for comparison purposes. The comparison results are tabulated in Table 4. The highest deviation between the present numerical results with those of Tekir et al. [20] is approximately 4%. In addition, the result of 1% hybrid nanofluid from this present numerical study for circular duct was also validated against the numerical data obtained by Takabi and Shokouhmand [9] as depicted in Fig. 3. As observed from Fig. 3, the present results are in good agreement with that of Takabi and Shokouhmand [9].
Furthermore, the simulated results of this present study for 0.1% Al 2 O 3 -Cu/H 2 O hybrid nanofluid in circular duct were also validated with the experimental result of Suresh et al. [14] and the comparison is depicted in Fig. 4. From  Fig. 4a, the Nusselt number of 0.1% Al 2 O 3 -Cu/H 2 O hybrid nanofluid obtained from this present study agrees well with that of Suresh et al. [14] with the maximum deviation of 12.6%. The largest deviation between the friction factor of 0.1% Al 2 O 3 -Cu/H 2 O hybrid nanofluid in the circular duct of this present result and Suresh et al. [14] is approximately 9%. From the validation done for hybrid nanofluid in circular and square duct, it was justified that the assumptions, the applied turbulence model, thermo-physical properties of hybrid nanofluid used, numerical setting, and numerical solution strategy have acceptable reliability and accuracy.

Effect of hybrid nanofluid concentration and Reynolds number
The effect of Reynolds number on the Nusselt number of mono nanofluid and hybrid nanofluid is depicted in Fig. 5.    Suresh et al. [14] b Fig. 4 Comparison of a Nusselt number and b friction factor between present result and literature data for 0.1% Al 2 O 3 -Cu/H 2 O hybrid nanofluid in circular duct number for circular duct flow increases. Similar trends are also observed for the square and rectangular ducts. The Nusselt number enhancement of 2% hybrid nanofluid is the highest, followed by 1%, 0.5% and 0.1%. The Nusselt number of 2% hybrid nanofluid is 37.47% higher than the value of water at the Reynolds number of 10000. In addition, the Nusselt number of hybrid nanofluid also increases with the increase of the Reynolds number. This is due to the increment of fluid inlet velocity when the Reynolds number increases (Table 5). From Fig. 7, for the Reynolds number of 10000, the maximum convective heat transfer coefficient of 58.6% higher than that of water is achieved by 2% hybrid nanofluid. Similar to Nusselt number, the convective heat transfer coefficient ratio ( h hnf ∕h bf ) is also improving with the increase of hybrid nanofluid concentration. These occur because when the concentration of hybrid nanofluid increases, the fluid properties such as thermal conductivity and dynamic viscosity increase. The high thermal conductivity of nanofluid allows it to capture, gather and release heat energy easier to the surrounding fluid, hence enhancing the heat convection rate. These findings are also reported by Takabi and Shokouhmand [9].
Moreover, when the dynamic viscosity increases with the concentration increment, the velocity of fluid at a particular Reynolds number increases. This can be proven from the velocity profile of hybrid nanofluid at the outlet of the circular duct ( R e =100000) shown in Fig. 8. The x-axis is the radial distance between the tube centre and tube wall, in which 0 mm refers to the exact point at the tube center and 5 mm refers to the point at the tube wall. Higher velocity enhances the energy transfer rate within the fluid. It can also be observed that despite the outlet velocities of hybrid nanofluid are different for different concentration, they share the same trend. The fluid velocity is higher at the centre. This trend is reported for the square and rectangular ducts as well. The increased random motion of nanoparticles at the centre of the duct would induce temperature profile flattening, contributing to steeper temperature gradient between fluid and tube wall. This can reduce the thermal boundary layer thickness, delay the boundary layer development process and hence enhance the heat transfer rate [12]. Table 6 shows the pressure drop of hybrid nanofluid with respect to that of the base fluid ( ΔP hnf ∕ΔP bf ) for the circular duct. The pressure drop of hybrid nanofluid is observed to be larger than the pressure drop of pure water. The trend of pressure drop is similar for all the geometries studied. The highest pressure drop enhancement in circular duct is attained by 2% hybrid nanofluid. The pressure drop of 2% Al 2 O 3 -Cu/H 2 O hybrid nanofluid is 4.785 times larger with respect to that of pure water. The increment in pressure drop of hybrid nanofluid is mainly due to the increment of viscosity when concentration increases. The increment in fluid viscosity leads to the augmentation of wall shear stress (Fig. 9). Higher wall shear stress is an indicator for higher pressure drop. Furthermore, it could be noticed that the pressure drop enhancement of hybrid nanofluid of a particular concentration decreases with a Reynolds number increment. These occur as the increment of fluid velocity due to increasing Reynolds number reduces the thermal and velocity boundary layer thickness, thus restricting the frictional effect caused by hybrid nanofluid when flowing across the tube. Similar finding was also reported by Tekir et al. [20].
As compared to results of Takabi and Shokouhmand [9], they claimed that the friction factor enhancement of 1% Al 2 O 3 -Cu/water hybrid nanofluid and Al 2 O 3 /water nanofluid for circular flow were approximately 8.45% and 6.45%, respectively at the Reynolds number of 20000. The friction factor enhancement of Al 2 O 3 -Cu/water hybrid nanofluid of this present study is 7.92%, which is close to that of Takabi and Shokouhmand [9] under the same

Effect of geometry
The Nusselt number of 1% Al 2 O 3 -Cu/H 2 O hybrid nanofluid in different geometries is depicted in Fig. 10. As observed in Fig. 10, the circular duct possesses better heat transfer performance, followed by rectangular duct and square duct. At the Reynolds number of 100000, the Nusselt number of 1% hybrid nanofluid are 681.89, 673.3 and 659.95 for circular, rectangular and square ducts, respectively. These occur as circular duct consists of a smooth curve body which allows particle to collide and transfer heat and energy equally, while rectangular and square ducts consist of sharp edges which may restrict the flow and hence reduce the heat transfer ability of the fluid. Figure 11 shows the temperature contours of 2% Al 2 O 3 -Cu/ H 2 O hybrid nanofluid at the outlet of different geometries at the Reynolds number of 10000. It can be noticed from Fig. 11 that the temperature is more distributed in the circular duct when compared to those of the non-circular ducts. Besides, higher temperature regions (hot spots) are observed near the corners of the square and rectangular   condition. Therefore, it can be deduced that hybrid nanofluid does consist of higher frictional effect as compared to mono nanofluid. ducts. It is postulated that the flow near the sharp edges (or corners) of the non-circular ducts is suppressed by the bulk flow field due to its higher velocity. Interaction between the fluid at the corner and the main stream is poor. As a result, the heat from the heated walls cannot be distributed effectively by the flow to the bulk fluid and the heat transfer performance of the non-circular ducts is reduced to a certain extent if compared to the circular duct. Whereas, the flow in the circular duct is distributed more uniformly throughout the pipe due to the absence of sharp edges and hence, higher velocity can be attained at the near-wall region. This contributes to a more uniform heat transfer between the tube wall and the bulk fluid. For a more uniform heat transfer, the temperature difference between the tube wall and the bulk fluid is smaller. According to Eq. (13), the heat transfer coefficient is greater with smaller difference between the wall and bulk temperature, hence, heat transfer rate is also greater. Similar finding are obtained by other researchers like Kumar et al. [30], Obot and Adu-Wusu [31], Ray and Misra [32], and Daschiel et al. [33]. Figure 12 shows the pressure drop of 1% Al 2 O 3 -Cu/H 2 O hybrid nanofluid in different geometries. It can be clearly seen from Fig. 12 that the pressure drop of hybrid nanofluid in circular duct is larger than those of the non-circular ducts while the pressure drop in both the non-circular ducts are comparable. The pressure drop across the circular duct for 1% hybrid nanofluid is 696 kPa at Reynolds number of 100000 while they are 661 kPa and 653 kPa for the square and rectangular ducts, respectively, at the same Reynolds number and concentration. The large pressure drop experienced in the circular duct is a result of its high wall shear stress. The wall shear stress of 1% hybrid nanofluid at the Reynolds number of 100000 are 570 Pa, 551 Pa and 536 Pa for the circular, square and rectangular ducts, correspondingly. The high wall shear stress in the circular duct is induced by its high near-wall velocity gradient [29]. The velocity gradient of 1% Al 2 O 3 -Cu/H 2 O hybrid nanofluid in the circular, square and rectangular ducts are 3169.4/s, 2270.1/s, and 1064.8/s, respectively, at the Reynolds number 100000. Besides, the difference in pressure drop for the geometries studied becomes more significant when the Reynolds number increases. This is because the pressure drop of hybrid nanofluid increases as the Reynolds number increases, as discussed in Sect. 4.2. These present numerical results agree with the one in literature stating that the ducts with non-circular cross section has lower heat transfer and pressure drop compared to circular pipe [24,[34][35][36].

Thermal performance factor
It was proven from the previous section that the use of hybrid nanofluid in circular and non-circular tubes can significantly enhance the heat transfer performance. However, it is achieved at the expense of higher pressure drop across the tubes, especially utilizing hybrid nanofluid with high volume concentration. As the penalty from the increased pressure drop might affect the promising benefit offered by hybrid nanofluid in heat transfer application, it would be necessary to evaluate its thermal performance factor. Thermal performance factor is analyzed in terms of performance evaluation criteria (PEC) which is used to determine whether the heat transfer performance of hybrid nanofluid is able to surpass the increased pressure drop penalty imposed. The PEC is a ratio between the effect of heat transfer enhancement and friction factor increment and it can be determined from Eq. (20). The PEC ratio of water at different Reynolds number are equal to one, hence hybrid nanofluid with PEC ratio greater than 1 indicates the dominance of overall heat transfer  The flow resistance or friction factor dominates the heat transfer enhancement, hence the overall thermal hydraulic performance is lower than water when PEC is lower than 1 for hybrid nanofluid flow in the rectangular duct. The PEC ratio for all ducts are discovered to typically increase with the increment of concentration. This is attributed to higher heat transfer augmentation than the friction factor increment as the concentration of hybrid nanofluid increases. In addition, the PEC ratio obtained for all the geometries is comparable. The highest PEC of 1.39 is achieved by 2% Al 2 O 3 -Cu/H 2 O hybrid nanofluid flow in the square duct at the Reynolds number of 60000. Greater PEC value indicates that the heat transfer performance of hybrid nanofluid in the square duct is able to surpass the increased pressure drop penalty imposed. The pressure drop of a non-circular ducts like square duct is lower in comparison to circular duct, but due to that, the heat transfer performance is also low [34][35][36]. The simulation results from this present study shows that the dispersion of hybrid nanoparticles in water can help to augment the heat transfer characteristic in the non-circular ducts to the level that is comparable to circular pipe. Besides the benefit of having a lower pressure drop, this heat transfer enhancement offered by hybrid nanofluid can provide more opportunity for the use of non-circular geometries for heat transfer applications in industries.

Conclusion
In this present study, turbulent thermal performance factor of Al 2 O 3 -Cu/water hybrid nanofluid in ducts with various cross-section (circular, square and rectangular ducts) was numerically investigated. The following conclusions can be outlined based on the present numerical results: • The Al 2 O 3 -Cu/water hybrid nanofluid possesses greater heat transfer capability compared to Al 2 O 3 /water nanofluid and pure water. This heat transfer enhancement of hybrid nanofluid is postulated to be owing to the higher thermal conductivity of hybrid nanofluid when compared to those of mono nanofluid and pure water. • The maximum heat transfer coefficient and Nusselt number enhancement are attained by 2% hybrid nanofluid. At the Reynolds number of 10000, the Nusselt number and heat transfer coefficient of 2% hybrid nanofluid are approximately 37.47% and 58.6% larger than the values of pure water, respectively. • The pressure drop of Al 2 O 3 -Cu/water hybrid nanofluid is found to be larger than the pressure drop of water. The pressure drop of hybrid nanofluid in the circular duct can increase to be 4.79 times larger than the value of water. The pressure drop increment of hybrid nanofluid is caused by the viscosity increment of the hybrid nanofluid. • Circular pipe is discovered to have better heat transfer performance, but higher pressure drop than the noncircular ducts. The absence of sharp edges on circular pipe provided smoother flow path, thus promoting the heat transfer rate. • The hybrid nanofluid is efficient and possessed superior thermal performance factor than water. The maximum PEC is determined to be 1.39 which was obtained by 2% hybrid nanofluid in the square duct at the Reynolds Number of 60000.
In overall, the adoption of hybrid nanofluid aids to resolve the poor heat transfer performance in non-circular ducts which have the advantage of low pressure drop, leading to more future industrial heat transfer application. This study is limited to single-phase model and temperature-independent properties for Al 2 O 3 -Cu/water hybrid nanofluid. Therefore, it is recommended to consider the temperature-dependent hybrid nanofluid properties and the multiphase model in the future numerical study of heat transfer and flow behaviour of hybrid nanofluid for different geometries.

Conflict interest The authors declare that they have no conflict of interest
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