**2. Numerical Method**

The numerical model based on computational fluid dynamics (CFD) is employed to analyze the heat transfer characteristics including thermal and flow attributes of single particle and hybrid nanofluids flow in uniformly heater tube. The tube with water, single particle and hybrid nanofluids as working fluids is considered under the constant heat flux condition. The constant heat flux applied on a tube is symmetrically distributed on the surface of tube to analyze the heat transfer characteristics of single particle and hybrid nanofluids. The 3D computational domain of tube is shown in Figure 1. The 3D computational domain is considered to account the effect of uniform heat flux boundary condition [28]. The tube has length of 1500 mm, inner diameter of 16 mm and outer diameter of 19 mm. The tube is made up of copper. The physical properties of copper considered for the numerical analysis as, density of 8940 kg/m<sup>3</sup> , specific heat of 376.8 J/kg·K and thermal conductivity of 401 W/m·K [29]. The constant heat flux of 7957 W/m<sup>2</sup> is applied uniformly on the tube outer surface [30]. To analyze the heat transfer characteristics of water and nanofluids, the tube under constant heat flux condition is simulated in ANSYS commercial software. The meshing with tetrahedron mesh elements and five different sizes is generated for the considered computational domain to show the mesh independency of the simulated results [31]. The Nusselt number and friction factor are simulated for five different mesh element numbers ranging from 100,000 to 700,000. The variation of Nusselt number (Nu) and friction factor (f) for various mesh element numbers is presented in Figure 2. The variation of the simulated results of Nusselt number and friction factor are within ±1% beyond the mesh element number of 425,691. After this mesh element number, the simulated results are independent of number of mesh elements. Hence, the mesh element number of 425,691 corresponding to the sizing of 2 mm is selected as the final meshing configuration for the numerical analysis on the considered computational domain. The inflation layers are employed on the fluid domain to account the effect of boundary layer.

The continuity, momentum and energy equations are solved for the considered tube with flow of various working fluids to simulate the thermal and flow characteristics [32,33].

Continuity equation

$$\nabla \cdot (\rho \mathcal{U}) = 0 \tag{1}$$

Momentum equation

$$\nabla \cdot (\rho \mathcal{U} \mathcal{U}) = -\nabla P + \nabla \tau \tag{2}$$

Stress tensor *τ* is expressed in terms of strain rate as follows

$$
\pi = \mu (\nabla \mathcal{U} + (\nabla \mathcal{U})^T - \frac{2}{3} \delta \nabla \cdot \mathcal{U}) \tag{3}
$$

Energy equation

$$\nabla \cdot (\rho \mathcal{U} h) = \nabla \cdot (k \nabla T) + \tau : \nabla \mathcal{U} \tag{4}$$

Here, *ρ* (kg/m<sup>3</sup> ) is density, *U* (m/s) is velocity, *P* (Pa) is static pressure, *µ* (Pa·s) is viscosity, *h* (J) is enthalpy, *k* (W/m·K) is thermal conductivity and ∇*T* is temperature gradient. *Symmetry* **2021**, *13*, x FOR PEER REVIEW 4 of 19 *Symmetry* **2021**, *13*, x FOR PEER REVIEW 4 of 19

**Figure 1.** 3D computational domain of tube. **Figure 1.** 3D computational domain of tube. **Figure 1.** 3D computational domain of tube.

**Figure 2.** Variation of Nusselt number and friction factor for various mesh element numbers. **Figure 2.** Variation of Nusselt number and friction factor for various mesh element numbers. **Figure 2.** Variation of Nusselt number and friction factor for various mesh element numbers.

The continuity, momentum and energy equations are solved for the considered tube with flow of various working fluids to simulate the thermal and flow characteristics The continuity, momentum and energy equations are solved for the considered tube The k-ε turbulence model could be expressed by Equations (5) and (6), respectively [34].

$$\frac{\partial(\rho \mathcal{U}\_l k)}{\partial \mathbf{x}\_l} = \frac{\partial}{\partial \mathbf{x}\_l} \left[ \left( \upsilon + \frac{\upsilon\_l}{\sigma\_k} \right) \frac{\partial k}{\partial \mathbf{x}\_l} \right] + P\_k - \rho \varepsilon \tag{5}$$

∇ ∙ () = −∇ + ∇ (2)

∇ ∙ () = −∇ + ∇ (2)

∇ ∙ (ℎ) = ∇ ∙ (∇)+ : ∇ (4)

∇ ∙ (ℎ) = ∇ ∙ (∇)+ : ∇ (4)

∇ ∙ ) (3)

∇ ∙ ) (3)

$$\frac{\partial(\rho \mathcal{U}\_l \varepsilon)}{\partial \mathbf{x}\_l} = \frac{\partial}{\partial \mathbf{x}\_l} \left[ \left( v + \frac{v\_t}{\sigma\_\varepsilon} \right) \frac{\partial \varepsilon}{\partial \mathbf{x}\_l} \right] + \rho \mathcal{C}\_1 \mathcal{S} \varepsilon - \rho \mathcal{C}\_2 \frac{\varepsilon^2}{k + \sqrt{v\varepsilon}} \tag{6}$$

 − 2 3

 − 2 3

Stress tensor is expressed in terms of strain rate as follows

Stress tensor is expressed in terms of strain rate as follows

= (∇ + (∇)

= (∇ + (∇)

[32,33].

[32,33].

Energy equation

Energy equation

Momentum equation

Here, *k* presents the turbulence kinetic energy, *v<sup>t</sup>* presents the turbulent eddy viscosity, ε presents the dissipation rate of turbulence energy, *S* presents the modulus of the mean rate of strain tensor, *P<sup>k</sup>* stands for the shear production of turbulent kinetic energy *σ<sup>k</sup>* and *σ*<sup>ε</sup> are 1 and 1.2, respectively.

The boundary conditions considered in the numerical analysis are constant heat flux of 7957 W/m<sup>2</sup> at tube wall with symmetrical distribution, velocity inlet with various Reynolds number of working fluid and pressure outlet. The working fluids considered in the numerical analysis are conventional fluid water, single particle nanofluid namely, Al2O<sup>3</sup> and hybrid nanofluid namely, Al2O3/Cu. The Reynolds number is ranging from 2000 to 12000 and inlet fluid temperature is considered as 298.15 K in the numerical analysis. The thermophysical properties of single particle and hybrid nanofluids are evaluated using the thermal properties of base fluid and respective nanoparticles. Three volume fractions of 0.5%, 1.0% and 2.0% are considered in the present numerical analysis. In case of hybrid nanofluids, the composition of both nanoparticles is mixed in the proportions of 75/25%, 50/50% and 25/75%. While solving equations using the considered boundary conditions, it is assumed that the flow is uniform, steady and incompressible [35]. The laminar and standard k-ε turbulence models are considered for the numerical analysis. The convergence criteria are selected as 10−<sup>8</sup> for all equations.
