*2.2. Governing Equation and Meshing*

The continuity, momentum and energy equations are solved using the computational fluid dynamics approach to analyze the thermodynamic attributes of the microplate heat exchanger with single-particle and hybrid nanofluids [44–46]. The continuity Equation (1) is expressed per unit of surface area (m<sup>2</sup> ). The governing equations are considered for the single pair microplates in the heat exchanger based on symmetrical heat transfer. While solving the equations, it is assumed that the flow is three-dimensional, steady, turbulent and incompressible. Additionally, the working fluids are assumed to be Newtonian [47].

Continuity equation

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

Momentum equation

$$\nabla \cdot (\rho L U) = -\nabla p + \nabla \cdot (\mu \nabla \mathbf{U}) \tag{2}$$

Energy equation for fluid domains

$$\nabla \cdot (\rho \mathcal{U} \mathcal{H}) = \nabla \cdot (\lambda \nabla T) \tag{3}$$

Energy equation for solid domains

$$
\nabla^2 T = 0 \tag{4}
$$

The working fluids in the microplate heat exchanger are exposed to heat exchange which results in entropy generation. The volumetric total entropy generation rate is the sum of the volumetric thermal entropy generation rate and the volumetric friction entropy generation rate as presented by Equation (5) [48].

$$
\dot{S}\_T = \dot{S}\_{Th} + \dot{S}\_{Fr} \tag{5}
$$

The volumetric thermal entropy generation rate is calculated using Equation (6) as the summation of volumetric thermal entropy generations due to average and fluctuating temperature gradients [48].

$$\dot{S}\_{Th} = \frac{\lambda}{\overline{T}^2} [ (\frac{\partial \overline{T}}{\partial x})^2 + (\frac{\partial \overline{T}}{\partial y})^2 + (\frac{\partial \overline{T}}{\partial z})^2] + \frac{\lambda}{\overline{T}^2} [ (\frac{\partial T'}{\partial x})^2 + (\frac{\overline{\partial T'}}{\partial y})^2 + (\frac{\overline{\partial T'}}{\partial z})^2 ] \tag{6}$$

The first term at the right side of Equation (6) presents the volumetric thermal entropy generation rate due to time–mean temperature gradients. Whereas, the second term at right side of Equation (6) presents the volumetric thermal entropy generation rate due to fluctuating temperature gradients, which could also be expressed as Equation (7) [48].

$$\dot{S}\_{Th'} = \frac{\lambda\_t}{\overline{T}^2} [(\frac{\partial \overline{T}}{\partial x})^2 + (\frac{\partial \overline{T}}{\partial y})^2 + (\frac{\partial \overline{T}}{\partial z})^2] \tag{7}$$

Therefore, the volumetric thermal entropy generation rate could be reduced to Equation (8) after manipulating *λe f f* = *λ* + *λ<sup>t</sup>* .

$$\dot{S}\_{Th} = \frac{\lambda\_{eff}}{\overline{T}^2} [(\frac{\partial \overline{T}}{\partial \mathbf{x}})^2 + (\frac{\partial \overline{T}}{\partial y})^2 + (\frac{\partial \overline{T}}{\partial z})^2] \tag{8}$$

The volumetric friction entropy generation rate is evaluated using Equation (9) as the summation of direct dissipation due to the average velocity gradient and indirect dissipation due to the fluctuating velocity gradient [48].

$$\begin{array}{c} \dot{S}\_{F} = \frac{\mu}{T} \{ 2[(\frac{\partial \overline{v}}{\partial x})^{2} + (\frac{\partial \overline{v}}{\partial y})^{2} + (\frac{\partial \overline{v}}{\partial z})^{2}] + (\frac{\partial \overline{v}\_{x}}{\partial y} + \frac{\partial \overline{v}\_{y}}{\partial x})^{2} + (\frac{\partial \overline{v}\_{x}}{\partial z} + \frac{\partial \overline{v}\_{z}}{\partial x})^{2} + (\frac{\partial \overline{v}\_{y}}{\partial z} + \frac{\partial \overline{v}\_{z}}{\partial y})^{2} \} \\ \qquad + \frac{\mu}{T} \mathbf{2}[(\frac{\partial \overline{v}\_{x}}{\partial x})^{2} + (\frac{\overline{v} \overline{v}\_{y}}{\partial y})^{2} + (\frac{\partial \overline{v}\_{z}}{\partial z})^{2}] + (\frac{\partial \overline{v}\_{x}}{\partial y} + \frac{\partial \overline{v}\_{y}}{\partial x})^{2} \\ \qquad + (\frac{\partial \overline{v}\_{x}}{\partial z} + \frac{\partial \overline{v}\_{z}}{\partial x})^{2} + (\frac{\partial \overline{v}\_{y}}{\partial z} + \frac{\partial \overline{v}\_{z}}{\partial y})^{2} \end{array} \tag{9}$$

The first term on right side of Equation (9) indicates the direct entropy generation due to dissipation in the mean flow field, which is commonly denoted as direct dissipation. Whereas, the second term on the right side of Equation (9) indicates turbulent or indirect dissipation due to fluctuating velocity gradients, which is also expressed as Equation (10) [48].

$$
\dot{S}\_{Fr'} = \frac{\rho \beta^\* kO}{\overline{T}} \tag{10}
$$

Therefore, the volumetric friction entropy generation rate could be reduced to Equation (11) [48].

$$\dot{S}\_{Fr} = \frac{\mu}{\overline{T}} \left\{ 2[(\frac{\partial \overline{w}}{\partial x})^2 + (\frac{\partial \overline{w}}{\partial y})^2 + (\frac{\partial \overline{w}}{\partial y})^2] + (\frac{\partial \overline{w}\_x}{\partial y} + \frac{\partial \overline{w}\_y}{\partial x})^2 + (\frac{\partial \overline{w}\_x}{\partial z} + \frac{\partial \overline{w}\_z}{\partial x})^2 + (\frac{\partial \overline{w}\_y}{\partial z} + \frac{\partial \overline{w}\_z}{\partial y})^2 \right\} + \frac{\rho \beta^\* k O}{\overline{T}} \tag{11}$$

Here, *β* ∗ is the model constant with value of 0.09.

The Bejan number is evaluated to quantify the contribution of the volumetric thermal entropy generation rate in the volumetric total entropy generation rate. The Bejan number (Be) is defined as the ratio of the volumetric thermal entropy generation rate to the volumetric total entropy generation rate, as presented by Equation (12) [48].

$$Be = \frac{\dot{\mathcal{S}}\_{Th}}{\dot{\mathcal{S}}\_T} \tag{12}$$

The governing equations are solved using the finite volume method and second order approach. The tetrahedron mesh elements are considered for the computational geometry of the heat exchanger and fluid domains. The inflation layers are provided on the fluid domains to consider the effect of boundary layers at the walls of the heat exchanger. The mesh independency test is carried out by generating five different mesh element numbers on the computational geometry. The results of performance index are evaluated for different mesh element numbers by considering the flow of water as hot and cold fluids. The inlet temperature and mass flow rate on the hot side are 90 ◦C and 20 kg/h, and those on the cold side are 20 ◦C and 20 kg/h. The mesh independency results for five different mesh elements are presented in Table 1. The temperature and pressure drop results for hot and cold fluids are significantly varying when the mesh element numbers are ranging from 157,649 to 732,993. However, beyond the mesh element number of 732,993, the simulated results of outlet temperatures and pressure drops of hot and cold fluids vary within ±1% [49]. Therefore, the computational geometry with a mesh element number of 732,993 is considered for the further numerical investigations. The mesh configuration of computational geometry with selected final mesh elements is depicted in Figure 2. The SIMPLE scheme with velocity-pressure coupling and the convergence criteria of 10−<sup>8</sup> are considered for solving the governing equations.



**Table 4.** Parameters for calculating viscosity of blade, platelet, cylinder and brick nanoparticle

732,993, the simulated results of outlet temperatures and pressure drops of hot and cold fluids vary within ±1% [49]. Therefore, the computational geometry with a mesh element number of 732,993 is considered for the further numerical investigations. The mesh configuration of computational geometry with selected final mesh elements is depicted in Figure 2. The SIMPLE scheme with velocity-pressure coupling and the convergence

> **Cold Fluid-Temperature (°C)**

157,649 88.669 29.841 2.413 1.321 489,478 86.295 26.877 2.637 1.444 732,993 85.180 24.888 2.738 1.502 1,142,485 85.182 24.086 2.742 1.505 1,588,899 85.181 23.865 2.751 1.507

**Hot Fluid-Pressure Drop (bar)** 

**Cold Fluid-Pressure Drop (bar)** 

(13)

(15)

criteria of 10−8 are considered for solving the governing equations.

**Hot Fluid-Temperature (°C)** 

*Symmetry* **2021**, *13*, x FOR PEER REVIEW 10 of 33

**Table 1.** Mesh independency test results.

**Mesh Elements** 

shape-based nanofluids

**Figure 2.** Mesh configuration for computational geometry. **Figure 2.** Mesh configuration for computational geometry. **Table 5.** Properties of base fluid and nanoparticles

#### **3. Thermophysical Properties of Nanofluids with Nanoparticle Shapes 3. Thermophysical Properties of Nanofluids with Nanoparticle Shapes Property Water Alumina Copper**

The thermophysical properties of single-particle and hybrid nanofluids with different nanoparticle shapes are evaluated using the models presented in Sections 3.1 and 3.2, respectively. The different nanoparticle shapes considered are Sp, OS, PS1, PS2, PS3, PS4, BL, PL, CY and BR, as depicted in Figure 3. The thermophysical properties of single-particle and hybrid nanofluids with different nanoparticle shapes are evaluated using the models presented in Sections 3.1 and 3.2, respectively. The different nanoparticle shapes considered are Sp, OS, PS1, PS2, PS3, PS4, BL, PL, CY and BR, as depicted in Figure 3. Density (kg/m3) 997.1 3050 8933 Specific heat (J/kg∙K) 4179 618.3 385 Thermal conductivity (W/m∙K) 0.613 30 400 Viscosity (Pa∙s) 0.001003 - -

**Figure 3.** Different nanoparticle shapes considered in the present study. **Figure 3.** Different nanoparticle shapes considered in the present study.
