*2.1. Computational Geometry and Boundary Conditions*

The numerical investigation based on the computational fluid dynamics (CFD) approach is conducted to evaluate the first and second law characteristics of the microplate heat exchanger with single-particle and hybrid nanofluids. The three-dimensional model of the microplate heat exchanger is depicted in Figure 1. The microplate heat exchanger comprises of two flow lines in the counter flow direction, the nanofluid as hot fluid and water as cold fluid. There are three flow passages for the nanofluid and three flow passages for water. Each flow passage is divided into 17 microchannels with a cross section of 0.25 mm × 0.32 mm for the nanofluid and that of 0.25 mm × 0.42 mm for water. Hence, there are total 51 microchannels with a length of 12.5 mm and wall thickness of 0.52 mm for the nanofluid and water. The microplate heat exchanger is made up of copper material. The microplate heat exchanger is symmetrical, which results in symmetrical heat transfer and fluid flow through the microchannel, hence one pair of microplates for hot and cold fluids is considered as the computational geometry to reduce the computational time. The single-particle nanofluid of Al2O<sup>3</sup> and hybrid nanofluid of Al2O3/Cu with volume fractions of 0.5%, 1.0% and 2.0% are considered as hot fluid. The nanofluids are flowing with inlet temperatures of 90 ◦C, 80 ◦C and 70 ◦C and inlet mass flow rates of 10 kg/h, 20 kg/h and 30 kg/h. The cold fluid water is flowing with inlet temperatures of 10 ◦C, 20 ◦C and 30 ◦C and inlet mass flow rates of 10 kg/h, 20 kg/h and 30 kg/h. The pressure outlet boundary condition is applied at the outlet of the heat exchanger. The nanofluids and water enter with uniform velocity and uniform temperature. The external surfaces of the heat exchanger are assumed to be insulated and a no-slip condition is assumed at all walls of the heat exchanger. The conjugate heat transfer is considered in the present study in that the solid domain is subjected to a conduction mechanism and fluid domains are subjected to conduction and convection mechanisms. The computational geometry of the microplate heat exchanger presents a similar pattern and symmetry boundary and forms the mirror pattern and thermal and flow characteristics. Therefore, the symmetry boundary conditions are applied on the computational geometry of the microplate heat exchanger. *Symmetry* **2021**, *13*, x FOR PEER REVIEW 5 of 33

**Figure 1.** Three-dimensional model of the microplate heat exchanger. **Figure 1.** Three-dimensional model of the microplate heat exchanger.

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 (m2). 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].

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

The volumetric thermal entropy generation rate is calculated using Equation (6) as the summation of volumetric thermal entropy generations due to average and fluctuating

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

் + ሶ

డ௭)ଶቃ + <sup>ఒ</sup> ் <sup>మ</sup> [(డ்ᇲ డ௫ )ଶ തതതതതതതത

∇ ∙ () = 0 (1)

∇ ∙ () = −∇ + ∇ ∙ (∇U) (2)

+ (డ்ᇲ డ௬ )ଶ തതതതതതതത

∇ ∙ (ℎ) =∇∙ (∇) (3)

∇ଶ=0 (4)

ி (5)

+ (డ்ᇲ డ௭ )ଶ തതതതതതതത

] (6)

*2.2. Governing Equation and Meshing* 

Continuity equation

Momentum equation

temperature gradients [48]. ሶ ் <sup>=</sup> <sup>ఒ</sup> ் <sup>మ</sup> [(డ்

Energy equation for fluid domains

Energy equation for solid domains

generation rate as presented by Equation (5) [48].

డ௫)ଶ + (డ்

ሶ ் = ሶ

డ௬)ଶ + ቀడ்
