*3.2. Hybrid Nanofluid Properties*

The open literature lacks presentation of the concrete models for evaluating the thermophysical properties of hybrid nanofluids with different nanoparticle shapes. In the present study, the updated equations for calculating the thermophysical properties of hybrid nanofluids with different nanoparticle shapes are proposed based on equations presented in Section 3.1 and references [54,55]. The density, specific heat, thermal conductivity and viscosity of hybrid nanofluids with different nanoparticle shapes are evaluated using Equations (24)–(30).

Volume fraction of hybrid nanofluid

$$O\_{hnf} = O\_{np1} + O\_{np2} \tag{24}$$

Density of hybrid nanofluid

$$
\rho\_{\text{hf}f} = \mathcal{O}\_{\text{np1}}\rho\_{\text{np1}} + \mathcal{O}\_{\text{np2}}\rho\_{\text{np2}} + \left(1 - \mathcal{O}\_{\text{hf}f}\right)\rho\_{\text{bf}} \tag{25}
$$

Specific heat of hybrid nanofluid

$$\mathbf{C}\_{p,lnf} = \frac{\mathbf{O}\_{np1}\rho\_{np1}\mathbf{C}\_{p,np1} + \mathbf{O}\_{np2}\rho\_{np2}\mathbf{C}\_{p,np2} + \left(1 - \mathbf{O}\_{lnf}\right)\rho\_{bf}\mathbf{C}\_{p,bf}}{\rho\_{lnf}}\tag{26}$$

Thermal conductivity of hybrid nanofluid

For Sp-, OS-, PS1-, PS2-, PS3- and PS4-shaped nanoparticles

$$\frac{k\_{\rm hnf}}{k\_{\rm bf}} = \frac{\frac{\text{O}\_{\rm npf}k\_{\rm npf} + \text{O}\_{\rm npf}k\_{\rm n2}}{\text{O}\_{\rm hnf}} + (n-1)k\_{\rm bf} + (n-1)\left(\text{O}\_{\rm npf}k\_{\rm npf} + \text{O}\_{\rm npf}k\_{\rm n2}\right) - (n-1)\text{O}\_{\rm hnf}k\_{\rm bf}}{\frac{\text{O}\_{\rm npf}k\_{\rm npf} + \text{O}\_{\rm npf}k\_{\rm n2}}{\text{O}\_{\rm hnf}} + (n-1)k\_{\rm bf} - (n-1)\left(\text{O}\_{\rm npf}k\_{\rm npf} + \text{O}\_{\rm npf}k\_{\rm n2}\right) + \text{O}\_{\rm hnf}k\_{\rm bf}}\tag{27}$$

For BL-, PL-, CY-, and BR-shaped nanoparticles

$$\frac{k\_{\rm Inf}}{k\_{bf}} = 1 + \left(\mathsf{C}\_{k}^{shape} + \mathsf{C}\_{k}^{surface}\right) \mathsf{O}\_{\rm Inf} = 1 + \mathsf{C}\_{k} \mathsf{O}\_{\rm Inf} \tag{28}$$

Viscosity of hybrid nanofluid

For Sp-, OS-, PS1-, PS2-, PS3- and PS4-shaped nanoparticles

$$\frac{\mu\_{nf}}{\mu\_{bf}} = \left(1 - \frac{O\_{hnf}}{O\_m}\right)^{-2} \tag{29}$$

For BL-, PL-, CY-, and BR-shaped nanoparticles

$$
\mu\_{nf} = \mu\_{bf} \left( 1 + A\_1 O\_{\text{lmf}} + A\_2 O\_{\text{lmf}} \, ^2 \right) \tag{30}
$$

The parameters for calculating thermal conductivity and viscosity of nanofluids with Sp-, OS- and PS-shaped nanoparticles are shown in Table 2 [53]. The parameters for calculating thermal conductivity and viscosity of nanofluids with BL-, PL-, CY- and BR-shaped nanoparticles are shown in Tables 3 and 4, respectively [53].

Table 5 depicts the properties of the considered nanoparticles of Boehmite alumina (Al2O3) and copper (Cu) and the base fluid of water [19,50]. These properties are computed in equations of Sections 3.1 and 3.2 to evaluate the density, specific heat, thermal conductivity and viscosity for Al2O<sup>3</sup> and Al2O3/Cu nanofluids with different nanoparticle shapes.


**Table 3.** Parameters for calculating thermal conductivity of blade, platelet, cylinder and brick nanoparticle shape-based nanofluids.

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


**Table 5.** Properties of base fluid and nanoparticles.

