*3.1. Single-Particle Nanofluid Properties*

[56,57].

(33) [58].

**4. Data Reduction**  The amount of heat released from the hot fluid (Nanofluid) and amount of heat The volume fraction of nanoparticles in nanofluid is calculated using Equation (13) [50].

$$O = \frac{V\_{np}}{V\_{bf} + V\_{np}} \tag{13}$$

The heat absorbed by the working fluid is calculated using Equation (18) [36]. ሶ = ሶ ,(, − ,) (31) The density and specific heat of single-particle nanofluids are not affected by nanoparticle shapes and are calculated using Equations (14) and (15), respectively [51,52].

$$
\rho\_{nf} = (1 - O)\rho\_{bf} + O\rho\_{np} \tag{14}
$$

$$\mathcal{C}\_{p,nf} = \frac{(1 - O)\rho\_{bf}\mathcal{C}\_{p,bf} + O\rho\_{np}\mathcal{C}\_{p,np}}{\rho\_{nf}} \tag{15}$$

The nanoparticle shape affects the thermal conductivity and viscosity of single-particle nanofluids. The thermal conductivity of single-particle nanofluids with Sp-, OS-, PS1-, PS2-,

ሶ

PS3- and PS4-shaped nanoparticles are calculated using the model proposed by Hamilton– Crosser, as presented by Equation (16). The model proposed by Timofeeva et al. [4], as presented by Equation (17), is used to calculate the thermal conductivity of single-particle nanofluids with BL-, PL-, CY-, and BR-shaped nanoparticles [53].

$$\frac{k\_{nf}}{k\_{bf}} = \frac{k\_{np} + (n-1)k\_{bf} - (n-1)\left(k\_{bf} - k\_{np}\right)O}{k\_{np} + (n-1)k\_{bf} + O\left(k\_{bf} - k\_{np}\right)}\tag{16}$$

$$\frac{k\_{nf}}{k\_{bf}} = 1 + \left(\mathcal{C}\_k^{shape} + \mathcal{C}\_k^{surface}\right) \mathcal{O} = 1 + \mathcal{C}\_k \mathcal{O} \tag{17}$$

Here, *n* is shape factor = <sup>3</sup> Ψ . The values *n* for OS, PS1, PS2, PS3 and PS4 are calculated using the sphericity parameter Ψ, whose values for given nanoparticle shapes are reflected in Table 2. *C shape k* is the nanoparticle shape contribution to thermal conductivity, *C sur f ace k* is surface resistance that affects the thermal conductivity of nanofluid and influences by the solid/liquid interface.

**Table 2.** Parameters for calculating thermal conductivity and viscosity of spherical, oblate spheroid and prolate spheroid nanoparticle shape-based nanofluids.


The viscosity of single-particle nanofluids with Sp-, OS-, PS1-, PS2-, PS3- and PS4 shaped nanoparticles is evaluated using Equation (18). This equation is proposed by Muller et al., based on the mathematical model presented by Maron and Pierce. For calculating the viscosity of single-particle nanofluids with BL-, PL-, CY-, and BR-shaped nanoparticles, Timofeeva et al.'s [4] model, as presented by Equation (19), is used [53].

$$\frac{\mu\_{nf}}{\mu\_{bf}} = \left(1 - \frac{O}{O\_m}\right)^{-2} \tag{18}$$

$$
\mu\_{\rm nf} = \mu\_{\rm bf} \left( 1 + A\_1 O + A\_2 O^2 \right) \tag{19}
$$

Here, *O* is the volume fraction of nanoparticles in nanofluid and *O<sup>m</sup>* is the packing fraction, which is calculated using Equation (20) [53]. *A*1, *A*<sup>2</sup> are coefficients proposed by Timofeeva et al., and values of these coefficients are extracted from references [4,53].

$$O\_m = \frac{2}{(0.321\delta + 3.02)}\tag{20}$$

The aspect ratio of spheroids is expressed as Equation (21). The parameters a and c are denoted as lengths of spheroid semi-axes, which could be seen in Figure 3. For OSand PS-shaped nanoparticles, these parameters are presented by Equations (22) and (23), respectively [53].

$$
\delta = \frac{c}{a} \tag{21}
$$

$$OS = 1 - (\frac{c}{a})^2, \ c < a \tag{22}$$

$$PS = 1 - (\frac{a}{c})^2, \ c > a \tag{23}$$
