• For gold nanoparticles

The mathematical expressions that describe the thermophysical properties of water and gold nanofluids are given as:

$$
\overline{\rho}\_{nf} = (1 - \phi)\overline{\rho}\_f - \phi\overline{\rho}\_p \tag{17}
$$

$$\left(\overline{\rho}\_{nf}\right)\_{\text{water/gold}} = 997.1(1-\phi) - 19300\phi\tag{18}$$

$$
\overline{\mu}\_{nf} = \left( 1.013 + 0.092\phi - 0.015\phi^2 \right) \overline{\mu}\_f \tag{19}
$$

$$\left(\overline{\mu}\_{nf}\right)\_{\text{water/gold}} = 0.89 \left(1.013 + 0.092\phi - 0.015\phi^2\right) \tag{20}$$

$$(\overline{\rho}\mathbb{C}\_P)\_{nf} = (\mathbb{C}\_P)\_f \overline{\rho}\_f (1 - \phi) - (\mathbb{C}\_P)\_p \phi \overline{\rho}\_p \tag{21}$$

$$\left(\overline{p}C\_P\right)\_{\text{water/gold}} = (4179)(997.1)(1 - \phi) - (126)(19300)\phi \tag{22}$$

$$
\vec{k}\_{nf} = (1.0204 + 0.0249\phi)\vec{k}\_f \tag{23}
$$

$$\left(\tilde{k}\_{nf}\right)\_{\text{water/gold}} = 0.569(1.0204 + 0.0249\phi) \tag{24}$$

where the gold density is 19,300 kg·m−<sup>3</sup> and heat capacity and thermal conductivity are 126 J/kg m and 317 W·m−1·K<sup>−</sup>1, respectively.

• For silver nanoparticles

The thermophysical properties of water and silver nanofluids are given as:

$$\left(\overline{\rho}\_{nf}\right)\_{\text{water/silver}} = 997.1(1 - \phi) - 10490\phi \tag{25}$$

$$\left(\overline{\mu}\_{nf}\right)\_{\text{water/silver}} = 0.89 \left(1.013 + 0.092\phi - 0.015\phi^2\right) \tag{26}$$

$$(\overline{p}\mathbb{C}\_P)\_{\text{water/silver}} = (4179)(997.1)(1 - \phi) - (233)(10490)\phi \tag{27}$$

$$\left(\tilde{k}\_{nf}\right)\_{\text{water/silver}} = 0.569(1.0204 + 0.0249\phi) \tag{28}$$

where the density of silver is 10,490 kg·m−<sup>3</sup> while heat capacity and thermal conductivity are respectively 233 J/Kg m and 429 W·m−1·K−<sup>1</sup>
