Table 1 Thermophysical relations of nanoparticles and base fluid⁵³.

Density

\({\rho }_{nf}=(1-{(\phi }_{1})){\rho }_{f}+{\phi }_{1}{\rho }_{s1}, {A}_{1}=\frac{{\rho }_{nf}}{{\rho }_{f}}\)

Dynamic viscosity

\({\mu }_{nf}=\frac{{\mu }_{f}}{{[1-\left({\phi }_{1}\right)]}^{5/2}}={K}_{1}\)

Heat capacity

\({\left(\rho {C}_{p}\right)}_{nf}=\left[1-\left({\phi }_{1}\right)\right]{\left(\rho {c}_{p}\right)}_{f}+{{\phi }_{1}(\rho {c}_{p})}_{s1}, {A}_{2}=\frac{{\left(\rho {C}_{p}\right)}_{nf}}{{\left(\rho {C}_{p}\right)}_{f}}\)

Thermal conductivity

\(\frac{{k}_{nf}}{{k}_{f}}=\frac{{\phi }_{1}{k}_{s1}+2{k}_{f}{\phi }_{1}+2{{\phi }_{1}}^{2}{k}_{s1}-2{{\phi }_{1}}^{2}{k}_{f}}{{\phi }_{1}{k}_{s1}+2{k}_{f}{\phi }_{1}-{{\phi }_{1}}^{2}{k}_{s1}+{{\phi }_{1}}^{2}{k}_{f}}, {A}_{3}=\frac{{k}_{nf}}{{k}_{f}}\)

Electrical conductivity

\(\frac{{\sigma }_{nf}}{{\sigma }_{f}}=1+\frac{3\left({\sigma }_{s1}-{\sigma }_{f}\right)}{\left({\sigma }_{s1}+2{\sigma }_{f}\right)-\left({\sigma }_{s1}-{\sigma }_{f}\right){\phi }_{1}}, {A}_{5}=\frac{{\sigma }_{nf}}{{\sigma }_{f}}\)

Thermal expansion

\({\left(\rho {B}_{t}\right)}_{nf}=(1-{(\phi }_{1})){\rho {B}_{t}}_{f}+{\phi }_{1}{\rho {B}_{t}}_{s1}, {A}_{4}=\frac{{\left(\rho {B}_{t}\right)}_{nf}}{{\left(\rho {B}_{t}\right)}_{f}}\)

Density

\({\rho }_{hnf}=(1-{(\phi }_{1}+{\phi }_{2})){\rho }_{f}+{\phi }_{1}{\rho }_{s1}+{\phi }_{2}{\rho }_{s2}, {B}_{1}=\frac{{\rho }_{hnf}}{{\rho }_{f}}\)

Dynamic viscosity

\({\mu }_{hnf}=\frac{{\mu }_{f}}{{[1-\left({\phi }_{1}+{\phi }_{2}\right)]}^{5/2}}={K}_{2}\)

Heat capacity

\({\left(\rho {C}_{p}\right)}_{hnf}=\left[1-\left({\phi }_{1}+{\phi }_{2}\right)\right]{\left(\rho {c}_{p}\right)}_{f}+{{\phi }_{1}(\rho {c}_{p})}_{s1}+{{\phi }_{2}(\rho {c}_{p})}_{s2}, {B}_{2}=\frac{{\left(\rho {C}_{p}\right)}_{hnf}}{{\left(\rho {C}_{p}\right)}_{f}}\)

Thermal conductivity

\({b}_{1}={\phi }_{1}{k}_{s1}+{\phi }_{2}{k}_{s2}+2{k}_{f}\left({\phi }_{1}+{\phi }_{2}\right)+2\left({\phi }_{1}+{\phi }_{2}\right)\left({\phi }_{1}{k}_{s1}+{\phi }_{2}{k}_{s2}\right)-2{\left({\phi }_{1}+{\phi }_{2}\right)}^{2}{k}_{f}\)

\({b}_{2}={\phi }_{1}{k}_{s1}+{\phi }_{2}{k}_{s2}+2{k}_{f}\left({\phi }_{1}+{\phi }_{2}\right)-\left({\phi }_{1}+{\phi }_{2}\right)\left({\phi }_{1}{k}_{s1}+{\phi }_{2}{k}_{s2}\right)+{\left({\phi }_{1}+{\phi }_{2}\right)}^{2}{k}_{f}\)

\(\frac{{k}_{hnf}}{{k}_{f}}=\frac{{b}_{1}}{{b}_{2}}={B}_{3}, {B}_{3}=\frac{{k}_{hnf}}{{k}_{f}}\)

Thermal expansion

\({\left(\rho {B}_{t}\right)}_{hnf}=(1-{(\phi }_{1}+{\phi }_{2})){\left(\rho {B}_{t}\right)}_{f}+{\phi }_{1}{\left(\rho {B}_{t}\right)}_{s1}+{\phi }_{2}{\left(\rho {B}_{t}\right)}_{s2}, {B}_{4}=\frac{{\left(\rho {B}_{t}\right)}_{hnf}}{{\left(\rho {B}_{t}\right)}_{f}}\)

Electrical conductivity

\(\frac{{\sigma }_{hnf}}{{\sigma }_{f}}=1+\frac{3\left[\frac{{\sigma }_{s1}{\phi }_{1}-{\sigma }_{s2}{\phi }_{2}}{{\sigma }_{f}}-{(\phi }_{1}+{\phi }_{2})\right]}{\left(2+\frac{{\sigma }_{s1}+{\sigma }_{s2}}{{\sigma }_{f}}\right)-\left[\frac{{\sigma }_{s1}{\phi }_{1}-{\sigma }_{s2}{\phi }_{2}}{{\sigma }_{f}}\right]+{(\phi }_{1}+{\phi }_{2})}, {B}_{5}=\frac{{\sigma }_{hnf}}{{\sigma }_{f}}\)