Table 2 Mathematical relations of thermo-physical characteristics^47,55.

Density

\(\frac{{\rho }_{thnf}}{{\rho }_{f}}=(1-{\phi }_{3})\left[\left(1-{\phi }_{2}\right)\left\{\left(1-{\phi }_{1}\right)+{\phi }_{1}\frac{{\rho }_{s1}}{{\rho }_{f}}\right\}+{\phi }_{2}\frac{{\rho }_{s2}}{{\rho }_{f}}\right]+{\phi }_{3}\frac{{\rho }_{s3}}{{\rho }_{f}}\)

Dynamic viscosity

\(\frac{{\mu }_{tnf}}{{\mu }_{f}}=\frac{1}{{(1-{\phi }_{1})}^{2.5}{(1-{\phi }_{2})}^{2.5}{(1-{\phi }_{3})}^{2.5}}\)

Electrical conductivity

\(\frac{{\sigma }_{tnf}}{{\sigma }_{hnf}}=\frac{{(1+{2\phi }_{3})\sigma }_{s3}+{(1-{2\phi }_{3})\sigma }_{hnf}}{{(1-{\phi }_{3})\sigma }_{s3}+{(1+{\phi }_{3})\sigma }_{hnf}},\)

\(\frac{{\sigma }_{hnf}}{{\sigma }_{nf}}=\frac{{\left(1+{2\phi }_{2}\right)\sigma }_{s2}+{\left(1-{2\phi }_{2}\right)\sigma }_{nf}}{{\left(1-{\phi }_{2}\right)\sigma }_{s2}+{\left(1+{\phi }_{2}\right)\sigma }_{nf}},\)

\(\frac{{\sigma }_{nf}}{{\sigma }_{f}}=\frac{{(1+{2\phi }_{1})\sigma }_{s1}+{(1-{2\phi }_{1})\sigma }_{f}}{{(1-{\phi }_{1})\sigma }_{s1}+{(1+{\phi }_{1})\sigma }_{f}}\)

Thermal conductivity

\(\frac{{k}_{tnf}}{{k}_{hnf}}=\frac{{k}_{s3}+2{k}_{hnf}-{2\phi }_{3}\left({k}_{hnf}-{k}_{s3}\right)}{{k}_{s3}+2{k}_{hnf}+{\phi }_{3}\left({k}_{hnf}-{k}_{s3}\right)},\)

\(\frac{{k}_{hnf}}{{k}_{f}}=\frac{{k}_{s2}+2{k}_{nf}-{2\phi }_{2}\left({k}_{nf}-{k}_{s2}\right)}{{k}_{s2}+2{k}_{nf}+{\phi }_{2}\left({k}_{nf}-{k}_{s2}\right)},\)

\(\frac{{k}_{nf}}{{k}_{f}}=\frac{{k}_{s1}+2{k}_{f}-{2\phi }_{1}\left({k}_{f}-{k}_{s1}\right)}{{k}_{s1}+2{k}_{f}+{\phi }_{1}\left({k}_{f}-{k}_{s1}\right)}\)

Heat capacitance

\(\frac{{(\rho {c}_{p})}_{tnf}}{{\left(\rho {c}_{p}\right)}_{f}}=\left(1-{\phi }_{3}\right)\left[\left(1-{\phi }_{2}\right)\left\{\left(1-{\phi }_{1}\right)+{\phi }_{1}\frac{{\left(\rho {c}_{p}\right)}_{s1}}{{\left(\rho {c}_{p}\right)}_{f}}\right\}+{\phi }_{2}\frac{{\left(\rho {c}_{p}\right)}_{s2}}{{\left(\rho {c}_{p}\right)}_{f}}\right]+{\phi }_{3}\frac{{\left(\rho {c}_{p}\right)}_{s3}}{{\left(\rho {c}_{p}\right)}_{f}}\)