Table 3 Modified Tiwari–Das thermal conductivity, nanoparticles properties and electrical conductivity models.

From: Thermal management in annular fin using ternary nanomaterials influenced by magneto-radiative phenomenon and natural convection

Characteristics

Empirical correlation

Thermal conductivity

\(\frac{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{k}_{{\left( {Al_{2} O_{3} - CuO - Cu} \right)w}} }}{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{k}_{{\left( {Al_{2} O_{3} - CuO} \right)w}} }} = \frac{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{k}_{Cu,p3} + 2\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{k}_{{\left( {Al_{2} O_{3} - CuO} \right)w}} - 2\phi_{Cu,p3} \left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{k}_{{\left( {Al_{2} O_{3} - CuO} \right)w}} - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{k}_{Cu,p3} } \right)}}{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{k}_{Cu,p3} + 2\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{k}_{{\left( {Al_{2} O_{3} - CuO} \right)w}} + \phi_{Cu,p3} \left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{k}_{{\left( {Al_{2} O_{3} - CuO} \right)w}} - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{k}_{Cu,p3} } \right)}}\)

\(\frac{{k_{{\left( {Al_{2} O_{3} - CuO} \right)w}} }}{{k_{nf} }} = \frac{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{k}_{CuO,p2} + 2\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{k}_{nf} - 2\phi_{CuO,p2} \left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{k}_{nf} - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{k}_{CuO,p2} } \right)}}{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{k}_{CuO,p2} + 2\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{k}_{nf} + \phi_{CuO,p2} \left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{k}_{nf} - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{k}_{CuO,p2} } \right)}}\)

\(\frac{{k_{nf} }}{{k_{f} }} = \frac{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{k}_{{Al_{2} O_{3} ,p1}} + 2\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{k}_{{H_{2} O}} - 2\phi_{{Al_{2} O_{3,p1} }} \left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{k}_{{H_{2} O}} - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{k}_{{Al_{2} O_{3} ,p1}} } \right)}}{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{k}_{{Al_{2} O_{3,p1} }} + 2\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{k}_{{H_{2} O}} + \phi_{{Al_{2} O_{3,p1} }} \left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{k}_{{H_{2} O}} - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{k}_{{Al_{2} O_{3,p1} }} } \right)}}\)

Electrical conductivity

\(\frac{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\sigma }_{{\left( {Al_{2} O_{3} - CuO - Cu} \right)H_{2} O}} }}{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\sigma }_{{\left( {Al_{2} O_{3} - CuO} \right)H_{2} O}} }} = \frac{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\sigma }_{Cu,p3} + 2\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\sigma }_{{\left( {Al_{2} O_{3} - CuO} \right)w}} - 2\phi_{Cu,p3} \left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\sigma }_{{\left( {Al_{2} O_{3} - CuO} \right)w}} - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\sigma }_{Cu,p3} } \right)}}{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\sigma }_{Cu,p3} + 2\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\sigma }_{{\left( {Al_{2} O_{3} - CuO} \right)w}} + \phi_{Cu,p3} \left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\sigma }_{{\left( {Al_{2} O_{3} - CuO} \right)w}} - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\sigma }_{Cu,p3} } \right)}}\) where

\(\frac{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\sigma }_{{\left( {Al_{2} O_{3} - CuO} \right)w}} }}{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\sigma }_{nf} }} = \frac{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\sigma }_{CuO,p2} + 2\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\sigma }_{nf} - 2\phi_{CuO,p2} \left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\sigma }_{nf} - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\sigma }_{CuO,p2} } \right)}}{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\sigma }_{CuO,p2} + 2\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\sigma }_{nf} + \phi_{CuO,p2} \left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\sigma }_{nf} - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\sigma }_{CuO,p2} } \right)}}\)

\(\frac{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\sigma }_{nf} }}{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\sigma }_{{H_{2} O}} }} = \frac{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\sigma }_{{Al_{2} O_{3} }} ,p1 + 2\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\sigma }_{{H_{2} O}} - 2\phi_{{Al_{2} O_{3} ,p1}} \left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\sigma }_{{H_{2} O}} - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\sigma }_{{Al_{2} O_{3} }} ,p1} \right)}}{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\sigma }_{{Al_{2} O_{3} ,p1}} + 2\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\sigma }_{{H_{2} O}} + \phi_{{Al_{2} O_{3} ,p1}} \left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\sigma }_{{H_{2} O}} - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\sigma }_{{Al_{2} O_{3} ,p1}} } \right)}}\)

Basic components

\(\hat{\rho }\;\left( {{\text{kg/m}}^{3} } \right)\)

\(\hat{c}_{p} \;\left( {{\text{J/Kg}}\;{\text{K}}} \right)\)

\(\hat{k}\;\left( {\text{W/mk}} \right)\)

\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\sigma } \;\left( {\Omega {\text{m}}} \right)^{ - 1}\)

CuO

6500

540

18

\(6.9\times {10}^{-2}\)

H2O

997.1

\(4180\)

\(0.6071\)

\(5.5\times {10}^{-6}\)

Cu

8933

385

400

\(59.6\times {10}^{6}\)

Al2O3

\(3970\)

\(765\)

\(40\)

\(35\times {10}^{6}\)

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