# Thermal Conductivity of Nanofluids: A Review on Prediction Models, Controversies and Challenges
## 4 Prediction Models for Thermal Conductivity and Other Properties of the NFs
### Thermal Conductivity Empirical Models
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Researchers** & **Mathematical Expressions** & **Included Parameters/Observations** \\ \hline Maxwell [73] & \(k_{nf}=K_{f}\bigg{(}\frac{k_{nr}+2K_{g/t}+2\sigma\left(K_{nr}-K_{g}\right)}{k_{nr} +2K_{g/t}-2\sigma\left(K_{nr}-K_{g/t}\right)/\big{)}}\) & Includes the thermal conductivities of BF \\  & & and NPs \\ \hline Hamilton and & & Includes the particle shape and \\  & Crosser [72] & \(k_{nf}=K_{bf}\bigg{(}\frac{K_{nr}+(n-1)K_{bf}-\varphi(n-1)\left(K_{bf}-K_{g} \right)}{K_{g/t}+(n-1)K_{g}+\varphi\left(K_{gf}-K_{g}\right)}\bigg{)}\) & \\ \hline Wasp [74] & \(k_{nf}=K_{bf}\bigg{(}\frac{K_{nr}+(n-1)K_{bf}-\varphi(n-1)\left(K_{bf}-K_{g} \right)}{K_{g/t}+(n-1)K_{bf}+\varphi\left(K_{gf}-K_{g}\right)}\bigg{)}\) & Includes the thermal conductivities of BF \\  & & and NPs and the sphericity of the \\  & & particles assumption \\ \hline Xue [71] & \(k_{nf}=K_{bf}\bigg{(}\frac{1-\varphi+2\sigma\frac{K_{nr}}{K_{nr}-K_{bf}}}{1- \varphi+2\sigma\frac{K_{nr}}{K_{nr}-K_{bf}}ln\bigg{(}\frac{K_{nr}+K_{bf}}{2K _{g/t}}\bigg{)}}\bigg{)}\) & Includes the particle shape and \\  & & composition and the logarithmic \\  & & progressions of the thermal \\  & & conductivities of BF and NPs \\ \hline [PERSON] [75] & \(k_{nf}=K_{nfMaxwell}+\frac{1}{2}\rho_{np}\sigma_{np}\varphi\sqrt{2D_{B}}\) & Includes the temperature and viscosity of \\  & & of the clusters, and the Brownian motion \\ \hline [PERSON] [76] & \(k_{nf}=K_{nfMaxwell}+\frac{5\times 10^{4}}{k_{bf}}\beta(\varphi)\rho_{bf}c_{npbf}f (T,\varphi)\sqrt{\frac{K_{g}l}{\rho_{pd}d_{rp}}}\) & Includes the temperature and viscosity of \\  & & the BF, the volume fraction and type of \\  & & NPs, and the Brownian diffusion \\ \hline [PERSON] [77] & \(k_{nf}=1+3\Theta_{\varphi T}+\frac{3\Theta_{\varphi^{2}T}^{2}}{1-\Theta_{ \varphi T}}\) & Includes the nanolayer thickness, the NPs \\  & & volume fraction and radius and the \\  & & thermal conductivity ratio between \\  & & the NPs and the BF \\ \hline [PERSON] et al. [78] & \(k_{nf}=k_{bf}\bigg{[}\frac{k_{nr}+(n-1)k_{bf}-(n-1)\sigma_{nf}\left(k_{nr}-k_{bf }\right)}{k_{nr}+(n-1)k_{bf}+\varphi\left(k_{nr}-k_{bf}\right)}\bigg{]}\) & Includes the liquid layer thickness, \\  & & thermal conductivities of BF and NPs, but \\  & & not the particle size and the interface \\  & & between the particles \\ \hline Jang and & & Includes the thermal conductivity and the \\ [PERSON] [32] & \(k_{nf}=k_{bf}(1-\varphi)+0.01k_{nam}\varphi+18\times 10^{6}\frac{d_{bf}}{d_{rp}}\). \(k_{bf}k_{bf}R_{d}^{2}Pr_{bf}\varphi\) & \\  & & involving the Kapitza resistance (surface \\  & & resistance),and the numbers of Reynolds \\ \hline Pak and Cho & & Includes the geometry, diameter, and the \\  & [79] & \(k_{nf}=1+7.47\varphi\) & \\  & & surface resistance of the NPs \\ \hline Timofeeva & & Includes the geometry, agglomeration \\  & [33] & \(k_{nf}=k_{f}(1+3\varphi)\) & state, and the surface resistance of \\  & & the NPs \\ \hline Yu and Choi & \(k_{nf}=k_{bf}\bigg{[}\frac{k_{nr}+2k_{bf}+2(k_{pr}-k_{bf})(1+\beta)^{3}\varphi}{ k_{nr}+2k_{bf}-(k_{nr}-k_{bf})(1+\beta)^{3}\varphi}\bigg{]}\) & Modified Maxwell model.Includes the \\  & \(k_{pe}=k_{mp}\bigg{[}\frac{[2(1-\gamma)+(1+\beta)^{3}(1+2\gamma)]_{\gamma}}{-(1 -\gamma)+(1+\beta)^{3}(1+2\gamma)}\bigg{]}\) & Includes nanolayer thickness, particle \\ \hline [PERSON] et al. [81] & \(k_{nf}=k_{bf}\bigg{[}1+\frac{\frac{\gamma_{f}(\varphi)}{\eta_{0}}}{1-\frac{ \gamma_{f}(\varphi)}{\eta_{0}}}\bigg{]}\) & Includes nanolayer thickness, particle \\  & & size, temperature, volume fraction, and \\  & & interaction between adjacent particles \\ \[PERSON] & \(MODEL\)\(I:k_{nf}=k_{bf}\bigg{[}\bigg{(}\frac{\sigma_{n},\sigma_{f}}{\zeta_{r}}\bigg{)} \bigg{(}\frac{\rho_{d}}{\rho}\bigg{)}^{133}\left(\frac{M}{M_{M}}\right)^{0.33} \bigg{]}MODEL\)\(II:k_{nf}\) & Model I is applicable over a wide range of particlen, fraction and size, and different base fluids. Model II help to determine the contribution layer thickness, particle shape, and Brownian motion \\ \hline Corcione [83] & \(k_{nf}=1+4.4R\rho^{0.46}\left(\frac{T}{t_{fp}}\right)^{10}\cdot\left(\frac{k_{nr}} {k_{bf}}\right)^{0.03}\cdot\varphi^{0.66}\) & Applicable for the temperature range of \\  & & 294–324 K, nanoparticle diameter of \\  & & 10–150 nm and volume fraction of \\  & & 0.002–0.9 \\ \hline \hline \end{tabular}
\end{table}