# Table 1 Theoretical expressions of existing thermal conductivity models for TiO2 nanofluids

Authors Year Model expressions Note
Murshed et al. [59]: 2008 $${k}_{\mathrm{eff}}=\frac{\left({k}_p-{k}_{lr}\right){\varphi}_p{k}_{lr}\left(2{\beta}_1^3-{\beta}^3+1\right)+\left({k}_p+2{k}_{lr}\right){\beta}_1^3\left[{\varphi}_p{\beta}^3\left({k}_{lr}-{k}_f\right)+{k}_f\right]}{\beta_1^3\left({k}_p+2{k}_{lr}\right)-\left({k}_p-{k}_{lr}\right){\varphi}_p\left({\beta}_1^3+{\beta}^3-1\right)}$$ β 1 = 1 + t/R β 2 = 1 + t/(2R)
k lr and t are thermal conductivity and thickness of the interfacial layer.
This model considers the interfacial layer and has been validated for TiO2, Al2O3, and Al nanofluids.
Duangthongsuk and Wongwises [77] 2009 k nf  = k f (a + )
At 15 °C: a = 1.0225, b = 0.0272
At 25 °C: a = 1.0204, b = 0.0249
At 35 °C: a = 1.0139, b = 0.0250
It is a fitted linear equation for TiO2 nanofluids within 2 vol.%.
Corcione [73] 2011 $$\frac{k_{\mathrm{eff}}}{k_f}=1+4.4 R{e}_p^{0.4}{ \Pr}_f^{0.66}{\left(\frac{T}{T_{f r}}\right)}^{10}{\left(\frac{k_p}{k_r}\right)}^{0.03}{\varphi}^{0.66}$$
$$R{e}_d=\frac{u_B{d}_p}{\upsilon}=\frac{2{\rho}_f{k}_B T}{\pi {\mu}_f^2{d}_p}$$
where T fr are the freezing point of the base fluid (about 273.16 K for water), Re p is the nanoparticle Reynolds number.
This model considers Brownian motion and has been validated for TiO2, Al2O3, and CuO nanofluids.
Applied range: 0.2% < φ < 9%,
10 nm < d < 150 nm, 294 K < T < 324 K.
Okeke et al. [76] 2011 (1 − φ nc )(k f  − k nc )/(k f  + 2k nc ) + φ nc (k p  − k nc )/(k f  + 2k nc ) = 0  where k nc is the thermal conductivity of the imaginary medium with backbones.
φ nc is taken as the volume fraction of the particles which belong to dead ends.
This model considers the aggregate sizes, particle loading, and interfacial resistance based on fractal and chemical dimensions. And it has been validated for Al2O3, CuO, and TiO2 nanofluids.
Azmi et al. [78] 2012 $$\frac{k_{\mathrm{eff}}}{k_f}=0.8938{\left(1+\frac{\varphi}{100}\right)}^{1.37}{\left(1+\frac{T}{70}\right)}^{0.2777}{\left(1+\frac{d}{150}\right)}^{-0.0336}{\left(\frac{\alpha_f}{\alpha_p}\right)}^{0.01737}$$ This model has been validated for water based Al2O3, ZnO, and TiO2 nanofluids.
Applied range: φ < 4%, 20 nm < d < 150 nm, 293 K < T < 343 K.
Reddy and Rao [88] 2013 k nf  = k f (a + )
Regression constants a and b at various temperatures for water, 40%:60% and 50%:50% EG/W.
It is a fitted expression for TiO2 nanofluids.
Applied range: 30 °C < T < 70 °C,
0.2% < φ < 1%.
Zerradi et al. [79] 2014 k nf  = k s  + k b
$${k}_s=\frac{\frac{k_p}{k_f}+\psi +\psi \varphi \left(1-\frac{k_p}{k_f}\right)}{\frac{k_p}{k_f}+\psi +\varphi \left(1-\frac{k_p}{k_f}\right)}$$
$${k}_B= G\frac{k_f{\varepsilon}_f{\displaystyle \sum_{i=1}^N\frac{A_i{ \Pr}^p}{d_i}\left[\alpha \varphi +\left(\beta +\chi \varphi \right){\left(\frac{1}{\nu}\sqrt{\frac{18{k}_b T}{\pi \rho {d}_i}}\right)}^q+\delta \right]}}{ \Pr {A}_T}$$
where ψ is a shape factor defined by
$$\psi =2{\varphi}^{0.2}\frac{l_p}{d_p}$$ for cylindrical particles
ψ = 2φ 0.2 for spherical particles
α, β, and χ are thermophysic coefficients.
This model is based on the Monte Carlo simulation combined with a new Nusselt number correlation. It has been validated for Al2O3–H2O, CuO–H2O, TiO2–H2O, and CNT–H2O nanofluids.
Abdolbaqi et al. [80] 2016 $$\frac{k_{\mathrm{eff}}}{k_f}=1.308{\left(\frac{\varphi}{100}\right)}^{0.042}{\left(\frac{T}{80}\right)}^{0.011}$$ It is nonlinear model for BioGlycol/water-based TiO2 nanofluids based on the aggregation theory using analysis of variance. Applied temperature range: 30 °C < T < 80 °C.
Shukla et al. [74] 2016 $$\frac{k_{nf}}{k_f}=\left(1-\varphi \right)+\pi {\left(\frac{6}{\pi}\right)}^{1/3}{\varphi}^{4/3}{\left[\frac{1+0.5{\left(\frac{6\varphi}{\pi}\right)}^{1/3}}{2}\left(\frac{k_{bf}}{k_p}\right)+\frac{\psi}{Nu}\right]}^{-1}$$ where ψ is the sphericity of nanoparticle. This model considers Brownian motion. And it has been validated for water and EG-based TiO2 and Al2O3 nanofluids.
Wei et al. [81] 2017 100(k nf  − k nf )/k f  = 0.443 + 2.636φ It is a linear fit of the measured values for diathermic oil-based TiO2 nanofluids. Applied range: φ < 1%.
Pryazhnikov et al. [82] 2017 $$\frac{k_{nf}}{k_f}=1+4.82\varphi -23.1{\varphi}^2$$ It is a fitted expression based on the measured values for 150 nm particles of TiO2.
Yang et al. [75] 2017 $${k}_{\mathrm{eff}}=\frac{1}{\pi}{\displaystyle \underset{0}{\overset{\pi}{\int }}{\left({k}_z^2{ \sin}^2\omega +{k}_x^2{ \cos}^2\omega \right)}^{1/2} d}\omega$$
$${k}_z=\frac{R{k}_p{k}_f}{\left( H+ R\right)\left({k}_p-\varphi {k}_p+\varphi {k}_f\right)}+\frac{ H\varphi {k}_p+ H\left(1-\varphi \right){k}_f}{H+ R}$$ $${k}_x={k}_f\frac{k_p+{k}_f+\varphi \left({k}_p-{k}_f\right)}{k_p+{k}_f-\varphi \left({k}_p-{k}_f\right)}$$
where k x and k z are the effective thermal conductivity in radial and axial directions, respectively.
This model considered the particle aspect ratio and has been validated for cylindrical TiO2 and Bi2Te3 nanofluids.
Yang et al. [72] 2017 $${k}_{\mathrm{eff}}=\frac{\left( H+2 t\right){k}_{\mathrm{eff}\_ x}+\left( R+ t\right){k}_{\mathrm{eff}\_ z}}{H+ R+3 t}$$
$${k}_{\mathrm{eff}\_ x}=\frac{A{\varphi}_p{k}_p+\left(\alpha B+\beta C\right){\varphi}_p{k}_{lr}+\left(1+\alpha +\beta \right){\varphi}_p{k}_f-{k}_f}{A{\varphi}_p+\left(\alpha B+\beta C\right){\varphi}_p+\left(1+\alpha +\beta \right){\varphi}_p-1}$$ $${k}_{\mathrm{eff}\_ z}=\frac{H+2 t}{H}{\varphi}_p\left(\frac{\left( H+2 t\right){k}_{lr}{k}_p}{2 t{k}_p+ H{k}_{lr}}+\alpha {k}_{lr}\right)+{k}_f-\frac{\left( H+2 t\right)\left(1+\alpha \right){k}_f{\varphi}_p}{H}$$  where $$\alpha =\frac{{\left( R+ t\right)}^2-{R}^2}{R^2},$$ $$\beta =\frac{2 t{\left( R+ t\right)}^2}{H{ R}^2},$$ $$t=\sqrt{2\pi}\sigma$$
$$A=-\frac{2{k}_{lr}}{k_p+{k}_{lr}},$$ $$B=\frac{R}{R+ t}\cdot \frac{k_p-{k}_{lr}}{k_{lr}+{k}_p}-1,$$ $$C=-\frac{2{k}_f}{k_{lr}+{k}_f}$$
$${k}_{lr}=\frac{k_p{\left(1+ t/ R-{k}_f/{k}_p\right)}^{{}^2}}{\left[\right(1+ t/ R-{k}_f/{k}_p-\left( t{k}_f\right)/\left( R{k}_p\right)\Big] \ln \left[\left(1+ t/ R\right){k}_p/{k}_f\right]+\left(1+ t/ R-{k}_f/{k}_p\right) t/ R}$$
This model considered the interfacial layer and particle shape and has been validated for rod-like TiO2 and Bi2Te3 nanofluids.