Table 1 Theoretical expressions of existing thermal conductivity models for TiO2 nanofluids
From: Toward TiO2 Nanofluids—Part 2: Applications and Challenges
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 + bφ) 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 + bφ) 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. |