# Preparation Techniques of TiO\({}_{2}\) Nanofluids and Challenges: A Review
## 3 Thermal Conductivity  
The thermal conductivity of the nanofluids has been measured by using different classical models, including the Hamilton and Crosser [89] model. \[k_{eff}=k_{bf}\left[\frac{k_{np}+(n-1)k_{bf}-(n-1)\varphi\Big{(}k_{bf}-k_{np} \Big{)}}{k_{np}+(n-1)k_{bf}+\varphi\Big{(}k_{bf}-k_{np}\Big{)}}\right] \tag{2}\]  
where \(n=\frac{3}{\omega}\) in which \(n\) and \(\omega\) represent the empirical shape factor and sphericity respectively. The factor \(\omega\) is defined as the ratio of the surface area of a sphere (equal in volume of the given particle) to the surface area of the particle. For nanofluids in which the thermal conductivity of nanoparticles is larger than a factor of 100, the above equation should be used for the value of \(n\) and for all other nanofluids \(n\) may be taken as 3. The H-C (Hamilton and Crosser) model is an improved form of the Maxwell model [1] that was used to predict the thermal conductivity of well dispersed solutions of solid-liquid at low volume concentration of nanoparticles. The Maxwell model considered the thermal conductivity and volume concentration of the base fluid and nanoparticles while the H-C model also considers the effects of particles shape. Equation (3) represents the Maxwell model used to compute the thermal conductivity of nanofluids. \[k_{eff}=k_{bf}\left[\frac{k_{np}+2k_{bf}+2\varphi\Big{(}k_{np}-k_{bf}\Big{)}}{ k_{np}+2k_{bf}-\varphi\Big{(}k_{np}-k_{bf}\Big{)}}\right] \tag{3}\]  
Figure 15: TEM image of the TiO\({}_{2}\) nanofluids with dispersed nanoparticles of average diameter. The H-C model is used for cylindrical and spherical particles and the Maxwell model is used only for spherical particles. [PERSON] et al. [90] generalized the Maxwell model by introducing the influence of interfacial thermal resistance with particle shape, size, and orientation. The prescribed model for spherical particles is expressed by:  
\[k_{eff}=k_{bf}\left[\frac{2k_{bf}+(1+2\alpha)k_{np}+2\varphi\Big{[}(1-\alpha)k_ {np}-k_{bf}\Big{]}}{2k_{bf}+(1+2\alpha)k_{np}-\varphi\Big{[}(1-\alpha)k_{np}- k_{bf}\Big{]}}\right] \tag{4}\]  
In this expression the dimensionless parameter \(\alpha\) describes the particle-fluid interaction and is expressed as follows:  
\[\alpha=\frac{Rk_{bf}}{r_{np}}\]  
where \(R\) is the thermal boundary resistance coefficient, and \(r\) represents the radius of the dispersed nanoparticle. The limitation of the particle volume fraction can be avoided by using the Bruggeman model [91] used for a binary mixture of randomly dispersed homogeneous spherical nanoparticles. \[k_{eff}=\frac{k_{bf}}{4}\Bigg{[}(3\varphi-1)\frac{k_{np}}{k_{bf}}+(2-3\varphi) +\frac{k_{bf}}{4}\sqrt{\Delta}\Bigg{]} \tag{5}\]  
\[\Delta=\Bigg{[}(3\varphi-1)^{2}\Bigg{(}\frac{k_{np}}{k_{bf}}\Bigg{)}+(2-3 \varphi)^{2}+2\Big{(}2+9\varphi^{2}\Big{)}\Bigg{(}\frac{k_{np}}{k_{bf}}\Bigg{)} \Bigg{]}\]  
The updated H-C model is the Wasp model [92], most commonly used for convection heat transfer problems established by considering the unity empirical shape factor. \[k_{eff}=k_{bf}\left[\frac{k_{np}+2k_{bf}-2\varphi\Big{(}k_{bf}-k_{np}\Big{)}} {k_{np}+2k_{bf}+\varphi\Big{(}k_{bf}-k_{np}\Big{)}}\right] \tag{6}\]  
The discussed classical models were unable to get the exact value of thermal conductivity for nanofluids because these models do not consider the effect of temperature, size of the particle, particle distribution, interfacial layer, cluster formation and Brownian motion of particles. [PERSON] et al. [93] considered the effects of nanolayer thickness, volume fraction, nanoparticle size and thermal conductivity ratio of the nanoparticle to the base fluid in the derived model to measure the enhanced thermal conductivity of nanofluids. Equation (7) represents the Xie thermal conductivity model.