# 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  
There are a number of prediction models for the conductivity of an NF, based essentially in a weighted average between the properties of the NPs and the base fluid(BF), giving the existing fraction between both [71]. Some of those models were selected to identify the behavior of NFs constituted by spherical particles, with diameters in the order of magnitude that is found in the scope of this work, being the equations presented subsequently. However, many other models can be considered for more specific applications, as it can be found in Lamas [72]. The Maxwell model [73], Equation (1) consists of an empirical model of good approximation and simplicity, for spherical particles in small concentrations. Being one of the first models of this kind, it was widely used as a baseline to other models where additional parameters or corrections were applied. \[K_{nfMaxwell}=K_{f}\left(\frac{K_{np}+2K_{bf}+2\varphi\Big{(}K_{np}-K_{bf}\Big{)} }{K_{np}+2K_{bf}-2\varphi\Big{(}K_{np}-K_{bf}\Big{)}}\right) \tag{1}\]  
In this equation, as well as in the following ones, \(K_{nf}\) corresponds to the NFs thermal conductivity, \(K_{np}\) to the NPs thermal conductivity, \(K_{bf}\) to the thermal conductivity of the BF,\(\varphi\) to the NP concentration and \(K_{nf}\) to the predicted NF thermal conductivity by the used model. Presented by [PERSON] and [PERSON] [72], Equation (2) is based on the same principle, however, it also considers the parameter relative to the shape of the particle. Thus, the shape factor, \(n\), is determined considering the sphericity of the particle, \(w\), as verified in Equation (3) Therefore, the sphericity of the particle varies with the shape, being for instance, for a sphere, \(w=3\), or for a cylinder, \(w=0.5\). \[K_{nfH\&C}=K_{bf}\left(\frac{K_{np}+(n-1)K_{bf}-\varphi(n-1)\Big{(}K_{bf}-K_{ np}\Big{)}}{K_{np}+(n-1)K_{bf}+\varphi\Big{(}K_{bf}-K_{np}\Big{)}}\right) \tag{2}\]\[n=\frac{3}{w} \tag{3}\]  
The Wasp prediction model [74], widely used for convection heat transfer problems, is a particular case of the Hamilton and Crosser model, where the particles are perfectly spherical (\(w=1\)). Consequently, the Wasp model serves as a good practical approach when there is uncertainty regarding the shape of the particle and can be given by Equation (4):  
\[K_{nf\,wap}=k_{bf}\left[\frac{k_{np}\,+2k_{bf}-2\varphi\Big{(}k_{bf}-k_{np} \Big{)}}{k_{np}+2k_{bf}+\varphi\Big{(}k_{bf}-k_{np}\Big{)}}\right] \tag{4}\]  
Other model is introduced by [PERSON] [71], that presents a relation between the logarithmic progressions, becoming that way, the most pessimist model for the conductivity gain, as verified in Equation (5):  
\[K_{nfXue}=K_{bf}\left(\frac{1-\varphi+2\varphi\frac{K_{np}}{K_{np}-K_{bf}}!In \Big{(}\frac{K_{np}+K_{bf}}{2K_{f}}\Big{)}}{1-\varphi+2\varphi\frac{K_{f}}{K_{ np}-K_{bf}}!In\Big{(}\frac{K_{np}+K_{bf}}{2K_{bf}}\Big{)}}\right) \tag{5}\]  
The mentioned models are considered static since they ignore the effect of the random movement of the particles in the fluid designated by the Brownian motion. Therefore, to complement the static models, other dynamic models were developed with the purpose of improving the description of mechanisms like the Brownian motion, temperature, size and distribution of the nanoparticles, interfacial layer and cluster formation and morphology that, as we have seen, have crucial effects in thermal conductivity. The dynamic Xuan [75] model uses the Maxwell conductivity model expressed by Equation (1), as a baseline, adding the dynamic effect of the thermal conductivity increase.