## 3 Nanofluid Thermal Conductivity Models
### Effective Medium Theory  
A precise calculation to predict the thermal conductivity of nanofluids is nearly impossible due to the mixture's inherent complex properties. The effective medium theory is a static model that does not account for any dynamic or random motion within the mixture (see Figure 27). Predictions of a mixture's physical properties can be theoretically solved for by using the effective medium theory. The component's material properties, along with the specific composition ratios can be used to calculate the mixture's combined effective properties [69].  
Figure 27: A representation of the effective medium theory showing the base fluid in grey, and the nanoparticles depicted as black circles.  
Most derived equations use the effective medium theory as the core theory to base equations off of. The effective medium theory yields the best results when predicting dilute suspensions containing nanoparticles having sizes within the micrometer and millimeter range. Without the incorporation of dynamic elements within the nanofluid, the theoretical results usually yield an underestimate for the overall thermal conductivity enhancement.  
[PERSON]'s [70] theoretical work was the foundation behind most models for predicting the thermal conductivity of nanofluids. His model assumes that no particles are moving, and there are no interactions in-between the suspended nanoparticles. The [PERSON] model only takes into consideration the thermal conductivity of the base fluid, the thermal conductivity of the nanoparticle and the nanoparticle volume fraction. The model is most accurate for suspensions with a low volume fraction and nanoparticles within the micrometer to millimeter size range. However, the practicality behind nanoparticles on the micrometer and millimeter size lead to issues concerning the stability of the nanofluid with the rapid settling of the nanoparticles. With an unstable nanofluid, the nanoparticles can settle out of the mixture. [PERSON]'s model usually underestimates the thermal conductivity enhancement due to nanofluids, however is still used as a comparison due to the model's simplicity.  
A mathematical approach by [PERSON] [71] attempted to solve for the heat conduction for a two-phase material. His mathematical model studied two spheres and the heat transfer between them. The model concludes that when perfectly conducting spheres are densely packed together, the effective heat transfer approaches infinity logarithmically. Unfortunately, the heat conduction between two spheres cannot be expanded to three or more spheres.  
[PERSON]'s work was further expanded on by [PERSON] and [PERSON] [72] who considered a factor for the shape of nanoparticles. [PERSON] and [PERSON] introduced the term sphericity to describe how closely the nanoparticle's shape resembles that of a perfect sphere. The nanoparticle's shape only makes a significant impact when the thermal conductivity of the nanoparticle is significantly larger than the thermal conductivity of the base fluid. Regularly the shape of the nanoparticle has no major significance for the overall thermal conductivity when only using the effective medium theory. The only exception to the importance of the nanoparticle's shape is when the nanoparticle is highly porous.