## 3 Nanofluid Thermal Conductivity Models
### Brownian Model  
Brownian motion describes the random, chaotic movements of particles within a fluid. This chaotic movement within the fluid is due to the continuous collisions between nanoparticles and the molecules of the base fluid. This chaotic movement within the nanofluid can induce localized convection, which can further enhance the heat transfer. Brownian motion is more prominent with smaller nanoparticles, where the collisions have a larger effect on the nanoparticles. The Brownian motion can lead to increased aggregation of nanoparticles (see Figure 30) [79; 80].  
[PERSON] et al. [81] combined the effects of convection and conduction in their thermal model. The Brownian motion in nanofluids can lead to convection like effects, which not only mix the particles but also can break apart formed clusters. Their model has the flexibility to allow for larger particle sizes that can be simulated through [PERSON]'s model where conduction is the driving mechanism. When the particle size decreases, other forces such as Brownian motion dominate. Kinetic theory was linked to temperature in addition to stationary heat transfer mechanics by [PERSON] et al. [82]. The stationary model was based on [PERSON]'s law of diffusion through the nanoparticles and the base fluid. The Stokes-Einstein formula was used to calculate the average particle velocity based on the viscosity of the base fluid.  
Through an order of magnitude analysis, [PERSON] et al. [83] concluded that convection induced by Brownian motion was the driving factor in the enhancement of the thermal conductivity of nanofluids. Due to the small size of nanoparticles, surface forces are predominant and use different energy modes for heat transfer within the nanofluid. It was theorized that the interparticle potential between nanoparticles can result in a phonon-like form of heat transfer. Phonon energy is where energy is passed as vibrational energy that oscillates atoms within a crystal. The interfacial layer aids phonon-like heat transfer, which has a significant influence on the thermal conductivity of nanofluids.  
One argument against the importance of Brownian motion was studied by [PERSON] et al. [80]. It was determined that Brownian motion does not solely lead to significant thermal conductivity enhancements due to an increase in temperature. It was suggested that the temperature of the nanofluid plays a more significant role. Increased temperature leads to a decrease in the base fluid's viscosity, thus resulting in reduced agglomeration and increased Brownian motion.  
[PERSON] and [PERSON] [84] looked into several different methods of heat transfer within a nanofluid. The different methods being: base fluid molecule collisions, collisions between nanoparticles, thermal diffusion of the nanoparticles, and finally the collisions between nanoparticles and base fluid molecules. It was then determined that heat transfer due to Brownian motion particle collisions with the base fluid molecules, was an order of magnitude lower and is negligible. The nanofluid does not have an overall bulk flow, and exhibits conduction due to the constant particle interaction. A combination of  
Figure 30: An illustration of the random motion of a singular nanoparticle caused by Brownian motion.  
Brownian motion and aggregation structures were considered by [PERSON] et al. [85] for the development of their model. The cluster fractal structures along with the radius of gyration had significant effects on the effective thermal conductivity of the nanofluid. The radius of gyration is the distribution of the nanoparticles around an axis of the cluster. When there was a reduction in the radius of gyration, there was an increase in the thermal conductivity. With reduced cluster size, the particles can travel faster and farther, which increases the effective thermal conductivity.