# Thermal conductivity relationship with particle size in nanofluids

Is thermal conductivity of nano sized particles and bulk particles the same? If not, what's the difference and is there any relation between thermal conductivity and and particle size?

You should add more details to your question but since you added the fluid mechanics tag I'll assume you mean nanofluids. Yes, of course there is a difference between thermal conductivity of solid particles (nano) and base fluid. The addition of nano-sized particles to the main fluid has been found to enhance the thermophysical properties of the base fluid (the main concept of nanofluids) [a].

It was found that the thermal conductivity of the nanofluid depends on many factors [b]:

• Thermal conductivity of the base fluid and nanoparticles $k_f$ and $k_p$
• Volume fraction $\phi$
• Surface area (shape of the nanoparticles suspended in the base fluid)

The effective thermal conductivity $k_{eff}$ of the nanofluid can be calculated using Maxwell and Hamilton and Crosser equations.

Maxwell equation (Valid for a fluid having a lower thermal conductivity than solid nanoparticles, and you can see that the effective thermal conductivity depends only on both $k$ values of fluid and solid nanoparticles and volume fraction):

$$k_{eff} = k_f (1 + \frac{3(k_p - k_f)\phi}{(k_p - k_f)-(k_p - k_f)\phi})$$

Hamilton and Crosser equation: $$k_{eff} = k_f (\frac{k_p + k_f (n-1) + (n-1)(k_p - k_f)\phi}{k_p + k_f (n-1)- (k_p - k_f)\phi})$$

This equation is valid for non-spherical particles, and $n$ is the shape factor that is obtained experimentally for different materials, you may refer to this paper for more details.

[a] Vincenzo Bianco, Oronzio Manca, Sergio Nardini and K Vafai -Heat Transfer Enhancement With Nanofluids (2015).

[b] D. C. Hernandez Aita - Design and Optimization of Volumetric Solar Receivers based on Nanoparticles with Supercritical CO2 (2014).