A new general fractional-order derivataive with Rabotnov fractional-exponential kernel applied to model the anomalous heat transfer

In this paper, we consider a general fractional-order derivataive of the Liouville-Caputo type with the non-singular kernel of the Rabotnov fractional-exponential function for the first time. A new general fractional-order derivataive heat transfer model is discussed in detail. The general fractional-order derivataive formula is a new mathematical tool proposed to model the anomalous behaviors in complex and power-law phenomena.

Theoretical Conclusions About the Claims of Anomalous Heat Transfer Enhancement Associated With Nanofluids

A theoretical answer to the controversial issue on the anomalous convective heat transfer in nanofluids has been provided, exploiting the Buongiorno model for convective heat transfer in nanofluids with modifications to fully account for the effects of nanoparticle volume fraction distributions on the continuity, momentum and energy equations. Firstly, a set of exact solutions have been obtained for hydrodynamically and thermally fully developed laminar nanofluid forced convection flows in channels and tubes, subject to constant heat flux. From the solutions, it has been concluded that the anomalous heat transfer rate, exceeding the rate expected from the increase in thermal conductivity, is possible in such cases as titania-water nanofluids in a channel, alumina-water nanofluids in a tube and also titania-water nanofluids in a tube. Moreover, the maximum Nusselt number based on the bulk mean nanofluid thermal conductivity is captured when the ratio of Brownian and thermophoretic diffusivities is around 0.5, which can be exploited for designing nanoparticles for high-energy carriers. Secondly, another set of exact solutions have been obtained for free convection in a vertical channel filled with a nanofluid, exploiting the Buongiorno model with nanoparticle volume fraction modifications. The effects of the bulk mean volume fraction of nanoparticles, the ratio of Brownian and thermophoretic diffusivities and the buoyancy ratio on both velocity and temperature profiles has been investigated in depth for the first time. The volume fraction of nanoparticles increases exponentially towards the cold wall, which makes the velocity and temperature gradients steeper near the hot wall than those near the cold wall. Unlike the case of forced convection, no anomalous heat transfer enhancement has been observed in this case of free convection, so that the Nussult number based on the thermal conductivity of nanofluid stays close to unity as in pure base fluid free convection.