The thermal entrance region in a porous medium saturated by a nanofluid: analysis of the Brinkman's model

The Darcy-Graetz problem for a channel filled by a nanofluid saturated porous medium is studied. The flow is assumed to be fully developed and described through Brinkman's model. For the model of the nanofluid, both thermophoresis and Brownian diffusion are taken into account. After an adiabatic preparation region, a boundary temperature linearly varying with the longitudinal coordinate is prescribed. A study of the thermal behaviour of the nanofluid is performed by solving numerically the fully-elliptic coupled equations, with reference both to the thermal entrance region and to the fully developed region. With reference to the fully developed region the solution has been obtained analytically, while for the thermal entrance region it has been obtained numerically, by a Galerkin finite element method implemented through the software package Comsol Multiphysics (© Comsol, Inc.). The analysis shows that, for physically interesting values of the Peclet number, the concentration field depends very weakly on the temperature distribution, for any given value assumed by the Darcy number. Indeed, since the effects of thermophoresis and Brownian diffusion are negligible, the homogeneous model could be employed effectively.