On the heat flux vector for flowing granular materials - part II: derivation and special cases

Heat transfer plays a major role in the processing of many particulate materials. The heat flux vector is commonly modelled by the Fourierʹs law of heat conduction and for complex materials such as non-linear fluids, porous media, or granular materials, the coefficient of thermal conductivity is generalized by assuming that it would depend on a host of material and kinematical parameters such as temperature, shear rate, porosity or concentration, etc. In Part I, we will give a brief review of the basic equations of thermodynamics and heat transfer to indicate the importance of the modelling of the heat flux vector. We will also discuss the concept of effective thermal conductivity (ETC) in granular and porous media. In Part II, we propose and subsequently derive a properly frame-invariant constitutive relationship for the heat flux vector for a (single phase) flowing granular medium. Standard methods in continuum mechanics such as representation theorems and homogenization techniques are used. It is shown that the heat flux vector in addition to being proportional to the temperature gradient (the Fourierʹs law), could also depend on the gradient of density (or volume fraction), and D (the symmetric part of the velocity gradient) in an appropriate manner. The emphasis in this paper is on the idea that for complex non-linear materials it is the heat flux vector which should be studied; obtaining or proposing generalized form of the thermal conductivity is not always appropriate or sufficient.