On the heat flux vector for flowing granular materials—Part I: effective thermal conductivity and background

Abstract

SUMMARYHeat transfer plays a major role in the processing of many particulate materials. The heat ux vector is commonly modelled by the Fourier's law of heat conduction and for complex materials such as nonlinear uids, porous media, or granular materials, the coe cient 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 ux vector. We will also discuss the concept of e ective 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 ux vector for a (single phase) owing granular medium. Standard methods in continuum mechanics such as representation theorems and homogenization techniques are used. It is shown that the heat ux vector in addition to being proportional to the temperature gradient (the Fourier's 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 ux vector which should be studied; obtaining or proposing generalized form of the thermal conductivity is not always appropriate or su cient.

Entropy Analysis for a Nonlinear Fluid with a Nonlinear Heat Flux Vector

“…This is simply the well-known constraint anticipated by [18,19], that the thermal conductivity a 1 ≤ 0. In the general case, the thermal conductivity is a second-order tensor.…”

“…For q, we follow Massoudi [18,19]. From mechanics-based arguments, he suggested that the heat flux vector depends not only on the temperature gradient, but also on the motion and the density (or volume fraction) gradient.…”

“…Massoudi [18,19] set a 1 = −k, where k is the thermal conductivity, m = ∇φ, and n = ∇θ. Equation (12) can then be re-written as:…”

“…Section 3 discusses the constitutive equations used in the analysis. For the stress tensor, we use the equation derived by Rajagopal and Massoudi [12], and for the heat flux vector we use the constitutive equations derived by Massoudi [18,19] and consider the consequences of the entropy inequality (the Clausius-Duhem inequality) on the material properties. Section 4 develops the constraints imposed by the entropy inequality.…”

Effects of Anisotropic Thermal Conductivity and Lorentz Force on the Flow and Heat Transfer of a Ferro-Nanofluid in a Magnetic Field

“…For complex materials, K can also depend on the concentration, the temperature, the shear rate, etc. (see [72,73]). When there is no magnetic field or no nanoparticles present in the fluid, the second order tensor, K, reduces to a scalar and Equation (10) becomes, q = −k∇θ (11) where k is the isotropic thermal conductivity.…”