Heat transfer in granular materials: effects of nonlinear heat conduction and viscous dissipation

Abstract

In this paper, we study the heat transfer in a one‐dimensional fully developed flow of granular materials down a heated inclined plane. For the heat flux vector, we use a recently derived constitutive equation that reflects the dependence of the heat flux vector on the temperature gradient, the density gradient, and the velocity gradient in an appropriate frame invariant formulation. We use two different boundary conditions at the inclined surface: a constant temperature boundary condition and an adiabatic condition. A parametric study is performed to examine the effects of the material dimensionless parameters. The derived governing equations are coupled nonlinear second‐order ordinary differential equations, which are solved numerically, and the results are shown for the temperature, volume fraction, and velocity profiles. Copyright © 2013 John Wiley & Sons, Ltd.

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

“…Through the homogenization process, we can think of the granular media as a single phase continuum [20]. If there are no chemical and electromagnetic effects, the basic governing equations are the conservation of mass:…”

“…Note that (9) has certain similarities to the stress tensor for the Korteweg fluids, where density gradient plays an important role (see [47,48]). Additional details along with some applications of this model are given in [13,20,49].…”

“…Yang et al [20] used (9) and (15) to study the heat transfer in granular materials. However, they did not consider the Clausius-Duhem inequality.…”

Effects of Shear Dependent Viscosity and Variable Thermal Conductivity on the Flow and Heat Transfer in a Slurry

“…It is noted that Wang [61] also derived a general expression for the heat flux vector for a fluid where heat convection is also important; he assumed that = ( , ∇ , , , ) where f is a vector-valued function, the temperature, ∇ is the gradient of temperature, v the velocity vector, L its gradient, and X designates other scalar-valued thermophysical parameters. A simplified form of Equation (13) was used in a recent study by Yang et al [62]. In general, this method is very difficult (If we assume that q can depend explicitly on the temperature gradient, concentration gradient, etc., then clearly the problem would become more non-linear.…”

“…When = 2 , the above equation reduces to the usual Fourier's law) as it requires knowledge of the various coefficients in Equation (14) and in the absence of many physical experiments, the only option is to do a parametric/numerical study. This was the case in [62]. In the present study, we take a different, and perhaps a more practical approach, namely we use the results of experiments to obtain correlations for the shear-dependent thermal conductivity.…”