On the heat flux vector for flowing granular materialspart II: derivation and special cases
Abstract
Heat transfer plays a major role in the processing of many particulate materials. The heat flux vector is commonly modelled by the Fourier's law of heat conduction and for complex materials such as nonlinear 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 frameinvariant 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 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. Themore »
 Authors:
 Publication Date:
 Research Org.:
 National Energy Technology Laboratory (NETL), Pittsburgh, PA, Morgantown, WV, and Albany, OR
 Sponsoring Org.:
 USDOE  Office of Fossil Energy (FE)
 OSTI Identifier:
 938571
 Report Number(s):
 DOE/NETLIR2006164; NETLTPR1575
Journal ID: ISSN 01704214; TRN: US200820%%87
 DOE Contract Number:
 None cited
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Mathematical Methods in the Applied Sciences; Journal Volume: 29; Journal Issue: 13
 Country of Publication:
 United States
 Language:
 English
 Subject:
 42 ENGINEERING; GRANULAR MATERIALS; HEAT FLUX; HEAT TRANSFER; PARTICULATES; SOLIDS FLOW; FLOW MODELS; TEMPERATURE GRADIENTS; THERMAL CONDUCTIVITY; granular materials; heat flux vector; constitutive relation; conduction; Fourier’s law
Citation Formats
Massoudi, Mehrdad. On the heat flux vector for flowing granular materialspart II: derivation and special cases. United States: N. p., 2006.
Web. doi:10.1002/mma.745.
Massoudi, Mehrdad. On the heat flux vector for flowing granular materialspart II: derivation and special cases. United States. doi:10.1002/mma.745.
Massoudi, Mehrdad. 2006.
"On the heat flux vector for flowing granular materialspart II: derivation and special cases". United States.
doi:10.1002/mma.745.
@article{osti_938571,
title = {On the heat flux vector for flowing granular materialspart II: derivation and special cases},
author = {Massoudi, Mehrdad},
abstractNote = {Heat transfer plays a major role in the processing of many particulate materials. The heat flux vector is commonly modelled by the Fourier's law of heat conduction and for complex materials such as nonlinear 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 frameinvariant 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 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 nonlinear 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.},
doi = {10.1002/mma.745},
journal = {Mathematical Methods in the Applied Sciences},
number = 13,
volume = 29,
place = {United States},
year = 2006,
month = 9
}

Heat transfer plays a major role in the processing of many particulate materials. The heat flux vector is commonly modelled by the Fourier’s law of heat conduction and for complex materials such as nonlinear fluids, porous media, or granular materials, the coeffcient 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.more »

Thermal performance of multilayer insulation; Part 1, Derivation of a predictionbased heatflux equation
How to reduce the heat loss at cryogenic temperatures in superconducting magnets used for MRI (Magnetic Resonance Imaging), magnetically levitated trains, nuclear fusion, etc., is an important subject. The topic is important in providing a superconducting magnet with highperformance thermal insulation. A predictionbased equation for heat flux through a multilayer insulator was derived from comparison of experimental results between room temperature and liquidnitrogen temperature. The employed multilayer insulator was a laminated material with a polyester net inserted between aluminized Mylar films. The prediction equation consists of one thermalradiation and two thermalconduction terms. The first conduction term is that of ordinarymore »