Modeling Granular Materials as Compressible Non-Linear Fluids: Heat Transfer Boundary Value Problems
Abstract
We discuss three boundary value problems in the flow and heat transfer analysis in flowing granular materials: (i) the flow down an inclined plane with radiation effects at the free surface; (ii) the natural convection flow between two heated vertical walls; (iii) the shearing motion between two horizontal flat plates with heat conduction. It is assumed that the material behaves like a continuum, similar to a compressible nonlinear fluid where the effects of density gradients are incorporated in the stress tensor. For a fully developed flow the equations are simplified to a system of three nonlinear ordinary differential equations. The equations are made dimensionless and a parametric study is performed where the effects of various dimensionless numbers representing the effects of heat conduction, viscous dissipation, radiation, and so forth are presented.
- Authors:
- Publication Date:
- Research Org.:
- National Energy Technology Laboratory (NETL), Pittsburgh, PA, Morgantown, WV, and Albany, OR (United States)
- Sponsoring Org.:
- USDOE - Office of Fossil Energy (FE)
- OSTI Identifier:
- 912705
- Report Number(s):
- DOE/NETL-IR-2007-031
TRN: US200801%%981
- DOE Contract Number:
- None cited
- Resource Type:
- Journal Article
- Journal Name:
- Mathematical Problems in Engineering
- Additional Journal Information:
- Journal Volume: 2006; Journal Issue: aritcle ID 56046; Related Information: This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
- Publisher:
- Hindawi Publishing Corporation
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 99 GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; BOUNDARY-VALUE PROBLEMS; DIFFERENTIAL EQUATIONS; DIMENSIONLESS NUMBERS; GRANULAR MATERIALS; HEAT TRANSFER; SOLIDS FLOW; NONLINEAR PROBLEMS
Citation Formats
Massoudi, M C, and Tran, P X. Modeling Granular Materials as Compressible Non-Linear Fluids: Heat Transfer Boundary Value Problems. United States: N. p., 2006.
Web. doi:10.1155/MPE/2006/56046.
Massoudi, M C, & Tran, P X. Modeling Granular Materials as Compressible Non-Linear Fluids: Heat Transfer Boundary Value Problems. United States. https://doi.org/10.1155/MPE/2006/56046
Massoudi, M C, and Tran, P X. 2006.
"Modeling Granular Materials as Compressible Non-Linear Fluids: Heat Transfer Boundary Value Problems". United States. https://doi.org/10.1155/MPE/2006/56046. https://www.osti.gov/servlets/purl/912705.
@article{osti_912705,
title = {Modeling Granular Materials as Compressible Non-Linear Fluids: Heat Transfer Boundary Value Problems},
author = {Massoudi, M C and Tran, P X},
abstractNote = {We discuss three boundary value problems in the flow and heat transfer analysis in flowing granular materials: (i) the flow down an inclined plane with radiation effects at the free surface; (ii) the natural convection flow between two heated vertical walls; (iii) the shearing motion between two horizontal flat plates with heat conduction. It is assumed that the material behaves like a continuum, similar to a compressible nonlinear fluid where the effects of density gradients are incorporated in the stress tensor. For a fully developed flow the equations are simplified to a system of three nonlinear ordinary differential equations. The equations are made dimensionless and a parametric study is performed where the effects of various dimensionless numbers representing the effects of heat conduction, viscous dissipation, radiation, and so forth are presented.},
doi = {10.1155/MPE/2006/56046},
url = {https://www.osti.gov/biblio/912705},
journal = {Mathematical Problems in Engineering},
number = aritcle ID 56046,
volume = 2006,
place = {United States},
year = {Sun Jan 01 00:00:00 EST 2006},
month = {Sun Jan 01 00:00:00 EST 2006}
}
Works referenced in this record:
On the fully developed flow of a dense particulate mixture in a pipe
journal, October 1999
- Massoudi, M.; Rajagopal, K. R.; Phuoc, T. X.
- Powder Technology, Vol. 104, Issue 3
R APID G RANULAR F LOWS
journal, January 2003
- Goldhirsch, Isaac
- Annual Review of Fluid Mechanics, Vol. 35, Issue 1
Constitutive relations that imply a generalized Mohr-Coulomb criterion
journal, March 1974
- Cowin, S. C.
- Acta Mechanica, Vol. 20, Issue 1-2
Flow of a non-Newtonian fluid between heated parallel plates
journal, January 1985
- Szeri, A. Z.; Rajagopal, K. R.
- International Journal of Non-Linear Mechanics, Vol. 20, Issue 2
Flow and heat transfer due to natural convection in granular materials
journal, March 1999
- Massoudi, M.; Phuoc, T. X.
- International Journal of Non-Linear Mechanics, Vol. 34, Issue 2
Flow of granular materials between rotating disks
journal, November 1994
- Rajagopal, K. R.; Massoudi, M.; Wineman, A. S.
- Mechanics Research Communications, Vol. 21, Issue 6
A fluid mechanistic approach to slow, frictional flow of powders
journal, June 1997
- Tardos, Gabriel I.
- Powder Technology, Vol. 92, Issue 1
Combined Conduction and Radiation heat Transfer in Plane-Parallel Packed beds with Variable Porosity
journal, March 1999
- Wu, Jhy-Wen; Chu, Hsin-Sen
- Journal of Quantitative Spectroscopy and Radiative Transfer, Vol. 61, Issue 4
On the heat flux vector for flowing granular materials—Part I: effective thermal conductivity and background
journal, January 2006
- Massoudi, Mehrdad
- Mathematical Methods in the Applied Sciences, Vol. 29, Issue 13
Reflections on the mechanics of granular matter
journal, December 1998
- de Gennes, P. G.
- Physica A: Statistical Mechanics and its Applications, Vol. 261, Issue 3-4
On the heat flux vector for flowing granular materials—part II: derivation and special cases
journal, January 2006
- Massoudi, Mehrdad
- Mathematical Methods in the Applied Sciences, Vol. 29, Issue 13
A continuum model for granular materials: Considering dilatancy and the Mohr-Coulomb criterion
journal, March 2001
- Massoudi, M.; Mehrabadi, M. M.
- Acta Mechanica, Vol. 152, Issue 1-4
Stress calculations for assemblies of inelastic speres in uniform shear
journal, November 1986
- Walton, O. R.; Braun, R. L.
- Acta Mechanica, Vol. 63, Issue 1-4
Experiments on the conductivity of suspensions of ionically-conductive spheres: Experiments on the Conductivity of Suspensions of Ionically-Conductive Spheres
journal, March 1990
- Bashir, Y. M.; Goddard, J. D.
- AIChE Journal, Vol. 36, Issue 3
Plane steady shear flow of a cohesionless granular material down an inclined plane: A model for flow avalanches Part I: Theory
journal, November 1986
- Hutter, K.; Szidarovszky, F.; Yakowitz, S.
- Acta Mechanica, Vol. 63, Issue 1-4
Instability in the evolution equations describing incompressible granular flow
journal, January 1987
- Schaeffer, David G.
- Journal of Differential Equations, Vol. 66, Issue 1
Temperature profiles in composite plates subject to time-dependent complex boundary conditions
journal, February 2003
- Miller, J. R.; Weaver, P. M.
- Composite Structures, Vol. 59, Issue 2
Flow of a fluid—solid mixture between flat plates
journal, January 1991
- Johnson, G.; Massoudi, M.; Rajagopal, K. R.
- Chemical Engineering Science, Vol. 46, Issue 7
A theory for the flow of granular materials
journal, February 1974
- Cowin, S. C.
- Powder Technology, Vol. 9, Issue 2-3
Plane steady shear flow of a cohesionless granular material down an inclined plane. A model for flow avalanches Part II: numerical results
journal, January 1987
- Hutter, K.; Szidarovsky, F.; Yakowitz, S.
- Acta Mechanica, Vol. 65, Issue 1-4
Natural convection flow of a non-Newtonian fluid between two vertical flat plates
journal, March 1985
- Rajagopal, K. R.; Na, T. Y.
- Acta Mechanica, Vol. 54, Issue 3-4
Constitutive relations for compressible granular materials using non-Newtonian fluid mechanics
journal, October 1998
- Elaskar, Sergio A.; Godoy, Luas A.
- International Journal of Mechanical Sciences, Vol. 40, Issue 10
Dissipative materials as models of thixotropy and plasticity
journal, January 1984
- Goddard, J. D.
- Journal of Non-Newtonian Fluid Mechanics, Vol. 14
On the entropy inequality
journal, January 1967
- Müller, Ingo
- Archive for Rational Mechanics and Analysis, Vol. 26, Issue 2
The Meaning of Viscometry in Fluid Dynamics
journal, January 1974
- Truesdell, C.
- Annual Review of Fluid Mechanics, Vol. 6, Issue 1
A continuum theory for granular materials
journal, January 1972
- Goodman, M. A.; Cowin, S. C.
- Archive for Rational Mechanics and Analysis, Vol. 44, Issue 4
Unsteady Compressible Flow of Granular Materials †
journal, April 1999
- Astarita, Tommaso; Ocone, Raffaella
- Industrial & Engineering Chemistry Research, Vol. 38, Issue 4
A Nonlinear Constitutive Model for Granular Materials: Application to Gravity Flow
journal, June 1982
- McTigue, D. F.
- Journal of Applied Mechanics, Vol. 49, Issue 2
The Flow of Cohesionless Grains in Fluids
journal, December 1956
- Bagnold, R. A.
- Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 249, Issue 964
Convective heat transfer to rapidly flowing, granular materials
journal, August 1986
- Patton, J. S.; Sabersky, R. H.; Brennen, C. E.
- International Journal of Heat and Mass Transfer, Vol. 29, Issue 8
Flow of granular materials down a vertical pipe
journal, January 1994
- Gudhe, R.; Yalamanchili, R. C.; Massoudi, M.
- International Journal of Non-Linear Mechanics, Vol. 29, Issue 1
On the conduction/radiation heat transfer problem in a body with wavelength-dependent properties
journal, September 2004
- Saldanha da Gama, Rogério Martins
- Applied Mathematical Modelling, Vol. 28, Issue 9
Two problems in the gravity flow of granular materials
journal, January 1971
- Goodman, M. A.; Cowin, S. C.
- Journal of Fluid Mechanics, Vol. 45, Issue 02
Application of thermomechanical principles to the modelling of geotechnical materials
journal, September 1997
- Collins, I. F.; Houlsby, G. T.
- Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, Vol. 453, Issue 1964
Effects of heat absorption and thermal radiation on heat transfer in a fluid–particle flow past a surface in the presence of a gravity field
journal, May 2000
- Chamkha, Ali J.
- International Journal of Thermal Sciences, Vol. 39, Issue 5
A Mathematical Theory of Dilatancy
journal, July 1945
- Reiner, M.
- American Journal of Mathematics, Vol. 67, Issue 3
Gravity flow of cohesionless granular materials in chutes and channels
journal, May 1979
- Savage, Stuart B.
- Journal of Fluid Mechanics, Vol. 92, Issue 1
Fully developed flow of granular materials down a heated inclined plane
journal, March 1994
- Gudhe, R.; Rajagopal, K. R.; Massoudi, M.
- Acta Mechanica, Vol. 103, Issue 1-4
On flows of granular materials
journal, January 1994
- Hutter, K.; Rajagopal, K. R.
- Continuum Mechanics and Thermodynamics, Vol. 6, Issue 2
Conduction Through a Random Suspension of Spheres
journal, November 1973
- Jeffrey, D. J.
- Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 335, Issue 1602