Advertisement

Wärme - und Stoffübertragung

, Volume 29, Issue 3, pp 185–193 | Cite as

Natural convection of non-Newtonian fluids in a horizontal porous layer

  • B. Amari
  • P. Vasseur
  • E. Bilgen
Originalarbeiten

Abstract

The buoyancy-induced flows of non-Newtonian fluids in a horizontal fluid saturated porous layer is studied analytically and numerically using the power-law model to characterize the non-Newtonian fluid behavior. A constant heat flux is applied for heating and cooling the two opposing walls of the layer while the other two walls are insulated. On the basis of a modified Darcy equation the problem is solved analytically, in the limit of a thin layer, using a parallel flow approximation and an integral form of the energy equation. Solutions for the flow and temperature fields, and Nusselt numbers are obtained explicitly in terms of the modified Rayleigh numberR and the power-law indexn. A numerical study of the same phenomenon, obtained by solving the complete system of governing equations, is also conducted. A good agreement is found between the analytical prediction and the numerical simulation.

Keywords

Heat Flux Nusselt Number Natural Convection Porous Layer Parallel Flow 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Nomenclature

A

aspect ratio of the cavity,L′/H′

a, b

bottom heating:a=1,b=0; side wall heating:a=0,b=1

C

dimensionless temperature gradient along thex direction

g

acceleration due to gravity

h

the consistency index

H′

thickness of the cavity

k

thermal conductivity of fluid-saturated porous medium

K

permeability of the porous medium

L′

length of the cavity

n

power-law index

Nu

Nusselt number, Eq. (27)

p′

pressure

q′

constant heat flux

R

modified Rayleigh number,KϱgβΔT′(H′/α) n /ε

Ra

Rayleigh number for a Newtonian-fluid,KϱgβΔT′H′/αμ

T′

temperature

T′0

reference temperature at the geometric center of the cavity

\(\vec V'\)

superficial velocity

u′

superficial velocity in thex′ direction

v′

superficial velocity in they′ direction

x′

horizontal coordinate

y′

vertical coordinate

Greek symbols

α

effective thermal diffusivity

β

coefficient of thermal expansion

ϑ

dimensionless temperature, Eq. (15)

μ

dynamic viscosity for a Newtonian fluid

μ′a

apparent viscosity for a power-law fluid, Eq. (2)

ϱ

density

Ψ′

stream function

ø

porosity of the porous medium

ε

parameter in power-law model, Eq. (3)

Superscript

dimensional quantities

Subscrints

max

maximum value

0

condition at the origin of the coordinate system

Natürliche Konvektion nicht-Newtonscher Fluide in einer horizontalen porösen Schicht

Zusammenfassung

Die durch Auftriebskräfte induzierten Strömmungen nicht-Newtonscher Fluide in einer horizontalen, flüssigkeitsgetränkten porösen Schicht werden analytisch und numerisch unter Verwendung des Potenzgesetz-Modells für das nicht-Newtonsche Fluidverhalten untersucht. Von einer beheizten Wand fließt ein konstanter Wärmestrom an die gekühlte gegenüberliegende, während die beiden übrigen Wände adiabat sind. Das Problem wird auf der Basis einer modifizierten Darcy-Gleichung analytisch gelöst, und zwar für den Grenzfall geringer Schichtdicke, wobei näherungsweise Parallelströmung unterstellt wird und die Energiegleichung in Integralform Verwendung findet. Lösungen für das Strömungs- und Temperaturfeld sowie Nusselt-Zahlen werden in expliziter Form als Funktionen der modifizierten Rayleigh-ZahlR und des Potenzgesetz-Indexn erhalten. Ferner wird eine numerische Untersuchung des gleichen Problems unter Lösung des vollständigen Systems der bestimmenden Gleichungen durchgeführt. Analytische Vorausberechnung und numerische Simulation stimmen gut überein.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Nield, D. A.; Bejan, A.: Convection in porous media. Springer Verlag 1992Google Scholar
  2. 2.
    Chen, H. T.; Chen, C. K.: Natural convection of non-Newtonian fluids about a horizontal surface in a porous medium. J. Energy Ress. Tech. 109 (1987) 119–123Google Scholar
  3. 3.
    Christopher, R. V.; Middleman, S.: Power-law flow through a packed tube. Ind. Eng. Chem. Fundls. 4 (1965) 422–426Google Scholar
  4. 4.
    Dharmadhikari, R. V.; Kale, D. D.: Flow of non-Newtonian fluids through porous media. Chem. Eng. Sci. 40 (1985) 527–529Google Scholar
  5. 5.
    Chen, H. T.; Chen, C. K.: Free convection flow of non-Newtonian fluids along a vertical plate embedded in a porous medium. J. Heat Transfer 110 (1988a) 257–260Google Scholar
  6. 6.
    Chen, H. T.; Chen, C. K.: Natural convection of a non-Newtonian fluid about a horizontal cylinder and a sphere in a porous medium. Int. Comm. in Heat and Mass Transfer 15 (1988 b) 605–614Google Scholar
  7. 7.
    Nakayama, A.; Koyama, H.: Buoyancy-induced flows of non-Newtonian fluids over a non-isothermal body of arbitrary shape in a fluid-saturated porous medium. Applied Scientific Research 48 (1991) 55–70Google Scholar
  8. 8.
    Pascal, H.: Rheological behavior effect of non-Newtonian fluids on steady and unsteady flow through porous media. In. J. Numerical and Analytical Methods in Geomechanics 7 (1983) 207–224Google Scholar
  9. 9.
    Pascal, H.: Rheological effects of non-Newtonian behavior of displacing fluids on stability of a moving interface in radial oil displacement mechanism in porous media. Int. J. Engin. Science 24 (1986) 1465–1476Google Scholar
  10. 10.
    Pascal, H.; Pascal, J. P.: Nonlinear effect of non-Newtonian fluids on natural convection in a porous medium. Physical 40 (1989) 393–402Google Scholar
  11. 11.
    Cormack, D. E.; Leal, L. G.; Imberger, J.: Natural convection in a shallow cavity with differentially heated end walls. Part I, Asymptotic theory. J. Fluid Mech. 65 (1974) 209–230Google Scholar
  12. 12.
    Vasseur, P.; Satish, M. G.; Robillard, L.: Natural convection in a thin, inclined, porous layer exposed to a constant heat flux. Int. J. Heat Mass Transfer 30 (1988) 537–549Google Scholar
  13. 13.
    Vasseur, P.; Wang, C.; Sen, M.: The Brinkman model for natural convection in shallow porous cavity with uniform heat flux. Num. Heat Transfer, Part A, 15 (1989) 221–242Google Scholar
  14. 14.
    Bejan, A.: The boundary layer regime in a porous layer with uniform heat flux from the side. Int. J. Heat Mass Transfer 100 (1978) 191–198Google Scholar

Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • B. Amari
    • 1
  • P. Vasseur
    • 1
  • E. Bilgen
    • 1
  1. 1.Ecole PolytechniqueMontrealCanada

Personalised recommendations