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Transport in Porous Media

, Volume 11, Issue 3, pp 219–241 | Cite as

Darcy-Forchheimer natural, forced and mixed convection heat transfer in non-Newtonian power-law fluid-saturated porous media

  • A. V. Shenoy
Article

Abstract

The governing equation for Darcy-Forchheimer flow of non-Newtonian inelastic power-law fluid through porous media has been derived from first principles. Using this equation, the problem of Darcy-Forchheimer natural, forced, and mixed convection within the porous media saturated with a power-law fluid has been solved using the approximate integral method. It is observed that a similarity solution exists specifically for only the case of an isothermal vertical flat plate embedded in the porous media. The results based on the approximate method, when compared with existing exact solutions show an agreement of within a maximum error bound of 2.5%.

Key words

non-Newtonian fluids power-law fluids Darcy-Forchheimer flow natural convection forced convection mixed convection 

Nomenclature

A

cross-sectional area

bi

coefficient in the chosen temperature profile

B1

coefficient in the profile for the dimensionless boundary layer thickness

C

coefficient in the modified Forchheimer term for power-law fluids

C1

coefficient in the Oseen approximation which depends essentially on pore geometry

Ci

coefficient depending essentially on pore geometry

CD

drag coefficient

Ct

coefficient in the expression forK*

d

particle diameter (for irregular shaped particles, it is characteristic length for average-size particle)

fp

resistance or drag on a single particle

FR

total resistance to flow offered byN particles in the porous media

g

acceleration due to gravity

gx

component of the acceleration due to gravity in thex-direction

\(Gr_{K^* }\)

Grashof number based on permeability for power-law fluids

K

intrinsic permeability of the porous media

K*

modified permeability of the porous media for flow of power-law fluids

lc

characteristic length

m

exponent in the gravity field

n

power-law index of the inelastic non-Newtonian fluid

N

total number of particles

Nux,D,F

local Nusselt number for Darcy forced convection flow

Nux,D-F,F

local Nusselt number for Darcy-Forchheimer forced convection flow

Nux,D,M

local Nusselt number for Darcy mixed convection flow

Nux,D-F,M

local Nusselt number for Darcy-Forchheimer mixed convection flow

Nux,D,N

local Nusselt number for Darcy natural convection flow

Nux,D-F,N

local Nusselt number for Darcy-Forchheimer natural convection flow

\(\bar p\)

pressure

p

exponent in the wall temperature variation

Pec

characteristic Péclet number

Pex

local Péclet number for forced convection flow

Pe′x

modified local Péclet number for mixed convection flow

Rac

characteristic Rayleigh number

Rax

local Rayleigh number for Darcy natural convection flow

Ra′x

local Rayleigh number for Darcy-Forchheimer natural convection flow

Re

convectional Reynolds number for power-law fluids

\(\operatorname{Re} _{K^* }\)

Reynolds number based on permeability for power-law fluids

T

temperature

Te

ambient constant temperature

Tw,ref

constant reference wall surface temperature

Tw(X)

variable wall surface temperature

δTw

temperature difference equal toTw,refTe

T1

term in the Darcy-Forchheimer natural convection regime for Newtonian fluids

T2

term in the Darcy-Forchheimer natural convection regime for non-Newtonian fluids (first approximation)

TN

term in the Darcy/Forchheimer natural convection regime for non-Newtonian fluids (second approximation)

u

Darcian or superficial velocity

u1

dimensionless velocity profile

ue

external forced convection flow velocity

us

seepage velocity (local average velocity of flow around the particle)

uw

wall slip velocity

UcM

characteristic velocity for mixed convection

UcN

characteristic velocity for natural convection

x, y

boundary-layer coordinates

x1,y1

dimensionless boundary layer coordinates

X

coefficient which is a function of flow behaviour indexn for power-law fluids

α

effective thermal diffusivity of the porous medium

α

shape factor which takes a value ofΜ/4 for spheres

Β

shape factor which takes a value ofΜ/6 for spheres

Β0

expansion coefficient of the fluid

δT

boundary-layer thickness

δT1

dimensionless boundary layer thickness

ε

porosity of the medium

η

similarity variable

θ

dimensionless temperature difference

λ

coefficient which is a function of the geometry of the porous media (it takes a value of 3Μ for a single sphere in an infinite fluid)

Μ0

viscosity of Newtonian fluid

Μ*

fluid consistency of the inelastic non-Newtonian power-law fluid

ξ

constant equal toX(2ε2−nλ′)/α

ρ

density of the fluid

Φ

dimensionless wall temperature difference

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Copyright information

© Kluwer Academic Publishers 1993

Authors and Affiliations

  • A. V. Shenoy
    • 1
  1. 1.Department of Energy and Mechanical EngineeringShizuoka UniversityHamamatsuJapan

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