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

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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%.

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Abbreviations

A :

cross-sectional area

b i :

coefficient in the chosen temperature profile

B 1 :

coefficient in the profile for the dimensionless boundary layer thickness

C :

coefficient in the modified Forchheimer term for power-law fluids

C 1 :

coefficient in the Oseen approximation which depends essentially on pore geometry

C i :

coefficient depending essentially on pore geometry

C D :

drag coefficient

C t :

coefficient in the expression forK *

d :

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

f p :

resistance or drag on a single particle

F R :

total resistance to flow offered byN particles in the porous media

g :

acceleration due to gravity

g x :

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

l c :

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

Pe c :

characteristic Péclet number

Pe x :

local Péclet number for forced convection flow

Pe′ x :

modified local Péclet number for mixed convection flow

Ra c :

characteristic Rayleigh number

Ra x :

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

T e :

ambient constant temperature

T w,ref :

constant reference wall surface temperature

T w(X):

variable wall surface temperature

δT w :

temperature difference equal toT w,ref−T e

T 1 :

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

T 2 :

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

T N :

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

u :

Darcian or superficial velocity

u 1 :

dimensionless velocity profile

u e :

external forced convection flow velocity

u s :

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

u w :

wall slip velocity

U c M :

characteristic velocity for mixed convection

U c N :

characteristic velocity for natural convection

x, y :

boundary-layer coordinates

x 1,y 1 :

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

Β 0 :

expansion coefficient of the fluid

δ T :

boundary-layer thickness

δ T 1 :

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)

Μ 0 :

viscosity of Newtonian fluid

Μ * :

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

ξ :

constant equal toX(2ε 2−n λ′)/α′

ρ :

density of the fluid

Φ :

dimensionless wall temperature difference

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  1. Department of Energy and Mechanical Engineering, Shizuoka University, 3-5-1 Johoku, 432, Hamamatsu, Japan

    A. V. Shenoy

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Shenoy, A.V. Darcy-Forchheimer natural, forced and mixed convection heat transfer in non-Newtonian power-law fluid-saturated porous media. Transp Porous Med 11, 219–241 (1993). https://doi.org/10.1007/BF00614813

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  • DOI: https://doi.org/10.1007/BF00614813

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