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Transport in Porous Media

, Volume 76, Issue 3, pp 391–405 | Cite as

Mixed Convection Heat Transfer in the Annulus Between Two Concentric Vertical Cylinders Using Porous Layers

  • Mohammad Sebty Al Zahrani
  • Suhil KiwanEmail author
Article

Abstract

A numerical investigation of the steady-state, laminar, axi-symmetric, mixed convection heat transfer in the annulus between two concentric vertical cylinders using porous inserts is carried out. The inner cylinder is subjected to constant heat flux and the outer cylinder is insulated. A finite volume code is used to numerically solve the sets of governing equations. The Darcy–Brinkman–Forchheimer model along with Boussinesq approximation is used to solve the flow in the porous region. The Navier–Stokes equation is used to describe the flow in the clear flow region. The dependence of the average Nusselt number on several flow and geometric parameters is investigated. These include: convective parameter, λ, Darcy number, Da, thermal conductivity ratio, K r, and porous-insert thickness to gap ratio (H/D). It is found that, in general, the heat transfer enhances by the presence of porous layers of high thermal conductivity ratios. It is also found that there is a critical thermal conductivity ratio on which if the values of Kr are higher than the critical value the average Nusselt number starts to decrease. Also, it found that at low thermal conductivity ratio (K r ≈ 1) and for all values of λ the porous material acts as thermal insulation.

Keywords

Mixed convection Porous media Darcy–Brinkman–Forchheimer model Heat transfer Critical thermal conductivity ratio 

Nomenclature

CF

Forchheimer coefficient , \({C_{\rm F} ={1.75}/{\sqrt{150\phi ^{5}}}}\)

Cp

Specific heat (J kg−1 K−1)

D

Gap between the two concentric cylinders (r o  − r i) [m]

Da

Darcy number (K/D 2)

g

Gravity acceleration (9.81 m/s2)

H

Porous layer thickness (m)

\({\overline h}\)

Average heat transfer coefficient (W m−2 K−1)

hz

Local heat transfer coefficient (W m−2 K−1)

K

Permeability (m2)

kf

Thermal conductivity for fluid (W m−1 K−1)

Kr

Thermal conductivity ratio (k eff /k f)

L

Length of the cylinder (m)

\({\overline {{\rm Nu}_D }}\)

Average Nusselt number \(({\overline {h}D/k_{\rm f} )}\)

\({\overline {{\rm Nu}_{\rm m}}}\)

Modified average Nusselt number \(({\overline {h} L/k_{\rm f} )}\)

Nuz

Local Nusselt number (h z z/k f)

Pr

Prandtl number (υ/α)

q′′

Heat flux (W/m2)

R

Non-dimensional Radial coordinate (r/D)

Ra*

Modified Rayleigh number \(({g \beta q^{\prime\prime}D^{4}/k_{\rm f} \alpha _{\rm f} \upsilon_{\rm f} )}\)

Re

Reynolds number, \({{\rm Re}=\frac{\rho _{\rm f} u_\infty D}{\mu _{\rm f}}}\)

r

Radial coordinate (m)

ri

Inner radius (m)

ro

Outer radius (m)

T

Temperature (K)

To

Ambient Temperature (K)

u

Velocity in x-direction (m/s)

U

Non-dimensional velocity in x-direction (uD/α f)

u

Inlet velocity (m/s)

V

Non-dimensional velocity in y-direction (vD/α f)

v

Velocity in y-direction (m/s)

Z

Non-dimensional axial coordinate (z/D)

z

Axial coordinate (m)

Greek Symbols

Λ

Forchheimer number \({(C_{\rm F} /\sqrt{\rm Da})}\)

α

Thermal diffusivity (m2/s)

β

Thermal expansion coefficient (1/K)

\({\phi}\)

Porosity

θ *

Non-dimensional temperature ((T − T 0 )k f /q′′D)

λ

Convective parameter, \({\lambda =\frac{{\rm Ra}^{\ast}}{{Pr Re}^{2}}}\)

μ

Viscosity (kg m−1 s−1)

υ

Kinematic viscosity (m2/s)

ρ

Density (kg/m3)

Subscripts

1

Fluid domain

2

Porous domain

eff

Effective

f

Fluid

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Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Mechanical Engineering DepartmentKing Saud UniversityRiyadhKingdom of Saudi Arabia
  2. 2.Jordan University of Science and TechnologyIrbidJordan

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