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Transport in Porous Media

, Volume 76, Issue 3, pp 345–362 | Cite as

Thermally Developing Forced Convection in a Porous Medium Occupied by a Rarefied Gas: Parallel Plate Channel or Circular Tube with Walls at Constant Heat Flux

  • A. V. KuznetsovEmail author
  • D. A. Nield
Article

Abstract

An adaptation of the classical Graetz methodology is applied to investigate the thermal development of forced convection in a parallel plate channel or a circular tube filled by a porous medium saturated by a rarefied gas, with walls held at constant heat flux. The Brinkman model is employed. The analysis leads to expressions for the local Nusselt number Nu as functions of the dimensionless longitudinal coordinate and the Darcy number. It is found that an increase in the velocity slip coefficient generally increases Nu by a small or moderate amount (but the circular tube at large Darcy number is an exception) while an increase in the temperature slip coefficient reduces Nu by a more substantial amount. These trends are uniform as the longitudinal coordinate varies.

Keywords

Forced convection Thermal development Rarefied gas Graetz problem Parallel plate channel and circular tube 

Nomenclature

cP

Specific heat at constant pressure

C0

Constant defined by Eq. 44

Cn

Coefficients defined by Eq. 38 for a channel and by Eq. 77 for a circular tube

Da

Darcy number defined as K/H 2 for a channel and K/\({r_{0}^{2}}\) for a circular tube

f (r)

Temperature perturbation function defined by Eq. 69 for a circular tube

f (y)

Temperature perturbation function defined by Eq. 30 for a channel

G

Negative of the applied pressure gradient

H

Half channel width

km

Effective thermal conductivity of the porous medium

K

Permeability

Kn

Knudsen number

M

Viscosity ratio, \({\tilde{\mu}/\mu}\)

Nu

Local Nusselt number defined as \({\frac{2H{q}^{\prime \prime}}{k_{\rm m} (T_{\rm m}^{\ast}-T_{\rm w}^{\ast})}}\) for a channel and \({\frac{2r_0 {q}^{\prime \prime}}{k_{\rm m}(T_{\rm m}^{\ast}-T_{\rm w}^{\ast})}}\) for a circular tube

\({\overline{Nu}}\)

Mean Nusselt number defined by Eq. 46

Pe

Péclet number defined as ρc P HU*/k m for a channel and ρc P r 0 U*/k m for a tube

q′′

Wall heat flux

r

r*/r 0

r*

Radial coordinate

r0

Circular tube radius

Rn(y)

Eigenfunctions for a circular tube

S

\({({\it MDa})^{-1/2}}\)

\({T_{\rm m}^{\ast}}\)

Bulk mean temperature

T*

Temperature

\({\hat{T}}\)

\({\frac{T^{\ast}-T_{\rm w}^{\ast}}{T_{\rm m}^{\ast}-T_{\rm w}^{\ast}}}\)

\({T_{\rm IN}^{\ast}}\)

Inlet temperature

\({T_{\rm w}^{\ast}}\)

Wall temperature

T+

Perturbation temperature, \({T^{\ast}-T_{\rm FD}^{\ast}}\)

u

\({\tilde{\mu}u^{\ast}/{\it GH}^{2}}\) for a channel and \({\tilde{\mu}u^{\ast}/Gr_{0}^{2}}\) for a circular tube

u*

Filtration velocity

û

u*/U*

U*

Mean filtration velocity

\({\tilde{x}}\)

x/Pe

x

x*/H

x*

Longitudinal coordinate

y

y*/H

y*

Transverse coordinate

Yn(y)

Eigenfunctions for a channel

Greek symbols

α

Velocity slip coefficient

β

Temperature slip coefficient

γ

Parameter defined by Eq. 8 for a channel and by Eq. 51 for a circular tube

θ

Dimensionless temperature, defined by Eq. 10

θ+

\({\frac{T^{+}}{H{q}^{\prime \prime}/k_{\rm m}}}\) for a channel and \({\frac{T^{+}}{r_0 {q}^{\prime \prime}/k_{\rm m}}}\) for a circular tube

λn

Eigenvalues

μ

Fluid viscosity

\({\tilde{\mu}}\)

Effective viscosity for the flow in the porous medium

ρ

Fluid density

Subscripts

FD

fully developed

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Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Department of Mechanical and Aerospace EngineeringNorth Carolina State UniversityRaleighUSA
  2. 2.Department of Engineering ScienceUniversity of AucklandAucklandNew Zealand

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