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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

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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.

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Abbreviations

c P :

Specific heat at constant pressure

C 0 :

Constant defined by Eq. 44

C n :

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

k m :

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

r 0 :

Circular tube radius

R n (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

Y n (y):

Eigenfunctions for a channel

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

λ n :

Eigenvalues

μ :

Fluid viscosity

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

Effective viscosity for the flow in the porous medium

ρ :

Fluid density

FD:

fully developed

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Kuznetsov, A.V., Nield, D.A. 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. Transp Porous Med 76, 345–362 (2009). https://doi.org/10.1007/s11242-008-9250-8

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