# 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.

## Keywords

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

*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

## 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

*λ*_{n}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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