, Volume 54, Issue 3, pp 505–524 | Cite as

Simulation of conjugate radiation–forced convection heat transfer in a porous medium using the lattice Boltzmann method

  • Yousef Kazemian
  • Saman Rashidi
  • Javad Abolfazli EsfahaniEmail author
  • Nader Karimi


In this paper, a lattice Boltzmann method is employed to simulate the conjugate radiation–forced convection heat transfer in a porous medium. The absorbing, emitting, and scattering phenomena are fully included in the model. The effects of different parameters of a silicon carbide porous medium including porosity, pore size, conduction–radiation ratio, extinction coefficient and kinematic viscosity ratio on the temperature and velocity distributions are investigated. The convergence times of modified and regular LBMs for this problem are 15 s and 94 s, respectively, indicating a considerable reduction in the solution time through using the modified LBM. Further, the thermal plume formed behind the porous cylinder elongates as the porosity and pore size increase. This result reveals that the thermal penetration of the porous cylinder increases with increasing the porosity and pore size. Finally, the mean temperature at the channel output increases by about 22% as the extinction coefficient of fluid increases in the range of 0–0.03.


Lattice Boltzmann Conjugate radiation–forced convection Porous medium Porosity Conduction radiation ratio 

List of symbols

\( {\text{c}}_{\mathrm L} \)

Speed of light


Lattice streaming speed


Parameter defined


Parameter defined


Speed of sound


Lattice speed discreted in the main directions of the lattice


Specific heat capacity


Darcy number, \( = {\text{K}}/{\text{H}}^{2} \)


Amount of each data per each mesh size


Pore size diameter


Discrete particle velocity in LBE model

\( {\text{e}}_{a}^{21} \)

Approximate relative error

\( e_{ext}^{21} \)

Extrapolated relative error


Total body force vector


Density distribution function in LBE model

\( {\text{f}}_{{\mathrm{k}}}^{{{\mathrm{eq}}}} \)

Equilibrium density distribution function in LBE model


Discrete body force in LBE model

\( {\text{F}}_{\upvarepsilon} \)

Geometric function


Temperature distribution function in LBE model

\( {\text{g}}_{{\mathrm{k}}}^{{{\mathrm{eq}}}} \)

Equilibrium temperature distribution function in LBE model


Incident radiation

\( GCI_{fine}^{21} \)

Fine-grid convergence index


Channel height


Direction index in LBE model (D2Q8 model)


Unit matrix


Radiation intensity of the black body


Radiation distribution function in LBE model

\( {\text{I}}_{{\mathrm{i}}}^{{{\mathrm{eq}}}} \)

Equilibrium radiation distribution function in LBE model


Kinematic viscosity ratio


Direction index in LBE model (D2Q9 model)


Absorbing coefficient


Channel length


Number of node

\( \dot{m}_{in} \)

Mass flow rate at the inlet of the channel

\( \dot{m}_{section} \)

Mass flow rate at each section of the channel


Conduction radiation ratio \( = \frac{{{\text{k}}_{\text{e}}\upbeta_{\text{e}} }}{{4\upsigma\left( {{\text{T}}_{\text{w}} - {\text{T}}_{\text{in}} } \right)^{3} }} \)


Nusselt number




Apparent order


Scattering phase function


Prandtl number,\( = \frac{{\upupsilon_{\mathrm{e}} }}{{\upalpha_{\mathrm{e}} }} \)

\( \hbox{q}^{\mathrm{r}} \)

Dimensionless radiative heat flux


Ratio of the number of nodes

\( \vec{\hbox{r}}_{\mathrm{n}} \)

Position vector


Radiation parameter \( = \frac{{\upbeta_{\rm f}^{2} \left( {1 - \upomega } \right)\hbox{H}^{2} }}{4\hbox{N}} \)


Reynolds number \( = \frac{{\hbox{U}_{\rm in} \hbox{H}}}{{\upupsilon_{\rm f} }} \)


Constant parameter

\( \vec{s} \)

Geometric distance






Inlet temperature


Wall temperature


Fluid velocity vector


Fluid velocity in x direction


Velocity at outlet


Non-dimensional fluid velocity, = u/Uin


Inlet fluid velocity

\( u_{section} \)

Local velocity in each section

\( URF \)

Under relaxation factor

\( \varDelta u^{ + } \)

Velocity difference with inlet section of the channel


Temporal velocity vector defined in Eq. (20)


Fluid velocity in y direction


Weighting coefficient for radiation


Weighting coefficient for energy and momentum


Axial coordinate


Non-dimensional axial coordinat, = x/H


Transversal coordinate


Non-dimensional transversal coordinat, = y/H

Greek symbols

\( \upalpha_{\mathrm f} \)

Thermal diffusivity coefficient for fluid

\( \upalpha_{\mathrm s} \)

Thermal diffusivity coefficient for solid

\( \upalpha_{\mathrm e} \)

Effective Thermal diffusivity coefficient

\( \Delta \hbox{t} \)

Time step

\( \Delta \hbox{x} \)

Space step

\( \upvarepsilon \)


\( \upepsilon \)

Difference of data

\( \upbeta_{\mathrm e} \)

Effective extinction coefficient

\( \upbeta_{\mathrm s} \)

Extinction coefficient for solid

\( \upbeta_{\mathrm f} \)

Extinction coefficient for fluid

\( \upupsilon_{\mathrm e} \)

Effective kinematical viscosity

\( \uptau_{{\upalpha_{\mathrm{e}} }} \)

Effective relaxation time for temperature distribution

\( \uptau_{{\upupsilon_{\mathrm e} }} \)

Effective relaxation time for velocity distribution

\( \upupsilon_{\mathrm{f}} \)

Kinematical viscosity for fluid

\( \upgamma \)

Thermal diffusivity ratio

\( \upsigma \)

Stefan–Boltzmann constant

\( \upsigma_{\mathrm s} \)

Scattering coefficient

\( \upvarepsilon_{\mathrm{e}} \)

Emissivity coefficient

\( \upomega \)

Scattering albedo

\( \uptheta \)

Dimensionless temperature, \( = \frac{{\hbox{T} - \hbox{T}_{\mathrm{in}} }}{{\hbox{T}_{\mathrm{w}} - \hbox{T}_{\mathrm{in}} }} \)

\( \theta_{\mathrm m} \)

Non-dimensional mean temperature

\( \theta_{\mathrm w} \)

Non-dimensional temperature on the channel wall

\( \hbox{K}_{\mathrm a} \)

Absorbing coefficient

\( \uprho_{\mathrm{in}} \)

Density at inlet

\( \uprho_{\mathrm o} \)

Density at outlet

\( \Omega \)

Space angle

\( \varvec{\upgamma} \)

Polar angle

\( \updelta \)

Horizon angle

\( {{\upvarphi }}_{ext}^{21} \)

Extrapolated values


Compliance with ethical standards

Conflict of interest

All authors declare that they have neither conflict of interest, nor external funding.


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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringFerdowsi University of MashhadMashhadIran
  2. 2.School of EngineeringUniversity of GlasgowGlasgowUK

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