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Lattice Boltzmann Pore Scale Simulation of Natural Convection in a Differentially Heated Enclosure Filled with a Detached or Attached Bidisperse Porous Medium

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Abstract

In this research, pore scale simulation of natural convection in a differentially heated enclosure filled with a conducting bidisperse porous medium is investigated using the thermal lattice Boltzmann method. For the first time, the effect of connection of the bidisperse porous medium to the enclosure walls is studied by considering the attached geometry in addition to the detached one. Effect of most relevant parameters on the streamlines and isotherms as well as hot wall average Nusselt number is studied for two of the bidisperse porous medium configurations. It is observed that effect of geometrical and thermo-physical parameters of the bidisperse porous medium on the heat transfer characteristics is more complicated for the attached configuration. To assess the validity of the local thermal equilibrium condition in the micro-porous media, the pore scale results are used to compute the percentage of the local thermal non-equilibrium for two of the bidisperse porous medium configurations. It is concluded that for the detached configuration, the local thermal equilibrium condition is confirmed in the entire micro-porous media for the ranges of the parameters studied here. However, for the attached geometry, it is shown that departure from the local thermal equilibrium condition is observed for the higher values of the Rayleigh number, micro-porous porosity, solid–fluid thermal conductivity ratio, and the smaller values of the macro-pores volume fraction.

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Abbreviations

D :

Block size in the macro-pores (m)

\(D^{{*}}\) :

Dimensionless block size in the macro-pores, D / H [defined by Eq. (7)]

d :

Block size in the micro-porous (m)

\(d^{{*}}\) :

Dimensionless block size in the micro-porous, d / H [defined by Eq. (9)]

\(f_{i}\) :

Density distribution function

\(f_{i}^{\mathrm{eq}}\) :

Equilibrium distribution function of \(f_{i}\)

\(g_{i}\) :

Fluid energy distribution function

\(g_{i}^{\mathrm{eq}}\) :

Equilibrium distribution function of \(g_{i}\)

\(g_{\mathrm{si}}\) :

Solid blocks energy distribution function

\(g_{\mathrm{si}}^{\mathrm{eq}}\) :

Equilibrium distribution function of \(g_{\mathrm{si}}\)

H :

Enclosure side (m)

\(k_{\mathrm{f}}\) :

Fluid thermal conductivity (W/mK)

\(k_{\mathrm{s}}\) :

Solid thermal conductivity (W/mK)

\(N_{\mathrm{mac}}^{2}\) :

Number of blocks in the macro-pores

\(N_{\mathrm{mic}}^{2}\) :

Number of blocks in the micro-porous media

\(\overline{{Nu}}\) :

Average Nusselt number at the hot wall of the enclosure

\(\overline{{Nu}}_{\mathrm{mac}}\) :

Macro-pores hot wall average Nusselt number

\(\overline{{Nu}}_{\mathrm{mic}}\) :

Micro-porous media hot wall average Nusselt number

\(\Pr \) :

Prandtl number

Ra :

Rayleigh number [defined by Eq. (5)]

\(T_{\mathrm{f}}\) :

Fluid temperature (K)

\(T_{\mathrm{s}}\) :

Solid temperature (K)

\(\mathbf{V}=(u,v)\) :

Velocity vector (m/s)

\(\mathbf{V}^{{*}}=(u^{{*}},v^{{*}})\) :

Dimensionless velocity vector

x, y :

Cartesian coordinates (m)

\(x^{{*}},y^{{*}}\) :

Dimensionless coordinates

\({\alpha }\) :

Thermal diffusivity \(({\hbox {m}^{2}}\)/s)

\({\delta }\) :

Clearance between the blocks and the enclosure walls (m)

\({\delta } x\) :

Lattice spacing

\({\delta } t\) :

Time step

\({\delta }^{{*}}\) :

Dimensionless clearance between the blocks and the enclosure walls

\({\varepsilon }_{\mathrm{mic}}\) :

Micro-porous porosity

\({\theta }\) :

Dimensionless temperature, [defined by Eq. (5)]

\({\theta }_{\mathrm{f}}\) :

Fluid dimensionless temperature

\({\theta }_{\mathrm{s}}\) :

Solid dimensionless temperature

\(\langle {\theta }_{\mathrm{s}}\rangle ^{\mathrm{s}}\) :

Solid intrinsic volume-averaged dimensionless temperature

\(\langle {\theta }_{\mathrm{f}}\rangle ^{\mathrm{f}}\) :

Fluid intrinsic volume-averaged dimensionless temperature

\({\lambda }\) :

Solid–fluid thermal conductivity ratio, \(k_{\mathrm{s}}/k_{\mathrm{f}}\)

\({\nu }_{\mathrm{f}}\) :

Fluid kinematic viscosity \((\hbox {m}^{2}/\hbox {s})\)

\({\rho }_{\mathrm{f}}\) :

Fluid density \(({\hbox {kg}}/{\hbox {m}^{{3}}})\)

\(({\rho } c_{p})_{\mathrm{f}}\) :

Fluid volumetric heat capacity \((\hbox {J}/{\hbox {Km}^{3}})\)

\(({\rho } c_{p})_{\mathrm{s}}\) :

Solid volumetric heat capacity \((\hbox {J}/{\hbox {Km}^{3}})\)

\({\tau }_{v}\) :

Fluid hydrodynamic dimensionless relaxation time

\({\tau }_{g}\) :

Fluid energy dimensionless relaxation time

\({\tau }_{gs}\) :

Solid dimensionless energy relaxation time

\({\varphi }_{\mathrm{mac}}\) :

Macro-pores volume fraction

f:

Fluid

s:

Solid

mac:

Macro-pores

mic:

Micro-porous

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Authors and Affiliations

  1. Department of Mechanical Engineering, Persian Gulf University, Bushehr, 75168, Iran

    Gholamreza Imani

  2. School of Mechanical and Mining Engineering, The University of Queensland, Bldg 50, Room S 406, St. Lucia Campus, Brisbane, QLD, 4072, Australia

    Kamel Hooman

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  1. Gholamreza Imani
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  2. Kamel Hooman
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Correspondence to Gholamreza Imani.

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Imani, G., Hooman, K. Lattice Boltzmann Pore Scale Simulation of Natural Convection in a Differentially Heated Enclosure Filled with a Detached or Attached Bidisperse Porous Medium. Transp Porous Med 116, 91–113 (2017). https://doi.org/10.1007/s11242-016-0766-z

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