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Modeling and Simulation of Local Thermal Non-equilibrium Effects in Porous Media with Small Thermal Conductivity

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Abstract

The two-equation model in porous media can describe the local thermal non-equilibrium (LTNE) effects between fluid and solid at REV scale, with the temperature differences in a solid particle neglected. A multi-scale model has been proposed in this study. In the model, the temperature differences in a solid particle are considered by the coupling of the fluid energy equation at REV scale with the heat conduction equation of a solid particle at pore scale. The experiments were conducted to verify the model and numerical strategy. The multi-scale model is more suitable than the two-equation model to predict the LTNE effects in porous media with small thermal conductivity. The effects of particle diameter, mass flow rate, and solid material on the LTNE effects have been investigated numerically when cryogenic nitrogen flows through the porous bed with small thermal conductivity. The results indicate that the temperature difference between solid center and fluid has the same trend at different particle diameters and mass flow rates, while the time to reach the local thermal equilibrium is affected by solid diameter dramatically. The results also show that the temperature difference between solid center and surface is much greater than that between solid surface and fluid. The values of \( \rho {\text{c}} \) for different materials have important influence on the time to reach the local thermal equilibrium between solid and fluid.

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Abbreviations

\( \varepsilon \) :

Porosity (Saad and Saad 2013)

\( \rho \) :

Density (kg/m3)

\( c_{\text{pf}} \) :

Specific heat capacity of fluid (J/kg K)

\( c_{\text{s}} \) :

Specific heat capacity of solid (J/kg K)

\( \vec{u} \) :

Velocity of fluid (m/s)

\( t \) :

Time (s)

\( h \) :

Specific enthalpy (J/kg)

\( \dot{m} \) :

Mass flow rate of fluid (kg/m2 s)

\( Q \) :

Heat exchanged per unit volume (J/m3 s)

\( T \) :

Temperature (K)

\( p \) :

Pressure (Pa)

\( V_{0} \) :

Volume of a sphere (m3)

\( \mu \) :

Dynamic viscosity (kg/ms)

\( K \) :

Permeability (Saad and Saad 2013)

\( k \) :

Heat conductivity (W/mK)

\( Re \) :

Reynolds number (Saad and Saad 2013)

\( L \) :

Length of porous region (m)

\( d_{\text{p}} \) :

Diameter of a sphere (m)

\( R \) :

Radius of a sphere (m)

\( \alpha \) :

Aspect ratio (Saad and Saad 2013)

\( q \) :

Heat flux (J/m2 s]

\( h_{\text{c}} \) :

Convective heat transfer coefficient (W/m2 K)

S :

Solid

f :

Fluid

0:

Initial time

in:

Inlet

out:

Outlet

wall:

Sphere surface

eff:

Effective

*:

Previous iteration

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Acknowledgements

The work was supported by the Fundamental Research Funds for the Central Universities (China University of Mining and Technology) (No: 2015XKMS060).

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Authors and Affiliations

  1. School of Electrical and Power Engineering, China University of Mining and Technology, Xuzhou, 221116, China

    Lang Lu, Xueping Du & Chengyun Xin

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  1. Lang Lu
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  2. Xueping Du
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  3. Chengyun Xin
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Correspondence to Chengyun Xin.

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Lu, L., Du, X. & Xin, C. Modeling and Simulation of Local Thermal Non-equilibrium Effects in Porous Media with Small Thermal Conductivity. Transp Porous Med 124, 553–575 (2018). https://doi.org/10.1007/s11242-018-1084-4

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  • DOI: https://doi.org/10.1007/s11242-018-1084-4

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