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Thermal Convection in a Partially Porous Rotating Chamber Using Local Thermal Non-Equilibrium Models

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Abstract

The effect of a porous layer on thermal convection in a closed rotating square chamber has been studied in this paper. The left border of the chamber is heated up, the right one is cooled, and other walls are thermally insulated. The governing relations with the initial and boundary conditions written employing stream function and vorticity variables are worked out by the finite difference technique. Two approaches are considered and analyzed for setting the heat border conditions at the internal interface between clear liquid and porous material. The first approach assumes that the thermal flux is divided between two phases based on their effective conductivities and temperature gradients. The second model states that both phases at the interface obtain the same thermal flux as the wall. The performed analysis has shown possible differences between these models and the range of governing parameters when these models allow to obtain the similar results.

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Abbreviations

C :

Specific heat capacity, (Jkg−1K−1)

Da:

Darcy number, (–)

g :

Gravity acceleration, (ms−2)

h :

Size of the porous layer, (m)

\(\tilde{h}\) :

Solid/fluid heat transfer coefficient, (Wm−3K−1)

K :

Porous insertion permeability, (m2)

L :

Size of the cavity, (m)

\(\overline{Nu}\) :

Average Nusselt number at heater, (–)

p :

Dimensional pressure due to the fluid motion, (Pa)

Pr:

Prandtl number, (–)

Ra:

Rayleigh number, (–)

T :

Dimensional temperature, (K)

T c :

Dimensional cold wall temperature, (K)

T h :

Dimensional hot wall temperature, (K)

T 0 :

Initial temperature, (K)

t :

Dimensional time, (s)

Ta:

Taylor number, (–)

\(\overline{u},\overline{v}\) :

Dimensional velocity components, (ms−1)

u, v :

Dimensionless velocity components, (–)

\(\overline{x},\overline{y}\) :

Dimensional coordinates, (m)

x, y :

Dimensionless coordinates, (–)

α :

Thermal diffusivity, (m2s−1)

β :

Thermal expansion coefficient, (K−1)

γ :

Fluid/solid thermal capacity ratio, (–)

δ :

Dimensionless porous layer thickness, (–)

δ s :

Porosity-scaled conductivity ratio, (–)

ε :

Porosity, (–)

ζ :

Interface parameter, (–)

θ :

Dimensionless temperature, (–)

Λ :

Thermal conductivity ratio, (–)

λ :

Thermal conductivity, (W⋅m−1⋅K−1)

μ :

Dynamic viscosity, (Pas)

ν :

Kinematic viscosity, (m2s−1)

ξ :

Nield number, (–)

ρ :

Fluid density, (kgm−3)

τ :

Dimensionless time, (–)

ϕ :

Angle of rotation, (–)

\(\overline{\psi }\) :

Dimensional stream function, (m2s−1)

ψ :

Dimensionless stream function, (–)

\(\overline{\omega }\) :

Dimensional vorticity, (s−1)

ω :

Dimensionless vorticity, (–)

ω 0 :

Angular velocity, (s−1)

c :

Cold wall

clear fluid:

Clear fluid

f:

Fluid

h:

Hot wall

max:

Maximum value

porous:

Porous medium

s:

Solid matrix of porous medium

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Acknowledgements

This research was supported by the Tomsk State University Development Programme (Priority-2030).

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  1. Laboratory on Convective Heat and Mass Transfer, Tomsk State University, Tomsk, Russia, 634050

    Stepan A. Mikhailenko & Mikhail A. Sheremet

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  1. Stepan A. Mikhailenko
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Mikhailenko, S.A., Sheremet, M.A. Thermal Convection in a Partially Porous Rotating Chamber Using Local Thermal Non-Equilibrium Models. Transp Porous Med 143, 619–637 (2022). https://doi.org/10.1007/s11242-022-01801-8

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