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Darcy–Boussinesq convective flow in a trapezoidal enclosure with thermal stratification

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Abstract

We explore numerically the Darcy–Boussinesq convective flow of water, kerosene, and engine oil through the glass ball, aluminum foam and sandstone porous medium inside a right-angle trapezoidal enclosure taking into consideration of thermal stratification. The bottom and left walls of the enclosure are uniformly heated, whereas thermal insulation is considered for the upper wall of it. Thermal stratification is applied to the right wall. The governing nondimensional partial differential equations are simulated using the Galerkin weighted residual finite element method with the help of COMSOL Multiphysics. We explored the time evolution of solutions and investigate the effects of the Rayleigh number, thermal stratification parameter, porosity parameter, types of the solid matrix and working fluids, and on the average Nusselt numbers, streamlines and isotherms. The simulated results confirm that the heat transfer rate amplifies with the increase in Rayleigh number and diminishes with the thermal stratification, porosity and aspect ratio of the enclosure. The results further guarantee that engine oil and aluminum foam perform better for heat transfer intensification. A two-cell flow is obtained for a stronger thermal stratification.

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Abbreviations

A :

Aspect ratio (–)

\(b\) :

Thermal stratification parameter (–)

\(B\) :

Thermal gradient (K m−1)

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

Specific heat (J kg−1 K−1)

\(g\) :

Gravitational acceleration (ms−2)

\({\text{Gr}}\) :

Grashof number (–)

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

Thermal conductivity of fluid (W m−1 K−1)

\(k_{\text{m}}\) :

Effective thermal conductivity (W m−1 K−1)

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

Thermal conductivity of solid (W m−1 K−1)

\(K\) :

Permeability (m2)

\(l\) :

Length of the upper wall (m)

\(L\) :

Length of the cavity (m)

\({\text{Nu}}_{\text{ave}}\) :

Average Nusselt number (–)

\({\text{Nu}}_{\text{L}}\) :

Local Nusselt number (–)

\(p\) :

Dimensional pressure (Pa)

\(P\) :

Nondimensional pressure (–)

\(\Pr\) :

Prandtl number (–)

\(q_{\text{w}}\) :

Heat flux (W m−2)

\({\text{Ra}}\) :

Rayleigh number (–)

\(t\) :

Time (s)

\(T\) :

Temperature (K)

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

Temperature of the inclined wall (K)

\(T_{\text{h}}\) :

Temperature of the hot wall (K)

\(T_{0}\) :

Temperature of the cold wall (K)

\(u\), \(v\) :

Dimensional velocity components (ms−1)

\(U\), \(V\) :

Nondimensional velocity components (–)

\(x\), \(y\) :

Cartesian coordinates (m)

\(X\), \(Y\) :

Nondimensional Cartesian coordinates (–)

\(\alpha\) :

Thermal diffusivity (m2 s−1)

\(\beta\) :

Coefficient of thermal expansion (K−1)

\(\rho\) :

Density (kg m−3)

\(\varepsilon\) :

Porosity (–)

\(\mu\) :

Viscosity (Pa s)

\(\gamma\) :

Thermal conductivity ratio (–)

\(\lambda\) :

Thermal diffusivity ratio (–)

\(\psi\) :

Stream function (m2 s−1)

\(\theta\) :

Nondimensional temperature (–)

\(\tau\) :

Dimensionless time (–)

c:

Cold

h:

Hot

f:

Fluid

m:

Effective

p:

Particle

s:

Solid

ave:

Average

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Acknowledgements

This work of M.M. Rahman was supported by the Sultan Qaboos University through the grants IG/SCI/DOMS/18/10 and IG/SCI/MATH/20/03, while the work of I. Pop has been supported from the Grant PN-III-P4-ID-PCE-2016-0036, UEFISCDI, Romania.

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Authors and Affiliations

  1. Department of Mathematics, College of Science, Sultan Qaboos University, PO Box 36, PC 123 Al-Khod, Muscat, Sultanate of Oman

    Mohammad M. Rahman & Kawkab A. Al Amri

  2. Department of Mathematics, Faculty of Mathematics and Computer Science, Babeş-Bolyai University, 400084, Cluj-Napoca, Romania

    Ioan Pop

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  1. Mohammad M. Rahman
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Correspondence to Ioan Pop.

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Rahman, M.M., Al Amri, K.A. & Pop, I. Darcy–Boussinesq convective flow in a trapezoidal enclosure with thermal stratification. J Therm Anal Calorim 145, 3325–3337 (2021). https://doi.org/10.1007/s10973-020-09912-4

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  • DOI: https://doi.org/10.1007/s10973-020-09912-4

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