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Three-Dimensional Free Convective Heat Transmission Flow of Copper–Water Nanofluid in a Glass Bead Permeable Matrix within a Right Trapezoidal Cavity in Consideration of Thermal Non-Equilibrium Conditions

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Abstract

This work focuses on the impacts of varying penetrability and porosity through the natural convective heat transmission flow of copper–water in a glass bead permeable matrix within a right trapezoidal cavity in consideration of thermal non-equilibrium conditions among the permeable medium, nanoparticles, and the base fluid using the Darcy–Brinkman–Forchheimer model. The model equations are simulated using the Galerkin weighted residual finite element strategy. We analyze the influences of the various model factors particularly, the critical Rayleigh number, the porosity factor, the nanoparticles volume fraction, the interface heat transmission parameters, and the bead diameter in the realms of flow and heat. Furthermore, we investigate the effects of the aspect ratios of the trapezoidal cavity and various thermal boundary situations on the rate of heat transmission for base fluid, nanoparticles, and porous matrix in detail. The results show that the critical Rayleigh number for the commencement of local thermal nonequilibrium states reduced with the enhancement of the bead diameter and the porosity parameter. The average Nusselt number in the base fluid, nanoparticles, and solid matrix increased with the increase of the bead diameter for about 11.7%, 11.6%, and 1.4%, respectively, when it rises from 0.4 to 0.6. The trapezoidal cavity exhibits the greatest heat transmission rate for the base fluid, nanoparticles, and solid matrix in comparison with the cube and the rectangular cavity.

Article Highlights

  • Glass bead diameter and porosity parameter control the state of thermal nonequilibrium.

  • Heat transmission in porous medium enhanced significantly with the glass bead diameter.

  • Heat transmission in the trapezoidal cavity is highest compared to the heat transmission in the cube and rectangular cuboid.

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Abbreviations

\(A_{1}\) :

Aspect ratio

\(A_{2}\) :

Aspect ratio

\(A_{3}\) :

Aspect ratio

\({\text{c}}_{1}\) :

Empirical constant

\({\text{c}}_{2}\) :

Empirical constant

\({\text{C}}_{{\text{F}}}\) :

Forchheimer coefficient

\({\text{C}}_{{\text{p}}}\) :

Specific heat

\({\text{D}}a\) :

Darcy number

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

Solid particles diameter

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

Dimensionless solid particles diameter

\(g\) :

Acceleration due to gravity

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

Interface heat transmission factor amid the base fluid and nanoparticles

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

Interface heat transmission factor amid the base fluid and porous medium

\(H\) :

Height of the cavity

\(K\) :

Permeability

\(l\) :

Upper wall length

\(L\) :

Lower wall length

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

Nield number of the base fluid/nanoparticle interface

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

Nield number of the base fluid/permeable medium interface

\(N{\text{u}}\) :

Nusselt number

\(p\) :

Dimensional pressure

\({\text{P}}\) :

Dimensionless pressure

\(\Pr\) :

Prandtl number

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

Heat flux

\(Ra\) :

Rayleigh number

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

Critical Rayleigh number

\(t\) :

Dimensional time

\(T\) :

Temperature

\(T_{0}\) :

Reference temperature

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

Dimensional Darcy velocity components

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

Dimensionless Darcy velocity components

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

Dimensional coordinates

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

Dimensionless coordinates

\(\alpha\) :

Thermal diffusivity

\(\beta\) :

Coefficient of volume expansion

\(\delta\) :

Ratio of conductivities

\(\tau\) :

Dimensionless time

\(\rho\) :

Fluid density

\(\mu\) :

Dynamic viscosity

\(\upsilon\) :

Kinematic coefficient of viscosity

\(\theta\) :

Dimensionless temperature

\(\phi\) :

Nanoparticle volume fraction

\(\kappa\) :

Thermal conductivity

\(\lambda\) :

Ratio of diffusivities

\(\varepsilon^{*}\) :

Porosity

\(\varepsilon_{\infty }\) :

Uniform porosity

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

Cold wall

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

Base fluid

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

Hot wall

\({\text{nf}}\) :

Nanofluid

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

Solid particle

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

Solid matrix

\(\rho\) :

Fluid density

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Acknowledgements

We would like to thank the anonymous referees for their valuable comments for the further improvement of the paper. M.M. Rahman is thankful to the Ministry of Higher Education, Research and Innovation (Oman), for funding through the research grant RC/RG-SCI/MATH/20/01.

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Authors and Affiliations

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

    Sheikha M. Al-Weheibi & M. M. Rahman

  2. Department of Mechanical and Industrial Engineering, Toronto Metropolitan University, Toronto, M5B 2K3, Canada

    M. Ziad Saghir

  3. Department of Mathematics, University of Central Florida, Orlando, FL, 32816, USA

    K. Vajravelu

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  1. Sheikha M. Al-Weheibi
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Correspondence to M. M. Rahman.

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Al-Weheibi, S.M., Rahman, M.M., Saghir, M.Z. et al. Three-Dimensional Free Convective Heat Transmission Flow of Copper–Water Nanofluid in a Glass Bead Permeable Matrix within a Right Trapezoidal Cavity in Consideration of Thermal Non-Equilibrium Conditions. Transp Porous Med 145, 653–681 (2022). https://doi.org/10.1007/s11242-022-01867-4

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