# Investigation of the Local Thermal Nonequilibrium Conditions for a Convective Heat Transfer Flow in an Inclined Square Enclosure Filled with Cu-Water Nanofluid

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## Abstract

In this paper, the local thermal nonequilibrium conditions between the base fluid and the nanoparticles inside an inclined square enclosure have been investigated and analyzed numerically. The effects of magnetic field intensity and the geometry inclination angle on the heat exchange between base fluid and nanoparticles are also taken into account. Two opposite walls of the enclosure are insulated, and the other two walls are kept at different temperatures. A PDE solver, Comsol Multiphysics which uses the Galerkin weighted residual finite element technique, has been employed to solve the governing nonlinear dimensionless equations. Comparisons with previously published works are performed, and excellent agreement is obtained. Numerical simulations are accomplished to calculate the dimensionless temperature profiles along lines \(X=0.05\) and \(Y=X\) inside the enclosure. A single-phase approach with two temperature equations is applied in this work for the first time. The results indicate that the local thermal nonequilibrium conditions are highly controlled by the Nield number and the nanoparticles volume fraction. The domains at which base fluid and nanoparticles are at local thermal equilibrium or local thermal nonequilibrium have been calculated. These findings may open a door for the researchers to choose the suitable model in analyzing the dynamics of nanofluids and will be helpful in investigating the heat transfer rate based on fluid/particle interface. It will also provide the basis for the future research on the entropy generation investigation during natural convection in order to improve the energy efficiency which may be applicable for different renewable energy systems.

## Keywords

Convection Thermal nonequilibrium Heat transfer Inclined square enclosure Nanofluids Finite element method## List of Symbols

- \(B_0\)
Magnetic field strength (\(\hbox {kg}\,\hbox {s}^{-2}\,\hbox {A}^{-1}\))

*g*Gravitational acceleration (\(\hbox {m s}^{-2}\))

*h*Volumetric heat transfer coefficient (\(\hbox {W}\,\hbox {m}^{-2}\hbox {K}^{-1}\))

*Ha*Hartmann number (–)

*k*Thermal conductivity (\(\hbox {W m}^{-1}\,\hbox {K}^{-1}\))

*L*Enclosure length (m)

*Nu*Nusselt number (–)

*p*Dimensional fluid pressure (Pa)

*P*Dimensionless fluid pressure (–)

*t*Time (s)

*Pr*Prandtl number (–)

*Ra*Rayleigh number (–)

*T*Temperature (K)

*u*,*v*Dimensional velocity components (\(\hbox {m s}^{-1}\))

*U*,*V*Dimensionless velocities (–)

*x*,*y*Dimensional coordinates (m)

*X*,*Y*Dimensionless coordinates (–)

## Greek Symbols

- \(\delta \)
Geometry inclination angle (\(^\circ \))

- \(\beta \)
Thermal expansion coefficient (\(\hbox {K}^{-1}\))

- \(\alpha \)
Thermal diffusivity (\(\hbox {m}^{2}\,\hbox {s}^{-1}\))

- \(\gamma \)
Magnetic field inclination angle (\(^\circ \))

- \(\mu \)
Dynamic viscosity (Ns \(\hbox {m}^{-2}\))

- \(\upsilon \)
Kinematic viscosity (\(\hbox {m}^{2}\,\hbox {s}^{-1}\))

- \(\rho \)
Density (\(\hbox {kg m}^{-3}\))

- \(\sigma \)
Electrical conductivity (\(\hbox {S m}^{-1}\))

- \(\theta \)
Dimensionless temperature (–)

- \(\phi \)
Nanoparticle volume fraction (–)

- \(\tau \)
Dimensionless time (–)

## Subscripts

- \({\hbox {av}}\)
Average

- \({\hbox {C}}\)
Cold wall

- \({\hbox {f}}\)
Base fluid

- \({\hbox {H}}\)
Hot wall

- \({\hbox {nf}}\)
Nanofluid

- \({\hbox {p}}\)
Solid particle

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## Notes

### Acknowledgements

The authors would like to thank the anonymous referees for their constructive comments for the further improvement of the paper. M. M. Rahman is thankful to The Research Council (TRC) of Oman for funding under the Open Research Grant Program ORG/SQU/CBS/14/007 and College of Science for the Grant No. IG/SCI/DOMS/16/15. K. S. Al Kalbani is grateful to TRC for a Doctoral Sponsorship.

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