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Investigation of the entropy generation during natural convection of Newtonian and non-Newtonian fluids inside the L-shaped cavity subjected to magnetic field: application of lattice Boltzmann method

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Abstract

In the present paper, free heat convection and entropy generation of Newtonian and two types of non-Newtonian fluids, shear-thickening and shear-thinning, inside an L-shaped cavity subjected to a magnetic field have been investigated by the finite difference lattice Boltzmann method. The power-law model was used for modeling the rheology of the fluids. The bottom and left walls of the cavity have been kept at a uniform high temperature. Internal walls are also kept cold. The remaining walls have been insulated against heat and mass transfer. The Boussinesq approximation is used to take the temperature dependency of density into account. The distribution functions of energy and density are modeled through the use of the nine-velocity two-dimensional scheme. The effects of Hartmann number (Ha), aspect ratio, power-law index, and Rayleigh number (Ra), on the flow field, temperature distribution, and entropy distributions are studied. The results show that the magnetic field and the power-law index have an ever-decreasing effect on the heat transfer rate and the entropy generation, while the Ra number has an ever-increasing effect. The maximum heat transfer enhancement of 71% happens at the lowest and the highest values of power-law index and Ra number, respectively, for the case with no magnetic field. The maximum heat transfer deterioration of 77% happens at the highest and lowest values of power-law index and Ra number, respectively, in the presence of the highest magnetic field strength. It is interesting that the sensitivities of heat transfer rate and the entropy generation to the Ha number become significant for shear-thinning fluids. It is found that there is an everlasting interplay between conduction and convection contributions to the irreversibilities, so that, for the Newtonian and shear-thinning fluids, thermal irreversibilities, characterized by Be number, increase with Ha number, reaching to a maximum, and then decline. The heat transfer rate and the total entropy generation for the Newtonian and shear-thinning fluids increase, monotonically, by raising the aspect ratio, while the figure for the shear-thickening case is different. It is decreased first and then increased.

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Abbreviations

AR:

Enclosure aspect ratio

\( B \) :

Strength of magnetic field

\( C_{p} \) :

Specific heat \( \left( {{\text{J}}\,{\text{kg}}^{ - 1} \,{\text{k}}^{ - 1} } \right) \)

\( F \) :

External forces

\( f \) :

Functions of density distribution

\( f_{\text{eq}} \) :

Functions of equilibrium density distribution

\( g \) :

Functions of internal energy distribution

\( g_{\text{eq}} \) :

Functions of equilibrium internal energy distribution

g :

Gravitational acceleration \( \left( {{\text{m}}\,{\text{s}}^{ - 2} } \right) \)

H :

Length of the inner walls \( \left( {\text{m}} \right) \)

\( {\text{Ha}} \) :

Hartmann number

\( K \) :

The coefficient of consistency

L :

Width of the enclosure\( \left( {\text{m}} \right) \)

\( n \) :

Index of power law

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

Nusselt number

\( P \) :

Pressure

\( { \Pr } \) :

Prandtl number

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

Rayleigh number

S :

Entropy generation

\( T \) :

Temperature (K)

\( t \) :

Time (s)

\( u \) :

Velocity in x-direction \( \left( {{\text{m}}\,{\text{s}}^{ - 1} } \right) \)

\( v \) :

Velocity in y-direction \( \left( {{\text{m}}\,{\text{s}}^{ - 1} } \right) \)

\( x,y \) :

Cartesian coordinates \( \left( {\text{m}} \right) \)

\( \sigma \) :

The electrical conductivity \( \left( {\varOmega \,{\text{m}}} \right) \)

\( \phi \) :

Relaxation time

\( \tau \) :

Shear stress \( \left( {{\text{N}}\,{\text{m}}^{ - 2} } \right) \)

\( \zeta \) :

Speeds of discrete particle

\( \Delta x \) :

Lattice spacing

\( \Delta t \) :

Time increment

\( \alpha \) :

Thermal diffusivity \( \left( {{\text{m}}^{2} \,{\text{s}}^{ - 1} } \right) \)

\( \rho \) :

Density \( \left( {{\text{kg}}\,{\text{m}}^{ - 3} } \right) \)

\( \mu \) :

Dynamic viscosity \( \left( {{\text{kg}}\,{\text{m}}^{ - 1} \,{\text{s}}^{ - 1} } \right) \)

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

Apparent viscosity (Pa.s)

\( \psi \) :

Stream function \( \left( {{\text{m}}^{2} \,{\text{s}}^{ - 1} } \right) \)

\( \varPsi \) :

Dimensionless stream function

\( \chi \) :

Irreversibility distribution ratio

GenT:

Thermal generation

\( m \) :

Average

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

Cold

\( {\text{H}} \) :

Hot

\( x,y \) :

Cartesian coordinates

\( \alpha \) :

The node number

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Acknowledgements

This research is partially supported by the National Science Foundation of China (61503284).

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Authors and Affiliations

  1. School of Automotive and Transportation, Tianjin University of Technology and Education, Tianjin, 300222, China

    Rui Zhang

  2. Department of Mechanical Engineering, Najafabad Branch, Islamic Azad University, Isfahan, Iran

    Saeed Aghakhani

  3. Department of Mechanical Engineering, Shahrekord University, Shahrekord, Iran

    Ahmad Hajatzadeh Pordanjani

  4. Department of Mechanical Engineering, Semnan University, Semnan, Iran

    Seyed Masoud Vahedi

  5. Department of Mechanical Engineering, Kermanshah University of Technology, Kermanshah, Iran

    Amin Shahsavar

  6. Laboratory of Magnetism and Magnetic Materials, Advanced Institute of Materials Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Masoud Afrand

  7. Faculty of Applied Sciences, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Masoud Afrand

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  1. Rui Zhang
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  2. Saeed Aghakhani
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  3. Ahmad Hajatzadeh Pordanjani
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  4. Seyed Masoud Vahedi
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  5. Amin Shahsavar
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  6. Masoud Afrand
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Correspondence to Masoud Afrand.

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Zhang, R., Aghakhani, S., Hajatzadeh Pordanjani, A. et al. Investigation of the entropy generation during natural convection of Newtonian and non-Newtonian fluids inside the L-shaped cavity subjected to magnetic field: application of lattice Boltzmann method. Eur. Phys. J. Plus 135, 184 (2020). https://doi.org/10.1140/epjp/s13360-020-00169-2

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  • DOI: https://doi.org/10.1140/epjp/s13360-020-00169-2

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