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Entropy generation analysis for thermomagnetic convection of paramagnetic fluid inside a porous enclosure in the presence of magnetic quadrupole field

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Abstract

The entropy generation characteristics for thermomagnetic convection in a porous cavity filled with paramagnetic air in the presence of a magnetic quadrupole field are investigated numerically using a finite volume method. The vertical walls of the porous cavity are maintained at different temperatures, whereas the horizontal walls are insulated. The Darcy–Brinkman–Forchheimer model is used for mathematical formulation of the fluid flow in porous media. Thermal, frictional and total entropy generation and Bejan number for a selected range of magnetic force number (γ = 1–100), Darcy number (Da = 5 × 10−4–5 × 10−2) and Rayleigh number (Ra = 104–106) are examined for both cases: (1) with gravity and (2) without gravity. The results indicate that the magnetic field had little or no effect on the total entropy generation for the lower values of Darcy number and the process is dominated by the heat transfer irreversibility. In contrast, the magnetic force provokes various irreversibilities to rise as a result of improved convection at higher Darcy numbers, and the irreversibility is dominated by the viscous effects. This work may give an insight into the design-related concept of entropy generation for various thermal systems and clarifying energy loss.

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Abbreviations

b :

Magnetic flux density (T)

b 0 :

Reference magnetic flux density, b0 = Br (T)

B :

Dimensionless magnetic flux

Be :

Bejan number

Br :

Magnetic flux density of permanent magnets (T)

Da :

Darcy number

g :

Gravitational acceleration (m s−2)

H :

Magnetic field intensity

k :

Thermal conductivity (W m−1 K−1)

L :

Length of the enclosure (m)

p :

Pressure (Pa)

P :

Dimensionless pressure

Pr :

Prandtl number

Ra :

Rayleigh number

S :

Dimensionless entropy generation

T :

Temperature (K)

u, v :

Velocity components (m s−1)

U, V :

Dimensionless velocity components

x, y :

Cartesian coordinates (m)

X, Y :

Dimensionless Cartesian coordinates

\(\alpha\) :

Thermal diffusivity (m s−1)

\(\beta\) :

Thermal expansion coefficient (K−1)

\(\gamma\) :

Dimensionless magnetic strength parameter

κ :

Permeability (m2)

θ :

Dimensionless temperature, \((T - T_{ 0} )/(T_{\text{h}} - T_{\text{c}} )\)

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

Magnetic permeability (H m−1)

\(\nu\) :

Kinematic viscosity (m2 s−1)

ε :

Porosity

\(\rho\) :

Density (kg m−3)

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

Dynamics viscosity (kg m−1 s−1)

\(\sigma\) :

Electrical conductivity (Ω−1 m−1)

\(\phi\) :

Irreversibility distribution ratio

\(\chi\) :

Mass magnetic susceptibility (m3 kg−1)

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

Scalar magnetic potential

0:

Reference value

av:

Spatial average

c:

Cold

h:

Hot

total:

Summation over the domain

θ:

Heat transfer

ψ:

Fluid friction

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Acknowledgements

The authors thank the financial support of the National Natural Science Foundation of China (No. 11572056), the Natural Science Foundation of Hunan Province (No. 2018JJ3533), the Education Department of Hunan Province (No. 17B008) and the China Scholarship Council under the research Grant of 201700800002.

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Authors and Affiliations

  1. School of Energy and Power Engineering, Changsha University of Science and Technology, Changsha, 410114, China

    Er Shi, Xuzhong Zang & Changwei Jiang

  2. Hunan Province 2011 Collaborative Innovation Center of Clean Energy and Smart Grid, Changsha, 410114, China

    Xuzhong Zang & Changwei Jiang

  3. Department of Physics, Temple University, Philadelphia, 19122, USA

    Er Shi

  4. Faculty of Chemical and Petroleum Engineering, University of Tabriz, Tabriz, 5166616471, Iran

    Mousa Mohammadpourfard

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Shi, E., Zang, X., Jiang, C. et al. Entropy generation analysis for thermomagnetic convection of paramagnetic fluid inside a porous enclosure in the presence of magnetic quadrupole field. J Therm Anal Calorim 139, 2005–2022 (2020). https://doi.org/10.1007/s10973-019-08556-3

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