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Two-phase simulation of magnetic field effect on the ferrofluid forced convection in a pipe considering Brownian diffusion, thermophoresis, and magnetophoresis

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Abstract

In this paper, a two-phase model is used to study Fe3O4/H2O magnetic nanofluid laminar forced convection in a uniformly heated pipe and under the influence of a quadrupole magnetic field. The extended model considers the three primary transport mechanisms of magnetic nanoparticles in the carrier liquid, namely Brownian diffusion, thermophoresis, and magnetophoresis. Governing equations are solved numerically utilizing a SIMPLE-based finite volume method. Computations are performed for various important parameters including magnetic and Reynolds numbers, particle size and volume fraction, and the magnetic source length. Numerical results show that some vortices are formed due to magnetic field which leads to heat transfer augmentation. It is found that in the sections where the magnetic field is applied, the particle distribution is almost uniform, and also the radial component of Brownian, thermophoresis, and magnetophoresis fluxes are comparable. Far away from the magnetic field, there is an obvious non-uniform particle distribution because of the thermophoretic diffusion. Numerical results indicate that the heat transfer enhancement is an increasing function of the magnetic number, particle size, and volume fraction, while a decreasing function of the magnetic source length and Reynolds number. Multiple magnetic sources provide higher heat transfer rates and hydro-thermal performance. The obtained simulation results reveal that such a magnetic field can increase the heat transfer rate more than three times.

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Abbreviations

c :

Specific heat at constant pressure (J kg−1 K−1)

D :

Pipe diameter (m)

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

Brownian diffusivity (m2 s−1)

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

Particle diameter (m)

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

Thermophoretic diffusivity (m2 s−1)

H :

Magnetic field intensity (A m−1)

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

Particle mass flux vector (kg m−2 s−1)

k :

Thermal conductivity (W m−1 K−1)

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

Boltzmann constant (\( 1.380648 \times 10^{ - 23} \) J K−1)

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

Magnetic source length (m)

M s :

Saturation magnetization (A m−1)

\( {\text{Mn}} = \frac{{\mu_{\text{o}} H_{\text{o}}^{2} D^{2} }}{{\rho_{\text{f}} \alpha_{\text{f}}^{2} }} \) :

Magnetic number (−)

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

Magnetic moment of nanoparticles (A m2)

\( {\text{Nu}} = \frac{1}{{T_{\text{w}}^{ *} \left( {z^{ *} } \right) - T_{\text{m}}^{ *} \left( {z^{ *} } \right)}} \) :

Nusselt number (−)

p :

Pressure (Pa)

\( { \Pr } = \frac{{\left( {\mu c} \right)_{\text{f}} }}{{k_{\text{f}} }} \) :

Prandtl number (−)

q″:

Heat flux (W m−2)

\( q^{ *} = \frac{{q_{\text{w}}^{''} D}}{{k_{\text{f}} T_{\text{in}} }} \) :

Non-dimensional heat flux (−)

r :

Radial coordinate (m)

\( {\text{Re}} = \frac{{V_{\text{in}} D}}{{\upsilon_{\text{f}} }} \) :

Reynolds number (−)

T :

Temperature (K)

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

Mean fluid temperature (K)

\( V_{\text{r}} , V_{\text{z}} \) :

Velocity components (m s−1)

z :

Axial coordinate (m)

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

Thermal diffusivity (m2 s−1)

\( \mu \) :

Dynamic viscosity (Pa s−1)

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

Bohr magneton (\( 9.274 \times 10^{ - 24} \) A m2)

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

Vacuum permeability (TmA−1)

\( \xi = \frac{{\mu_{\text{o}} m_{\text{p}} H}}{{k_{\text{B}} T}} \) :

Langevin parameter (−)

\( \rho \) :

Density (kg m−3)

\( \phi \) :

Particle volume fraction (−)

\( \chi \) :

Magnetic susceptibility (−)

f:

Base fluid

ff:

Ferrofluid

in:

Inlet

p:

Nanoparticle

w:

Wall

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Authors and Affiliations

  1. Department of Mechanical Engineering, Urmia University of Technology, P.O.B. 57155-419, Urmia, Iran

    Hossein Soltanipour

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Soltanipour, H. Two-phase simulation of magnetic field effect on the ferrofluid forced convection in a pipe considering Brownian diffusion, thermophoresis, and magnetophoresis. Eur. Phys. J. Plus 135, 702 (2020). https://doi.org/10.1140/epjp/s13360-020-00725-w

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