Advertisement

Investigation of solid/liquid interface evolution in the solidification process of liquid metal in an annulus crucible at the presence of static magnetic field: numerical study

  • Vahid AhmadpourEmail author
  • Iraj Mirzaei
  • Sajad Rezazadeh
  • Nima Ahmadi
Review Paper
  • 7 Downloads

Abstract

This paper studies the effect of constant magnetic field on solidification of a conductive liquid that filled a vertical three-dimensional annulus-shaped crucible. The computational domain of the proposed geometry is a vertical annulus cavity; the inner and outer sides of the mentioned cavity have constant temperature, but top and bottom sides have been thermally insulated. The governing equations have been solved numerically using finite volume method, so obtained results illustrated acceptable conformity with available reported experimental and numerical data. The investigation has been carried out for various vital parameters, such as Hartmann and Rayleigh numbers. The magnetic field effect on the melted flow field and interface of solid/liquid form and position has been discussed with more details. The numerical results have shown a strong relation between interface shape, Rayleigh number and magnetic field strength. Then, the applied magnetic field leads to vanishing axisymmetric condition of solid/liquid interface. Finally, it has been conducted that strongest stabilization of the flow field and solid/liquid interface occurs at the xz plane. The proposed predictions are very helpful in developing methods to improve solidification and casting systems under magnetic field effects.

Keywords

Magnetic field Solidification Electric field Natural convection Solid/liquid interface 

List of symbols

A = L/D

Aspect ratio of annulus

\(\overrightarrow {{B_{o} }}\)

External magnetic field, Tesla (Wb/m2)

Cp

Heat capacity (J/kg K)

D = (ro − ri)

Annulus width (reference length) (m)

\(\overrightarrow {E}\)

Electric field (V/m)

\(\overrightarrow {F}\)

Body force (N/m3)

g

Acceleration coefficient of gravity (m/s2)

L

Annulus height (m)

fl

Liquid fraction

hf

Latent heat of fusion

\(Ha = D B_{0} \sqrt {\frac{\sigma }{\rho \nu }}\)

Hartmann number

\(\overrightarrow {J}\)

Electric current density (A/m2)

\(Nu = \frac{D}{{T_{H} - T_{C} }}\left. {\frac{\partial T}{\partial r}} \right|_{{r = r_{o} }}\)

Nusselt number

\(\overline{Nu} = \frac{1}{2\pi }\int\limits_{0}^{2\pi } {\int\limits_{0}^{H} {Nu \partial \theta \partial z} }\)

Average Nusselt number

P

Pressure (pa)

\(Pr = \frac{\nu }{\alpha }\)

Prandtl number

\(\overrightarrow {r}\)

Radial unit vector

\(Ra = \frac{{g\beta (T_{H} - T_{m} )(r_{o} - r_{i} )^{3} \ }}{\nu \alpha }\)

Rayleigh number

\(R_{\text{m}} = \mu_{0} \sigma VD\)

Magnetic Reynolds number

Ste = Cp(TH − Tm)/Lf

Stefan number

T

Temperature (K)

Tc

Cold wall temperature (K)

TH

Hot wall temperature (K)

Tini

Initial temperature (K)

Tm

Phase change temperature (K)

To

Reference temperature (K)

u

Radial velocity vector (m/s)

v

Angular velocity vector (rad/s)

w

Axial velocity vector (m/s)

z

Coordinate (m)

Greek letters

α

Thermal diffusivity (m2/s)

β

Thermal expansion coefficient (K−1)

θ

Coordinate (rad)

\(\overrightarrow {\theta }\)

Angular unit vector

λ = ro/ri

Annulus radial aspect ratio (m)

μ

Dynamic viscosity of fluid (kg/m s)

μ0

Magnetic permeability (m/s Ω)

\(\nu\)

Kinematic viscosity of fluid (m2/s)

ρ

Density (kg/m3)

ρo

Density at T = To (kg/m3)

σ

Electric conductivity of melt (Ω m−1)

Notes

References

  1. 1.
    Gau C, Viskanta R (1986) Melting and solidification of a pure metal on a vertical wall. J Heat Transf 108:174–181CrossRefGoogle Scholar
  2. 2.
    Wolff F, Viskanta R (1988) Solidification of a pure metal at a vertical wall in the presence of liquid superheat. Int J Heat Mass Transf 31(8):1735–1744CrossRefGoogle Scholar
  3. 3.
    Szekely J, Chhabra PS (1970) The effect of natural convection on the shape and movement of the melt-solid interface in the controlled solidification of lead. Metall Trans 1(5):1195–1203CrossRefGoogle Scholar
  4. 4.
    Campbel TA, Koster JN (1994) visualization of liquid–solid interface morphologies in gallium subject to natural convection. Cryst Growth 140(3–4):414–425CrossRefGoogle Scholar
  5. 5.
    Ben-David O, Levy A, Mikhailovich B, Azulay A (2013) 3D numerical and experimental study of gallium melting in a rectangular container. Int J Heat Mass Transf 67:260–271CrossRefGoogle Scholar
  6. 6.
    Rady MA, Mohanty AK (1996) Natural convective during melting and solidification of pure metals in a cavity. Numer Heat Transf Part A 29(1):49–63CrossRefGoogle Scholar
  7. 7.
    Franke S, Buttner L, Czarske J, Rabiger D, Eckert S (2010) Ultrasound Doppler system for two-dimensional flow mapping in liquid metals. Flow Meas Instrum 21(3):402–409CrossRefGoogle Scholar
  8. 8.
    Davidson PA (2001) Introduction to magneto hydrodynamic. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  9. 9.
    Garandet JP, Alboussiere T, Moreau R (1992) Buoyancy driven convection in a rectangular enclosure with a transverse magnetic field. Int J Heat Mass Transf 35:741–748CrossRefGoogle Scholar
  10. 10.
    Sankar M, Venkatachalappa M, Shivakumara IS (2006) Effect of magnetic field on natural convection in a vertical cylindrical annulus. Int J Eng Sci 44:1556–1570MathSciNetCrossRefGoogle Scholar
  11. 11.
    Xu B, Li BQ, Stock DE (2006) An experimental study of thermally induced convection of molten gallium in magnetic fields. Int J Heat Mass Transf 49:2009–2019CrossRefGoogle Scholar
  12. 12.
    Teimouri H, Afrand M, Sina N, Karimipour AIsfahani AHM (2015) Natural convection of liquid metal in a horizontal cylindrical annulus under radial magnetic field. Int J Appl Electromagn Mech 49:453–461CrossRefGoogle Scholar
  13. 13.
    Sankar M, Venkatachalappa M, Do Y (2011) Effect of magnetic field on the buoyancy and thermo capillary driven convection of an electrically conducting fluid in an annular enclosure. Int J Heat Fluid Flow 32:402–412CrossRefGoogle Scholar
  14. 14.
    Wrobel WA, Wajs EF, Szmyd JS (2012) Analysis of the influence of a strong magnetic field gradient on convection process of paramagnetic fluid in the annulus between horizontal concentric cylinders. J Phys Conf Ser 395:012124.  https://doi.org/10.1088/1742-6596/395/1/012124 CrossRefGoogle Scholar
  15. 15.
    Kakarantzas SC, Sarris IE, Vlachos NS (2011) Natural convection of liquid metal in a vertical annulus with lateral and volumetric heating in the presence of a horizontal magnetic field. Int J Heat Mass Transf 54:3347–3356CrossRefGoogle Scholar
  16. 16.
    Afrand M, Toghraie D, Karimipour A, Wongwises S (2017) A numerical study of natural convection in a vertical annulus filled with gallium in the presence of magnetic field. J Magn Magn Mater 430:22–28CrossRefGoogle Scholar
  17. 17.
    Dulikravich GS, Ahuja V, Lee S (1994) Modeling three-dimensional solidification with magnetic fields and reduced gravity. Int J Heat Mass Transf 37:837–853CrossRefGoogle Scholar
  18. 18.
    Sampath R, Zabaras N (2001) Numerical study of convection in the directional solidification of a binary alloy driven by the combined action of buoyancy, surface tension, and electromagnetic forces. J Comput Phys 168:384–411CrossRefGoogle Scholar
  19. 19.
    Colaco MJ, Dulikravich GS, Martin TJ (2003) Reducing convection effects in solidification by applying magnetic fields having optimized intensity distribution. In: Proceedings of HT2003 ASME summer heat transfer conference Las Vegas NV July, 21–23Google Scholar
  20. 20.
    Colaço MJ, Dulikravich GS (2005) A multilevel hybrid optimization of magnetohydrodynamic problems in double-diffusive fluid flow. In: Proceedings of the third meeting of the study of matter at extreme conditions (SMEC) edSaxena S Miami Beach FL April, 17–21Google Scholar
  21. 21.
    Farsani RY, Raisi A, Nadooshan AA, Vanapalli S (2017) The effect of a magnetic field on the melting of gallium in a rectangular cavity. Heat Transf Eng 40:1–13Google Scholar
  22. 22.
    Bondareva NS, Sheremet MA (2015) Study of melting of a pure gallium under influence of magnetic field in a square cavity with a local heat source. IOP Conf Ser Mater Sci Eng 93:012004CrossRefGoogle Scholar
  23. 23.
    Mechighel F, Kadja M (2007) External horizontally uniform magnetic field applied to steel solidification. J Appl Sci 7:903–912CrossRefGoogle Scholar
  24. 24.
    Charmchi M, Zhang H, Li W, Faghri M (2004) Solidification and melting of gallium in the presence of magnetic field: experimental simulation of low gravity environment. In: ASME 2004 international mechanical engineering congress and exposition. American Society of Mechanical Engineers Digital Collection January, pp 581–589Google Scholar
  25. 25.
    Bouabdallah S, Bessaih R (2012) Effect of magnetic field on 3D flow and heat transfer during solidification from a melt. Int J Heat Fluid Flow 37:154–166CrossRefGoogle Scholar
  26. 26.
    Ben-David O, Levy A, Mikhailovich B, Azulay A (2014) Magneto hydrodynamic flow excited by rotating permanent magnets in an orthogonal container. Phys Fluids 26:1–18CrossRefGoogle Scholar
  27. 27.
    Avnaim MH, Mikhailovich B, Azulay A, Levy A (2018) Numerical and experimental study of the traveling magnetic field effect on the horizontal solidification in a rectangular cavity part 2: acting forces ratio and solidification parameters. Int J Heat Fluid Flow 69:9–22CrossRefGoogle Scholar
  28. 28.
    Vives CH, Perry CH (1986) solidification of pure metal in the presence a stationary magnetic field. Int Commun Heat Mass Transf 13:253–263CrossRefGoogle Scholar
  29. 29.
    Wang X, Fautrelle Y, Moreau R, Etay J, Bianchi A, Baltaretu F, Na X (2015) Flow, heat and mass transfers during solidification under traveling/rotating magnetic field. Int J Energy Environ Eng 6(4):367–373CrossRefGoogle Scholar
  30. 30.
    Swaminathan CR, Voller VR (1992) A general enthalpy method for modeling solidification processes. Metall Trans B 23(5):651–664CrossRefGoogle Scholar
  31. 31.
    Brent AD, Voller VR, Reid KJ (1988) Enthalpy porosity technique for modeling convection-diffusion phase change: application to the melting of a pure metal. Numer Heat Transf 13(3):297–318.  https://doi.org/10.1080/10407788808913615 CrossRefGoogle Scholar
  32. 32.
    Sin K, Anghaie S, Chen G (2001) A fixed-grid two-phase numerical model for convection-dominated melting and solidification. In: Proceedings first MIT conference on computational fluid and solid mechanics. MIT Cambridge, MA, 12–14 JuneGoogle Scholar

Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2019

Authors and Affiliations

  • Vahid Ahmadpour
    • 1
    • 4
    Email author
  • Iraj Mirzaei
    • 2
  • Sajad Rezazadeh
    • 1
  • Nima Ahmadi
    • 3
  1. 1.Faculty of Mechanical EngineeringUrmia University of TechnologyUrmiaIran
  2. 2.Mechanical Engineering DepartmentUrmia UniversityUrmiaIran
  3. 3.Department of Mechanical Engineering, Faculty of Shaheed BeheshtiTechnical and Vocational University (TVU)UrmiaIran
  4. 4.TabrizIran

Personalised recommendations