The Effect of a Magnetic Field on the Melting of Gallium in a Rectangular Cavity

Figures & data

Figure 1. Physical model of the storage cavity and the boundary conditions.

Figure 2. Control volume for U-velocity in a staggered grid.

Figure 3. Cavity liquid fraction (CLF) as a function of dimensionless time (Fo) for various time steps (a) and grid sizes (b).

Figure 4. Numerical code validation of the present study with the numerical results of Brent, Voller, and Reid [19], and the experimental study of Gau and Viskanta [13].

Table 1. Thermophysical properties of gallium and other non-dimensional parameters used in this study.

Parameter	Value
Density, ρf	6093 kg.m^− 3
Volumetric thermal expansion coefficient of liquid, β	1.2 × 10^− 4°C^− 1
Thermal conductivity, k	32.0 W.m^− 1.°C ^− 1
Melting point, Tl	29.78°C
Latent heat fusion, hla	80160 J.kg^− 1
Thermal diffusivity, ′af	1.38 × 10^− 5 m^2.s^− 1
Acceleration due to gravity, g	9.81 m.s^− 2
Aspect ratio, R	0.714
Prandtl number, Pr	0.0216
Ratio of heat capacity of solid and liquid, cs/cf	1.0
Ratio of thermal conductivity of solid and liquid, ks/kf	1.0

Figure 5. Streamlines at three different dimensionless times (Fo) for various Rayleigh numbers (Ra) and Ha = 0.

Figure 6. Isotherms at three different dimensionless times (Fo) for various Rayleigh numbers (Ra) and Ha = 0.

Figure 7. The solid–liquid phase interface at three different dimensionless times (Fo) for various Rayleigh numbers (Ra) and Ha = 0.

Figure 8. Evolution of the CLF in dimensionless time (Fo) for various Rayleigh numbers (Ra) and Ha = 0.

Table 2. The dimensionless time (Fo) for approaching steady state, and the corresponding cavity liquid fraction for various Rayleigh numbers (Ra).

Ra	Fo	Cavity liquid fraction
10^3	17.6	0.826
10^4	15.4	0.864
6.5 × 10^5	7.3	0.807
10^6	7.5	0.820

Figure 9. Average Nusselt number at the hot wall as a function of dimensionless time (Fo) for different Rayleigh numbers (Ra) and Ha = 0.

Figure 10. Streamlines at three different dimensionless times (Fo) for various Hartmann numbers (Ha) and Ra = 6.5 × 10⁵.

Figure 11. Isotherms at three different dimensionless times (Fo) for various Hartmann numbers (Ha) at Ra = 6.5 × 10⁵.

Figure 12. Solid–liquid interface at three different dimensionless times (Fo) for various Hartmann numbers (Ha) and Ra = 6.5 × 10⁵.

Figure 13. Evolution of the CLF in dimensionless time (Fo) for different Hartmann numbers (Ha) and Ra = 6.5 × 10⁵.

Table 3. The dimensionless time (Fo) for approaching steady state, and the corresponding cavity liquid fraction for various Hartmann numbers (Ha). The Rayleigh number in all these cases is Ra = 6.5 × 10⁵.

Ha	Fo	Cavity liquid fraction
0	7.3	0.807
25	7.7	0.745
50	8.2	0.710
100	9.6	0.621

Figure 14. Average Nusselt number at the hot wall versus dimensionless times (Fo) for different Hartmann numbers (Ha) and Ra = 6.5 × 10⁵.

A. D. Brent, V. R. Voller, and K. T. J. Reid, “Enthalpy-porosity technique for modeling convection-diffusion phase change: application to the melting of a pure metal,” Numer. Heat Transfer, Part, vol. 13, no. 3, pp. 297–318, 1988.

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C. Gau, and R. Viskanta, “Melting and solidification of a pure metal on a vertical wall,” J. Heat Transfer, vol. 108, no. 1, pp. 174–181, 1986. DOI:10.1115/1.3246884.

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