Effect of Uniform Magnetic Field on Melting at Various Rayleigh Numbers

Melting phenomena occurs in various industrial applications, such as metal castings of turbine blades, environmental engineering, PCM-based thermal storage devices, etc. During the design of these devices, they are designed for efficient heat transfer rate. To improve the heat transfer rate, understanding of the important flow processes during the melting (and solidification) is necessary. An objective of the present work is to study the effect of natural convection and magnetic field on interface morphology and thereby on melting rate. In this work, therefore, an effect of uniform transverse magnetic field on the melting inside a cavity, filled initially with solid gallium, at various Rayleigh numbers (Ra=3×10, 6×10, and 9×10) is presented. A 2D unsteady numerical simulation, with the enthalpy-porosity formulation, is performed using ANSYS-Fluent. The magnetic field is characterized by the Hartmann number (Ha) and the results are shown for the Ha = 0, 30 and 50. The horizontal walls of the cavity are considered insulated and vertical walls are respectively considered hot and cold. It is observed that the role of natural convection during the melting is significant on the temperature distribution and solid-liquid interface. The increased magnetic field (Ha = 30 and 50) found to have a suppressing effect on the dominance of natural convection at all Rayleigh numbers (Ra=3×10, 6×10, and 9×10).


1-Introduction
Melting occurs in metal castings of turbine blades, environmental engineering, PCM-based thermal storage devices, nuclear reactor systems, materials processing, solar energy systems, etc. [1]. To improve the melting rate and heat transfer efficiency, understanding of the important flow processes is necessary. The natural convection, for example, can affect the liquid-solid interface morphology. If the liquid is electrically conducting then the interface morphology also depend on the interaction of the buoyancy force and Lorentz force. Thus, the imposed magnetic field can control the melting rate in the electrically conducting fluid. Hence, there is considerable interest in studying the physics that affect the rate of melting (or solidification) preferably inside a rectangular enclosure.
The effect of natural convection on the solid-interface morphology during the melting of gallium has been experimentally studied by Gau and Viskanta (1986) and Dadzis et al. (2016) [2,3]. They considered a classical differentially heated rectangular cavity and observed a strong effect of natural convection on the interface and the heat transfer rate. Numerical work has been conducted, for instance, by Dadzis et al. (2016), Brent et al. (1988), Bertrand et al. (1999) and Farsani et al. (2017) [3][4][5][6]. Also confirmed a strong effect of natural convection on the interface and the heat transfer rate. Various geometry configurations and various boundaries and initial conditions have been studied, for instance, by Rady and Mohanty (1996) [7][8][9][10], to study the effect of natural convection.
The effect of magnetic field on the solid-interface morphology during the melting of gallium inside a rectangular enclosure is studied, for instance, by Farsani [6,[11][12][13][14][15][16][17][18]. The study of convection has not been limited to the external static magnetic field only, but also electric and electromagnetic field [19] and traveling magnetic field [3,10,17,18,20] have been studied. From the above literature, it is interesting to note that the effect of a longitudinal or transverse magnetic field substantially affect the heat and fluid-flow structures.
In the modeling of magneto-hydrodynamic flows, it is important to consider the effect of three-dimensionality and in the homogeneity of the strong magnetic field. Despite the limitation offered in the physical significance, two-dimensional modeling of the flows is often considered as it offers simplification and possibility of analyses over a wide range of nondimensional parameters in lesser time. In this work, a 2D unsteady numerical simulation, using ANSYS-Fluent with the enthalpy-porosity formulation, are performed to study an effect of a uniform horizontal magnetic field on melting inside a cavity filled initially with solid gallium at various Rayleigh numbers (Ra=3×10 5 , 6×10 5 , and 9×10 5 ). By varying the Rayleigh numbers, the buoyancy force is varied. The horizontal walls of the cavity are considered insulated and vertical walls are, respectively, considered hot and cold. The magnetic field and the resulting Lorentz force are characterized by the Hartmann number (Ha) and the results are shown for the Ha = 0, 30 and 50. Figure 1 shows the classical rectangular cavity geometry with boundary conditions. The left and right walls are maintained at a uniform isothermal temperature of T h =311.15K and T c =301.15K, respectively. A horizontal uniform magnetic field, as shown in the figure, is imposed in the horizontal direction, i.e. perpendicular to hot and cold walls. The aspect ratio of the cavity is 1.0 and the cavity is filled with the pure solid gallium. The initial temperature of the cavity is 302.85 K i.e. less than the melting point of gallium. The flow properties, except density, are assumed to be constants, as the maximum temperature difference (i.e., 10) is very small. To perform 2-D unsteady, laminar flow numerical simulations, following governing equations with enthalpy-porosity formulation are solved. The governing equations are:

Continuity equation:
Momentum in y-direction: Where  In the current problem gravity force along with Lorentz force (due to the external magnetic field) is considered. Uniform magnetic field with varying strength (as characterized by the Hartmann number) has been applied normal to the right wall as shown in the figure. The electrically conducting gallium interacts with an external horizontal uniform magnetic field of constant magnitude, B 0 . The magnetic field produced by the movement of liquid gallium has been assumed to be negligible as compared to externally applied magnetic field (low magnetic Reynolds number approximation) and cavity walls are electrically insulating. With this approximation, electromagnetic force reduces to ⁄ where q ′ is the heat flux, ∆Y is the change in length at the point of estimation and ∆T is the temperature change with the reference temperature and k is the thermal conductivity of fluid. In the non-dimensional parameters, g is the acceleration due to gravity, C p is the specific heat, β is the coefficient of thermal expansion, L y is the length of the side of the square cavity, ν is the kinematic viscosity, α is the thermal diffusivity, σ is the electrical conductivity, and ρ is the density.
To solve the governing equations, the SIMPLE method and the second-order accurate scheme in discretizing the momentum and energy conservation equations is used. The pressure is solved using PRESTO!. The stopping criterion for the solution of energy, momentum and continuity, is 10 −6 , 10 −4 and 10 −4 , respectively. Simulations are performed on Rayleigh numbers (Ra=3×10 5 , 6×10 5 , and 9×10 5 ). At fixed Ra, the Hartmann number (Ha) is varied from 0, 30 and 50. The thermo-physical properties of pure gallium considered in the present simulation are mentioned in Table 1. The above discussed methodology is shown in the form of the flowchart in Figure 2.

3-Results and Discussions
To perform the numerical simulation, it is important to perform grid-size independence, time-step size independence and validation and verification of the model. Therefore, grid-size independence, time-step size independence and validation and verification are shown first with results at Rayleigh numbers (Ra=3×10 5 , 6×10 5 , and 9×10 5 ) and the Hartmann number (Ha = 0, 30 and 50) next, followed by the average Nusselt number on the hot wall.

3-2-Effect of the Rayleigh Number and the Hartmann Number
To understand the interactions between natural convection and conduction heat transfer in the melt region, under the influence of imposed uniform magnetic field, the effect of Ra and Ha in terms of solid-liquid interface, contours of streamlines, temperature and velocity magnitude and average Nusselt number on the hot wall is studied. The results are presented in this and next section.  Figure 7 shows the streamline contours at steady-state (i.e., τ = 0.15) for Ra = 3×10 5 , 6×10 5 , and 9×10 5 and Ha = 0, 30 and 50. In the liquid region clockwise rotating vortices are observed as a result of the natural convection that occurred due to the temperature difference between the hot wall and the cooler solid boundary. The liquid near the hot wall rise up and descend near the top insulated wall and the interface and affect the morphology of the interface. Since the top region of the interface always is in contact with the heated molten liquid, as compared to the lower region, the morphology of the interface turns into a convex shape. From Figure 7, it can be observed that the streamlines are much denser at higher Rayleigh numbers. This shows the increase in the strength of the circulation due to natural convection. However, at higher Ha, the denseness of the streamlines decreases and indicates the suppression of natural convection. Figure 8 shows the temperature contours at steady-state (i.e., = 0.15) for Ra=3×10 5 , 6×10 5 , and 9×10 5 and Ha = 0, 30 and 50. From Figure 8, it can be observed that the isotherms are curvy at higher Rayleigh numbers. This shows the increase in natural convection. However, at higher Ha, the isotherms are less curvy indicating a departure from heat convection.
Contours of velocity magnitude at Ra= 6x10 5 and Ha=0, 30, 50 at τ= 0.08, and 0.15 are shown in Figure 9. At fixed Ra, the velocity magnitudes reduce at the steady-state with the increase in Ha. This indicates that increase in the magnetic field suppresses the strength of the circulation and therefore the natural convection.

3-3-Averaged Nusselt Number versus Dimensionless Time
The heat transfer rate at the hot wall is calculated by using the dimensionless averaged Nusselt number ((Nu) avg = (q ′ • ∆Y) (∆T • k) ⁄ ). Figure 10 shows the average Nusselt number on the hot wall as a function of non-dimensional time, for various values of Ra and Ha. During the initial dimensionless time, the Nusselt number is higher at all Ra. The higher values are due to very thin layer of the melt and sharp temperature gradients near the hot wall. As the melt layer increases, the Nusselt number decreases at the later time due to increase in the thermal resistance. From the figure, it can be noticed that the Nusselt number increases with the increase in Ra due to increase in the local velocities and heat transfer rate. On the other hand, the Nusselt number decreases with the increase in Ha due to suppress in the local velocities and the heat transfer rate.

4-Conclusion
Two-dimensional unsteady numerical simulations are performed to study an effect of a uniform horizontal magnetic field on melting inside a cavity filled initially with a solid gallium at various Rayleigh numbers (Ra=3×10 5 , 6×10 5 , and 9×10 5 ) the Hartmann number (Ha = 0, 30 and 50) using ANSYS-Fluent with the enthalpy-porosity formulation. A cavity of aspect ratio one is used in the simulation. The horizontal walls of the cavity are considered insulated. The vertical walls are subjected to differential heating. The simulation results are produced in terms of solid-liquid interface, contours of streamlines, temperature and velocity magnitude and average Nusselt number on the hot wall. It is observed that the natural convection increases with the increase in the strength of buoyancy. On the other hand, the natural convection activity decreases with the increase in the strength of the magnetic field. The Average Nusselt number on the hot wall increases with the increase in Ra and decreases with an increase in Ha. In conclusion the melting rate increases with Ra and decreases with Ha.

5-Conflict of Interest
The authors declare no conflict of interest.