Mekanika : majalah ilmiah mekanika 53 Volume 16 Nomor 2 September 2017 NATURAL CONVECTION NUMERIC SIMULATION ON METAL FREEZING USING DIFFERENTIAL METHOD

The research of modeling of natural convection in metal solidification process with finite different method was conducted to determine temperature distribution and fluid flow profil with variations value Rayleigh number. The research conducted by solving governing equation of natural convection with finite difference approximation. Governing equation of natural convection consist of continuity equation, momentum equations, and energy equation. The ADI (Alternating Directional Implicit) method was used to discriteze for governing equation of natural convection. Finite difference method was written in Fortran language whereas the temperature distribution and fluid flow profile were visualized with Matlab software. The results of this research was validated by comparing the results obtained with Rajiv Sampath research. Comparison of the results of research showed good agreement. The result showed that solidification process occurs faster at Ra compared with and


INTRODUCTION
Numerous things related to the heat transfer are found in everyday life, especially in the field of industry.Heat transfer can occur in three ways, namely conduction, convection, and radiation.
Convection is a heat transfer that occurs between solid surface to the moved fluid which caused by the temperature difference within.Convection which is based on the fluid flow origins are categorized into two categories, namely the forced convection and natural convection.
Forced Convection is heat transfer fluid flow convection which happened influenced by external tools, such as fans, pumps, and others.While natural convection is heat transfer fluid flow convection which is caused by the differences in fluid density caused by heating and cooling.
Natural convection plays an important role in the engineering industry, one of them in the metal solidification process.Research which concerned on natural convection freezing problem is extremely crucial, because the fluid flow which caused by natural convection in liquid state, it changing the shape of the liquid/ solid interface and temperature distribution during freezing (Yinheng, 1994).
Physical phenomena that control the solid/ liquid interface shape during freezing are becoming necessary in numerous industrial processes.Its main characteristic is that the interface moves to separate the two phases with different physical properties.Differences in temperature cause the buoyancy liquid produces significant convection currents.Natural convection has a major influence on the morphology of its interface, freezing rate, and temperature distribution (Mohammad 2009).
Research about the natural convection issue of metals freezing has been carried out both experimental and numerically.Experimental laboratory research requires a significant financial cost and the process is quite complicated.Therefore, numerically study was developed which much cheaper.Various methods of numerical approach to determine the natural convection phenomenon has been done, using a mathematical model of the continuity equation, momentum equation, and energy equation.
Numerical study on natural convection issue of metal freezing was growing rapidly from year to year.
McDaniel and Zabaras (1994) made a 2D numerical modeling on the basis of natural convection phase transformation on the issues of freezing and thawing of pure metals using the finite element method.Chen and Yoo (1995) analyzed the natural convection of aluminum freezing process by applying finite element method.Sampath and Zabaras (1999) created 2D and 3D numerical modeling on it utilizing the finite element method.Sampath and Zabaras research studied it on pure metals and alloys.Mohammad (2010) examined the numerical simulations in freezing water in a square mold in natural convection by employing finite volume method.Balhamadia, Kane, and Fortin (2012) made a phase transition modeling by natural convection in the water freezing and thawing gallium.

SCOPE
This study is aimed to make a natural convection modeling in the metal solidification process with finite difference methods.The velocity vector and temperature distribution were included.

LITERATURE REVIEW
McDaniel and Zabaras (1994) made a 2D numerical modeling on the basis of natural convection phase transformation on the freezing and thawing issues of pure metal using the finite element method.There were two cases analyzed.It was 105 and 106 of Rayleigh numbers.The boundary conditions that used were top, bottom, and right side insulation while the left side was the convection.
Chen and Yoo (1995) analyzed the freezing process aluminum in natural convection with the finite element method.
Sampath and Zabaras (1999) created 2D and 3D numerical modeling on the basis of natural convection phase transformation using the finite element method.Sampath and Zabaras Research investigated the freezing and thawing problem of pure metals and alloys.Basically, this study was continued by Zabaras research (1994).
Mohammad (2010) studied the numerical simulations in freezing water in a square mold in atural convection by using finite volume method.Balhamadia, Kane, and Fortin (2012) fabricated the phase transition modeling by natural convection in the water freezing and thawing gallium.

RESEARCH METHODOLOGY 4.1 Research Procedure
Research was done by making the program implementation to resolve the momentum equation, energy equation, and the continuity equation with the ADI method.
An outline of the research can be made following flow chart:  ( ) .θ.

Y-axis Momentum Equation
Therefore those equations, would be collected into unknown variable on left side and known variable on right side: Discretization of momentum equation used y-axis momentum equation y direction without including the pressure element.Therefore, the equation becomes:   Visually, nowadays research shows similarities of flow temperature distribution and velocity with Rajiv Sampath study (1999).Thereforem, it can be said using the finite difference method has a good fit.

Natural Convection Simulation on Metal Solidification Process
Natural convection simulation cases on metal solidification process on a square mold is shown with the 81x81 grid, 0.0149 of Prandlt number (Pr), and dt= 0.001 of time step, and Rayleigh number (Ra) variations were 104.105 and 106.The simulation results can be seen in the following image: (c) t=40 (d) t=60 Temperature tranformation can be seen in the temperature distribution graph which represented by a point (x, 0.1), wherein x were 0.1, 0.2, 0.3, 0.4, 0.5, 0.6 , 0.7, 0.8, 0.9, and 1.A temperature distribution graph is as follows: Temperature distribution can be seen in the temperature distribution graph which performed by a point (x, 0.1), wherein x is 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7 , 0.8, 0.9, and 1. Temperature distribution graph is as follows:  The initial solidification comparison which according to the Ra variations can be seen in Figure 4.13.It portrays that the initial freezing at Ra= 104 is faster than Ra= 105 and 106.It can be concluded that rayleigh number greatly affects the freezing process, the greater rayleigh number the longer it will reverse the freezing process so shall.

CONCLUSION
From this research and discussion that has been done.It produced a numerous conclusions: Research comparison of Rajiv Sampath study (1999) for natural convection problem in metal solidification process on a square mold indicates that the method which was used in this study can provide acceptable results in those cases.

Figure 4 . 1 .
Research flowchart4.2Control EquationNatural convection equation set consists of the continuity equation, momentum equation, and energy equation.In order to set the natural convection equations can be applied in the programming language.Firstly, discretization equation set was set up.This study implementation, the discretization equation solved by the method set ADI.By defining ( ) unknown variable (such as, u, v, and θ).
axis Momentum Equation Discretization of momentum equation used x-axis momentum equation x direction without including the pressure element.Therefore, the equation becomes: ( ) θ Discretization for each equation above can be explained: a. X-Sweep ❖ X-sweep equation for momentum equation x-axis is ❖ [ .( -) .] .( -) .] -By substituting equations above and multiplying with -❖ -, can be obtained matrix tridiagonal coefficient: example , so, the equation above becomes: .θ. Arrangement above idencally with equation with components ai, bi, ( ) θ a. X-Sweep X-sweep equation for momentum equation x-axis is: ci and di each is tridiagonal matrix components Mekanika : majalah ilmiah mekanika 56 Volume 16 Nomor 2 September 2017 -By substitu ting equations above and multiplying with , can be obtained: tridiagonal matrix coefficient: θ ( -)θ .θ] By substituting equations above and multiplying with -By substituting equations above and multiplying with , can be obtained: tridiagonal matrix coefficient: Isothermal comparison results between the current study to Sampath research above shows good accuracy with 5862 for the 61x61 grid and 2892 for 81x81 of a maximum error.It shows for the 81x81 5. RESULT AND DISCUSSION 5.1 Program Validation Validation program was done by comparing the current research to the Rajiv Sampath research (1999).Rajiv Sampath research domain (1999) was the resolution of natural convection case in metal solidification process on a square mold with 1: 1 of aspect ratio, with the walls below, above, and right side condition were insulating, while the left Wall was convection.

Figure 5 . 1
Figure 5.1 Boundary and require research condition Rajiv Sampath research (1999) was using finite element method with the same boundary conditions to the present study boundary conditions.Isothermal visualization and velocity vector results will be compared with Rajiv Sampath research (1999) at Ra = 105.Isothermal comparison results is shown in followingTable 4.1.and Table 4.2: Table 4. 1. Isotermal comparison result at t=20

Figure 4 .Figure 5 .Figure 5 .
Figure 5.10.Velocity vector at Re= 10 4 a. Cold fluid moved down and the hot fluid moving upwards.Cold fluid movement was influenced by their gravity and density changes due to its temperature transformation which caused the density went up, while the movement of hot fluid was affected by Buoyancy force due to it has a smaller density than the cold fluid.b.Rayleigh number greatly affects the metal solidification speed.The smaller of Rayleigh, the faster the solidification process and the greater Rayleigh number clots more slowly.