199 mathematical modeling of mold-filling and solidification - Acta ...
199 mathematical modeling of mold-filling and solidification - Acta ... 199 mathematical modeling of mold-filling and solidification - Acta ...
Acta Metallurgica Slovaca, 15, 2009, 3 (199 - 207) 204 explicitly, but are computed by solving the flow equations and these flow equations are solved in terms of mixture variables. Since densities of liquid metal and air differ greatly (e.g., by a factor of 7000 for steel), the mixture velocity may not always be an accurate measure of the relative motion of metal and air [16]. Kim et al. [17] proposed an interesting flow chart of the mold filling computational procedure, as shown in Fig. 1. In the first stage of the numerical analysis, an initial grid is generated and values for the order of surface refinement, material properties, boundary conditions, and initial volume fractions are the model input. Then, the time step is increased and the filling pattern is selected for each element, and the free surface is predicted by the filling pattern. In the next stage, through the refinement and mergence procedure, an adaptive grid is generated. According to the volumetric fraction at a given time during filling, the domain element for the FEM (Finite Element Method) analysis is created. Then, by the FEM analysis, the velocity and pressure fields are obtained and the flow rate in each element is calculated. Subsequently, the procedure of advection treatment is accomplished and the volume fraction in each element is obtained. These procedures are iterated until the current filling time reaches the total time. Fig.1 Flow chart of the computational procedure [17] One of the requirements that should be satisfied in the VOF method for mold filling, is the conservation of the F(r,t) function even though with the presence of convection. This method uses a minimum of stored information and it follows regions rather than boundaries, which avoids the logic problems associated with intersecting surfaces. The derivatives of this function can be used to estimate the orientation of the fluid surface, improving the computational efficiency [18].
Acta Metallurgica Slovaca, 15, 2009, 3 (199 - 207) 205 2.3 Equations for solidification The mathematical formulation of heat transfer to predict the temperature distribution during solidification is based on the general equation of unsteady state heat conduction, which is given by [1; 15; 19; 20; 21; 22]: ∂ ( ρ . c. T ) = ∇.( k∇T ) + Q& (7) ∂ t where ρ is the density [kgm -3 ]; c is the specific heat [J kg -1 K -1 ]; k is the thermal conductivity [Wm -1 K -1 ]; ∂T/∂t is the cooling rate [K s -1 ], T is the temperature [K], t is time [s], x, y and z are the space coordinates [m] and Q & represents the term associated to the internal heat generation due to the phase change. For this equation, it was assumed that the thermal conductivity, density and specific heat vary with temperature [22]. The boundary condition at the mold external surface is given by: ∂T − -k = h ( T - T∞ ) (8) ∂n Where h is the ambient surface convection heat transfer coefficient associated with free convection, n is the unit vector outward normal to the sand mold, T ∞ is the environment air temperature and T is the temperature at the outer mold wall. 2.3.1. Phase change When there is a phase change from liquid to solid, a heat source term arises, which is given by an explicit solid fraction–temperature relationship. The solid fraction depends on a number of parameters; however, it is quite reasonable to assume f s varying only with temperature, where f s is the solid fraction and L is the latent heat of fusion [J kg -1 ]. So, we can write: c ’ = c–L.∂f s /∂T (9) The term c’ can be considered a pseudo-specific heat. The sub-indices S and L indicate solid and liquid, respectively [22]. In the current system, no external heat source is applied and the only heat generation is due to the latent heat of solidification, L (Jkg -1 ) or ∆H (Jm -3 ). Q & is proportional to the rate of change of the solid fraction, f s , as it follows [21; 22; 23]: f f Q & ∂ H s ∂ = ∆ = ρL s (10) ∂t ∂t In many systems, especially when the undercooling is small, the solid fraction may be assumed as being dependent on temperature only. Different forms have been proposed to take into account the evolution of solid fraction with temperature. The simplest approach is that given by a linear relationship [23]: f s ⎧0 ⎪ = ⎨( Tl ⎪ ⎩ − T ) /( Tl − Ts ) 1 T > Tl Ts ≤ T ≤ Tl T < Ts (11)
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<strong>Acta</strong> Metallurgica Slovaca, 15, 2009, 3 (<strong>199</strong> - 207) 205<br />
2.3 Equations for <strong>solidification</strong><br />
The <strong>mathematical</strong> formulation <strong>of</strong> heat transfer to predict the temperature distribution<br />
during <strong>solidification</strong> is based on the general equation <strong>of</strong> unsteady state heat conduction, which is<br />
given by [1; 15; 19; 20; 21; 22]:<br />
∂ ( ρ . c.<br />
T ) = ∇.(<br />
k∇T<br />
) + Q&<br />
(7)<br />
∂ t<br />
where ρ is the density [kgm -3 ]; c is the specific heat [J kg -1 K -1 ]; k is the thermal conductivity<br />
[Wm -1 K -1 ]; ∂T/∂t is the cooling rate [K s -1 ], T is the temperature [K], t is time [s], x, y <strong>and</strong> z are<br />
the space coordinates [m] <strong>and</strong> Q & represents the term associated to the internal heat generation<br />
due to the phase change. For this equation, it was assumed that the thermal conductivity, density<br />
<strong>and</strong> specific heat vary with temperature [22].<br />
The boundary condition at the <strong>mold</strong> external surface is given by:<br />
∂T<br />
− -k = h ( T - T∞<br />
)<br />
(8)<br />
∂n<br />
Where h is the ambient surface convection heat transfer coefficient associated with<br />
free convection, n is the unit vector outward normal to the s<strong>and</strong> <strong>mold</strong>, T ∞ is the environment air<br />
temperature <strong>and</strong> T is the temperature at the outer <strong>mold</strong> wall.<br />
2.3.1. Phase change<br />
When there is a phase change from liquid to solid, a heat source term arises, which is<br />
given by an explicit solid fraction–temperature relationship. The solid fraction depends on a<br />
number <strong>of</strong> parameters; however, it is quite reasonable to assume f s varying only with<br />
temperature, where f s is the solid fraction <strong>and</strong> L is the latent heat <strong>of</strong> fusion [J kg -1 ]. So, we can<br />
write:<br />
c ’ = c–L.∂f s /∂T (9)<br />
The term c’ can be considered a pseudo-specific heat. The sub-indices S <strong>and</strong> L<br />
indicate solid <strong>and</strong> liquid, respectively [22].<br />
In the current system, no external heat source is applied <strong>and</strong> the only heat generation<br />
is due to the latent heat <strong>of</strong> <strong>solidification</strong>, L (Jkg -1 ) or ∆H (Jm -3 ). Q & is proportional to the rate <strong>of</strong><br />
change <strong>of</strong> the solid fraction, f s , as it follows [21; 22; 23]:<br />
f f<br />
Q & ∂<br />
H<br />
s ∂<br />
= ∆ = ρL<br />
s<br />
(10)<br />
∂t<br />
∂t<br />
In many systems, especially when the undercooling is small, the solid fraction may be<br />
assumed as being dependent on temperature only. Different forms have been proposed to take<br />
into account the evolution <strong>of</strong> solid fraction with temperature. The simplest approach is that given<br />
by a linear relationship [23]:<br />
f s<br />
⎧0<br />
⎪<br />
= ⎨(<br />
Tl<br />
⎪<br />
⎩<br />
− T ) /( Tl<br />
− Ts<br />
)<br />
1<br />
T > Tl<br />
Ts<br />
≤ T ≤ Tl<br />
T < Ts<br />
(11)
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