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) 200 model for simulation of mold-filling and solidification of a binary alloy. For the mold filling process, turbulence models in addition to the Fractional Volume (Volume of Fluid - VOF) algorithm is used to model the computational fluid dynamics (CFD). The approach allows one to handle, with at least second order accuracy, the continuously moving boundary between the injected fluid and the air that initially is in the mold cavity. The liquid metal assumes Newtonian fluid and incompressible fluid conditions. For the solidification process, the mathematical formulation of heat transfer is based on the general equation of unsteady state heat conduction. Three forms of latent heat release during solidification are presented: linear, exponential and sinusoid behavior as a function of the solid fraction. Keywords: numerical simulation, finite-element method, mold-filling, solidification, heat generation 1. Introduction In the last years a perceptible evolution of numerical modeling in all science areas can be easily detected [1]. Particularly for casting processes, a great progress happened due to the evolution of both hardware and software used for simulation purposes. The stages of casting can be defined as: heating-melting, mold-filling and solidification. Each stage has a great influence on the final casting quality. Mold-filling can be classified as a problem of fluid mechanics with many phenomena involved such as, molten metal turbulence; heat transfer along the metal/mold system; pressure variation inside the casting, etc. [2]. The modeling of solidification associated to the fluid dynamics permits many phenomena and parameters which are important to assure a free defect product to be analyzed, e.g. [3]: heat transfer in the whole system (casting and mold), thermal gradients, macrosegregation and microsegregation [4], thermal stresses in metal and mold, porosity formation, etc. Despite its importance, the simulation of the casting process becomes a challenge due to the complexity of the simultaneous fluid-thermal-mechanics phenomena which are involved. Besides, they are transient phenomena and experimental measurements permitting fluid motion to be characterized are not an easy task [2]. In this paper a numerical model is developed permitting the simulation of both mold filling and solidification of binary alloys castings. The moving boundary liquid metal/air is modeled using the VOF method to trace the free surface during mold filling. 2. Governing equations The mathematical modeling of mold-filling and solidification of metallic alloys will be described in the present study by the following equations: 2.1 Mold filling In the simulation of mold filling, the fluid was considered to be incompressible and having Newtonian characteristics. The mass, momentum and energy conservation equations that govern fluid flow and heat transfer can be expressed in vector calculus notation, as it follows: a) Mass conservation equation (continuity) for incompressible fluids: ∂ u ∂ v ∂ w + + = 0 ∂ x ∂ y ∂ z (1)
Acta Metallurgica Slovaca, 15, 2009, 3 (199 - 207) 201 where v r =(u, v, w) is the velocity. b) The momentum conservation equation (Navier–Stokes) can be expressed as [5]: r ⎧∂ v r r⎫ 2 r r ρ ⎨ + v ∇. v⎬ = -∇ P + µ ∇ v + ρg ⎩ ∂ t ⎭ (2) where, ρ is the density assumed to be constant, P r is the pressure, µ is the dynamic viscosity constant and g r is the gravity vector. c) The energy conservation equation for the liquid region is given by [6]: ⎛ ∂T r ⎞ 2 L ∂φ ρc⎜ + v. ∇T ⎟ = k∇ T − ⎝ ∂t ⎠ c ∂t (3) where c is the specific heat, k is the thermal conductivity, L is the latent heat, φ is the volumetric liquid fraction and T is the temperature. When the molten metal starts to fill the mold, the turbulence phenomenon begins. Turbulence means that the instantaneous velocity is fluctuating at every point in the flow field. Many mathematical models existing in the literature are used to simulate the turbulence,, i.e., a total of eight turbulence models are available [5; 7, 8; 9; 10; 11; 12]. These models acronyms and names are: Standard k-ε Model, Zero Equation Model, RNG - (Re-normalized Group Model), NKE - (New κ-ε Model), GIR, SZL, Standard k-ω Model, SST. The κ-ε model and its extensions entail solving partial differential equations for the turbulent kinetics energy κ and its dissipation rate ε. d) The turbulent kinetics energy equation for NKE - (New κ-ε Model) is given by: ∂( ρκ ) ∂( ρuκ) ∂( ρvκ ) ∂( ρwκ) ∂ ⎛ µ + + + = ⎜ t ∂t ∂x ∂y ∂z ∂x ⎝ σ k C βµ ⎛ ∂T ∂T ∂T ⎞ − ρε + 4 t ⎜ g x + g y + g z ⎟ σ t ⎝ ∂x ∂y ∂z ⎠ ∂κ ⎞ ∂ ⎛ µ ⎟ + ⎜ t ∂x ∂ ⎠ y ⎝ σ k ∂κ ⎞ ∂ ⎛ µ ⎟ + ⎜ t ∂y ∂ ⎠ z ⎝ σ k ∂κ ⎞ ⎟ + µ tΦ ∂z ⎠ (4) The viscous dissipation term in tensor notation is: ⎛ ⎜ ∂u Φ = µ i ⎜ ⎝ ∂xk ∂u ⎞ k ∂u + ⎟ i ∂x j ⎟ ⎠ ∂xk where σ t , is the turbulent Prandtl (Schmidt) number; g x , g y and g z , are the components of acceleration due to gravity and u i , the magnitude of the velocity vector. The turbulent viscosity is calculated as a function of the turbulent kinetics energy parameter, κ, and its dissipation rate ε, that is: 2 κ µ t = ρCµ ε where, C µ , is the turbulent constant; κ is the turbulent kinetics energy parameter and ε is the turbulent kinetics energy dissipation rate.
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<strong>Acta</strong> Metallurgica Slovaca, 15, 2009, 3 (<strong>199</strong> - 207) 200<br />
model for simulation <strong>of</strong> <strong>mold</strong>-<strong>filling</strong> <strong>and</strong> <strong>solidification</strong> <strong>of</strong> a binary alloy. For the <strong>mold</strong> <strong>filling</strong><br />
process, turbulence models in addition to the Fractional Volume (Volume <strong>of</strong> Fluid - VOF)<br />
algorithm is used to model the computational fluid dynamics (CFD). The approach allows one to<br />
h<strong>and</strong>le, with at least second order accuracy, the continuously moving boundary between the<br />
injected fluid <strong>and</strong> the air that initially is in the <strong>mold</strong> cavity. The liquid metal assumes Newtonian<br />
fluid <strong>and</strong> incompressible fluid conditions. For the <strong>solidification</strong> process, the <strong>mathematical</strong><br />
formulation <strong>of</strong> heat transfer is based on the general equation <strong>of</strong> unsteady state heat conduction.<br />
Three forms <strong>of</strong> latent heat release during <strong>solidification</strong> are presented: linear, exponential <strong>and</strong><br />
sinusoid behavior as a function <strong>of</strong> the solid fraction.<br />
Keywords: numerical simulation, finite-element method, <strong>mold</strong>-<strong>filling</strong>, <strong>solidification</strong>, heat<br />
generation<br />
1. Introduction<br />
In the last years a perceptible evolution <strong>of</strong> numerical <strong>modeling</strong> in all science areas can<br />
be easily detected [1]. Particularly for casting processes, a great progress happened due to the<br />
evolution <strong>of</strong> both hardware <strong>and</strong> s<strong>of</strong>tware used for simulation purposes. The stages <strong>of</strong> casting can<br />
be defined as: heating-melting, <strong>mold</strong>-<strong>filling</strong> <strong>and</strong> <strong>solidification</strong>. Each stage has a great influence<br />
on the final casting quality. Mold-<strong>filling</strong> can be classified as a problem <strong>of</strong> fluid mechanics with<br />
many phenomena involved such as, molten metal turbulence; heat transfer along the metal/<strong>mold</strong><br />
system; pressure variation inside the casting, etc. [2]. The <strong>modeling</strong> <strong>of</strong> <strong>solidification</strong> associated<br />
to the fluid dynamics permits many phenomena <strong>and</strong> parameters which are important to assure a<br />
free defect product to be analyzed, e.g. [3]: heat transfer in the whole system (casting <strong>and</strong> <strong>mold</strong>),<br />
thermal gradients, macrosegregation <strong>and</strong> microsegregation [4], thermal stresses in metal <strong>and</strong><br />
<strong>mold</strong>, porosity formation, etc. Despite its importance, the simulation <strong>of</strong> the casting process<br />
becomes a challenge due to the complexity <strong>of</strong> the simultaneous fluid-thermal-mechanics<br />
phenomena which are involved. Besides, they are transient phenomena <strong>and</strong> experimental<br />
measurements permitting fluid motion to be characterized are not an easy task [2]. In this paper<br />
a numerical model is developed permitting the simulation <strong>of</strong> both <strong>mold</strong> <strong>filling</strong> <strong>and</strong> <strong>solidification</strong><br />
<strong>of</strong> binary alloys castings. The moving boundary liquid metal/air is modeled using the VOF<br />
method to trace the free surface during <strong>mold</strong> <strong>filling</strong>.<br />
2. Governing equations<br />
The <strong>mathematical</strong> <strong>modeling</strong> <strong>of</strong> <strong>mold</strong>-<strong>filling</strong> <strong>and</strong> <strong>solidification</strong> <strong>of</strong> metallic alloys will<br />
be described in the present study by the following equations:<br />
2.1 Mold <strong>filling</strong><br />
In the simulation <strong>of</strong> <strong>mold</strong> <strong>filling</strong>, the fluid was considered to be incompressible <strong>and</strong><br />
having Newtonian characteristics. The mass, momentum <strong>and</strong> energy conservation equations that<br />
govern fluid flow <strong>and</strong> heat transfer can be expressed in vector calculus notation, as it follows:<br />
a) Mass conservation equation (continuity) for incompressible fluids:<br />
∂ u ∂ v ∂ w<br />
+ + = 0<br />
∂ x ∂ y ∂ z<br />
(1)
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