199 mathematical modeling of mold-filling and solidification - Acta ...

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.