Abstract
A two-phase volume averaging model for globular equiaxed solidification is presented. Treating both liquid and solid (disperse grains) as separated but highly coupled interpenetrating continua, we have solved the conservation equations for mass, momentum, species mass fraction, and enthalpy for both phases. We also consider the conservation of grain density. Exchange or source terms take into account interactions between the melt and the solid, such as mass transfer (solidification and melting), friction and drag, solute redistribution, release of latent heat, and nucleation. An ingot casting with a near globular equiaxed solidification alloy (Al-4 wt pct Cu) is simulated. Results including grain evolution, melt convection, sedimentation, solute transport, and macrosegregation formation are obtained. The mechanisms producing these results are discussed in detail.
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Abbreviations
- c 0 :
-
initial concentration
- c l, c s :
-
species concentration
- c*:
-
interface species
- C ls(= −C sl :
-
species exchange rate
- C dls (= −C dsl ):
-
species diffusional flux
- C pls (= −C psl ):
-
species exchange due to phase change
- c mix :
-
mix concentration
- c p(l), c p(s) :
-
specific heat
- D l, D s :
-
diffusion coefficient
- d s :
-
grain diameter
- f l, f s :
-
volume fraction
- f cs :
-
grain packing limit
- g :
-
gravity
- g α :
-
growth factor
- H :
-
heat-transfer coefficient
- H*:
-
heat-transfer coefficient at the s/l interface
- h l, h s :
-
enthalpy
- h*:
-
interface enthalpy
- Δh f :
-
latent heat (heat of fusion)
- K :
-
permeability of liquid in porous medium
- K sl(= K ls):
-
momentum exchange coefficient
- k :
-
distribution coefficient of phase diagram
- k l, k s :
-
thermal conductivity
- L :
-
character size of the fluid domain
- M ls(= −M sl :
-
mass-transfer rate
- M :
-
slope of liquidus in phase diagram
- N :
-
grain production rate
- n :
-
grain density
- n max :
-
maximum grain density
- p :
-
pressure
- p 0 :
-
initial pressure
- Q ls(= −Q sl):
-
energy exchange rate
- Q dls (= −Q dsl ):
-
energy exchange by heat transfer
- Q pls (= −Q psl ):
-
energy change due to phase change
- T 0 :
-
initial temperature
- T, T l, T s :
-
temperature
- T E :
-
eutectic temperature
- T f :
-
melting point of pure metal (Al)
- T wl :
-
boundary temperature
- T ref :
-
reference temperature for enthalpy definition
- ΔT :
-
undercooling
- ΔT N :
-
Gaussian distribution width of nucleation law
- ΔT σ :
-
undercooling for maximum grain production rate
- t :
-
time
- U ls (= −U sl ):
-
momentum exchange rate
- U d ls (= −U d sl ):
-
momentum change due to drag force
- U p ls (= −U p sl ):
-
momentum exchange due to phase change
- u l, u s :
-
velocity component in x direction
- u l , u s :
-
velocity vector
- u ls , u sl :
-
interphase velocity
- u*:
-
interface velocity
- Δu :
-
relative velocity between two phases
- v l, v s :
-
velocity component in y direction
- y :
-
length of boundary layer
- Δδ :
-
boundary mesh size
- ρ l, ρ s :
-
density
- μ l, μ s :
-
viscosity
- μ mix :
-
mix viscosity
- \(\overline{\overline \tau } _{_l } ,\overline{\overline \tau } _{_s } \) :
-
stress-strain tensors
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Ludwig, A., Wu, M. Modeling of globular equiaxed solidification with a two-phase approach. Metall Mater Trans A 33, 3673–3683 (2002). https://doi.org/10.1007/s11661-002-0241-z
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DOI: https://doi.org/10.1007/s11661-002-0241-z