Abstract
A model of a casting is presented which describes the freezing of a mushy zone, growing with a dynamically calculated undercooling at the dendrite tips. Equiaxed grains are introduced ahead of a columnar front. The model resembles a combination of the Stefan and mushy zone problems.
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Abbreviations
- Symbol:
-
Meaning
- c 0 :
-
composition
- C :
-
heat capacity per unit volume
- C 1, C 2, C 3 :
-
parameters
- D :
-
solutal diffusion coefficient
- g :
-
weight fraction
- G :
-
temperature gradient
- h :
-
heat transfer coefficient
- H :
-
heat content or enthalpy per unit volume
- I :
-
nucleation rate per unit volume
- k :
-
distribution coefficient
- k i :
-
thermal conductivity (i=s or l)
- K i :
-
thermal conductivity of a single phase (i=s or l)
- L :
-
latent heat per unit volume evolved at the front
- m :
-
liquidus slope
- N :
-
number of grain per unit volume
- N 0 :
-
number of substrate particles per unit volume
- r :
-
extended radius of a grain
- t :
-
time
- T :
-
temperature at time t
- v :
-
velocity
- x :
-
distance
- α :
-
width of control volume immediately ahead of the columnar front
- β :
-
width of control volume immediately behind the columnar front
- δ :
-
boundary layer thickness
- δt :
-
time discretisation interval
- δx :
-
spatial discretisation interval
- ΔH :
-
latent heat per unit volume
- ΔT :
-
undercooling
- κ :
-
pseudo heat transfer coefficient
- ω :
-
extended volume fraction of grains
- Ω :
-
volume fraction of grains
- Subscript:
-
Meaning
- e :
-
east face of the control volume
- E :
-
centre of the control volume to the east
- ex :
-
of the mould
- L :
-
liquidus
- l :
-
liquid
- 0:
-
Centre of the control volume
- s :
-
solid
- w :
-
west face of the control volume
- W :
-
centre of the control volume to the west
- Superscript:
-
Meaning
- B :
-
bulk
- I :
-
columnar front
- m :
-
modified
- p :
-
pure
- *:
-
equiaxed
- ′:
-
at time t+δt
- -:
-
average
References
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Flood, S.C., Hunt, J.D. A model of a casting. Applied Scientific Research 44, 27–42 (1987). https://doi.org/10.1007/BF00412004
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DOI: https://doi.org/10.1007/BF00412004