Abstract
In this article, the columnar-to-equiaxed transition (CET) in directionally solidified castings is investigated. Three CET prediction methods from the literature that use a simulation of the growing columnar front are compared to the experimental results, for a range of Al-Si alloys: Al-3 wt pct Si, Al-7 wt pct Si, and Al-11 wt pct Si. The three CET prediction methods are the constrained-to-unconstrained criterion, the critical cooling rate criterion, and the equiaxed index criterion. These methods are termed indirect methods, because no information is required for modeling the equiaxed nucleation and growth; only the columnar solidification is modeled. A two-dimensional (2-D) front-tracking model of columnar growth is used to compare each criterion applied to each alloy. The constrained-to-unconstrained criterion and a peak equiaxed index criterion agree well with each other and some agreement is found with the experimental findings. For the critical cooling rate criterion, a minimum value for the cooling rate (between 0.07 and 0.11 K/s) is found to occur close to the CET position. However, this range of values differs from those cited in the literature (0.15 to 0.16 K/s), leading to a considerable difference in the prediction of the CET positions. A reason for this discrepancy is suggested, based on the fundamental differences in the modeling approaches.
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Abbreviations
- A :
-
dendrite growth coefficient
- a 0 :
-
polynomial coefficient
- a 1 :
-
polynomial coefficient
- a 2 :
-
polynomial coefficient
- C CV :
-
specific heat for a control volume
- C L :
-
specific heat for liquid
- C S :
-
specific heat for solid
- c p :
-
specific-heat capacity
- D :
-
diameter of ingot
- d :
-
volume fraction of a control volume
- e :
-
error
- g s :
-
solid fraction
- H :
-
height of the ingot
- I :
-
equiaxed index
- i :
-
grid coordinate
- j :
-
grid coordinate
- K :
-
thermal conductivity
- K CV :
-
thermal conductivity of a control volume
- K D :
-
derivative gain
- K I :
-
integral gain
- K L :
-
thermal conductivity of liquid
- K P :
-
proportional gain
- K S :
-
thermal conductivity of solid
- L :
-
latent heat
- n :
-
dendrite growth law exponent
- ncols :
-
number of columns in a grid
- nrows :
-
number of rows in a grid
- P :
-
general polynomial value
- Q :
-
growth restriction
- q :
-
heat flux at the chill surface
- q loss :
-
heat flux at the free liquid surface
- s :
-
Laplace coordinate
- T :
-
temperature
- T E :
-
eutectic temperature
- T init :
-
initial temperature
- T L :
-
liquidus temperature
- T M :
-
melting temperature of solvent material
- t :
-
time coordinate
- t q :
-
cutoff time for heat loss
- U b :
-
undercooled bulk liquid
- V CV :
-
control volume size
- V m :
-
volume of mush in a control volume
- v t :
-
dendrite growth velocity
- ΔT :
-
undercooling
- ΔT n :
-
nucleation undercooling
- Δt :
-
time-step
- Δx :
-
grid spacing
- Δy :
-
grid spacing
- ρ :
-
density
- τ :
-
first-order lag constant
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Acknowledgments
The authors acknowledge the support of the European Space Agency through Columnar-to-Equiaxed Transition in Solidification Processing (CETSOL), a project of the Microgravity Application Promotions program. One of the authors (SMF) expresses his gratitude to Professor Emeritus Annraoi De Paor, for providing illuminating discussions on Control and Filter Theory.
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Manuscript submitted June 11, 2008.
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McFadden, S., Browne, D. & Gandin, CA. A Comparison of Columnar-to-Equiaxed Transition Prediction Methods Using Simulation of the Growing Columnar Front. Metall Mater Trans A 40, 662–672 (2009). https://doi.org/10.1007/s11661-008-9708-x
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DOI: https://doi.org/10.1007/s11661-008-9708-x