A Comparison of Columnar-to-Equiaxed Transition Prediction
Methods Using Simulation of the Growing Columnar Front
S. MCFADDEN, D.J. BROWNE, and CH.-A. GANDIN
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.
DOI: 10.1007/s11661-008-9708-x
Ó The Minerals, Metals & Materials Society and ASM International 2009
I.
INTRODUCTION
THE columnar-to-equiaxed transition (CET) is a
phenomenon that is sometimes revealed by macroetching a section of a cast component. The columnar
region consists of elongated grains with a preferred
growth direction. The equiaxed region consists of many
grains with low aspect ratios and random orientations.
A CET is formed at the shared boundaries at which the
two zones meet. Sometimes the transition is more
gradual and spread over a mixed columnar-equiaxed
zone.
There is a technical advantage to knowing if and when
a CET may occur in a cast component. For example,
directionally solidified turbine blades are manufactured
to promote a columnar zone (thus avoiding the CET),
because a columnar structure along the length of the
turbine blade improves the creep resistance at high
temperatures. In other applications, small equiaxed
grains can give a higher yield strength and an improved
liquid feeding in castings. Recently, Ares et al.[1] demonstrated how CET affects the resistance of a Zn-Al
alloy to corrosion. They measured the charge-transfer
resistance for a group of Zn-Al alloys. It was shown
that, in some cases, the equiaxed zone could have better
S. MCFADDEN, Postdoctoral Researcher, and D.J. BROWNE,
Senior Lecturer, are with the School of Electrical, Electronic, and
Mechanical Engineering, University College Dublin, Dublin 4, Ireland.
Contact e-mail: shaun.mcfadden@ucd.ie CH.-A. GANDIN, Senior
Research Scientist, is with the CEMEF - UMR CNRS 7635, MINES
ParisTech, Sophia Antipolis 06904, France.
Manuscript submitted June 11, 2008.
Article published online January 15, 2009
662—VOLUME 40A, MARCH 2009
corrosion resistance than the columnar zone. Thus,
much attention has been given to the phenomenon of
CET. Spittle[2] gave a recent review of CET experimental
and modeling work.
Columnar dendrites, which form the columnar grains,
typically nucleate at a mold or chill surface. Initially,
during a competitive growth phase, the dendrites grow
under a moderately high temperature gradient. The
preferred crystallographic direction for columnar dendrites is for h100i dendrite arms to grow in the direction
opposite to the heat flow. In contrast, the equiaxed
dendrites prefer to grow in the bulk undercooled liquid,
where the temperature gradients are much lower.
Equiaxed dendrites nucleate with seemingly no preferred
crystallographic orientation. Equiaxed grains can originate in different ways. Hutt and StJohn[3] gave a
summary of the origins of equiaxed grains. Typically,
equiaxed grains nucleate at the site of an existing
particle in the melt. Alternatively, equiaxed dendrites
can originate from fragments of the columnar dendrites,
and the detached arms grow to become equiaxed grains
(Jackson et al.[4]).
Regardless of the origin of the dendrites, it is clear
that the condition of the bulk liquid determines whether
a columnar zone or an equiaxed zone prevails. The
growth restriction for a binary alloy is quantified in the
literature as a factor Q,[5] where it is shown that Q helps
establish the relationship between the undercooling at
the columnar dendrite tip and the growth rate. Specifically, the undercooling at the columnar dendrite tip
and the temperature profile within the liquid determine
the extent of the undercooled liquid region ahead of
the columnar front. The temperature gradients in the
undercooled liquid play an important role in the
METALLURGICAL AND MATERIALS TRANSACTIONS A
macrostructure formation. This relationship between
the growth rate and the temperature gradient in forming
the CET is succinctly described in the CET diagram of
Hunt.[6] Hunt’s diagram is a plot of the dendrite tip
growth velocity vs the temperature gradient. From the
diagram, one can see the combinations of growth
velocity and temperature gradient that give a fully
columnar, fully equiaxed, or mixed columnar-equiaxed
structure. Modifications to Hunt’s original CET diagram are available. Gäumann et al.[7] developed a CET
diagram based on a more sophisticated dendrite growth
law, while Martorano et al.[8] have included the effects
of solutal interactions between dendrites, to give a
modified CET diagram with solutal blocking effects.
Quested and Greer[5] append the Hunt diagram with
contours of equal grain size in the equiaxed region of the
diagram.
The CET is also predicted by simulating the growth of
the solidification macrostructure. As discussed in Reference 9, two approaches are possible, namely, direct
and indirect methods. In the direct approach, the
nucleation and growth of both a columnar and an
equiaxed mushy zone are simulated. Models exist that
simulate the CET at various length scales, for example,
at the macro scale[10–16] and at the micro scale.[17–19] In
the indirect approach, only the evolution of the columnar mushy zone is modeled, that is, the nucleation and
growth of the equiaxed zone is omitted. In this case, the
condition ahead of the columnar front is analyzed with
respect to the possibility of an equiaxed zone forming
there.
This article focuses on indirect CET prediction
techniques. A two-dimensional (2-D) front-tracking
model of columnar growth[14,15] is used and comparisons between various indirect CET prediction methods
are assessed. An advantage of using this type of indirect
approach is the reduced simulation run times: the final
results are gathered expeditiously. Furthermore, reliable
detailed information about the nucleation criteria of
equiaxed grains, which is usually difficult to obtain, is
not required for the simulation runs.
II.
INDIRECT CET PREDICTION
In this article, we discuss three published indirect CET
criteria: the constrained-to-unconstrained columnar
growth criterion, the critical cooling rate criterion, and
the equiaxed index.
A. Constrained-to-Unconstrained Criterion
Gandin[20] developed a one-dimensional front-tracking model of columnar growth using Landau transforms
to explicitly track the dendritic tip and eutectic interface
positions. The growth of the dendrite tip was determined by the undercooling at the tip relative to the
liquidus temperature; the growth of the eutectic interface was determined by the undercooling at the interface
relative to the eutectic temperature. Between the columnar dendrite tip and the eutectic interface, the mushy
METALLURGICAL AND MATERIALS TRANSACTIONS A
zone followed a truncated Scheil solidification
path.[21,22] The enthalpy was calculated for the system
based on both the heat transfer by conduction and the
latent heat released during solidification. The thermophysical properties varied with the temperature and the
solid fraction.
Gandin compared the results of the model to the
results from experiments on directionally solidified
alloys.[23] Agreement was found between the model
and the experimental cooling curves for the majority of
the solidification times. Any differences in the cooling
curves were attributed to the omission of the equiaxed
solidification from the model.
The location of the CET was measured from the
experimental results. The model was able to provide the
growth rate of the columnar front and the temperature
gradient in the liquid for every position of the front. It
was observed that, when the columnar front reached the
position of the observed CET, the columnar growth rate
had reached a local maximum and the temperature
gradient in the liquid had switched to become slightly
negative. Thus, the growth regime at the dendrite tips
changed from constrained to unconstrained (a positiveto-negative temperature gradient). Gandin proposed
that the CET coincided with the point at which the
local maximum dendrite growth rate and the constrained-to-unconstrained transition occur. It was suggested that these growth conditions could favor a
fragmentation mechanism as the source of the equiaxed
grains, as was previously observed.[4]
Ares et al.[24,25] estimated the temperature gradients at
the CET position for a series of directionally solidified
binary alloys. They reported temperature gradients close
to zero at the CET position, with some gradients being
slightly negative.
B. Critical Cooling Rate Criterion
Siqueira et al.[26,27] conducted directional solidification experiments on Al-Cu and Sn-Pb alloys, to
analyze the CET. Their furnace was instrumented to
establish the heat-transfer coefficient at the chill surface. A model of the columnar solidification using the
heat-conduction energy equation and the Scheil equation was developed, to analyze the cooling conditions
achieved in the furnace. It is important to point out
that the model of Siqueira et al. assumed that solidification begins at the liquidus temperature. No distinction was made between columnar and equiaxed zones
in this heat-flow analysis.
Siqueira et al. used their model to study the thermal
conditions at the position of the liquidus isotherm. They
plotted the isotherm velocity and temperature gradient
for each position of the liquidus isotherm. Furthermore,
by multiplying the isotherm velocity and gradient for
each position, the cooling rate at the instantaneous
position of the liquidus isotherm was plotted. It is
important to clarify that the calculated cooling rate was
the cooling rate at the position of the isotherm and not
the cooling rate of the isotherm, which is, obviously,
zero. When comparing the cooling rates to the measured
CET positions, the authors concluded that the CET
VOLUME 40A, MARCH 2009—663
occurrences coincided with a critical cooling rate. After
additional investigation, it was also concluded that the
critical cooling rate remained constant for each alloy
regardless of the thermal parameters (initial temperature, heat-transfer coefficient, etc.). For Al-Cu systems,
the critical cooling rate was measured as 0.2 K/s;[26] for
Sn-Pb alloys, it was 0.014 K/s.[27] The critical cooling
rates given were specific to the alloy system but
independent of composition. Later, Peres et al.[28] performed similar experiments with Al-Si alloys and
showed that, for this alloy system, the critical cooling
rate varied. The critical cooling rates for CET in their
Al-Si ingot varied from 0.15 to 0.2 K/s, with variations
in silicon content ranging from 3 to 9 wt pct. Most
recently, the critical cooling rates for Al-Ni and Al-Sn
systems were found to be invariant and were 0.3 and
0.16 K/s, respectively.[29]
C. Equiaxed Index
Browne[30] developed an indirect method for CET
prediction based on a columnar-front-tracking model.
As already mentioned, the columnar-front-tracking
model distinguishes between dendritic columnar mush
and undercooled liquid. In this CET prediction method,
the bulk undercooled liquid ahead of the columnar front
was analyzed. To demonstrate, consider a square
computational grid made up of orthogonal control
volumes, with dimensions Dx and Dy. The number of
rows in the grid is nrows and the number of columns is
ncols. An equiaxed index, I(t), was calculated at each
time, t, in the simulation; it was defined as
IðtÞ ¼
nrows
cols
X nX
the alloy composition and heat-transfer coefficient; the
peak equiaxed index increased with an increasing solute
content and a decreasing heat-transfer coefficient. Qualitatively, it was proposed that this trend was supported
by foundry experience, because it was shown elsewhere
that an increased solute content[31] and a lower heattransfer coefficient[32] increased the extent of the equiaxed zone.
Hence, it was proposed that the peak equiaxed index
be used as an indicator to give the relative likelihood of
an equiaxed zone forming in a casting-mold cavity of
fixed dimensions. The results of Browne[30] were recently
directly compared to those of a front-tracking model of
combined columnar and equiaxed growth.[33] It was
shown that the simulated equiaxed zone increased in
qualitative agreement with the peak equiaxed index. In
addition to the peak equiaxed index, the undercooling
dwell time, which is the accumulated time during which
a control volume remains undercooled, was suggested as
a good indicator for the location of the equiaxed
zone.[34] Preliminary investigations[35] showed that the
index could increase when the effects of the natural
thermal convection were included in the simulation.
No previous direct quantitative experimental validation of the equiaxed index was performed; rather, its
predictions were shown[30] to be in qualitative agreement
with a range of experimental studies from the literature.
This article aims to test, for the first time, the proposed
equiaxed index criterion against the experimental
results.
III.
Ub ði; jÞDxDyjt¼const
½1
i¼1 j¼1
where Ub(i,j) was the level of the undercooling in the
control volume with coordinates (i,j) and
8
if Tði; jÞ>TL
<0
Ub ði; jÞ ¼ TL Tði; jÞ if Tði; jÞ TL and dði; jÞ<0:5
:
0
if Tði; jÞ TL and dði; jÞ 0:5
½2
where T(i,j) is the node-center temperature at a control
volume, TL the liquidus temperature, and d(i,j) the
volumetric fraction of any control volume captured by
the columnar mushy zone. The bulk undercooling was
calculated, as described, only in undercooled liquid
control volumes; elsewhere, Ub(i,j) = 0. The index I(t)
can be interpreted as the discrete integral of the bulkliquid undercooling across the domain.
The equiaxed index plotted as a function of time
(where it started at zero for a superheated liquid)
increased as the extent of the undercooled liquid zone
increased, reached a peak value, and diminished to zero
as the bulk liquid was consumed by mush. Browne
investigated three compositions of an Al-Cu alloy cast in
similar molds, but with each alloy simulated three times
with different heat-transfer coefficients at the walls. The
peak value of the equiaxed index changed depending on
664—VOLUME 40A, MARCH 2009
EXPERIMENTAL CASES
Gandin’s experimental data[23] for a series of directionally solidified Al-Si alloys are used in this analysis.
The experimental procedure consisted of melting the
alloys in a crucible and pouring the liquid into the mold
that is maintained in a furnace to achieve a uniform
temperature. Removing the apparatus from the furnace
and applying a water-cooled chill to the bottom surface
of the cylindrical mold allowed solidification to proceed
in the upward direction. The final ingot size was 7 cm in
diameter and approximately 17 cm in height. The Al-Si
alloy system is not prone to solutal-driven natural
convection, because both aluminum and silicon have
similar densities. Cooling from below ensures that the
liquid is also stable with regard to thermally derived
density gradients in the liquid. It is usually assumed that
the effect of the natural convection in the liquid can be
neglected in this case. The three alloys studied were Al-3
wt pct Si, Al-7 wt pct Si, and Al-11 wt pct Si. The
apparatus had seven thermocouples along the central
axis, which were uniformly spaced by 2 cm. The CET
positions measured for Al-3 wt pct Si, Al-7 wt pct Si,
and Al-11 wt pct Si alloys were 12.0, 11.8, and 10.9 cm,
respectively, measured from the bottom of the ingot.
A tolerance of ±1 mm was quoted with each CET
position. Additional data from the experiments, namely,
additional cooling curves and averaged equiaxed radii,
are available in Martorano et al.[8]
METALLURGICAL AND MATERIALS TRANSACTIONS A
IV.
COLUMNAR-FRONT-TRACKING MODEL
A 2-D model of the furnace was developed using
columnar front tracking. Additional details about the
front-tracking algorithm are available in Reference 15;
additional details about front tracking with an implementation of the Scheil algorithm are available in
Reference 14. Equivalence with other models from the
literature is to be emphasized,[8,10–12,20,21] because all
these models use a tracking method to follow the
position of the columnar growth front and the undercooling at the position of the front.
The thermophysical properties for the alloys changed
as a function of temperature and phase fraction,
according to the following equations:
Vm
Vm
KS ðTÞ ½3
KL ðTÞ þ gs
KCV ðT; gs Þ ¼ 1 gs
VCV
VCV
Fig. 1—Furnace schematic, with a flowchart for the inverse-heattransfer algorithm.
and
CCV ðT; gs Þ ¼
Vm
Vm
1 gs
CS ðTÞ ½4
CL ðTÞ þ gs
VCV
VCV
where KCV (T,gs) is the thermal conductivity of the
control volume as a function of the temperature, T, and
the solid fraction, gs; KL(T) and KS(T) are the thermal
conductivities of the liquid and solid, respectively; and
Vm is the mushy or captured volume in a control volume
of size VCV. The specific heat at a control volume is
CCV(T,gs); CL(T) and CS(T) are the specific heats of the
liquid and solid, respectively.
The properties for the liquid and solid are approximated with quadratic polynomials, which are based on
best fits for the available data, as in
PðTÞ ¼ a2 T 2 þ a1 T þ a0
½5
where P(T) is the property of interest (either the specific
heat or the conductivity) and a2, a1, and a0 are the
coefficients. The temperature is given in degrees Celsius.
The following dendrite growth law governs the
columnar front growth rate:
vt ¼ ADT n
½6
where A is a growth constant, DT the undercooling at
the tip (given by TL – T), and n a growth exponent.
The vertical mold walls are assumed adiabatic.
During the experiments, there was a gap between the
liquid-free surface and the top of the mold. Similar to
Gandin[20] and Martorano et al.,[8] the heat flux across
the free surface was estimated at some value up to a
cutoff time; after this time, the heat flux was assumed to
be zero.
The heat flux at the chill surface was an unknown and
could not be estimated as simply. An inverse-heattransfer method was developed based on a modified
proportional integral derivative (PID) control algorithm. Figure 1 shows a schematic for this inverseheat-transfer scheme. The model is allowed to run and
the simulated temperature at a position of 2 cm from the
chill surface is returned to the controller; this is the
calculated temperature. The measured temperature from
METALLURGICAL AND MATERIALS TRANSACTIONS A
the thermocouple at the 2-cm position is provided to the
controller; this is the desired temperature. The error in
the model, e, is the difference between the calculated and
desired temperatures at the 2-cm position. This error
value is then used to change the heat flux at the chill
surface. The heat flux at the chill, q, is given by the
following expression written in the Laplace domain, s,
instead of the time domain:
KI
KD s
½7
qðsÞ ¼ eðsÞ KP þ þ
1 þ ss
s
where KP, KI, and KD are the coefficient of the PID
controller, and s the break period of a first-order lag
that helps to filter any noise on the differential channel. The PID terms KP, KI, and KD were selected by
tuning the system response with the Ziegler–Nichols
method.[36] The value of s was selected as
s ¼ 0:1
KP
KD
½8
Thus, Eq. [7] describes a negative feedback control
system. The controller algorithm gives a dynamic
response, that is, it changes with time, so that no
iteration is required during the explicit time-stepping
algorithm. The algorithm is robust; it can allow for the
phase change and changes in thermophysical properties
without having to make any adjustments to the algorithm parameters. The temperature of the sample at the
chill face is simulated; thus, the model simulates the
heterogeneous nucleation of the columnar front from
the chill face. Reference 16 explains an approach to
obtaining the boundary conditions for a microgravity
experiment on solidification that is similar to that used
in a model of a PID controller
V.
ALLOY PROPERTIES
Table I gives the properties used in the simulations for
the three alloys. The data from the phase diagram, the
VOLUME 40A, MARCH 2009—665
Table I.
Alloy
Al-3 Wt Pct Si
Al-7 Wt Pct Si
Al-11 Wt Pct Si
640
0.12
577
660
1
1.7 9 103
2.7
969
618
0.13
577
660
1
2.9 9 104
2.7
1064
590
0.14
577
660
1
1.0 9 104
2.7
1159
Liquidus temperature, TL (°C)
Segregation coefficient, k
Eutectic temperature, TE (°C)
Pure aluminium melting temperature, TM (°C)
Columnar nucleation undercooling, DTn (°C)
Dendrite growth constant, A (cm/s °Cn)
Dendrite growth exponent, n
Latent heat, L (J/cm3)
Specific-heat polynomial
coefficients, q Cp (J/cm3 °C)
a0
a1
a2
Liquid
Solid
Liquid
Solid
Liquid
solid
2.775
3.040 9 104
0.0
2.422
9.598 9 104
0.0
3.06
3.247 9 104
0.0
2.349
9.720 9 104
0.0
3.083
2.330 9 104
0.0
2.422
9.598 9 104
0.0
Thermal conductivity polynomial
coefficients, K (W/cm °C)
a0
a1
a2
Alloy Properties
Liquid
Solid
Liquid
Solid
Liquid
Solid
0.5000
3.125 9 104
0.0
1.5581
2.284 9 103
3.84 9 106
0.4400
2.747 9 104
0.0
1.3581
2.284 9 103
3.84 9 106
0.3670
3.330 9 104
0.0
2.4125
1.050 9 103
1.262 9 107
dendritic growth kinetics, and the thermal conductivity
are based on the values given by Gandin.[20] The latent
heat data used here are based on measured latent heat
data. It is well known that silicon, which has a latent
heat value 4.6 times that of aluminum, has an important
effect related to increasing the latent heat released from
the alloy.[37,38] Djurdjevic et al.[38] investigated the
increase in latent heat in commercial alloys, in terms
of the chemistry and cooling rate. They showed that the
cooling rate had a very small effect on the measured
latent heat and that a silicon addition had the most
significant effect.
The specific-heat data for Al-7 wt pct Si and Al-11
wt pct Si are based on data given for commercial alloys
in Reference 39. The commercial alloy Al-LM25 is
quoted in Reference 39 with 7 wt pct silicon, 91.1 wt pct
aluminum, and no more than 0.5 wt pct of other
elements. Hence, the specific-heat values quoted for
the liquid and solid LM25 are used here for Al-7 wt pct
Si. Similarly the specific-heat data quoted for Al-LM13
are used because of the closeness of the alloy content to
Al-11 wt pct Si. The LM13 constituents quoted in
Reference 39 are 12 wt pct Si, 84.3 wt pct Al, and no
more than 1.2 wt pct of other constituents. For Al-3 wt
pct Si, we use the specific-heat data given by Gandin,[20]
which are based on pure aluminum. This estimate is
assumed reasonable because of the dilute nature of the
3 pct Si alloy.
A comparison was made between the thermal conductivity data quoted for LM25 and LM13[39] and those
quoted by Gandin[20] for Al-7 wt pct Si and Al-11 wt pct
Si. These values for conductivity are in close agreement;
thus, with confidence, the data given by Gandin are
used.
666—VOLUME 40A, MARCH 2009
VI.
MODEL PARAMETERS
The model uses an explicit time-stepping algorithm on
a fixed orthogonal grid. Table II gives the size of the
grid and the time-step. The initial temperatures for each
simulation are listed here. As already explained, the heat
flux at the bottom surface is estimated by using an
adaptation of a PID controller algorithm. The PID
controller gains are listed in Table II. These gains were
established by tuning the Al-7 wt pct Si model to a
Ziegler–Nichols criterion. The same gains work well for
the Al-3 wt pct Si and Al-11 wt pct Si systems, thus
demonstrating the robustness of the PID algorithm as
an inverse-heat-transfer method.
Table II.
Simulation Parameters
Simulation Run
Domian size, D 9 H (cm)
Grid size, Dx, Dy (cm)
Time-step, Dt (s)
Initial temperature,
Tinit (°C)
Controller
proportional gain, KP
Controller
integral gain, KI
Controller
derivative gain, KD
Heat flux from top,
qloss (W/cm2)
Cutoff time, tq (s)
Al-3
Al-7
Al-11
Wt Pct Si Wt Pct Si Wt Pct Si
7 9 17
0.1
7.5 9 104
776.9
7 9 17
0.1
7.5 9 104
764.0
7 9 17
0.1
7.5 9 104
768.9
6.66
6.66
6.66
1.23
1.23
1.23
8.991
8.991
8.991
0.4
0.4
1.0
900
900
900
METALLURGICAL AND MATERIALS TRANSACTIONS A
Fig. 2—Cooling curves, isotherm and front positions, and temperature gradients for each alloy case: Al-3 wt pct Si, 7 wt pct Si, and 11 wt pct Si.
Similar to previous work,[8,20] the heat flux, qloss, from
the top surface is estimated to be constant up to a cutoff
time, tq, after which time the heat flux is assumed to be
zero. Table II gives the estimated heat loss and the
cutoff times selected for our simulations.
VII. RESULTS
Figure 2 shows the first set of results in a 3 9 3
montage of images. The results for each alloy are
arranged in rows: the first row shows the results for Al-3
wt pct Si; the center row, for Al-7 wt pct Si; and the last
row, for Al-11 wt pct Si. Figure 2 shows the cooling
curves in the column on the left, the liquidus isotherm
and columnar front positions in the column in the
center, and the simulated temperature gradients in the
column on the right.
Additionally, the graphs for the cooling curves in the
column on the left in Figure 2 show the simulated
cooling curves at the thermocouple positions and the
experimentally measured cooling curves. Thus, a comparison can be made between the experiment and the
model, for each alloy and for each simulation run.
METALLURGICAL AND MATERIALS TRANSACTIONS A
The graphs for the liquidus isotherm and the columnar front position, in the center column in Figure 2,
show the simulated liquidus isotherm position and the
measured liquidus isotherm position. Thus, a comparison can again be made between the experiment and the
model, for each case. The columnar front position was
not measured during the experiment; therefore, only the
simulated columnar front position may be shown. It
should be noted that the columnar front nucleates on
the chill surface and then travels away from it. It is the
vertical dimension from the columnar front to the chill
that is quoted.
The temperature gradients, shown in the column on
the right in Figure 2, give the simulated temperature
gradients at the position of the columnar fronts. Two
gradients are given, one in the liquid ahead of the
columnar front and the other in the mush behind the
columnar front. The temperature gradients are given as
a function of the calculated position of the columnar
front. The vertical dashed line in this set of graphs shows
the measured CET position from the postmortem
analysis. Overall, Figure 2 gives a summary of the
simulated thermal conditions with comparisons to the
measured thermal conditions from the experiment.
VOLUME 40A, MARCH 2009—667
Fig. 3—Simulated columnar growth rates, cooling rates, and equiaxed index for each alloy case: Al-3 wt pct Si, 7 wt pct Si, and 11 wt pct Si.
Figure 3 shows the second, and final, set of results for
the simulation cases in another 3 9 3 montage of
images. Similar to Figure 2, the results for each alloy
in Figure 3 are arranged in rows: the first row gives Al-3
wt pct Si; the center row, Al-7 wt pct Si; and the last
row, Al-11 wt pct Si. The column on the left shows the
growth rate of the columnar front as a function of the
distance of the columnar front from the chill surface;
the column in the center shows the instantaneous
cooling rate at the position of the liquidus isotherm vs
the distance of the isotherm from the chill; and the
column on the right shows the equiaxed index for each
position of the columnar front measured from the chill.
The data in Figure 3 will be used to give a summary of
the three CET prediction criteria. For reference, the
measured CET positions are superimposed onto each
graph as a vertical dashed line.
VIII.
DISCUSSION
The column on the left in Figure 2 shows the cooling
curves from the model compared to those measured on
the experimental apparatus. The thermocouple positions
are indicated on each cooling curve. From the first row,
we see that agreement between the model and the
experiment for Al-3 wt pct Si is quite good, especially
668—VOLUME 40A, MARCH 2009
in the early stages of solidification. There is some small
amount of deviation in the results toward the end of the
experiment, at the 14-cm position. This location is well
within the equiaxed zone; hence, this slight temperature
deviation may be partly due to the absence of an equiaxed
solidification model and, possibly, to the absence of a
model of convection in the liquid. Also, we must consider
that the 14-cm position is close to the free surface of the
ingot, where a simple but crude approximation of the
boundary condition was made. However, the general
agreement is good. Similar agreement is found in the row
in the center, for the cooling curves of Al-7 wt pct Si.
Furthermore, direct comparisons can be made between
this set of modeling results and those given by Gandin’s
columnar solidification model,[20] offering some verification status to the present front-tracking model with the
inverse-heat-transfer method. In the last row, we see
more pronounced deviations between the model and the
experiment for the Al-11 wt pct Si alloy. Deviations in
this result may be due to the same reasons as cited earlier;
however, general agreement is found during the majority
of the cooling phase.
The column in the center in Figure 2 shows the
simulated position of the liquidus isotherm from the
surface of the chill. The measured isotherm positions are
compared to the predictions. Agreement between the
model and the experiments is reasonably good; this
METALLURGICAL AND MATERIALS TRANSACTIONS A
reinforces confidence in the ability of the model to
predict the correct thermal conditions. In addition,
Figure 2 (the column in the center) shows the simulated
columnar front positions. It is clear that the columnar
front lags behind the liquidus isotherm; the liquidus
isotherm reaches the top of the ingot before the
columnar front. The extent of the bulk undercooled
liquid zone is simply the difference between the liquidus
isotherm and the columnar front positions. It is clear
that, during the first stage of solidification, this extent is
small; it increases, however, until the liquidus isotherm
reaches the top of the ingot, at which time the extent
begins to decrease. It is this extent of the undercooled
bulk liquid that plays an important role in the development of the equiaxed zone.
The column on the right in Figure 2 shows the
temperature gradients at the columnar front positions in
the liquid ahead of the columnar front and in the mush
behind. In the case of the Al-7 wt pct Si (in the center
row), we can compare the experimental results to the
modeling results reported in Reference 20. In this
respect, the agreement between the present model and
the results in Reference 20 is good. On the temperature
gradient graphs, the vertical dashed line indicates the
measured CET position. For the cases of Al-3 wt pct Si
and Al-7 wt pct Si, the temperature gradient in the liquid
becomes slightly negative at the point at which the CET
occurred. Thus, in agreement with Gandin, a constrained-to-unconstrained transition takes place close to
the CET position. However, applying the constrainedto-unconstrained criterion for the CET to the results for
the Al-11 wt pct Si alloy shows that the CET should
occur at approximately 12.5 cm from the chill surface.
The measured CET was at 10.9 cm; thus, in this case, we
see a 1.6-cm overestimate for the columnar length.
Figure 3 gives a summary of the indirect CET prediction criteria vs the measured CET. The column on the left
shows the simulation results for the columnar growth
rate. As discussed by Gandin,[20] there is typically a local
maximum in the growth rate close to the constrained-tounconstrained transition, which is also the predicted CET
position. The column in the center gives the results for the
critical cooling rate CET criterion of Siqueira et al.[26]
The column on the right shows the results for the
equiaxed index CET criteria of Browne.[30]
For the Al-3 wt pct Si and the Al-7 wt pct Si cases, we
see close agreement between the local maximum in the
columnar front growth rate and the measured CET
positions. When these results for columnar growth rates
are considered in conjunction with the temperature
gradient (Figure 2), we conclude that the constrainedto-unconstrained theory gives a good prediction for the
CET position. For the case of Al-11 wt pct Si, the local
maximum in the growth rate occurs at approximately
1.6 cm past the measured CET position; thus, as
discussed earlier, the constrained-to-unconstrained criteria overestimate the CET position by 14.6 pct of the
actual CET distance.
The column in the center in Figure 3 summarizes the
results for the critical cooling rate CET criteria. If we
first consider the case of Al-3 wt pct Si (the first row), we
see that the minimum cooling rate is approximately
METALLURGICAL AND MATERIALS TRANSACTIONS A
0.07 K/s and that it occurs approximately 1.5 cm before
the measured CET position. By using the minimum
critical cooling rate of Peres et al. (0.15 K/s), the
predicted CET position would be 5 cm from the chill.
The error associated with this prediction is approximately 7 cm. Similarly, for the Al-7 wt pct Si, the
cooling rate reaches a minimum of approximately
0.11 K/s, just before the CET measured position. This
minimum is below that quoted in the literature. To use
the value from the literature, the columnar length for
Al-7 wt pct Si is predicted to be approximately 6 cm,
which is approximately 6 cm too short. For the case of
Al-11 wt pct Si, we see a local minimum of approximately 0.1 K/s in the cooling rate at the measured CET
position. Notably, if we used the critical cooling rate of
0.15 K/s (Peres et al.), we would have predicted a CET
at approximately 1 cm from the chill; in other words,
the prediction would give an almost fully equiaxed
structure.
Finally, the column on the right in Figure 3 summarizes the results for the equiaxed index CET criteria. For
the Al-3 wt pct Si alloy, a peak value of 120 cm2 K is
reached at the measured CET position. For the case of
Al-7 wt pct Si, the equiaxed index peak value is
195 cm2 K. It is clear that, in this case, the CET
occurred just prior to the point at which the equiaxed
index reached the peak value. For the Al-11 wt pct Si
alloy, a peak value of 180 cm2 K occurs at approximately 12.5 cm. Thus, the peak equiaxed index value, in
the case of Al-11 wt pct Si, corresponds closely with the
constrained-to-unconstrained transition; however, the
estimate is approximately 1.6 cm too long for the
columnar zone. Nevertheless, it should be remembered
that the equiaxed index function gives the relative
likelihood of a CET occurring. The chances of a CET
occurring are reduced after the peak equiaxed index has
occurred; therefore, the CET is most likely to occur
before or at the peak equiaxed index. In this case of Al11 wt pct Si, the CET occurs just before the peak
equiaxed index. It seems that the peak equiaxed index
criterion is more robust and is less sensitive to errors in
the thermal calculations than the critical cooling rate
criterion, as demonstrated in the case of Al-11 wt pct Si.
In Figure 3, the parameters are plotted as a function
of the distance from the chill surface. This spatial
plotting method was performed so that direct comparisons could be made with the measured position of the
CET from the chill. All of the information in Figure 3
can also be plotted against time. No information about
the temporal evolution of the CET was available for
these experiments. Hence, it was impossible to draw a
comparison between the simulated timing of a CET vs
the measured times. As described in the literature, a realtime, in-situ observation of CET development requires
200-lm-thick samples and a powerful X-ray
source.[40,41] However, comparisons between the simulated timing of events for each criterion were made. For
Al-3 wt pct Si, the peak columnar growth rate occurred
at 915 seconds of simulated time; the peak equiaxed
index occurred at 930 seconds. These values are in
reasonably good agreement with each other, considering
that a 15-second difference corresponds to a 3-mm
VOLUME 40A, MARCH 2009—669
difference in the position of the columnar growth front,
when it is growing at approximately 200 lm/s. In
contrast, the critical cooling rate of 0.16 K/s occurred
at the earlier time of 520 seconds; the minimum cooling
rate achieved in the simulation occurred at 790 seconds.
Similar trends were observed for the Al-7 wt pct Si and
Al-11 wt pct Si alloys. For Al-7 wt pct Si, the peak
columnar growth rate occurred at 950 seconds, the peak
equiaxed index at 940 seconds, the critical cooling rate
of 0.15 K/s at 570 seconds, and the minimum cooling
rate at 775 seconds. For Al-11 wt pct Si, the peak
columnar growth rate occurred at 1300 seconds, the
peak equiaxed index at 1290 seconds, the critical cooling
rate (0.15 to 0.2 K/s) at approximately 345 seconds, and
the minimum cooling rate at 1110 seconds. Considering
that the critical cooling rate criterion is based on the
liquidus isotherm and not the columnar front position, it
is no surprise that the critical cooling rate criterion
predicts that the CET will occur earlier than the other
criteria. The column in the center of figure 2 shows that
the liquidus isotherm can be quite far ahead of the
columnar front during the latter stages of solidification.
IX.
SUMMARY OF CET PREDICTION RESULTS
Overall, using the columnar-front-tracking model, we
see the following trends in the results at the CET
position: (1) a local peak in the columnar velocity that
also corresponds to a constrained-to-unconstrained
transition, 2) a minimum value in the cooling rate at
the position of the liquidus isotherm, and 3) a peak
value in the equiaxed index. These trends seem to be in
general
agreement
with
previously
published
work.[20,26,30] However, the critical cooling rates calculated here seem lower than those from earlier work.[28]
Our calculated critical cooling rates for CET are
between 0.07 to 0.11 K/s, which, when compared with
0.15 and 0.16 K/s, is lower than expected. A reason for
these discrepancies may come from the alternative
modeling approaches. As mentioned, Peres et al.[28] uses
a macroscopic model that is based on the modified
specific-heat approach, which is similar to an enthalpy
model. In their model, no distinction is made between
columnar and equiaxed mush. A combined columnarequiaxed mushy zone is modeled with a Scheil approximation. In the columnar-front-tracking approaches of
Gandin[20] and Browne,[30] a distinction is made for the
columnar mushy zone. In the absence of an equiaxed
solidification component to the model, these approaches
distinguish between columnar mush and undercooled
bulk liquid ahead of the columnar front. The modified
specific-heat approach or enthalpy method used by
Peres et al. does not allow for bulk undercooled liquid.
(Refer to Banaszek et al.[42] for a detailed discussion of
this distinction in the models.) Thus, a fundamental
difference among the modeling approaches may account
for the differences found in these experiments between
the critical cooling rates required for CET and the
critical cooling rates modeled here. However, because
the CET is formed when the columnar front is blocked
and because the columnar tips are undercooled, it seems
670—VOLUME 40A, MARCH 2009
clear that the most successful CET criteria are based on
models that allow for undercooling at the columnar
dendrites. Most macroscopic models that solve the
energy conservation with a unique solidification path do
not distinguish the undercooled columnar dendrite tips
within the mushy zone.
X.
CONCLUSIONS
A columnar-front-tracking model combined with an
inverse-heat-transfer method was applied to a series of
directional solidification experiments on Al-Si alloys.
The experiments in question were on Al-3 wt pct Si, Al-7
wt pct Si, and Al-11 wt pct Si binary alloys. Three CET
criteria from the literature were tested to see how well
the observed CET was predicted. The criteria involved
were the constrained-to-unconstrained transition criterion, the critical cooling rate criterion, and the peak
equiaxed index criterion.
For Al-3 wt pct Si and Al-7 wt pct Si, the constrainedto-unconstrained criterion and the peak equiaxed index
criterion predictions agreed well with the CET position.
For the Al-11 wt pct Si alloy, the constrainedto-unconstrained and peak equiaxed index criterion
overestimated the columnar length by 1.6 cm. This
overestimation may be due to some discrepancies between
the predicted and measured thermal conditions.
A minimum cooling rate at the position of the
isotherm was observed close to the CET position, for
each alloy. However, the values of the critical cooling
rate for a CET found here (0.07 to 0.11 K/s for Al-3 wt
pct Si and Al-7 wt pct Si, respectively) differ considerably from those quoted in the literature (0.16 to 0.15 K/s
for Al-3 wt pct Si and Al-7 wt pct Si, respectively[28]),
leading to a large deviation in the predictions of the
CET positions. It is proposed that the differences come
from a fundamental difference in the modeling approaches used to obtain the values.
The constrained-to-unconstrained criterion and the
peak equiaxed index criterion may be used in conjunction with a columnar-front-tracking model to predict the
CET position in noninoculated directional castings.
Whether or not a CET occurs depends on the level of
undercooling ahead of the columnar front. As suggested
in Reference 34, the peak level of the bulk-liquid
undercooling across the domain should also be considered, to determine whether a threshold value is reached.
This threshold value could be the same as the mean
value for the heterogeneous nucleation of the equiaxed
dendrites. The critical cooling rate criteria should only
be used in conjunction with a full enthalpy or a modified
specific-heat model. A columnar solidification model, as
used here, should not be used with the critical cooling
rate criterion of Siqueira et al.[26]
The indirect CET prediction criteria discussed here
offer the significant advantage of giving reasonable
CET predictions without the computational expense of
modeling the development of an equiaxed zone. Thus,
much computational run time can be saved when gleaning
CET information for many casting cases. The equiaxed
index method of Browne can be used for predicting the
METALLURGICAL AND MATERIALS TRANSACTIONS A
CET in shape castings. It should be possible to define a
lower bound on the equiaxed index required for the CET
to occur in a particular casting. This level will be
dependent upon the size and the geometry of the mold
cavity, from casting to casting.
It is yet to be established whether such criteria are valid
for casting cases with rapid solidification or with the
inoculation of the melt to enhance heterogeneous nucleation, or when strong liquid convection takes place.
In the case of inoculated alloys, it is expected that
nucleation of the equiaxed dendrites takes place before
the temperature gradient in the liquid vanishes and well
before the equiaxed index or columnar growth rate
reaches its peak. The determination of a CET in this
case requires information on the temperature gradient,
the growth undercooling, and the equiaxed nucleation
undercooling. A Hunt analysis should give good predictions of the CET in cases in which the inoculant
details are known. Otherwise, direct modeling of the
equiaxed structure is needed (as performed in other
modeling approaches, for example, in References 14
and 43). In other work, the experiments discussed here
are simulated with models that include details of an
equiaxed mushy zone nucleating ahead of the columnar
zone (References 8 and 14). Thus, these models predict
the full extent of the combined mushy zone with both
columnar mush and equiaxed mush.
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.
NOMENCLATURE
A
a0
a1
a2
CCV
CL
CS
cp
D
d
e
gs
H
I
i
j
K
KCV
dendrite growth coefficient
polynomial coefficient
polynomial coefficient
polynomial coefficient
specific heat for a control volume
specific heat for liquid
specific heat for solid
specific-heat capacity
diameter of ingot
volume fraction of a control volume
error
solid fraction
height of the ingot
equiaxed index
grid coordinate
grid coordinate
thermal conductivity
thermal conductivity of a control volume
METALLURGICAL AND MATERIALS TRANSACTIONS A
KD
KI
KL
KP
KS
L
n
ncols
nrows
P
Q
q
qloss
s
T
TE
Tinit
TL
TM
t
tq
Ub
VCV
Vm
vt
DT
DTn
Dt
Dx
Dy
q
s
derivative gain
integral gain
thermal conductivity of liquid
proportional gain
thermal conductivity of solid
latent heat
dendrite growth law exponent
number of columns in a grid
number of rows in a grid
general polynomial value
growth restriction
heat flux at the chill surface
heat flux at the free liquid surface
Laplace coordinate
temperature
eutectic temperature
initial temperature
liquidus temperature
melting temperature of solvent material
time coordinate
cutoff time for heat loss
undercooled bulk liquid
control volume size
volume of mush in a control volume
dendrite growth velocity
undercooling
nucleation undercooling
time-step
grid spacing
grid spacing
density
first-order lag constant
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