AN INVESTIGATION INTO BRIDGMAN FURNACE
SOLIDIFICATION OF TITANIUM ALLOYS
R.P. Mooney¹ and S. McFadden1
1. Department of Mechanical & Manufacturing Engineering,
Trinity College Dublin, Dublin 2, Ireland.
ABSTRACT
Titanium alloys are emerging as a candidate to replace nickel superalloys
in the aerospace industry due to their very low density and exceptional
mechanical properties at high temperatures. Casting these materials can be
difficult due to the tendency of molten titanium to react with other elements, and
due to a lack of understanding of the material solidification process. This paper
provides an overview of a body of research being carried out as part of the
European Space Agency backed research project GRADECET (GRAvity
DEpendance of Columnar to Equiaxed Transition in Ti-Al alloys). Four key
activities of the work are identified: a Bridgman furnace solidification model;
model verification; furnace characterisation; and columnar to equiaxed transition
experiments by power down technique. A method outline for each activity is
given. A front tracking model for Bridgman furnace solidification is
demonstrated. Model verification with an analytical model, for pure Titanium, is
discussed. A possible procedure for furnace characterisation is presented. Finally,
some preliminary modelling work of power down experiments is given.
KEYWORDS: ‘Titanium Aluminide’ ‘Casting’ ‘Columnar to Equiaxed
Transition’
1.
INTRODUCTION
Titanium alloys – specifically peritectic titanium aluminide alloys – have
emerged as a candidate to replace nickel superalloys in the aerospace industry [1].
This is on account of their very low density (approximately half that of nickel
superalloys [2]) and their exceptional mechanical properties at high temperatures.
The main application of interest is in the production of turbine blades for aero
engines and stationary gas turbines [3]. The material offers clear advantages, in
terms of engine efficiency and power-to-weight ratio improvements, for engine
designers. However, casting this alloy is made difficult due to its high liquidus
temperature (approx. 1550°C) and because of the high reactivity of molten Ti.
Other common casting problems include; shrinkage porosity, hot tearing, and
misrun [4]. Numerical modelling of the casting process can help to guide
experimental research, and industry, so that these problems can be predicted and
avoided as necessary.
One particular phenomenon of interest is the occurrence of Columnar to
Equiaxed Transition (CET) in castings. A CET is said to occur when the progress
of constrained (columnar) grain growth is blocked by the nucleation and
subsequent growth of unconstrained (equiaxed) grains [5]. Fig. 1 shows a CET
achieved in a Ti-Al based alloy, taken from a study Lapin and Gabalcová [6].
Fig. 1 CET in a Ti-Al based alloy [6]
In normal casting scenarios either a fully columnar or fully equiaxed grain
structure is desired, so that consistent mechanical properties are achieved
throughout the casting. For example, castings with a columnar grain structure are
used in directionally solidified turbine blades in order to reduce creep at high
temperatures. Whereas, fully equiaxed cast components are used in applications
where strength is important, to improve feeding, or reduce the possibility of hot
tearing. It is therefore vital to understand the conditions that produce a CET in
order to avoid it as necessary [7].
Microgravity experiments are unique in that the complicating effects of
gravity on solidification are suppressed. In terrestrial (1-g) casting experiments
thermal and solutal transport, due to natural convection in the melt, produce
sedimentation and macro-segregation. By carrying out microgravity experiments
and comparing to identical ground based experiments it is possible to distinguish
the effect of these phenomena from others – common to both environments – and
therefore refine and improve theoretical models for casting [8]. The European
Space Agency (ESA) backed research project IMPRESS (Intermetallic Materials
Processing in Relation to Earth and Space Solidification) looked in detail at the
effect of casting Ti-Al in microgravity environments [9] as part of the MAXUS 8
sounding rocket mission. Notably, a clear CET was not present in any of the four
samples processed in microgravity.
The work presented in this paper began when the IMPRESS project was
coming to an end, and another ESA backed project GRADECET (GRAvity
DEpendance of Colunmar to Equaixed Transition in Ti-Al Alloys) was beginning.
GRADECET has similar objectives to IMPRESS in that another MAXUS
sounding rocket experiment is planned (early 2015) where the experiment
designers hope to achieve a CET in the microgravity processed sample. As part of
the GRADECET team, we are engaged to carry out mathematical modelling to
aid the experiment design process through simulation of the solidification, using
the front tracking model (FTM) of McFadden and Browne [10]. More
specifically, we were asked to model a set of terrestrial Bridgman furnace
experiments – carried out by a fellow GRADECET collaborator – where a CET
was achieved in a Ti-Al based alloy. This paper outlines a summary of our
activities, thus far, in respect of this request.
2.
BUILDING A BRIDGMAN FURNACE FRONT TRACKING
MODEL (BFFTM)
2.1
The Bridgman & Power Down Methods
Front tracking has been used to model the solidification of Ti-Al in a ‘power
down’ scenario as part of the IMPRESS project [11]. In power down experiments
a cylindrical sample is fixed in a stationary crucible with two (or sometimes
more) tubular heaters positioned along the length of the sample – see Fig. 2(b).
Once the sample is melted, the heaters begin to cool in a controlled manner, at
fixed cooling rates (dTH/dt in the hot zone, and dTC/dt in the cold zone). A
solidification interface (solid-liquid) travels at a velocity, v, from the cold region,
through an adiabatic zone, to the hot region. In traditional Bridgman solidification
a pointed bottom crucible is drawn through the furnace, at some velocity, u,
where the hot heater temperature (TH) and cold heater temperature (TC) are fixed –
thereby constraining the temperature gradient in the sample during solidification,
as shown in Fig. 2(a). In this case, in steady state solidification, the solid-liquid
interface appears stationary in the adiabatic zone – however, relative to the
crucible, it is moving at some velocity, v. In principle both methods are different
forms of directional solidification; however, the movement of the sample adds a
complexity to the modelling task not previously tackled using an FTM. For alloys
an additional mushy zone – containing solid columnar dendrites surrounded by
liquid – will form in the sample between the fully liquid and fully solid regions
[12]. In this case the FTM tracks the location of the columnar dendrite tips (a
columnar front).
Fig. 2 Directional solidification by (a) the Bridgman method and (b) the power
down method
2.2
Fundamentals of the BFFTM for Binary Alloys
A detailed account of a BFFTM for binary alloys, as applied to an Al-7wt.%Si
alloy, was given by Mooney et al. [13]. The fundamental workings of the model
are briefly explained here. Considering one dimensional heat flow in a long
cylindrical rod (the sample), with radius r, cross sectional area, A, perimeter, p,
moving at velocity, u, and transferring heat laterally with its surroundings with a
heat transfer coefficient, h, we get Eq. (1) – as adapted from Carslaw and Jaeger
[10].

C pT     k T   uC p T  hp T  T   E
t
x  x 
x
A
hLc
k
Bi 
(1)
(2)
Where, Cp, and k are the sample density, specific heat capacity at constant
pressure, and thermal conductivity, respectively. Axial position and axial
temperature are given by x and T, respectively, T∞ is the temperature in the hot or
cold zones, and finally E is a term that deals with latent heat released during
solidification. This equation was solved using an explicit finite difference control
volume (CV) formulation where the sample was free to translate through a fixed
domain of CVs. A Dirichlet boundary condition was applied at the extremities of
the domain. Thermophysical properties were evaluated by polynomial functions
of temperature, taken from a study by McFadden et al. [14]. The dendrite growth
kinetics for the alloy were given by an exponential function of dendrite tip
undercooling, values for which were taken from the same study. The Biot
number, Bi in Eq. (2), relates the thermal resistance to heat flow at the surface of
a body (h), to the thermal resistance to heat flow inside that body (Lc/k), where Lc
is the characteristic length of the body. If Bi<0.1 it can be said that the
temperature inside the body is approximately equal to the temperature at the
surface of the body, for any axial location [15]. This number was used to justify
the assumption that the majority of heat flow in the sample was in the axial
direction. The latent term in Eq. (1), E, is the summation of the latent heat
released due to; advancement of the columnar front, Ea, and thickening of the
mush behind the columnar front, Et, as given by Eq. (3) and Eq. (4).
Ea 
Et 
L
VCV
gs
Vm
t
g
L
Vm s
VCV
t
(3)
(4)
Where L is the latent heat of fusion for the alloy per unit volume, VCV is the
volume of one CV, Vm is the captured volume of mush in a CV, and gs is the
fraction of solid in a CV, calculated as a function of temperature using the Scheil
equation [16]. The model assumes that the eutectic solidification of Al-Si occurs
in equilibrium using a conservative enthalpy model based on isothermal freezing,
as given by Swaminathan and Voller [17]. Note that this model can be adapted
easily for use with titanium alloys – or any alloy – once a there is an available
function of temperature for the calculation of solid fraction, gs.
2.3
Simulation using the BFFTM for Binary Alloys
Two notional Bridgman furnace scenarios were numerically simulated for a
100mm long rod of Al-7wt.%Si. The temperatures in the hot and cold zones were
set to values 50°C above and 50°C below the equilibrium liquidus and solidus
temperatures for the alloy, respectively. The first scenario demonstrated how a
fixed sample found a steady state temperature profile, initially having a constant
temperature in the hot and cold zones, and linear temperature profile in the
adiabatic zone.
Fig. 3 Thermal history and temperature profile evolution for scenario 2
In the second scenario the sample underwent two step changes in pulling velocity
– initially having the steady state temperature profile from the first scenario. In
each scenario the position of the columnar front and the evolution of temperature
profile were recorded – Fig. 3 shows the thermal history with front position, and
evolution of temperature profile for the second scenario.
3.
MODEL VERIFICATION
A one-dimensional analytical model for heat flow in a rod solidified in a
Bridgman furnace was given by Naumann [18]. The solution of which is
applicable where there is no mushy zone present in the sample, rather the Stefan
condition for a planar front (liquid-solid, non-dendritic interface) is used to solve
a dimensionless form of Eq. (1). Planar fronts normally occur with pure materials
and using slow pulling velocities [5]. It is possible, therefore, to verify the
BFFTM with the Naumann analytical model using pure titanium as the sample
material. The key equation in the Naumann solution is given by Eq. (5). Where, k
is the thermal conductivity of the sample, Pe is the Peclet number, TH and TC are
the temperatures of the hot and cold zones, TM is the alloy melting temperature, LF
is latent heat of fusion,  and  are constants, the subscripts ‘L’ and ‘S’ refer to
liquid and solid, respectively, and ±X1 defines the limits of the adiabatic zone.
The solution for the front position, X0, in this transcendental equation, requires an
iterative technique achievable using mathematical software such as Matlab®. Note
the solution is only valid for front positions within the adiabatic zone limits.
k L PeL TH  TM exp( PeL X 0 )
k S PeS TM  TC exp( PeS X 0 )
 Lf 
 PeL 
 Pe

exp( PeS X 1 ) S  1  exp( PeS X 0 )
exp(  PeL X 1 )
 1  exp( PeL X 0 )

*


 * 
4.
(5)
FURNACE CHARACTERISATION
The BFFTM relies heavily on knowledge of the heat transfer coefficient, h,
between the furnace heater and the sample curved surface. In order to apply the
model in a useful way – for example, using the model to predict growth velocity
and temperature gradient at the mush-liquid interface for a particular
experimental procedure – we must first determine the heat transfer coefficients in
the furnace. This can be done using steady state temperature measurements from
the crucible wall, at various positions along its length, in an experiment where the
crucible is static in the furnace. The temperature at the outside edge of the
crucible can be used to calculate the radiative heat transfer coefficient, hrad. The
thermal effect of the crucible can then be added, using Eq. (6) [19], to get a
combined heat transfer coefficient, hcomb, which can be inserted directly into the
BFFTM as h. Where ri and ro are the inside and outside radii, respectively, of the
hollow cylindrical crucible, and kcru is the thermal conductivity for the crucible
material.
hcomb
 ln ro / ri 
1  
ri 
 

k
h
r
cru
rad
i

 

1
(6)
Where the Bridgman furnace chamber is rarefied, it would be expected that
radiative heat transfer would be dominant [20] – in other words, the convective
heat transfer at the crucible wall becomes negligible. In this case, the radiative
heat transfer coefficient, and hence the combined heat transfer coefficient, would
be a function of temperature [21]. It is the intention of the authors to investigate
this by application of an inverse heat transfer method to a set of static Bridgman
furnace experiments carried out by a fellow GRADECET research partner at the
Slovak Academy of Sciences (SAS). The results of this exercise will
‘characterise’ the furnace so that the BFFTM can be applied to other experimental
data from the same furnace.
5.
POWER DOWN EXPERIMENTS
Power down experiments have been completed by the SAS where a CET has been
achieved in samples of GRADECET alloy 445 (Ti-44.5at.%Al-4.5at.%Nb0.2at.%B-0.2at.%C). The experimental procedure combined traditional Bridgman
solidification with the power down method, by initially pulling the sample into
the cold zone (to initiate columnar grain growth) and then arresting the sample,
before cooling the heaters at a constant rate. The procedure was repeated at three
other cooling rates. CET occurred in all four experiments. The BFFTM has been
applied to this experiment scenario using an estimate for the heat transfer
coefficient based on initial calculations taken from the characterisation exercise
referred to in section 4. It has been possible to generate a Hunt plot [22] (growth
velocity versus temperature gradient) for each cooling rate, and to overlay the
CET locations (achieved by experiment) onto this. Initial calculations show that
the fully equiaxed region in each sample occurred in an unconstrained growth
regime. This means that the equiaxed grains grew in a negative temperature
gradient [23]. This result is interesting from a CET theory perspective. Similar
results have been observed in directional solidification of Al-Si alloys [24].
However, the results are preliminary.
6.
CONCLUSION
A brief and clear summary of our work to date, as part of the ESA GRADECET
research project, has been given here. The fundamental workings of a Bridgman
furnace front tracking model (BFFTM) have been explained with a simple
example simulation outlined. A method for verification of the model was
presented. A detailed study on a method for Bridgman furnace characterisation
has been carried out, and some key concepts to determine the furnace heat
transfer coefficient have been briefly discussed here. It is our intention to apply a
verified BFFTM to a set of experimental data (provided by a GRADECET coworker) where a power down technique was used to achieve a CET at four
different cooling rates in samples of Ti-44.5at.%Al-4.5at.%Nb-0.2at.%B0.2at.%C. Some initial results point towards an unconstrained growth regime for
fully equiaxed grain growth. This may be an important result from a CET theory
point of view, however results are preliminary. The entire body of work here will
provide valuable data – in terms of growth velocity and temperature gradient
required for CET – to the designers of the GRADECET microgravity
solidification experiment, due to fly on the MAXUS 9 sounding rocket mission in
2015.
ACKNOWEDGEMENTS
The authors would like to thank the European Space Agency PRODEX
programme for funding this work via Enterprise Ireland. Thanks go to Juraj Lapin
and Zuzana Gabalcová of the Slovak Academy of Sciences for sharing their
experimental data with us. Special thanks go to Ulrike Hecht of ACCESS e.V.
Ulrike’s guidance has been invaluable and much appreciated.
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