Seetharaman S. Fundamentals of metallurgy
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It is also assumed that when solidification is initiated, the interface temperature
immediately attains T
int
as calculated in equation 10.37. Of course, there must
also be a kinetic undercooling ( T
k
) to allow heat transfer and there can also be
a pressure undercooling (T
p
); however, both of these corrections are usually
small and are normally ignored.
9
A schematic of a thermal profile for under-
cooled solidification is given in Fig. 10.10 where the various undercoolings are
marked.
An exampl e of undercooled growth can be seen in Fig. 10.11 where liquid
iron is undercooled until solidi fication occurs on an alumina substrate. In this
10.10 Schematic of thermal profile in liquid and solid for undercooled growth
of an alloy.
10.11 Example of undercooled solidification.
15
Photograph by H. Shibata.
416 Fundamentals of metallurgy
case the undercooling for initiation of solidification is 211ëC and the droplet
fully recalesces in the time of 1 frame of the video camera.
Transient moving boundary problems of this type are Stefan problems and, if
one looks at the situation where a semi-infinite solution of the transient heat
transfer equation is appropriate and can be used to determine thermal gradients:
@T
@t
@
2
T
@x
2
10:44
it follows that the growth rate for a plane front growing into an undercooled
liquid is:
R
t
r
10:45
where
p
exp
2
erfc
C
p
T
t
H
St 10:46
where erfc is the error function complement, T
t
is the thermal undercooling of
the liquid and St is the Stefan number. It should be noted that when the Stefan
number equals or is greater than 1, this function has no solution. If the Stefan
number equals 1, C
p
T H and this means that the heat given out during
solidification can recalesce the liquid to its equilibrium point. At Stefan numbers
greater than 1, the solidification process cannot recalesce the liquid to its
equilibrium point and this is termed hypercooling. Under this condition the
interface temperature would not be easily determined and this solution is not
appropriate.
Upon nucleation of a solid the first solid formed is a sphere and the interface
conditions must include the fact that the interfacial area is increasing. Equation
10.40, the interface heat balance must be rewritten for spherical growth as
follows:
k
s
G
s
ÿ k
l
G
l
H R ÿ
1
A
dA
dt
H ÿ R 10:47
Where the growth rate is increased due to the formation of surface, i.e. less
energy needs to be conducted as it is consumed in surface creation. Mullins
showed that spheres in undercooled liquids become unstable very quickly and
rather than grow as spheres begin to grow as cells or dendrites.
If we assume that the sphere is growing in an undercooled liquid of
temperature T
1
, then the growth rate of the sphere will be:
R
dr
dt
rr
p
ÿ
k
l
H
dT
dr
rr
p
k
l
H
T
i
ÿ T
1
1
r
p
1
t
p
10:48
where T
i
is the interface temperature. At longer times equation 10.48 will tend to
the steady state solution where:
Solidification and steel casting 417
R
k
l
H
T
i
ÿ T
1
r
p
10:49
The progression of radius with time will then be:
r
p
r
p;o
K
l
T
i
ÿ T
1
H
t
s
10:50
where r
p;o
is the starting radius. To determine the length of time it takes to reach
a steady state solution, the characteristic time () can be determined where
r
2
p
l
10:51
in metals where
l
is approx 10
ÿ5
m
2
/sec, for a of less than 0.1 sec, r
p
must be
less than 0.1 mm. Thus for very small spheres the steady state solution would be
appropriate; otherwise the full transient solution is necessary.
As we have already noted, T
i
is a function of r for very small radii and:
T
i
T
mpt
ÿ
2ÿ
r
10:52
Therefore equation 10.49 can be rewritten as:
R
k
l
H
T ÿ
2ÿ
r
p
r
p
10:53
where T T
Mpt
ÿ T
1
. Thus t he growth rate will become zero when
T 2ÿ=r
p
as T
i
will tend to T
1
and the driving force for heat transfer will
be eliminated.
As previously noted, Mullins has shown that spherical growth is unstable,
perturbations form and grow where these perturbations develop with a tip radius
that is constant. A general problem in solidification is to determine the relation-
ship between the radius and growth rate. If one considers this problem as a
cylinder growing with a hem ispherical cap into an undercooled liquid, one
develops a relationship very similar to equation 10.49 for steady state growth:
rR
2
C
p
T
i
ÿ T
1
H
10:54
which, if taking into account the effect of radius on the interface temperature, is
written as:
Pe
T
1 ÿ
T
r
T
10:55
where Pe
T
is the thermal peclet number and is the inverse of the Stefan
number. Thus in equation 10.54 or 10.55, if the radius is known the growth rate
can be calculated and vice versa. A more complete solution for this diffusion
418 Fundamentals of metallurgy
problem was first given by Ivantsov for a geometry that is closer to that of a real
dendrite where, if we correct for capillarity:
IP 1 ÿ
T
r
T
10:56
where
IP Pe
T
exp Pe
T
E
1
Pe
T
and
E
1
Pe
T
ÿ0:577 ÿ lnPe
T
4Pe
T
Pe
T
4
The above discussion concerns undercooled growth; however, solidification is
often initiated by cooling against a mold. A schematic of the thermal profile for
normal solidification is given in Fig. 10.12.
Normal solidification is the most common mode of growth and, if one
assumes perfect contact at the mold±shell interface and thermal gradients in both
the shell and the mold, Schwartz's semi-infinite solution of the Stefan problem
results, where:
R
shell
t
r
10:57
and is calculated as follows:
T
m
ÿ T
o
C
s
p
H
p
exp
2
kC
p
shell
kC
p
mold
s
erf
" #
10:58
And the mold shell interface temperature is calculated from the following relation:
10.12 Schematic of the temperature profile for normal solidification.
Solidification and steel casting 419
T
s
T
o
T
m
ÿ T
o
kC
p
shell
kC
p
mold
s
kC
p
shell
kC
p
mold
s
erf
2
6
6
6
6
6
4
3
7
7
7
7
7
5
10:59
Of course, reality is always more complicated than these simplifications and, if
one considers the situation of undercooling a liquid against a mold, there would
be heat transfer in both directions at the interface. In this case, equation 10.60
gives the expression for the growth rate where can be determined as
follows:
7
St
s
p
exp
2
erf
St
initial
p
exp
2
erf
1 10:60
and
St
s
c
ps
T
m
ÿ T
s
H
and St
initial
c
pl
T
m
ÿ T
initial
H
s
l
r
The situation in Fig. 10.11 is typical of an undercooled solidification problem
where the system cools to a temperature before solidification is initiated and
then when solidification occurs the growing front immediately attains the
equilibrium interface temperature and, heat is conducted into the liquid and
into the shell. In this case there is heat transfer through the shell and the
alumina pedestal that the iron droplet rests upon, thus this solution must be a
combination of a Schwartz solution and the undercooled solution as shown in
Fig. 10.13.
Experimentally measured results versus this simplified model are shown in
Fig. 10.14 where fair agreement between experimentally measured results is
observed at higher undercoolings but not at lower undercoolings due to complete
recalescence of the droplet before completion of solidification.
Another more practical condition for undercooled growth would be against a
cooled metal mold where, if we assume that solidification occurs by nucleation,
then initially the liquid against the mold would be undercooled until the
temperature reaches the level necessary for activation of a heterogeneous nuclei.
Once the front is nucleated there will be a transfer of heat into the liquid and
through the steel. Once the undercooled liquid has been reheated, all thermal
gradients will be through the shell and there will be a transition from under-
cooled growth to normal solidification. There is no analytical method to
represent this situation therefore a numerical solu tion is necessary. Results from
such a calculation are given in Fig. 10.15 where as the critical undercooling for
nucleation is increased there is a longer time before solidification is initiated.
420 Fundamentals of metallurgy
10.13 Schematic of the temperature profile for undercooled growth.
15
10.14 Growth of a liquid iron shell as a function of undercooling.
15
Solidification and steel casting 421
This initiation time is followed by a rapid rise in growth rate due to the
combined modes of heat transfer. As the undercooled liquid is recalesced, the
solution becomes that for normal solidification. It can be seen that as the ability
to undercool increases, the time to solidification is decreasing. This occurs
because the efficiency of heat transfer is greates t when the liquid contacts the
mold. Once the shell is formed the total heat flux decreases. Clearly this effect is
most obvious when casting small sections.
The assumption of perfect contact at the shell±mold interface is not normally
accurate and in reality there is a complex situation that is found where the shell
is in intermi ttent contact with the mold leading to a complex conduction±
convection and radiation problem. This complex situation is overcome by
assuming that Newton's law can be invoked and that a heat transfer coefficient
for the mold can be determined.
In actual molds it is common that all of the resistances to heat transfer are
important and it is common to write the heat flux equation as a sum of
resistances using Newton's law and assuming an analog to electrical resistance,
the heat balance can be written as:
Q hAT
p
ÿ T
o
T
p
ÿ T
o
<
s
HR 10:61
where T
p
is the temperature of the liquid and T
o
is the temperature of the
medium remo ving heat from the mold. Resistance for heat transfer coefficients
will be 1=hA and for conduction will be L=kA where l is the distance over which
heat is conducted.
0 0.1 0.2 0.3 0.4 0.5
0.0010
0.0008
0.0006
0.0004
0.0002
0
50 C critical undercooling
100 C critical undercooling
150 C critical undercooling
200 C critical undercooling
250 C critical undercooling
300 C critical undercooling
o
o
o
o
o
o
Time (s)
Shell thickness (m)
10.15 Shell profile calculations as a function of the critical undercooling for
nucleation against a mold wall.
422 Fundamentals of metallurgy
Thus, for a water-cooled mold where there is a mold thickness and a growing
shell with intermittent contact, the total resistance to heat transfer < would be
calculated as follows:
<
1
h
water
A
L
mold
k
mold
A
1
h
interface
A
L
shell
k
shell
A
1
h
liquid
A
10:62
In the above equations the heat transfer coefficients for the fluid against the
mold and the liquid against the shell must be determined from a knowledge of
fluid flow in the system and h
interface
is a complicated heat transfer coefficient
that includes conduction, convection and radiation terms. The conduction terms
come from point contacts with the mold, the convection and radiation terms are
due to heat transfer across the valleys between the point contacts. In radiation
heat transfer the heat flux is described by Stefan' s law, where:
Q / T
4
1
ÿ T
4
2
10:63
The radiation con tribution to heat tran sfer can be significant for high
temperature metals. It is often assumed that the interface h is a simple sum of
heat transfer coefficients that account for the three modes of heat transfer:
h
interface
h
conduction
h
convection
h
radiation
10:64
It is therefore no surprise that h
interface
is very difficult to calculate as it is a
function of time, local mold geometry and the detailed thermal conditions within
the mold and analytical solutions of the real casting problems are not possible.
If the interface resistance dominated heat transfer, then
m
h
int
T
M
ÿ T
o
t
H
10:65
R
h
int
T
M
ÿ T
o
H
10:66
In the case where resistance to heat transfer is in the shell and at the interface,
Adams developed the following solution to this problem:
m ÿ
k
s
h
k
2
s
h
2
2k
s
T
M
ÿ T
o
a H
t
s
10:67
where
a
1
2
1
4
C
ps
T
M
ÿ T
s
3H
r
Now if h k
s
m ÿ
k
s
h
2k
s
T
M
ÿ T
o
a H
t
s
10:68
Solidification and steel casting 423
and
R
k
s
T
M
ÿ T
o
a H t
s
10:69
In alloys where there is a significant freezing range where solidification begins
at the interface temperature (T
i
) and ends at the effective solidus temperature
(T
s
) there is zone where the fraction of solid varies from 0 at T
i
to 1 at T
s
. In this
area of the changing fraction of solid, cells and dendrites develop in normal and
undercooled solidification. It is normal to view this area as a separate zone of
solidification and assume that in the mus hy zone the total enthalpy consists of an
enthalpy of the liquid H
l
and an enthalpy of the solid H
s
and to partition the
release of latent heat to correspond to the temporal evolution of the fraction of
solid (f), where:
@H
@T
H
@f
s
@t
f
l
@H
l
@t
f
s
@H
s
@t
10:70
and equation 10.43 can be rewritten as follows:
1 ÿ
H
c
p
@f
s
@t
@T
@t
@
2
T
@x
2
10:71
For equilibrium solidification the use of this approach is straightforward;
however, in all other conditions, it is strongly influenced by segregation and
numerical solution is necessary.
If equation 10.43 is written to include an energy source term such that:
@T
@t
@
2
T
@x
2
_
Q
C
p
10:72
and, assuming
_
Q H
@f
s
@t
that, then:
9
@T
@t
@
2
T
@x
2
H
C
p
@f
s
@t
10:73
which is equival ent to equation 10.70, as
@f
s
@t
@f
s
@T
@T
@t
The partitioning of the latent heat across the actual liquidus to solidus gap leads
to a change in the gradient of cooling in the mushy zone. If internal temperature
gradients in the shell are small, then the cooling rate of a casting can be
calculated as follows:
@T
@t
ÿ
hAT
p
ÿ T
o
C
p
1 ÿ
H
C
p
@f
s
@T
10:74
424 Fundamentals of metallurgy
Even in heat transfer dominated processe s, at the initiation of solidification, the
potential heat flux and therefore growth rate is very high. The ability to grow a
front by heat transfer cannot be faster than the ability to add atoms to the front;
therefore, the growth rate at very small times must be limited by this issue. This
is schematically shown in Fig. 10.16.
10.3.3 Mass transfer dominated growth
In conditions where dissipation of heat is not controlling the solidification rate or
during isothermal solidification, it is possible to find conditi ons where transport
of mass in the liquid controls the growth rate.
The mass balance at a planar interface in one dimension can be written as:
D
s
dC
s
dx
xx
i
ÿD
l
dC
l
dx
xx
i
RC
i
l
ÿ C
i
s
RC
i
l
1 ÿ k
eq
10:75
where C
i
l
and C
i
s
are the interface composition of the interface and the interface
composition of the solid respectively and k
eq
is the equilibrium partition coefficient
determined from a phase diagram and it is assumed that at all growth rates
interfacial equilibrium is maintained. Equation 10.75 is similar in form to equation
10.40, the heat balance at the interface. However, there is one significant difference,
in the heat flux equation all heat energy must travel from liquid to solid to complete
solidification and after solidification a casting can be cooled to one isothermal
temperature; however, in the transport of mass, it is normal that the time to cool a
casting to an isothermal temperature is much shorter than the time to ensure
chemical homogenization of a casting. This means that gradients in composition
that arise due to the solidification process will remain in the casting, unless some
thermal treatment is carried out after casting. Another difference is that composition
can be stored in the liquid where heat cannot; thus during solidification the
composition of the liquid can increase of decrease, depending upon the phase
10.16 Schematic of growth rate as a function of time.
Solidification and steel casting 425