Solidification structure
5.1 Heat transfer
5.1.1 Resistances to heat transfer
Chapter 5
In many instances. resistance (5) is also negligible in practice. For instance, for normal sand moulds the environment of the mould does not affect solidification, since the mould becomes hardly warm on its outer surface by the time the casting has solidified inside. However, there are, of course, a number of exceptions to this general rule, all of which relate to various kinds of thin-walled moulds, which, because of the thinness of the mould shell, are somewhat sensitive to their environment. Iron castings made in Croning shell moulds (the Croning shell process is one in which the sand grains are coated with a thermosetting resin, which is cured against a hot pattern to produce a thin, biscuit-like mould) solidify faster when the shell is thicker, or when the shell is thin and backed up with steel shot. Conversely, the freezing of investment shell castings in steel is delayed by a backing to the shell of granular refractory material preheated to high temperature, but is accelerated by being allowed to radiate heat away freely to cool the surroundings.
Iron and steel dies for the casting of aluminium alloys cool faster when the backs of the dies are cooled by water.
Nevertheless, despite such useful ploys for coaxing greater productivity, it remains essential to understand that in general the major fundamental resistances to heat flow from castings are items (2), (3) and (4). For convenience we shall call these resistances 1, 2 and 3.
The effects of all three simultaneously can nowadays be simulated with varying degrees of success by computer. However, the problem is both physically and mathematically complex, especially for castings of complex geometry.
There is therefore still much understanding and useful guidance to be obtained by a less ambitious approach, whereby we look at the effect of each resistance in isolation, considering only one dimension (i.e. unidirectional heat flow). In this way we can define some valuable analytical solutions that are surprisingly good approximations to casting problems. We shall continue to follow the approach by Flemings.
5.1.1.1 Resistance 1 : The casting
It has to be admitted that this type of freezing regime is not common for metal castings of high thermal conductivity such as the light alloys or Cu-based alloys.
However, it would nicely describe the casting of Pb-Sb alloy into steel dies for the production of battery grids and terminals; the casting of steel into a copper mould; or the casting of hot wax into metal dies as in the injection of wax patterns for investment casting. It would be of wide application in the plastics industry.
For the unidirectional flow of heat from a metal
poured exactly at its melting point T , against a mould wall initially at temperature To, the transient heat flow problem is described by the partial differential equation, where a, is the thermal diffusivity of the solid:
(5.1) The boundary conditions are x = 0, T = To; at x = S, T = T,,,, and at the solidification front the rate of heat evolution must balance the rate of conduction down the temperature gradient, Le.:
(5.2) where K , is the thermal conductivity of the solid, H is the latent heat of solidification, and for which the solution is:
s
= 2 y K t (5.3)The reader is referred to Flemings for the rather cumbersome relation for y. The important result to note is the parabolic time law for the thickening of the solidified shell. This agrees well with experimental observations. For instance, the thickness S of steel solidifying against a cast iron ingot mould is found to be:
(5.4) where the constants a and b are of the order of 3 and 25 respectively when the units are millimetres and seconds. The result is seen in Figure 5.2.
T h e apparent delay in the beginning of solidification shown by the appearance of the constant b is a consequence of the following: (i) the turbulence of the liquid during and after pouring, resulting in the loss of superheat from the melt, and so slowing the start of freezing, and (ii) the finite interface resistance further slows the initial rate of heat loss. Initially the solidification rate will be linear, as described in the next section (and hence giving the initial curve in Figure 5.2 because of this plot using the square root of time). Later, the resistance of the solidifying metal becomes dominant, giving the parabolic relation (shown, of course, as a straight line in Figure 5.2 because of the plot using the square root plot of time).
5.1.1.2 Resistance 2: The metal/mould interface In many important casting processes heat flow is controlled to a significant extent by the resistance at the metallmould interface. This occurs when both the metal and the mould have reasonably good rates of heat conductance, leaving the boundary between the two the dominant resistance. The interface
Solidification structure 1 19 Time (min)
0 4 16 36 64 100
300
-g 200
-
ED c ._
n - 8
v)
a, Y
c .-
+ 100
I I I I
0 /' 2 4 6 8 10
/ / G ( r n i n ' / * )
Figure 5.2 Unidirectional solidification of pure iron against a cast iron mould coated vbsith a protective wa.rh (from Flemings 1974).
becomes overriding in this way when an insulating mould coat is applied, or when the casting cools and shrinks away from the mould (and the mould heats up, expanding away from the metal), leaving an air gap separating the two. These circumstances are common in the die casting of light alloys.
For unidirectional heat flow the rate of heat released during solidification of a solid of density ps and latent heat of solidification H is simply:
(5.5) This released heat has to be transferred to the mould.
The heat transfer coefficient h across the metal/
mould interface is simply defined as the rate of transfer of energy q (usually measured in watts) across unit area (usually a square metre) of the interface, per unit temperature difference across the interface. This definition can be written:
(5.6) assuming the mould is sufficiently large and conductive not to allow its temperature to increase
= - hA(T,,, - TO)
significantly above To, effectively giving a constant temperature difference (T,, - To) across the interface.
Hence equating 5.5 and 5.6 and integrating from S
= 0 at t = 0 gives:
(5.7) It is immediately apparent that since shape is assumed not to alter the heat transfer across the interface, Equation 5.7 may be generalized for simple-shaped castings to calculate the solidification time tf in terms of the volume V to cooling surface areaA ratio (the geometrical modulus) of the casting:
P\ H V
h ( T , - T " )
x
tf =
All of the above calculations assume that I7 is a constant. As we shall see later, this is perhaps a tolerable approximation in the case of gravity die (permanent mould) casting of aluminium alloys where an insulating die coat has been applied. In most other situations h is highly variable, and is particularly dependent on the geometry of the casting.
The air gap
As the casting cools and the mould heats up, the two remain in good thermal contact while the casting interface is still liquid. When the casting starts to solidify, it rapidly gains strength, and can contract away from the mould. In turn, as the mould surface increases in temperature it will expand. Assuming for a moment that this expansion is homogeneous, we can estimate the size of the gap d as a function of the diameter D of the casting:
where a is the coefficient of thermal expansion, and subscripts c and m refer to the casting and mould respectively. The temperatures T are Tt the freezing point, Tmi the mould interface. and To the original mould temperature.
The benefit of the gap equation is that it shows how straightforward the process of gap formation is. It is simply a thermal contraction-expansion problem, directly related to interfacial temperature.
It indicates that for a casting a metre across which is allowed to cool to room temperature the gap would be expected to be of the order of 10 mm at each of the opposite sides. This is a substantial gap by any standards!
Despite the usefulness of the elementary formula in giving some order-of-magnitude guidance on the dimensions of the gap, there are a number of interesting reasons why this simple approach requires further sophistication.
In a thin-walled aluminium alloy casting of section only 2 mm the room temperature gap would be only 10 pm. This is only one-twentieth of the size of an average sand grain of 200 p m diameter.
Thus the imagination has s o m e problem in visualizing such a small gap threading its way amid the jumble of boulders masquerading as sand grains.
It really is not clear whether it makes sense to talk about a gap in this situation.
Woodbury and co-workers (2000) lend support to this view for thin wall castings. In horizontally sand cast aluminium alloy plates of 300 mm square and up to 25 mm thickness, they measured the rate of transfer of heat across the metal/mould interface.
They confirmed that there appeared to be no evidence for an air gap. Our equation would have predicted a gap of 2.5 pm. This small distance could easily be closed by the slight inflation of the casting because of two factors: (i) the internal metallostatic pressure provided by the filling system (no feeders were used), and (ii) the precipitation of a small amount of gas; for instance, it can be quickly shown that 1 per cent porosity would increase the thickness of the plate by at least 70 pm. Thus the plate would swell by creep under the combined internal pressure due to head height and the growth of gas pores with minimal difficulty. The 25 p m movement from thermal contraction would be so comfortably overwhelmed that a gap would probably never have chance to form.
Our simple air gap formula assumes that the mould expands homogeneously. This may be a reasonable assumption for the surface of a greensand mould, which will expand into its surrounding cool bulk material with little resistance. A rigid, chemically bonded sand will be subject to more restraint, thus preventing the surface from expanding so freely. The surface of a metal die will, of course, be most constrained of all by the surrounding metal at lower temperature, but the higher conductivity of the mould will raise the temperature of the whole die more uniformly, giving a better approximation once again to homogeneous expansion.
Also, the sign of the mould movement for the second half of the equation is only positive if the mould wall is allowed to move outwards because of small mould restraint (i.e. a weak moulding material) or because the interface is concave. A rigid mould and/or a convex interface will tend to cause inward expansion, reducing the gap, as shown in Figure 5.3. It might be expected that a flat interface will often be unstable, buckling either way. However, Ling and co-workers (2000) found that both theory and experiment agreed that the walls of their cube- like mould poured with white cast iron distorted outwards in the case of greensand moulds, but inwards in the case of the more rigid chemically bonded moulds.
There are further powerful geometrical effects
Figure 5.3 Movement of mould walls, illustrating the principle of inward expansion in convex regions and outward expansion in concave regions.
to upset our simple linear temperature relation.
Figure 5.4 shows the effect of linear contraction during the cooling of a shaped casting. Clearly, anything in the way of the contraction of the straight lengths of the casting will cause the obstruction to be forced hard against the mould. This happens in the corners at the ends of the straight sections.
Gaps cannot form here. Similarly, gaps will not occur around cores that are surrounded with metal, and on to which the metal contracts during cooling.
Conversely, large gaps open up elsewhere. The situation in shaped castings is complicated and is only just being tackled with some degree of success by computer models.
Figure 5.4 Variable air gap in a shaped casting: arrows denote the probable sires of zero gap.
Solidification m u c t u r c I 2 I
Richmond and Tien (1971) and Tien and Richmond (1982) demonstrate via a theoretical model how the formation of the gap is influenced by the internal hydrostatic pressure in the casting, and by the internal stresses that occur within the solidifying solid shell. In Richmond et al. (1990) Richmond goes on to develop his model further, showing that the development of the air gap is not uniform. but is patchy. He found that air gaps were found to nucleate adjacent to regions of the solidified shell that were thin, because, as a result of stresses within the solidifying shell, the casting/mould interface pressure first dropped to zero at these points. Conversely. the casting/mould interface pressure was found to be raised under thicker regions of the solid shell, thereby enhancing the initial non- uniformity in the thickness of the solidifying shell.
Growth becomes unstable, automatically moving away from uniform thickening. This rather counter- intuitive result may help to explain the large growth perturbations that are seen from time to time in the growth fronts of solidifying metals. Richmond reviews a considerable amount of experimental evidence to support this model. All the experimental data seem to relate to solidification in metal moulds.
It is possible that the effect is less severe in sand moulds.
Attempts to measure the gap formation directly (Isaac et ul. 1985; Majumdar and Raychaudhuri 198 1) are extremely difficult to carry out accurately.
Results averaged for aluminium cast into cast iron dies of various thickness reveal the early formation of the gap at the corners of the die where cooling is fastest. and the subsequent spread of the gap to the centre of the die face. A surprising result is the reduction of the gap if thick mould coats are applied.
(The results in Figure 5.5 are plotted as straight lines. The apparent kinks in the early opening of the gap reported by these authors may be artefacts of their experimental method.)
It is not easy to see how the gap can be affected by the thickness of the coating. The effect may be the result of the creep of the solid shell under the internal hydrostatic pressure of the feeder. This is more likely to be favoured by thicker mould coats as a result of the increased time available and the increased temperature of the solidified skin of the casting. If this is true then the effect is important because the hydrostatic head in these experiments was modest, only about 2 0 0 m m . Thus f o r aluminium alloys that solidify with higher heads and times as long or longer than a minute or so, this mechanism for gap reduction will predominate.
It seems possible, therefore, that in gravity die casting of aluminium the die coating will have the major influence on heat transfer, giving a large and stable resistance across the interface. The air gap will be a small and variable contributor. For computational purposes, therefore, it is attractive
Corner
0 Centre r7
Time (s)
Figure 5.5 Results civeraged from varioii.c die.% ( I S L I ~ K ('1
al. 1985). illustrating the .start of the air gap ut the corners, and its spread to the centre o f t h e inoiild ,film.
Increased thickness of mould coating is .seen to d e l q solidification and to reduce the growth of'the gap.
to consider the great simplification of neglecting the air gap in the special case of gravity die casting of aluminium.
In conclusion, it is worth mentioning that the name 'air gap' is perhaps a misnomer. The gap will contain almost everything except air. As we have seen previously, mould gases are often high in hydrogen, containing typically 50 per cent. At room temperature the thermal conductivity of hydrogen is approximately 6.9 times higher than that of air, and at 500°C the ratio rises to 7.7. Thus, the conductivity of a gap at the casting/mould interface containing a 5050 mixture of air and hydrogen at 500°C can be estimated to be approximately a factor of 4 higher than that of air. In the past, therefore, most investigators in this field have probably chosen the wrong value for the conductivity of the gap, and by a substantial margin!
The heat-transfer coefficient
The authors Ho and Pehlke (1984) from the University of Michigan have reviewed and researched this area thoroughly. We shall rely mainly on their work in this section.
When the metal first enters the mould the macroscopic contact is good because of the conformance of the molten metal. Gaps exist on a microscale between high spots as shown in Figure 5.6. At the high spots themselves, the high initial heat flux causes nucleation of the metal by local severe undercooling (Prates and Biloni 1972). The solid then spreads to cover most of the surface of the casting. Conformance and overall contact between the surfaces is expected to remain good during all of this early period, even though the
122
produce analytical equations for each of these contributors to the total heat flux. We can summarize their findings as follows:
(b)
Figure 5.6 MetaWmould interface at an early stage when solid is nucleating at points of good thermal contact.
Overall macroscopic contact is good at this stage (a).
Later (bj the casting gains strength, and casting and mould both deform, reducing contact to isolated points at greater separations on non-conforming rigid surfaces.
mould will now be starting to move rapidly because of distortion.
After the creation of a solidified layer with sufficient strength, further movements of both the casting and the mould are likely to cause the good fit to be broken, so that contact is maintained across only a few widely spaced random high spots (Figure 5.6b).
The total transfer of heat across the interface may be written as the sum of three components:
h, = h,
+
h,+
h,where h, is the conduction through the solid contacts, h, is the conduction through the gas phase, and h, is that transferred by radiation. Ho and Pehlke Table 5.1 Mould and metal constants
While the casting surface can conform, the contribution of solid-solid conduction is the most important. In fact, if the area of contact is enhanced by the application of pressure, then values of h, up to 60 000 Wm-2K-' are found for aluminium in squeeze casting. Such high values are quickly lost as the solid thickens and conformance is reduced, the values fallin to more normal levels of 100-1000 Wm- K (Figure 5.7).
When the interface gap starts to open, the conduction through solid contacts becomes negligible. The point at which this happens is clear in Figure 5.7b. (The actual surface temperature of the casting and the chill in this figure are reproduced from the results calculated by Ho and Pehlke.) The rapid fall of the casting surface temperature is suddenly halted, and reheating of the surface starts to occur. An interesting mirror image behaviour can be noted in the surface temperature of the chill, which, now out of contact with the casting, starts to cool. The estimates of heat transfer are seen to simultaneously reduce from over 1000 to around
100 Wm-*K-' (Figure 5 . 7 ~ ) .
8
- 1J . After solid conduction diminishes, the important mechanism for heat transfer becomes the conduction of heat through the gas phase. This is calculated from:
h, = Wd
where k is the thermal conductivity of the gas and d is the thickness of the gap. An additional correction is noted by Ho and Pehlke for the case where the
Material Melting Liquid- Specific heat Densiy Thermal conductivity
point solid (J.Kg K ) (kglm 1 (Jlrn K s)
("C) contraction
(%I Solid Liquid Solid Liquid Solid Liquid
20°C m.p. m.p 20°C m.p. m.p. 20°C m.p. m.p.
Pb 327 3.22 130 (138) 152 11680 11020 10678 39.4 (29.4) 15.4
Zn 420 4.08 394 (443) 481 7140 (6843) 6575 119 95 9.5
Mg 650 4.2 1038 (1300) 1360 1740 (1657) 1590 155 (90)? 78
A1 660 7.14 917 (1200) 1080 2700 (2550) 2385 238 - 94
cu
1084 5.30 386 (480) 495 8960 8382 8000 397 (235) 166Fe 1536 3.16 456 (1130) 795 7870 7265 7015 73 14)? -
Graphite - - 1515 - - 2200 - - 147
Silica sand - 1130 - 1500 - - 0.0061 -
(Mullite) 750 - - 1600 - - 0.0038 - -
- -
Investment
- -
Plaster - 840 - - 1100 - 0.0035 - -
References: Wray (1976); Brandes ( 1 99 I ); Fleming5 ( 1 974)