In many important casting processes heat flow is controlled to a significant extent by the resistance at the metal–mold interface. This occurs when both the metal and the mold have reasonably good rates of heat conductance, leaving the boundary between the two the dominant resistance. The interface becomes overriding in this way when an insulating mold coat is applied, or when the casting cools and shrinks away from the mold (and the mold 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 densityrs and latent heat of solidificationHis simply:
q ẳ rsHAvS
vt (5.5)
This released heat has to be transferred to the mold. The heat transfer coefficienthacross the metal–
mold interface is simply defined as the rate of transfer of energyq(usually measured in watts) across unit area (usually a square meter) of the interface, per unit temperature difference across the interface.
This definition can be written:
q ẳ hAðTmT0ị (5.6) assuming the mold is sufficiently large and conductive not to allow its temperature to increase significantly above T0, effectively giving a constant temperature difference (Tm – T0) across the interface. Hence equating 5.5 and 5.6 and integrating fromS[0 att[0 gives:
S ẳ hðTmT0ị
rsH $t (5.7)
It is immediately apparent that since shape is assumed not to alter the heat transfer across the interface, Equation 5.7 may be generalized for simple-shaped castings to calculate the solidification timetfin terms of the volumeVto cooling surface areaAratio (the geometrical modulus) of the casting:
tf ẳ rsH hðTmT0ị$V
A (5.8)
All of the above calculations assume that h is a constant. As we shall see later, this is perhaps a tolerable approximation in the case of gravity die (permanent mold) casting of aluminum alloys where an insulating die coat has been applied. In most other situationsh is highly variable, and is particularly dependent on the geometry of the casting.
The air gap
As the casting cools and the mold heats up, the two remain in good thermal contact while the casting is still liquid. When the casting starts to solidify, it rapidly gains strength, and can contract away from the mold. In turn, as the mold surface increases in temperature it will expand. Assuming for a moment that 5.1 Heat transfer 191
this expansion ishomogeneous, we can estimate the size of the gapdas a function of the diameterDof the casting:
d=D ẳ acfTf Tg ỵamfTmiT0g
whereais the coefficient of thermal expansion, and subscriptscandmrefer to the casting and mold, respectively. The temperaturesTareTfthe freezing point,Tmithe mold interface, andT0the original mold 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 steel casting (acẳ14106K–1) of 1-meter diameter 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 simply from the contraction of the steel, neglecting any expansion of the mold. 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 aluminum alloy casting of section only 2 mm the room temperature gap would be only 10mm. This is only one-twentieth of the size of an average sand grain of 200mm diameter. Thus the imagination has some 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 aluminum alloy plates of 300 mm square and up to 25 mm thickness, they measured the rate of transfer of heat across the metal–mold interface. They confirmed that there appeared to be no evidence for an air gap. Our equation would have predicted a gap of 25mm. This small distance could easily be closed by the slight inflation of the casting because of two factors. (i) The internal metal- lostatic pressure provided by the filling system (no feeders were used). (ii) The precipitation of a small amount of gas; for instance it can be quickly shown that 1% porosity would increase the thickness of the plate by at least 70mm. Thus the plate would swell by creep with minimal difficulty under the combined internal pressure due to head height and the growth of gas pores. The 25mm 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 mold expands homogeneously. This may be a reasonable assumption for the surface of a greensand mold, which will expand into its surrounding cool bulk material with little resistance. A more rigid chemically bonded sand would be subject to rather 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 mold will raise the temperature of the whole die more uniformly, giving a better approximation once again to homogeneous expansion.
Also, the sign of the mold movement for the second half of the equation is only positive if the mold wall is allowed to move outwards because of small mold restraint (i.e. a weak molding material) or because the interface is concave. A rigid mold and/or a convex interface will tend to cause inward expansion, reducing the gap, as shown inFigure 5.3. It might be expected that a flat interface will often be unstable, buckling either way. However, Ling, Mampaey, and co-workers (2000) found that both 192 CHAPTER 5 Solidification structure
theory and experiment agreed that the walls of their cube-like mold poured with white cast iron distorted always outwards in the case of greensand molds, but always inwards in the case of the more rigid chemically bonded molds.
There are further powerful geometrical effects 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 mold. 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 onto 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. Even so, Kron (2004) is skeptical of the computer models to date. Few attempt to allow for the mechanical factors influencing the air gap. Even where such allowance is attempted, there is no agreement on a suitable constitutive equation for mechanical deformation of solids at temperatures near their melting points. The swelling of the casting in the mold due to the hydrostatic pressure in the liquid, and the friction of the rather soft casting against the mold are additional complicating factors.
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
Sand mold
Liquid alloy
FIGURE 5.3
Movement of mold walls, illustrating the principle of inward expansion in convex regions and outward expansion in concave regions.
5.1 Heat transfer 193
internal stresses that occur within the solidifying solid shell. Richmond (1990) 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 initiate adjacent to regions of the solidified shell that were thin, because, as a result of stresses within the solidifying shell, the casting–mold interface pressure first dropped to zero at these points. Conversely, the casting–mold 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 solid- ifying 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 molds. It is possible that the effect is less severe in sand molds.
Lukens et al. (1991) confirmed that increased feeder height increased the heat transfer at the base of the casting. Furthermore, for a horizontal cylinder 91 mm diameter in a chromite greensand mold, as the cylinder contracted away from the mold, gravity caused the cylinder to sit at the bottom of the mold, therefore contacting the mold purely along the line of contact along its base. Thus for this rather larger casting than the 25-mm-thick plate cast by Woodbury, it seems an air gap definitely occurs in a sand mold. This result confirms Lukens’ earlier result (1990) in which most heat appears to be extracted through the drag, and increased head pressure enhances metal–mold contact.
Attempts to measure the gap formation directly (Isaac et al. 1985, Majumdar and Raychaudhuri 1981) are difficult to carry out accurately. Results averaged for aluminum 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 center of the die face. A surprising result is the reduction of the gap if thick mold coats are applied. (The results inFigure 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 be 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 favored by thicker mold 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
Air gap Casting
Mold
FIGURE 5.4
Variable air gap in a shaped casting: arrows denote the probable sites of zero gap.
194 CHAPTER 5 Solidification structure
because the hydrostatic head in these experiments was modest, only about 200 mm. Thus for aluminum 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 aluminum 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 to consider the great simplification of neglecting the air gap in the special case of gravity die casting of aluminum.
It is probably helpful to draw attention to the fact that the name ‘air gap’ is perhaps a misnomer.
The gap will contain almost everything except air. As we have seen previously, mold gases are often high in hydrogen, containing typically 50%. At room temperature the thermal conductivity of hydrogen is approximately 5.9 times higher than that of air, and at 500C the ratio rises to 7.7. Thus, the conductivity of a gap at the casting–mold interface containing a 50:50 mixture of air and hydrogen at 500C 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!
This effect has been used by Doutre (1998) who injected helium gas, with a conductivity nearly identical to hydrogen (approximately 7 times the conductivity of air), into the air gap of a large Al alloy intake manifold cast in a permanent mold. The gas was introduced 30 s after pouring at a rate in the region of 10 ml s–1. In a full-scale works trial the production rate of the casting was increased by 25%. An even larger potential productivity increase, 45%, was found by Grandfield’s team (2007) when casting horizontal ingots in an open iron ingot mold. Gebelin and Griffiths (2007) found a reduced effect when attempting to cast resin-bonded sand molds in an evacuated chamber back-filled
Corner Center 0.5
0.4
0.3
0.2
0.1
0
)suidar tnec rep(pag riA
Zero coating thickness
100 μm coating
200 μm coating
50 100
Time (s) FIGURE 5.5
Results averaged from various dies (Isaac et al. 1985), illustrating the start of the air gap at the corners, and its spread to the center of the mold face. Increased thickness of mold coating is seen to delay solidification and to reduce the growth of the gap.
5.1 Heat transfer 195
with He. The explanation of the poor result in this latter case is almost certainly the result of the outgassing of sand molds under vacuum. This outward wind of volatiles would prevent the ingress of the He.
In passing, it seems worth commenting that He is expensive, and world supplies are limited.
Naturally pure hydrogen introduced into the gap would perform similarly, but involve an unacceptable danger in the work place. Almost the same effect would be expected if steam, or better still, a water mist, were introduced into the air gap of a permanent mold. It would react with the metal to form hydrogen in situ, precisely where it is needed. Alternatively, many sand molds, such as greensand or sodium-silicate-bonded sands, have sufficient water that the hydrogen atmosphere is provided automatically, and free of charge.
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 analytical approach for an understanding of the heat transfer problem.
When the metal first enters the mold the macroscopic contact is good because of the conformance of the molten metal to the detailed shape of the mold. Gaps exist on a microscale between high spots as shown inFigure 5.6. At the high spots themselves, the high initial heat flux causes nucleation of the solid metal by local severe undercooling (Prates and Biloni 1972). The nucleated solid then spreads to cover most of the surface of the casting because the thin layer of liquid adjacent to the cool mold surface will be expected to be undercooled. Conformance and overall contact between the surfaces is expected to remain good during all of this early period, even though the mold will now be starting to move rapidly because of distortion.
Solid
(a)
Mold Liquid Heat flux
Solid
(b)
Mold Liquid
FIGURE 5.6
Metal–mold 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 (b) the casting gains strength, and casting and mold both deform, reducing contact to isolated points at greater separations on non-conforming rigid surfaces.
196 CHAPTER 5 Solidification structure
After the creation of a solidified layer with sufficient strength, further movements of both the casting and the mold 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 interfacehtmay now be written as the sum of three components:
ht ẳ hsỵhcỵhr
wherehsis the conduction through thesolidcontacts,hcis the conduction through thegasphase, andhr is that transferred by radiation. Ho and Pehlke produce analytical equations for each of these contributors to the total heat flux. We can summarize their findings as follows:
1. 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 ht up to 60 000 Wm–2K–1are found for aluminum in squeeze casting. Such high values are quickly lost as the solid thickens and conformance is reduced, the values falling to more normal levels of 100–1000 Wm–2K–1(Figure 5.7).
2. When the interface gap starts to open, the conduction via any remaining 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 behavior 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–2K–1(Figure 5.7c).
3. After solid conduction diminishes, the important mechanism for heat transfer becomes the conduction of heat through the gas phase. This is calculated from:
hc ẳ k=d
wherekis 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 gap is smaller than the mean free path of the gas molecules, which effectively reduces the conductivity. Thus heat transfer now becomes
Transducers
Water cooling coils TC 1
TC 2
mm 051
TC 3 TC 4
Al casting 127 mm ứ Copper chill
(a)
TC 4 3
TC 1 2 0 5 10 15 20 25
Time (min) 700
600 500 400 300 200 100 0
( erutarepmeT°)C
(b)
Calculated from inverse solution
Conduction and radiation calculation from experimentally measured gap
0 5 10 15 20 25 Time (min) 2500
2000 1500 1000 500 0 feoc refsnart taeHtneicif m/W(2)K/
(c) FIGURE 5.7
Results from Ho and Pehlke (1984) illustrating the temperature history across a casting–chill interface, and the inferred heat transfer coefficient.
5.1 Heat transfer 197
a strong function of gap thickness. As we have noted above, it will also be a strong function of the composition of the gas. Even a small component of hydrogen will greatly increase the conductivity.
Note also that it is assumed, almost certainly accurately, that the gas is stationary, providing heat flow by conduction only, not by convection.
For the case of light alloys, Ho and Pehlke find that the contribution to heat transfer from radiation is of the order of 1% of that due to conduction by gas. Thus radiation can be safely neglected at these temperatures.
For higher-temperature metals, results by Jacobi (1976) from experiments on the casting of steels in different gases and in vacuum indicate that radiation becomes important to heat transfer at these higher temperatures.
Turning now to experimental work on the effect of die coatings on permanent mold on heat transfer coefficients, a comprehensive review has been made by Nyamekye et al. (1994). Chiesa (1990) found that the conductance of a black coat was roughly twice that of a white coat of moderate thickness in the region of 120mm. Also, the insulating effect of a white coat increased only marginally with thickness.
Their findings that coats with high surface roughness were more effective insulators have been confirmed by the calculations of Hallam and Griffiths (2000) for the case of Al alloy castings. They demonstrate excellent predictions based on the assumption that the resistance of the die coating is mainly due to the gas voids between the casting and the coating surface. Thus the character of the coating surface was a highly influential factor in determining the heat transfer across the casting–mold interface.
The effect of gravity on the contact between the casting and mold has already been discussed above. Woodbury (1998) finds that for a lightweight horizontal plate casting of only 6 mm thickness the heat transfer settles to a constant level of 70 Wm–2K–1. However, for a 25-mm-thick plate the heat transfer from the top is approximately unchanged at 70, but the value from the underside of this heavier plate is now approximately doubled at 140 Wm–2K–1.
For sand molds the use of pressure to enhance metal–mold contact is of course limited by pene- tration of the metal into the sand. The use of pressure has no such limitation in the case of metal molds.
Tadayon (1992) reports the freezing time of a squeeze casting to be 84 s at zero applied pressure, but reducing to 56 s at 5 MPa. A further increase of pressure to 10 MPa reduced the time only minimally further to 54 s.
Finally, it seems necessary to draw attention to the comprehensive review by Woolley and Woodbury (2007). These authors critically assess the vast literature on the determination of heat transfer coefficients, concluding that they are skeptical of the accuracy of all of the published data, and cite a number of key reasons for unreliability. In a later paper (2009), for instance, these authors find that the use of thermocouples to measure heat flow in these experiments alone introduces an error of 65%. Clearly, we still have a long way to go to achieve reliable heat transfer coefficients.