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Modelling of rapid solidification by melt spinning: effect of heat transfer in the cooling substrate
Materials Science and Engineering, A136 (1991 ) 85-97
85
Modelling of rapid solidification by melt spinning: effect of heat transfer in the cooling substrate G.-X. Wang and E. F. Matthys Department of Mechanical Engineering, University of California, Santa Barbara, CA 93106 (U.S.A.)
(ReceivedApril 9, 1990; in revisedform September 11, 1990)
Abstract An improved control volume integral method has been developed for the numerical modelling of heat transfer during the melt-spinning process. The heat transfers both inside the melt and in the substrate are incorporated directly in the numerical models. Several parametric studies have been conducted to investigate the effect of the heat transfer in the wheel, of the wheel material, of the melt material and of the superheat level on the solidification characteristics, and in particular on the interface velocity and on the cooling rate at the interface. The calculations show also that the local substrate surface temperature may increase under the solidification puddle by several hundred kelvins, even for a copper wheel, and this surface temperature increase is found to have a significant impact on the solidification process. We believe that it is essential to include the surface heating factor in numerical models of melt spinning and other rapid solidification processes relying on direct contact with a solid substrate.
1. Introduction Melt spinning is one of the most commonly used processes for the production of rapidly solidified thin metal foils. Two variants of this process are free-jet melt spinning (FJMS) and planar flow casting (PFC). The main difference between these two techniques is that in the FJMS the crucible nozzle is located relatively far away from the wheel, whereas it is located very close for the PFC case. It is believed that the small gap between the crucible and the wheel in the PFC case results in the formation of an extensive melt puddle that damps perturbations in the melt flow and thereby leads to improved foil uniformity, The ribbon is dragged out of the melt puddle by the relative substrate motion and may emerge in solidified, semisolidified, or fully liquid form, depending on the heat transfer, nucleation and crystal growth characteristics of the given process. Two different mechanisms of ribbon formation are often mentioned for rapid solidification by melt spinning: thermal control and momentum transport control. These are defined according to the relative efficiency of the heat and momentum transport phenomena taking place in the puddle. 0921-5093/91/$3.50
If the ribbon is dragged out of the puddle already in solid form (i.e. if the solidification of the ribbon is completed in the puddle itself), the ribbon formation mechanism is said to be subject to thermal control. On the contrary, if solidification in the puddle is delayed for some reason (e.g. because of large superheat, poor thermal contact at the puddle-wheel interface or large nucleation barrier for glass-forming materials), a layer of liquid melt will be dragged out of the puddle, and solidification will take place further downstream. In this situation, the ribbon formation process is far more dependent on the fluid flow in the puddle, and the ribbon formation mechanism is said to be subject to momentum control [1]. In the FJMS case, Kavesh [2] postulated a thermal control mechanism for amorphous alloys. Subsequent fluid flow and heat treatment analyses [3, 4] suggested, however, that the melt temperature at the exit of the puddle is still greater than the glass transition temperature of the materials spun. It may be then concluded that the ribbon formation mechanism is dominated by momentum transport in the puddle and that the solidification is completed out of the puddle [1]. For crystalline materials, on the contrary, Katger© Elsevier Sequoia/Printed in The Netherlands
86
man [5] suggested that heat transfer may make a substantial contribution to the ribbon formation mechanism as well. For PFC of crystalline materials, Vogt [6] measured the temperature of the ribbon top surface and--using this information in a heat transfer analysis of the process--concluded that the ribbon formation mechanism is thermally controlled in the case of good thermal contact at the puddle-wheel interface. A similar conclusion was reached by Muhlbach et al. [7]. Several mathematical models have been developed in order to study and predict the ribbon formation process during melt spinning. Most of the earlier modelling work focused on the FJMS process and addressed the formation of metallic glasses for which the release of latent heat of solidification .need not be taken into consideration (see for example refs. 2, 3, 8 and 9). More recently, however, several studies have been conducted on the rapid solidification of crystalline materials by PFC. Yu [10], for example, developed a fluid flow mathematical model for PFC at low Reynolds numbers. In this model, lubrication theory is used in the puddle under the nozzle, and film theory is applied to the liquid portion of the puddle downstream of the nozzle. This model uses an average solidification front velocity for the calculations. Another excellent model based on lubrication theory that also couples phase change and heat transfer in the puddle has been developed for PFC by Gutierrez and Szekely [ 11 ]. In order to avoid some of the low Reynolds numbet constraints that are inherent in this lubrication theory, however, we have also recently developed a two-dimensional boundary layer theory model for PFC which includes coupled fluid flow, heat transfer and solidification phenomena[12, 13]. In most of the previous models, the focus has been on the fluid flow and heat transfer in the puddle and ribbon, but little work appears to have been conducted on the effect of the heat transfer in the wheel on the ribbon formation and solidification. Indeed, a constant and uniform wheel temperature during casting has been assumed by most researchers. Given the unsteady nature of the conductive heat transfer in the substrate and the very short durations involved, it is clear that this assumption may be of limited generality, however. In addition, recent measurements of the top surface temperature of the ribbon downstream of the crucible by Vogt [6] and Muhlbach et al. [7] have suggested that a
significant increase in the surface temperature of steel wheels may exist. It appears therefore that a detailed reexamination of the validity of the constant-wheel-temperature assumption in models is necessary, and in particular in the region directly underneath the solidification puddle. Accordingly, this article addresses the issue of the heat transfer in the wheel and of its impact on the solidification characteristics. A one-dimensional heat transfer and phase change numerical model including both the puddle and the wheel has been developed for this purpose. This model is based on a control volume integral (CVI) method rather than on the usual enthalpy method and provides a precise tracking of the solid-liquid interface position. The effect of different wheel and melt materials and other parameters on the heat transfer and solidification in the puddle has also been investigated. In the present article we are addressing more specifically the PFC process, but the main results and conclusions are applicable to the FJMS process as well. 2. Problem statement A detailed description of our model of the PFC process and of the numerical scheme used can be found elsewhere [14], but a brief summary is given here for convenience. A schematic diagram of the idealized PFC geometry used in our model is also shown in Fig. 1 for reference. Using high speed movies of the solidification puddle, we have shown [13] that the downstream meniscus usually detaches from the crucible bottom well downstream of the nozzle. Given the large aspect ratio of the typical PFC melt puddle and the high axial velocities, it is then reasonable to approximate the puddle by a thin rectangular strip of
uclble
y=H
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y=0 Surfoco) x=0 (Wheel
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used inournumericalmodel.
87 liquid. The main assumption used in the present model is to consider that there is no velocity gradient in the puddle. Clearly, this is not true in the actual case, but our primary objective here is to study the effect of the conduction in the wheel on the solidification, and we believe that this effect can be adequately evaluated without introducing at this point the complications of a twodimensional model of the flow field, A number of assumptions have also been made to simplify the model. We assume that the wheel is initially at room temperature ( To = 300 K). We limit ourselves here also to the study of the melt puddle under the crucible and the wheel below the puddle. We neglect convective heat transfer normal to the wheel surface. Heat conduction parallel to the substrate surface is also neglected, We consider solidification of a pure metal. We assume that local thermodynamic equilibrium exists at the solid-liquid interface (i.e. no undercooling). The temperature at the top surface of the puddle is assumed to be maintained at the pouring temperature Tp. The heat loss through the menisci of the puddle is neglected. The wheel surface is assumed to be horizontal and to exhibit a no-slip condition. Distinct values of material properties are used in the liquid and solid regions respectively but are otherwise assumed to be independent of temperature. Shrinkage during solidification is not taken into account, Because of the assumption that there is no relative motion within the melt puddle, the twodimensional steady state boundary layer problem can then be reduced to a one-dimensional unsteady state heat conduction problem. At the solid-liquid interface within the puddle, energy conservation can be described by the usual interface condition involving the interface velocity and the temperature gradients on each side of the interface. Because of our assumption of local thermodynamic equilibrium at the solid-liquid interface, the interface temperature is considered
to be fixed at the melting temperature TM. The non-perfect thermal contact at the interface between the melt puddle and the wheel surface is quantified by the usual heat transfer coefficient h, based on the temperature difference between the puddle bottom and the wheel surface. A symmetric temperature condition is used at the center of the wheel. We also assume for simplicity that the heat transfer coefficient is constant throughout the puddle. In the present work, an improved CVI method is used to generate finite difference equations for this one-dimensional phase change with moving solid-liquid interface. In our approach, the resolution of the interface tracking is greatly increased by the implementation of a special treatment of the phase change volume element, which is particularly useful for the modelling of solidification with large levels of undercooling and recalescence [14]. 3. Numerical results and discussion
The computations described in this article are based on experimental data that were obtained recently in our laboratory during an investigation of the PFC solidification puddle dynamics. The experimental apparatus and procedures have been described in detail elsewhere [15]. The parameters used in these calculations are as follows: wheel speed, 23 m s-1; puddle height and crucible-wheel gap, 350 pro; superheat, 50 K; puddle length (approximated by the distance between the nozzle and the meniscus detachment point), 5.45 ram. The corresponding ribbon thickness was 68 pm. In our computations we have used these geometrical and process parameters to model the spinning of various melts (aluminum, nickel and titanium) on different wheel materials (copper, nickel and stainless steel). The physical properties used in the calculations are shown in Table 1. The number of nodes
TABLE 1 Material propertyvaluesused in the computations Material
TM (K)
L (kJkg J)
CpL (Jkg JK -I)
Cps (Jkg-~K-j)
AI Ni Ti Cu Steel
933 1725 1940
395 300 365
1200 620 700
1060 600 528 503 550
kL
(Wm-lK -I) 100 30 22
ks (Wm-lK-l)
PL (kgm-3)
Ps (kgm-3)
200 60 22 393 20
2340 8500 4510
2700 8900 4510 9000 7650
88
used is 200 both in the puddle and in the wheel, The calculation domain in the wheel was 1000 # m for the copper wheel and 500 # m for the stainless steel wheel and the nickel wheels. These wheel domains were found to be large enough for accurate results, given the very small thermal penetration depth that is achieved in the short duration corresponding to the wheel passage under the puddle area (about 0.25 ms only). (Larger domains were used for the analysis of overall wheel heating, however; see Section 3.4.) The effect of the heat transfer coefficient on the solidified layer growth and on the interface velocity and interface cooling rate has been investigated previously, and the results have been presented elsewhere [14]. It was shown in particular that the solidified layer at the crucible detachment point was approximately equal to the ribbon thickness for h = 106 W m -2 K -1, i.e. the ribbon is fully solidified when it leaves the puddle. The solidification is then "thermally controlled". The increase in the cooling rate of the liquid melt at the interface and in the interface velocity with increasing heat transfer coefficients was also quantified. The cooling rate of the melt at the interface (or "interface cooling rate" in short) is defined here as OT[ Oyh,k where the temperature gradient is computed at the interface on the liquid side, and V~is the interface velocity. It was also seen that both the interface velocity and the interface cooling rate decrease significantly as the interface moves far away from the wheel, especially for high heat transfer coefficients. The interface velocity was found to exhibit a maximum very close to the wheel surface, however. This maximum results very probably from the superposition of the diminishing need for removal of liquid superheat and the increasing thermal resistance of the solidified layer as the interface penetrates further into the melt. The interface velocity and cooling rate were also shown to be much lower for a wheel of finite thermal diffusivity than for a hypothetical ideal (isothermal) wheel. The differences were shown to be particularly significant for large heat transfer coefficients. As mentioned above, it is commonly assumed in melt-spinning (and other)models that the substrate temperature remains constant. Our
preliminary results suggested, however, that numerical models for rapid solidification should indeed include wheel heating for more accurate calculations of cooling rates and interface velocities. Accordingly, a discussion of this effect and some results on the magnitude of the surface heating are given below.
3.1. Effect of the nature of the substrate on the solidification characteristics As mentioned in the previous section, the wheel material has a greater effect on the solidification characteristics when there is good thermal contact between the puddle and the wheel. We shall therefore limit this discussion to the case of large heat transfer coefficients. Figure 2 shows the solid-liquid interface position in the puddle as a function of distance from the upstream meniscus for various wheel materials, including a hypothetical ideal wheel which is assumed to have an infinite heat capacity and thermal conductivity (i.e. the wheel maintains a constant and uniform temperature during the process). For better comparison, we are using in these computations the same heat transfer coefficient (10 6 W m 2 K 1) for the various wheel materials, but in practice the thermal contact may be affected by the melt-wheel material combination [7]. (Recent experimental results suggest that a value of 106 W m -2 K -1 c a n indeed be achieved before the ribbon is dragged out of the puddle [6, 7, 16].) The numerical model corn-
~" ~o 60 -~ g ~ ~- 40
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-
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.~ 20 u~,, ~
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Fig. 2. Vertical location of the interface as a function of distance from the u p s t r e a m meniscus for a l u m i n u m s p u n on ideal, copper, nickel and stainless steel wheels ( h = 106 W m 2 K t; gap, 350 /~m; V w = 2 3 m s t ; T p - T m = 5 0 K ; T,,= 300 K; XE) = 5.45 ram; ribbon thickness, 68 #m).
89
putes the heat transfer and temperature in the wheel as well as the increase in temperature at the wheel surface, together with the temperature field in the puddle itself. The two are coupled, of course, and it is the heat transfer in the wheel that is the source of the variation in solidification rate shown here. It can be seen that the rate at which the liquid melt solidifies is significantly affected by the heat conduction in the wheel. As expected, a higher resistance to conduction in the wheel results in a Aluwlirlum
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Fig. 3. Interface velocity as a function of distance from the wheel for a l u m i n u m spun on ideal, copper, nickel and stainless steel wheels ( h = 1 0 6 W m - - ' K i; gap, 350 /am; Vw = 23 m s-% T p - T m = 50 K; T. = 300 K; X D = 5.45 m m ;
ribbon thickness, 68 pro).
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~,, =o ~ ~-
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6 Interface (10 5
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Fig. 4. Local melt cooling rate at the solidification interface as a function of distance from the wheel for a l u m i n u m spun on ideal, copper, nickel and stainless steel wheels ( h = 10 ~ W m z K ~; gap, 3 5 0 / a m ; V w = 2 3 m s - l ; T p - Tm = 5 0 K; T. = 300 K; X D = 5.45 ram; ribbon thickness, 68/am).
decrease in heat flux from the puddle and therefore in a decrease in solidification rate as well (see above). For materials with a high thermal conductivity such as copper, the reduction with respect to an ideal wheel is moderate, but for materials with poorer thermal conductivity such as stainless steel, for example, a large reduction in solidification rate with respect to an ideal wheel takes place. It can be seen that the solidified layer is about twice as thick for the copper wheel than it is for the steel wheel at the same distance downstream. The interface velocity and the interface cooling rate are shown in Figs. 3 and 4 respectively. As mentioned, the interface velocity (Fig. 3) reaches a maximum close to the wheel before decreasing as the interface penetrates further in the puddle. This is true for all the materials discussed here, but the maximum is closer to the surface for a wheel of low thermal conductivity. The decrease in interface velocity is also more rapid for the wheel of lowest thermal conductivity. This is because in a wheel of low thermal diffusivity the surface temperature increases significantly with time, which in turn decreases greatly the heat flux and therefore the interface velocity. The interface cooling rate is also seen in Fig. 4 to be significantly affected by the wheel material. As discussed previously, the interface cooling rate decreases rapidly as the interface penetrates inside the puddle. These results confirm that the heat transfer in the wheel has a large effect on the solidification characteristics and thermal history of the ribbon. Clearly, the surface temperature of the wheel is likely to increase significantly for a wheel of low thermal diffusivity. To illustrate this effect, Fig. 5 shows the increase in surface temperature during the passage of the wheel under the solidification puddle. The calculations have been conducted for the same wheel materials as in the previous figures. Naturally, an ideal wheel would see its surface temperature remain constant. For the parameters considered here, the wheel will remain only approximately 0.2 ms under the puddle, during which a large amount of energy is nevertheless transferred to the wheel (because of the large temperature difference between the wheel surface and the puddle bottom, and also because the heat transfer coefficient is large as w e l l ) . Given this short duration and the limited thermal diffusivity of non-ideal materials, the energy can only penetrate a short distance in the
90
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Aluminum mell 2
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A l u m ' h u m m e l t on a Copper wheel
300
450
(ram)
Fig. 5. T e m p e r a t u r e of the wheel surface as a function of distance from u p s t r e a m meniscus for a l u m i n u m s p u n on copper, nickel and stainless steel wheels (h = 106 W m -2 K ~; gap, 350 ,um; V w = 2 3 m s 1; T p _ T m = 5 0 K ; T , = 3 0 0 K ; X D = 5.45 mm; ribbon thickness, 68 Hm).
wheel (the classic "penetration depth" concept in conductive heat transfer) which results in a higher temperature near and at the wheel surface. The magnitude of this temperature increase depends on the wheel material, of course. Even for a wheel made out of copper (which has a high heat conductivity), the temperature of the wheel surface is seen to increase by 170 K over the length of the puddle. (As discussed above, we are limiting our calculations to the puddle up to the detachment point, but in reality the ribbon is still cooled further downstream, which will increase the wheel temperature even more.) For stainless steel which has a lower thermal conductivity, on the contrary, an increase in surface temperature of over 400 K is seen to occur. Obviously, such large variations in the substrate surface temperature should not be neglected in the models used for this process. A similar increase in surface temperature will also probably take place in other rapid solidification processes using a solid substrate as heat sink (e.g. splat cooling, FJMS, hammer-and-anvil system, melt overflow and two-roll quencher), The temperature profiles in the solid and liquid regions of an aluminum melt puddle as well as in the copper wheel on which the aluminum is solidified are all shown for various downstream locations in Fig. 6. It is interesting to look at the change in temperature difference between the puddle bottom and the wheel surface over time.
_
L__ 600
750
900
Temperature (K)
Fig. 6. T e m p e r a t u r e profile in several locations for a l u m i n u m where the profiles are shown for u p s t r e a m meniscus ( h = 1 0 ~ W Vw = 23 m s ~; T p - Tm = 50 K; ribbon thickness, 68 Hm).
~~o ~ ~ ~
300
the puddle and wheel at spun on a copper wheel, 1, 3 and 5.45 m m from the m 2 K ~; gap, 350 Hm; T, = 300 K; X~ = 5.45 ram;
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150 Puddle
•
0
~ -150 Eo ~- -300 o ~:~
--450
Aluminum melt
300
450
600 750 Temperoture (K)
900
Fig. 7. T e m p e r a t u r e profile in the puddle and wheel at the puddle d e t a c h m e n t point (XD = 5.45 m m ) f o r a l u m i n u m spun on copper, nickel and stainless steel wheels ( h = 1()~ W m -2 K ~; gap, 350 Hm; V w = 2 3 m s '; T ~ - T m = 5 0 K ; T,,= 300 K; X a = 5.45 ram; ribbon thickness, 68 Hm).
Clearly, a larger temperature difference exists at X = 1.0 mm than at X = 5.45 m m = X b (the detachment point): 5 1 0 K vs. 355 K. This decrease results from the combination of a decreasing puddle bottom temperature and an increasing wheel surface temperature. The decrease in temperature jump at the wheel surface results in turn in a decrease in heat transfer into the wheel. (It should be noted that we have
91
assumed in these calculations that the heat transfer coefficient remains constant over the whole length of the puddle, but in reality it is likely that this coefficient decreases with distance as the melt solidifies and shrinks, which would then lead to lower heat fluxes than those computed here.) A better understanding of the spatial distribution in resistance to heat transfer can perhaps be gained from Fig. 7 which shows the temperature profiles in the puddle and wheel for three different wheel materials. All are computed at the detachment point measured experimentally for aluminum spun on a copper wheel (XD = 5.45 mm) and for a heat transfer coefficient of 106 W m -2 K 1. The relative temperature drops in the puddle, in the wheel and at the puddle-wheel interface are indicative of the distribution of heat resistances across the field. For a wheel of high thermal conductivity, such as copper, the largest temperature drop takes place at the puddlewheel interface, with smaller temperature drops in the wheel and in the puddle (which is aluminum here, also a high conductivity metal). The main thermal resistance is therefore "located" at this interface, even though the heat transfer coefficient used is relatively high in this case (h = 10c'W m - 2 K-~). On the contrary, and as expected, for a wheel of low thermal conductivity metal, such as stainless steel, the largest temperature drop exists in the wheel, with smaller temperature drops in the puddle and at the puddle-wheel interface. The main thermal resistance is therefore the wheel itself inthiscase. If we compare the temperature drops across the puddle-wheel interface at X = 5.45 mm, we find values of 355 K for a copper wheel, 280 K for nickel and 180 K for stainless steel. Given the constant-h assumption used here, the heat fluxes would be directly proportional to these temperature differences. In other words, the heat flux would be twice as large for a copper wheel as it is for a steel wheel. This difference will, of course, also result in larger interface cooling rates for the copper wheel, The increase in wheel surface temperature results from the fact that a high heat transfer takes place over a short time. If the wheel has a finite thermal diffusivity, the energy will not have the time to diffuse much into the wheel, and the surface will heat up significantly. This concept of limited thermal penetration depth is illustrated clearly in Fig. 7 as well. It can be seen that, after
the same time, the energy has visibly "penetrated" a much greater vertical distance in the copper wheel (about 500 /~m) than in the steel wheel (150 /~m). These values are consistent with the relative magnitudes of the characteristic lengths computed as the square root of the product of the thermal diffusivity by the time (i.e. the classic diffusion length). This diffusion length is indeed approximately four times greater for copper than for steel.
3.2. Effect of the nature of the melt on solidification characteristics and substrate temperature The discussion above was concerned with an aluminum melt of high thermal conductivity for which the relative resistances to heat transfer in the melt and ribbon are small and would have comparatively little effect on the solidification characteristics. In this case, either the thermal contact resistance (for wheels of high conductivity) or the heat conduction in the wheel (for wheels of low conductivity) may limit the overall heat transfer process. On the contrary, if a low conductivity material is cast, such as titanium for example, the solidification characteristics may be somewhat different. For illustration, some calculated results are shown hereafter for three different melts: titanium, nickel and aluminum. The process parameters are assumed to be the same in all cases, including the level of superheat, but all the material properties used are those corresponding to each specific melt. Figure 8 shows the solid-liquid interface velocity for the three different melts spun on a copper wheel. The same thermal contact condition at the puddle-wheel interface (h-- 106 W m- 2 K- 1) and the same detachment distance (5.45 mm) are assumed in all cases. The variation in interface velocity across the ribbon reflects the effect of the thermal conductivity of the melt on the solidification rate. For aluminum which has a high thermal conductivity, the latent heat and superheat can be easily transferred from the solidification interface to the wheel, and the interface velocity changes relatively little as the interface penetrates further in the puddle. In the titanium case, on the contrary, the latent heat and superheat will be less readily conducted to the wheel when the solid layer thickens because of titanium's smaller conductivity. A larger variation in interface velocity across the ribbon thickness is therefore seen as the interface grows inside the
92 i
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I
Copper wheel 60
~
0
60
Copper wheel
0 v
S
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o_
Ap
40
~
Q
40
o
20
Ni
~
20
u~
0 0.0
0 0.2
0.4
0.6
i
0.8
50
Interface Velocity ( m / s e c )
Fig. 8. Interface velocity as a function of distance from the wheel for three different melts (aluminum, nickel and titanium) spun on a copper wheel (h = 106 W m- "- K ~; gap, 350 /~m; V w = 2 3 m s ~; T p - T m = 5 0 K ; T0= 300 K; XD= 5.45 ram; ribbon thickness, 68 ~m).
Overall, the interface velocity is calculated to be about twice as large for titanium than it is for aluminum close to the wheel. It decreases then significantly away from the wheel and is seen to become in fact smaller than that for aluminum (which remains nearly constant across the ribbon). The interface velocities are in all cases of the order of 0.1-1 m s- 1 throughout the ribbon, Figure 9 shows the cooling rate upon solidification for the three melts. As expected, the cooling rate is much higher at the bottom of the ribbon for titanium ( 10 7 K s- ~) and nickel (5 x 106 K s - ~) than for aluminum (8 x 105 K s l ) which has a much lower melting temperature. Towards the top of the ribbon, however, the cool-
50
Melt Cooling Rote at Interface (I0 s K / s e c )
Fig. 9. Interface cooling rate as a function of distance from the wheel for three different melts (aluminum, nickel and titanium) spun on a copper wheel (h = 106 W m ' K '; gap, 350 ¢tm; V w = 2 3 m s ~; T p - T,,= 50 K; T , = 3 0 0 K ; Xt~ = 5.45 mm: ribbon thickness, 68/~m). ,
700
puddle. For nickel, which has a thermal conductivity somewhat larger than titanium, an intermediate level of variation across the ribbon is seen. It is interesting to note that the numerical results suggest that the interface velocity for titanium is much greater than for aluminum close to the wheel, but that it becomes smaller further away. Nickel, on the contrary, which has a melting temperature close to that of titanium is seen to exhibit an interface velocity similar to that of aluminum close to the wheel. These relative magnitudes are not easy to predict with back-of-theenvelope calculations because of the combined effects of the varying temperature gradients and different material properties for the three metals.
_
100
r
Copper wheel
~. o
?~
600
~
500
~_ 400 ~300
, 2 Dow~r~o~
Di~ton~
4 (~0-~ ~)
6
Fig. 10. Wheel surface temperature increase as a function of distance from the u p s t r e a m meniscus for three different melts (aluminum, nickel and titanium) spun on a copper wheel ( h = 1 0 ~ W m 2 K - l ; gap, 350 # m ; V w = 2 3 m s-~; r~- Tm = 50 K; T0 = 300 K; X~ = 5.45 ram; ribbon thickness, 68/~m).
ing rates are about the same for all three melts (approximately 2 × 105 K s-l), the result of a combination of relative influences of interface velocities and temperature gradients in the liquid. It should be noted that for titanium, for example, the variation in cooling rate across the ribbon is approximately two orders of magnitude, from 10 7 to about 105 K s- l, a rather large variation. Figures 10 and 11 show the temperature increase of the wheel surface for the three melts
93
spun on copper and stainless steel wheels respectively. It is perhaps surprising to see that the wheel surface temperature has increased less for titanium than for nickel toward the end of the puddle in the case of the copper wheel (despite a lower melting temperature for nickel), whereas the increase is larger for titanium i~a the case of the steel wheel. A possible explanation might be as follows. In the case of the copper wheel, the resistance to heat transfer in the wheel is rather low because of the high thermal conductivity of copper. In these computations the thermal resistance at the puddle-wheel interface is also small (h=106 W m -2 K-t). The main resistance to heat transfer is therefore likely to be in the solidified melt layer for the case of the copper wheel, Since nickel has a conductivity that is three times that of titanium and a similar melting temperature, it exhibits a smaller resistance to conduction in the solidified layer and therefore a proportionally higher heat flux than titanium, which in turn results in a higher wheel temperature increase. For the steel wheel, however, the main resistance to heat transfer is probably in the wheel itself and no longer in the solidified layer, and the relative magnitude of the conductivity of this layer plays a smaller role. Titanium having a somewhat larger melting temperature than nickel may then exhibit a higher heat flux which in turn would lead to higher wheel temperatures. Regarding the necessity of the inclusion of the 1500
--
~
o
F Sfainless Steel Wheel
1200
wheel heat transfer in a rapid solidification model, one should consider the magnitude of the wheel surface temperature increase calculated here. In the case of melt spinning of a high melting temperature metal such as titanium or nickel, we see that the wheel surface temperature may increase by as much as 200 or 300 K before puddle detachment, even for a copper wheel. If one assumes an isothermal wheel in calculations, the predicted heat flux could be easily overestimated by 100%. Naturally, the situation would be even worse for a wheel of lower conductivity. Assuming that one would be interested in spinningtitanium, say, onasteelwheel, the wheel surface would easily increase to 1200 K according to these calculations, a situation likely to result in some experimental difficulties!
3.3. The effect of the melt superheat on the solidification mechanism Figures 12 and 13 show the effect of the degree of superheat on the solid-liquid interface velocity for aluminum and titanium spun on a copper wheel. In both cases, the interface velocity decreases as the level of superheat increases. This is to be expected because more sensible heat in the liquid must then be removed by conduction to enable the interface to move further into the liquid. Comparing the results for the two metals, it can be seen that the effect of the superheat on the
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=
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4
"120 o~
Tp-T : 1 0 0 m
K
50
10 I
0 0.0
0.2
0.4
0.6
0.8
Downstream Distance (10 -3 m)
Fig. 11. Wheel surface temperature increase as a function of distance from the upstream meniscus for three different melts (aluminum, nickel and titanium) spun on a stainless steel wheel (h = 1 0 6 W m - 2 K-[; gap, 350 /am; Vw=23 m s-I; T p - T m = 5 0 K ; T 0 = 3 0 0 K ; XD=5.45 mm; ribbon thickness, 68/am).
Interface Velocity (m/sec)
Fig. 12. Interface velocity as a function of distance from the wheel for three different levels of superheat (10, 50 and 100 K) for aluminum spun on a copper wheel (h = 106 W m -2 K-J; gap, 350 /am; Vw=23 m s-J; T0=300 K; Xo = 5.45 mm; ribbon thickness, 68/am).
94 60
- - ~
E
-
~
i
--
Titanium rnell on copper wheel
~
i
~
750
-
-
r
746 K (SS)
~- -
Aluminum nlelt
v
o
F O 40
o o_
~ J:: \\\
Tp
Tin=
K
I00
50 K
_c •--
20
\\~,k
600
"o ~
10K
~ L
J
E
i l I ~ I
450
~.BS K
L ~
s
s
(Cu)
steel wheel
2
o G)
0 0.0
: 0.2
0.4
0.6
I n t e r f a c e Velocity
0.8
1.0
(m/sec)
Fig. 13. Interface velocity as a function of distance from the wheel for three different levels of superheat (10, 50 and 100 K) for titanium spun on a copper wheel (h = 106 W m -~ K - I ; gap, 350 /~m; V w = 2 3 m s ~; T . = 3 0 0 K; XD=5.45 mm; ribbon thickness, 68/~m).
interface velocity appears to be larger in the case of aluminum. This difference may be related to the fact that aluminum has 10 times the thermal conductivity of titanium. Indeed, it should be noted that the upper boundary condition for the puddle is that the temperature is constant and equal to the pouring temperature. It is therefore possible for energy to be transferred from the bottom of the crucible to the melt. For aluminum, which has a high thermal conductivity, a large heat flux can be generated in this manner if the superheat is large. Naturally, this energy has to be removed by the wheel in addition to the original superheat in the liquid and to the latent heat released at the interface. A significant effect of the pouring temperature can then be expected, as seen in Fig. 12. For titanium, which has a low thermal conductivity, the temperature gradient in the liquid will be much more localized near the interface, with only a small gradient near the top of the puddle. The heat flux from the crucible will therefore be smaller and its effect on the interface velocity much more limited than for aluminum, The effect of the level of superheat on the increase of wheel temperature was also investigated, but the calculations showed this to be a relatively small effect,
3.4. Changes in wheel surface temperature during continuous me# spinning In all the calculations discussed above, we have assumed that the wheel surface is at room tern-
300 ~ 0
250
~ 500
75(
Distance f r o m meit c o n t a c t (10 -s m)
Fig. 14. Temperature of the wheel surface as a function of distance from melt contact location during the first revolulion after contact, for aluminum spun on copper and stainless steel wheels (heat transfer coefficients, h = I(P W m 2 K before separation and h = 0 after separation of ribbon from wheel; gap, 350 /~m; Vw=23 m s ~; 7 ] , - T ~ = 5 ( ) K; T,= 300 K: A's = 8 mm; ribbon thickness, 68 #m; wheel
diameter,().28 m).
perature T, before the first melt contact. Naturally, the initial room temperature assumption is no longer valid for the following wheel revohitions. Tens to hundreds of revolutions may indeed be common even during relatively short laboratory PFC experiments. During that time, the wheel will be subjected not only to conductive heat transfer from the solidifying metal but also to radiative and convective heat transfer from the crucible. In that case, significant "long-term" increases in surface temperature may take place if the wheel is not cooled externally. We have conducted some calculations to investigate the degree to which this temperature increase may become a significnt factor. Figure 14 shows the wheel surface temperature calculated over one full wheel revolution (the wheel diameter is 0.28 m), and assuming an equivalent ribbon separation distance X, (i.e. puddle length here) of 8 mm. (The actual separation distance is likely to be much greater but would probably involve a sharply decreasing heat transfer coefficient towards the end of the puddle and beyond, whereas we assume here a constant (high) value of this coefficient. Also, the upper surface of the ribbon would not be heated in reality (and could even be significantly cooled by radiation and convection) whereas we assume contact with the hot crucible. It is therefore hoped that a reduced
95 equivalent separation distance is a reasonable substitute for the actual case.) After separation, we assume that the heat transfer coefficient becomes zero. We also neglect both convective cooling of the wheel by the ambient gas and heating by the crucible. The calculated surface temperatures are shown in this figure for both a copper and a steel wheel. As expected, it can be seen that the surface temperature reaches rapidly high levels (746 K for the steel wheel and 485 K for copper in this case) and then decreases after ribbon separation, the energy being conducted away toward the inside of the wheel. The surface temperature appears then to have decreased significantly by the end of the first revolution after melt contact. This is the new surface temperature
~ o E ~ ~ -looo
that the melt will effectively "see" at the beginning of the next pass of the wheel under the puddle, and it is also the temperature that should be used as the initial condition for the computations
Fig. 15. Temperature profiles in the wheel at the ribbon separation point (As),at 35 mm downstreamof the meniscus, and after one fullrevolutionfor aluminumspun on a copper wheel (gap, 350 ~m; Vw=23 m s-t; Tv-Tm=5OK; T0=300 K; Xs=8 mm; h=106 W m 2 K-1 before separation and h=0 after separation; ribbon thickness, 68 /xm; wheel diameter,0.28 m).
thereafter, For copper, we see an estimated 10 K increase in surface temperature at the end of the first revolution after melt contact, whereas the increase is about 25 K for a steel wheel. This discrepancy is not that large despite the significant difference in wheel conductivity because there is enough time for most of the energy transferred into the wheel to diffuse inward. Also, the heat capacity per unit volume of copper is somewhat greater than that of stainless steel, so that a copper wheel can store more energy for a given ternperature increase than a stainless steel of the same dimension, which alleviates the fact that more energy is transferred into the cooler copper wheel. The use of a solid wheel assumption in these computations has a significant impact on the results, and the end-of-revolution temperature increase would be smaller if we were to model a hollow wheel cooled internally by water, Temperature profiles inside a copper wheel subject to aluminum spinning are shown in Fig. 15, both at the ribbon separation distance and at the end of the first revolution after melt contact, An intermediate location (X= 35 mm) is also shown. As expected, a high temperature at the surface (500 K) and a short penetration depth (0.8 mm) are seen to exist at separation. A much greater penetration depth and a correspondingly smaller surface temperature peak is seen for larger times because the same amount of energy (we assume no additional heat transfer after separation) has spread by diffusion over a much
~
(/3
/
'
x~- 35 m m
End of r e v o l u t i o n -2000
E o 8 -3000
L
~
A~om~. . . . . It p p o r wheel , , 35o 400 4so soa Temperofure in the Wheel (K) . . . .
-4000 300
larger domain. Nevertheless, if the wheel is not cooled externally, the surface temperature is seen to be about 10 K above room temperature at the second contact with the puddle. Naturally, the temperature of the surface will then keep increasing with the subsequent revolutions as the wheel slowl~¢ heats up. If we assume as a first approximation a 10 K increase per revolution, it can be seen that the surface temperature would soon become rather high. Of course, in reality, other heat losses may reduce this effect, but the advanrage of a water-cooled wheel over a solid one in this regard is obvious. The inclusion of long-term surface heating in numerical models for this type of process appears therefore to be desirable for uncooled wheels but is probably not necessary for water-cooled wheels. The main point of this article was to evaluate the larger localized increases in temperature directly under the puddle, however. It should be emphasized then that--because of the small penetration depth typical of the short wheel residence time under the puddle--such a localized increase would very likely take place even for a water-cooled wheel, unless the thickness of the rim could be made much smaller than a millimeter. (The thermal penetration depth at ribbon separation is about 0.8 mm as seen in Fig. 15.)
96
4. Conclusions
Acknowledgments
A n i m p r o v e d C V I a p p r o a c h was used to develop a numerical heat transfer model of rapid solidification by the PFC process. T h e models developed include in particular the effect of the heat transfer within the substrate and allowed us to investigate the influence of the thermal contact between the wheel and the melt, of the wheel material, of the melt material and of the superheat level on the solidification history, on surface ternperature increases, on interface velocity and on the interface cooling rate. T h e s e parameters have been discussed and analyzed in terms of the various thermal mechanisms such as local thermal resistances due to thermal conductivity,
We would like to acknowledge gratefully the support of the National Science Foundation (Grant M S S - 8 9 5 7 7 3 3 ) and of the Defense A d v a n c e d Research Projects Agency (Contract N 0 0 0 1 4 - 8 6 - K - 0 7 5 3 ) . We would also like to thank Professor C. Levi (University of California, Santa Barbara, C A ) for m a n y valuable discussions. Mr. Wang is currently a graduate student at T h e University of California, Santa Barbara, CA.
energy releases and transfers, and relative magnitudes of the individual material properties. O f particular significance is the fact that the wheel surface t e m p e r a t u r e is seen to increase greatly under the solidification puddle, even for a c o p p e r wheel. In this case, typical surface temperature increases would be of the order of 250 K at the end of a puddle 5 m m long for spun titanium, and about 175 K for spun aluminum. If the wheel is m a d e out of a low conductivity material, e.g. steel, the local increase could be as large as 1000 K. Naturally, the impact of these increases on the solidification characteristics was found to be very significant. For example, the heat flUX into the wheel could be easily overestimated by 100% if an isothermal c o p p e r wheel is assumed, It should also be noted that the wheel surface heating is linked to the fact that rapid solidification processes involve large amounts of energy transferred over small periods of time. Interestingly, using a water-cooled wheel would therefore generally not prevent this large surface heating. This is because the wheel heating occurs only over a very thin layer near the surface in a wheel of finite thermal diffusivity, and this thin layer may not be able to "see" m u c h of the effect of t h e w a t e r c o o l i n g . In conclusion, it is essential in most cases to include substrate heat transfer and local surface tbmperature increases in numerical models of PFC. Furthermore, the results and conclusions presented here should be directly applicable to the FJMS process as well. It is also likely that the modelling of m a n y other rapid solidification processes based on direct contact with a substrate would benefit f r o m a similar approach.
References 1 H. A. Davies, Solidification mechanisms in amorphous and crystalline ribbon casting, in S. Steeb and H. Warlimont (eds.), Rapidly Quenched Metals, Vol. 1, NorthHolland, Amsterdam, 1985, pp. 101 - 106. 2 s. Kavesh, Principles of fabrication, in J. J. Gilman and H.J. Leamy (eds.), Metallic Glasses, American Society for Metals, Metals Park, OH, 1978, pp. 36-73. 3 K. Takeshita and P. H. Shingu, An analysis of the ribbon formation process by the single roller rapid solidification technique, Trans. Jpn. lnst. Met., 24(1983) 529-536. 4 z. Sun and H. A. Davies, Computer modelling of combined heat and momentum transfer in the melt spinning of amorphous and crystalline metals, in S. Kou and R. Mehrabian (eds.), Modeling and Control of Casting and Welding Processes, The Metallurgical Society, Warrendale, PA, 1986, pp. 179-194. 5 L. Katgerman, Continuous products in rapid solidification, in P. R. Sahm, H. Jones and C. M. Adam (eds.), Science and Technology of the Undercooled Melt,
Martinus Nijhoff, Dordrecht, 1986, pp. 121-135. 6 E. Vogt, On the heat transfer mechanism in the melt spinning process, Int. J. Rapid Solidif., 3 (1987) 131-146. 7 H. Muhlbach, G. Stephani, R. Sellger and H. Fiedler, Cooling rate and heat transfer coefficient during planar flow casting of microcrystalline steel ribbons, Int. J. RapidSolidif, 3(1987)83-94.
8 T. R. Anthony and H. E. Clyne, On the uniformity of amorphous metal ribbon formed by a cylindrical jet impinging on a flat moving substrate, J. Appl. Phys., 49 (1978)829-837. 9 P. den Decker and A. Drevers, Model calculations on the solidification and crystallization processes during melt spinning, in C. Hargital, I. Bakonyi and T. Kemeny (eds.), Metallic Glasses: Science and Technology, Kultura, Budapest, 1981,pp. 181-188. 10 H. Yu, A fluid mechanics model of planar flow melt spinning process under low Reynolds number conditions, Metall. Trans. B, 18(1987) 557-563. 11 E. M. Gutierrez and J. Szekely, A mathematical model of the planar flow melt spinning process, Metall. Trans. B, 17(1986) 695-703. 12 M. I. Eskenazi and E. E Matthys, Modelling of planar flow melt-spinning using the boundary layer equations, Rep. UCSB-ME-88-4, 1988 (University of California). 13 z. Gong, P. Wilde and E. F. Matthys, Numerical model-
97 ling of the planar flow melt-spinning process, and experimental investigation of its solidification puddle dynamics, Int. J. Rapid Solidf., (1990), in the press, 14 G.-X. Wang and E. F. Matthys, Numerical modelling of phase change and heat transfer during rapid solidification processes: use of control volume integrals with element subdivision, to be published.
15 P. Wilde and E. E Matthys, Experimental investigation of the planar flow melt-spinning process: development and free surface characteristics of the solidification puddle dynamics, to be published. 16 K. Takeshita and P. H. Shingu, Thermal contact during the cooling by the single roller chill block casting, Trans. Jpn. Inst. Met., 27 (1986) 454-462.
85
Modelling of rapid solidification by melt spinning: effect of heat transfer in the cooling substrate G.-X. Wang and E. F. Matthys Department of Mechanical Engineering, University of California, Santa Barbara, CA 93106 (U.S.A.)
(ReceivedApril 9, 1990; in revisedform September 11, 1990)
Abstract An improved control volume integral method has been developed for the numerical modelling of heat transfer during the melt-spinning process. The heat transfers both inside the melt and in the substrate are incorporated directly in the numerical models. Several parametric studies have been conducted to investigate the effect of the heat transfer in the wheel, of the wheel material, of the melt material and of the superheat level on the solidification characteristics, and in particular on the interface velocity and on the cooling rate at the interface. The calculations show also that the local substrate surface temperature may increase under the solidification puddle by several hundred kelvins, even for a copper wheel, and this surface temperature increase is found to have a significant impact on the solidification process. We believe that it is essential to include the surface heating factor in numerical models of melt spinning and other rapid solidification processes relying on direct contact with a solid substrate.
1. Introduction Melt spinning is one of the most commonly used processes for the production of rapidly solidified thin metal foils. Two variants of this process are free-jet melt spinning (FJMS) and planar flow casting (PFC). The main difference between these two techniques is that in the FJMS the crucible nozzle is located relatively far away from the wheel, whereas it is located very close for the PFC case. It is believed that the small gap between the crucible and the wheel in the PFC case results in the formation of an extensive melt puddle that damps perturbations in the melt flow and thereby leads to improved foil uniformity, The ribbon is dragged out of the melt puddle by the relative substrate motion and may emerge in solidified, semisolidified, or fully liquid form, depending on the heat transfer, nucleation and crystal growth characteristics of the given process. Two different mechanisms of ribbon formation are often mentioned for rapid solidification by melt spinning: thermal control and momentum transport control. These are defined according to the relative efficiency of the heat and momentum transport phenomena taking place in the puddle. 0921-5093/91/$3.50
If the ribbon is dragged out of the puddle already in solid form (i.e. if the solidification of the ribbon is completed in the puddle itself), the ribbon formation mechanism is said to be subject to thermal control. On the contrary, if solidification in the puddle is delayed for some reason (e.g. because of large superheat, poor thermal contact at the puddle-wheel interface or large nucleation barrier for glass-forming materials), a layer of liquid melt will be dragged out of the puddle, and solidification will take place further downstream. In this situation, the ribbon formation process is far more dependent on the fluid flow in the puddle, and the ribbon formation mechanism is said to be subject to momentum control [1]. In the FJMS case, Kavesh [2] postulated a thermal control mechanism for amorphous alloys. Subsequent fluid flow and heat treatment analyses [3, 4] suggested, however, that the melt temperature at the exit of the puddle is still greater than the glass transition temperature of the materials spun. It may be then concluded that the ribbon formation mechanism is dominated by momentum transport in the puddle and that the solidification is completed out of the puddle [1]. For crystalline materials, on the contrary, Katger© Elsevier Sequoia/Printed in The Netherlands
86
man [5] suggested that heat transfer may make a substantial contribution to the ribbon formation mechanism as well. For PFC of crystalline materials, Vogt [6] measured the temperature of the ribbon top surface and--using this information in a heat transfer analysis of the process--concluded that the ribbon formation mechanism is thermally controlled in the case of good thermal contact at the puddle-wheel interface. A similar conclusion was reached by Muhlbach et al. [7]. Several mathematical models have been developed in order to study and predict the ribbon formation process during melt spinning. Most of the earlier modelling work focused on the FJMS process and addressed the formation of metallic glasses for which the release of latent heat of solidification .need not be taken into consideration (see for example refs. 2, 3, 8 and 9). More recently, however, several studies have been conducted on the rapid solidification of crystalline materials by PFC. Yu [10], for example, developed a fluid flow mathematical model for PFC at low Reynolds numbers. In this model, lubrication theory is used in the puddle under the nozzle, and film theory is applied to the liquid portion of the puddle downstream of the nozzle. This model uses an average solidification front velocity for the calculations. Another excellent model based on lubrication theory that also couples phase change and heat transfer in the puddle has been developed for PFC by Gutierrez and Szekely [ 11 ]. In order to avoid some of the low Reynolds numbet constraints that are inherent in this lubrication theory, however, we have also recently developed a two-dimensional boundary layer theory model for PFC which includes coupled fluid flow, heat transfer and solidification phenomena[12, 13]. In most of the previous models, the focus has been on the fluid flow and heat transfer in the puddle and ribbon, but little work appears to have been conducted on the effect of the heat transfer in the wheel on the ribbon formation and solidification. Indeed, a constant and uniform wheel temperature during casting has been assumed by most researchers. Given the unsteady nature of the conductive heat transfer in the substrate and the very short durations involved, it is clear that this assumption may be of limited generality, however. In addition, recent measurements of the top surface temperature of the ribbon downstream of the crucible by Vogt [6] and Muhlbach et al. [7] have suggested that a
significant increase in the surface temperature of steel wheels may exist. It appears therefore that a detailed reexamination of the validity of the constant-wheel-temperature assumption in models is necessary, and in particular in the region directly underneath the solidification puddle. Accordingly, this article addresses the issue of the heat transfer in the wheel and of its impact on the solidification characteristics. A one-dimensional heat transfer and phase change numerical model including both the puddle and the wheel has been developed for this purpose. This model is based on a control volume integral (CVI) method rather than on the usual enthalpy method and provides a precise tracking of the solid-liquid interface position. The effect of different wheel and melt materials and other parameters on the heat transfer and solidification in the puddle has also been investigated. In the present article we are addressing more specifically the PFC process, but the main results and conclusions are applicable to the FJMS process as well. 2. Problem statement A detailed description of our model of the PFC process and of the numerical scheme used can be found elsewhere [14], but a brief summary is given here for convenience. A schematic diagram of the idealized PFC geometry used in our model is also shown in Fig. 1 for reference. Using high speed movies of the solidification puddle, we have shown [13] that the downstream meniscus usually detaches from the crucible bottom well downstream of the nozzle. Given the large aspect ratio of the typical PFC melt puddle and the high axial velocities, it is then reasonable to approximate the puddle by a thin rectangular strip of
uclble
y=H
I ]
'
',
'
', Nozzle
, • , ' ' PUDBLE
y=0 Surfoco) x=0 (Wheel
I L I R~bbo, --> ×=x0 vw
s/L L,terfoce WHEEL (L= Liquid region of puddle; S= Solid region) Fig. 1. Schematic diagram of the idealized PFC geometry
used inournumericalmodel.
87 liquid. The main assumption used in the present model is to consider that there is no velocity gradient in the puddle. Clearly, this is not true in the actual case, but our primary objective here is to study the effect of the conduction in the wheel on the solidification, and we believe that this effect can be adequately evaluated without introducing at this point the complications of a twodimensional model of the flow field, A number of assumptions have also been made to simplify the model. We assume that the wheel is initially at room temperature ( To = 300 K). We limit ourselves here also to the study of the melt puddle under the crucible and the wheel below the puddle. We neglect convective heat transfer normal to the wheel surface. Heat conduction parallel to the substrate surface is also neglected, We consider solidification of a pure metal. We assume that local thermodynamic equilibrium exists at the solid-liquid interface (i.e. no undercooling). The temperature at the top surface of the puddle is assumed to be maintained at the pouring temperature Tp. The heat loss through the menisci of the puddle is neglected. The wheel surface is assumed to be horizontal and to exhibit a no-slip condition. Distinct values of material properties are used in the liquid and solid regions respectively but are otherwise assumed to be independent of temperature. Shrinkage during solidification is not taken into account, Because of the assumption that there is no relative motion within the melt puddle, the twodimensional steady state boundary layer problem can then be reduced to a one-dimensional unsteady state heat conduction problem. At the solid-liquid interface within the puddle, energy conservation can be described by the usual interface condition involving the interface velocity and the temperature gradients on each side of the interface. Because of our assumption of local thermodynamic equilibrium at the solid-liquid interface, the interface temperature is considered
to be fixed at the melting temperature TM. The non-perfect thermal contact at the interface between the melt puddle and the wheel surface is quantified by the usual heat transfer coefficient h, based on the temperature difference between the puddle bottom and the wheel surface. A symmetric temperature condition is used at the center of the wheel. We also assume for simplicity that the heat transfer coefficient is constant throughout the puddle. In the present work, an improved CVI method is used to generate finite difference equations for this one-dimensional phase change with moving solid-liquid interface. In our approach, the resolution of the interface tracking is greatly increased by the implementation of a special treatment of the phase change volume element, which is particularly useful for the modelling of solidification with large levels of undercooling and recalescence [14]. 3. Numerical results and discussion
The computations described in this article are based on experimental data that were obtained recently in our laboratory during an investigation of the PFC solidification puddle dynamics. The experimental apparatus and procedures have been described in detail elsewhere [15]. The parameters used in these calculations are as follows: wheel speed, 23 m s-1; puddle height and crucible-wheel gap, 350 pro; superheat, 50 K; puddle length (approximated by the distance between the nozzle and the meniscus detachment point), 5.45 ram. The corresponding ribbon thickness was 68 pm. In our computations we have used these geometrical and process parameters to model the spinning of various melts (aluminum, nickel and titanium) on different wheel materials (copper, nickel and stainless steel). The physical properties used in the calculations are shown in Table 1. The number of nodes
TABLE 1 Material propertyvaluesused in the computations Material
TM (K)
L (kJkg J)
CpL (Jkg JK -I)
Cps (Jkg-~K-j)
AI Ni Ti Cu Steel
933 1725 1940
395 300 365
1200 620 700
1060 600 528 503 550
kL
(Wm-lK -I) 100 30 22
ks (Wm-lK-l)
PL (kgm-3)
Ps (kgm-3)
200 60 22 393 20
2340 8500 4510
2700 8900 4510 9000 7650
88
used is 200 both in the puddle and in the wheel, The calculation domain in the wheel was 1000 # m for the copper wheel and 500 # m for the stainless steel wheel and the nickel wheels. These wheel domains were found to be large enough for accurate results, given the very small thermal penetration depth that is achieved in the short duration corresponding to the wheel passage under the puddle area (about 0.25 ms only). (Larger domains were used for the analysis of overall wheel heating, however; see Section 3.4.) The effect of the heat transfer coefficient on the solidified layer growth and on the interface velocity and interface cooling rate has been investigated previously, and the results have been presented elsewhere [14]. It was shown in particular that the solidified layer at the crucible detachment point was approximately equal to the ribbon thickness for h = 106 W m -2 K -1, i.e. the ribbon is fully solidified when it leaves the puddle. The solidification is then "thermally controlled". The increase in the cooling rate of the liquid melt at the interface and in the interface velocity with increasing heat transfer coefficients was also quantified. The cooling rate of the melt at the interface (or "interface cooling rate" in short) is defined here as OT[ Oyh,k where the temperature gradient is computed at the interface on the liquid side, and V~is the interface velocity. It was also seen that both the interface velocity and the interface cooling rate decrease significantly as the interface moves far away from the wheel, especially for high heat transfer coefficients. The interface velocity was found to exhibit a maximum very close to the wheel surface, however. This maximum results very probably from the superposition of the diminishing need for removal of liquid superheat and the increasing thermal resistance of the solidified layer as the interface penetrates further into the melt. The interface velocity and cooling rate were also shown to be much lower for a wheel of finite thermal diffusivity than for a hypothetical ideal (isothermal) wheel. The differences were shown to be particularly significant for large heat transfer coefficients. As mentioned above, it is commonly assumed in melt-spinning (and other)models that the substrate temperature remains constant. Our
preliminary results suggested, however, that numerical models for rapid solidification should indeed include wheel heating for more accurate calculations of cooling rates and interface velocities. Accordingly, a discussion of this effect and some results on the magnitude of the surface heating are given below.
3.1. Effect of the nature of the substrate on the solidification characteristics As mentioned in the previous section, the wheel material has a greater effect on the solidification characteristics when there is good thermal contact between the puddle and the wheel. We shall therefore limit this discussion to the case of large heat transfer coefficients. Figure 2 shows the solid-liquid interface position in the puddle as a function of distance from the upstream meniscus for various wheel materials, including a hypothetical ideal wheel which is assumed to have an infinite heat capacity and thermal conductivity (i.e. the wheel maintains a constant and uniform temperature during the process). For better comparison, we are using in these computations the same heat transfer coefficient (10 6 W m 2 K 1) for the various wheel materials, but in practice the thermal contact may be affected by the melt-wheel material combination [7]. (Recent experimental results suggest that a value of 106 W m -2 K -1 c a n indeed be achieved before the ribbon is dragged out of the puddle [6, 7, 16].) The numerical model corn-
~" ~o 60 -~ g ~ ~- 40
, -,dro, Whee,
-
cu N~
~
s~oo~
.~ 20 u~,, ~
0 2 Do~,~o~
4 Oisfance (I 0 -3
m)
Fig. 2. Vertical location of the interface as a function of distance from the u p s t r e a m meniscus for a l u m i n u m s p u n on ideal, copper, nickel and stainless steel wheels ( h = 106 W m 2 K t; gap, 350 /~m; V w = 2 3 m s t ; T p - T m = 5 0 K ; T,,= 300 K; XE) = 5.45 ram; ribbon thickness, 68 #m).
89
putes the heat transfer and temperature in the wheel as well as the increase in temperature at the wheel surface, together with the temperature field in the puddle itself. The two are coupled, of course, and it is the heat transfer in the wheel that is the source of the variation in solidification rate shown here. It can be seen that the rate at which the liquid melt solidifies is significantly affected by the heat conduction in the wheel. As expected, a higher resistance to conduction in the wheel results in a Aluwlirlum
E ° g ~ a_
]
I|
me,
60
I
, ' co, ~hee,
cu
40
,
,
o .~ z0 ~"
/
S~o~o,es~ \
0 0.00
\
\
,
0.15 o.30 0.45 Interface Velocity (m/see)
Fig. 3. Interface velocity as a function of distance from the wheel for a l u m i n u m spun on ideal, copper, nickel and stainless steel wheels ( h = 1 0 6 W m - - ' K i; gap, 350 /am; Vw = 23 m s-% T p - T m = 50 K; T. = 300 K; X D = 5.45 m m ;
ribbon thickness, 68 pro).
,E" ~o 60
. \/"./i
.
.
. A,o~. . . . . ,1
~,, =o ~ ~-
k/~, ,0
.~ 20 ~-. ~3
',, ,( ~o, wh~, ,,,
\ ,A \ , ~ ~ ' - . S,o~o,~
"-.
0
"" 3 Melt Cooling Rate at
6 Interface (10 5
s K/see)
Fig. 4. Local melt cooling rate at the solidification interface as a function of distance from the wheel for a l u m i n u m spun on ideal, copper, nickel and stainless steel wheels ( h = 10 ~ W m z K ~; gap, 3 5 0 / a m ; V w = 2 3 m s - l ; T p - Tm = 5 0 K; T. = 300 K; X D = 5.45 ram; ribbon thickness, 68/am).
decrease in heat flux from the puddle and therefore in a decrease in solidification rate as well (see above). For materials with a high thermal conductivity such as copper, the reduction with respect to an ideal wheel is moderate, but for materials with poorer thermal conductivity such as stainless steel, for example, a large reduction in solidification rate with respect to an ideal wheel takes place. It can be seen that the solidified layer is about twice as thick for the copper wheel than it is for the steel wheel at the same distance downstream. The interface velocity and the interface cooling rate are shown in Figs. 3 and 4 respectively. As mentioned, the interface velocity (Fig. 3) reaches a maximum close to the wheel before decreasing as the interface penetrates further in the puddle. This is true for all the materials discussed here, but the maximum is closer to the surface for a wheel of low thermal conductivity. The decrease in interface velocity is also more rapid for the wheel of lowest thermal conductivity. This is because in a wheel of low thermal diffusivity the surface temperature increases significantly with time, which in turn decreases greatly the heat flux and therefore the interface velocity. The interface cooling rate is also seen in Fig. 4 to be significantly affected by the wheel material. As discussed previously, the interface cooling rate decreases rapidly as the interface penetrates inside the puddle. These results confirm that the heat transfer in the wheel has a large effect on the solidification characteristics and thermal history of the ribbon. Clearly, the surface temperature of the wheel is likely to increase significantly for a wheel of low thermal diffusivity. To illustrate this effect, Fig. 5 shows the increase in surface temperature during the passage of the wheel under the solidification puddle. The calculations have been conducted for the same wheel materials as in the previous figures. Naturally, an ideal wheel would see its surface temperature remain constant. For the parameters considered here, the wheel will remain only approximately 0.2 ms under the puddle, during which a large amount of energy is nevertheless transferred to the wheel (because of the large temperature difference between the wheel surface and the puddle bottom, and also because the heat transfer coefficient is large as w e l l ) . Given this short duration and the limited thermal diffusivity of non-ideal materials, the energy can only penetrate a short distance in the
90
F___ ~
750-
I
E
I
Stainless Steel Wheel
°~
/
® I0 ~
o
300
150
°°
2
I
o 600
"~
.~
I 1
o
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X= 5.45 Puddle
~ o
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~: -150
o
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450 0~ o -300
c~ ~00
~'-0
I
-450
Aluminum mell 2
4
Downstream Distance
A l u m ' h u m m e l t on a Copper wheel
300
450
(ram)
Fig. 5. T e m p e r a t u r e of the wheel surface as a function of distance from u p s t r e a m meniscus for a l u m i n u m s p u n on copper, nickel and stainless steel wheels (h = 106 W m -2 K ~; gap, 350 ,um; V w = 2 3 m s 1; T p _ T m = 5 0 K ; T , = 3 0 0 K ; X D = 5.45 mm; ribbon thickness, 68 Hm).
wheel (the classic "penetration depth" concept in conductive heat transfer) which results in a higher temperature near and at the wheel surface. The magnitude of this temperature increase depends on the wheel material, of course. Even for a wheel made out of copper (which has a high heat conductivity), the temperature of the wheel surface is seen to increase by 170 K over the length of the puddle. (As discussed above, we are limiting our calculations to the puddle up to the detachment point, but in reality the ribbon is still cooled further downstream, which will increase the wheel temperature even more.) For stainless steel which has a lower thermal conductivity, on the contrary, an increase in surface temperature of over 400 K is seen to occur. Obviously, such large variations in the substrate surface temperature should not be neglected in the models used for this process. A similar increase in surface temperature will also probably take place in other rapid solidification processes using a solid substrate as heat sink (e.g. splat cooling, FJMS, hammer-and-anvil system, melt overflow and two-roll quencher), The temperature profiles in the solid and liquid regions of an aluminum melt puddle as well as in the copper wheel on which the aluminum is solidified are all shown for various downstream locations in Fig. 6. It is interesting to look at the change in temperature difference between the puddle bottom and the wheel surface over time.
_
L__ 600
750
900
Temperature (K)
Fig. 6. T e m p e r a t u r e profile in several locations for a l u m i n u m where the profiles are shown for u p s t r e a m meniscus ( h = 1 0 ~ W Vw = 23 m s ~; T p - Tm = 50 K; ribbon thickness, 68 Hm).
~~o ~ ~ ~
300
the puddle and wheel at spun on a copper wheel, 1, 3 and 5.45 m m from the m 2 K ~; gap, 350 Hm; T, = 300 K; X~ = 5.45 ram;
. . . . .--__ CuNW ~ hee,w"ee' .. S,o~o~essStee~Whee,
150 Puddle
•
0
~ -150 Eo ~- -300 o ~:~
--450
Aluminum melt
300
450
600 750 Temperoture (K)
900
Fig. 7. T e m p e r a t u r e profile in the puddle and wheel at the puddle d e t a c h m e n t point (XD = 5.45 m m ) f o r a l u m i n u m spun on copper, nickel and stainless steel wheels ( h = 1()~ W m -2 K ~; gap, 350 Hm; V w = 2 3 m s '; T ~ - T m = 5 0 K ; T,,= 300 K; X a = 5.45 ram; ribbon thickness, 68 Hm).
Clearly, a larger temperature difference exists at X = 1.0 mm than at X = 5.45 m m = X b (the detachment point): 5 1 0 K vs. 355 K. This decrease results from the combination of a decreasing puddle bottom temperature and an increasing wheel surface temperature. The decrease in temperature jump at the wheel surface results in turn in a decrease in heat transfer into the wheel. (It should be noted that we have
91
assumed in these calculations that the heat transfer coefficient remains constant over the whole length of the puddle, but in reality it is likely that this coefficient decreases with distance as the melt solidifies and shrinks, which would then lead to lower heat fluxes than those computed here.) A better understanding of the spatial distribution in resistance to heat transfer can perhaps be gained from Fig. 7 which shows the temperature profiles in the puddle and wheel for three different wheel materials. All are computed at the detachment point measured experimentally for aluminum spun on a copper wheel (XD = 5.45 mm) and for a heat transfer coefficient of 106 W m -2 K 1. The relative temperature drops in the puddle, in the wheel and at the puddle-wheel interface are indicative of the distribution of heat resistances across the field. For a wheel of high thermal conductivity, such as copper, the largest temperature drop takes place at the puddlewheel interface, with smaller temperature drops in the wheel and in the puddle (which is aluminum here, also a high conductivity metal). The main thermal resistance is therefore "located" at this interface, even though the heat transfer coefficient used is relatively high in this case (h = 10c'W m - 2 K-~). On the contrary, and as expected, for a wheel of low thermal conductivity metal, such as stainless steel, the largest temperature drop exists in the wheel, with smaller temperature drops in the puddle and at the puddle-wheel interface. The main thermal resistance is therefore the wheel itself inthiscase. If we compare the temperature drops across the puddle-wheel interface at X = 5.45 mm, we find values of 355 K for a copper wheel, 280 K for nickel and 180 K for stainless steel. Given the constant-h assumption used here, the heat fluxes would be directly proportional to these temperature differences. In other words, the heat flux would be twice as large for a copper wheel as it is for a steel wheel. This difference will, of course, also result in larger interface cooling rates for the copper wheel, The increase in wheel surface temperature results from the fact that a high heat transfer takes place over a short time. If the wheel has a finite thermal diffusivity, the energy will not have the time to diffuse much into the wheel, and the surface will heat up significantly. This concept of limited thermal penetration depth is illustrated clearly in Fig. 7 as well. It can be seen that, after
the same time, the energy has visibly "penetrated" a much greater vertical distance in the copper wheel (about 500 /~m) than in the steel wheel (150 /~m). These values are consistent with the relative magnitudes of the characteristic lengths computed as the square root of the product of the thermal diffusivity by the time (i.e. the classic diffusion length). This diffusion length is indeed approximately four times greater for copper than for steel.
3.2. Effect of the nature of the melt on solidification characteristics and substrate temperature The discussion above was concerned with an aluminum melt of high thermal conductivity for which the relative resistances to heat transfer in the melt and ribbon are small and would have comparatively little effect on the solidification characteristics. In this case, either the thermal contact resistance (for wheels of high conductivity) or the heat conduction in the wheel (for wheels of low conductivity) may limit the overall heat transfer process. On the contrary, if a low conductivity material is cast, such as titanium for example, the solidification characteristics may be somewhat different. For illustration, some calculated results are shown hereafter for three different melts: titanium, nickel and aluminum. The process parameters are assumed to be the same in all cases, including the level of superheat, but all the material properties used are those corresponding to each specific melt. Figure 8 shows the solid-liquid interface velocity for the three different melts spun on a copper wheel. The same thermal contact condition at the puddle-wheel interface (h-- 106 W m- 2 K- 1) and the same detachment distance (5.45 mm) are assumed in all cases. The variation in interface velocity across the ribbon reflects the effect of the thermal conductivity of the melt on the solidification rate. For aluminum which has a high thermal conductivity, the latent heat and superheat can be easily transferred from the solidification interface to the wheel, and the interface velocity changes relatively little as the interface penetrates further in the puddle. In the titanium case, on the contrary, the latent heat and superheat will be less readily conducted to the wheel when the solid layer thickens because of titanium's smaller conductivity. A larger variation in interface velocity across the ribbon thickness is therefore seen as the interface grows inside the
92 i
i
r
I
Copper wheel 60
~
0
60
Copper wheel
0 v
S
S .:E
o_
Ap
40
~
Q
40
o
20
Ni
~
20
u~
0 0.0
0 0.2
0.4
0.6
i
0.8
50
Interface Velocity ( m / s e c )
Fig. 8. Interface velocity as a function of distance from the wheel for three different melts (aluminum, nickel and titanium) spun on a copper wheel (h = 106 W m- "- K ~; gap, 350 /~m; V w = 2 3 m s ~; T p - T m = 5 0 K ; T0= 300 K; XD= 5.45 ram; ribbon thickness, 68 ~m).
Overall, the interface velocity is calculated to be about twice as large for titanium than it is for aluminum close to the wheel. It decreases then significantly away from the wheel and is seen to become in fact smaller than that for aluminum (which remains nearly constant across the ribbon). The interface velocities are in all cases of the order of 0.1-1 m s- 1 throughout the ribbon, Figure 9 shows the cooling rate upon solidification for the three melts. As expected, the cooling rate is much higher at the bottom of the ribbon for titanium ( 10 7 K s- ~) and nickel (5 x 106 K s - ~) than for aluminum (8 x 105 K s l ) which has a much lower melting temperature. Towards the top of the ribbon, however, the cool-
50
Melt Cooling Rote at Interface (I0 s K / s e c )
Fig. 9. Interface cooling rate as a function of distance from the wheel for three different melts (aluminum, nickel and titanium) spun on a copper wheel (h = 106 W m ' K '; gap, 350 ¢tm; V w = 2 3 m s ~; T p - T,,= 50 K; T , = 3 0 0 K ; Xt~ = 5.45 mm: ribbon thickness, 68/~m). ,
700
puddle. For nickel, which has a thermal conductivity somewhat larger than titanium, an intermediate level of variation across the ribbon is seen. It is interesting to note that the numerical results suggest that the interface velocity for titanium is much greater than for aluminum close to the wheel, but that it becomes smaller further away. Nickel, on the contrary, which has a melting temperature close to that of titanium is seen to exhibit an interface velocity similar to that of aluminum close to the wheel. These relative magnitudes are not easy to predict with back-of-theenvelope calculations because of the combined effects of the varying temperature gradients and different material properties for the three metals.
_
100
r
Copper wheel
~. o
?~
600
~
500
~_ 400 ~300
, 2 Dow~r~o~
Di~ton~
4 (~0-~ ~)
6
Fig. 10. Wheel surface temperature increase as a function of distance from the u p s t r e a m meniscus for three different melts (aluminum, nickel and titanium) spun on a copper wheel ( h = 1 0 ~ W m 2 K - l ; gap, 350 # m ; V w = 2 3 m s-~; r~- Tm = 50 K; T0 = 300 K; X~ = 5.45 ram; ribbon thickness, 68/~m).
ing rates are about the same for all three melts (approximately 2 × 105 K s-l), the result of a combination of relative influences of interface velocities and temperature gradients in the liquid. It should be noted that for titanium, for example, the variation in cooling rate across the ribbon is approximately two orders of magnitude, from 10 7 to about 105 K s- l, a rather large variation. Figures 10 and 11 show the temperature increase of the wheel surface for the three melts
93
spun on copper and stainless steel wheels respectively. It is perhaps surprising to see that the wheel surface temperature has increased less for titanium than for nickel toward the end of the puddle in the case of the copper wheel (despite a lower melting temperature for nickel), whereas the increase is larger for titanium i~a the case of the steel wheel. A possible explanation might be as follows. In the case of the copper wheel, the resistance to heat transfer in the wheel is rather low because of the high thermal conductivity of copper. In these computations the thermal resistance at the puddle-wheel interface is also small (h=106 W m -2 K-t). The main resistance to heat transfer is therefore likely to be in the solidified melt layer for the case of the copper wheel, Since nickel has a conductivity that is three times that of titanium and a similar melting temperature, it exhibits a smaller resistance to conduction in the solidified layer and therefore a proportionally higher heat flux than titanium, which in turn results in a higher wheel temperature increase. For the steel wheel, however, the main resistance to heat transfer is probably in the wheel itself and no longer in the solidified layer, and the relative magnitude of the conductivity of this layer plays a smaller role. Titanium having a somewhat larger melting temperature than nickel may then exhibit a higher heat flux which in turn would lead to higher wheel temperatures. Regarding the necessity of the inclusion of the 1500
--
~
o
F Sfainless Steel Wheel
1200
wheel heat transfer in a rapid solidification model, one should consider the magnitude of the wheel surface temperature increase calculated here. In the case of melt spinning of a high melting temperature metal such as titanium or nickel, we see that the wheel surface temperature may increase by as much as 200 or 300 K before puddle detachment, even for a copper wheel. If one assumes an isothermal wheel in calculations, the predicted heat flux could be easily overestimated by 100%. Naturally, the situation would be even worse for a wheel of lower conductivity. Assuming that one would be interested in spinningtitanium, say, onasteelwheel, the wheel surface would easily increase to 1200 K according to these calculations, a situation likely to result in some experimental difficulties!
3.3. The effect of the melt superheat on the solidification mechanism Figures 12 and 13 show the effect of the degree of superheat on the solid-liquid interface velocity for aluminum and titanium spun on a copper wheel. In both cases, the interface velocity decreases as the level of superheat increases. This is to be expected because more sensible heat in the liquid must then be removed by conduction to enable the interface to move further into the liquid. Comparing the results for the two metals, it can be seen that the effect of the superheat on the
~
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60
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=
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Tp-T : 1 0 0 m
K
50
10 I
0 0.0
0.2
0.4
0.6
0.8
Downstream Distance (10 -3 m)
Fig. 11. Wheel surface temperature increase as a function of distance from the upstream meniscus for three different melts (aluminum, nickel and titanium) spun on a stainless steel wheel (h = 1 0 6 W m - 2 K-[; gap, 350 /am; Vw=23 m s-I; T p - T m = 5 0 K ; T 0 = 3 0 0 K ; XD=5.45 mm; ribbon thickness, 68/am).
Interface Velocity (m/sec)
Fig. 12. Interface velocity as a function of distance from the wheel for three different levels of superheat (10, 50 and 100 K) for aluminum spun on a copper wheel (h = 106 W m -2 K-J; gap, 350 /am; Vw=23 m s-J; T0=300 K; Xo = 5.45 mm; ribbon thickness, 68/am).
94 60
- - ~
E
-
~
i
--
Titanium rnell on copper wheel
~
i
~
750
-
-
r
746 K (SS)
~- -
Aluminum nlelt
v
o
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o o_
~ J:: \\\
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_c •--
20
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~ L
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E
i l I ~ I
450
~.BS K
L ~
s
s
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2
o G)
0 0.0
: 0.2
0.4
0.6
I n t e r f a c e Velocity
0.8
1.0
(m/sec)
Fig. 13. Interface velocity as a function of distance from the wheel for three different levels of superheat (10, 50 and 100 K) for titanium spun on a copper wheel (h = 106 W m -~ K - I ; gap, 350 /~m; V w = 2 3 m s ~; T . = 3 0 0 K; XD=5.45 mm; ribbon thickness, 68/~m).
interface velocity appears to be larger in the case of aluminum. This difference may be related to the fact that aluminum has 10 times the thermal conductivity of titanium. Indeed, it should be noted that the upper boundary condition for the puddle is that the temperature is constant and equal to the pouring temperature. It is therefore possible for energy to be transferred from the bottom of the crucible to the melt. For aluminum, which has a high thermal conductivity, a large heat flux can be generated in this manner if the superheat is large. Naturally, this energy has to be removed by the wheel in addition to the original superheat in the liquid and to the latent heat released at the interface. A significant effect of the pouring temperature can then be expected, as seen in Fig. 12. For titanium, which has a low thermal conductivity, the temperature gradient in the liquid will be much more localized near the interface, with only a small gradient near the top of the puddle. The heat flux from the crucible will therefore be smaller and its effect on the interface velocity much more limited than for aluminum, The effect of the level of superheat on the increase of wheel temperature was also investigated, but the calculations showed this to be a relatively small effect,
3.4. Changes in wheel surface temperature during continuous me# spinning In all the calculations discussed above, we have assumed that the wheel surface is at room tern-
300 ~ 0
250
~ 500
75(
Distance f r o m meit c o n t a c t (10 -s m)
Fig. 14. Temperature of the wheel surface as a function of distance from melt contact location during the first revolulion after contact, for aluminum spun on copper and stainless steel wheels (heat transfer coefficients, h = I(P W m 2 K before separation and h = 0 after separation of ribbon from wheel; gap, 350 /~m; Vw=23 m s ~; 7 ] , - T ~ = 5 ( ) K; T,= 300 K: A's = 8 mm; ribbon thickness, 68 #m; wheel
diameter,().28 m).
perature T, before the first melt contact. Naturally, the initial room temperature assumption is no longer valid for the following wheel revohitions. Tens to hundreds of revolutions may indeed be common even during relatively short laboratory PFC experiments. During that time, the wheel will be subjected not only to conductive heat transfer from the solidifying metal but also to radiative and convective heat transfer from the crucible. In that case, significant "long-term" increases in surface temperature may take place if the wheel is not cooled externally. We have conducted some calculations to investigate the degree to which this temperature increase may become a significnt factor. Figure 14 shows the wheel surface temperature calculated over one full wheel revolution (the wheel diameter is 0.28 m), and assuming an equivalent ribbon separation distance X, (i.e. puddle length here) of 8 mm. (The actual separation distance is likely to be much greater but would probably involve a sharply decreasing heat transfer coefficient towards the end of the puddle and beyond, whereas we assume here a constant (high) value of this coefficient. Also, the upper surface of the ribbon would not be heated in reality (and could even be significantly cooled by radiation and convection) whereas we assume contact with the hot crucible. It is therefore hoped that a reduced
95 equivalent separation distance is a reasonable substitute for the actual case.) After separation, we assume that the heat transfer coefficient becomes zero. We also neglect both convective cooling of the wheel by the ambient gas and heating by the crucible. The calculated surface temperatures are shown in this figure for both a copper and a steel wheel. As expected, it can be seen that the surface temperature reaches rapidly high levels (746 K for the steel wheel and 485 K for copper in this case) and then decreases after ribbon separation, the energy being conducted away toward the inside of the wheel. The surface temperature appears then to have decreased significantly by the end of the first revolution after melt contact. This is the new surface temperature
~ o E ~ ~ -looo
that the melt will effectively "see" at the beginning of the next pass of the wheel under the puddle, and it is also the temperature that should be used as the initial condition for the computations
Fig. 15. Temperature profiles in the wheel at the ribbon separation point (As),at 35 mm downstreamof the meniscus, and after one fullrevolutionfor aluminumspun on a copper wheel (gap, 350 ~m; Vw=23 m s-t; Tv-Tm=5OK; T0=300 K; Xs=8 mm; h=106 W m 2 K-1 before separation and h=0 after separation; ribbon thickness, 68 /xm; wheel diameter,0.28 m).
thereafter, For copper, we see an estimated 10 K increase in surface temperature at the end of the first revolution after melt contact, whereas the increase is about 25 K for a steel wheel. This discrepancy is not that large despite the significant difference in wheel conductivity because there is enough time for most of the energy transferred into the wheel to diffuse inward. Also, the heat capacity per unit volume of copper is somewhat greater than that of stainless steel, so that a copper wheel can store more energy for a given ternperature increase than a stainless steel of the same dimension, which alleviates the fact that more energy is transferred into the cooler copper wheel. The use of a solid wheel assumption in these computations has a significant impact on the results, and the end-of-revolution temperature increase would be smaller if we were to model a hollow wheel cooled internally by water, Temperature profiles inside a copper wheel subject to aluminum spinning are shown in Fig. 15, both at the ribbon separation distance and at the end of the first revolution after melt contact, An intermediate location (X= 35 mm) is also shown. As expected, a high temperature at the surface (500 K) and a short penetration depth (0.8 mm) are seen to exist at separation. A much greater penetration depth and a correspondingly smaller surface temperature peak is seen for larger times because the same amount of energy (we assume no additional heat transfer after separation) has spread by diffusion over a much
~
(/3
/
'
x~- 35 m m
End of r e v o l u t i o n -2000
E o 8 -3000
L
~
A~om~. . . . . It p p o r wheel , , 35o 400 4so soa Temperofure in the Wheel (K) . . . .
-4000 300
larger domain. Nevertheless, if the wheel is not cooled externally, the surface temperature is seen to be about 10 K above room temperature at the second contact with the puddle. Naturally, the temperature of the surface will then keep increasing with the subsequent revolutions as the wheel slowl~¢ heats up. If we assume as a first approximation a 10 K increase per revolution, it can be seen that the surface temperature would soon become rather high. Of course, in reality, other heat losses may reduce this effect, but the advanrage of a water-cooled wheel over a solid one in this regard is obvious. The inclusion of long-term surface heating in numerical models for this type of process appears therefore to be desirable for uncooled wheels but is probably not necessary for water-cooled wheels. The main point of this article was to evaluate the larger localized increases in temperature directly under the puddle, however. It should be emphasized then that--because of the small penetration depth typical of the short wheel residence time under the puddle--such a localized increase would very likely take place even for a water-cooled wheel, unless the thickness of the rim could be made much smaller than a millimeter. (The thermal penetration depth at ribbon separation is about 0.8 mm as seen in Fig. 15.)
96
4. Conclusions
Acknowledgments
A n i m p r o v e d C V I a p p r o a c h was used to develop a numerical heat transfer model of rapid solidification by the PFC process. T h e models developed include in particular the effect of the heat transfer within the substrate and allowed us to investigate the influence of the thermal contact between the wheel and the melt, of the wheel material, of the melt material and of the superheat level on the solidification history, on surface ternperature increases, on interface velocity and on the interface cooling rate. T h e s e parameters have been discussed and analyzed in terms of the various thermal mechanisms such as local thermal resistances due to thermal conductivity,
We would like to acknowledge gratefully the support of the National Science Foundation (Grant M S S - 8 9 5 7 7 3 3 ) and of the Defense A d v a n c e d Research Projects Agency (Contract N 0 0 0 1 4 - 8 6 - K - 0 7 5 3 ) . We would also like to thank Professor C. Levi (University of California, Santa Barbara, C A ) for m a n y valuable discussions. Mr. Wang is currently a graduate student at T h e University of California, Santa Barbara, CA.
energy releases and transfers, and relative magnitudes of the individual material properties. O f particular significance is the fact that the wheel surface t e m p e r a t u r e is seen to increase greatly under the solidification puddle, even for a c o p p e r wheel. In this case, typical surface temperature increases would be of the order of 250 K at the end of a puddle 5 m m long for spun titanium, and about 175 K for spun aluminum. If the wheel is m a d e out of a low conductivity material, e.g. steel, the local increase could be as large as 1000 K. Naturally, the impact of these increases on the solidification characteristics was found to be very significant. For example, the heat flUX into the wheel could be easily overestimated by 100% if an isothermal c o p p e r wheel is assumed, It should also be noted that the wheel surface heating is linked to the fact that rapid solidification processes involve large amounts of energy transferred over small periods of time. Interestingly, using a water-cooled wheel would therefore generally not prevent this large surface heating. This is because the wheel heating occurs only over a very thin layer near the surface in a wheel of finite thermal diffusivity, and this thin layer may not be able to "see" m u c h of the effect of t h e w a t e r c o o l i n g . In conclusion, it is essential in most cases to include substrate heat transfer and local surface tbmperature increases in numerical models of PFC. Furthermore, the results and conclusions presented here should be directly applicable to the FJMS process as well. It is also likely that the modelling of m a n y other rapid solidification processes based on direct contact with a substrate would benefit f r o m a similar approach.
References 1 H. A. Davies, Solidification mechanisms in amorphous and crystalline ribbon casting, in S. Steeb and H. Warlimont (eds.), Rapidly Quenched Metals, Vol. 1, NorthHolland, Amsterdam, 1985, pp. 101 - 106. 2 s. Kavesh, Principles of fabrication, in J. J. Gilman and H.J. Leamy (eds.), Metallic Glasses, American Society for Metals, Metals Park, OH, 1978, pp. 36-73. 3 K. Takeshita and P. H. Shingu, An analysis of the ribbon formation process by the single roller rapid solidification technique, Trans. Jpn. lnst. Met., 24(1983) 529-536. 4 z. Sun and H. A. Davies, Computer modelling of combined heat and momentum transfer in the melt spinning of amorphous and crystalline metals, in S. Kou and R. Mehrabian (eds.), Modeling and Control of Casting and Welding Processes, The Metallurgical Society, Warrendale, PA, 1986, pp. 179-194. 5 L. Katgerman, Continuous products in rapid solidification, in P. R. Sahm, H. Jones and C. M. Adam (eds.), Science and Technology of the Undercooled Melt,
Martinus Nijhoff, Dordrecht, 1986, pp. 121-135. 6 E. Vogt, On the heat transfer mechanism in the melt spinning process, Int. J. Rapid Solidif., 3 (1987) 131-146. 7 H. Muhlbach, G. Stephani, R. Sellger and H. Fiedler, Cooling rate and heat transfer coefficient during planar flow casting of microcrystalline steel ribbons, Int. J. RapidSolidif, 3(1987)83-94.
8 T. R. Anthony and H. E. Clyne, On the uniformity of amorphous metal ribbon formed by a cylindrical jet impinging on a flat moving substrate, J. Appl. Phys., 49 (1978)829-837. 9 P. den Decker and A. Drevers, Model calculations on the solidification and crystallization processes during melt spinning, in C. Hargital, I. Bakonyi and T. Kemeny (eds.), Metallic Glasses: Science and Technology, Kultura, Budapest, 1981,pp. 181-188. 10 H. Yu, A fluid mechanics model of planar flow melt spinning process under low Reynolds number conditions, Metall. Trans. B, 18(1987) 557-563. 11 E. M. Gutierrez and J. Szekely, A mathematical model of the planar flow melt spinning process, Metall. Trans. B, 17(1986) 695-703. 12 M. I. Eskenazi and E. E Matthys, Modelling of planar flow melt-spinning using the boundary layer equations, Rep. UCSB-ME-88-4, 1988 (University of California). 13 z. Gong, P. Wilde and E. F. Matthys, Numerical model-
97 ling of the planar flow melt-spinning process, and experimental investigation of its solidification puddle dynamics, Int. J. Rapid Solidf., (1990), in the press, 14 G.-X. Wang and E. F. Matthys, Numerical modelling of phase change and heat transfer during rapid solidification processes: use of control volume integrals with element subdivision, to be published.
15 P. Wilde and E. E Matthys, Experimental investigation of the planar flow melt-spinning process: development and free surface characteristics of the solidification puddle dynamics, to be published. 16 K. Takeshita and P. H. Shingu, Thermal contact during the cooling by the single roller chill block casting, Trans. Jpn. Inst. Met., 27 (1986) 454-462.