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HEAT TRANSFER DURING PREPARATION OF AMORPHOUS
METALLIC ALLOY RIBBON
M.Geller, E.Brook-Levinson.and V.Manov
Advanced Metal Technologies Ltd., Even Yehuda, Israel
ABSTRACT
The maximum thickness of an amorphous ribbon strongly depends on the ribbon-todrum heat transfer. By theoretical analysis of the heat transfer the thickness of the
ribbon is determined as a function of the thermophysical properties of the melt and
drum material, melt and vitrification temperatures of the ribbons material, the melt
cooling rate. The theoretical formula for the maximum thickness of the amorphous
ribbon is in good agreement with known thickness-cooling time empirical correlation
and reflects main factors of heat transfer between the melt and drum during cooling of
the ribbon on the drum.
1. INTRODUCTION
The melt-spinning process is most commonly procedure of amorphous metallic alloy
ribbon production. It is well known that the cooling rate of the ribbon on the drum
must be high enough to achieve the amorphous solidification. The previous
estimations showed that the average cooling rate must range 105 K/s to 106 K/s [3].
Theoretical analysis of heat transfer in this case is complicated and requires solution
of conjugated problem of heat transfer. Several papers [1-5] are devoted to theoretical
investigation of the problem. Some of them use the numerical modeling of heat
transfer during melt spinning process. The work [2] is the good example of such
approach. The heat transfer both inside the melt and in the substrate is incorporated
directly in the numerical solutions. The calculations have been conducted to
investigate the effect of the heat transfer in the drum, of the drum material, of the melt
material and of the superheat level on solidification characteristics, and in particular
on the interface velocity and on the cooling rate at the interface. The results of
investigation are very useful for more deep understanding of influence of heat transfer
on service properties of amorphous metallic alloy ribbons. But in the same time they
are complicated for engineers for practical use. The next series of the investigation
uses analytical approach based on the well known solutions of heat conductivity
equation. The works [1,3-5] are typical in this way. For example, [1,3] assumes a
stable puddle under the nozzle. The cooling process leads formally to the problem of
contacting a hot plate with a cold body. The thermal contact between the puddle and
the wheel describes by Newton’s law of ribbon-to-drum heat transfer. The main
limitation of this approach relates to determination of the ribbon-to-drum heat transfer
coefficient. The accuracy of the experimental methods is very low and precise of heat
transfer coefficient measurements can achieve 50-100.
The purpose of the present work is to develop a heat transfer model more appropriate
for melt-spinning procedure and to get the simple engineering solution of the problem
which will be reflect main features of the process.
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2. MATHEMATICAL FORMULATION OF THE PROBLEM
The heat transfer from the puddle to the drum is analyzed on the basis of the model
schematically shown in Fig. 1.
q=0
puddle
drum
Fig. 1. Heat transfer model.
The following simplifications are made to analyze the problem:
1. The cylindrical geometry of the problem is replaced by the flat one because of
the low drum curvature value (the drum diameter ranges normally 300 to 800
mm);
2. No heat loss is assumed from the puddle surface to the environment due to the
short process duration(heat flux q is equal to zero}
3. No heat transfer is assumed along the ribbon length;
4. The ideal heat contact is assumed between the puddle and the drum;
5. The thermophysical properties of the puddle and cooling drum do not depend on
the temperature;
6. Uniform temperature of the puddle is assumed because of the very low thickness
of the ribbon (normally 15 to 30 m);
7. The drum is infinite along the radius because the drum diameter is essentially
larger then the puddle thickness.
These assumptions allow formulating the following mathematical model:
thermal conductivity problem in the drum can be described by the unsteady heat
transfer differential equation:
T2
k2  2 T2

 2
 2  c2 x
(1)
where k2 is the thermal conductivity of the drum, c2 is the specific heat capacity of the
drum, 2 is the density of the drum, T2 is the temperature of the drum,  is the time, x
is the radial coordinate.
The initial and boundary conditions can be written as follows:
T2(x,0) = T0
(2)
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T2(0,) = T0 + 2
(3)
T2 (  )
0
x
(4)
where T0 is the initial temperature of the drum, 2 is the cooling rate of the melt.
Boundary condition (3) assumes that the ribbon cooling rate is constant. Such
assumption can be made because of the very short cooling time. The cooling time of
the ribbon can be estimated as 10-3 seconds. For such a short cooling time the cooling
rate can be considered as constant with high accuracy. Boundary condition (4) implies
that the heat flux on opposite side of the drum is zero. The heat transfer problem (1-4)
is well known in the literature 6. The solution for the temperature gradient of the
ribbon-drum contact temperature is:
T2
1 2 

x
 2
(5)
where 2 = k2/c22 is the thermal diffusivity
Now the boundary conditions at the ribbon - drum interface are of the form:
T1(0,) = T2(0,) = T(0,)
(6)
T
T

c111 
- k2 x
(7),
where  is the ribbon thickness, subscript 1 relates to the ribbon, and subscript 2
T
relates to the drum. The cooling rate  is equal to 2 in accordance with equation
(3). The cooling time  can be determined through the following equation:
 = (Tm - Tgl) /2
(8),
where Tm is the melt temperature, Tgl is the vitrification temperature. The vitrification
temperature is the temperature of the ribbon solidification. The cooling till this
temperature must be very fast to prevent crystallization .
Using equations (5 - 8) the thickness of the amorphous metallic alloy ribbon can be
expressed by the formula:

Tm  T gl )

k 2c 2  2
c1  1  2
(9)
It must be underlined that the cooling time can be determined also from the
hydrodynamic conditions at the contact between the melt and drum.
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3. RESULTS
Equation (9) was used to estimate the maximum thickness  of the amorphous
metallic alloy ribbon as a function of the thermophysical properties of the melt and
cooling drum. The calculations were carry out for different vitrification temperatures
and for different values of the melt cooling ratio 2. In all calculations, the drum
material was copper, the melt temperature was 13500C. The results of the calculations
are presented in Fig. 2.
One can see that the maximum ribbons thickness grows with rising of the temperature
difference between melt and vitrification temperatures, dropping of the cooling ratio.
As can be seen from the equation (9) the thermal activity of the drum K=(k2c22)1/2
strongly influence on the maximum thickness of amorphous ribbon. The higher K the
bigger maximum thickness of the amorphous metallic alloy ribbon.
The parameter of the thermal activity is very important because checking of the drums
material to supply the needing cooling ratio. The different materials were analyzed in
accordance to their thermal activity. The results of K calculations are presented in the
Table 1.
Ribbons thickness, m
1200
Tgl=973K
Tgl=873K
Tgl=773K
Tgl=673K
1000
800
600
400
200
0
1.E+04
1.E+05
1.E+06
Cooling ratio, K/sec
Fig.2. Quenching of the Ribbon on the Drum.
As can be seen copper is the best material for the drum production. Aluminum,
tungsten and molybdenum are close to copper regarding their heat activity K. Bronze
heat activity is essentially lower compared copper however bronze is widely used for
the drum construction. It means that in addition to the above mentioned parameters,
few additional factors effect the heat transfer process such as viscosity of the melt,
drums roughness, wettability of the melt-drum boundary. The above described
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analysis gives the maximum possible value of the ribbons thickness where the
quenching material remains amorphous.
Table 1. K for some materials
Parameter K10-4
3.70
2.43
2.06
1.90
1.66
1.60
1.55
1.46
1.40
1.33
Material
Copper
Aluminum
Wolfram
Molybdenum
Cobalt
Zinc
Chromium
Nickel
Bronze
Steel
4. COMPARISON WITH EXPERIMENTAL DATA
The thickness of the ribbon as a function of the cooling time was experimentally
determined in [7]. The experiments were carry out for the Fe80B20 ribbon cooled on
the rotated water cooled copper drum. The experimental results were described by the
following correlation:
 = 3150 
(10),
As can be seen from (10) the experimental thickness in square root proportion with
time , where  - is the contact time of the melt with the drum . Coefficient 3150
m/s0.5 has been found after the treatment of experimental data for Fe80B20 ribbon. The
equation (10) in different cases may be written in the following form:
=A 
(11),
where A the experimental coefficient determined by experimental data.
If now to back to the theoretical expressions (8 and (9) easy to see that (9) can be
transformed to the new form:
=
1
K2 C2 2

C1 1

(12)
As can be seen the theoretical equation (9) gives the same function  = () that
experimental one represented in [7]. Beside this the theoretical value of coefficient A
is:
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A=
1
K2 C2 2

C1 1
(13)
The coefficient A was estimated for cooling of the melt on a copper drum for the
following parameters C1 = 700 j/kg0C, 1 = 7000 kg/m3 and (k2c22)1/2 = 3.7.104 . It is
easy to get that A is equal to 4228m/s0.5 and for this case the equation (12) can be
written in the form:
 = 4228 
(14)
One can see from comparison of (10) and (14) the difference between them only 30
It means that the above theoretical analysis reflects right the main features of the heat
transfer between the ribbon and the drum.
5. CONCLUSIONS
1. The maximum thickness of amorphous metallic alloy ribbon depends on the
temperature difference between the melt and vitrification points of the material,
thermal activity of the drum and cooling ratio of the melt. The bigger the temperature
difference and the heat activity is bigger the thickness of amorphous metallic alloy
ribbon.
2. The bigger the cooling ratio is the lower the ribbon thickness.
3. The heat activity of the material is very important parameter in the choice of
material for the drum.
4. The results of heat transfer investigation will be useful for production bulk
amorphous materials.
5. The equation (12) can be recommended for engineering estimations of heat transfer
conditions between melt and cooling drum.
REFERENCES
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2. G.-X. Wang ,E.F. Matthys, Materials Science and Engineering, A136(1991) 85-97
3. L. Granasy, A. Ludwig, Materials Science and Engineering, A133(1991) 751-754
4. H. Muhlbach, G. Stefani, R.Sellger and H. Fiedler. Int. J. Rapid Solidif., 3(1987)
83-94.
5. E. Vogt, On the heat transfer mechanism in the melt spinning process, Int. J. Rapid.
Solidif. 3 (1987) 131-146.
6. J.P. Holman. Heat Transfer, McGraw-Hill Book Company, 1989, 676.
7. R. E. Maringer and C. E. Mobley. Rapidly Quenched . Metals 3. Proceedings of the
Third International Conference on Rapidly Quenched Metals, Brighton, 3-7 July 1978,
44-49.
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