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An analysis of heat and mass transfer and ribbon formation by rapid quenching from the melt
Materials Science and Engineering, A l lO (1989) 131 - 138
131
An Analysis of Heat and Mass Transfer and Ribbon Formation by Rapid Quenching from the Melt L. A. ANEST1EV
lnst#ute J'or Metal Science and Technology, Bulgarian Academy of Sciences, 1574 Sofia (Bulgaria) (Received June 3.1988)
Abstract
The process of ribbon formation by the singleroller technique and by rapid quenching from the melt has been studied with the aid of the boundary layer theory. An easy calculation method has been proposed for the solution of heat and mass transfer equations for the case in which the melt is rapidly quenched onto a substrate. The proposed method takes into account the viscosity change due to the temperature reduction and the heat resistance on the substrate surface. The thicknesses of the ribbon cah'ulated with the aid of the proposed method show good correlation with the experimental results'.
1. Introduction
Recently, much attention has been paid to the formation and processing of amorphous alloys because of their possible important applications [1-12]. Among the various rapid solidification techniques, the single-roller chill-block casting methods--planar flow casting (PFC) and chillblock melt spinning (CBMS), in which continuously supplied melt is cooled by a wheel rotating at a high revolution rate--are most common at present because of their relative simplicity and ability to produce a large amount of rapidly solidified material in the form of amorphous ribbon. The main technologically important characteristics of this ribbon are its width w and thickness t. Use of the PFC method solves in practice the problem of production of amorphous ribbons with a defined width, since in this case the width of the ribbon equals the length of the crucible slot. The problem of the production of ribbons with a defined thickness, however, is still unresolved as the thickness depends on the pro0921-5093/89/$3.50
cesses which take place in the formation of the ribbon. The aim of this paper is to examine the processes which occur during rapid quenching from the melt and the influence which these processes have on the ribbon formation and on the geometry of the amorphous metal ribbon. In both the PFC and the CBMS methods a liquid melt pool (puddle) (Fig. 1) forms on the wheel surface. In effect the puddle spreads to a size such that the resultant ribbon thickness and width (for CBMS) satisfy the mass balance condition. The main characteristic of the puddle is its length I, and analysis of the experimental data shows that the thickness of the ribbon strongly depends on the length of the melt pool. Taking
lout I=
=-7
Vs
-~ =X
Ires Fig. 1. Schematic diagram of the molten alloy jet and the puddle formed on the cold substrate. The thickness 6 s of the material solidified in the puddle volume, the thickness 6M of the flow boundary layer, the heat flows l~,,, l.u ~ and 1~, and their directions are shown.
© Elsevier Sequoia/Printed in The Netherlands
132
this into account and also considering the experimental evidence obtained by several workers [1, 2, 8-11], we can draw the conclusion that the main ribbon-forming processes occur inside the melt pool. As the liquid metal is brought into contact with the cold surface of the moving substrate, simultaneous heat and momentum transfer arises. Owing to the adhesion forces a thin layer of the material is dragged out from the puddle volume; this layer cools on the substrate and finally forms the amorphous ribbon. As in the present case we have a continuous process and as some of the final product can be formed within the volume of the puddle, the thickness t of the ribbon is given by 6M
t =-
1 f
j
V,~ &
vx(y,l)dy+as(l)
(1)
where v~ is the surface velocity of the substrate, 6s(l ) is the thickness of the amorphous layer solidified in the volume of the puddle, 6 M is the flow layer induced from the moving substrate and v x is the x component of the liquid velocity as a function of x and y at x = 1. So far v,(x, y) and 6s(X ) have to be known in order to define the ribbon thickness. The processes of heat and mass transfer interact very closely with each other, and it is very difficult from a mathematical point of view to obtain an explicit solution which takes into account the influence of both processes on the ribbon formation. That is why one of these processes is assumed to be the limiting process, which historically led to the concept of the socalled "thermal" and "hydrodynamical" mechanisms of ribbon formation. The first mechanism assumes that the first term in eqn. (1) is insignificantly small relative to 6s(1), i.e. the ribbon is formed within the volume of the puddle and consequently cools on the surface of the substrate. The hydrodynamical mechanism of ribbon formation on the contrary assumes that the final product is formed from the liquid layer dragged out from the puddle by the moving substrate in the form of a thin film, which freezes prior to the substrate departure. In both cases the theory leads to identical functional relationships between t and the technological parameters [1, 11, 12]. To reveal the real nature of the ribbon formation, it is necessary to
solve together the equations describing both processes which take part in the ribbon formation, namely the processes of heat and mass transfer. Mathematically the description of these two interacting processes is given by the Navier-Stokes and Fourier equations for a viscous incompressible liquid and the Fourier equation for heat transfer in the solid phase (the frozen amorphous layer)[13]:
Ov 1 - - + (v'V) v = ---r. Vp + V'(vV)v, Or p 0T - - + v . V T = D L V2T
V.v=O (2)
OT s V2 T 0 r D~ where /9 L and v are the density and kinematic viscosity of the melt, Dr s and DTL a r e the heat diffusion coefficients of the solid and liquid phases, v is the velocity vector, T is the temperature, r is the time, and V and V 2 are the Hamilton and Laplace operators. This is a system of non-linear partial differential equations which are solved in explicit form only for particular cases [13]. Since the case that we are discussing at present cannot be reduced to one of the explicitly solvable cases, it is necessary to make some simplifications in order to obtain a solution for the required quantities: vx(x, y) and
as(X). In the present work, in order to simplify the system (2), the following assumptions, which are believed to be in agreement with experiment, are made. (a) The process is time independent. (b) The puddle width is constant; hence the flow can be treated as two dimensional. (c) The heat diffusion coefficients Dr s and D L are the temperature independent and D S = DTL= DT. (d) The heat loss due to the radiation from the free surfaces of the puddle is negligible. (e) The process of mass transfer is laminar. (f) In the volume of the melt pool, owing to the high velocity of the substrate surface ( 10 m s- 1 or more), and the large temperature difference between the melt and the substrate temperatures, the gradients of v and T along the y axis are much greater than along the x axis (0 v/Ox ~. Ov/Oy; OT/Ox *. O T/Oy).
133 The last assumption is a necessary condition for application of the boundary layer theory [14] which when applied to the system (2) reduces it to the following set of equations: a v,
c3,!, _
V'ax + v" ay
ay
ay ]
av,.+a,,,=o ax
(3b)
Oy aT
(3a)
O:T ay 2
aT
v,..~ + v~,yy=D, aT O-~T t, ax = D~ o f
(3c)
(3d)
with the boundary conditions y-. oo: t,, =0, T = TB y=ds:t,,=%
V,.= 0 , T = T G
y=0:;t 0T =a(T-Tsr,) ay [,=, where TB is the temperature of the melt, TG is the temperature where the melt viscosity is infinitely high in the sense of the Vogel-Fulcher-Tamman equation, TSD is the temperature of the substrate, a is the heat transfer coefficient and 2 is the thermal conductivity of the quenched material. Usually two methods are used to integrate the system of partial differential equations so obtained. The first is the explicit finite difference method [3, 15] and the second is through additional transformations to reduce the system of partial differential equations (3) to an equation or a system of equations using the one-variable method of Blasius. Although from the mathematical point of view the Blasius method is simpler to solve, it has nowadays been abandoned because the melt substrate boundary condition )c(OT/aY)ly_o = a( T - TSD) cannot be transformed to an ordinary differential equation. That is why at the present time the explicit finite difference method is used to analyse the ribbon formation processes, despite the difficulties arising from its solution and its low accuracy [15]. In order to avoid the difficulties arising from using the explicit finite difference method, we propose in this paper a method which, although based on the Blasius "similarity transformations"
takes into account the existence of thermal resistance on the material-substrate contact area. The analysis of the results obtained so far shows that the integration of the system of equations (3) with the aid of the explicit finite difference method [3, 5] leads to the interesting result that, in the melt-substrate contact area at y = 0 along the x axis, a gradient is obtained with a mean value of (gradx T)y=0 = 104 K s -j In contrast, the experimental data published recently [6] shows that the temperature along most of the contact length is constant. This difference between the theory and the experiment is probably a consequence of the application of the boundary layer theory to the system (2), where Dr L OeT/Ox 2 and Dr s O 2 T / a x 2 of the system of equations (2) are neglected as insignificantly small relative to Dr L 02T/Oy 2 and Dr s 02 T/Of-. In the case of viscous liquids near the substrate surface, however, these terms are the only ones which describe the heat transfer along the x axis and their omission leads to the appearance of a gradient along the x axis, which really does not exist or is much smaller than that calculated in refs. 3 and 5. In order to bring the model close to the experimentally observed data, the temperature of the contact area is postulated as a constant in the present work and is denoted as 7~u. Tsu is an unknown quantity as also is 6s(X), and is subject to determination in the calculation process. As mentioned before, the method proposed here is based on the Blasius similarity transformations [16]
vx=
a~'
vY= - O x ' ~=Y
(4)
q,=(xv, O,) '/:f(~) where ~o is the stream function, ~ is a new independent variable and f(~) is a function of ~. Applying these transformations to the system of equations (3) yields a new system of ordinary differential equations [7]: Oir
(vy")'+~ff
.
=0
(5a)
h"+ ~fh' = 0
(5b)
h"+ ½~h '= 0
(5c)
134 with the boundary conditions
needed, which has to be physically equivalent to the boundary condition
= 0: h = hsu •
l
~=~s.f=~s,f =l,h=l,
OT
hs'=hL '
~--, m:f'--,0, h--'0, h'--'0 where h denotes the so-called normalized temperature h = ( T B - T ) / ( T B - TG), ~s is the profile of the material frozen in the thickness 5s(X) of the puddle as a ~ variable, and hsu=(TB - Tsv)/ (TB TG). The mass balance equation (1), which describes the ribbon thickness as a function of Vx(X, y) and 5s(X) is transformed to the following relationship: -
i
1
t= f
~
\ Vsl
(6)
where the expressions
If we bear in mind that this condition has the units watts per square metre (the heat flow through a square unit), the balance of heat flows entering and leaving the volume of the puddle may be used in order to obtain the required fourth equation. In this work the balance of the powers transmitted into and out of the puddle are used instead of the heat flow balance for convenience. This is physically equivalent to the heat flow balance, as both are based on the energy conservation law. For example the heat flow entering the substrate given by Ires=a(Tsu - TSD) is related to the power transmitted from the volume of the puddle to the substrate by the following equation:
y= ~ (XDT]~/2 Ires =
\Vsl
(7)
Pre,~ ls
= a( Tsu - TSD)
Vx(X, Y) = Vsf'(~ ) obtained from Blasius transformations have been used• If we bear in mind that ~M
(9)
where s is the puddle width. According to the energy conservation law, the power balance equation obeys the following expression:
Pr~s=Pm-Pout
f f ' ( ~ ) d ~ =f(~M)--f(~s)
where f(~M) and f(~s) are substituted [14] from eqn. (6), the following expression is obtained:
''2
=a(T u- TsD)
'~ ~YY y=0
I,,
(10)
where Pin is the power transmitted to the puddle volume by the melt entering the puddle and Pout is the power transmitted out of the puddle from the material, both molten and solid, which is dragged out by the moving substrate. The power transmitted to the puddle by the molten material entering it is given by
Pin=pLCpLvisTnd which is equivalent to eqn. (1). Thus far, only the value of f(~) at ~ --, m is needed in order to define the thickness of the amorphous ribbon. The system (5) is undetermined because, of the required seven boundary conditions, we have only six as the value of hsu is undetermined. The only way to solve it in its present form is to assume that the value of hsu equals hsD, which is almost the case where there is ideal contact between the melt and the substrate [7]. Unfortunately this case does not happen in practice and is interesting only from the methodological point of view. To account for the existence of the heat transfer resistance at the melt-substrate contact area, an additional equation to system (5) is
(11)
where CpL is the specific heat capacity of the melt, d is the width of the melt flow entering the puddle and vi is the velocity of this flow. According to the mass balance equation for continuous processes, the following equations are valid:
vid = tvs (12)
\Vsl
where t is substituted from eqn. (8). Substituting eqn. (12) into eqn. (11 ), we finally obtain
[ ID ~1/2
Pin= pL CpLsTBf( °° ) l ~ ~
(13)
135 The power Pout consists of two terms: first the power Pout S transmitted from the puddle with the solidified material and second the power Po~,L transmitted with the liquid layer: Pout = Pouts + PoreL
ere = spG v,( 7; - To) x
h(~)d~ +
h(~)f'(~) d~
ts
lD \V~l (19)
(14)
Taking into account that
Finally, substituting eqn. (19) into eqn. (9) results in the required fourth equation:
P,,u,S=sp
=(hsD - hsu)
f cSr(l,y)dy
is
1)
ds
=spSC,,Sv, f T(l,y)dy
+ f h(~)f'(~)d~
(15)
is
0
(20)
and Po~,L=sp L f
Equation (20) together with the system of equations (5) yields a system of integral-differential equations:
c~,LG(I,y)T(I,y)dy
b~ OM
=sRLCpL f G(l,y)T(l,y)dy 6~
DT ,, (vf")'+~ff = 0
(21a)
h" +½fh'=0
(21b)
h" +½~h'=0
(21c)
(16)
we obtain for the full power transmitted from the puddle
a(hsD--
Pout=psCp G
t( t12
hsu)=pC,
h(~) d~
T(l,y)dy+ f vx(l,y)T(l,y)d bs + f h(~)f'(~)d~
(17)
~-
(21d)
is
where in eqns. (15)-(17) it is assumed that CL and C. s are temperature independent and that ps = p [ = p. The transformation of eqn. (17) from x and y to ~ variables according to the Blasius transformations (4) yields the expression
with the boundary conditiop ~. = O: h = hsu --- ~s: f = ~s, f ' = 1, h = 1, hs '= h L' ~
P,,ut=spCpv~[TBf(°o)-( TB- To) x f
o
f
~
\ V~l (18)
where the ~s
f d~=~s 0
oo:f'--,0, h--'0, h'-"0
This system solves the problem formulated at the beginning of the present paper. The integration of system of equations (21) yields the values of t h e functions f(~) and h(~) as well as the values of the unknown quantities ~s (the thickness of the solidified material) and hsv (the normalized substrate-material contact area temperature). Except for eqn. (21c) the system of equations so obtained cannot be solved analytically (see Appendix A). For this purpose an algorithm has been created which enables us to treat the system under discussion numerically.
r= rB--(TB-- ro)h( ) together with eqn. (7) have been used in the process. Substituting eqns. (13) and (18) in eqn. (10), we obtain
2. Mathematical procedure The algorithm used for integration of the system of eqns. (21) is based on the Runge-Kutta
136 TABLE 1
Selected materials properties
Parameter (units')
Value for the following alloys
Heat capacity Cp(J kg- ] K ~) Densityp (kg m 3) Thermal diffusioncoefficientDT (m2 s J) Viscosity % (m2s- ]) SolidificationtemperatureT~ (K) Melt temperatureTa (K) E(K) Heat transfer coefficienta (W m- 2K- ~) Substrate temperatureTso (K)
method for numerical solution of ordinary differential equations. A short description of the algorithm used is given below. (1) The first stage is the introduction of the initial data (see Table 1 ) needed for the numerical treatment of the system (2 1 ). (2) The starting parameters for integration of eqns. (21a) and (21b) are introduced: f(~s), f'(~s), /"(~s), h(~s) and h'(~s). As f"(~s) and h'(~s) are unknown, they are chosen arbitrarily and their exact value is determined during the process of computation. (3) The values of the functions f(~), f'(~), f"(~), h(~) and h'(~) for the i+ 1 step of integration are determined. (4) After every i+ 1 step the newly obtained values for f'(~), h(~) and h'(~) are tested in order to determine whether they fulfil the boundary conditions (f'(~)~0, h(~)--'0, h'(~)--'0; ~--,oo) or not. If they are fulfilled, the program passes to the next stage. If at ~ --' 00 these conditions are not fulfilled, the starting parameters f"(~s) and h'(~s) are changed and stages (3) and (4) are performed anew.
(5) In this stage, s~s
f h(~)d~ 0
~M
and
f h(~)f'(~)d~ ~s
are numerically integrated, using the calculated values for h(~), f'(~), ~M and ~s obtained in the previous stages. If the difference between the lefthand and the right-hand sides of the integral equation (2 ld) is smaller than a selected quantity, the values for f(~), f'(~), f"(~), h(~) and h'(~) are a solution of the system (21). Using the value for f(~) at ~--,oo the thickness of the ribbon is obtained with the aid of eqn. (8).
Fe4oNi4oPt4Be,
Fe,s.(P~,C7
545 7.7 × 103 5.0 × 10 -6 6 . 6 × 1 0 -9 668 1303 828 4.2 × 10 5 300
545 7,7 × 103 5,0 × 10 -6 3 , 0 × 1 0 -Ij 613 1268 4600 1.0× 10 ~ 300
TABLE 2 Values of the calculated and the experimentally obtained thicknesses as a function of I and vs Alloy
z~
l x 10 .~
t~, × 10 ~
t¢~t × 10 ~
(ms-')
(m)
(m)
(m)
Fe40Ni411Pj4B(, [8] Fe40Ni411PI4B(, [8] Fe40Ni4.Pi4B~ [8] Fe411Ni41)PI4B~,[8] Fe~oNi411Pj4B~ [8] Fe40Ni40P]4B(, [8]
15 20 30 40 50 60
5.3 4.8 3.6 3.2 2.85 2.7
56 44.5 31 24 19 16.5
56.5 44.5 29 22.5 18.5 16
Fe8.PI3C 7 [9] Fe~,P13C7 [9] fesoPi3C 7 [9]
20 30 40
3,5 3.1 3.0
37 26 20
37 24 19
(6) If this difference is larger than this quantity, the value of hsu is changed and stages (2)-(6) are performed anew. The value of hsv is changed until a value which satisfies eqn. (2 ld) is obtained. 3. Testing of the method To test the method described in the present work, two alloys--Fe40Ni40P14B 6 and FesoPI3C 7 were chosen; the physical properties and ribbon geometry dependences of these alloys have been of interest to many workers [8, 9, 1 1, 17]. All the constants used during computation are listed in Table 1. It is assumed that all the constants are temperature independent, except for the kinematic viscosity of the melt, which obeys the Vogel-Fulcher-Tammann equation for the undercooled melts: v = v 0 e x p ~ff~-E /
(22)
The results obtained with the method discussed in the present paper are compared in Table 2 with
137 the experimental results. As can be seen, the values for the ribbon thickness computed with the aid of the system of equations (21) correlate well with those experimentally obtained. It is interesting to note that for both alloys the total amount of material leaving the puddle volume is in the liquid form, i.e. the m o m e n t u m transport mechanism is dominant when rapid quenching from the melt of the amorphous metallic alloys is applied. T h e good correlation between theory and experiment is, in the present author's opinion, a consequence of the assumption that the temperature of the contact area is constant, which is in agreement with the experimental evidence [6]. T h e high accuracy of the applied method, equal to that of the R u n g e - K u t t a method, probably plays a significant role too.
4. Conclusion In the present paper a convenient method which is easy to apply is proposed to investigate the processes which take place in rapid quenching from the melt. T h e amorphous ribbon thicknesses calculated with the aid of the system of equations (21) agree well with the experimentally obtained results. W h e n this is taken into account a general conclusion can be drawn that the present method describes correctly the processes of the ribbon formation by rapid quenching from the melt. This method can be successfully used to investigate the influence of the casting parameters and the physical properties of the melt ( Tu, a and v) on the geometry of the ribbon. T h e method can also be used to assess the casting parameter values (e.g. a), which cannot be measured by standard methods. A more detailed investigation of the influence of the different technological parameters and of the physical properties of the alloy quenched on the geometry of the ribbon, using the method proposed in the present paper, will be the subject of another paper. It should be noted that, as the experimental data of Hillmann and Hiizinger [8] were obtained for the CBMS method, the values for the ribbon thickness calculated by the method discussed above are multiplied by a normalization coefficient of 0.8 7 [12] and are listed in Table 2 as tc~.
References 1 S. Kavesh, in J. J. Gilman and H. J. Leamy (eds.), Metallic (;lasses, American Society for Metals, Metals Park. Ohio, 1976, p. 36. 2 H.H. Lieberman, Mater. Sci. Eng., 43 (1980) 2(13. 3 K. Takeshita and P. H. Shingu, Trans. Jpn. Inst. Met., 24 (7)(1983) 529. 4 T. R. Antony and H. E. Cline, J. AppL Phys., 50 (1 I) (1979) 239. 5 H.A. Davies,in S. Steeb and H. Warlimont (eds.), Rapidly Quenched Metals, Vol. 1, North-Holland, Amsterdam, 1984, p. 101. 6 M. J. Tenwick and H. A. Davies, in S. Steeb and H. Warlimont (eds.), Rapidly Quenched Metals', Vol. 1, North-Holland, Amsterdam, 1984, p. 67. 7 P. den Decker and A. Drevers, in C. Hargitai, 1. Bakonyi and T. Kemeny (eds.), Proc. Conj. on Metallic Glasses: Seience and Technology, Budapest, 1980, Vol. 1, Organizing Committee of the Central Research Institute of Physics, Budapest, 1981, p. 181. 8 H. Hillmann and H. R. Hilzinger,in B. Cantor (ed.), Proc. 3rd Int. Conf on Rapidly Quenched Metals, Brighton, July 3-7, 1978, Vol. 1, Metals Society,London, 1978, p. 22. 9 J. H. Vincent and H. A. Davies, Proc. Con]i on Solidification 7eehniques it, Foundr), and Casthouse, Warwiek, 1980, The Metals Society,London, 1983, p. 153. 10 H. H. Liebermann, in F. E. Luborsky (ed.), Amorphous Metal#c Alloys, Butterworths, London, 1983, p. 38.
11 J. H. Vincent, J. G. Herbertson and H. A. Davies, in T. Masumoto and K. Suzuki (eds.), l'roc. 4th Int. Conf. on Rapidly Quenched Metals. Sendai, August 1981, Vol. 1, Japan Institute of Metals, Sendai, 1982, p. 77. 12 L. A. Anestiev and K. A. Russev, Mater. Sci. Eng., 95 (1987) 281. 13 L. D. Landau and E. M. Lifshitz, Hydrodynamics, Nauka i Izkustvo,Sofia, 1978 (in Bulgarian). 14 H. Schlichting, Bounda O' Layer Theory, McGraw-Hill, New York, 1966. 15 D. Potter, Computational Methods" in Physics, Mir, Moscow (in Russian). 16 T. Y. Na, Computational Methods in Engineering: Bounda O, Value Problems, Academic Press, New York, 1979. 17 J. Steinberg, S. Tyagi and A. E. Lord. Aeta Metall., 29 ~,1981)13(}9.
Appendix A Analytically, eqn. (2 lc) is solved by separation of the variables, which yields for h'(~) and h(~) the expressions h'(~) = h'(0) exp( - ¼~2)
(A1)
h(~) = h(0)+ h'(0) f e x p ( - ~ qz) dr/ 0
:
+
err
where err denotes the error function.
,A2,
138 With the aid of (A1) and (A2) the boundary condition at the melt-solid interface hL'(~s) = hs'(~s) becomes in analytical form hL'(~s) = {h(~s)- h(0)} exp( - ¼~s2) n i l 2 eff(~s/2 ) = ( 1 - h s v ) exp(-1 ~r1/2 eff(~s/2)
2)
on substituting for h(~s) and h(0).
(i3)
131
An Analysis of Heat and Mass Transfer and Ribbon Formation by Rapid Quenching from the Melt L. A. ANEST1EV
lnst#ute J'or Metal Science and Technology, Bulgarian Academy of Sciences, 1574 Sofia (Bulgaria) (Received June 3.1988)
Abstract
The process of ribbon formation by the singleroller technique and by rapid quenching from the melt has been studied with the aid of the boundary layer theory. An easy calculation method has been proposed for the solution of heat and mass transfer equations for the case in which the melt is rapidly quenched onto a substrate. The proposed method takes into account the viscosity change due to the temperature reduction and the heat resistance on the substrate surface. The thicknesses of the ribbon cah'ulated with the aid of the proposed method show good correlation with the experimental results'.
1. Introduction
Recently, much attention has been paid to the formation and processing of amorphous alloys because of their possible important applications [1-12]. Among the various rapid solidification techniques, the single-roller chill-block casting methods--planar flow casting (PFC) and chillblock melt spinning (CBMS), in which continuously supplied melt is cooled by a wheel rotating at a high revolution rate--are most common at present because of their relative simplicity and ability to produce a large amount of rapidly solidified material in the form of amorphous ribbon. The main technologically important characteristics of this ribbon are its width w and thickness t. Use of the PFC method solves in practice the problem of production of amorphous ribbons with a defined width, since in this case the width of the ribbon equals the length of the crucible slot. The problem of the production of ribbons with a defined thickness, however, is still unresolved as the thickness depends on the pro0921-5093/89/$3.50
cesses which take place in the formation of the ribbon. The aim of this paper is to examine the processes which occur during rapid quenching from the melt and the influence which these processes have on the ribbon formation and on the geometry of the amorphous metal ribbon. In both the PFC and the CBMS methods a liquid melt pool (puddle) (Fig. 1) forms on the wheel surface. In effect the puddle spreads to a size such that the resultant ribbon thickness and width (for CBMS) satisfy the mass balance condition. The main characteristic of the puddle is its length I, and analysis of the experimental data shows that the thickness of the ribbon strongly depends on the length of the melt pool. Taking
lout I=
=-7
Vs
-~ =X
Ires Fig. 1. Schematic diagram of the molten alloy jet and the puddle formed on the cold substrate. The thickness 6 s of the material solidified in the puddle volume, the thickness 6M of the flow boundary layer, the heat flows l~,,, l.u ~ and 1~, and their directions are shown.
© Elsevier Sequoia/Printed in The Netherlands
132
this into account and also considering the experimental evidence obtained by several workers [1, 2, 8-11], we can draw the conclusion that the main ribbon-forming processes occur inside the melt pool. As the liquid metal is brought into contact with the cold surface of the moving substrate, simultaneous heat and momentum transfer arises. Owing to the adhesion forces a thin layer of the material is dragged out from the puddle volume; this layer cools on the substrate and finally forms the amorphous ribbon. As in the present case we have a continuous process and as some of the final product can be formed within the volume of the puddle, the thickness t of the ribbon is given by 6M
t =-
1 f
j
V,~ &
vx(y,l)dy+as(l)
(1)
where v~ is the surface velocity of the substrate, 6s(l ) is the thickness of the amorphous layer solidified in the volume of the puddle, 6 M is the flow layer induced from the moving substrate and v x is the x component of the liquid velocity as a function of x and y at x = 1. So far v,(x, y) and 6s(X ) have to be known in order to define the ribbon thickness. The processes of heat and mass transfer interact very closely with each other, and it is very difficult from a mathematical point of view to obtain an explicit solution which takes into account the influence of both processes on the ribbon formation. That is why one of these processes is assumed to be the limiting process, which historically led to the concept of the socalled "thermal" and "hydrodynamical" mechanisms of ribbon formation. The first mechanism assumes that the first term in eqn. (1) is insignificantly small relative to 6s(1), i.e. the ribbon is formed within the volume of the puddle and consequently cools on the surface of the substrate. The hydrodynamical mechanism of ribbon formation on the contrary assumes that the final product is formed from the liquid layer dragged out from the puddle by the moving substrate in the form of a thin film, which freezes prior to the substrate departure. In both cases the theory leads to identical functional relationships between t and the technological parameters [1, 11, 12]. To reveal the real nature of the ribbon formation, it is necessary to
solve together the equations describing both processes which take part in the ribbon formation, namely the processes of heat and mass transfer. Mathematically the description of these two interacting processes is given by the Navier-Stokes and Fourier equations for a viscous incompressible liquid and the Fourier equation for heat transfer in the solid phase (the frozen amorphous layer)[13]:
Ov 1 - - + (v'V) v = ---r. Vp + V'(vV)v, Or p 0T - - + v . V T = D L V2T
V.v=O (2)
OT s V2 T 0 r D~ where /9 L and v are the density and kinematic viscosity of the melt, Dr s and DTL a r e the heat diffusion coefficients of the solid and liquid phases, v is the velocity vector, T is the temperature, r is the time, and V and V 2 are the Hamilton and Laplace operators. This is a system of non-linear partial differential equations which are solved in explicit form only for particular cases [13]. Since the case that we are discussing at present cannot be reduced to one of the explicitly solvable cases, it is necessary to make some simplifications in order to obtain a solution for the required quantities: vx(x, y) and
as(X). In the present work, in order to simplify the system (2), the following assumptions, which are believed to be in agreement with experiment, are made. (a) The process is time independent. (b) The puddle width is constant; hence the flow can be treated as two dimensional. (c) The heat diffusion coefficients Dr s and D L are the temperature independent and D S = DTL= DT. (d) The heat loss due to the radiation from the free surfaces of the puddle is negligible. (e) The process of mass transfer is laminar. (f) In the volume of the melt pool, owing to the high velocity of the substrate surface ( 10 m s- 1 or more), and the large temperature difference between the melt and the substrate temperatures, the gradients of v and T along the y axis are much greater than along the x axis (0 v/Ox ~. Ov/Oy; OT/Ox *. O T/Oy).
133 The last assumption is a necessary condition for application of the boundary layer theory [14] which when applied to the system (2) reduces it to the following set of equations: a v,
c3,!, _
V'ax + v" ay
ay
ay ]
av,.+a,,,=o ax
(3b)
Oy aT
(3a)
O:T ay 2
aT
v,..~ + v~,yy=D, aT O-~T t, ax = D~ o f
(3c)
(3d)
with the boundary conditions y-. oo: t,, =0, T = TB y=ds:t,,=%
V,.= 0 , T = T G
y=0:;t 0T =a(T-Tsr,) ay [,=, where TB is the temperature of the melt, TG is the temperature where the melt viscosity is infinitely high in the sense of the Vogel-Fulcher-Tamman equation, TSD is the temperature of the substrate, a is the heat transfer coefficient and 2 is the thermal conductivity of the quenched material. Usually two methods are used to integrate the system of partial differential equations so obtained. The first is the explicit finite difference method [3, 15] and the second is through additional transformations to reduce the system of partial differential equations (3) to an equation or a system of equations using the one-variable method of Blasius. Although from the mathematical point of view the Blasius method is simpler to solve, it has nowadays been abandoned because the melt substrate boundary condition )c(OT/aY)ly_o = a( T - TSD) cannot be transformed to an ordinary differential equation. That is why at the present time the explicit finite difference method is used to analyse the ribbon formation processes, despite the difficulties arising from its solution and its low accuracy [15]. In order to avoid the difficulties arising from using the explicit finite difference method, we propose in this paper a method which, although based on the Blasius "similarity transformations"
takes into account the existence of thermal resistance on the material-substrate contact area. The analysis of the results obtained so far shows that the integration of the system of equations (3) with the aid of the explicit finite difference method [3, 5] leads to the interesting result that, in the melt-substrate contact area at y = 0 along the x axis, a gradient is obtained with a mean value of (gradx T)y=0 = 104 K s -j In contrast, the experimental data published recently [6] shows that the temperature along most of the contact length is constant. This difference between the theory and the experiment is probably a consequence of the application of the boundary layer theory to the system (2), where Dr L OeT/Ox 2 and Dr s O 2 T / a x 2 of the system of equations (2) are neglected as insignificantly small relative to Dr L 02T/Oy 2 and Dr s 02 T/Of-. In the case of viscous liquids near the substrate surface, however, these terms are the only ones which describe the heat transfer along the x axis and their omission leads to the appearance of a gradient along the x axis, which really does not exist or is much smaller than that calculated in refs. 3 and 5. In order to bring the model close to the experimentally observed data, the temperature of the contact area is postulated as a constant in the present work and is denoted as 7~u. Tsu is an unknown quantity as also is 6s(X), and is subject to determination in the calculation process. As mentioned before, the method proposed here is based on the Blasius similarity transformations [16]
vx=
a~'
vY= - O x ' ~=Y
(4)
q,=(xv, O,) '/:f(~) where ~o is the stream function, ~ is a new independent variable and f(~) is a function of ~. Applying these transformations to the system of equations (3) yields a new system of ordinary differential equations [7]: Oir
(vy")'+~ff
.
=0
(5a)
h"+ ~fh' = 0
(5b)
h"+ ½~h '= 0
(5c)
134 with the boundary conditions
needed, which has to be physically equivalent to the boundary condition
= 0: h = hsu •
l
~=~s.f=~s,f =l,h=l,
OT
hs'=hL '
~--, m:f'--,0, h--'0, h'--'0 where h denotes the so-called normalized temperature h = ( T B - T ) / ( T B - TG), ~s is the profile of the material frozen in the thickness 5s(X) of the puddle as a ~ variable, and hsu=(TB - Tsv)/ (TB TG). The mass balance equation (1), which describes the ribbon thickness as a function of Vx(X, y) and 5s(X) is transformed to the following relationship: -
i
1
t= f
~
\ Vsl
(6)
where the expressions
If we bear in mind that this condition has the units watts per square metre (the heat flow through a square unit), the balance of heat flows entering and leaving the volume of the puddle may be used in order to obtain the required fourth equation. In this work the balance of the powers transmitted into and out of the puddle are used instead of the heat flow balance for convenience. This is physically equivalent to the heat flow balance, as both are based on the energy conservation law. For example the heat flow entering the substrate given by Ires=a(Tsu - TSD) is related to the power transmitted from the volume of the puddle to the substrate by the following equation:
y= ~ (XDT]~/2 Ires =
\Vsl
(7)
Pre,~ ls
= a( Tsu - TSD)
Vx(X, Y) = Vsf'(~ ) obtained from Blasius transformations have been used• If we bear in mind that ~M
(9)
where s is the puddle width. According to the energy conservation law, the power balance equation obeys the following expression:
Pr~s=Pm-Pout
f f ' ( ~ ) d ~ =f(~M)--f(~s)
where f(~M) and f(~s) are substituted [14] from eqn. (6), the following expression is obtained:
''2
=a(T u- TsD)
'~ ~YY y=0
I,,
(10)
where Pin is the power transmitted to the puddle volume by the melt entering the puddle and Pout is the power transmitted out of the puddle from the material, both molten and solid, which is dragged out by the moving substrate. The power transmitted to the puddle by the molten material entering it is given by
Pin=pLCpLvisTnd which is equivalent to eqn. (1). Thus far, only the value of f(~) at ~ --, m is needed in order to define the thickness of the amorphous ribbon. The system (5) is undetermined because, of the required seven boundary conditions, we have only six as the value of hsu is undetermined. The only way to solve it in its present form is to assume that the value of hsu equals hsD, which is almost the case where there is ideal contact between the melt and the substrate [7]. Unfortunately this case does not happen in practice and is interesting only from the methodological point of view. To account for the existence of the heat transfer resistance at the melt-substrate contact area, an additional equation to system (5) is
(11)
where CpL is the specific heat capacity of the melt, d is the width of the melt flow entering the puddle and vi is the velocity of this flow. According to the mass balance equation for continuous processes, the following equations are valid:
vid = tvs (12)
\Vsl
where t is substituted from eqn. (8). Substituting eqn. (12) into eqn. (11 ), we finally obtain
[ ID ~1/2
Pin= pL CpLsTBf( °° ) l ~ ~
(13)
135 The power Pout consists of two terms: first the power Pout S transmitted from the puddle with the solidified material and second the power Po~,L transmitted with the liquid layer: Pout = Pouts + PoreL
ere = spG v,( 7; - To) x
h(~)d~ +
h(~)f'(~) d~
ts
lD \V~l (19)
(14)
Taking into account that
Finally, substituting eqn. (19) into eqn. (9) results in the required fourth equation:
P,,u,S=sp
=(hsD - hsu)
f cSr(l,y)dy
is
1)
ds
=spSC,,Sv, f T(l,y)dy
+ f h(~)f'(~)d~
(15)
is
0
(20)
and Po~,L=sp L f
Equation (20) together with the system of equations (5) yields a system of integral-differential equations:
c~,LG(I,y)T(I,y)dy
b~ OM
=sRLCpL f G(l,y)T(l,y)dy 6~
DT ,, (vf")'+~ff = 0
(21a)
h" +½fh'=0
(21b)
h" +½~h'=0
(21c)
(16)
we obtain for the full power transmitted from the puddle
a(hsD--
Pout=psCp G
t( t12
hsu)=pC,
h(~) d~
T(l,y)dy+ f vx(l,y)T(l,y)d bs + f h(~)f'(~)d~
(17)
~-
(21d)
is
where in eqns. (15)-(17) it is assumed that CL and C. s are temperature independent and that ps = p [ = p. The transformation of eqn. (17) from x and y to ~ variables according to the Blasius transformations (4) yields the expression
with the boundary conditiop ~. = O: h = hsu --- ~s: f = ~s, f ' = 1, h = 1, hs '= h L' ~
P,,ut=spCpv~[TBf(°o)-( TB- To) x f
o
f
~
\ V~l (18)
where the ~s
f d~=~s 0
oo:f'--,0, h--'0, h'-"0
This system solves the problem formulated at the beginning of the present paper. The integration of system of equations (21) yields the values of t h e functions f(~) and h(~) as well as the values of the unknown quantities ~s (the thickness of the solidified material) and hsv (the normalized substrate-material contact area temperature). Except for eqn. (21c) the system of equations so obtained cannot be solved analytically (see Appendix A). For this purpose an algorithm has been created which enables us to treat the system under discussion numerically.
r= rB--(TB-- ro)h( ) together with eqn. (7) have been used in the process. Substituting eqns. (13) and (18) in eqn. (10), we obtain
2. Mathematical procedure The algorithm used for integration of the system of eqns. (21) is based on the Runge-Kutta
136 TABLE 1
Selected materials properties
Parameter (units')
Value for the following alloys
Heat capacity Cp(J kg- ] K ~) Densityp (kg m 3) Thermal diffusioncoefficientDT (m2 s J) Viscosity % (m2s- ]) SolidificationtemperatureT~ (K) Melt temperatureTa (K) E(K) Heat transfer coefficienta (W m- 2K- ~) Substrate temperatureTso (K)
method for numerical solution of ordinary differential equations. A short description of the algorithm used is given below. (1) The first stage is the introduction of the initial data (see Table 1 ) needed for the numerical treatment of the system (2 1 ). (2) The starting parameters for integration of eqns. (21a) and (21b) are introduced: f(~s), f'(~s), /"(~s), h(~s) and h'(~s). As f"(~s) and h'(~s) are unknown, they are chosen arbitrarily and their exact value is determined during the process of computation. (3) The values of the functions f(~), f'(~), f"(~), h(~) and h'(~) for the i+ 1 step of integration are determined. (4) After every i+ 1 step the newly obtained values for f'(~), h(~) and h'(~) are tested in order to determine whether they fulfil the boundary conditions (f'(~)~0, h(~)--'0, h'(~)--'0; ~--,oo) or not. If they are fulfilled, the program passes to the next stage. If at ~ --' 00 these conditions are not fulfilled, the starting parameters f"(~s) and h'(~s) are changed and stages (3) and (4) are performed anew.
(5) In this stage, s~s
f h(~)d~ 0
~M
and
f h(~)f'(~)d~ ~s
are numerically integrated, using the calculated values for h(~), f'(~), ~M and ~s obtained in the previous stages. If the difference between the lefthand and the right-hand sides of the integral equation (2 ld) is smaller than a selected quantity, the values for f(~), f'(~), f"(~), h(~) and h'(~) are a solution of the system (21). Using the value for f(~) at ~--,oo the thickness of the ribbon is obtained with the aid of eqn. (8).
Fe4oNi4oPt4Be,
Fe,s.(P~,C7
545 7.7 × 103 5.0 × 10 -6 6 . 6 × 1 0 -9 668 1303 828 4.2 × 10 5 300
545 7,7 × 103 5,0 × 10 -6 3 , 0 × 1 0 -Ij 613 1268 4600 1.0× 10 ~ 300
TABLE 2 Values of the calculated and the experimentally obtained thicknesses as a function of I and vs Alloy
z~
l x 10 .~
t~, × 10 ~
t¢~t × 10 ~
(ms-')
(m)
(m)
(m)
Fe40Ni411Pj4B(, [8] Fe40Ni411PI4B(, [8] Fe40Ni4.Pi4B~ [8] Fe411Ni41)PI4B~,[8] Fe~oNi411Pj4B~ [8] Fe40Ni40P]4B(, [8]
15 20 30 40 50 60
5.3 4.8 3.6 3.2 2.85 2.7
56 44.5 31 24 19 16.5
56.5 44.5 29 22.5 18.5 16
Fe8.PI3C 7 [9] Fe~,P13C7 [9] fesoPi3C 7 [9]
20 30 40
3,5 3.1 3.0
37 26 20
37 24 19
(6) If this difference is larger than this quantity, the value of hsu is changed and stages (2)-(6) are performed anew. The value of hsv is changed until a value which satisfies eqn. (2 ld) is obtained. 3. Testing of the method To test the method described in the present work, two alloys--Fe40Ni40P14B 6 and FesoPI3C 7 were chosen; the physical properties and ribbon geometry dependences of these alloys have been of interest to many workers [8, 9, 1 1, 17]. All the constants used during computation are listed in Table 1. It is assumed that all the constants are temperature independent, except for the kinematic viscosity of the melt, which obeys the Vogel-Fulcher-Tammann equation for the undercooled melts: v = v 0 e x p ~ff~-E /
(22)
The results obtained with the method discussed in the present paper are compared in Table 2 with
137 the experimental results. As can be seen, the values for the ribbon thickness computed with the aid of the system of equations (21) correlate well with those experimentally obtained. It is interesting to note that for both alloys the total amount of material leaving the puddle volume is in the liquid form, i.e. the m o m e n t u m transport mechanism is dominant when rapid quenching from the melt of the amorphous metallic alloys is applied. T h e good correlation between theory and experiment is, in the present author's opinion, a consequence of the assumption that the temperature of the contact area is constant, which is in agreement with the experimental evidence [6]. T h e high accuracy of the applied method, equal to that of the R u n g e - K u t t a method, probably plays a significant role too.
4. Conclusion In the present paper a convenient method which is easy to apply is proposed to investigate the processes which take place in rapid quenching from the melt. T h e amorphous ribbon thicknesses calculated with the aid of the system of equations (21) agree well with the experimentally obtained results. W h e n this is taken into account a general conclusion can be drawn that the present method describes correctly the processes of the ribbon formation by rapid quenching from the melt. This method can be successfully used to investigate the influence of the casting parameters and the physical properties of the melt ( Tu, a and v) on the geometry of the ribbon. T h e method can also be used to assess the casting parameter values (e.g. a), which cannot be measured by standard methods. A more detailed investigation of the influence of the different technological parameters and of the physical properties of the alloy quenched on the geometry of the ribbon, using the method proposed in the present paper, will be the subject of another paper. It should be noted that, as the experimental data of Hillmann and Hiizinger [8] were obtained for the CBMS method, the values for the ribbon thickness calculated by the method discussed above are multiplied by a normalization coefficient of 0.8 7 [12] and are listed in Table 2 as tc~.
References 1 S. Kavesh, in J. J. Gilman and H. J. Leamy (eds.), Metallic (;lasses, American Society for Metals, Metals Park. Ohio, 1976, p. 36. 2 H.H. Lieberman, Mater. Sci. Eng., 43 (1980) 2(13. 3 K. Takeshita and P. H. Shingu, Trans. Jpn. Inst. Met., 24 (7)(1983) 529. 4 T. R. Antony and H. E. Cline, J. AppL Phys., 50 (1 I) (1979) 239. 5 H.A. Davies,in S. Steeb and H. Warlimont (eds.), Rapidly Quenched Metals, Vol. 1, North-Holland, Amsterdam, 1984, p. 101. 6 M. J. Tenwick and H. A. Davies, in S. Steeb and H. Warlimont (eds.), Rapidly Quenched Metals', Vol. 1, North-Holland, Amsterdam, 1984, p. 67. 7 P. den Decker and A. Drevers, in C. Hargitai, 1. Bakonyi and T. Kemeny (eds.), Proc. Conj. on Metallic Glasses: Seience and Technology, Budapest, 1980, Vol. 1, Organizing Committee of the Central Research Institute of Physics, Budapest, 1981, p. 181. 8 H. Hillmann and H. R. Hilzinger,in B. Cantor (ed.), Proc. 3rd Int. Conf on Rapidly Quenched Metals, Brighton, July 3-7, 1978, Vol. 1, Metals Society,London, 1978, p. 22. 9 J. H. Vincent and H. A. Davies, Proc. Con]i on Solidification 7eehniques it, Foundr), and Casthouse, Warwiek, 1980, The Metals Society,London, 1983, p. 153. 10 H. H. Liebermann, in F. E. Luborsky (ed.), Amorphous Metal#c Alloys, Butterworths, London, 1983, p. 38.
11 J. H. Vincent, J. G. Herbertson and H. A. Davies, in T. Masumoto and K. Suzuki (eds.), l'roc. 4th Int. Conf. on Rapidly Quenched Metals. Sendai, August 1981, Vol. 1, Japan Institute of Metals, Sendai, 1982, p. 77. 12 L. A. Anestiev and K. A. Russev, Mater. Sci. Eng., 95 (1987) 281. 13 L. D. Landau and E. M. Lifshitz, Hydrodynamics, Nauka i Izkustvo,Sofia, 1978 (in Bulgarian). 14 H. Schlichting, Bounda O' Layer Theory, McGraw-Hill, New York, 1966. 15 D. Potter, Computational Methods" in Physics, Mir, Moscow (in Russian). 16 T. Y. Na, Computational Methods in Engineering: Bounda O, Value Problems, Academic Press, New York, 1979. 17 J. Steinberg, S. Tyagi and A. E. Lord. Aeta Metall., 29 ~,1981)13(}9.
Appendix A Analytically, eqn. (2 lc) is solved by separation of the variables, which yields for h'(~) and h(~) the expressions h'(~) = h'(0) exp( - ¼~2)
(A1)
h(~) = h(0)+ h'(0) f e x p ( - ~ qz) dr/ 0
:
+
err
where err denotes the error function.
,A2,
138 With the aid of (A1) and (A2) the boundary condition at the melt-solid interface hL'(~s) = hs'(~s) becomes in analytical form hL'(~s) = {h(~s)- h(0)} exp( - ¼~s2) n i l 2 eff(~s/2 ) = ( 1 - h s v ) exp(-1 ~r1/2 eff(~s/2)
2)
on substituting for h(~s) and h(0).
(i3)