T K S K
KFKI- 1982-32
L , GRÁNÁSY T . KEMÉNY
COMMENT ON THE KAVESH'S MODEL OF RIBBON FORMATION DURING
MELT-SPINNING
Hungarian Academy of‘ Sciences
C EN TR A L RESEARCH
IN S TITU TE FOR P H Y S IC S
B U D A PEST
KFKI-1982-32
COMMENT ON THE KAVESH'S MODEL OF RIBBON FORMATION DURING MELT-SPINNING
L. Gránásy and T. Kemény
Central Research Institute for Physics H-1525 Budapest 114, P.O.B. 49, Hungary
HU ISSN 0368 5330 ISBN 963 371 918 6
ABSTRACT
The Kavesh's model of ribbon formation during the melt-spinning process has been discussed. It has been shown that a good agreement between the
theoretical and the experimental results for both the variation of the ribbon dimensions and the behaviour of the melt puddle can be obtained only after the correction of the original equations.
АННОТАЦИЯ
Нами изучалось предложенное Кавешем описание лентообразующего процесса в течение "melt-spinning". Показано, что теоретические и экспериментальные результаты изучения изменения размера ленты и поведения расплавленной капли совпадают только после коррекции оригинальных уравнений.
KI VONAT
Megvizsgáltuk a "melt-spinning" eljárás során fellépő szalagképződési folyamat Kavesh által javasolt leírását. Megmutattuk, hogy a szalagméretek változására és az olvadéktócsa viselkedésére vonatkozó elméleti és kísérleti eredmények csak az eredeti egyenletek korrekciója után hozhatók összhangba.
Several phe n o m e n o l o g i c a l relations are available for the dep e n d e n c e of the ribbon geometry on the t e c h n o l o g i c a l pa r a m e t e r s d u r i n g the melt-spinning process [1-4]. The m o s t o f ten quoted
re l a t i o n s h i p s are o b t a i n e d from the thermal t r a n sport c o n t r o l l e d r i b b o n formation m o d e l of Kavesh [5]:
- 1 q° ' ^ Q ° ’ ^^
SR = ^ З Л 5 and WR “ c" Г 0 Л 5 (la-b>
v o v o
w h e r e JR and w R are the average thickness and the w i d t h of the ribbon, respectively, Q is the v o l u metric rate of m e l t flow, V o is the surface v e l o c i t y of the cooling substrate and c" is a p r o p o r t i o n a l i t y constant independent of the casting parameters, Q and V . These expre s s i o n s are in a fair agreement w i t h the exper i m e n t a l results [3,5-7]. However, this agreement is s u r p r i s ing because the d e r i v a t i o n of Eqs. (la,b) is based on a melt p u d d l e behav i o u r w h i c h is in clear c o n t r a d i c t i o n w i t h the e x p e r i m e n t a l findings [6]. W e will show that the m a t h e m a t i c a l l y cor r e c t t r e a tment of the K a v e s h ' s m o del changes s i g n i f i c a n t l y the r e l a tions be t w e e n the r ibbon dimensions and the casting parameters.
Impinging to the surface of the m o v i n g substrate the m e l t jet forms a puddle (see Fig. 1), under w h i c h the solid layer thickens according to the relation:
«.<?> = cQ (^-)m (2)
о
w h e r e 6 is the thickness of the solid layer, c and m are the
s J о
parameters c h a r a c t e r i s t i c to the m e c h a n i s m of solidification, w h i l e £ is the length m e a s u r e d from the front edge of the p uddle
in the d i r e ction of the m o v e m e n t of the substrate surface.
- 2 -
The c r o s s - s e c t i o n geometry of the ribbon is determined by the form of the puddle base. It is reasonable to assume that the latter can be d e s c r i b e d b y a function con t a i n i n g two parameters:
w h e r e l is the length and w is the w i d t h of the puddle, w h i c h is s u p p o s e d to be equal to that of the ribbon w = w R . The x - y plane c o r r e
sponds to the surface of the substrate, w h i l e the origo is fixed to the midp o i n t of the puddle base, and the Fig. 1. The melt puddle
formed by the melt jet on the surface of the substrate a/ front view b/ lateral view
x-axis is in the d i r e c t i o n of the m o v e m e n t of the substrate.
C o m b i n i n g Eq. (2) w i t h Eq. (3) the thickness m e a s u r e d at point у a long the w i d t h of the r i b b o n can be g i ven as:
6R (y) = 6S U = 2x(yj) (w / 2 } m
(4)
T a k i n g into account the m a s s flo w balance, the v o l u m e t r i c rate of flo w in the jet and in the ribbon is equal:
3
(5)
w h e r e <5 is the a verage r i b b o n thickness d e f i n e d by:
w/2
(6)
w h i l e ő0 (y) has to be t aken from Eq. (4).
The equation of K a v e s h for the average r i b b o n thickness (Eq. (4) of [5]) d o e s not c o n t a i n the 2/w n o r m a l i z a t i o n factor, w h i c h makes 6 d i m e n s i o n a l l y incorrect. This short a g e in 2/w can be follo w e d all along his c a l c u l a t i o n s (see E q s . (5-17) of [5]).
We w i l l show that it res u l t s in a sign i f i c a n t c h a n g e of the c a s t i n g parameter d e p e n d e n c e of the r i b b o n dimensions.
T o determine the c o n n e c t i o n b e t w e e n the cas t i n g parameters (Q and V ) and the ribbon d i m e n s i o n s (6 and w ) w e need a further
О Jc\ H
relation, namely that of the de p e n d e n c e of the form of the puddle upon the values of Q and V Q :
In the frame of the above p h e n o m e n o l o g i c a l f o r m u l a t i o n the p r o p e r ties of the m a t e r i a l are r e p r e s e n t e d by C Q ,m and Eq. (7). If these
informations are g i v e n we c a n predict the r i bbon g e o m e t r y for a g i v e n technological p a r a m e t e r set (Q and V Q ) .
In his w o r k K a v e s h a p p l i e d c q and m v alues d e r i v e d from a t h e o r e t i c a l model r efered to as heat t r a n s p o r t c o n t r o l l e d ribbon formation. He supposed a p u d d l e b e h a v i o u r d e s c r i b e d by the f o l l o w ing fo r m of Eq. (7) :
w h i c h means, that the p uddle base area is i n d e p e n d e n t of Q and V Q . It mus t be m e n t i o n e d tha t this s u p p o s i t i o n is in clear c o n t r a d i c t i o n with the e x p e r i m e n t a l o b s e r v a t i o n s of b o t h Hill m a n n and H i l z inger and V i n c e n t et al. [6,7]. The y found that under cons t a n t Q the p u d d l e length d e c r eased for incre a s e d V . Taking into account that under such conditions an increase in V cannot
i = M w ; Q , V o ) (7)
w a ■
2 2 ~ const (8)
о
4
enlarge the width of the puddle, Eq. (8) p r e s cribes a variation of l opposite to the observed case. The original expressions obtained by Kavesh for the ribbon d i m e nsions are the following:
Q
1- m
2- m
V2-m
and w
R C
ti 42-m 1-m V 2-m
о
(9a,b)
w h i c h give Eq. (la,b) w i t h the parameters of the hea t transport c o n t rolled ribbon formation (m=0.67, [5]).
In the following short calc u l a t i o n w e intend to show how the proper n o r m a l i z a t i o n of 6 changes the results of Kavesh.
F i r s t we recall a special form of the e q u a t i o n for the v o l u metric flow balance proposed by him:
c" w 2 m m = Q Eq. (13) of [5]
The proper n o r m a l i z a t i o n contributes and e x t r a 2/w factor. It can be seen by s u bstituting Eqs. (4,6) and Eq. (8) to Eq. (5) that:
2c"(w V Q )1_m = Q (10)
As this relation should hold for every V v a lue for a given Q
-1 °
this expression requires w ^ Vq . C o m b i n i n g this r esult with Eq. (5) we can realize that 6 turns out to be e n t i r e l y inde- pe n d e n t of Vq .* This w a y we have shown that starting from the u n p h ysical assumption csf Eq. (8) the correct equations lead to relations c ontradicting the experiments. It means that the
m i s s i n g 2/w factor is the only reason w h y the u n p h ysical puddle b e h a v i o u r leads to acceptable casting p a r a m e t e r d e p e ndence of the r i b b o n dimensions w i t h i n the frame of the m o d e l of Kavesh.
*The independence of <5_ of V is a straight forward consequence of E q . (8) irrespectively of the value of m, as it can be seen from the form of Eq. (10) .
5
Let us test the correct e q u a tions by using a m o r e realistic rela t i o n for the desc r i p t i o n of the puddle b e h a v i o u r e.g. by the simplest a p proximation suggested by V incent et al. [7], which d e s c ribes the trends well:
l
w K, ( I D
w h e r e is a d i m e n s ionless constant, independent of Q , Vq ,w and A. A more deta i l e d expression for the average ribbon thickness can be presented by substituting Eq. (4) into Eq. (6):
T T. # £ t m
6 R ” K o ° о (\ Г>
О
(12)
w h e r e K Q = / f m (c)dc is a di m e n s i o n l e s s constant, independent о
of Q , Vq ,w and A. Eva l u a t i o n of the v o l umetric flow b a l a n c e by inserting Eq. (11) and Eq. (12) into Eq. (5) yields the following formulae for the ribbon dimensions:
= К
m
^m+1 2m Vm+1о
and w_ = —1 Q ‘ К
m + 1 1-m .m+l
(13a,b)
w h ere K = Ko c qK^. These e x p ressions coincide w i t h those derived by V i n c e n t et al. [7]. A c c o r d i n g to this fact the theoret i c a l l y d erived limiting values of m lead to the following results:
a/ Ideal cooling, m=0.5 [8]
Q0 -33 б л, У---
R v 0.67
0.67
and W R * (14a,b)
b/ N e w t o n i a n cooling, m = l [9]:
О 0 '5
5 ^ У--
R V w R ^ Q0.5
and (15a,b)
6
demonstrating that choosing an a p p ropriate value of m (e.g.
m = 0.7) the r e l a tions following from E q s . (13a,b) are able to describe the a p p roximate casting p a r a m e t e r dependence of the ribbon geometry.
ACKNOWLEDGEMENT
We thank dr. I. V incze for the crit i c a l reading of the m a n u s c r i p t .
REFERENCES
[1] H.H. Liebermann, C.D. Graham, IEEE Trans. Mag. M A G - 1 2 , 921 /1976/
[2] H.H. Liebermann, Mater. Sei. Eng. 4_3, 203 /1980/
13] S.J.B. Charter, D.R. Mooney, R. Cheese, B. Cantor, J. Mater.
Sei. 15, 2658 /1980/
[4] S. Takayama, T.Oi, J. Appl. Phys. 50, 4962 /1979/
[5] S. Kavesh, "Metallic Glasses", ed. J.J. Gilman, H.J. Leamy, /1976/ Metals Park, Ohio, The A m e r i c a n Society for Metals, p. 36
[6] H.H. Hillmann. H.R. Hilzinger, "Rapidly Quen c h e d Metals III", ed. B. Cantor, /The Metals Society, London, 1978/ V o l . 1, p. 22
[7] J.H. Vincent, H .A . Davies, J.G. Herbertson, Proc. Symp. on Continuous Cas t i n g of Small Cross-Sections, Pittsburgh 1980, in the press a n d J.H. Vincent, J.G. Herbertson, H.A. Davies, Proc. 4th Inti. Conf. Rapidly Q u e n c h e d Metals, Sendai, Japan 1981, in the press
[8] H.S. Carslaw, J.C. Jaeger, C o n d u c t i o n of Heat in Solids, Oxford U n i v e r s i t y Press, Oxford U.K. /1959/
[9] P.H. Shingu, R. O z a k i , Met. Trans. A, 6A, 33 /1975/
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