Kinetics of the diffusion controlled peritectic reaction during solidification of iron-carbon-alloys
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taken to e x i s t at the i n t e r f a c e s , that is the c o n c e n t r a t i o n s c* and c 5 as well as c* and c L a r e the c o n c e n t r a t i o n s for the 5/V or v / l i q u i d e q u i l i b r i a as given by the phase d i a g r a m , Fig. i . Diffusion a c r o s s the Yphase is g o v e r n e d by
a--t-8c=
[1]
D ~X_,2_,82C xl(t) < x < xz(t)
The b o u n d a r y conditions at the i n t e r f a c e s depend on t i m e b e c a u s e of the cooling. The cooling b e h a v i o r is known f r o m e x p e r i m e n t a l l y d e t e r m i n e d cooling c u r v e s .
,5o[ -~ 1500-
C ~
It
"~ 1450-
14000
0.2
0.4 0.6 0.8 1.0 carbon content c (wt. %)
1.2
Z4
Fig. 1--Iron-carbon phase diagram used in the present computations.
a)
rp
~
~
-~_.c.~
/
.
..
.
rL
-.-."-'. :-L /2
~0~,,,0o!em.~",.).:...i..i..i
.
.
..
.
.o
.
.
.
*
.
..
9
.
.
.
.
.
.
c (x, t) 0
C,~ C4~
Cz*
Ct
L/2 Fig. 2--Solidification model used (a) position of volume element in mushy zone (b) magnified volume element with coneentration profile. VOLUME 6A, JANUARY 1975-235
c
=
c*(T(t)),
x
=
x,(t)
[2]
c
=
c*(T(t)),
x
=
x2(t)
[3]
The unknown p o s i t i o n s of the m o v i n g boundaries are to be d e t e r m i n e d by the r e l a t i o n s h i p s e x p r e s s i n g the m a t e r i a l balance a c r o s s the interface as follows:
dx,
-(c*
c 6) dt
-
(CL -- C*) ~
-
_ xz(t) d e 6
--37-- + D \ a x / x ,(t)
[4]
) \ Ox/xz(t
[5]
= (--~ -- Xz) ~
c6 c*
)
-0"6068
[6]
k~, = c~._~* = 0.3"/58 + ( 5 . 6 2 7 . 1 0 - 4 -
3.981.10-'~T)
[7]
and the t e m p e r a t u r e dependence of c* and c* is e x p r e s s e d as =
2202'00-
~'lliquid
interface
~ 100. 00"
.T
eL
C*~(T)
[11]
_ 14.75.o;
2 -
260"
Eqs. [1] to [5] d e s c r i b e the p r o g r e s s i o n of the p e r i t e c t i c r e a c t i o n c o r r e s p o n d i n g to the s p e c i f i c cooling rate T(t). Above the p e r i t e c t i c t e m p e r a t u r e , the s o l i d ification f o l l o w s v i r t u a l l y the c o u r s e of equilibrium f r e e z i n g , as v e r i f i e d in a r e c e n t publication. 4 For subsequent computations, the partition c o e f f i c i e n t s k ~ and k2 are d e r i v e d f r o m the c o r r e s p o n d i n g equilibrium l i n e s of the phase d i a g r a m . ~'~ k,-
k
w h e r e L is in ~ and Of in s e e . Of is defined as the s o l -
a(ac'~
+
p e r t m e n t a l l y for a s t e e l with 0.6 pet carbon and 2 pct m a n g a n e s e s and is r e l a t e d to the local solidification t i m e Of as
60" .^
~,/~
interface
20.
1C-p~ (T - 1392)
[8]
O~
o'2
-
--
o:,
- -
&
do
,o
dimenM~less position in volume element L-~
c*(T)
= 1526.3 -
T
1 6.9
[9]
with Cp = c/(1494~
= c*(1404~
= 0.1826
Fig. 4 - - P r o g r e s s i o n of "//6 and "//liquid i n t e r f a c e s as a funcLion of time during solidification of i t slab with 0.39 pct c a r bon as computed in the p r e s e n t work.
[10]
T h e c ' s a r e in w t p e t a n d T i s in ~
The dendrite a r m spacing has been d e t
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