Thermodynamic prediction of the ae 3 temperature of steels with additions of Mn, Si, Ni, Cr, Mo, Cu
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T H E c o n s t r u c t i o n of T T T and CCT d i a g r a m s and the a c c u r a t e c o n t r o l of the heat t r e a t m e n t of alloy s t e e l s r e q u i r e s a knowledge of the Ae3 t e m p e r a t u r e as a function of composition. B e c a u s e these e q u i l i b r i u m p a r a m e t e r s a r e difficult to m e a s u r e it is u s u a l to quote i n s t e a d the Ac3 or Ar3 t e m p e r a t u r e s (i.e., the apparent transition temperatures from ferrite ~ austenite on heating and cooling, r e s p e c t i v e l y . T h e s e transitions are usually determined dilatometrically. A n d r e w s ~ and Kunitake and Ohtani 2 have developed l i n e a r r e g r e s s i o n f o r m u l a e for Ac3 t e m p e r a t u r e s f r o m a v o l u m i n o u s file of e x p e r i m e n t a l data. A n d r e w s ~ and Grange 3 also have developed Ae3 r e g r e s s i o n f o r m u l a e f r o m the l a t t e r ' s e x p e r i m e n t a l Ae3 data. 3 A l l these r e g r e s s i o n f o r m u l a e a r e employed on a r a t h e r l a r g e scale. D u r i n g the past few y e a r s a l a r g e a m o u n t of t h e r m o d y n a m i c data on F e - X and F e - C - X s y s t e m s have been a c c u m u l a t e d making it completely f e a s i b l e to c a r r y out a r i g o r o u s t h e r m o d y n a m i c c a l c u l a t i o n of the Ae3 t e m p e r a t u r e in m u l t i c o m p o n e n t a l l o y s . While a p p r o x i mate methods of doing this were outlined some t i m e ago by B r i o n et al, 4 and A n d r e w s s we p r o c e e d h e r e with the r i g o r o u s t h e r m o d y n a m i c e x p r e s s i o n s . THE THERMODYNAMIC RELATIONS
-3"
RT aoc? -v RT
-
.
Xo~
i n -
= In
x? xy
z
+
In
~oa [1]
v
+ In ' ~ , i = 1 to 7
[2]
~Y l,
J. S. KIRKALDYis Professor, McMaster University, Hamilton, Ontario, Canada, where E. A. BAGANISwas a former Graduate Student. Manuscript submitted February 1, 1977. METALLURGICAL TRANSACTIONS A
In T0 = - 1/2
~ EikXiX k i,/~ - 1
[3]
and
In Vi = ~
QkXk
[4]
k=l
w h e r e the m a t r i x of Wagner i n t e r a c t i o n p a r a m e t e r s , elk, is s y m m e t r i c . W r i t i n g Eqs. [1] and [2] as AOG~ - 3" - - +
I n X ~ + I n y oy = I n X o ~ + i n y ~
RT
[5]
and AOGa -3"
RzT
+ In
Xi3' + in Yi3" =
in X~
+
In Yi
and s u b s t i t u t i n g Eqs. [3] and [4], Eqs. [5] and [6] can for c o m p u t a t i o n a l p u r p o s e s be w r i t t e n in the f o r m So(T, e~k, X~) - ~7o(E~k, X~)
The g o v e r n i n g equations a r e the equality of c h e m i cal p o t e n t i a l s between f e r r i t e and a u s t e n i t e for each of the components in the s y s t e m . H e r e we d e s i g n a t e F e as 0 and C, Mn, Si, Ni, Cr, Mo and Cu as 1 to 7, r e s p e c t i v e l y . Sulfur and phosphorous a r e not taken into account. The eight e q u a l i t i e s can be w r i t t e n as a~
where A~ -V = 0G7 _ ~ ' the X ' s a r e mole f r a c t i o n s and the V's a r e the a c t i v i t y coefficients. F o r r e l a t i v e l y s m a l l alloy additions we can use T a y
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