Kinetics of the internal nitridation of austenitic Fe-Cr-Ni-Ti alloys
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solute, it was only s u c c e s s f u l when the two r e a c t i o n f r o n t s were close together, i . e . , the p r e c i p i t a t e d oxides had n e a r l y the s a m e s o l u b i l i t y . The s i m p l i f i e d t r e a t m e n t given l a t e r by W a g n e r ~1 c o n s i d e r e d only a single solute; for p r o p e r c h a r a c t e r i z a t i o n of the m u l t i p l e i n t e r n a l n i t r i d a t i o n in this study it was found n e c e s s a r y to extend W a g n e r ' s concepts. Such a m o d i f i c a t i o n has been given by Mack, lz who t r e a t e d the case of i n t e r n a l oxidation b e n e a t h an a d v a n c i n g s u r f a c e oxide scale. In the m a t h e m a t i c a l d e v e l o p m e n t to follow it has b e e n a s s u m e d that one can s u b s t i t u t e the s l o w e r moving
I
I
CrN + T i N ! TiN SUBSCALE ONLY I UNREACTED BASE IUBSCALE I I METAL
I I
2
I I
I I
to z
I I
I b-~_
NTi
u. _~ O 2;
k F
y
I Jq •
DEPTH OF PENETRATION,x-,Fig. 1--Schematic representation of internal nitrJdation in Fe-Cr-Ni-Ti alloys. VOLUME I,JANUARY 1970-163
subscale bolically advance Using control, cally in
during double internal nitridation for a paraa d v a n c i n g s u r f a c e s c a l e , i.e., t h e r a t e o f o f b o t h s u b s c a l e s i s a s s u m e d to b e p a r a b o l i c . the above concepts, u'12 and assuming diffusion one can represent X a n d Y, s h o w n s c h e m a t i F i g . 1, a s :
X = 2 R ( D N t ) '/2
[1]
Y = (2kct) 112
[2]
erfc [x/2(DTit)l/2]t eryc 1
NTi = N~i { 1 -
where
[6]
where
R = a dimensionless
constant
D N = diffusion of nitrogen
kc
limiting solubility N~ is actually moving into the material with time, because of precipitation of chromium nitride along the second reaction front. Similarly, the relation for the concentration profile of the solute metal component in the base metal beyond point X is given by:
in the base
alloy
NTi = mole fraction
of metal
solute in solution
N~i = mole fraction
of m e t a l
solute at infinity
= a rate constant
~b = DN/DTi
t = time Taking the boundary N N = N SN
for
conditions
By assuming stoichiometric TiN and taking a mass b a l a n c e a t X (in t h e m a n n e r o f W a g n e r u a n d M a a k TM) the following relation is obtained:
as:
x = Y, t > 0
[aa] NSND~2 e x p ( _ R 2)
N N= 0 Nwi = 0
for
x->X,
for
N T i = N TOi
t > 0
[3b]
x 0
for
x>--0,
erfR
-
t =0
[4b] relation
Using the mathematical
the relation
[51
where
I~N
---
of nitrogen
in solution
NSN = s o l u b i l i t y l i m i t o f c h r o m i u m as a mole fraction
nitride
at point x
[7]
erfc(Rdp lz2)
b)/~
u'~4
1/2R(~ 1/~
to:
e x p ( R 2)
Ferf R
L
by combining
Eqs.
/k
-
\~/q
j
[8]
[1] a n d [2] o n e f i n d s :
~ kc ~~'2 = R yY
[9]
expressed which yields the relation:
NSN -Tr1'2R exp(R2)[erfR-erf(RY)]
This relation is in effect the solution for diffusion into a n i n f i n i t e s y s t e m , 13 m o d i f i e d to a c c o u n t f o r r e m o v a l of the diffusing specie through
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