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Reply to “comments on a mathematical model describing the effects of microvoids upon the diffusion of hydrogen in iron and steel”
Scripta METALLURGICA
Vol. 8, pp. 769-772, 1974 Printed in the United States
Pergamon
Press,
REPLY TO "COMMENTS ON A MATHEMATICAL MODEL DESCRIBING THE EFFECTS OF MICROVOIDS UPON THE DIFFUSION OF HYDROGEN
IN IRON AND STEEL".
David Allen-Booth Sheffield Polytechnic Department of Applied Physics Sheffield, England (Received May 13, 1974)
In the original paper the following equation deduced by NcNabb and Foster is presented:2H=
I
DH
"~
~H+HX~
~
- -
7
This is the exact equation of diffusion for any hydrogen trapping model. When applied to the void theory both H and H* are functions of the equilibrium pressure and consequently functions of each other. It is not possible to rewrite the equation in terms of H alonge, although it can be rewritten in terms of either P or H*. In either case a complex, non linear second order differential equation is produced, which is unstable when numerical analysis is attempted. However, by using a combination of iteration and net analysis numerical solutions have been obtained and a description of the method, together with the results, will be published shortly, as indicated on the original paper. The approximation,
that
H H-~ = r
, a constant,
was based upon
the fact that for the range of hydrogen gas pressure up to 5 x 109 -2 dynes cm , the variation of H with H*, whilst extremely complex in theory, approximates reasonably to a linear one when plotted as shown in figure I. Such a pressure range is of great technological interest as it would correspond to a total hydrogen concentration of about 6ml/1OOg for a steel with a void volume concentration of 0.01~. In the treatment proposed by Kass, however, the simple Seivert Law Variation of interestitial hydrogen concentration with pressure is employed, namely:H = K~-~ Compared with equation (I) in the original paper:-
769
[nc
770
REPLY TO "COMMENTS ON EFFECTS OF MICROVOIDS UPON DIFFUSION OF H IN F e "
VoI.
8,
No.
7
The two equations only converge at low pressure w h e n the gas b e h a v i o u r a p p r o x i m a t e s to that of an ideal gas. At higher pressures, there is c o n s i d e r a b l e divergence, as can be seen from figure I.
L)N A~ ~ P P ~ ) M ~ T , oN
7
.21~
APP~ox)~Tno~ k,VF"
//
Fac~u~
Vol.
8,
No.
7
REPLY TO "COMMENTS ON E F F E C T S OF M I C R O V O I D S UPON D I F F U S I O N OF II IN F e "
Whilst the s o l u t i o n p r e s e n t e d by Kass, is u s e f u l at low h y d r o g e n c o n c e n t r a t i o n s or for large void concentrations, at h i g h e r pressures is likely to be less accurate than the linear a p p r o x i m a t i o n presented in the original paper. The use of this a p p r o x i m a t i o n canrlot, of course, be held to ckamge the f u n d a m e n t a l p h y s i c a l basis of the model, nor is it n e c e s s a r y at this s~age to introduce two models of d i f f u s i o n w h e n the void model is capable of e x p l a i n i n g b o t h the w i d e l y r e p o r t e d evolution of h y d r o g e n w i t h ~ a p p a r e n t l y constsz~t d i f f u s i o n rate and the v a r i a t i o n of the apparent d i f f u s i o n rate during evolution. That this is so can be seen clearly f r o m fmgure 2 where the v a r i a t i o n in e v o l u t i o n curve shape has been plotted as a f u n c t i o n of void volume concentration. The shape p a r a m e t e r has been m e a s u r e d :in terms of the ratio of the time for 25~J of the h y d r o g e n initially present to be r e m a i n i n g to the time for' oct'responding 50~ to be remaining. For a fixed d i f f u s i o n rate process, this should always be const
771
772
REPLY TO "COMMENTS ON EFFECTS OF MICROVOIDS UPON DIFFUSION OF H IN F e "
Vol.
8,
No.
z O O
>o tl
®
t
r
6
000"~
~ug
ol
,4" If
6
c~ 2
O Vt il.
A
Z
6
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7
Vol. 8, pp. 769-772, 1974 Printed in the United States
Pergamon
Press,
REPLY TO "COMMENTS ON A MATHEMATICAL MODEL DESCRIBING THE EFFECTS OF MICROVOIDS UPON THE DIFFUSION OF HYDROGEN
IN IRON AND STEEL".
David Allen-Booth Sheffield Polytechnic Department of Applied Physics Sheffield, England (Received May 13, 1974)
In the original paper the following equation deduced by NcNabb and Foster is presented:2H=
I
DH
"~
~H+HX~
~
- -
7
This is the exact equation of diffusion for any hydrogen trapping model. When applied to the void theory both H and H* are functions of the equilibrium pressure and consequently functions of each other. It is not possible to rewrite the equation in terms of H alonge, although it can be rewritten in terms of either P or H*. In either case a complex, non linear second order differential equation is produced, which is unstable when numerical analysis is attempted. However, by using a combination of iteration and net analysis numerical solutions have been obtained and a description of the method, together with the results, will be published shortly, as indicated on the original paper. The approximation,
that
H H-~ = r
, a constant,
was based upon
the fact that for the range of hydrogen gas pressure up to 5 x 109 -2 dynes cm , the variation of H with H*, whilst extremely complex in theory, approximates reasonably to a linear one when plotted as shown in figure I. Such a pressure range is of great technological interest as it would correspond to a total hydrogen concentration of about 6ml/1OOg for a steel with a void volume concentration of 0.01~. In the treatment proposed by Kass, however, the simple Seivert Law Variation of interestitial hydrogen concentration with pressure is employed, namely:H = K~-~ Compared with equation (I) in the original paper:-
769
[nc
770
REPLY TO "COMMENTS ON EFFECTS OF MICROVOIDS UPON DIFFUSION OF H IN F e "
VoI.
8,
No.
7
The two equations only converge at low pressure w h e n the gas b e h a v i o u r a p p r o x i m a t e s to that of an ideal gas. At higher pressures, there is c o n s i d e r a b l e divergence, as can be seen from figure I.
L)N A~ ~ P P ~ ) M ~ T , oN
7
.21~
APP~ox)~Tno~ k,VF"
//
Fac~u~
Vol.
8,
No.
7
REPLY TO "COMMENTS ON E F F E C T S OF M I C R O V O I D S UPON D I F F U S I O N OF II IN F e "
Whilst the s o l u t i o n p r e s e n t e d by Kass, is u s e f u l at low h y d r o g e n c o n c e n t r a t i o n s or for large void concentrations, at h i g h e r pressures is likely to be less accurate than the linear a p p r o x i m a t i o n presented in the original paper. The use of this a p p r o x i m a t i o n canrlot, of course, be held to ckamge the f u n d a m e n t a l p h y s i c a l basis of the model, nor is it n e c e s s a r y at this s~age to introduce two models of d i f f u s i o n w h e n the void model is capable of e x p l a i n i n g b o t h the w i d e l y r e p o r t e d evolution of h y d r o g e n w i t h ~ a p p a r e n t l y constsz~t d i f f u s i o n rate and the v a r i a t i o n of the apparent d i f f u s i o n rate during evolution. That this is so can be seen clearly f r o m fmgure 2 where the v a r i a t i o n in e v o l u t i o n curve shape has been plotted as a f u n c t i o n of void volume concentration. The shape p a r a m e t e r has been m e a s u r e d :in terms of the ratio of the time for 25~J of the h y d r o g e n initially present to be r e m a i n i n g to the time for' oct'responding 50~ to be remaining. For a fixed d i f f u s i o n rate process, this should always be const
771
772
REPLY TO "COMMENTS ON EFFECTS OF MICROVOIDS UPON DIFFUSION OF H IN F e "
Vol.
8,
No.
z O O
>o tl
®
t
r
6
000"~
~ug
ol
,4" If
6
c~ 2
O Vt il.
A
Z
6
[
|
|
!
,4-
o~
[
O
7