Application of commercial computer codes to modeling the carburizing kinetics of alloy steels
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0 = -- (-J,) Ot Ox
their concentration gradients are zero), because the effect of the carbon gradient is too small to cause longrange diffusion of substitutional atoms. In this case, the only property needed for the diffusion equation is D~I, the diffusivity of carbon in the steel of interest. The concentration dependence of D~I has been characterized by several authors for binary Fe-C systems Is-HI but not for alloys. For the case of a steel that forms carbides (i.e, that becomes multiphase) during carburizing, it is possible for significant gradients to develop in the solute concentrations. These gradients are caused by carbide precipitation which depletes the matrix of solute, as illustrated in Figure 1. Therefore, more than one term in Eq. [2] must be considered. Equation [2] can be simplified by writing it in terms of an effective diffusivity defined by the equation: Jl
= _ n e f f Oc~
"--'l
n- I
OC'~
o~ff = j =l
[41
Oc7 Ox
Equation [4] can be simplified further by assuming local equilibrium and no long-range diffusion by substitutional atoms. Under these conditions, there is only one independent concentration variable. Accordingly, each matrix carbon concentration is associated with a specific total carbon concentration as illustrated in Figure 2 for a hypothetical Fe-M-C system. With only one degree of freedom, Eq. [4] can be simplified to
n--1 Of? Lllneff Dlt + E Dij j=2 Oc~ c~ =
n--I
Jl = - - E D l j j=l
Ocj
[5]
Equation [5] reflects that D~ ff h a s a unique value for each carbon concentration and that it is a function of both kinetic, D u, and phase diagram, Oct~Oct, factors.
[2]
OX
in which component 1 is carbon and components 2, 3 . . . . . n - 1 refer to substitutional alloying elements present, for example, Ni, Cr, Mo, etc. The effect of other interstitial elements is assumed to be negligible. For the case of a steel that is single phase during carburizing, it can be assumed that the concentration of substitutional atoms remains uniform during carburizing (i.e.,
J.E. MORRAL, Professor, and B.M. DUPEN, Graduate Student, are with the Department of Metallurgy and Institute of Materials Science, University of Connecticut, Storrs, CT 06269-3136. C.C. LAW, Senior Research Engineer, is with Pratt & Whitney, United Technologies Corporation, East Hartford, CT 06108. Manuscript submitted November 27, 1991. METALLURGICAL TRANSACTIONS A
[3]
Combining Eqs. [2] and [3] gives t121
[l]
in which x is distance and J~ is the flux of carbon. Assuming that the volume fraction of precipitate is small enough to be neglected, the flux of carbon in a multicomponent steel can be written generally as 17l
0X
tot
~/+
MxC
c2
/4
Solute concentration
0
X
Fig. 1 - - T h e total and matrix concentrations, c2~ and c~", of a hypothetical solute in a carburized zone where M~C carbides form. The carbides draw solute from the matrix, reducing the concentration that is soluble and creating solute concentration gradients in the matrix. VOLUME 23A, JULY 1992--2069
C1 m
Carbon concentration
_
C1
Y tot
c
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