Page 181 - Materials Science and Engineering An Introduction

5.5 Factors that Influence Diffusion • 153

3
1
Thus, from Figure 5.8, at 1/T 1 0.8 10 (K) , log D 1 12.40, whereas for 1/T 2 1.1
3
1
10 (K) , log D 2 15.45, and the activation energy, as determined from the slope of the line
segment in Figure 5.8, is
log D 1 - log D 2
Q d = -2.3R
£ 1 1 §
-
T 1 T 2
# -12.40 - (-15.45)
= -2.3(8.31 J/mol K)c -3 -1 -3 -1 d
0.8 * 10 (K) - 1.1 * 10 (K)
= 194,000 J/mol = 194 kJ/mol
Now, rather than try to make a graphical extrapolation to determine D 0 , we can obtain a
more accurate value analytically using Equation 5.9b, and we obtain a specific value of D (or
log D) and its corresponding T (or 1/T) from Figure 5.8. Because we know that log D 15.45
1
3
at 1/T 1.1 10 (K) , then
1
Q d
log D 0 = log D + a b
2.3R T
-1
(194,000 J/mol)(1.1 * 10 -3 [K] )
= -15.45 + #
(2.3)(8.31 J/mol K)
= -4.28
2
5
2
Thus, D 0 10 4.28 m /s 5.2 10 m /s.
DESIGN EXAMPLE 5.1

Diffusion Temperature–Time Heat Treatment Specification

The wear resistance of a steel gear is to be improved by hardening its surface. This is to be
accomplished by increasing the carbon content within an outer surface layer as a result of car-
bon diffusion into the steel; the carbon is to be supplied from an external carbon-rich gaseous
atmosphere at an elevated and constant temperature. The initial carbon content of the steel is
0.20 wt%, whereas the surface concentration is to be maintained at 1.00 wt%. For this treatment
to be effective, a carbon content of 0.60 wt% must be established at a position 0.75 mm below
the surface. Specify an appropriate heat treatment in terms of temperature and time for tem-
peratures between 900 and 1050 C. Use data in Table 5.2 for the diffusion of carbon in g-iron.
Solution
Because this is a nonsteady-state diffusion situation, let us first employ Equation 5.5, using the
following values for the concentration parameters:
C 0 = 0.20 wt% C
C s = 1.00 wt% C
C x = 0.60 wt% C
Therefore,

C x - C 0 0.60 - 0.20 x
= = 1 - erfa b
C s - C 0 1.00 - 0.20 21Dt
and thus,
x
0.5 = erfa b
21Dt