Page 176 - Materials Science and Engineering An Introduction

148 • Chapter 5 / Diffusion

0.80 wt% at a position 0.5 mm below the surface? The diffusion coefficient for carbon in iron
at this temperature is 1.6 10 11 m /s; assume that the steel piece is semi-infinite.
2
Solution
Because this is a nonsteady-state diffusion problem in which the surface composition is held
constant, Equation 5.5 is used. Values for all the parameters in this expression except time t are
specified in the problem as follows:
C 0 = 0.25 wt% C
C s = 1.20 wt% C
C x = 0.80 wt% C
-4
x = 0.50 mm = 5 * 10 m
2
D = 1.6 * 10 -11 m >s
Thus,
-4
C x - C 0 0.80 - 0.25 (5 * 10 m)
= = 1 - erf £ §
2
C s - C 0 1.20 - 0.25 22(1.6 * 10 -11 m /s)(t)
62.5 s 1/2
0.4210 = erf a b
1t
Tutorial Video
We must now determine from Table 5.1 the value of z for which the error function is 0.4210.
An interpolation is necessary, as

z erf(z)
0.35 0.3794
z 0.4210
0.40 0.4284

z - 0.35 0.4210 - 0.3794
=
0.40 - 0.35 0.4284 - 0.3794
or

z = 0.392
Therefore,
62.5 s 1/2
= 0.392
1t
and solving for t, we find
62.5 s 1/2 2
t = a b = 25,400 s = 7.1 h
0.392

EXAMPLE PROBLEM 5.3

Nonsteady-State Diffusion Time Computation II
The diffusion coefficients for copper in aluminum at 500 C and 600 C are 4.8 10 14 and 5.3
2
10 13 m /s, respectively. Determine the approximate time at 500 C that will produce the same
diffusion result (in terms of concentration of Cu at some specific point in Al) as a 10-h heat
treatment at 600 C.