An FCC iron-carbon alloy initially containing 0.20 wt% C is carburized at an elevated temperature and in an atmosphere wherein the surface carbon concentration is maintained at 1.0 wt%.

If after 50 h the concentration of carbon is 0.35 wt% at a position 3.5 mm below the surface, determine the temperature at which the treatment was carried out.

Fick's second Law posits that the rate of change of concentration of diffusing species is directly proportional to the second derivative of the concentration.

Using the expression of the Fick's second Law:

[tex]\mathbf{\frac{C_x-C_o}{C_s-C_o} = 1- erf(\frac{x}{2\sqrt{Dt} })}[/tex]

where;

[tex]C_o[/tex] = initial concentration

[tex]C_x[/tex] = the depth of the concentration

[tex]C_s[/tex] = surface concentration

[tex]erf(\frac{x}{2\sqrt{Dt} })}[/tex] = Gaussian error function.

Let variable z be used for the expression of the Gaussian error function. [tex]erf(\frac{x}{2\sqrt{Dt} })}[/tex]

Then, from the above equation: replacing [tex]C_x[/tex] with 0.35 ; [tex]C_o[/tex] with 0.2 and [tex]C_s[/tex] with 1.0; we have:

[tex]\mathbf{\frac{0.35-0.2}{1.0-0.2} = 1- erf( z)}[/tex]

erf (z) = 0.8125

we obtain the error function value close to 0.8125 from the error function table and we did the interpolation to obtain the exact value of variable corresponding to 0.8125.

The table below shows the tabular form of the error function value close to 0.8125 .

Value for z Value for erf (z)

0.9 0.797

z 0.8125

0.950 0.8209

From above; we can find the value of variable corresponding to the error function 0.8125 .

i.e

[tex]\frac{z-0.9}{0.95-0.9} =\frac{0.8125-0.797}{0.8209-0.797}[/tex]

z = 0.932

However, the temperature dependence relation for the diffusion coefficient D can be expressed as:

[tex]z = \frac{x}{\sqrt{Dt} }[/tex]

where;

z = 0.932

x = 3.5 mm = 0.0035 m

t = 50 h = 180000 sec

[tex]0.932 = \frac{0.35}{2\sqrt{D*180000} }[/tex]

D = [tex]1.958*10^{-11} m^2/s[/tex]

Finally, the temperature T at which the treatment is carried is out is calculated as:

[tex]\mathbf{D=D_o \ exp \ (-\frac{Q_d}{RT}) }[/tex]

From the table ‘Diffusion data’, we obtain the values of temperature-independent pre exponential and activation energy for diffusion of carbon in FCC Fe.

[tex]D_o = 2.3*10^{-5} \ m^2/s[/tex]

[tex]Q_d = 148, 000 \ J/mol[/tex]

Replacing all values needed for the above equation; we have:

[tex]1.958*10^{-11}= (2.3*10^{-5})exp(\frac{-148,000}{(8.31)T})[/tex]

[tex]8.51*10^{-7}=exp(\frac{-17,810}{T})[/tex]

[tex]In(8.51*10^{-7})=(\frac{-17,810}{T})[/tex]

-13.977 = -17,810/T

T = -17,810/ - 13.977

T = 1274.24 K

Hence, the temperature T at which the treatment is carried is out is 1274.24 K