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Valentin [98]
3 years ago
6

An FCC iron-carbon alloy initially containing 0.20 wt% C is carburized at an elevated temperature and in an atmosphere wherein t

he 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.
Engineering
1 answer:
weeeeeb [17]3 years ago
5 0

Answer:

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

Explanation:

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:

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

where;

C_o = initial concentration

C_x = the depth of the concentration

C_s = surface concentration

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

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

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

\mathbf{\frac{0.35-0.2}{1.0-0.2} = 1- erf( z)}

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

\frac{z-0.9}{0.95-0.9} =\frac{0.8125-0.797}{0.8209-0.797}

z = 0.932

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

z = \frac{x}{\sqrt{Dt} }

where;

z = 0.932

x = 3.5 mm = 0.0035 m

t = 50 h = 180000 sec

0.932 = \frac{0.35}{2\sqrt{D*180000} }

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

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

\mathbf{D=D_o \ exp \ (-\frac{Q_d}{RT}) }

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

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

Q_d = 148, 000 \ J/mol

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

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

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

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

-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

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