Carburization" of pure iron (Fe) is carried out at 950∘ C. It is desirable to achieve a carbon content of 0.9% at a depth of 0.1 mm below the surface. A constant supply of carbon at the surface maintains a surface concentration of 1.2%. Assuming the diffusivity of carbon in Fe is 10−10 m2/s at this temperature; calculate the time (in seconds) required for this process.
To calculate the time required for carburization of pure iron, we need to consider the diffusion of carbon atoms into the material.
The carburization process involves the diffusion of carbon atoms from the surface into the iron with the goal of achieving a carbon content of 0.9% at a depth of 0.1 mm below the surface. We are given that the surface concentration of carbon is maintained at 1.2%.
We can use Fick's second law of diffusion to solve this problem. The equation is as follows:
dC/dt = D * d^2C/dx^2
where dC/dt is the rate of change of carbon concentration with respect to time, D is the diffusivity of carbon in iron, d^2C/dx^2 is the second derivative of carbon concentration with respect to depth.
In this case, we need to find the time required for the carbon concentration to reach 0.9% at a depth of 0.1 mm. Let's solve this step by step:
1. Convert the depth of 0.1 mm to meters:
0.1 mm = 0.1 * 10^-3 m = 1 * 10^-4 m
2. Calculate the concentration gradient at the desired depth:
d^2C/dx^2 = (0.009 - 0.012) / (1 * 10^-4) = -3 * 10^4
3. Plug in the given values to the diffusion equation:
-3 * 10^4 = (10^-10) * d^2C/dt
4. Rearrange the equation to solve for d^2C/dt:
d^2C/dt = -3 * 10^4 * 10^10 = -3 * 10^6
5. Integrate the equation with respect to time:
∫(d^2C/dt) dt = ∫(-3 * 10^6) dt
Integrating both sides gives us:
dC/dt = -3 * 10^6t + C1
6. Apply the initial condition that at t = 0, C = 1.2%:
1.2 = -3 * 10^6 * 0 + C1
C1 = 1.2
7. Substitute the value of C1 back into the equation:
dC/dt = -3 * 10^6t + 1.2
8. Integrate both sides again to solve for C(t):
∫(dC/dt) dt = ∫(-3 * 10^6t + 1.2) dt
Integrating both sides gives us:
C(t) = (-3 * 10^6 / 2) * t^2 + 1.2t + C2
9. Apply the initial condition that at t = 0, C = 0:
0 = (-3 * 10^6 / 2) * 0^2 + 1.2 * 0 + C2
C2 = 0
10. Substitute the values of C1 and C2 back into the equation:
C(t) = (-3 * 10^6 / 2) * t^2 + 1.2t
11. Now, set C(t) equal to the desired carbon content of 0.9%:
0.009 = (-3 * 10^6 / 2) * t^2 + 1.2t
12. Rearrange the equation to solve for t:
(-3 * 10^6 / 2) * t^2 + 1.2t - 0.009 = 0
At this point, we can solve this quadratic equation for t using the quadratic formula. The values obtained will be the time required for the carburization process.