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For a steel alloy it has been determined that a carburizing heat treatment of 11.3 h duration at Temperature T1 will raise the carbon concentration to 0.44 wt% at a point 1.8 mm from the surface. A separate experiment is performed at T2 that doubles the diffusion coefficient for carbon in steel.
Estimate the time necessary to achieve the same concentration at a 4.9 mm position for an identical steel and at the same carburizing temperature T2.
Answer:
the required time to achieve the same concentration at a 4.9 is 83.733 hrs
Explanation:
Given the data in the question;
treatment time t₁ = 11.3 hours
Carbon concentration = 0.444 wt%
thickness at surface x₁ = 1.8 mm = 0.0018 m
thickness at identical steel x₂ = 4.9 mm = 0.0049 m
Now, Using Fick's second law inform of diffusion
[tex]x^2[/tex] / Dt = constant
where D is constant
then
[tex]x^2[/tex] / t = constant
[tex]x^2_1[/tex] / t₁ = [tex]x^2_2[/tex] / t₂
[tex]x^2_1[/tex] t₂ = t₁[tex]x^2_2[/tex]
t₂ = t₁[tex]x^2_2[/tex] / [tex]x^2_1[/tex]
t₂ = ([tex]x^2_2[/tex] / [tex]x^2_1[/tex])t₁
t₂ = [tex]([/tex] [tex]x_2[/tex] / [tex]x_1[/tex] [tex])^2[/tex] × t₁
so we substitute
t₂ = [tex]([/tex] 0.0049 / 0.0018 [tex])^2[/tex] × 11.3 hrs
t₂ = 7.41 × 11.3 hrs
t₂ = 83.733 hrs
Therefore, the required time to achieve the same concentration at a 4.9 is 83.733 hrs