"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.

Question

"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.

Answer 1

To calculate the time required for carburization of pure iron (Fe) to achieve a carbon content of 0.9% at a depth of 0.1 mm below the surface, we can use Fick's second law of diffusion.

Fick's second law of diffusion states that the rate of change of concentration with respect to time is proportional to the rate of change of concentration with respect to position. Mathematically, it can be represented as:

∂C/∂t = D * (∂^2C/∂x^2)

Where:
∂C/∂t is the rate of change of concentration with respect to time,
D is the diffusion coefficient,
∂^2C/∂x^2 is the rate of change of concentration with respect to position.

In this case, we are given the diffusion coefficient D as 10^-10 m^2/s, the desired concentration at the surface Cs as 1.2%, and the desired concentration at a depth of 0.1 mm as Cd as 0.9%.

To find the time required, we need to determine the distance over which the carbon needs to diffuse. In this case, it is the depth of 0.1 mm, which can be converted to meters as 0.1 * 10^-3 m.

We can rearrange Fick's second law of diffusion to solve for time:

∂C/∂t = D * (∂^2C/∂x^2)
∂C/∂t = D * (Cs - C) / x^2

Integrating this equation, we get:

∫ (∂C/∂t) dt = D ∫ (Cs - C) / x^2 dx

Integrating both sides, the left side becomes just time (t), and the right side becomes:

t = D * ∫ (Cs - C) / x^2 dx

Now, we can substitute the given values:

D = 10^-10 m^2/s
Cs = 1.2%
C = 0.9%
x = 0.1 * 10^-3 m

t = (10^-10 m^2/s) * ∫ ((1.2% - 0.9%) / (0.1 * 10^-3 m)^2) dx

Evaluating the integral:

t = (10^-10 m^2/s) * ((1.2% - 0.9%) / (0.1 * 10^-3 m)^2) * x

Plugging in the values:

t = (10^-10 m^2/s) * ((1.2% - 0.9%) / (0.1 * 10^-3 m)^2) * (0.1 * 10^-3 m)

Calculating this expression,

t ≈ 2.4 seconds.

Therefore, the time required for the carburization process is approximately 2.4 seconds.