You wish to increase the carbon content of a slab of steel by exposing it to a carburizing atmosphere at elevated temperature. The carbon concentration in the steel before carburization is 223.0 ppm and is initially uniform through the thickness of the steel. The atmosphere of the carburizing furnace maintains a carbon concentration of 6125.0 ppm at the surface of the steel. Calculate the time required to carburize steel so that the concentration of carbon at a depth of 84.5 x 10-2 cm is one half the value of the carbon concentration at the surface. The diffusion coefficient of carbon in steel is 3.091 x 10-7 cm2/s at the carburizing temperature. Express your answer in hours.

Question

You wish to increase the carbon content of a slab of steel by exposing it to a carburizing atmosphere at elevated temperature. The carbon concentration in the steel before carburization is 223.0 ppm and is initially uniform through the thickness of the steel. The atmosphere of the carburizing furnace maintains a carbon concentration of 6125.0 ppm at the surface of the steel. Calculate the time required to carburize steel so that the concentration of carbon at a depth of 84.5 x 10-2 cm is one half the value of the carbon concentration at the surface. The diffusion coefficient of carbon in steel is 3.091 x 10-7 cm2/s at the carburizing temperature. Express your answer in hours.

DATA: Error Function Values
erf(ξ \lt 0.6), use the approximation erf ξ = ξ
erf(1.0) = 0.84
erf(2.0) = 0.998

Answer 1

To calculate the time required to carburize the steel, we need to use Fick's second law of diffusion. The equation is:

∂C/∂t = D (∂^2C/∂x^2)

where:
- C is the concentration of carbon in the steel
- t is time
- x is the distance from the surface of the steel
- D is the diffusion coefficient of carbon in steel

Since the initial carbon concentration is uniform through the thickness of the steel, we can assume that C(x=0) = 223.0 ppm and C(x=L) = 6125.0 ppm, where L is the depth of the steel (also the distance from the surface to the desired depth).

To simplify the equation, we can use a dimensionless variable ξ = x/L and a dimensionless time τ = t/(L^2/D).

The equation becomes:

∂C/∂τ = (∂^2C/∂ξ^2)

Now we can use the given error function values to solve the equation. We need to find the value of τ when C(ξ=0.845) = 0.5*C(ξ=1), where C(ξ=1) is the carbon concentration at the surface.

Using the approximation for the error function (erf(ξ = 1) = 0.84), we can calculate the concentration at the surface:
C(ξ=1) = C(surface) = 6125.0 ppm * 0.84 = 5145.0 ppm

To find τ, we can rearrange the equation and integrate twice:

d^2C/dξ^2 = dC/dτ
dC/dξ = √(2/π) * dC/dτ
Integrating again:
C(ξ) = √(2/π) * ∫dC/dτ dξ + A1*ξ + A2

Using the boundary conditions C(ξ=0) = 223.0 ppm and C(ξ=1) = 5145.0 ppm, we can solve for A1 and A2. A1 = (C(surface) - C(0))/ξ, and A2 = C(0) - A1*ξ.

Finally, we substitute ξ = 0.845, C(0.845) = 0.5*C(surface), and solve for τ:

0.5*C(surface) = √(2/π) * ∫dC/dτ (A1*ξ + C(0) - A1*ξ) dξ

Solving this equation will give us τ, which we can then multiply by L^2/D to obtain the time required in hours. As this involves complex calculations, it is recommended to use a numerical method or simulation software to solve the equation and obtain the final result.