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 206.5 ppm and is initially uniform through the thickness of the steel. The atmosphere of the carburizing furnace maintains a carbon concentration of 6475.0 ppm at the surface of the steel. Calculate the time required to carburize steel so that the concentration of carbon at a depth of 17.0 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 206.5 ppm and is initially uniform through the thickness of the steel. The atmosphere of the carburizing furnace maintains a carbon concentration of 6475.0 ppm at the surface of the steel. Calculate the time required to carburize steel so that the concentration of carbon at a depth of 17.0 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

Answer 1

To calculate the time required for carburizing the steel, we need to use Fick's second law of diffusion, which relates the change in concentration with time and diffusion coefficient. The equation is as follows:

∂C/∂t = D(∂²C/∂x²)

Where:
∂C/∂t is the change in concentration with time,
D is the diffusion coefficient of carbon in steel, and
∂²C/∂x² is the second derivative of concentration with respect to the position.

Given that the carbon concentration at the surface is 6475.0 ppm and the carbon concentration at a depth of 17.0 x 10^-2 cm is half the surface concentration, we can assign the following values:

C0 = 6475.0 ppm (surface concentration)
C1 = 0.5 * C0 (concentration at depth)

Now, we need to find the time required (t) for the carbon concentration to change from C0 to C1 at a distance of x = 17.0 x 10^-2 cm.

First, let's find the concentration difference ∆C (C0 - C1):
∆C = C0 - C1 = C0 - 0.5 * C0 = 0.5 * C0

Next, we need to find the distance traveled by carbon (x) using Fick's first law of diffusion:
x = √(4 * D * t)

Rearranging the equation, we solve for time (t):
t = x^2 / (4 * D)

Now, we can substitute the known values into the equation and calculate:

x = 17.0 x 10^-2 cm = 0.17 cm
D = 3.091 x 10^-7 cm^2/s

t = (0.17 cm)^2 / (4 * 3.091 x 10^-7 cm^2/s)
t ≈ 3.303271 hours

Therefore, it would take approximately 3.303271 hours to carburize the steel so that the carbon concentration at a depth of 17.0 x 10^-2 cm is half the value of the carbon concentration at the surface.