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 359.5 ppm and is initially uniform through the thickness of the steel. The atmosphere of the carburizing furnace maintains a carbon concentration of 6695.0 ppm at the surface of the steel. Calculate the time required to carburize steel so that the concentration of carbon at a depth of 38.0 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 359.5 ppm and is initially uniform through the thickness of the steel. The atmosphere of the carburizing furnace maintains a carbon concentration of 6695.0 ppm at the surface of the steel. Calculate the time required to carburize steel so that the concentration of carbon at a depth of 38.0 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.

DATA: Error Function Values
erf(î < 0.6), use the approximation erf î = î
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. This law relates the rate of change of the concentration of a diffusing species to the diffusion coefficient and the second derivative of the concentration with respect to distance.

The equation for Fick's second law in one dimension is given by:

∂C/∂t = D * (∂²C/∂x²)

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

In this problem, we are interested in finding the time required for the carbon concentration at a depth of 38.0 x 10^-2 cm to be half the concentration at the surface. Let's denote the surface concentration as Cs and the concentration at depth as Cd.

According to the problem, Cs = 6695.0 ppm and Cd = 0.5 * Cs = 0.5 * 6695.0 ppm.

Since the initial carbon concentration of the steel before carburization is uniform through its thickness (359.5 ppm), we can assume that the initial condition at t=0 is a uniform concentration of 359.5 ppm throughout the steel.

We can use the error function to solve the boundary condition at the surface. The error function is defined as:

erf(î) = (1/√π) * ∫(from 0 to î) e^-x² dx

where î is the dimensionless distance given by:

î = x / √(4 D t)

To solve for t, we need to find the dimensionless distance î when the concentration at depth Cd is reached.

We can rearrange the equation for î as:

î = x / √(4 D t)

Solving for t:

t = (x²) / (4 D (î²))

Substituting the given values:
x = 38.0 x 10^-2 cm = 0.38 cm
D = 3.091 x 10^-7 cm²/s

Let's calculate î:
î = x / √(4 D t)
î = 0.38 cm / √(4 * 3.091 x 10^-7 cm²/s * t)

Now, we can solve for t using the approximation given for the error function (erf î ≈ î):

t = (x²) / (4 D (î²))
t = (0.38 cm)² / (4 * 3.091 x 10^-7 cm²/s * (0.38 / √(4 * 3.091 x 10^-7 cm²/s * t))²)
t = (0.38 cm)² / (4 * 3.091 x 10^-7 cm²/s * (0.38)² / (4 * 3.091 x 10^-7 cm²/s * t))
t = t / (0.38 / (4 * 3.091 x 10^-7 cm²/s))

Simplifying further:
1 = 4 * 3.091 x 10^-7 cm²/s * t / 0.38
t = 0.38 / (4 * 3.091 x 10^-7 cm²/s)

Finally, let's calculate the value of t:

t = 0.38 / (4 * 3.091 x 10^-7 cm²/s)
t ≈ 309844.5 seconds

To convert this to hours, we divide by 3600 seconds/hour:

t ≈ 86.07 hours

Therefore, the time required to carburize the steel so that the carbon concentration at a depth of 38.0 x 10^-2 cm is one half the value of the carbon concentration at the surface is approximately 86.07 hours.