The carbon concentration in the steel before carburization is 366.5 ppm and is initially uniform through the thickness of the steel. The atmosphere of the carburizing furnace maintains a carbon concentration of 8040.0 ppm at the surface of the steel. Calculate the time required to carburize steel so that the concentration of carbon at a depth of 88.0 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(ξ [mathjaxinline]\lt[/mathjaxinline] 0.6), use the approximation erf ξ = ξ
erf(1.0) = 0.84
erf(2.0) = 0.998
To solve this problem, we can use Fick's second law of diffusion, which states that the rate of change of concentration with respect to time is proportional to the second derivative of the concentration with respect to position.
The equation for Fick's second law is:
∂C/∂t = D * (∂^2C/∂x^2)
Where:
- C is the concentration of carbon
- t is time
- D is the diffusion coefficient of carbon in steel
- x is the distance from the surface of the steel
We are given the initial concentration of carbon (C0 = 366.5 ppm) and the carbon concentration at the surface (Cs = 8040.0 ppm).
We need to find the time required for the carbon concentration at a depth of 88.0 x 10^-2 cm (x = 88.0 x 10^-2 cm) to be half the value of the carbon concentration at the surface (Cs/2).
Let's first calculate the concentration gradient (∂C/∂x) at the surface (x = 0). We can use the approximation erf ξ = ξ when ξ < 0.6.
∂C/∂x = (Cs - C0) * erf(x/√(4Dt))
= (8040.0 ppm - 366.5 ppm) * erf(0/√(4 * 3.091 x 10^-7 cm^2/s * t))
Using the approximation erf(0) = 0, the concentration gradient becomes:
∂C/∂x = 7673.5 ppm * 0 = 0 ppm/cm
Since the concentration gradient at the surface is zero, the carbon concentration remains constant throughout the steel at time t. Therefore, we need to solve for the time t when the concentration at a depth of 88.0 x 10^-2 cm (x = 88.0 x 10^-2 cm) is equal to Cs/2.
∂C/∂x = -C0 * erf((x - 0)/√(4Dt))
= -366.5 ppm * erf((88.0 x 10^-2 cm - 0)/√(4 * 3.091 x 10^-7 cm^2/s * t))
Setting this equal to half the concentration at the surface:
0.5 * (Cs - C0) = -366.5 ppm * erf((88.0 x 10^-2 cm)/√(4 * 3.091 x 10^-7 cm^2/s * t))
Rearranging and substituting the values:
-1836.75 ppm = -366.5 ppm * erf(0.88 cm/√(4 * 3.091 x 10^-7 cm^2/s * t))
Now, we can use the given values for the error function to find erf(0.88 cm/√(4 * 3.091 x 10^-7 cm^2/s * t)).
Using the given approximation, erf(ξ) = ξ when ξ < 0.6, and the value of the error function for ξ = 1.0, we have:
-1836.75 ppm = -366.5 ppm * 0.88 cm/√(4 * 3.091 x 10^-7 cm^2/s * t)
Solving for t:
t = ((0.88 cm)^2 * 4 * 3.091 x 10^-7 cm^2/s)/(0.88 cm * 0.6)^2
t ≈ 0.04019 s
To express the result in hours, we need to convert seconds to hours:
t ≈ 0.04019 s * (1 min/60 s) * (1 h/60 min)
t ≈ 4.13 x 10^-5 h
Therefore, the time required to carburize the steel so that the carbon concentration at a depth of 88.0 x 10^-2 cm is one-half the value of the carbon concentration at the surface is approximately 4.13 x 10^-5 hours.