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 235.0 ppm and is initially uniform through the thickness of the steel. The atmosphere of the carburizing furnace maintains a carbon concentration of 7345.0 ppm at the surface of the steel. Calculate the time required to carburize steel so that the concentration of carbon at a depth of 62.5 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.
Step 1:
ξ= (0.5 * carbon concentration at the surface of the steel - carbon concentration at the surface of the steel)/ (carbon concentration in the steel before carburization - carbon concentration at the surface of the steel)
(0.5* 7345.0 - 7345.0)/ ( 235.0 - 7345.0)= erf(ξ)
Step 2:
erf(ξ < 0.6), use the approximation erf ξ = ξ
Step 3:
t= d^2/4*D*ξ^2
d=62.5 x 10-2 cm
D=3.091 x 10-7 cm2/s
Step 4:
t in sec -> hours
Conversion seconds to hours is total/(60x60)
To calculate the time required to carburize the steel, we can use Fick's second law of diffusion, which relates the concentration profile of a diffusing species to the diffusion coefficient and time.
Fick's second law of diffusion is given by the equation:
∂C/∂t = D * (∂^2C/∂x^2)
Where:
∂C/∂t is the rate of change of carbon concentration with time
D is the diffusion coefficient of carbon in steel
∂^2C/∂x^2 is the second derivative of carbon concentration with respect to distance
Assumptions:
1. One-dimensional diffusion (only changes in the x-direction)
2. Steady-state diffusion (constant carbon concentration at the surface)
3. Negligible carbon concentration change in the y and z directions
To solve this problem, we need to find the time it takes for the carbon concentration at a depth of 62.5 x 10^-2 cm to reach half the value of the carbon concentration at the surface.
Let's define some variables:
Cs = Carbon concentration at the surface (7345.0 ppm)
Cd = Carbon concentration at the desired depth (62.5 x 10^-2 cm) (unknown)
C0 = Initial carbon concentration (235.0 ppm)
D = Diffusion coefficient (3.091 x 10^-7 cm^2/s)
t = Time (unknown)
First, we need to find the concentration gradient (∂C/∂x) between the surface and the desired depth:
∂C/∂x = (Cs - Cd) / x
= (7345.0 ppm - (1/2) * Cs) / (62.5 x 10^-2 cm)
Next, we can plug this into Fick's second law of diffusion and solve for the time (t):
∂C/∂t = (D * (∂^2C/∂x^2))
∂C/∂t = D * (∂^2C/∂x^2)
∂C/∂t = (D * (∂/∂x(∂C/∂x)))
∂C/∂t = (D * (∂/∂x((Cs - Cd) / x)))
Assuming Cs is much greater than Cd, and Cd is much greater than C0, we can approximate (∂C/∂t) as 0. Therefore, we have:
0 = D * (∂/∂x((Cs - Cd) / x))
Integrating both sides with respect to x gives:
0 = D * ln(Cs - Cd) - D * ln(Cs - C0)
D * ln(Cs - Cd) = D * ln(Cs - C0)
ln(Cs - Cd) = ln(Cs - C0)
Cs - Cd = Cs - C0
Simplifying:
Cd = C0
Substituting the given values:
Cd = 235.0 ppm
Therefore, to reach half the value of the carbon concentration at the surface, the carbon concentration at a depth of 62.5 x 10^-2 cm needs to be 235.0 ppm.
No time is required to reach Cd = C0, as it is already the initial concentration at that depth.