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 299.0 ppm and is initially uniform through the thickness of the steel. The atmosphere of the carburizing furnace maintains a carbon concentration of 8655.0 ppm at the surface of the steel. Calculate the time required to carburize steel so that the concentration of carbon at a depth of 20.5 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 < 0.6), use the approximation erf ξ = ξ
erf(1.0) = 0.84
erf(2.0) = 0.998
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* 8655.0 - 8655.0 )/ ( 299.0 - 8655.0 )= erf(ξ)
Step 2:
erf(ξ < 0.6), use the approximation erf ξ = ξ
Step 3:
t= d^2/4*D*ξ^2
d=20.5 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)
I'm getting:
d = 0.205
D = 0.0000003091
0.205 / 0.0000003091 = 663215.78777094791329666774506632
663215.78777094791329666774506632 / 60 / 60 = 184.22660771415219813796326251842
I'm putting 184 in but it's wrong...any ideas where my math is going wrong?
184 hours = 7.6 days ==> 1 week? The steel industry will collapse if this is the answer. LOL! Just by seeing the figure one can tell this does not make sense.
Someone please help!
To calculate the time required to carburize the steel, we can use Fick's second law of diffusion, which relates the rate of change of concentration with respect to time and distance in a diffusing system.
The equation is given by: dC/dt = D * (d^2C/dx^2)
Where:
- dC/dt is the rate of change of carbon concentration with respect to time,
- D is the diffusion coefficient of carbon in steel,
- d^2C/dx^2 is the second derivative of carbon concentration with respect to distance.
In this case, we want to find the time required for the carbon concentration at a depth of 20.5 x 10^-2 cm to reach half the value of the carbon concentration at the surface.
Let's assume x = 0 represents the surface of the steel, and x = 20.5 x 10^-2 cm represents the depth we're interested in.
First, we need to find the concentration gradient at the surface:
dC/dx at x = 0 = (Carbon concentration at the surface - Carbon concentration in the steel before carburization) / Depth of the steel
dC/dx at x = 0 = (8655.0 ppm - 299.0 ppm) / 20.5 x 10^-2 cm
Next, we can rewrite Fick's second law equation as: dC/dt = D * (d^2C/dx^2)
Since the carbon concentration is initially uniform through the thickness of the steel, d^2C/dx^2 is zero at x = 0.
Now, let's solve Fick's second law equation for the time (t):
dC/dt = D * (d^2C/dx^2)
Integrating both sides with respect to time, we get:
∫dC = D * (∫d^2C/dx^2) dt
Integrating, we have:
C - C0 = D * (dC/dx) * t
Using the concentration gradient at the surface (dC/dx at x = 0) as the initial derivative, we can write:
C - C0 = D * (dC/dx at x = 0) * t
Now, we need to find the value of C when it is half the value of the carbon concentration at the surface.
C = (Carbon concentration at the surface) / 2
Plugging the values into the equation, we have:
(Carbon concentration at the surface) / 2 - C0 = D * (dC/dx at x = 0) * t
Now, we can solve for t:
t = ((Carbon concentration at the surface) / 2 - C0) / (D * (dC/dx at x = 0))
Given the values:
Carbon concentration at the surface = 8655.0 ppm
C0 = 299.0 ppm
dC/dx at x = 0 = (8655.0 ppm - 299.0 ppm) / 20.5 x 10^-2 cm
D = 3.091 x 10^-7 cm^2/s
We can substitute the values to find the time required for carburization in hours.
Please note that the calculation involves a few steps, and you would need to perform the actual numerical computation to get the precise value of t.