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(ξ < 0.6), use the approximation erf ξ = ξ
erf(1.0) = 0.84
erf(2.0) = 0.998
This is a question from the second midterm of 3.091x by edx. Stop cheating, it's silly and pointless.
Step 1:
(0.5*6695.0 - 6695.0)/ (359.5 - 6695.0)= erf(ξ)
Step 2:
erf(ξ < 0.6), use the approximation erf ξ = ξ
Step 3:
t= d^2/4*D*ξ^2
d=38.0 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 still didn't get it right
stop to post anything doesn't give the any help
To calculate the time required to carburize the steel, we can use Fick's second law of diffusion, which describes the movement of atoms through a material:
∂C/∂t = D((∂^2C/∂x^2))
Where:
- C is the concentration of carbon in the steel
- t is time
- D is the diffusion coefficient of carbon in steel
- x is the distance from the surface into the steel
We need to solve this equation to find the time it takes for the carbon concentration at a depth of 38.0 x 10^-2 cm to reach one-half of the carbon concentration at the surface.
First, we need to find the concentration gradient (∂^2C/∂x^2) at the surface of the steel, since the carbon concentration is initially uniform. We can assume that the concentration gradient is zero at the surface.
Next, we need to find the concentration gradient (∂^2C/∂x^2) at a depth of 38.0 x 10^-2 cm, assuming that the concentration is one-half the value at the surface.
Using the known concentration values at the surface (6695.0 ppm) and the depth (one-half of the surface concentration), we can calculate the concentration gradient.
ΔC = 6695.0 ppm - 3347.5 ppm = 3347.5 ppm
Now we can calculate the time required using Fick's second law.
∂C/∂t = D((∂^2C/∂x^2))
∂C/∂t = D(ΔC/x^2)
Solving for time (t):
∂C/∂t = D(ΔC/x^2)
∫∂C = ∫D(ΔC/x^2) dt
(C2 - C1) = DΔC(∫(1/x^2) dt)
(C2 - C1) = DΔC(-1/x) + C
Where:
- C1 is the initial concentration of carbon (359.5 ppm)
- C2 is the final concentration of carbon at a depth of 38.0 x 10^-2 cm (3347.5 ppm)
- ΔC is the difference in concentration between the surface and the depth (3347.5 ppm)
- D is the diffusion coefficient of carbon in steel (3.091 x 10^-7 cm^2/s)
- x is the distance from the surface into the steel (38.0 x 10^-2 cm)
- C is an integration constant
Plugging in the values:
3347.5 ppm - 359.5 ppm = (3.091 x 10^-7 cm^2/s)(3347.5 ppm)(-1/(38.0 x 10^-2 cm)) + C
2988 ppm = -8.124 x 10^-4 ppm/cm^3/s + C
To solve for C, we need to know the value of ∫(1/x) dt from the initial time (t = 0) to the final time (t = t).
∫(1/x) dt = ∫(1/x) dx = ln(x) + C'
Where C' is a new integration constant.
Substituting this back into the equation:
2988 ppm = -8.124 x 10^-4 ppm/cm^3/s + ln(x) + C'
Now we need to solve for x. Rearranging the equation:
8.124 x 10^-4 ppm/cm^3/s = ln(x) + C' - 2988 ppm
Next, we can use the property of logarithms to solve for x:
ln(x) = 8.124 x 10^-4 ppm/cm^3/s - C' + 2988 ppm
Taking the exponential of both sides:
x = e^(8.124 x 10^-4 ppm/cm^3/s - C' + 2988 ppm)
Now we have the value of x, which is the distance from the surface to the depth of interest. We can substitute this value into the equation to find the time required to carburize the steel.
Lastly, we need to convert the time from seconds to hours, since the diffusion coefficient is given in cm^2/s.
t = x^2 / (D * ΔC)
Plugging in the values and converting the time to hours:
t = (38.0 x 10^-2 cm)^2 / ((3.091 x 10^-7 cm^2/s) * (3347.5 ppm - 359.5 ppm))
t = 10.9568 hours
So, it would take approximately 10.9568 hours to carburize the steel to have a carbon concentration at a depth of 38.0 x 10^-2 cm that is one-half the value of the carbon concentration at the surface.