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 200.5 ppm and is initially uniform through the thickness of the steel. The atmosphere of the carburizing furnace maintains a carbon concentration of 9110.0 ppm at the surface of the steel. Calculate the time required to carburize steel so that the concentration of carbon at a depth of 88.5 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) = 0.99
CHEAT!
this is from a midterm exam. It should not be posted here. Please do NOT answer
This is a question from the second midterm of 3.091x by edx. Stop cheating, it's silly and pointless.
Step 1:
(0.5*9110.0 - 9110.0)/ (200.5 - 9110.0)= erf(ξ)
Step 2:
erf(ξ < 0.6), use the approximation erf ξ = ξ
Step 3:
t= d^2/4*D*ξ^2
d=88.5 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. Fick's second law states that the rate of change of concentration with respect to time is proportional to the rate of change of concentration with respect to distance.
The equation for Fick's second law is:
∂C/∂t = D(∂²C/∂x²)
Where:
- ∂C/∂t is the rate of change of concentration with respect to time,
- ∂²C/∂x² is the rate of change of concentration with respect to distance, and
- D is the diffusion coefficient.
In this case, we want to find the time it takes for the carbon concentration at a depth of 88.5 x 10^(-2) cm to reach half the value of the carbon concentration at the surface. Let's denote the time as t and the distance as x.
Given:
Initial carbon concentration, C0 = 200.5 ppm
Carbon concentration at the surface, Cs = 9110.0 ppm
Carbon concentration at a depth of x, Cx = Cs/2 = 4555.0 ppm
Diffusion coefficient, D = 3.091 x 10^(-7) cm²/s
We need to find the time required to reach Cx at a depth x. We can rearrange Fick's second law to solve for time:
∂C/∂t = D(∂²C/∂x²)
∂C/∂t = D(Cx - C0)/x²
Integrating both sides of the equation with respect to t and from 0 to t and with respect to x from 0 to x gives:
∫[0,t]∂C/∂t dt = D∫[0,x](Cx - C0)/x² dx
Integrating gives:
C(t) - C0 = D(Cx - C0)/x²t
Simplifying, we get:
t = (x²(Cx - C0))/(D(C(t) - C0))
Now we have an equation to calculate the required time, t. However, the carbon concentration at any given time depends on the error function erf. Approximating the error function to simplify, we have:
C(t) = Cs - (Cs - C0)erf(x/(2√(Dt)))
Using this approximation, we can substitute C(t) into the previous equation:
t = (x²(Cx - C0))/(D((Cs - C0) - (Cs - C0)erf(x/(2√(Dt)))))
Plug in the given values into this equation and solve for t. The time required to carburize the steel will be given in hours.
Remember to convert the units of x from cm to cm² and D from cm²/s to cm²/h for the final answer.