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 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) = 0.998
CHEAT!
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)
Thanks mate!!!
To calculate the time required for carburization, we can use Fick's second law of diffusion. The equation is as follows:
δC/δt = D * δ²C/δx²
Where:
- δC/δt is the rate of change of carbon concentration with respect to time,
- D is the diffusion coefficient of carbon in steel,
- δ²C/δx² is the second derivative of carbon concentration with respect to depth.
In this case, we want to find the time required for the carbon concentration at a depth of 88.5 x 10^-2 cm to be half the value of the carbon concentration at the surface.
Let's break down the steps to solve this problem:
Step 1: Calculate the concentration gradient.
We know that the initial carbon concentration is 200.5 ppm, and the concentration at the surface of the steel is 9110.0 ppm. The concentration gradient, ∆C/∆x, can be calculated as:
∆C/∆x = (C_s - C_i) / x
Where:
- C_s is the carbon concentration at the surface,
- C_i is the initial carbon concentration,
- x is the depth.
Plugging in the values:
∆C/∆x = (9110.0 ppm - 200.5 ppm) / (0.885 m)
Step 2: Calculate the diffusion flux.
The diffusion flux, J, is given by Fick's first law:
J = -D * ∆C/∆x
Where:
- D is the diffusion coefficient of carbon in steel.
Plugging in the values:
J = -3.091 x 10^-7 cm²/s * (∆C/∆x)
Step 3: Calculate the time required.
The time required, t, can be calculated using the following equation:
t = (x²) / (2D)
Where:
- x is the depth,
- D is the diffusion coefficient of carbon in steel.
Plugging in the values:
t = (0.885 m)² / (2 * 3.091 x 10^-7 cm²/s)
Note: Since the units are different (m² and cm²/s), we need to convert the depth to cm:
t = (0.885 m * 100 cm/m)² / (2 * 3.091 x 10^-7 cm²/s)
Finally, convert the time to hours by dividing by 3600 seconds:
t (in hours) = t (in seconds) / 3600
Simplify and solve the equation to find the time required in hours.