"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 165.0 ppm and is initially uniform through the thickness of the steel. The atmosphere of the carburizing furnace maintains a carbon concentration of 7715.0 ppm at the surface of the steel. Calculate the time required to carburize steel so that the concentration of carbon at a depth of 61.5 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"
I keep getting 2.127 hours, but apparently it's wrong. I've checked and checked and I can't find what's wrong. Accuse me for cheating or whatever; I've done it with other data and I do get the right answer.
**We are here to help each other on this forum. Hope the info below helps'!
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* 7715.0 - 7715.0 )/ ( 165.0 - 7715.0 )= erf(ξ)
Step 2:
erf(ξ < 0.6), use the approximation erf ξ = ξ
Step 3:
t= d^2/4*D*ξ^2
d=61.5 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)
Yawnnnn. What a kind-hearted person you are Odesa I am touched by your way of cheating and helping each other. LOL.
To solve this problem, we can use Fick's second law of diffusion, which describes how the concentration of a diffusing species changes with time and distance.
The equation for Fick's second law is:
∂C/∂t = D * (∂²C/∂x²)
Where:
- ∂C/∂t represents the change in concentration with time
- ∂²C/∂x² represents the change in concentration with distance
- D is the diffusion coefficient
In this case, we want to find the time required to reach a certain concentration at a specific depth in the steel. We know the initial concentration is 165.0 ppm and the surface concentration is 7715.0 ppm. We want to find the time when the concentration at a depth of 61.5 x 10^-2 cm is half the surface concentration.
Let's break down the problem into steps:
Step 1: Calculate the concentration gradient (∂C/∂x)
The concentration gradient at the surface is given by:
∂C/∂x = (7715.0 ppm - 165.0 ppm) / 0
Since the initial concentration is uniform throughout the steel, the gradient is constant and equal to (7715.0 ppm - 165.0 ppm).
Step 2: Calculate the diffusion coefficient in cm²/s
Given the diffusion coefficient is 3.091 x 10^-7 cm²/s, we can use this value directly.
Step 3: Calculate the distance
The depth at which we want to find the concentration is given as 61.5 x 10^-2 cm.
Step 4: Plug the values into the equation
∂C/∂t = D * (∂²C/∂x²)
Since ∂²C/∂x² is zero in this case (the concentration is uniform in the beginning), the equation becomes:
∂C/∂t = D * 0
This means that the concentration at the depth of interest will remain constant over time.
Step 5: Calculate the time required
Since the concentration at the depth of interest remains constant, the time required is infinite.
Therefore, the correct answer is that it will take an infinite amount of time for the concentration at a depth of 61.5 x 10^-2 cm to reach half the value of the carbon concentration at the surface.
I apologize for any initial confusion, but in this specific case, the concentration will not change at the desired depth, regardless of the time.