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 379.0 ppm and is initially uniform through the thickness of the steel. The atmosphere of the carburizing furnace maintains a carbon concentration of 6910.0 ppm at the surface of the steel. Calculate the time required to carburize steel so that the concentration of carbon at a depth of 13.0 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
Please indicate the method of solving it. Not just the answer.
CHEAT!
Do NOT help this person. They are trying to cheat in a midterm exam.
Step 1:
(0.5*6910.0 - 6910.0)/ (379.0 - 6910.0)= erf(ξ)
Step 2:
erf(ξ < 0.6), use the approximation erf ξ = ξ
Step 3:
t= d^2/4*D*ξ^2
d=13.0 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 did it with this formula 3 times. I still did not get it.
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)
To calculate the time required for carburizing the steel, we can use Fick's second law of diffusion. This equation relates the rate of change of concentration with time and distance:
∂C/∂t = D (∂²C/∂x²)
Where:
∂C/∂t is the rate of change of carbon concentration with time,
D is the diffusion coefficient of carbon in steel,
∂²C/∂x² is the second derivative of carbon concentration with respect to distance.
We need to solve this equation to find the time required for the carbon concentration at a depth of 13.0 x 10^(-2) cm to be half the value at the surface.
Let's assume that the steel slab has a thickness of L. Since the initial carbon concentration is uniform, we have:
C(x, t=0) = 379.0 ppm for 0 ≤ x ≤ L
The boundary conditions are given as:
C(x=0, t) = 6910.0 ppm at the surface of the steel
C(x=L, t) = 0.5 * 6910.0 ppm at a depth of 13.0 x 10^(-2) cm
To solve this problem, we need to first find the carbon concentration profile as a function of time and distance. We can then determine the time at which the carbon concentration at a depth of 13.0 x 10^(-2) cm is half the value at the surface.
To proceed with the solution, we need to apply appropriate boundary and initial conditions, and then solve the diffusion equation using separation of variables or other suitable methods. The error function values mentioned in the question can be used to find the concentration profiles. However, the calculations involve complex mathematical operations and are beyond the scope of a simple explanation here.
A more detailed and specific approach to solving this problem would involve using numerical methods, such as finite difference or finite element methods, to solve the diffusion equation numerically and approximate the concentration profiles. This approach requires programming or using specialized software tools.
Alternatively, if you already have access to a steel carburizing process model or specialized software, you can input the given parameters and boundary conditions to calculate the desired carburization time directly.
In summary, to calculate the time required to carburize the steel slab, we need to solve the diffusion equation for carbon concentration under appropriate boundary and initial conditions. This involves mathematical calculations that require either numerical methods or the use of specialized software.