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 211.5 ppm and is initially uniform through the thickness of the steel. The atmosphere of the carburizing furnace maintains a carbon concentration of 6155.0 ppm at the surface of the steel. Calculate the time required to carburize steel so that the concentration of carbon at a depth of 26.5 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
how to find erf??? ??? error function(z)
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)
Step 1:
(0.5* 6155.0 - 6155.0)/ ( 211.5 - 6155.0)= erf(ξ)
Step 2:
erf(ξ < 0.6), use the approximation erf ξ = ξ
Step 3:
t= d^2/4*D*ξ^2
d=59.0 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)
how to do step 2??
To calculate the time required to carburize the steel, we can use Fick's second law of diffusion. This law describes the diffusion of a substance in a material and relates the concentration gradient to the diffusion coefficient and time.
The equation for Fick's second law is given by:
∂C/∂t = D * (∂²C/∂x²)
Where:
∂C/∂t is the rate of change of concentration with respect to time,
D is the diffusion coefficient,
∂²C/∂x² is the second derivative of the concentration with respect to distance.
In this case, we want to find the time required for the carbon concentration at a depth of 26.5 x 10^-2 cm to be one half the value of the carbon concentration at the surface. Let's denote the carbon concentration at the surface as C_s and the carbon concentration at the desired depth as C_d.
∂C/∂t = D * (∂²C/∂x²)
Since the carbon concentration is initially uniform throughout the steel, the concentration gradient is given by:
∂C/∂x = (C_s - C_0) / x_s
Where:
C_0 is the initial carbon concentration in the steel before carburization,
x_s is the thickness of the steel.
At the surface, the concentration gradient is given by:
∂C/∂x = (C_s - C_0) / x_s
At the desired depth, the concentration gradient is:
∂C/∂x = (C_d - C_0) / x_d
Since the concentration at the desired depth is one half the value at the surface, we can write:
(C_d - C_0) / x_d = (C_s - C_0) / x_s * 0.5
Rearranging the equation, we get:
(C_d - C_0) / (C_s - C_0) = x_d / x_s * 0.5
Now, we can substitute the values into this equation:
(C_d - 211.5 ppm) / (6155.0 ppm - 211.5 ppm) = (26.5 x 10^-2 cm) / x_s * 0.5
Simplifying further, we get:
(C_d - 211.5 ppm) / 5943.5 ppm = (26.5 x 10^-2 cm) / x_s * 0.5
Now, we can solve this equation to find the value of C_d. Once we know C_d, we can solve for the time using Fick's second law of diffusion.
However, since we don't have specific values for x_s and x_d in the problem statement, we cannot provide the exact time required for carburization.