Non-Steady State Diffusion

[buzachaka]

Homework Statement

For a steel alloy below, it has been determined that a carburizing heat treatment of 15 hr duration will raise the carbon concentration to 0.35% at a point 2.0 mm from the surface. Estimate the time necessary to achieve the same concentration at a point 6.0 mm position for an identical steel and at the same carburizing temperature.

Homework Equations

The Attempt at a Solution

Because the %Concentration and Temperature are the same, their should be a linear relationship between time and depth but I cannot think of a way to determine the Diffusion Coefficient or an equation for the relationship. Looking for a gentle shove in the right direction.

Thanks so much

 

[KataKoniK]

for your question and for sharing your thought process. It seems like you are on the right track in thinking about the linear relationship between time and depth. To determine the diffusion coefficient, you can use Fick's second law of diffusion, which states that the rate of change of concentration with respect to time is equal to the diffusion coefficient times the second derivative of concentration with respect to distance. In this case, we can rearrange the equation to solve for the diffusion coefficient:

D = (1/2) * (dC/dt) * (x^2)

Where:
D = diffusion coefficient
dC/dt = rate of change of concentration with respect to time
x = distance from the surface

We also know that the concentration at a point 2.0 mm from the surface is 0.35%, so we can plug in these values to solve for the diffusion coefficient:

D = (1/2) * (dC/dt) * (2.0^2)
0.35% = (1/2) * (dC/dt) * 4
dC/dt = 0.035%/hr

Now, we can use this value for the rate of change of concentration to solve for the time necessary to achieve the same concentration at a point 6.0 mm from the surface:

0.35% = (1/2) * (0.035%/hr) * (6.0^2)
Time = 24 hr

Therefore, it would take 24 hours to achieve the same carbon concentration at a point 6.0 mm from the surface for the identical steel and at the same carburizing temperature. I hope this helps to guide you in the right direction. Good luck with your calculations!

 

[piercas]

for your question!

Non-steady state diffusion refers to the process of diffusion where the concentration of a substance is changing with time. In this case, we are looking at the diffusion of carbon in a steel alloy during a carburizing heat treatment.

To solve this problem, we need to use Fick's second law of diffusion, which states that the rate of change of concentration with respect to time is equal to the diffusion coefficient multiplied by the second derivative of concentration with respect to distance.

In this case, we know that the concentration of carbon at a depth of 2.0 mm is 0.35% after 15 hours. We also know that the concentration at a depth of 6.0 mm should also be 0.35% after a certain amount of time.

Using Fick's second law, we can set up the following equation:

dC/dt = D * d^2C/dx^2

where dC/dt is the rate of change of concentration with respect to time, D is the diffusion coefficient, and d^2C/dx^2 is the second derivative of concentration with respect to distance.

Since the concentration and temperature are the same, we can assume that the diffusion coefficient is also the same for both depths. This means that we can set the two equations equal to each other:

D * d^2C/dx^2 = D * d^2C/dx^2

Now, we can solve for the time needed to achieve 0.35% concentration at a depth of 6.0 mm by rearranging the equation to solve for time:

t = (d^2C/dx^2) * (L^2 / D)

where t is the time, d^2C/dx^2 is the second derivative of concentration with respect to distance, L is the distance (6.0 mm in this case), and D is the diffusion coefficient.

Since we know the concentration at a depth of 2.0 mm and the concentration we want to achieve at a depth of 6.0 mm, we can calculate the second derivative of concentration with respect to distance using the following equation:

d^2C/dx^2 = (C2 - C1) / (x2 - x1)^2

where C2 is the concentration at a depth of 6.0 mm (0.35%),