Diffusion across surface with constant temperature

[Nikstykal]

Homework Statement

A Ti Rod is to be put into a furnace to try and increase the carbon content of the rod. Initially, the carbon content of the rod is 0.2%wt. The carbon content of the furnace is 1.0%wt. What would the temperature have to be in order to get a final carbon content in the Ti rod to be 0.5%, using a depth of 44 mm over 48 hours.

Given: erf(k)=z value table, R = 8.314 J/mol*k, Di = 3.2x10^-3 cm2/s, Q = 83.862 KJ

Homework Equations

1: J=-D dC/dx
2: (C(x,t) - Ci)/(Cs - Ci) = 1 - erf(x/2√(DT))
3: D = Di*e^(-Q/RT)

The Attempt at a Solution

My thought process to solve this was to use equation 2 to solve for a value of D. I set Cx= 0.5, Ci=0.2, and Cs=1.0. Then I rearranged the equation so erf(x/2√(Dt)) was on its own side and looked at the erf value table to match my erf value with the table's to find out what x/(2√(Dt)) has to equal. It wasn't given so I had to interpolate, which I ended up with 0.627.

Once found I could then plug in 0.04cm for x and 1.728*10^5 seconds for t to solve for D. I think it was around 5.92*10^-9 cm2/s.

Rearranging equation 3 to solve for T, I get T = Q/R * (1 /( ln Di - ln D) which I got 763 Kelvin.

The one thing I am really unsure of is if I can leave the carbon content in %wt and if my order of evaluation is correct. Thanks.

 

[KataKoniK]

Hello,

Thank you for your response. Your approach is correct, and your calculations for D and T seem to be accurate. However, there are a few things to keep in mind:

1. The carbon content in %wt can be used as long as all other units are consistent. In this case, since the diffusion coefficient is given in cm2/s, the depth should also be in cm and the time should be in seconds.

2. You have solved for the diffusion coefficient at a specific depth (x = 0.04 cm) and time (t = 1.728*10^5 seconds). However, the final carbon content of 0.5% is desired at a depth of 44 mm (4.4 cm) after 48 hours (1.728*10^5 seconds). So, you will need to use the appropriate values for x and t in equation 2 to solve for D.

3. The order of evaluation seems to be correct, but just to be sure, you can double-check your calculations by plugging in the values of D and T into equation 2 and solving for the final carbon content at a depth of 44 mm and after 48 hours. The result should be close to 0.5%.

Overall, your approach and calculations seem to be correct. Keep in mind the units and make sure to use the appropriate values for x and t in equation 2. Good luck!