Question 5.8: Design of a More Economical Heat Treatment We find that 10 h......
Design of a More Economical Heat Treatment
We find that 10 h are required to successfully carburize a batch of 500 steel gears at 900 °C, where the iron has the FCC structure. We find that it costs $1000 per hour to operate the carburizing furnace at 900 °C and $1500 per hour to operate the furnace at 1000 °C. Is it economical to increase the carburizing temperature to 1000 °C? What other factors must be considered?
We again assume that we can use the solution to Fick’s second law given by Equation 5-7:
\frac{c_{s} \ – \ c_{x}}{c_{s} \ – \ c_{0}}= erf ( \frac{x}{2\sqrt{Dt}}) (5-7)
The temperatures of interest are 900 °C = 1173 K and 1000 °C = 1273 K. To achieve the same carburizing treatment at 1000 °C as at 900 °C:
D_{1273}t_{1273} = D_{1173}t_{1173}
For carbon diffusing in FCC iron, the activation energy is 32,900 cal/mol. Since we are dealing with the ratios of times, it does not matter whether we substitute for the time in hours or seconds. It is, however, always a good idea to use units that balance out. Therefore, we will show time in seconds. Note that temperatures must be converted into kelvin.
D_{1273}t_{1273} = D_{1173}t_{1173}
D = D_{0} \exp(-Q/RT)
t_{1273}= \frac{D_{1173}t_{1173}}{D_{1273}}
t_{1273} = \frac{\exp(-14.1156)(10)(3600)}{ \exp(-13.0068)}
= 11,878 \ s
t_{1273} = 3.299 \ h = 3 \ h and 18 \ min
Notice, we did not need the value of the pre-exponential term D_{0} since it canceled out.
At 900 °C, the cost per part is ($1000/h) (10 h)/500 parts = $20/part. At 1000 °C, the cost per part is ($1500/h) (3.299 h)/500 parts = $9.90/part.
Considering only the cost of operating the furnace, increasing the temperature reduces the heat-treating cost of the gears and increases the production rate. Another factor to consider is if the heat treatment at 1000 °C could cause some other microstructural or other changes. For example, would increased temperature cause grains to grow significantly? If this is the case, we would be weakening the bulk of the material. How does the increased temperature affect the life of the other equipment such as the furnace itself and any accessories? How long would the cooling take? Will cooling from a higher temperature cause residual stresses? Would the product still meet all other specifications? These and other questions should be considered. The point is, as engineers, we need to ensure that the solution we propose is not only technically sound and economically sensible, it should recognize and make sense for the system as a whole. A good solution is often simple, solves problems for the system, and does not create new problems.