Question 5.7: Design of a Carburizing Treatment The surface of a 0.1% C st......

Since the boundary conditions for which Equation 5-7 was derived are assumed to be valid, we can use this equation:
\frac{c_{s} \ – \ c_{x}}{c_{s} \ – \ c_{0}}= erf ( \frac{x}{2\sqrt{Dt}})
We know that c_{s} = 1.2% C, c_{0} = 0.1% C, c_{x} = 0.45% C, and x = 0.2 cm. From Fick’s second law:
\frac{c_{s} \ – \ c_{x}}{c_{s} \ – \ c_{0}}=  \frac{1.2 \% \ C \ – \ 0.45 \% \ C}{1.2 \% \ C \ – \ 0.1 \% \ C} =0.68 =erf ( \frac{0.2 \ cm}{2\sqrt{Dt}}) = erf ( \frac{0.1 \ cm}{\sqrt{Dt}})
From Table 5-3, we find that
\frac{0.1 \ cm}{\sqrt{Dt}} = 0.71 or Dt =( \frac{0.1}{0.71})^2 = 0.0198 cm^2
Any combination of D and t with a product of 0.0198 cm² will work. For carbon diffusing in FCC iron, the diffusion coefficient is related to temperature by Equation 5-4:
D = D_{0} \exp(\frac{-Q}{RT})
From Table 5-1:
D = 0.23 \exp[\frac{-32,900 \ cal/mol}{(1.987 \ \frac{cal}{mol  ⋅  K})T} ] = 0.23 \exp(\frac{-16,558}{T})
Therefore, the temperature and time of the heat treatment are related by
t = \frac{0.0198 \ cm^2}{D} =\frac{0.0198 \ cm^2}{0.23 \exp(-16,558/T)} =\frac{0.0861}{\exp (-16,558/T)}
Some typical combinations of temperatures and times are
If T = 900 °C = 1173 K, then t = 116,273 s = 32.3 h
If T = 1000 °C = 1273 K, then t = 38,362 s = 10.7 h
If T = 1100 °C = 1373 K, then t = 14,876 s = 4.13 h
If T = 1200 °C = 1473 K, then t = 6,560 s = 1.82 h
The exact combination of temperature and time will depend on the maximum temperature that the heat treating furnace can reach, the rate at which parts must be produced, and the economics of the tradeoffs between higher temperatures versus longer times. Another factor to consider is changes in microstructure that occur in the rest of the material. For example, while carbon is diffusing into the surface, the rest of the microstructure can begin to experience grain growth or other changes.