1. **Problem statement:** Calculate the time necessary to increase the carbon content to 0.35 wt% at 0.45 mm below the surface of 1018 steel carburized at 977°C. 2. **Given data:** - Initial carbon content, $C_0 = 0.18$ wt% - Surface carbon content, $C_s = 1.15$ wt% - Desired carbon content at depth, $C_x = 0.35$ wt% - Depth, $x = 0.45$ mm = 0.045 cm - Temperature = 977°C - Steel structure: FCC (austenite) 3. **Formula used:** The diffusion of carbon in steel during carburizing can be modeled by Fick's second law for a semi-infinite solid with constant surface concentration: $$\frac{C_x - C_0}{C_s - C_0} = 1 - \text{erf}\left(\frac{x}{2\sqrt{Dt}}\right)$$ where: - $C_x$ is the carbon content at depth $x$ after time $t$ - $C_0$ is the initial carbon content - $C_s$ is the surface carbon content - $D$ is the diffusion coefficient of carbon in austenite at 977°C - $t$ is the time in seconds - $\text{erf}$ is the error function 4. **Find diffusion coefficient $D$:** For carbon in FCC iron (austenite), $D$ is given by: $$D = D_0 \exp\left(-\frac{Q}{RT}\right)$$ Typical values for carbon in austenite: - $D_0 = 0.2$ cm$^2$/s - Activation energy $Q = 142000$ J/mol - Gas constant $R = 8.314$ J/mol·K - Temperature $T = 977 + 273 = 1250$ K Calculate $D$: $$D = 0.2 \times \exp\left(-\frac{142000}{8.314 \times 1250}\right)$$ Calculate exponent: $$\frac{142000}{8.314 \times 1250} = \frac{142000}{10392.5} \approx 13.66$$ So: $$D = 0.2 \times e^{-13.66} = 0.2 \times 1.16 \times 10^{-6} = 2.32 \times 10^{-7} \text{ cm}^2/\text{s}$$ 5. **Calculate the error function argument:** Rearranged formula: $$\text{erf}\left(\frac{x}{2\sqrt{Dt}}\right) = 1 - \frac{C_x - C_0}{C_s - C_0}$$ Calculate the right side: $$1 - \frac{0.35 - 0.18}{1.15 - 0.18} = 1 - \frac{0.17}{0.97} = 1 - 0.1753 = 0.8247$$ So: $$\text{erf}(z) = 0.8247$$ where $$z = \frac{x}{2\sqrt{Dt}}$$ 6. **Find $z$ from erf table or calculator:** $\text{erf}(0.95) \approx 0.8209$ and $\text{erf}(0.96) \approx 0.8249$ So $z \approx 0.96$ 7. **Solve for time $t$:** $$z = \frac{x}{2\sqrt{Dt}} \Rightarrow \sqrt{Dt} = \frac{x}{2z}$$ Square both sides: $$Dt = \left(\frac{x}{2z}\right)^2$$ Solve for $t$: $$t = \frac{x^2}{4z^2 D}$$ Substitute values: $$t = \frac{(0.045)^2}{4 \times (0.96)^2 \times 2.32 \times 10^{-7}}$$ Calculate numerator: $$0.045^2 = 0.002025$$ Calculate denominator: $$4 \times 0.9216 \times 2.32 \times 10^{-7} = 4 \times 0.9216 \times 2.32 \times 10^{-7} = 8.55 \times 10^{-7}$$ So: $$t = \frac{0.002025}{8.55 \times 10^{-7}} = 2368 \text{ seconds}$$ Convert to hours: $$\frac{2368}{3600} \approx 0.66 \text{ hours}$$ **Final answer:** The time necessary to increase the carbon content to 0.35 wt% at 0.45 mm depth is approximately **0.66 hours** (about 40 minutes).