Understanding Carbon Diffusion in Steel

The problem involves calculating the depth ($x$) in a steel specimen where a specific carbon concentration ($C(x,t)$) is reached after carburizing. We are given the initial carbon concentration ($C_0$), the surface carbon concentration ($C_s$), the carburizing temperature ($T$), time ($t$), and diffusion parameters ($D_0$, $Q$).

The relevant formula for diffusion under constant surface concentration in a semi-infinite solid is derived from Fick's second law:

$ \frac{C_s - C(x,t)}{C_s - C_0} = \text{erfc}\left(\frac{x}{2\sqrt{Dt}}\right) $

• $C_s = 1.2$ wt.% C (Surface concentration)
• $C_0 = 0.2$ wt.% C (Initial concentration)
• $C(x,t) = 0.4$ wt.% C (Target concentration at depth $x$, time $t$)
• $t = 10 \text{ h} = 10 \times 3600 \text{ s} = 36000 \text{ s}$ (Carburizing time)

Calculating Diffusion Coefficient (D)

The diffusion coefficient ($D$) depends on temperature ($T$) and is given by the Arrhenius equation:

$ D = D_0 \exp\left(-\frac{Q}{RT}\right) $

• $D_0 = 2.0 \times 10^{-5} m^2/s$
• $Q = 142 \text{ kJ/mol} = 142000 \text{ J/mol}$
• $R = 8.314 \text{ J/(mol·K)}$ (Gas constant)
• $T = 870^\circ C = 870 + 273.15 = 1143.15 \text{ K}$

First, calculate the term $Q/(RT)$:

$ \frac{Q}{RT} = \frac{142000 \text{ J/mol}}{8.314 \text{ J/(mol·K)} \times 1143.15 \text{ K}} \approx 14.938 $

Now, calculate $D$:

$ D = (2.0 \times 10^{-5} m^2/s) \times \exp(-14.938) $ $ D \approx (2.0 \times 10^{-5} m^2/s) \times (2.186 \times 10^{-7}) $ $ D \approx 4.372 \times 10^{-12} m^2/s $

Applying Fick's Second Law

Calculate the concentration ratio term:

$ \frac{C_s - C(x,t)}{C_s - C_0} = \frac{1.2 - 0.4}{1.2 - 0.2} = \frac{0.8}{1.0} = 0.8 $

We need to find the argument $z = \frac{x}{2\sqrt{Dt}}$ such that $\text{erfc}(z) = 0.8$.

Using the provided table data, which seems to represent $z$ vs $\text{erfc}(z)$:

z	erfc(z)
0.85	0.7707
0.90	0.7970
0.95	0.8209

Our target value $\text{erfc}(z) = 0.8$ lies between $0.7970$ (at $z=0.90$) and $0.8209$ (at $z=0.95$). We interpolate to find $z$:

$ z = 0.90 + \frac{0.8 - 0.7970}{0.8209 - 0.7970} \times (0.95 - 0.90) $ $ z = 0.90 + \frac{0.0030}{0.0239} \times 0.05 \approx 0.90 + 0.0063 = 0.9063 $

Determining Diffusion Depth (x)

Now we can calculate the depth $x$ using $z = \frac{x}{2\sqrt{Dt}}$.

First, calculate $2\sqrt{Dt}$:

$ 2\sqrt{Dt} = 2\sqrt{(4.372 \times 10^{-12} m^2/s) \times (36000 s)} $ $ 2\sqrt{Dt} = 2\sqrt{1.5739 \times 10^{-7} m^2} \approx 2 \times (3.967 \times 10^{-4} m) $ $ 2\sqrt{Dt} \approx 7.934 \times 10^{-4} m $

Calculate the depth $x$:

$ x = z \times (2\sqrt{Dt}) $ $ x \approx 0.9063 \times (7.934 \times 10^{-4} m) $ $ x \approx 7.187 \times 10^{-4} m $

Convert the depth to micrometers ($µm$):

$ x \approx 7.187 \times 10^{-4} m \times \frac{10^6 µm}{1 m} \approx 718.7 µm $

Note: Based on standard calculations and interpolation from the provided table, the depth is approximately $719 µm$. The provided options might imply a different interpretation or data set. Proceeding with the calculated value based on the given information.