A steel specimen containing 0.2 wt.% C is carburized in an atmosphere that maintains a carbon content of 1.2 wt.% C at the surface of the specimen. Given:
For carbon diffusion in austenite: $D_0=2.0\times10^{-5} m^2/s$
Activation energy for diffusion, $Q=142 kJ/mol$y erf(y) 0.85 0.7707 0.90 0.7970 0.95 0.8209
Diffusion Analysis
This problem uses Fick's second law to model carbon diffusion in steel during carburization. The governing equation for a semi-infinite solid with constant surface concentration ($C_s$) and initial concentration ($C_0$) is:
$ \frac{C(x,t) - C_0}{C_s - C_0} = \text{erfc}\left(\frac{x}{2\sqrt{Dt}}\right) $
Where:
- $C(x,t)$ is the carbon concentration at depth $x$ and time $t$.
- $C_0 = 0.2$ wt.% C (initial concentration).
- $C_s = 1.2$ wt.% C (surface concentration).
- $C_x = 0.4$ wt.% C (target concentration at depth $x$).
- $D$ is the diffusion coefficient.
- $t$ is the time.
- $x$ is the depth.
Calculating Concentration Term
First, determine the normalized concentration ratio:
$ \frac{C_x - C_0}{C_s - C_0} = \frac{0.4 - 0.2}{1.2 - 0.2} = \frac{0.2}{1.0} = 0.2 $
This means we need $\text{erfc}(Z) = 0.2$, where $Z = \frac{x}{2\sqrt{Dt}}$.
Determining Diffusion Parameter Z
From $\text{erfc}(Z) = 0.2$, we find $\text{erf}(Z) = 1 - 0.2 = 0.8$. We use the provided table values and interpolation to find $Z$.
| Argument ($y$) | erf($y$) |
| 0.90 | 0.7970 |
| 0.95 | 0.8209 |
Interpolating to find $Z$ when $\text{erf}(Z) = 0.8$:
$ Z = 0.90 + (0.95 - 0.90) \times \frac{0.8 - 0.7970}{0.8209 - 0.7970} $
$ Z = 0.90 + 0.05 \times \frac{0.0030}{0.0239} \approx 0.90 + 0.05 \times 0.1255 \approx 0.9063 $
Time for Doubled Depth Calculation
The relationship $Z = \frac{x}{2\sqrt{Dt}}$ implies $x = 2Z\sqrt{Dt}$. Since $Z$ is constant for a given concentration, $x \propto \sqrt{Dt}$. Assuming constant temperature, $D$ is constant, so $x \propto \sqrt{t}$.
Let $t_1$ be the time to reach depth $x_1$ with 0.4 wt.% C.
Let $t_2$ be the time to reach depth $x_2 = 2x_1$ with 0.4 wt.% C.
From $x \propto \sqrt{t}$, we get:
$ \frac{x_2}{x_1} = \sqrt{\frac{t_2}{t_1}} $
Given $x_2 = 2x_1$, the ratio $\frac{x_2}{x_1} = 2$. So,
$ 2 = \sqrt{\frac{t_2}{t_1}} $
Squaring both sides yields:
$ 4 = \frac{t_2}{t_1} \quad \text{or} \quad t_2 = 4 t_1 $
This shows that the time required to reach double the depth for the same concentration is four times the original time.
Final Answer
The question asks for the total time in hours required to double the depth. Based on the derived relationship $t_2 = 4 t_1$, the time needed is 40 hours.
Important Questions from Diffusion Fick's Second Law Concentration Profile
During carburizing of a steel, the surface concentration is kept constant at 1.4 wt.% carbon. Diffusivity of carbon for the steel at 950 $^\circ$C is $6.25 \times 10^{-11}$ m$^2$/s. At 950 $^\circ$C, the time required to carburize the steel with an initial composition of 0.2 wt.% carbon to 0.8859 wt.% carbon at a depth of 0.2 mm is ______________ seconds (approximate to the nearest integer).
Use the nearest value of the error function from the table given below for your calculation.
z erf (z) 0.3 0.3268 0.4 0.4284 0.5 0.5205 - What is the depth (in $µm$) from the surface of the specimen at which a composition of 0.4 wt.% C is obtained after carburizing at $870^\circ C$ for 10 h?
For self-diffusion in polycrystalline copper with a lattice diffusion coefficient $D_L$, grain boundary diffusion coefficient $D_{GB}$, and surface diffusion coefficient $D_S$, the correct relationship is
- The concentration $C$ of a solute (in units of atoms$\cdot\text{mm}^{-3}$) in a solid along $x$direction (for $x > 0$) follows the expression
$C = a_1x^2 + a_2x$
where $x$ is in mm, $a_1$ and $a_2$ are in units of atoms$\cdot\text{mm}^{-5}$ and atoms$\cdot\text{mm}^{-4}$,respectively. Assuming $a_1= a_2= 1$, the magnitude of flux at $x = 2 \text{ mm}$ is________ $\times 10^{-3} \text{ atoms} \cdot \text{mm}^{-2} \cdot \text{s}^{-1}$ (answer rounded off to the nearest integer).
Given: diffusion coefficient of the solute in the solid is $3 \times 10^{-3} \text{ mm}^2 \cdot \text{s}^{-1}$. - Determine the correctness or otherwise of the following Assertion [a] and the Reason [r]
Assertion [a]: The rate of homogenization in a dilute substitutional solid solution of B in A is controlled by the diffusivity of B.
Reason [r]: Atomic migration cannot occur along dislocations and grain boundaries.
