Given: Diffusivity of carbon in $\gamma$-Fe at 950 °C = $1.6 \times 10^{-11}$ $m^2$ $s^{-1}$
Error function table:z 0.35 0.40 0.45 0.50 0.55 0.60 erf(z) 0.3794 0.4284 0.4755 0.5205 0.5633 0.6039
Carburization Diffusion Depth Calculation
This problem involves calculating the depth of carbon penetration into a steel plate after carburization. This is a classic diffusion problem governed by Fick's second law, solved using the error function (erf) for a semi-infinite solid with a constant surface concentration.
Diffusion Equation for Carburization
The concentration of carbon $C(x,t)$ at a depth $x$ and time $t$ can be described by the following equation:
$ \frac{C(x,t) - C_0}{C_s - C_0} = \text{erf}\left(\frac{x}{2\sqrt{Dt}}\right) $Where:
- $C(x,t)$: Carbon concentration at depth $x$ and time $t$ (0.6 wt.%)
- $C_0$: Initial carbon concentration in the plate (0.1 wt.%)
- $C_s$: Surface carbon concentration (1.1 wt.%)
- $x$: Depth below the surface (in meters)
- $D$: Diffusivity of carbon in $\gamma$-Fe (1.6 x 10-11 m2/s)
- $t$: Time of diffusion (9 hours)
- erf: The error function
Calculating the Concentration Ratio
First, calculate the ratio term on the left side of the equation:
$ \frac{C(x,t) - C_0}{C_s - C_0} = \frac{0.6 \text{ wt.%} - 0.1 \text{ wt.%}}{1.1 \text{ wt.%} - 0.1 \text{ wt.%}} = \frac{0.5}{1.0} = 0.5 $Determining the Dimensionless Parameter 'z'
We need to find the value of $z = \frac{x}{2\sqrt{Dt}}$ such that $\text{erf}(z) = 0.5$. Using the provided error function table:
| z | 0.35 | 0.40 | 0.45 | 0.50 | 0.55 | 0.60 |
|---|---|---|---|---|---|---|
| erf(z) | 0.3794 | 0.4284 | 0.4755 | 0.5205 | 0.5633 | 0.6039 |
The value 0.5 for erf(z) lies between erf(0.45) = 0.4755 and erf(0.50) = 0.5205. We interpolate to find $z$:
$ z = 0.45 + (0.50 - 0.45) \times \frac{0.5 - 0.4755}{0.5205 - 0.4755} $ $ z = 0.45 + 0.05 \times \frac{0.0245}{0.0450} \approx 0.45 + 0.05 \times 0.5444 \approx 0.4772 $Calculating the Diffusion Parameter $2\sqrt{Dt}$
First, convert time to seconds:
$ t = 9 \text{ hours} \times 3600 \text{ seconds/hour} = 32400 \text{ seconds} $Now, calculate $2\sqrt{Dt}$:
$ 2\sqrt{Dt} = 2\sqrt{(1.6 \times 10^{-11} \text{ m}^2/\text{s}) \times (32400 \text{ s})} $ $ 2\sqrt{Dt} = 2\sqrt{5.184 \times 10^{-6} \text{ m}^2} = 2 \times (2.2768 \times 10^{-3} \text{ m}) $ $ 2\sqrt{Dt} \approx 4.5536 \times 10^{-3} \text{ m} = 4.5536 \text{ mm} $Calculating the Depth (x)
Using the relationship $z = \frac{x}{2\sqrt{Dt}}$, we solve for $x$:
$ x = z \times (2\sqrt{Dt}) $ $ x \approx 0.4772 \times 4.5536 \text{ mm} $ $ x \approx 2.173 \text{ mm} $Rounding off to 2 decimal places, the calculated depth is approximately 2.17 mm.
Note: The calculated value of approximately 2.17 mm does not fall within the provided answer range of 0.65 to 0.75 mm, indicating a potential discrepancy in the problem's parameters or expected outcome.
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.
