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
Carburizing Steel Time Calculation
This problem involves calculating the time required for carburizing a steel sample based on diffusion principles. We will use Fick's second law and the provided error function table.
Given Parameters
- Surface Carbon Concentration ($C_s$): 1.4 wt.%
- Initial Carbon Concentration ($C_0$): 0.2 wt.%
- Target Carbon Concentration ($C_x$): 0.8859 wt.%
- Depth ($x$): 0.2 mm = $0.2 \times 10^{-3}$ m
- Diffusivity ($D$) at 950 $^\circ$C: $6.25 \times 10^{-11}$ m$^2$/s
Error Function Table
| z | erf(z) |
|---|---|
| 0.3 | 0.3268 |
| 0.4 | 0.4284 |
| 0.5 | 0.5205 |
Calculating Diffusion Time
Step 1: Apply Fick's Second Law Formula
For diffusion into a semi-infinite solid with a constant surface concentration, the concentration profile is described by:
$ \frac{C_x - C_0}{C_s - C_0} = 1 - \text{erf}\left(\frac{x}{2\sqrt{Dt}}\right) $
Step 2: Calculate Concentration Ratio
Substitute the given concentration values into the left side of the equation:
$ \text{Ratio} = \frac{0.8859 \text{ wt}\% - 0.2 \text{ wt}\%}{1.4 \text{ wt}\% - 0.2 \text{ wt}\%} = \frac{0.6859}{1.2} \approx 0.5716 $
Step 3: Find the Error Function Argument (z)
Equate the calculated ratio to the right side of Fick's equation:
$ 0.5716 = 1 - \text{erf}(z) $
Solve for $\text{erf}(z)$:
$ \text{erf}(z) = 1 - 0.5716 = 0.4284 $
Using the provided table, $\text{erf}(z) = 0.4284$ corresponds to $z = 0.4$.
Step 4: Solve for Time (t)
Now, use the definition of the argument $z$:
$ z = \frac{x}{2\sqrt{Dt}} $
Substitute the values for $z$, $x$, and $D$:
$ 0.4 = \frac{0.2 \times 10^{-3} \text{ m}}{2\sqrt{(6.25 \times 10^{-11} \text{ m}^2/\text{s})t}} $
Rearrange to solve for $\sqrt{Dt}$:
$ \sqrt{Dt} = \frac{0.2 \times 10^{-3}}{2 \times 0.4} = \frac{0.2 \times 10^{-3}}{0.8} = 0.25 \times 10^{-3} \text{ m/s}^{0.5} $
Square both sides to find $Dt$:
$ Dt = (0.25 \times 10^{-3})^2 = 0.0625 \times 10^{-6} \text{ m}^2 $
Finally, calculate the time $t$:
$ t = \frac{0.0625 \times 10^{-6} \text{ m}^2}{6.25 \times 10^{-11} \text{ m}^2/\text{s}} $
$ t = \frac{6.25 \times 10^{-8}}{6.25 \times 10^{-11}} \text{ s} = 1 \times 10^3 \text{ s} $
$ t = 1000 \text{ seconds} $
Conclusion
The calculated time required for carburizing the steel to the specified depth and composition is 1000 seconds. This value falls within the provided range of 990 to 1010 seconds.
Important Questions from Diffusion Fick's Second Law Concentration Profile
- 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. - A species can diffuse through the lattice (diffusion coefficient, $D_L$), along grain boundaries (diffusion coefficient, $D_{GB}$), and along free surfaces (diffusion coefficient, $D_S$). Which of the following relations is CORRECT?
