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Chapter 5: Problem 12

An FCC iron-carbon alloy initially containing 0.55 wt \(\%\) C is exposed to an oxygen-rich and virtually carbon-free atmosphere at \(1325 \mathrm{K}\) \(\left(1052^{\circ} \mathrm{C}\right) .\) Under these circumstances the carbon diffuses from the alloy and reacts at the surface with the oxygen in the atmosphere that is, the carbon concentration at the surface position is maintained essentially at 0 wt \(\%\) C. (This process of carbon depletion is termed decarburization.) At what position will the carbon concentration be 0.25 wt\% after a 10-h treatment? The value of \(D\) at \(1325 \mathrm{K}\) is \(4.3 \times 10^{-11} \mathrm{m}^{2} / \mathrm{s}\).

Short Answer

Expert verified
Based on the given information and calculations, the position at which the carbon concentration will be 0.25 wt% C after a 10-hour treatment can be found using Fick's second law of diffusion, error function solution, and the provided boundary conditions. By rearranging the equation and solving for the dimensionless parameter (η), we can obtain the value of η. Once we have the value of η, we can substitute the known values for the diffusion coefficient (D) and time (t) to find the desired position (x). Answer: To find the position at which the carbon concentration will be 0.25 wt% C after a 10-hour treatment, we need to first find the value of η and then substitute the known values for D and t to find the desired position x.

Step by step solution

01

Use Fick's Second Law of Diffusion

From Fick's second law of diffusion, which states: $$\frac{\partial C(x,t)}{\partial t}=D \frac{\partial^2 C(x,t)}{\partial x^2}$$ where \(C(x,t)\) is the carbon concentration at position \(x\) and time \(t\), and \(D\) is the diffusion coefficient.
02

Apply Boundary Conditions

As the carbon concentration at the surface position is maintained essentially at 0 wt% C, and the initial carbon concentration is 0.55 wt% C, the boundary conditions are given by: 1. \(C(x,0)=0.55\) wt% 2. \(C(0,t)=0\) wt%
03

Use the Error Function Solution

To solve the partial differential equation with the given boundary conditions, we can use the error function solution for diffusion, which is given by: $$C(x,t) = C_{\ell} - C_s \left[ 1 - \frac{2}{\sqrt{\pi}} \int_0^{\eta} e^{-u^2}\,du \right]$$ where \(C_s\) is the concentration at the surface, \(C_{\ell}\) the initial concentration, and \(\eta=\frac{x}{2\sqrt{Dt}}\) is the dimensionless parameter. In our case, \(C_{\ell}=0.55\), \(C_s=0\), and \(D=4.3 \times 10^{-11}\,\mathrm{m}^2/\mathrm{s}\).
04

Determine the Position

We need to determine the position \(x\) at which the carbon concentration will be 0.25 wt% after a 10-hour treatment. Thus, we insert the known values into the error function solution and solve for \(x\): $$0.25 = 0.55 \left[ 1 - \frac{2}{\sqrt{\pi}} \int_0^{\eta} e^{-u^2}\,du \right]$$ Let the treatment time be \(t=10 \cdot 3600\,\mathrm{s}\). We can rearrange the above equation to find the value of \(\eta\): $$\frac{0.25}{0.55} = 1 - \frac{2}{\sqrt{\pi}} \int_0^{\eta} e^{-u^2}\,du$$ $$\eta=\frac{x}{2\sqrt{Dt}} \Rightarrow x=\eta \cdot 2\sqrt{Dt}$$ Now solve for \(\eta\) and then substitute the known values for \(D\) and \(t\) to find the desired position \(x\).

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Most popular questions from this chapter

(a) Briefly explain the concept of a driving force. (b) What is the driving force for steady-state diffusion?

(a) Compare interstitial and vacancy atomic mechanisms for diffusion. (b) Cite two reasons why interstitial diffusion is normally more rapid than vacancy diffusion.

The purification of hydrogen gas by diffusion through a palladium sheet was discussed in Section \(5.3 .\) Compute the number of kilograms of hydrogen that pass per hour through a 6 -mm-thick sheet of palladium having an area of \(0.25 \mathrm{m}^{2}\) at \(600^{\circ} \mathrm{C}\). Assume a diffusion coefficient of \(1.7 \times 10^{-8} \mathrm{m}^{2} / \mathrm{s},\) that the concentrations at the high- and low-pressure sides of the plate are 2.0 and \(0.4 \mathrm{kg}\) of hydrogen per cubic meter of palladium, and that steady-state conditions have been attained.

Self-diffusion involves the motion of atoms that are all of the same type; therefore it is not subject to observation by compositional changes, as with inter-diffusion. Suggest one way in which self-diffusion may be monitored.

Consider a diffusion couple composed of two semi-infinite solids of the same metal, and that each side of the diffusion couple has a different concentration of the same elemental impurity; furthermore, assume each impurity level is constant throughout its side of the diffusion couple. For this situation, the solution to Fick's second law (assuming that the diffusion coefficient for the impurity is independent of concentration), is as follows: $$C_{x}=\left(\frac{C_{1}+C_{2}}{2}\right)-\left(\frac{C_{1}-C_{2}}{2}\right) \operatorname{erf}\left(\frac{x}{2 \sqrt{D t}}\right).$$In this expression, when the \(x=0\) position is taken as the initial diffusion couple interface, then \(C_{1}\) is the impurity concentration for \(x<0\) likewise, \(C_{2}\) is the impurity content for \(x>0\).A diffusion couple composed of two platinum-gold alloys is formed; these alloys have compositions of \(99.0 \mathrm{wt} \% \mathrm{Pt}-1.0 \mathrm{wt} \%\) Au and 96.0 wt \(\%\) Pt- 4.0 wt\% Au. Determine the time this diffusion couple must be heated at \(1000^{\circ} \mathrm{C}(1273 \mathrm{K})\) in order for the composition to be 2.8 wt \(\%\) Au at the \(10 \mu \mathrm{m}\) position into the \(4.0 \mathrm{wt} \%\) Au side of the diffusion couple. Preexponential and activation energy values for Au diffusion in \(\mathrm{Pt}\) are \(1.3 \times 10^{-5}\) \(\mathrm{m}^{2} / \mathrm{s}\) and \(252,000 \mathrm{J} / \mathrm{mol},\) respectively.
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