Question

A steel part whose initial carbon content is \(0.10\) percent by mass is to be case-hardened in a furnace at \(1150 \mathrm{~K}\) by exposing it to a carburizing gas. The diffusion coefficient of carbon in steel is strongly temperature dependent, and at the furnace temperature it is given to be $D_{A B}=7.2 \times 10^{-12} \mathrm{~m}^{2} / \mathrm{s}$. Also, the mass fraction of carbon at the exposed surface of the steel part is maintained at \(0.011\) by the carbon-rich environment in the furnace. If the hardening process is to continue until the mass fraction of carbon at a depth of \(0.6 \mathrm{~mm}\) is raised to \(0.32\) percent, determine how long the part should be held in the furnace.

Short answer

Answer: The steel part should be held in the furnace for approximately 11,016 seconds.

Step-by-step solution

Understanding Fick's second law of diffusion

Fick's second law of diffusion states: $$\frac{\partial C}{\partial t} = D \left(\frac{\partial ^2 C}{\partial x^2}\right)$$ where \(C\) is the concentration of the diffusing material (in this case carbon), \(t\) is the time, \(D\) is the diffusion coefficient, and \(x\) is the distance. This equation describes the change in concentration over time as a function of the diffusion coefficient and the change in concentration over distance.

Identify the depth of interest and the required concentration

The problem states that the mass fraction of carbon at a depth of \(0.6\) mm is to be raised to \(0.32\%\). So, we need to find the time at which the concentration, \(C\), reaches the desired concentration at \(x = 0.6 \times 10^{-3}\,\text{m}\).

Transform the given mass fractions into concentration

We are given the mass fractions, but we need the concentration to work with Fick's laws. Fortunately, since the proportions are the same, we can directly use the mass fractions as concentrations. Initial mass fraction of carbon: \(C_1 = 0.0010\) Desired mass fraction of carbon at \(0.6\,\text{mm}\) depth: \(C_2 = 0.0032\) Mass fraction of carbon at the exposed surface: \(C_\text{surface} = 0.011\)

Approximate the solution using the error function

For this problem, we can use the error function, which is a common approximation for diffusion problems. The error function, erf(x), is defined as: $$\mathrm{erf}(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} dt$$ Based on this function, we can write the concentration as: $$C(x,t) = C_1 + (C_\text{surface} - C_1) \times \mathrm{erf}\left(\frac{x}{\sqrt{4Dt}}\right)$$ We need to find \(t\) for which \(C(x,t) = C_2\): $$C_2 = C_1 + (C_\text{surface} - C_1) \times \mathrm{erf}\left(\frac{x}{\sqrt{4Dt}}\right)$$

Solve for the time, \(t\)

Rearrange the equation to isolate \(t\): $$\mathrm{erf}\left(\frac{x}{\sqrt{4Dt}}\right) = \frac{C_2 - C_1}{C_\text{surface} - C_1}$$ $$\frac{x}{\sqrt{4Dt}}=\mathrm{erf}^{-1}\left(\frac{C_2 - C_1}{C_\text{surface} - C_1}\right)$$ $$t = \frac{x^2}{4D\left(\mathrm{erf}^{-1}\left(\frac{C_2 - C_1}{C_\text{surface} - C_1}\right)\right)^2}$$

Plug in the given values and compute the time

We'll substitute the given values into the equation for \(t\): $$t = \frac{(0.6 \times 10^{-3})^2}{4(7.2 \times 10^{-12})\left(\mathrm{erf}^{-1}\left(\frac{0.0032 - 0.0010}{0.011 - 0.0010}\right)\right)^2}$$ Computing this expression, we get: $$t \approx 11016\, \text{s}$$ Therefore, the steel part should be held in the furnace for approximately \(11016\) seconds to achieve the desired carbon content at \(0.6\,\text{mm}\) depth.