Question

A steel part whose initial carbon content is \(0.12\) 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 \(0.7 \mathrm{~mm}\) 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 6944 seconds or around 1.93 hours to achieve the desired carbon content at a depth of 0.7 mm.

Step-by-step solution

Write down the given information

We are given the following data: - Initial mass fraction of carbon in steel, \(C_{i} = 0.12\)% - Mass fraction of carbon at the surface, \(C_{s} = 0.011\) - Desired mass fraction of carbon at depth \(0.7 \mathrm{~mm}\), \(C_{0.7} = 0.32\)% - Diffusion coefficient of carbon in steel at the furnace temperature, \(D = 7.2 \times 10^{-12} \mathrm{~m}^{2} / \mathrm{s}\)

Use Fick's second law of diffusion

According to Fick's second law, the diffusive flux, \(J\), can be related to the concentration gradient by the relation: \(J = -D \cdot \frac{\partial C}{\partial x}\). Integrating over the distance, \(x\), from the surface to the depth of \(0.7 \mathrm{~mm}\), we get: $$ C_{0.7} - C_i = -\int_0^x J \, dx $$

Find the diffusive flux and time taken for diffusion

From the previous equation, we can rearrange to find the diffusive flux, \(J\): $$ J = -\frac{C_{0.7} - C_{i}}{x} $$ Integrating over time, we get the time taken for diffusion, \(t\): $$ t = -\int_0^x \frac{1}{D} \cdot \frac{\partial C}{\partial x} \, dx = -\frac{1}{D} \int_0^x \frac{C_{0.7} - C_{i}}{x} \, dx $$

Calculate the time taken for diffusion

Substituting the given values in the equation and converting the distance to meters, we get: $$ t = -\frac{1}{7.2 \times 10^{-12} \, \mathrm{m^2/s}} \int_0^{0.7 \times 10^{-3} \, \mathrm{m}} \frac{0.0032 - 0.0012}{0.7 \times 10^{-3} \, \mathrm{m}} \, \mathrm{dx} $$ The integral becomes: $$ t = -\frac{1}{7.2 \times 10^{-12}} \cdot \frac{1}{0.7 \times 10^{-3}} \cdot (0.0032 - 0.0012) \times (0.7 \times 10^{-3}) \, \mathrm{m} $$ Now we can solve for the time, \(t\): $$ t \approx 6944 \, \mathrm{s} $$ So, the steel part should be held in the furnace for about \(6944\) seconds, or around \(1.93\) hours, to achieve the desired carbon content at a depth of \(0.7 \mathrm{~mm}\).

Key concepts

Fick's Second Law of Diffusion

Understand Fick's second law of diffusion is essential for analyzing how atoms and molecules travel through materials. At high temperatures, the atoms inside a metal such as steel have enough energy to move from areas of high concentration to areas of low concentration, a process known as diffusion.

Fick's second law particularly helps to predict the change in concentration of these atoms over time. The law is mathematically expressed as: $$ \frac{\partial C}{\partial t} = D \frac{\partial^2 C}{\partial x^2} $$where \( C \) is the concentration of diffusing species at time \( t \) and position \( x \), and \( D \) is the diffusion coefficient, which can vary with temperature. For the case-hardening of steel, Fick's second law allows the calculation of how long it would take for carbon to reach a certain concentration at a specific depth within the steel when exposed to a carbon-rich environment.

Carbon Diffusion in Steel

Carbon plays a pivotal role in defining the properties of steel, specifically its hardness. When steel is exposed to a carbon-rich atmosphere at high temperatures, carbon atoms diffuse into the steel's surface.

The depth to which the carbon diffuses and the resulting concentration gradient within the steel is crucial to the overall strength and wear resistance of the component. The rate of carbon diffusion in steel correlates with temperature and the chemical potential difference between the high-carbon surface and the lower-carbon interior. Using Fick's laws, specifically the second law, one can describe this carbon diffusion process quantitatively. The diffusion coefficient of carbon in steel, a temperature-dependent value, is a key parameter for these calculations.

Mass Transfer in Heat Treatment

Heat treatment involves controlled heating and cooling to alter the microstructure of a material such as steel to change its mechanical properties. Mass transfer during heat treatment, like case-hardening, is the movement of atoms or molecules - in this case, carbon - into the surface layer of steel.

This transformation is driven by the concentration gradient established between the carbon-rich surface and the carbon-deficient core, leading to diffusion. The objectives of such treatments are to improve characteristics like hardness, strength, and toughness. By controlling factors like temperature, time, and the environment to which the material is exposed, engineers can tailor the properties of steel for specific applications. Time calculations, like those in the given exercise, are crucial for ensuring that the steel achieves the desired carbon profile depth throughout the heat treatment process.