Question

Consider a piece of steel undergoing a decarburization process at \(925^{\circ} \mathrm{C}\). The mass diffusivity of carbon in steel at \(925^{\circ} \mathrm{C}\) is \(1 \times 10^{-7} \mathrm{~cm}^{2} / \mathrm{s}\). Determine the depth below the surface of the steel at which the concentration of carbon is reduced to \(40 \%\) from its initial value as a result of the decarburization process for \((a)\) an hour and \((b)\) ten hours. Assume the concentration of carbon at the surface is zero throughout the decarburization process.

Short answer

Based on the given information about carbon diffusivity in steel and the decarburization process, calculate the depth below the surface where the carbon concentration is reduced to 40% of its initial value for (a) one hour and (b) ten hours.

Step-by-step solution

Write down the Fick's second law of diffusion

Fick's second law of diffusion is given by the following formula: \[ \frac{\partial C}{\partial t} = D \frac{\partial^2 C}{\partial x^2}, \] where \(C\) is the concentration of the species, \(t\) is the time, \(D\) is the mass diffusivity, and \(x\) is the distance from the surface.

Identify the boundary conditions

We are given two boundary conditions: 1. The concentration of carbon at the surface is always zero, i.e., \(C(0,t)=0\). 2. The concentration of carbon at an infinite distance from the surface remains unchanged, i.e., \(C(\infty,t)=C_0\).

Apply the concentration reduction criteria

We need to find the depth \(x\) where the concentration of carbon is reduced to 40% of its initial value, i.e., \(C(x,t) = 0.4C_0\).

Find the analytic solution of Fick's second law for the given conditions

For the given boundary conditions, the analytic solution of Fick's second law can be written as: \[ C(x,t) = C_0\left[1 - \mathrm{erf}\left(\frac{x}{2\sqrt{Dt}}\right)\right], \] where \(\mathrm{erf}(z)\) is the error function, defined as \(\mathrm{erf}(z) = \frac{2}{\sqrt{\pi}} \int_0^z e^{-u^2} du\).

Set the carbon concentration reduction to 40%

Now apply the required condition of carbon concentration reduction to 40% of the initial value: \[ 0.4C_0 = C_0\left[1 - \mathrm{erf}\left(\frac{x}{2\sqrt{Dt}}\right)\right]. \]

Solve for x

We need to solve for \(x\): \[ 0.4 = 1 - \mathrm{erf}\left(\frac{x}{2\sqrt{Dt}}\right). \]

Calculate x for given times#a)

For \((a)\) 1 hour, we have \(t = 3600 \mathrm{~s}\). Plug this into the equation and solve for \(x\) using the given diffusivity value: \[ 0.4 = 1 - \mathrm{erf}\left(\frac{x}{2\sqrt{(1 \times 10^{-7} \mathrm{~cm}^{2} / \mathrm{s}) \cdot 3600 \mathrm{~s}}\right). \] Solving for \(x\), we get \(x \approx 2.22 \mathrm{~cm}\).

Calculate x for given times#b)

For \((b)\) 10 hours, we have \(t = 36000 \mathrm{~s}\). Plug this into the equation and solve for \(x\) using the given diffusivity value: \[ 0.4 = 1 - \mathrm{erf}\left(\frac{x}{2\sqrt{(1 \times 10^{-7} \mathrm{~cm}^{2} / \mathrm{s}) \cdot 36000 \mathrm{~s}}\right). \] Solving for \(x\), we get \(x \approx 7.03 \mathrm{~cm}\).

Summary

In this exercise, we have found the depth below the surface at which the carbon concentration in the steel piece is reduced to 40% of its initial value for two decarburization durations: 1 hour and 10 hours. The results obtained were \(x \approx 2.22 \mathrm{~cm}\) for 1 hour and \(x \approx 7.03 \mathrm{~cm}\) for 10 hours.

Key concepts

Fick's second law of diffusion

Understanding Fick's second law of diffusion is crucial when analyzing processes such as decarburization in materials like steel. This law mathematically describes how the concentration of a diffusing species changes with time. It states that the change in concentration at any point is proportional to the second derivative of concentration with respect to position.

Visualize this concept by imagining a sugar cube dissolving in a cup of coffee. Initially, a high concentration of sugar is at the cube itself, but with time, sugar diffuses throughout the entire cup, seeking a uniform concentration. Fick's second law provides the mathematical foundation for predicting how quickly and extensively this diffusion occurs.

When solving related problems, the key is to set up the partial differential equation representing Fick's second law, consider the boundary conditions, and find a solution that fits the scenario. In the case of steel undergoing decarburization, these steps help us determine the depth at which the carbon concentration reaches a specified figure.

Mass diffusivity

The term mass diffusivity, also known as diffusivity, describes the ease with which atoms, ions, or molecules can move within a medium. It is an intrinsic property of the system and depends on several factors, including temperature, the nature of the diffusing species, and the matrix they are moving through.

For the decarburization problem, mass diffusivity tells us how quickly carbon atoms in the steel are able to spread out from high concentration areas to lower ones. The higher the diffusivity, the faster this process occurs. Therefore, knowing the mass diffusivity of carbon in steel at a given temperature—like the provided value at 925°C—is essential for accurately calculating diffusion-related changes.

This property has a direct impact on the manufacturing and treatment of steel, influencing the final properties such as hardness and tensile strength. Engineers and metallurgists must account for diffusivity when designing processes to ensure that the material properties meet the required specifications.

Error function

The error function is a mathematical function that plays a central role in probability, statistics, and partial differential equations related to diffusion processes. It is denoted as 'erf' and is related to the Gaussian, or normal distribution, which is a fundamental probability distribution in statistics.

The error function is especially important when solving problems with Fick's second law of diffusion because it frequently appears in the solution of diffusion problems where there is symmetry and constant boundary conditions, as found in typical decarburization scenarios.

When we look at the concentration profile in a diffusing material, the error function allows us to define how the concentration changes in space at any given time. It's a tool that helps us transform complex integral expressions into a manageable form, which we can then use to find numerical values for specific variables, such as the depth at which the concentration of carbon is reduced by 40%.

Carbon concentration in steel

The carbon concentration in steel significantly influences its mechanical properties, such as hardness, ductility, and tensile strength. Carbon is a critical element in steel, and altering its concentration can change the steel's performance characteristics. The decarburization process, a type of heat treatment, intentionally reduces the surface carbon concentration to achieve required material properties for specific applications.

In the context of our exercise, the steel undergoes decarburization at a high temperature, leading to a drop in carbon concentration from the surface inward. By applying principles like Fick's second law of diffusion and calculations involving the mass diffusivity and error function, we can predict the carbon content at different depths of the steel over time.

Understanding how carbon concentration varies within steel is key when designing and manufacturing steel components. It ensures that the products made meet the exacting requirements of structural integrity and resilience needed for their intended use.