Question

A heated piece of steel, with a uniform initial carbon concentration of \(0.20 \%\) by mass, was exposed to a carburizing atmosphere for an hour. Throughout the entire process, the carbon concentration on the surface was \(0.70 \%\). If the mass diffusivity of carbon in steel in this process was uniform at \(1 \times\) \(10^{-11} \mathrm{~m}^{2} / \mathrm{s}\), determine the percentage of mass concentration of carbon at \(0.2 \mathrm{~mm}\) and \(0.4 \mathrm{~mm}\) below the surface after the process.

Short answer

Based on the given information and by applying Fick's second law of diffusion, we found that the percentage of mass concentration of carbon after the carburizing process at 0.2 mm and 0.4 mm below the surface is approximately 0.46% and 0.22%, respectively.

Step-by-step solution

Note down the given information

We are given the following information: - Initial carbon concentration, \(C_i = 0.20 \%\) - Surface carbon concentration during the process, \(C_s = 0.70 \%\) - Mass diffusivity of carbon in steel, \(D = 1 \times 10^{-11} \mathrm{m}^2 / \mathrm{s}\) - Time duration of the process, \(t = 1 \text{ hour}\) - The desired depth below the surface: \(x_1 = 0.2 \mathrm{mm}\) and \(x_2 = 0.4 \mathrm{mm}\) Before proceeding with the calculations, let's convert the time from hours to seconds, and both the depths from millimeters to meters: - \(t = 1 \text{ hour} \times 3600 \frac{\text{s}}{\text{hour}} = 3600 \text{ s}\) - \(x_1 = 0.2 \mathrm{mm} \times 10^{-3} \frac{\text{m}}{\text{mm}} = 2 \times 10^{-4} \text{ m}\) - \(x_2 = 0.4 \mathrm{mm} \times 10^{-3} \frac{\text{m}}{\text{mm}} = 4 \times 10^{-4} \text{ m}\)

Apply Fick's second law of diffusion

The diffusion process can be described by Fick's second law, which for this case, can be simplified and evaluated in the error function complementary form. Since there's no dependence on steels shape and size, the equation becomes: \(C(x, t) = C_i + (C_s - C_i) \cdot \mathrm{erfc} \left(\frac{x}{2 \sqrt{Dt}}\right)\) Where \(C(x, t)\) is the carbon concentration at depth \(x\) after time \(t\), and \(\mathrm{erfc}(x)\) is the complementary error function.

Calculate the carbon concentrations at the desired depths

Now, we will use the equation from Step 2 to find the percentage of mass concentration of carbon after the process at the desired depths below the surface, \(x_1 = 2 \times 10^{-4} \text{ m}\) and \(x_2 = 4 \times 10^{-4} \text{ m}\). For \(x_1\): \(C(x_1, t) = 0.20 + (0.70 - 0.20) \cdot \mathrm{erfc} \left(\frac{2 \times 10^{-4}}{2\sqrt{(1 \times 10^{-11})(3600)}}\right)\) For \(x_2\): \(C(x_2, t) = 0.20 + (0.70 - 0.20) \cdot \mathrm{erfc} \left(\frac{4 \times 10^{-4}}{2\sqrt{(1 \times 10^{-11})(3600)}}\right)\)

Calculate the error function complementary values

For our convenience, we can use a calculator, a mathematical software or online tool to find the values of the complementary error function. For \(x_1\): \(\mathrm{erfc} \left(\frac{2 \times 10^{-4}}{2\sqrt{(1 \times 10^{-11})(3600)}}\right) \approx 0.5203\) For \(x_2\): \(\mathrm{erfc} \left(\frac{4 \times 10^{-4}}{2\sqrt{(1 \times 10^{-11})(3600)}}\right) \approx 0.8427\)

Substitute the erfc values back into the equation

Now we will substitute the erfc values back into the equation from Step 3 to find the percentage of mass concentration of carbon after the process. For \(x_1\): \(C(x_1, t) = 0.20 + (0.70 - 0.20) \times 0.5203 \approx 0.46 \%\) For \(x_2\): \(C(x_2, t) = 0.20 + (0.70 - 0.20) \times 0.8427 \approx 0.22 \%\)

Write down the final results

After the carburizing process, the percentage of mass concentration of carbon at 0.2 mm and 0.4 mm below the surface is as follows: - At \(0.2 \mathrm{mm}\) depth: \(C(x_1, t) \approx 0.46 \%\) - At \(0.4 \mathrm{mm}\) depth: \(C(x_2, t) \approx 0.22 \%\)

Key concepts

Fick's Second Law of Diffusion

Fick's second law of diffusion is a fundamental equation that describes the evolution of concentration within a material over time, as particles, such as carbon atoms in steel, move from regions of high concentration to lower concentration areas. This law is mathematically consolidated through the diffusion equation, which takes into account the varying concentration gradients over time.
When dealing with diffusion in solid materials, like carbon in heated steel, this law provides a predictive model for how concentration changes at a point inside the material, not just at the interface. This is particularly useful in metallurgy for processes such as carburizing, where the surface carbon concentration is different from the interior.
In practice, to apply Fick's second law and calculate the carbon concentration at a certain depth within steel after some time, the knowledge of initial concentration, surface concentration, mass diffusivity, and time are crucial. The solution often involves the use of mathematical functions, such as the error function or its complement, to simplify and solve the equation.

Mass Diffusivity

Mass diffusivity, often represented by the symbol 'D', is crucial in the study of diffusion processes. It is a material-specific constant that quantifies the rate at which particles can spread through a material. In the context of carbon diffusion in steel, mass diffusivity determines how rapidly carbon atoms can migrate within the steel structure when there is a non-uniform distribution.
Mass diffusivity depends on several factors including temperature, the presence of other elements, and the crystal structure of the material. Diffusivity is measured in square meters per second (m^2/s). In our problem, the uniform mass diffusivity is given as 1 × 10^-11 m^2/s, which means that under the specified conditions, carbon atoms in steel diffuse at this stated rate. This value is crucial for calculations using Fick's second law as it directly impacts the extent to which carbon penetrates the steel over time.

Complementary Error Function

The complementary error function, often abbreviated as erfc, is a mathematical function derived from the integral of the Gaussian distribution and is related to the error function erf. In the context of diffusion problems, the erfc is particularly useful because it simplifies the expressions obtained when solving the diffusion equations, such as Fick's second law.
The utilization of the complementary error function in diffusion problems allows for the description of how concentration profiles change over time within a material. For instance, in the step-by-step solution provided, it is used to calculate the carbon concentration at specific depths in the steel. The erfc tells us the fraction of carbon that remains below a certain depth after a period of exposure to a carburizing atmosphere. The lower the value of erfc, the higher the concentration of carbon at that depth. For practical purposes, computing erfc often requires a scientific calculator or software as it's not a straightforward operation like basic arithmetic.

Carbon Concentration

Carbon concentration in steel is a critical measure that determines the material's characteristics such as hardness, strength, and ductility. It refers to the percentage of carbon by mass within the steel composition. Carbon concentration varies throughout a piece of steel when exposed to a process like carburizing; the surface might be saturated with carbon while the core remains at its original concentration.
Understanding and controlling carbon concentration is key in material engineering as it helps define the steel's final properties. In computational terms, within the diffusion context, carbon concentration is a variable that changes with position and time, and its value can be calculated using Fick's second law provided the necessary parameters are known. In the exercise, we calculated the carbon concentration at different depths after carburizing to determine the effect of the process on steel's properties. This is essential for applications requiring specific performance characteristics from the steel.