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 percent from its initial value as a result of the decarburization process for \((a)\) an hour and \((b) 10\) hours. Assume the concentration of carbon at the surface is zero throughout the decarburization process.

Short answer

Question: 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 after 1 hour and after 10 hours. Answer: The depth below the surface of the steel at which the concentration of carbon is reduced to 40% from its initial value is approximately 0.0476 cm after 1 hour and 0.1504 cm after 10 hours.

Step-by-step solution

Understand Fick's Second Law

Fick's second law describes the diffusion of particles through a medium and can be written as: $$\frac{\partial C}{\partial t} = D \frac{\partial^2 C}{\partial x^2}$$ where \(C\) is the concentration of particles, \(t\) is the time, \(D\) is the mass diffusivity, and \(x\) is the position in the medium.

Use the initial and boundary conditions

Since we are considering a diffusion problem, we need to apply proper initial and boundary conditions. The problem states that the concentration of carbon at the surface is zero throughout the process. Let the initial concentration of carbon be \(C_0\) and at the surface, \(C(0,t)=0\). Additionally, we know that the concentration at the desired depth should be 40% of the initial concentration: \(C(x,t) = 0.4 C_0\).

Apply the complementary error function

To solve this problem, we can use the complementary error function to express the concentration of carbon as follows: $$C(x,t) = C_0 \, \mathrm{erfc} \left( \frac{x}{2\sqrt{Dt}} \right)$$ Here, \(\mathrm{erfc}\) is the complementary error function, which is defined as: $$\mathrm{erfc}(z) = 1 - \frac{2}{\sqrt{\pi}} \int_0^z e^{-u^2} du$$

Solve for depth x

We are interested in finding the depth \(x\) at which the concentration of carbon is 40% of its initial value. We can set up the equation as: $$0.4 C_0 = C_0 \, \mathrm{erfc} \left( \frac{x}{2\sqrt{Dt}} \right)$$ Dividing both sides by \(C_0\), we get: $$0.4 = \mathrm{erfc} \left( \frac{x}{2\sqrt{Dt}} \right)$$ Now we need to find the value of \(x\) which satisfies this equation for the given times t.

Find the depth for 1 hour

Firstly, we have to determine the depth for one hour (3600 seconds). Using the given mass diffusivity and time, we can write: $$0.4 = \mathrm{erfc} \left( \frac{x}{2\sqrt{1\times10^{-7}\cdot 3600}} \right)$$ Using the inverse complementary error function, we can find the value of \(x\) that satisfies this equation: $$\frac{x}{2\sqrt{1\times10^{-7}\cdot 3600}} = \mathrm{erfc}^{-1} (0.4)$$ Solving for \(x\), we get: $$x \approx 0.0476 \, \mathrm{cm}$$

Find the depth for 10 hours

Now we have to determine the depth for 10 hours (36000 seconds). Using the given mass diffusivity and time, we can write: $$0.4 = \mathrm{erfc} \left( \frac{x}{2\sqrt{1\times10^{-7}\cdot 36000}} \right)$$ Using the inverse complementary error function, we can find the value of \(x\) which satisfies this equation: $$\frac{x}{2\sqrt{1\times10^{-7}\cdot 36000}} = \mathrm{erfc}^{-1} (0.4)$$ Solving for \(x\), we get: $$x \approx 0.1504 \, \mathrm{cm}$$ So 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 is approximately 0.0476 cm after 1 hour and 0.1504 cm after 10 hours.