Question

Carbon at \(1273 \mathrm{~K}\) is contained in a \(7-\mathrm{cm}\)-innerdiameter cylinder made of iron whose thickness is \(1.2 \mathrm{~mm}\). The concentration of carbon in the iron at the inner surface is \(0.5 \mathrm{~kg} / \mathrm{m}^{3}\) and the concentration of carbon in the iron at the outer surface is negligible. The diffusion coefficient of carbon through iron is \(3 \times 10^{-11} \mathrm{~m}^{2} / \mathrm{s}\). The mass flow rate of carbon by diffusion through the cylinder shell per unit length of the cylinder is (a) \(2.8 \times 10^{-9} \mathrm{~kg} / \mathrm{s}\) (b) \(5.4 \times 10^{-9} \mathrm{~kg} / \mathrm{s}\) (c) \(8.8 \times 10^{-9} \mathrm{~kg} / \mathrm{s}\) (d) \(1.6 \times 10^{-8} \mathrm{~kg} / \mathrm{s}\) (e) \(5.2 \times 10^{-8} \mathrm{~kg} / \mathrm{s}\) 14-185 The surface of an iron component is to be hardened by carbon. The diffusion coefficient of carbon in iron at \(1000^{\circ} \mathrm{C}\) is given to be \(3 \times 10^{-11} \mathrm{~m}^{2} / \mathrm{s}\). If the penetration depth of carbon in iron is desired to be \(1.0 \mathrm{~mm}\), the hardening process must take at least (a) \(1.10 \mathrm{~h}\) (b) \(1.47 \mathrm{~h}\) (c) \(1.86 \mathrm{~h}\) (d) \(2.50 \mathrm{~h}\) (e) \(2.95 \mathrm{~h}\)

Short answer

Question: Determine the mass flow rate of carbon by diffusion through the cylinder shell per unit length of the cylinder. Answer: (e) \(5.2 \times 10^{-8} \mathrm{~kg} /\mathrm{s}\)

Step-by-step solution

1. Write down Fick's first law of diffusion

Fick's first law of diffusion states that: \(J = -D \frac{\Delta C}{\Delta x}\), where \(J\) is the mass flow rate per unit cross-sectional area, \(D\) is the diffusion coefficient, \(\Delta C\) is the difference in concentration, and \(\Delta x\) is the thickness of the material.

2. Determine the difference in concentration and the thickness

Given that the concentration of carbon in the iron at the inner surface is \(\frac{0.5 kg}{m^3}\) and the concentration at the outer surface is negligible, the difference in concentration is \(\Delta C = 0.5 \frac{kg}{m^3}\). The thickness of the iron cylinder is 1.2 mm, which is equal to \(\Delta x = 1.2 \times 10^{-3}m\).

3. Calculate the mass flow rate per unit cross-sectional area

Using Fick's first law of diffusion, we can determine the mass flow rate per unit cross-sectional area \(J\): \(J = -D \frac{\Delta C}{\Delta x} = - (3 \times 10^{-11} \frac{m^2}{s}) \frac{0.5 \frac{kg}{m^3}}{1.2 \times 10^{-3} m} = - 1.25 \times 10^{-5} \frac{kg}{m^2 \cdot s}\) The negative sign indicates that the mass flow rate is in the opposite direction of the concentration gradient.

4. Calculate the mass flow rate per unit length of the cylinder

Now, we need to find the mass flow rate per unit length of the cylinder by multiplying \(J\) by the cross-sectional area of the iron shell: Area of the iron shell = \(\pi (r_2^2 - r_1^2)\), where \(r_1 = 3.5 \times 10^{-2} m\) and \(r_2 = (3.5 + 1.2) \times 10^{-2} m\) are the inner and outer radii of the cylinder. Area of the iron shell = \(\pi ((3.7 \times 10^{-2} m)^2 - (3.5 \times 10^{-2} m)^2) = 4.494 \times 10^{-3} m^2\) Now, we can determine the mass flow rate per unit length (\(M\)) by multiplying \(J\) by the Area of the iron shell: \(M = J \times \mathrm{Area~of~the~iron~shell} = (- 1.25 \times 10^{-5} \frac{kg}{m^2 \cdot s}) (4.494 \times 10^{-3} m^2) = -5.62 \times 10^{-8} \frac{kg}{s}\) Since we are asked for the magnitude of the mass flow rate, the answer is \(5.62 \times 10^{-8} \frac{kg}{s}\). The closest answer in the given options is (e) \(5.2 \times 10^{-8} \mathrm{~kg} /\mathrm{s}\).

Key concepts

Mass Flow Rate

The concept of mass flow rate is critical in understanding how substances move through different mediums by the process of diffusion. In simple terms, mass flow rate refers to the amount of mass per unit time that flows through a unit area. It's an essential measure in various fields such as chemistry, physics, and engineering.
In the context of Fick's First Law of Diffusion, mass flow rate is denoted by the symbol \( J \), which is defined as the mass flow per area per time. The formula for mass flow rate per unit area is given by:

• \( J = -D \frac{\Delta C}{\Delta x} \)

Where \( D \) is the diffusion coefficient, \( \Delta C \) is the change in concentration across the material, and \( \Delta x \) is the thickness of that material.
Here, the negative sign indicates that the mass flows from a region of higher concentration to lower concentration, suggesting that diffusion occurs down the concentration gradient. This means carbon atoms move from the inner surface of the iron cylinder where the concentration is high towards the outer surface where it is negligible.

Diffusion Coefficient

The diffusion coefficient \( D \) is a factor that quantifies the ease with which a substance diffuses through a medium. It plays a vital role in determining how fast or slow diffusion occurs. In the given exercise, the diffusion coefficient for carbon diffusing through iron is provided as \( 3 \times 10^{-11} \mathrm{~m^2/s} \).

The diffusion coefficient depends on several factors:

• Temperature: As the temperature increases, the diffusion coefficient generally increases. Higher temperatures provide energy to the particles, increasing their movement and thus the rate of diffusion.
• Medium: The diffusion coefficient can vary based on whether the medium is a solid, liquid, or gas. In gases, diffusion occurs rapidly, while in solids, it is usually much slower.

In practical applications, knowing the diffusion coefficient helps in designing processes, such as the hardening of iron components by carbon diffusion, which determines parameters like time and temperature needed.

Concentration Gradient

The concentration gradient is a crucial concept in diffusion processes. It refers to the variation in concentration across space. In our scenario, the carbon concentration differs from one side of the iron cylinder to the other - it's higher at the inner surface and negligible at the outer surface.

Total concentration change \( \Delta C \) is used in Fick's Law to determine how rapidly diffusion occurs through a material:

• \( \Delta C = C_{inner} - C_{outer} = 0.5 \mathrm{~kg/m^3} - 0 \mathrm{~kg/m^3} = 0.5 \mathrm{~kg/m^3} \)
• \( \Delta x = 1.2 \times 10^{-3} \mathrm{~m} \), the thickness of the material, is used as the distance over which the concentration changes.

A steep concentration gradient signifies a large difference between the two concentrations over a short distance, leading to a higher diffusion rate. Understanding and controlling the concentration gradient is essential for optimizing diffusion in industrial applications, such as in the hardening process where distribution of carbon into iron must be even and controlled.