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) \(5.2 \times 10^{-8} \mathrm{~kg} / \mathrm{s}\)

Short answer

Answer: To find the mass flow rate of carbon per unit length through the iron cylinder, we used Fick's first law of diffusion and calculated the mass flow rate per unit length as: \(Q = - (3 \times 10^{-11} \tfrac{\text{m}^2}{\text{s}}) \tfrac{0.5 \text{ kg}/\text{m}^3}{0.0012 \text{ m}} (\pi (0.0362)^2 - \pi (0.035)^2)\). Comparing the calculated value with the given options, we can determine the closest option to the correct mass flow rate.

Step-by-step solution

Convert the given dimensions to SI units

The diameter and thickness are given in centimeters and millimeters, respectively. We need to convert these values to meters. Inner diameter = 7 cm = 0.07 m Iron thickness = 1.2 mm = 0.0012 m

Calculate the area through which diffusion occurs

To find the area of the annulus (ring-like) shape, we need to first find the area of the outer circle and then subtract the area of the inner circle. Outer radius = (Inner diameter / 2) + Iron thickness = (0.07 / 2) + 0.0012 = 0.0362 m Inner radius = Inner diameter / 2 = 0.07 / 2 = 0.035 m Area of the outer circle, \(A_o = \pi R_o^2 = \pi (0.0362)^2\) Area of the inner circle, \(A_i = \pi R_i^2 = \pi (0.035)^2\) Annulus area, A = \(A_o - A_i\)

Apply Fick's first law of diffusion

Fick's first law: \(j = -D \frac{\Delta C}{\Delta x}\) Here, \(j\) is the mass flux of carbon through iron, D is the diffusion coefficient, \(\Delta C\) is the change in concentration of carbon, and \(\Delta x\) is the thickness of the iron. ∆C = 0.5 kg/m³ (concentration at inner surface) - 0 kg/m³ (concentration at outer surface) = 0.5 kg/m³ ∆x = 0.0012 m (thickness of iron) Now we can substitute the given values into Fick's first law: \(j = - (3 \times 10^{-11} \tfrac{\text{m}^2}{\text{s}}) \tfrac{0.5 \text{ kg}/\text{m}^3}{0.0012 \text{ m}}\)

Calculate the mass flow rate per unit length

The mass flow rate per unit length, Q can be found using the following equation: \(Q = j \times A\) Substitute the expression for \(j\) and the area A to calculate the mass flow rate per unit length: \(Q = - (3 \times 10^{-11} \tfrac{\text{m}^2}{\text{s}}) \tfrac{0.5 \text{ kg}/\text{m}^3}{0.0012 \text{ m}} (\pi (0.0362)^2 - \pi (0.035)^2)\)

Compare the calculated value with the given options

Finally, compare the calculated mass flow rate per unit length with the given options and choose the closest one: (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) \(5.2 \times 10^{-8} \mathrm{~kg} / \mathrm{s}\) Find the closest option to the calculated value.