Question

A packed distillation column, with vapor having an average molecular weight of \(45~\text{kg}\cdot\text{kmol}^{-1}\), density of \(2~\text{kg}\cdot\text{m}^{-3}\) and a molar flow rate of \(0.1~\text{kmol}\cdot\text{s}^{-1}\), has a flooding velocity of \(0.15~\text{m}\cdot\text{s}^{-1}\). The column is designed to operate at \(60%\) of the flooding velocity. Which one of the following is the CORRECT value for the column diameter (in m)?

• A. \(\dfrac{5}{\sqrt{\pi}}\)
• B. \(5\sqrt{\pi}\)
• C. \(4\pi\)
• D. \(\dfrac{10}{\sqrt{\pi}}\)

Hint:

Use \(\dot V=\dot n/C\) with \(C=\rho/M\) to convert molar flow to volumetric flow.
For packed columns, size using the design superficial velocity: \(A=\dot V/u\).
Finally, \(D=\sqrt{4A/\pi}\).

Answer 1

The Correct Option is D

Step 1: Convert the given mass density to molar density using the molecular weight \(M=45~\text{kg}\cdot\text{kmol}^{-1}\): \[ C = \frac{\rho}{M}=\frac{2}{45}~\text{kmol}\cdot\text{m}^{-3}. \] Volumetric flow rate of vapor: \[ \dot V = \frac{\dot n}{C}=\frac{0.1}{2/45}=2.25~\text{m}^3\!\cdot\!\text{s}^{-1}. \] Step 2: Design superficial vapor velocity (at \(60%\) of flooding): \[ u = 0.6\,u_f = 0.6\times 0.15 = 0.09~\text{m}\cdot\text{s}^{-1}. \] Step 3: Required cross-sectional area and diameter: \[ A=\frac{\dot V}{u}=\frac{2.25}{0.09}=25~\text{m}^2,\qquad D=\sqrt{\frac{4A}{\pi}}=\sqrt{\frac{100}{\pi}}=\frac{10}{\sqrt{\pi}}~\text{m}. \] Thus, the correct diameter is \(\dfrac{10}{\sqrt{\pi}}\) m.