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GATE 2021 CH – Question 39

Mass Transfer · Stage-wise and continuous contacting, stage efficiency, HTU and NTU · 2 marks · Multiple choice

In a solvent regeneration process, a gas is used to strip a solute from a liquid in a countercurrent packed tower operating under isothermal condition. Pure gas is used in this stripping operation. All solutions are dilute and Henry’s law, $y^*=mx$, is applicable. Here, $y^*$ is the mole fraction of the solute in the gas phase in equilibrium with the liquid phase of solute mole fraction $x$, and $m$ is the Henry’s law constant. Let $x_1$ be the mole fraction of the solute in the leaving liquid, and $x_2$ be the mole fraction of solute in the entering liquid. When the ratio of the liquid-to-gas molar flow rates is equal to $m$, the overall liquid phase Number of Transfer Units, $NTU_{OL}$, is given by

  1. $\frac{x_2-x_1}{x_1}$
  2. $\frac{x_2+x_1}{x_2-x_1}$
  3. $\ln\left(\frac{x_2}{x_1}\right)$
  4. $\ln\left(\frac{x_2+x_1}{x_2-x_1}\right)$

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Correct answer: (A) $\frac{x_2-x_1}{x_1}$

Explanation

For dilute streams the liquid and gas molar flow rates L and G are constant. A balance from the bottom, where the entering gas has $y_1=0$, gives $y=(L/G)(x-x_1)$.

With $L/G=m$, $y=m(x-x_1)$, so the liquid composition in equilibrium with the local gas is $x^*=y/m=x-x_1$. The stripping driving force is therefore **constant**: $x-x^*=x_1$.

Consequently
$$NTU_{OL}=\int_{x_1}^{x_2}\frac{dx}{x-x^*}=\frac{x_2-x_1}{x_1}.$$

**Answer: A.**