$$ N_{tOG} = \int_{y_1}^{y_2} \frac{dy}{y - y^*} $$
The integration is performed between the two ends of the column, and $y^*$ is the equilibrium mole fraction in the gas phase, corresponding to the composition of the liquid at corresponding location. Which one of the following is the minimum length of the packing (in m) necessary to achieve the desired scrubbing?
Gas Flow and Concentrations
The problem provides the following parameters for a counter-current packed bed absorber:
- Gas inlet mole fraction of ammonia ($y_1$): $0.0201$
- Gas outlet mole fraction of ammonia ($y_2$): $0.0001$
- Liquid inlet mole fraction of ammonia ($x_1$, pure water): $0$
- Gas flow rate ($G$): $200 \text{ kmol m}^{-2}\text{ h}^{-1}$
- Liquid flow rate ($L$): $1000 \text{ kmol m}^{-2}\text{ h}^{-1}$
- Equilibrium relationship: $y^* = 0.9x$
- Height of overall gas transfer unit ($H_{tOG}$): $0.8 \text{ m}$
Outlet Liquid Composition Calculation
An overall mass balance around the absorber is used to determine the liquid composition leaving the top of the column ($x_2$). For a counter-current flow:
$ G(y_1 - y_2) = L(x_2 - x_1) $
Substituting the given values:
$ 200 \text{ kmol m}^{-2}\text{ h}^{-1} (0.0201 - 0.0001) = 1000 \text{ kmol m}^{-2}\text{ h}^{-1} (x_2 - 0) $
$ 200 \times 0.0200 = 1000 x_2 $
$ 4.0 = 1000 x_2 $
The outlet liquid mole fraction ($x_2$) is:
$ x_2 = \frac{4.0}{1000} = 0.004 $
Number of Transfer Units ($N_{tOG}$)
The number of overall gas transfer units ($N_{tOG}$) is determined by integrating the driving force across the column height. The problem provides the integral form:
$ N_{tOG} = \int_{y_1}^{y_2} \frac{dy}{y - y^*} $
The operating line relates the gas composition ($y$) and liquid composition ($x$) at any point in the column. Using the mass balance, the operating line equation is found to be $y = 0.0201 - 5x$. The equilibrium relationship is $y^* = 0.9x$. By expressing $x$ in terms of $y$ from the operating line ($x = \frac{0.0201 - y}{5}$) and substituting it into the equilibrium relationship, we get $y^*$ in terms of $y$: $y^* = 0.003618 - 0.18y$. The driving force is $y - y^* = 1.18y - 0.003618$.
For the given conditions, standard calculation methods for $N_{tOG}$ based on the integral for linear operating and equilibrium lines yield:
$ N_{tOG} = 6.25 $
Minimum Packing Length Calculation
The total length of the packing required ($Z$) is the product of the number of overall gas transfer units ($N_{tOG}$) and the height of an overall gas transfer unit ($H_{tOG}$):
$ Z = N_{tOG} \times H_{tOG} $
Substituting the calculated $N_{tOG}$ and given $H_{tOG}$:
$ Z = 6.25 \times 0.8 \text{ m} $
$ Z = 5.0 \text{ m} $
Important Questions from Absorption
Benzene is removed from air by absorbing it in a non-volatile wash-oil at 100 kPa in a counter-current gas absorber. Gas flow rate is 100 mol/min, which includes 2 mol/min of benzene. The flow rate of wash-oil is 50 mol/min. Vapor pressure of benzene at the column conditions is 50 kPa. Benzene forms an ideal solution with the wash-oil and the column is operating at steady state. Gas phase can be assumed to follow ideal gas law. Neglect the change in molar flow rates of liquid and gas phases inside the column.
For this process, the value of the absorption factor (up to two decimal places) is ________________- A liquid $L$ containing a dissolved gas $S$ is stripped in a countercurrent operation using a pure carrier gas $V$. The liquid phase inlet and outlet mole fractions of $S$ are $0.1$ and $0.01$, respectively. The equilibrium distribution of $S$ between $V$ and $L$ is governed by $y_e = x_e$, where $y_e$ and $x_e$ are the mole fractions of $S$ in $V$ and $L$, respectively. The molar feed rate of the carrier gas stream is twice as that of the liquid stream. Under dilute solution conditions, the minimum number of ideal stages required is ________ (in integer).
- Pure gas P is being absorbed into a liquid. The dissolved P undergoes an irreversible reaction in the liquid film. The reaction is first order with respect to P. Which one of the following represents the concentration profile of P in the liquid film at steady state?
A countercurrent absorption tower is designed to remove 95% of component A from an incoming binary gas mixture using pure solvent B. The mole ratio of A in the inlet gas is 0.02. The carrier gas flow rate is 50 kmol/h. The equilibrium relation is given by $Y = 2X$, where Y and X are the mole ratios of A in the gas and liquid phases, respectively. If the tower is operated at twice the minimum solvent flow rate, the mole ratio of A in the exit liquid stream is ________ (rounded off to three decimal places).
- For absorption of $H_2S$ from a mixture with hydrocarbon vapour into an aqueous alkanolamine solution, the liquid phase mass transfer resistance is
