A column packed with \(5-\mathrm{cm}\) polypropylene saddles \(\left(a=55 \mathrm{m}^{2} / \mathrm{m}^{3}\right)\) is being designed for the removal of chlorine from a gas stream \((G=100\) mol/s \(\cdot \mathrm{m}^{2}, 2.36 \% \mathrm{Cl}_{2}\) ) by countercurrent contact with an \(\mathrm{NaOH}\) solution \(\left(L=250 \mathrm{mol} / \mathrm{s} \cdot \mathrm{m}^{2}, 10 \% \mathrm{NaOH}, C_{\mathrm{B}}=2736 \mathrm{mol} / \mathrm{m}^{3}\right)\) at about \(40-45^{\circ} \mathrm{C}\) and 1 atm. How high should the tower be for \(99 \%\) removal of chlorine? Double the calculated height to take care of deviations from plug flow. Data The reaction \(\mathrm{Cl}_{2}+2 \mathrm{NaOH} \rightarrow\) product is very very fast and irreversible. For these very high flow rates (close to the limits allowed) an extrapolation of the correlations in Perry 6 th ed., section \(14,\) gives $$\begin{array}{rlrl} k_{g} a & =133 \mathrm{mol} / \mathrm{hr} \cdot \mathrm{m}^{3} \cdot \mathrm{atm} & H_{\mathrm{A}} & =125 \times 10^{6} \mathrm{atm} \cdot \mathrm{m}^{3} / \mathrm{mol} \\ k_{l} \mathrm{a} & =45 \mathrm{hr}^{-1} & \mathscr{D} & =1.5 \times 10^{-9} \mathrm{m}^{2 / \mathrm{s}} \end{array}$$

When it comes to designing a packed column for applications such as gas absorption, several factors must be taken into consideration to ensure efficient and effective operation. Designing a packed column involves selecting suitable packing material that provides a high surface area for mass transfer between the gas and the liquid phases.

For instance, polypropylene saddles are a common packing choice due to their large surface area and open geometry, which facilitate good contact between the two phases. The specific surface area, denoted by 'a' in the given problem, is a critical parameter, reflecting the accessible surface area per unit volume of the packing.

The packed column's height is another critical design parameter. It's not just about accommodating the necessary volume of packing but ensuring that the packing is tall enough to achieve the desired level of mass transfer. This is calculated by determining the number of transfer units (NTU) and the height of a transfer unit (HTU), which are based on mass transfer coefficients and the solubility of the gas in the liquid, governed by Henry's Law.

Finally, operational conditions such as temperature and pressure also play a role in the packed column design. In the case of chlorine absorption, a temperature around 40-45°C and atmospheric pressure are specified, which directly affect the reaction kinetics and equilibrium relationships.

The mass transfer coefficient is a measure of the transfer rate of a species from one phase to another, often used to design and analyze processes like gas absorption in a packed column.

In the provided exercise, the overall gas-phase mass transfer coefficient, denoted as \( k_g a \) for chlorine gas, is extrapolated from existing correlations. This coefficient represents the product of the individual gas-phase mass transfer coefficient \( k_g \) and the specific surface area \( a \).

To isolate \( k_g \) itself, we divide the overall coefficient by the packing's surface area. Understanding and accurately calculating \( k_g \) is crucial as it determines how quickly the solute can move from the gas phase to the gas-liquid interface.

The effectiveness of the process heavily depends on the mass transfer resistances, which are represented by the mass transfer coefficients in both gas and liquid phases. Choosing the correct type of packing and operating conditions can help optimize \( k_g \) and enhance the efficiency of the column.

Gas absorption is a process where a solute is transferred from the gas phase to the liquid phase, often with the aim of removing a particular component from a gas stream.

In our exercise, chlorine gas is being removed from a gas stream by absorption into an \( \text{NaOH} \) solution. This is a classic example of a unit operation in chemical engineering where a packed column is used to facilitate the process.

The degree of absorption can be quantified by the efficiency of solute removal, which in this case, is 99%. To achieve the desired removal efficiency, the packed column needs to be designed to provide sufficient contact time and surface area, enabling the transfer of chlorine from the gas to the liquid phase where it reacts with \( \text{NaOH} \) to form the product.

The design and operation of the gas absorption column are closely linked to the concept of the mass transfer coefficient and Henry's Law. The latter gives insights into solubility and helps determine the driving force for mass transfer.

Henry's Law is fundamental in processes like gas absorption, as it describes the solubility of a gas in a liquid under equilibrium conditions. According to Henry's Law, at a constant temperature, the amount of dissolved gas is proportional to its partial pressure in the gas phase.

In mathematical terms, the law can be stated as \( C = H_{A} P \), where \( C \) is the solute concentration in the liquid, \( H_{A} \) is the Henry's law constant for the solute, and \( P \) is the partial pressure of the gas.

In the problem, the Henry's Law constant \( H_{A} \) for chlorine is a critical value provided for the calculations. It is used in tandem with the gas-phase mass transfer coefficient to determine the height of a transfer unit (HTU) for the gas phase, a parameter needed to estimate the required column height. The value of \( H_{A} \) reflects the affinity of chlorine for the liquid phase and is influenced by factors such as temperature and the nature of the liquid solvent, here being an \( \text{NaOH} \) solution.