Analyzing the Concentration Profile
The problem describes the absorption of a pure gas P into a liquid, where P undergoes an irreversible, first-order reaction within the liquid film. We need to determine the concentration profile of P in this film at steady state.
Key Factors Influencing Concentration
- Absorption: Gas P enters the liquid at the gas-liquid interface. Its concentration is highest at the interface (let's denote it as $C_{P,i}$).
- Diffusion: P diffuses from the interface into the bulk liquid film.
- Reaction: P is consumed by a first-order irreversible reaction ($R_P = k_1 C_P$) within the film. This reaction removes P, decreasing its concentration.
- Steady State: The system has reached a steady state, meaning the concentration profile is constant over time. The rate of P diffusion towards any point in the film is balanced by the rate of reaction consuming P at that point and diffusion away from it.
Expected Concentration Behavior
- The concentration of P must be maximum at the interface ($x=0$), i.e., $C_P = C_{P,i}$.
- As P diffuses into the liquid film (increasing $x$), its concentration decreases due to consumption by the reaction.
- The rate of decrease is influenced by both the diffusion rate and the reaction rate. Since the reaction consumes P, the concentration gradient will be steeper than in the absence of a reaction.
- The concentration profile will therefore decrease from $C_{P,i}$ at the interface towards a lower value in the bulk liquid. The profile is typically non-linear due to the combination of diffusion and reaction kinetics.
Evaluating the Options
- Option 1 shows concentration increasing into the film, which is incorrect for absorption.
- Option 3 shows a linear decrease, representing diffusion without reaction or a zero-order reaction, not a first-order reaction.
- Option 4 shows concentration increasing from zero at the interface, which is incorrect.
- Option 2 shows a concentration profile that starts at a maximum at the interface and decreases curvilinearly into the film. This shape accurately reflects the combined effects of diffusion and a first-order chemical reaction consuming the absorbed species. The concentration decreases as it moves away from the interface and gets consumed.
Governing Equation (for context)
The concentration profile $C_P(x)$ in the film (where $x$ is the distance from the interface) is governed by the diffusion-reaction equation:
$D_P \frac{d^2 C_P}{dx^2} - k_1 C_P = 0$With boundary conditions typically being $C_P(0) = C_{P,i}$ and $C_P(L) = C_{P,b}$ (concentration in the bulk liquid). The solution to this equation results in a decaying concentration profile, consistent with Option 2.
Important Questions from Absorption
- A gas stream with $2.01 \text{ mol }\%$ ammonia is to be scrubbed in a counter-current isothermal packed bed absorber using pure water to reduce its concentration to $0.01 \text{ mol }\%$. Assume that dilute conditions apply, operating line is linear and the mass transfer coefficients are constant throughout the column. The liquid and the gas flows inside the absorber (in $\text{kmol m}^{-2}\text{ h}^{-1}$) are 1000 and 200, respectively. The equilibrium relationship is $y = 0.9 x$, where, $y$ is the mole-fraction of ammonia in the gas phase and $x$ is that in the liquid. Under these conditions, the height of overall gas transfer unit ($H_{tOG}$) is $0.8 \text{ m}$ and the number of overall gas transfer units ($N_{tOG}$) is given by the following integral.
$$ N_{tOG} = \int_{y_1}^{y_2} \frac{dy}{y - y^*} $$
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? 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).
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
