A gas that contains CO₂ is contacted with liquid water in an agitated batch absorber. The equilibrium solubility of CO₂ in water is given by Henry’s law (Section 6.4b)

C_A = p_A/H_A

where C_A(mol/cm³) = concentration of CO₂ in solution, p_A(atm) = partial pressure of CO₂ in the gas phase, and H_A[atm/(mol/cm³)] = Henry’s law constant. The rate of absorption of CO₂ (i.e., the rate of transfer of CO₂ from the gas to the liquid per unit area of gas–liquid interface) is given by the expression

where C_A = actual concentration of CO₂ in the liquid, C*_A = concentration of CO₂ in the liquid that would be in equilibrium with the CO2 in the gas phase, and k(cm/s) = a mass transfer coefficient.

The gas phase is at a total pressure P(atm) and contains y_A(mol CO₂/mol gas), and the liquid phase initially consists of V(cm³) of pure water. The agitation of the liquid phase is sufficient for the composition to be considered spatially uniform, and the amount of CO² absorbed is low enough for P, V, and y_A to be considered constant throughout the process.

(a) Derive an expression for dC_A/dt and provide an initial condition. Without doing any calculations, sketch a plot of C_A versus t, labeling the value of C_A at t = 0 and the asymptotic value at t → 1.

Give a physical explanation for the asymptotic value of the concentration.

(b) Prove that

where S(cm²) is the effective contact area between the gas and liquid phases.

(c) Suppose the system pressure is 20.0 atm, the liquid volume is 5.00 liters, the tank diameter is 10.0 cm, the gas contains 30.0 mole% CO₂, the Henry’s law constant is 9230 atm/(mole/cm³), and the mass transfer coefficient is 0.020 cm/s. Calculate the time required for C_A to reach 0.620 mol/L if the gas-phase properties remain essentially constant.

(d) If A were not CO₂ but instead a gas with a moderately high solubility in water, the expression for C_A given in Part (b) would be incorrect. Explain where the derivation that led to it would break down.

Transcribed Image Text:

ra = k(C - CA) [mol/(cm² s)]