Vapor-liquid equilibrium is crucial for understanding absorption and stripping processes. Henry's law, Raoult's law, and K-values help predict how components distribute between phases. Absorption and stripping factors quantify the ease of separating components in these operations.

Mass transfer in gas-liquid systems involves concentration gradients and film resistances. The two-film theory models interface behavior, while overall mass transfer coefficients combine individual resistances. Understanding these concepts is essential for designing efficient absorption and stripping columns.

Vapor-Liquid Equilibrium and Mass Transfer in Absorption/Stripping

Vapor-liquid equilibrium in absorption

Henry's law for dilute solutions describes solute concentration in vapor phase proportional to liquid phase
- Expression: $y = Hx$ relates mole fractions in vapor (y) and liquid (x) phases
- $H$ : Henry's constant varies with temperature, pressure (atm⋅L/mol)
Raoult's law for ideal solutions predicts vapor pressure of components in liquid mixture
- Expression: $y_i = x_i P_i^*/P$ calculates vapor composition from liquid composition
- $P_i^*$ : vapor pressure of pure component $i$ at system temperature (kPa)
- $P$ : total system pressure (kPa)
K-values (vapor-liquid distribution ratios) measure tendency of component to vaporize
- $K_i = y_i/x_i$ represents ratio of mole fractions in vapor and liquid phases
- Related to equilibrium constant, indicates relative volatility of components
Absorption factor quantifies ease of absorption process
- $A = L/(mG)$ compares liquid flow rate to product of gas flow rate and equilibrium line slope
- $L$ : liquid flow rate (mol/h), $G$ : gas flow rate (mol/h), $m$ : slope of equilibrium line
Stripping factor measures difficulty of removing solute from liquid phase
- $S = mG/L = 1/A$ inverse of absorption factor
- Higher stripping factor indicates easier removal of solute from liquid

Mass transfer in gas-liquid systems

Two-film theory models mass transfer resistance at gas-liquid interface
- Gas film resistance limits transfer of sparingly soluble gases (O₂ in water)
- Liquid film resistance controls transfer of highly soluble gases (NH₃ in water)
Overall mass transfer coefficient combines individual film resistances
- $1/K_G = 1/k_G + m/k_L$ relates overall coefficient to gas and liquid film coefficients
- $K_G$ : overall gas-phase coefficient, $k_G$ : gas-phase coefficient, $k_L$ : liquid-phase coefficient
Concentration gradients drive mass transfer between phases
- Steeper gradients lead to faster mass transfer rates
Diffusion in gas and liquid phases follows Fick's law
- Flux proportional to concentration gradient and diffusion coefficient
Interfacial area affects absorption/stripping efficiency
- Larger area increases mass transfer rate (packed columns, spray towers)

Mass transfer coefficients for columns

Gas-phase mass transfer coefficient ( $k_G$ $k_{G}$ ) estimated using dimensionless correlations
- Sherwood number relates mass transfer to fluid flow and diffusion (Re, Sc)
Liquid-phase mass transfer coefficient ( $k_L$ $k_{L}$ ) predicted by theoretical models
- Penetration theory assumes unsteady-state diffusion into liquid elements
- Surface renewal theory considers continuous replacement of liquid surface
Overall mass transfer coefficient ( $K_G$ $K_{G}$ or $K_L$ $K_{L}$ ) combines individual resistances
- $K_G$ used for gas-phase controlled systems, $K_L$ for liquid-phase controlled
Specific interfacial area ( $a$ $a$ ) measures available surface for mass transfer
- Defined as surface area per unit volume of column (m²/m³)
Volumetric mass transfer coefficient ( $K_Ga$ $K_{G} a$ or $K_La$ $K_{L} a$ ) crucial for column design
- Combines mass transfer coefficient and interfacial area
Empirical correlations estimate coefficients for different packing types
- Random packing (Raschig rings, Pall rings) and structured packing (corrugated sheets)

Theoretical stages for separation

McCabe-Thiele method graphically determines number of equilibrium stages
- Operating line represents material balance between phases
- Equilibrium curve shows composition relationship at equilibrium
- Step-wise construction between operating line and equilibrium curve
Kremser equation for absorbers calculates theoretical stages analytically
- $N = \frac{\log[(A-1)(y_1/y_{N+1} - 1/A) + 1]}{\log A}$
- $N$ : number of theoretical stages, $y_1$ : inlet gas composition, $y_{N+1}$ : outlet gas composition
Kremser equation for strippers determines stages for liquid purification
- $N = \frac{\log[(S-1)(x_N/x_0 - 1/S) + 1]}{\log S}$
- $x_N$ : inlet liquid composition, $x_0$ : outlet liquid composition
Height equivalent to a theoretical plate (HETP) relates packed height to stages
- HETP = total packed height / number of theoretical stages (m)
Height of a transfer unit (HTU) and number of transfer units (NTU) alternative approach
- HTU represents height of column in which concentration change equals driving force
- NTU measures difficulty of separation, analogous to number of stages

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