5.2 Equilibrium and mass transfer in absorption/stripping

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

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• 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 ($k_G$) estimated using dimensionless correlations
  • Sherwood number relates mass transfer to fluid flow and diffusion (Re, Sc)
• Liquid-phase mass transfer coefficient ($k_L$) 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 ($K_G$ or $K_L$) combines individual resistances
  • $K_G$ used for gas-phase controlled systems, $K_L$ for liquid-phase controlled
• Specific interfacial area ($a$) measures available surface for mass transfer
  • Defined as surface area per unit volume of column (m²/m³)
• Volumetric mass transfer coefficient ($K_Ga$ or $K_La$) 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