🌬️Heat and Mass Transport Unit 9 – Interphase Mass Transfer

Key Concepts and Definitions

• Interphase mass transfer involves the movement of mass between two distinct phases (gas-liquid, liquid-liquid, or gas-solid)
• Mass transfer occurs due to differences in chemical potential or concentration gradients across the interface
• Diffusion is the molecular transport of mass driven by concentration gradients (Fick's law)
• Convection is the transport of mass by bulk fluid motion, which can be natural (buoyancy-driven) or forced (externally induced)
• Mass transfer coefficient ($k_c$) quantifies the rate of mass transfer across the interface per unit area and concentration difference
  • Depends on fluid properties, flow conditions, and geometry
• Concentration boundary layer is a thin region near the interface where the concentration gradient is steep
  • Its thickness affects the rate of mass transfer
• Equilibrium is reached when the chemical potentials of a species are equal in both phases, and no net mass transfer occurs

Fundamentals of Interphase Mass Transfer

• Interphase mass transfer is governed by the principles of conservation of mass and species continuity
• Fick's first law describes diffusive mass flux ($J$) as proportional to the concentration gradient ($dC/dx$): $J = -D \frac{dC}{dx}$
  • $D$ is the diffusion coefficient, which depends on temperature, pressure, and molecular properties
• Convective mass transfer is described by Newton's law of cooling analogy: $N_A = k_c (C_{A,s} - C_{A,\infty})$
  • $N_A$ is the molar flux of species A, $k_c$ is the mass transfer coefficient, $C_{A,s}$ and $C_{A,\infty}$ are concentrations at the interface and bulk fluid
• Dimensionless numbers, such as Schmidt number ($Sc = \mu/\rho D$) and Sherwood number ($Sh = k_c L/D$), characterize mass transfer processes
  • $\mu$ is viscosity, $\rho$ is density, $L$ is characteristic length
• Interfacial area ($a$) and exposure time are crucial factors affecting the overall mass transfer rate
• Mass transfer resistances can exist in both phases and are additive, similar to electrical resistances in series

Mass Transfer Coefficients

• Mass transfer coefficients ($k_c$) are essential parameters for quantifying interphase mass transfer rates
• They are determined experimentally or estimated using empirical correlations based on dimensionless numbers (Sherwood, Reynolds, Schmidt)
• Correlations often take the form: $Sh = a Re^b Sc^c$, where $a$, $b$, and $c$ are constants specific to the geometry and flow conditions
• Mass transfer coefficients are affected by factors such as fluid properties, flow velocity, surface roughness, and interfacial turbulence
• Individual phase mass transfer coefficients ($k_c$) can be combined to obtain an overall mass transfer coefficient ($K_c$) using the resistance-in-series model:
  • $\frac{1}{K_c} = \frac{1}{k_{c,1}} + \frac{1}{k_{c,2}}$ for two-phase systems
• Higher mass transfer coefficients indicate faster mass transfer rates and more efficient interphase mass transfer processes
• Enhancing mass transfer coefficients can be achieved through techniques such as increasing interfacial area, promoting turbulence, or reducing boundary layer thickness

Diffusion vs. Convection in Mass Transfer

• Diffusion and convection are two fundamental mechanisms of mass transfer in interphase systems
• Diffusion is driven by concentration gradients and occurs at the molecular level
  • It is the dominant mechanism in stagnant or laminar flow conditions
  • Diffusive mass transfer is relatively slow compared to convective mass transfer
• Convection involves the transport of mass by bulk fluid motion and is driven by external forces or density differences
  • It is the dominant mechanism in turbulent or well-mixed flow conditions
  • Convective mass transfer is generally faster than diffusive mass transfer
• The relative importance of diffusion and convection is characterized by the Péclet number ($Pe = uL/D$)
  • $u$ is the fluid velocity, $L$ is the characteristic length, and $D$ is the diffusion coefficient
  • High $Pe$ (> 1) indicates convection-dominated mass transfer, while low $Pe$ (< 1) indicates diffusion-dominated mass transfer
• In most practical applications, both diffusion and convection contribute to the overall mass transfer process
  • The concentration boundary layer is a region where diffusion and convection are both significant
• Understanding the interplay between diffusion and convection is crucial for designing and optimizing mass transfer equipment and processes

Concentration Gradients and Driving Forces

• Concentration gradients are the spatial variations in the concentration of a species within a phase
• They serve as the driving force for diffusive mass transfer, as molecules move from regions of high concentration to regions of low concentration
• Concentration gradients can be linear (steady-state diffusion) or nonlinear (unsteady-state diffusion)
  • Linear gradients are observed in simple geometries (plane wall) and steady-state conditions
  • Nonlinear gradients are common in complex geometries (packed beds) and transient conditions
• The steepness of the concentration gradient affects the rate of diffusive mass transfer
  • Steeper gradients result in higher diffusive fluxes
• In convective mass transfer, the concentration difference between the interface and the bulk fluid acts as the driving force
  • This concentration difference is often expressed as $(C_{A,s} - C_{A,\infty})$ in Newton's law of cooling analogy
• Maintaining a high concentration difference is essential for efficient interphase mass transfer
  • This can be achieved by continuously removing the transferred species from the bulk fluid or replenishing the depleted species at the interface
• Concentration gradients can be manipulated by altering process conditions (temperature, pressure) or using mass transfer enhancement techniques (surface renewal, turbulence promoters)

Mass Transfer Equipment and Operations

• Various types of equipment are used to facilitate interphase mass transfer in industrial processes
• Packed columns are widely used for gas-liquid mass transfer operations (absorption, stripping, distillation)
  • They consist of a vertical column filled with packing materials (random or structured) that provide a large interfacial area for mass transfer
  • Examples include Raschig rings, Pall rings, and structured packings (Mellapak, Flexipac)
• Tray columns are another common type of gas-liquid mass transfer equipment
  • They feature a series of horizontal trays or plates that promote intimate contact between the gas and liquid phases
  • Examples include sieve trays, valve trays, and bubble-cap trays
• Stirred tanks are used for liquid-liquid and gas-liquid mass transfer operations (extraction, fermentation, hydrogenation)
  • They provide efficient mixing and high interfacial area through the use of impellers and baffles
• Membrane contactors are emerging as a compact and efficient alternative to traditional mass transfer equipment
  • They use selective membranes to create a large interfacial area and control the contact between phases
  • Examples include hollow fiber membrane contactors and flat sheet membrane contactors
• The choice of mass transfer equipment depends on factors such as the nature of the phases, the desired mass transfer rate, and the process scale and constraints

Modeling and Calculations

• Mathematical modeling and calculations are essential for designing, analyzing, and optimizing interphase mass transfer processes
• The two-film theory is a simple and widely used model for describing interphase mass transfer
  • It assumes that mass transfer resistances are confined to thin films on either side of the interface
  • The model provides a framework for calculating mass transfer coefficients and fluxes based on film thicknesses and concentration differences
• The penetration theory is another model that accounts for the unsteady-state nature of mass transfer
  • It considers the periodic renewal of the interfacial surface and the penetration of solute into the bulk fluid
  • The model is particularly useful for describing mass transfer in turbulent or agitated systems
• Computational fluid dynamics (CFD) simulations are increasingly used to model complex mass transfer processes
  • CFD tools solve the governing equations of fluid flow and mass transport numerically, providing detailed insights into local concentration fields and mass transfer rates
• Dimensionless correlations, such as the Sherwood number ($Sh$) and the Stanton number ($St$), are used to predict mass transfer coefficients based on system parameters and fluid properties
  • These correlations are derived from experimental data and theoretical analysis and are specific to certain geometries and flow conditions
• Mass transfer calculations often involve solving differential equations (e.g., Fick's second law) or using numerical methods (finite differences, finite elements) to determine concentration profiles and mass transfer rates
• Proper modeling and calculations are crucial for optimizing mass transfer processes, scaling up laboratory findings, and troubleshooting industrial operations

Real-World Applications and Case Studies

• Interphase mass transfer finds numerous applications in various industries, including chemical, petrochemical, environmental, and biochemical sectors
• Absorption is widely used for removing pollutants from gas streams (CO2 capture, SO2 scrubbing) and recovering valuable products (acid gas removal, natural gas sweetening)
  • Case study: Amine-based CO2 capture from power plant flue gases using packed columns
• Distillation is a common separation process that relies on interphase mass transfer between liquid and vapor phases
  • It is used for purifying liquids, separating azeotropic mixtures, and fractionating petroleum products
  • Case study: Crude oil fractionation in a petroleum refinery using a series of distillation columns
• Liquid-liquid extraction is employed for separating components based on their relative solubilities in two immiscible liquids
  • Applications include the recovery of antibiotics, the purification of vegetable oils, and the treatment of radioactive wastes
  • Case study: Extraction of penicillin from fermentation broth using butyl acetate in a countercurrent extraction column
• Gas-liquid mass transfer is crucial in biological processes, such as fermentation and wastewater treatment
  • Oxygen transfer from air to the liquid phase is often the rate-limiting step in aerobic bioprocesses
  • Case study: Optimization of oxygen transfer in a stirred-tank bioreactor for the production of baker's yeast
• Adsorption is a gas-solid or liquid-solid mass transfer process used for purification, separation, and catalysis
  • Applications include air and water purification, gas separation, and heterogeneous catalysis
  • Case study: Removal of toxic heavy metals from industrial wastewater using activated carbon adsorption columns
• Understanding and optimizing interphase mass transfer is essential for improving the efficiency, sustainability, and economics of various industrial processes