🍳Separation Processes Unit 3 – Mass Transfer and Diffusion

Fundamentals of Mass Transfer

• Mass transfer involves the movement of a substance from a region of higher concentration to a region of lower concentration
• Occurs due to a concentration gradient, which is the driving force for mass transfer
• Can happen in various systems, including gas-liquid, liquid-liquid, and solid-liquid interfaces
• Plays a crucial role in separation processes, such as absorption, adsorption, and extraction
• Influenced by factors like temperature, pressure, and the properties of the substances involved
• Requires an understanding of the thermodynamic and kinetic aspects of the system
• Governed by the principles of conservation of mass and energy

Diffusion Mechanisms and Fick's Laws

• Diffusion is the spontaneous movement of molecules from a region of higher concentration to a region of lower concentration
• Fick's first law describes the steady-state diffusion flux ($J$) as proportional to the concentration gradient ($\frac{dC}{dx}$): $J = -D \frac{dC}{dx}$
  • $D$ is the diffusion coefficient, which depends on the properties of the diffusing species and the medium
• Fick's second law describes the change in concentration over time ($\frac{\partial C}{\partial t}$) due to diffusion: $\frac{\partial C}{\partial t} = D \frac{\partial^2 C}{\partial x^2}$
• Molecular diffusion occurs due to the random motion of molecules (Brownian motion)
• Eddy diffusion occurs in turbulent flows and enhances the mixing of substances
• Effective diffusion coefficient ($D_{eff}$) accounts for the combined effects of molecular and eddy diffusion
• Diffusion in porous media is influenced by factors like porosity, tortuosity, and pore size distribution

Steady-State and Unsteady-State Diffusion

• Steady-state diffusion occurs when the concentration profile does not change with time ($\frac{\partial C}{\partial t} = 0$)
  • Concentration gradient remains constant
  • Flux is uniform throughout the system
• Unsteady-state diffusion occurs when the concentration profile changes with time ($\frac{\partial C}{\partial t} \neq 0$)
  • Concentration gradient varies with position and time
  • Flux is non-uniform and changes with time
• Analytical solutions for unsteady-state diffusion can be obtained using Fick's second law and appropriate boundary conditions
• Numerical methods (finite difference, finite element) are often used to solve complex unsteady-state diffusion problems
• Quasi-steady-state approximation assumes that the concentration profile adjusts quickly to changes in boundary conditions
• Transient diffusion is important in processes like adsorption, drying, and membrane separation

Mass Transfer Coefficients

• Mass transfer coefficient ($k$) relates the mass transfer rate to the concentration driving force
• Depends on the geometry of the system, fluid properties, and flow conditions
• Can be determined experimentally or estimated using empirical correlations
• Overall mass transfer coefficient ($K$) accounts for the resistances in both phases (gas and liquid, or liquid and solid)
  • Calculated using the resistance-in-series model: $\frac{1}{K} = \frac{1}{k_1} + \frac{1}{k_2}$
• Dimensionless numbers (Sherwood, Schmidt, Reynolds) are used to correlate mass transfer coefficients
• Analogous to heat transfer coefficients in heat transfer processes
• Higher mass transfer coefficients indicate faster mass transfer rates

Convective Mass Transfer

• Convective mass transfer involves the transport of a substance due to the bulk motion of a fluid
• Can be either natural convection (driven by density differences) or forced convection (driven by external forces)
• Characterized by the Sherwood number ($Sh$), which relates the convective mass transfer to the diffusive mass transfer
  • $Sh = \frac{k L}{D}$, where $L$ is a characteristic length and $D$ is the diffusion coefficient
• Correlations for the Sherwood number are available for various geometries and flow conditions (laminar flow, turbulent flow)
• Boundary layer theory is used to analyze convective mass transfer in external flows
• Penetration theory and surface renewal theory are used to model convective mass transfer in gas-liquid systems
• Convective mass transfer is enhanced by factors like turbulence, surface roughness, and high fluid velocities

Interphase Mass Transfer

• Interphase mass transfer occurs between two immiscible phases (gas-liquid, liquid-liquid, or gas-solid)
• Governed by the equilibrium distribution of the transferring component between the phases
• Equilibrium distribution is described by partition coefficients or solubility data
• Mass transfer rate is determined by the concentration driving force and the mass transfer coefficients in each phase
• Two-film theory assumes that the resistance to mass transfer is concentrated in thin films near the interface
• Penetration theory considers the unsteady-state diffusion of the transferring component into the bulk phases
• Danckwerts' surface renewal theory assumes that the interface is continuously replaced by fresh fluid elements
• Interphase mass transfer is crucial in separation processes like absorption, stripping, and extraction

Mass Transfer Equipment and Operations

• Various types of equipment are used to facilitate mass transfer in separation processes
• Packed columns contain a bed of packing material (random or structured) to provide a large interfacial area for mass transfer
  • Used in absorption, stripping, and distillation processes
• Tray columns use a series of perforated trays to create stages for mass transfer
  • Commonly used in distillation and extraction processes
• Falling film contactors employ a thin film of liquid flowing over a surface to enhance gas-liquid mass transfer
• Spray columns disperse one phase into droplets or a spray to increase the interfacial area
• Membrane contactors use a permeable membrane to separate the phases and facilitate selective mass transfer
• Design of mass transfer equipment involves considerations like capacity, efficiency, pressure drop, and material compatibility

Applications in Separation Processes

• Mass transfer principles are applied in various separation processes to purify or recover valuable components
• Absorption involves the transfer of a solute from a gas phase to a liquid phase (CO2 capture, gas sweetening)
• Stripping is the reverse of absorption, where a solute is transferred from a liquid phase to a gas phase (air stripping, steam stripping)
• Distillation separates components based on their relative volatilities, using vaporization and condensation (petroleum refining, alcohol production)
• Liquid-liquid extraction transfers a solute between two immiscible liquid phases (metal extraction, pharmaceutical purification)
• Adsorption involves the selective binding of a solute onto a solid surface (water treatment, gas purification)
• Ion exchange uses solid resins to exchange ions between a liquid and a solid phase (water softening, chemical purification)
• Membrane separation processes (reverse osmosis, ultrafiltration, pervaporation) rely on selective mass transfer through a membrane