Heat and Mass Transfer Unit 6 Review: Mass Transfer Fundamentals

Key Concepts and Definitions

• Mass transfer involves the transport of a substance from a region of higher concentration to a region of lower concentration
• Diffusion is the movement of molecules from an area of high concentration to an area of low concentration driven by a concentration gradient
• Convection is the transport of mass due to the bulk motion of a fluid (liquid or gas)
• Mass flux is the rate of mass transfer per unit area perpendicular to the direction of transfer, typically expressed in units of $kg/(m^2 \cdot s)$
• Concentration gradient is the change in concentration of a species per unit distance in a particular direction
• Steady-state diffusion occurs when the concentration at any point does not change with time
• Transient diffusion occurs when the concentration at a point changes with time
• Equimolar counterdiffusion is the diffusion of two species in opposite directions with no net molar flux

Mechanisms of Mass Transfer

• Molecular diffusion is the transport of mass due to the random motion of molecules, driven by a concentration gradient
  • Occurs in gases, liquids, and solids
  • Governed by Fick's laws of diffusion
• Convective mass transfer is the transport of mass due to the bulk motion of a fluid
  • Can be natural convection driven by density differences (hot air rising) or forced convection driven by external forces (wind, pumps)
  • Enhances mass transfer compared to molecular diffusion alone
• Turbulent diffusion is the transport of mass due to the chaotic motion of fluid particles in a turbulent flow
  • Increases mixing and mass transfer rates compared to laminar flow
• Eddy diffusion is the transport of mass due to the motion of large-scale eddies in a turbulent flow
• Surface diffusion is the transport of adsorbed molecules along a solid surface
• Knudsen diffusion occurs when the mean free path of molecules is comparable to or larger than the pore size in a porous medium (low-pressure gases in small pores)

Fick's Laws of Diffusion

• Fick's first law relates the diffusive flux to the concentration gradient: $J = -D \frac{dC}{dx}$, where $J$ is the diffusive flux, $D$ is the diffusion coefficient, and $\frac{dC}{dx}$ is the concentration gradient
  • The negative sign indicates that diffusion occurs in the direction of decreasing concentration
• Fick's second law describes the change in concentration with time due to diffusion: $\frac{\partial C}{\partial t} = D \frac{\partial^2 C}{\partial x^2}$ for one-dimensional diffusion
  • Assumes constant diffusion coefficient and no convection
• The diffusion coefficient $D$ is a measure of the ease with which a species diffuses through a medium
  • Depends on temperature, pressure, and the properties of the diffusing species and the medium
  • Can be estimated using empirical correlations or measured experimentally
• Steady-state diffusion occurs when $\frac{\partial C}{\partial t} = 0$, simplifying Fick's second law to $\frac{d^2 C}{dx^2} = 0$
• Transient diffusion occurs when the concentration changes with time, requiring the use of Fick's second law and appropriate initial and boundary conditions

Mass Transfer Coefficients

• The mass transfer coefficient $k$ relates the mass flux to the concentration difference between a surface and the bulk fluid: $J = k(C_s - C_\infty)$
  • $C_s$ is the concentration at the surface, and $C_\infty$ is the concentration in the bulk fluid
• Mass transfer coefficients are used to quantify the rate of mass transfer in convective mass transfer problems
• The overall mass transfer coefficient $K$ accounts for resistance to mass transfer in both the fluid and the solid phases: $\frac{1}{K} = \frac{1}{k_f} + \frac{1}{k_s}$
  • $k_f$ is the fluid-side mass transfer coefficient, and $k_s$ is the solid-side mass transfer coefficient
• Mass transfer coefficients can be determined experimentally or estimated using empirical correlations based on dimensionless numbers (Sherwood, Schmidt, Reynolds)
• The Chilton-Colburn analogy relates mass transfer coefficients to heat transfer coefficients: $\frac{h}{k} = \rho c_p (Le)^{2/3}$, where $h$ is the heat transfer coefficient, $\rho$ is the fluid density, $c_p$ is the specific heat, and $Le$ is the Lewis number

Convective Mass Transfer

• Convective mass transfer occurs when mass is transported by the bulk motion of a fluid
• The convective mass flux is given by: $J = k(C_s - C_\infty)$, where $k$ is the mass transfer coefficient
• Natural convection mass transfer is driven by density differences caused by concentration gradients (salt dissolution in water)
• Forced convection mass transfer is driven by external forces such as pumps, fans, or wind (drying of clothes in a dryer)
• The Sherwood number $Sh$ is a dimensionless number that relates the convective mass transfer coefficient to the diffusive mass transfer coefficient: $Sh = \frac{kL}{D}$, where $L$ is a characteristic length
• The Schmidt number $Sc$ is a dimensionless number that relates the viscous diffusion rate to the molecular diffusion rate: $Sc = \frac{\mu}{\rho D}$, where $\mu$ is the dynamic viscosity
• Empirical correlations for the Sherwood number are often expressed in terms of the Reynolds number $Re$ and the Schmidt number $Sc$ (Dittus-Boelter correlation for turbulent flow in pipes: $Sh = 0.023 Re^{0.8} Sc^{1/3}$)

Boundary Layer Theory in Mass Transfer

• The boundary layer is a thin region near a surface where the velocity and concentration gradients are significant
  • The velocity boundary layer is the region where the fluid velocity changes from zero at the surface to the free-stream velocity
  • The concentration boundary layer is the region where the concentration changes from the surface concentration to the bulk concentration
• The thickness of the concentration boundary layer $\delta_c$ is defined as the distance from the surface where the concentration difference is 99% of the total concentration difference
• The mass transfer Biot number $Bi_m$ is a dimensionless number that relates the external mass transfer resistance to the internal diffusion resistance: $Bi_m = \frac{kL}{D}$, where $L$ is a characteristic length
  • For $Bi_m \gg 1$, the external mass transfer resistance is negligible, and the process is controlled by internal diffusion
  • For $Bi_m \ll 1$, the internal diffusion resistance is negligible, and the process is controlled by external mass transfer
• The analogy between heat and mass transfer allows the use of heat transfer correlations for estimating mass transfer coefficients (Chilton-Colburn analogy)
• The Sherwood number can be expressed as a function of the Reynolds number and the Schmidt number for flow over a flat plate: $Sh_x = 0.332 Re_x^{1/2} Sc^{1/3}$ for laminar flow and $Sh_x = 0.0292 Re_x^{4/5} Sc^{1/3}$ for turbulent flow, where $Sh_x$ and $Re_x$ are based on the distance $x$ from the leading edge

Mass Transfer Equipment and Applications

• Packed bed columns are used for gas-liquid mass transfer operations such as absorption and stripping
  • Consist of a vertical column filled with packing material (Raschig rings, Berl saddles) to increase the interfacial area for mass transfer
  • The height of a transfer unit (HTU) and the number of transfer units (NTU) are used to design and analyze packed bed columns
• Tray columns are used for gas-liquid mass transfer operations and distillation
  • Consist of a vertical column with a series of horizontal trays that promote mixing and mass transfer between the gas and liquid phases
  • The efficiency of a tray is characterized by the Murphree tray efficiency, which compares the actual concentration change to the ideal concentration change
• Membrane separations use selective membranes to separate mixtures based on differences in permeability
  • Examples include reverse osmosis, ultrafiltration, and gas separations (CO2 capture)
  • The performance of a membrane is characterized by its selectivity and permeability
• Adsorption processes use solid adsorbents to selectively remove components from a fluid phase
  • Examples include activated carbon for water treatment and zeolites for gas purification
  • The adsorption capacity and selectivity of an adsorbent depend on its surface area, pore size distribution, and surface chemistry
• Drying processes involve the removal of moisture from a solid material by evaporation
  • Can be achieved through various methods such as hot air drying, vacuum drying, and freeze-drying
  • The drying rate depends on the properties of the material, the drying conditions (temperature, humidity), and the mass transfer coefficients

Problem-Solving Techniques

• Identify the type of mass transfer problem: steady-state or transient, one-dimensional or multi-dimensional, diffusion or convection
• Determine the appropriate governing equations: Fick's laws for diffusion, convective mass transfer equations, or a combination of both
• Specify the initial and boundary conditions based on the physical situation
  • Examples of boundary conditions include constant concentration, constant flux, or convective mass transfer at a surface
• Simplify the problem by making reasonable assumptions, such as constant properties, no chemical reactions, or negligible convection
• Solve the governing equations analytically or numerically
  • Analytical solutions are possible for simple geometries and boundary conditions (one-dimensional, steady-state diffusion with constant concentration boundaries)
  • Numerical methods (finite difference, finite element) are required for more complex problems
• Use dimensionless numbers (Sherwood, Schmidt, Biot) to characterize the mass transfer process and to obtain mass transfer coefficients from empirical correlations
• Apply the analogy between heat and mass transfer to adapt heat transfer solutions and correlations to mass transfer problems
• Verify the solution by checking the units, the limiting cases (zero flux at an impermeable boundary), and the overall mass balance
• Interpret the results in terms of the physical situation and the design requirements (mass transfer rate, concentration profiles, equipment size)