Heat and Mass Transfer Unit 9 Review: Convective Mass Transfer

Key Concepts and Definitions

• Mass transfer involves the transport of a species within a mixture due to concentration gradients
• Convective mass transfer occurs when fluid motion enhances the mass transport process
• Diffusion is the movement of species from a region of high concentration to a region of low concentration
  • Diffusion is driven by the random motion of molecules (Brownian motion)
  • Diffusion occurs in gases, liquids, and solids
• Mass flux ($J$) represents the rate of mass transfer per unit area perpendicular to the direction of transfer
• Concentration gradient ($\frac{dC}{dx}$) is the change in concentration of a species with respect to distance
• Fick's law relates the mass flux to the concentration gradient: $J = -D \frac{dC}{dx}$, where $D$ is the diffusion coefficient
• Diffusion coefficient ($D$) is a measure of the ease with which a species can diffuse through a medium
  • Diffusion coefficients are higher in gases compared to liquids and solids

Governing Equations and Principles

• Conservation of mass is a fundamental principle in mass transfer, stating that mass cannot be created or destroyed
• The continuity equation for species $A$ in a binary mixture: $\frac{\partial C_A}{\partial t} + \nabla \cdot (C_A \mathbf{v}) = -\nabla \cdot \mathbf{J}_A$
  • $C_A$ is the concentration of species $A$
  • $\mathbf{v}$ is the velocity vector
  • $\mathbf{J}_A$ is the mass flux vector of species $A$
• Fick's second law describes the transient diffusion process: $\frac{\partial C}{\partial t} = D \nabla^2 C$
• Maxwell-Stefan equations describe multicomponent diffusion in mixtures with more than two species
• Film theory assumes that mass transfer occurs through a thin, stagnant film near the interface
  • The film thickness ($\delta$) is a key parameter in film theory
• Penetration theory considers the unsteady-state diffusion process and the periodic renewal of the interface
• Surface renewal theory combines aspects of film theory and penetration theory

Boundary Layer Theory in Mass Transfer

• Concentration boundary layer develops when a fluid with a different concentration flows over a surface
• The concentration boundary layer thickness ($\delta_c$) is the distance from the surface where the concentration reaches 99% of the freestream value
• Boundary layer thickness depends on factors such as fluid velocity, diffusion coefficient, and distance from the leading edge
• Analogous to the velocity boundary layer in fluid mechanics and thermal boundary layer in heat transfer
• Mass transfer coefficient ($h_m$) relates the mass flux to the concentration difference: $J = h_m (C_s - C_\infty)$
  • $C_s$ is the concentration at the surface
  • $C_\infty$ is the concentration in the freestream
• Sherwood number ($Sh$) is a dimensionless number that represents the ratio of convective mass transfer to diffusive mass transfer: $Sh = \frac{h_m L}{D}$
  • $L$ is a characteristic length
• Higher Sherwood numbers indicate a greater influence of convection on mass transfer

Dimensionless Numbers and Their Significance

• Reynolds number ($Re$) represents the ratio of inertial forces to viscous forces: $Re = \frac{\rho v L}{\mu}$
  • $\rho$ is the fluid density
  • $v$ is the fluid velocity
  • $\mu$ is the dynamic viscosity
• Schmidt number ($Sc$) is the ratio of momentum diffusivity to mass diffusivity: $Sc = \frac{\nu}{D}$
  • $\nu$ is the kinematic viscosity ($\nu = \frac{\mu}{\rho}$)
• Sherwood number ($Sh$) represents the ratio of convective mass transfer to diffusive mass transfer: $Sh = \frac{h_m L}{D}$
• Péclet number for mass transfer ($Pe_m$) is the product of Reynolds and Schmidt numbers: $Pe_m = Re \cdot Sc$
  • Péclet number represents the ratio of advective transport to diffusive transport
• Stanton number for mass transfer ($St_m$) relates the mass transfer coefficient to the fluid velocity: $St_m = \frac{h_m}{\rho v}$
• Correlations between dimensionless numbers are used to predict mass transfer coefficients in various geometries and flow conditions
  • Example: $Sh = f(Re, Sc)$ for flow over a flat plate

Convective Mass Transfer Coefficients

• Mass transfer coefficient ($h_m$) relates the mass flux to the concentration difference: $J = h_m (C_s - C_\infty)$
• Convective mass transfer coefficients depend on factors such as fluid properties, flow conditions, and geometry
• Correlations for mass transfer coefficients are often expressed in terms of dimensionless numbers
  • Example: $Sh = 0.664 Re^{0.5} Sc^{0.33}$ for laminar flow over a flat plate
• Analogy between heat and mass transfer allows the use of heat transfer correlations to estimate mass transfer coefficients
  • Chilton-Colburn analogy: $\frac{h_m}{v} = \frac{h}{\rho c_p v} (\frac{Sc}{Pr})^{-2/3}$, where $h$ is the heat transfer coefficient and $Pr$ is the Prandtl number
• Experimental techniques, such as the naphthalene sublimation method, can be used to measure mass transfer coefficients
• Mass transfer coefficients are important in the design of mass transfer equipment (absorbers, extractors, and membrane separators)

Analogies Between Heat and Mass Transfer

• Heat and mass transfer share many similarities in their governing equations and transport mechanisms
• Fick's law for mass transfer is analogous to Fourier's law for heat transfer
  • Fick's law: $J = -D \frac{dC}{dx}$
  • Fourier's law: $q = -k \frac{dT}{dx}$, where $q$ is the heat flux and $k$ is the thermal conductivity
• The continuity equation for species concentration is analogous to the energy equation in heat transfer
• Dimensionless numbers in mass transfer have counterparts in heat transfer
  • Schmidt number ($Sc$) in mass transfer is analogous to Prandtl number ($Pr$) in heat transfer
  • Sherwood number ($Sh$) in mass transfer is analogous to Nusselt number ($Nu$) in heat transfer
• Chilton-Colburn analogy relates heat and mass transfer coefficients: $\frac{h_m}{v} = \frac{h}{\rho c_p v} (\frac{Sc}{Pr})^{-2/3}$
• Lewis number ($Le$) is the ratio of thermal diffusivity to mass diffusivity: $Le = \frac{\alpha}{D}$, where $\alpha$ is the thermal diffusivity
  • For $Le = 1$, heat and mass transfer rates are equal

Common Applications and Examples

• Drying processes involve the removal of moisture from solids through convective mass transfer
  • Examples: drying of food products, pharmaceuticals, and textiles
• Humidification and dehumidification processes involve the addition or removal of water vapor from air
  • Applications in air conditioning, greenhouse control, and industrial processes
• Evaporative cooling relies on the convective mass transfer of water to cool air or surfaces
  • Used in cooling towers, evaporative coolers, and human sweating
• Mass transfer in chemical reactions, such as the absorption of gases in liquids (CO2 absorption in water)
• Separation processes, such as distillation and extraction, involve convective mass transfer between phases
• Convective mass transfer in biological systems, such as oxygen transport in blood and nutrient uptake in cells
• Environmental applications, such as the dispersion of pollutants in air and water
  • Modeling the spread of contaminants in rivers, lakes, and groundwater

Problem-Solving Techniques

• Identify the type of mass transfer problem (steady-state or transient, one-dimensional or multi-dimensional)
• Determine the appropriate governing equations and boundary conditions
  • Apply conservation of mass, Fick's laws, and continuity equation
• Simplify the problem using assumptions, such as constant properties, incompressible flow, or negligible chemical reactions
• Non-dimensionalize the equations using appropriate dimensionless numbers (Re, Sc, Sh)
• Solve the equations analytically or numerically, depending on the complexity of the problem
  • Analytical solutions are possible for simple geometries and boundary conditions (1D steady-state diffusion)
  • Numerical methods, such as finite difference or finite element, are used for complex problems
• Use empirical correlations or analogies to estimate mass transfer coefficients
  • Select appropriate correlations based on the geometry and flow conditions (laminar or turbulent, internal or external flow)
• Interpret the results and validate them using experimental data or literature values
• Perform sensitivity analysis to identify the most influential parameters on mass transfer performance
• Optimize the design of mass transfer equipment or processes based on the analysis results