🌬️Heat and Mass Transport Unit 8 – Convective Mass Transfer

Key Concepts and Definitions

• Convective mass transfer involves the transport of a species within a fluid due to both molecular diffusion and bulk fluid motion (convection)
• Concentration gradient is the driving force for mass transfer and represents the change in concentration of a species per unit distance
  • Expressed mathematically as $\frac{dC}{dx}$ where $C$ is concentration and $x$ is distance
• Mass flux quantifies the rate of mass transfer per unit area and is proportional to the concentration gradient according to Fick's law
  • Fick's law: $J = -D \frac{dC}{dx}$ where $J$ is mass flux, $D$ is diffusion coefficient, and $\frac{dC}{dx}$ is concentration gradient
• Diffusion coefficient characterizes the ease with which a species can diffuse through a medium and depends on factors such as temperature, pressure, and molecular properties
• Convective mass transfer coefficient relates the mass flux to the concentration difference between the surface and the bulk fluid
  • Defined as $h_m = \frac{J}{\Delta C}$ where $h_m$ is mass transfer coefficient, $J$ is mass flux, and $\Delta C$ is concentration difference
• Schmidt number is a dimensionless number that relates the viscous diffusion rate to the molecular diffusion rate
  • Defined as $Sc = \frac{\nu}{D}$ where $\nu$ is kinematic viscosity and $D$ is diffusion coefficient
• Sherwood number is a dimensionless number that represents the ratio of convective mass transfer to diffusive mass transfer
  • Defined as $Sh = \frac{h_mL}{D}$ where $h_m$ is mass transfer coefficient, $L$ is characteristic length, and $D$ is diffusion coefficient

Fundamentals of Convective Mass Transfer

• Convective mass transfer occurs when a fluid in motion transports a species from a region of high concentration to a region of low concentration
• Two main types of convective mass transfer: forced convection and natural convection
  • Forced convection occurs when an external force (pump or fan) drives fluid motion
  • Natural convection occurs when fluid motion is induced by density differences caused by concentration gradients
• Convective mass transfer is influenced by fluid properties (density, viscosity, diffusivity), flow characteristics (velocity, turbulence), and geometry (surface roughness, shape)
• Boundary layer develops near the surface where the fluid velocity and concentration vary from their values in the bulk fluid
  • Concentration boundary layer is the region where the concentration gradient is significant
  • Velocity boundary layer is the region where the velocity gradient is significant
• Mass transfer rate is enhanced by increasing the fluid velocity, reducing the boundary layer thickness, and promoting turbulence
• Analogy between heat and mass transfer allows the use of heat transfer correlations to predict mass transfer coefficients
  • Chilton-Colburn analogy: $\frac{h_m}{U} = \frac{h}{\rho c_p U} = \frac{C_f}{2}$ where $h_m$ is mass transfer coefficient, $U$ is fluid velocity, $h$ is heat transfer coefficient, $\rho$ is fluid density, $c_p$ is specific heat, and $C_f$ is skin friction coefficient

Governing Equations and Boundary Layer Theory

• Conservation of mass (continuity equation) describes the balance between the rate of change of concentration and the net flux of species due to convection and diffusion
  • $\frac{\partial C}{\partial t} + \nabla \cdot (\mathbf{v}C) = \nabla \cdot (D\nabla C)$ where $C$ is concentration, $t$ is time, $\mathbf{v}$ is velocity vector, and $D$ is diffusion coefficient
• Conservation of momentum (Navier-Stokes equations) governs the fluid motion and couples with the mass conservation equation through the velocity field
• Boundary layer theory simplifies the governing equations by assuming that mass transfer occurs primarily in a thin layer near the surface
  • Concentration boundary layer thickness $\delta_c$ is defined as the distance from the surface where the concentration difference is 99% of the total difference
  • Velocity boundary layer thickness $\delta$ is defined similarly for the velocity profile
• Boundary layer equations are obtained by scaling the governing equations based on the boundary layer assumptions
  • $\frac{\partial C}{\partial t} + u\frac{\partial C}{\partial x} + v\frac{\partial C}{\partial y} = D\frac{\partial^2 C}{\partial y^2}$ where $u$ and $v$ are velocity components in $x$ and $y$ directions, respectively
• Similarity solutions can be obtained for the boundary layer equations under certain conditions (constant fluid properties, simple geometries)
  • Example: Blasius solution for laminar flow over a flat plate gives the concentration profile as a function of the similarity variable $\eta = y\sqrt{\frac{U_\infty}{x\nu}}$

Mass Transfer Coefficients

• Mass transfer coefficient is a key parameter that quantifies the rate of convective mass transfer and relates the mass flux to the concentration difference
• Depends on factors such as fluid properties, flow conditions, and geometry
• Can be determined experimentally, analytically (for simple cases), or using empirical correlations
• Experimental methods involve measuring the mass flux and concentration difference under controlled conditions
  • Example: Naphthalene sublimation technique measures the mass loss of a naphthalene surface exposed to a fluid flow
• Analytical solutions are available for simple geometries and flow conditions (laminar flow over a flat plate, stagnation point flow)
  • Example: Levèque solution for laminar flow in a circular tube gives $Sh = 1.62(ReSc\frac{d}{L})^{1/3}$ where $Re$ is Reynolds number, $Sc$ is Schmidt number, $d$ is tube diameter, and $L$ is tube length
• Empirical correlations are based on experimental data and dimensional analysis
  • Typically expressed in terms of dimensionless numbers (Reynolds, Schmidt, Sherwood)
  • Example: Dittus-Boelter correlation for turbulent flow in a circular tube: $Sh = 0.023Re^{0.8}Sc^{0.4}$
• Mass transfer coefficients can be used to calculate the mass transfer rate, concentration distribution, and other relevant quantities in convective mass transfer problems

Dimensionless Numbers in Mass Transfer

• Dimensionless numbers are used to characterize the relative importance of different physical phenomena in convective mass transfer
• Reynolds number ($Re$) represents the ratio of inertial forces to viscous forces and determines the flow regime (laminar, transitional, turbulent)
  • Defined as $Re = \frac{UL}{\nu}$ where $U$ is characteristic velocity, $L$ is characteristic length, and $\nu$ is kinematic viscosity
• Schmidt number ($Sc$) relates the viscous diffusion rate to the molecular diffusion rate and characterizes the relative thickness of the velocity and concentration boundary layers
  • Defined as $Sc = \frac{\nu}{D}$ where $\nu$ is kinematic viscosity and $D$ is diffusion coefficient
  • High $Sc$ indicates that the concentration boundary layer is much thinner than the velocity boundary layer
• Sherwood number ($Sh$) represents the ratio of convective mass transfer to diffusive mass transfer and is analogous to the Nusselt number in heat transfer
  • Defined as $Sh = \frac{h_mL}{D}$ where $h_m$ is mass transfer coefficient, $L$ is characteristic length, and $D$ is diffusion coefficient
• Péclet number ($Pe$) is the product of Reynolds and Schmidt numbers and represents the ratio of advective transport to diffusive transport
  • Defined as $Pe = ReSc = \frac{UL}{D}$
• Stanton number ($St$) represents the ratio of the mass transfer rate to the advective transport rate
  • Defined as $St = \frac{h_m}{U} = \frac{Sh}{RePr}$
• These dimensionless numbers are used to develop empirical correlations, perform scaling analysis, and compare mass transfer performance across different systems

Analogies Between Heat and Mass Transfer

• Heat and mass transfer share many similarities due to the analogous governing equations and physical mechanisms involved
• Fourier's law for heat conduction is analogous to Fick's law for mass diffusion
  • $q = -k\frac{dT}{dx}$ (Fourier's law) and $J = -D\frac{dC}{dx}$ (Fick's law)
• Convective heat transfer coefficient ($h$) is analogous to convective mass transfer coefficient ($h_m$)
  • $q = h\Delta T$ and $J = h_m\Delta C$
• Prandtl number ($Pr$) in heat transfer is analogous to Schmidt number ($Sc$) in mass transfer
  • $Pr = \frac{\nu}{\alpha}$ and $Sc = \frac{\nu}{D}$ where $\alpha$ is thermal diffusivity
• Nusselt number ($Nu$) in heat transfer is analogous to Sherwood number ($Sh$) in mass transfer
  • $Nu = \frac{hL}{k}$ and $Sh = \frac{h_mL}{D}$ where $k$ is thermal conductivity
• Chilton-Colburn analogy relates heat and mass transfer coefficients for turbulent flow over a flat plate
  • $\frac{h_m}{U} = \frac{h}{\rho c_p U} = \frac{C_f}{2}$ where $\rho$ is fluid density, $c_p$ is specific heat, and $C_f$ is skin friction coefficient
• Reynolds analogy relates heat, mass, and momentum transfer for turbulent flow in a circular tube
  • $\frac{h}{\rho c_p U} = \frac{h_m}{U} = \frac{f}{2}$ where $f$ is the Darcy friction factor
• These analogies allow the use of heat transfer correlations and solutions to predict mass transfer behavior, and vice versa

Applications and Real-World Examples

• Convective mass transfer plays a crucial role in various engineering applications and natural processes
• Chemical processing: Separation and purification of mixtures, gas absorption, distillation, and extraction
  • Example: Packed bed absorbers use convective mass transfer to remove pollutants (CO2, SO2) from gas streams
• Biomedical engineering: Drug delivery, artificial organs, and tissue engineering
  • Example: Transdermal drug delivery patches rely on convective mass transfer to deliver medication through the skin
• Environmental engineering: Pollutant dispersion, air and water quality control, and climate modeling
  • Example: Atmospheric dispersion models use convective mass transfer principles to predict the spread of pollutants from sources (power plants, factories)
• Heat and mass exchanger design: Optimization of heat and mass transfer performance in compact exchangers
  • Example: Cooling towers use convective mass transfer to cool water by evaporating a portion of the water into the air stream
• Food processing: Drying, freezing, and packaging of food products
  • Example: Convective drying of fruits, vegetables, and grains involves the removal of moisture from the surface by a flowing air stream
• Corrosion and materials degradation: Prediction and control of corrosion rates in metals and alloys
  • Example: Convective mass transfer of oxygen and corrosive species (chlorides) affects the corrosion rate of steel structures in marine environments
• Meteorology and oceanography: Evaporation, precipitation, and transport of moisture and pollutants in the atmosphere and oceans
  • Example: Convective mass transfer at the air-sea interface influences the exchange of gases (CO2, oxygen) and the global climate

Problem-Solving Techniques

• Identify the type of convective mass transfer problem (steady-state or transient, forced or natural convection, internal or external flow)
• Determine the relevant physical properties (density, viscosity, diffusivity) and flow conditions (velocity, temperature, pressure)
• Simplify the problem by making appropriate assumptions (constant properties, incompressible flow, dilute solutions)
• Select the appropriate governing equations (conservation of mass, momentum, and species) and boundary conditions
• Non-dimensionalize the equations using relevant dimensionless numbers (Reynolds, Schmidt, Sherwood) to identify the key parameters and simplify the analysis
• Solve the equations analytically, numerically, or using empirical correlations
  • Analytical methods: Separation of variables, similarity solutions, Laplace transforms
  • Numerical methods: Finite difference, finite element, finite volume, computational fluid dynamics (CFD)
  • Empirical correlations: Dittus-Boelter, Sieder-Tate, Chilton-Colburn, Sherwood-Rayleigh
• Interpret the results and validate them against experimental data or known solutions
• Perform sensitivity analysis to assess the impact of uncertainties in input parameters and assumptions
• Use the results to design, optimize, or troubleshoot convective mass transfer systems
  • Example: Determine the required length of a packed bed absorber to achieve a specified removal efficiency of a pollutant
• Consider the analogies between heat and mass transfer to adapt existing solutions or correlations from one domain to the other
• Break down complex problems into simpler sub-problems that can be solved individually and combined to obtain the overall solution