🌬️Heat and Mass Transport Unit 8 – Convective Mass Transfer
Convective mass transfer is a crucial process in engineering and natural systems. It involves the transport of species within fluids due to both molecular diffusion and bulk fluid motion. Understanding this phenomenon is essential for designing efficient chemical processes, environmental systems, and biomedical applications.
Key concepts include concentration gradients, mass flux, and transfer coefficients. These principles are applied using governing equations, boundary layer theory, and dimensionless numbers. Real-world applications range from chemical processing to environmental engineering, showcasing the widespread importance of convective mass transfer in various fields.
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 dxdC 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=−DdxdC where J is mass flux, D is diffusion coefficient, and dxdC 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 hm=ΔCJ where hm is mass transfer coefficient, J is mass flux, and ΔC is concentration difference
Schmidt number is a dimensionless number that relates the viscous diffusion rate to the molecular diffusion rate
Defined as Sc=Dν where ν 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=DhmL where hm 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: Uhm=ρcpUh=2Cf where hm is mass transfer coefficient, U is fluid velocity, h is heat transfer coefficient, ρ is fluid density, cp is specific heat, and Cf 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
∂t∂C+∇⋅(vC)=∇⋅(D∇C) where C is concentration, t is time, 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 δc is defined as the distance from the surface where the concentration difference is 99% of the total difference
Velocity boundary layer thickness δ is defined similarly for the velocity profile
Boundary layer equations are obtained by scaling the governing equations based on the boundary layer assumptions
∂t∂C+u∂x∂C+v∂y∂C=D∂y2∂2C 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 η=yxνU∞
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(ReScLd)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=0.023Re0.8Sc0.4
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=νUL where U is characteristic velocity, L is characteristic length, and ν 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=Dν where ν 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=DhmL where hm 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=DUL
Stanton number (St) represents the ratio of the mass transfer rate to the advective transport rate
Defined as St=Uhm=RePrSh
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=−kdxdT (Fourier's law) and J=−DdxdC (Fick's law)
Convective heat transfer coefficient (h) is analogous to convective mass transfer coefficient (hm)
q=hΔT and J=hmΔC
Prandtl number (Pr) in heat transfer is analogous to Schmidt number (Sc) in mass transfer
Pr=αν and Sc=Dν where α is thermal diffusivity
Nusselt number (Nu) in heat transfer is analogous to Sherwood number (Sh) in mass transfer
Nu=khL and Sh=DhmL where k is thermal conductivity
Chilton-Colburn analogy relates heat and mass transfer coefficients for turbulent flow over a flat plate
Uhm=ρcpUh=2Cf where ρ is fluid density, cp is specific heat, and Cf is skin friction coefficient
Reynolds analogy relates heat, mass, and momentum transfer for turbulent flow in a circular tube
ρcpUh=Uhm=2f 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