Heat and Mass Transport

🌬️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.

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 dCdx\frac{dC}{dx} where CC is concentration and xx 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=DdCdxJ = -D \frac{dC}{dx} where JJ is mass flux, DD is diffusion coefficient, and dCdx\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 hm=JΔCh_m = \frac{J}{\Delta C} where hmh_m is mass transfer coefficient, JJ is mass flux, and ΔC\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=νDSc = \frac{\nu}{D} where ν\nu is kinematic viscosity and DD is diffusion coefficient
  • Sherwood number is a dimensionless number that represents the ratio of convective mass transfer to diffusive mass transfer
    • Defined as Sh=hmLDSh = \frac{h_mL}{D} where hmh_m is mass transfer coefficient, LL is characteristic length, and DD 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: hmU=hρcpU=Cf2\frac{h_m}{U} = \frac{h}{\rho c_p U} = \frac{C_f}{2} where hmh_m is mass transfer coefficient, UU is fluid velocity, hh is heat transfer coefficient, ρ\rho is fluid density, cpc_p is specific heat, and CfC_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
    • Ct+(vC)=(DC)\frac{\partial C}{\partial t} + \nabla \cdot (\mathbf{v}C) = \nabla \cdot (D\nabla C) where CC is concentration, tt is time, v\mathbf{v} is velocity vector, and DD 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\delta_c is defined as the distance from the surface where the concentration difference is 99% of the total difference
    • Velocity boundary layer thickness δ\delta is defined similarly for the velocity profile
  • Boundary layer equations are obtained by scaling the governing equations based on the boundary layer assumptions
    • Ct+uCx+vCy=D2Cy2\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 uu and vv are velocity components in xx and yy 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=yUxν\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(ReScdL)1/3Sh = 1.62(ReSc\frac{d}{L})^{1/3} where ReRe is Reynolds number, ScSc is Schmidt number, dd is tube diameter, and LL 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=0.023Re0.8Sc0.4Sh = 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 (ReRe) represents the ratio of inertial forces to viscous forces and determines the flow regime (laminar, transitional, turbulent)
    • Defined as Re=ULνRe = \frac{UL}{\nu} where UU is characteristic velocity, LL is characteristic length, and ν\nu is kinematic viscosity
  • Schmidt number (ScSc) 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=νDSc = \frac{\nu}{D} where ν\nu is kinematic viscosity and DD is diffusion coefficient
    • High ScSc indicates that the concentration boundary layer is much thinner than the velocity boundary layer
  • Sherwood number (ShSh) represents the ratio of convective mass transfer to diffusive mass transfer and is analogous to the Nusselt number in heat transfer
    • Defined as Sh=hmLDSh = \frac{h_mL}{D} where hmh_m is mass transfer coefficient, LL is characteristic length, and DD is diffusion coefficient
  • Péclet number (PePe) is the product of Reynolds and Schmidt numbers and represents the ratio of advective transport to diffusive transport
    • Defined as Pe=ReSc=ULDPe = ReSc = \frac{UL}{D}
  • Stanton number (StSt) represents the ratio of the mass transfer rate to the advective transport rate
    • Defined as St=hmU=ShRePrSt = \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=kdTdxq = -k\frac{dT}{dx} (Fourier's law) and J=DdCdxJ = -D\frac{dC}{dx} (Fick's law)
  • Convective heat transfer coefficient (hh) is analogous to convective mass transfer coefficient (hmh_m)
    • q=hΔTq = h\Delta T and J=hmΔCJ = h_m\Delta C
  • Prandtl number (PrPr) in heat transfer is analogous to Schmidt number (ScSc) in mass transfer
    • Pr=ναPr = \frac{\nu}{\alpha} and Sc=νDSc = \frac{\nu}{D} where α\alpha is thermal diffusivity
  • Nusselt number (NuNu) in heat transfer is analogous to Sherwood number (ShSh) in mass transfer
    • Nu=hLkNu = \frac{hL}{k} and Sh=hmLDSh = \frac{h_mL}{D} where kk is thermal conductivity
  • Chilton-Colburn analogy relates heat and mass transfer coefficients for turbulent flow over a flat plate
    • hmU=hρcpU=Cf2\frac{h_m}{U} = \frac{h}{\rho c_p U} = \frac{C_f}{2} where ρ\rho is fluid density, cpc_p is specific heat, and CfC_f is skin friction coefficient
  • Reynolds analogy relates heat, mass, and momentum transfer for turbulent flow in a circular tube
    • hρcpU=hmU=f2\frac{h}{\rho c_p U} = \frac{h_m}{U} = \frac{f}{2} where ff 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


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© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.