❤️‍🔥Heat and Mass Transfer Unit 9 – Convective Mass Transfer

Convective mass transfer is a crucial process in many engineering applications. It involves the transport of species within mixtures due to concentration gradients, enhanced by fluid motion. Understanding this phenomenon is essential for designing efficient systems in industries like chemical processing and environmental engineering. Key concepts include mass flux, concentration gradients, and Fick's law. Governing equations, boundary layer theory, and dimensionless numbers play vital roles in analyzing convective mass transfer. The analogy between heat and mass transfer allows for the application of similar principles and problem-solving techniques across both fields.

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 (JJ) represents the rate of mass transfer per unit area perpendicular to the direction of transfer
  • Concentration gradient (dCdx\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=DdCdxJ = -D \frac{dC}{dx}, where DD is the diffusion coefficient
  • Diffusion coefficient (DD) 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 AA in a binary mixture: CAt+(CAv)=JA\frac{\partial C_A}{\partial t} + \nabla \cdot (C_A \mathbf{v}) = -\nabla \cdot \mathbf{J}_A
    • CAC_A is the concentration of species AA
    • v\mathbf{v} is the velocity vector
    • JA\mathbf{J}_A is the mass flux vector of species AA
  • Fick's second law describes the transient diffusion process: Ct=D2C\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 (δ\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 (δc\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 (hmh_m) relates the mass flux to the concentration difference: J=hm(CsC)J = h_m (C_s - C_\infty)
    • CsC_s is the concentration at the surface
    • CC_\infty is the concentration in the freestream
  • Sherwood number (ShSh) is a dimensionless number that represents the ratio of convective mass transfer to diffusive mass transfer: Sh=hmLDSh = \frac{h_m L}{D}
    • LL is a characteristic length
  • Higher Sherwood numbers indicate a greater influence of convection on mass transfer

Dimensionless Numbers and Their Significance

  • Reynolds number (ReRe) represents the ratio of inertial forces to viscous forces: Re=ρvLμRe = \frac{\rho v L}{\mu}
    • ρ\rho is the fluid density
    • vv is the fluid velocity
    • μ\mu is the dynamic viscosity
  • Schmidt number (ScSc) is the ratio of momentum diffusivity to mass diffusivity: Sc=νDSc = \frac{\nu}{D}
    • ν\nu is the kinematic viscosity (ν=μρ\nu = \frac{\mu}{\rho})
  • Sherwood number (ShSh) represents the ratio of convective mass transfer to diffusive mass transfer: Sh=hmLDSh = \frac{h_m L}{D}
  • Péclet number for mass transfer (PemPe_m) is the product of Reynolds and Schmidt numbers: Pem=ReScPe_m = Re \cdot Sc
    • Péclet number represents the ratio of advective transport to diffusive transport
  • Stanton number for mass transfer (StmSt_m) relates the mass transfer coefficient to the fluid velocity: Stm=hmρvSt_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)Sh = f(Re, Sc) for flow over a flat plate

Convective Mass Transfer Coefficients

  • Mass transfer coefficient (hmh_m) relates the mass flux to the concentration difference: J=hm(CsC)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=0.664Re0.5Sc0.33Sh = 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: hmv=hρcpv(ScPr)2/3\frac{h_m}{v} = \frac{h}{\rho c_p v} (\frac{Sc}{Pr})^{-2/3}, where hh is the heat transfer coefficient and PrPr 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=DdCdxJ = -D \frac{dC}{dx}
    • Fourier's law: q=kdTdxq = -k \frac{dT}{dx}, where qq is the heat flux and kk 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 (ScSc) in mass transfer is analogous to Prandtl number (PrPr) in heat transfer
    • Sherwood number (ShSh) in mass transfer is analogous to Nusselt number (NuNu) in heat transfer
  • Chilton-Colburn analogy relates heat and mass transfer coefficients: hmv=hρcpv(ScPr)2/3\frac{h_m}{v} = \frac{h}{\rho c_p v} (\frac{Sc}{Pr})^{-2/3}
  • Lewis number (LeLe) is the ratio of thermal diffusivity to mass diffusivity: Le=αDLe = \frac{\alpha}{D}, where α\alpha is the thermal diffusivity
    • For Le=1Le = 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


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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.