6.3 Convective Mass Transfer

Convective mass transfer is all about how chemicals move in fluids. It's key to understanding how stuff spreads in liquids and gases. This topic digs into the nuts and bolts of how it works, from basic principles to real-world applications.

We'll look at the math behind convective mass transfer and how to solve problems. We'll also explore important concepts like boundary layers and dimensionless numbers that help engineers design better systems.

Convective Mass Transfer Mechanisms

Transport of Chemical Species

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• Convective mass transfer involves the transport of chemical species within a fluid due to the bulk motion of the fluid
  • Forced convection occurs when external forces cause fluid motion (pumps, fans)
  • Natural convection occurs when density differences drive fluid motion (buoyancy-driven flow)
• The rate of convective mass transfer is influenced by various factors
  • Fluid properties such as density, viscosity, and diffusivity affect the transport of species
  • Flow characteristics including velocity, turbulence, and boundary layer development impact mass transfer rates
  • Geometric considerations like surface area and shape determine the extent of mass transfer

Advection and Diffusion

• The two primary mechanisms of convective mass transfer are advection and diffusion
  • Advection is the transport of species by the bulk motion of the fluid, characterized by the fluid velocity
    • Example: The transport of dissolved sugar in a stirred tank
  • Diffusion is the transport of species due to concentration gradients, governed by Fick's law
    • Example: The spread of dye in a stagnant fluid
• Convective mass transfer plays a crucial role in various engineering applications
  • Chemical reactors rely on convective mass transfer to bring reactants together and remove products
  • Heat exchangers utilize convective mass transfer to enhance heat transfer through phase change processes
  • Absorption and desorption processes, such as gas scrubbing or stripping, depend on convective mass transfer
  • Separation techniques like distillation and extraction employ convective mass transfer to separate components based on their relative volatilities or solubilities

Conservation Equations for Species Transport

Species Continuity Equation

• The conservation of species in a convective mass transfer system is described by the species continuity equation

  • The equation accounts for the accumulation, convection, and diffusion of species

  • The general form of the species continuity equation for species A in a binary mixture is: $\frac{\partial C_A}{\partial t} + \nabla \cdot (\mathbf{u}C_A) = \nabla \cdot (D_{AB} \nabla C_A) + R_A$

    where $C_A$ is the concentration of species A, $t$ is time, $\mathbf{u}$ is the velocity vector, $D_{AB}$ is the binary diffusion coefficient, and $R_A$ is the rate of generation or consumption of species A

• Simplifications to the species continuity equation can be made based on the specific problem
  • Steady-state conditions assume that concentrations and velocities do not change with time
  • Incompressible flow assumes constant fluid density, simplifying the continuity equation
  • Negligible diffusion in certain directions may be assumed when diffusion is much slower than advection

Coupling with Other Conservation Equations

• The species continuity equation is coupled with other conservation equations to fully describe the convective mass transfer system
  • The conservation of mass (continuity equation) ensures that the total mass of the system is conserved
    • Example: The continuity equation for an incompressible fluid is $\nabla \cdot \mathbf{u} = 0$
  • The conservation of momentum (Navier-Stokes equations) describes the motion of the fluid
    • Example: The Navier-Stokes equations for an incompressible Newtonian fluid are $\rho (\frac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u}) = -\nabla p + \mu \nabla^2 \mathbf{u} + \mathbf{f}$
• Solving the coupled equations simultaneously provides a complete description of the convective mass transfer system
  • Numerical methods, such as finite difference or finite element methods, are often employed to solve the coupled equations
  • Simplifying assumptions and appropriate boundary conditions are applied to make the problem tractable

Solving Convective Mass Transfer Problems

Boundary Conditions

• Solving convective mass transfer problems requires specifying appropriate boundary conditions
  • Boundary conditions describe the concentration or flux of species at the system boundaries (walls, inlets, outlets)
  • Common boundary conditions include:
    • Specified concentration: The concentration of a species is known at the boundary (constant concentration at a wall)
    • Specified flux: The mass flux of a species is known at the boundary, which can be due to surface reactions, adsorption, or desorption (constant flux at a catalytic surface)
    • Symmetry: The concentration gradient or mass flux is zero at the boundary due to symmetry (centerline of a pipe)
• Choosing appropriate boundary conditions is crucial for accurately solving convective mass transfer problems
  • Boundary conditions should reflect the physical reality of the system
  • Incorrect or incomplete boundary conditions can lead to erroneous solutions

Simplifying Assumptions and Solution Methods

• Simplifying assumptions can be made to reduce the complexity of convective mass transfer problems
  • Steady-state conditions assume that concentrations and velocities do not change with time, eliminating time derivatives
  • Incompressible flow assumes constant fluid density, simplifying the continuity equation
  • Negligible diffusion in certain directions may be assumed when diffusion is much slower than advection (plug flow reactor)
• Analytical solutions to convective mass transfer problems are available for simple geometries and boundary conditions
  • Example: The Graetz problem for laminar flow in a circular tube with a constant wall concentration has an analytical solution
  • Analytical solutions provide insight into the fundamental behavior of the system and can serve as benchmarks for numerical solutions
• Numerical methods are often employed to solve more complex convective mass transfer problems
  • Finite difference methods discretize the domain into a grid and approximate derivatives using finite differences
  • Finite element methods divide the domain into elements and approximate the solution using basis functions
  • Numerical methods allow for the solution of problems with complex geometries, non-linear equations, and coupled phenomena

Dimensionless Numbers in Convective Mass Transfer

Schmidt Number

• The Schmidt number (Sc) is a dimensionless number that characterizes the relative thickness of the velocity and concentration boundary layers

  • It is defined as the ratio of momentum diffusivity (kinematic viscosity) to mass diffusivity: $Sc = \frac{\nu}{D}$

    where $\nu$ is the kinematic viscosity and $D$ is the mass diffusivity

• The Schmidt number provides insight into the relative importance of momentum and mass transfer
  • For Sc » 1, the concentration boundary layer is much thinner than the velocity boundary layer (mass transfer is slower than momentum transfer)
  • For Sc « 1, the concentration boundary layer is much thicker than the velocity boundary layer (mass transfer is faster than momentum transfer)
• The Schmidt number is used to develop empirical correlations for convective mass transfer coefficients
  • Example: The Sherwood number (Sh) is often correlated with the Reynolds number (Re) and Schmidt number (Sc) as $Sh = f(Re, Sc)$

Sherwood Number

• The Sherwood number (Sh) is a dimensionless number that represents the ratio of the convective mass transfer rate to the diffusive mass transfer rate

  • It is defined as: $Sh = \frac{h_m L}{D}$

    where $h_m$ is the convective mass transfer coefficient, $L$ is a characteristic length, and $D$ is the mass diffusivity

• The Sherwood number is analogous to the Nusselt number (Nu) in heat transfer
  • Higher values of the Sherwood number indicate a greater influence of convection on mass transfer compared to diffusion
• The Sherwood number is used to characterize the performance of convective mass transfer systems
  • Example: In a packed bed reactor, the Sherwood number can be used to determine the effectiveness factor, which relates the actual reaction rate to the ideal reaction rate without mass transfer limitations
• Empirical correlations for the Sherwood number are developed based on experimental data and dimensionless analysis
  • These correlations relate the Sherwood number to other dimensionless numbers, such as the Reynolds number (Re) and Schmidt number (Sc)
  • Example: The Dittus-Boelter correlation for mass transfer in turbulent flow through a circular pipe is $Sh = 0.023 Re^{0.8} Sc^{0.4}$

Significance in Analysis and Design

• Dimensionless numbers help in the analysis, design, and scaling of convective mass transfer systems
  • They allow for the comparison of different systems and the identification of limiting factors
  • Dimensionless numbers can be used to develop empirical correlations for mass transfer coefficients, which are essential for design calculations
• The Schmidt and Sherwood numbers are particularly useful in the development of analogies between heat and mass transfer
  • The Chilton-Colburn analogy relates the Nusselt number (Nu) for heat transfer to the Sherwood number (Sh) for mass transfer, enabling the use of heat transfer correlations for mass transfer problems
  • Example: The Chilton-Colburn analogy states that $Sh = Nu (Sc/Pr)^{1/3}$, where $Pr$ is the Prandtl number (ratio of momentum diffusivity to thermal diffusivity)
• Dimensionless numbers also facilitate the scaling and optimization of convective mass transfer processes
  • By maintaining similar dimensionless numbers, small-scale experiments can be used to predict the performance of large-scale systems
  • Optimization can be performed by manipulating the dimensionless numbers to achieve desired mass transfer rates or system efficiency