8.2 Forced convection mass transfer

Forced convection mass transfer is all about moving stuff through fluids using external forces. It's like blowing bubbles - the air you blow (forced convection) carries soap molecules (mass) away from the wand. This process combines bulk fluid motion and molecular diffusion to transport mass.

Understanding forced convection mass transfer is key to many real-world applications. From drug delivery in the body to industrial chemical processes, it's the driving force behind how substances move and mix in flowing fluids. Mastering this concept opens doors to solving complex mass transfer problems.

Forced Convection Mass Transfer

Mechanisms of Mass Transfer

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• Forced convection mass transfer involves the transport of a species (mass) in a fluid due to an externally imposed velocity field or flow
• The primary mechanisms of mass transfer in forced convection are advection and diffusion
  • Advection transports mass due to the bulk motion of the fluid, driven by the imposed velocity field (e.g., flow through a pipe)
  • Diffusion transports mass due to concentration gradients within the fluid, driven by random molecular motion (e.g., diffusion of a solute in a solution)
• The relative importance of advection and diffusion in forced convection mass transfer is determined by the Péclet number (Pe), which is the ratio of the advective transport rate to the diffusive transport rate
  • High Pe indicates advection-dominated mass transfer
  • Low Pe indicates diffusion-dominated mass transfer

Role of Boundary Layers

• Boundary layers play a crucial role in forced convection mass transfer, as they determine the concentration gradient and mass transfer rate at the surface
  • The concentration boundary layer is the region near the surface where the concentration of the species changes from the surface concentration to the freestream concentration
  • The thickness of the concentration boundary layer depends on factors such as the fluid velocity, fluid properties, and surface geometry
• Thinner concentration boundary layers result in higher concentration gradients and increased mass transfer rates (e.g., high-velocity flow over a flat plate)
• Thicker concentration boundary layers result in lower concentration gradients and reduced mass transfer rates (e.g., low-velocity flow through a packed bed)

Factors Influencing Mass Transfer Rates

Fluid Properties and Flow Conditions

• The fluid velocity has a significant impact on the mass transfer rate in forced convection. Higher velocities lead to thinner concentration boundary layers and increased mass transfer rates
• The physical properties of the fluid, such as density, viscosity, and diffusivity, influence the mass transfer rate
  • Higher diffusivity leads to faster mass transfer, as it promotes the transport of species due to concentration gradients (e.g., gases have higher diffusivities than liquids)
  • Higher viscosity can reduce the mass transfer rate by dampening turbulent fluctuations and increasing the thickness of the concentration boundary layer (e.g., viscous oils)
• Temperature variations can affect the mass transfer rate by altering the fluid properties and the diffusion coefficient of the species
  • Higher temperatures generally increase diffusion coefficients and reduce viscosity, enhancing mass transfer rates

Surface Geometry and Reactions

• The geometry of the surface affects the mass transfer rate by influencing the flow pattern and the development of the concentration boundary layer
  • Surfaces with sharp edges or protrusions can promote turbulence and enhance mass transfer (e.g., roughened surfaces, fins)
  • Streamlined surfaces or surfaces with gradual changes in geometry may result in lower mass transfer rates (e.g., smooth pipes, airfoils)
• The presence of chemical reactions at the surface can significantly enhance or limit the mass transfer rate, depending on the nature of the reaction and its kinetics
  • Fast reactions can consume the species at the surface, maintaining a high concentration gradient and enhancing mass transfer (e.g., catalytic reactions)
  • Slow reactions or surface saturation can limit the mass transfer rate by reducing the concentration gradient (e.g., adsorption processes)

Solving Mass Transfer Problems

Dimensionless Numbers and Correlations

• 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 analogous to the Nusselt number in heat transfer
  • $Sh = \frac{hL}{D}$, where $h$ is the mass transfer coefficient, $L$ is the characteristic length, and $D$ is the diffusion coefficient
• The Reynolds number (Re) is a dimensionless number that characterizes the flow regime (laminar, transitional, or turbulent) and is used to determine the appropriate correlation for mass transfer
  • $Re = \frac{\rho uL}{\mu}$, where $\rho$ is the fluid density, $u$ is the velocity, $L$ is the characteristic length, and $\mu$ is the dynamic viscosity
• The Schmidt number (Sc) is a dimensionless number that relates the momentum diffusivity (kinematic viscosity) to the mass diffusivity. It is used to characterize the relative importance of momentum and mass transfer
  • $Sc = \frac{\nu}{D}$, where $\nu$ is the kinematic viscosity and $D$ is the diffusion coefficient
• Empirical correlations, such as the Dittus-Boelter correlation or the Chilton-Colburn analogy, can be used to estimate the mass transfer coefficient or the Sherwood number for specific flow geometries and conditions
  • Dittus-Boelter correlation for turbulent flow in pipes: $Sh = 0.023Re^{0.8}Sc^{0.4}$
  • Chilton-Colburn analogy for laminar flow over a flat plate: $Sh = 0.664Re^{0.5}Sc^{1/3}$

Calculating Mass Transfer Rates

• The mass transfer coefficient (h_m) relates the mass transfer rate to the concentration difference between the surface and the freestream. It can be determined using the Sherwood number and the characteristic length of the system
  • $h_m = \frac{ShD}{L}$
• The mass transfer rate can be calculated using the mass transfer coefficient, the concentration difference, and the surface area available for mass transfer
  • $\dot{m} = h_mA(C_s - C_\infty)$, where $\dot{m}$ is the mass transfer rate, $A$ is the surface area, $C_s$ is the surface concentration, and $C_\infty$ is the freestream concentration
• The concentration difference is the driving force for mass transfer, and it can be determined based on the specific problem conditions (e.g., solubility, partial pressure, chemical equilibrium)

Mass Transfer vs Heat Transfer

Similarities

• Forced convection mass transfer and heat transfer share similarities in their governing mechanisms, as both involve the transport of a quantity (mass or energy) due to fluid motion and gradients
• The Péclet number in mass transfer is analogous to the Péclet number in heat transfer, as it determines the relative importance of advection and diffusion
• The Sherwood number in mass transfer is analogous to the Nusselt number in heat transfer, as both represent the ratio of the convective transfer rate to the diffusive transfer rate
• The Schmidt number in mass transfer is analogous to the Prandtl number in heat transfer, as both relate the momentum diffusivity to the diffusivity of the transported quantity (mass or heat)

Differences

• Mass transfer involves the transport of a specific species, while heat transfer involves the transport of energy
  • Mass transfer is driven by concentration gradients, while heat transfer is driven by temperature gradients
  • The diffusion coefficient in mass transfer depends on the specific species being transported (e.g., oxygen in air, salt in water), while the thermal diffusivity in heat transfer is a property of the fluid
• Chemical reactions can have a direct impact on mass transfer rates, while they do not directly affect heat transfer rates (although they may indirectly influence heat transfer through changes in fluid properties or the generation/consumption of heat)
  • Exothermic reactions can enhance mass transfer by consuming the species at the surface and maintaining a high concentration gradient
  • Endothermic reactions can limit mass transfer by producing the species at the surface and reducing the concentration gradient
• The boundary conditions for mass transfer can be more complex than those for heat transfer, as they may involve multiple species, chemical equilibrium, or surface reactions (e.g., catalysis, adsorption)