8.3 Natural convection mass transfer

Natural convection mass transfer occurs when density differences in fluids, caused by concentration gradients, drive mass movement without external forces. This process relies on buoyancy forces and forms concentration boundary layers near surfaces where mass transfer happens.

Key dimensionless parameters like the Grashof, Sherwood, and Schmidt numbers help describe and analyze natural convection mass transfer. These numbers relate buoyancy forces, convective mass transfer, and fluid properties, enabling us to solve complex mass transfer problems using empirical correlations.

Natural convection mass transfer

Principles and driving forces

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• Natural convection mass transfer transports mass due to density differences in a fluid caused by concentration gradients, without the aid of external forces (pumps or fans)
• The buoyancy force, which arises from density variations in the fluid due to concentration differences, drives natural convection mass transfer
• In natural convection mass transfer, the concentration gradient induces fluid motion, leading to the formation of boundary layers near the surface where mass transfer occurs
• The concentration boundary layer, a region near the surface where the concentration gradient exists, has a thickness that depends on the fluid properties and the strength of the buoyancy force
• The balance between the buoyancy force and the viscous force acting on the fluid determines the rate of mass transfer in natural convection

Dimensionless parameters

• The Grashof number (Gr) represents the ratio of buoyancy forces to viscous forces in natural convection mass transfer
  • Gr = $\frac{g \beta (C_s - C_\infty) L^3}{\nu^2}$, where $g$ is the gravitational acceleration, $\beta$ is the volumetric expansion coefficient, $C_s$ and $C_\infty$ are the surface and bulk concentrations, $L$ is the characteristic length, and $\nu$ is the kinematic viscosity
• The Sherwood number (Sh) represents the ratio of convective mass transfer to diffusive mass transfer
  • Sh = $\frac{h_m L}{D}$, where $h_m$ is the mass transfer coefficient, $L$ is the characteristic length, and $D$ is the mass diffusivity
• The Schmidt number (Sc) is the ratio of momentum diffusivity to mass diffusivity
  • Sc = $\frac{\nu}{D}$, where $\nu$ is the kinematic viscosity and $D$ is the mass diffusivity

Factors affecting mass transfer rates

Concentration difference and fluid properties

• A higher concentration difference between the surface and the bulk fluid leads to a stronger buoyancy force and enhanced mass transfer rates
• Fluid properties (density, viscosity, and diffusivity) play a crucial role in determining the natural convection mass transfer rate
  • Lower fluid viscosity promotes faster fluid motion and increases the mass transfer rate
  • Higher diffusivity of the species being transferred facilitates faster mass transport through the boundary layer

Geometry and surface characteristics

• The geometry of the system (shape and orientation of the surface) affects the flow patterns and the mass transfer rate
  • Vertical surfaces generally experience higher mass transfer rates compared to horizontal surfaces due to the unobstructed upward flow of the fluid
  • Rough surfaces enhance mass transfer by promoting turbulence and disrupting the boundary layer
• The presence of external factors (vibrations or surface waves) can enhance the mass transfer rate by inducing additional fluid motion and mixing

Solving mass transfer problems

Dimensionless correlations

• Empirical correlations based on dimensionless numbers estimate the mass transfer coefficients in natural convection
• The most common correlation for natural convection mass transfer is the Sherwood-Rayleigh-Schmidt (Sh-Ra-Sc) correlation, which relates the Sherwood number to the Rayleigh number (Ra) and Schmidt number
  • The Rayleigh number is the product of the Grashof number and Schmidt number (Ra = Gr × Sc)
  • The Sh-Ra-Sc correlation takes the form: Sh = C × (Ra)^n × (Sc)^m, where C, n, and m are constants that depend on the geometry and flow conditions

Problem-solving approach

• To solve natural convection mass transfer problems, select the appropriate correlation based on the system geometry and flow regime
• Calculate the mass transfer coefficient using the dimensionless numbers (Sh, Ra, Sc)
• Example: For a vertical plate in laminar flow, the correlation is Sh = 0.59 × (Ra)^(1/4) × (Sc)^(1/3)
• Example: For a horizontal cylinder in turbulent flow, the correlation is Sh = 0.53 × (Ra)^(1/4) × (Sc)^(1/3)

Mass transfer vs heat transfer

Similarities

• Both processes involve the transport of a quantity (mass or heat) due to buoyancy-driven fluid motion
• The driving force is the density difference in the fluid, caused by either concentration gradients (mass transfer) or temperature gradients (heat transfer)
• The development of boundary layers (concentration boundary layer for mass transfer and thermal boundary layer for heat transfer) is a common feature

Differences

• In mass transfer, the transported quantity is mass or species, while in heat transfer, it is thermal energy
• The driving potential in mass transfer is the concentration difference, whereas in heat transfer, it is the temperature difference
• The dimensionless numbers used in mass transfer (Sh, Sc) are analogous to those in heat transfer (Nu, Pr), but they represent different physical quantities
  • Nusselt number (Nu) represents the ratio of convective heat transfer to conductive heat transfer
  • Prandtl number (Pr) is the ratio of momentum diffusivity to thermal diffusivity