Transport phenomena

From Chemepedia

Transport Phenomena are irreversible processes that arise from the random continuous motion of molecules. The three types of transport phenomena are momentum, mass of a component in a multicomponent mixture, and thermal energy. A complete description of the transport phenomena is embodied in the following two relationships:

  • Conservation laws
  • Constitutive molecular flux laws

The conservation laws are continuity equations; these equations dictate that the accumulation of a quantity in a control volume equals the transport of into and out of the control volume plus any generation or consumption of in the control volume due to non-conservative mechanisms.

In other words,

The conservation laws can be formulated as either microscopic (differential) balances or as macroscopic (integral) balances.

Microscopic balance

In the microscopic description, the balance equation is formulated on a per volume basis. For instance, if is a transport quantity is on a per volume basis, the differential balance is

            (1)

where

  • is the rate of accumulation of on a per volume basis. is the mass density.
  • is the convective inflow and outflow of per unit volume due to both advective (bulk flow) and molecular (diffusive) mechanisms.
  • is the generation or consumption of on a per volume basis due non-conservative mechanisms. This may be an external force in the momentum balance or a reaction in a mole balance.

Macroscopic balance

In the macroscopic description and there are three types mass, momentum and energy, , and the integral balance equation is

            (2)

where

  • is the rate of accumulation of in the control volume.
  • is the convective inflow and outflow of due to both advective (bulk-flow) and molecular (diffusive) mechanisms.
  • is the generation or consumption of per unit area at the control surface.
  • is the generation or consumption of per unit volume at the control surface.
  • is the unit normal that defines the surface of the the control volume.

Flux

Flux describes the movement of a property. For this class specifically, we will only be dealing with transport properties of Momentum, Energy and Mass. Flux is a vector that has both magnitude and direction and usually has units of the property per unit area per unit time.

For example,

  • Mole Flux has units of moles per unit area per unit time.
  • And Energy Flux has units of energy per unit area per unit time.


Different mechanisms allow for flux to occur.

  1. Bulk Flow or Advective Mechanism
  2. Molecular or Diffusive / Conductive Mechanism


Advective Flux

Advection is the bulk movement of a property due to a velocity field. That is, the advective movement occurs in the direction of the velocity field.

For example,

After perfume is sprayed in a room with the wind blowing, someone standing in the path of the wind will smell the perfume, because the wind carries the perfume particles along with it until the particles are evenly distributed. This all occurs due to the advective movement of the particles.

Convective Flux

Convective flux is the movement of a property due to both advective and molecular mechanisms. It has dimensions of "property per unit time per unit area" and has the general form

            (3)

where

  • is the advective flux of due to a velocity field
  • is the molecular flux of due to the gradient in .

Constitutive molecular flux laws

The constitutive equations describe how the flux of , or is related to a particular gradient (Cartesian, Cylindrical, Spherical). The three most common flux laws are

  • Newton's law of viscosity in momentum transfer:
  • Fourier's law of conduction in heat transfer :
  • Fick's law of diffusion in mass transfer:

Diffusivity and the analogy between the three types of molecular transport

In the absence of external bulk flow and non-conservative mechanisms, the microscopic balance reduces to

            (4)

Substituting the constitutive molecular flux laws into the above equation for momentum, heat, and mass, each equation can be written explicitly as

            (5)

In each case, the coefficient before the Laplacian operator has dimensions of length2 per unit time and is known as the diffusivity. The diffusivities for the three cases are

  • Momentum: Kinematic viscosity
  • Thermal energy: Thermal diffusivity
  • Mass: Diffusion coefficient

Hence, the diffusivities give the characteristic rates at which the velocity field, temperature field, and density field spreads in the absence of advective (bulk flow) mechanisms.

Summary of transport relationships

Momentum Mass Thermal Energy
Transport quantity (Internal energy) or (Enthalpy)
Transport quantity per unit volume or
Solution of microscopic balance Velocity field Density or concentration field Temperature field
Flux law
Diffusivity (Length2/time)
Convective transfer coefficient
Convective boundary condition
Dimensionless convection/diffusion ratio Reynolds number Sherwood number Nusselt number
Corresponding macroscopic balance Mechanical energy balance Component mass balance Internal energy balance
Key dimensionless numbers Friction factor, Reynolds number Sherwood number, Reynolds number, Schmidt number Nusselt number, Reynolds number, Grashof number, Prandtl number
Correlation relationship (forced convection) or (natural convection)
Other transport modes Radiation