Diffusion in Mass Transfer

 Mass Transfer is the net movement due to a concentration gradient of a component in a mixture from one location to another. Generally, this transfer occurs between two phases across an interface. Mass transfer can also be defined as the selective permeation through a non-porous polymeric material of a component of a gas mixture.

Mass transfer models can be used to describe processes like the passage of a species through a gas to the outer surface of a porous adsorbent particle and into the pores of the adsorbent, where the species is adsorbed on the porous surface.

Mass transfer occurs through two basic mechanisms:

(a) Molecular diffusion by random and spontaneous microscopic movement of individual molecules in a gas, liquid, or solid due to thermal motion

(b) Eddy (turbulent) diffusion by random macroscopic fluid motion.

Molecular Diffusion vs. Eddy Diffusion

Both diffusion types involve different species moving in opposite directions. When a net flow occurs in one of these directions, the total rate of mass transfer of the individual species is increased or decreased by this bulk flow or convection effect, which is a third mechanism of mass transfer.

Molecular diffusion is extremely slow, while eddy diffusion, when it occurs, is orders of magnitude more rapid.

Molecular diffusion typically occurs in solids and fluid involving stagnant, laminar or turbulent flow while eddy diffusion occurs in fluids with turbulent flow.

• In a binary mixture, molecular diffusion occurs due to one or more driving forces like:
• differences in concentration (ordinary diffusion)
• pressure (pressure diffusion). This requires a large pressure gradient which is achieved for gas mixtures with a centrifuge.
• temperature (thermal diffusion). Temperature gradients can be achieved by using thermal diffusion columns to separate liquid and gas mixtures
• external force fields (forced diffusion) that act unequally on the different chemical species present. This is achieved by using an electric field to cause ions of different charges to move in different directions at different speeds

When both molecular diffusion and eddy diffusion occur, they take place in parallel and are additive. Furthermore, they take place because of the same concentration gradient. 

Describing diffusion quantitatively

Assume that molecule A is diffusing between boundary 1 and 2 with fixed concentrations c_A,1 and c_A,2 respectively. The rate of diffusional mass transfer (moles/time) across an area, A, can be determined. A represents an area of y by z in the following illustration:

Fick’s Law of Diffusion

The concentration profile is linear for any system with only pure diffusion (no convection, no reactions, constant properties). Thus, Fick’s law can be approximated as:

The driving gradient for diffusional mass transfer is the difference in concentrations i.e., c_A,1 - c_A,2

Variable	Definition	Units
ṄA,x	moles of species A transferred per unit time from location 1 to location 2	mol/s
DAB	the diffusion coefficient of species A through medium B	m2/s
A	the area through which transfer occurs	m2
cA,1 - cA,2	the concentration gradient between locations 2 and 1	mol/m3
x2-x1	The distance between locations 2 and 1	m

NOTE: the area for mass transfer is not the ‘edge-view’ area but the ‘face-view’ area. This is the area used in Fick’s Law and Fourier’s Law calculations:

In some cases, mass transport occurs through a porous membrane with a pore fraction of ε_pore. In such cases, the actual area available for mass transfer is only the porous fraction of the total, and thus,

A = A_apparent ε_pore

Diffusion Coefficient

The diffusion coefficient, D_AB, is a proportionality constant between the molar flux due to molecular diffusion and the gradient in the concentration of the species.

Theoretically, the diffusion coefficient is proportional to the mean squared displacement divided by the time elapsed: 

The value of the diffusion coefficient is determined through experiments, theory and estimation

The properties that influence the diffusion coefficient of molecule A in solution B are:

• molecular weight of A and/or B,
• molecular size of A and/or B,
• molecular properties like charge, ionic strength, dipole moment of A and/or B,
• temperature,
• pressure.

Equimolecular counter diffusion

This occurs when the mass transfer rates of the two components are equal and opposite.

It occurs in the case of the box with a movable partition and also in a distillation column when the molar latent heats of the two components are the same (λA = λB).