Details

,

,

where is the total flux (), and , are the fluxes of and ().

The previous equation can be rearranged in the form of Fick's law by solving for :

.

The term accounts for the bulk flow effect, where

is the total molar concentration (),

is the ideal gas constant (),

is temperature (K),

is diffusivity (),

is the liquid mole fraction of

and is the height of the tube (m).

At quasi-steady state and with constant molar density, in integral form:

.

Upon integration,

,

where is the mole fraction at , and is the mole fraction at .

The mole fraction variation as a function of is:

.

A more useful form of can be derived from the definition of the log mean:

.

Combining and gives:

.

From Dalton's law, assuming equilibrium at the liquid octane/air interface:

,

,

where is the saturation pressure (atm).

The saturation pressure is calculated from the Antoine equation:

,

where , and are Antoine constants.

Solve the diffusive flux equation using the boundary conditions and :

.

The concentration profile is

;

solving for ,

.

If the diffusive flux is greater than 90% of the total flux, then .

Finally, the molar flow rate through the top of the tube is calculated. Given that is the cross-sectional area of the tube:

.

The mass flow rate is then

,

where is the molecular weight of the compound.

Reference

[1] J. D. Seader, E. J. Henley and D. K. Roper, Separation Process Principles: Chemical and Biochemical Operations, 3rd ed., Hoboken, NJ: John Wiley & Sons, 2011 pp. 88–89.