Page 103 - Separation process principles 2

68 Chapter 3 Mass Transfer and Diffusion

lower to the upper region. After a long time, a dynamic equi- where, for convenience, the z subscript on J has been
librium will be achieved and the concentration of dye will be dropped, c = total molar concentration or molar density
uniform throughout the solution. Based on these observa- (c = 1 /v = p / M) , and XA = mole fraction of species A.
tions, it is clear that: Equation (3-4) can also be written in the following equiv-
alent mass form, where jA is the mass ilux of A by ordinary
1. Mass transfer by ordinary molecular diffusion occurs
molecular diffusion relative to the mass-average velocity of
because of a concentration, difference or gradient; that the mixture in the positive z-direction, p is the mass density,
is, a species diffuses in the direction of decreasing
and WA is the mass fraction of A:
concentration.
2. The mass-transfer rate is proportional to the area normal
to the direction of mass transfer and not to the volume
of the mixture. Thus, the rate can be expressed as a flux.
Velocities in Mass Transfer
3. Net mass transfer stops when concentrations are
uniform. It is useful to formulate expressions for velocities of chemi-
cal species in the mixture. If these velocities are based on
the molar flux, N, and the molar diffusion flux, J, the molar
Fick's Law of Diffusion
average velocity of the mixture, VM, relative to stationary
The above observations were quantified by Fick in 1855, who coordinates is given for a binary mixture as
proposed an extension of Fourier's 1822 heat-conduction
theory. Fourier's first law of heat conduction is

Similarly, the velocity of species i, defined in terms of Ni, is
relative to stationary coordinates:
where q, is the heat flux by conduction in the positive z-
direction, k is the thermal conductivity of the medium, and
dT/dz is the temperature gradient, which is negative in the
direction of heat conduction. Fick's first law of molecular Combining (3-6) and (3-7) with xi = ci/c gives
diffusion also features a proportionality between a flux and a
gradient. For a binary mixture of A and B,
Alternatively, species diffusion velocities, viD, defined in
terms of Ji, are relative to the molar-average velocity and are
defined as the difference between the species velocity and
and the molar-average velocity for the mixture:

where, in (3-3a), JA2 is the molar flux of A by ordinary mol-
ecular diffusion relative to the molar-average velocity of the When solving mass-transfer problems involving net
mixture in the positive z direction, DAB is the mutual diffu- movement of the mixture, it is not convenient to use fluxes
sion coefficient of A in B, discussed in the next section, c~ is and flow rates based on VM as the frame of reference. Rather,
the molar concentration of A, and dcA/dz is the concentra- it is preferred to use mass-transfer fluxes referred to station-
tion gradient of A, which is negative in the direction of ordi- ary coordinates with the observer fixed in space. Thus, from
nary molecular diffusion. Similar definitions apply to (3-3b). (3-9), the total species velocity is
The molar fluxes of A and B are in opposite directions. If the
Vi = VM + ViD
gas, liquid, or solid mixture through which diffusion occurs
is isotropic, then values of k and DAB are independent of di- Combining (3-7) and (3- lo),
rection. Nonisotropic (anisotropic) materials include fibrous
and laminated solids as well as single, noncubic crystals.
The diffusion coefficient is also referred to as the diffusivity
Combining (3-11) with (3-4), (3-6), and (3-7),
and the mass diffusivity (to distinguish it from thermal and
momentum diffusivities).
Many alternative forms of (3-3a) and (3-3b) are used,
depending on the choice of driving force or potential in the
gradient. For example, we can express (3-3a) as and