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Introduction to Mass Transfer

Mass Transfer
The term mass transfer is used to denote the transference of a component in a mixture
from a region where its concentration is high to a region where the concentration is lower.
Mass transfer process can take place in a gas or vapour or in a liquid, and it can result
from the random velocities of the molecules (molecular diffusion) or from the circulating
or eddy currents present in a turbulent fluid (eddy diffusion).
In processing, it is frequently necessary to separate a mixture into its components and,
in a physical process, differences in a particular property are exploited as the basis for
the separation process. Thus, fractional distillation depends on differences in volatility,
gas absorption on differences in solubility of the gases in a selective absorbent and,
similarly, liquid- liquid extraction is based on on the selectivity of an immiscible liquid
solvent for one of the constituents. The rate at which the process takes place is dependent
both on the driving force (concentration difference) and on the mass transfer resistance.
In most of these applications, mass transfer takes place across a phase boundary where
the concentrations on either side of the interface are related by the phase equilibrium
relationship. Where a chemical reaction takes place during the course of the mass transfer
process, the overall transfer rate depends on both the chemical kinetics of the reaction and
on the mass transfer resistance, and it is important to understand the relative significance
of these two factors in any practical application.
In this chapter, consideration will be given to the basic principles underlying mass
transfer both with and without chemical reaction, and to the models which have been
proposed to enable the rates of transfer to be calculated. The applications of mass transfer
to the design and operation of separation processes are discussed in Volume 2, and the
design of reactors is dealt with in Volume 3.
A simple example of a mass transfer process is that occurring in a box consisting of
two compartments, each containing a different gas, initially separated by an impermeable
partition. When the partition is removed the gases start to mix and the mixing process
continues at a constantly decreasing rate until eventually (theoretically after the elapse of
an infinite time) the whole system acquires a uniform composition. The process is one
of molecular diffusion in which the mixing is attributable solely to the random motion
of the molecules. The rate of diffusion is governed by Pick's Law, first proposed by
FlCK(I) in 1855 which expresses the mass transfer rate as a linear function of the molar
concentration gradient. In a mixture of two gases A and B, assumed ideal, Pick's Law
for steady state diffusion may be written as:

where NA is the molar flux of A (moles per unit area per unit time),
CA is the concentration of A (moles of A per unit volume),
DAB is known as the diffusivity or diffusion coefficient for A in B, and
y is distance in the direction of transfer

The condition for the pressure or molar concentration to remain constant in such a system
is that there should be no net transference of molecules. The process is then referred to
as one of equimolecular counterdiffusion, and:
Equation 10.4, which describes the mass transfer rate arising solely from the random
movement of molecules, is applicable to a stationary medium or a fluid in streamline
flow. If circulating currents or eddies are present, then the molecular mechanism will be
reinforced and the total mass transfer rate may be written as:
Whereas D is a physical property of the system and a function only of its composition,
pressure and temperature, ED, which is known as the eddy diffusivity, is dependent on the
flow pattern and varies with position. The estimation of ED presents some difficulty, and
this problem is considered in Chapter 12.
The molecular diffusivity D may be expressed in terms of the molecular velocity um
and the mean free path of the molecules A.m. In Chapter 12 it is shown that for conditions
where the kinetic theory of gases is applicable, the molecular diffusivity is proportional to
the product um"km. Thus, the higher the velocity of the molecules, the greater is the distance
they travel before colliding with other molecules, and the higher is the diffusivity D.
Because molecular velocities increase with rise of temperature 71, so also does the
diffusivity which, for a gas, is approximately proportional to T raised to the power of
1.5. As the pressure P increases, the molecules become closer together and the mean free
path is shorter and consequently the diffusivity is reduced, with D for a gas becoming
approximately inversely proportional to the pressure.
A method of calculating D in a binary mixture of gases is given later (equation 10.43). For
liquids, the molecular structure is far more complex and no such simple relationship exists.
although various semi-empirical predictive methods, such as equation 10.96, are useful.
In the discussion so far, the fluid has been considered to be a continuum, and distances
on the molecular scale have, in effect, been regarded as small compared with the dimen
sions of the containing vessel, and thus only a small proportion of the molecules collides
directly with the walls. As the pressure of a gas is reduced, however, the mean free path
may increase to such an extent that it becomes comparable with the dimensions of the
vessel, and a significant proportion of the molecules may then collide directly with the
walls rather than with other molecules. Similarly, if the linear dimensions of the system
are reduced, as for instance when diffusion is occurring in the small pores of a cata
lyst particle (Section 10.7), the effects of collision with the walls of the pores may be
important even at moderate pressures. Where the main resistance to diffusion arises from
collisions of molecules with the walls, the process is referred to Knudsen diffusion, with
a Knudsen diffusivity DKH which is proportional to the product uml, where / is a linear
dimension of the containing vessel.
Each resistance to mass transfer is proportional to the reciprocal of the appropriate diffu
sivity and thus, when both molecular and Knudsen diffusion must be considered together,
the effective diffusivity De is obtained by summing the resistances as:

In liquids, the effective mean path of the molecules is so small that the effects of Knudsen
type diffusion need not be considered.

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