Page 342 - Academic Press Encyclopedia of Physical Science and Technology 3rd Chemical Engineering

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Encyclopedia of Physical Science and Technology EN009H-407 July 18, 2001 23:34

Mass Transfer and Diffusion 179

not equal. The free energies are equal, but free energy is sort of fictional concentration difference designed for our
a considerably more difficult concept than concentration. convenience.
The result of this chemistry is that the mass flux across
an interface from one phase into the other is not directly
proportional to the concentration difference between the IV. CONCLUSIONS
two phases. Instead, it is proportional to the concentration
in the one phase minus the concentration that would exist Diffusion, dispersion, and mass transfer are three ways to
in the other phase if it were in equilibrium. In the example describe molecular mixing. Diffusion, the result of molec-
just given, this concentration difference is the value in ular motions, is the most fundemental, and leads to pre-
water minus the value in hypothetical water in equilibrium dictions of concentration as a function of position and
with benzene. This concentration difference makes the time. Dispersion can follow the same mathematics used
study of mass transfer coefficients difficult. for diffusion, but it is due not to molecular motion but
To make these ideas more quantitative, imagine that we to flow. Mass transfer, the description of greatest value to
are absorbing sulfur dioxide from a flue gas stream into the chemical industry, commonly involves solutes moving
an aqueous stream. The flux of sulfur dioxide is given by across interfaces, most commonly, fluid-fluid interfaces.
the equations Together, these three methods of analysis are important
tools for chemical engineering.
N 1 = k p (p 1 − p 1i ) (39)
where k p is the form of mass transfer coefficients based
on partial pressure differences, p 1 is the partial pressure NOTATION
of the SO 2 in the bulk gas, and p 1i is the partial pressure in
the gas at the gas/liquid interface. This flux is also given a Surface area per volume
by A Area
c 1 Concentration of species “1”
N 1 = k x (x 1i − x 1 ) (40)
d Pipe diameter
where x 1i is the mole fraction of SO 2 at the gas/liquid in- D Diffusion coefficient
terface but in the liquid, and x 1 is the mole fraction of SO 2 D ij Diffusion coefficients in multicomponent
in the bulk liquid. While these interfacial concentrations systems
are almost always unknown, they are related by a Henry’s E Dispersion coefficient
law constant H: H Henry’s law constant
Diffusion flux of species “1”
j 1
(41)
p 1i = Hx 1i
k, k p , k x Mass transfer coefficients
When we combine Eqs. (35)–(37), we obtain the relation- k B Boltzman’s constant
ship K p Overall mass transfer coefficient
l Length or thickness
 
M Total solute mass in pulse
1
 
N 1 =   (p 1 − Hx 1 ) (42) n 1 Total flux of species “1”
 1
H  Interfacial flux of species “1”
N 1
+
k p k x p Total pressure
This result is frequently written as p 1 Partial pressure of species “1”
S Amount solute emitted per time

N 1 = K p p 1 − p ∗ (43)
1 T Temperature
where the overall mass transfer coefficient K p is equal to t Time
the quantity in square brackets in Eq. (42) and the hypo- v 1 , v 0 Velocity of species “1” and of reference,
thetical partial pressure p is simply equal to Hx 1 . This respectively
∗
1
p is the partial pressure that would exist in the gas if the V Volume
∗
1
gas were in equilibrium with the liquid. x Velocity direction
This analysis is difficult, and takes careful thought to x 1 , y 1 Mole fractions of species “1” in liquid
understand. The key test is to constantly ask what hap- and gas, respectively
pens at equilibrium. At equilibrium, the partial pressure z Position
difference, or the mole fraction difference, or the concen- γ 1 Activity coefficient of species “1”
tration difference must be zero. The only question is does µ Viscosity
that difference represent an actual concentration or some µ 1 Chemical potential