Page 139 - Separation process principles 2

104 Chapter 3 Mass Transfer and Diffusion

gaseous A. Because the gas is pure A at total pressure the bulk liquid. If the diffusivity of SOz in water is
1.7 x cm2/s, determine the mass-transfer coefficient, kc, and
P = PA, there is no resistance to mass transfer in the gas
phase. At the gas-liquid interface, phase equilibrium is as- the film thickness, neglecting the bulk-flow effect.
sumed so the concentration of A, CA, , is related to the partial
pressure of A, PA, by some form of Henry's law, for exam- SOLUTION
ple, cA, = HApA In the thin, stagnant liquid film of thick-
0.027(1,000) mol
ness 6, molecular diffusion only occurs with a driving force Nsoz = (3,600)(100)2 = 7.5 lo-7- cm2-s
of CA, - CA~. Since the film is assumed to be very thin, all of
the diffusing A passes through the film and into the bulk liq- For dilute conditions, the concentration of water is
uid. If, in addition, bulk flow of A is neglected, the concen-
tration gradient is linear as in Figure 3.18a. Accordingly,
Fick's first law, (3-3a), for the diffusion flux integrates to
From (3-188),

If the liquid phase is dilute in A, the bulk-flow effect can be 7.5 10-7
- = 6.14 x cmls
-
neglected and (3-187) applies to the total flux: 5.55 x 10-2(0.0025 - 0.0003)
Therefore,

If the bulk-flow effect is not negligible, then, from (3-31),
which is very small and typical of turbulent-flow mass-transfer
processes.

Penetration Theory
where
A more realistic physical model of mass transfer from a
fluid-fluid interface into a bulk liquid stream is provided by
the penetration theory of Higbie [59], shown schematically
in Figure 3.18b. The stagnant-film concept is replaced by
Boussinesq eddies that, during a cycle, (1) move from the
In practice, the ratios DA~/8 in (3-188) and DAB/
bulk to the interface; (2) stay at the interface for a short,
S(l - xA)LM in (3- 189) are replaced by mass transfer coeffi-
fixed period of time during which they remain static so that
cients kc and ki, respectively, because the film thickness, 6,
molecular diffusion takes place in a direction normal to the
which depends on the flow conditions, is not known and the
interface; and (3) leave the interface to mix with the bulk
subscript, c, refers to a concentration driving force.
The film theory, which is easy to understand and apply, is stream. When an eddy moves to the interface, it replaces an-
other static eddy. Thus, the eddies are alternately static and
often criticized because it appears to predict that the rate of
moving. Turbulence extends to the interface.
mass transfer is directly proportional to the molecular diffu-
sivity. This dependency is at odds with experimental data, In the penetration theory, unsteady-state diffusion takes
place at the interface during the time the eddy is static. This
which indicate a dependency of Dn, where n ranges from
process is governed by Fick's second law, (3-68), with
about 0.5 to 0.75. However, if DAB/^ is replaced with kc, boundary conditions
which is then estimated from the Chilton-Colburn analogy,
Eq. (3-165), we obtain kc proportional to ~ : 8 / ~ , which is in CA = CA~ at t = 0 for 0 5 z 5 oo;
better agreement with experimental data. In effect, 6 de- CA = CA~ at z = 0 for t > 0; and
pends on DAB (or NSc) Regardless of whether the criticism
CA = CA~ at z = oo for t > 0
of the film theory is valid, the theory has been and continues
to be widely used in the design of mass-transfer separation These are the same boundary conditions as in unsteady-state
equipment. diffusion in a semi-infinite medium. Thus, the solution can
be written by a rearrangement of (3-75):
EXAMPLE 3.17

Sulfur dioxide is absorbed from air into water in a packed absorp-
tion tower. At a certain location in the tower, the mass-transfer flux where tc = "contact time" of the static eddy at the interface
is 0.0270 kmol S021m2-h and the liquid-phase mole fractions are during one cycle. The corresponding average mass-transfer
0.0025 and 0.0003, respectively, at the two-phase interface and in flux of A in the absence of bulk flow is given by the