Page 340 - 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 177

shown in Fig. 5b. The concentration of water in the exiting mass transfer like a first-order chemical reaction, but a re-
air c 1 will be given by (Cussler, 1997) versible reaction with an equilibrium constant of one. The
equilibrium constant equals one because diffusion is the
c 1 −ka(z/v)
= 1 − e (34) same in both directions. Nonetheless, the mass transfer
c 1(sat) coefficient is unlike a chemical reaction because it does
not describe chemical change. It describes changes with
where z is now the distance from the entry of the bed
position or time.
and v is the velocity of air flowing through the bed. This
equation is essentially equivalent to the previous one, but
with the residence time (z/v) replacing the actual physical
A. Mass Transfer Coefficients
time. Again, it suggests a way in which we can organize
data using a mass transfer coefficient k. Experimental values of mass transfer coefficients can be
But what exactly is being done? We are replacing our collected as dimensionless correlations. One collection of
detailed description of diffusion of the water with a much these correlations is in Table II (Cussler, 1997). Because
more approximate analysis. We are assuming that the bulk heat transfer is mathematically so similar to mass trans-
of the air is mixed enough to give it a constant concen- fer, many assert that other correlations can be found by
tration. We are assuming that the only significant con- adapting results from the heat transfer literature. While
centration change occurs close to the water/air interface. this is sometimes true, the analogy is frequently overstated
This type of analysis and the equations it implies treat because mass transfer coefficients normally apply across

TABLE II Useful Correlations of Mass Transfer Coefficients for Fluid–Fluid Interfaces
Physical situation Basic equation b Key variables Remarks
Liquid in a packed 1 1/3 ν 0 0.67 D 0.50 a = Packing area per bed Probably the best available correlation for
tower k = 0.0051 (ad) 0.4 volume liquids; tends to give lower values than
νg aν ν
d = Nominal packing size other correlations
0.5
kd dν 0 0.45 ν
= 25 d = Nominal packing size The classical result, widely quoted;
D ν D
probably less successful than above
k dν 0 −0.3 D 0.5
= α d = Nominal packing size Based on older measurements of height of
ν 0 ν ν
transfer units (HTUs); α is of order one
Gas in a packed k ν 0 0.70 ν 1/3 −2.0 a = Packing area per bed Probably the best available correlation for
tower = 3.6 (ad) volume gases
aD aν D
d = Nominal packing size
1/3
kd dν 0 0.64 ν
= 1.2(1 −
) 0.36 d = Nominal packing size Again, the most widely quoted classical
D ν D
ε = Bed void fraction result
Pure gas bubbles in a kd (P/V ) d 4 1/4 ν 1/3 d = Bubble diameter Note that k does not depend on bubble size
stirred tank D = 0.13 ρν 3 D P/V = Stirrer power per
volume
Pure gas bubbles in an kd d g ρ/ρ 1/3 ν 1/3 d = Bubble diameter For small swarms of bubbles rising in a
3
unstirred liquid = 0.31 2 ρ = Density difference liquid
D ν D
between gas and liquid
Large liquid drops kd d ρg 1/3 ν 0.5 d = Bubble diameter
3
rising in unstirred D = 0.42 ρν 2 D ρ = Density difference Drops 0.3-cm diameter or larger
solution between bubbles and
surrounding fluid
Small liquid drops kd dν 0 0.8 d = Drop diameter These small drops behave like rigid
rising in unstirred = 1.13 v = Drop velocity spheres
0
D D
solution
Falling films kz zν 0 0.5 z = Position along film Frequently embroidered and embellished
= 0.69 0
D D v = Average film velocity
a
Notes : The symbols used include the following: D is the diffusion coefficient; g is the acceleration due to gravity; k is the local mass transfer
0
coefficient; v is the superficial fluid velocity; and ν is the kinematic viscosity.
b 3 2
Dimensionless groups are as follows: dv/ν and v/aν are Reynolds numbers; ν/D is the Schmidt number; d g( ρ/ρ)ν is the Grashoff number.
kd/D is the Sherwood number; and k/(νg) 1/3 is an unusual form of Stanton number.