Page 341 - Academic Press Encyclopedia of Physical Science and Technology 3rd Chemical Engineering

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178 Mass Transfer and Diffusion

fluid-fluid interfaces. They describe mass transfer from a TABLE III Common Forms of Mass Transfer Coefficients
liquid to a gas or from one liquid to another liquid. Heat
Basic Typical units
transfer coefficients normally describe transport from a a b
equation of k Remarks
solid to a fluid. This makes the analogy between heat and
mass transfer less useful than it might at first seem. N 1 = k c 1 cm/sec Common in the older literature;
The correlations in Table II are most often written in used here because of its
simple physical significance
dimensionless numbers. The mass transfer coefficient k, 2
which most frequently has dimensions of velocity, is in- N 1 = k p p 1 mol/cm -sec Common for a gas absorption;
− atm equivalent forms occur
corporated into a Sherwood number Sh in biological problems
2
kd N 1 = k x x 1 mol/cm -sec Preferred for practical calculations,
Sh = (35) especially in gases
D
Ni = k c 1 cm/sec Used in an effort to include
where d is some characteristic length, like a pipe diameter + c 1 v 0 diffusion-induced convection
or a film thickness, and D is the same diffusion coeffi-
2
a
cient which we talked about earlier. The mass transfer Notes: In this table, N 1 is defined as moles/L t, and c 1 as
3 2
coefficient is most frequently correlated as a function of moles/L . Parallel definitions where N 1 is in terms of M/L t and c 1
3
is M/L t are easily developed. Definitions mixing moles and mass are
velocity, which often appears in a Reynolds number Re infrequent.
dv b For a gas of constant molar concentration c, k = RT k p = k y /c.For
Re = (36) a dilute liquid solution k = (M 2 /ρ)k x , where M 2 is the molecular weight
ν
of the solvent, and ρ is the solution density.
where v is the fluid velocity and ν is the kinematic viscos-
ity; in a Stanton number St
in a gas phase which was an equilibrium with the blood at
k the experimental conditions.
St = (37)
v Each of these units of concentration may be used to
or as a Peclet number Pe define a different mass transfer coefficient, as exemplified
by the definitions in Table III. It is not a difficult task to
dv
Pe = (38) convert a value from one form of coefficient into another
D
form of coefficient (Cussler, 1997; Treybal, 1980). How-
The variation of mass transfer coefficients with other pa- ever, it is complicated and requires care. It’s like balancing
rameters, including the diffusion coefficient, is often not a check book: it doesn’t always work out the first time you
well studied, so the correlations may have a weaker exper- try it. Still, we normally find that with the definitions like
imental basis than their frequent citations would suggest. those in Table III held firmly in mind, we can readily con-
vert from one form of coefficient to another.
The second reason that mass transfer coefficients are
B. Problems with Mass Transfer Coefficients
considered difficult happens when mass transfer occurs
Mass transfer coefficients are frequently regarded as a from one fluid phase into another. This is a genuine source
difficult subject, not because the subject is inherently of difficulty, where confusion is common. To see why the
difficult, but because of different definitions and be- difficulty occurs, imagine we are extracting bromine from
cause of complexities for mass transfer from one solution water into benzene. When we begin, the bromine is at
into a second solution. These differences merit further a higher concentration in the water than in the benzene
discussion. (Cussler, 1997). Later on, the concentrations in water and
The complexities of definitions occur primarily because benzene become equal. Still later, the concentration in the
concentration can be expressed in so many different vari- water will have dropped well below that in the benzene.
ables. In the above, we have assumed that it is expressed in Even then, bromine can still be diffusing from its low con-
mass per volume or moles per volume. The concentration centration in the water into its much higher concentration
can equally be well expressed as a mole fraction, which in the benzene.
in the liquid phase is commonly indicated by the symbol The reason that this occurs is that bromine is much more
x 1 and in a gas phase is written as y 1 . In gases, one can soluble in benzene than it is in water. It partitions from
also express concentrations as partial pressures. In some water into benzene. At equilibrium, the concentration in
cases, especially in medicine, the concentration can be ex- benzene divided by that in water will be a constant much
pressed in other more arcane units. For example, “oxygen greater than one, and almost independent of the initial
tension” measures the amount of oxygen present in blood, concentration of the bromine in the water. Phrased in other
but it is expressed as the partial pressure that would exist terms, in the eventual equilibrium, the concentrations are