Mass Transfer
443
Table 10.6.
Some important dimensionless groups for mass transfer correlations8
Definition
Name
Significance
rk td~
lA ,J
Sherwood number
mass transfer velocity relative to diffusion velocity
Sc =
* %
Q i
.
Schmidt number
momentum diffusivity relative to mass diffusivity
Re =
•§* ^
3
1.....
....
1
Reynolds number
inertial forces relative to viscous forces
Gr
=
d 3gPi(Pi - P g)
Grashof number
bouyancy forces relative to viscous forces
Pe =
du
V
\
Peclet number
flow velocity relative to diffusion velocity
V is a length scale characteristic of the system which is studied, i.e. bubble, cell or cell aggregate.
DA
is the
diffusion coefficient for the considered species in the continuous phase;
u
is the linear velocity of bubble,
cell, etc. relative to the continuous phase.
Also in the absence of a known governing equation it is, however, possible to derive dimensionless
variables upon which to base empirical correlations. This can be done by
dimensional analysis
, i.e.
by checking that correlations are dimensionally correct (see Example 10.5). The basis for this
analysis is the Buckingham ^-theorem, which is explained in detail in many chemical engineering
textbooks (see e.g. Geankoplis, 1993).
Some of the more well-known dimensionless groups used in mass transfer correlations are
summarized in Table 10.6.
Example 10,5. Deriving dimensionless groups by dimensional analysis
The procedure for arriving at correlations based on dimensionless groups can be illustrated by the following
example. Suppose that we need a correlation for the mass transfer coefficient,
kh
for a compound A, present
in a bubble which moves rapidly with respect to its surrounding medium. To make a dimensional analysis,
all relevant variables must be known, and these must, furthermore, be independent of each other. In the
current example the following variables are regarded as important; diffusivity of A,
DA,
bubble diameter,
db,
relative velocity of bubble,
ub,
viscosity of liquid,
rj,
and the density of the liquid,
p.
Our hypothesis is that
we can express the mass transfer coefficient as a function of these variables, i.e.
k,xD‘'u;dinlp‘
(i)
where the oi|, p, y, 5, and e are unknown exponents to be determined.
Eq. 1 must be dimensionally correct.
kt
has dimensions length (L) time (t)'1,
db
has dimension L, u,, has