Mass Transfer
445
Reported correlations for the Sherwood number are often of the kind
Sh = 2 + const
ScaR ep
(10.28)
If the exponents a and p have the same value, the Schmidt and Reynolds number can be multiplied
together forming the
Peclet number, Pe.
In cases where free convection dominates, the Grashof
number appears instead of the Reynolds number (see Table 10.7). It can be seen from Eq 10.28 that
as the Reynolds number approaches 0, i.e. for a bubble which is stagnant relative to the surrounding
liquid, the Sherwood number will approach 2. This is in accordance with the analytical steady-state
solution of the diffusion equation for a sphere in stagnant medium (see Note 10.2). For high values
of
Re
the first term may be neglected compared to the second term. A key question is furthermore
whether the bubble behaves as a rigid surface or not (see Table 10.7). For a rigid bubble, with an
immobile interface, the boundary conditions at the surface will state a zero relative velocity
between the liquid and the bubble surface. For a mobile interface, however, this boundary condition
does not apply, and the mass transfer characteristics will be different (for a further discussion see
e.g. Blanch and Clark, 1997).
Note 10.2. Derivation of the Sherwood number for a sphere in stagnant medium
For the case of steady-state mass transfer from a spherical particle with radius,
Rp = dJ2,
in a stagnant
medium, it is possible to derive an analytical value for the Sherwood number. The diffusion equation for
species A at steady state is given by
V2c,= 0
(1)
Due to the symmetry of the problem, it is convenient to use spherical coordinates, and with the radial
distance denoted
Eq. 1 can be expressed
with the boundary conditions
H
= 0
C A ~ CA,s
for
<t-~Rp
and
cA —>
0 for
%
—> 03
Integrating Eq. 2 twice gives
=
c
A,s
£
(
2
)
(3)
(4)
(5)