Mass Transfer
437
a = 71.97 1 O
'3 N m"1
n= 1.00 103 kg m"1
s'
First we want to calculate the initial bubble diameter using Eq. (10.20);
d
b,i
6- 71.97 - IQ"3
-103
9.82 -(997 -1.285)
= 3.53 • 10”3 m
(
1
)
From visual inspection of the system, however, we observe that a jet stream is formed at the orifices. We
therefore search in the literature for a correlation for the initial bubble diameter, which may be more
suitable for the high gas flow rate applied in the system. Bhavaraju
et al.
(1978) states that the correlation in
Eq. (2) holds for gas flow rates up to 2 10"4 m3
s'1
(which is close to the 4.2 10'4 m3 s'1
used in the present
system). Re0 is the Reynolds number for the gas stream at the orifice [given by Eq. (3)] and
Fra
is the
Froude number
for the gas stream at the orifice [given by Eq. (4)]:
dbi
=
3.23d Re^
0,1
0
Q
Fr:
(
2
)
4 p p s
Tjd
'
Fr0
(4)
With the operational values specified in Table 10.3, we find
Re0 =
5.3T04 and
Fr0
= 1.8T05 (the total gas
flow is equally distributed to the ten orifices in the sparger), and therefore
dbi
=(3.23-10_3)-(5.3-104)-°‘1(l.8-105)0'21 =13.8-10“3 m
(5)
This is a larger initial bubble diameter than found by using Eq. (10.20), and it corresponds better with the
bubble size observed in the bioreactor when there is no agitation. Note that the correlation in Eq. (2) is
insensitive to even large variations in the orifice diameter, and a change of the hole size of the sparger
therefore has little effect on the initial bubble diameter. The two completely different values obtained tell us
that correlations (both empirical and theoretically derived) should always be used with some caution, i.e.,
one should always check the range of validity for the correlation.
With the specified power input, we calculate the maximum stable bubble diameter, using Eq. (10.23). First
we take
k
to be 1.93 m.
^,=1.93-
(71.97-IQ"3)0
(75/25 •10”3)°'4(997)c
= 4.07-lO-
te)
Next we assume that the gas holdup is 0.1 (as is reasonable for the examined system) and use Eq. (10.24) to
find
k =
1.34. Thus from Eq. (10.23) we now find
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