Journal of Bioscience and Bioengineering
VOL. 107 No. 4, 412 – 418, 2009
www.elsevier.com/locate/jbiosc
Culture scale-up studies as seen from the viewpoint of oxygen supply and dissolved
carbon dioxide stripping
Naoki Matsunaga,1,⁎ Kenjiro Kano,1 Yasuyuki Maki,2 and Toshiaki Dobashi2
Bio Process Research and Development Laboratories, Kyowa Hakko Kirin Company, Limited, Hagiwara-cho, Takasaki, Gunma 370-0013, Japan 1 and
Department of Chemistry and Chemical Biology, Faculty of Engineering, Gunma University, Tenjin-cho, Kiryu, Gunma 376-8515, Japan 2
Received 17 June 2008; accepted 24 December 2008
Oxygen supply and dissolved carbon dioxide (dCO2) stripping are two of the most important control parameters in cell
culture. In this study, we investigated the effect of scale-up on the volumetric gas transfer coefficient with bioreactors of
different sizes (working volume: 80 L, 500 L, 2000 L, and 10,000 L; aspect ratio: 1.0–1.6). Sparging air into water increased
the volumetric oxygen transfer coefficient (kLa), an index of oxygen supply efficiency, by scale-up roughly in proportion to
the depth of the water. A corresponding increase in kLa was found in a real cell culture of Chinese hamster ovary cells.
dCO2 stripping efficiency was evaluated in water tests using changes in kLaco2, an index defined in relation to kLa. kLaco2
increased following surface aeration, but the rate of increase was reduced by scale-up, which was attributed to a decrease
in the liquid surface-to-volume ratio. A similar decrease in efficiency was observed in a 2000 L bioreactor by increasing
the liquid volume at constant liquid surface area. The observed scale-up effects are discussed based on a simple theoretical
consideration.
© 2009, The Society for Biotechnology, Japan. All rights reserved.
[Key words: Dissolved carbon dioxide; Scale-up; Bioreactor; Volumetric oxygen transfer coefficient; Volumetric carbon dioxide transfer coefficient;
Gas transfer]
Research, development, and commercialization of antibody
medicines have recently been conducted within the pharmaceutical
industry. Various kinds of antibody medicines are generated from
cultures of animal cells such as Chinese hamster ovary (CHO) cells.
Of all the animal cells, CHO cells are preferred for the mass
production of recombinant proteins, since they are ideal hosts with
respect to rapid growth rate, high stability and efficient foreign gene
expression (1). Over the last decade, many studies promoting cell
growth or an increase in the productivity of proteins have been
conducted (2–5).
In general, antibody medicines have the advantage of low toxicity;
however, they require large dosages to be effective. Therefore, it is
necessary to establish a high-density culture system in large-scale
bioreactors. The scale-up of culture starts from a few liters in the
laboratory and goes through many hundreds of liters, up to several
thousand according to the stage of development. The culture volume
may finally reach the order of a 10,000 L scale at the commercial
manufacturing stage.
One of the most important factors in operating cell culture
bioreactors is mass transfer, which includes oxygen supply and
dissolved carbon dioxide (dCO2) stripping. In the scale-up of
industrial cell culture, dCO2 buildup is one of the most serious
⁎ Corresponding author. Tel.: +81 027 353 7152; fax: +81 027 353 2094.
E-mail address: [email protected] (N. Matsunaga).
problems. The accumulation of dCO2 is implicated in reduced
growth and productivity, in addition to adverse effects on
glycosylation (6–8). Therefore, it is very important to remove
dCO2 in large-scale cultures. Another important parameter is oxygen
supply. Oxygen supply and dCO2 stripping, the key operations for
maintaining the metabolism of cultured cells, must be properly
controlled. Conventionally, oxygen supply and dCO2 stripping are
performed by gas sparging and agitation in large-scale cultures,
because of their large mass transfer rate and operational simplicity.
However, both gas sparging and agitation rates are restricted to low
levels, because animal cells are very sensitive to shear stress and are
easily damaged (9–12). These restrictions result in dCO2 accumulation in large bioreactors. In addition, the liquid surface-to-volume
ratio decreases gradually as the bioreactor enlarges. This also
induces dCO2 accumulation in large-scale cultures. Thus, it is
difficult to control gas transfer in bioreactors, because the effects
of scale-up must be matched.
Many studies on oxygen supply, such as the scale-up strategy to
maintain constant kLa have been actively conducted to date. However,
oxygen transfer and dCO2 stripping over a wide range of volumes
from pilot to plant scale have not been rigorously examined using
some real bioreactors, except by process modeling and simulations
(13–15).
In this study, we experimentally examined the effect of scale-up on
oxygen transfer and dCO2 stripping with different working-volume
bioreactors (80 L, 500 L, 2000 L, and 10,000 L). The results are
1389-1723/$ - see front matter © 2009, The Society for Biotechnology, Japan. All rights reserved.
doi:10.1016/j.jbiosc.2008.12.016
VOL. 107, 2009
CULTURE SCALE-UP STUDIES FOCUSING ON GAS TRANSFER
FIG. 1. Configuration of the 80 L, 500 L, 2000 L, and 10,000 L bioreactors and the 2000 L
agitation test tank. The 80 L and 500 L bioreactors have only one wide impeller inclined
near the bottom; the 2000 L and 10,000 L bioreactors, and the 2000 L agitation test tank
have an impeller not inclined near the bottom, and an impeller inclined at the middle.
All bioreactors and the 2000 L agitation tank have four baffles. The aspect ratios are
from 1.0 to 1.6. The liquid depths are 520 mm (80 L), 905 mm (500 L), 1530 mm (2000 L
and 2000 L agitation test tank), and 2285 mm (10,000 L).
discussed in relation to the theoretical considerations. We obtained
some important findings about scale-up strategies.
FIG. 3. Time course of [dCO2] with volume = 500 L, tip speed= 0.75 (m/s), and surface air
supply = 0 (L min− 1) at various sparging gas volume per liquid volume per min (vvm).
sparged into the water from the bottom until [DO] reached about 0. Then, air sparging
was started with agitation under the desired operational conditions, and [DO] was
measured continuously. The liquid surface was not aerated during the measurement
period. The experimental results of [DO] obtained under typical conditions are shown in
Fig. 2.
According to the recognized theory (16), kLa is expressed as follows:
MATERIALS AND METHODS
kL a =
Bioreactors
All data in this paper were obtained using 80 L, 500 L, 2000 L,
10,000 L bioreactors and a 2000 L agitation test tank, which have impellers, baffles, and
spargers, as shown in Fig. 1. Here, the volume of the bioreactors is expressed as the basic
working volume. The 80 L and 500 L bioreactors did not have a middle impeller, and the
2000 L agitation test tank did not have a cover. The aspect ratios (L/D) were in the range
1.0–1.6. The liquid surface-to-volume ratios of the bioreactors were 0.020 cm− 1,
0.010 cm− 1, 0.007 cm− 1, and 0.005 cm− 1 for the 80 L, 500 L, 2000 L, and 10,000 L
bioreactors, respectively. Each bioreactor was equipped with sensors (temperature, pH,
DO, and dCO2).
Measurement of kLa using water tests
kLa was measured as follows. The
bioreactors were filled with water to the basic working volume, and nitrogen was
413
ln ½DOs ½DO0 ln ½DOs ½DOt
t
ð1Þ
where [DO]s, [DO]0 and [DO]t are DO (dissolved oxygen concentration) (mol L− 1)
saturated in water, at the beginning of air sparging (t = 0), and at sparging time t. kLa
was obtained from the slope of the linear plot of ln([DO]s − [DO]t) against t.
kLa was measured continuously at 24–28 °C. The DO sensor used for this study was
“InPro6800” (Mettler Toledo), which can compensate automatically with the built-in
Resistance Temperature Detector. The DO sensor was calibrated after water was poured
into the tank to consider the influence of water pressure.
CHO cell culture
Animal cell cultures were performed with the 80 L, 500 L, and
2000 L reactors. The cell line derived from a dihydrofolate reductase deficient (DHFR−)
CHO DUKX B11 host (17) was used in the experiment. EX-CELL 325 (SAFC Biosciences)
is a protein-free, animal component-free dry powder medium, which has been
developed for the growth of CHO cells. The main culture conditions (e.g., cell clone;
medium temperature, 37 °C; initial viable cell density, 2.0 × 105 cells/mL; gas supply to
liquid surface, 5% CO2/air; tip speed, 0.75 m/s; and culture time, 3 days) were
controlled to equal between the bioreactors. During cell culture [DO] / [DO]s was kept at
100 ± 10% by oxygen sparging, and the oxygen supply was managed by continuous
proportional–integral–derivative control.
Evaluation of dCO2 stripping using water tests
The dCO2 sensor used for this
study was “YSI8500” (YSI) (18). The sensor was calibrated with water saturated by 5%
CO2 gas. The bioreactors or the agitation test tank were filled up with water to the basic
working volume and CO2 gas was sparged into the water until dCO2 reached about 20%.
Then, air sparging was started with agitation under the desired operational conditions,
e.g., agitation, air sparging, pore size of the sparger, air supply to the liquid surface etc.,
and dCO2 was measured continuously at 24–28 °C. The experimental results of dCO2 (%)
obtained under typical conditions are clearly expressed by the linear plot of ln
([dCO2]t − [dCO2]s) against t, as shown in Fig. 3. Therefore, corresponding to Eq. 1, we
defined
kL aCO2 =
ln ½dCO2 0 ½dCO2 s ln ½dCO2 t ½dCO2 s
t
ð2Þ
as an index for evaluating the efficiency of stripping of dCO2. Here, [dCO2]s, [dCO2]0,
and [dCO2]t are dCO2 at a steady state after a long sparging period (∼ 0), at the
beginning of air sparging (t = 0), and at sparging time t, respectively.
RESULTS AND DISCUSSION
FIG. 2. Time course of 100[DO]/[DO]s with volume = 500 L and tip speed = 0.75 (m/s),
and surface air supply = 0 (L min− 1) at various sparging gas volume per liquid volume
per min (vvm).
In general, cell culture systems are very complex. The flow profiles
of gas and liquid are complicated, and affected by culture tank
geometry, agitation, sparging etc. Therefore, it is laborious to analyze
414
MATSUNAGA ET AL.
J. BIOSCI. BIOENG.,
tip speed of 0.5 m/s, but less effectively at high vvm at middle and
high tip speeds of 0.75 and 1.0 m/s.
As shown in Figs. 4 and 5, kLa and kLaco2 increased roughly linearly
in proportion to vvm, although saturation points were found in large
bioreactors (2000 L and 10,000 L). To determine the characteristic
points, we approximated the plots in Fig. 4a–c and Fig. 5a–c to straight
lines, and defined the slopes as ϕO2 and ϕCO2, respectively. As shown
in Fig. 6, ϕO2 increased linearly in proportion to liquid depth in 80 L–
10,000 L bioreactors, except for the largest bioreactor (10,000 L) at
low tip speed. On the other hand, ϕCO2 increased linearly in proportion
to liquid depth only in a narrow range for depths less than 1000 mm
(80 L–500 L bioreactors) and saturated at a roughly constant level,
even at high tip speed. The difference in the behavior of ϕO2 and ϕCO2
FIG. 4. kLa as a function of vvm in bioreactors with various sizes and at impeller tip
speeds of 0.5 m/s (a), 0.75 m/s (b), and 1.0 m/s (c).
all effects of the control parameters to scale-up the system properly. In
chemical engineering, it is important to find a methodology for
determining the working equation for each system. It is also important
to clarify the effects of scale-up on key factors in the culture system. In
this study, we focused on the effect of scale-up on kLa and kLaco2 and
examined these experimentally using bioreactors with a wide range of
working volumes.
Evaluation of kLa and kLaco2 by sparging using water tests Fig.
4a–c shows kLa as a function of sparging gas volume, per liquid
volume, per min (vvm) at various tip speeds and bioreactor volumes.
kLa clearly increased in proportion to vvm, except that it tended to
saturate at low and middle tip speed conditions (0.5 m/s and 0.75 m/
s, respectively) in large bioreactors. kLa also increased with scale-up
and tip speed.
Fig. 5a–c shows the effect of tip speed, vvm of sparged aeration,
and bioreactor volume on kLaco2 with no aeration of the liquid surface.
kLaco2 increased with increasing vvm and increasing bioreactor volume
at all tip speeds; kLaco2 increased roughly in proportion to vvm at a low
FIG. 5. kLaco2 as a function of vvm in bioreactors with various sizes and at impeller tip
speeds of 0.5 m/s (a), 0.75 m/s (b), and 1.0 m/s (c)).
VOL. 107, 2009
CULTURE SCALE-UP STUDIES FOCUSING ON GAS TRANSFER
415
FIG. 7. Change in sparging velocity of bubbles as a function of vvm in various bioreactor
scales.
FIG. 6. Average slope of kLa and kLaco2 against vvm in Figs. 4 and 5 as a function of liquid
depth.
indicates that it is not always valid to adjust the culture conditions for
dCO2 stripping based only on the constant kLa, which is conventional
in real cultures.
As shown in the Appendix, in the case that the rate of gas transfer
between a bubble and the liquid is slow, the relationship between the
volumetric mass transfer coefficient K and sparging volume per liquid
volume per minute vvm = Q/V is derived as
aQ s
aaL
=
X
ðA12Þ
V
vf
where V is the liquid volume; Q is the bubble volume sparged per unit
time to the bioreactor; τ is the lifetime of a bubble; L is the liquid depth;
vf is the terminal velocity of bubbles; α is a coefficient due to agitation
with a = 3k
r ; k is the rate of mass transfer between a bubble and the
medium liquid, defined as the ratio of the diffusion constant of the gas
and the length of the boundary layer surrounding the bubble; and r is
the radius of a bubble. Ω is defined as Q/V (=vvm). ϕO2 and ϕCO2 are
estimated from K/Ω. If k, r, and vf can be approximated as constants, the
proportional increase of ϕ versus depth is derived as shown in Eq. A12.
Therefore, the behavior of ϕO2 observed in Fig. 6a is roughly explained by
the above theoretical considerations, though a deviation was found at
the upper end of the bioreactor size range. On the other hand, the
increase in ϕCO2 was clearly suppressed by scale-up, and ϕCO2 saturation
occurs at a much lower depth of ∼1000 mm, as shown in Fig. 6b.
Although it is very difficult to identify the reason for this difference in gas
exchange behavior, the time required for the gas transfer for carbon
dioxide may be smaller than that for oxygen because the solubility of
carbon dioxide in water is larger than that of oxygen. Therefore, we could
attribute the saturation of ϕCO2 to this effect as shown by Eq. A14 in the
case of fast gas transfer. However, we cannot specify a reason for
saturation at this stage, since the parameters k, r, and vf, which we
estimated as constant, could be changed by operations such as agitation
K=
through the parameters of the bioreactor (e.g., tip speed, or vvm). In
addition, the sparging velocity of bubbles could be increased by scale-up
even if vvm remains the same, as shown in Fig. 7. This tendency could
also affect the gas transfer efficiency through the parameters.
CHO cell culture
Table 1 shows the total oxygen supply volume
per culture volume in each tank (80, 500, and 2000 L) during CHO cell
culturing. These cultures started with the same initial viable cell
density, 2.0 × 105cells/mL, and DO was controlled at 100 ± 10% during
the cultures. The cell growth rate was almost the same, irrespective of
the tank volume. As shown in Table 1, the total oxygen supply volume
per culture volume during culture decreased according to the scale-up
of the bioreactor. This behavior is consistent with the behavior of kLa
during water tests.
Evaluation of dCO2 stripping from the liquid surface using
water tests Fig. 8 shows the relationship between aeration to liquid
surface and dCO2 accumulation by scale-up at a fixed tip speed of
0.75 m/s. Where there was no surface aeration, the sparging aeration
volume was controlled, so that kLaco2 was within 0.15–0.20. The
sparging aeration volumes were 0.5 L min− 1 (80 L), 1.5 L min− 1
(500 L), 3.0 L min− 1 (2000 L) and 15 L min− 1 (10,000 L), which
correspond to the value of vvm at 0.00625, 0.003, 0.0015 and 0.0015 L
min− 1 L− 1, respectively. Ten and twenty times surface aeration of the
sparging aeration in volume was then added while sparging aeration
was maintained. As shown in Fig. 8, the efficiency of dCO2 stripping by
surface aeration in 80 and 500 L bioreactors was much greater than
that in 2000 L and 10,000 L bioreactors. kLaco2 increased with surface
aeration roughly proportionally in 80 L and 500 L bioreactors in the
experimental range of surface aeration (20 times surface aeration of
the sparging aeration corresponds to the value of vvm at 0.125 and
0.06 L min− 1 L− 1, respectively, for 80 L and 500 L bioreactors). On the
other hand, the rate of the increase of kLaco2 with surface aeration
decreased and was saturated at the higher surface aeration in 2000 L
and 10,000 L bioreactors (20 times surface aeration of the sparging
aeration in volume corresponds to the value of vvm at 0.03 L
min− 1 L− 1 for both 2000 L and 10,000 L bioreactors). This is
attributed to the decrease in liquid surface-to-volume ratios and
increasing bioreactor volumes.
TABLE 1. Total oxygen supply volume per culture volume during CHO cell culturing in
80 L, 500 L, and 2000 L bioreactors
Tank scale
80 L
500 L
2000 L
Culture volume (L)
Total oxygen supply/culture volume (L/L)
71
403
1896
647
468
297
416
MATSUNAGA ET AL.
J. BIOSCI. BIOENG.,
FIG. 8. Effect of surface aeration on kLaco2 in bioreactors of different sizes with a sparger pore size of 100 μm.
Fig. 9 shows the effect of liquid depth on dCO2 stripping, observed
using the same 2000 L agitation test tank. kLaco2 decreased with the
increasing liquid surface-to-volume ratio. In typical cultures of animal
cells in 2000 L fed-batch reactions, the liquid volume starts at 1500 L
and reaches 2000 L at the end of culture. Fig. 9 indicates that this
increase in volume in the real culture also reduced the liquid surfaceto-volume ratio, resulting in a decrease in kLaco2. On the other hand,
the sparger pore size had no significant effect on kLaco2.
FIG. 9. kLaco2 observed using the same 2000 L agitation test tank at the indicated sparger pore size, liquid depth, and liquid volume at sparged aeration volumes of 5.7 L min− 1 or
0 L min− 1 (no sparge) (a) and kLaco2 as a function of liquid surface area/volume ratio at different sparging pore sizes (b).
VOL. 107, 2009
CULTURE SCALE-UP STUDIES FOCUSING ON GAS TRANSFER
CONCLUSION
In this study, the effect of scale-up on the volumetric gas transfer
coefficient in bioreactors was measured for a wide range of volumes from
80 L to 10,000 L. Both kLa and kLaco2 increased roughly in proportion to
vvm, and the proportional constant of these values increased with the
increasing depth of liquid at lower depths, which is consistent with a
simple theoretical consideration, as shown in the Appendix. However,
this linear relationship was not satisfied at greater depths, and saturation
behavior was observed, especially for kLaco2. This result suggests that it is
not always valid to adjust the culture conditions based on only the
constant kLa, which is the conventional approach to real cultures. The
evidence observed in CHO cultures shows that the average vvm of
sparged oxygen at constant DO during the culture, which decreased
during the scale-up of bioreactors, is consistent with the roughly linear
increase in kLa vs. vvm in the scale-up using water tests. This result
suggests that water tests are useful for evaluating oxygen supply in real
cultures. It was also demonstrated that a decrease in the liquid surfaceto-volume ratio by scale-up (or increasing liquid volume in bioreactors)
affected kLaco2 and resulted in dCO2 accumulation.
APPENDIX A. RELATIONSHIP BETWEEN VOLUMETRIC MASS
TRANSFER COEFFICIENT AND SPARGING VOLUME PER LIQUID
VOLUME PER UNIT TIME
The rate of change of the concentration in the medium by sparging
with bubbles is expressed as
dC
= K Cf C
dt
ðA1Þ
Then, the change of gas concentration in the medium by sparging
with bubbles is given as
Cf C ðt Þ
= eKt
Cf C0
ðA2Þ
where K is the volumetric mass transfer coefficient (e.g., kLa, kLaco2);
Cf, C0, and C(t) are the gas concentrations in saturated medium liquid,
at the beginning of air sparging (t = 0), and at sparging time t.
We will define the average mass (oxygen or carbon dioxide in the
present study) transfer from a bubble into the medium liquid during
unit time at sparging time t as q(t). Where the rate of mass transfer is
slow, this process is not completed during the lifetime τ of the bubble
in the tank. In this case, q(t) is expressed as
qðt Þ =
1
s
Z
0
s
qðt; t VÞdt V
ðA3Þ
where q(t, t′) is the rate of mass transfer between a bubble and the
medium liquid at the elapsed time t′ since the beginning of sparging
(0 b t′ b τ), and is given by
qðt; t VÞ = 4pr 2 kðCi ðt; t VÞ C ðt ÞÞ
ðA4Þ
where k is the rate of mass transfer from a bubble to the medium
liquid, defined as the ratio of the diffusion constant of the gas and the
length of the boundary layer surrounding the bubble. Ci is the mass
concentration at the gas–liquid interface, which depends on the mass
concentration in the bubble and the solubility in the liquid, and is
given by
4 3 dCi ðt; t VÞ
pr
= qðt; t VÞ
3
dt V
ðA5Þ
When C(t) varies more slowly than Ci(t, t′), Eqs. A4 and A5 give
Ci ðt; t VÞ = C ðt Þ + Cf C ðt Þ eat V
ðA6Þ
417
Here, we put a = 3k
r . We note that Ci(t,0) is independent of time t
and should be equal to Cf .
By substituting Eqs. A4 and A6 in Eq. A3, we have
qðt Þ =
4pr3 Cf C ðt Þ ð1 eas Þ
3s
ðA7Þ
If we assume that τ is much smaller than the time for change of C
(t), Eq. A7 is reduced to
qðt Þfavb Cf C ðt Þ
ðA8Þ
3
where vb is the volume of a bubble 4πr /3. Then, the rate of mass
transfer is given by
dC
qðt ÞN
qðt ÞQ s
=
=
dt
V
Vvb
ðA9Þ
where V is the liquid volume; Q is the bubble volume sparged during
unit time to the bioreactor; and N is the number of bubbles in the
bioreactor, i.e., N = Qs
vb . Comparing Eqs. A8 and A9, we have
dC aQ s f
Cf C ðt Þ
dt
V
ðA10Þ
The lifetime of a bubble, τ, is estimated as
s=a
L
vf
ðA11Þ
where L is the liquid depth, vf is the terminal velocity of bubbles, and
α is a coefficient due to agitation. Thus, combining Eqs. A1, A10 and
A11, Ci (0) = Cf we have
K=
aQ s
aaL
=
X
V
vf
ðA12Þ
where Ω is defined by Q/V (=vvm). ϕO2 and ϕCO2 are defined in the
text and are expressed as K/Ω.
In the case of time te for the completion of mass transfer is fast, i.e.,
te b b τ, q(t) is expressed as
qðt Þ =
1
s
Z
0
te
qðt; t VÞdt V =
uvb Cf C ðt Þ
s
ðA13Þ
where u is a constant. Then, we have
K fuX
ðA14Þ
in place of Eq. A12.
Nomenclature
C0, gas concentration in liquid at the beginning of air sparging
[mol L− 1]
Cf, gas concentration in saturated liquid [mol L− 1]
Ci, concentration at the gas–liquid interface [mol L− 1]
C(t), gas concentration in liquid at sparging time t [mol L− 1]
[dCO2]0, ratio of mole of dissolved carbon dioxide at the
beginning of air sparging to mole of total gas dissolved
in medium [%]
[dCO2]s, ratio of mole of dissolved carbon dioxide at a steady state
after a long sparging period to mole of total gas dissolved in
medium [%]
[dCO2]t, ratio of mole of dissolved carbon dioxide at sparging time t to
mole of total gas dissolved in medium [%]
[DO]0, dissolved oxygen concentration at the beginning of air
sparging [mol L− 1]
[DO]s, dissolved oxygen concentration saturated in water [mol L− 1]
[DO]t, dissolved oxygen concentration at sparging time t [mol L− 1]
418
MATSUNAGA ET AL.
K, volumetric mass transfer coefficient [h− 1]
kLa, volumetric oxygen transfer coefficient [h− 1]
kLaco2, volumetric dissolved carbon dioxide transfer coefficient [h− 1]
k, rate of mass transfer between bubble and liquid [m/s]
L, liquid depth [mm]
N, number of bubbles in bioreactor [-]
Q, bubble volume sparged per unit time to bioreactor [L min− 1]
q(t, t′), rate of mass transfer between bubble and liquid at time t′
[mol/s]
r, radius of bubble [mm]
V, liquid volume [L]
vb, volume of a bubble [mm3]
vf, terminal velocity of bubble [mm/s]
vvm, gas volume per liquid volume per min [L min− 1 L− 1]
α, coefficient due to agitation [-]
Ω Q/V, [L min− 1 L− 1]
τ, lifetime of bubble [s]
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