A simple parameterization for the rising velocity of bubbles in a liquid pool
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A simple parameterization for the rising velocity of bubbles in a liquid pool. This article suggests a simple parameterization for the gas bubble rising velocity as a function of the volume-equivalent bubble diameter; this parameterization does not require prior knowledge of bubble shape. This is more convenient than previously suggested parameterizations because it is given as a single explicit formula.
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A simple parameterization for the rising velocity of bubbles in a liquid pool. This article suggests a simple parameterization for the gas bubble rising velocity as a function of the volume-equivalent bubble diameter; this parameterization does not require prior knowledge of bubble shape. This is more convenient than previously suggested parameterizations because it is given as a single explicit formula..
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between the molten core and the bubbles is highly dependent spherical-cap regimes were visualized in the diagram. More-
on the bubble size, shape, and rising velocity. Determination over, the condition for the direct conversion of spherical
of the bubble shape and rising velocity for a given bubble bubbles to spherical-cap bubbles, without passing through the
volume, therefore, is an important procedure in the analysis spheroidal regime, was given simply as “M > 10.” Never-
of radioactive aerosol emissions in terms not only of the pool theless, the determination of Re using this diagram may be
scrubbing efficiency but also of the fission product release troublesome and liable to error because an interpolation for M
rate. All the postulated severe accident analysis codes is required.
currently used worldwide contain their own schemes to In this article, a simple parameterization that expresses Re
determine the bubble shape and rising velocity. The devel- as an explicit function of Eo and M is suggested. This is more
opment of an efficient scheme to determine the bubble shape convenient than the previously suggested parameterizations
and rising velocity is one of the key issues in enhancing the because it is given as a single explicit formula. In addition, this
performance of the accident analysis code in terms of the parameterization can be used to produce a bubble-shape di-
fission product release. agram similar to Grace's diagram.
Several different theories for determining bubble rising
velocity are available in the literature [6e13]. Most of those
theories deal with a particular bubble shape type, e.g., sphere,
2. Theories of bubble rising in a liquid pool
spheroid, and spherical (or spheroidal) cap; the bubble shape
must be determined in advance to decide which theory to use.
Gas bubbles rising in a liquid pool can be categorized based on
The problem, however, is that the bubble shape cannot be
their shape into one of the following three groups: sphere,
determined without information on the bubble rising velocity.
spheroid, and spherical cap [10,17]. Bubbles are spherical
This implies that iteration is needed to simultaneously
when they are so small that the inertial force is much smaller
determine both the bubble shape and the rising velocity.
than the surface tension or the viscous force. As the bubble
Wallis  suggested 10 different bubble rising velocity
sizedand hence, the rising velocitydincrease, the bubbles
equations that depend on the size, shape, and rigidity of the
change into oblate spheroid shapes because of the resistance
bubbles. The study of Jamialahmadi et al  was apparently
imposed by the liquid medium. When the bubbles are suffi-
the first effort to suggest a universal formula to determine the
ciently large, they tend to have flat and often indented bases,
bubble rising velocity, but they neglected the effect of inertial
breaking the upedown symmetry of the bubble shape. This
force for spherical bubbles. Bozzano and Dente  suggested
shape is called the spherical cap.
a method to determine the bubble shape and rising velocity
The shape and rising velocity of a bubble given a volume-
simultaneously without iteration. They determined the bub-
equivalent diameter have long been an important subject of
ble drag coefficient by assuming that a rising bubble would
fluid mechanics. It is well known that the bubble rising ve-
have such a shape that the total energy (potential
locity depends on the bubble shape, which in turn is deter-
energy þ surface energy þ kinetic energy) is minimized. The
mined according to three dimensionless numbers [9,17]: the
results of numerical minimization were approximated into a gr d2
Eotvos number Eo ¼ sLL e , the Reynolds number Re ¼ rL vmbL de , and
parameterization, which was a function of two dimensionless gm4
the Morton number M ¼ r sL3 , where g is the gravitational ac-
parameters: the Eotvos number Eo and the Morton number M. L L
celeration, rL is the density of the liquid medium, de is the
Using the drag coefficient, the bubble rising velocity was given
volume-equivalent diameter of the bubble, sL is the surface
as a solution of a second-order equation.
tension of the liquid medium, vb is the terminal rising velocity
By analyzing experimental data obtained from 21 different
of the bubble, and mL is the viscosity of the liquid medium. Eo is
liquids with a very wide range of physical properties, Grace 
the ratio between body forces and surface tension forces and
showed that the size, shape, and rising velocity of a single
Re is the ratio between inertial forces and viscous forces. M,
bubble in infinite liquid can be deduced from a diagram in
roughly speaking, increases with increasing viscous forces
which the relation among three dimensionless numbers, Eo ,
and decreasing surface tension forces.
M, and the Reynolds number Re , is given graphically. In this
When the inertial force is negligible compared to the
diagram (hereafter referred to as Grace's diagram), Re (repre-
viscous force ðRe < 1Þ, the terminal rising velocity of a
senting bubble velocity) was plotted as a function of Eo (rep-
spherical bubble is given by [6,18]:
resenting bubble size) for different values of M (representing
liquid properties). Dividing the particle shape into three re- gðrL rG Þd2e 1 þ k
vb;vis ¼ ; (1)
gimes (sphere, spheroid, and spherical cap), Grace  con- 6mL 2 þ 3k
verted the relationships between particle size and rising
where rG is the density of the gas, mG is the viscosity of the gas,
velocity in the spherical regime (i.e., for small particles) and in
and k ¼ mG =mL . Because for most liquids and gases mG ≪mL (kz0)
the spherical-cap regime (large particles) into relationships
and rG ≪rL , Eq. (1) becomes:
between Eo and Re . Then, for the spheroidal regime (i.e., be-
tween those 2 size limits), cross-plotting was used to fill the grL d2e
vb;vis ¼ : (2)
gap. Grace et al  extended the work of Grace  to single 12mL
liquid drops moving in another liquid medium.
Eq. (2) is not valid when Re is significantly larger than 1
Grace's diagram was very useful and provided great insight
because inertial force is not negligible. Wallis  suggested
into the bubble behavior. The boundaries between the
the following formula for spherical bubbles with non-
spherical and spheroidal regimes, between the spheroidal and
negligible inertial force in an Re range of 1 < Re < 100:
spherical-cap regimes, and between the spherical and
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vb;in ¼ 0:14425g5=6 e :
For spheroidal bubbles, vb is determined by [7,17]:
vb;spheroid ¼ þ 0:505gde : (4)
For spherical-cap bubbles, the following formula was sug-
gested for vb [8,17]:
vb;cap ¼ 0:721 gde : (5)
Fig. 1 compares Eqs. (2e5) as a function of de . Pure water
and air were chosen as the liquid and gas for preparing this
figure and the following similar figures in which vb is plotted
against de . The rising velocity of a spherical bubble increases
with increasing bubble size because increased body force
(buoyancy) dominates over increased friction in this shape
regime. As the bubble shape changes to spheroid, however,
the rising velocity begins to decrease with increasing bubble
size because increased friction becomes greater than
increased buoyancy owing to the effect of flattening. After the
aspect ratio, the ratio of the longer axis length to the shorter
axis length, becomes sufficiently large, no further flattening Fig. 2 e General formula for the rising velocity of bubbles
occurs and the rising velocity begins to increase again with with internal circulation compared to experimentally
increasing bubble size. When the bubbles become too large, measured data.
the bubbles finally change into the spherical cap shape.
vb ¼ 0:711 gde , which is almost the same as Eq. (5). Actually,
3. Parameterization for bubble rising velocity vb ¼ 0:711 gde was suggested for spherical-cap bubbles by
for entire bubble shape range Clift et al  when Eo 40 and Re 150. Therefore, it is sug-
gested to use Eq. (4) not for spheroidal bubbles only but for all
In this section, a new parameterization that involves all the nonspherical bubbles.
formulas for the three bubble shape regimes (Eqs. 2e5) is Second, to let Eq. (2), (3), or (4) be selected automatically for
suggested. appropriate bubble size and shape, the following equation is
The first step is to unify the two regimes for nonspherical suggested for bubbles with arbitrary size and shape.
bubbles. Fig. 1 shows that Eqs. (4) and (5) exhibit very similar
trends for large bubble size. When de is sufficiently large vb ¼ min vb;vis ; vb;in ; vb;spheroid : (6)
rL de ,
i.e., Eo [4:24) Eq. (4) converges to The reason for using the minimum value in Eq. (6) can be
easily seen in Fig. 1. One important advantage of using Eq. (6)
is that it is not necessary to identify the bubble shape in
advance. Rather, the bubble shape is identified automatically
when the bubble rising velocity is determined.
One shortcoming of Eq. (6), however, is that there is an
abrupt change in the derivative of the bubble rising velocity
when the bubble sizeeshape regime changes (e.g., from
sphere to spheroid). Actually, the transition from sphere to
spheroid happens gradually, and the boundary between the
two shape regimes is defined somewhat arbitrarily, e.g., by an
Sphere in creeping flow aspect ratio of about 1.1 [9,10,17], indicating that the abrupt
Sphere with inertial force bending at the shape regime boundaries is not natural.
Spheroid Therefore, we suggest the following equation to bridge Eqs. (2),
(3), and (4) smoothly:
vb ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ :
þ v21 þ v2 1 144m2L
g2 r2 d4
þ 2 5=3 4=3 3
þ 2:14sL 1
L e 0:14425 g rL d e rL de
Fig. 1 e Comparison of the bubble rising velocities for three (7)
bubble shape regimes: sphere, spheroid, and spherical cap.
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Eq. (7) can be regarded as a combination of the two for- no internal circulation occurs, whereas it is 1 when the bub-
mulas for spherical and nonspherical bubbles: bles are not contaminated or are sufficiently large and hence
internal circulation fully develops. The value of fsc in real sit-
vb ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ; (8) uations varies between 1 and 1.5 depending on the specific
contaminants present and their concentrations.
In the same way, the effect of surface contaminants needs
where vb;sp and vb;nonsp are the rising velocities of spherical
to be taken into account also for spherical bubbles with non-
and nonspherical bubbles, respectively, given by:
negligible inertial force, resulting in the following equation:
vb;sp ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
ﬃ (9) 1 1
þ 1 144m2L
mL vb;sp ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ : (12)
vb;in þ 1
þ v21 144m2L
fsc g2 r2 d4 þ
g2 r2 d L e
0:14425 g rL d e 3
v2 b;vis b;in 2 5=3 4=3 3
L e 0:14425 g rL de
and vb;nonsp ¼ vb;spheroid .
Applying this method to Eq. (7), we have:
Fig. 2 compares the bubble rising velocity formulas for
spherical and nonspherical bubbles with the general formula 1 1
vb ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
(Eq. 7) as well as with experimentally measured data 1
144m 2 mL
v2b;sp v2b;nonsp 2
[8,10,19e28] reproduced from Clift et al . Despite the gen- f sc g2 r2 dL4 þ 4=3 3 þ 2:14s
0:14425 g rL de L
eral agreement, it is observed that the scatteredness of the
data is large and that Eq. (7) tends to overestimate the bubble
rising velocity, especially for small bubbles. This can be The value of fsc must have the following three properties.
attributed, at least partly, to the effect of surface contami- First, it must be 1.5 for very small bubbles, i.e., for very small
nants contained in the liquid, which is the subject of the next Eo . For instance, Bond and Newton  argued that internal
section. gas circulation does not occur when Eo 4. However, the
measurements shown in Fig. 2 indicate that internal gas cir-
culation does not vanish completely for Eo as small as 0.01
(deq z0:3 mm). Therefore, we assume here that fsc ¼ 1:5 for
4. Effects of surface contaminants Eo 0:001. Second, it must be 1 for very large bubbles. It is
assumed here that fsc ¼ 1 for Eo 10; i.e., bubbles always have
Eqs. (2e4) and their combination, Eq. (7), are based on the internal circulation when body forces dominate over surface
assumption that internal gas circulation is fully developed tension. Third, it must decrease monotonously with
when a bubble rises by momentum transfer through the liq- increasing particle size (i.e., with increasing Eo ) from 1.5 to 1 in
uidegas interface. It has often been observed in experiments, the range of 0:001 Eo 10. Although several different highly
however, that small spherical bubbles move with a lower ve- sophisticated methods to determine the value of fsc as a
locity given by the following Stokes equation, which indicates function of Eo were suggested previously [31e33], differences
that they behave like rigid bodies with no internal circulation: among the methods are relatively large, and their agreements
with experimental data are only qualitative. Therefore, a
vb;vis ¼ : (10) much simpler parameterization for fsc is suggested here:
The bubble rising velocity predicted by Eq. (10) is 33% lower fsc ¼ 1 þ : (14)
log Eo þ1
than that predicted by Eq. (2) because the suppression of gas 1 þ exp 0:38
circulation inside the bubble increases the friction imposed by
liquid on the gas bubbles. Frumkin and Levich  and Levich
and Technica  attributed this to the presence of surface-
active substances in the liquid medium. According to their 1.5
explanation, the surface-active substances accumulating at
the liquidegas interface (bubble surface) reduce the surface
tension. As the bubbles rise, the surface-active substances are
dragged to the bubble bottom, building a surface tension
gradient. This gradient creates tangential stress, which sup- 1.3
presses the fluid motion at the interface. The strength of the
effect of the surface tension gradient increases with
decreasing particle size.
By taking this effect of surface contaminants into account,
the rising velocity of spherical bubbles can be expressed by: 1.1
1 grL d2e
vb;vis ¼ $ ; (11)
fsc 12mL 1.0
0.001 0.01 0.1 1 10
where fsc is a factor accounting for the suppressed internal gas
circulation due to surface contaminants. The value of fsc is 1.5 Eo
when the bubbles are very small and contaminated and hence
Fig. 3 e Parameterization for f sc as a function of Eo .
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The value of fsc calculated using Eq. (14) is plotted as a
function of Eo in Fig. 3. It can be clearly seen that Eq. (14)
satisfies all the above-mentioned three properties.
Eq. (13) combined with Eq. (14) is the general formula for
bubble rising velocity suggested to be approximately valid for
any bubble size and shape. Fig. 4 compares it with Eq. (12) (for
spherical bubbles), Eq. (4) (for nonspherical bubbles), and Eq.
(7) (general formula with fsc ¼ 1) as well as with measured data
(the same as those shown in Fig. 2) as a function of de . For Eq.
(12), fsc ¼ 1 or 1:5 was assumed depending on whether internal
circulation was taken into account. Consideration of the effect
of surface contaminants led to better agreement between the
parameterization and the measured data obtained with
contaminated water (represented by the symbols located
lower in the scatter plot), whereas Eq. (7) gives better agree-
ment with the measured data obtained with pure water
(represented by the symbols located higher in the scatter plot).
Therefore, the scatteredness of the measured data indicates
Fig. 5 e Comparison of the parameterization suggested in
that real systems can fall into various degrees of
the present study with previous parameterizations found
in the literature.
Eq. (13) combined either with fsc ¼ 1 or with Eq. (14) can be
used for “uncontaminated” and “highly contaminated” liq-
uids, respectively. The bubble rising velocity in “slightly parameterizations show reasonable agreement with
contaminated” liquid may lie between those two limits measured data. However, the parameterization in the present
depending on the degree of contamination. Unfortunately, study has a couple of distinct advantages over the others.
there is no theory available to represent the factor fsc as a First, it is more convenient than the parameterizations sug-
function of the specific contaminants and their concentra- gested by Wallis  and by Bozzano and Dente  because it
tions. In practical applications of pool scrubbing, water is is given as a single explicit formula. Second, it can be used to
inevitably contaminated. In addition, polar liquids are known very easily produce a bubble-shape diagram similar to Grace's
to be more sensitive to the effect of contamination than diagram that was suggested based on experimental observa-
nonpolar liquids . Therefore, it is expected that Eq. (13) tions (without theoretical justification). This will be the sub-
combined with Eq. (14) can be used in most pool scrubbing ject of the next section.
The parameterization suggested in this study is compared
with the previous parameterizations found in the literature
[12,14,15] in Fig. 5. Except that of Jamialahmadi et al ,
which significantly overestimates the rising velocity of
In the work of Grace , the boundaries among the spherical,
spherical bubbles because it neglects the inertial force, all the
spheroidal, and spherical-cap regimes were delineated in the
Eo eRe plane, in which the bubble size changes for different M
values were depicted by parallel lines. Grace's diagram can be
Sphere without internal circulation
produced easily using the results of this study. To do so, it is
Sphere with internal circulation
Nonsphere required to express the equations shown in the previous
General formula with internal circulation sections in terms of dimensionless numbers.
General formula with fsc
By multiplying both sides of Eqs. (12) and (4) by rL de =mL and
rearranging the equations, the following two equations are
obtained for spherical and nonspherical bubbles, respectively:
Re ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ for spherical bubbles; (15)
þ M 2 5=2
fsc E3 o 0:14425 Eo
Re ¼ 2:14 þ 0:505Eo for nonspherical bubbles: (16)
Applying the method of bridging two bubble shape regimes
to Eqs. (15) and (16), we have:
Re ¼ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
Fig. 4 e Comparison of the general formula [Eq. (13) 2
f sc 144M
5=2 þ 1=2
combined with Eq. (14)] with the formulas for different 0:144252 Eo Eo ð2:14þ0:505Eo Þ
shape regimes as well as with measured data.
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It should be noted that Eq. (17) can be directly obtained bubbles increases with decreasing Eo because strong surface
from Eq. (13) by multiplying both sides by rL de =mL . tension forces tend to minimize the bubble surface area, and
In Fig. 6, Eq. (17) is plotted for different values of M ranging with decreasing Re because deformation (from spherical
from 1014 to 108. The lines plotted in this figure using Eq. (17) shape) is caused by inertial forces and bubbles tend to be
are almost the same as those created by Grace . Besides this, spherical under strong viscous forces (relative to inertial
the boundaries between the bubble shape regimes are forces) .
included in this figure. Detailed discussion of these bound-
aries is given below. 5.2. Boundary between spheroid and spherical cap
5.1. Boundary between sphere and spheroid In Section 3, Eq. (5) for spherical-cap bubbles was regarded as a
limiting case of Eq. (4) (for Eo [4:24). The same is obtained
With the analogy that was used to unify the formulas for from Eq. (16) when the second term in the square root is much
bubble rising velocity for spherical and nonspherical bubbles, larger than the first term. Eo ¼ 40 suggested by Grace  based
the boundary between sphere and spheroid can be defined as on observations agrees very well with the result of this study,
the point where Eq. (16) (rising velocity of spheroid) begins to where Eo [4:24. The boundary between spheroid and spher-
be smaller than Eq. (15) (rising velocity of sphere). When Re is ical cap in Fig. 6 was plotted using Eo ¼ 40.
sufficiently large (at the boundary between sphere and
spheroid regimes), Eq. (15) converges to: 5.3. Boundary between sphere and spherical cap
0:14425Eo Considering that the spherical-cap regime can be regarded as
Re ¼ : (18)
fsc M5=12 a limiting case of the nonspherical regime, with Eo [4:24, as
Therefore, by equating the right-hand sides of Eqs. (18) and mentioned above, the boundary between sphere and spherical
(16), rearranging the resulting equation in terms of M, and cap can be expressed by a limiting case of Eq. (19) with a very
combining it with either Eq. (18) or Eq. (16), the following large value of Eo , which would result in:
relation between Re and Eo is obtained.
Re ¼ 7:77: (20)
4:24 This result is somewhat different from that suggested by
Re ¼ 7:77fsc
1þ : (19)
Eo Grace  (Re ¼ 1:2), but the two cases are similar in that the
boundary between sphere and spherical cap is given as a
The curve appearing as the boundary between sphere and
constant Re value (i.e., it does not depend on Eo ).
spheroid in Fig. 6 was plotted using Eq. (19). The sphericity of
Another important aspect of the boundary between sphere
and spherical cap is that it exists only with a value of M larger
than a certain value (e.g., ~10, suggested by Grace). When M is
smaller than this value, the bubble passes through the
spheroid region. This phenomenon can be explained using
Fig. 7, in which there are two different cases of the intersec-
tion of the rising velocity lines for spherical bubbles and
nonspherical bubbles. In the first case (with low M), the rising
velocity line for spherical bubbles intersects with that for
Fig. 7 e Intersection of the rising velocity line for spherical
Fig. 6 e Diagram showing different bubble shape regimes bubbles with low M and high M with that of nonspherical
in the Eo eRe plane. bubbles.
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nonspherical bubbles, where Eo is smaller than 40. In this case, Acknowledgments
the bubble passes through spheroid when it grows. In the
second case (with high M), by contrast, the spherical bubble This work was supported by the Nuclear Research & Devel-
line intersects with the nonspherical bubble line where Eo is opment of the Korea Institute of Energy Technology and
larger than 40. In this case, the bubble shape converts directly Planning (KETEP) grant funded by the Korea government
from sphere to spherical cap. Therefore, whether or not a Ministry of Trade, Industry and Energy (No. 2011T100200045).
spheroidal shape appears when a bubble grows can be
determined based on the value of Eo at the intersection of the
two lines represented by Eqs. (15) and (16). Again, we use Eq. references
(18) instead of Eq. (15) because the conversion from sphere to
nonsphere occurs at Re [1 (see Fig. 5). In addition, fsc can be
assumed to be 1 because here we are interested in the phe-  A.T. Wassel, A.F. Mills, D.C. Bugby, R.N. Oehlberg, Analysis of
nomenon for Eo of around 40. By equating the right-hand sides radionuclide retention in water pools, Nucl. Eng. Des. 90
of Eqs. (18) and (16), we have: (1985) 87e104.
 S.M. Ghiaasiaan, G.F. Yao, A theoretical model for deposition
E2o 24:27M1=3 Eo 102:8M1=3 ¼ 0: (21) of aerosols in rising spherical bubbles due to diffusion,
convection, and inertia, Aerosol Sci. Technol. 26 (1997)
The solution of Eq. (21) is dependent on the value of M and 141e153.
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