A simplified correlation for bubble volume estimation.
Shyu, Jin-Cherng ; Chang, Chih-Wei ; Chen, Ping-Hei 等
In order to propose a simplified correlation to predict the
detached bubble volume without considering the gas velocity, an
experimental work has been performed to measure the detached bubble
volume through a micro-hole submerged in liquid. Several micro-holes
with diameters of 60, 90, 126, 220, 580, and 1200 [micro]m are
respectively used for bubble formation, while liquid in a test chamber
is continuously drained at a constant rate of 0.006 ml/s. The predicted
results by the present simplified correlation agree well with the
measured values. Predictions of the detached bubble volume from
previously published correlations using a model to compute the gas
velocity are also compared with experimental data.
Afin de proposer une correlation simplifiee pour predire le volume
de bulles detachees sans tenir compte de la vitesse de gaz, on a realise
un travail experimental pour mesurer le volume de bulles detachees dans
un micro-trou submerge dans un liquide. Plusieurs micro-trous de 60, 90,
126, 220, 580 et 1200 [micron]m sont utilises pour la formation des
bulles, tandis que le liquide dans la chambre d'essai est draine en
continu a un debit constant de 0,006 ml/s. Les resultats predits par la
presente correlation simplifiee concordent bien avec les valeurs
mesurees. Les predictions du volume de bulles detachees venant de
correlations publiees anterieurement et utilisant un modele pour
calculer la vitesse de gaz sont egalement comparees avec les donnees
experimentales.
Keywords: bubble formation, micro-hole, driving pressure
difference, detached bubble volume
Gas injection into liquids is a common operation employed in many
chemical and process engineering applications, such as bubble formation
at spargers and single orifices in many different gas-liquid contactors.
It is well known that both the bubble size and gas held up in a bubble
column are important factors, which influence the gas-liquid mass
transfer, liquid mixing, and residence time of gas and liquid. It is
thus essential to estimate the volume of bubbles formed at an orifice and to determine factors that can influence the bubble size. Many
theories on bubble formation have been proposed with experimental
verifications, most of these experiment set-ups consisted of a test
chamber with an orifice on the bottom plate and a gas chamber attach
beneath the orifice. Gas is forced to form bubbles at the orifice via
the gas chamber. It is well known that the gas chamber has a significant
effect on bubble formation. The operating conditions can be divided into
three regions, namely constant flow, intermediate and constant pressure
conditions, based on a dimensionless parameter, [N.sub.c] (Hughes et
al., 1955).
In order to predict the detached bubble volume without considering
the gas chamber effect, many models have also been derived and verified
experimentally, such as the works of Davidson and Schuler (1960),
Ramakrishnan et al. (1969), Takahashi and Miyahara (1976), Miyahara et
al. (1983), and Gaddis and Vogelpohl (1986). The range of the orifice
diameters used lies between 0.2 mm and 6.02 mm at various gas flow rates
and operating conditions. The gas velocity is involved in these
correlations, except for the constant pressure condition. However, for
the bubble formation at a micro-sized hole, the constant flow condition
should be more practical due to the relatively large value of
L/[D.sub.h] (Clift et al., 1978, Takahashi and Miyahara, 1976). As noted
by McCann and Prince (1971), the equation used for bubble volume
prediction at smaller orifice diameter ranged from 1.6 to 2800 [micro]m
in the work of Blanchard and Syzdek (1977), was only applicable in the
static bubbling regime. Moreover, the measurement of the gas velocity
through a micro-hole is so difficult that it is not convenient to
calculate bubble volume by those correlations. Therefore, the present
study proposes to simplify these correlations by using L/Dh, detailed
discussion will be part of these pages.
Since either a gas chamber effect or gas flow rate needed to be
taken into account in the calculation of the detached bubble volume, the
present work aims to realize the characteristics of a bubble formed at a
hole of micro-meter size (denoted as the micro-hole hereafter) without
an attached gas chamber under pressure variation, and to derive a simple
correlation without coupling the gas velocity. Hence, an experiment,
which is similar to bubble generation used in an ink cartridge of a
thermal bubble inkjet printer, was conducted to model the generation of
bubbles at a micro-hole on the bottom plate of a test chamber, as shown
in Figure 1.
[FIGURE 1 OMITTED]
EXPERIMENTAL PROCEDURES
The experimental apparatus consists of four major components,
namely a test chamber, a PC-controlled transverse system, a pressure
measuring and recording system, and an image processing system. Two
holes are drilled in the bottom plate of the test chamber; one with
diameter of 4.5 mm for liquid drainage from the test chamber and the
other being a micro-hole for bubble generation. The bottom plate is
replaceable with test pieces containing micro-hole diameters ranging
from 1200, 580, 220, 126, 90 and 60 [micro]m.
A high-speed video camera (NAC colour HSV-1000) with a capture rate
of 1000 frames/s and a shutter speed of 1/2500 s, and a high power
halogen lamp was used to take high quality pictures. The recorded images
were then transmitted to a personal computer for image analysis to
evaluate the volume of detached bubbles, [V.sub.b]. Through proper image
processing procedures, the boundaries of bubbles can be found and thus
the bubble volume can be determined by integration. The detached bubble
volume is determined by taking an average value of 10 bubbles just
detached from the hole, with maximum deviation of approximately 3%. The
maximum uncertainty of calculating bubble volume from the images is
about 7%, depending on the measured accuracy and pixel resolution.
Since the experimental set-up and the measurements in this study
are to develop a correlation, we follow the system and process of the
previous study (Chen et al., 2002), and the detailed description of the
experiment referred to. The experimental conditions and the properties
of the fluids used are summarized in Table 1.
The initial height of liquid level in the test chamber is kept at
3.5 cm before each measurement. The liquid is then drained at a fixed
rate during measurement. Since the liquid drained rate has no
significant effect on the bubble volume (Shyu et al., 2002), one preset liquid drained rate ([Q.sub.d]), 0.006 ml/s, is tested for each
micro-hole, and a total amount of 10 ml is drained for each test. When
the air pressure in the test chamber drops to a threshold value during
the continuous drainage, a bubble starts to form at the air-liquid
interface of the micro-hole. Once the bubble detaches, it floats through
the liquid and breaks at the liquid surface. Due to the supplement of
air bubbles, the air pressure in the chamber will continuously rise to a
value, which will terminate the formation process. Due to continuous
drainage, bubbles are continuously generated at the micro-hole, which
makes the air pressure in the test chamber varies with time. (Chen et
al., 2002)
During measurement, the time-dependent pressure differences value
of ([P.sub.atm]-[P.sub.a]) is obtained from a differential pressure
transmitter, and the height of liquid level can be deter mined from the
amount of liquid drained out of the test chamber.
THEORETICAL DEDUCTION
Many correlations have been developed to predict the detached
bubble volume for various flow conditions and expressed by dimensionless
parameters (Bo, Fr and [N.sub.w] = BoFr), which include (Tsuge and
Hibino, 1983):
(1) For constant flow condition:
[V.sub.b] = 0.89[pi][([[mu].sub.l]/[[mu].sub.water]).sup.0.15][D.sup.3.sub.h]/Bo: for small [N.sub.w] (1)
[V.sub.b] = 1.1[Q.suo.12][g.sup.-0.6]: for medium [N.sub.w] and
liquid of relatively low viscosity (2)
(2) For intermediate condition:
[V.sub.b] = ([pi]+1.31[N.sub.w])[D.sup.3.sub.h]/Bo: for medium
[N.sub.w] ([N.sub.w] < 16) (3)
(3) For constant pressure condition (Nw < 16):
[V.sub.b] = 28.8[D.sup.3.sub.h]/Bo (4)
(4) For high gas flow rate ([N.sub.w] > 16):
[V.sub.b] = 906[N.sup.-1.3.sub.w][D.sup.3.sub.h]/Bo (5)
In order to verify the measured values and to calculate bubble
volume correlations previously published, a theoretical model proposed
by Shyu et al. (2002) is used to model the present condition as
simplified in Figure 1. As observed from Equations (2), (3) and (5),
knowledge of gas flow rate into a bubble during formation is required to
employ these correlations. The equations that describe the bubble
formation model of Shyu et al. (2002) are expressed as:
dV/dt = [K.sub.h][[dP - 2[sigma]/[(3V/4[pi]).sup.1/3] +
[[rho].sub.l]gs].sup.1/2] (6)
and
d/dt (M ds/dt) = ([[rho].sub.l] - [[rho].sub.g]) 4/3 [pi][r.sup.3]g
+ [[rho].sub.g][u.sup.2][A.sub.h] - 6[pi][[mu].sub.l] ds/dt r (7)
Solving Equations (6) and (7) with initial conditions at r =
[r.sub.h], s = 0 and ds/dt = 0 at t = 0. The bubble detachment criterion
is set as s = r + [r.sub.h].
The driving pressure difference (dP in Equation (6)) is defined as:
dP = [P.sub.atm] - [P.sub.a] - [[rho].sub.t]gH
The constant [K.sub.h], in Equation (6) is the corresponding
orifice constant expressed as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8a)
where f is the Darcy friction factor for entrance flow, expressed
as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8b)
For a detailed derivation refer to the work of Shyu et al. (2002).
RESULTS AND DISCUSSIONS
Figure 2 shows the results comparing the bubble volume formed in
water with that of 10% wt. isopropanol solution. It is found that the
bubbles formed in water are larger. The ratio of bubble volume generated
in water and isopropanol solution lies between 2.41 and 1.14 at liquid
drained rate of 0.006 ml/s at various micro-holes. An equation used to
predict the detached bubble volume in both liquids (Davidson and
Schuler, 1960), expressed as
[V.sub.b] = [pi][D.sub.h][sigma]/([[rho].sub.l] - [[rho].sub.g])g
(9)
[FIGURE 2 OMITTED]
It is found that the predicted bubble volume is more precise for
bubbles generated at a hole diameter less than 220 [micro]m for both
liquids. Equation (9) is valid only when surface tension force is
predominant with all forces acting in the vertical direction. Bubble
formation at hole diameters less than 220 [micro]m is slow enough to
neglect all other resistant forces such as viscous drag, inertia force
and gas momentum effects. Equation (9) also implies that bubbles
formation at micro-hole diameters less than 220 [micro]m can be regarded
as spherical in this system. As observed from Equation (9), the major
reason that causes the larger bubbles generated in water compared to
isopropanol solution is attributed to the different surface tension
between two liquids.
In the present system, one end of the micro-hole is exposed to the
atmosphere, which is the gas reservoir, thus the volume effect of gas
chamber can be neglected. Moreover, the pressure inside the test chamber
is time-dependent. Bubble seems to be formed either under constant flow
condition or under intermediate condition. As shown in Figure 3, bubble
formation in the present situation can be regarded as under constant
flow conditions, especially for micro-hole diameters less than 220
[micro]m. The major pressure drop is due to the gas flow through a
relatively long path at the small hole diameters, thus the assumption of
constant flow condition is reasonable (Clift et al., 1978). The
criterion used to define the bubble formation under constant flow
condition is either L/[D.sub.h.sup.4] > [10.sup.12] [m.sup.-3]
(Takahashi and Miyahara, 1976) or a pressure drop in the micro-hole
greater than 4[sigma]/[D.sub.h] (Terasaka and Tsuge, 1993). The same
condition is shown in this experiment only for hole diameters less than
or equal to 220 [micro]m, when the value of L/[D.sub.h.sup.4] is larger
than [10.sup.12] [m.sup.-3]. As observed, the effect of L/[D.sub.h] is
very significant. Therefore, coupling this variable into a parameter,
such as [K.sub.h] an entrance flow assumption, is necessary.
[FIGURE 3 OMITTED]
In Figure 3, the detached bubble volume predicted by Equation (3)
is precise over all hole diameters. The bubble volumes predicted by
Equations (3) to (5) are coupled together at hole diameters between 1500
[micro]m and 2000 [micro]m in both liquids. This implies that for the
bubble formation in both liquids, the critical condition [N.sub.w] = 16,
occurs at hole diameters between 1500 [micro]m and 2000 [micro]m. Hence,
the value of [u.sub.crit] can be obtained and the gas flow rate can be
assumed to be a function of [D.sub.h]/L. Noted that for the detached
bubble volume calculated by the present model (Equations (6) and (7)), a
hole length of 3 mm is used for hole diameters larger than 1200
[micro]m.
Due to the difficulties in measuring the gas velocity through a
micro-hole, a simplified correlation without considering gas velocity is
proposed based on Equation (3) for bubble volume prediction in liquid of
low viscosity. The gas velocity is assumed to be a function of
[D.sub.h]/L according to the observation of Equations (6) and (8).
Equation (3) can thus be further derived for the present test
conditions, and expressed as:
[V.sub.b] = [D.sub.h.sup.3][[pi]/Bo + 1.31/[square root of
([D.sub.h]g)] [u.sub.crit] [square root of ([D.sub.h]/L/R)] ([m.sup.3])
(10)
The empirical constant, R, is defined as the ratio of critical hole
diameter to hole length. In this work, [u.sub.crit] is the gas velocity
corresponding to [N.sub.w] = 16. For water and isopropanol solution, R =
0.6 and 0.5, [u.sub.crit] = 4.819 m/s and 3.445 m/s, respectively.
Comparison of the predicted values obtained by the correlation
(Equation (10)) with the empirical data of Takahashi and Miyahara (1976,
1981), Miyahara et al. (1983), Terasaka and Tsuge (1993) and the present
experimental results is shown in Figure 4. Noted that the present
correlation is valid at a specific gas flow rate induced by pressure
difference, dP (in Equation (6)), which is equal or a little larger than
4[sigma]/[D.sub.h] at each diameter. As observed from the results of
Takahashi and Miyahara (1981), the effect of gas chamber on bubble
volumes is less significant as gas flow is induced at a dP value that is
just a little larger than 4[sigma]/[D.sub.h] at a given [D.sub.h]/L.
[FIGURE 4 OMITTED]
As shown in Figure 4a, the results denoted by symbols of triangle
and square show larger deviations. The reason for the former one should
be caused by the inaccuracies in acquiring exact values due to the small
scale of the original figures (Takahashi and Miyahara, 1976), while the
later one should be the large [D.sub.h]/L value of approximately 1.88
used in their work. It is also well known that the hole length may
affect the bubble size (Kumar and Kuloor, 1970). For the micro-hole in
the present experimental system, the gas reservoir maintains a constant
pressure, which is the atmospheric pressure. Therefore, it can be
speculated that bubble is formed nearly under constant pressure
condition. This will be a limiting case that induces deviation from the
intermediate condition and results in inaccuracy. However, a satisfying
agreement is obtained in most cases of Figure 4.
The standard deviation based on
|[V.sub.b,pred]-[V.sub.b,exp]/[V.sub.b,pred]| of all data is about
25.12%. Note that the absence of liquid viscosity effect in Equation
(10) will cause deviation in predicting bubble volume for liquid with
very high or low viscosity. However, no significant limitation on gas
flow rate is showed. Moreover, the gas flow rate is found to be directly
proportional to [square root of ([D.sub.h]/L)]. The predicted result is
precise either at hole diameter less than 6500 [micro]m or [D.sub.h]/L
< 1.8 over all compared data. The present correlation is valid to
predict the detached bubble volume under the following condition:
micro-holes of diameter ranges from micrometer to millimetre, which are
submerged in liquid of viscosity less than 2 mPa x s and; the pressure
difference across the micro-hole is a little larger or equal to
4[sigma]/[D.sub.h].
CONCLUSIONS
An experiment work of generating bubbles at a micro-hole submerged
in liquid by continuous drainage in a test chamber was conducted. Bubble
volume increases with increase of hole diameter under the present test
condition, which is approximately constant flow condition due to
micro-scale hole's diameter with large L/[D.sub.h]. A simplified
correlation was derived to predict detached bubble volume in liquids
with low viscosity under intermediate condition for the pressure
difference across the micro-hole a little larger or equal to
4[sigma]/[D.sub.h] as:
[V.sub.b] = [D.sub.h.sup.3][[pi]/Bo + 1.31/[square root of
([D.sub.h]g)] [u.sub.crit] [square root of ([D.sub.h]/L/R)]]
where R = 0.6, [u.sub.crit] = 4.819 m/s for viscosity of liquid
less than 1 mPa x s, and R = 0.5, [u.sub.crit] = 3.445 m/s for viscosity
of liquid between 1 and 2 mPa x s.
The predicted values agree well with experimental data for bubble
volume estimation at all measured micro-hole's diameters in this
experiment. The prediction of a detached bubble volume without measuring
gas velocity through a micro-hole can thus be achieved by the derived
correlation and complicate computational model (Equations (6) and (7))
becomes evitable. Since this discussion point is valid for liquid with
viscosity lower than 2 mPa x s, it is suggested that the values of
[u.sub.crit] and the empirical constant, R, should be predetermined as
the same procedure done in this work for any further application to
liquid of high viscosity in related work.
ACKNOWLEDGEMENTS
The authors greatly appreciate the financial support by the
National Science Council R.O.C. (NSC 89-2212-E-002-117).
NOMENCLATURE
[A.sub.h] cross-sectional area of micro-hole (m)
Bo Bond number, = [[rho].sub.l][D.sub.h.sup.2]g/[sigma]
[D.sub.h] diameter of micro-hole [sigma] (m)
dP driving pressure difference (Pa)
[d.sub.b] diameter of bubble (m)
f Darcy friction factor
Fr Froude number, = u/[square root of ([D.sub.h]g)]
H liquid height inside test chamber (m)
[K.sub.h] corresponding orifice constant in this article,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
L the length of micro-hole (m)
[L.sup.+] dimensionless hole's length, = L ([D.sub.h]Re)
M virtual mass of bubble, = (11/16 [[rho].sub.l]) 4/3
[pi][r.sup.3] (kg)
[N.sub.c] dimensionless capacitance number,
= 4g([[rho].sub.l] - [[rho].sub.g])/[pi][D.sub.h.sup.2]
[[rho].sub.g][c.sup.2]
[N.sub.w] dimensionless gas flow rate, = Bo x Fr
[P.sub.a] air pressure inside test chamber (Pa)
[P.sub.atm] atmospheric pressure (Pa)
Q volumetric gas flow rate, = u x [A.sub.h] ([m.sup.3]/s)
[Q.sub.d] liquid drained rate (ml/s)
R empirical constant of [D.sub.h]/L corresponding to
[N.sub.w] = 16
r radius of bubble (m)
[r.sub.h] radius of micro-hole (m)
s distance between bubble centre and orifice plate (m)
u gas velocity through micro-hole (m/s)
V bubble volume ([m.sup.3])
[V.sub.b] detached bubble volume ([m.sup.3])
Greek Symbols
[sigma] surface tension (N/m)
[rho] density (kg/[m.sup.3])
[mu] viscosity (Pa x s)
[summation]K total minor loss coefficients,
= 0.5 + 0.8333[([d.sub.b]/[D.sub.h]).sup.2] - 1.8333
(d [d.sub.b]/[D.sub.h]) + 1
Subscripts
crit the critical value corresponding to [N.sub.w] = 16
exp experimental value
g gas
l liquid
pred predictive value by Equation (10)
Manuscript received May 25, 2005; revised manuscript received
November 21, 2005; accepted for publication December 9, 2005.
REFERENCES
Blanchard, D. C. and L. D. Syzdek, "Production of Air Bubbles
of a Specified Size," Chem. Eng. Sci. 32, 1109-1112 (1977).
Clift, R., J. R. Grace and M. E. Weber, "Bubbles, Drops, and
Particles," Academic Press, NY (1978), pp. 321-330.
Chen, P.-H., J.-C. Shyu and W.-F. Cheng, "Valveless Pressure
Regulation with a Submerged Micro-hole," J. Chin. Inst. Eng. 25,
653-661 (2002).
Davidson, J. F. and B. O. G. Schuler, "Bubble Formation at an
Orifice in a Viscous Liquid," Tran. Inst. Chem. Engr. 38, 144-154
(1960).
Gaddis, E. S. and A. Vogelpohl, "Bubble Formation in Quiescent Liquids under Constant Flow Conditions," Chem. Eng. Sci. 41, 97-105
(1986).
Hughes, R. R., A. E. Handlos, H. D. Evans and R. L. Maycock,
"The Formation of Bubbles at Simple Orifices," Chem. Eng.
Prog. 51, 557-563 (1955).
Kumar, R. and N. R. Kuloor, "The Formation of Bubbles and
Drops," in "Advances in Chemical Engineering," T. B.
Drew, G. R. Cokelet, J. W. Hoopes and T. Vermeulen, Eds., Academic
Press, New York (1970), Vol. 8, pp. 257-334.
McCann, D. J. and R. G. H. Prince, "Regimes of Bubbling at a
Submerged Orifice," Chem. Eng. Sci. 26, 1505-1512 (1971).
Miyahara, T., N. Haga and T. Takahashi, "Bubble Formation from
an Orifice at High Gas Flow Rates," Int. Chem. Eng. 23, 524-531
(1983).
Ramakrishnan, R., R. Kumar and N. R. Kuloor, "Studies in
Bubble Formation I Bubble Formation under Constant Flow
Conditions," Chem. Eng. Sci. 24, 731-747 (1969).
Shyu, J.-C., P.-P. Ding, W.-F. Cheng and P.-H. Chen, "Air
Bubble Generation through a Submerged Micro-Hole," Chem Eng. Res.
Des. 25, 355-363 (2002).
Tadaki, T. and S. Maeda, "The Size of Bubbles from Single
Orifices," Kagaku Kogaku 27, 147-155, (1963).
Takahashi, T. and T. Miyahara, "Bubble Volume Formed at
Submerged Nozzles: Constant Gas Flow Condition," Kagaku Kogaku
Ronbunshu 2, 138-143, (1976).
Takahashi, T. and T. Miyahara, "Volume of a Bubble Formed at a
Single, Circular, Submerged Orifice: Effect of the Volume of the Gas
Chamber," Int. Chem. Eng. 21, 224-228, (1981).
Terasaka, K. and H. Tsuge, "Bubble Formation under
Constant-Flow Conditions," Chem. Eng. Sci. 48, 3417-3422, (1993).
Tsuge, H. and S. Hibino, "Bubble Formation from an Orifice
Submerged in Liquids," Chem. Eng. Commun. 22, 63-79 (1983).
Jin-Cherng Shyu (1), Chih-Wei Chang (2) and Ping-Hei Chen (2) *
(1.) Mechanical Industry Research Laboratories, Industry Technology
Research Institute, Tainan City, Taiwan 70955, Republic of China
(2.) Department of Mechanical Engineering, National Taiwan
University, Taipei, Taiwan 10617, Republic of China
* Author to whom correspondence may be addressed. E-mail address:
phchen@ntu.edu.tw
Table 1. Experimental conditions
Items Conditions
Gas properties: Air
Dynamic viscosity of gas, 0.018
[micro]g (mPa x s)
Density of gas, [rho]g (kg/m3) 1.205
Liquids properties: Pure water
Dynamic viscosity, 0.89 x [10.sup.-3]
[[micro].sub.1] (Pa x s)
Density, [[rho].sub.1] (kg/[m.sup.3]) 1000
Surface tension, [sigma](N/m) 72 x [10.sup.-3]
Test conditions:
Hole diameters, [D.sub.h] (PM) 12,005,802,201,269,100
Test plate thickness, L ([micro]m) 3000
570
Liquid drained rate, [Q.sub.d] (ml/s) 0.006
Environmental temperature ([degrees]C) 24 [+ or -] 1
Items
Gas properties:
Dynamic viscosity of gas,
[micro]g (mPa x s)
Density of gas, [rho]g (kg/m3)
Liquids properties: 10% w.t. Isopropanol sol.
Dynamic viscosity, 1.51 x [10.sup.-3]
[[micro].sub.1] (Pa x s)
Density, [[rho].sub.1] (kg/[m.sup.3]) 995.90
Surface tension, [sigma](N/m) 39 x [10.sup.-3]
Test conditions:
Hole diameters, [D.sub.h] (PM)
Test plate thickness, L ([micro]m) for [D.sub.h] = 1200,
580, and 220
for [D.sub.h] = 126, 90,
and 60
Liquid drained rate, [Q.sub.d] (ml/s)
Environmental temperature ([degrees]C)