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A simplified correlation for bubble volume estimation.

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)
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Author:Shyu, Jin-Cherng; Chang, Chih-Wei; Chen, Ping-Hei
Publication:Canadian Journal of Chemical Engineering
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Date:Apr 1, 2006
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