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Bubble formation at a single orifice in non-Newtonian liquids
Chemical Engineering Science, Vol. 46, No. Printed in Great Britain.
I, pp. 85-93.
BUBBLE
1991.
coos2m9/91 $3.00 + o.oLl Q 1990 Pcrgamon Press plc
FORMATION AT A SINGLE ORIFICE NON-NEWTONIAN LIQUIDS
IN
KOICHI TERASAKA and HIDEKI TSUGEt Department of Applied Chemistry, Faculty of Scienceand Technology, Keio University,Yokohama 223, Japan (First
received 2 November 1989; accepted in revised form 25 January 1990)
Abstract-The effectsof various factors on the volumes and shapes of bubbles formed at a single orifice submerged in non-Newtonian liquids, such as physical properties of liquids, gas chamber volume, orifice dtimeter and gas flow rate were studied. To clarify the bubble formation mechanism, the bubble volume, bubble shape and gas chamber pressure during the bubble growth were measured simultaneously. A revised non-spherical bubble formation model was proposed to describe the bubble formation mechanism in nonNewtonian liquids. The bubble volume, bubble shape and pressurechangein the gas chamber calculated by this model agreed well with the experimental results over a wide range of rheological characteristics of liquids.
complex that bubble formation models describing the bubble formation phenomena well have not yet been proposed. Rlbiger and Vogelpohl (1986) measured the volumes of bubbles formed in relatively low viscous aqueous solutions of carboxymethyl cellulose (CMC) and polyacrylamide (PAA) shown in Table 1 under the constant-flow condition, and classified the bubbling behavior into single bubbles, double bubbles and jet regimes. However, they did not discuss theoretically the bubble formation in nonNewtonian liquids. experimentally the Acharya et al. (1978) obtained volumes of the bubbles formed in CMC aqueous solutions and proposed a correlation equation of the bubble volume under the constant-flow condition. Ghosh and Ulbrecht (1989) proposed the nonspherical bubble formation model under the constantflow condition and obtained relatively good agreement between the calculated and experimental results in aqueous solutions of CMC, PAA and polyvinylpyrrolidone (PVP). Miyahara et al. (1988) measured the bubble volumes for relatively large m under the intermediate condition and proposed a two-stage spherical bubble formation model. In the case of highly apparent viscosity, however, it is not able to estimate the bubble volume well by their model. The experimental works described above have been done on the volumes of bubbles formed at an orifice submerged in power-law model liquids, while few works have been done on bubble formation mechanism; that is, the changes of bubble shape and bubble volume, and the pressure fluctuation in the gas chamber during the bubble growth. The objectives of the present work are as follows. (1) To make clear experimentally the effects of the various factors, such as power-law model coefficient and index, gas chamber volume, orifice diameter and gas flow rate, on the volume of bubble formed at an
INTRODUCTION
For the fundamental design and operation of gas-liquid contacting equipment, such as chemical reactors or fermentors, it is essential to clarify the effects of various factors on the volume and shape of bubble formed att a gas distributor. The most fundamental gas dispersion method is gas bubbling through a single orifice in liquids. The effects of various factors on the volume of bubble formed at an orifice in low viscous Newtonian liquids have been studied extensively and summarized (for example, Kumar and Kuloor, 1970). Recently, it has been shown that the rheological characteristics of liquids change with the progress of reactions and show viscous non-Newtonian behavior, especially in gas-phase polymerization reactors and bioreactors. Therefore, it is necessary to clarify the bubble formation in viscous non-Newtonian liquids. Table 1 shows the list of previous works on the bubble formation at an orifice in non-Newtonian liquids. They used power-law model liquids, whose rheological parameters are written as follows: t = my” = m7”-l-J
(I)
where T, 7;. m and n are shear stress, shear rate. powerlaw model coefficient and index, respectively. my- 1 is called the apparent viscosity and increases with increasing n if 9 is larger than 1 s-l. Bubble formation is classified into three conditions: constant-flow, constant-pressure and intermediate. Many works on bubble formation in non-Newtonian liquids have been done under the constant-flow condition, in which the gas flow rate through an orifice is constant, or the constant-pressure condition, in which the gas chamber pressure is constant. Bubble formation under the intermediate condition, in which the pressure fluctuation exists in a gas chamber, is so ‘Author
to whom
correspondence
should
be addressed. 85
86
KOICHI
TERASAKAand HIDEKI TSUGE
Table 1. Previous works on bubble formation in non-Newtonian
Researchers Acharya et al. (1978) Rsbiger and Vogelpohl(1986) Miyahara et al. (1988) Ghosh and Ulbrecht (1989)
m (Pas”) 0.18-1.4 0.012-0.52 0.275-31.3 0.0345-0.600
Non-Newtonian liquid
n 0.72-0.86 0.52-0.97 0.403-0.95
EXPERIMENTAL PROCEDURE AND CONDITIONS Figure 1 shows the experimental apparatus. The bubble column has 0.15 m square cross-section and is 0.4 m high. By assuming that the bubble size is uniform, a single bubble volume was calculated by dividing gas flow rate by the frequency of bubble formation measured by a photo-transistor. The bubble shapes during bubble formation were photographed by a 16 mm high speed camera (LOCAM Model 51: Red Lake Co., Ltd.), and the bubble volumes were analyzed as a function of time with a film analyzer (FILMOTION: Bell & Howell Co., Ltd.). On the other hand, the pressure in the gas chamber was measured with a pressure transducer (Model 239: Setra Systems Inc.) and recorded with an electromagnetic oscillograph (Model 2931 Photocorder: Yokogawa Electric Works Co., Ltd.). The operating conditions were as follows: gas flow rate Q, = 0.1-10 x 1O-6 m3/s, orifice diameter D, = 0.8-2.0 mm, gas chamber volume V, = 34-300 x 10v6 m3 and liquid height H, = 0.30 m. The liquids used were PAA (Aldrich Chemical Co., Inc.), aqueous solutions and CMC (a)-(d) (Daiichi Kogyo Seiyaku Co. Ltd.), aqueous solutions as nonNewtonian liquids, and glycerol and glycerol aqueous solutions as Newtonian liquids, which correspond to non-Newtonian liquids of n = 1. Molecular weights and degrees of polymerization of the PAA and CMC (a)-(d) polymers are listed in Table 2, which are data from their production companies. The physical properties of the solutions are shown in Table 3. Their power-law model parameters, density and surface tension were measured experimentally by a cone-plate type viscometer (visconic ED-type viscometer: Tokyo Keiki Co., Ltd.), a hydrometer
Model -
CMC CMC, PAA CMC
two-stage spherical
PAA, PVP
0.67-0.95
orifice submerged in non-Newtonian liquids under the intermediate condition over a wide range of apparent liquid viscosity. (2) To discuss the bubble formation mechanism by measuring simultaneously the bubble volume, the bubble shape and the gas chamber pressure during the bubble growth. (3) To compare these experimental results with the calculated ones by the non-spherical bubble formation model for non-Newtonian liquids obtained by modifying the model for Newtonian liquids proposed by Terasaka and Tsuge (1990).
liquids (power-law model fluids)
non-spherical (under constant-flow condition)
3 Gas flow meter 4 Hgmanometer 5 Needle valve 6 Gas chamber 7 Orifice plate I3 l?utbIe cdumn 9 Lamp
10 11 12 13 14 15 16 17 16
Phatotmnslstor Electronic counter Persoml computer Flood lamp 16mmhighspeedoomem Ccnst. temp reguhtor RessuretHP manometer Eledt-omagnettc OsciuOgmph
Fig. 1. Experimental apparatus.
Table 2. Properties of polymers
Polymer PAA CMC CMC
(a) (b)
CMC (c) CMC (d)
Molecular weight 56 x 10s 3.85 x 10’
8.0 x 10’ 3.9 x lo5 3.0 x lo5
Degree of polymerization
1800
3600 1330 1440
(Nippon Keiryoki Kogyo Co., Ltd.), and a CBVPtype surface tension meter (CBVP Al-type: Kyowa Scientific Equipment Co., Ltd.), respectively. The elasticity of the PAA aqueous solution considered generally to be viscoelastic was measured experimentally by a coaxial cylinder-type rheometer (MR-3 Soliquid Meter: Rheology Co., Ltd.) based on the principle of the forced oscillation resonance method (Nakagawa, 1952). The elasticity of the 0.7wt% PAA aqueous solution used in this experiment was 9.98 x 10e3 Pa and much smaller than those of viscoelastic fluids such as rubber-xylene solutions, extract of root of Hibiscus manihot L. and Hg solutions; that is, sulfosalicylic acid aqueous 0.63-10.5, 2.76 and 2.78-3.5 Pa (Nakagawa, 1952),
Bubble formation
at a single orifice in non-Newtonian
Table 3. Physical
Liquid PAA
properties
liquids
87
of liquids MO
Cont. (wt%)
PI x 10-s (kg/ma)
(PI&
’
0.7 0.7
1.001 1.001
0.140 0.154
0.734 0.63 1
70.5 72.6
1.37 x 1o-4 3.45 x 1o-5
293.0 298.1
1.36 2.34
0.663 0.669
70.3 63.9
3.81 x 10-l 4.69
302.4 298.0
0.428 0.412 0.379
68.0 67.7 64.3
2.81 x 10’ 1.86 x 102 4.63 x 10’
298.0 298.0 298.0
2.07 x 10-z
295.2
CMC
(a)
1.3 1.6
1.003 1.006
CMC
(b)
1.2
1.006 1.015 1.038
1.4
1.6
10.2 17.6 24.8
(m&m)
(2)
CMC
(c)
1.3
1.003
0.454
0.746
67.4
CMC
(d)
2.8
1.012
1.35
0.737
66.1
1.181
1.37
0.611
63.8
1.71 x 10-r
302.7
1.238 1.261
0.154 1.35
1 1
61.1 62.0
1.95 x 10-z 1.09 x IO’
295.0 290.0
Mixed solution
t
Glycerot
‘CMC
92 100 (b)-glycerol-H,O
(a)
1.45
302.3
= 0.25: 71.00: 28.75 (wt%).
(b)
Fig. 2. Bubble formation with V, = 42.5 x 10e6m3, D, = 1.47 mm and QS = 1.10 x 10-6m3 photographed at 200 frames/s by high speed camera. (a) N&.7 wt% PAA aqueous solution (m = 0.140 Pas” and n = 0.734) system. (b) N,-1.3 wt% CMC aqueous solution (m = 0.454 Pas” and n = 0.746) system.
88
KOICHI
TERASAKA and HIDEKI
TSUGE
respectively; so that the effect of elasticity can be neglected in this concentration of PAA aqueous solution. EXPERIMENTAL
Bubble
RESULTS
shape change during bubble
growth
Figure 2(a) and (b) shows the bubble growth processes under the conditions of constant n, D,, V, and 0.7 wt% solution (m PAA aqueous Qp in = 0.140 Pas“) and in 1.3 wt% CMC (c) aqueous solution (m = 0.454 Pas”), respectively. It was observed that the bubble shape deviates from spherical with increasing bubble volume V, or apparent viscosity. Bubble
I
1
1
I
III,
I
0.1
,I..,,
10 Gigx
i@
I
[m’/s]
Fig. 4. Effect of m on the relation between V, and Q..
volume and gas chamber pressure changes
of power-law model coeflcient m. Figure 3 shows bubble volume and gas chamber pressure changes during bubble growth as a parameter of m under the conditions of constant n, D,, V, and Q,. The period, from the start of bubble formation (t = 0) to the time occurring the rapid increase of bubble volume, increases with increasing m. Figure 4 shows the effect of m on the relation between final bubble volume V, and Q, under the conditions of constant n, D, and V,. When n is nearly constant, V, increases with increasing m. Furthermore, the effect of Q, on V, becomes large with increasing m. (1)
Efict
(2) EJ’kct of power-law model index n. Figure 5 shows the bubble growth curve and the gas chamber pressure change as a parameter of n under the condi-
Fig. 5. Effect of n on the gas chamber pressure change and bubble
0
D,:l.L7mm 0.5 -V, : 288 dOp6m3
\\ ’
Fig. 3. ElTect of m on the gas chamber pressure change and bubble growth curve.
growth
curve.
tion of constant m, D,, V, and Q,. The bubble formation time increases with increasing n; that is, with increasing apparent viscosity. Figure 6 shows the effect of n at constant m ‘+ 1.36 Pas”. V,, increases with increasing n in the case of constant m. (3) Eflect of gas chamber volume V,. Figure 7 shows the bubble growth curve and the gas chamber pressure change as a parameter of V, under the condition of constant D,, Q, and power-law model parameters. Vb increases more rapidly and P, reduces more slowly with increasing V,. Furthermore, it was observed that V, decreases and P, increases after V, and P, achieved maximal and minimal values, respectively, in the case of small V, ( = 39.8 x 10e6 m”). Figure 8 shows the effect of V, on the relation between V, and Qs. V, increases with increasing V,.
Bubble formation at a single orifice in non-Newtonian
0.1
liquids
10
1 Q, x lo6 [m3/s I
Fig. 6. Effect of n on the relation between V, and QS.
t ISI Fig. 9. Effect of D, on the gas chamber pressure change and bubble growth curve.
D,: 1.47mm
’
& “g 1
,L
w/i
1
0.02
0
$0.1
I
0.04
Fig. 7. Effect of V, on the gas chamber pressure change and bubble growth curve.
q
x
10’ [m’/sl
Fig. 8. Effect of V, on the relation between Vb and QS.
While the difference of V, caused by V. decreases with increasing Q8, and V, is nearly constant under the condition of large V, ( = 288 x lo-” m”).
1
0.1
tlsl
[m3/sl
Fig. 10. Effect of D, on the relation between V, and QE.
change as a parameter of D, under the condition of constant V,, Q. and power-law model parameters. Initial P, increases with decreasing D,. Figure 10 shows the effect of D, on the relation between V, and Q,. At relatively low gas flow rate (Qp 6 5 x 10m6 m3/s under this condition), V, decreases with increasing D,, while V, is not strongly influenced by D, at high gas flow rate (Q, 3 5 x 1O-6 m3/s). (5) E#kct ofgasflow rare Q,. Figure 11 shows the bubble growth curve and chamber pressure change as a parameter of Qs in the conditions of constant D,, V,, m and n. It was observed that V, increases rapidly with increasing Q, after P, achieved a maximal value. The
final volume
increases
BUBBLE
(4) E&ct oforifice diameter D,. Figure 9 shows the bubble growth curve and the gas chamber pressure
10
Q, x lo6
with increasing
FORMATION
Q,.
MODEL
physically the bubble formation, two types of bubble formation model, the spherical and To
describe
KOICHI TERASAKAand HI~EKI TSUGE
90
Fig. 12. Schematic diagram of the non-spherical bubble formation model.
t Is1 Fig. 11. Effect of Q. on the gas chamber pressurechange and bubble growth curve.
non-spherical models, are presented. As shown in Fig. 2, most bubble shapes at the orifice in nonNewtonian liquids are inverted teardrop-like rather than spherical, so that the non-spherical mode1 is applied in this work. Terasaka and Tsuge (1990) presented a non-spherical bubble formation mode1 by revising the mode1 of Pinczewski (1981) for highly viscous Newtonian liquids. By modifying the term of viscous resistance in the previous mode1 (Terasaka and Tsuge, 1990), a revised bubble formation model available for both Newtonian and non-Newtonian liquids is proposed. Concept of the revised non-spherical model In this non-spherical model, the bubble surface is divided into many elements as shown in Fig. 12. Gas flows into a gas chamber at the constant flow rate Q, so that P, increases. When the pressure in the gas chamber becomes larger than the sum of hydrostatic pressure and surface tension, gas in the gas chamber flows through an orifice and bubbles begin to grow. Two motion equations in the radial and vertical directions for each element are solved, so that radial and vertical velocities are estimated and then the positions of the bubble surface elements are determined. This estimation is finished when the bubble detaches from an orifice; that is, the bubble neck is closed. Throughout bubble formation, it is assumed that the bubble grows symmetrically about the vertical axis on the center of an orifice and the bubble motion is not affected by the presence of other bubbles.
The equations describing the above phenomena formulated in the following manner.
are
Pressure change in the gas chamber As the behavior of gas in the gas chamber is assumed to be polytropic (Terasaka and Tsuge, 1990), the pressure change in the gas chamber is expressed as: dP,ldt
=
@c(Qg - QcNK
= @c(Qg - dV,lWl
K (2)
where Q, and K are gas flow rate through an orifice and polytropic coefficient, respectively. K obtained experimentally in this study is 1.1, which agrees with the experimental result of Park et al. (1977). Orifice equation The pressures in the bubble P, and P, are assumed to be related by the orifice equation as follows: IP, -
P,I = (QJk,)’
The orifice constant mentally as follows:
k,
= (d V,/dt)*/k:.
(3)
determined
experi-
was
k, = O-8130,2
(4)
where the units of k, and D, are m7i2/kg1J2 and m, respectively. Dejinition
of equivalent
radii
The bubble surfaceis divided into a number of twodimensional axisymmetric elements which are characterized by two principal radii of curvature R and R’ as shown in Fig. 12. R’ is the radius of the circle which passes through the elementsj - 1,j and j + 1, and has a center point 0. Another radius R is the distance from the bubble symmetrical axis to the element j through the point 0.
91
Bubble formation at a single orifice in non-Newtonian liquids Pressure
balance on the gas-liquid
Procedures
interface
Pinczewski (1981) used the equivalent radius a defined by eq. (5) as the characteristic radius: l/R = (l/R
+ l/R’)/2.
(5)
However, the forces, with the exception of surface tension, depend on the volume or mass of the bubble rather than the curvature of the bubble surface, so that E is used for surface tension and R is used for inertia and viscous forces as the characteristic radius in the present model. On the element j, it is supposed that the bubble shape is a sphere which has radius R. The expansion of the gas-Newtonian liquid interface is written by the following modified Rayleigh equation (Terasaka and Tsuge, 1990):
(6) where Pa is the hydrostatic pressure at the elementj and the three terms on the r.h.s. of eq. (6) represent inertia, surface tension and viscous forces. For power-law model liquids, the term of viscous resistance in eq. (6) is modified as follows:
to solve
the model
The simultaneous differential equations described above are solved by the following procedures. (1) Determining the initial gas chamber pressure by the orifice diameter and surface tension. (2) Estimating the gas chamber pressure by the mass balance equation in the gas chamber, eq. (2). (3) Estimating the pressure in the bubble by solving the orifice equation, eq. (3). (4) Calculating the equivalent radii R and R at any element on the bubble surface by eq. (5). (5) Evaluating the radial acceleration and velocity of the bubble surface by solving the pressure balance between the inside and outside of the bubble surface, eq. (7). (6) Evaluating the vertical acceleration and velocity of the bubble by solving the motion equation of the rising bubble, eq. (8). (7) This calculation finishes when the detachment condition is satisfied. DWCUSSLON Comparison
between
experimental
and calculated
re-
Sllh
In the present model, the motion equation of a rising bubble is described by inertial, buoyancy and viscous drag forces, the vertical component of surface tension acting on bubble surface and gas momentum rate through an orifice as follows:
Figure 13 shows a comparison of the bubble shapes traced from photographs with those calculated by the present model during bubble formation in 0.7 wt% PAA aqueous solution. The present model estimates well the experimental bubble shapes. In Figs 3-11, the results calculated by the present model are expressed as lines. The final bubble volumes and the bubble growth curves calculated by the present model agree welt with the experimental values. Figure 14 shows a comparison between the experimental and calculated results under the conditions of larger m and smaller n than those in Figs 3-11. It is confirmed in Fig. 14 that V, estimated for large m and small n, that is m = 24.8 Pas” and n = 0.379, by the present model agrees well with the experimental results. The present model is applicable to estimate
where M’ and D, are virtual mass [( = ps + 11 p,/ 16) V,] and maximum horizontal bubble diameter. As shown in Fig. 12,0 is the angle between the normal to the bubble surface at any element j and the vertical axis, and ON is angle 8 at the element N. CD is the drag coefficient assumed to be written by the function of Reynolds number Re modified for powerlaw model fluids as foHows:
Experimental Calculated Liquid : 0.7wtXpAA m:O.l40Fbs” n:0.734 D,:1.47mm V,:398-lO*m’ Q$.&lO”rnk
p, - p, = PI [Rs+;($y]+; +8m
1dR
( 2R dt >
n
(7)
where the viscosity term is applicable for Newtonian liquids by equating n = 1 and m = p. Motion
equation
Re
of a rising bubble
=
M’XWd~)2-” m
(9)
Fig. 13. Comparison between experimental and calculated bubble shapes during bubble growth.
KOICHI
92
TERASAKA and HIDEKI
TSUGE
10
o.o5l0.1
* * * ‘*...I
’ ’ “,,I.’ 1 10 Q, x 10’ [m3/s 1
I
Fig. 14. Relation between F’, and Q, in the condition of larger M and smaller n than those in Figs 3-l 1.
Liquid:CSlC aqueous scbtion m:l.6&3pO~5” n : 0.768
0.1
0.1
1 10 Q, x 10” [m3/s]
Fig. 16. Comparison between the model of Miyahara et al. and the present model. CONCLUSION
_&z&N
-This model -;--Miyubetal.
Q, x 10’ tm3/s3 Fig. 15. Comparison between the model of Miyahara et al. and the present model.
bubble volume wide ranges of m over = 0.14-24.8 Pas” and n = 0.379-l. On the other hand, the gas chamber pressures obtained experimentally correspond with those estimated by the present model. However, in the final period of bubble growth, as shown in Fig. 11, the experimental P, is smaller than the calculated P, because of the reduction of the static pressure of liquid surrounding the bubble by the wake of the previous rising bubble. Comparison between the calculated and the other researcher’s results Figures 15 and 16 show a comparison of bubble volume between the experimental and calculated results by Miyahara et al. (1988) and the results computed by the present model. Under the condition of m = 1.688 Pas” and R = 0.768 shown in Fig. 15, the bubble volumes calculated by both the model of Miyahara et al. and this model agree well with the experimental results. However, for larger m shown in Fig. 16 than in Fig. 15, the experimental Vb values of Miyahara et al. are much smaller than those calculated by their model, while these are estimated well by the present model. It is possible to estimate the volume of bubbles formed for a wider range of apparent viscosity of liquids, including the other researchers’ experimental results.. bv_ this model.
The effects of power-law model coefficient, index, gas flow rate, gas chamber volume and orifice diameter on the volumes and shapes of bubbles at an orifice submerged in non-Newtonian liquids, and the gas chamber pressure were investigated experimentally over the wide range of rheological characteristics, that is, m = 0.14-24.8 Pas” and n = 0.379-l. The final bubble volumes increased with increasing JL Qs, m and n. Under the present condition, Vb increased with decreasing D, at relatively low gas flow rate, while DO did not influence V, at high gas flow rate. The bubble formation time increased with an increase of the apparent viscosity, that is increases of m and/or n. By modifying the authors’ model for bubble formation in Newtonian liquids, the non-spherical bubble formation model for non-Newtonian liquids was presented. The present model is able to estimate well the volume and shape of a bubble and the gas chamber pressure over a wide experimental range of rheological parameters of non-Newtonian liquids. Furthermore, the present model estimates well the other researchers’ experimental results. NOTATION
drag coefficient maximum horizontal bubble diameter, m orifice diameter, m gravitational acceleration, m/s2 height of liquid, m orifice constant, m7/‘/kg’1’ virtual mass [ = (llp,/16 + p,) V,], kg power-law model coefficient, Pas” modified Morton number ( = g3”-ZmA/ z-na2+n Pt
1
power-law model index pressure in bubble, Pa hydrostatic pressure at any element, Pa hydrostatic pressure at orifice plate, Pa atmospheric pressure, Pa flow rate of gas into gas chamber, m’/s flow rate of gas through an orifice, m”/s R, R’ principal radii of curvature, m R mean of equivalent radius defined by eq. (5), m
Bubble formation at a single orifice in non-Newtonian
[ = p,D”,(dr/ modified Reynolds number dt)z -“/ml temperature of liquid, K bubbling time, s bubble volume, m3 gas chamber volume, m3 vertical distance from orifice plate, m shear rate, l/s angle between the normal to bubble surface and vertical axis, rad polytropic coefficient liquid viscosity, Pas gas density, kg/m3 liquid density, kg/m3 surface tension, N/m shear stress, Pa REFERENCES
Acharya, A., Mashelkar, R. A. and Ulbrecht, J. J.. 1978, Bubble formation in non-Newtonian
Chem. Fundam. 17, 230-232.
liquids. Ind. Engng
93
liquids
Ghosh, A. K. and Ulbrecht, I. J., 1989, Bubble formation from a sparger in polymer solutions-I. Stagnant liquid. Chem. Eagng Sci. 44.957-968. Kumar, 8. and Kuloor, N. R., 1970, The formation of bubbles and drops. Adv. Chem. Engng 8, 255. Miyahara, T., Wang, W.-H. and Takahashi, T., 1988,Bubble formation at a submerged orifice in non-Newtonian and highly viscous Newtonian liquids. J. them. Engng Japan 21,620426. Nakagawa,
T., 1952, Spinnability of liquid. A viscoelastic
state. II. Spinnability and viscoelastic property. Bull. them. Sot. Japan 25,93-97. Park, Y., Tyler, A. L. and de Nevers, N., 1977,The chamber orifice interaction in the formation of bubbles. Chem. Engng Sci. 32,907-916. Pinczcwski, W. V., 1981, The formation and growth of bubbles at a submerged orifice. Chem. Engng Sci. 36,
405411. Riibiger, N. and Vogelpohl. A., 1986, Encyclopedia Mechanics, Bubble Formation Newtonian and Non-Newtonian Houston, TX.
o/Fluid and its Movement in Liquids, pp. 59-88. Gulf,
Terasaka, K. and Tsuge, H., 1990, Bubble formation at a single orifice in highly viscous liquids. J. them. Engng Japan 23, 160-165.
I, pp. 85-93.
BUBBLE
1991.
coos2m9/91 $3.00 + o.oLl Q 1990 Pcrgamon Press plc
FORMATION AT A SINGLE ORIFICE NON-NEWTONIAN LIQUIDS
IN
KOICHI TERASAKA and HIDEKI TSUGEt Department of Applied Chemistry, Faculty of Scienceand Technology, Keio University,Yokohama 223, Japan (First
received 2 November 1989; accepted in revised form 25 January 1990)
Abstract-The effectsof various factors on the volumes and shapes of bubbles formed at a single orifice submerged in non-Newtonian liquids, such as physical properties of liquids, gas chamber volume, orifice dtimeter and gas flow rate were studied. To clarify the bubble formation mechanism, the bubble volume, bubble shape and gas chamber pressure during the bubble growth were measured simultaneously. A revised non-spherical bubble formation model was proposed to describe the bubble formation mechanism in nonNewtonian liquids. The bubble volume, bubble shape and pressurechangein the gas chamber calculated by this model agreed well with the experimental results over a wide range of rheological characteristics of liquids.
complex that bubble formation models describing the bubble formation phenomena well have not yet been proposed. Rlbiger and Vogelpohl (1986) measured the volumes of bubbles formed in relatively low viscous aqueous solutions of carboxymethyl cellulose (CMC) and polyacrylamide (PAA) shown in Table 1 under the constant-flow condition, and classified the bubbling behavior into single bubbles, double bubbles and jet regimes. However, they did not discuss theoretically the bubble formation in nonNewtonian liquids. experimentally the Acharya et al. (1978) obtained volumes of the bubbles formed in CMC aqueous solutions and proposed a correlation equation of the bubble volume under the constant-flow condition. Ghosh and Ulbrecht (1989) proposed the nonspherical bubble formation model under the constantflow condition and obtained relatively good agreement between the calculated and experimental results in aqueous solutions of CMC, PAA and polyvinylpyrrolidone (PVP). Miyahara et al. (1988) measured the bubble volumes for relatively large m under the intermediate condition and proposed a two-stage spherical bubble formation model. In the case of highly apparent viscosity, however, it is not able to estimate the bubble volume well by their model. The experimental works described above have been done on the volumes of bubbles formed at an orifice submerged in power-law model liquids, while few works have been done on bubble formation mechanism; that is, the changes of bubble shape and bubble volume, and the pressure fluctuation in the gas chamber during the bubble growth. The objectives of the present work are as follows. (1) To make clear experimentally the effects of the various factors, such as power-law model coefficient and index, gas chamber volume, orifice diameter and gas flow rate, on the volume of bubble formed at an
INTRODUCTION
For the fundamental design and operation of gas-liquid contacting equipment, such as chemical reactors or fermentors, it is essential to clarify the effects of various factors on the volume and shape of bubble formed att a gas distributor. The most fundamental gas dispersion method is gas bubbling through a single orifice in liquids. The effects of various factors on the volume of bubble formed at an orifice in low viscous Newtonian liquids have been studied extensively and summarized (for example, Kumar and Kuloor, 1970). Recently, it has been shown that the rheological characteristics of liquids change with the progress of reactions and show viscous non-Newtonian behavior, especially in gas-phase polymerization reactors and bioreactors. Therefore, it is necessary to clarify the bubble formation in viscous non-Newtonian liquids. Table 1 shows the list of previous works on the bubble formation at an orifice in non-Newtonian liquids. They used power-law model liquids, whose rheological parameters are written as follows: t = my” = m7”-l-J
(I)
where T, 7;. m and n are shear stress, shear rate. powerlaw model coefficient and index, respectively. my- 1 is called the apparent viscosity and increases with increasing n if 9 is larger than 1 s-l. Bubble formation is classified into three conditions: constant-flow, constant-pressure and intermediate. Many works on bubble formation in non-Newtonian liquids have been done under the constant-flow condition, in which the gas flow rate through an orifice is constant, or the constant-pressure condition, in which the gas chamber pressure is constant. Bubble formation under the intermediate condition, in which the pressure fluctuation exists in a gas chamber, is so ‘Author
to whom
correspondence
should
be addressed. 85
86
KOICHI
TERASAKAand HIDEKI TSUGE
Table 1. Previous works on bubble formation in non-Newtonian
Researchers Acharya et al. (1978) Rsbiger and Vogelpohl(1986) Miyahara et al. (1988) Ghosh and Ulbrecht (1989)
m (Pas”) 0.18-1.4 0.012-0.52 0.275-31.3 0.0345-0.600
Non-Newtonian liquid
n 0.72-0.86 0.52-0.97 0.403-0.95
EXPERIMENTAL PROCEDURE AND CONDITIONS Figure 1 shows the experimental apparatus. The bubble column has 0.15 m square cross-section and is 0.4 m high. By assuming that the bubble size is uniform, a single bubble volume was calculated by dividing gas flow rate by the frequency of bubble formation measured by a photo-transistor. The bubble shapes during bubble formation were photographed by a 16 mm high speed camera (LOCAM Model 51: Red Lake Co., Ltd.), and the bubble volumes were analyzed as a function of time with a film analyzer (FILMOTION: Bell & Howell Co., Ltd.). On the other hand, the pressure in the gas chamber was measured with a pressure transducer (Model 239: Setra Systems Inc.) and recorded with an electromagnetic oscillograph (Model 2931 Photocorder: Yokogawa Electric Works Co., Ltd.). The operating conditions were as follows: gas flow rate Q, = 0.1-10 x 1O-6 m3/s, orifice diameter D, = 0.8-2.0 mm, gas chamber volume V, = 34-300 x 10v6 m3 and liquid height H, = 0.30 m. The liquids used were PAA (Aldrich Chemical Co., Inc.), aqueous solutions and CMC (a)-(d) (Daiichi Kogyo Seiyaku Co. Ltd.), aqueous solutions as nonNewtonian liquids, and glycerol and glycerol aqueous solutions as Newtonian liquids, which correspond to non-Newtonian liquids of n = 1. Molecular weights and degrees of polymerization of the PAA and CMC (a)-(d) polymers are listed in Table 2, which are data from their production companies. The physical properties of the solutions are shown in Table 3. Their power-law model parameters, density and surface tension were measured experimentally by a cone-plate type viscometer (visconic ED-type viscometer: Tokyo Keiki Co., Ltd.), a hydrometer
Model -
CMC CMC, PAA CMC
two-stage spherical
PAA, PVP
0.67-0.95
orifice submerged in non-Newtonian liquids under the intermediate condition over a wide range of apparent liquid viscosity. (2) To discuss the bubble formation mechanism by measuring simultaneously the bubble volume, the bubble shape and the gas chamber pressure during the bubble growth. (3) To compare these experimental results with the calculated ones by the non-spherical bubble formation model for non-Newtonian liquids obtained by modifying the model for Newtonian liquids proposed by Terasaka and Tsuge (1990).
liquids (power-law model fluids)
non-spherical (under constant-flow condition)
3 Gas flow meter 4 Hgmanometer 5 Needle valve 6 Gas chamber 7 Orifice plate I3 l?utbIe cdumn 9 Lamp
10 11 12 13 14 15 16 17 16
Phatotmnslstor Electronic counter Persoml computer Flood lamp 16mmhighspeedoomem Ccnst. temp reguhtor RessuretHP manometer Eledt-omagnettc OsciuOgmph
Fig. 1. Experimental apparatus.
Table 2. Properties of polymers
Polymer PAA CMC CMC
(a) (b)
CMC (c) CMC (d)
Molecular weight 56 x 10s 3.85 x 10’
8.0 x 10’ 3.9 x lo5 3.0 x lo5
Degree of polymerization
1800
3600 1330 1440
(Nippon Keiryoki Kogyo Co., Ltd.), and a CBVPtype surface tension meter (CBVP Al-type: Kyowa Scientific Equipment Co., Ltd.), respectively. The elasticity of the PAA aqueous solution considered generally to be viscoelastic was measured experimentally by a coaxial cylinder-type rheometer (MR-3 Soliquid Meter: Rheology Co., Ltd.) based on the principle of the forced oscillation resonance method (Nakagawa, 1952). The elasticity of the 0.7wt% PAA aqueous solution used in this experiment was 9.98 x 10e3 Pa and much smaller than those of viscoelastic fluids such as rubber-xylene solutions, extract of root of Hibiscus manihot L. and Hg solutions; that is, sulfosalicylic acid aqueous 0.63-10.5, 2.76 and 2.78-3.5 Pa (Nakagawa, 1952),
Bubble formation
at a single orifice in non-Newtonian
Table 3. Physical
Liquid PAA
properties
liquids
87
of liquids MO
Cont. (wt%)
PI x 10-s (kg/ma)
(PI&
’
0.7 0.7
1.001 1.001
0.140 0.154
0.734 0.63 1
70.5 72.6
1.37 x 1o-4 3.45 x 1o-5
293.0 298.1
1.36 2.34
0.663 0.669
70.3 63.9
3.81 x 10-l 4.69
302.4 298.0
0.428 0.412 0.379
68.0 67.7 64.3
2.81 x 10’ 1.86 x 102 4.63 x 10’
298.0 298.0 298.0
2.07 x 10-z
295.2
CMC
(a)
1.3 1.6
1.003 1.006
CMC
(b)
1.2
1.006 1.015 1.038
1.4
1.6
10.2 17.6 24.8
(m&m)
(2)
CMC
(c)
1.3
1.003
0.454
0.746
67.4
CMC
(d)
2.8
1.012
1.35
0.737
66.1
1.181
1.37
0.611
63.8
1.71 x 10-r
302.7
1.238 1.261
0.154 1.35
1 1
61.1 62.0
1.95 x 10-z 1.09 x IO’
295.0 290.0
Mixed solution
t
Glycerot
‘CMC
92 100 (b)-glycerol-H,O
(a)
1.45
302.3
= 0.25: 71.00: 28.75 (wt%).
(b)
Fig. 2. Bubble formation with V, = 42.5 x 10e6m3, D, = 1.47 mm and QS = 1.10 x 10-6m3 photographed at 200 frames/s by high speed camera. (a) N&.7 wt% PAA aqueous solution (m = 0.140 Pas” and n = 0.734) system. (b) N,-1.3 wt% CMC aqueous solution (m = 0.454 Pas” and n = 0.746) system.
88
KOICHI
TERASAKA and HIDEKI
TSUGE
respectively; so that the effect of elasticity can be neglected in this concentration of PAA aqueous solution. EXPERIMENTAL
Bubble
RESULTS
shape change during bubble
growth
Figure 2(a) and (b) shows the bubble growth processes under the conditions of constant n, D,, V, and 0.7 wt% solution (m PAA aqueous Qp in = 0.140 Pas“) and in 1.3 wt% CMC (c) aqueous solution (m = 0.454 Pas”), respectively. It was observed that the bubble shape deviates from spherical with increasing bubble volume V, or apparent viscosity. Bubble
I
1
1
I
III,
I
0.1
,I..,,
10 Gigx
i@
I
[m’/s]
Fig. 4. Effect of m on the relation between V, and Q..
volume and gas chamber pressure changes
of power-law model coeflcient m. Figure 3 shows bubble volume and gas chamber pressure changes during bubble growth as a parameter of m under the conditions of constant n, D,, V, and Q,. The period, from the start of bubble formation (t = 0) to the time occurring the rapid increase of bubble volume, increases with increasing m. Figure 4 shows the effect of m on the relation between final bubble volume V, and Q, under the conditions of constant n, D, and V,. When n is nearly constant, V, increases with increasing m. Furthermore, the effect of Q, on V, becomes large with increasing m. (1)
Efict
(2) EJ’kct of power-law model index n. Figure 5 shows the bubble growth curve and the gas chamber pressure change as a parameter of n under the condi-
Fig. 5. Effect of n on the gas chamber pressure change and bubble
0
D,:l.L7mm 0.5 -V, : 288 dOp6m3
\\ ’
Fig. 3. ElTect of m on the gas chamber pressure change and bubble growth curve.
growth
curve.
tion of constant m, D,, V, and Q,. The bubble formation time increases with increasing n; that is, with increasing apparent viscosity. Figure 6 shows the effect of n at constant m ‘+ 1.36 Pas”. V,, increases with increasing n in the case of constant m. (3) Eflect of gas chamber volume V,. Figure 7 shows the bubble growth curve and the gas chamber pressure change as a parameter of V, under the condition of constant D,, Q, and power-law model parameters. Vb increases more rapidly and P, reduces more slowly with increasing V,. Furthermore, it was observed that V, decreases and P, increases after V, and P, achieved maximal and minimal values, respectively, in the case of small V, ( = 39.8 x 10e6 m”). Figure 8 shows the effect of V, on the relation between V, and Qs. V, increases with increasing V,.
Bubble formation at a single orifice in non-Newtonian
0.1
liquids
10
1 Q, x lo6 [m3/s I
Fig. 6. Effect of n on the relation between V, and QS.
t ISI Fig. 9. Effect of D, on the gas chamber pressure change and bubble growth curve.
D,: 1.47mm
’
& “g 1
,L
w/i
1
0.02
0
$0.1
I
0.04
Fig. 7. Effect of V, on the gas chamber pressure change and bubble growth curve.
q
x
10’ [m’/sl
Fig. 8. Effect of V, on the relation between Vb and QS.
While the difference of V, caused by V. decreases with increasing Q8, and V, is nearly constant under the condition of large V, ( = 288 x lo-” m”).
1
0.1
tlsl
[m3/sl
Fig. 10. Effect of D, on the relation between V, and QE.
change as a parameter of D, under the condition of constant V,, Q. and power-law model parameters. Initial P, increases with decreasing D,. Figure 10 shows the effect of D, on the relation between V, and Q,. At relatively low gas flow rate (Qp 6 5 x 10m6 m3/s under this condition), V, decreases with increasing D,, while V, is not strongly influenced by D, at high gas flow rate (Q, 3 5 x 1O-6 m3/s). (5) E#kct ofgasflow rare Q,. Figure 11 shows the bubble growth curve and chamber pressure change as a parameter of Qs in the conditions of constant D,, V,, m and n. It was observed that V, increases rapidly with increasing Q, after P, achieved a maximal value. The
final volume
increases
BUBBLE
(4) E&ct oforifice diameter D,. Figure 9 shows the bubble growth curve and the gas chamber pressure
10
Q, x lo6
with increasing
FORMATION
Q,.
MODEL
physically the bubble formation, two types of bubble formation model, the spherical and To
describe
KOICHI TERASAKAand HI~EKI TSUGE
90
Fig. 12. Schematic diagram of the non-spherical bubble formation model.
t Is1 Fig. 11. Effect of Q. on the gas chamber pressurechange and bubble growth curve.
non-spherical models, are presented. As shown in Fig. 2, most bubble shapes at the orifice in nonNewtonian liquids are inverted teardrop-like rather than spherical, so that the non-spherical mode1 is applied in this work. Terasaka and Tsuge (1990) presented a non-spherical bubble formation mode1 by revising the mode1 of Pinczewski (1981) for highly viscous Newtonian liquids. By modifying the term of viscous resistance in the previous mode1 (Terasaka and Tsuge, 1990), a revised bubble formation model available for both Newtonian and non-Newtonian liquids is proposed. Concept of the revised non-spherical model In this non-spherical model, the bubble surface is divided into many elements as shown in Fig. 12. Gas flows into a gas chamber at the constant flow rate Q, so that P, increases. When the pressure in the gas chamber becomes larger than the sum of hydrostatic pressure and surface tension, gas in the gas chamber flows through an orifice and bubbles begin to grow. Two motion equations in the radial and vertical directions for each element are solved, so that radial and vertical velocities are estimated and then the positions of the bubble surface elements are determined. This estimation is finished when the bubble detaches from an orifice; that is, the bubble neck is closed. Throughout bubble formation, it is assumed that the bubble grows symmetrically about the vertical axis on the center of an orifice and the bubble motion is not affected by the presence of other bubbles.
The equations describing the above phenomena formulated in the following manner.
are
Pressure change in the gas chamber As the behavior of gas in the gas chamber is assumed to be polytropic (Terasaka and Tsuge, 1990), the pressure change in the gas chamber is expressed as: dP,ldt
=
@c(Qg - QcNK
= @c(Qg - dV,lWl
K (2)
where Q, and K are gas flow rate through an orifice and polytropic coefficient, respectively. K obtained experimentally in this study is 1.1, which agrees with the experimental result of Park et al. (1977). Orifice equation The pressures in the bubble P, and P, are assumed to be related by the orifice equation as follows: IP, -
P,I = (QJk,)’
The orifice constant mentally as follows:
k,
= (d V,/dt)*/k:.
(3)
determined
experi-
was
k, = O-8130,2
(4)
where the units of k, and D, are m7i2/kg1J2 and m, respectively. Dejinition
of equivalent
radii
The bubble surfaceis divided into a number of twodimensional axisymmetric elements which are characterized by two principal radii of curvature R and R’ as shown in Fig. 12. R’ is the radius of the circle which passes through the elementsj - 1,j and j + 1, and has a center point 0. Another radius R is the distance from the bubble symmetrical axis to the element j through the point 0.
91
Bubble formation at a single orifice in non-Newtonian liquids Pressure
balance on the gas-liquid
Procedures
interface
Pinczewski (1981) used the equivalent radius a defined by eq. (5) as the characteristic radius: l/R = (l/R
+ l/R’)/2.
(5)
However, the forces, with the exception of surface tension, depend on the volume or mass of the bubble rather than the curvature of the bubble surface, so that E is used for surface tension and R is used for inertia and viscous forces as the characteristic radius in the present model. On the element j, it is supposed that the bubble shape is a sphere which has radius R. The expansion of the gas-Newtonian liquid interface is written by the following modified Rayleigh equation (Terasaka and Tsuge, 1990):
(6) where Pa is the hydrostatic pressure at the elementj and the three terms on the r.h.s. of eq. (6) represent inertia, surface tension and viscous forces. For power-law model liquids, the term of viscous resistance in eq. (6) is modified as follows:
to solve
the model
The simultaneous differential equations described above are solved by the following procedures. (1) Determining the initial gas chamber pressure by the orifice diameter and surface tension. (2) Estimating the gas chamber pressure by the mass balance equation in the gas chamber, eq. (2). (3) Estimating the pressure in the bubble by solving the orifice equation, eq. (3). (4) Calculating the equivalent radii R and R at any element on the bubble surface by eq. (5). (5) Evaluating the radial acceleration and velocity of the bubble surface by solving the pressure balance between the inside and outside of the bubble surface, eq. (7). (6) Evaluating the vertical acceleration and velocity of the bubble by solving the motion equation of the rising bubble, eq. (8). (7) This calculation finishes when the detachment condition is satisfied. DWCUSSLON Comparison
between
experimental
and calculated
re-
Sllh
In the present model, the motion equation of a rising bubble is described by inertial, buoyancy and viscous drag forces, the vertical component of surface tension acting on bubble surface and gas momentum rate through an orifice as follows:
Figure 13 shows a comparison of the bubble shapes traced from photographs with those calculated by the present model during bubble formation in 0.7 wt% PAA aqueous solution. The present model estimates well the experimental bubble shapes. In Figs 3-11, the results calculated by the present model are expressed as lines. The final bubble volumes and the bubble growth curves calculated by the present model agree welt with the experimental values. Figure 14 shows a comparison between the experimental and calculated results under the conditions of larger m and smaller n than those in Figs 3-11. It is confirmed in Fig. 14 that V, estimated for large m and small n, that is m = 24.8 Pas” and n = 0.379, by the present model agrees well with the experimental results. The present model is applicable to estimate
where M’ and D, are virtual mass [( = ps + 11 p,/ 16) V,] and maximum horizontal bubble diameter. As shown in Fig. 12,0 is the angle between the normal to the bubble surface at any element j and the vertical axis, and ON is angle 8 at the element N. CD is the drag coefficient assumed to be written by the function of Reynolds number Re modified for powerlaw model fluids as foHows:
Experimental Calculated Liquid : 0.7wtXpAA m:O.l40Fbs” n:0.734 D,:1.47mm V,:398-lO*m’ Q$.&lO”rnk
p, - p, = PI [Rs+;($y]+; +8m
1dR
( 2R dt >
n
(7)
where the viscosity term is applicable for Newtonian liquids by equating n = 1 and m = p. Motion
equation
Re
of a rising bubble
=
M’XWd~)2-” m
(9)
Fig. 13. Comparison between experimental and calculated bubble shapes during bubble growth.
KOICHI
92
TERASAKA and HIDEKI
TSUGE
10
o.o5l0.1
* * * ‘*...I
’ ’ “,,I.’ 1 10 Q, x 10’ [m3/s 1
I
Fig. 14. Relation between F’, and Q, in the condition of larger M and smaller n than those in Figs 3-l 1.
Liquid:CSlC aqueous scbtion m:l.6&3pO~5” n : 0.768
0.1
0.1
1 10 Q, x 10” [m3/s]
Fig. 16. Comparison between the model of Miyahara et al. and the present model. CONCLUSION
_&z&N
-This model -;--Miyubetal.
Q, x 10’ tm3/s3 Fig. 15. Comparison between the model of Miyahara et al. and the present model.
bubble volume wide ranges of m over = 0.14-24.8 Pas” and n = 0.379-l. On the other hand, the gas chamber pressures obtained experimentally correspond with those estimated by the present model. However, in the final period of bubble growth, as shown in Fig. 11, the experimental P, is smaller than the calculated P, because of the reduction of the static pressure of liquid surrounding the bubble by the wake of the previous rising bubble. Comparison between the calculated and the other researcher’s results Figures 15 and 16 show a comparison of bubble volume between the experimental and calculated results by Miyahara et al. (1988) and the results computed by the present model. Under the condition of m = 1.688 Pas” and R = 0.768 shown in Fig. 15, the bubble volumes calculated by both the model of Miyahara et al. and this model agree well with the experimental results. However, for larger m shown in Fig. 16 than in Fig. 15, the experimental Vb values of Miyahara et al. are much smaller than those calculated by their model, while these are estimated well by the present model. It is possible to estimate the volume of bubbles formed for a wider range of apparent viscosity of liquids, including the other researchers’ experimental results.. bv_ this model.
The effects of power-law model coefficient, index, gas flow rate, gas chamber volume and orifice diameter on the volumes and shapes of bubbles at an orifice submerged in non-Newtonian liquids, and the gas chamber pressure were investigated experimentally over the wide range of rheological characteristics, that is, m = 0.14-24.8 Pas” and n = 0.379-l. The final bubble volumes increased with increasing JL Qs, m and n. Under the present condition, Vb increased with decreasing D, at relatively low gas flow rate, while DO did not influence V, at high gas flow rate. The bubble formation time increased with an increase of the apparent viscosity, that is increases of m and/or n. By modifying the authors’ model for bubble formation in Newtonian liquids, the non-spherical bubble formation model for non-Newtonian liquids was presented. The present model is able to estimate well the volume and shape of a bubble and the gas chamber pressure over a wide experimental range of rheological parameters of non-Newtonian liquids. Furthermore, the present model estimates well the other researchers’ experimental results. NOTATION
drag coefficient maximum horizontal bubble diameter, m orifice diameter, m gravitational acceleration, m/s2 height of liquid, m orifice constant, m7/‘/kg’1’ virtual mass [ = (llp,/16 + p,) V,], kg power-law model coefficient, Pas” modified Morton number ( = g3”-ZmA/ z-na2+n Pt
1
power-law model index pressure in bubble, Pa hydrostatic pressure at any element, Pa hydrostatic pressure at orifice plate, Pa atmospheric pressure, Pa flow rate of gas into gas chamber, m’/s flow rate of gas through an orifice, m”/s R, R’ principal radii of curvature, m R mean of equivalent radius defined by eq. (5), m
Bubble formation at a single orifice in non-Newtonian
[ = p,D”,(dr/ modified Reynolds number dt)z -“/ml temperature of liquid, K bubbling time, s bubble volume, m3 gas chamber volume, m3 vertical distance from orifice plate, m shear rate, l/s angle between the normal to bubble surface and vertical axis, rad polytropic coefficient liquid viscosity, Pas gas density, kg/m3 liquid density, kg/m3 surface tension, N/m shear stress, Pa REFERENCES
Acharya, A., Mashelkar, R. A. and Ulbrecht, J. J.. 1978, Bubble formation in non-Newtonian
Chem. Fundam. 17, 230-232.
liquids. Ind. Engng
93
liquids
Ghosh, A. K. and Ulbrecht, I. J., 1989, Bubble formation from a sparger in polymer solutions-I. Stagnant liquid. Chem. Eagng Sci. 44.957-968. Kumar, 8. and Kuloor, N. R., 1970, The formation of bubbles and drops. Adv. Chem. Engng 8, 255. Miyahara, T., Wang, W.-H. and Takahashi, T., 1988,Bubble formation at a submerged orifice in non-Newtonian and highly viscous Newtonian liquids. J. them. Engng Japan 21,620426. Nakagawa,
T., 1952, Spinnability of liquid. A viscoelastic
state. II. Spinnability and viscoelastic property. Bull. them. Sot. Japan 25,93-97. Park, Y., Tyler, A. L. and de Nevers, N., 1977,The chamber orifice interaction in the formation of bubbles. Chem. Engng Sci. 32,907-916. Pinczcwski, W. V., 1981, The formation and growth of bubbles at a submerged orifice. Chem. Engng Sci. 36,
405411. Riibiger, N. and Vogelpohl. A., 1986, Encyclopedia Mechanics, Bubble Formation Newtonian and Non-Newtonian Houston, TX.
o/Fluid and its Movement in Liquids, pp. 59-88. Gulf,
Terasaka, K. and Tsuge, H., 1990, Bubble formation at a single orifice in highly viscous liquids. J. them. Engng Japan 23, 160-165.