A Theoretical Model for the Size Prediction of Single Bubbles Formed under Liquid Cross-flow

FLUID FLOW AND TRANSPORT PHENOMENA Chinese Journal of Chemical Engineering, 18(5) 770ü776 (2010)

A Theoretical Model for the Size Prediction of Single Bubbles Formed under Liquid Cross-flow* LIU Changjun (ࡎ䮵ߑ)1, LIANG Bin (ặᯂ)1,**, TANG Shengwei (୆ⴑՕ)1, ZHANG Haiguang (ᕖ⎭ᒵ)1 and MIN Enze (䰫᚟⌳)2 1 2

Multi-phases Mass Transfer and Reaction Engineering Laboratory, College of Chemical Engineering, Sichuan University, Chengdu 610065, China SINOPEC CORP. Research Institute of Petroleum Processing, Beijing 100083, China

Abstract The size of initial bubbles is an important factor to thedeveloped bubble size distriEution in a gas-liquid contactor. A liquid cross-flow over a sparger can produce smaller bubbles, and hereby enhance the performance of contactor. A one stage model by balancing the forces acting on a growing bubble was developed to describe the formation of the bubble from an orifice exposed to liquid cross-flow. The prediction with this model agrees with the experimental data available in the literatures, and show that orifice size strongly affects the bubble size. It is showed that the shear-lift force, inertia force, surface tension force and buoyancy force are major forces, and a simplified mathematical model was developed, and the detachment bubble diameter can be predicted with accuracy of <±21%. Keywords bubble formation, liquid cross-flow, one stage model, force balance

1

INTRODUCTION

Bubble columns are widely used in industrial practice. The bubble size and its distribution are important design parameters of bubble column. Millies and Mewes [1] distinguished the column space into four different regions: (1) the primary bubble region, near the gas distributor. (2) the secondary bubble region where the primary bubbles break-up. (3) the region of dynamic equilibrium between coalescence and disruption of the bubbles. (4) the separation region, at the top of the liquid layer. Liang et al. [2] investigated the influence of spargers on the bubble size in a bubble column and found that the initial bubbles formed on the orifice opening of a conventional sparger are large bubbles, and then these bubbles break up into the final stabilized ones as they rise up. It means that the bubble size can not be reduced under maximum stabilized size. So, it is important to form bubbles smaller than the maximum stabilized size in the primary stage, and a microporous sparger can produce bubbles much smaller than the stabilizing size. The size of initial bubbles depends on the time of bubble adhered over the orifice and the gas flow rate in the orifice. Many researches [37] showed that a liquid cross-flow over the orifice can greatly reduce the initial bubble size. In cross-flow liquid, a drag force by the liquid flow results in early detachment of a forming bubble, and hence a smaller bubble size and a narrow size distribution [8]. The cross-flow not only accelerates the detaching of emerging bubbles, but also sweeps the bubbles away from the orifice region

and reduces the coalescence possibility. Although plenty of works on the bubble formation from a submerged single orifice in stagnant liquid are available in the literature, only a few considered the effect of liquid phase flow on the initial bubble size. Some theoretical models for bubble formation with liquid cross-flow also were developed [912], but most of them used empirical parameters. Kim et al. [13] developed a model for bubble and drop formation. They assumed that the vertical velocity of bubble center equals to the evolving speed of bubble diameter, and neglected the buoyancy of the bubble especially under normal gravity. Nahra and Kamotani [14] developed a two-dimensional one-stage model based on a global force balance and validated it with the experimental data on 0.3
MODEL DEVELOPMENT Physical model As shown in Fig. 1, during the formation of an

Received 2009-06-21, accepted 2010-05-21. * Supported by the National Natural Science Foundation of China (20736009). ** To whom correspondence should be addressed. E-mail: [email protected]

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bubble is 1 3 Sd  U L  Ug g (2) 6 which is vertically upward, and ȡg and ȡL are the density of gas and liquid phase, respectively. The gas is pushed into the liquid phase by the pressure difference of pg and pL at the opening of the orifice, and this can be viewed as an additional force Fp (pressure force): S 2 Fp d 0  pg  pL (3) 4 FB

Figure 1 bubble

The balance of the forces acting on a growing

initial bubble, air is pressurized into the liquid phase from an orifice and the bubble expands along with air flowing in. During such a process, the bubble diameter evolves, and the bubble center moves away from the orifice with the combined action of the bubble expansion and the shift of bubble base away from the orifice center. As the bubble moves away from the orifice, it connects with the opening of the orifice through an umbilical neck until the neck broken. The bubble is assumed to keep a spherical shape during the whole process [18]. As sketched in Fig. 1, for simplification, we also assumed that the bubble is formed independently without mutual-interaction with other ones. The vertical orifice has a diameter of d0 and the gas flow rate Qg flowing out from the orifice was assumed always constant. The liquid flow was assumed to flow over the orifice horizontally with a uniform velocity UL. Similar to the work of Nahra and Kamotani [14], we assumed that the bubble begins to drift away from the orifice axis when the diameter of the bubble hemisphere becomes equal to d0. A mathematical model can be developed by balancing the forces acting on the bubble to describe the subsequent process of bubble evolution. As the bubble growing, the velocity of the bubble centre in the vertical direction is denoted as dy/dt, and that in horizontal direction is dx/dt. The angle T tan 1 ( x(t ) / y (t )) . Time t 0 corresponds to the instant when the bubble hemisphere diameter equals to d0, and the terminal time for the bubble departure is the instant as the distance of bubble center to orifice center exceeds the sum of the diameters of bubble sphere (d) and the orifice (d0), then the umbilical cord breaks and the bubble detached from the orifice. As a consequence of constant gas flow rate, the bubble volume is then 1 VB Qg t  Sd 03 (1) 12 2.2

VB  U L  Ug g

Mathematical model

The effective buoyancy force (FB) acting on the

pg  pL

1 C02 Re

2 1 16 Ug Qg C02 S2 d 04

K

(4)

32 L0 Re d 0

(5)

4 Ug Qg

(6)

Sd 0 Pg

in which L0 is the length of the orifice, C0 is the orifice coefficient and K is constant which is experimentally determined by Marshall [19]. If the gas flow rate is slow enough the pressure force is often negligible [20]. When the liquid flows around the bubble, a shear-lift force FSL exists due to the velocity difference between the gas and liquid. The force can be expressed as S 2 U LU L2 d CL (7) 4 2 where CL is the shear-lift coefficient, and Legendre and Magnaudet [21] suggested the CL can be calculated with FSL

CL

1 1  16 ReB1 , 2 1  29 ReB1

5  ReB  500

(8)

where ReB is the Reynolds number based on bubble diameter, ReB U LU eff d / PL , and PL is the viscosity of liquid phase. The relative velocity, Ueff, of the bubble to liquid is 2

U eff

dx · § dy · § ¨U L  ¸  ¨ ¸ dt ¹ © dt ¹ ©

2

(9)

Nahra and Kamotani [14] compared the CL with the drag coefficient CD over a typical bubble Reynolds number range (5
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Chin. J. Chem. Eng., Vol. 18, No. 5, October 2010

5 m·s1). Surface tension force Fı is adopted in the literature [20, 22];

FV

Sd 0V

(11)

where ı is the surface tension coefficient. The surface tension force acts along the orifice rim, and anchors the bubble onto the orifice. It can be divided into two components, namely in the negative y-direction:

FV y

Sd 0V cos T

(12)

added mass coefficient used here is 11/16 [23]. In y-direction the component of inertia force is dª dy º Ug  CMC U L VB » (19)  « dt ¬ dt ¼ and in x-direction the component of inertia force is FIy

FIx

dª dx § dx ·º UgVB  CMCVB U L ¨  U L ¸ » « dt ¬ dt © dt ¹¼

Then the force balance in the x-direction is

and in the positive x-direction:

FV x

(13)

Drag force FD is due to the relative velocity between the bubble and liquid: U 2 S 2 (14) FD d CD L U eff 4 2 In a small Reynolds number range (ReBİ0.1), the drag force of a sphere is mainly due to the viscous effect. CD is the drag coefficient of a bubble moving through an infinite liquid phase. By assuming the solid sphere bubble model and neglecting its wall effects, the drag coefficient used in this model is given by [22]

CD

­ 24 ° Re °° B ® 18.5 ° Re0.6 ° B °¯0.44

for

ReB  2

for

2 İ ReB İ 500

for

500 İ ReB

(15)

2

2

(16)

and in x-direction the component of the drag force is, 2

U § S 2 dx · § dy · § dx · d CD L ¨ U L  ¸  ¨ ¸ ¨ U L  ¸ 4 2 © dt ¹ © dt ¹ © dt ¹ (17) Inertia force FI is also an important force acting on the bubble: FDx

FI

Substituting Eqs. (13), (17), (20) into Eq. (21), it reads 2

2

U § S 2 dx · § dx · § dx · d CD L ¨ U L  ¸  ¨ ¸ ¨ U L  ¸ 4 2 © dt ¹ © dt ¹ © dt ¹ dª dx § dx ·º UgVB  CMCVB U L ¨  U L ¸ » « dt ¬ dt © dt ¹¼ (22) After simplification and rearrangement, we obtain, Sd 0V sin T 

d2 x dt 2

1 UgVB  CMCVB U L

2 2 ªS U « d 2 CD L §¨ U L  dx ·¸  §¨ dx ·¸ §¨ U L  dx ·¸  2 © dt ¹ © dt ¹ © dt ¹ ¬« 4

º

dx

U § S 2 dx · § dy · dy d CD L ¨ U L  ¸  ¨ ¸ 4 2 © dt ¹ © dt ¹ dt 2

(21)

 Ug Qg  CMC ULQg dt   CMC ULQgU L  Sd0V sin T »¼

As the bubble velocity in the x-direction, dx/dt, is always smaller than the liquid velocity, the surrounding liquid pulls the bubble front away from the orifice. It means the drag force acts as a detaching force. In the y-direction the component of the drag force is, FDy

FV x  FIx

FDx

Sd 0V sin T

(20)

d§ dS · d ª § dS ·º  U L iˆ ¸ » (18) ¨ UgVB ¸  « U L CMCVB ¨ dt © dt ¹ dt ¬ © dt ¹¼

where dS / dt is the velocity of bubble center away from the origin located at the orifice center and CMC is the added mass coefficient. Eq. (18) shows that inertia consists of two parts. The first term is the inertia due to bubble mass, called bubble inertia. The second term occurs because the liquid is pushed by the accelerating and expanding bubble, called liquid inertia [22]. The

(23) Force balance in the y-direction is FV y  FDy  FIy

Fp  FSL  FM  FB

(24)

Substituting Eqs. (2), (3), (7), (10), (12), (16), (19), into Eq. (24) leads to 2

2

U § S dx · § dy · § dy · Sd 0V cos T  d 2CD L ¨ U L  ¸  ¨ ¸ ¨ ¸  4 2 © dt ¹ © dt ¹ © dt ¹ dª dy º Ug  CMC U L VB » VB  U L  Ug g   « dt ¬ dt ¼ U U2 S S 2 S d 0  pg  pL  d 2 CL L L  d02 UgU g2 4 4 2 4 (25) After simplification and rearrangement, we obtain d2 y dt 2

1  Ug  CMC UL VB

2 2 ­° S dx dy dy ® d 2 CD U L §¨ U L  ·¸  §¨ ·¸  dt ¹ © dt ¹ dt © °¯ 8 dy ª Qg  Ug  CMC U L  «VB  U L  Ug g  dt ¬

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Chin. J. Chem. Eng., Vol. 18, No. 5, October 2010

formation, we calculated the departure radii of the initial bubbles when gas velocity ranges from 1.50 m·s1 to 50.0 m·s1, liquid velocity ranges from 1 m·s1 to 5.10 m·s1, and the orifice diameters range from 0.100 mm to 1.00 mm (see Fig. 2). Results indicate that the liquid velocity (UL) has strong effect on the bubble departure radii, specially, for larger orifices. The bubble departure radii decrease with the increasing liquid velocity, and increase with the increasing orifice diameter (d0) and increasing gas velocity (Ug). In other words, smaller orifice, lower gas flow rate and higher liquid velocity will result in smaller bubble departure radii.

S 2 S º½ d 0 pg  pL  UgU g2  d 2CL U LU L2  Sd 0V cos T » ¾ 4 8 ¼¿ (26) Eqs. (23) and (26) are to be solved simultaneously using the fourth-order Runge Kutta method with the initial conditions: dx dy t 0, x 0; y 0 (27) dt dt The detachment time was defined as the instant as the distance between the bubble centre and the origin exceeds the sum of the diameters of the orifice and the bubble which is adopted from the experiments of Marshall [19], i.e.,





S 3

x 2  y 2 ı d  d0

(28)

RESULTS AND DISCUSSION

The model was used to predict the initial bubble diameters of water-air system and the results were compared with the limited experimental data available in Ref. [12] in Table 1. The bubble departure radii predicted from the model for air-water system agree well with the literature data, namely, within ±16% deviations. 3.1

Figure 2 Bubble departure radii under various Ug, UL and d0

Effects of operating parameters

Meanwhile, at low gas velocity (Ug), the initial size of bubbles formed from the orifice seems to be less sensitive to the orifice diameter. At higher gas

With the foregoing established model of bubble Table 1

Comparison between the predicted and experimental results Ref. [12]

This work

Run

d0/mm

UL/m·s1

Qg/ml·s-1

Rexp/mm

texp/ms

Rpre/mm

RSD/%

Rpre/mm

tpre/ms

RSD/%

1

0.25

1.20

0.57

0.65

2.3

0.70

7.23

0.55

1.21

15.74

2

0.25

1.20

1.11

0.82

1.6

0.69

16.34

0.74

1.50

10.26

3

0.25

1.20

1.74

0.97

3.6

0.74

24.23

0.89

1.70

8.15

4

0.25

0.60

1.11

1.06

4.9

ü

ü

1.01

3.90

4.58

5

0.35

1.20

1.10

0.85

2.6

0.83

2.94

0.77

1.71

9.91

6

0.35

1.20

2.23

1.10

2.8

0.83

24.91

1.02

2.01

7.09

7

0.35

1.20

1.72

1.00

2.4

0.83

17.40

0.92

1.91

7.84

8

0.35

2.41

2.22

0.85

1.7

0.62

26.59

0.75

0.81

11.43

9

0.35

0.60

2.52

1.47

4

1.05

28.78

1.49

5.50

1.39

10

0.50

1.20

1.72

1.04

3.2

1.08

4.13

0.95

2.12

8.20

11

0.50

1.20

2.26

1.15

4.6

1.08

6.09

1.08

2.31

6.36

12

0.50

1.20

3.29

1.31

3.2

1.08

17.48

1.27

2.61

3.03

13

0.50

2.40

2.27

0.89

1.5

0.83

6.63

0.82

1.01

7.96

14

1.00

1.20

2.27

1.24

3.1

1.93

55.56

1.22

3.32

1.96

15

1.00

1.20

5.28

1.67

3.8

1.88

12.63

1.71

3.95

2

3

3

3

5

2.25 2

Note: g 9.8 m·s , ȡL 1000 kg·m , ȡg 1.20 kg·m , ȝL 1.0087×10 Pa·s, ȝg 1.82×10 Pa·s, ı 7.275×10 N·m1; Rexp, experimental bubble radii; Rpre, predicted bubble radii; RSD, relative standard deviations of bubble radius; texp, bubble formation time: movie record; tpre, bubble formation time: model predicted.

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Chin. J. Chem. Eng., Vol. 18, No. 5, October 2010

flow rate, such dependence becomes more significant. As to the bubble formed under given liquid flow rate from a given orifice, the bubble diameter is proportional to the gas velocity. However, the departure radii of the initial bubbles are not reduced further by the much higher liquid velocity (see Fig. 3). This suggests that a critical liquid velocity exists, and beyond that point the departure radii is independent of liquid velocity. Fig. 4 illustrates the velocities of the bubble in both x and y axial directions (vx, vy) and the results show that the vx quickly reaches its maximum in 0.5 ms and then tends to stable velocity of UL. It means that the bubble quickly leaves the orifice. However, vy increases almost linearly with the time. When the liquid velocity higher than the critical value, it does not reduce the bubble diameter significantly.

Figure 5 The bubble radii evolving within the formation time (d0 1.00 mm, UL 1.2 m·s1, Qg 5.28 ml·s1) ƻ bubble radii

taches from the orifice. If the bubble detaches from the orifice in a time much longer than the acceleration period (about 0.5 ms), the gas volume flowed in during the 0.5 ms interval is a small part of the final bubble volume. The initial volume V0 normally contributes less than 1/9 to the total bubble volume. But the influence of V0 becomes more and more important when the departure time of bubble shortens with the increase of UL, especially, when the bubble detached sooner after the acceleration period 0.5 ms is over. 3.3 Force development during the bubble formation

Figure 3

Extended the prediction to much higher UL

The forces acting on the emerging bubble also evolves with the time. During the bubble formation, the attaching force in the x-direction is merely the x-component of surface tension force, and Fig. 6 shows it slightly decreases as the bubble evolves. The detaching forces include the x-component of inertia force and the drag force. The drag force in the x-direction is much small compared with the inertia force as shown in Fig. 7. The drag force in the x-direction sharply drops as the difference between the velocities of bubble and the liquid cross-flow becomes smaller within the acceleration period.

Figure 4 The bubble velocity comparison between x- and y-direction (d0 1.00 mm, UL 1.2 m·s1, Qg 5.28 ml·s1) ƶ velocity in y-direction;ͪvelocity in x-direction

3.2

Bubble movement during the bubble formation

During the bubble formation process, bubble moves in both x and y directions. Fig. 5 shows that bubble enlarges before it moves away from the orifice. Because the velocity of bubble center in x-direction is much higher than that in the y-direction, the detachment time of the new-born bubble mainly depends on the motion in the x-direction. From Fig. 4, we know that during the bubble formation, the bubble accelerates first and then flows with the liquid before it de-

Figure 6 The development of the inertia force and surface tension force in x-direction (d0 1.00 mm, UL 1.2 m·s1, Qg 5.28 ml·s1) ƾ inertia force in x-direction; ƶ surface tension force in x-direction

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Chin. J. Chem. Eng., Vol. 18, No. 5, October 2010

Figure 7 The development of the drag force in x-direction (d0 1.00 mm, UL 1.2 m·s1, Qg 5.28 ml·s1) ƻdrag force in x-direction

In y-direction, both the gas momentum force and pressure force are constant during the whole process (see Fig. 8) as the gas flow is constant and are relatively small while compared to the shear-lift force (see Fig. 9). Although the buoyant force becomes more and more important as the bubble evolves, Fig. 9 shows that in the y-direction the main detaching force is the shear-lift force and the main attaching force is the inertia force. Fig. 10 indicates that both the surface tension force and the drag force are negligible.

Figure 9 The development of the shear-lift force and inertia force in y-direction (d0 1.00 mm, UL 1.2 m·s1, Qg 5.28 ml·s1) ƻshear-lift force; ƶinertia force in y-direction (-FIy)

Figure 10 The development of the drag force and surface tension force in y-direction (d0 1.00 mm, UL 1.2 m·s1, Qg 5.28 ml·s1) ƻdrag force in y-direction; ƶsurface tension force in y-direction

In x-direction: FV x  FIx

0

(30)

Then the simplified model is written as: Figure 8 The development of the gas momentum force, buoyancy force and pressure force (d0 1.00 mm, UL 1.2 m·s1, Qg 5.28 ml·s1) + buoyance force; gas momentum force;ƶpressure force

3.4

3 CL U LU L2 Qg dy  U L  Ug g 4    (31) VB dt Ug  CMC U L  Ug  CMC U L d d2 x dt 2

Simplified model

Comparing the forces during the formation process, we can found that some forces are very small, such as, the drag force in the x-direction, which is only about one percent of the inertia force or the surface force. In y-direction, the sum of gas momentum force, pressure force, drag force and surface tension force is about three percents of the inertia force or the shear-lift force. So, simplification of the force balance model can be made by omitting the minor force items. In y-direction: FIy

d2 y dt 2

FSL  FB

(29)

Qg dx CMC U L QgU L  Sd 0V  VB dt  Ug  CMC UL VB

(32)

Eqs. (31) and (32) were solved simultaneously by the fourth-order Runge Kutta method with the same initial conditions aforementioned. The bubble radii were predicted by Eqs. (31) and (32) and compared with the experimental data in Table 1 (see Fig. 11). The simplified model can also predict the bubble radii well. The radii obtained from the simplified model are within ±21% deviation, which is slightly larger than those of the complete model. So, the simplified model is valid in the conditions we concerned, which can save the calculation time.

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Chin. J. Chem. Eng., Vol. 18, No. 5, October 2010 ı

surface tension coefficient, N·m1

exp g L pre x y

experimental value gas liquid model prediction x-direction y-direction

Subscripts

REFERENCES 1

Figure 11 The comparison of the predicted bubble radii with the experimental ones Ƶ simplified model;ƽoriginal model

2

4

3

CONCLUSIONS

By balancing forces acting on the growing bubbleˈwe obtained a one stage model to describe the process of bubble formation from the orifice exposed under a liquid cross-flow. And the model predicates the bubble diameter with an accuracy of ±16%. The orifice diameter has a strong effect on the bubble diameter. By considering only the major forces like shear-lift force, inertia force, surface tension force and buoyancy force, a simplified mathematical model was derived to predict the detachment diameter of bubble with an accuracy of ±21%. NOMENCLATURE CD CL CMC C0 d d0 FB FD FI FM Fp FSL Fı g K L0 p Qg R Re ReB S t Ueff Ug UL VB v ș μ ȡ

drag coefficient shear-lift coefficient added mass coefficient orifice coefficient bubble diameter, m orifice diameter, m buoyancy force, N drag force, N inertia force, N gas momentum force, N pressure force, N shear-lift force, N surface tension force, N acceleration of gravity, m·s2 constant in Eq. (5) orifice length, m pressure, Pa gas flow rate, m3·s1 bubble radius, m orifice Reynolds number bubble Reynolds number bubble translation distance, m bubble formation time, s relative velocity of bubble to liquid, m·s1 gas velocity, m·s1 liquid velocity, m·s1 bubble volume, m3 bubble velocity, m·s1 bubble translation angle (ș tan1(x(t)/y(t))) viscosity, Pa·s density, kg·m3

4 5 6

7

8

9 10

11 12

13

14

15

16 17

18

19 20 21 22

23

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