Gas Dispersion Phenomena and Bubble Motion in Agitated … · Gas Dispersion Phenomena and ... Fig. 4.1-2 The plot of Pg/Po vs.. aeration number ... One of the shortcomings of the

MULTIPLE IMPELLER GAS-LIQUID CONTACTORS48

4Gas Dispersion Phenomena and

Bubble Motion in Agitated VesselsThe performance of gas dispersion determines the bubble size and exerts a great

influence to the mass transfer rate within the gas-liquid contactor, which severely affects the

reaction rate and product yield as a result. However, the mechanism of gas dispersion by

impeller in a mechanically agitated vessel was quite ambiguous until the hydrodynamics of

liquid flow behind the blade has been examined in late sixties. In this chapter, the gas

dispersion phenomena of different impellers and the bubble motion within the single and

multiple impeller systems are discussed in detail to offer a guideline for the impeller

selection.

4.1 States of gas dispersion and their transitions within stirred vessel

From the visual observation, Nienow et al. (1977, 1985) delineated the five states of gas

dispersion within the single Rushton gas-liquid stirred vessel as shown in Fig. 4.1-1 and

categolized them as: (a) no gas dispersion is achieved, large bubbles escape along along the

impeller axis; (b) sufficient dispersion for the upper part of the vessel to act as a bubble

column; (c) gas circulation can be seen in the upper part of the vessel with some movement

into the lower region; (d) circulation of bubbles occurs throughout the vessel; (e) at a still

higher impeller rotational speed, secondary circulation loop of bubbles appears at the location

close to the surface.

In the operation of gassed mechanically stirred tanks, it has long been found that the

flooding phenomenon occurs when the power input by agitation is too low to disperse a

certain gassing flow rate. Below the flooding point, gas dispersion becomes inefficient, so that

both the gas holdup and the gas-liquid mass transfer rate decrease sharply, which is an

undesirable condition and need to be avoided in gas-liquid mixing operations. Designers

usually need a specific set of data of the minimum impeller speed at which flooding

phenomenon is overcome.

Gas Dispersion Phenomena and Bubble Motion in Agitated Vessels 49

Fig. 4.1-1 The five characteristic stages of gas dispersion.(Nienow et al. 1977).

Various methods for estimating the flooding condition are summarized in Table 4.1-1. It

was after 1970, that more attention was paid to this subject when the mechanism of the

sparged gas dispersion was of primarily interest. It is interesting to compare the changes in the

definitions and the correlation equations before and after 1970. Before 1970, a minimum

impeller rotational speed was observed below which the impeller exerts no influence on the

transport properties such as the overall gas holdup, the specific gas-liquid interfacial area, and

the overall volumetric mass transfer coeffcient. The correlations for the minimum speed

presented in this period contain the effect of the physical properties (surface tension, density),

but not the gas sparged rate. On the contrary, the gas sparged rate has been one of the most

important factors to the correlations presented lately and the physical properties play no role

in the recent correlations. The most common method used recently to determine the flooding

condition is the impeller power consumption curve method developed by Nienow et al.

(1977).

Fig. 4.1-2 The plot of Pg/Po vs.. aeration number (Calderbank, 1958).


Table 4.1-1 Summary of impeller flooding condition for Rushton turbine impeller.

Author )(cmTTD Metho

dResults

Westerterpet Al. (1963)

14-90 0.2-0.7 SPAR( )( )4

1ρσgDTbaDNM += a=1.22 b=1.25, for turbine

a=2.25 b=0.68, for paddleDierendonck et al.

(1968)15.6-92.3 1/3 HOUP 25.1219.0 −= DTNM for pure liquid and T<1.0 m

( ) ( ) 241

21

2 −= DghTNM ρσ for pure liquid and T>1.0 m

( )41

2 ρσgTDN M−= for ionic solution

Smith et al. (1977) 44-183 1/3-1/2 CAVY21

99.0 DN F =Nienow et al.

(1977)29 1/3-2/3 PWRO

241

21

4 −= DTQN FGreaves and

Kobbacy (1981)20.3 0.375-2/3 SMEM 72.12.029.0 −= DTaQN F a=1.52, for pure liquid

a=1.66, for ionic solutionWiedmann et al.

(1980, 1981, 1982)45 1/3 PWAR 207.1283.0694.2 −= DQN F

Ismail et al. (1984) 40 1/3 PWRO 146.1

22

22

051.0⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎟⎠

⎞⎜⎜⎝

⎛=

inA

F

DDD

WDN

gDN

b

Nienow et al.(1985)

29-120 0.22-0.58 PWAR ( ) ( ) 35.331

40322. DTDQgN F =Lu and Chen

(1985)28.8 1/3 VELO 8.13.054.0 −= DQN F

Warmoeskerkenand Smith (1985)

44-120 0.4 MPRO ( ) 314941.0 DQgN F =Wong et al. (1986) 29 1/3 PWAR 8.13.075.0 −= DQN F

CAVY: large cavity unstable condition.HOUP: overall gas holdup.PWAR: power curve under constant impeller speed and varying gas rate.PWRO: power curve under constant gas rate and varying impeller speed.SPAR: specific gas-liquid interfacial area.SMEM: semi-empirical technique.VELO: volume averaged turbulent fluctuating velocity.MPRO: mini propeller.

It is not clear who was the first person introduced the word "FLOODING" to aerated

stirred tank operations, but we know that Calderbank (1958) first mentioned the word

"FLOODING" in his paper as shown in Fig.4.1-2.

Westerterp et al. (1963) measured the gas absorption rates as shown in Fig.4.1-3, using

reactions between a sulfite solution and absorbed oxygen and between a sodium hydroxide

solution and absorbed carbon dioxide in a gas-liquid contactor agitated with the turbine, the

paddle, and the propeller impellers of various ratio of tanks to impeller diameter. They found

that the influence of the impeller rotational speed on the specific absorption rate could be

divided into two regions. (a) A region without agitation effect-at very low impeller speeds the

specific absorption rate does not improve due to the stirring until a certain minimum speed is


surpassed. In this region the specific absorption rate depends on the gas load and the type of

sparger, but not on the impeller speed. (b) A region with agitation effect-as the impeller speed

surpasses the minimum speed, the specific absorption rate increases quickly and linearly with

increase in the impeller speed. They pointed out that at very low impeller speeds the gas is not

dispersed at all. In the neighborhood of the minimum impeller speed some effect was seen;

the amount of gas bubbles below the stirrer is still very low, and fine bubbles are only found

in the region above the impeller. Once the minimum speed has been surpassed, the dispersion

gets better and more homogeneous. In accordance with this result, they proposed the idea of

the “minimum impeller speed”. The minimum impeller speed was obtained by linear

extrapolating the specific absorption rate towards zero and finding the corresponding impeller

speed. From the result of the influence of the impeller diameter on the specific absorption rate,

it was found that at increasing impeller diameters the value of the minimum impeller speed

becomes lower. The “minimum agitation rate” was then defined as the product of the

minimum impeller speed and the impeller diameter. They presented the first correlation for

the minimum agitation rate as :

NoD = (A+BT/D)(gσ/ρ)0.25 (4.1-1)

where A=1.22 and B=1.25 for turbine impeller; A=2.25 and B=0.68 for paddle impeller.

Fig. 4.1-3 The effect of impeller speed on oxygen absorption into sulfite solution(Westerterp et al., 1963).

One of the shortcomings of the above correlation is that it does not take the influence of

the gas sparging rate on the minimum agitation rate into account. Van Dierendonck et al.

(1968) determined the minimum impeller speed from the measurement of the overall gas

holdup as shown in Fig. 4.1-4. They pointed out that the gas bubbles from a gas distributor


were dispersed and the overall gas holdup started to increase with impeller speed as the

impeller speed passed a certain value. They obtained the minimum impeller speed by linearly

extrapolating the overall gas holdup towards zero and proposed the following correlations.

(1) For pure liquid

NoD2 =0.07gT1.5 for T<1.0 (4.1-2)

NoD2 =2.0(H’T)0.5(gσ/ρ)0.25 for T>1.0 (4.1-3)

where H’ is the liquid height above the impeller.

(2) For electrolyte solutions

NoD = (aD+bT)(gσ/ρ)0.25 (4.1-4)

where a is very small figure and b is nearly equal to unity.

Fig. 4.1-4 The plot of N vs. gas holdup (Van Dierendonck et al., 1968).

Their correlations did not take into account the influence of the sparging rate on the

minimum impeller speed either. A strict definition of "IMPELLER FLOODING" first

appeared in Rushton and Bimbinets’ work in 1965. Their definition is one of the most

well-known ones and has been referred to very often by later investigators. They found that at

a given total power input, gas holdup increases with air flow rate up to a critical value beyond

which gas holdup decreases sharply and the sparged gas is not well distributed. At a constant

air rate, if (Pg/V) is decreased, larger and larger bubbles will be formed; the impeller is less

and less able to disperse the air throughout the tank properly. When (Pg/V) has decreased to a

critical value, the flow pattern changes suddenly; it is no longer a horizontal flow of gas

bubbles starting from the impeller and reaching the walls.There are no bubbles present in the

large space below and around the impeller. Very large bubbles rise to the surface along the

impeller axis. They stated this state as "FLOODED". They also pointed out the hysterisis

phenomenon of flooding. When the impeller speed is decreased from a high point, flooding


occurs at a lower (Pg/V) value below which dispersion is not likely; tthen the impeller speed is

increased from a low point, flooding occurs at a higher (Pg/V) value beyond which flooding is

also unlikely; Both critical (Pg/V) values were found to be proportional to (D/T) for (D/T)

between 0.26 and 0.44. The ratio of these two critical values was found to be 1.33. These two

critical (Pg/V) values increase with increasing air rate. It is really a pity that no correlation was

given in their paper.

The mechanisms of gas dispersion and the developments of various kinds of gas cavities

has been intensively studied by Bruijn et al.(1974) They found that large cavities can exist

only when the Froude number exceeds 0.1; otherwise, large bubbles will escape from the

cavities as a result of buoyancy forces. This result has been used by Vant Riet and

NF= (0.1g / D)0.5 (4.1-5)

No influence of air flow rate on the minimum speed is included in this correlation. By plotting

Pg/ Po vs. aeration number under constant air rate (Fig. 4.1-5) shows a minimum and a

maximum in the curves, and comparing the result of this plot with the observed phenomena,

Nienow et al.(1985) stated that the minimum point in the curves correspond very well with

the transition from state (d) to state (c) as shown in Fig.4.1-1. So they defined this transition

as the flooding condition and presented the following correlation.

NF= rQS0.5T0.25 D-2 (4.1-6)

This is the first correlation that includes the influence of air flow rate on the miminum

(flooding) speed.

Fig. 4.1-5 The plot of Pg/Po vs. aeration number for various gas flow rates (Nienow et al.1985).

Although Nienow et al. (1977) have depicted the five states of gas dispersions and

considered the definition of Rushton and Bimbinet to correspond to the transition between

state (c) and state (b), it is not easy to differentiate state (b) from state (c) in practice and in


many cases the differentiation between these two states is not possible. However, the

transition from state (a) to state (b) is always very obvious. In this respect, the definition of

impeller flooding given by Rushton and Bimbient may be considered to be equivalent to the

transition from state (b) to state (a), though Nienow et al. considered that to correspond to the

transition from state (c) to state (b).

Later, Nienow et al. (1985) (Fig.4.1-6) found that a plot of the ratio of gassed-to-

ungassed power against aeration number under constant impeller rotational speed shows a

step jump at the point at which flooding occurred. With this method, they measured the

flooding conditions and presented the following correlation.

NF= 0.322(QS/D4)0.333 (T/D)1.67 (4.1-7)

Fig. 4.1-6 The plot of Pg/Po vs. aeration number under a constant impeller speed(Nienow et al. 1985).

They also found that the flooding conditions are essentially identical for water and the

non-coalescing electrolyte solution. By comparing the results of Eq.4.1-6 and Eq.4.1-7, it can

be found that there exists a large deviation between them. This might simply suggest that the

impeller power consumption curve method is not very accurate for the determination of

flooding conditions. From the impeller power consumption curve as shown in Fig.4.1-7,

Greaves and Kobbacy (1981) divided the dispersion states into three regions of recirculation,

efficient mixing, and flooding, and defined an impeller dispersion efficiency to describe them.

The region encompassed by the minimum and the maximum in the plot of gassed-to-ungassed

power against aeration number under constant air rate is called the efficient dispersion zone

with unity impeller dispersion efficiency. To the right of the minimum is the flooding region

with an impeller dispersion efficiency less than unity, and the left of the maximum is the

recirculation region wnh an impeller dispersion efficiency greater than unity.


Fig. 4.1-7 The plot of Pg/Po vs. NA for various gassing rates (Greaves and Kobbacy,1981).

They derived semi-empirically the following correlation for the following condition. For

pure water,

NF= 1.52T0.2QS0.29D-1.72 (4.1-8)

For ionic solutions,

NF= 1.66T0.2QS0.29D-1.72 (4.1-9)

The impeller power consumption curve method was also employed by Ismail et al. (1984)

to determine the flooding condition They plotted the ratio of gassed-to-ungassed power

against the impeller speed, instead of against the aeration number and calculated the flooding

conditions from the maximum of the curves.

Fig. 4.1-8 Variation of average turbulent fluctuating velocity by changing gassing ratesunder a given impeller speed.


Lu and Chen (1986) were the first ones to attempt to measure the mean and the root-

mean-square (rms) velocities of the liquid in an aerated stirred tank using hot film

anemometry with a conical film probe. At constant rotational speed, the power consumpnon

per unit mass always decreases as the au flow rate increases. The volume average velocity,

<u’>, was found to have a curious relationship with P/V in an aerated agitated system. Figure

4.1-8 shows a dumping behavior occurred at some critical value of Pg/V for various rotational

speeds.

When they gradually reduced the air flow rate at a constant rotational speed, the Pg/V

gadually increased and the <u’> vs. Pg/V proceeded along the lower line. Further decrease in

the air flow rate to some critical value, the value of <u’> abruptly jumped from the lower line

to the upper line, and the transition of a poor dispersion of air in the system to a

well-dispersed status could be observed in this instance. The magnitude of critical Pg/V

increased as N increased. This is denoted by the vertical dash line in this figure. The figure

strongly suggests that at a given Pg/V, there exists a critical revolutional speed N, or an

appropriate aeration number to have good dispersion of air in the system.

They defined this jump as the flooding condition and gave a correlation as follows:

NF= QS0.3D-1.8 (4.1-10)

The use of the rms fluctuating velocity to determine the flooding condition is an effective

tool. It is not difficult to realize how the liquid velocity will change when the impeller is

flooded. All the previous investigations confirmed the fact that the impeller does not disperse

the sparged gas and eases to produce a radial liquid velocity to carry the sparged gas bubbles

to the vessel wall horizontally when flooding occurs. This implies that the impeller pumps

nothing out of it. Under this circumstance, it is reasonable to expect that there will exist a

drastic change in the liquid velocity when the impeller is operated in the vicinity of flooding

condition. On the other hand, the use of the volume averaged value of the liquid velocity may

be inappropriate because impeller flooding condition is, in essence, a feature pertaining to the

impeller itself, though it may influence the overall performance in the tank.

The use of the liquid velocity at locations close to the impeller periphery will show a

more pronounced response to the transition from flooding to dispersion. In this respect, Lu

and Ju (1987) improved the velocity measurement technique for air-water flow in aerated

stirred tanks and developed a calibration technique for a conical film probe. The result given

in as shown in Fig. 4.1-9 quantitatively illustrates the relation between the liquid velocities

and the impeller rotational speed under a constant air rate. The figure clearly shows the

hysterical phenomenon of flooding which has been pointed out by Rushton and Bimbinet.


Fig. 4.1-9 The hysterisis phenomenon of impeller discharge velocity between floodingstage and dispersion stage.

From the intersections of these two curves, two flooding conditions can be obtained; one

is the lowest limit of flooding below which dispersion is unlikely to happen; the other is the

highest limit of flooding beyond which flooding is also unlikely to occur. The flooding

conditions obtained with this method are plotted in Fig. 4.1-10 in which a comparison with

the published data is made. The data from thisresearch fall between those of Nienow and

coworkers (1977, 1985) and are in good agreement with those of Nienow et al. (1985). Fig.

4.1-11 shows the correlative relation between N and the number of the impeller blade. The

results of Figs. 4.1-10 and 4.1-11 are well represented by the following correlation.

NF= 0.066(T/D)3.5/3(QSD4)1/3(4.07+1.21nb-0.147nb2) (4.1-11)

Fig. 4.1-10 The correlation for the critical flooding condition for Rushton turbineimpeller.


Fig. 4.1-11 The correlation of the critical impeller flooding velocity interns of D, NA andnumber of blades.

Transition conditions determined by bubble size distribution against gas flow number

In examining bubble size distribution in aerated stirred vessels, Lu et al. (1993) found

that the transition rotational speed of impeller can be determined from the plots of Sauter

mean diameter of bubbles vs. Fl (aeration or gas flow number). The typical plots are shown in

Figs.4.1-12, 4.1-13 and 4.1-14, for the upper circulation region of the impeller. At a given QS,

32D first decreases to a minimum value as Fl decreases (that is, increases in N), then

increases to higher value as N continuously increases when the impellers have four and six

blades.

This fact demonstrates that if the value of N is smaller than that corresponding to the

minimum 32D , the negative pressure generated behind the blade is not strong enough to

capture all the sparged gas, and the pumping capacity of the impeller is not high enough to

recirculate of the dispersed bubbles, thus the most dispersed bubbles rise directly to the free

surface. Consequently, the bubble sizes at this stage are dominated by the dispersion ability of

the impeller, and the value of D32 decreases as N increases. As observed, this status of gas

dispersion is within stage (b) and stage (c) as shown in Fig.4.1-1. If the value of N is larger

than the value of N corresponding to the minimum 32D , the pumping capacity of the

impeller will reach a value to at which stronger circulation of both liquid and gas is generated,

whose flow pattern corresponds to stage (d) in Fig 4.1-1. Thus the size of the bubbles will

increase because the circulation of bubbles will enhance their coalescence. From the

description given here, it is seen that the value of the rotational speed corresponding to the

minimum 32D appearing in the plots of 32D vs. Fl is a transition speed at which the status

of gas dispersion of the system shifts from stage (c) to stage (d) and can be defined as NC. Due


to the circulation flow of fluid, the major control factor for the size of bubbles in this region

will be changed from gas dispersion by impeller to the degree of coalescence.

Fig. 4.1-12 32D vs. Fl for nb=4 in the upper circulation region at various gassing rates.




The values of NC defined here are quite different from the values of Ncd proposed by

Nienow et al. (1977) as shown in Table 4.1-2. The difference is probably caused by the

difference in layout of the impeller and design of the spargers. Under experimental conditions

in this study, the characteristics of gas dispersion of the eight-blade impeller system has

overpassed stage (c); i e. most data are beyond stage (d), and the curves of NC vs., Fl show a

different trend if they are compared with the plots of the four- And six-blade impellers. At a

given QS, 32D in smaller N or a larger value of Fl tends to increase to a maximum value,

then to decrease as N increases. Comparing this result with the relationship between Pg/Po and

Fl, it is noted that the maximum 32D is found near the other transition point, NR a

rectrculation point defined by Nienow et. al. (1977)(i.e. the transition region point from stage

(d) to stage (e) in Fig. 4.1-1), NR lines are also given in Figs 4.1-12 and 4.1-13. Therefore, the

change of 32D in this region at a given QS can be explained as that, even at the lower N, the

eight-blade impeller still pumps enough fluid to provide a circulatory flow, which can provide

32D up to the maximum value. When the increase of N goes beyond the maximum 32D or

the point of NR, part of the recycle bubbles are sucked into the impeller and redispersed by the

impellers which reduce 32D . Therefore, the mechanism of gas dispersion by the trailing

vortex again controls the size of bubbles.

Table 4.1-2 Comparison of transitional speed for gas dispersion stage (c) to stage (d) inFig.4.1-1.

smQg 3510× 3.33 5.05 7.60 10.00 12.50)(rpsNcd 1.383 2.25 2.75 3.17 3.53)(rpsNc 4.70 4.75 5.12 5.18 5.63

( ) ( ) 5.05.020.0: FrT

DFlNcd = defined by Nienow et al. 12)

:cN defined in this study

For the lower circulation region, Since bubbles scarcely appear in this region until the

status of gas dispersion reaches stage (c) shown in Fig. 4.1-1, only smaller bubbles can be

seen at lower N. To examine the role of recirculation in bubble-size distribution in this region,

plots of 32D vs. Fl determined for this region are shown in Fig.4.1-15. For a given QS and

lower N, the increase in N provides a stronger recirculation of fluids which enhances the

coalescence of the bubbles; thus 32D increases to a maximum value. Beyond this maximum

point, 32D will decrease because when N increases, the enhanced circulation flow will

promote redispersion of bubbles by the impeller. These trends coincide with the results of


Zhang et al.(1989). Therefore, the bubble sizes in this region are dominated be the strength of

the circulation of fluid, and NR corresponding to the maximum 32D can be seen as a

transitional speed of the gas dispersion from stage (d) and (e) in Fig. 4.1-1.

Fig. 4.1-15 32D vs. Fl for various impellers in the upper circulation region at variousgassing rates.

4.2 Transitions of ventilated cavities

4.2.1 Cavity structures behind the blade of the Rushton trubine impeller

The rotation of a disc turbine generates a pair of rolling vortices behind each blade

(Takeda and Hoshino,1966 Rennie and Valentin, 1968; Nienow and Wisdom, 1974; Van't

Riet and Smith, 1973, 1975). In low gas flow rates, these vortices suck in the sparged gas to

form gas cavities and then disperse the gas through their dispersive ends as small bubbles.

The cavities play a very important role in gas-phase mixing as well as in gasphase dispersion.

The cavities were classified into three categories: vortex cavity, clinging cavity, large cavity

as shown in Fig.4.2-1 by Bruijn et al. (1974) according to the shape and size of the cavities.

Fig. 4.2-1 Transition of cavities behind the blades (Bruijin et al. ,1974).


This figure also illustrates how these cavities related to power drawn by the impeller and

gas flow number. They pointed out that the number of large cavities goes from one through

two, three, .... to the number of blades as the sparged gas rate increases. However, recent

investigators such as Warmoeskerken et al. (1981, 1984, 1985a), Ismail et al. (1984), and

Nienow et al. (1985) argued that the sequence of cavity formation for a 6-blade turbine should

be as follows: (a) at a low aeration rate, six vortex cavities predominate; (b) clinging cavities

emerge as the aeration rate increases; (c) as the aeration rate increases further, there is a

transition from the 6-clinging cavity configuration to a 3-large-cavity 3-clinging-cavity

configuration with one large and one clinging in alternative arrangement; (d) continue to

increase in gas flow rate, the clinging cavities turn into large cavities, but there exist two sizes

of large cavities; (e) further increase aeration number, the neighboring cavities coalesces to

three ragged cavities. The course of these transition is shown in Fig.4.2-2.

(a) (b) (c) (d) (e)

Fig. 4.2-2 Sequence of cavity formation for a six-blade turbine impeller.

Under this 3-3 configuration as shown in Fig.4.2-2(c), a position change between large

cavities and clinging cavities occurs. In large cavities, gas loss from the cavities is associated

with a turbulent break up of the rear surface with little or no evidence of elongational spinning

flow that is so important behind vortex cavities.

Determination of Cavity Configuration

To examine the cavity configuration, a conical hot film probe was placed very close to

the impeller tip, and the sampling rate of the computer software program for digital data

acquisition was designed to synchronize with the impeller rotational speed for an adequate

duration of time. The results shown in Fig.4.2-3 were obtained for aeration number,

NA=(QS/ND3) form 0.022 to 0.102 at two impeller rotational speeds; 5 and 8.33 rev/s. The

data were poltted continuously form left to right and from up to down. Each trace (curve) in

these figures corresponds to one impeller revolution. The numbers specified on the traces

denote the serial number of the blade. Figure 4.2-3(a) indicates that one single large cavity

exists steadily at impeller no. l under the operating conditions of N=5 rev/s and NA=0.022.


(a) (b)

Fig. 4.2-3(a) The CTA signal at N=5 rps and NA=0.022 (one large cavity).Fig. 4.2-3(b) The CTA signal at N=8.33 rps and NA=0.04 (one or two large cavities).

These cavities does not shift its position to other blades. Figure 4.2-3(a) also indicates

that small cavities(as denoted by characters a and b) continuously change their position. Due

mainly to the absence of replotting the end point of the last trace as the starting point of the

following trace, and a minute difference between the digital sampling rate and the impeller

rotational speed, the angular position of (say) cavity no.l show a forward shift. The number of

large cavities under the operating conditions of N=8.33 rev/s and NA= 0.045 is one or two, as

shown in Fig.4.2-3(b). A position change for the large cavities can easily be seen in the results


of Fig.4.2-3(b). The configurations of two large cavities and the associated operating

conditions are shown in Figs.4.2-4(a) and 4.2-4(b).

(a) (b)

Fig. 4.2-4(a) The CTA signal at N=5 rps and NA=0 028 (two large cavities).Fig. 4.2-4(b) The CTA signal at N=8.33 rps and NA=0.05 (two or three large cavities).

At the lower rotational speed, the large cavities stay attached to blades no.2 and 5. At the

higher rotation speed, however, the large cavities change their resident positions. Fig.4.2-5(a)

indicates that a three large cavity configuration appears at NA=0.04 for N=5 rev/s. The

configuration of the cavities is one large cavity and one small cavity in alternate arrangement.


(a) (b)

Fig. 4.2-5(a) The CTA signal at N=5 rps and NA=0.04 (incipient three large cavities).Fig. 4.2-5(b) The CTA signal at N=8.33 rps and NA=0.058 (two or three large cavities).

No position change for the large cavities is observed. Figure 4.2-5(b) shows that the

number of large cavities is two or three at NA=0.058 and N=8.33 rev/s. The positions of the

large cavities are still not stable. At a still higher aeration rate, the configuration of the

cavities is three large cavities and three small cavities with one large and one small in

alternate arrangement as shown in Figs.4.2-6(a) and 4.2-6(b).


(a) (b)

Fig. 4.2-6(a) The CTA signal at N=5 rps and NA=0.098 (well developed three largecavities).

Fig. 4.2-6(b) The CTA signal at N=8.33 rps and NA= 0.102 (well developed three largecavities).

The large cavities do not change their resident positions, but the small cavities still

change their positions indefinitely. The above results indicate that situations of one large

cavity and two large cavities do exist. These large cavities do not shift their resident positions

at low rotational speed but they shift positions at high rotational speed before the most stable

configuration of three large cavities and three small cavities in alternate arrangement is

reached (for a 6-blade Rushton turbine). Once a three large cavities and three small cavities

condition is established, the large cavities and the small cavities are always arranged in an

alternate way. This arrangement is usually referred to as a stable 3-3 structure. The shift of


position of the large cavities in this structure was not observed at either rotational speed. The

conditions for various cavity configurations are summarized in Table 4.2-1.

Table 4.2-1 The formation condition of the various number of large cavities and theirconfiguration.

N=5 rev s-1 N=8.33 rev s-1No. ofcavities NA Position NA Position

0 <0.016 <0.0411 0.016-0.022 1 0.041-0.045 change2 0.022-0.028 1-5 0.045-0.05 change3 0.028-0.04 2-4-6 0.05-0.058 change3 0.04-0.098 1-3-5 0.058-0.102 1-3-5

It can be seen from the result of this table that the aeration number alone cannot dictate

the state of the cavity structure. The statement that a 3-3 configuration occurs at NA = 0.03

made by Warmoeskerken et al. (1981) is obviously specific to their equipment condition.

Approximation of Cavity Structure through CFD Simulation

Due to the complexity of the fluid hydrodynamics in stirred vessels, the experimental

determination of the flow field becomes a very difficult job. Although there are many

advantages coming with the LDA and it was adopted extensively to obtain the detailed

velocity distribution within stirred vessels (Shoots and Calabrese, 1995; Yianneskis et

a1.,1987; Lu and Yang,1998, etc), it is so time-consuming, which limits its application. The

commercial available software, such as ‘CFX 4.1’ (AEA technology company) could be

adopted to calculate the flow pattern, pressure distribution and some relevant turbulent

characteristics, i.e. the deformation rate and shear stress, by using a fine grid distribution

around the Rushton turbine impeller under an ungassed condition. The locus and

conformation of the trailing vortex were then determined according to the simulated results.

During the simulation, apart from the intrinsic boundary conditions, e.g. shaft, periodic plane,

free surface and mass transfer boundary, etc., the only extra-needed boundary condition is the

rotational speed, and the sliding grid facility was adopted to differentiate the impeller region

and the stationary part of the stirred vessel. Using the Reynolds’ Stress model to combine the

SIMPLE scheme to link the relationship between pressure and velocity, the flow field within

the stirred vessel is calculated and transferred over the interface between the impeller core and

the motionless part. The details of the simulation can be referred to Wu’s thesis (2000).

Van’t Riet and Smith (1973,1975) had pointed out that the low-pressure region comes

with the vortex zone, therefore the locus of the trailing vortex can be determined by

connecting the lowest pressure point at each azimuthal slice. Furthermore, since the steepest

pressure gradient exists at the interface between the vortex zone and non-vortex zone, the


conformation of the trailing vortex can be depicted along the pressure contour with the largest

gradient. The change in the vortex diameter can be also determined through the pressure

contour shape change at each azimuthal slice.

As seen in previous experimental works, although “the zero axial velocity connection

method” can depict the vortex locus reasonably, there are three main disadvantages coming

with this method. First, the criteria used for the zero axial velocity point judgment at each

azimuthal slice are different from person to person, which may induce the stretching

tangential angle of trailing vortex inconsistent from each other under the same operating

condition. Secondly, as pointed out by Van’t Riet and Smith (1973, 1975), an abrupt change

in direction appears to the vortex locus as it approaches the leading blade, which implies that

a finite axial velocity exists there. Therefore, it is not appropriate to depict the vortex locus by

using this method at the location close to the leading blade. Finally, nothing as far was proven

that this method could be used to depict the conformation of the trailing vortex. All of these

constraints limit the application of this method. To avoid the disadvantages described above, a

method named “the minimum pressure point connecting method” was proposed by Van’t

Riet & Smith, (1973 and 1975) was proposed and used to depict the locus and conformation

of the trailing vortex. The locus of the trailing vortex obtained in this study was compared

with others’ results depicted by using the zero axial velocity connection method.

With a lower sparged gas rate, gas is sucked into the negative-pressure zone behind the

leading blade and then dispersed completely, under which most of the dispersed bubbles will

circulate with the liquid flow. Under this circumstance, the structure of the trailing vortex is

less destroyed by the sparged gas and the structures of the vortex cavity and clinging cavity

can be deemed as close to the trailing vortex, which can be approximated by the single-phase

flow simulation.

1. Pressure distributions around the disk turbine impeller

Prior to depict the locus and conformation of the trailing vortex, it is necessary to acquire

the precise pressure distribution around the impeller. Choosing the vessel bottom as the

referential position and set the pressure as 1 atm (101,325 Pa). The pressure distribution

around the impeller was calculated precisely. Figures 4.2-7(a) and 4.2-7(b) show the top view

of the calculated pressure contour at Z*=2Z/W=0.63 and the pressure contour at the tangential

plane with θ=5° behind the leading blade with N=4.17rps under an ungassed condition,

respectively. From the picture shown in Fig. 4.2-7(a), it is clearly found that the negative

pressure region always appears behind the leading blade, which stretches backward and

outside and terminates at a certain tangential angle behind the leading blade. In the boundary


of the trailing vortex, the variation in the pressure value becomes erratically large, i.e. a steep

pressure gradient exists in the interface between the vortex and non-vortex zones. The value

of the pressure at the edge of the trailing vortex changes from a large negative value to a

positive value within a small displacement, which can be used to distinguish the

circumference of the trailing vortex as a result. In Fig. 4.2-7(b), it is seen that there are two

symmetrical negative pressure zones located at the upper and lower sides of the disk plate for

the Rushton turbine impeller. It indicates that a pair of the trailing vortex develops at the

backside of the leading blade for the Rushton turbine impeller.

(a) Top view

Fig. 4.2-7(a) Top view of the calculated pressure distribution for the Rushton turbineimpeller with N=4.17rps.

(b) Tangential plane

Fig. 4.2-7(b) The calculated pressure distribution at the tangential plane with 5o behindthe leading blade for the Rushton turbine impeller with N=4.17rps.


2. Vortex locus behind the blade of the impeller

Once the pressure distribution around the impeller was calculated, the locus of trailing

vortex between two neighboring blades can be determined using the minimum pressure point

connecting method. Figure 4.2-8 shows the top view of vortex locus for the Rushton turbine

impeller with N=4.17rps. The results after the LDA experiment of Lu & Yang (1998) and

Yianneskis et al. (1987) by using the zero axial velocity connecting method were also shown

in this figure for comparison. It can be seen that the vortex originates from the leading blade

initially and stretches back and outside the impeller, and finally terminates at about 40o behind

the leading blade. Comparing these three loci, it is found that the locus obtained in this study

is more confined and closer to the impeller center than those obtained by using the zero axial

velocity connecting method based on the LDA experimental results.

Fig. 4.2-8 Comparison of the top view of the vortex loci for the Rushton turbineimpeller obtained in this study and those by Lu & Yang (1998) andYianneskis et al. (1987) with N=4.17rps.

3. Vortex Conformation around the Rushton turbine impeller

Since the calculated values of pressure change from large to negative values to positive

values at interface between the vortex zone and non-vortex zone, the conformation of trailing

vortex can also delineated along the calculated pressure contour behind the leading blade. Fig.

4.2-9 shows an example of the vortex conformation depicted according to the calculated

pressure contour at each azimuthal slice around the Rushton turbine impeller with N=4.17 rps,

where the vortex tail was drawn based on the experimental observation. From the picture

shown in this figure, it is found that: (1) a pair of vortices clings to each leading blade, which

streches back and away from the impeller. They possess intense but opposite-direction rotary

motions, which may tear gas into small bubbles; (2) vortex develops close to the leading blade

and grows in diameter along vortex axis initially, where it changes direction from axial to


horizontal and sweeps outside the impeller. After passing the maximum diameter, the

diameter of the vortex becomes smaller and smaller along the vortex axis and finally breaks

into small eddies. Table 4.2-2 lists the variation of the vortex diameter along the vortex axis.

The trailing vortex grows in diameter within the range of 0o to 9o behind the leading blade.

After passing the largest diameter (2.01 cm at 9o for this case), the vortex shrinks and finally

disappears at 40o behind the leading blade.

Fig, 4.2-9 Conformation of the Trailing Vortex behind the Rushton Turbine blade withN=4.17rps.

Table 4.2-2 Variation of the vortex diameter along the vortex axis for the Smith turbineimpeller and Rushton turbine impeller with N=4.17 rps.

(1) (2) (3) (4)( )Rrr =* 0.5 0.5-0.7 0.93 1.15

θ 0° 1-3° 9° 40°

( )WZZ 2* = 0 1± 6.0± 4.0±

Vortex diameter 0.81 cm 1.20 cm 2.01 cm ---

As gassing rate reaches to VS=1.69x10-3m/s, the whole blade is embraced by large cavity,

and the cavity structure has changed into large cavity. And the intense rotary motion

disappeared.

4.2.2 Pressure distribution behind the blade of Smith turbine impeller

From the experimental observation, it was found that the cavity structure behind the

blade of the Smith turbine impeller changes gradually from the vortex cavity to clinging

cavity by decreasing the rotational speed and/or increasing the aeration rate. However, the

formation of the large cavity behind the blade of the Smith turbine impeller is still not seen

even with an extremely large aeration rate (N=3.33rps, QS=1.07vvm, i.e. NA=1.085), under

which the flooding condition had happened to the Rushton turbine impeller.


Comparison of pressure distributions behind the blades of different disk turbines

Figure 4.2-10 shows the calculated pressure contours for the Smith turbine impeller and

the Rushton turbine impeller behind the blade with θ=-3o and N=4.17rps under an ungassing

condition. There are two symmetrical negative pressure zones located at the upper and lower

sides of the disk plate for the Rushton turbine impeller. However, for the Smith turbine

impeller, apart from the single large negative pressure zone, two small negative pressure

zones locate at the upper and lower sides of the backside of leading blade. This result implies

that there is a pair of trailing vortex attaching to the leading blade of the Rushton turbine

impeller, while for the Smith turbine impeller, apart from the single main trailing vortex, there

are two extra small vortices cling to the leading blade.

(a) Rushton turbine impeller

(b) Smith turbine impeller

Fig. 4.2-10 The calculated pressure shaded contours for the Rushton turbine impellerand the Smith turbine impeller with N=4.17rps.

100

100

50

3030

50

0-200

300

300

-100300

-200-100 0

Unit: Pa

Unit: Pa300

300

100

50

50

3030

00

0

0

-200-50

-300

-300

100300


Loci and conformations of trailing vortices for disk turbine impellers

By adopting the method proposed by Lu et al, (2000a), the loci of trailing vortexes for

the Smith turbine impeller and the Rushton turbine impeller could be depicted by connecting

the minimum pressure point at each tangential slice. Figure 4.2-11 shows the azimuthal view

of the vortex loci for these two impellers with N=4.17rps. There is a pair of trailing vortex

developing close to the leading blade of the Rushton turbine impeller, which changes

direction from axial to horizontal and sweeps outside from the impeller and terminates at

about 40º after the leading blade. For the Smith turbine impeller, in addition to the single

main vortex, which terminates at about 55º after the leading blade, two small vortices cling to

upper and lower sides of backside of the leading blade and they disappear at about 5º after

commencement. The locus of the single main vortex initiates at the center of the disk and

stretches backward and outside in a completely horizontal way.

0O 10O 20O 30O 40O 50O 60O 0O 10O 20O 30O 40O 50O 60O

r

L e

Locus of main vortex

L x

C e

(a) Rushton turbine impeller (b) Smith turbine impelle

Fig. 4.2

Ba

can be d

the resu

impeller

from the

The var

compare

grows in

maximu

Smith t

Vortex locus

-1

s

r

l

i

d

m

u

eading blad

1 Azimuthal view of the loci of the N=4.17rps.

ed on the calculated pressure distribution,

awn by connecting the pressure contour w

ts are shown in Fig.4.2-12. It can be s

or the Rushton turbine impeller, the trail

impeller center and terminate at a certain

ations of vortex diameters along the v

and also shown in Table 4.2-2. For the R

diameter along vortex axis initially and b

diameter, and finally breaks into small

rbine impeller always has the largest

ocus of small vorte

tra

th

ith

een

ing

tan

ort

ush

eco

ed

dia

oncaved blad

iling vortexes for the impellers with

e conformations of the trailing vortexes

zero value at each tangential slice and

that no matter for the Smith turbine

vortexes stretch backward and outside

gential angle behind the leading blade.

ex axis of these two impellers were

ton turbine impeller, the trailing vortex

mes smaller and smaller after passing a

dies. However, the main vortex of the

meter close to the leading blade and


continuously shrinks along the vortex axis and breaks into small eddies at the vortex tail.

Comparing the vortex sizes for these two impellers, it is found that the main vortex of the

Smith turbine impeller has a smaller maximum vortex diameter than the Rushton impeller

with a given rotational speed.

Fig

4.2

var

Fig

gra

mu

Main vortex

Small vortex

r
(a) Rushton turbine impeller (b) Smith turbine impelle
. 4.2-12 Conformations of the trailing vortices for the Rushton turbine impeller andSmith turbine impeller with N=4.17rps.

.3 Mapping of ventilated cavities

By combining several relationship between rotational speed and gas flow rate for

ious cavity transition, Wannoeskerken et al.(1986) presented a flow map as shown in

.4.2-13. for the stirred vessel with flat blade disk turbine. This map enables an engineer to

sp the performance of gas dispersion. A very similar flow map as shown in Fig.4.2-14 for

ltiple impeller system was given by Smith et al. (1985).

Fig. 4.2-13 The flow map for the Rushton turbine (Warmoeskerken et al., 1986).


Fig. 4.2-14 The flow map for multi-turbines vessel of T/D=2.5 (Smith,1985).

4.3 Machanism of Gas Dispersion behind Impeller Blades

To discuss the gas dispersion mechanisms of various impellers, the values of

deformation rates, shear stresses and the bubble size distribution at different locations around

the cavities should be estimated.

4.3.1 Forces related to gas dispersion around the trailing Vortex of Rushton turbine

The deformation rate and shear stress play important roles in gas dispersion. For the

cylindrical coordinates, the deformation rate tensor can be written as (Lu and Yang, 1998):

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

∂∂

∂∂

+∂∂

∂∂

+∂∂

∂∂

+∂∂

⎟⎠⎞

⎜⎝⎛ +

∂∂

∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

+∂∂

∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

=∆

zV2 1

zV

1z

V r12 1

rr

z

Vz

V 1r

r 2

z

rz

θ

θθθ

θ

θ

θθθ

θ

zrz

zrr

rr

Vrz

Vz

V

Vrr

VVVrr

V

Vrr

Vr

V

(4.3-1)

and the average deformation rate at each location can be calculated as:

∆∆= :21δ (4.3-2)

With the determined locus and conformation of trailing vortex (approximate to the vortex

cavity or clinging cavity), the forces related to gas dispersion were calculated and plotted

along the vortex axis. The deformation rates were calculated at the edge and central line of

the trailing vortex with N=4.17rps and the results were shown in Fig. 4.3-1. In this figure, Zv

denotes the position coordinate along the cavity axis and Zc is the full length of cavity axis,


therefore the value of Zv/Zc is zero at the commencement of the vortex and equal to 1 at the

vortex tail (Van’t Riet and Smith, 1973). At the core of the trailing vortex, the deformation

rate increases monotonically along the vortex tail, while at edge of the vortex, it decreases

along with the vortex axis initially, after passing a minimum value at ZV/ZC=0.4, it increases

along with the vortex axis and gives a maximum value at vortex tail.

Fig. 4.3-1 Variation in deformation rate along the vortex axis at the core and edge ofthe trailing vortex with N=4.17rps.

Fig. 4.3-2 Variation in shear stress along the vortex axis at the core and edge of thetrailing vortex with N=4.17rps.

Similarly, the shear stress at each location was obtained by calculating the determinant


of the following shear stress tensor.

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎦⎤

⎢⎣⎡ ⋅∇−

∂∂

⎥⎦⎤

⎢⎣⎡

∂∂

+∂∂

⎥⎦⎤

⎢⎣⎡

∂∂

+∂∂

⎥⎦⎤

⎢⎣⎡

∂∂

+∂∂

⎥⎦⎤

⎢⎣⎡ ⋅∇−+

∂∂

⎥⎦

⎤⎢⎣

⎡∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

⎥⎦⎤

⎢⎣⎡

∂∂

+∂∂

⎥⎦

⎤⎢⎣

⎡∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

⎥⎦⎤

⎢⎣⎡ ⋅∇−

∂∂

=

)(32

zV2- 1

zV- -

1z

V- )(32)1(2- 1

r-

- 1r

- )(322-

z VVrz

Vz

V

Vr

Vr

VVr

Vrr

Vr

zV

zVV

rrVrV

rV

zrz

zrr

rzrr

µθ

µµ

θµ

θµ

θµ

µθ

µµ

τ

θ

θθθ

θ

(4-3-3)

And the results were compared for the circumference and central line of the trailing vortex as

shown in Fig. 4.3-2. It is found that (1) the largest shear stress always appears at vortex tail;

(2) a definite value of shear stress exists in the circumference of the vortex cavity, which

implies that the intense rotary motion happens to the vortex cavity or clinging cavity.

Yamamoto and Nishino (1992) had attempted to estimate the values of various dispersed

forces around the trailing vortex and the results were tabulated in Table 4.3-1 along with the

results obtained from this work. The measured mean bubble sizes at the circumference and

tail of the vortex cavity are also shown in this table for comparison. Both the results obtained

by Yamamoto and Nishino (1992) and this study show that: (1) the dispersing force resulting

from the turbulent eddy at the vortex tail was much larger than those provided by the rotary

motion of the vortex and the impeller blade itself; (2) the gas sucking force produced by the

negative pressure at the blade backside was the largest. Although all the results shown in this

table were obtained from the simulation for a single-phase systems, the structures of the

vortex and clinging cavities can be approximated. Therefore, it can be concluded that the

sparged gas was dispersed into small bubbles through two steps when the cavity structure

falls into the vortex or clinging cavities. First, gas is sucked into the vortex, and then

dispersed into small bubbles through the turbulent eddies around the vortex or turbulent eddy

at the vortex tail. Comparing the dispersed bubble sizes shown in this table, it was also found

that the bubbles appearing at the cavity tail result in a smaller mean bubble size.

Table 4.3-1 The values of various dispersed forces around the trailing vortex obtainedfrom this work and those proposed by Yamamoto and Nishino (1992).

Y. * & N. * This work This work32D

(300 rpm) (300 rpm) (250 rpm) (250 rpm)Impeller blade tip 0.45 mmH2O 0.96 mmH2O 0.42 mmH2O ---Vortex edge 0.089 mmH2O 0.49 mmH2O 0.47 mmH2O 1.107 mmVortex core 50 mmH2O 117 mmH2O 83 mmH2O ---Vortex tail 43.9-43.9 mmH2O 161.2 mmH2O 41.4 mmH2O 1.047 mm*Yamamoto and Nishino (1992)


4.3.2 Forces around the cavities for Smith turbine impeller

The deformation rate and shear stress around trailing vortex were approximated and

compared through the single-phase flow simulation for the Rushton turbine impeller and

Smith turbine impeller with Pg/V=557.8 W/m3 to examine the gas dispersion mechanism of

the vortex cavity. Figures 4.3-3 and 4.3-4 shows the variations in the deformation rate and the

shear stress along the vortex axis, respectively. For the Smith turbine impeller, the

deformation rate increases along with the vortex axis monotonically. However, for the

Rushton turbine impeller, it decreases along with the vortex axis initially. After passing a

minimum value at ZV/ZC=0.4, it increases along with the vortex axis and gives a maximum

value at vortex tail. Relating the deformation rate to the vortex diameter, it is interesting to

find that the deformation rate becomes smaller as the diameter of the vortex increases. For

both impellers, the largest deformation rate always appears at the vortex tail, which implies

that most of the entering gas may be dispersed there. With a given Pg/V, the Rushton turbine

impeller gives a little larger deformation rate at the vortex tail than the Smith turbine impeller,

therefore it may possess a better gas dispersion capability under an extremely low aeration

condition (i.e. vortex cavity). The high shear stress values at the trailing vortex edge as

shown in Fig.4.3-4 clearly point out the existence of the intense rotary motion of trailing

vortex.

0.00 0.20 0.40 0.60 0.80 1.00dimensionless distance along axis ZV/ZC

0.0E+0

2.5E+1

5.0E+1

7.5E+1

1.0E+2

1.3E+2

Def

orm

atio

n ra

te (1

/s)

QS=0 L/min

QS=1.07vvm

QS=0 L/min

QS=1.07 vvm

Smith turbine impeller

Rushton turbine impeller

Fig. 4.3-3 Variations in deformation rates in the cavity edges along vortex axis for theimpellers with Pg/V=557.8 W/m3 and different aeration rates.


Fig. 4.3-4 Variation in shear stresses in the cavity edges along vortex axis for theimpellers with Pg/V=557.8W/m3 and different aeration rates.

Adjusting the aeration rate QS to 1.07vvm and keep Pg/V as 557.8 W/m3, the cavity of

the Rushton turbine impeller has changed into the large cavity, while the vortex cavity still

clings to the Smith turbine impeller. The deformation rate and shear stress at the cavity edge

were calculated and the results are also shown in Fig.4.3-3 and Fig.4.3-4 to examine the gas

dispersion mechanisms of these two impellers under a higher aeration condition. For the

Rushton turbine impeller, once the cavity falls into the large cavity, the deformation rate

always increases smoothly with approaching the cavity tail and no minimum point exists in

the deformation rate curve as what is seen for the vortex cavity at ZV/ZC=0.4. Since the

maximum deformation rate at the tail of a large cavity is only approximately half of that for a

vortex cavity, the vortex cavity has a much better gas dispersion capability than the large

cavity. The large shear stress dose not appear at the periphery of a large cavity, which

indicates that the intense swirling motion does not exist for a large cavity any more and the

entering gas is dispersed at the tail of the large cavity only.

Comparing the deformation rate and shear stress for these two impellers with

QS=1.07vvm and Pg/V=557.8 W/m3, it is found that the Smith turbine impeller always gives

larger deformation rates and shear stresses. This fact demonstrates the better gas dispersion

capability of the Smith turbine impeller under a higher aeration condition. From the results

obtained here, one may conclude that no matter for the Smith turbine impeller or the Rushton

turbine impeller, the entering gas is mostly dispersed at either the tail and/or the

circumference of cavity through the larger shear force.

0.00 0.20 0.40 0.60 0.80 1.00dimensionless distance along axis ZV/ZC

0.0E+0

2.0E-7

4.0E-7

6.0E-7

8.0E-7

1.0E-6

1.2E-6

1.4E-6

1.6E-6

Shea

r stre

ss (k

g/m

s2 )


Smith turbine impeller


To examine the effect of the two small vortices appearing at the upper and lower edge of

leading blade of the Smith turbine impeller on the gas dispersion, the deformation rate and

shear stress around the small vortices are calculated under different operating conditions. It is

surprising to find out that the values of deformation rates and shear stress around the small

vortexes are guite large, even 30% higher than those at the tail of the cavity, which may

enhance the gas dispersion capability of the Smith turbine impeller.

4.3.3 Bubble size distributions around the cavity


To illustrate how the forces related to the gas dispersion behind the blade, the bubble

sizes and bubble size distribution were experimentally determined at various locations around

different type cavities.

Fig. 4.3-5 The schematic diagram of the experimental setup and the distribution ofmeasured points for the bubble size measurements.

(a) Vortex core (b) Vortex tail (c) Vortex edge

Fig. 4.3-6 Distributions of bubble size at various locations around vortex cavity for thestandard Rushton turbine impeller with N=4.17rps and QS=0.8 L/min.


(a) Vortex cavity

The sampling points for the bubble size measurements according to the simulation

results were shown in Fig. 4.3-5. Due to the limitation for locating the suction tube, the

measured locations are all confined to the regions outside the impeller. Figure 4.3-6 shows

the measured mean bubble sizes and bubble size distributions at different locations around

the trailing vortex with N=4.17rps and QS=0.8 L/min, under which the cavity structure falls

into a vortex cavity. From the plots shown in this figure, it can be seen that more small

bubbles were produced at the tail of the vortex cavity and larger bubbles at the vortex tail is

rare as compare to those appear at other locations, which is consistent with the results

discussed in the previous section. The sizes of the bubbles appearing at the vortex tail are

more concentrated around the mean bubble size and the distribution is narrower, i.e. the

variance for the frequency curve is smaller.

To examine how the power drawn by the impeller affects the gas dispersion, the mean

bubble size and the bubble size distribution at the vortex tail were determined under three

different rotational speed conditions, i.e. three different power input levels, and the results are

shown in Fig. 4.3-7. It can be seen that since the larger power drawn by the impeller provides

a stronger gas dispersed force, it produces a smaller mean bubble size and a narrower

distribution of bubble size. From the above results, one may conclude that the dispersed

forces at the tail of the vortex are the largest, if the cavity behind the impeller blade falls into

the vortex cavity or the clinging cavity.

(a) N=4.17 rps, Ps/V=506.4 W/m3 (b) N=3.83 rps, Ps/V=442.4 W/m3 (c) N=3.33 rps, Ps/V=378.4 W/m3

Fig. 4.3-7 The measured bubble size distributions at the vortex cavity tail with QS=1.5L/min under three different power input levels.

To examine the effectiveness of the gas dispersion at various locations around the vortex

cavity (as the positions shown in Fig. 4.3-9), the fractions of the total gas dispersed at various

locations are determined. These fractions are calculated through the product of the mean

bubble volume and the frequency of the bubble appearance within an enough long time

(exceed 2 hour) and then over it with the total amount of gas passing through the impeller


(sum of the sparged and recirculated gas). Where the gas recirculation rate can be estimated

by using the method proposed by Lu et al. (2000). From the results shown in Table 4.3-2, it is

found that more than 60% of the total gas is dispersed at the vortex tail and approximately

15% is dispersed in the circumference of the trailing vortex. However, almost no gas is

dispersed at the vortex core, where gas is sucked into. This result quantitatively confirms the

conclusion which it was made in above.

Table 4.3-2 Comparison of the fraction of the sparged gas dispersed at the center,circumference and tail of the vortex under N=4.17rps and QS=0.8 L/min.

Position Vortex tail Vortex edge Vortex core Other placesFraction of the dispersed gas [%] 63.5 15.2 0.01 19.3

(b) Large cavity

Keep the cavity structure as a large cavity and change the rotational speed of impeller

and aeration rate into 6.67rps and QS=20.0 L/min=1.07vvm to give the same energy

dissipation density Pg/V=577.8 W/m3 as the vortex cavity. The mean bubble size and bubble

size distribution at the tail of a large cavity was determined and compared with that produced

at the tail of a vortex cavity as shown in Fig. 4.3-8. From the plots shown in this figure, it can

be seen that a large cavity produces much larger mean bubble size than that given by a vortex

cavity with the same energy dissipation density. This fact indicates that the capability of gas

dispersion of a large cavity is much worse than that of a vortex cavity.

(a) Large cavity (b) Vortex cavity

Fig. 4.3-8 Comparison of the bubble size distribution at the tail of a large cavity and avortex cavity with Pg/V=557.8W/m3.

Similarly, the effectiveness of gas dispersion at various locations around a large cavity is

determined with a constant rotational speed as well as the same energy dissipation density as

the vortex cavity. The results are listed in Table 4.3-3 along with the gas dispersion fraction

obtained at the tail of a vortex cavity (N=4.17rps and QS=0.8L/min). It can be seen that no


matter under what operating conditions, much more gas is dispersed at the tail of a large

cavity than that for a vortex cavity, which is more than 80% as compare to 60% for a vortex

cavity. However, almost no gas is dispersed at the edge of a large cavity, which clearly

indicates the diminution of the intense swirling motion of a large cavity.

Table 4.3-3 The fraction of the sparged gas dispersed at the center, circumference andtail of a large cavity with N=4.17rps and QS=20.0L/min along with thatobtained at the tail of a vortex cavity with N=4.17rps and QS=0.8L/min.

Large cavity Vortex tailPosition Tail Edge Core Other placesOperating conditions 4.17

[rps]0.35

[vvm]

557.8[W/m3]

1.07[vvm]

4.17[rps]

557.8[W/m3]

4.17[rps]

557.8[W/m3]

4.17[rps]

557.8[W/m3]

4.17[rps]

Fraction ofthe dispersed gas [%]

82.5 83.1 0.09 0.08 0.03 0.02 16.3 15.9 63.5

Effect of blade number on bubble size around the impellers

According to the observed conclusion reached by Takahashi and Nienow (1992) and

Barigou (1992), the bubbles near the location (r1, z2) or (r1, z4) are freshly generated and

underisturbed, therefore the discussion here will also concentrate on the values of bubble

diameters of determined at these points. Fig.4.3-9(a) and Fig.4.3-9(b) shows the size of

Sauter mean diameter of bubble diameter of bubbles around the impellers having 4, 6 and 8

blade under the same rotational speed of 4.17 rps or under the power input per each blade

respectively. Among the impellers having 4, 6 and 8 blades, the impeller with four blades has

not only the strongest mean deformation rates and the largest turbulent kinetics enemy, it also

has the smallest value of Sauter mean bubble diameter under a low gassing rate condition.The

next is the impeller with six blades. The impeller having eight blades performs the worst in

both cases. It is noticed that the bubbles appearing in the upper vortex are always smaller

than the bubbles dispersed by the lower vortex. This fact shows that the diameter of the upper

trailing vortex is usually smaller- (The size of vortex can be seen from the plot of mean

velocity in azimuthal planes, more details are available in Yang’s thesis (1995))-due to the

existence of hub on the upper side of tile impeller disk. Here, the impeller with two blades is

excluded from the discussion because the status of gas dispersion is different and a more

coalescent effect is possible involved due to low coverage of the vortex (lower value of the

ratio of azimuthal angle of the vortex to the azimuthal angle between two blades) as well as

the smaller pumping rate which makes it very difficult to obtain a good comparison.


(a) (b)

Fig. 4.3-9(a) Comparison of 32D at r*=1.29 for various impellers with Qg=0.1L/min/blade and N=250 rpm.

(b) Comparison of D32 at *r =1.29 for various impellers with Qg=0.1L/min/blade and Pg=0.52 W/blade.

Comparison of the gas dispersion for different impellers

In accordance with the depicted conformations of the trailing vortices, the sizes of

dispersed bubbles at the core, edge and tail of the trailing vortices for the Rushton turbine

impeller and Smith turbine impeller were measured. Figure 4.3-10 shows the bubble size

distributions at the vortex tails around the Smith turbine impeller and Rushton turbine

impeller with QS=9.8L/min=0.5vvm and the same power input Pg/V=557.8 W/m3. Comparing

the plots shown in these two figures, one can find that the Rushton turbine impeller gives a

little smaller mean bubble size and result in a narrower bubble distribution about this mean

value. To accentuate the prevalence of the Smith turbine impeller under higher aeration

conditions, the bubble size and bubble size distributions for these two impellers were also

measured with QS=20L/min=1.07vvm and Pg/V=557.8 W/m3. Figure 4.3-11 compares the

measured bubble size distributions at the cavity tail for these two impellers with QS=1.07vvm

and Pg/V=557.8 W/m3. From the plots shown in this figure, one can see that the Smith turbine

impeller disperses gas more effectively under higher gassing rate condition and gives a much

smaller mean bubble size than the Rushton turbine impeller. The mean bubble size and

bubble size distribution around the small vortex attaching to the Smith turbine impeller are

also determined with QS=1.07vvm and Pg/V=557.8W/m3. It is found that the small vortexes

always result in much smaller bubbles ( mm95.0D32 = ) than those produced at other

locations. These results prove that the Smith turbine impeller possesses a better gas

dispersion capability and is proper to be used for the higher gassing condition.


(a)Rushton turbine impeller (b) Smith turbine impeller

Fig. 4.3-10 Distribution of bubble size at the tail of vortex for the Rushton turbineimpeller and Smith turbine impeller with QS=0.5vvm and Pg/V=557.8W/m3.

(a) Rushton turbine impeller (b) Smith turbine impeller

Fig. 4.3-11 Distribution of bubble size at the tail of vortex for the Rushton turbineimpeller and Smith turbine impeller with QS=1.07vvm and Pg/V=557.8W/m3.

The effectiveness of gas dispersion is quantified by determining the fraction of the total

gas dispersed at various locations around the ventilated cavities for these two impellers

through the method proposed by Lu et al. (2000a) and the results are listed in Table 4.3-4 for

different operating conditions. Under a gas completely dispersed condition (QS=0.5vvm), no

matter for the Smith turbine impeller or the Rushton turbine impeller, more than 60% and

0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7

D32 (mm)

0

5

10

15

20

25

30

35

40

45

50

Bubb

le n

umbe

r Nb

D32=1.05mm

0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7

D32 (mm)

0

5

10

15

20

25

30

35

40

45

50

Bubb

le n

umbe

r Nb

D32=1.09mm

0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7

D32 (mm)

0

5

10

15

20

25

30

35

40

45

50

Bubb

le n

umbe

r Nb

D32=2.01mm

0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7

D32 (mm)

0

5

10

15

20

25

30

35

40

45

50

55

Bubb

le n

umbe

r Nb

D32=1.13mm


about 15% of the total gas are dispersed at the tail and circumference of the vortex,

respectively. Increase the sparged gas rate to 20L/min=1.07vvm and keep Pg/V=557.8W/m3,

the fraction of the total gas dispersed at various locations around the cavities for these two

impellers are also measured and tabulated in Table 4.3-3. For the Rushton turbine impeller,

once the cavity structure changes into the large cavity, most of the total gas is dispersed at the

cavity tail and the dispersed bubbles become much larger as compare to the vortex cavity.

This result indicates that the gas dispersion capability of the vortex cavity is stronger. For the

Smith turbine impeller, in additional to the gas dispersed at the tail and edge of the main

vortex, approximately 5% of the total gas is also torn by the pair of the smaller vortexes,

which enhances the gas dispersion capability of the Smith turbine impeller.

Table 4.3-4 Fraction of the sparged gas dispersed at various locations around thecavities for different impellers with QS=0.5vvm or 1.07vvm and Pg/V=557.8W/m3.

Fraction of dispersed gas (%) Bubble size (mm)Operating

condition

Pg/V=557.8W/m3

Locations

ImpellerVortexedge

Vortexcore

Vortextail

Otherplaces

Smallvortex

Vortextail

Smallvortex

Rushton turbine 15.2 0.01 63.5 19.3 1.05QS=0.5vvmSmith turbine 13.4 0.02 60.3 21.2 5.12 1.09 0.92

Rushton turbine 0.08 0.02 83.1 15.9 2.01QS=1.07vvmSmith turbine 11.4 0.02 62.1 22.1 4.21 1.13 0.95

4.4 Motion and Coalescence of Small Bubbles in A Laboratory Scale Vessel

Method used for simulation of multi-phase flow in a stirred vessel can be classified into

four approaches. They are (1) the network of zone model,(2) the Material balance model, (3)

the Eulerian-Eulerian method, and (4) the Eulerian-Lagrangian method.

The network of zone model was proposed by Mann in 1977 to study a liquid flow

pattern in a stirred tank. In this model, a representative plane of the system is divided into

several zones to construct a network, according to its macroscopic flow pattern. Each small

region is considered to be a complete mixing system, and the transfer of momentum and/or

mass transfer between neighboring zones are calculated according to their flow. The model

was further extended to predict mixing time and local gas hold-up Mann (1982, 1988) and

suspended particle concentration by Brucato et al. (1988, 1990).

Applying the simulated liquid flow pattern, Bakker(1992) devised a material balance

model and a computer model (Gas Hold-up Simulation Tool) to predict local gas hold-up,

bubble size distribution, mass transfer rate and total gas hold-up. His simulated results agreed

quite well with experimental data.

32D


The Eulerian-Eulerian approach was used by Issa and Gosman (1981) to study the

motion of gas and liquid in a stirred vessel. In their study, only a simplified momentum

equation of gas was considered. By considering turbulent phenomena, Looney et al. (1985)

estimated the locus of particle flow in a stirred vessel and had quite good results. Using the

Eulerian-Eulerian approach, it is possible to obtain the value of average velocity and the

turbulent characteristics of the dispersed phase in a stirred vessel system.

The Eulerian-Lagrangian method is the most popular approach used by researchers in

this field. Table 4.4-1 summarizes the results of studies using the E-L method in simulation

of multi-phase flow in stirred vessels. Patterson (1991) had discussed the effect of gas flow

on liquid flow, and has pointed out that liquid flow is not affected by the existence of gas

flow except when the gas flow rate is large. In this chapter, the E-L approach was adopted to

design a program to simulate the motion of small bubbles in a laboratory scale mechanically

stirred vessel with multiple impellers. The simple mechanism for coalescence and break-up

of bubbles is also considered in this simulation

Table4.4-1 Summary of results and boundary conditions in different E-L models.Zeitlin(1972) Ambeganontar(

1977)Fort et al.

(1986)Hutchings(1989

)Patterson(1991)

Dimension 2D 2D 2D 3D 3DNo. of dispersedparticle phase

Approximately1000

Multi-particles Multi-bubbles Single-bubblec=T/3

Multi-bubbles

No. and locationof impellers

Single stageturbine c=T/3


Single stageturbine or A315c=T/3



Liquid flowimpeller

Tip velocity Seven regionanalytical solid

Eight zoneanalytical solid

Obtained byusing fluent

Obtained byusing fluent

Decision ofparticle velocity

See**

Coalescence Considered * Neglected Neglected NeglectedBreak up Considered * Neglected Neglected NeglectedFeed rate ofdispersed phase

Fixed gas hold-up

* Fixed gas flowrate

Fixed gas flowrate

Termination ofcalculation

Gas hold-up notchange

* Gas hold-up notchange

Objective results Local particleconcentration

Local particleconcentration

Local gas hold-up

Bubble motion Bubble motion

**:

Method to calculate Up

Zeitlen and Ambanontar Up =Uf +Ut, if random number ≧ U'/UUp =Ur, if random number ≦U'/U

Fort et al. Up =Uf +Ubax+Ur

Hutchings and Paterson Integrating force balance equation with respect to time


4.4.1 Equation for motion of bubbles

For simplicity, it is assumed that the concentration of the bubbles in the mixing tank in

small. It can be assumed that the flow of the liquid is not affected by the presence of bubbles

and that interaction among bubbles is negligible. In addition, neither heat transfer nor mass

transfer is considered. Then, the motion of bubbles can be computed by considering a force

balance on the bubbles; i.e., the mass acceleration product is equal to the sum of the forces

acting on the bubble. The various forces acting on a bubble are as follows: (1) drag force, (2)

pressure gradient force, (3) buoyancy force, (4) added mass force, (5) Basset history force, (6)

lift force, and (7) wall force presented by Antal et al. (1991). Then, the equation of motion for

a bubble having mass M moving at velocity Vb, in a liquid with velocity UL can be expressed

as follows (Chiu, 1994):

)(''

)()(1)()23(

)()()()()43(

021

2

'0

2

yRCC

RUm

UVUmCdtttdt

VUdCdm

dtdV

dtdUmCgmMU

DtDUmUVVVC

dm

dtdVM

bWW

bLbLL

tbLL

n

bLVLL

LLbLbd

b

++×∇⋅−+−

−+

−+−+∇−+−−−=

∫πν

ν

(4.4-1)

where Cd, the drag coefficient, is estimated from universal correlations of the bubble rise

speed in a large extent of liquid following Wallis (1974). The liquid velocity is given by UL =

uL+u’L, where uL is obtained from the mean flow of liquid calculated by FLUENT, u’L is

chosen randomly from an isotropic Gaussian distribution with mean square deviation 2κ/3,

and κis the turbulent kinetic energy of the liquid estimated by FLUENT. The bubble

location at any instant of time is given by dXb/dt =Vb.

4.4.2 Bubble diameter in the pressure field

If it is assumed that the bubble is spherical and the mass and heat transfer between the

two phases are not considered, then the change of bubble diameter can be considered in a

quasi-steady-state. The equation of the change of bubble diameter in terms of the pressure

difference, ambient static pressure of the liquid and the internal pressure of the bubble can be

given as (Chiu, 1994):

(p-pv) d3+4σd3-podo3=0 (4.4-2)

where the liquid ambient static pressure p can be estimated as a function of position, pv, po, do

denote the saturated vapor pressure, the initial values of the pressure in the bubble and the

initial bubble diameter, respectively.

4.4.3 Mechanism of bubble coalescence

The coalescence of a pair of bubbles was found to consist of the following three


different stages. (1) Two bubbles are close to each other but separated by a certain distance,

hj. (2) The film of liquid separating the two bubbles is drained until the liquid film reaches a

critical thickness, hj. (3) Rupture of the film of liquid leads to their coalescence. During the

drainage stage, the turbulent fluctuation, which drains the liquid film might also separate the

bubbles. Since not every collision between two bubbles will result in coalescence, it is

necessary to estimate the probability of coalescence to determine whether the collision will

result in coalescence. The probability of coalescence is estimated based on the film rupture,

which occurs if the interaction time (or contact time, τcont) exceeds the coalescence timeτ

cont is the time required for drainage from the film thickness hj to hf. The probability of

coalescence will be a function of the contact time and the coalescence time. Since the

function satisfying the probability of coalescence Pcoal has the form Pcoal=exp(-τcoal /τcont),

the value of Pcoal tends to zero for large value of the ratio (-τcoal /τcont) and to unity for small

ones. An equation to estimate the probability of bubble coalescence for an air-water system

has been given by Prince and Blanch (1990) as:

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=

3/1

3/2

2/13

)ln()16

(exp

ε

σ

ρ

ij

j

iij

coalR

hhR

P (4.4-3)

4.4.4 Mechanism of bubble breakup

The break-up of a bubble is caused by hydrodynamic stresses within the system. In

turbulent flow, bubble break-up occurs through interactions between bubbles, turbulent

eddies and when there is sufficient energy to cause rupture. Hinze (1995) suggested that a

bubble would break up if the ratio of the Weber number of the bubble to the Reynolds

stresses and surface tension forces exceeds a critical value. The critical Weber number can

lead to the prediction of the maximum bubble diameter, dmax = σWecrit / τ. An expression

for the Reynolds stress at the bubble surface, τL=ρLu-2 can be obtained by considering

Kolmogoroff’s local isotropic turbulence; here, u2 is the mean square velocity difference over

a distance equal to the maximum bubble diameter. Batchelor (1951) has given an expression

to determine u2 by the rate of energy dissipation per unit mass εas u2=2(εdmax)2/3.

Combining these expressions results in the following equation for dmax:

4.06.06.0max )()

2( −= ε

ρσ

L

critl

Wed (4.4-4)


Equation (4.4-4) can be used to estimate the maximum bubble diameter when the critical

Weber number is known. Since the eddies responsible for break-up have to contain sufficient

energy to cause rupture, not every collision between a bubble and eddy will result in bubble

break-up. Prince and Blanch (1990) provided an expression to estimate the probability of

break-up as:

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−= 3/2max )(2

)3.2(exp

d

RP b

breakε

σ

(4.4-5)

4.4.5 Description of the numerical simulation

Applying the equation of motion, coalescence, and break-up of bubbles described in the

previous section, a numerical program to simulate the 3-D motion of multi-bubble in a stirred

tank can be devised, and the procedure of the simulation is illustrated in Fig.4.4-1. Some

details of the simulation method are given as follows:

Fig. 4.4-1 Flow chart for simulation.


Input data

The calculation begins with input of operation conditions (the rotational velocity of the

impeller and the gas flow rate), geometry (four various agitated systems), the data of flow

field of the liquid (the mean and fluctuating velocities of the liquid, which variables are

assumed to not change when bubble are present), the physical properties of two phases, and

the model parameters of the equation of the bubble motion, bubble breakup and bubble

coalescence. In addition, the experimental data of the initial bubble diameter distribution near

the impeller tip are obtained from data of Lin (1994).

The initial value of bubbles

It has been shown that gas is supplied by a ring sparger beneath the lowest impeller and

is dispersed into bubbles by the Rushton impeller. In this model, the bubble size distribution

found by measurement (Lin, 1994) near the impeller region is adopted to determine the input

of new bubbles. A random procedure is employed to determine the initial size (in accordance

with prescribed distribution) and position of new bubbles added to the system in the region

near the impeller at each time increment. The bubbles are continually fed into the system

until the sum of their volumes reaches the value of the given gas flow rate and the time of one

iteration cycle. The initial velocity of a bubble is assumed to be the same as that of the liquid

at the same location.

Bubble motion

The trajectory of each bubble is computed by the equations of motion of the bubble (Eq.

4.4-1) using the Lagrangian approach. The equation of motion of the bubbles, which is an

ordinary differential equation, is divided into its radial, tangential and axial components of

bubble acceleration using a second-order finite-difference algorithm. The new velocity and

the position of the bubble can be numerically calculated by integration of the equations of the

acceleration components after each time increment since Eq. 4.4-2 is solved by use of the

New Raphson method to determine the bubble diameter at any point.

Bubble break-up

In this model, the bubble break-up mechanism varies according to the region in which it

is located.

1. Impeller region

The large bubble, which enters into the impeller region, is dispersed into many small

bubbles. The sizes of the small bubbles are determined by using of the experimental bubble

size distribution near the impeller region a random mechanism. The bubbles are generated

until the sum of the volumes of the small bubbles exceeds the volume of the larger bubble,


which was dispersed.

2. Bulk region

When the size of the bubble is near the maximum bubble size, a random procedure is

used to determine whether the bubble will break according to the probability of bubble break-

up. If bubble breakage occurs, the bubble is assumed to break into two small ones of equal

size. No breakage occurs when the size of the bubble is smaller than the maximum bubble

size.

Bubble coalescence

If a bubble collides with another bubble, a random procedure is used to determine the

probability of whether the collision will result in bubble coalescence.

Escape of bubbles

If a bubble moves through the liquid to the free surface, the bubble is assumed to leave

the system.

Output data

The program continues to calculate the bubble motion after another bubble is examined.

After the motion of all the bubbles at this iteration are examined, the program records the

results and decides whether to continue or terminate the simulations. If the program decides

to continue, it returns to the stage of input of new bubbles.

4.4.6 Simulated bubble motion and its effect on power consumption of the impellers

To examine the effect of the motion of bubbles within the system on the power

consumption of each impeller in multiple impeller systems, simulation of bubble motion was

carried out for four different geometrical systems as described in part I and shown in Table

4.4-2. Differential gas flow rates were also imposed to determine how the gas flow rate

would affect the bubble size, which is caused by the coalescence and break-up of the bubble.

In Fig. 4.4-2, variations of bubble size with time for various tank designs and various

flow rates are shown. The average size of bubbles increases with time due to coalescence of

bubbles. Comparing curves (4), (3) and (2) it can be seen that the gas flow rate clearly affects

the size of bubbles because the increase in the gas flow rate naturally enhances the chance of

coalescence of bubbles. From a comparison of curves (4) and (5) no significant difference in

average bubble size is observed between the systems of the dual and the triple Rushton

turbine. Although greater liquid depth and longer retention time should cause more

coalescence, the additional stage of impeller redisperses the sucked gas and break-up some of

larger bubble. Finally, it results in only a slight difference is seen in the average size of

bubbles. Comparing curves (3) and (6), the effect of the different flow patterns on the growth


of bubbles can be examined. The axial flow stream created by the A310 impeller for system 5

seems to increase the coalescence activity of bubbles and results in more large bubbles. In

this figure, it is also interesting to note that the bubble size in a single Rushton turbine system

reaches a flat value very quickly, which may be caused by the simple flow pattern of the

system.

Table 4.4-2 Six tank systems.Gas flow rateTank

systemImpellers (280

rpm) m3/sec l/min1 One Rushton 51005.0 −× 32 dual-Rushton 51005.0 −× 33 dual-Rushton 5100334.0 −× 24 dual-Rushton 5100167.0 −× 15 tri-Rushton 5100334.0 −× 26 A310 and

Rushton5100334.0 −× 2

Fig. 4.4-2 Variation of the average bubblediameters in the mixing tank vs.time for different systems.

In Fig. 4.4-3, the variation of total gas hold-up for various designs and gas flow rates are

shown. By comparing curves (2), (3) and (4) in this figure, the effect of gas flow rate on the

total gas hold-up is seen: The effect of the axial flow stream on gas hold-up seems less clear

from comparison of curves (3) and (6), although a somewhat larger gas hold-up is seen as

time elapses. Examining the variation of curves (1), (4) and (5), the effect of the stage of the

Rushton turbine can be found. Since the triple Rushton impeller system has the best

capability for extending the retention time of bubbles, it also results in a greater gas hold-up

as time elapse while the gas hold-up in a single Rushton impeller system reaches a flat value

in a very short time.

In Fig. 4.4-4, a comparison of the cumulative bubble size distribution for system 1-6 is

shown. The effect of the gas flow rate in the bubble size distribution can be seen by

comparing curves (2), (3) and (4) while the effect of the stage numbers can be examined by

comparing curves (4) and (5). There seems to be no significant difference in the size

distribution between the dual and triple Rushton turbine design. The smallest size distribution

of the bubbles found in both the dual and triple impeller systems implies that the multiple


Rushton impeller design can provide better gas dispersion than can any combination of axial

and radial flow impeller design.

Fig. 4.4-3 Variation of total the gas hold-up in the mixing tank vs. thetime for different systems.

Fig. 4.4-4 Comparison of the cumulativebubble diameter distribution indifferent systems.

Fig. 4.4-5 Spatial distribution of the local average bubble diameter for a dual-Rushtonimpellers system.


Fig. 4.4-5 illustrates the spatial distribution of local average bubble diameter for the

system with dual Rushton turbines. The simulated result clearly shows that the local bubble

diameter is heavily dependent on the local strength of the turbulent and flow pattern. The

smallest bubbles are found near the impeller discharge region and the center core of the

vessel. Larger bubbles are found near the wall and baffle plates due to the greater opportunity

for coalescence in these regions. The bubble size increases along the discharge stream

because the turbulent eddies of the discharge flow enhance coalescence.

In Fig. 4.4-6 comparisons between simulated mean bubble sizes at discharge region in

system 1 and system 2 are presented. Although the values are not quite the same, the order

and the magnitude are in acceptable deviation.System 1

System 2

Fig. 4.4-6 Comparison between simulated 32D and experimental 32D .

Fig. 4.4-7(a), 4.4-7(b) and 4.4-7(c) show examples of bubble distribution for system 2

(Dual Rushton Turbines) at three different vertical planes, i.e., at the 45° mid-plane, the

plane in front of the baffle (relative to the fluid flow direction) and the plane behind the baffle.

Figure 4.4-8(a) (b) and (c) show the similar picture for the system 5, which was triple

Rushton Turbine. These pictures clearly demonstrate that the bubble distribution of the tank


is far from homogeneous. Most bubbles were driven very quickly to the wall due to the high

radial velocity of the impeller discharge stream. From picture Fig. 4.4-7(b) and 4.4-8(b) it can

be seen that bubbles in front of the baffle plates tend to collide with each other, become

larger bubbles and rise along the baffle plates while bubbles at other planes tend to recirculate

along the fluid flow in the tank.

(a) (b) (c)

Fig. 4.4-7 Diagram of the bubbles in the stirred tank with a dual Rushton impellerssystem(gas flow rate =3 l /min, rotational speed =250 rpm).(a) Midway between the baffles(b) In front of the baffles(c) Behind the baffles

(a) (b) (c)

Fig. 4.4-8 Diagram of bubbles in a triple impeller system (Qg=1 l/min, N=250 rpm).


Table 4.4-3 summarizes the estimated results of the volumetric percentage of gas

(originally fed to the lower impeller through the sparger) passing through the upper impeller

as found by adding the volume of bubbles passing through the cross sector (r<5.5 cm,

z=26cm) of the upper impeller for the dual Rushton turbine system. The estimated value of

α is 25.5, 20.75 and 20.25% for gas flow rates of 3, 2 and 1 L/min, respectively. The

available experimental data (Lin, 1994) are listed for comparison. In the Table 4.4-4, the

same estimation for the triple-Rushton turbine system is shown along with available

experimental data. Since the gas flow adopted in the simulation was restricted to a very low

gas flow rate due to the capacity of the available computer, only the case of Qg=1 L/min was

simulated. The large deviation between the simulated result and experimental data is due to

the difference in gas flow rate. The increase in gas flow rate usually increase the chance of

gas recirculation, and results in a higher value of α.

Table 4.4-3 Comparison of the simulated and experimental value of α for dualturbine system (system 2,3,4).

Systemdesign

Original gasflow rate

Qg: m3/sec(Ql: l/min)

α(% of Qg)Simulated

result

Experimental

System 2System 3System 4

5105 −×51034.3 −×51067.1 −×

25.58%20.75%20.25%

27.0%*

*Lin (1994) Qg =3.3 l/min, N=250 rpm.

Table 4.4-4 Comparison of the simulated and experimental value of αfor a tripleturbine system (system 5).

System design Original gasflow rate

(Ql: l/min)

α(% of Qg)Simulated result

Experimental

Middle impellerUpper impeller

1.011.01

10~15%10~15%

41%*

33%*

*Lu and Yao (1992) Qg = sec1025 35 m−× (15 l/min), N=250 rpm.

From the results of the simulation discussed here, a model to simulate the bubble motion

in a 3-D multiple-impeller mixing tank is presented, and the model has been extended to

calculate the local properties such as bubble diameter and gas hold-up in a multiple-impeller

mixing tank. By summing the volume of bubbles passing through the upper impellers, the

fraction of gas flow to the originally feed has been estimated, and the simulated results agree

fairly well with the available experiment results.


NOTATION

B Baffle width [m]

D Impeller diameter [m]

Di Bubble size for each measurement point [m]

D32 Sauter mean bubble diameter [m]

Fl Gas flow number(=Qg/ND3) [-]

Fr Froud number [-]

g Gravitation factor [m/s2]

H Liquid level of stirred tank [m]

h0 Height of lower impeller from bottom [m]

k Dimensionless turbulent kinetic energy [-]

L Turbulence macroscale [m]

Lres Resultant turbulence scale [m]

N Impeller rotational speed [1/s]

NA Aeration number(=Q/ND3) [-]

Nc Transitional impeller rotating speed for stage(c) to stage(d) in Fig.3.2-1 [s-1]

NF Impeller rotating speed at flooding point [s-1]

No Minimum impeller speed which can disperse gas completely [s-1]

Nc Transitional impeller rotating speed for stage(d) to stage(f) in Fig.3.2-1 [s-1]

nb Baffle number [-]

nI Impeller number [-]

np Number of impeller blade [-]

P Power consumption [W]

P0 The ambient pressure for the cavity [kg/s2m2]

Pg Power consumption with aeration [kgm2/s3]

Po Power consumption without aeration [kgm2/s3]

QS Sparged gas flow rate [m3/s]

qi,l,g Liquid volumetric flow rate between cells in the I direction [m3/s]

r Radial coordinate [m]

r* Dimensionless radial coordinate(=2r/D) [-]

s Time [s]

T Tank diameter [m]

tM Mixing time [s]

U Rotating frame of reference mean velocity [m/s]

Ul,g Liquid phase velocity with aeration [m/s]

Ul,u Liquid phase velocity without aeration [m/s]

u’,v’ Fluctuation velocity [m/s]

V Liquid volume in the tank [m3]

v Stationary frame of reference mean velocity [m/s]

Vtip Impeller tip velocity [m/s]

W Mean tangential velocity [m/s]

w Impeller blade width [m]

Z Axial coordinate [m]


z* Dimensionless axial coordinate(=2z/w) [-]

<Greeks Letters>Δ Mean rate of deformation tensor [1/s]

α Defined in Eq.(6-1)(=nbB/T) [-]

β Exponent adopted in Eq.(6.1-2) [-]

δ Mean deformation rate [1/s]

ε Energy dispersion rate [m2/s3]

εg Local gas hold-up [-]

θ Shaft angular position [degree]

κ Turbulent kinetic energy [m2/s2]

ρ Density [kg/m3]

σ Surface tension [N/m]

ω Mean rotation rate [1/s]

<Subscripts>cd Completely dispersed condition

g Gassed condition

i Initial condition

l Liquid phase

Pro Projective

r,θ,z Radial, tangential, axial

u Ungassed condition

∞ Final condition

<Superscripts>* Nondimensional

- Average quantity

~ Instantaneous quantity

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