Pressure drop in gas-liquid-solid three-phase slug flow in vertical pipes

Pressure Drop in Gas-Liquid-Solid Three-Phase Slug Flow in Vertical Pipes

T. Sakaguchi H. Minagawa A. Tomiyama Department of Mechanical Engineering, Faculty of Engineering, Kobe University, Kobe 657, Japan

H. Shakutsui Department of Mechanical Engineering, Kobe City College of Technology, Kobe 651-21, Japan

• Experimental results related to the pressure drop in steady and fully developed gas-liquid-solid three-phase slug flows in vertical pipes are presented. The experiment was carried out in pipes of 20.9, 30.6, and 50.8 mm I.D. and with solid particles of 2.57 mm mean diameter. Air and water were used as the gas and liquid phases, respectively. An estimating method for the pressure drop in three-phase slug flow is proposed based on a model that divides a slug unit into six regions. A momentum equation is applied to each region and to each boundary between regions. The volumetric fraction and the velocity of each phase in each region are estimated, using the volume balance equations and some correlations. The pressure drop is estimated, using these values. The estimated results of the pressure drop are compared with the measured values, and it is confirmed that this method is useful in estimating such a pressure drop.

Keywords: pressure drop, gas-liquid-solid three-phase flow, slug flow, three-phase slug flow model, multiphase flow, vertical pipe

INTRODUCTION

Gas-liquid-solid three-phase flows in vertical pipes are encountered in air-lift pumps for mining manganese nod- ules, preheaters of coal liquefaction plants, and so on. To design and control such equipment efficiently, the values of pressure drops and volumetric fractions must be determined. However, few experimental data have been available [1-6]. Thus, in the present study, the pressure drop was measured in three-phase slug flows in vertical pipes in which large bubbles with liquid film around them and liquid slugs containing solid particles flowed alternately.

Most estimating methods for the pressure drop in three-phase flows were not derived by considering the flow situation but by modifying the existing methods for gas-liquid or liquid-solid two-phase flows [3, 6, 7]. In this study, an estimating method for the pressure drop is proposed that is based on a three-phase slug flow model. The estimated results are compared with measured values. It is confirmed by the comparison that this method is useful for estimating such a pressure drop.

EXPERIMENTAL APPARATUS AND PROCEDURE

The pressure drop in gas-liquid-solid three-phase slug flows was measured in 10 m long vertical pipes of 20.9, 30.6, and 50.4 mm I.D. made of acrylic resin. Aluminum ceramic particles of 2.57 mm mean diameter and 2380 kg /m 3 density were used as the solid phase. Water and air

at room temperature and atmospheric pressure were used as the liquid and gas phases, respectively.

A schematic diagram of the experimental apparatus is shown in Fig. 1. The air from a compressor was fed to the test section through a cooler, a filter, a regulator valve, and a critical flow nozzle. The flow rate of the air was controlled and measured, with the pressure at the upstream side of the critical flow nozzle calibrated in ad- vance. Solid particles that accumulated in the hopper of an electromagnetic feeder were supplied with water from a tank by Mohno pump (A). Additional water was supplied from the bottom by a second Mohno pump (B) to prevent the particles from settling and piling up on the bottom. Excess water was drained away by a third Mohno pump (C). The flow rate of the solid particles was controlled by the rotating speeds of the three pumps.

We had two measuring sections along the pipe. Two pressure taps connected with manometers were installed on each measuring section to measure the static pressure. The volumetric fraction of each phase in the measuring section was measured by the quick-closing valve method [5]. The flow rates of the solid and liquid phases were measured with graduated cylinders. That of the gas phase was corrected to the value corresponding to the static pressure at the centers of the two measuring sections, assuming that the gas phase isothermally experienced volume changes.

The experimental ranges were limited to the slug flow region, and the volumetric fluxes were (j~.) = 0.282-0.893 m/s , (J/) = 0.385-0.978 m/s, and (Js) = 0.00429-0.0635

Address correspondence to Dr. Tadashi Sakaguchi, Department of Mechanical Engineering, Faculty of Engineering, Kobe University, Rokkodai, Nada, Kobe 657, Japan.

Experimental Thermal and Fluid Science 1993; 7:49-60 © 1993 by Elsevier Science Publishing Co., Inc., 655 Avenue of the Americas, New York, NY 10010 0894-1777/93/$6.00

49

50 T. Sakaguchi et al

Figure 1. Experimental apparatus.

A

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C los ing Valve No,2 W

(2way) ~,L~:l =. A i r S u p p l y L l n e W W&S

Crlt l o a I F lo w N o z z l e

[ ~ R s g u l a t o r Va I,.,-, \ , . , - , / F i l t e r

Quiok C l o s i n g / Valve No.1 . Compressor'

rate~ Dre~ i Mohno --""~ '

w W:Water S : S o l i d

,~ ,c '(B) ~ A :A i r Water S u p p l y L i n e

-Solid & Water Supply L ine

L t L 0 L I 2 L23 L34 (mm) (ram) (mm) (mm) (mm)

D=20.9mm 9820 599 3853 2002 2020 D=30.6mm 9965 716 3841 2003 2002 D=50.4mm 9954 875 3706 2003 1995

m/s. Here, ( ) denotes the area-averaging operator [8] and the subscripts g, l, and s denote gas, liquid, and solid phase, respectively.

Generally the pressure drop of a multiphase flow consists of gravitational, accelerational, and frictional pressure drops. As for slug flow in pipes, a further pressure drop is known to occur at the tail of each large bubble or

in the mixing region at the nose of each liquid slug [9-15], which we henceforth refer to as the "bubble-tail pressure drop." The bubble-tail pressure drop has been estimated relative to the change in momentum, that is, the accelerational pressure drop between the regions upstream and downstream of the bubble tail [9-14] or the pressure drop due to the sudden enlargement of the pipe [15]. Thus, the

Pressure Drop in Gas-Liquid-Solid Slug Flow 51

total pressure gradient (dP/dz ) T for vertical slug flows is expressed by

Here, (dP/dz) H is the gravitational, (dP/dz) A the accelerational, and (dP/dz) F the frictional pressure gradient and (dP/dz) t is the pressure gradient due to the bubble-tail pressure drop.

As the flow situation in this experiment is steady and fully developed, the time-averaged value of (dP/dz) A for a slug unit becomes zero, as will be shown later. The value of (dP/dz) T was obtained by dividing the measured pressure difference between the two pressure taps by the distance between them. The value of (dP/dz) H was calculated from the measured volumetric fraction and the density of each phase as

= (Og( ag> + pl(at> + Os( ce=>)g (2)

where a and p are the volumetric fraction and density, respectively, and g is gravitational acceleration. There- fore, from Eq. (1), we can obtain the sum of the frictional and bubble-tail pressure gradients by subtracting (dP/dz) H from (dP/dz)T. This sum will be denoted by (dP/dz )rr When we obtained the values of (dP/dZ)Ft from Eq. (1), an evaluation of (dP/dz)n was performed, using the calculated value of the volumetric fraction of each phase through equations that were obtained by applying the method of least squares to the experimental results [5].

Experimental Uncertainty

The slug flows are characterized by pressure oscillation, which causes errors in the pressure difference measurement. We cannot afford to make the distance between two quick-closing valves long enough to capture many slug units. This induces errors in the volumetric fraction measurement. The inevitable random error introduced by the physical system, which includes the random error introduced by the instrumentation [16], was evaluated for each volumetric fraction and pressure gradient experimentally. Their relative values were 3.68%, 2.48%, and 10.6% for the volumetric fractions of the gas, liquid, and solid phases, respectively. On the other hand, the source of fixed error was not detected. Hence, the uncertainties estimated at the 95% confidence level were _+7.36%, _+4.96%, and + 21.2% of the observed values, respectively. Those of the pressure gradients were 0.492% for (dP/dz) r, 2.19% for (dP/dz) H, and 13.1% for (dP/dz)Ft. Therefore, their estimated uncertainties were _+0.984%, _+4.38%, and _+ 26.2% of the observed values, respectively.

EXPERIMENTAL RESULTS

The measured pressure gradients are shown in Figs. 2-6. The mean values of volumetric fluxes of the plotted data

are shown by (Ji> in the figures. Solid lines show the values estimated by the method described later. Bold lines show those for a gas-liquid two-phase slug flow. Dotted lines show the pressure gradients for gas and liquid

12.0

0.0

~ 8.0

v

~ 6.0

4.

O. O~ 0.0

I I I I i I I I I

= i l b ~ , ' I V ....

_ A.! . . . . . . . . . / .... " AI Liquid Single Phase Flow ~ , ~ v

_O :0(G-L TPF) ~ ~ _ J , ~ - ~ v :0. 011 ~ J ] l ~ " ~ , ' ~ i ~ . ,,i~1~¢"~ o

v :0. 019 A ~l ( .~ '~__,r l~J{ , -"J~ " :0 . 031 o

<>:0.043 v ~ / ~ -

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• :0(L-S TPF) d--2 57 mm o :0. 314 "- " 3

S m :0. 479 Os=2380 kg/m ~.~ o :0.652 Gas Single Phase Flow "T" /

I I I I I I I I I

1.0 2.0 <Jr> (m/s)

Figure 2. Relation between (dP/dz) r and (JT).

single-phase flow calculated with the Darcy-Weisbach equation, using the Blasius equation for the friction coefficient. The measured pressure gradients for the gas-liquid two-phase slug flow and a liquid-solid two- phase flow are also shown.

From Figs. 2-4, the characteristics of each pressure gradient under a constant value of ( J r ) are investigated. In these figures, the different symbol shapes--circle, tri-

angle, and so on--represent different values of (j=), and the differences within each shape--solid, vertical line, and

so on--correspond to differences in (jg). For instance, diamonds with a vertical line are used in Fig. 2 for

three-phase flow data of ( j= )= 0.043 m / s and (jg)= 0.479 m/s . From Fig. 2, we can see that all the values of (dP/dz) v increase with ( J r ) . Their rates of increase for the three-phase flow tend to increase with (Jr ) under constant values of ( jg) and (j=). The values of (dP/dz) r for liquid-solid two-phase flow are the largest, with those of the liquid single-phase flow, three-phase slug flow, and gas-liquid two-phase slug flow following in that order, and those of the gas single-phase flow are the smallest. Thus, the values of (dP/dz) T increased when solid particles were added into the liquid single-phase flow and decreased considerably when gas was added to the liquid-solid two-phase flow within this experimental range. From another point of view, (dP/dz)T increased steeply when liquid was added to the gas single-phase flow and increased further when solid particles were added to it.

The values of (dP/dz) H for the gas-liquid two-phase slug flow and the three-phase slug flow increased with (Jr) , and their rates of increase decreased with increasing (Jr ) under constant values of ( jg) and (j=), whereas the values of (dP/dz) H for the liquid-solid two-phase flow decreased, and their rates of decrease decreased with


- ~ 8 .

~ 0 N o

e" ,

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12,0 . . . . I . . . .

D=20. 9rm Aluminum Ceramic Particles - ds=2.57 me ps=2380 kg/m 3

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- < T , > ( m / s ) _ . -

0 _o :O(G-L TPF) ~ ~ - - - - ~ o A :0.011 -_.~v"'~'~ ::r,,~ ~ ' v :0.019 ~ x ~ m / O

-0 :0 .031 ~ ~ ~ _ ~ a

O - _ - ~

- (h> (ffs) • 7 ~" • O(L-S I P F I o - -

4. - o O. 314 @ 0. 479 ~_.,

0 . 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I I I I I I I l

0.0 1.0 2.0 < J ? (m/s)

Figure 3. Relation between (dP/dz). and (JT)"

increasing (iT) if the values of ( is ) were kept constant as shown in Fig. 3. The order of their values was identical with that for (dP/dz)T, because the values of (dP/dZ)Fl were very small compared with other pressure gradients. The reasons for these characteristics will be discussed later.

From Fig. 4, it is found that all the values of (dP/dz) m and their rates of increase increased with (JT) under constant values of (j~) and (j•,). Almost all the values of (dP/dz) m for the three-phase slug flow, gas-liquid two- phase slug flow, liquid-solid two-phase flow, and liquid single-phase flow were dispersed within narrow ranges. It was confirmed that the frictional pressure gradient of the liquid-solid two-phase flow was almost equal to that of the liquid single-phase flow under this flow condition [17]. However, the value of (dP/dZ)Ft for the three-phase flow were a little larger than those for the liquid single-phase flow when (jg) was small and a little smaller when (j~) was large. Therefore, by adding the solid into the liqmd single-phase flow, the values of (dP/dz)rt were kept almost constant. However, by adding the gas into the liquid-solid two-phase flow, the values of (dP/dz)Ft were first increased but then decreased. On the other hand, (dP/dZ)Ft increased when liquid was added into the gas single-phase flow and increased still further when solid particles were added.

Effects of the volumetric flux of each phase on the pressure gradient are now investigated for the case where the volumetric fluxes of the other phases were kept constant. All the pressure gradients increased with ( is ) as shown in Fig. 5. The rates of increase of (dP/dz) T and (dP/dz) H with respect to (Js) were large when (Jl) was small and decreased as (Jl) increased. The values of (dP/dz) r and (dP/dz)H decreased with increasing (j~,), but the effect of ( jg) o n (dP/dz)rt was not obvious in this experimental range.

1.5

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O. 005 -

I I I I I ' I I " I

- D:20.9ram <>\,v o ¢,~v ¢ ~ v9 - Aluminum Ceramic Particles\/'

ds=2.57 mm // 0s=2380 kg/m 3 / / /

<Tg> Ira/s) IJ/< • :OiL-S TPF) ~)/~ o :0. 314 /,,~¢,ll • :0. 479 /z~t?' -- e :0. 652 / / ~

( , / , ) - , - , /~ / .~ ' / 0 :O(G-L TPF)-

.v ~ :0. 043

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Gas Single Phase Flow

I Ii,..l,.-I . . . . . . . . . I . . . . . . . . . . . . . . . . . . I .... I l

0.0 1.0 <JT> (m/s)

Figure 4. Relation between (dP/dz) m and (j~-).

J

2 .0

All the pressure gradients also increased with (Jl) as shown in Fig. 6. The values of (dP/dz) m w e r e large for smaller D, whereas the effect of D on (dP/dz), was not obvious. As a result, the values of (dP/dz)T were slightly larger for smaller D.

Physical Interpretations

The values of ( a s) increased with (Js}, and those of ( a / ) increased with (Jl) when other flow conditions were kept constant [5]. Therefore, (dP/dz)H also increased with (j~} or (J/) because the density of the solid or liquid phase was greater than that of the gas phase. In contrast, the value of ( ag ) increased with (jg), and (dP/dz)H decreased. These are the main reasons for the increase of (dP/dz) T with (j.~) and (Jr) and for its increase with decreasing (jg) under a constant value of ( J r ) . The relationship between (dP/dz)Ft and ( j l ) or D was quali- tatively the same as that of the frictional pressure gradient in single-phase flow. However, it is difficult to determine why (dP/dz) m increased with (j.~) and why the effect of ( jg) on (dP/dz)Ft was not obvious• It must be possible to explain these facts by considering more precise structures of the three-phase slug flow such as the local velocity and local volumetric fraction of each phase and the pressure drops in each region and in each boundary region.

Therefore, three-phase slug flow models are presented in the following section to investigate more precise structures of the flow.

ESTIMATION OF PRESSURE DROP WITH THREE-PHASE SLUG FLOW MODEL

Three-Phase Slug Flow Model

We presented a three-phase slug flow model in an earlier paper [5] to estimate the volumetric fraction of the gas phase. This model is improved here to derive an estimate


10.0

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<j~>=O. 499 (m/s) D=20. 9m

ds=2. 57m

ps=2380kg~ 3

.Aluminum Ceramic Part ic les

m - /

O. 04

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<jL>=0. 703 (m/s)

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, I '0 I I , 0.0 0 .04' 0

<Ja> (m/s)

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<jjt>=O. 888 (m/s)

o : (dP/dz) T (kPa/m) • : (dP/dz) H (kPa/m}

: (dP/dz) Ft (kPa/m) m -

A , I , I , I ,

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Part ic les

0.0 0.01 0 0.01 <is> (m/s)

b

2.0

<Jo> (m/a) o :0. 358 a :0. 403 v :0. 540

o : (dP/dz) r (kPa/m) • : (de/dz) H (kPa/m)-

: (dP/dz) Ft (kPa/m)

0 O. 01 O. 02 Figure 5. Relation between (dP/dz) and <Js>. (a) D = 20.9 mm; (b) D = 50.4 m m .

of the pressure drop. Although the models proposed here are based on observations of the flow, they are formed by considering the applicability to various flow conditions of three-phase slug flows. Therefore, although some regions and some phases constituting a region were not confirmed by the observations, they may exist under other flow conditions. Hence, we must select the regions and phases corresponding to each flow when the model is applied.

The three-phase slug flow models proposed here are shown in Fig. 7. One slug unit consists of a liquid slug and a large bubble. For the case of Fig. 7a, the liquid slug is divided into four regions:

Region l - - T h e region just above the large bubble where the velocity and the phase distributions in the flow direction are affected by the large

bubble and the liquid contains only solid particles

Region 2--the region where the liquid contains only solid particles and the influence of the large bubble is negligible

Region 3-- the region that has both small bubbles and solid particles in the liquid phase

Region 4-- the region just below the large bubble. The core of this region contains a bubble swarm and solid particles. The cross-sectional area of the core is equal to that of the large bubble tail. The liquid film part of this region contains solid particles.

Under some flow conditions, bubbles are contained in regions 1 and 2. Therefore, we also consider the case of


s=:

N

e,~

v

10 ' I ' I ' I ' I ' I klueinum Ceramic Par t ic les

ds=2. 5 7 R os=2380kg/m 3

0 :D:20. gm 0 :(dP/dz)T zx :D:30. 6wn • : (dP/dz)H [] :D:50. 4wn ¢ : (dP/dz)Ft

<ig>=0. 5 2 + 5 % ( ~ s ) <is>=0. 01---.0. 02 (~s )

0.4 0.6 0.8 <j.~ > (m/s)

Figure 6. Relation between dP/dz and (j;).

Fig. 7b. In this case the liquid slug is divided into three regions because the difference between regions 2 and 3 disappears. Small bubbles can travel through the slug unit. Therefore, they can exist in region 1 and in the liquid film part of region 4.

The large bubble is divided into two regions:

Region 5- - the region where the thickness of the liquid film around the large bubble is approximately constant

Region 6-- the region near the nose of the large bubble where the thickness of the liquid film changes gradually and the velocities and phase distributions in the flow direction change.

Solid particles and /o r liquid drops can exist in the large bubble, although they were not observed in the experimental range [5]. For case b, both small bubbles and solid particles can exist in the film, whereas for case a, only solid particles exist in it.

Henceforth, L k, k = 1 ,2 , . . . ,6 , denotes the length of region k; V, k and aik denote the local velocity and volumetric fraction of phase i in region k. The subscripts c and f denote the core part and the liquid film part, respectively. For example, Vs5 f is the local velocity of the solid phase in the liquid film in region 5. The upstream and downstream ends of each region are denoted by - and +, respectively.

Derivation of Momentum Equations in Each Region

In order to develop a method to estimate the pressure drop based on the three-phase slug flow model described above, the following assumptions are made, referring to the flow situations in this experimental range [5].

Assumption 1. The flow is isothermal. There is no heat and mass transfer between phases.

6

5 "

+ o 0

3 o o " o 0 o •

+ • • e

2 • _ •

11

• 0 0

0 0 o •

6 +

(:1 c DI •

, 5 0

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...ir&.....Q.O_~_...~,... o: _ • 0 +

0 • © g

0 o o 3

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b

Figure 7. Three-phase slug flow model.

Assumption 2. The flow is fully developed and in the steady state. The length of each region does not change. Each region rises at the same velocity as the rising velocity of the large bubble V~.

Assumption 3. In each region, the local velocity and volumetric fraction of each phase are constant in the flow direction.

Assumption 4. At the tail of the large bubble, the bubble-tail pressure drop occurs.

Applying a steady-state momentum balance [18] and the above assumptions to region k and solving for the difference in pressure between the upstream and downstream ends of the region yields

= L k ( P ) k - - (P)k+ ~-z Fk (3)

+ g E [ Pi((aikc) + (aikf))Lk] i

where Y;i denotes Ei g,t,s. This formula holds only for regions 4, 5, and 6, which are divided into core and film parts. For regions 1, 2, and 3, (ai/,c) + (aik r) must be replaced with (aik). Thus, the pressure difference is expressed as the sum of the frictional and gravitational pressure drops.

The accelerational pressure difference, A P A, is expressed as the change of momentum between two adja-


cent regions. For example, ApA between regions 3 and 4, APA3_4, is given by

APA3-4 = E ( Pi°ti3Vi3(l/ii3 - Vb)) i

-- E {( Pi°li4cVii4c(Vi4c - Vb)) ( 4 ) i

-F'( PiOii4fVif4( Vi4f - Vb )) }

According to Assumption 3, the volumetric fraction and velocity of each phase change only at the boundary of each region. Hence, the pressure difference across the boundary is expressed by Eq. (5) if we also consider the bubble-tail pressure drop here,

( P ) 3 + - ( P ) 4 - = APA3-4 + AP, (5)

As described before, the bubble-tail pressure drop occurs at the bubble tail. With reference to the studies for the gas-liquid two-phase slug flows [11-14], the bubble-tail pressure drop A P t is estimated, in this study, by multiply- ing a coefficient s c by the absolute value of the accelerational pressure difference at the bubble tail, that is,

Ap, = ~:IAPA3_nl (6)

Solving the simultaneous equations (3) and equations for pressure difference across the boundaries such as Eq. (5) for ( ( P ) l - - (P}7) yields

(P)1 - (P)7-

= k=lZ ~Z Fk Lk "]- k=l ~ g~i (pi(Olik))Lk

(7)

Here, the subscript 7 refers to the first region of the slug unit going ahead. The terms of the acceleration pressure differences offset one another. Thus, the pressure drop for one slug unit consists of the frictional, gravitational, and bubble-tail pressure drops.

In actual slug flows, the lengths of regions, and the velocities and volumetric fractions of the phases in each part (core or film part) in each region are distributed statistically [19]. However, the time-averaged pressure drop for one slug unit can be evaluated by substituting the time-averaged values into Eq. (7). The time-averaged pressure gradient is also obtained by dividing the pressure drop by the time-averaged slug unit length. Hence, their time-averaged values are required.

Mean Velocities and Volumetric Fractions of Each Phase in Each Region

In this section, a method for estimating the mean velocity and the mean volumetric fraction in each part in each region will be developed by the volume balance in the slug flow model. In order to solve the simultaneous equations of volume balance, additional assumptions are made.

Assumption 5. Solid particles are distributed uniformly in the liquid phase in any region.

Assumption 6. The mean velocity of the gas phase remains constant in any region. That is to say, the rising velocities of the small gas bubbles in the liquid slug have the same value as that of the large bubble.

Assumption 7. In the core part of region 4, liquid and solid particles as well as small bubbles rise at the same velocity as that of the large bubble.

Assumption 8. The effects of the region around the nose of the large gas bubble on the pressure drop for the slug unit are assumed to be negligible.

Here, some regions and phases that constitute each region are deleted according to the observations of the three- phase slug flow [4, 5] in the experimental range described above. Region 2 was not recognized in this experimental range. Hence, model b is adopted. Solid particles and liquid droplets were not observed in any large bubble, and the amount of small bubbles in the liquid films was negligible. Hence, the values of atsc, %5(, ag4f, and agsf are set to zero. Moreover, the lengths of regions 1 and 6 are set to zero according to Assumption 8.

Now the slug flow model consists of three regions: 3, 4, and 5. The volume balance equations for the slug unit [20] are as follows. For each region,

E (Olik > = 1 (8) i

Z ( ( Olik ) Viik ) = ( JTk ) = ( JT ) ( 9 ) i

where ~ denotes the volumetric fraction-weighted mean velocity [8] of phase i.

For the slug unit,

5

Y'~ ( ( a i k ) L k ) k=3

5

Lk k=3

= ( o , i ) (I0)

5

E ((Olik~VikLk) k=3

5

Lk k=3

= <Ji) ( l l )

For each phase,

(Piik -- Vb ) ( aik ) = constant (12)

In addition, the volume balance for regions 4 and 5 with core and film parts is given by

(a ik ) = (aikc) + (~ik:) (13)

The sum of the volumetric fractions in the core part of region 4 is equal to that of the large bubble,

(ag4c ~ -I- (Ols4c~ -]- (Oll4c~ = (Olg5c ~ ( 1 4 )


Using Assumption 5,

( O/S31 1 - ( ag 31

< Ols4 c > < Ols4f > 1 - ( O/g4 > - < O/s4f > - ( O/14f > 1 - (O/gs,,)

( asS[ ) ( a s ) (15)

1 -- (o/g5c > 1 - (o/g)

Akagawa and Sakaguchi's [20] correlation for the void fraction in the liquid slug in gas-l iquid two-phase slug flows is extended to the three-phase slug flow as

(o /go ) / ( 1 - ( % L ) ) = [ (o /g ) / (1 - (c~,))] ~' (16)

where ( % L ) and ( a sL) are the mean volumetric fractions in the liquid slug and 3' is an empirical constant. From the volume balances for the liquid slug, ( % L ) and (%1~) can be expressed as

(OlgL) = (L3(Otg3) + L4(o/g4c)) / (L3 + L4)

(O/sL) = {L3(o/s31 q- L4((Ols4c)

+ < O/s4f))}/(L3 -~- Z4) (18)

As no correlations are available for the volumetric fraction of the gas phase in the bubble swarm part, we define the ratio of the volumetric fractions of the gas phase in regions 3 and 4 as

(Olg4c>/( Ogg3> = to (19)

The values of Y and to are discussed later. Now the relations for the mean velocities are found.

Assumptions 6 and 7 yield

Vg4c = Vgs, = Vg3 = Vl4c = Vs4,. = Vb (20)

V b is related to the liquid velocity V/3 just ahead of the large bubble [21] by

Vb = CoV13 + Vb ' (21)

where C O is a coefficient and Vb, is the terminal rising velocity of a large bubble in an imaginary stationary liquid-solid mixture. For two-phase flow, there are some correlations for C o and Vbt [5]. Here, these correlations are modified by considering the presence of solid particles in the liquid ahead of the large bubble, that is,

C o = 1.2 + 0.8{1 + [ (Re 3 - 600)/5851 '1 }

where Re 3 is the Reynolds number defined by

p/D(<o/g3)Vg 3 -]-< O/131V/3 )

1 ( 2 2 )

(23)

and

0.25 )

V m = 0.35 - [( B°°'5 _ 1.91/2.1212.67 + 1 (24)

× [ g D ( l _ <ce~3)) o.5 P t - P g ] ° S p '

where B o 3 is the Bond number defined by

B o 3 = ptgD2(1 _ ( O/s31)0"5/0 . (25)

/z t is the liquid viscosity, and ~ is the surface tension. The flow in region 3 can be regarded_approximately as a

three-phase bubbly flow. Therefore, V,3 is estimated by the drift-flux correlation of the solid velocity for the three-phase bubbly flow [22]. Thus,

Vs3 = C* ( ( O/s3>~s3 Jr ( 0Q3>~13 ) s3 1 -- (O/g3) -- ( j l )~us

where

(26)

<o/.)P~3 ) C~* 3 = m exp - 0.03((%3)P~3 + (o/t31~3) + 1.1 (27)

X I

(29) = - 1 2 8 + 23.6 D - 1.14

( j t )~vs = g~t [1 - ( % 3 1 / ( 1 - (c~g3))lV (30) [ ( o / , ~ } / ( 1 - (%~))]~

Y = a [ ( % 3 ) / ( l - ( % 3 ) ) ] ,.2 (31)

a = 34.9 - 6 . 5 3 (o/g3)p~ 3

(32)

- 0 . 4 5 7 ( ~ ) 1 1 " 7 + 1.31

( < O/g3 >~g3 )

- 0.797 d" + 0.732 D

• :¢ 1 ( j t ) sus = (J t ) sus( - (o/g3)) (34)

V~ is the terminal settling velocity of a single particle in a stationary liquid. It can be calculated by solving Eqs. (35)-(37) simultaneously.

~ , = {[4d~( p, - p t ) g ] / 3 C o p t } °5 (35)

R% = plVstds/i.zl (36)

24 C o = ~T-o (1 + 0.15 R% 0.687)

0.42 (37)

q 1 + 4.25 X 104 Re~ 1.16 ( f rom [23]) RE 3 m ( 1 _ (O/ . ) )0 .5


The mean lengths of the large bubble and the liquid slug are affected by the structure of the mixing section of the apparatus. Hence, these values cannot be predicted by the correlations obtained for other apparatus. Conse- quently, the lengths are calculated by using the correlations obtained based on their measured results in this study. They were measured with the aid of electrodes and a high-speed videotape recorder system. The time- averaged values of the lengths are correlated by

L 5 = a l ( j g ) / ( j r ) + b 1 (38)

L 3 @ L 4 = a 2 ( J g > / ( j T > + b 2 (39)

L4 = a3L5 (40)

The constants a I, a 2, b~, and b 2 are summarized in Table 1 for different combinations of D and d s. Here, the length of the bubble swarm region, L4, is assumed to be proportional to the large bubble length L 5. Of course, it may not be adequate if the large bubble is long enough for the film flow around it to be fully developed. Mao and Dukler [24], however, reported that the liquid film was not fully developed even if the large bubble length was about 1.4 m (27D) for their gas-liquid two-phase slug flow measurements. In this experimental range, the large bubble was not long enough either. It was confirmed by the measurement of liquid velocity in the film by laser-Dop- pler velocimetry that the liquid in the film is still acceler- ating toward the bubble tail, and therefore the liquid velocity at the bubble tail increased with L 5 in this range [25]. According to a study on the surface gas entrainment caused by a water jet [26], it was found that the length of entrainment is approximately proportional to the jet velocity in the low jet velocity region. The velocity of liquid jet flowing out of the film part and impinging into the liquid slug depends on L 5. Hence, L 4 also depends on Ls, and we assume Eq. (40) for this experimental range. The mean value of the L 4 / L 5 data measured using the VTR system was 0.284. This value is used for the value of a 3. The details of the experiments with respect to such slug characteristics are presented in another report [25].

The following correlation is developed for 3':

= p l ( J T ) D / I J , i ) - . 3' 350( " 0512 (41)

The value 2.6 is adopted for to. For these values, however, more investigation will be needed in the future.

There are 28 unknowns: 14 volumetric fractions, (ag) , (0/l>, (O/s) , (O/g3> , (0/13), (ORS3), <Olg4c>, (Ofl4C> , <OfS4C) , (OQ4f> , (Ols4f>, (O~go:.Sc >, (..Oll5f > , (Ols5f ) ; 11 mean velocities, v~3, v~3, v~3, v~.c, v.c, V~.c, v~4, Vs4r v~sc, v~sr, vs~I; and three lengths, L3, L4, L 5. There are also 28 independent relations: 13 by the volume balance equations (8)-(13); one by Eq. (14); three for the volumetric fractions of the solid phase by Eq. (15); five for the mean velocities by Eqs. (20)-(25); one for ~3 by Eqs. (26)-(37); one by Eqs.

Table 1. Constants in Eqs. (38) and (39)

al a2 bl b2

D = 20.9mm, d s = 2.57mm 0.642 -0 .1140 .5910 .026 D = 30.6mm, ds = 2.57mm 1.080 -0 .2830.411 0.272 D = 50.4mm, d s = 2.57mm 1.161 -0.291 0.5950.297

(16)-(18) and (41); and one by Eq. (19); and oJ = 2.6 and 3 for the lengths of each region by Eqs. (38)-(40) and Table 1. Thus, the simultaneous equations are solved, and the mean velocity and volumetric fraction of each phase in each part of each region can be estimated.

Frictional Pressure Drop

The frictional pressure gradient due to the wall shear stress in region 3 is evaluated, using the correlation we obtained for the three-phase bubbly flow [22]:

where

- - 2

(dP)~ F3 = I~I3PI((°l13)V13)f~22D 3 (42)

qi)23 = (1 -- (Ols3>-4"95) ( 1 Z < ~---~-3) 1 + 350 Rel3 ( a g 3 ) ) ( 4 3 ) F r t ~

Here A is a friction coefficient and qb 2 the two-phase multiplier. Rel3 and the Froude number Frl3 are defined by

Re/3 = Pt ( ( o% ) V/3 ) O / t X l (44)

and

Frl3 = ((al3)Vt3)2/gD

Further, A/3 in Eq. (42) can be evaluated by

(45)

64/Re13

0.3164 Re;~ °25

for laminar flow; Re/3 < 2300

for turbulent flow; Ret3 > 2300 (46)

~'13 =

The frictional pressure gradient for the regions with liquid film containing solid particles is evaluated using the correlations for liquid-solid two-phase flows. In region 4,

,4,,

Now /~/4 is evaluated based on the Reynolds number of the liquid film, p t ( ( a t 4 f ) ~ 4 f ) D / l z t , whereas ~2 is evaluated by using the following empirical correlation for the liquid-solid two-phase flow [17]:

cI~42 = 1 +

( ) × 1 - ( a g S f )

() 400 ~4f

[ ( d s / D ) / O . 0 3 8 ] 362 + 1

- 2 . 8

(48)

Then (dP /dz )F5 is obtained only by replacing the subscript 4 in Eqs. (47) and (48) with 5. The mean film velocity around the large bubble ~sf may have negative values. We set the frictional pressure gradient (dP/dZ)F5 negative under such conditions.

Bubble-Tail Pressure Drop

The bubble-tail pressure drop is assumed to occur at the boundary between regions 3 and 4 in this study, because


the liquid and solid phases flow out of a narrowed liquid film into region 3, large turbulent vortices are observed around here, and the mean velocities of the liquid and solid phases should change considerably in this boundary region. It is evaluated by Eq. (6). Applying Eqs. (6) and (4) in this boundary region, we obtain

Ap, = ~IAPA3 41 = ~IPl<°~t3>Vt3(V13 -- Vb)

+ & ( a ~ 3 ) K 3 ( K 3 - Fb)

-- p l ( O Q 4 f ) V l 4 f ( V l 4 f - Vb) (49)

-- Ps<Ols4f>~,4f (~ '4 f - - V b ) ]

where we use Assumptions 2 and 7, and the further assumption that the influence of the non-flat distribution of each phase in regions 3 and 4f on the bubble-tail pressure drop can be neglected. For the value of ~ in Eq. (49), we use 0.35, which was obtained to correlate the measured data well. Thus, the bubble-tail pressure drop is also computable•

Then the time-averaged pressure drop for the three- phase slug flow can be evaluated if the flow conditions--the pipe diameter, solid-particle diameter, the properties of each phase, and the volumetric flux of each phase- -a re given.

PRACTICAL USEFULNESS

The estimated results are shown in Figs. 2-6 by solid lines. From these figures, it is confirmed that all the qualitative characteristics described before can be estimated without contradictions.

Several values calculated in the estimation process are shown in Fig. 8. The mean velocity of each phase increases with (A). In this case, Vl5 has a negative value when ( is) is small and increases from negative to positive

(ie, from downward to upward) as (Js) increases. The volumetric fractions of the gas and liquid phases except for (c~g 3) decrease with increasing (Js). The volumetric fraction of the solid phase and (c~ 3) increase with (A)

• . g . . "

Thus, the unknown velocmes and volumetric fractions can be calculated in the course of the estimation. Of course, because these values are estimated, their characteristics must be confirmed by measurement in the future.

The results of the quantitative comparison of the estimated and measured pressure gradients are shown in Figs. 9-11 for (dP/dz) T, (dP/dz) H, and (dP/dz)Ft. The mean values of the ratio of the measured and estimated values are 0.998, 0.991, and 0.982, and the standard deviations around the mean values are 1.98%, 3.94%, and 22.5% for (dP/dz)T, (dP/dz)H, and (dP/dz)F,, respectively. The value of (dP/dz) m can also be estimated by an existing method [6]. The mean value of the ratio of the measured and estimated values is 2.83, and the standard deviation around the mean value is 89.4%. Through these values and the measuring uncertainties described earlier, it is verified that this method is useful for estimating the pressure drop. The applicable range of the estimation is limited to experimental conditions at present as some parameters are obtained from experiments. But the model and the estimation process can be applied to a wide range of three-phase slug flows.

CONCLUSIONS

Experiments were carried out to clarify the characteristics of three-phase slug flow, and the measured results of the pressure drop in a three-phase slug flow in vertical pipes were presented. Their characteristics were discussed. A method for estimating the pressure drop was proposed based on three-phase slug flow models. It can be used to predict the time-averaged gravitational, frictional, local, and total pressure drops under given flow conditions. The estimated pressure drop was compared with the measured values, and it was verified that this method is useful.

Figure 8. Values obtained in the estimation process.

2 . 0

1 .6

1 .2 O3

E "--" 0 . 8

<-'-"" 0 . 4

0

- 0 . 4 0

-[ T T 1~ D=2o.9 mm / 2 Aluminum Ceramic Pa r t i c l e s / ~

- ds=2.57 mm V -V -V =V "4A" . . . . . 3 / b- g ~ - g4o g3 /

K ,m / --------,,

- / <j~>:o.5 (m/s) v "q0

0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 8

<is> (m/s)

, s m

v ~ n v

• ~ / 'x

v v

8 0 . 0 8

4 0 . 0 4

0


or_

8

--,1

CD :=E

10 , ~ , ~ ,

Aluminum Ceramic Particles . . ~ ds=2. 57me ~ ,o s=2380kg/m 3 ~ 1 7

D=20. 9, 30. 6, 50. 4m ~ a k ~ "

<as> o :0. 005.---0. 02 o.oI _.-0.o,3 v :0. 03 ~0. 07

6 ~ <og> j , m 0 : 0 . 2-~-0. 3

/ ~ i (P :0. 3,--0. 4 ~ r • :0, 4-,-0. 6

44 , t , i , 6 8 10

Estimated (dP/dz) x (kPa/m) Figure 9. Measured and estimated values of (dP/dz) r.

E

¢ o e~

N

v

Sw

: = E

1. v/~ , , , i , , i A / I

Aluminum Ceramic Particles J _] ds=2.57mm • / I

1. Os=2380kg/m3 ~ -~

D:2o. 9, 30. e, 50. 4 . I 1

g ;

O. 8 _ ~ A :0. o2 --.0. o3 v : 0 . 0 3 , -~0 .07

<a0> o :0.2---0.3

0 . 4 (P :0.3~0.4 • :0.4--0.6

I I I I I I I 0 0 .4 0 . 8 1 .2

Estimated (dP/dz)Ft (kPa/m)

<as> o :0. 005~0. 02 -

Figure 11. Measured and estimated values of (dP/dz) m.

1 . 6

R E C O M M E N D A T I O N S AND F U T U R E R E S E A R C H N E E D S

Because the history of the study of three-phase flows in pipes is much shorter than that of two-phase flows, and many parameters appear when the third phase is added to two-phase flows, there is little information on precise characteristics of three-phase flow [25, 27]. For this rea- son, the three-phase slug flow model and the estimation methods presented are only tentative and should be improved in the future. For example, some parameters in the estimation process are obtained empirically to fit the

10

e-i

8 :3=

C I -

CD

" - I

¢ O

' I ' I ' j

klueinum Ceramic Particles / ds=2.57mm

'°s:2380kg/m3

D:20. 9, 30. 6, 50. 4m A , ~

~ * <as> O :0. 005"--0. 02

v :0.03~0.07 6 ~ <=g>

o :0. 2-~0. 3 , , i l m o :0. 3,--0. 4

• :0.4"--0.6

44 ' 61 ' 81 ,

Est imated (dP/dz)H (kPa/m)

Figure 10. Measured and estimated values of (dP/dz) , .

measured data or tentatively quoted from studies for two-phase flows. They should be determined by more precise measurement or by modeling in the future.

We wish to acknowledge the financial support of the Technical Research Center of the Kansai Electric Power Co., Inc., and also to express our thanks to Messrs. K. Sahara, T. Saibe, K. Hashimoto, M. Ushio, K. Yamakoshi, S. Tadakuma, and H. Sugimura for their assistance in the experiments.

NOMENCLATURE

D diameter, m d s particle diameter, m g gravitational acceleration, m / s 2 j volumetric flux, m3/ (m 2- s) = m / s

L length, m P pressure, kPa V velocity, m / s z axial distance, m

G r e e k S y m b o l s

c~ volumetric fraction, dimensionless y exponent in Eq. (16), dimensionless A Darcy friction coefficient, dimensionless /~ viscosity, Pa • s

bubble-tail pressure coefficient, dimensionless p density, k g / m 3 o- surface tension, N / m

( I ) 2 two-phase multiplier of frictional pressure drop, dimensionless

w ratio of volumetric fraction defined in Eq. (19), dimensionless

( ) area-averaging operator volumetric fraction-weighted mean value

mean values of the data shown in each figure

S u b s c r i p t s

A acceleration b bubble c core part


F fr ic t ion f film part g gas phase

H gravity i phase, i = g , l , s

k region number , k = 1 . . . . , 7 l l iquid phase s solid phase

S U S suspension condi t ion T total t tail, te rminal

REFERENCES

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2. Weber, M., and Dedegil, Y., Transport of Solids According to the Air-Lift Principle, 4th. Int. Conf. Hydraulic Transport of Solids in Pipes, Banff, Alberta, Canada, pp. H1-1-23, May 1976.

3. Toda, M., Harada, E., Kuriyama, M., Saruta, S., and Konnno, H., Transport of Solids by Gas-Liquid Upward Flow in Vertical Pipes, Trans. Chem. Eng., 8(4), 380-386, 1982.

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8. Zuber, N., and Findlay, J. A., Average Volumetric Concentration in Two-Phase Flow Systems, Trans. ASME, J. Heat Transfer, 87(4), 453-468, 1965.

9. Laird, A. D. K., and Chisholm, D., Pressure and Forces Along Cylindrical Bubbles in a Vertical Tube, Ind. Eng. Chem., 48(8), 1361-1364, 1956.

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Chem. Fundam., 14(4), 337-347, 1975. 12. Fernandes, R. C., Semiat, R., and Dukler, A. E., Hydrodynamic

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13. Sylvester, N. D., A Mechanistic Model for Two-Phase Vertical Slug Flow in Pipes, Trans. ASME, 109, 206 213, 1987.

14. Maron, D. M., Yacoub, N., and Brauner, N., Lett. Heat Mass TransJ~r, 9, 333-342, 1982.

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17. Sakaguchi, T., Minagawa, H., Tomiyama. A., and Shakutsui, H., Characteristics of Pressure Drop for Liquid--Solid Two-Phase Flow in Vertical Pipes, Mere. Fac. Eng. Kobe Unit:., 36, 63-90, 1989.

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20. Akagawa, K., and Sakaguchi, T., Studies on Fluctuation of Void Fraction in Gas-Liquid Slug Flow II, Trans. JSME, 31(2241, 594-607, 1965.

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Received August 5, 1992; revised January 9, 1993

Documents

Pressure drop in gas-liquid-solid three-phase slug flow in vertical pipes