numerical simulation of gas-liquid slug flow along vertical pipes using the slug tracking model

8/12/2019 NUMERICAL SIMULATION OF GAS-LIQUID SLUG FLOW ALONG VERTICAL PIPES USING THE SLUG TRACKING MODEL

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Proceedings of ASME-JSME-KSME Joint Fluids Engineering Conference 2011AJK2011-FEDJuly 24-29 2011 Hamamatsu Shizuoka JAPAN

AJK2011-10045NUMERICAL SIMULATION OF GAS-LIQUID SLUG FLOW ALONG VERTICAL PIPES

USING THE SLUG TRACKING MODEL

Alex Pachas N.LACIT/PPGEM/UTFPR

Curitiba, PR, Brazil

Csar Perea M.LACIT/PPGEM/UTFPR


Rigoberto E. M. MoralesLACIT/PPGEM/UTFPR


Cristiane Cozin

LACIT/PPGEM/UTFPRCuritiba, PR, Brazil

Eugnio E. Rosa

FEM/UNICAMPCampinas, SP, Brazil

Ricardo A. Mazza

FEM/UNICAMPCampinas, SP, Brazil

ABSTRACTThe intermittent gas-liquid flow, or slug flow, in vertical

tubes occurs over a wide range of gas and liquid flow rates,with many applications, such as oil industry. Predicting the

properties of this kind of flow is important to design properlypumps, risers and other components involved. In the presentwork, vertical upward slug flow is studied through a one-dimensional and lagrangian frame referenced model called slugtracking. In this model, the mass and the momentum balance

equations are applied in control volumes constituted by the gasbubble and the liquid slug, which are propagated along thepipe. The flow intermittency is reproduced through theconditions at the entrance of the pipe, which are analyzed instatistical terms. These entrance conditions are given by asequence of flow properties for each unit cell. The objective ofthe present work is to simulate the slug flow and itsintermittency through the slug tracking model. The numericalresults are compared with experimental data obtained by2PFG/FEM/UNICAMP for air-water flow and good agreementis observed.

INTRODUCTION

The slug flow occurs over a wide range of gas and liquidflow rates. This flow pattern is characterized by an intermittentsequence of liquid slugs and elongated bubbles distributedirregularly over time and space. The liquid slug may containdisperse bubbles and the elongated bubble flows underneath orinside a thin liquid film. The prediction of the gas-liquid

properties (lengths, frequency and velocity as well as thepressure drop) is necessary to design facilities operating withslug flow pattern.

There are a number of slug flow models based on the unit-cell concept. Fernandes et al. [1] proposed one hydrodynamicmodel to predict the flow properties of a gas-liquid slug flow invertical tubes, Taitel and Barnea [2] reported one generamodel for horizontal, inclined and vertical flows and morerecently Abdul-Mejeed and Al-Mashat [3] developed onemechanistic model to predict the flow behavior for upwardvertical and inclined two-phase slug flow.

These models are the so-called steady state models, because

they consider that all bubbles and slugs are equal in time andspace (periodic flow). The steady state models are easy to useand predict properly the mean values of the importantvariables, such as pressure drop and bubble velocity. Howeverthe main characteristic of slug flow its intermittency is nocaptured. During the 90s a new class of models called slugtracking emerged. Computationally, these models are moreexpensive, but are capable of simulating the transient behaviorof the flow.

Taitel and Barnea [4] and Al-Safran et al. [5] presented slugtracking models as an evolution of the earlier Taitel and Barnea[2] work, still considering incompressible fluids. Ujang et al[6]presented a model based on mass conservation in each unit

cell, but the authors considered both phases as incompressibleGrenier [7] and Rodrigues [8] used integral control volumes toobtain mass and momentum conservation equations for eachunit cell.

Slug tracking models, in principle, capture some transieneffects of the flow, such as the effect of bubbles entrance andexit of the pipe, the bubble to bubble interaction and theinstantaneous change of the mass and momentum fluxes of

both phases. However, these models are strongly dependent o

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the conditions imposed at the pipe inlet, which are related tointermittence.

The geometric parameters and the main variables in eachregion show intermittence along the pipe. Therefore, it is notenough to know their average behavior, but also thedistribution. For that reason, slug flow variables should be

described in statistical terms [9]. The flow intermittence can bereproduced by the use of the probability density function(PDF).

One of the first works on intermittence of slug tracking waspresented by Barnea and Taitel [9]. A model was developed forthe slug length probability distribution highlighting theimportance of the maximum slug length along the pipe. Theyalso developed two types of slug length distributions at theentrance: a random and a uniform distribution. It wasconcluded that the slug length distribution at the entrance doesnot affect its distribution along the pipe.

The present work uses the slug tracking model presented inRodrigues [8] for vertical flow considering the gas

compressibility and aerated slugs with an intermittent inletcondition. Therefore, each bubble and slug entering the pipehas a different length and velocity. The values of the bubblelengths and velocities are based on a Normal distribution whilethe slug lengths are based on a log-Normal distribution [9]. Themean values and standard deviation are obtained fromexperimental results.

NOMENCLATURED Pipe diameterLS Slug length.LB Bubble length.R

S Liquid holdup in the liquid slug.

RB Mean void fraction in the bubble region.US Mean liquid slug velocity.UGS Mean velocity of the gas bubbles dispersed in the slugULB Mean velocity of the liquid filmUGB Mean velocity of the Taylor bubbleUT Translational velocity of the front of the bubbleUDS Drift velocity

PG Pressure in the gas bubble.jL Liquid superficial velocityjG Gas superficial velocityJ Mixture velocity (jL+jG)hLB Film height in the bubble region

x Position of the back of the bubbley Position of the front of the bubble Density Shear stress Inclination angle = 90

Subscriptsj jthunit cellk Number in the entrance conditions list

L Liquid phase

G Gas phaseS Slug region

B Bubble regioni Interface

THE SLUG TRACKING MODELThe model is based on the one-dimensional integral formof the mass and momentum conservation equations applied toeach of the components of the unit cell. As a result, twoequations are obtained as a function of the bubble pressure andthe liquid slug velocity. These two equations are discretizednumerically in order to be solved in a linear equations system.

The control volumes are deformable and follow eachbubble and slug along the pipe. Figure 1 presents the jth unicell, in which coordinates xj and yj represent the front of theslug and the bubble, respectively.

FIGURE 1. VERTICAL SLUG FLOW UNIT

Mathematical modelThe liquid mass conservation equation is applied to the

slug defined by the pipe wall and the x and coordinatesy(Fig. 1). The resulting equation yields:

Sj Sj

Sxj Syj

Sj

L dRU U

R dt (1)

The liquid velocities at the slug boundaries (USxjand USyj)must be expressed as function of the mean liquid slug velocityUSj. Defining USj as the arithmetic mean of the boundaries




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velocity, 2Sj Sxj SyjU U U , the liquid velocities may bewritten as

,2 2

Sj Sj Sj Sj

Sxj Sj Syj Sj

Sj Sj

L dR L dRU U U U

R dt R dt (2)

Similarly for the gas balance, but also considering the gascompressibility, the gas velocities at the slug boundaries (UGSxjand UGSyj) can be calculated by:

1 1

2 1

1 1

2 1

Sj Sj Gj

GSxj GSj

GjSj

Sj Sj Gj

GSyj GSj

GjSj

L dR dU U

dt dt R

L dR dU U

dt dt R

(3)

Mass balance is applied to the liquid phase inside thecontrol volume encompassing the bubble region and defined bythe pipe wall and the boundaries y and 1x . Using Eq. (1),

it is obtained:

11

1 11 1

1 1

2 2

j j Bj

Sj Bj Sj Bj Bj

Sj Sj Sj Sj

Sj Sj Sj Sj

dy dx dRR R R R L

dt dt dt

L dR L dRR U R U

dt dt

(4)

The mass balance equation is also applied to the gas phasein the bubble region, considering the results of Eq. (3):

11

1 1

1 1 11 1

1 1

1 1

1

2 2

1 11

2 1

1 11

2 1

j j Bj

Sj Bj Sj Bj Bj

Sj Sj Sj Sj Gj

Bj Bj

Gj

Sj Sj Gj

Sj GSj

Sj Gj

Sj Sj Gj

Sj GSj

Sj Gj

dy dx dRR R R R L

dt dt dt

L dR L dR dL R

dt dt dt

L dR dR U

R dt dt

L dR dR U

R dt dt

(5)

It can be observed that the LHS of Eqs. (4) and (5) arethe same, so they can be merged in one equation. Consideringideal gas, the gas density is expressed in pressure terms. Inaddition, the velocity of the dispersed bubbles in the slug UGScan be written as the superposition of the mixture velocity Jand the drift velocity UDS, which can be calculated throughcorrelations found in the literature [2]. That way, it can bewritten as one equation that represents the total mass balance inthe unit cell:

11 1

1

11

1

1 1

1 1

2 2

Sj Sj

Sj Sj DSj DSj

Sj Sj

Sj SjBj Bj Sj Sj

Bj

Bj Bj Bj

R RU U U U

R R

R RdP R L LL

dt P P P

(6)

Now, momentum balance is applied in the liquid slug:

S S Sj Sj L Sj Sj Sjyj xjj

Sxj L Sj Sxj Syj L Sj Syj

dP P A DL AR L U

dt

dx dyU AR U U AR U

dt dt

(7)

Pressure in the momentum equation is evaluated in theliquid, but in the mass balance equation (6) it appears asfunction of the pressure in the bubble PB. In that context

pressures in Eq. (7) must be expressed as pressure in thebubble, through the application of the balance equations in theinterface.

Due to the smooth shape of the front of the bubble, thepressure drop can be considered negligible between the front othe bubble and the tail of the slug. Thus, it is obvious that:

S BjyjP P (8)

On the other hand, pressure drop at the back of theelongated bubble can not be negligible. It is modeled by

performing a stationary momentum balance in the controvolume specified in Fig. FIGURE 2 considering that the

pressure along the bubble remains constant.

1 1 11

LBj LBj Bj

S Bjxj

S LP P

A

(9)

Finally, momentum equation for the liquid slug is given byreplacing Eq. (8) and Eq. (9) in Eq. (7)

1 1 1

1

1 11

1

2

Bj Bj Bj Sj Sj

Bj Bj

L Sj Sj Bj Bj

Sj Sj j j

L Sj Sj Sj Sj

S L DLP P

A

R L R L gSen

dU dR dx dyL R L U

dt dt dt dt

(10)

Equation (10) shows that the pressure drop between twoadjacent unit cells occurs due to the shear stress, thegravitational weight and a term related to the local accelerationof the slug.




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FIGURE 2. CONTROL VOLUME FOR THE PRESSURECOUPLING

Numerical discretizationEquations (6) and (10) form the coupled system that is

solved at each time step for USj and PBj. The system is

discretized with the finite differences method using the semi-implicit Crank-Nicholson scheme and using the Darcycorrelation for the shear stress. The discretized forms of Eqs.(6) and (10) become:

1 1

11

1 1

1

1

1 12

1 12

2

Sj Sj Sj SjBj BjN N N

Sj Sj Bj O O O

Bj Bj GBj

O

Sj Sj Sj Sj GBjBj BjO O

Sj Sj O

GBj

Dj

L R L RL RU U P

tP tP tP

L R L R PL RU U

t t tP

U

(11)

and

1

1 1

2 4

22

Sj Sj LSjN N O N

Bj L Sj Sj Sj Bj

O

Sj S j Sj LO O

Bj Bj Sj j

R L CP U L U P

t D

R L UP P P I

t

(12)

where

11

1

21 1 1

1 1

2

2

Sj Sj

DSj DSj DSj

LSj LSj

L LBj LBj

Sj Bj LBj

Sj j j LSj

Sj LSj L

R RU U U

R RC S

P L UD

L dx dy dRI L U

dt dt dt

(13)

The super-indexes N and O indicate the new and oldvalues of the variables, respectively.

METHODOLOGYIn the previous section, the slug tracking model was

presented. In this section, the methodology for the solution othe slug tracking model is detailed.

A couple of two equations, (11) and (12), are written foreachjunit cell (1 j n). If nis the number of unit cells inside

the pipe, there would be 2n equations. That way, there is anequation system in terms of the mean velocity in the liquid slugregion USjand the pressure inside thej

thbubblePBj. The set ofunits cells produce a linear system which can be written as A.= B, whereA, is a tridiagonal matrix, is the unknownvectorand B the source term vector. Finally, the TDMA method isused to solve the equation system. One system is solved at eachtime step.

For the application of this method, some boundaryconditions must be known for the first and the last cell. In thelast cell (j = n), the value PBn+1 is used, which represents the

pressure at the exit. Commonly, the atmospheric pressurewritten as 1Bn atmP P , is used at the exit.

In the first cell (j = 1), the value US0 is used, whichrepresents the instantaneous velocity of the liquid in the firstslug, calculated by the Eq. (14):

0

0 00

1 S

S L G DS

S

RU j j U

R

(14)

In order to initialize the simulation, initial conditions at t=0must be established. In the present work, it is considered thathe pipe is full of liquid with initial velocity US0 and the firs

bubble is positioned in 0z (See Figure 3).However, whenever a unit cell needs to be inserted at the

pipe entrance, the superficial velocities must be known tocalculate US0. In addition, the bubble and slug lengths also musbe known to calculate the coefficients in Eqs. (11) and (12). Inorder to reproduce the flow intermittence, each of the unit cellsentering the pipe should be different. Thus, a list of unit cellsneeds to be generated, which is called entrance conditions. Thecalculation of these entrance conditions is presented in the nextsection. Once the unit cells list is generated, they are saved in afile, which will be read line by line by the program when thesimulation begins.

When the simulation starts (at t=0), two unit cell arerequired from the entrance conditions. The first has its bubblenose at z= 0and the second is behind the first one still outsidethe pipe. One time step later, the parameters of the first unit celare updated through the solution of the tridiagonal systemTime steps are increased and tridiagonal systems are solveduntil the first bubble is completely inside the pipe. At thatmoment, the second bubble starts entering the pipe and a thirdunit cell from the entrance conditions is required. This thirdunit cell is positioned behind the second. This procedure isrepeated for every single unit cell entering the pipe. Simulationfinishes when a number of unit cells specified by the userleaves the pipe.




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FIGURE 3. BOUNDARY AND INITIAL CONDITIONS

The lagrangian slug tracking model presented isimplemented in an object-oriented computational programwritten in FORTRAN language, using Intel Visual Fortran ascompiler. In this approach, bubbles and slugs are discreteobjects which are propagated along the pipe through thegoverning equations.

ENTRANCE CONDITIONSIn the slug tracking method, the unit cells set as entrance

conditions are propagated along the pipe. Thus, coherententrance conditions determine an accurate simulation of theslug flow intermittency. For the initialization of the slugtracking, physical and geometric parameters of the pipe and theunit cells are needed. In order to determine the characteristicsof the unit cells used as entrance condition, it is necessary toknow the variables listed on Table 1.

TABLE 1. INPUT DATA FOR THE GENERATION OF ENTRANCECONDITIONS

Description Description

Gj Mean gassuperficial velocity

BL

Bubble length standarddeviation

Lj

Liquid superficialvelocity SL

Liquid slug lengthstandard deviation

BL Mean bubble length

TU Bubble translational

velocity standard deviation

SL

Mean liquid sluglength

f Frequency

Each of the mean values and standard deviations will beused to generate a sequence of distributed values. Each of theterms in the sequence of data will be denominated with thesubscript k. In the case of LB and jG, normal distributions areapplied. In the case ofLSa log-normal distribution is used [10]In the case of jL, it is not necessary to build a distribution as i

is assumed constant along the simulation. Also, the slug liquidholdup SR and void fraction of the elongated bubble BR are

calculated through the bubble design model presented in thesection below.

Statistical distributionsFor the reproduction of LS, using the log-norma

distribution, the following parameters need to be calculated:

2 2

1 , exp / 2SL S

Ln L

(15)

where, and are statistical distribution parameters.In order to obtain the distributed values sequence, the

transformation proposed by Box and Muller (1958) is used. Inthis transformation, a sequence of random data with normaldistribution can be generated through two lists of independentrandom (1and 2) values with a uniform distribution (between0 and 1). If sequences data generated by the Box-Mullerfunction, is used, a third sequence of random data is calculated

by:

1 2

1 2

1 1 1

2 cos 2

2 sen 2

k k k

k k k

Ln

Ln

(16)

where 1k and

2k are independent random variables that are

uniformly distributed in the interval , k is an

independent random variable. These random variablesconstitute a set of results close to the principal mean and withstandard deviation equals to 1.

Then, the data set with normal distribution is used to obtainthe slug flow parameters. In other words, it is obtainedsequences with normal distribution for the gas superficiavelocity (jGk) and bubble length (LBk) and sequences with log-normal distribution for the slug length. These sequences are

function of the mean values and standard deviation, calculatedthrough the following equations:

0/TGk G k U L G Dj j j j V C (17)

1BBk B k L

L L (18)




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expSk kL Ln (19)

where VD is the drift velocity and C0 is the flow distributioncoefficient related to the translational bubble velocity UT.

Bubble design model

When the parametersjGk,LBkandLSkare known, the volumefractionsRSkandRBkcan be calculated using the bubble designmodel presented by Taitel and Barnea [4]:

22

1 1

cos1

G GL Li i L G

L G G LLB

G GBL LB B BL G

B LB B LB

SSS gsen

A A A Ah

z VV dR dRg

R dh R dh

(20)

where =90 for vertical flow, VLB=UT - ULBand VGB=UT - UGB.Eq. (20) represents the geometric shape of the Taylor

bubble expressed as the variation of the film height along the zcoordinate.

The most influential parameter in the bubble design modelis the translational velocity UT [8] as it determines thedistribution of the unit cell components. Its expression is given

by an empirical correlation:

(1 )T o DU C J V (21)

where DV C gD is the drift velocity of the elongated

bubble, J is the mixture velocity, is the factor of thewake behind the bubble, and C is a coefficient related to the

slope pipe. The coefficients C and oC are calculated byBendiksens correlations [12], as a function of the ReynoldsnumberReM, Froude numberFrMand Etvs numberEo. Notethat VDdepends onJ, which depends onjG, so it is not constant.

The wake factor in Eq. (21) is calculated as

exp SLW w Da b [11]. For vertical pipes 8.0Wa and1.06Wb [8].

In order to integrate Eq. (20), the initial height isconsidered as a function of the initial volumetric fraction

BR .

Once calculated the slope and the conditions initials in the film,closure relationships for the bubble design equations areneeded.

Eq. (20) is integrated numerically from z= 0toz = LBitoobtain the bubble geometry, assuming a value forRSj. Then, themean void fraction in the bubble RBi is calculated through theheight distribution (hLB) along the bubble length.

The calculated values of RSj and RGj must satisfy theconservation of mass. Thus, Eqs. (22) and (23) are obtainedfrom the stationary mass balance in the unit cell [2]. The liquidholdup in the slug RSi is recalculated through Eq. (22), The

void fraction in the bubble is also recalculated through Eq(23)

1 BG T

S

GS T

j R UR

U U

(22)

1 GS TGB ST T

U UjR R

U U

(23)

where = LB/(LS+LB). Then the recalculated values arecompared with the old ones until a convergence is attained.

Algori thm for the entrance condi tionsThis section describes the calculation algorithm, which is

used to generate the entrance conditions for the slug trackingmodel. The input data required to apply this method is shownin Table 1. This procedure is organized in six steps:Generation of the data sequence

1) Generate as many random values as unit cells requiredthrough Eq.(16).

2) Apply Eqs.(15), (17), (18) and (19) to each generatedvalue to calculatejGi,LBiandLSi.

3) Assume a value for RS. Use the Eq. (20) and theprocedures described in the bubble design model sectionto calculateRBi.

4) CalculateRSandRB through Eqs. (22) and (23).5) Compare RSand RB from 4) with the assumed RS in 3)

andRBfrom the bubble design. The convergence criteriaare: 0.01% forRSand 0.1% forRB.

6) If the values in 5) dont converge, steps 3) and 4) arerepeated using theRSfound in 5).

It is important to mention that in the data sequencegenerated, some values of RS are not physically possible

because they are higher than 1. Thus, the entire unit cell mustbe removed from the sequence. This filter makes that theaverage value from the obtained sequence is not equals to thevalue provided as input.

In this section, the methodology to reproduce entranceconditions was presented. These entrance conditions reproducethe intermittence through distributions obtained as function ofstatistical variables and random values. Generated slug and

bubble length are evaluated using the liquid mass balance inorder to calculate the volume fraction. Thus, all the generated

cells satisfy the mass balance.

RESULTS AND DISCUSSIONOnce the methodology is implemented in the program, the

simulation through the slug tracking model can take placeSimulations are performed in a processor PC Intel CoreTM2CPU 2.13 GHz with 2.00 GB RAM. For the slug trackingmodel, the stop condition is the exit of 600 bubbles which takean average of 20 minutes. Table 2 shows the geometric and




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physical characteristics of the experiments performed by Rosaand Altemani [10] for air-water flow.

The experimental setup consists in a vertical tube with twopoints for observation. The first one, called probe 1, is near theentrance and the second one, probe 2, is near the exit. Two setof data with different superficial gas velocities were selected to

be compared with the simulation results. The first (A@W#1)has a low gas flow rate and the second (A@W#2) has a highgas flow rate. Analyzed parameters are: the bubble translationalvelocity, the slug and the bubble length and the pressure.

TABLE 2. Characteristics of the experimental data.

Pipe length, [m.] 5.81Diameter, [m.] 0.026Location of probe #1 [m.] 0.10 (3.85D)Location of probe #2 [m.] 4.693 (180.5D)Wake effect factor [aw] 8.0Wake effect factor [bw] 1.06

Liquid density [kg/m3

] 999Liquid Viscosity [Pa.s] 0.000855A@W#1

Superf liquid velocity[m/s]

0.33

Superf gas velocity [m/s] 0.464A@W#2

Superf liquid velocity[m/s]

0.33

Superf gas velocity [m/s] 1.42

As the flow is intermittent, the variables will have differentvalues at each time-step. In order to evaluate the average flow

behavior, time averaged values are obtained. In Figure 4 andFigure 5, the mean values along the pipe for A@W#1 andA@W#2 are presented. For the bubble length, it is observedthat the model reproduces correctly the expansion of the bubbledue to the pressure drop for both cases. Best agreement isobtained at the entrance.

In the numerical results, the slug length almost remainsconstant along the pipe, showing a slight expansion.Comparing with the experimental data, the values at theentrance are higher and at the exit are lower, so the numericalvalues stay in the middle of the experimental points. In otherwords, the total mean slug length coincides with theexperimental.

For the translational velocity, it can be observed a goodagreement at the entrance. However, along the pipe the modeltends to slightly overestimate its value, which can be related tothe selected model to quantify the wake effect.

The pressure is well estimated for both cases. In verticalflow the pressure drop is extremely related to the gravitationalterm, which is why the simulated pressure is very close to theexperimental, as the gravitational term is independent from theslug (or bubble) length.

FIGURE 4. MEAN VALUES ALONG THE PIPE FOR A@W#1

FIGURE 5. MEAN VALUES ALONG THE PIPE FOR A@W#2




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FIGURE 6.PROBABILITY DENSITY FUNCTIONS FOR A@W#1

In Figure 6 and Figure 7 the probability density functionsfor the unit cell properties are presented. This statisticalmeasure represents the probability of a variable given aninterval. The numerical distribution for the bubble and liquidslug length are similar with the experimental data at theentrance for both cases, coinciding in the mean. At the exit, theresults adjusts better for A@W#1, for A@W#2 there is adislocation in the mean and in the distribution, as theexperimental values are more disperse than the numerical. Thisconcentration around the mean for the numerical results may becaused by the adjustment of the wake factors, which are relatedto the magnitude of the wake effect and the slug length.

For the translational bubble velocity, the data shows lowdispersion for the probe at the entrance. In this case,Bendiksens correlations for C0 and C are important due to

its dependence with the gas superficial velocity and the fluidproperties. We are using a distribution for the gas superficiavelocity jG at the entrance, which is directly related to thetranslational velocity UT. The slug tracking model, propagatesthis parameter along the pipe which may cause a higher gradeof dispersion in its distribution.

In general, it can be observed that at the entrance all theparameters are more concentrated around the mean. In additiona better agreement with the experimental data is observed at theentrance. This shows that the intermittency is reproducedcorrectly by the methodology presented in the entranceconditions. On the other hand, at the exit the values are moredispersed.

FIGURE 7.PROBABILITY DENSITY FUNCTIONS FOR A@W#2

As the slug tracking model is lagrangian, it allows followthe evolution of a unit cell along its passage through the duct.In order to guarantee a stable unit cell, it is chosen the 200th

unit cell that enters the pipe. Figure 8 shows the translationa




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velocity and the pressure of the 200 thunit cell measured from areference frame travelling together with the bubble. It can beobserved that the intermittent condition introduces instabilities,which causes oscillations. For the translational velocity, it can

be observed that in spite of the unstable behavior, theinstantaneous values oscillate around the mean value. In the

case of the pressure, instability is also observed, but showing adecreasing tendency.

FIGURE 8.CAPTURED PROPERTIES FROM A TRACKED

BUBBLE

CONCLUSIONSSimulation of two-phase vertical slug flow through the

slug tracking model was presented. This transient model, basedon the balance equations, considers aerated slugs and theexpansion of the gas in the elongated bubble due to the

pressure drop. The model consists in two governing equationsfor each unit cell, whose solution through TDMA allows thecalculation the slug velocity and the bubble pressure at eachtime step. The intermittency of the flow was reproduced by theentrance conditions, based on stationary models and the theoryof statistical distributions.

The numerical results were compared with experimentaldata for air-water flow. The analyzed results were the meanvalues and the statistical distributions (PDF) of the main flow

parameters:LB, LS and UT. Evolution of the mean values anddistributions along the pipe presented qualitative similarities asthe experimental. It is observed that the model is able tocapture the intermittence of the unit cell length and the bubbletranslational velocity. Results show that the model reproduces

the experimental data with good agreement in the mean and inthe distributions

ACKNOWLEDGMENTSThe authors acknowledge the financial support from the

National Agency for Petroleum, Natural Gas and Biofuels

(ANP) through its Human Resources Program in UTFPR(PRH-10) and from TE/CENPES/PETROBRAS.

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Hydrodynamic model for gas-liquid slug flow in verticatubes.AIChe Journal, vol. 29, no. 6, pp. 981-989.

[2] Taitel, Y. and Barnea, 1990, D., "Two phase slug flow"Advances in Heat Transfer, Hartnett J.P. and Irvine Jr. T.Fed., vol. 20, 83-132, Academic Press (1990a).

[3] Abdul-Majeed, G. H., Al-Mashat, A. M., 2000, Amechanistic model for vertical and inclined two-phase slugflow, Journal of Petroleum Science and Engineering, 27

pp. 5967.[4] Taitel, Y. and Barnea, D.,1998, Effect of gascompressibility on a slug tracking model. Chem. EngScience, Vol. 53, No. 11, pp. 2089-2097.

[5] Al-Safran, E. M., Taitel, Y. and Brill, J. P., 2004, Predictionof slug length distribution along a hilly terrain pipelineusing slug tracking model. Journal of Energy ResourcesTechnology, Vol. 126, No. 1, pp. 54-62.

[6] Ujang, P. M., Lawrence, C. J. and Hewitt, G. F., 2006Conservative incompressible slug tracking model for gasliquid flow in a pipe.BHR group multiphase technology,5373-388.

[7] Grenier,P., 1997, Evolution des longueurs de bouchons en

coulement intermittent horizontal, PhD thesis,. Institut deMcanique des Fluides de Toulouse, Toulouse, France.[8] Rodrigues, H. T., 2009 Simulao Numrica de

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[11] Moissis, R., Griffith, P., 1962 Entrance effects in a two phase slug flow.Journal Heat Transfer, Vol. 85, pp. 29-39

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numerical simulation of gas-liquid slug flow along vertical pipes using the slug tracking model

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