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ARTICLE IN PRESSG ModelOCS-225; No. of Pages 14

Journal of Computational Science xxx (2013) xxx–xxx

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Journal of Computational Science

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odeling and simulation of severe slugging in air-water systemsncluding inertial effects

.L. Balinoepartmento de Engenharia Mecânica, Escola Politécnica, Universidade de São Paulo, Av. Prof. Mello Moraes, 2231, CEP 05508-900, Cidade Universitária,ão Paulo, SP, Brazil

r t i c l e i n f o

rticle history:eceived 31 January 2013eceived in revised form 29 July 2013ccepted 25 August 2013vailable online xxx

eywords:evere sluggingipeline-riser systemir-water flowtability

a b s t r a c t

A mathematical model and numerical simulations corresponding to severe slugging in air-water pipeline-riser systems are presented. The mathematical model considers continuity equations for liquid and gasphases, with a simplified momentum equation for the mixture. A drift-flux model, evaluated for the localconditions in the riser, is used as a closure law. In many models appearing in the literature, propagationof pressure waves is neglected both in the pipeline and in the riser. Besides, variations of void fraction inthe stratified flow in the pipeline are also neglected and the void fraction obtained from the stationarystate is used in the simulations. This paper shows an improvement in a model previously published by theauthor, including inertial effects. In the riser, inertial terms are taken into account by using the rigid water-hammer approximation. In the pipeline, the local acceleration of the water and gas phases are included inthe momentum equations for stratified flow, allowing to calculate the instantaneous values of pressure

umped and distributed parameter systemswitched systemsetroleum production technology

drop and void fraction. The developed model predicts the location of the liquid accumulation front inthe pipeline and the liquid level in the riser, so it is possible to determine which type of severe sluggingoccurs in the system. A comparison is made with experimental results published in literature including achoke valve and gas injection at the bottom of the riser, showing very good results for slugging cycle andstability maps. Simulations were also made assessing the effect of different strategies to mitigate severeslugging, such as choking, gas injection and increase in separation pressure, showing correct trends.

. Introduction

Severe slugging is a terrain dominated phenomenon, character-zed by the formation and cyclical production of long liquid slugsnd fast gas blowdown. Severe slugging may appear for low gas andiquid flow rates when a section with downward inclination anglepipeline) is followed by another section with an upward inclina-ion (riser). This configuration is common in off-shore petroleumroduction systems. Main issues related to severe slugging are:a) High average back pressure at well head, causing tremendousroduction losses, (b) High instantaneous flow rates, causing insta-ilities in the liquid control system of the separators and eventuallyhutdown, and (c) Reservoir flow oscillations.

For steady state and low flow rates, the flow pattern in theipeline may be stratified, while it may be intermittent in the riser,s shown in Fig. 1(a).

Please cite this article in press as: J.L. Balino, Modeling and simulationJ. Comput. Sci. (2013), http://dx.doi.org/10.1016/j.jocs.2013.08.006

A cycle of severe slugging can be described as taking placeccording to the following stages [1]. Once the system destabi-izes and gas passage is blocked at the bottom of the riser, liquid

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continues to flow in and gas already in the riser continues to flowout, being possible that the liquid level in the riser falls belowthe top level at the separator. As a consequence, the riser columnbecomes heavier and pressure at the bottom of the riser increases,compressing the gas in the pipeline and creating a liquid accumu-lation region. This stage is known as slug formation (Fig. 1(b)).

As the liquid level reaches the top while the gas passage is keptblocked at the bottom, pressure reaches a maximum and there isonly liquid flowing in the riser. This is the slug production stage(Fig. 1(c)).

Since gas keeps flowing in the pipeline, the liquid accumulationfront is pushed back until it reaches the bottom of the riser, startingthe blowout stage (Fig. 1(d)).

As the gas phase penetrates into the riser the column becomeslighter, decreasing the pressure and then rising the gas flow. Whengas reaches the top of the riser, gas passage is free through thestratified flow pattern in the pipeline and the intermittent/annularflow pattern in the riser, causing a violent expulsion and a rapiddecompression that brings the process to slug formation again. This

of severe slugging in air-water systems including inertial effects,

stage is known as gas blowdown (Fig. 1(e)).Fig. 1(f) shows the different stages in the pressure history at the

bottom of the riser corresponding to an experiment under labora-tory conditions [2].

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ARTICLE IN PRESSG ModelJOCS-225; No. of Pages 14

2 J.L. Balino / Journal of Computational Science xxx (2013) xxx–xxx

ere sl

o

•

•

•

•

t

Fig. 1. Stages for sev

A classification of severe slugging can be made, according to thebserved flow regime, as follows:

Severe Slugging 1 (SS1): the liquid slug length is greater to orequal to one riser length and maximum pipeline pressure is equalto the hydrostatic head of the riser (neglecting other pressuredrop terms).Severe Slugging 2 (SS2): the liquid length is less than one riserlength, with intermittent gas penetration at the bottom of theriser.Severe Slugging 3 (SS3): there is continuous gas penetration at thebottom of the riser; visually, the flow in the riser resembles nor-mal slug flow, but pressure, slug lengths and frequencies revealcyclic variations of smaller periods and amplitudes compared toSS1.Oscillation (OSC): there are cyclic pressure fluctuations without


the spontaneous vigorous blowdown.

Most of the models for severe slugging were developed for ver-ical risers and assume one-dimensional, isothermal flow and a

ugging (from [1,2]).

mixture momentum equation in which only the gravitational termis important.

In [3] a model was presented considering constant mean val-ues for the gas density and void fraction in the riser, allowing tocalculate time variations of pipeline pressure, position of the accu-mulation region, flow rate into the riser and mean holdup. It wasfound that as the operation point moves closer to the stabilityline the numerical procedure did not converge, giving gas massflows going to infinite as the spatial discretization was decreased.Experimental data were obtained from a facility for different buffervolumes (simulating equivalent pipeline lengths) and a comparisonwas made with the simulation results, showing good agreementexcept for the blowout/blowdown stage. Setting apart the non-convergence problems, lumped parameter models seem to workfine for short risers, where the local variations of variables are small,but are not successful in long risers, typical of offshore systems.


In [4] a model with a distributed parameter formulation forthe riser was presented. Considering continuity equations for theliquid and gas without phase change and a gravity-dominant mix-ture momentum equation, the model was capable of handling

dx.doi.org/10.1016/j.jocs.2013.08.006

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ARTICLEOCS-225; No. of Pages 14

J.L. Balino / Journal of Comput

iscontinuities such as liquid accumulation in the piping and liquidevel in the riser. The resulting equations were solved by using the

ethod of characteristics. A comparison of simulations with differ-nt experimental data showed reasonable agreement, although theodel also suffered from non-convergence in the unstable region.In [5] a model for severe slugging valid for risers with vari-

ble inclination was presented. The model was used to simulateumerically the air-water multiphase flow in a catenary riser for thexperimental conditions reported in [6]. Stability and flow regimeaps in the system parameter space for the multiphase flow in a

atenary pipeline-riser system were built.In [7] the model developed in [5] was extended to investigate the

ynamics of gas, oil and water flow in a pipeline-riser system. Massransfer between the oil and gas phases was calculated using thelack oil approximation. The properties of fluids were calculatedy analytical correlations based on experimental results and fieldata.

The stationary solution for a given point in the system parame-er space is given as initial condition for the numerical simulation;f the numerical solution does not go away from the initial condi-ion with time, the stationary solution is stable and it is the systemteady state. If the numerical solution goes away with time, thetationary state is unstable, there is no steady state and an inter-ittent solution develops with time. By changing the point in the

ystem parameter space and repeating this process, the stabilityap can be built. For unstable flow, the analysis of the limit cycle

eads to the determination of the flow regime map, showing theegions corresponding to the different types of intermittency.

A recent monograph [8] reviews different issues related toevere slugging for air-water systems.

In all the models reviewed above, propagation of pressure wavess neglected both in the pipeline and in the riser; this constituteshe no-pressure-wave (NPW) approximation [9]. As a result of thePW approximation, pressure changes are felt instantaneously atny point in the domain (pressure waves travel at infinite speed).

Also in the models reviewed above, a stratified flow patterns assumed at the pipeline and variations of void fraction areeglected. The void fraction is obtained from a momentum bal-nce in the gas and liquid phases, resulting an algebraic relationetween the mean variables [10]. The void fraction determined inhe stationary state is assumed as constant in the simulations, sohe momentum balance equation is not satisfied and variations ofhe void fraction cannot be calculated in the transients.

As stated in [5], a riser model with a simplified momentumquation in which only the gravity force is considered has limi-ations to deal with general boundary conditions. In particular, foruch a model to handle a discontinuous pressure boundary con-ition, it should react with a discontinuous time variation in theoid fraction distribution; to do so, distributional (Dirac’s delta)uperficial velocities would be necessary. A gravity dominant riserehaves like an ideal mechanical spring, which reacts with a dis-ributional velocity (whose time integral is a displacement) to aiscontinuous applied force. This limitation probably gave rise tohe non-convergence problems reported in [3,4] as “infinite gasenetration at the bottom of the riser”.

A discontinuous pressure boundary condition occurs when ahoke valve is located at the top of the riser and the valve openings changed discontinuously. For a constant valve opening, a discon-inuity in pressure also occurs in the severe slugging slug formationtage (see Fig. 1(b)): pressure drop across the choke valve is negli-ible for gas flow only, but becomes important when the mixtureevel reaches the top of the riser. For these cases, discontinuous


uperficial velocities appear, giving unrealistic results.In [5] simulations were made, in which the separation pressure

as constant and there was no choke valve; in this case, continu-ty in pressure (and also in pressure time derivative) at the bottom

PRESSl Science xxx (2013) xxx–xxx 3

of the riser was assured by suitable pipeline switching conditionsbetween the two possible configurations (see Section 2.1). Thesimplified friction term implemented in the model improved theaccuracy of the results and had an stabilizing effect. The implicitnumerical procedure used to solve the model equations (includingthe time step and the gas velocity used to displace the nodes in thenonlinear iterations) also contributed to the numerical stability andconvergence.

It becomes evident that a more sophisticated riser model isnecessary to deal with general boundary conditions, taking intoaccount additional terms in the dynamic equations. In addition togravitational and friction, inertial and compressibility effects canbe taken into account.

Compressibility effects are necessary for rapid transients inwhich discontinuous velocity boundary conditions exist, such as asudden closure of the choke valve, giving rise to pressure waves andincreasing the number of the characteristic values and the complex-ity of the numerical solving procedure. Inertial effects, associated tothe fluid acceleration, can deal with discontinuous pressure bound-ary conditions and can be added as a good approximation for slowtransients.

This paper shows an improvement in the model previously pub-lished by the author [5], including inertial effects.

In the riser, inertial terms are taken into account by using therigid water-hammer approximation [11]. In this approximation,the acceleration terms for the liquid and gas phases are taken intoaccount in the momentum equation, but compressibility of the liq-uid phase is neglected in the mass conservation equation. In thepipeline, convective acceleration terms are neglected but the localacceleration terms for the water and gas phases are included in themomentum equations for stratified flow, allowing to calculate theinstantaneous values of pressure drop and void fraction.

The model includes additional devices, such as a valve locatedat the top of the riser and a gas injection line at the bottom of theriser. In this way it is possible to evaluate valve closure and gas liftas mitigating actions for severe slugging.

2. Model

The model considers one-dimensional flow in both pipeline andriser subsystems. The liquid phase is assumed as incompressible,while the gas phase is considered as an ideal gas. Both phases flow inisothermal conditions. The flow pattern in the pipeline is assumedas stratified. The model is capable of handling discontinuities in theflow, such as liquid accumulation in the pipeline, liquid level in theriser and void fraction waves.

2.1. Pipeline

The pipeline, shown in Fig. 2, can be either in a condition of liquidaccumulation (x > 0) or in a condition of continuous gas penetration(x = 0), where x is the position of the liquid accumulation front. Theexistence of a buffer vessel with volume �e is considered in orderto simulate an equivalent pipeline length Le = �e

A , where A is theflow passage area (A = 1

4 �D2, where D is the inner diameter).Three locations in the pipeline can be identified, with corre-

sponding pressures and superficial velocities for the liquid and gasphases: the bottom (Ppb, jlb and jgb), the stratified region located atthe liquid accumulation front (Pps, jls and jgs) and the top (Ppt, jlt andjgt).

Mass and momentum conservation equations are applied for the


control volumes corresponding to the liquid and gas phases. In themass equation, a mean pressure Pg(t) = 1

2 (Ppt + Pps) is assumed. Inmass and momentum equations, variations of the void fraction withposition are neglected and a mean void fraction ˛p(t) is considered.

dx.doi.org/10.1016/j.jocs.2013.08.006



varia

Tm

a

j

j

wvt

2

i

j

Ai

wl

−

j

g

j

vt

P

wis

Fig. 2. Definition of

hese approximations are made in order to get a lumped parameterodel for the pipeline.The superficial velocities at the top of the pipeline can be written

s:

lt = Ql0

A(1)

gt = mg0RgTg

PptA(2)

here mg0 and Ql0 are respectively the gas mass flow and the liquidolumetric flow injected in the pipeline and Rg and Tg are respec-ively the gas constant and temperature.

.1.1. Condition x > 0For this condition there is no gas flowing out the pipeline, result-

ng:

gb = 0 (3)

pplying mass conservation equation for the liquid and gas phases,t is obtained:

d˛p

dt= jlb − Ql0

A + ˛pdxdt

L − x(4)

dPg

dt=

−Pg

(jlb − Ql0

A

)+ Rg Tg

A mg0

(L − x) ˛p + Le(5)

here L is the pipeline length and t is time. A mass balance for theiquid in the stratified region yields:

(L − x)d˛p

dt+ jls − Ql0

A= 0 (6)

From Eqs. (4) and (6), it results:

ls = jlb + ˛pdx

dt(7)

From the kinematic condition at the liquid penetration front, weet:

gs = −˛pdx

dt(8)

Applying the momentum conservation equation in the contrololume of liquid region at the penetration front and including fric-ion and inertial terms, we get:

pb = Pps + �lx(

g sin − 2fll jlb|jlb| − djlb)

(9)


D dt

here fll is the Fanning friction factor (assuming that only liquids filling the cross sectional area), g is the gravity acceleration con-tant, �l is the liquid density and is the pipeline inclination angle

bles at the pipeline.

(positive when downwards). The Fanning friction factor is calcu-lated by using the correlation from [12].

2.1.2. Condition x = 0For this condition there is no liquid penetration front, resulting:

jgs = jgb (10)

jls = jlb (11)

Pps = Ppb (12)

Applying mass conservation equation for the liquid and gasphases, it is obtained:

d˛p

dt= 1

L

(jlb − Ql0

A

)(13)

dPg

dt=

−Pg

(jgb + jlb − Ql0

A

)+ Rg Tg

A mg0

L˛p + Le(14)

2.1.3. Switching between conditionsThe superficial velocities must be determined when the pipeline

commutes between the states of liquid accumulation and continu-ous gas penetration.

Assuming that the commutation from the state x > 0 to the statex = 0 happens at time t0, it can be shown that the superficial veloc-ities at times immediately before t−

0 and immediately after t+0 can

be determined as follows:

dx+

dt= 0 (15)

j−gb

= 0 (16)

j+gb

= −˛pdx−

dt(17)

j+lb

= j−lb

+ ˛pdx−

dt(18)

Assuming that commutation from the state x = 0 to x > 0 happensat time t0, it can be shown that the superficial velocities at timesimmediately before t−

0 and immediately after t+0 can be determined

as follows:

j−gb

= j+gb

= 0 (19)

j+lb

= j−lb

(20)

+


dx

dt= 0 (21)

In Eqs. (15)–(21) the superscripts − and + denote variables eval-uated respectively at t−

0 and t+0 . In this way, in any commutation

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J.L. Balino / Journal of Computational Science xxx (2013) xxx–xxx 5

tp

2

msTtr

u

u

�

j

j

wfctrt

cp

Fi

P

Imat

Fig. 3. Stratified flow at the pipeline.

he total superficial velocity, pressure and time derivative of theressure at the bottom of the riser are continuous.

.1.4. Local equilibrium condition for stratified flowIn previous models, the void fraction at the pipeline is deter-

ined from an algebraic relationship evaluated from the stationarytate, derived from the momentum balance in stratified flow [10].his relationship can be generalized by including the local iner-ial terms. Assuming stratified flow (see Fig. 3), the generalizedelationship can be written as:

�wgSg

˛p− �wl

Sl

1 − ˛p+ �iSi

(1

1 − ˛p+ 1

˛p

)

+ (�l − �g)Ag sin − A

(�l

dul

dt− �g

dug

dt

)= 0

(22)

g = jg˛p

(23)

l = jl1 − ˛p

(24)

g = Pg

RgTg(25)

g = 12

(jgt + jgs) (26)

l = 12

(Ql0

A+ jls

)(27)

here jg and jg are respectively the mean superficial velocitiesor gas and liquid, Sg, Si and Sl are respectively the gas, interfa-ial and liquid wetted perimeters, �wg, �i and �wl are respectivelyhe wall-gas, interface and wall-liquid shear stresses, ug and ug areespectively the mean phase velocities for gas and liquid and �g ishe mean gas density.

The pressure gradient along the stratified region in the pipelinean be obtained from the linear momentum equation for the liquidhase as:

∂P

∂s= −�wl

Sl

Al+ �i

Si

Al+ �l

(g sin − dul

dt

)(28)

inally, the pressure at the top of the pipeline can be obtained byntegrating Eq. (28) as:

pt = Pps −∫ L−x

0

∂P

∂sds = Pps − ∂P

∂s(L − x) (29)


n Eqs. (22)–(29) the wetted and interfacial perimeters are deter-ined considering a stratified geometry, while the shear stresses

re related to the superficial velocities of the phases through fric-ion factors based on the hydraulic diameters for each phase [13].

Fig. 4. Definition of variables at the riser.

2.2. Riser

Continuity equations for the phases are considered at the riser(see Fig. 4). This results in the following set of equations:

−∂˛

∂t+ ∂jl

∂s= 0 (30)

∂∂t

(P˛) + ∂∂s

(Pjg) = 0 (31)

where jg, jl and j are respectively the gas, liquid and total superficialvelocities, P is the pressure, s is the position along the riser, is thevoid fraction and ug and ul are respectively the gas and liquid phasevelocities.

The superficial velocities for the phases are determined by usinga drift flux correlation, assumed to be locally valid:

jg = ug = ˛(Cdj + Ud) (32)

jl = j − jg = ul(1 − ˛) = (1 − ˛Cd)j − ˛Ud (33)

It will be assumed that the drift parameters Cd and Ud dependat most on the local flow conditions and inclination angle � = � (s),this is, Cd = Cd

(˛, P, j, �

)and Ud = Ud

(˛, P, j, �

)[14,15].

The drift coefficients used in the model are [14]:

• For Frj < 3.5:

Cd = 1.05 + 0.15 sin � (34)

Ud =√

gD(

0.35 sin � + 0.54 cos �)

(35)

• For Frj ≥ 3.5:

Cd = 1.2 (36)

Ud = 0.35√

gD sin � (37)

where the Froude number Frj is defined as:

Frj = j√gD

(38)

Considering as the state variables in the riser the void fraction,pressure and total superficial velocity (functions of position and


time), Eqs. (30) and (31) can be finally rewritten as:

Dg˛

Dt+ ˛

∂∂s

(Cdj + Ud) − ∂j

∂s= 0 (39)

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IN PRESSG ModelJ

6 ational Science xxx (2013) xxx–xxx

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rifttro


J.L. Balino / Journal of Comput

DgP

Dt+ P

∂j

∂s= 0 (40)

here:

Dg

Dt= ∂

∂t+ ug

∂∂s

(41)

A mixture momentum equation is considered, in which thenertial terms corresponding to the liquid and gas phases werencluded:

∂P

∂s= −�m

(g sin � + 2

fmD

j∣∣j∣∣) − ˛�g

Dgug

Dt− (1 − ˛)�l

Dlul

Dt(42)

m = �l (1 − ˛) + P

RgTg (43)

Dl

Dt= ∂

∂t+ ul

∂∂s

(44)

here fm is the Fanning friction factor (dependent on the Reynoldsumber and the relative roughness �/D, where � is the pipeoughness), calculated from [12] using a homogeneous mixturewo-phase model and �m is the mixture density.

.2.1. Gas liftGas lift is a process used to artificially lift fluids from wells where

here is insufficient reservoir pressure. By injecting gas, it is possibleo aerate the liquid column and to reduce the pressure gradient. Dueo this effect, it is acknowledged that gas lift can stabilize the flown severe slugging, although relatively large gas flow rates are nec-ssary [16]. In the model it is considered the possibility of injectingas in a position sl along the riser. Considering the balance con-ervation equations with a gas mass source term, it can be shownhat the gas injection introduces the following discontinuities inosition in the gas superficial velocity, pressure and void fractions:

+g =

Rg TgA mgl + P−j−g

P+ (45)

− =P+

[1 − 1

Rg Tg

(j2g˛

)+]− �lj

2l

(1

1−˛+ − 11−˛−

)

1 − 1Rg Tg

(j2g˛

)− (46)

+ = j+g(Cdj + Ud)+ (47)

here mgl is the injected gas lift mass flow and the superscripts and + denote evaluation respectively at location upstream andownstream of position sl.

.2.2. Gas regionIn transients where the liquid level falls below the top of the

iser (su < st) a gas region is formed, as shown in Fig. 5. This regions modeled considering a constant mean pressure Pgr = 1

2 (Pt + Pu)or the mass balance equation and friction, gravitational and inertialerms in the momentum balance equation; Pt and Pu are respec-ively the pressure at the top and at the liquid level in the riser. Theesulting equations for the mass balance depends on the locationf the gas lift injection sl compared to the liquid level position su.

If su > sl, we get:

dPgr

dt= − Pgr

st − su(jgt − ju) (48)


If su ≤ sl, we get:

dPgr

dt= − Pgr

st − su

(jgt − ju − RgTg

PgrAmgl

)(49)

Fig. 5. Definition of variables at the gas region.

where jgt is the gas superficial velocity at the top of the riser, ju isthe total superficial velocity at the liquid level and st is the positionof the top of the riser.

The momentum conservation equation results, for both cases:

Pu = Pt + �g

[g(zt − zu) +

(2fgg

Djgr

∣∣jgr

∣∣ − djgr

dt

)(st − su)

](50)

jgr = 12

(ju + jgt) (51)

where fgg is the Fanning friction factor considering only gas flowing,jgr and �g are respectively the mean gas superficial velocity and themean gas density at the gas region; zt and zu are respectively thevertical positions at the top of the riser and at the liquid level.

2.2.3. Riser geometryThe riser geometry is characterized by the coordinates X and Z

corresponding to the top of the riser and a set of functions furnish-ing the ordinate z and local inclination angle � as a function of thelocal position s along the riser (see Fig. 4). For a constant angle riser,for instance, it results:

� = arctan(

Z

X

)(52)

st = (X2 + Z2)1/2

(53)

z = s sin � (54)

The definition of geometry for a catenary riser can be seen in [5].

2.2.4. Choke valveIn normal operation in petroleum production systems the choke

valve controls the flow, allowing a production compatible with thereservoir characteristics. In severe slugging, it is acknowledged that


choking can stabilize the flow by increasing the back pressure [2,1].For low pressures, typical of air-water laboratory systems, the valveoperates in subcritical condition; in this case, the flow depends onthe pressure difference across the valve (see Fig. 6). The model

dx.doi.org/10.1016/j.jocs.2013.08.006



cm

P

�

wvac

u

P

wCi

2

tc

P

j

j

f

˛

spf

Fig. 6. Definition of variables at the choke valve.

onsiders a valve characteristic based on the homogeneous flowodel:

t − Ps = Kv12

�mtjt∣∣jt∣∣ (55)

mt = ˛t�gt + (1 − ˛t)�l (56)

here jt, ˛t, �mt, �gt are respectively the total superficial velocity,oid fraction, mixture density and gas density, all of them evaluatedt the top of the riser, Ps is the separator pressure and Kv is the valveonstant.

Another expression for the pressure drop across the valve wassed in [16]:

t − Ps = Cjlt∣∣jlt∣∣ (57)

here jlt is the liquid superficial velocity at the top of the riser and is a dimensional valve constant. According to this relationship, it

s neglected the contribution of the gas phase to pressure drop.

.3. Coupling between pipeline and riser

Assuming the same flow passage area for the pipeline and riser,he pressure and superficial velocities at the bottom of the riser areontinuous:

(s = 0, t) = Pb (t) (58)

g (s = 0, t) = jgb (t) (59)

l (s = 0, t) = jlb (t) (60)

The boundary condition for the void fraction can be obtainedrom Eq. (32) evaluated at the bottom of the riser:

(s = 0, t) = ˛b (t) = jgb

Cdbjb + Udb(61)


Fig. 7 shows the state variables and the coupling between theubsystems. State variables for the pipeline are the average gasressure, void fraction and position of the liquid accumulationront, while for the riser they are the local pressure, void fraction

Fig. 7. Coupling betwe

Fig. 8. Discretization along the characteristic directions.

and total superficial velocity. The pipeline imposes the superfi-cial velocities for the gas and liquid phases at the bottom of theriser, while the riser imposes the pressure to the pipeline; thesevariables are the boundary conditions for the corresponding sub-systems. Additional boundary conditions are the liquid volumetricflow rate and the gas mass flow rate at the pipeline, as well as thegas lift mass flow rate and separation pressure at the riser.

As initial conditions, the stationary conditions were chosen, thisis, the solution of the system of equations obtained after settingequal to zero the time derivatives.

3. Discretization and numerical implementation

The system of equations corresponding to the stationary state,as well as the system of dynamic equations, were discretized andnumerically implemented using the software MATLAB [17].

In the riser a moving grid method was adopted (see Fig. 8),in which node i (1 ≤ i ≤ N − 1) moves with the corresponding gasvelocity (red lines), in order to calculate the directional derivativesof Eq. (41). Last node N moves with the liquid velocity if the liq-uid level falls below the top of the riser (su < st), or remains fixedat position st otherwise. The time step tk+1 is chosen as the timestep such that the characteristic propagated from the N − 1th nodeintersects the position su at time tk + tk+1 if the liquid level fallsbelow the top level in the riser, or as the time step such that thecharacteristic propagated from the N − 1th node intersects positionst otherwise. Values at time tk are interpolated in order to calculatethe directional derivatives of Eq. (44), corresponding to the liquidvelocity (blue lines). An implicit scheme was used, with a predictor-corrector method for treatment of the nonlinearities. Details of theprocedure can be seen in [5].

4. Simulations


In this section, simulations corresponding to experimental datafor a vertical riser are shown, considering the effects of the chokevalve and gas lift injection at the bottom of the riser.

en subsystems.

dx.doi.org/10.1016/j.jocs.2013.08.006



63 m/

nfiPa

j

j

4

eTraP

jslba

ictpvts

Fig. 9. Simulation results for jg0 = 0.0

After a nodalization study, the riser was discretized in N = 21odes. Input flow variables are defined in terms of the super-cial velocities jg0 and jl0 at standard conditions (pressure0 = 1.013 bara, temperature T0 = 293 K); these superficial velocitiesre related to the flows as:

g0 = RgT0mg0

P0A(62)

l0 = Ql0

A(63)

.1. Data from Taitel et al. [3]

The following parameters were chosen for a comparison withxperimental data of [3]: fluid parameters are l = 1 ×10−3 kg/m/s,g = 1.8 × 10−5 kg/m/s, �l = 1 ×103 kg/m3, Rg = 287 m2/s2/K and

g = 293 K; pipeline parameters are L = 9.1 m, Le = 1.69 m and = 5◦;iser height is Z = 3 m; common parameters for pipeline and riserre D = 2.54 × 10−2 m and � = 1.5 × 10−6 m; separation pressure iss = 1.013 bara. No choke valve or gas injection was considered.

Fig. 9 shows results of a simulation for jg0 = 0.063 m/s andl0 = 0.124 m/s for representative variables: gas (jg1 and liquid (jl1)uperficial velocities at the bottom of the riser, position (x) of theiquid penetration front, void fraction (˛1) and pressure (P1) at theottom of the riser, liquid level at the riser (su) and void fraction (˛p)nd pressure drop (Ppe − Ppt) in the stratified region at the pipeline.

It can be seen that, in this case, the stationary state used as thenitial condition is not stable and the system goes to a limit cycle. Itan be observed a large variation of superficial velocities, comparedo the initial stationary values. Variations in the void fraction at the


ipeline are relatively small, supporting the assumption of constantoid fraction made in previous models. On the other hand, varia-ions in pressure drop at the pipeline can be large compared to thetationary values, particularly in the blowdown stage. It can also be

s and jl0 = 0.124 m/s (case 1, Table 1).

observed a liquid penetration front and a plateau in the pressure atthe bottom of the riser, characterizing the flow regime SS1.

Many parameters corresponding to the transient can be calcu-lated from Fig. 9. Considering that the slugging cycle begins whenthe gas passage at the bottom of the riser is blocked, the severe slug-ging period and times corresponding to different stages describedin Section 1 can be calculated. In this case, it can be seen that theliquid level does not remain at the top of the riser. From the sim-ulations, it is also possible to determine the pressure amplitude atthe bottom of the riser, the maximum position of the liquid pene-tration front in the pipeline and the minimum position of the liquidlevel in the riser.

Table 1 shows a comparison of experimental and model simu-lated severe slugging periods for different gas and liquid superficialvelocities. The periods calculated with the model are in very goodagreement with the experimental ones. There are some cases (cases17–24, 26 and 27) in which the simulation predicts a unstable con-dition while the experiment reports a stable condition; as will beseen later in this Section, these points are located close to the sta-bility curve.

In Table 1 are also shown simulation results using the modelsof [5] and [3]. As stated before, in [5] inertia was not taken intoaccount; besides, it was considered a constant void fraction andnegligible pressure drop across the stratified region at the pipeline.In [3] a further simplification was made, considering the riser as alumped parameter system with a mean void fraction and ignoringthe friction term.

The model of [5] also predicts as unstable some stable cases closeto the stability curve (cases 17, 18, 21–23) and shows overall errorsslightly higher than the present model. The model of [3] captures


better the stability curve, although predicts as stable two pointsthat are unstable (cases 11 and 15); the overall errors are higher.

In summary, it can be observed that the model with inertiashows slightly better overall performance than the model of [5]

dx.doi.org/10.1016/j.jocs.2013.08.006



Table 1Comparison with experimental results [3].

Case Experiment Simulation Simulation [5] Simulation [3]

jg0 (m/s) jl0 (m/s) Texp (s) Tsim (s) Error (%) Tsim (s) Error (%) Tsim (s) Error (%)

1 0.063 0.124 24 25.4 5.8 24.2 0.8 22 −8.32 0.064 0.209 20 19.5 −2.5 18.4 −8.0 17 −153 0.123 0.183 15 15.5 3.3 14.7 −2.0 14 −6.74 0.124 0.212 14 14.6 4.3 13.7 −2.1 13 −7.15 0.062 0.679 6 7.36 22.7 6.6 10 6 0.06 0.063 0.367 13 13.3 2.3 12.0 −7.7 11 −15.47 0.063 0.679 9 7.35 −18.3 6.5 −27.8 6 −33.38 0.064 0.535 10 9.27 −7.3 8.3 17.0 7 −30.09 0.065 0.226 19 18.7 −1.6 17.3 −8.9 16 −15.810 0.122 0.374 11 10.8 −1.8 9.8 −10.9 9 −18.111 0.123 0.621 8 7.36 −8.0 6.8 −15.0 Stable NA12 0.126 0.228 13 14.0 7.7 13.0 0.0 12 −7.713 0.187 0.226 11 11.9 8.2 10.8 −1.8 10 −9.114 0.188 0.466 8 8.22 2.8 7.3 −8.8 7 −12.515 0.188 0.502 7 7.86 12.3 7.0 0.0 Stable NA16 0.19 0.312 10 10.1 1.0 9.2 −8.0 8 −20.017 0.058 0.705 Stable 7.19 NA 6.4 NA Stable NA18 0.063 0.698 Stable 7.21 NA 6.5 NA Stable NA19 0.122 0.730 Stable 6.45 NA Stable NA Stable NA20 0.126 0.673 Stable 14.0 NA Stable NA Stable NA21 0.126 0.085 Stable 18.2 NA 18.2 NA Stable NA22 0.184 0.127 Stable 4.24 NA 13.0 NA Stable NA23 0.185 0.161 Stable 12.3 NA 12.0 NA Stable NA24 0.187 0.551 Stable 7.36 NA Stable NA Stable NA25 0.188 0.755 Stable Stable NA Stable NA Stable NA26 0.19 0.685 Stable 6.24 NA Stable NA Stable NA27 0.313 0.433 Stable 14.0 NA Stable NA Stable NA28 0.314 0.347 Stable Stable NA Stable NA Stable NA29 0.319 0.614 Stable Stable NA Stable NA Stable NA30 0.321 0.744 Stable Stable NA Stable NA Stable NA31 0.43 0.604 Stable Stable NA Stable NA Stable NA

N

wo

ttfavfAucr

sacpvS

csdovsfstat

and 7 in Table 1, for instance, show a large discrepancy in experi-mental period for almost the same values of superficial velocities,indicating that these or other variables were not kept constant inthe experiment.

32 0.433 0.701 Stable Stable

hich, in turn, shows better overall performance than the modelf [3].

The errors appearing in the comparison are reasonable for awo-phase dynamic system, considering the simplifying assump-ions and that the model does not have any fitting constant adjustedrom the experimental data to be predicted. Among the simplifyingssumptions, it can be mentioned the use of a drift flux correlationalid for constant flow in ducts [14] and the determination of theriction pressure drop with a homogeneous two-phase flow model.s far as the author knows, there is no data for drift flux coefficientsnder accelerating flow, but a more sophisticated model to cal-ulate the friction pressure drop might improve the simulationesults.

With the simulations, it is also possible to obtain the numerictability curve by keeping constant a value of liquid or gas flow ratend varying the other in fixed increments until passing from oneondition (stable or unstable) to another; when this happens, therocedure is repeated with half the increment until achieving con-ergence. The procedure is laborious and computationally costly.tability maps generated for catenary risers can be seen in [5,7].

Fig. 10 shows the numerically generated stability map for theonditions corresponding the experimental conditions of [3]. In theame figure the experimental data points are shown; these unstableata points were classified as “unstable fall” (su < st in the transient)r “unstable no fall” (su = st always in the transient), based on aisual observation. It can be seen that the numerically generatedtability curve includes all the unstable data points. Data pointsor which there is a discrepancy between experiment and stability


imulation prediction are located close to the stability curve; forhese points pressure amplitudes and liquid penetration lengthsre small and liquid level in the riser remains at the top or closeo it, making difficult to differentiate between the severe slugging

A Stable NA Stable NA

instability condition and the fluctuations associated to the intermit-tent flow based only on a visualization. Besides, the experimentaldetermination of the stability curve requires a very careful controlof variables such as separator pressure and input flows; cases 5


Fig. 10. Numerically generated stability map and data from [3].

dx.doi.org/10.1016/j.jocs.2013.08.006



Table 2Comparison with experimental results, choke valve, C = 1.2 × 105 Pa s2/m2 [16].

Case Experiment Simulation Simulation [16]

jg0 (m/s) jl0 (m/s) Texp (s) Tsim (s) Error (%) Tsim (s) Error (%)

1 0.0753 0.0959 46.6 54.9 17.8 44.9 −3.62 0.1147 0.0949 37.6 43.7 16.2 38.1 1.33 0.1739 0.0959 31.8 32.8 3.1 Stable NA4 0.0781 0.0497 47.5 56.2 18.3 50.7 6.75 0.1181 0.0487 38.5 43.7 13.5 Stable NA6 0.0809 0.1704 45 50.4 12.0 35 −22.07 0.1209 0.1693 39.9 42.4 6.3 31 −22.38 0.1713 0.0497 31.5 31.8 1.0 Stable NA9 0.1209 0.2365 Stable 43.0 NA Stable NA10 0.1734 0.2354 Stable 36.7 NA Stable NA11 0.2493 0.2386 Stable 30.6 NA Stable NA12 0.1698 0.1693 Stable 35.5 NA Stable NA13 0.2474 0.1704 Stable 28.5 NA Stable NA14 0.2502 0.0959 Stable 25.0 NA Stable NA

4

uˇ

io

lvamcswus

oaa((

TC

15 0.251 0.0497 Stable

.2. Data from Jansen et al. [16]

In [16] it was used the same test facility as in [3]. The sim-lations parameters were the same, except for Le = 10 m and

= 1◦.A choke valve with C = 1.2 × 105 Pa s2/m2, see Eq. (57), was

ntroduced at the top of the riser in order to study the influencef choking on severe slugging; no gas injection was considered.

Table 2 shows a comparison of experimental and model simu-ated severe slugging periods for different gas and liquid superficialelocities. As in Section 4.1, the experimental and simulated periodsre in very good agreement, with some cases (9–15) in which theodel predicts as unstable some stable cases close to the stability

urve. According to the model presented in Section 2.2.4, pres-ure drop at the choke valve is not dependent on the gas flow,hat seems to be a too simplistic assumption and a source ofncertainties; a more sophisticated valve model could improve theimulation results.

In Table 2 are also shown simulation results using the modelf [16], which is a modification of the model of [3] (without inertia


nd friction) to take into account the choke valve. It can be observed better overall performance for small liquid superficial velocitiescases 1, 2 and 4) and a better capture of the stability conditioncases 9–15). On the other hand, higher errors appear for higher

able 3omparison with experimental results, gas injection, jg0gl = 0.091 m/s [16].

Case Experiment

jg0 (m/s) jl0 (m/s) Texp (s)

1 0.0808 0.2528 21.5

2 0.1153 0.2549 18.9

3 0.1702 0.2571 14.9

4 0.2515 0.2582 13.4

5 0.0791 0.152 24.1

6 0.1147 0.152 20.5

7 0.3125 0.1542 10.8

8 0.1695 0.1013 16.8

9 0.248 0.0949 13.2

10 0.1129 0.0487 27

11 0.1737 0.0476 18.8

12 0.3215 0.0916 9.5

13 0.2489 0.0465 14

14 0.366 0.1552 Stable

15 0.4115 0.1552 Stable

16 0.369 0.0981 Stable

17 0.3141 0.0444 Stable

22.0 NA Stable NA

liquid superficial velocities (cases 6 and 7) and some cases exist inwhich a unstable flow is predicted as stable (cases 3, 5 and 8).

In another experimental campaign, a constant gas superficialvelocity at standard condition jg0gl = 0.091 m/s was injected at thebottom of the riser in order to study the influence of gas lift onsevere slugging; no choke valve was considered. The gas superficialvelocity and the corresponding gas mass flow are related by:

jg0gl = RgT0mgl

P0A(64)

Table 3 shows a comparison of experimental and model simu-lated severe slugging periods for different gas and liquid superficialvelocities. Compared to the former simulations, errors are higherfor the cases with low gas superficial velocities, comparable to theinjection gas superficial velocity (cases 1–3, 5 and 6). Again, thereare some stable cases close to the stability line (14–17) in whichthe model prediction is unstable. In general, the agreement withthe experimental data is pretty good.

In Table 3 are also shown simulation results using the model of[16], which is a modification of the model of [3] (without inertia


and friction) to take into account the gas injection. It can also beobserved higher errors for low gas superficial velocities. Besides,there are some cases in which a unstable flow is predicted as stable(cases 7–13).

Simulation Simulation [16]

Tsim (s) Error (%) Tsim (s) Error (%)

29.8 38.6 29.1 35.323.8 25.9 21.8 15.318.1 21.5 16.1 8.113.2 −1.5 11.6 −13.436.9 53.1 33.8 40.227.3 33.2 24.4 19.011.0 1.9 Stable NA19.4 15.5 Stable NA13.9 5.3 Stable NA28.0 3.7 Stable NA19.7 4.8 Stable NA11.1 16.8 Stable NA13.6 −2.9 Stable NA9.82 NA Stable NA8.96 NA Stable NA9.41 NA Stable NA10.4 NA Stable NA

dx.doi.org/10.1016/j.jocs.2013.08.006



0.063

4

sgae

4

eo

csilap

Fig. 11. Simulation results for jg0 =

.3. Severe slugging mitigation strategies

In this section some simulations are shown, assessing differenttrategies used to mitigate or suppress severe slugging. Choking,as injection and separation pressure increase are studied through

comparison between the case shown in Fig. 9 (considered as ref-rence) and simulations in which suitable parameters are varied.

.3.1. ChokingFig. 11 shows simulation results for the conditions of Fig. 11,

xcept that a choke valve with kv = 100 was added. The followingbservations can be made by making a comparison with Fig. 9.

It can be observed that the flow is also unstable and a limitycle develops from the stationary state. As choking (as well as anytabilizing action) modifies the stability map and shifts the operat-


ng point closer to the stability line, it takes longer to develop theimit cycle. As the flows of the phases are kept constant, there isn increase in pressure in the stationary state and also in the meanressure during the transient. This pressure increase gives rise to a

Fig. 12. Simulation results for jg0 = 0.063

m/s, jl0 = 0.124 m/s and kv = 100.

decrease in the severe slugging period and in the fluctuations of thevariables of interest. For this condition, it can be observed that theliquid level remains at the top of the riser. Besides, there is a muchsmaller liquid penetration front but the pressure at the bottom ofthe riser does not show any plateau, indicating that the riser is notcompletely filled with liquid and characterizing the flow regimeSS2.

Fig. 12 shows simulation results for the conditions of Fig. 11,except that a choke valve coefficient was set to kv = 150. It can beobserved that the small numerical perturbations (coming from thesimulation with a dynamic computer code using an initial conditioncalculated with a stationary computer code) fade away, reaching asteady state. In this case, the flow is stable.


4.3.2. Gas injectionFig. 13 shows simulation results for the conditions of Fig. 11,

except that gas was injected at the bottom of the riser with a super-ficial velocity jg0gl = 0.1 m/s.

m/s, jl0 = 0.124 m/s and kv = 150.

dx.doi.org/10.1016/j.jocs.2013.08.006

Please cite this article in press as: J.L. Balino, Modeling and simulation of severe slugging in air-water systems including inertial effects,J. Comput. Sci. (2013), http://dx.doi.org/10.1016/j.jocs.2013.08.006



Fig. 13. Simulation results for jg0 = 0.063 m/s, jl0 = 0.124 m/s and jg0gl = 0.1 m/s.

Fig. 14. Simulation results for jg0 = 0.063 m/s, jl0 = 0.124 m/s and jg0gl = 0.3 m/s.

Fig. 15. Simulation results for jg0 = 0.063 m/s, jl0 = 0.124 m/s and Ps = 1.3 bara.

dx.doi.org/10.1016/j.jocs.2013.08.006



.063 m

tatiigvtj

4

e

wcptop

5

iplaistbi

lavmraotw

[

Fig. 16. Simulation results for jg0 = 0

Although the quantity of injected gas in the riser is greater thanhe quantity of gas in the pipeline, the flow is still unstable. Making

comparison with Fig. 9, it can be observed a longer time to achievehe limit cycle and a shorter severe slugging period. Although theres a liquid penetration front in the pipeline, the gas superficial veloc-ty and void fraction are always positive in the riser due to theas injection. Pressure drop across the riser and fluctuations of theariables of interest are reduced; the liquid level remains at theop of the riser. Increasing the gas injection superficial velocity tog0gl = 0.3 m/s the flow stabilizes, as shown in Fig. 14.

.3.3. Separation pressureFig. 15 shows simulation results for the conditions of Fig. 11,

xcept that the separation pressure was increased to Ps = 1.3 bara.For this condition, the flow is still unstable. Making a comparison

ith Fig. 9, it can be observed a longer time to achieve the limitycle and a shorter severe slugging period. There is a reduced liquidenetration front in the pipeline, while the liquid level remains athe top of the riser. Pressure drop across the riser and fluctuationsf the variables of interest are reduced. Increasing the separationressure to Ps = 1.7 bara the flow stabilizes, as shown in Fig. 16.

. Conclusions

A mathematical model and numerical simulations correspond-ng to severe slugging in air-water pipeline-riser systems, areresented. The model is an improvement of the one previously pub-

ished by the author [5], including inertial effects. Inertial effectsre taken into account by using the rigid water-hammer approx-mation, which was numerically implemented without increasingubstantially the complexity of the model. The inclusion of iner-ial effects allows the riser for dealing with discontinuous pressureoundary conditions without showing unrealistic discontinuities

n the superficial velocities and void fraction.A comparison is made with experimental results published in

iterature for vertical risers including the effect of a choke valvet the top and gas injection at the bottom of the riser, showingery good results for slugging cycles and stability maps. The presentodel predicts the period of severe slugging with reasonable accu-

acy. In some cases, the model predicts as unstable, cases that


re stable; in this way, the unstable region in the stability map isverestimated. Other simpler models, not taking into account iner-ia and friction [3,16], predict some unstable conditions as stable,hich is not safe for design purposes. Simulation performance can

[

[

/s, jl0 = 0.124 m/s and Ps = 1.7 bara.

be improved by using more sophisticated models for the frictionterm in the riser, as well as for the choke valve. An important issuein making a comparison between experimental data and simulationis the measurement and control of the inlet flows, in order to haveaccurate and constant parameters used as boundary conditions.

Simulations were made assessing the effect of different strate-gies to mitigate severe slugging, such as choking, gas injection andincrease in separation pressure, showing correct trends. Throughthese studies it is possible to built stability and flow pattern mapsfor different sets of system parameters.

Acknowledgements

This work was supported by Petróleo Brasileiro S. A. (Petrobras).The financial support of Fundac ão de Amparo à Pesquisa do Estadode São Paulo (FAPESP) to attend IMAACA 2012 (where a shorterversion of this paper received the Best Paper Award) is deeplyacknowledged.

References

[1] Y. Taitel, Stability of severe slugging, International Journal of Multiphase Flow12 (2) (1986) 203–217.

[2] Z. Schmidt, Experimental study of two-phase slug flow in a pipeline-riser sys-tem, The University of Tulsa, Tulsa, USA, 1977 (Ph.D. thesis).

[3] Y. Taitel, S. Vierkand, O. Shoham, J.P. Brill, Severe slugging in a riser system:experiments and modeling, International Journal of Multiphase Flow 16 (1)(1990) 57–68.

[4] C. Sarica, O. Shoham, A simplified transient model for pipeline-riser systems,Chemical Engineering Science 46 (9) (1991) 2167–2179.

[5] J.L. Balino, K.P. Burr, R.H. Nemoto, Modeling and simulation of severe sluggingin air-water pipeline-riser systems, International Journal of Multiphase Flow36 (2010) 643–660.

[6] C. Wordsworth, I. Das, W.L. Loh, G. McNulty, P.C. Lima, F. Barbuto, MultiphaseFlow Behavior in a Catenary Shaped Riser. CALTEC Report No.: CR 6820, Bedford,UK, 1998.

[7] R.H. Nemoto, J.L. Balino, Modeling and simulation of severe slugging with masstransfer effects, International Journal of Multiphase Flow 40 (2012) 144–157.

[8] S. Mokhatab, Severe Slugging in Offshore Production Systems, Nova SciencePublishers, Inc., New York, 2010.

[9] J.M. Masella, Q.H. Tran, D. Ferre, C. Pauchon, Transient simulation of two-phaseflows in pipes, International Journal of Multiphase Flow 24 (1998) 739–755.

10] Y. Taitel, A.E. Dukler, A model for predicting flow regime transitions in hor-izontal and near horizontal gas–liquid flow, American Institute of Chemical


Engineers Journal 22 (1) (1976) 47–55.11] M.H. Chaudhry, Applied Hydraulic Transients, Van Nostrand Reinhold, New

York, USA, 1987.12] N.H. Chen, An explicit equation for friction factor in pipe, Industrial and Engi-

neering Chemical Fundamentals 18 (1979) 296–297.

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http://refhub.elsevier.com/S1877-7503(13)00102-6/sbref0005



















































































































































































































































































ING ModelJ

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Instituto Balseiro (1985–2000) in Argentina and Institutode Pesquisas Energéticas e Nucleares (2001–2003) at SãoPaulo, Brazil. Since 2004 he is Professor at Universidade


4 J.L. Balino / Journal of Comput

13] S.L. Kokal, J.F. Stanislav, An experimental study of two-phase flow in slightlyinclined pipes – I. Flow patterns, Chemical Engineering Science 44 (1989)665–679.

14] K.H. Bendiksen, An experimental investigation of the motion of long bubbles ininclined tubes, International Journal of Multiphase Flow 10 (4) (1984) 467–483.

15] B. Chexal, G. Lellouche, J. Horrowitz, J. Healer, A void fraction correlation forgeneralized applications, Progress in Nuclear Energy 27 (4) (1992) 255–295.


16] F.E. Jansen, O. Shohan, Y. Taitel, The elimination of severe slugging – experi-ments and modeling, International Journal of Multiphase Flow 22 (6) (1996)1055–1072.

17] E.B. Magrab, S. Azarm, B. Belachandran, J.H. Duncan, K.H. Herold, G.C. Walsh,An Engineer ’s Guide to MATLAB, Pearson Prentice Hall, New Jersey, USA, 2010.

PRESSl Science xxx (2013) xxx–xxx

Jorge Luis Balino was born in Buenos Aires, Argentina.He graduated (1983) and made a PhD (1991) in NuclearEngineering at Instituto Balseiro, Argentina. He workedfor echint S.A. (1983–1984), Centro Atómico Bariloche and


de São Paulo. His research interests are Fluid Dynamics,Heat Transfer and Multiphase Flow.

dx.doi.org/10.1016/j.jocs.2013.08.006

































































































































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