Chemical Engineering Science 142 (2016) 165–179

Contents lists available at ScienceDirect

Chemical Engineering Science

http://d0009-25

n CorrE-m

journal homepage: www.elsevier.com/locate/ces

Prediction of the liquid film distribution in stratified-dispersedgas–liquid flow

M. Bonizzi n, P. AndreussiTEA Sistemi, Pisa, Italy

H I G H L I G H T S

� A model for liquid film distribution in gas–liquid stratified dispersed flows has been derived.

� The model allows the numerical calculation of the local axial liquid film height and velocity profiles.� Droplet deposition, gravitational drainage and wave spreading are relevant.� The strength of each mechanism depends on the underlying flow conditions.� The wave spreading affect is modelled as function of a modified Froude number.

a r t i c l e i n f o

Article history:Received 30 July 2015Received in revised form28 October 2015Accepted 9 November 2015Available online 14 December 2015

Keywords:Stratified dispersed gas–liquid flowsLiquid film distributionMultiphase flow modellingEntrainmentDepositionWave spreading

x.doi.org/10.1016/j.ces.2015.11.04409/& 2015 Elsevier Ltd. All rights reserved.

esponding author. Tel.: +390506396140ail address: marco.bonizzi@tea-group.com (M

a b s t r a c t

A mathematical model for predicting the circumferential liquid film distribution in stratified-dispersedflow is presented. Objective of the model is to describe the typical flow conditions of wet gas trans-portation in long, near-horizontal pipelines. In these applications, depending on the gas velocity and pipediameter, a large asymmetry of the liquid film distribution may arise. The model is based on theassumption that in stratified-dispersed flow, liquid droplets can only be entrained by the gas from thethick liquid layer flowing at pipe bottom. It is also assumed that the deposition of smaller droplets isrelated to an eddy diffusivity mechanism and regards the entire pipe circumference, while larger dro-plets deposit by gravitational settling on the pipe bottom. These assumptions explain the formation of athin, non-atomizing film in the upper part of the pipe. The presence and flow structure of this filmappreciably affect the pressure gradient and the liquid hold-up in the pipe and are of great importance inflow assurance studies. The model has been validated against i) the experimental observations recentlypublished by Pitton et al. (2014), the data collected by ii) Laurinat (1982), iii) Dallman (1978), and iv) thepredictions of three-dimensional CFD simulations conducted by Verdin et al. (2014). It is shown that therelevant mechanisms which are responsible for the liquid film distribution are the gravitational filmdrainage, droplet entrainment/deposition and wave spreading. In particular, at high gas velocities and/orsmall pipe diameters, the asymmetry of the liquid film diminishes owing to the wetting mechanism ofwave spreading which makes the distribution of the film more uniform in the circumferential direction.As the gas velocity diminishes and/or for larger pipe diameters, wave spreading is less effective and forthese flow conditions only gravitational drainage and droplet entrainment/deposition are responsible forthe more asymmetric shape of the liquid film.

& 2015 Elsevier Ltd. All rights reserved.

1. Introduction

Pipeline transportation over long distances of natural gas orsaturated steam in presence of a liquid phase is a common practicein the oil and the geothermal industry and can be extremelychallenging when major flow assurance issues, such as corrosion

. Bonizzi).

or solid formation and deposition on pipe wall arise. In near-horizontal pipes, stratified flow conditions are encountered atmoderate phase velocities. At increasing the gas velocity, only partof the liquid flows at the pipe wall, while the remaining liquid isentrained by the gas in the form of droplets which tend to depositback onto the wall layer. The competing phenomena of dropletentrainment and deposition determine the liquid hold-up in thepipe and appreciably affect the pressure gradient. In large pipes

M. Bonizzi, P. Andreussi / Chemical Engineering Science 142 (2016) 165–179166

the resulting flow pattern is usually classified as stratified-dispersed flow, while in smaller pipes as horizontal annular flow.

The critical flow parameter to be measured in stratified-dispersed flow is the flow rate and thickness distribution of theliquid layer flowing at pipe wall. This is because the split of theliquid phase determines the overall liquid hold-up in the pipe andthe value of the frictional pressure losses. Besides to the fluid-dynamic issue, a better knowledge of the flow behavior of the walllayer has many implications in flow assurance studies. In parti-cular, the effectiveness of the inhibitors usually adopted to preventpipe corrosion depends on the formation of a liquid film aroundthe pipe wall.

In stratified-dispersed flow, the flow field presents strong 3-Dfeatures. This makes difficult to describe this flow pattern intransient 1-D flow simulators, such as the model proposed byBonizzi et al. (2009). In industrial applications, these simulatorsare widely adopted for flow assurance studies, but often theirpredictions are poor. The main objective of the present work is todevelop a detailed model of stratified-dispersed flow. This modelcan then be coupled with a 1-D flow simulator and provide acomplete picture of this flow pattern.

Stratified-dispersed or horizontal annular flow is more com-plicated than annular flow in a vertical pipe, due to the gravityforce, which typically causes an asymmetrical liquid film dis-tribution around the pipe circumference. For instance, Paras andKarabelas (1991) and Williams et al. (1996) observed significantgradients of the liquid film height in the circumferential directionand large vertical gradients of the droplet concentration. Theseauthors used a sampling probe to measure droplet concentrationand a conductance technique to measure the local film heights.

One of the pioneering investigations on horizontal annular flowhas been carried out by Butterworth (1969), who measured thefilm thickness distribution of air/water flow in a 3.18 cm horizontalpipe with a conductance method. This author argued that fivemechanisms may contribute to the asymmetrical film distribution:

� Gravitational drainage.� Spreading of the film by wave motion.� Liquid transfer because of atomization and deposition effects.� Interfacial stresses due to the gas secondary flow.� Surface tension effects.

At the end of his analysis, Butterworth concluded that the filmthickness distribution was determined by a balance between thefilm drainage due to gravitational effects and the upward liquidmovement associated with the lateral spreading of largedisturbance waves.

A similar investigation was carried out by Lin et al. (1985) whoanalyzed the film distribution in a 2.69 cm I.D. pipe using a needleprobe approach and performed a modelling analysis based on thefundamental conservation equations of mass and momentumwritten for the liquid film. These authors suggest a relevant effectof the term associated with the gas secondary flow.

Laurinat (1982) conducted an experimental study of air–waterhorizontal annular flow in a 5.08 cm I.D. pipe. In these experi-ments the liquid film height was measured at 7 different cir-cumferential locations using conductivity probes. Laurinat et al.(1985) developed a 2-D model of liquid flow based on momentumconservation equations, where both normal and tangential stres-ses were considered. These authors found that a good agreementwith the experimental data could be obtained by acting on thenormal shear stress along the circumferential coordinate, while intheir model the effect of gas secondary flow was negligible.

In both afore mentioned models, the direction of the gas sec-ondary flows was modelled to be upwards, namely with flowdirected downward along the vertical pipe diameter and upward

at the walls. Nonetheless, it should be remarked that some con-troversy exists on the role of secondary gas flows in horizontaltwo-phase flows. For instance, Fisher and Pearce (1993) deter-mined the liquid film distributions for horizontal annular flow in a5 cm I.D. pipe and developed a model that neglected the second-ary flow effect; yet they report a fair agreement between modelpredictions and the corresponding experimental measurements.

Secondary flows have been extensively investigated in the lit-erature, and contradictory findings were published. The firstdetailed observations of turbulent secondary flows were made byNikuradse (1930) and Prandtl (1927). The first used both flowvisualization with a red dye and Pitot tube measurements to mapthe gas velocity profiles. The second suggested that the shape ofthe measured velocity contours implied the existence of secondarymotion. According to Prandtl, turbulent velocity fluctuations existtangent to the curved contours of constant mean axial velocity (i.e.isotach) surfaces, and these fluctuations increase with increasedcurvature of the isotachs. Hence, the resulting Reynolds stresseswill generate forces on the convex side of the isotachs, which giverise to the secondary flows. According to this observation, con-sidering the case of a gas–liquid flow in a circular pipe, a cir-cumferential disturbance such as the asymmetric distribution ofthe liquid film (which would then lead to an asymmetrical inter-facial roughness) might be sufficient in order to get secondaryflows initiated under turbulent gas flow conditions.

As mentioned above, in a circular duct the gas secondary flowmay be directed downwards along the vertical diameter orupwards. Darling and McManus (1969) conducted an experimentusing a pipe with an eccentric thread, being deeper at the bottomthan the top. In this way they could simulate the conditions of anon-uniform liquid film. Using hot-wire velocity measurements,they found that the gas velocity profile was skewed toward thebottom of the pipe. This indicates the presence of secondary flowsdirected downwards along the vertical diameter.

Similar observations were reported by Andreussi and Persen(1987) and by Vlachos et al. (2003). The latter authors adopted aLaser Doppler method to measure the time-averaged gas flow fieldin 5 cm and 2.4 cm pipes for gas–liquid stratified flow and con-firmed the presence of secondary flows, directed downwardsalong the vertical diameter. It has to be remarked that the range ofgas superficial velocities investigated by Vlachos et al. (2003) wasbelow 12 m/s.

Dykhno et al. (1994) took detailed velocity measurements inair–water stratified/annular horizontal flows for a 9.5 cm pipe.Using Prandtl's interpretation of curved isotachs, they confirmedthe existence of secondary flows. These authors were the first toidentify conditions under which the direction of the secondaryflows changed: while at lower gas velocities (typically o20 m/s)the motion of secondary flows was directed downward at thecenter, at higher gas velocities the secondary flows appeared to bedirected upwards. Dykhno et al. (1994) argued that the atomiza-tion of the liquid film was responsible for the change in directionof the gas secondary flows.

Dallman (1978) investigated air–water annular flows in a2.3 cm inner diameter pipe, and, from his measurements of localliquid film thickness, proposed to correlate the film height with amodified Martinelli flow parameter. Hurlburt and Newell (1997)proposed a simplified model for estimating the liquid film dis-tribution, based on the Laurinat et al. (1985) derivation. Theseauthors analyzed available experimental measurements of theliquid film heights for gas–liquid stratified/dispersed flows gath-ered by different researchers for horizontal pipes, and proposed acorrelation for predicting the degree of asymmetry of the liquidfilm, based on a modified Froude number, which represents thesquare root of the ratio between the gas kinetic energy and thework required to pump the liquid from the bottom to the top of

Fig. 1. Adopted frame of reference for model development.

M. Bonizzi, P. Andreussi / Chemical Engineering Science 142 (2016) 165–179 167

the pipe. Hurlburt and Newell (1997) succeeded in eliminating thelarge scatter in the data plotting the ratio of the average liquid filmheight to liquid film height at the pipe bottom versus the modifiedFroude number defined as aforementioned. Moreover, the authorsmade model assumptions which further simplified the mathe-matical liquid film model envisaged by Laurinat et al. (1985); inparticular the term associated to the gas secondary flows wasaltogether dropped from model equations. Nonetheless Hurlburtand Newell (1997) found good agreement with the experimentaldata, despite having to tune a coefficient which altered the mag-nitude of the normal shear stress term associated to the wavespreading effect. These authors concluded that the two relevantmechanisms driving the distribution of the liquid film around thepipe walls are the turbulent normal shear stresses in the cir-cumferential direction (which is related to the wave spreading)and the drainage of the liquid film from the top of the tube due togravity.

During the latest years, some interesting work, conductedusing three-dimensional Computational Fluid Dynamics (i.e. CFD)packages, has been published in the literature. McCaslin andDesjardins (2014) simulated three-dimensional liquid–gas annularflows using CFD methods, and, by applying dimensional analysis,these authors selected the relevant governing parameters of themulti-phase flow problem under investigation. From the obtainedCFD results, McCaslin and Desjardins (2014) showed that, for thecases in which the liquid is not significantly drained (in otherwords when the asymmetry of the liquid film distribution is nottoo pronounced and the gradient of the liquid film height alongthe circumferential coordinate is not particularly high), the actionof the surface waves tends to drive the liquid up the pipe walls. Byreducing the magnitude of the gas stream kinetic energy, the CFDresults indicated more significant asymmetry of the liquid filmdistribution. Quite interestingly, under these conditions, McCaslinand Desjardins (2014) noticed an almost total absence of theupwards liquid film motion by wave spreading.

Verdin et al. (2014) conducted CFD studies for large diameterpipes (380 0 inner diameter), and their main finding was that a verysignificant asymmetry of the liquid film distribution resulted,which became far more evident as the gas velocity diminished.These authors found that almost all the continuous liquid phase(which the authors called the liquid pool) was sitting in the bot-tom part of the tube, and a very thin liquid film (with a thicknessnever greater than 300 μm at the top of the pipe) draining fromthe top of the tube. These authors concluded that dropletdeposition was the physical mechanisms responsible for sustain-ing the thin liquid film draining along the inner perimeter.

In a recent paper, Pitton et al. (2014) report an experimentalinvestigation of stratified-dispersed flow in a horizontal pipe,7.9 cm I.D. diameter operating under an appreciable pressure.These authors measured the circumferential liquid film distribu-tion with an array of conductance probes, which were also used tomeasure the liquid entrainment and the rates of droplet entrain-ment and deposition by the tracer method, originally developedby Quandt (1965) to study vertical annular flows. The main resultsof this investigation have been a fairly accurate measurement ofthe liquid entrainment and a good estimate of the rate of dropletentrainment, which has been found to be one order of magnitudelarger than the values usually adopted in 1-D simulation tools andof the values determined in vertical annular flow.

The experimental observations of tracer mixing along the pipe,led Pitton et al. (2014) to conclude that a simple two-field modelfor describing the underlying physics within a one-dimensionalmodelling framework was not feasible. They suggested to modelthe flow structure using a three-field model, whereby the liquidphase is split between a continuous liquid film flowing at pipe walland two distinct droplet fields: smaller droplets able to interact

with gas turbulent motions (and eventually with the gas second-ary flow) and larger droplets which move on a trajectory flight andre-deposit on the pipe bottom by gravitational settling. In verticalannular flows, the simultaneous presence of two differentmechanisms of droplet transfer (eddy diffusion and trajectorymotion) was reported by Andreussi and Azzopardi (1983).

Another interesting observation reported by Pitton et al. (2014)is that the liquid film tends to be wavy only for angles up to about70° from the bottom for a significant range of flow parameters. Theremaining part of the wall layer appears to be smooth, with thetotal absence of a large disturbance waves. This result can becoupled with the experimental and theoretical work by Andreussiet al. (1985), who found that the critical film flow rate for theinitiation of large waves closely corresponds to the film flow ratebelow which no entrainment occurs. One may then conclude thatthe atomization process would only be relevant for the bulk of theliquid film sitting at the pipe bottom, whereas the thinner liquidfilm wetting the remaining part of the pipe wall would only becharacterized by the two phenomena: the deposition of smallerdroplets and the subsequent drainage of the deposited liquid fromthe top to the bottom of the pipe.

The experimental work reported by Pitton et al. (2014) did notinclude direct measurements of the gas secondary flow, but,according to these authors, the tracer distribution in the wall layerappears to be more consistent with the assumption of a weak gassecondary motion directed downwards along the pipe walls ratherthan upwards. In what follows a unified laminar-turbulent two-dimensional model of liquid flow in horizontal stratified-dispersedflow is presented and it is shown that this model provides a fairlygood fit to the experimental measurements of Pitton et al. (2014).For these data, the main mechanism which is able to counteractthe drainage of the liquid film appears to be the droplet deposi-tion. It is also shown that the normal shear stress gradient due tothe velocity fluctuations of the liquid layer in the circumferentialdirection (term associated to the wave spreading effect) can berelevant at large gas velocities and/or small pipe diameters, suchas the flow conditions investigated by Dallman (1978) and Laurinat(1982).

2. Model derivation

The mathematical model developed in the present work is amodified version of that proposed by Laurinat et al. (1985). Withreference to Fig. 1, let θ denote the angle measured from thebottom of the pipe, R the pipe radius, x, y and z the circumferential,

M. Bonizzi, P. Andreussi / Chemical Engineering Science 142 (2016) 165–179168

radial and axial coordinates respectively. Eqs. (52)–(54) of theLaurinat et al. (1985) work represent the momentum equationwritten in the axial coordinate, circumferential coordinate, and theconservation of mass respectively:

I1τþyz;hþ

I2Rþ

∂τþxz

∂θ¼Γþ

z ð1Þ

I1τþyx;hþ

I2Rþ

∂τþxx

∂θ� I2Rþ Fr

sin θþ cos θRþ

dhþ

dθ

� �¼Γþ

x ð2Þ

dΓþx

Rþdθ¼ Rþ

D �RþA ð3Þ

The derivation of these equations is reported in Appendix A.These equations are written in non-dimensional form and thesuperscript þ represents the corresponding quantity made non-dimensional. In particular, the shear stresses are normalized withrespect to the gas-wall shear stress:

τG ¼ 12f GsρGu

2G ð4Þ

In the equation above the gas-wall shear stress is computedusing the standard Fanning correlation for turbulent flow in asmooth pipe (f Gs ¼ 0:046Re�0:2

Gs ). The τþyz;h term is the ratio of the

interfacial shear stress to the gas wall shear stress:

τþyz;h ¼

τyz;hτG

¼ f intf Gs

ð5Þ

The term τþyx;h in Eq. (2) denotes the shear stress exerted by the

gas secondary flow on the liquid film.Eq. (4) allows to introduce the friction velocity u�,

u� ¼ffiffiffiffiffiτGρL

rð6Þ

The non-dimensional liquid film height and the pipe radius arethen defined as:

hþ ¼ hu�

υL;Rþ ¼ Ru�

υLð7Þ

and the Froude number as:

Fr¼ τGρLgR

ð8Þ

The terms on the right hand side of Eqs. (1) and (2), Γþz and

Γþx , denote the non-dimensional axial mass flow rate per unit

circumferential length and the circumferential mass flow rate peraxial unit length, respectively:

Γþz ¼Γz

μL¼Z hþ

0uþz dyþ ¼ uþ

z

� �hþ ð9Þ

Γþx ¼Γx

μL¼Z hþ

0uþx dyþ ¼ uþ

x

� �hþ ð10Þ

As shown above, the mass flow rates per unit length are madenon-dimensional with respect to the liquid dynamic viscosity μL.As usual, the average value of a variable in the radial direction isdefined as:

φ� �¼

R hþ

0 φdyþ

hþ ð11Þ

The terms on the right hand side of Eq. (3) denote the non-dimensional droplets deposition and atomization fluxes(kg= m2s

� �):

RþD ¼ RD

1ρLu� ð12Þ

and

RþA ¼ RA

1ρLu� ð13Þ

respectively. Provided that adequate closure laws are found for theshear stress terms (τþ

yz;h; τþxz ; τ

þyx;h; τ

þxx ), and likewise for the

deposition and atomization fluxes, Eqs. (1)–(3) represent a systemof 3 equations in 3 variables (Γþ

z ;Γþx ;hþ ).

The model developed by Andreussi et al. (1985) to predict theliquid film flow rate at the onset of the large disturbance waveregime can be used to calculate the critical mass flow rate, belowwhich atomization of the liquid film shall not occur:

Γz;c ¼ μL

4Rez;c; ð14Þ

where the critical liquid Reynolds number can be computed as:

Rez;c ¼ 4Γz;c

μL¼ 7:3 log 10ω

� �3þ4:22 log 10ω� �2�263 log 10ω

� �þ439

ð15ÞIn this correlation, the dimensionless group ω¼ μL=μG

� �ffiffiffiffiffiffiffiffiffiffiffiffiffiρG=ρL

pwas indicated by Andreussi et al. (1985) as the only

quantity upon which the critical liquid Reynolds number depends.As stated by Pan and Hanratty (2002), measurements in hor-

izontal flows by Laurinat (1982) indicate that the critical Reynoldsnumber is around 480, or in other terms

Γz;c ¼ 120μL ð16ÞUnder conditions where

Γz ¼ ρL uzh ih� �oΓz;c ¼ 120μL ð17Þ

atomization of the liquid film shall not occur and the equationswhich express the mass and momentum conservation along thecircumferential direction can be simplified as follows:

dΓx

Rdφ¼ RD φ

� � ð18Þ

and

μL∂2ux

∂y2¼ ρLg sinφ ð19Þ

Eqs. (18) and (19) are written in dimensional form, the angle φis taken from the top of the tube (φ¼ π�θ), and the Dirichlet andNeumann boundary conditions related to the x-momentum Eq.(19) are as follows:

ux;y ¼ 0 ¼ 0;∂ux

∂y y ¼ h ¼ 0 ð20Þ

Eq. (19) is the typical laminar film motion equation and can beeasily integrated taking into account the boundary conditionsexpressed in Eq. (20). This allows the circumferential velocityvariation to be determined as function of the radial coordinate:

ux y;φ� �¼ ρLg

μL

y2

2�hðφÞy

�sinφ ð21Þ

Eq. (21) can then be used to compute the circumferential massflow rate per unit axial length:

Γx φ� �¼ Z y ¼ h φð Þ

0ρLux y;φ

� �dy¼ �ρ2

L gh φ� �3 sinφ3μL

ð22Þ

If the circumferential mass flow rate per unit axial length isknown at the given angle, Eq. (22) allows the liquid film height tobe determined as

h φ� �¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3μL jΓxðφÞjρ2L g sinφ

3

sð23Þ

M. Bonizzi, P. Andreussi / Chemical Engineering Science 142 (2016) 165–179 169

3. Closure equations

The system of Eqs. (1)–(3) requires empirical closures for theshear stresses and the fluxes of droplets deposition and entrain-ment. The first term requiring closure is that related to the inter-facial gas–liquid shear stress τþ

yz;h. When the liquid film is turbu-lent, following the recommendations of Laurinat (1982), the shearstress is assumed to be given by the equation

τþyz;h ¼

τyz;hτG

¼ f intf Gs

¼ C3þC4Γþz ð24Þ

where, according to Laurinat (1982), C3 � 2 and C4 � 10�3. Undera laminar liquid flow regime (i.e. when Eq. (17) is satisfied), theequation recommended by Andreussi et al. (1985) can be adopted,

τþyz;h ¼

τyz;hτG

¼ f intf Gs

¼ 1þ0:023 hþG �10:9

� � ð25Þ

where hþG ¼ hu�

GνG, u�

G ¼ffiffiffiffiffiffiffiτyz;hρG

q.

The equation used to predict τþyx;h is based on the experiments

of Darling and McManus (1969), who measured the film roughnessdistribution for a horizontal air–water annular flow in an eccen-trically threaded pipe. The following correlation has been derivedfrom the analysis of their measurements:

τþyx;h ¼

τyx;hτG

¼ C5τI sinθ ð26Þ

In Eq. (26) τI denotes the (non-dimensional) average interfacialshear stress between the gas core and the liquid film, while theconstant C5 � 3:0� 10�2.

The normal shear stress in the circumferential direction isassumed to be characterized by flow oscillations (in the x-direc-tion) taken as angular deviations of the main flow (in the axialdirection), and therefore to scale with the square of the local meanaxial velocity:

τþxx ¼

τxxτG

¼ �C1uþz 2¼ �C1

Γþz

hþ

!2

ð27Þ

Eq. (27) is similar to the representation of the wave-spreadingeffect proposed by Butterworth (1969). According to this author,the wave spreading mechanism is based on the assumption thatlarge disturbance waves drive the liquid film in front of each waveup the tube walls. In their model, Laurinat et al. (1985) tuned thecoefficient C1 in order to fit the experimental data. The valueadopted for this term (C1 ¼O 10�1

� ) makes the normal shear

stress to be the main term able to balance the gravitational forcesacting on the liquid film, and the authors speculated that themechanisms associated to entrainment and deposition were ofsecondary order.

The present model shows that the competing physicalmechanisms which affect the resulting liquid film distribution area balance between gravitational drainage, droplet deposition andwave spreading. The dominant terms are problem dependent. Infact the C1 coefficient needs an adequate tuning in order to matchthe measured data with fair accuracy. It will be shown that,depending on the specific flow conditions under investigation, thewave spreading effect might be altogether dropped from themodel equations. Moreover, in the Section 6, where the modelresults are compared against the data collected by Dallman (1978)and Laurinat (1982), a novel correlation for estimating the mag-nitude of the C1 coefficient is proposed.

The last term requiring a closure is the dispersion of the z-momentum in the x-direction, which is modelled following the

recommendations by Townsend (1970):

τþxz ¼

τxzτG

¼ C2duþ2

z

dθð28Þ

In the above equation the constant has an order of magnitudearound C2 � 10�2. If Eqs. (24)–(28) are inserted into Eqs. (1) and(2), the momentum equations in the z and x direction can bewritten as follows:

I1τþyz;hþ

C2

Rþ I2d2uþ

z 2

dθ2 ¼Γþz ð29Þ

I1C5τI sin θ� I2RþC1

duþ2z

dθ� I2Rþ Fr

sin θþ cos θRþ

dhþ

dθ

� �¼Γþ

x ð30Þ

When the liquid film is turbulent, Eq. (30) can be greatly sim-plified if the derivative of the axial velocity squared is expressed asfunction of gradients along the circumferential direction of theaxial mass flow rate Γþ

z and the film height hþ :

duþ2z

dθ¼ ddθ

Γþz

hþ

!2

¼ 2Γþz

hþ2

dΓþz

dθ�Γþ

z

hþdhþ

dθ

" #ð31Þ

Eqs. (29) and (24) can then be deployed in order to derive arelation between the angular gradients of the axial mass flow rateand that of the liquid film height:

dΓþz

dθ¼φ

dhþ

dθð32Þ

If Eqs. (32) and (31) are inserted into Eq. (30), an equation forthe circumferential gradient of the liquid film height is immedi-ately derived:

dhþ

dθ¼Γþ

x � I1C5τI sin θþ I2Rþ Fr

sin θI2Rþ

2C1Γþz

hþ 2Γþz

hþ �ϕ�

� cos θRþ Fr

h i ð33Þ

Under turbulent flow conditions of the liquid film, the systemthat will be numerically solved is therefore composed by Eqs. (33),(29) and (3).

In order to close the system, the fluxes of droplets atomizationand deposition must be defined. The atomization flux is based onthe recommendation by Williams et al. (1996) and Pan and Han-ratty (2002):

RA ¼kAu2

G ρGρL

� �1=2σGL

ΓZ�ΓZ;C� � ð34Þ

The atomization constant in Eq. (34) has been taken to bekA � 2:0� 10�6. This value is directly derived from the experi-ments of Pitton et al. (2014) and is about one order of magnitudelarger than the value adopted by Laurinat et al. (1985) and in the1-D transient models used for flow assurance studies.

The flux of droplet deposition is usually expressed as:

RDh i ¼ kDCB ð35Þwhere kD denotes the deposition velocity and CB the bulk con-centration (as mass of liquid droplets per unit volume):

CB ¼EWL

QGS¼ ρL

αD

αGð36Þ

In the above equation E denotes the entrainment ratio (i.e. ratioof the mass flow carried by the liquid droplets to the overall liquidmass flow rate),WL is the total liquid mass flow in the pipe, QG thegas volumetric flow rate, S the slip ratio between the liquid dro-plets and the gas core velocity, αD the liquid droplets volumefraction and αG the gas volume fraction. Eq. (35) denotes anaverage over the pipe cross section; the local deposition can beexpressed assuming a concentration profiles of the droplets which

M. Bonizzi, P. Andreussi / Chemical Engineering Science 142 (2016) 165–179170

can be derived as proposed by Paras and Karabelas (1991):

C Yð Þ ¼ C0exp �uT

ϵY

� ð37Þ

In the above equation C0 denotes the droplets concentration atthe bottom of the pipe, uT is the droplet settling velocity, ε theturbulent diffusivity of the droplets, and Y the vertical distancefrom the bottom pipe (Y ¼ R 1� cosθ

� �). The following equation

arises for the local droplet deposition flux:

RD θ� �¼ kD θ

� �C0exp �kð Þexp ðk cosθÞ ð38Þ

where k¼ uTRε

� . The averaged droplets deposition flux RDh i is

given by the following integral of the local deposition value:

⟨RD⟩¼R π0 RD θ

� �dθ

πð39Þ

The deposition velocity can be computed as a blendingbetween two different deposition mechanisms: the gravitationaland turbulent droplet settling. As illustrated by Pan and Hanratty(2002), the gravitational droplet deposition can be expressed asfollows:

kD;g θ� �¼ 1

13:5ρLgd

1:6

ρG0:4μG

0:6

" #5=7cosθ ð40Þ

Assuming a Gaussian distribution for the radial turbulentvelocity fluctuations, the turbulent deposition coefficient can becomputed as:

kD;t ¼0:9ffiffiffiffiffiffiffiffiffi14π

p uG

ffiffiffiffiffiffiffiffiffiffif int;D2

sð41Þ

In the above equation f int;D is the interfacial friction factorrelation proposed by Dallman et al. (1979) for horizontal gas–liquid separated flows. The concentration at the pipe bottom, C0

can be obtained from the cross-sectional area average:

C0 ¼π2

CBexp kð ÞR π0 sin θ� �2exp k cos θ

� �h idθ

ð42Þ

Under flow conditions such that Eq. (17) is satisfied, Eq. (18) canused to calculate the angular mass flow rate per axial unit lengthby integrating the local deposition flux expressed from Eq. (38):

Γx φ� �¼ RkD;tC0exp �kð Þ

Z φ

0exp �k cos ϑ

� �dϑ ð43Þ

The transcendent integral in Eq. (43) can be numerically solvedusing the appropriate Taylor expansion of the integrand function.It has to be remarked that, for the portion of the liquid film dis-tribution that does not experience atomization, the turbulentdeposition velocity is deployed in Eq. (43).

Once the circumferential mass flow rate per axial unit length isknown, the film thickness immediately follows from Eq. (23).Table 1 indicates the equations employed by the model in the twodistinct regions which characterize the liquid film distributionprofile.

Table 1Adopted equations by liquid film model.

Γz4Γz;c Massequation

x-Momentum z-Momentum τþyz;h dimensionless� �

True (atomiz-ing film)

Eq. (3) Eq. (33) Eq. (29) Eq. (24)

False (onlydeposition)

Eq. (43) Eq. (23) Eq. (29) Eq. (25)

4. Numerical model

The system of equations illustrated in the earlier sectionsrepresents an elliptic set of equations that can be solved providedthat the appropriate boundary conditions are supplied. Theequations will be integrated along half the circumference peri-meter (i.e. for θA 0;π½ �). Therefore the first boundary conditions totake into consideration is the symmetry condition on the cir-cumferential mass flow rate per unit axial length:

Γx 0ð Þ ¼Γx πð Þ ¼ 0 ð44Þ

Inspection of Eq. (30) provides the constraints required tosatisfy the symmetry condition expressed by Eq. (44) above:

dhþ

dθ

θ ¼ 0

¼ dhþ

dθ

θ ¼ π

¼ 0 ð45Þ

duþ2z

dθ

θ ¼ 0

¼ duþ2z

dθ

θ ¼ π

¼ 0 ð46Þ

Besides the conditions above, the system of equations can beintegrated if the appropriate Dirichlet boundary condition is pre-scribed at one solution boundary, i.e. h θ ¼ 0 ¼ h 0ð Þ

.The equations have been numerically integrated adopting a

first-order Runge–Kutta (i.e. Euler's method) discretizationscheme:

dydx

¼Φ x; yð Þ ) yjþ1�yjxjþ1�xj

¼Φ xj; yj�

ð47Þ

Within the numerical solution scheme, Eq. (17) is checked foreach numerical point of the angular grid. When the equation issatisfied, in order to guarantee a smooth transition from the tur-bulent (i.e. Γz4Γz;c) to the laminar (i.e. ΓzrΓz;c) liquid filmregion, Eq. (43) is employed in order to back-calculate the turbu-lent deposition velocity as follows:

Γx;t θtr� �¼Γx;l π�θtr

� � ð48Þ

In Eq. (48) the subscripts t and l denote turbulent and laminarflow regime conditions for the liquid film, and θtr denotes theangle at which transition occurs (i.e. the angular node for whichEq. (17) is satisfied). Equation Γx;t θtr

� �is calculated from the

numerical integration of Eq. (3), while the Γx;lam π�θtr� �

termcomes from solution of Eq. (43). One then derives the followingexpression for the kD;t deposition coefficient:

kD;t ¼Γx;turb θtr

� �RC0exp �kð Þ R π�θtr

0 exp �k cos φ� �

dφð49Þ

Other details related to the adopted numerical scheme can befound in Appendix B.

5. Model validation

The experimental data by Pitton et al. (2014) relate to gas–liquid annular flow in a horizontal pipe having a 7.8 cm innerdiameter, an outlet pressure set around 5 bar, and liquid and gassuperficial velocities of 0.068 and 25.5 m/s, respectively. Table 2below summarizes the relevant experimental measurements.

In this Table, αLF denotes the film liquid hold-up, ΦA the filmatomization rate, ΦB the droplets deposition rate at the bottom ofthe pipe, ΦR the mass flow rate of the non-atomizing film flowingin the upper part of the pipe and E the entrainment ratio. Theliquid film holdup, knowing the circumferential distribution of the

Table 2experimental data-set related to problem under investigation.

WLkgs

h iWG

kgs

h iαLF ½�� h 0ð Þ½mm� dP

dz Pam½ � ΦA

kgm3s

h iΦB

kgm3s

h iΦR

kgm3s

h iE [dimensionless]

0.33 0.73 0.021 2.1 -940 68 60 8 0.47

Table 3Coefficients used for closing the shear-stress model.

C1 τþxx� �

C2 τþxz� �

C3 τþyz;h

h iC4 τþyz;h

h i

5:0� 10�3 1:0� 10�2 2:0 1:27� 10�3

M. Bonizzi, P. Andreussi / Chemical Engineering Science 142 (2016) 165–179 171

film, can be calculated by the following integral:

αLF ¼R π0 D�h θ

� �� �h θ� �

dθ

πR2 ð50Þ

The liquid film flow rate is related to the local axial flow rateper circumferential length as follows:

WLF ¼ 1�Eð ÞWL ¼ 2Z π

0Γz θ� �

Rdθ ð51Þ

The average axial film flow rate per unit length can be found bythe following averaging:

ΓLF ¼ ⟨Γz⟩¼R π0 Γz θ

� �dθ

πð52Þ

Eqs. (51) and (52) give

WLF ¼ πD Γz� �¼ πDΓLF ð53Þ

The rates of deposition/entrainment given by Table 2 above arelinked by the following identity:

ΦA ¼ΦD ¼ΦBþΦR ð54ÞThe drainage rate of the non-atomizing film corresponds to the

rate of deposition of the smaller droplets which are carried by thegas core in the upper part of the pipe tube. The relation betweenthe entrainment/deposition rate and flux is defined by the fol-lowing equation:

Φq ¼ Rq� �P

Að55Þ

In the above equation P and A denote the pipe perimeter andcross section area respectively. For circular tubes P

A¼ 4D, D being the

pipe diameter. If we apply Eq. (55) to the droplet deposition rate,and use the deposition flux which obeys Eq. (35), we can thenwrite the following equation:

ΦD ¼ 4DkDCB ð56Þ

According to the data analysis performed by Pitton et al. (2014),the deposition rate is composed by two terms which arise fromdifferent physical mechanisms: the deposition rate ΦB due to thegravitational settling of the larger droplets, and the deposition rateΦR on the upper pipe wall which is followed by the film drainagefrom the top to the bottom of the pipe. The deposition velocitykD;g , given by Eq. (40), characterizes the deposition coefficient forthe process occurring at the pipe bottom, whereas the velocity kD;t ,as given by Eq. (41), is responsible for the deposition of the dro-plets which subsequently form the draining liquid film. Eq. (40)involves a reference droplet diameter which can be predicted withthe equation proposed by Pan and Hanratty (2002):

d� 3:60:765

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0:0091σGLD

ρGu2G

sð57Þ

Eqs. (40) and (41) allow the calculation of the referencedeposition coefficients:

kD;g � 1:4ms

h ið58Þ

kD;t � 0:2ms

h ið59Þ

Let β denote the angle around the pipe perimeter whichencloses the base film (i.e. the portion of the liquid film whichundergoes the atomization process and therefore characterized bya turbulent regime), then an averaged value for the depositioncoefficient can be estimated as follows:

kD � βkD;gþ π�β� �

kD;tπ

ð60ÞPitton et al. (2014) report that the β angle is around 70°; from

Eq. (60) kD � 0:7 ms

� �is calculated. From Eq. (56) the bulk liquid

droplet concentration can be then estimated,

CB ¼D4ΦD

kD� 1:9

kgm3

� �ð61Þ

Eq. (42), which expresses the relation between the bulk andbottom liquid droplets concentration, can be numerically solvedand the concentration of the liquid droplets at the bottom is foundto be C0 � 9:3 kg

m3

h i. From the experimental data, assuming a C3

coefficient (see Eq. (24)) of 2.0, we can determine the C4 coeffi-cient from the experimental measurement of the pressure gra-dient. The momentum equation in the z-direction, assumingsteady-state conditions can be written as:

� 1�αLFð ÞdPdz

� τIR2 D�2 h

� �� �¼ 0 ð62Þ

The average liquid height is related to the average liquid filmholdup by the following equation:

⟨h⟩¼R π0 h θ� �

dθπ

¼D2

1�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�αLF

p� ð63Þ

For the specific problems under examination we find that h� �

¼ 409 μm� �

and τI �: Using Eq. (4) for calculating the gas shearstress and Eq. (53) for computing the averaged axial mass flow rateper unit circumferential length, leads to the calculation ofC4 ¼ 1:27� 10�3. Table 3 below reports the values of C1 and C2

used in the present model in order to attain the best agreementwith the measured data. While the C2 coefficient, proportional tothe diffusion of the axial momentum in the circumferentialdirection τþ

xz , is set at a value which has the same order of mag-nitude as that deployed by Laurinat et al. (1985) (the values takenare C2 ¼ 1:0� 10�2 in the current model and C2 ¼ 1:7� 10�2 inthe Laurinat et al. (1985) work), it should be remarked that the C1

coefficient, related to the wave spreading term τþxx as expressed by

Eq. (27), was taken two order of magnitudes smaller than thevalues (around O 10�1

� ) typically assumed by Laurinat et al.

(1985). In fact larger values led to erroneous model results,whereas smaller values led to numerical instability issues. At thisstage, the gas secondary flows are not taken into consideration(C5 ¼ 0).

In Fig. 2 The resulting liquid film profile is compared with theexperimental measurements.

Fig. 3. Local axial velocity and cumulative liquid film holdup as predicted by themodel without secondary flows.

M. Bonizzi, P. Andreussi / Chemical Engineering Science 142 (2016) 165–179172

This figure indicates a fair agreement between the model pre-dictions and the measurements. It should be remarked that theclear inflection point at 35° is due to the discontinuity of the eddydiffusivity model between the fully turbulent and the transitionregions, as explained in Appendix A. From the data collected bythe model, it is possible to numerically solve the integrals of Eqs.(50) and (51), which allow to determine the liquid film holdup andoverall liquid film mass flow rate respectively. Table 4 shows thecomparison between the experimental measurements and thenumbers obtained from the model results. Fig. 3 shows thecumulative liquid holdup (Eq. (50) is integrated in discrete stepsfrom the pipe bottom to the pipe top) associated to the liquid filmand the local axial velocity. From this graph one can appreciatehow the bulk of the liquid film is carried by the portion of the filmsitting at the pipe bottom.

It is interesting to notice that, for the problem under examination,transition from the atomizing to the non-atomizing liquid film (cor-responding to the threshold dictated by Eq. (17)) occurs for an anglearound θtr � 961, which is larger than the experimentally determinedθtr � 701. This information is readily extracted from the output of thedeveloped model in that, once the numerical model converges, foreach angular numerical cell, the value of the local axial mass flow rateper circumferential unit length is known and the angle of transitionfrom turbulent to laminar liquid film regime is determined frominspection of Eq. (17).

In order to evaluate the potential effects of gas secondary flows,two opposite values for the constant C5 were taken(C5 ¼ 73:0� 10�2). It has to be remarked that the absolutemagnitude of C5 is consistent with the values suggested by Darlingand McManus (1969). A positive sign, in accordance with theconsidered frame of reference, would imply secondary flows withupward direction (up the wall and down the vertical center line); anegative sign would instead dictate that the secondary flows aremodelled having a downward direction (down the walls and upalong the vertical center line).

The model results with inclusion of the τþyx;h shear stress, with

opposite direction, are illustrated in Fig. 4, where, for sake of clarity,

Fig. 2. Resulting liquid film profile from the model and comparison with experi-mental data points.

Table 4model predictions for relevant flow variables.

C5 τþyx;h

h iθtr [°] αLF ½�� WLF

kgs

h i

0 96 0.022 0.197

þ3:0� 10�2 102 0.023 0.211

�3:0� 10�2 71 0.020 0.179

also the results obtained neglecting the secondary flows are shown.As shown in this figure, the effects of gas secondary flows are limited.The most interesting result is related to the newly determined angleθtr , at which transition to a non-atomizing liquid film takes place inaccordance with Eq. (17). In fact, while for the case of a positivecoefficient C5, the transition is predicted to occur at an angleθtr � 1021, for the case with negative coefficient C5, the transition ispredicted to occur at smaller angles (θtr � 711), which is in excellentagreement with the experimental observations. Other effects of theadopted closure for the τþ

yx;h on some relevant flow variables areshown in Table 4 above.

E [dimensionless] ΦAkgm3s

h iΦB

kgm3s

h iΦR

kgm3s

h i

0.4 56 50 60.35 60 54 6

0.45 52 45 7

Fig. 4. Resulting liquid film profile from the model with or without the shear stressaccounting for the gas secondary flows and comparison with experimental datapoints.

Table 5Selected test cases from Laurinat (1982) and Dallman (1978).

Case D [m] WLkgs

h iWG

kgs

h iuG

ms

� �ρG

kgm3

h iE [dimensionless] h 0ð Þ½mm� hh i

h 0ð Þ½�� ReLF ¼ 4ΓLFμL

[dimensionless]

A (Laurinat) 0.0508 0.09 0.073 18.0 2.05 0.1 1.9 0.21 2030B (Laurinat) 0.0508 0.09 0.139 35.0 2.05 0.39 0.49 0.47 1376C (Laurinat) 0.0508 0.09 0.236 57.0 2.05 0.63 0.16 0.73 834D (Laurinat) 0.0508 0.09 0.292 70.0 2.05 0.75 0.10 0.83 564E (Laurinat) 0.0508 0.09 0.554 130.0 2.09 0.83 0.04 0.88 385F (Dallman) 0.023 0.076 0.025 43.0 1.4 0.34 0.84 0.32 2777G (Dallman) 0.023 0.076 0.038 63.0 1.45 0.67 0.23 0.59 1388

Table 6Calculated Froude numbers, concentration law coefficients k and droplets con-centrations (bulk and bottom values) for the selected test cases from Laurinat(1982) and Dallman (1978).

Case Fr�G [dimensionless] k� [dimensionless] CBkgm3

h iC0

kgm3

h i

A (Laurinat) 1.17 3.96 0.32 3.54B (Laurinat) 2.18 2.11 0.54 2.7C (Laurinat) 3.68 1.25 0.50 1.45D (Laurinat) 4.54 1.01 0.48 1.17E (Laurinat) 8.50 0.54 0.28 0.47F (Dallman) 3.55 1.30 1.48 4.44G (Dallman) 5.18 0.89 1.96 4.34

Table 7Deployed coefficients for the test cases under investigation.

Case kA �½ � C2 τþxz� �

C5 τþyx;h

h iC4 τþyz;h

h iC1 τþxx� �

A (Laurinat) 2:0� 10�6 1:0� 10�2 7:0� 10�3 3:2� 10�3 2:0� 10�2

B (Laurinat) 2:0� 10�6 1:0� 10�2 7:0� 10�3 2:8� 10�3 8:0� 10�2

C (Laurinat) 2:0� 10�6 1:0� 10�2 7:0� 10�3 2:9� 10�3 2:0� 10�1

D (Laurinat) 2:0� 10�6 1:0� 10�2 7:0� 10�3 2:4� 10�3 2:8� 10�1

E (Laurinat) 2:0� 10�6 1:0� 10�2 7:0� 10�3 1:8� 10�3 1:5� 100

F (Dallman) 2:0� 10�6 1:0� 10�2 7:0� 10�3 2:9� 10�3 6:0� 10�3

G (Dallman) 2:0� 10�6 1:0� 10�2 7:0� 10�3 3:0� 10�3 7:0� 10�2

M. Bonizzi, P. Andreussi / Chemical Engineering Science 142 (2016) 165–179 173

6. Discussion

The current model is based on the assumption that the liquidfilm distribution around the pipe walls entirely depends on thelocal liquid film regime, either laminar or turbulent. Hence, thegoverning equations are chosen, as illustrated in Table 1, after theaxial film flow rate per unit circumferential length is determinedby Eq. (17). Under fully turbulent liquid film conditions (Γz4Γz;c),the model coincides with that proposed by Laurinat et al. (1985)while it deviates from it under laminar liquid film flow conditions,in that, as far as the governing equations along the circumferentialcoordinate are concerned, only the deposition flux is accounted forin the mass conservation equation, and the Navier–Stokes equa-tions written for a laminar film apply for the conservation ofmomentum. The assumptions made under laminar liquid film flowconditions allow the immediate calculation (once the depositionlaw has been determined) of the Γx term (Eq. (43)), from whichthe local liquid film height follows from Eq. (23). Another impor-tant difference is that, under fully laminar liquid film flow, thenormal shear stress term τþ

xx associated to the disturbance wavespreading effect is taken to be absent from the model equations.Objective of the current section is to validate the proposed modelagainst additional data, taken from the experimental work ofDallman (1978), Laurinat (1982), and the three-dimensional CFDinvestigation carried out by Verdin et al. (2014).

Table 5 summarizes the flow conditions for the selected testcases taken from Laurinat (1982) and Dallman (1978).

In order to deploy the proposed model, the droplets con-centration at the pipe bottom C0 must be estimated, as suchparameter is explicitly required by the droplets deposition fluxexpressed in Eq. (38). Assuming that the droplet concentration lawcoefficient k (i.e. C Yð Þ ¼ C0exp �kY=R

� �) has been computed, the C0

parameter is evaluated using Eq. (36) (which provides the relationbetween the averaged droplets concentration CB and the dropletsentrainment E) and Eq. (42), which expresses the ratio of thebottom to the averaged droplets concentration. In order to calcu-late the k coefficient, we herein speculate that the relevantdimensionless group, which affects the droplets concentration law,is the Froude number expressed in the following form:

FrG ¼ffiffiffiffiffiffiffiffiffiffiffiρGu

2G

ρLgD

sð64Þ

Such Froude number is defined as the square root of the ratio ofthe gas dynamic pressure to the gravitational effects acting on theliquid phase. As the magnitude of the Froude number defined inEq. (64) increases, one would expect the vertical liquid dropletsconcentration to become more homogeneous (i.e. diminishing kcoefficients); on the contrary, as the gas kinetic energy is furtherreduced and/or the liquid gravitational effects become more pro-nounced (i.e. as the pipe diameter gets larger or the liquid phaseheavier), one would expect the droplets concentration to be lesshomogeneous (i.e. increasing k coefficients). This assumption isdirectly related to the experimental observations reported by

Pitton et al. (2014), who showed in their experiments that largedisturbance waves were only present on the turbulent liquid layerflowing at pipe bottom, while the residual liquid film flowingaround the pipe presented the typical ripple structure of a laminarfilm. We then propose the use of Eq. (65) below in order to esti-mate the liquid droplet concentration coefficient k:

k¼ FrþGFrG

kþ ð65ÞIn the equation above, the coefficient kþ and the Froude num-

ber FrþG relate to flow conditions already characterized (the liquiddroplet concentration profile is assumed to be known). In thepresent analysis, such reference values are taken from Pitton et al.(2014) work, for which the values of the concentration law coef-ficient and Froude number are kþ ¼ 2 and FrþG ¼ 2:31 respectively.Table 6 lists the Froude numbers, the concentration profile coef-ficients, and the bulk and bottom liquid droplets concentrationsthat have been calculated using Eq. (65).

As shown in Table 7, the selected test cases from Laurinat(1982) and Dallman (1978) were simulated maintaining constantthe atomization flux coefficient kA, and the parameters C2 and C5,which are proportional to the normalized shear-stresses τþ

xz andτþyx;h respectively. As explained in a previous section, the C4 coef-

ficient, related to the interfacial shear τþyz;h, is back-calculated

knowing the measured pressure gradients, entrainment ratio andliquid film averaged holdup. For each test case, a sensitivity ana-lysis on the C1 coefficient (related to the τþ

xx waves-spreading

Fig. 5. Comparison between model predictions and Laurinat (1982) data for case A.

Fig. 6. Comparison between model predictions and Laurinat (1982) data for case B.

Fig. 7. Comparison between model predictions and Laurinat (1982)data for case C.

Fig. 8. Comparison between model predictions and Laurinat (1982)data for case D.

Fig. 9. Comparison between model predictions and Laurinat (1982) data for case E.

Fig. 10. Comparison between model predictions and Dallman (1978) data forcase F.

M. Bonizzi, P. Andreussi / Chemical Engineering Science 142 (2016) 165–179174

term) was conducted and the determined optimum values aregiven in Table 7. Figs. 5–9 and Figs. 10 and 11 illustrate the modelresults compared against the liquid film heights measured byLaurinat (1982) and Dallman (1978) respectively.

The C1 values reported in Table 7 clearly demonstrate that theselection of the optimum coefficient is problem dependent: dif-ferent flow conditions lead to different values; in other words, themodel seems to indicate that the weight of the disturbance wavespreading effect is greatly affected by the underlying flow condi-tions. To this regard Laurinat et al. (1985) noticed, from their best

fit values investigation, that the C1 coefficient was dependent onthe ratio of the gas Reynolds number ReGs ¼ ρGuGD=μG

� �to the

liquid film Reynolds number ReLF ¼ 4ΓLF=μL, and they proposedthe following equation in order to estimate the magnitude of theC1 term:

C1 ¼ 3:36� 10�6 ReGsReLF

� �1:74ð66Þ

Fig. 11. Comparison between model predictions and Dallman (1978) data forcase G.

Fig. 12. C1 correlation proposed by Laurinat et al. (1985) compared against best fitvalues found in the current investigation.

Fig. 13. C1 correlation given by Eq. (69) compared against best fit values found inthe current investigation.

M. Bonizzi, P. Andreussi / Chemical Engineering Science 142 (2016) 165–179 175

Fig. 12 compares the values computed from Eq. (66) with thecoefficients determined in the current investigation. Althoughqualitatively fairly good, Eq. (66) leads to significant data scatter-ing which might be reduced. Effort was then put in order toidentify a dimensionless number that might improve the predic-tions of the wave spreading coefficient. Since it is herein specu-lated that the liquid film distribution is mainly driven by gravita-tional effects associated to droplets entrainment/deposition andwave spreading, it would then make sense to consider the fol-lowing parameter, which was introduced by Hurlburt and Newell(1997) in their work:

Υ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiWGu2

G

WLgD

s¼

ffiffiffiffiffiffiffiffiWG

WL

s� uGffiffiffiffiffiffi

gDp ð67Þ

as the number upon which the wave spreading mechanism ismostly dependent. In fact Eq. (67) above can be viewed as amodified Froude number, representing the square root of the ratioof the kinetic energy carried by the gas stream to the workrequired to transport the liquid from the bottom to the top of the

pipe. As theffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiWGu2

G=WLgDq

number increases, one would expectthe wetting mechanism of wave spreading to be relevant inredistributing the liquid around the circumferential perimeter ofthe tube. On the contrary, when this number is low, one wouldexpect the mechanisms associated to gravitational drainage anddroplet entrainment/deposition to be primarily responsible for amore asymmetric liquid film circumferential redistribution. Hurl-burt and Newell (1997) proposed Eq. (68) below in order to

estimate the symmetry parameter h� �

=h 0ð Þ (defined as the ratio ofthe averaged liquid film height, defined in Eq. (63), to the filmheight at the pipe bottom):

h� �h 0ð Þ ¼ 0:2þ0:7 1:�exp �

ffiffiffiffiffiffiffiffiffiffiWGu2

GWLgD

q�20

75

0@

1A

24

35 ð68Þ

Eq. (68) indicates that as the parameterffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiWGu2

G=WLgDq

increases, the symmetry number h� �

=h 0ð Þ becomes larger, which istrue for a more symmetrical liquid film distributions. Fig. 13 plotsthe C1 coefficients, listed in Table 7 in a semi-logarithmic plane

having theffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiWGu2

G=WLgDq

number on the horizontal axis. Thequadratic interpolation

C1 ¼ 5:62� 10�6�

Υ 2þ 7:15� 10�4�

Υ�1:78� 10�2 ð69Þ

is found to fit the coefficients fairly well, even if more data at lowvalues of Υ would help to improve the correlation. Inspection ofFigs. 13 and 12 (where the C1 coefficients were compared againstthe predictions of Eq. (66)) suggests that the dimensionlessnumber introduced by Hurlburt and Newell (1997) helps to reducethe data scattering; in particular the standard deviation of the C1

coefficients is smaller when Eq. (69), rather than Eq. (66), isdeployed (2% against 16% respectively). Eq. (69) satisfies theinequality C1Z0 for values of the parameterffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiWGu2

G=WLgDq

4 � 21, which represents the lower limit of theproposed equation. Therefore the wave spreading effect would not

be relevant for flow conditions which satisfyffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiWGu2

G=WLgDq

o21.The last case that was analyzed is taken from the CFD investi-

gation of Verdin et al. (2014); the relevant flow conditions are

given in Table 8. The modified Froude numberffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiWGu2

G=WLgDq

iscalculated to be less than 3: since, according to Eq. (69), the wavespreading coefficient C1 should be equal to zero for the presentcase, we assume that the normal shear stress in the circumfer-ential direction τþ

xx does not play any role in the liquid filmredistribution. Using the model settings specified in Table 9, Fig. 14shows the predicted circumferential liquid film height comparedagainst that resulting from the CFD work of Verdin et al. (2014). Ithas to be remarked that Verdin et al. (2014) did not simulate thebulk liquid film region sitting at the pipe bottom, which insteadwas assumed to be a moving wall (characterized by the velocity ofthe liquid film bulk region). For this reason, the two physicalmechanisms that were implicitly accounted for by their CFD modelwere droplet entrainment/deposition and secondary gas flows.This is because wave spreading and pumping action could not betaken into consideration within the envisaged CFD model. In the

Table 8Relevant flow data for selected numerical test case from Verdin et al. (2014).

WLFkgs

h iWG

kgs

h iαLF dimensionless� �

h 0ð Þ½mm� dPdz Pa

m½ � σGLNm

� �ρG

kgm3

h iρL

kgm3

h iμL

kgms

h i

56.0 403.4 0.0419 80.0 �33 7:2� 10�3 88.1 685.0 3:5� 10�4

Table 9Selected model coefficients for the selected test case from Verdin et al. (2014).

kA �½ � C2 τþxz� �

C5 τþyx;h

h iC4 τþyz;h

h iC1 τþxx� �

5:0� 10�7 1:0� 10�2 7:0� 10�3 7:� 10�6 0

Fig. 14. Comparison between model predictions and selected test case from Verdinet al. (2014).

Fig. 15. Algorithm flowchart.

M. Bonizzi, P. Andreussi / Chemical Engineering Science 142 (2016) 165–179176

proposed model, the C1 coefficient was taken to be zero (from Eq.(69)); hence wave spreading effects were a priori discarded. Sincethe model results previously presented highlighted the scarcerelevance of the gas secondary motion, it is clear that, according tothe proposed model, the only physical mechanism which acts inreplenishing the draining liquid film from the pipe walls has to beassociated to droplet deposition. Inspection of Fig. 14 reveals thatthe liquid film height abruptly drops at an angle of θtr � 351, whichroughly corresponds to the transition from the turbulent to lami-nar film region. Hence, for the selected test case, the thin laminarliquid filmwetting the inner periphery of the tube origins from thebalance between gravitational drainage and droplets deposition.

7. Conclusions

In the present paper a mathematical model for predicting thecircumferential liquid film distribution in stratified-dispersed flowis proposed. In general, the present work confirms that the rele-vant competing mechanisms which define the circumferentialliquid film distribution are gravitational drainage, dropletentrainment/deposition and wave spreading. The current analysisindicates that the intensity associated to the wave spreading

mechanism diminishes as the parameterffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiWGu2

G=WLgDq

(origin-ally introduced by Hurlburt and Newell (1997)) attains valuesapproaching 21. In the latter case, the present work entirely agreeswith the physical model proposed by Fisher and Pearce (1993),who suggested that the asymmetrical liquid film distributionaround the pipe circumference be related to the phenomena of

droplet entrainment and deposition and to the drainage of the thinliquid film wetting the upper part of the tube.

Under fully turbulent liquid film conditions (i.e. Γz4Γz;c), thepresent film flow model coincides with that developed by Laurinatet al. (1985), while it deviates from it for laminar flow. The maindifference between the two models is that according to theseauthors the normal shear stress, τþ

xx is able to balance the grav-itational force acting on the film under all flow conditions, while inthe present model wave spreading mechanism strongly depends

on the magnitude of the parameterffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiWGu2

G=WLgDq

and completelyvanishes when the liquid film becomes laminar. At the same time,Laurinat et al. (1985) assumed a value for the atomization constantkA which is about one order of magnitude less than the experi-mental value measured by Pitton et al. (2014) and adopted in thepresent model.

The circumferential liquid film distributions predicted with thepresent model have been compared with the data collected byPitton et al. (2014), Laurinat et al. (1985), Dallman (1978) and withthe results of a multi-dimensional CFD investigation conducted byVerdin et al. (2014). The model was found to agree quite well withthese data. A preliminary investigation into the effects of gassecondary flows has also been conducted. It has been concludedthat the direction of such flows should be downward (i.e. downthe pipe walls and up the vertical center line). Inclusion of theshear stress term associated to the gas secondary flows slightlymodifies the model results when compared to the model withoutit. If any conclusion can be drawn, it appears that the gas sec-ondary flows are directed downwards (i.e. down the wall and upthe pipe center line), as reported by Dykno et al. (1994) for stra-tified flow in presence of liquid entrainment.

The present paper highlights the complexity of stratified-dispersed flow in a horizontal pipe. It can be added that theavailable data are scarce and do not cover the cases of relevant

M. Bonizzi, P. Andreussi / Chemical Engineering Science 142 (2016) 165–179 177

industrial interest, such as large pipe diameters, gas densities andliquid viscosities. Our main recommendation can only be toaddress future work towards the acquisition of new data, whichcan then be used to improve, in particular, the model proposed topredict the wave spreading effect.

Appendix A

With reference to the geometry and coordinates system illu-strated in Fig. 1, if local steady-state conditions are assumed, themomentum equations for the x, y, and z coordinates can berespectively written as follows:

� ∂PR∂θ

�ρLg sin θþ∂τxxR∂θ

þ∂τyx∂y

¼ 0 ðA� 1Þ

�∂P∂y

�ρLg cosθ¼ 0 ðA� 2Þ

∂τyz∂y

þ∂τxzR∂θ

¼ 0 ðA� 3Þ

In the above equations P denotes the pressure, g the gravityacceleration, ρL the liquid density, and τ the shear stress actingalong the direction as specified by the corresponding indexing. Eq.(A-2) can be integrated across the film to the interface; suchintegration yields:

�P hþP y�ρLg h�yð Þ cosθ¼ 0 ðA� 4Þ

The pressure at the interface is related to the bulk gas pressureP0 and the radius of curvature RC by the following relation:

Pjh ¼ P0þσGL

RCðA� 5Þ

Insert Eq. (A-5) into (A-4) and differentiate along the cir-cumferential coordinate and obtain:

∂PR∂θ

þσGL∂

R∂θ1RC

��ρLg cosθ

dhR∂θ

¼ 0 ðA� 6Þ

Laurinat (1982) investigated the magnitude of the surfacetension effects and he concluded that only for the smallest pipediameters that term could have significant effect; nonetheless, forthe pipe diameters where available experimental measurementswere collected, the term could altogether be dropped. Eq. (A-1)can then be simplified using Eq. (A-6) as follows:

∂τxxR∂θ

þ∂τyx∂y

�ρLg sinθ�ρLg cosθdhR∂θ

¼ 0 ðA� 7Þ

If the shear stresses are made non-dimensional through divi-sion by the gas-wall shear stress as expressed in Eq. (4), and if theradial coordinate is taken non-dimensional by the followingequation:

yþ ¼ yu�

υLðA� 8Þ

then the momentum equations along the circumferential andaxial direction, expressed by Eqs. (A-7) and (A-3) respectively, canbe rearranged as follows:

∂τþxx

Rþ ∂θþ∂τþ

yx

∂yþ � 1Rþ Fr

sin θ� 1

Rþ2Frcos θ

dhþ

dθ¼ 0 ðA� 9Þ

∂τþyz

∂yþ þ ∂τþxz

Rþ ∂θ¼ 0 ðA� 10Þ

The Froude number Fr is defined in Eq. (8). Eqs. (A-9) and (A-10) above are integrated along the radial coordinate from yþ to

hþ to obtain:

hþ �yþ� � ∂τþxx

Rþ ∂θþτþyx jhþ �τþ

yx jyþ

� hþ �yþ� �Rþ Fr

sin θþ 1Rþ cos θ

dhþ

dθ

� �¼ 0 ðA� 11Þ

τþyz jhþ �τþ

yz jyþ

!þ hþ �yþ� �

Rþ∂τþ

xz

∂θ¼ 0

ðA� 12Þ

The shear stress terms τyxþ hþ and τyz þ hþ

represent the shearstress at the gas-film interface due to the gas secondary flows andthe gas–liquid interfacial shear stress in the axial directionexpressed both in non-dimensional forms. The shear stress at anyradial position can be expressed using expressions for the eddyviscosity as follows:

τijþ jyþ ¼ 1þϵT ijþ� �∂uþ

j

∂xþi

ðA� 13Þ

The non-dimensional eddy viscosity ϵT ijþ is defined as follows:

ϵT ijþ ¼ ϵT ijυL

¼μT ij

ρL

υL¼ 1υL

� u0iu

0j

D E∂uj∂xi

8<:

9=; ðA� 14Þ

The eddy-viscosity follows the transformation of the Reynoldsstress dictated by the equation below:

�ρL u0iu

0j

D E¼ μT

∂uj

∂xiðA� 15Þ

Butterworth (1969) proposed, for the shear stresses τyxþ yþ and

τyz þ yþ the following formulations:

τyxþ jyþ ¼ 1þϵT yz þ� �∂uþ

x

∂yþ ðA� 16Þ

τyz þ jyþ ¼ 1þϵT yz þ� �∂uþ

z

∂yþ ðA� 17Þ

The author also recommended the following expressions forthe turbulent viscosity:

ϵT þyz ¼

0 for yþ τþyz;h

� 12r5

yþ τ þyz;h

� 12

5 �1

0B@

1CAfor 5oyþ τþ

yz;h

� 12r30

yþ τ þyz;h

� 12

2:5 �1

0B@

1CAfor yþ τþ

yz;h

� 12430

8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

ðA� 18Þ

Eq. (A-16) can be inserted into Eq. (A-11) which yields:

1þϵþTyz

� ∂uþx

∂yþ ¼ hþ �yþ� � ∂τþxx

Rþ ∂θþτþ

yx jhþ

� hþ �yþ� �Rþ Fr

sin θþ 1Rþ cos θ

dhþ

dθ

� �ðA� 19Þ

Similarly, insertion of Eq. (A-17) into (A-12) yields:

1þϵT yz þ� �∂uþ

z

∂yþ ¼ τyz þ jhþ þ hþ �yþ� �Rþ

∂τxz þ

∂θðA� 20Þ

Integrate Eqs. (A-19) and (A-20) from 0 to yþ and obtain:

uþx jyþ ¼ ∂τþ

xx

Rþ ∂θ

Z yþ

0

hþ �yþ� �dyþ

1þϵþTyz

� þτþyx jhþ

Z yþ

0

dyþ

1þϵþTyz

�

� 1Rþ Fr

sin θþ 1Rþ cos θ

dhþ

dθ

� � Z yþ

0

hþ �yþ� �dyþ

1þϵþTyz

� ðA� 21Þ

M. Bonizzi, P. Andreussi / Chemical Engineering Science 142 (2016) 165–179178

uþz yþ ¼ τyz þ hþ

Z yþ

0

dyþ

1þϵT yz þ� �þ 1

Rþ∂τxz þ

∂θ

Z yþ

0

hþ �yþ� �dyþ

1þϵT yz þ� �

ðA� 22ÞEqs. (A-21) and (A-22) represent the local values of the non-

dimensional circumferential and axial velocity respectively alongthe non-dimensional radial coordinate yþ ; the above Eqs. (A-21)and (A-22) can eventually be inserted into the expressions of thecircumferential mass flow rate per unit axial length and that of theaxial mass flow rate per circumferential axial length, expressed byEqs. (9) and (10) and here re-written for sake of clarity:

Γþz ¼Γz

μL¼Z hþ

0uþz yþ dyþ ¼ uþ

z

� �hþ ðA� 23Þ

Γþx ¼Γx

μL¼Z hþ

0uþx yþ dyþ ¼ uþ

x

� �hþ ðA� 24Þ

Eqs. (A-21) and (A-22) are then inserted into (A-23) and (A-24)respectively and the following equations, making use of the inte-grals as defined above, are then derived:

Γþx ¼ ∂τþ

xx

Rþ ∂θ

Z hþ

0

Z yþ2

0

hþ �yþ1

� �dyþ

1 dyþ2

1þϵþTyz

� þτþyx j

hþ

Z hþ

0

Z yþ

0

dyþ1 dyþ

2

1þϵþTyz

�

� 1Rþ Fr

sin θþ 1Rþ cos θ

dhþ

dθ

� � Z hþ

0

Z yþ2

0

hþ �yþ1

� �dyþ

1 dyþ2

1þϵþTyz

� ðA� 25Þ

uþz 20 j j ¼

14 uþz 2j jþ1�uþ

z 2j j�1� �þ14 uþ

z 2j jþ2�uþz 2j j�2

� �þ6 uþz 2j jþ3�uþ

z 2j j�3� �þ uþ

z 2j jþ4�uþz 2j j�4

� �128 θj�θj�1

� � ðB� 3Þ

Γþz ¼ τþ

yz jhþ

Z hþ

0

Z yþ

0

dyþ1 dyþ

2

1þϵþTyz

� þ 1Rþ

∂τþxz

∂θ

Z hþ

0

�Z yþ

2

0

hþ �yþ1

� �dyþ

1 dyþ2

1þϵþTyz

� ðA� 26Þ

If the same terminology as that advocated by Laurinat et al.(1985) is deployed, then the following integrals can be defined:

I1 ¼Z hþ

0

Z yþ2

0

11þϵT yz þ

�dyþ

1 dyþ2 ðA� 27Þ

I2 ¼Z hþ

0

Z yþ2

0

hþ �yþ1

1þϵT yz þ

!dyþ

1 dyþ2 ðA� 28Þ

Eqs. (A-25) and (A-26) can then be rewritten as follows:

Γþx ¼ I2

∂τþxx

Rþ ∂θþ I1τþ

yx jhþ � I21

Rþ Frsin θþ 1

Rþ cos θdhþ

dθ

� �ðA� 29Þ

Γþz ¼ I1τþ

yz jhþ þ I21Rþ

∂τþxz

∂θðA� 30Þ

Inspection of Eqs. (A-29) and (A-30) shows that they areequivalent to Eqs. (1) and (2) defined in the model derivationsection.

Eq. (3) follows from the mass conservation equation writtenalong the circumferential coordinate:

dΓx

Rdθ¼ RD θ

� ��RA θ� � ðA� 31Þ

Using Eqs. (7) and (10) the mass conservation equation can beexpressed as follows:

dΓþx

Rþdθ¼ 1ρLu� RD θ

� ��RA θ� �� �¼ RD θ

� �ρLu� �RA θ

� �ρLu� ¼ RD θ

� �þ �RA θ� �þðA� 32Þ

Appendix B

The diffusion of the z-momentum in the circumferential x-direction (shear stress τþ

xz ) was found to trigger numericalinstabilities; in order to reduce the instability seeds, the term wasat first under-relaxed:

Χ i ¼ αiC2

Rþ I2d2uþ

z 2

dθ2 ðB� 1Þ

Besides, the second derivative of the square of the axial velocityhas been computed using a smooth noise-robust differentiator asproposed by Holoborodko (2015):

d2uþ2z

dθ2

j¼ ddθ

ddθ

uþ2z

j¼ uþ20

z j j�uþ20z j j�1

θj�θj�1ðB� 2Þ

In the above equation the derivative of the squared axialvelocity at node j is computed using the followingdifferentiator:

The z-momentum equation is therefore under-relaxed in thefollowing way:

I1τþyz;hþΧi ¼Γþ

z ðB� 4Þ

At the beginning of the solution procedure, for the first outeriteration of the loop, the Χ i term is neglected; once solution hasbeen found for the first outer iteration, the under-relaxation factoris incremented by the user prescribed tolerance, and the Χ i term isnewly computed with help of Eq. (B-3) accounting for the latestavailable value of the flow field variables; the sequence of theoperations shall then be repeated until the under-relaxation factorreaches the value of 1. It has to be remarked that, within eachouter iteration loop, the x-mass and x- and z-momentum equa-tions, shall be invoked until the following averaged equation issatisfied:

⟨RA⟩¼R π0 RA θ

� �dθ

π¼ ⟨RD⟩¼

R π0 RD θ

� �dθ

πðB� 5Þ

Eq. (B-5) dictates that under the assumed steady-state condi-tions, the fluxes of droplets deposition and atomization must beequivalent. Eq. (B-5) is satisfied, within each new outer iteration,by adjusting the deposition coefficient kD;g , whereby the guessedaveraged deposition coefficient shall be corrected (i.e. incrementedor decremented by a fixed small variation ΔkD;g) depending on therespective magnitude of the averaged fluxes of deposition andatomization. The algorithm is therefore recursive, in that, for eachnew outer iteration, the x-momentum equation, x-mass equation,and z-momentum equation shall always be solved in cascade untilEq. (B-5) is satisfied. Let i and k denote the index for the outer andinner iterations respectively, then Fig. 15 shows the adoptedalgorithm flow chart.

M. Bonizzi, P. Andreussi / Chemical Engineering Science 142 (2016) 165–179 179

References

Andreussi, P., Azzopardi, B.J., 1983. Mathematical modelling of droplet depositionand interchange in annular flows. Int. J. Multiph. Flow 9, 681–695.

Andreussi, P., Persen, L.N., 1987. Stratified gas–liquid flow in downwardly inclinedpipes. Int. J. Multiph. Flow 13, 565–575.

Andreussi, P., Asali, J.C., Hanratty, T.J., 1985. Initiation of roll waves in gas–liquidflows. AIChE J. 31, 119–126.

Bonizzi, M., Andreussi, P., Banerjee, S., 2009. Flow regime independent, highresolution, multi-field modelling of near horizontal gas–liquid flows in pipe-lines. Int. J. Multiph. Flow 35, 34–46.

Butterworth, D., 1969. Note on fully developed, horizontal, annular two-phase flow.Chem. Eng. Sci. 24, 1832.

Dallman, J.C., 1978. Investigation of separated flow model in annular-gas liquidtwo-phase flows (Ph.D. thesis). University of Illinois, Urbana, US.

Dallman, J.C., Jones, B.G., Hanratty, T.J., 1979. Interpretation of entrainment mea-surements in annular gas-liquid flows, Two-phase momentum, heat and masstransfer, 2, pp. 681–693.

Darling, R.S., McManus, H.N., 1969. Flow Patterns in Circular Ducts with Cir-cumferential Variation of Roughness: A Two-phase Analog. Developments inMechanics, 5. Iowa State University Press, Ames.

Dykhno, L.A., Williams, L.R., Hanratty, T.J., 1994. Maps of mean gas velocity forstratified flows with and without atomization. Int. J. Multiph. Flow 20, 691–702.

Fisher, S.A., Pearce, D.L., 1993. An annular flow model for predicting liquid carry-over into austenitic superheaters. Int. J. Multiph. Flow 19, 295–307.

Holoborodko, P., 2015. Available at: http://www.holoborodko.com/pavel/numerical-methods/numerical-derivative/smooth-low-noise-differentiators/.

Hurlburt, E.T., Newell, T.A., 1997. Prediction of the circumferential film thicknessdistribution in horizontal annular gas–liquid flow. University of Illinois, Urbana,US, Report ACRC TR-111.

Laurinat, J.E., 1982. Studies of the effects of pipe size on horizontal annular two-phase flows (Ph.D. thesis). University of Illinois, Urbana.

Laurinat, J.E., Hanratty, T.J., Jepson, W.P., 1985. Film thickness distribution for gas–liquid annular flow in a horizontal pipe. PCH Physicochem. Hydrodyn. 6,179–195.

Lin, T.F., Jones, O.C., Lahey, R.T., Block, R.C., Murase, M., 1985. Film thickness mea-surements and modelling in horizontal annular flows. PCH Physicochem.Hydrodyn. 6, 197–206.

McCaslin, J.O., Desjardins, O., 2014. Numerical investigation of gravitational effectsin horizontal annular liquid gas flow. Int. J. Multiph. Flow 67, 88–105.

Nikuradse, J., 1930. Untersuchungen uber turbulente stromunger in nicht kreis-formigen rohren. Ing.-Arch. 1, 306–332.

Pan, L., Hanratty, T.J., 2002. Correlation of entrainment for annular flow in hor-izontal pipes. Int. J. Multiph. Flow 28, 385–408.

Paras, S.V., Karabelas, A.J., 1991. Droplet entrainment and deposition in horizontalannular flow. Int. J. Multiph. Flow 17, 455–468.

Pitton, E., Ciandri, P., Margarone, M., Andreussi, P., 2014. An experimental study ofstratified-dispersed flow in horizontal pipes. Int. J. Multiph. Flow 67, 92–103.

Prandtl, L. 1927. Turbulent flow. Technical Memorandum 435, NACA.Quandt, E.R., 1965. Measurement of some basic parameters in two-phase annular

flow. AIChE J. 11, 311–318.Townsend, A.A., 1970. Entrainment and the structure of turbulent flow. J. Fluid

Mech. 41, 13.Verdin, P.G., Thompson, C.P., Brown, L.D., 2014. CFD modelling of stratified/atomi-

zation gas–liquid flow in large diameter pipes. Int. J. Multiph. Flow 67, 135–143.Vlachos, N.A., Paras, S.V., Mouza, A.A., Karabelas, A.J., 2003. Studies of stratified/

atomization gas–liquid flow in horizontal pipes. J. Energy Resour. Technol. 125,131–136.

Williams, L.R., Dykhno, L.A., Hanratty, T.J., 1996. Droplet flux distributions andentrainment in horizontal gas–liquid flows. Int. J. Multiph. Flow 22, 1–18.