FEASIBLE DOMAIN AND UNCERTAINTY ANALYSIS OF …

IV Journeys in Multiphase Flows (JEM2015)March 23-27, 2015 - Campinas, Brazil

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FEASIBLE DOMAIN AND UNCERTAINTY ANALYSIS OFMECHANISTIC SLUG MODELS

G. F. N. GonçalvesD. A. RodriguesA. P. Silva FreirePrograma de Engenharia Mecânica (COPPE/UFRJ)PC.P. 68503, 21945-970, Rio de Janeiro, [email protected]

Abstract. The present work discusses the mathematical and physical feasible domain of mechanistic slug models. Themodel of Dukler and Hubbard (1975) is particularly used as a reference model. A parameter sensitivity and uncertaintyanalysis is also carried out. The input frequency distribution is taken from the evolutionary model of Cook and Behnia(2000a). Results obtained with both models are compared to the experimental data of Ujang et al. (2006). The model ofCook and Behnia (2000a) is here specialized so as to furnish pressure loss results.

Keywords: Slug Model, Feasible Domain, Uncertainty Analysis, Unity Cell.

1. INTRODUCTION

Slug flow is the prevalent flow pattern in many industrial applications. In the production and transportation of hydro-carbons in horizontal and vertical pipelines, slug flow is certainly the dominant flow configuration. Be it due to naturaloccurrence or to artificially introduced producing enhanced techniques, flow patterns resulting from a sequence of longbubbles followed by liquid slugs permeated by small bubbles are largely encountered the oil industry.

Unfortunately, the modelling of slug flow presents some serious difficulties due to its chaotic features. Any casualobservation of slug flows shows that the characteristic lengths of the long bubbles and liquid slugs appear to be randomlydistributed in time and space.

A classical manner of treating slug flow is to postulate the existence of a typical cell that repeats itself moving down apipe. Provided there exists a reference frame where the liquid and gas phases are considered to travel with about the samevelocity and to attain a fully developed state, disregard of the flow randomness is justifiable to a certain point (Fabre andLine, 1992). A steady state hypothesis results in many simplifications that can then be used to develop a less complicate– but reliable – working model.

The simplifying assumptions can give origin to a model where the cells have a single fixed length, such as the modelof Dukler and Hubbard (1975), or a model where a stochastic distribution of the cell length is considered.

In the present work, unit cell models are considered. Two are the main purposes. First we discuss the mathematicaland physical feasible domain of a fixed cell length mechanistic model, the model of Dukler and Hubbard (1975). Next,a parameter sensitivity and uncertainty analysis is carried out for this model. To define the input frequency distributionfor the model of Dukler and Hubbard (1975) the model of Cook and Behnia (2000a) is used. The resulting slug lengthdistribution obtained with the model of Dukler and Hubbard (1975) is compared back with the slug length distribution ofCook and Behnia (2000a).

The use of any slug mechanistic model requires the assistance of a flow pattern map. Strictly speaking, models canonly be confidently used within the limits established by flow pattern maps. However, these maps are only available fora few pipe diameters and do no consider the state of development of flows, e.g., the frequency of observed unit cells. Inthe present work, it is shown that the mathematical and physical validity domain of steady mechanistic slug models arehighly dependent on statistical properties of the flow, in particular on the slug frequency. We also discuss the way in whichuncertainties associated to the input slug frequency propagate through the model of Dukler and Hubbard (1975) onto theprediction of other flow parameters including the length of the unit cell and the pressure drop.

The model of Cook and Behnia (2000a) is strongly based on the work of Barnea and Taitel (1993). Both modelsdetermine the statistical distribution of slug lengths from fundamentally an empirical expression that defines the rate ofcollapse of short slugs as a function of their lengths. The work of Cook and Behnia (2000a) uses its own data to propose acollapsing equation with coefficients different from those of Barnea and Taitel (1993). The model is purely evolutionary.Once the translational velocities of the front and rear of randomly distributed slugs are assigned through the workingempirical expressions, the motion of the slug can be described through a simple marching process. The models at noinstant resort to the first principles; thus, no dynamic equation of motion is solved.

Results obtained through the model of Cook and Behnia (2000a) are compared against the experimental data of Ujang

G. F. N. Gonçalves, D. A. Rodrigues and A. P. Silva FreireFeasible Domain of Mechanistic Slug Models

et al. (2006) and the mechanistic model of Dukler and Hubbard (1975). The former model is here specialized so as tofurnish pressure loss results.

2. UNIT CELL MODELS

A theory of vertical slug flow was advanced by Nicklin et al. (1962) after some early observations by Dumitresco(1949) and Davies and Taylor (1950). The theory basically discussed the motion of large bubbles moving in verticalliquid streams.

Wallis (1969) was possibly the first to discuss horizontal and vertical slug flows from the point of view of a unitcell. In his analysis, the bubble and liquid phase dynamics were discussed so that expressions for the void fraction andpressure drop became available. This work was followed by a comprehensive model proposed by Dukler and Hubbard(1975). Compared to the previous formulations, the model of D&H was much more detailed. The authors discussed theconditions of slug flow, its initiation and dissipation. The hydrodynamic model of pressure drop across a slug identifiestwo components: “the pressure drop that results from the acceleration of the slow moving liquid film to the slug velocity”and “the pressure drop required to overcome wall shear in the back section of the slug”.

Nicholson et al. (1978) suggested improvements to the model of Dukler and Hubbard (1975), in particular, newequations were proposed for the average translational velocity of the unit cell and the liquid fraction in the slug, aninput quantity. Methods to predict a second input quantity, the slug flow frequency, based on empirical and mechanisticarguments have been proposed by several authors. Zabaras (2000) describes in detail eight methods, comparing theirpredictions with data of his own and with further data available in the literature. His analysis covered different flowconditions in pipes with diameters varying from 1 to 8 inches. A total of 339 points were considered in his analysis.

More recently, the unit cell model has been given more simplified formulations. The work of Orell (2005), e.g.,incorporates a modified pressure loss term as suggested by Cook and Behnia (2000b). The effect of gas entrainment inthe slug on the wall shear stress is accounted for through an effective viscosity. This effect is carried over to the definitionof the Reynolds number employed in the evaluation of the friction coefficient.

2.1 The model of Dukler and Hubbard

The idealized slug model of Dukler and Hubbard (1975) consists of a long bubble passing above a liquid film thatis followed by a liquid plug with a strong mixing region at the front, the latter structure is referred to in literature as theliquid slug. Since in the film region pressure is essentially constant, pressure losses are confined to the slug region wherethey can be considered to arise from two effects: the acceleration of the slow moving liquid film to slug velocity and thepressure loss required to overcome the wall shear stress in the main part of the slug (Duckler and Hubbard, 1975).

The mean velocity of the slug (Vs) is very close to the mixture velocity,

Vs =1

A

(WL

ρL+WG

ρG

)(1)

where W is the mass flow rate and the subscripts L and G indicate the liquid and gas phases.The effective Reynolds number in the liquid slug can be evaluated from

Res = DVsρLRs + ρG(1−Rs)µLRs + µG(1−Rs)

(2)

where Rs is the liquid holdup and ρ stands for density. The subscripts L and G indicate the liquid and gas phases.The translational velocity of the slug nose is the mean velocity of the slug added to the apparent velocity gained by

fluid shedding from the preceding slug. Parameter CDH expresses the ratio of the rate of shedding to the rate of flow inthe slug. A mass balance performed over a cross section of the flow in the liquid slug region can be used to estimate theshedding rate so that CDH can be expressed as

CDH = 0.021 lnRes + 0.022 (3)

There follows immediately that the translational velocity of the slug nose can be found from

Vt = (1 + CDH)Vs (4)

To find the pressure drop due to the deceleration of the film, the fractional area occupied by the film at the rear of thelong bubble and the mean velocity at this position need to be known. The momentum and mass balances applied to thefilm region yield∫ Rs

Rfe

W (Rf )dRf =lfD

(5)


and

Vtνs− lf =

Vsνs(Rs −Rfe)

[WL

ρLAVs−Rfe + CDH(Rs −Rfe)

](6)

where, lf is the film length, Rfe the liquid holdup of the film just before pick-up, and

W (Rf ) =

C2DHR

2s

R2f

+ 1Fr

(π2Rf sin θ

2 +sin2 θ2 cos θ2

1−cos θ − 12 cos θ2

)ffB2 θ

π +RfFr sinβ

(7)

Rf =θ − sin θ

2π(8)

Fr =V 2s

gD(9)

Ref =2πBRf

θRes (10)

ff = 0.0791Re−0.25f (11)

The solution of Eqs. (5) and (6) furnishes lf and Rfe.The above equations can be rearranged to find θfe through an equation with the general form,

F(θfe) =

∫ θS

θfe

W ?(θ)dθ − G(θfe) = 0 (12)

With a known Rfe, parameter Vfe can be determined from:

Vfe = Vs

[1− CDH

(Rs −RfeRfe

)](13)

The rate of liquid pick-up, the length of the slug and the length of the mixing vortex region, can then be evaluatedfrom

x = CDHρLARfe(Vt − Vfe) (14)

ls =Vtνs− lf (15)

lm =0.3(Vs − Vfe)2

2g(16)

The total pressure loss is finally found from

∆Ps = ∆Pa + ∆Pf (17)

where

∆Pa =x

A(Vs − Vfe) (18)

∆Pf =fs[ρLRs + ρG(1−Rs)]V 2

s (ls − lm)

2D(19)

2.2 The model of Cook e Behnia (2000a)

The model of Cook and Behnia (2000a) is based on the evolution model of Barnea and Taitel (1993). The essence ofthe model is a single evolution equation that relates the velocities of the front and the rear of liquid slugs with their ownlength. The models of Barnea and Taitel (1993) and Cook e Behnia (2000a) differ through the specification of distinctevolutionary laws. Cook e Behnia (2000a) used their own experimental data whereas Barnea and Taitel (1993) fitted thedata of Moissis and Griffith (1962).


In both methods, the translational velocity of a bubble is furnished solely as a function of the length of the downstreamliquid slug. Cook and Behnia (2000a) propose

VBVt

= 1.0 + 0.56 exp

(−0.46

lsD

)(20)

where VT is the velocity of a bubble behind a stable slug.According to Nicklin et al. (1962), VT is given by

Vt = CVm + Vd (21)

where C = 1.2, Vm is the mixture velocity and Vd is the bubble drift velocity.A mass balance applied to the liquid phase relates the film length Lf to the gas superficial velocity according to

lf =VGSlS

(1−Rlf )Vt − VGS(22)

where we have considered RL ≈ 1.For long bubbles,

Vlfe ≈ 0 (23)

So that a mass balance across the bubble gives for the holdup at the end of the film

Rlfe ≈(Vt − Vm)

Vt(24)

Experimental evidence suggests

Rlf ≈ 1.4Rlfe (25)

The evolution of a slug is described by the displacement of the slug front (F ) and rear (B) through the propagationequations

Bt+∆ti = Bti + VBi∆t (26)

F t+∆ti = F ti + VFi∆t and VFi = VB(i−1). (27)

A slug collapses when Ri approaches Fi and the number of each slug behind it decreases by one (Cook and Behnia,2000a).

The results of Barnea and Taitel (1993) and of Cook and Behnia (2000a)show that for large enough distances form theentrance section, the distribution of the lengths of the slugs is insensitive to the initial random distribution.

2.3 Pressure drop for the model of Cook and Behnia (2000a)

Here we show briefly how the model of Cook and Behnia (2000a) can be used to evaluate the pressure drop in slugflow.

Taitel and Barnea (2000) show through a momentum balance that the pressure drop in a unit cell is given by

∆Pu = ρsg sinβls +τsπD

A+ ∆Pmix (28)

where the first term represents the hydrostatic head and the second and third terms represent respectively the wall shearlosses in the liquid slug and the losses in the fluid pick up region.

The third term can be written as

∆Pmix = ρlRs(Vt − Vs)(Vfi − Vfe) + ρLg cosβ

∫ hfe

hfi

Rf dhf (29)

For a horizontal flow and with the additional hypothesis, Rs ≈ 1, the above equation reduces to

∆Pu =τsπD

A+ ρl(Vt − Vs)(Vs − Vfe) + ρLg

∫ hfe

D

Rf dhf (30)

where Vfe, can be considered close to zero for long bubbles (Taitel and Barnea, 1990) and τs = fsρLV 2s /2.

The liquid height in any given section can be related to the hold-up through the geometric relation

R = 1− 1

π

arccos

(2h

D− 1

)+

(2h

D− 1

)√1−

(2h

D− 1

)2 (31)

Equations (24) and (31) permit the evaluation of hfe and ∆Pmix.


3. Comments on the pdf distributions of liquid slug lengths and frequencies

According to Taitel and Barnea (1990), the currently available experimental data support the notion that the averagelength of liquid slugs is insensitive to the gas and liquid superficial velocities. In horizontal flows, the average lengthof liquid slugs varies between 12 to 40D, whereas in vertical flow this range changes to 8-25D (Nicholson et al., 1978).Many authors (see, e.g., Nydal et al. (1992), van Hout et al. (2001) and Al-Safran et al. (2005)) note that the distributionof the liquid slug lengths is skewed and follow a log-normal distribution.

Al-Safran et al. (2005) suggests a Poisson probability distribution to describe the passing frequency of slugs inhorizontal pipes. That is to say that the passage interval between successive slugs follows an exponential behavior. Ujanget al. (2006) observed that a Poisson distribution can be used to correlate the process of slug initiation. However, onceslugs reach a developed state, the slug lengths and frequency are best represented by the log-normal distribution.

The typical pdf distributions used in the description of liquid slug lengths are shown in Table 1.

PDF Parameters Reference

Log-normal 1x√

2πσexp−

(ln x−µ)2

2σ2 µ, σ van Hout et al (2003), Brill et al. (1981), Nydal et al. (1992)

Inverse Gaussian(

λ2πx3

)exp

−λ(x−µ)2

2µ2x µ, λ Dhulesia et al. (1991),Al-Safran et al. (2005)Gamma βαxα−1 exp−xβ

Γ(α) α, β Nieckele et al. (2013)Table 1. PDF distributions used in the description of liquid slug lengths.

4. Flow maps for two-phase flows

The topology or the geometry of two-phase flows is normally expressed in terms of flow pattern maps. However,because there exists in the literature some disagreement in regard to the terms that best describe any observed phasedistribution, different flow maps can be found by different authors. The flow pattern maps of Mandhane et al. (1974) andTaitel and Dukler (1976) are typical examples (Fig. 1).

Figure 1. The flow pattern maps of Mandhane et al (1974) and Taitel and Dukler (1976) for pipes with D = 0.05 m.

In the following we will refer to the map of Mandhane et al (1974) due to its large acceptance in the literature.


5. Results

5.1 Mathematical and physical feasible domain of the D&H model

The model of Dukler and Hubbard (1975) was examined so that its mathematical and physical feasible domain couldbe determined. Flow pattern maps do not include information on the passage interval of long bubbles, which is an inputparameter of great importance to mechanistic models.

The mathematical feasible domain of the system of Eqs. (1) to (17) only requires that all mathematical operationsbe justifiable. For example, fraction denominators and logarithmic terms cannot respectively be zero or have a negativeargument. The physical domain requires that flow parameters comply to physical constraints. Lengths and frequency, forexample, cannot be negative numbers.

The mathematical and physical feasible domain of the D&H model is shown in Fig. 2 for two unit cell frequencies, νt= 0.1 and 10 s−1. The first noticed feature is that the model furnishes mathematical and physical – in the sense definedabove – feasible results well outside the experimentally observed region of elongated bubbles and slug flow. Evidently, thetransitions between one pattern and another are not clearly defined through solid lines, but rather through indeterminateregions. However, Fig. 2 shows that for the lower frequency a large region of feasible solutions is obtained whichoverlaps most of the elongated bubbles and slug regions. This large region, in particular, furnishes mathematical andphysical feasible solutions for the model of D&H from combinations of VSL and VSG that are known to yield stratifiedand bubbly flow patterns. For the higher frequency, the feasible domain is reduced but still solutions in the stratified andbubbly flow patterns are obtained.

Figure 2. Comparison between the mathematical and physical feasible domain of the model of Dukler and Hubbard(1975) and the flow pattern map of Mandhane (1974).

An increase in the cell frequency of unit cells reduces the mathematical and physical feasible domain. However, to therange of frequencies normally observed in laboratory experiments, the entire regions of elongated bubbles and slug flowseems to be covered by the model.

Of course, the frequency of passage of long bubbles is not an arbitrary quantity. Rather, νt is correlated to the super-ficial velocities of the liquid and gas phases, to the pipe diameter and the fluid properties. Several empirical expressionscan be found in the literature to express νt in terms of the other relevant flow parameters. Zabaras (2000) analized 339data points to propose

νt = 0.0226

(VSLgD

)1.2 [19.75

Vm+ Vm

]1.2

[0.836 + 2.75 sin0.25(α)] (32)

The feasible domain of the D&H model with an input νt given by Eq. (32) is shown in Fig. 3. The feasible domain in-cludes large portions of the regions of stratified and dispersed flows. However, a large portion of experimentally observed


slug flow is out of the range of the model. Figure 3 illustrates how much caution must be exerted on the choice of modelsand correlations for the description of slug flows.

Figure 3. Assessment of the mathematical and physical feasible domain of the model of Dukler and Hubbard (1975)complemented by the frequency expression of Zabaras (2000).

5.2 Evolutionary slug flow predictions

The evolution equation of Cook and Behnia (2000a) was tested against the data of Ujang et al. (2006). The experimentswere carried out for air-water flow in a pipe with 37 m in length and internal diameter of 0.078 m. Five different baseflow conditions were studied, with superficial velocities ranging in the intervals VSL = 0.22 − 0.61 ms−1 and VSG =2.27− 2.55 ms−1.

The present computations were performed for the condition VSL = 0.61 ms−1 and VSG = 2.55 ms−1.The inlet condition for the simulations were taken directly from the experiments. The lengths of liquid slugs followed

a log-normal distribution. Actually, Ujang et al. (2006) noticed that in the inlet region a log-normal distribution resultedin poor fittings. Near to the inlet, the exponential distribution provided a better representation of the lengths distributionwhich, however, rapidly degraded with the downstream distance.

In view of the above, the inlet conditions where taken at position x = 5.01 m, where a log-normal distribution wasobserved to hold for the slug lengths. It is of interest to note that if a log-normal distribution was to be assigned to the timeintervals between slugs at the first two measuring stations, non-physical results would come out of the model of Duklerand Hubbard (1975).

The mean and the variance of a given experimental data set can be related with parameters µLN and σLN of thelog-normal distribution through

µ = exp

(µLN +

σ2LN

2

)(33)

σ =√

exp (2µLN + σ2LN )[exp(σ2

LN )− 1] (34)

About 10,000 unit cells were considered in the simulation. The time step was 0.005 s. A convergence test showed thatan increase in the number of cells or a decrease in the time step resulted in a variation of less than 1% in the mean lengthof the liquid slugs.

Figure 4 shows a comparison between the mean and the root-mean square values of the liquid slug lengths for thesimulations and the experiments.

One clear pattern is the tendency of the computations to stabilize the slug lengths at a relatively short distance fromthe inlet. Up to 10 m, the experimental and the numerical data are relatively close. However, at the last position (x = 35.1m) the difference between experiments and numerical prediction is as high as 50%.


Figure 4. Evolution of the mean and the root-mean square values of the liquid slug lengths and time intervals betweenslugs. ◦: Experiments. +: Numerical.

Due to the merging process, the time intervals between slugs increases with the downstream distance. The simulationspredicted a linear growing rate for T of the order of 0.075 as compared with the value of 0.04 for the experiments.

The general agreement between the predicted and the simulated time interval between slugs is good, in particular afterthe 10 m mark. Near the inlet, the standard deviation of the experimental data is very high, which makes any comparisona difficult question.

The use of Eq. (30) allows the pressure drop to be estimated for the simulations of Cook and Behnia (2000a). Table 2presents results for ls/D and ∆P/lu [Pa/m].

x = 5.01 m x = 35.1 mQuantity Mean Variance Coef. Variance Mean Variance Coef. Variance1/T [s−1] 0.58 0.15 27% 0.26 0.13 50%ls/D 7.58 2.69 35% 14.16 3.77 27%∆P [Pa] 4024.36 189.54 4.7% 4488.86 265.98 5.9%∆P/lu [Pa/m] 636.20 197.76 31% 323.07 132.79 41%

Table 2. Log-Normal distribution parameters according to the model of Cook and Behnia (2000a).

5.3 Mechanistic slug flow predictions

The frequency distributions obtained through the experiments of Ujang et al. (2006) and the model of Cook andBehnia (2000a) for positions x = 5.01 m and x = 35.1 m were used as input conditions in the model of Dukler andHubbard (1975) (see, Fig. 5).

The results are shown in Fig. 6 and Table 3. The model of Cook and Behnia (2000a) used as an input condition for themodel of Dukler and Hubbard (1975) furnishes values of ls/D with a percentage error relative to the experiments of 18%.On the other hand, the percentage error of ls/D found with a directly application of model of Cook and Behnia (2000a)


Figure 5. Input frequency distributions for the model of Dukler and Hubbard (1975). Left: x = 5.01 m. Right: x = 35.1m.

is 77%.Predicted values of ∆P/lu vary between both models 132% and 36% at positions x = 5.01 m and x = 35.1 m.The standard deviation coefficients obtained in the predictions of ls/D and ∆P/lu for the model of Cook and Behnia

(2000a) were of the order of 32% and 36%. For the mechanistic model, these values changed to 67% and 6.2% respec-tively.

The conclusion is that for large downstream distances the evolutionary model furnishes results for mean ls/D that arewell off target. The mechanistic model performs better but with large standard deviation coefficients.

x=5.01m x=35.1mQuantity Mean Variance Coef. Variance Mean Variance Coef. Variancels/D 6.63 5.20 78% 32.77 18.37 56%∆P [Pa] 2023.98 505.31 25% 4268.63 1473.53 34%∆P/lu [Pa/m] 273.88 14.20 5.2% 236.91 17.07 7.2%

Table 3. Log-Normal distributions according to the model of Dukler and Hubbard (1975).

To decide if a data set follows a specific distribution, the Kolmogorov-Smirnov test can be used.The Kolmogorov-Smirnov test statistic is defined by

D = max1≤i≤N

(F (Yi)−

i

N,i

N− F (Yi)

)(35)

where F is the theoretical cumulative distribution of the continuous distribution being tested.One presumed distribution form must be rejected provided the test statistic, D, exceeds a certain critical value. The

critical value depends on the level of significance of the rejection of the null hypothesis.An assessment of the goodness-of-fit for four types of distribution are shown in Tables 4 to 7 for the results of Cook

and Behnia (2000a) and Dukler and Hubbard (1975). The distributions that best satisfy the confidence level test arehighlighted in bold cases.

At x = 5.01 m, the Inverse Gaussian is a better fit for parameters ν and ∆P/lu obtained through the evolution model.Further downstream, at x = 35.1 m, the Log-Normal distribution offers the best results.

Log-normal Normal Gamma Inverse Gaussianν 0.017 0.081 0.038 0.014∆P 0.063 0.073 0.067 0.063∆P/lu 0.011 0.068 0.028 0.009

Table 4. Kolmogorov-Smirnov statistic for the data of Cook and Behnia (2000a), x = 5.01 m.

The model of Dukler and Hubbard (1975) uses ν as an input parameter. Results obtained with the experimentaldistribution of ν at stations x = 5.01 and 35.1 m return as outputs distributions of ls/D and ∆P/lu that are best fittedthrough Normal and Gamma distributions (Tables 6 and 7).


Figure 6. Outputs from the model of Dukler and Hubbard (1975): ls/D, ∆P e ∆P/lu. Left: x = 5.01 m. Right:x = 35.1 m.

Log-normal Normal Gamma Inverse Gaussianls/D 0.134 0.114 0.118 0.169ν 0.059 0.149 0.091 0.063∆P 0.109 0.114 0.113 0.109∆P/lu 0.067 0.145 0.092 0.071

Table 5. Kolmogorov-Smirnov statistic for the data of Cook and Behnia (2000a), x = 35.1 m.

6. Final remarks

The present work has shown how the slug flow models of Cook and Behnia (2000a) and Dukler and Hubbard (1975)can be combined to yield good predictions of flow properties without the need of specification of the time interval between


Log-normal Normal Gamma Inverse Gaussianls/D 0.061 0.119 0.026 0.104∆P 0.058 0.097 0.081 0.057∆P/lu 0.049 0.043 0.046 0.049

Table 6. Kolmogorov-Smirnov statistic for the data of Dukler and Hubbard (1975), x = 5.01 m.

Log-normal Normal Gamma Inverse Gaussianls/D 0.106 0.047 0.061 0.174∆P 0.055 0.034 0.030 0.061∆P/lu 0.078 0.093 0.083 0.079

Table 7. Kolmogorov-Smirnov statistic for the data of Dukler and Hubbard (1975), x = 35.1m.

slugs. The idea is to use the evolutionary model of Cook and Behnia (2000a) to generate input frequency values for themechanistic model of Dukler and Hubbard (1975) to provide predictions of pressure drop.

In view of the present preliminary results, further comparisons between the evolutionary and mechanistic models andadditional experimental data is being carried out.

Acknowledgements. APSF is grateful to the Brazilian National Research Council (CNPq) for the award of a ResearchFellowship (Grant No 303982/2009-8). The work was financially supported by CNPq through Grants No 477293/2011-5and by the Rio de Janeiro Research Foundation (FAPERJ) through Grant E-26/102.937/2011.


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