Schulkes_Slug Frequencies revisited.pdf

Slug frequencies revisited

R Schulkes Statoil, Research Centre Porsgrunn, Norway

1 INTRODUCTION

Slug frequencies or slug lengths are typically required as one of the closure relations in slug tracking models. Due to the difficulties of modelling the highly nonlinear interface motion prior to the formation of slugs, details of slug formation are still poorly understood. As a consequence, predictions of slug frequencies based on mechanistic models have had a limited success so far. For this reason, many thousands of measurements have been performed in which slug frequencies have been determined experimentally.

Gregory & Scott (1969) were one of the first to perform systematic measurements of slug frequencies. Based on their measurements in horizontal pipes they proposed the following relation

2.1

75.190226.0⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+= mix

mix

SLs U

UgDUf (1.1)

in which sf denotes the slug frequency (measured in number of slugs passing a fixed observer per unit time), SLU denotes the superficial liquid velocity, SGSLmix UUU +=

and D denotes the pipe diameter. The above relation is found to give a good description of slug frequencies in systems similar to that in which the experiments of Gregory & Scott (1969) were performed. However, due to the fact that experimental data is obtained from experiments performed in 19 and 35mm pipes with CO2 and water as the working fluids, the accuracy of relation (1.1) when applied to other systems is limited. This relates in particular to the effects of viscosity and pipe inclination on the slug frequency. The limited extrapolation properties of (1.1) may also, to some extent, be related to the fact that the relation is dimensionally inconsistent.

Since Gregory & Scott (1969) many others have proposed slug frequency correlations which are summarised in the appendix to this paper. These correlations can be divided into three, more or less, separate categories. The first category of correlations is based on (or similar to) relation (1.1). Examples of such correlations are those proposed by Greskovich & Shrier (1972), Heywood & Richardson (1979) and Zabaras (2000). The

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second category of correlations is in some way based on the steady state liquid height (obtained by solving the holdup equation). Examples of these correlations are those proposed by Tronconi (1990) and Hill & Wood (1990). The final category of correlations is based on a correlating slug frequency data without taking as a starting point relation (1.1) or the steady state holdup. The correlation by Shea et al (2004) and Gokcal et al (2009) are examples. Zabaras (2000) has analysed the performance of seven different slug frequency relations against a slug frequency data set obtained from experiments with low-viscosity fluids. His conclusion was that none of the relations (mostly relations that fall in the first category) performed satisfactorily. Based on this observation Zabaras (2000) proposed a new relation that is an extension of (1.1) by including the effects of pipe inclinations. A similar analysis was performed by Gokcal et al (2009) in relation to slug frequencies for viscous oils. Slug frequency relations from the second and third category were tested against high-viscosity experimental data, showing that none of the correlations predicts the measured slug frequency with any degree of accuracy. Based on their experiments, Gokcal et al (2009) proposed a new slug frequency correlation but they state explicitly that this relation does not correlate well with slug frequency measurements obtained for low-viscosity fluids. However, exactly where the border lies between “high-viscosity” and “low-viscosity” systems is unclear, so that the range of validity of the correlation by Gokcal et al (2009) is uncertain. In this paper we take a new look at slug frequency data for hydro-dynamic slugging in gas-liquid systems. Published data comprising a wide range of experimental conditions has been collected. Subsequently, the data was analysed in a systematic manner with the aim to provide a slug frequency relation that is dimensionally correct and valid for all available (published) data. We will show that it is possible to correlate all the available data with a limited number of relevant dimensionless groups. 2 EXPERIMENTAL DATA No new experiments have been performed as part of the work published in this paper. The analysis is performed on the basis of available data in the open literature in addition to a limited number of proprietary Statoil in-house data sets. In the table below we summarise the main characteristics of the data that has been used. The data presented in the table given spans the following range of physical and fluid properties:

- Pipe diameter: 19 - 100mm - Fluid viscosity: 1 - 589cP - System pressure: 1 - 50bar (giving a gas density in the range 1.2-53kgm-3) - Inclination: -1 - 80 degrees

The range of physical and fluid parameters is large so that a unified correlation based on this comprehensive data set (comprising some 1200 points) should have reasonable extrapolation properties.

312 © BHR Group 2011 Multiphase 15

Table 1: Summary of experimental data used in the analysis presented in this paper.

# data points

Pipe diameter (mm)

Inclination (degrees)

Fluids Liquid viscosity (cP)

Pressure (bar)

Gregory & Scott (1969)

99 19, 35 0 CO2/water 1 1

Heywood & Richardson (1979)

78 42 0 Air/water 1 1

Nydal (1991)

56 53 0 Air/water 1 1

Manolis (1995)

230 78 0 Air/water and air/oil

1-50 1-14

Woods (1999)

56 76.3 0 Air/water 1 1

Van Hout (2003)

36 54 10-80 Air/water 1 1

Langsholt (2002)

68 100 1-45 SF6 / Exxsol

1.7 4

Statoil (2009)

64 77.9 0 Gas/oil 33-165 10-50

Kristiansen (2004)

253 69 -1 & +1 SF6 + air / Exxsol

2 1-8

Woods et al (2006)

48 95 0 Air/water 1 1

Gokcal et al (2009)

97 50.8 0 Air/oil 181-589 1

3 METHODOLOGY The starting point of our analysis is a dimensional argument where we (based on physical intuition) pose that the slug frequency sf is a function of 8 parameters via

),,,,,,,( θρρμ GLLSGSLs DgUUFf = (3.1)

The gas viscosity is not included in the parameter list since this parameter is relatively constant for different gases under different pressure conditions. Equation (3.1) shows that we have 8 parameters and 3 independent dimensions (length, time and mass). Standard arguments in dimensional analysis tell us that the dimensionless frequency defined via mixs UDf /=F is a function of 8-3=5 dimensionless groups. Note that we have used the mixture velocity to define the dimensionless frequency. This is motivated by the fact that the slug front propagation velocity is known to be proportional to the mixture velocity (Bendiksen, 1984). Dimensional analysis leads to the conclusion that we are searching for a function of the form


),,,,( 54321 GGGGGΩ=F (3.2)

There is, of course, no unique way in which to choose the 5 dimensionless groups 1G , ..,

5G . However, structures noted by previous workers do offer some guidelines. Namely, both Greskovich & Shrier (1972) and Heywood & Richardson (1979) note that the input liquid fraction and the Froude number correlate well the measured slug frequencies. The influence of viscous effects is included through a balance of gravitational and viscous forces by Gokcal et al (2009) but we find the liquid Reynolds number yields better correlating properties. Manolis et al (1995) did not find a significant influence of the system pressure on the measured slug frequencies. However, for completeness we will investigate how the gas-liquid density ratio (which we believe to be the relevant parameter) influences slug frequencies. Hence, we are led to consider the following five groups: The input liquid fraction: mixSL UU /=α ;

The Froude number: θcos/ DgUFr SL= ;

The Reynolds number: LSLLL DU μρ /Re =

The pipe inclination: θ

The density ratio: LG ρρρ /=

In what follows, the function ),,Re,,( ρθα LFrΩ=Ω will be constructed step-by-step in the following manner. We start with an analysis of slug frequency data from experiments in horizontal pipes with low-viscosity liquid (typically air-water systems). Since the liquid Reynolds number is typically large in this case while the density ratio is small, we may postulate

),( FrαΨ=F (3.3)

In the following section it will be shown that the influence of the Froude number on low-pressure horizontal slug frequency data is rather limited. Namely, including the Froude number does not lead to a correlation with a larger 2R -value so we are justified in working with the simplified function )(αΨ=Ψ . Once the function )(αΨ is determined we focus on the influence of viscosity by postulating that viscous effects can be included in the frequency function Ω via

)(Re)( LΦ×Ψ= αF (3.4)

The benefit of this approach is that input liquid fraction effects and viscous effects are separated. This allows a straightforward construction of the function Φ once the function Ψ is known. Proceeding to study the influence of the pipe inclination on the slug frequency, we again assume that the influence of the inclination on the slug frequency can be separated from the liquid fraction and the viscosity effects, enabling us to write

),()(Re)( FrL θα Θ×Φ×Ψ=F (3.5)

As before, the central simplifying assumption is that the effects of inclination can be separated such that the function Θ can be determined readily once the functions Ψ and


Φ are known. The final step involves determining the influence of pressure on the measured slug frequencies. Here we proceed as before by writing

)(),()(Re)( ρθα Π×Θ×Φ×Ψ= FrLF (3.6)

Several bold steps have been taken in reaching the final functional form as specified in equation (3.6). The data analysis presented in the following section will determine to what extend these steps are justified. 4 DATA ANALYSIS In what follows, the available experimental data will be analysed by following the steps outlined in the previous section. Hence, we start with low pressure air-water data obtained from experiments in horizontal pipes and subsequently we study the influence of viscosity. In the final steps we include the effects of inclination and pressure on the measured slug frequencies. 4.1 Data comprising horizontal, low-pressure, low-viscosity systems The aim of this section is to determine the function )(αΨ and this is achieved by considering those experimental points that have been obtained from experiments where the pressure was (near) atmospheric and the working fluids were air and water. From Table 1 we find that there are approximately 330 data points in this data set comprising 6 different pipe diameters. Plotting the measured frequencies made dimensionless as a function of the input liquid fraction α we obtain the result as shown in figure 1.

0.0000

0.0100

0.0200

0.0300

0.0400

0.0500

0.0600

0.0700

0 0.2 0.4 0.6 0.8 1

α

F

Figure 1: A plot of the dimensionless frequency F versus

the input liquid fraction α .

We observe in figure 1 that the input liquid fraction correlates well with the dimensionless frequency. It is reasonable to expect that other physical effects are important and we have in particular investigated the influence of the Froude number. Figure 2 shows one of the results of this investigation. There we have plotted the dimensionless frequency as a function of the input liquid fraction for separate Froude numbers. The particular data shown is the data set of Nydal (1991). The results in figure


2 indicate that the frequency increases as a function of the Froude number suggesting that we have to take ),( FrαΨ=F . However, a similar analysis of the other data sets does not display the same structure. This then motivates us to opt for the simpler function )(αΨ=F . A least-squares fit of the data in figure 1 yields

( )ααα 32016.0)( +=Ψ (4.1)

Before we proceed with the identification of the additional physical effects needed in the complete function for the dimensionless frequency, it is instructive to compare equation (4.1) with the original correlation of Gregory & Scott (1969) as given by equation (1.1). In dimensional from, equation (4.1) reads

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

mix

SLSLs U

UD

Uf 32016.0 (4.2)

While there are differences between the above relation and equation (1.1) it is worth noting the similarities. In the limit of large superficial liquid and gas velocities, we find that (4.2) predicts a slug frequency which is proportional to the superficial liquid velocity. This limiting behaviour is different from (1.1) but in line with experimental observations of Nydal (1991).

0.0000

0.0100

0.0200

0.0300

0.0400

0.0500

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

α

F

Fr 0,832Fr1,66 Fr 3,32Fr 4,85

Figure 2: A plot of the dimensionless frequency F versus the input liquid fraction α for the Froude numbers as indicated in the plot.

The data set used is that published by Nydal (1991). 4.2 The influence of viscous effects The next step in our analysis is to include effects of viscosity in the slug frequency correlation function ),,,Re,( ρθα FrLΩ=F . The data used in this part of the analysis, are all those used to obtain )(αΨ as well as the high-viscosity data of Manolis (1995), Statoil (2009) and Gokcal et al (2009) (in total approximately 500 points). We start by showing that the influence of viscosity is evident when we plot the high-viscosity data in a plot of the dimensionless frequency F versus the input liquid fractionα . We recall that the low-viscosity data correlated well with α (figure 1) but when the high-viscosity data is included it is evident the input liquid fraction is not the only relevant parameter, see figure 3. At this stage it is already interesting to note that the Manolis and Statoil


data (viscosity from 33-165cP) correlates rather well with the low-viscosity data while significant differences become apparent when the liquid viscosity exceeds 180cP. It is evident that an increase in the viscosity leads to an increase in the slug frequency, but clearly, the viscosity has to be above a certain threshold.

0.0000

0.0200

0.0400

0.0600

0.0800

0.1000

0.1200

0.1400

0.1600

0.1800

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

α

FWaterGokcal 181cpGokcal 589cp

Statoil dataManolis 50cP

Figure 3: A plot F as a function of the liquid fractionα

for the liquid viscosities as indicated in the plot. In section 3 we postulated that viscous effects could be included via the viscosity function )(ReLΦ , giving )(Re)( LΦ×Ψ= αF . This bold assumption is easily tested by analysing all the available low- and high-viscosity data in horizontal pipes. Namely, if our assumption turns out to be justified, we should see a clear trend by plotting )(/ αΨF versus LRe . The result of our analysis is shown in figure 4. First of all we note that the trend is very clear, despite a degree of scatter. It appears that we are justified in our assumption that liquid fraction effects and viscous effects can be separated.

0.1

1

10

1 10 100 1000 10000 100000 1000000

Re L

Φ

Water

Gokcal 181cP

Gokcal 589cP

Statoil 33-165cP

Manolis 50cP

Figure 4: A plot )(/ αΨ=Φ F as a function of the liquid Reynolds number LRe .


It is very interesting to observe that the function Φ has a shape not unlike that of a typical friction factor. Namely, for Reynolds numbers below 4000, the function Φ displays a type of power-law behaviour (straight line in a log-log plot) while for larger Reynolds numbers the function Φ is approximately unity. A least-squares fit of the data for Reynolds numbers below 4000 yields

37.0Re1.12)(Re −×=Φ LL (4.3)

Hence, if the function )(ReLΦ is defined according (4.3) for 4000Re <L while we take 1)(Re =Φ L for 4000Re ≥L we have a unified slug frequency correlation for low- and

high-viscosity data limited to horizontal pipes. The advantage of the present approach as compared with that of Gokcal et al (2009) is that it is immediately clear for which parameter range the low-viscosity correlation can be applied and when high-viscosity effects need to be included. 4.3 The influence of inclination In the next step of the development of the unified slug frequency correlation, we focus our attention on the effect of inclination. In order to determine the influence of pipe inclination on the slug frequency we start with the series of experiments performed by Van Hout et al (2003) with air and water.

1.00

10.00

100.00

1000.00

0.00 0.25 0.50 0.75 1.00 1.25 1.50

θ(rad)

F/ ψ

Fr=0.28

Fr=0.12

Fr=0.137

Fr=0.124

Fr=0.014

Figure 5: A plot of )(/ αΨ=Θ F as a function of the inclination angle. The different

symbols indicate different values of the Froude number gDUFr SL /0 = . Since we have assumed that the influence of inclination can be separated from liquid fraction and viscosity effects, the function ))(Re)(/( LΦ×Ψ=Θ αF is expected to show the influence of pipe inclination. In figure 5 we have plotted the function )(/ αΨ=Θ F as a function of the inclination angle (note that since we only have access to low-viscosity data, we have 1)(Re ≡Φ L ). We have used different symbols for the data points in order


to visualise the influence of the Froude number. We note first of all that the measured slug frequencies increase with decreasing Froude number. A detailed analysis of the data shows that the dependence of the dimensionless frequency on the mixture velocity is marginal. This is exemplified by the closeness of the data points where 12.00 ≈Fr but the mixture velocity lies in the range 73.022.0 ≤≤ mixU m/s. The data thus suggests that the dimensionless slug frequency is a function of the Froude number and the inclination angle of the pipe. The data in figure 5 can be correlated well by the following equation

( )226.08.1),( θθθ −+×=ΘFr

Fr (4.3)

Since the data set of Van Hout et al (2003) is obtained for inclination angles in the range [ ]oo 80,10∈θ , the above correlation can not be used for angles below o10=θ . Forthese smaller angles we used a data set obtained in the flow loop of IFE by Langsholt et al (2002). From this data set we obtain that for small inclinations o10<θ

θθθ )sgn(21),( ×+=ΘFr

Fr (4.4)

Although equations (4.3) and (4.4) correlate well low-viscosity data there is no high-viscosity data available for inclined flow. This means that the accuracy of the inclination function ),( FrθΘ is not verified for viscous fluids (that is, fluids in which 4000Re <L ). The same holds for slug frequencies in pipes with negative inclinations.

4.4 The influence of pressure The final physical effect we believe may influence the slug frequency is the system pressure. This effect enters the slug frequency correlation through the density ratio

LG ρρρ /= . The data available provides a range of density ratios

[ ]23 106.6,102.1 −− ××∈ρ .

In order to establish the influence of the system pressure we use the data by Manolis (1995). Our step-by-step approach so far has been successful in establishing the influence of viscosity and inclination and hence we anticipate that the influence of pressure may be made visible in the same way. The Manolis (1995) data is established in horizontal pipes so we do not need the inclination function ),( FrθΘ . Hence, the influence of pressure (or density) effects should become visible by plotting

))(Re)(/( LΦ×Ψ=Π αF as a function of the input liquid fraction α .

This plot is shown in figure 6. The plot does not show a clear trend – the data for different pressure levels does not show the clear structure we have seen in the previous analyses. This then suggests that the influence of pressure on the slug frequencies is small. Performing a similar analysis with the high-pressure data from Statoil and the high-pressure air-water data from Manolis (1995) yields the same conclusion: pressure effects on the slug frequency do not show clearly in the data, supporting a similar conclusion by Manolis et al (1995).


0.5 0.6 0.7 0.8 0.9

1 1.1 1.2 1.3

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

α

F 1bar

6.5bar

10.5bar

14.5bar

Figure 6: A plot of ))(Re)(/( LΦ×Ψ=Π αF as a function of the input

liquid fraction α based on the Manolis (1995) oil-air data. It may be that the influence of pressure is more convoluted than the influence of viscosity and/or inclination implying that the present type of analysis is not suitable to determine the pressure effect. Here the data set by Kristiansen (2004) has particular value. Kristiansen has performed experiments with SF6 at up to 8 bars yielding gas densities as high as 52kgm-3. Neither of these experiments show a significant influence of the system pressure on the slug frequencies. In line with findings of others, Kristiansen (2004) shows very clearly that the slug flow region is significantly reduced when the system pressure increases. In addition Kristiansen (2004) shows, that it becomes increasingly difficult to get good slug frequency data for high-pressure experiments due to the decreasing slug frequency: for increasing system pressures it becomes increasingly difficult to keep the standard deviation of the slug frequency measurements within reasonable bounds. The quality of some points in the Manolis (1995) data set appear to be influenced by this effect (for many frequency measurements there was on average less than one slug in the pipe line). In these cases the slug frequency becomes entirely dependent on the inflow conditions and data like this should not be included in the type of analysis as presented in this paper. 5 DISCUSSION, SUMMARY AND CONCLUSIONS In this paper we have shown how a step-by-step analysis of available slug frequency data enables us to establish a unified slug frequency correlation. Based on the analysis of some 1200 data points covering a wide range of physical parameters, we have arrived at the following slug frequency correlation

),()(Re)( FrL θα Θ×Φ×Ψ=F (5.1)

In which

( )ααα 32016.0)( +=Ψ

⎪⎩

⎪⎨⎧

≥<

=Φ−

4000Refor 14000Refor Re1.12)(Re

37.0

L

LLL


( )⎪⎪⎩

⎪⎪⎨

⎧

>−+×

≤+=Θ

17.0for 26.08.1

17.0||for ||)sgn(21),(

2 θθθ

θθθθ

Fr

FrFr

In the above equations, mixSL UU /=α denotes the input liquid fraction,

θcos/ DgUFr SL= denotes the Froude number, LSLLL DU μρ /Re = denotes the liquid Reynolds number and θ denotes the pipe inclination in radians. The influence of the system pressure on the slug frequency was found to be too small to give a significant contribution. Plotting all the data points against our unified slug frequency correlation yields the result as shown in figure 7. We observe that slug frequencies varying over almost three orders of magnitude obtained in systems that cover a wide range of physical parameters (diameter effects, fluid effects and pipe inclinations) correlate well with equation (5.1).

0.0001

0.0010

0.0100

0.1000

1.0000

0.0001 0.001 0.01 0.1 1

Ψ(α)Φ (Re )Θ(Fr ,θ)

F

Van Hout

Woods 2006

Heyw oods

Gregory

Gokcal

Statoil

Nydal

Woods 1999

Manolis A/W

Manolis A/O

Figure 7: All the slug frequency data plotted against correlation (5.1).

Different data set are denoted by different symbols. The data set by Kristiansen (2004) has not been used in the establishment of equation (5.1). It can thus be regarded as in independent data set which can be used to test the predictive quality of (5.1). The data set comprises low and high-pressure data with SF6 and oil obtained in a 69mm pipe which had an inclination of ± 0.1 degrees. In the experiments of Kristiansen (2004) the influence of different inflow configurations on the measured slug frequencies was tested. The data set of Kristiansen is special in that statistical properties of the measured slug frequencies were also established. When we select those points in the data set in which the standard deviation in the slug frequency measurements is less than 10% of the frequency measurement we obtain figure 8. We observe that the agreement with correlation (5.1) satisfactory although a fair degree of scatter is present. Including points in which the standard deviation was larger, increased


the scatter significantly. The published slug frequency data in the literature does not include statistical properties of the slug frequency measurements. It is not unlikely that the scatter in figure 7 would have been less if only those points were included in which the standard deviation was below a given bound.

0.0001

0.001

0.01

0.1

0.0001 0.001 0.01 0.1

Ψ(α)Φ(Re L)Θ(θ, Fr )

F

Figure 8: The slug frequency data obtained by Kristiansen (2004) plotted against

correlation (5.1) for the case where the standard deviation of the slug frequencies is less than 10% of the actual frequency measurement.

We conclude that a unified slug frequency correlation has been established where we have shown that the correlation performs well against a large set of experimental data. For horizontal flow with low and high-viscosity fluids it is expected that the correlation performs well. For upward inclined flows with low viscosity fluids we also expect the correlation to perform well. Since we have not been able to test the correlation against data for inclined flow with high-viscosity fluids, the extrapolation property of the established correlation is uncertain for this case. The available experimental data does not indicate a significant pressure-dependence of the slug frequency. ACKNOWLEDGMENTS Roman Shpak and Julia Marhotka were central during the start of this work. They digitised an enormous amount of data and performed the initial stages of the analysis. Without their endurance during the summer of 2010 it would have been difficult to move ahead with this work as quickly as has been the case. I am indebted to Olav Kristiansen for supplying me with the data set from his PhD. REFERENCES Bendiksen, K.H. 1984 An experimental investigation of the motion of long bubbles in

inclined tubes. Int. J. Multiphase Flow. 10(4), 467-483. Davies, S.R. 1992 Studies of two-phase intermittent flow in pipe lines. PhD Thesis,

Imperial College, London.


Gokcal, B., Al-Sarkhi, A.S., Sarica, C. and Al-Safran, E.M. 2009 Prediction of slug frequency for high viscosity oils in horizontal pipes. SPE 124057.

Gregory, G.A. & Scott, D.S. 1969 Correlation of liquid slug velocity and frequency in horizontal co-current gas-liquid slug flow. AIChE J., 15, 933-935.

Greskovich, E.J. & Shrier, A.L. 1972 Slug frequency in horizontal gas-liquid flow. AIChE J. 11, 317-318.

Heywood, N.I. & Richardson, J.F. 1979 Slug flow of air-water mixtures in a horizontal pipe: determination of liquid holdup by gamma-ray absorption. Chem. Engng. Sci. 34, 17-30.

Hill, T.J & Wood, D.G. 1990 A new approach to the prediction of slug frequency. SPE 20629.

Kristiansen, O. 2004 Experiments on the transition from stratified to slug flow in multiphase pipe flow. PhD NTNU, Trondheim.

Langsholt, M., Pettersen, B.H., Andersson, P. 2002 Pipe inclination effects on three-phase slug flow. Report IFE/KR/F/2002/064.

Manolis, I. 1995 High pressure gas-liquid slug flow. PhD Thesis, Imperial College, London.

Manolis, I., Meendes-Tatis, M. & Hewitt, G. 1995 The effect of pressure on slug frequency in two-phase horizontal flow. 2nd Int. Conf. Multiphase Flow. Kyoto, Japan.

Nydal, O.J. 1991 An experimental investigation of slug flow. PhD Thesis, University of Oslo, Oslo.

Shea, R.H., Eidesmoen, H., Nordsveen, M., Rasmussen, J., Xu, Z.G. & Nossen, J. 2004 Slug frequency prediction method comparisons. Proc. 4th North American Conference on Multiphase Technology, 227-237.

Statoil 2009: Kjeldby, T.B. Time series analysis of multiphase pipe flow. Project Thesis NTNU for Statoil.

Taitel, Y.& Dukler, A.E. 1977 A model for slug frequency during gas-liquid flow in horizontal and near horizontal pipes. Int. J. Multiphase Flow 3, 585-596.

Tronconi, E. 1990 Prediction of slug frequency in horizontal two-phase slug flow. AIChE 36, 701-709.

Van Hout, R., Shemer, L. & Barnea, D. 2003 Evolution of hydrodynamic and statistical parameters of gas-liquid slug flow along inclined pipes. Chem. Engng. Sci. 58, 115-133.

Woods, B.D. & Hanratty, T.J. 1999 Influence of Froude number on physical processes determining frequency of slugging in horizontal gas-liquid flows. Int. J. Multiphase Flow 25, 1195-1223.

Woods, B.D., Fan, Z. & Hanratty, T.J. 2006 Frequency and development of slugs in a horizontal pipe at large liquid flows. Int. J. Multiphase Flow 32, 902-925.

Zabaras, G. 1999 Prediction of slug frequency for gas-liquid flows. SPE 56542.

APPENDIX

In this appendix we give a summary of slug frequency correlations that have been proposed. We choose to group the slug frequency correlations into three, more or less separate, categories.

The first set of correlations, essentially derived from data obtained with low-viscosity fluids, is based on (or similar to) the well-known relation by Gregory & Scott (1969), namely


2.1

75.190226.0⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+= mix

mix

SLs U

UgDUf (A.1)

in which sf denotes the slug frequency (measured in number of slugs passing a fixed observed per unit time), SLU denotes the superficial liquid velocity and the mixture velocity is denoted by SGSLmix UUU += . Greskovich & Shrier (1972) suggested a modified form of (A.1), namely

2.1

202.20226.0⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+=

gDU

Df mix

s α (A.2)

in which mixSL UU /=α denotes the no-slip liquid holdup while Heywood & Richardson (1979) suggested a slight modification of (A.2), namely

06.1

202.20434.0⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+=

gDU

Df mix

s α (A.3)

Based on the data of Manolis (1995), Manolis et al (1995) proposed the relation

8.1

2250037.0

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ +=

mix

mixSLs U

UgD

Uf (A.4)

It is worth noting that the above correlation predicts a slug frequency which is almost an order of magnitude lower than any of the other correlations. There are indications (see main text) that this may be related to the experimental setup. Zabaras (2000) modified (A.1) by including pipe inclination effects

( ))(sin75.2836.075.190226.0 4/12.1

θ+×⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+= mix

mix

SLs U

UgDUf (A.5)

in whichθ denotes the inclination angle of the pipe. Based on air-water experiments in pipes with diameters in the range 31-90mm, Nydal (1991) proposed the relation

gDUf SL

s

2)5.1(088.0 += (A.6)

The second set of correlations is in some way related to solutions of the holdup equation. Tronconi (1990) proposed a relation in this category which was based on the assumption that the slug frequency is related to the frequency of the most unstable waves, giving

)(61.0

LL

GGs hD

vf−

=ρ

ρ (A.6)

in which Gv denotes the in-situ gas velocity and Lh denotes the thickness of the layer of liquid as obtained from the holdup equation. A relation of a similar form was proposed by Hill & Wood (1990), namely


Dvv

HHf LG

L

Ls

−×

−×=

)1(360074.2 (A.7)

in which LH denotes the equilibrium liquid holdup and Lv denotes the in-situ liquid velocity. A second relation proposed by Hill & Wood had the form

LHmixs e

DU

f 17.6

3600275.0

××= (A.8)

Taitel & Dukler (1976) approached the slug frequency problem by solving a dimensionless form of the combined (integral) momentum and continuity equations. These equations constitute essentially the dynamic extension of the well-known Lockhart-Martinelli equation. Taitel & Dukler show that the solution of this system of equations is dependent on five dimensionless groups: the usual Martinelli parameters X and Y (denoting the ratio of pressure gradients in the liquid and gas phase and the gravitational force scaled by the pressure gradient in the gas phase respectively), a new parameter Z (denoting the liquid inertia term scale by pressure gradient in the gas phase) as well as the dimensionless velocities θcos/ gDUV SLL ≡ and

)/(cos/ GLGSLG gDVV ρρρθ −×≡ . The slug frequency is found by solving the momentum and continuity equations until a critical liquid height is exceeded. The time to reach this critical height is taken to be the inverse of the slug frequency, expressed in the form

),,,,( GLSL

s VVZYXFD

Uf = (A.9)

The third category of slug frequency correlations is that obtained attempting to correlate slug frequency data without reference to the Gregory & Scott relation (A.1). Correlations in this category are, for example, the expression proposed by Shea et al (2004) based on field data

55.02.1

75.047.0

p

SLs LD

Uf = (A.10)

and the Shell slug frequency correlation (quoted by Zabaras, 1999)

( )( ) ⎥⎦⎤

⎢⎣⎡ −++×=

2064.01.081.0 17.1048.0 LGLLs FrFrFrAFrDgf (A.11)

in which gDUFr SGSLGL /,, = and 34.273.0 LFrA = . This last correlation is based on the low-viscosity data of Heywood & Richardson (1979). The final correlation in this category is that derived by Gokcal et al (2009) on the basis of experiments with high-viscosity fluids, giving

612.0816.2 μND

Uf SLs = (A.12)

In which ))(/( 2/3 gDN GLLL ρρρμμ −= denotes the ratio viscous and gravitational forces.


Schulkes_Slug Frequencies revisited.pdf

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