vertical flow

VERTICAL TWO-PHASE FLOW

Part III: Pressure Drop

P. L. SPEDDING (FELLOW), G. S. WOODS (GRADUATE), R. S. RAGHUNATHAN and J. K. WATTERSON

Department of Aeronautical Engineering, The Queen’ s University of Belfast, Belfast, UK

Data are presented on the total two-phase pressure drop for air-water vertical ¯ ow in a0.026 m i.d. pipe. The data exhibit a maximum value, at the end of the churn ¯ owregime, set between two minima at the end of the slug and annular ripple regimes

respectively. There was a noticeable diameter effect. It was possible to explain the form of thepressure loss data qualitatively. In the slug and allied regimes the rising gas bubble carried apocket of liquid in its wake resulting in a reduction in the ¯ uid head.

None of the current theories of pressure loss prediction proved to be useful. However, it waspossible to predict bubble ¯ ow at low gas rates and annular droplet ¯ ow at high gas rates.

Keywords: vertical; two-phase; pressure drop; prediction; up lift pressure

INTRODUCTION

Many studies have been carried out on the pressure drop invertical 1 90° two-phase gas-liquid ¯ ow. Table 1 details themajor literature on the subject. Not all the data has proven tobe reliable mainly because of the inclusion of entrance andexit effects into the measurements. Calvert and Williams5

showed the expected effect on the measurement of pressuredrop if either the design of, or the operation of, the apparatusdid not eliminate such end effects. In addition, Hewitt etal.23,52 ± 54 and Woods and Spedding55 showed that the methodused to premix the phases can exacerbate the problem. Thusthere exists a need to evaluate the data (e.g. in Table 1) toensure that the measured pressure drop truly conformed tothe straight pipe geometry.

Spedding and Ferguson56 have shown that prediction ofpressure drop for horizontal ¯ ow was ¯ ow regimedependent with either the Olujic57 model or the Speddingand Hand58 model achieving reliable prediction across theentire ¯ ow regime range. In general, gravity, can be expectedto have a more pronouncedeffect on pressure drop in vertical¯ ow. It is therefore highly unlikely that one model will beable to predict pressure drop over the whole range of vertical¯ ows despite the fact that many workers have attempted to doso1,7±11,13,14,17,18,25,28±31,34,36,42±44,46,48±51,59,60. Other workers have,by contrast, developed pressure prediction models dependenton ¯ ow regimes2,5,7,12,16,19±24,27,32,33,35,37,39,47,58±61. The three mostwidely recommended methods of vertical pressure lossprediction are:

(a) The general method of Hagedorn and Brown29±31,recommended by Gould44, Abdul-Majeed et al.46 andAggour et al.51.(b) The models of Beggs and Brill62 recommended byAggour et al.51.(c) The four regime dependent model or Orkiszewski35

recommended by De Gance and Atherton63 and MataIturbe64.

More recently Holt et al.65 have developed a ¯ ow pattern

speci® c model for pressure drop estimation in vertical two-phase ¯ ow using a number of theories to estimate thecomponent of acceleration, gravitational and frictionalpressure drop for the bubble, churn and annular ¯ owregimes.

RESULTS

Figure 1 gives the air-water vertical total two phasepressure drop for the 0.026m i.d. pipe. Also shown are datafor 0.0454m i.d. pipe by Nguyen41, and for 0.03818m i.d.pipe by the Harwell group19±24 in annular ¯ ow. Other data[e.g. Govier et al.7±9, Beggs37, etc] gave general averageagreement with Figure 1, but showed a degree of scatterwhich was no doubt accentuated by an unsystematicapproach to measurement of the data. In general, the totalpressure drop data presented as a maximum value setbetween two minima situated at medium and low gas ¯ owrates. The larger diameter pipe possessed a higher and lowervalue of pressure drop than the corresponding data for thesmaller diameter set at lower and medium gas ratesrespectively. The cross over point (arrowed on Figure 1)moved to lower gas ¯ ows as the liquid rate was increasedand approximately coincided with the neutral frictionalpressure drop point. Thus a signi® cant diameter effect wasin evidence which was particularly noticeable at lower gasrates.

Figures 2 to 5 present the 0.26 m i.d. data for four liquidrates and also show the average values for the measuredtotal pressure loss and the calculated head and frictionallosses. In addition, Figure 2 gives the range of the totalpressure drop.

The data were obtained by initially setting a liquid rateand systematically increasing the gas rate, and thenreversing the process. At ® rst, the total pressure loss wascomposed of a pure liquid head with a small frictionalcomponent. The liquid or gas only data gave agreementwith single phase calculations. At very low gas ¯ ows,

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0263±8762/98/$10.00+0.00� Institution of Chemical Engineers

Trans IChemE, Vol. 76, Part A, July 1998

629VERTICAL TWO-PHASE FLOW: PART III

Trans IChemE, Vol 76, Part A, July 1998

Table 1. Data and Models for Vertical Two Phase Flow.

Diameter VSL VSG

System cm ms2 1 ms 2 1 Comment Reference

diesel-air 0.21, 0.15 0±0.64 0.06±38.5 data correlation [1]kero-air 0.21, 0.15 0±0.19 0.07±35.7

oil-gas 5.07, 5.18, 6.20, 6.27, 0.05±0.73 0.10±2.75 data correlation [2]7.42, 7.60, 7.67, 7.79

water-air 1.32, 5.08 0±0.54 0.74±16.0 data [3]

water-steam 5.94 0.79±0.98 x = 0.315±0.0448 data [4]20±80 atm (a)

water-air 2.54, 2.82, 5.03 0±0.13 2±68.9 data, theory [5]

water-steam 2.21 0.42±0.93 x = 0.004±0.043 data [6]

water-air 1.6, 2.6, 3.81, 6.35 0.02±2.24 0±27.82 data correlation [7±9]

water-air 0.09, 1.95 0.09±0.66 0.16±20.3 data correlation [10±11]aq. soln-air 2.77

water-N2 0.53±0.70 0.05±1.20 4.3±215 data [12]aq soln-N2

water-steam 0.50 0.23±1.45 370±815

water-air 2.70 0.2±3.0 0.3±64 data correlation [13±14]

water-steam 2.21 0.30±0.90 2.75±30.2 data [15]

water-air 1.08 0.65±13.7 2±64 data, theory [16]

water-air 2.66 0.02±0.75 0.6±31 data correlation [17]

water-air 1.27, 1.91, 2.54 0.08±1.0 0.07±1.1 data, theory [18]

water-air 3.18, 3.45 0.±0.2 20±90 data correlation [19±20]

water-air 3.18 0.01±0.96 25.6±64.0 data correlation [21±24]

water-steam 2.54, 3.81 VT = 0.91±2.13 data correlation [25]

water-Argon 2.50 0±2.4 7.4±1324 data [26]aq. soln-Argon 2.50 0±1.2 23.5±565

oil-gas 5.96 0.05±0.62 0.01±4.52 data correlation [27]

water-air 2.66 0.02±0.73 0±32.7 data correlation [28]oil-air 2.66 0.03±0.85 0±32.7aq. soln-air 2.66 0.02±0.70 0±31.8

water-gas 2.66, 3.51 0.01±3.20 5.8±67.8 data correlation [29±31]oil-gas 4.09, 5.25

water-air 1.59, 3.18 0±0.6 16.3±136.1 data correlation [32]

water-air 1.28, 2.23 0.02±0.01 0±28.6 data correlation [33]

water-air 1.3±3.8 0.02±3.64 0.01±105 data [34]

oil-gas 1.27, 1.91, 2.54, 7.62, 0±1.0 0±2.75 data correlation [35]20.32

water-steam 0.5, 0.8, 0.92 0.2±2.58 0.52±26 data correlation [36]water-Argon 2.5 0.25±2.40 0.01±22water-N2 2.5 0.22±1.38 0.01±22EtOH-Argon 1.5 0.28±2.58 0.01±23

water-air 2.66, 4.09 0.02±2.53 1.21±95.9 data correlation [37]

water-air 3.81 0.03±0.32 0.01±0.03 data [38]

water-air 3.18 0.01±0.64 13.39±72.21 data [39]HCl-air 3.18 0.02±0.17 14.91±58.28

water-air 2.54 0.01±2.06 0±45.8 data [40]aq. soln-air

water-air 4.54 0.01±1.05 0.08±60.2 data [41]

water-steam 0.5±1.0 1±13.75 74±4833 data correlation [42]1.5, 2.5 3.5±4.9 40±84440.5±1.5 0.6±12.0 37±133701.2, 1.7 0.76±1.60 46±1852

water-N2 2.5, 5.08 0.1±3.0 2.4±2195

water-N2 7.79±10.5 0±0.96 0.65±175 data correlation [43]

water-steam 19.88 0.08±2 1.8±8 data correlation [44]

water-air 1.17 0.31±10.6 0.07±96.0 data [45]water-He 0.31±10.6 0.13±148

homogeneous conditions prevailed in the bubble regime. Inthis region the total pressure loss fell rapidly and could becalculated from the two phase density. These conditionswere shown to prevail until QG/QL = 0.1. Again in thisregion, the frictional component was a minor element in thetotal pressure drop. As the gas rate was increased, the totalpressure drop continued to fall rapidly along with the two-phase density, but the bubbles grew in size and formed a capor slug of gas over the whole cross section of the pipe. In thisregion, the Nicklin et al.66 model gave an accurateprediction of holdup right up until the onset of annular¯ ow. The limit of the bubble region was shown to follow theTaitel et al.67 criterion.

V SL = 3.0 V SG 2 1.15g(rL 2 rG)j

r2L

1/4

(1)

In both the large bubble and slug regimes the calculatedfrictional pressure drop became negative in magnitude. Thenormal explanation for the effect has been that the liquid¯ ows downward in an annular ring between the inside of thepipe wall and the rising gas Taylor bubble resulting in anegative frictional region. However, while the downwardliquid annulus ¯ ow can explain the wide ¯ uctuationsobserved in the total pressure drop, it cannot explain thenegative value of the overall average value of frictionalpressure drop.

Guedes de Carvalho et al.68 have shown that a risingTaylor bubble of gas carried a pocket of liquid in its wake

such as to effectively reduce the apparent head loss of thetwo-phase liquid to below that expected from the actualtwo-phase density. Thus the calculated negative friction wasdue to the upward lifting of the liquid in the wake of therising gas Taylor bubble. Indeed, it has been shown69 thatthe annular ring of liquid ¯ owing downward around therising bubble was supercritical and therefore would enwrapa pocket of liquid below the Taylor bubble to a depthprescribed by the onset of the hydraulic jump whichoccurred when the Fr in the annular liquid ® lm passedbelow unity. The calculation of the frictional pressure dropby subtracting the two-phase head from the total pressuredrop would be incorrect, as the rise effect in the Taylorbubble wake liquid has been ignored. Two things were ofimportance. Initially, frictional pressure loss was insignif-icant and the ® rst minimum in the calculated frictionpressure loss in Figures 2 to 5 corresponded to the transitionfrom slug to churn ¯ ow. With the onset of churn ¯ ow, the

630 SPEDDING et al.


Table 1. Continued

Diameter VSL VSG

System cm ms2 1 ms 2 1 Comment Reference

water-Freon 12 0.31±4.2 0.16±36.0 data [45]

oil-gas 2.54±16.5 0±2.50 0.02±77.7 data correlation [46]

water-air 1, 1.6, 2.6 0.006±0.1 20±60 data correlation [47]

water-steam [26], [36], [42], [43] [26], [36], [42], [43] [26], [36], [42], [43] correlation [48]

water-air 7.6 0.01±227 0.01±6.00 correlation [49]

aq. cmc-air 1.9 0.3±1.0 0.17±1.62 data correlation [50]

oil-gas 5.07, 16.17 0.03±144 0.05±2.71 data [51]

Figure 1. Total two phase pressure drop for air-water vertical ¯ ow in twopipes of different diameters.

Figure 2. Total two phase pressure drop for air-water vertical ¯ ow in a0.026m i.d. pipe for GSL = 10.97 kg m 2 2 s 2 1 . The bars indicate the¯ uctuations in the measured two phase pressure. Also shown is the head,frictional and up lift pressure.

frictional component started to become signi® cant while theuplifting component steadily reduced until at the onset ofannular ¯ ow, the destruction took place of the liquid wakefollowing the gas slug. The onset of churn ¯ ow reduced theuplift component and increased the frictional component,thus giving a minimum in the total pressure drop curvewhere the churn ¯ ow regime passed into the semi-annular

pattern. In this and the subsequent regimes the chaoticnature of the ¯ ow experienced in the churn regime wassteadily reduced along with the liquid holdup. The totalpressure drop was reduced as the gravitational, theinterfacial and the ® lm activity were also reduced as the® lm passed successively from annular roll wave to theannular ripple regime. However, the pressure drop com-menced to rise again with increased gas ¯ ow at the onset ofthe annular droplet regime. The momentum exchangebetween the gas and liquid phases increased and, as dropletswere increasingly being formed, the energy necessary wasextracted from the gas phase giving a substantial increase inthe frictional component of pressure drop. These competingeffects resulted in a second minimum being formed in thepressure drop characteristic with increasing gas rate.

Thus the formation of different ¯ ow patterns provided acoherent explanation of the two-phase pressure dropcharacteristics for vertical upward ¯ ow. The differentgeneral behaviour observed for the two sets of pressuredrop curves given in Figures 2 and 3 and Figures 4 and 5resulted from the super® cial liquid velocity being respec-tively below and above the rise velocity of the gas Taylorbubble for this system. For the latter case, the trailing liquidslug was not held securely behind the rising gas bubble dueto disturbance of the enclosing liquid down ¯ ow.

It is clear from the discussion that it is not possible toreliably predict pressure drop using one model alone.

Various prediction models were tested, using data forvarious diameters, Nguyen41, Hewitt et al.52,53, Woods55.The prediction methods tested by Spedding and Ferguson56



Figure 3. Total two phase pressure drop for air-water vertical ¯ ow in0.026m i.d. pipe for GSL = 62.67 kg m 2 2 s 2 1 . Also shown is the head,frictional and up lift pressures.

Figure 4. Total two phase pressure drop for air-water vertical ¯ ow in a0.026m i.d. pipe for GSL = 261.00kg m 2 2 s 2 1 . Also shown is the head,frictional and up lift pressures.

Figure 5. Total two phase pressure drop for air-water vertical ¯ ow in0.026m i.d. pipe for GSL = 376.00 kg m 2 2 s 2 1 . Also shown is the head,frictional and up lift pressures.

for the horizontal condition did not prove to be useful. Thisincluded the Olujic57 and Spedding and Hand58 modelswhich were successful for the horizontal case. The generalmethod of Hagedorn and Brown29±31 gave predictions wellin excess of data. The reason was that the system studied byHagedorn and Brown29±31 was a gas lift operation where ahead of liquid was imposed at the base of the ¯ ow rig. Inaddition, no consideration was given to possible changes of¯ ow regime up the column and there was no assurance thatthe column was indeed vertical. By contrast, the models ofBeggs and Brill62 and Orkisewski35 made provision fordifferent types of ¯ ow regimes. However, neither modelproved useful, with the Beggs and Brill method62 tending tounderpredict the data, while the Orkisewski35 methodshowed a wide scatter around the data. The method ofHolt et al. 65 while allowing for prediction over the wholerange of ¯ ow regimes, tended to overpredict the data.

Prediction has been achieved for low gas ¯ ow rate underQG/QL = 0.1 in the homogeneous regime. For the slug ¯ owregime the amount of liquid holdup which was uplifted bythe gas slug RLU was:

RLU

RL

= 2.34 ´ 10 2 4 d

dc

2 2.34 ´ 10 2 4 ReSG

2 1.638d

dc

1 1.638 (2)

The critical diameter dc = 0.05 m above which RLU = 0,and was a region where the hydraulic jump of the annularliquid ® lm could not enclose and uplift liquid in the gas slugtail. Equation (2) only applies for the condition that V SL wasless than the Taylor bubble rise velocity. When V SL was inexcess of the bubble rise velocity, RLU was small. Equation(2) predicted the slug data for 0.026 and 0.0454m i.d. pipeto within an average of 1% (2 10.2% to 1 4.3% range).

The annular regimes at high gas ¯ ow rates ReSG > 8 ´ 104

followed the relation

DP

D z TP

= 1.5457 ´ 10 2 14Re3.5683SL Re( 2 0.6315 log ReSL 1 3.1909)

SG

´ f0.0317

d

3

(3)

within 6 14%. Values of f are presented in Figure 6. Therationale underlying the model was that total two-phasepressure drop was mainly caused by friction in the annular¯ ow regime and that it varied approximately as 1/d3, thesame as with single phase ¯ ow.

Predictions in the intermediate gas ¯ ow region arepossible but will be the subject of future work.

CONCLUSIONS

The total pressure drop for vertical ¯ ow against gas ratepresented as a maximum value at the end of the churn ¯ owregime, set between two minima situated at low andmedium gas ¯ ow rates. There was a noticeable effect ofdiameter on the pressure drop. It was not possible to predictthe whole range of pressure drop values. However, thegeneral form of the total pressure drop curve was explainedqualitatively.

Calculation of the frictional pressure drop using onlyhold-up data resulted in negative values in the slug, churn

and semi-annular regimes. This was physically unrealisticand resulted from ignoring the pocket of liquid rising in thewake of the Taylor gas bubble.

None of the theories for predicting pressure loss provedeffective. This was particularly true of those models basedon air-lift operation. It was possible to predict the totalpressure drop at very low gas rates (QG/QL # 0.1) in thebubble regime and a high gas rates (GG > 30 kg m 2 2 s 2 1) inthe annular droplet regime.

NOMENCLATURE

D diameter, mg gravitational constant, m s2 2

p pressure, kg m 2 1 s 2 2

Q ¯ ow rate, m3 s 2 1

Re Reynolds NumberV velocity, ms 2 1

Greek lettersr density, kg m2 3

f factor equation (3)

Subscriptsc criticalG gasL liquidS super® cialT totalU uplift

REFERENCES

1. Lockhart, R. W. and Martinelli, R. C., 1949, Proposed correlation ofdata for isothermal two-phase, two-component ¯ ow in pipes, ChemEng Proc, 45: 39±48.

632 SPEDDING et al.


Figure 6. Frictional factor f values used in equation (2) for prediction oftotal two phase pressure drop for vertical air-water annular droplet ¯ ow.

2. Poettmann, F. H. and Carpenter, P. G., 1952, The multiphase ¯ ow ofgas, oil and water through vertical ¯ ow strings with application to thedesign of gas lift installations, Drill & Proc Pract, API 257±317.

3. Galegar, W. C., Stovall, W. B. and Huntington,R. L., 1954, More dataon two-phase vertical ¯ ow, Pet Ref 33: 208±11.

4. Schwarz, K., 1954, Investigation of density distribution, water andsteam velocities and frictional pressure drop in horizontal and verticalup¯ ow boiler tubes, Ver Deut Ing Forsch, B20(445): 1±41.

5. Calvert, S. and Williams, B., 1955, Upward co-current annular ¯ ow ofair and water in smooth tubes, AIChE J, 1: 78±86.

6. Isbin, H. S., Sher, N. C. and Eddy, K. C., 1957, Void fractions in twophase steam-water ¯ ow, AIChE J, 3: 135±142.

7. Govier, G. W., Radford, B. A. and Dunn, J. S. C., 1957, The upwardvertical ¯ ow of air water mixtures I, Can J Chem Eng, 35: 58±70.

8. Govier, G. W. and Short, W. L., 1958, The upward vertical ¯ ow of airwater mixtures II, Can J Chem Eng, 36: 195±202.

9. Brown, R. A. S., Sullivan, G. A. and Govier, G. W., 1960, Theupward vertical ¯ ow of air water mixtures III, Can J Chem Eng, 38:62±66.

10. Ueda, T., 1958, Studies on the ¯ ow of air-water mixtures, Bull JSocMech Eng, 1: 139±145.

11. Ueda, T., 1967, An upward ¯ ow of gas-liquid mixtures in verticaltubes, Bull JSoc Mech Eng, 10: 989±999, 1000±1007.

12. Silvestri, M., Finzi, S., Ruseo, L., Schiavon, M. and Zavattarelli, R.,1958, Gas-liquid and vapour-liquid mixtures as cooling agents, Proc22nd Int Conf Peaceful Uses for Atomic Energy, 7: 819±826.

13. Chisholm, D. and Laird, A. D. K., 1958, Two-phase ¯ ow in roughtubes, Trans Am Soc Mech Eng, 80C: 276±286.

14. Chisholm, D., 1983, Two-Phase Flow in Pipelines and HeatExchangers (Godwin).

15. Isbin, H. S., Larson, H. C., Moen, R. H., Mosher, D. R. and Wickey,R. C., 1959, Two-phase steam-water pressure drops, Chem Eng ProgSymp Ser, 55: 75±84.

16. Anderson, G. H. and Mantzouranis, B. G., 1960, Two-phase (gas-liquid) ¯ ow phenomena I Pressure drop and hold-up for two-phase ¯ owin vertical tubes, Chem Eng Sci, 12: 109±126.

17. Hughmark, G. A. and Pressburg, B. S., 1961, Holdup and Pressure dropwith gas-liquid ¯ ow in a vertical pipe, AIChE J, 7: 677±682.

18. Grif® th, P. and Wallis, G. B., 1961, Two phase slug ¯ ow, Trans Am SocMech Eng, 83C: 307±320.

19. Bennett, J. A. R. and Thornton, J. D., 1961, Data on vertical ¯ ow of air-water mixtures in the annular and dispersed ¯ ow regions Part Ipreliminary study, Trans Inst Chem Eng, 39: 101±112.

20. Collier, J. G. and Hewitt, G. F., 1961, Data on the vertical ¯ ow of air-water mixtures in the annular and dispersed ¯ ow regions Part II ® lmthickness and entrainment data and analysis of pressure dropmeasurements, Trans Inst Chem Eng, 39: 125±136.

21. Hewitt, G. F., King, I. and Lovegrove,P. C., 1961, Holdup and pressuredrop measurement in the two-phase annular ¯ ow of air-water mixtures,AERE, R3764.

22. Gill, L. E. and Hewitt, G. F., 1962, Further data on the upward annular¯ ow of air-water mixtures, AERE, R3935.

23. Gill, L. E., Hewitt, G. F., Hitchon, J. W. and Lacey, P. M. C., 1963,Sampling probe studies of the gas core in annular two-phase ¯ ow. Theeffect of length on phase and velocity distribution, Chem Eng Sci, 18:525±535.

24. Gill, L. E., Hewitt, G. F. and Lacey, P. M. C., 1964, II Studies of theeffect of phase ¯ ow rates on phase and velocity distribution,Chem EngSci, 19: 665±682.

25. Haywood, G. E. and Thom, J. R. S., 1961, An experimental study of the¯ ow conditions and pressure drop of steam-water mixtures at highpressure inheatedandunheated tubes,ProcInstMech Eng, 175:669±745.

26. Adorni, N., Casagrande, I., Cravarolo, A., Hassid, A. and Silvestri, M.,1961, Experimental data on two-phase adiabatic ¯ ow: Liquid ® lmthickness, phase and velocity distribution,pressure drop in vertical gas-liquid ¯ ow, CISE Rept R35.

27. Francher, G. H. and Brown, K. E., 1963, Prediction of pressuregradients for multiphase ¯ ow in tubing, Soc Pet Eng J, March, 59±69.

28. Hughmark, G. A., 1963, Pressure drop in horizontal and vertical co-current gas-liquid ¯ ow. IEC Fund, 2: 315±321.

29. Hagedorn, A. R., 1964, Experimental study of pressure gradientsoccurring during vertical two-phase ¯ ow in small diameter conduits,PhD Thesis, (University of Texas).

30. Hagedorn, A. R. and Brown, K. E., 1964, The effect of viscosity ontwo-phase vertical ¯ ow, J Pet Tech, 16: 203±210.

31. Hagedorn, A. R. and Brown, K. E., 1965, Experimental study of

pressure gradients occurring during continuous two-phase ¯ ow in smalldiameter vertical conduits, J Pet Tech, 17: 475±484.

32. Shearer, C. J. and Nedderman, R. M., 1965, Pressure gradient andliquid ® lm thickness in co-current upwards ¯ ow of gas/liquid mixtures.Application to ® lm-cooler design, Chem Eng Sci, 20: 671±683.

33. Willis, I. J., 1965, Upwards annular two-phase air/water ¯ ow in verticaltubes, Chem Eng Sci, 20: 895±902.

34. Shiba, M. and Yamazaki, Y., 1967, A comparative study on thepressure drop of air-water ¯ ow, Bull J Soc MechEng, 10: 290±298.

35. Orkiszewski, J., 1967, Predicting two-phase pressure drops in verticaltubes, J Pet Tech, July 829±838.

36. Lombardi, C. and Pedrocchi, E., 1972, A high pressure drop correlationin two-phase ¯ ows, Eng Nucl, 19: 91±99.

37. Beggs, H. D., 1976, An experimental study of two-phase ¯ ow ininclined pipes, PhD Thesis, (University of Tulsa).

38. Bhaga, D. and Weber, M. E., 1972, Holdup in vertical two and threephase ¯ ow Part II Experimental investigation, Can J Chem Eng, 50:329±336.

39. Whalley, P. B., Hewitt, G. F. and Hutchinson, P., 1973, Experimentalwave and entrainment measurements in vertical annular two-phase¯ ow, AERE, R7521.

40. Oshinowo, T. and Charles, M. E., 1974, Vertical two-phase ¯ ow Part IIHold up and pressure drop, Can J Chem Eng, 52: 438±448.

41. Nguyen, V. T., 1975, Two-phase gas-liquid cocurrent ¯ ow, PhDThesis, (University of Auckland).

42. Lombardi, C. and Ceresa, I., 1978, A generalized pressure dropcorrelation in two-phase ¯ ow, Eng Nucl, 25: 181±198.

43. Bonfanti, F., Ceresa, I. and Lombardi, C., 1979, Two-phase pressuredrop in the low ¯ ow rate region, Eng Nucl, 26: 481±492.

44. Gould, T. L., 1974, Vertical two-phase steam-water ¯ ow in geothermalwells, J Pet Tech, 26: 833±842.

45. Aggour, M. A. and Sims, G. E., 1984, Effect of the gas-phase density on¯ ow pattern and frictional pressure drop in two-phase, two-componentvertical ¯ ow, Process Tech Proc 1 (multi-phase ¯ ow heat transfer,3, PtA), 137±153 (Editors: Veziroglu, T. N. and Bergles, A. E.).

46. Abdul Majeed, G. H., Abu Al-Souf, N. B. and Alassal, J. R., 1989, Animproved revision of the Hagedorn and Brown liquid holdupcorrelation, J Can Pet Tech, 78(6): 25±8.

47. Fukano, T., Kawakami, Y., Ousaka, A. and Tominaga, A., 1991,Interfacial shear stress and holdup in an air-water annular two-phase¯ ow, Proc ASME-JSME Thermal Eng Joint Conf, 3(2): 217±22,(Editor Lloyd, J. R. Kurosahi, Y).

48. Lombardi, C. and Carsana, C. G., 1992, A dimensionless pressuredrop correlation for two-phase mixtures ¯ owing uphill in verticalducts covering wide parameter ranges, Heat Technology, 10: 125±141.

49. Chokshi, R. N., 1994, Prediction of pressure drop and liquid holdup invertical two-phase ¯ ow through large diameter tubing, PhD Thesis,(University of Tulsa).

50. Das, S. K. and Biswas, M. N., 1995, Pressure losses in two-phase gas-non-newtonian liquid ¯ ow in a vertical tube, Chem Eng Comm, 135:229±237.

51. Aggour, M. A., Al-Yousef, H. Y. and Al-Muraikhi, A. J., 1996,Vertical multiphase ¯ ow correlations for high production rates andlarge turbulence, SPE Prod Facil, 11(1): 41±8.

52. Gill, L. E., Hewitt, G. F. and Lacey, P. M. C., 1965, Data on theupwards annular ¯ ow of air-water mixture, Chem Eng Sci, 20: 71±88.

53. Gill, L. E. and Hewitt, G. F., 1968, Sampling probe studies on the gascore in annular two-phase ¯ ow III, Chem Eng Sci, 23: 677±686.

54. Hewitt, G. F., 1982, Applications of two-phase ¯ ow, Chem Eng Prog,38±46.

55. Woods, G. S. and Spedding, P. L., 1994, Vertical multiphase ¯ ow,IChemE Research Event, London, 1211±1213.

56. Spedding, P. L. and Ferguson, M. E. G., 1995, Measurement andprediction of pressure drop in two-phase ¯ ow, J Chem Tech Biotech,62: 262±278.

57. Olujic, Z., 1985, Predicting two-phase ¯ ow friction loss in horizontalpipes, Chem Eng, June 24, 45±50.

58. Spedding, P. L. and Hand, N. P., 1997, Prediction in strati® ed gas-liquid co-current ¯ ow in horizontal pipelines, Int J Multiphase Flow,40: 1923±1935.

59. Ros, N. C. J., 1961, Simultaneous ¯ ow of gas and liquid as encounteredin well tubing, J Pet Tech, 13: 1037.

60. Duns, H. and Ros, N. C. J., 1965, Vertical ¯ ow of gas and liquidmixtures from bore holes, Proc 6th World Pet Conf, Frankfurt, Sect. II,Paper 22-PD6, June.



61. Solovev, A. V., Preobrazhonskii, E. I. and Semenov, P. A., 1967,Hydraulic resistance in two-phase ¯ ow, Int Chem Eng, 7: 59±63.

62. Beggs, H. D. and Brill, J. P., 1973, A study of two-phase ¯ ow ininclined pipes, J Pet Tech, 607±617.

63. De Gance, A. E. and Atherton, R. W., 1970, Chemical Engineeringaspects of two-phase ¯ ow Part 6. Vertical and inclined ¯ ow correlation,Chem Eng, 5 Oct, 87±94.

64. Mata Iturbe, J. C., 1989, Program for calculating pressure drop invertical tubes with multiphase ¯ ow using an HP/41C calculator, RevInst Mex Pet, 21(2): 23±33.

65. Holt, A. J., Azzopardi, B. J. and Biddulph, M. W., 1998, Estimation oftwo-phase pressure drop for narrow passages by means of a ¯ owpattern speci® c model, to be published in Trans IChemE, Part A.

66. Nicklin, D. J., Wilkes, J. C. and Davidson, J. F., 1962, Two-phase ¯ owin vertical tubes, Trans Inst Chem Eng, 40: 61±68.

67. Taitel, Y., Barnea, D. and Duker, A. E., 1980, Modelling ¯ ow patterntransitions for steady upward gas-liquid ¯ ow in vertical tubes, AIChEJ,26: 345±354.

68. Guedes de Carvalho, J. R. F., Campos, J. B. L. M. and Teixeira, J. A. S.,1996, Physical modeling of axial mixing in slugging gas-liquidcolumns, Chapter 3, in Multiphase Reactor and Polymerization SystemHydrodynamics, (Editor, N. P. Cheremisinoff) (Gulf), p49±65.

69. Spedding,P. L., Chen, J. J. J. and Nguyen, V. T., 1982, Pressure drop intwo phase gas-liquid ¯ ow in inclined pipes, Int J Multiphase Flow, 8:427±431.

ADDRESS

Correspondence concerning this paper should be addressed to ProfessorP. L. Spedding, Department of Aeronautical Engineering, The Queen’ sUniversity of Belfast, David Keir Buildings, Stranmillis Road, BelfastBT9 5AG, UK.

The manuscript was received 4 August 1997 and accepted for publicationafter revision 3 February 1998.

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