Dev. Chem. Eng. Mineral Process. 12(1/2), pp. 179-188, 2004. Momentum Balance for Two-Phase Horizontal Pipe Flow Part 1: Friction Factors P.L. Spedding", J.S. Cole and G.F. Donnelly School of Aeronautical Engineering, The Queen 's University of Belfast, Stranmillis Road, Belfast BT9 SAG, Northern Ireland, UK Estimations of gas wall, liquid wall and interfacial fiiction factors for two-phase horizontal co-current pipe flow are discussed critically after being checked against reliable data obtained under a wide range of conditions. The use of equivalent diameters and the Blasius relation were shown to be valid for estimation of the gas wall fiiction. Prediction of liquid wall and interfacial friction factors proved to be more drfficcult but estimation improved if consideration was given to the efects of liquid holdup and interfacial liquid shape. Introduction The one dimensional momentum balance model has been proposed for the prediction of pressure loss, holdup and multiphase flow characteristics by considering the transfer of momentum between the gas and liquid phases. This mechanistic approach is not entirely theoretical since the development of representative wall and interfacial friction factors are necessary for closure of the derived equations. The approach also neglects any variations in the flow geometry. The model was originally developed by Johannessen [ 13 for turbulent flows and extended by others [2, 31. Taitel and Dukler [4] suggested a dimensionless form of the momentum balance which has been shown *Authorfor correspondence. I79 P.L. Spedding, J.S. CoIe and G.F. Donnelly to have a multivalued solution under certain circumstances [5-71. Successive modifications have been proposed for the relationships governing the interfacial and wall friction factors [3, 6, 8-24]. It is the purpose of this study to examine the friction factors to ascertain which are the most appropriate for use with the momentum balance two-phase model. Friction Factors (0 Gas Wall Friction Factor Andritsos [ 171 showed experimentally that the use of equivalent diameters was valid when estimating the gas shear stress while others have experimentally verified that the Blasius relation correctly predicted the gas wall friction [19, 20, 23, 241. There is perhaps a slight under prediction at higher gas Reynolds numbers in the annular regimes where surface waves and droplet formation were in evidence. (ii) Liquid Wall Friction Factor Figure 1 gives an example of the results of two-phase studies performed over a diameter range of 0.0259 to 0.0935 m [25-271 where both the Poiseuille and Blasius relations are shown to underpredict the liquid wall friction, except for certain of the stratified data. These findings are in agreement with other detailed work [3, 17, 19, 20, 23, 241. The deviations between experiment and theory increased steadily with gas velocity as evidenced by the St+R and A+D data. The failure to predict accurately arises because effects of the interfacial shear stress exerted on the liquid by the gas, and the shape of the gadliquid interface, have been ignored. In order to improve prediction performance a number of workers have proposed the use of liquid height [17, 191, holdup [20, 241 or interfacial friction [23, 281 in the relations for predicting liquid wall friction. In addition, some workers have given consideration to the degree of surface wetting by the liquid [23, 281 thus challenging the conventional smooth stratified flow assumption. Attempts to model data using two of the correlations, notably Equation (1) by Kowalski [20] and Equation (2) by Hand [26], were an improvement on the Blasius relation but were not entirely successful, as illustrated by Figure 2. I80 Momentum Balance for Two Phase Horizontal Pipe Flow: Friction Factors Figurel. Liquid wall friction against liquid Reynolds group compared to the Poiseuille and Blasius relations using the 0.0454 m diameter data of Nguyen [25]. .-8 5 .- -r; 5J ..-s ,J 0.1 -: --._ -... ST+R ST+RW+D 0 a - _ .P 0 A i.;-,rt" *$GH-.-$. x U 0.01 -: e a ......- Kowelski Q A A J -Hand ST+RW A+BTS .d-s...+---x.+*+ *. *x X x - - X.. ..- .._ ...--- . .---.. 5 0.001 9 181 P.L. Spedding, JS. Cole and G.F. Donneliy 600 500 400 300 200 100 0 0 100 200 300 400 500 Measured h Figure 3. Prediction of h; against measurement using the 0.0953 m diameter data of Andritsos [17/. Figure 4. Intevacial to superficial gas friction against superficial gas Reynolds number using the 0.0454 m diameter data of Nguyen [25]. 182 Momentum Balancefor Two Phase Horizontal Pipe Flow: Friction Factors f, = 0.263(gLRe,)-'" f, = 0.0262(gLResL)-o'139 The main reasons for poor performance of these models was that no consideration was given to the influence of interfacial friction and the actual shape of the wetted surface. Secondly the relations depend on accurate prediction of liquid holdup. Andritsos and Hanratty [18] proposed a relationship based on the work of Cheremisinoff and Davis [13J to describe the wall stress in terms of a dimensionless film height, hl ,as given by: d, = 4A, I S , (5) Figure 3 shows as an example that the model performed favourably for a large proportion of the data but was inadequate at lugher gas rates in the annular type regimes. Tests of other models against data gave similar trends with neglect of interfacial shape and friction contributing to a deterioration in prediction performance. (iii) Interfacial Friction Factor The interfacial friction is usually inferred from the measured axial flowing pressure gradient and liquid holdup, although other measurements have been used [8-101 such as velocity profiles, etc. The magnitude of the interfacial shear stress is influenced by both the surface roughness of the liquid and droplet generation. Johannessen [I] neglected A. altogether. Others assigned a constant value to interfacial friction [9, lo], sometimes only for lower gas rates [13, 151. Either way, such methods led to gross under predictions of two-phase parameters [29]. Taitel and Dukler [4]used the gas wall friction while others employed it only at low gas flows [13, 171. Generally more complex relations have been proposed [8, 12, 20, 22, 23, 281. Hagiwara et al. 183 P.L. Spedding, J.S.Cole and G.F. Donnelly 0.06 0.05 0.04 I ,2 0.03 z 0.02 0.01 0 0 0.01 0.02 0.03 M e a s d fi 0.04 0.05 0.06 Figure 5. Measured and predicted interfacial friction factor using the method of Hand [26] tested against data of Andritsos [17], d = 0.02515 m. - I": 0.01 0.1 Increased liquid loading ' """4 ' 1 """:' 10 "'UA loo Measured Figure 6. Measured and predicted interfacial fitiction factor using the method of Grolman [28] tested against data of Hand [26], d = 0.0935 m. I84 Momentum Balancefor Two Phase Horizontal Pipe Flow: Friction Factors [30] demonstrated experimentally that the gas wall shear stress increased with both wave amplitude at the liquid interface and associated turbulence, thus justifjmg an emphasis on relations using a function of interfacial to gas wall friction for prediction purposes [ 16, 17, 19,261. The profile given in Figure 4 highlights that as the gas rate was increased so as to generate roll waves, there was a corresponding sharp rise in the magnitude of the f,. / fsc ratio. In modelling f,. solely in terms of gas wall friction factor, Crowley and Rothe [6] recognised the variation in the magnitude of the interfacial shear stress associated with scale-up and suggested that data may be bounded within a f, I f s G ratio of 1 to 10. The data of Donnelly [27] was also located within these limits. Hand [26] indicated that a sizable selection of 0.0935 m diameter data was located on a f, / fsc loci of 4. For the larger diameter of 0.18 m data, Kawaji et al. [21] found a loci of 3 provided the most appropriate data fit. The interfacial friction factor proposed by Hand [26] is tested against data in Figure 5, and gave a generally poor performance but certainly the best fit for this general type of f, prediction method. Both Hart et al. [23] and Grolman [28] advanced the accuracy of prediction, as shown in Figures 6 to 8, by considering the effect of interfacial liquid shape when developing methods of interfacial shear estimation. Discussion The prediction of shear stress deteriorated in accuracy when proceeding progressively from ?wG , the gas wall value, to ,T for the liquid wall, through to ‘ti for the gadliquid interface. The value of the latter shear stress t i has a pronounced effect on the momentum balance and its accurate estimation presents the greatest contemporary challenge to successful phenomenological modelling of multiphase flow. It is clear that neither the liquid wall friction nor the interfacial shear can be accurately accounted for without proper consideration being given to both the liquid holdup value and interfacial shape present within the system. Accurate prediction of liquid holdup has been detailed by Spedding et al. [3 1-33] while both the interfacial shape 185 P.L.Spedding, J.S.Cole and G.F. Donnelly 100 10 B 1 B 0.1 L4 0.01 0.01 0.1 1 10 100 Measuxed Figure 7. Measured and predicted interfacial frication factor using the method of Grolman [28J tested against data of Nguyen [25], d = 0.0454m. Increased liquid loading 0.01 0.01 0.1 1 10 100 Measured Figure 8. Measured and predicted inte~acialpiction factor using the method of Grolman [28J tested against data of Donnelly [27]. d = 0.0259m. 186 Momentum Balance for Two Phase Horizontal Pipe Flow: Friction Factors and shear stress have been explored for low liquid holdup values [23, 291. A detailed examination of these factors and their relationship to the momentum balance modelling will be the subject of a future study. Conclusions The gas wall shear stress in two-phase flow can be accurately predicted using the Blasius relation in association with the equivalent diameter. Prediction of both liquid wall and interfacial shear exhibited scattered results particularly with the latter. Attention to the effects of liquid holdup and interfacial shape led to an improvement in prediction. Nomenclature A AL BTS d dL D f F hL h' IW LRW R R Annular regime Cross sectional liquid area (m') Blow through Slug regime diameter (m) Equivalent liquid diameter (m) Droplet regime Friction factor Film regime Centre line liquid height (m) Dimensionless film height (Eqn 3) Inertial wave regime Long roll wave regime Ripple regime Holdup RW Re SL St V P P '5 Roll wave regime Reynolds number, dVp / p Liquid wetted length (m) Stratified regime Velocity (m s-' Viscosity (kg m-' s-' ) Density (kg m-3) Shear stress (kg m" s-' ) Subscripts G Gas i L S W Interface Liquid Superficial Wall References I . Johannessen, T. 1972. A theoretical solution of the Lockhart and Martinelli flow model for calculating two phase flow pressure drop and holdup. Int. J. Heat Mass Transfer, 1443-1449. 2. Agrawal, S.S., Gregory, G.A. and Govier, G.W. 1973. An analysis of horizontal stratified two phase flow in pipes. Can. J. Chern. Eng., 280-2861. 3. Russell, T.W.F., Etchells, A.W., Jensen, R.H. and Armnda, T.J. 1974. Pressure drop and holdup in stratified gas-liquid flow. AIChEJ, 20,664-669. 4. Taitel, Y. and Dukler, A.E. 1976. A model for predicting flow regime transitions in horizontal and near horizontal gas liquid flow. AIChEJ, 2 , 4 7 - 5 5 . 5. Baker, A., Nielsen, K. and Gabb, A. 1988. Pressure loss, liquid holdup calculations developed. Oil & Gas J. Technology, 55-59, March 14"'. 6. Crowley, C.J. and Rothe, P.H. 1988. Assessment of mechanistic two phase analysis method for gascondensate pipelines. Ann. Meeting PSIG, Toronto, Oct 20-25. 7. Landrnan, M.J. 1991. Non-unique holdup and pressure drop in two-phase stratified inclined pipe flow. 377-394. Int. J. Multiphase Flow, u, a, u, 187 P.L. Spedding, J.S. Cole and G.F.Donnelly 8. Hidy, G.M. and Plate, E.J. 1966. Wind action on water standing in a laboratory channel. J. Fluid Mech., 2,651-687. 9. Cohen, L.S. and Hanratty, T.J. 1971. Effects of waves at a gadliquid interface on turbulent airflow. J. Fluid Mech., 467-479. 10. Miya, M., Wordmansee, D.E. and Hanratty, T.J. 1971. A model for roll waves in gas-liquid flow. Chem. Eng. S c i . , s , 1915-1931. 1 I . Aggour, M.A. and Sims, G.E. 1978. A theoretical solution of pressure drop and holdup in two phase stratified flow. Proc. Heat. Transfer Fluid Mech. Inst., 205-217. 12. Tsiklauri, G.V., Besfamiling, P.V. and Barysher, Y.V. 1979. Two phase momentum mass and heat transfer. Edit. Durst, F., Tsiklauri, G.V. and Afghan, N.H., 1,357-372, Hemisphere. 13. Cheremisinoff, N.P. and Davis, J.E. 1979. Stratified turbulent - turbulent, gas-liquid flow. AIChEJ, 2, 48-56. 14. Kadambi, V. 1981. Void friction and pressure drop in two-phase stratified flow. Can. J. Chem. Eng., 59,582-589. 15. Shoham, 0. and Taitel, Y. 1984. Stratified turbulent-turbulent gas-liquid flow in horizontal and inclined pipes. AIChEJ, 30,377-385. 16. Laurinat, J.E., Hanratty, T.J. and Dallman, J.C. 1984. Pressure drop and flow height measurements for annular gas-liquid flows. Int. J. Multiphase Flow, lo,341-356. 17. Andritsos, N. 1986. Effects of pipe diameter and liquid viscosity on horizontal stratified flow. Ph.D. a, Thesis, Univ. Illinois Urbana. 18. Andritsos, N. and Hanratty, T.J. 1987. Interfacial instabilities for horizontal gas-liquid flow in pipelines. Int. J. Multiphase Flow, 12,583-603. 19. Andreussi, P. and Persen, L.N. 1987. Stratified gas-liquid flow in downwardly inclined pipes. Int. J. Multiphase Flow, 1 1 , 5 6 5 6 7 . 20. Kowalski, J.E. 1987. Wall and interfacial shear stress in stratified flow in a horizontal pipe. AIChEJ, 274-281. 21. Kawaji, M.. Anoda, Y.,Nakamura, H. and Tasaka, T. 1987. Phase and velocity distributions and u, holdup in high pressure steadwater stratified flow in a large diameter honzontal pipe. Int. J. Multiphase Flow, 13,145-159. 22. Oilemans, R.V.A. 1987. Modelling of gas condensate flow in horizontal and inclined pipes. ASME Pipeline Eng. Symp.,ECTE, Dallas, 73-81. 23. Hart, J., Hamersma, P.J. and Fotuin, J.M. 1989. Correlations predicting frictional pressure drop and liquid hold-up during horizontal gas-liquid pipe flow with a small hold-up. Int. J. Multiphase Flow, u, 947-964. 24. Spedding, P.L. and Hand, N.P. 1997. Prediction in stratified gas-liquid co-current flow in horizontal pipelines. Int. J. Heat Mass Transfer, 40,1923-1935. 25. Nguyen, V.T. 1975. Two phase gas-liquid co-current flow. An investigation of holdup, pressure drop and flow pattern in a pipe at various inclinations. Ph.D. Thesis, Univ. Auckland. 26. Hand, N.P. 1991. Gas-liquid co-current flow in a horizontal pipeline. Ph.D. Thesis, The Queen’s Univ. of Belfast. 27. Donnelly, G.F. 1997. An analytical evaluation of horizontal multiphase flow. Ph.D. Thesis, The Queen’s Univ. of Belfast. 28. Grolman, E. 1994. Gas-Liquid flow with low liquid loading in slightly inclined pipes. Ph.D. Thesis, Univ. Amsterdam. 29. Crowley, C.J., Sam, R.G., Wallis, G.B. and Mehta, D.C. 1984. Slug flow in large diameter pipe - effect of fluid properties. AlChE Conf. Fund. Res. Multiphase Flow, Paper 105d, November. 30. Hagiwara, Y., Esmneilzadeh, E., Tsutsui, H. and Suzuki, K. 1989. Simultaneous measurement of liquid film thickness, wall shear stress and gas turbulence of horizontal wavy two phase flow. Int. J. Multiphase Flow, 421-431. 31. Spedding, P.L. 1997. Holdup prediction in vertical upwards to downwards flow. Dev. Chern. Eng. Min. Process., &43-60. 32. Spedding, P.L. and Cooper, R.K. 2002. A note on the prediction of liquid holdup with the stratified roll-wave regime for gadliquid co-current flow in horizontal pipes. Int. J. Heat Mass Transfer, 45,219- u, 222. 33. Watterson, J.K., Cooper, R.K. and Spedding, P.L. 2003. A theory of liquid holdup in stratified roll 107-112. wave horizontal flow. Dev. Chem. Eng. Min. Process., u, Received: 16 July 2002; Accepted afer revision: 1 May 2003. 188

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