Large amplitude waves in a slug tracking scheme

Transcription

1 Computationa Methods in Mutiphase Fow V 99 Lare ampitude waves in a su trackin scheme A. De Leebeeck & O. J. Nyda Department of Enery and Process Enineerin, Norweian University of Science and Technooy (NTNU), Norway Abstract Lare ampitude ro waves are incorporated into a previousy deveoped su trackin scheme for two phase as-iquid pipe fow. The appicabiity of the trackin scheme to are ampitude waves is demonstrated with a simpified mode for the waves. The waves are modeed anaoous to sus on a movin rid with correspondin wave veocities and a pressure variation determined usin an orifice type reation. Sus and waves in the trackin scheme are separated by reions of stratified fow, which are modeed on a stationary rid usin the two-fuid mode. The computationa scheme is described, compared to experimenta data on ro waves, and some wave dynamics such as waves deveopin to sus and sus decayin to waves are demonstrated. Keywords: ro waves, trackin, two phase pipe fow, modein. Introduction In two phase as-iquid pipe fow, different fow reimes occur dependin on as and iquid phase veocities, fuid properties, and pipe eometries. Various numerica strateies exist for the different fow reimes in dynamic modes. Su fow, for exampe, can be treated with unit ce modes (Bendiksen et a. [2]), in su capturin (Bonizzi and Issa [3], Issa and Kempf [7], Renaut [2]), or in trackin schemes [Taite and Barnea [3], Nyda and Banerjee []). Athouh capturin schemes (Issa and Kempf [7], Bonizzi and Issa [3]) can mode the initiation of sus and ro waves, they require the use of fine rids which are computationay expensive and the are computationa times are prohibitive for simuation in on pipeines. Trackin schemes, however, use WIT Transactions on Enineerin Sciences, Vo 63, ISSN (on-ine) doi:.2495/mpf99 29 WIT Press

2 Computationa Methods in Mutiphase Fow V orders of manitude fewer rid points. Trackin schemes can aso be suitabe for pu simuations (Kjøaas [9]). A combination of capturin and trackin has aso been tested by Renaut [2]. Lare ro waves can have simiar scaes and behavior to su fow in that they transport iquid and have a propaatin. A simpe mode treatin waves as movin objects in a simiar way as for sus is therefore desired in the trackin scheme. Simiarities between sus and ro waves such as a propaatin and a sopin tai have been observed experimentay in, for exampe, Johnson [8]. Pressure variations across wave s simiar to sus have aso been measured (De Leebeeck et a. [4]). These experiments are used to deveop a wave mode incudin the observed pressure variation. The su trackin scheme of Kjøaas [9] is the startin point for incorporatin wave trackin capabiities into a su trackin scheme. Su fow is modeed as aternatin iquid sus and bubbes with stratified fow. The two-fuid mode is soved on a stationary staered rid in bubbes, whie intera momentum equations are soved in sus on a movin rid. Before the addition of wave trackin, decayin sus were repaced immediatey with stratified fow. With the addition of wave trackin, sus can decay into waves, modeed anaoous to sus with intera momentum equations and their own and tai veocities. The trackin scheme of Hu et a. [6] incudes wave trackin with a iquid heiht profie behind waves and sus, sovin the two-fuid mode in combination with modein the wave as a hydrauic jump (Hu et a. [6]). Other modes for ro waves incude, for exampe, Johnson [8] and Homås [5] who sove the two-fuid mode with modified friction terms in ro waves. Johnson [8] assumes a sequence of repeatin maximum ampitude waves with a sharp and incudes a unique interfacia friction factor as part of a steady state soution. The mode of Homås [5] mode is dynamic and incudes increased turbuence at the wave usin a modified Biber friction mode. In our scheme, we simpify waves and sus as square objects that can be modeed dynamicay on a coarse rid. The dynamics of the waves and sus are determined from mass and momentum baances. The stratified as reions between waves and sus are soved with a two fuid mode. A are rid ives square shaped bubbes. Su and wave tais can be reproduced by refinin the rid in the bubbe reion. 2 Description of the mode The wave trackin mode buids on a su trackin scheme (Kjøaas [9]) which is coded in C++ usin object oriented prorammin techniques. Su fow is represented in one dimension with aternatin su objects that competey fi the pipe and stratified sections incudin both phases as shown in fiure A. In stratified sections, the two-fuid mode is soved on a stationary staered rid where phase veocities are determined at section borders whie pressure and masses are determined at section centers. Su sections are modeed as movin WIT Transactions on Enineerin Sciences, Vo 63, ISSN (on-ine) 29 WIT Press

3 Computationa Methods in Mutiphase Fow V objects where iquid phase veocity, su enth, and tai veocities are determined from mass and momentum baances. Waves, shown in fiure B, are modeed in a simiar way to sus as movin objects and incude a pressure variation due to iquid acceeration at the wave. In the as phase, the pressure variation across the wave is modeed with an orifice type reation. Assumin that the pressure variation is the same in both phases, the phase veocities in the wave can be determined from the momentum baance equations. Front and tai veocities and iquid hodup are aso determined in the wave assumin a fixed enth as opposed to sus which have a variabe enth. Fiure : Schematic of modes for A. su fow and B. wave fow in the trackin scheme. The arrows indicate direction of fow. Dashed ines represent section borders. Gray iquid phase. White as phase. Gas fow in a su is modeed usin a sip reation, however, in a wave the as phase fows throuh a ap between the iquid phase and the upper pipe wa. In this way, as fow over a are wave can be thouht of as simiar to as fow throuh an orifice, and therefore an orifice type reation is used in the as momentum baance eqn () for waves. The orifice type reation, the second term in eqn (), repaces the as wa and interfacia friction terms. Eqn () is then used to determine the as veocity in a wave by reatin it to the pressure variation across the wave. M U t H ( H ) A ρ C d H R = ( H ) A( P P ) M sinθ L R 2 ( U n + n U ) U U () Usin the same pressure variation across a wave as in eqn (), the iquid momentum baance eqn (2) is used to determine the iquid phase veocity in a wave. In the iquid phase, the main component ivin pressure variation is the WIT Transactions on Enineerin Sciences, Vo 63, ISSN (on-ine) 29 WIT Press

4 2 Computationa Methods in Mutiphase Fow V acceeration of the iquid at the wave. There is aso a contribution from iquid wa friction and ravity. M n n n+ U + Hρ A( U U )( UR U ) t (2) = HA( P L P ) + ( R 8 LS λ ρ U n U n+ ) M sinθ Sus are modeed as objects with movin boundaries, the of a su moves with a veocity determined from a mass baance across the whie the tai moves with a bubbe nose veocity. If the veocity is reater than the tai veocity, the su wi row in enth, otherwise its enth wi decrease. Simiary, waves are modeed as movin objects but they have a fixed enth of one to two pipe diameters and move with the wave veocity. The veocity of a wave is determined from the mass baance across the, eqn (3), and iven in eqn (4). n+ n H ( U U ) = H R ( U R U ) (3) H n+ H R n U = U U (4) R H H R H H R One of the aims of the wave trackin scheme was to have a simpified mode, therefore a simpe wave tai speed reationship was desired. The wave tai speed is iven in eqn (5). Utai =. 2U (5) The factor of.2 aows for continuous transition between wave and su fow. When the iquid hodup in a wave approaches unity, the iquid phase veocity in the wave approaches the mixture veocity. The bubbe nose veocity or wave tai veocity is commony reated to the mixture veocity by a factor of.2. The mass baance equations in a wave or su are the same, where the chane in mass in a iven time step is the difference in mass fux in and out. Eqns (6) and (7) are the iquid phase and as phase mass baance equations respectivey. The iquid hodup is iven in eqn (8). M n n n n ( M U Utai L M U U L) + = ( ) / ( + ) t / (6) M n n+ n n+ = ( M ( U Utai ) / L M ( U U ) / L) (7) t n+ M H = (8) ALρ Since wave s are modeed with a fixed enth and they move at the wave veocity, the wave tai veocity ony appears in the mass baance eqns (6) and (7). If the speed is arer than the tai speed the iquid mass in the wave wi increase and vice versa. Therefore waves can row or decay in ampitude. WIT Transactions on Enineerin Sciences, Vo 63, ISSN (on-ine) 29 WIT Press

5 Computationa Methods in Mutiphase Fow V 3 2. Transitions and wave insertion Waves can be formed from stratified fow or they can form from decayed sus. At the transition from stratified fow waves are inserted accordin to the inviscous Kevin Hemhotz stabiity criteria, eqn (9). Neectin surface tension and viscous effects, stratified fow is stabe if (Barnea and Taite [], Lin and Hanratty []): A A + ( ρ ) cos ( ) 2 > ρ θ U U (9) ρ Si ρ Si On the other hand, if a wave is dyin, it wi be removed when the hodup in the wave approaches the hodup in the stratified section in of it. A decayin su wi be converted to a wave when its enth oes beow a user defined minimum, i.e. one or two pipe diameters in enth. In the reverse case where a wave rows to a su, a wave wi be converted to a su if its hodup oes above a user defined maximum, e.. a hodup of.99. Sus can aso form if two waves mere or a su overtakes a sower movin wave. 3 Resuts and discussion The ro wave trackin mode has been compared with experiments that were conducted in the mutiphase fow aboratory at NTNU in a 6 m on,.6 m I.D. pipe usin air and water at atmospheric pressure (De Leebeeck et a. [4]). The experiments incuded pipe incinations from - to 3 derees, Us from 2 to.5 m/s, and Us from.4 to.52 m/s where a mixture of waves and sus occurred. Data for comparison incude iquid hodup and pressure time traces, averae wave veocities from cross correation between hodup time traces, and pressure drop. One purpose of doin the experiments was to measure the pressure variation over a wave, as this is an assumption in the trackin mode. This was confirmed in the experiments, and ed to an estimate of the dischare coefficient in the orifice reation in eqn () of C d =.2 to.4. The simuations discussed here used a fixed rid size of 2 pipe diameters in wave s, a minimum of 2 and maximum of pipe diameters in stratified sections. Pipe enth, diameter, fuid properties, and a simuation time of 8 sec were as in the experiments. A dischare coefficient ivin the best approximation of experimenta wave speed and pressure variation with vaue C d =.2 was used in the as momentum equation for waves. The waves were inserted at a simiar frequency to the experiments. Fiure 2 shows a pot of wave or su veocity for a experiments and wave veocity from the mode potted aainst mixture veocity. The arest veocities are associated with sus which have arer veocities than waves. The experiments with ower veocities contained more waves than sus. Lookin at the data quaitativey, the mode ives wave veocities in the same rane as the experiments. WIT Transactions on Enineerin Sciences, Vo 63, ISSN (on-ine) 29 WIT Press

6 4 Computationa Methods in Mutiphase Fow V Wave or su veocity (m/s) mode experimenta Mixture veocity (m/s) Fiure 2: 6 5 Experimenta veocities compared with wave veocities from dynamic wave trackin simuations. For Umix < 4 m/s: mainy su fow. Umix > 4 m/s: a mixture of waves and sus. mode experimenta Pressure drop (Pa/m) Mixture veocity (Us + Us) (m/s) Fiure 3: Experimenta pressure drop compared with averaed pressure drop from trackin simuations. In fiure 3, the experimenta and modeed pressure drops are compared. Athouh a of the simuations reproduced a pressure variation in waves, some of the pressure drops tended to be ow compared to the experimenta vaues. The WIT Transactions on Enineerin Sciences, Vo 63, ISSN (on-ine) 29 WIT Press

7 Computationa Methods in Mutiphase Fow V 5 manitude of the pressure drop depends on the number of waves and sus in the experiment or simuation. An experiment with more sus wi have a arer pressure drop than one with fewer. A the experiments contained a mixture of waves and sus but some of the simuations, especiay downward incined, reproduced waves which did not row to sus resutin in ow pressure drops. One advantae of modein waves in a trackin scheme is that coarse rids can be used aowin for oner pipe systems to be modeed. Usin a coarse rid means that waves and sus are modeed as square objects correspondin to the pots fiure 4A, usin a maximum stratified section enth of pipe diameters. Finer rids can be used, however, aowin for waves with more apparent tais, shown in fiure 4B, as occurs in the experimenta time traces, for exampe fiure 4C. Pressure (Pa).e5.e5 Pressure (Pa).e5.e5 Pressure (Pa).e5.e5 Hodup (-) Liquid veocity (m/s).5 6 Hodup (-) Liquid veocity (m/s) A B C Hodup (-).5 Fiure 4: Pressure, hodup and iquid veocity time traces where a wave passes at a iven ocation in the pipe. A. A coarse rid with maximum enth pipe diameters. B. A fine rid with maximum enth pipe diameters. C. Experimenta. Us = 8. m/s, Us =. m/s, θ = deree. Fiure 5 shows how an experimenta hodup time trace compares to a simuated time trace on a fine rid at the same Us, Us and pipe incination. The mode time trace in fiure 5B shows a mixture of sus and waves of various sizes occurrin in the pipe as we as the shape of the waves. Wave dynamics such as waves rowin to sus, or sus decayin to waves is inherent in the trackin mode. Exampes of a wave rowin to a su and a WIT Transactions on Enineerin Sciences, Vo 63, ISSN (on-ine) 29 WIT Press

8 6 Computationa Methods in Mutiphase Fow V A Hodup (-) B Hodup (-) Fiure 5: Hodup time traces A. experimenta and B. simuation usin a fine rid with maximum enth pipe diameters. Us = 6.9 m/s, Us =.8 m/s, θ = 2 derees. Pressure variation (Pa) Hodup (-) Liquid veocity (m/s) A B Fiure 6: Pressure variation, iquid hodup and iquid veocity in a wave vs. time for A. a wave rowin to B. a su. Us = 6. m/s, Us =.2 m/s, horizonta pipe. WIT Transactions on Enineerin Sciences, Vo 63, ISSN (on-ine) 29 WIT Press

9 Computationa Methods in Mutiphase Fow V 7 A B C Fiure 7: Pressure variation, iquid hodup, and iquid veocity in a wave vs. time for A. a wave rowin to B. a su and then decayin to C. a wave aain. Us = 5.87 m/s, Us =.3 m/s, θ = derees. wave which becomes a su and then decays into a wave aain are shown in fiures 6 and 7 respectivey. The pressure variation across the wave, iquid hodup and veocity in the wave object as it moves are potted aainst time in both fiures. When a wave becomes a su the pressure variation across it and the iquid veocity increase, and the hodup approaches one. In fiure 7, when the su decays aain, pressure variation, iquid veocity and hodup decrease. The time traces are cut off when the wave or su exits the pipe. 4 Concusions A mode for are ro waves has been impemented and tested in a su trackin scheme. The mode introduces an orifice type reation for pressure variation across the wave and a simpified reationship for wave speed in a simiar way as for su fow. Computations have been demonstrated in comparison to experimenta data on ro waves in two-phase air-water pipe fow at atmospheric pressure. The mode ives a reasonabe approximation of wave speed and pressure variations in waves. Lookin at pressure drops, modeed pressure drops WIT Transactions on Enineerin Sciences, Vo 63, ISSN (on-ine) 29 WIT Press

10 8 Computationa Methods in Mutiphase Fow V are sometimes ow compared to experiments due to a difference in the number of waves and sus in the pipe. The trackin scheme can run with a coarse rid which aows simuation in oner pipes but means that waves and sus are modeed as square objects without tais. A finer rid aows a more physica representation of waves with tais. The mode incudes wave dynamics such as a wave rowin to a su or a su decayin to a wave. 5 List of Symbos A area, m 2 C d dischare coefficient ravity, 9.8 m/s 2 H iquid hodup I.D. interna diameter L enth of section, m M mass, k P pressure, Pa S wetted perimeter, m t time, sec U veocity, m/s Umix mixture veocity, m/s Us superficia as veocity, m/s Us superficia iquid veocity, m/s Greek symbos chane in a iven quantity λ friction factor θ ane of pipe incination, derees ρ density, k/m 3 Superscripts n current time step n+ next time step Subscripts of a wave or su as phase i interface iquid phase L eft section R riht section tai tai of a wave of su WIT Transactions on Enineerin Sciences, Vo 63, ISSN (on-ine) 29 WIT Press

11 Computationa Methods in Mutiphase Fow V 9 Acknowedement Financia support from Tota E&P Nore is ratefuy acknoweded by A. De Leebeeck. References [] Barnea, D. & Taite, Y., Kevin-Hemhotz stabiity criteria for stratified fow: Viscous versus non-viscous (inviscid) approaches, Internationa Journa of Mutiphase fow, 9, pp , 993. [2] Bendiksen, K.H., Manes, D. & Nyda, O.J., On the modein of su fow, Chemica Enineerin Communications, 4, pp. 7-2, 996. [3] Bonizzi, M. & Issa, R.I., A mode for simuatin as bubbe entrainment in two-phase horizonta su fow, Internationa Journa of Mutiphase fow, 29, pp , 23. [4] De Leebeeck, A., Gaarder, A.H. & Nyda, O.J., Experiments on Ro Waves in Air-Water Pipe Fow, 6 th Austraasian Fuid Mechanics Conference, God Coast, Austraia, 27. [5] Homås, H., Numerica simuation of waves in two phase pipe fow usin D two-fuid modes, Doctora dissertation, University of Oso, 28. [6] Hu, B., Stewart, C., Manfied, P.D., Ujan, P.M., Hae, C.P., Lawrence, C.J. & Hewitt, G.F., A Mode for Trackin the Evoution of Sus and Waves in Straiht Pipeines, 6 th Internationa Conference on Muitphase Fow, Leipzi, Germany, 27. [7] Issa, R.I. & Kempf, M.H.W., Simuation of su fow in horizonta and neary horizonta pipes with the two-fuid mode, Internationa Journa of Mutiphase fow, 29, pp , 23. [8] Johnson, G.W., A Study of Stratified Gas-Liquid Pipe Fow, Doctora dissertation, University of Oso, 25. [9] Kjøaas, J., Pu propaation in mutiphase fow, Doctora thesis, Norweian University of Science and Technooy, 27. [] Lin, P.Y. & Hanratty, T.J., Prediction of the initiation of sus with inear stabiity theory, Internationa Journa of Mutiphase fow, 2, pp , 986. [] Nyda, O.J. & Banerjee, S., Dynamic su trackin simuation for as-iquid fow in pipes, Chemica Enineerin Communications, 4-42, pp. 3-39, 996. [2] Renaut, F., A Laranian su capturin scheme for as-iquid fows in pipes, Doctora thesis, Norweian University of Science and Technooy, 27. [3] Taite, Y. & Barnea D., Effect of as compressibiity on a su trackin mode, Chemica Enineerin Science, 53(), pp , 998. WIT Transactions on Enineerin Sciences, Vo 63, ISSN (on-ine) 29 WIT Press