Comparison between Empirical Correlation and Computational Fluid Dynamics Simulation for the Pressure Gradient of Multiphase Flow

Transcription

1 rocdings o World Congrss on nginring 2008 Vol III Comparison btwn mpirical Corrlation and Computational Fluid Dynamics Simulation or rssur Gradint o Multiphas Flow Yvonn S. H. Chang, T. Gansan and K. K. Lau Abstract - Th objctiv o is rsarch is to compar us o mpirical corrlation wi Computational Fluid Dynamics (CFD) simulation in dtrmining prssur gradint in two-phas low piplins at sam inlt condition. In is wor, mpirical modl o Bggs and Brill modl had bn usd whras turbulnt modl applid in CFD simulation was Rnormalization Group (RNG) -ε modl. A statistical analysis was conductd to dtrmin dviation in valus obtaind by CFD simulation as compard to os rom mpirical corrlation. It was ound at CFD rsults wr in a good agrmnt wi indings obtaind rom mpirical corrlation wi a dviation o lss an ±5%. Indx Trms - mpirical corrlation, Computational Fluid Dynamics, prssur gradints, two-phas low I. INTRODUCTION Multiphas low is dind as a low wi two or mor distinct phass, which in is cas is a liuid-gas low. Th prdiction o its charactristics has not bn asy sinc ach sgmnt o low map has signiicantly dirnt nrgy ruirmnts to sustain low and a ral-li low will jump rom on sgmnt to nxt at an unprdictabl tim. In gnral, mpirical corrlations ar usd to dduc prssur gradints in a multiphas low piplin [1]. For a singl-phas low, prssur gradint uation is dvlopd by using consrvation principls o mass and momntum. Th sam principls ar usd to calculat prssur gradints or a multiphas low. Howvr prsnc o an additional phas mas dvlopmnt much mor complicatd. Yvonn S. H. Chang, T. Gansan and K. K. Lau ar currntly wi Univrsiti Tnologi TRONAS, Chmical nginring Dpartmnt, Bandar Sri Isandar, Tronoh, ra, Malaysia (phon: ; ax: ; -mail: yvonn_csh@yahoo.com). Th Bggs and Brill modl [2] has bn slctd as mpirical corrlation usd in is wor bcaus it shows svral signiicant aturs at st it apart rom or multiphas low modls. In is modl, slip condition and low pattrn ar considrd in computing prssur gradints along piplins. Dpnding on stablishd low pattrn, liuid holdup and riction actor corrlations can also b dtrmind. Morovr, it is important to rcogniz at is modl can dal wi angls or an vrtical upward low. Consuntly, it can b applid or hilly trrain piplins and injction wlls which ar always ncountrd in ptrolum nginring. In short, Bggs and Brill modl is considrd classic in ild o multiphas low and has bn citd by svral paprs to b rliabl or calculations involving larg liuid mass input ractions and small diamtr pips at various orintation angls [2]. In CFD simulation, bo boundary and oprating conditions wr dtrmind. Thn, partial dirntial uations (D) or Rnormalization Group (RNG) -ε modl was slctd. Wi spciid boundary conditions, prssur proil was approximatd numrically and us solving D. Th convrgnc critrion and rsiduals wr adjustd accordingly to obtain bst possibl rsults. This was don wi aid o CFD commrcial sotwar. Th procss o crating mpirical uations or a spciic application always ruirs an itrativ xprimntation wor which is tdious and xpnsiv to prorm. Th CFD simulation, on or hand, is vry ast or at purpos and chapr in cost rlativ to procss o crating or dsigning mpirical corrlations [3]. Thus, comparison btwn rsults obtaind by mpirical corrlation and CFD simulation would dtrmin applicability o CFD simulation in rplacing mpirical corrlation to study prssur gradints in a multiphas low.

2 rocdings o World Congrss on nginring 2008 Vol III II. BGGS AND BRILL MODL Th Bggs and Brill modl has bn idntiid to b applicabl in is rsarch as it xhibits svral charactristics at st it apart rom or multiphas low modls: a) Slippag btwn phass is tan into account Du to two dirnt dnsitis and viscositis involvd in low, lightr phas tnds to travl astr an havir on trmd as slippag. This lads to largr liuid hold-up in practic an would b prdictd by trating mixtur as a homognous on. b) Flow pattrn considration Dpnding on vlocity and composition o mixtur, low bhaviour changs considrably, so at dirnt low pattrns mrg. Ths ar catgorizd as ollows:-sgrgatd, intrmittnt and distributd. Dpnding upon low pattrn stablishd, hold-up and riction actor corrlations ar dtrmind. c) Flow angl considration This modl dals wi lows at angls or an os in vrtical upwards dirction. Som assumptions had bn usd in dvlopmnt o is corrlation: 1. Th two spcis involvd do not ract wi on anor, us composition o mixtur rmains constant. 2. Th gasous phas dos not dissolv into liuid on, and vaporation o liuid into gas dos not occur. Th Bggs and Brill modl [2] has ollowing prssur-gradint uation or an inclind pip: d dl = 2 ρ n v m + ρ s g sin θ 2 d 1 K (1) Whr d/dl is prssur gradint, is riction actor, ρ n is ovrall gas and liuid dnsity rlativ to ir mass raction, v m is man vlocity, g is gravitational acclration and is a dimnsionlss trm. III. CFD SIMULATION OF MULTIHAS FLOWS A. Rnormalization Group ε Modl Th Rnormalization Group (RNG) ε modl [4] is drivd statistically rom Navir-Stos uation. It avrags highr nrgy lvls in low statistically and producs lowr nrgy lvl proprtis as a rsult. In act, is modl is similar in orm to Standard -ε modl [5], xcpt or som addd rinmnts in ε uation to improv accuracy or rapidly straind lows. As a rsult, RNG ε modl can b utilizd to obtain bo high and low Rynolds numbr low cts whras Standard ε modl can only acuir cts o high Rynolds numbr lows. Thror, RNG modl is mor accurat and abl to catr or a gratr rang o lows rlativ to Standard ε modl. Th transport uations o RNG ε modl ar as ollows: ( ) ( ui ) ρ + ρ = α μ + G + Gb ρε Ym + S t xi x j x j (2) ε ε 2 ( ρε) ( ρεu ) + i = α μ C1 ( G C3 Gb ) C2 ρ ε ε + + ε t x x x i j j (3) whr ρ is luid dnsity, is intic nrgy, ε is dissipation rat, u is man vlocity, μ is ctiv viscosity, α is invrs ctiv randlt numbr or trm, α ε is invrs ctiv randlt numbr or ε trm, G is gnration o turbulnt intic nrgy du to man vlocity gradints, G b is gnration o turbulnt intic nrgy du to buoyancy, Y m is contributions o luctuating dilation in comprssibl turbulnc to ovrall dissipation rat, S is usr dind sourc trms and C 1 (1.42) and C 2 (1.68) ar mpirical constants [6] B. Volum o Fluid (VOF) Modl Th Volum o Fluid (VOF) modl [7] is usd or simulation o luid charactristics in which r ar mor an on phass o luids prsnt in low. Th phass o luids must not b mixing, i.. not intrpntrating wi ach or. For vry additional phas o luid, a supplmntary variabl is ε (1)

3 rocdings o World Congrss on nginring 2008 Vol III introducd in volum raction o phas in cll rough init volum mod. Th VOF modl is suitabl to b applid in cass undr stady stat condition. Th proprtis o luid obtaind rom phass ar in orm o volum avragd. Howvr, s proprtis ar subjct to volum ractions at spciic locations. Th volum raction o luid is dnotd as α. In VOF modl, ollowing conditions apply: I α = 0, n r is no luid in cll. I α = 1, n cll is ull o luid. I 0 < α < 1, cll contains a mix o luid wi or luids. Hnc, by taing α as a basis, proprtis o luid can b dtrmind in control volum rough init volum mod. On or hand, ac luxs or all clls, including luxs at intrac, can b obtaind rough: n + 1 n α α n + 1 n + 1 V + ( U α, ) = 0 (4) Δ t whr Δ t is tim-stp and n+ 1 U is luid proprty at n+1. IV. RSULTS AND DISCUSSION A. Assumptions Usd In ordr to produc a valid comparison btwn bo o mods usd, i.. mpirical corrlation and CFD simulation, a w assumptions had bn mad. Th assumptions mad ar as ollows: 1. Th low is at stady stat. 2. Th low is adiabatic, i.. no hat transr roughout low 3. Th low is isormal, i.. no tmpratur chang during low. 4. Th low is stratiid and sgrgatd, i.. bo liuid and gas phass do not mix wi ach or 5. Th pip usd is a commrcial stl or wrought iron, wi grad N80 and a rlativ roughnss o Th inlt vlocity or low rat o liuid and gas is assumd to b sam. 7. Thr is no chmical raction taing plac roughout low. 8. Thr is no diusion or phas gnration roughout low. 9. Bo liuid and gas phass ar lowing wi a ully dvlopd turbulnt low Th luid proprtis usd in is rsarch ar as ollows [8]: Oprating rssur, = a Oprating Tmpratur = o C Dnsity o o C, ρ L = g/m 3 Dnsity o o C, ρ G = g/m 3 Viscosity o o C, μ L = Ns/m 2 Viscosity o o C, μ G = Ns/m 2 B. rssur Gradints rom mpirical Corrlation Th prssur gradints at various mass ractions o liuid ar as tabulatd in Tabl 1. Tabl 1: rssur gradints at various mass raction o liuid rom mpirical corrlation C. rssur Gradints rom CFD Simulation A summary o prssur gradints obtaind or ach mass raction along wi corrsponding rsiduals is as tabulatd in Tabl 2. Tabl 2: rssur gradints and normalization o rsiduals at various mass raction o liuid rom CFD simulation Liuid Mass Fraction (gliuid/g mixtur) λl d/dl rssur Gradint (a/m) Linar stimat: Norm o Rsiduals

4 rocdings o World Congrss on nginring 2008 Vol III Th linar stimat, which is rprsntd in orm o normalization o rsiduals, is usd to provid a consrvativ approximation o prssur gradints obtaind by CFD simulation (Tabl 2). Figur 1 shows rlationship btwn normalization o rsiduals and mass raction o liuid obtaind rough CFD simulation. Norm o rsiduals data linar Mass raction o liuid componnt Figur 1: Rlationship btwn normalization o rsiduals and mass raction o liuid From Figur 1, as mass raction o liuid incrass (which mans mass raction o gas rducs), normalization o rsiduals rducs, and vic vrsa. Whn mass raction o gas incrass, turbulnc lvl in low incrass bcaus gass ar gnrally mor turbulnt an ir liuid countrparts at sam vlocity. Consuntly, non-linar or irrgular prssur gradints wr gnratd, and us a gratr dviation rom linar stimat, in cass wi lowr mass raction o liuid. Tabl 3: Th prssur gradint and linar stimat, norm o rsiduals at ar obtaind rom Flunt sotwar and Matlab 7.1 sotwar or a variation o liuid mass raction. Liuid Mass Fraction (gliuid/g mixtur) CFD (a/m) (a/m) D (%) From Tabl 3, it is clar at D p o prssur gradints obtaind by CFD simulation rlativ to os obtaind by mpirical corrlation ar uit small, i.. lss an 5%. Th ngativ magnituds obtaind or CFD and dnot at prssur droppd progrssivly across tst sction. Howvr, wi mor boundary conditions at is mor data on natur o low a mor rliabl and accurat rsult can b obtaind. This is bcaus according to Von Numann critria stability o low is dpndnt on boundd solution. I solution is boundd n low is stabl, and hnc to obtain a boundd solution, suicint data rgarding systm or low is ruird. In CFD, init lmnt mod is utilizd to solv D numrically. Thus computational powr is ruird to obtain a solution. Computational powr plays a vital rol in incrasing accuracy and rliability o data, which dirctly mans at highr numbr o itrations prormd, highr accuracy. D. Comparison btwn mpirical corrlation and CFD simulation Th magnitud o dviation or prssur gradints obtaind rom bo mpirical corrlation and CFD simulation was calculatd by an rror analysis uation as givn blow: D = CFD 100 % (5) whr D is prcntag o dviation, CFD is prssur gradint obtaind by CFD simulation and is prssur gradint obtaind by mpirical corrlation. Th rsults ar as shown in Tabl 3. V. CONCLUSION Th dviation o prssur gradints obtaind rough CFD simulation wi rspct to Bggs and Brill mpirical corrlations was ound to b rlativly small, i.. lss an ±5%. Thror, it can b concludd at CFD simulation, which is mor icint and conomic, can b usd as an altrnativ to mpirical corrlations to obtain prssur gradints in twophas low piplins. Howvr, it is rcommndd at is rsarch is rpatd by comparing CFD rsults wi data rom xprimntal wor. This will provid a mor accurat analysis or multiphas piplin dsigns and constructions.

5 rocdings o World Congrss on nginring 2008 Vol III ACKNOWLDGMNT W wish to rcord our sincr gratitud to Univrsiti Tnologi TRONAS (UT). Apart rom inancial support, y mad availabl to us rsarch acilitis and rsourcs which wr intgral to is wor. Thans ar also xtndd to A.. Dr. Thanabalan Murugsn, who rviwd sctions o manuscript. [8] McCain, W. D. Jr. (1990). Th proprtis o ptrolum luids. 2 nd d. Olahoma: nnwll ulishing Company, p SYMBOLS prssur (a) L pip lng (m) d pip diamtr (m) riction actor ρ L liuid dnsity (g/m 3 ) ρ G gas dnsity (g/m 3 ) ρ n dnsity o gas-liuid (g/m 3 ) λ L liuid mass raction λ G gas mass raction μ L liuid dynamic viscosity (N s/m 2 ) μ G gas dynamic viscosity (N s/m 2 ) μ n dynamic viscosity o gas-liuid (N s/m 2 ) ν m man vlocity (m/s) Fr Fraud numbr liuid holdup H L(Ө) RFRNCS [1] Brill, J.. and Muhrj, H. (1999). Multiphas low in wlls. Monograph Volum 17, S, Hnry L.Dohrty Sris. [2] Bggs, H. D., and Brill, J.., (1973). A study o two-phas low in inclind pips. Trans. AIM, 255, p. 607 [3] Frzigr, J. H. and ric, M. (2002). Computational mods or luid dynamics. Springr. [4] Choudhury, D. (1993). Introduction to rnormalization group mod and turbulnc modling. Flunt Inc. Tchnical mmorandum TM-107. [5] Biswas, G. and swaran, V. (2002). Turbulnt lows: Fundamntals, xprimnts and modling, Alpha Scinc Intrnational Ltd. [6] Flunt Inc. (2001). Flunt 6 Usr s Guid. India: Flunt Documntation Sotwar. [7] Manninn, M., Taivassalo, V. and Kallio, S. (1996). On mixtur modl or multiphas low. VTT ublications 288, Tchnical Rsarch Cntr o Finland.