Numerical Simulation of Natural Gas Flow in Transmission Lines through CFD Method Considering Effective Parameters
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1 J. Bsi. Appl. Si. Res. (8) TexRod bliion ISSN Jonl of Bsi nd Applied Sienifi Reseh Nmeil Simlion of Nl Gs Flow in Tnsmission Lines hogh CFD Mehod Consideing Effeive mees Nikn Noobehesh Depmen of Chemil Engineeing Amikbi Univesiy of Tehnology (Tehn olyehni) Tehn In ABSTRACT Mos of gs soes e loed in long disnes fom onsmpion enes. Gs is sen o finl onsmpion poins someimes hosnds kilomees f fom hei soes fe exion. So oding o gs indsies developmen nd impovemen of is nsmission indsies ehnil developmens e neessy in ode o nlye he gs nsmission sysem. In his sdy fo he fis ime nl gs flow in nsmission pipe lines is modeled vi oniniy momenm nd enegy eqions nd lso idel se eqion s n xiliy eqion. Also he effe of blene is onsideed in D geomey. Resled pil diffeenil eqions ms be solved simlneosly hen ompionl flid dynmi (CFD) nd finie volme mehod e sed fo diseiion of diffeenil eqions. An ppoxime % eo shows deliy of his mehod. Finlly effe of viion in empee of inle gs gs s flow e nd mbien empee on pesse dop nd sysem s pmees e sdied. KEYWORDS: Gs flow; Mhemil model; Gs nsmission line; Compionl flid dynmi. INTRODUCTION In nsmission line sysems nl gs is nspoed sing hosnds of miles of pipelines h e pssing hogh wide viey of ein nd nde diffeen wehe ondiions. One of mjo sks of nl gs nsmission ompnies is onol of pesse nd qliy of nspoing gs sing ompesso nd gs blending sions. An insigh ino he behvio of omplex sysems of mesemen nd onol is neessy peeqisie when designing o impoving onol segies fo hese sysems. A ompe simlion of sysem s behvio is n ive wy o gin his insigh. In ode o be flexible when we e seing p he mhemil model sdied sysems e deomposed ino bsi modles like pipeline seions ompessos vlves mesemen devies nd onolles. Wh h we epo hee is modeling of gs flow hogh pipeline seions. The mhemil model of nsien gs flow in pipe n be s i is demonsed heoeilly pil diffeenil eqion o sysem of eqions. The fom of hese eqions vies wih ssmpions mde egds o opeing ondiions of pipeline. I is well-esblished f h flow in gs pipelines is nsedy. Condiions e lwys hnging wih ime no me how mh smll some of he hnges my be. When we e modeling sysems howeve i is someimes onvenien o mke his simplifying ssmpion h flow is sedy. Unde mny ondiions his ssmpion podes deqe engineeing esls. Thee hve been mny sdies on flow of ompessible gs in pipeline in exbooks ppes nd ehnil domens. In mh of liee flow of gs in pipes ws desibed by one-dimensionl ppoh. Also eihe n isoheml o n dibi ppoh is doped. Fom he fis sdies on ompessible flow in pipeline Fnno nd Ryleigh flow models (Shpio 95) n be menioned h ws one-dimensionl sedy se model fo flow of idel gs in pipe onsideing fiion nd he nsfe. Simplifying ssmpions sh s isoheml o dibi wee pplied someimes indsy bese of simlneos effe of she sess nd he nsfe on gs flow. Gs nspoion pipeline ws invesiged by Tylo e l. (96) nde isoheml ondiion wih negleing he ineil em o he nonline onveive em sing he mehod of heisis (MOC). Wylie e l. (97) pesened implii enl finie diffeene mehod nd omped i wih MOC fo he fis ime. They showed h implii mehod is moe e big ime seps. Osid (987) eded pil diffeenil eqions (FE) o odiny diffeenil eqions (ODE) wih ppliion of mehod of lines. Then he eqions wee solved sing Chebyshev Rnge-K mehod. The ineil em in momenm eqion ws negleed in ll he bove sdies nil Zho nd Adewmi (995) pesened new ehniqe fo solving he fll onsevion lws h goven one-dimensionl gs flow. I ws demonsed h eliminion of ineil em when hee e fs hnges in he line leds o signifin deesing of y whees hee is no impon effe in slow hnges. Ibheem nd Adewmi (999) sed ol viion diminishing (TVD) sheme in onjnion wih Rnge-K mehod o ge moe sble obs nd e esls. * Coesponding ho: Nikn Noobehesh. Depmen of Chemil Engineeing AmikbiUnivesiy of Tehnology (Tehn olyehni) Tehn In. Emil: nikn.nb@gmil.om 773
2 Noobehesh 0 Fo mny gs nspo ppliions his ssmpion h poess hs onsn empee o is dibi is no vlid. A ompison beween diffeen (isoheml nd non-isoheml) models ws pesened in he wok of Osid nd Chykowski (00). A signifin diffeene ws obseved in he pesse pofile long he pipeline beween isoheml nd non-isoheml poess. Simlions fo flows in il bes e vey spse espeilly by ompessible Nvie Sokes eqions. To he hos knowledge no wok hs been epoed h sed wo-dimensionl ompessible Nvie Sokes eqions in ylindil oodines fo blen flow of nl gs in nsmission lines nd sing CFD nd finie volme mehod fo solving se of pil diffeenil eqions. The objeive of he pesen sdy ws o develop n effiien sheme fo gs flows in nsmission lines bsed on he ompessible wo-dimensionl Nvie Sokes eqions long wih he ompessible oniniy enegy eqions nd se eqion in ylindil oodines. Compionl flid dynmi (CFD) nd finie volme mehod e sed in ode o solve opled diffeenil eqions. The ppose of sing ompessible fomlion is o enble simlions whih fo hem empee viion wihin he flow is vey signifin. Howeve in he pesen ppe sensiiviy nlysis of signifin pmees sh s inle gs nd mbien (soil) empees inle pesse nd mss flow e e invesiged. Mhemil model whih govens gs flow in sedy se The gs flow behvio in nsmission line is modeled by he onsevion lws of mss momenm nd enegy nd lso by gs s se eqion. This is ssmed h nsmission line hs onsn oss-seion e (A). The gs flow is onsideed highly blen. Consevion of mss: oniniy eqion Genelly he oniniy eqion is expessed s eqion () (Bid e l. 00): 0 () Newon s seond lw of moion: momenm eqion Fo gs flows in pipe he ompessible Nvie Sokes eqions n be expessed s following in xil nd dil dieions espeively oding o Bid e l. 00. ( p) ( ) ( ) () p ( ) ( ) (b) In hese eqions e he xil nd dil veloiy omponens espeively. The dimensionless visos sess ems e defined fo he enso omponens s follows.. V eff 3 eff. V eff 3. V eff 3 Whee. V In he bove eqions eff is he effeive visosiy ( whee is he blen visosiy defined oding o he sed blene model) (Blek 00). Consevion of enegy The bsi fom of enegy eqion oding o (Rd nd Bojedi 008) nd (Bid e l. 00) is s following: E p E p b b (3) E is he ol inenl enegy. The he flx nd he wok whih e done by fiionl foes ems in he enegy eqion e defined s follows. 77
3 J. Bsi. Appl. Si. Res. (8) eff eff T k b T k b In whih k eff is effeive ondiviy (k+k whee k is blen heml ondiviy defined oding o he sed blene model) (Blek 00). Se eqion Se eqion of gs is eled o vibles p ρ nd T. The ype of eqion whih ommonly is sed in nl gs indsy is (Klik e l. 988) (Wylie nd Seee 978). p ZRT () Whee he deviion fom he idel gs lw is bsobed in he ompessibiliy fo Z whih is fnion of p nd T. Tblene model Tblen visosiy nd heml ondiviy e explined by he blene model (Blek 00) C k C eff k k The kinei enegy of blene nd he dissipion em e omped vi solving wo nspo eqions s follows (Lnde nd Splding 97): S ) ( ) ( (5) S C C ) ( ) ( (6) In whih C ε = The igh sides of (5) nd (6) onin he podion nd he dissipion ems whih e ssigned s nd : 3 D S 3 D S Whee = 0.9 nd =.83. By definiion he following ems fo D nd in D n be sed: D The lssil model is vlid nde he hypohesis h he lol Reynolds nmbe is high. Theefoe i is no deqe o desibe egions lose o solid wll. The ide is o se wo-lye ppoh i.e. o ople he model o one-eqion model omilly. The mehod enbles s o ompe he flow p o he wll wih no empiil wok. Of ose his eqies moe ompionl esoes s fine mesh shold be sed. This mehod ompises inoding lol Reynolds nmbe y + s w w y y / whee sbsip w mens omped he poin loses o he wll nd y is he disne beween he en poin nd his poin. To ompe fo y + < 00 sing he following nspo eqion (Rd nd Bojedi 008): iss D. ) (. ).( ) ( (7) Whee l D iss / 3/ nd he dynmi visosiy of blene will be obined fom l in whih l μ nd l ε e wo lengh sles onining he dmping effes in he ne wll egions nd e defined s exp exp 3 / 3/ 3 3 / y y l y y l 775
4 Noobehesh 0 Whee 3 = 70 nd = 0.[5]. Nmeil ehniqe In he pesen ppe he CFD mehod is sed in ode o solve seies of diffeenil eqions nd lso finie volme mehod is sed o diseiion of eqions. CFD is nmeil solion mehod fo eqions of oniniy momenm nd enegy nd is eled phenomen sh s hemil eions. The genel fom of nspo eqions I is le h hee e signifin ommonliies beween he vios eqions. If we inode genel vible he onsevive fom of ll flid flow eqions n seflly be wien in he following fom: div div gd S (8) The eqion (8) is he so-lled nspo eqion fo popey. I lely highlighs he vios nspo poesses: he e of hnge em nd he onveive em on he lef hnd side nd he diffsive em (Γ=diffsion oeffiien) nd he soe em espeively on he igh hnd side. In ode o bing o ommon fees we hve of ose hd o hide he ems h e no shed beween he eqions in he soe ems. Eqlling o i (h o T) nd e. nd hoosing he pope qniies fo diffsion oeffiien (Γ) nd soe ems speil foms fo eh six pil diffeenil eqions n be obined. Key sge is ppliion of finie volme mehod whih is sed o diseiion of hese eqions. This mehod onins inegion of eqion (8) ove he onol volme (Veseeg nd Mllseke 995). div dv div gd dv S (9) CV dv CV CV Applying he Gss Divegene Theoem eq. (9) n be ewie s follows: n. ( ) da n. gd da SdV (0) A A CV Diseiion of govening eqions The fis sep in finie volme mehod is dividing he egion o disee onol volmes. A ommon poede is shown in fige. A genel node is showed by nd is neighbos in D ylindil geomey e s follows: E he esen node; W he wesen node; N he nohen node nd S he sohen node. The e in wes side of onol volme is lled w he esen e is e he nohen one is n nd he sohen e is s. Fige : A p of he ompionl domin In his fige he line veloiy ( ) nd dil veloiy ( ) e espeively simplified s nd v. The min sge in he finie volme mehod is inegion of govening eqions ove onol volme o obin disee eqion on node. Then fo onol volme whih is defined bove we hve (oding Eq 0): A e A w A n A s A A A n A s S S V e w () 776
5 J. Bsi. Appl. Si. Res. (8) Inegion ove oninosness Eq esls o: Ae Aw An As 0 () To hieve disee onveion-diffsion eqions he omponens esled fom eqion () shold be ppoximed. In finie volme mehod he soe em is linely ppoximed (nk 980). S S S p Applying enl diffeening sheme o illse he poion of diffsion ems nd seond-ode pwind sheme fo onveion ems of he ell es fo he omponens esled fom eq.0 he genel disee eqion is wien s follows (Chng 00): W W EE SS NN WW WW SSWW SV () E D e Fe W D w F w Fe N D n Fn S D s F s F n WW F w ss F s nb nb S V F e F A F A D x F nd D epesen he onveive mss flx nd diffsion ondne. x is genelly lled lengh sle nd shows he veloiy (in eh dieion). The solion lgoihm fo pesse-veloiy opling In he pesen eseh he SIMLER lgoihm is sed o obin pesse nd veloiy. Moe deils e disssed in (nk nd Splding 97) nd (nk 980). Solion of disee eqions In ode o solve poblem nd deeminion of disibion of popey disee eqion s eq. shold be pplied fo ll nodes. Fo onol volmes whih e loed ne o bondies genel disee eq. shold be modified fo pplying bondy ondiions his modifiion is s follows: oeffiien of bondy side will eh o eo nd bondy pssing flx s s mesh nd boh S nd S p will inese (Veseeg nd Mllseke 995). In his wy seies (o sysem) of lgebi eqions will obin whih shold be solved o speify disibion of popey in node poins. In he pesen eseh fo sl nd opled sysems Gss-Seidel nd Inomplee LU deomposiion (ILU) epeiive mehods (Axelsson 996) hve been sed espeively bese hei ppliion in ompe pogms is esy. B in omplied poblems onvegene of he eqions will hppen sofly. Mligid mehod is sed o oveome his poblem. Mligid mehod esming he f h onvegene e of Gss-Seidel nd ILU mehods deeses de o inese in eqions nmbes mligid mehod is ilied. The mligid sheme nsfoms se of eqions is given fo DE diseied on fine mesh ino seies of pogessively ose meshes solving he eqions flly on he oses mesh. The solion is hen solved on he seies of sessively fine meshes sing he solion fom he pevios (ose) mesh s n iniil esime of he solion finlly solving on he fines mesh (Chng 00). By nsfeing fom fine o ose mesh he medim wvelengh eos in he fine mesh solion e nsfomed ino sho wvelengh eos on he ose mesh whih e mh esie o smooh. The ompion os fo solving on he ose meshes is low nd he os fo solving on he fine meshes is eded by sing he ose mesh solion s n iniil esime. Bondy ondiion The bondy ondiions fo gs flow in pipeline in D geomey e s follows (fige ): Flid bondies: Inle: T e defined nd ρ is defined by se eqion. Ole: genel ondiion of flid whih is lmos ommonly pplied in finie volme mehod is s follows (Veseeg nd Mllseke 995): T / n 0 nd n / n 0 And defined o iming onsideed mss flow e nd n is he noml owd veo of ole sfe. w F n F s 777
6 Noobehesh 0 Solid bondy: No slip ondiion = w = 0 Consn empee on he wll T = T w Symmei bondy ondiion / n 0 Fige : Flow domin nd bondy ondiions RESULTS AND DISCUSSION The im of his sdy is he simlion of nl gs flow in gs nsmission line in ode o nlye he esponse of he sysem o he hnges in vios flow pmees. To do his p of exising el gs sysem ws onsideed fo boosing he nl gs in In s soh-noh line fo domesi onsmpion. This gs nspoion sysem (fige 3) onsiss of eigh ompesso sions nd is 80 km long. Fige 3: Se of gs nspoion sysem Fo he ppose of o invesigion one pipe beween ompesso sion nmbe nd 3 ws ken. Cllions wee ied o fo he following pmees: pipe dimee D = 56" (. mm) pipe wll hikness 9. mm; pipeline lengh L = 35 km; pipeline elevion diffeene ΔH = 0 m; inle pesse in = 7. b; ole pesse o = b; inle empee T in = 6.86 ºC; ole empee T o = 3 ºC; gs flow Q = 7.6 MMSMD; hysil popeies of gs en e obined by onsideing vege pesse nd empee beween inle nd ole nd wee ssmed onsn. molel weigh M w = 7.77; heml ondiviy k = W/m.K; he nsfe oeffiien h = 5 W/m.ºK; veged Compessibiliy fo Z ve = ; speifi he onsn pesse C p = 533 J/kg.ºC; speifi he onsn volme C v = 697 J/kg.ºC; visosiy of nl gs μ = p = kg/m.s; pipe oghness = 0.03 mm; R = J/kg.ºC; Compessibiliy fos wee lled fo evey diseiion seions of he pipeline sing SGERG 88 (ISO ) eqion. To sdy he gid independene simlions wee pefomed wih diffeen gid esolions nd finlly mesh onining ells ws seleed. Vlidion: Compison wih field d Vlidion of sed nmeil mehod is pefomed by omping simled esls wih l vles h wee obined fom Nionl Inin Gs Compny. Resled eos wee.% nd.8% h mens e llions e peise (fige nd 5). 778
7 J. Bsi. Appl. Si. Res. (8) Fige : Compison of simlion esls wih l d Fige 5: Compison of simlion esls wih l d esse empee densiy nd veloiy viion long he pipeline In figes 6 o 9 pesse empee densiy nd veloiy viions e illsed espeively long he pipeline. Inle pesse empee pssing flow e nd mbien empee e espeively 7. bs 6.86 C 7.6 MMSMD nd 5 ºC. Fige 6: Gs pesse viions long he pipeline 779
8 Noobehesh 0 Fige 7: Gs empee viions long he pipeline Fige 8: Gs densiy viions long he pipeline Fige 9: Gs veloiy viions long he pipeline As shown in figes gs pesse deeses o bs (ole pesse) linely s expeed de o fiionl esisne. Gs enes o he pipeline empee of 6.86 C nd he nsfe os wih pipe s wll. Then gs s empee edes o5 ºC (wll s empee) fe 0 km wy fom gs inle. Afe his poin gs empee emins onsn. Noe h e of empee edion (empee gdien) in fis 7 km is moe hn pesse edion e (pesse gdien); heefoe oding o eqion () gs densiy will inese. B fe his poin e of pesse edion beomes moe hn he e of empee 780
9 J. Bsi. Appl. Si. Res. (8) edion; heefoe densiy deeses. Afe 0 km sine empee will emin onsn densiy deeses linely s pesse deeses. Gs veloiy hs evese elion wih densiy oding o following eqion: m = ρa () The effe of flow e on he flow pmees Fige 0 shows gs pesse viion long pipeline fo nd 90 MMSMD flow es nd fige shows pesse dop vess flow e viion. I shold be menioned h flow es of 0 nd 90 e low nd high limis fo his pil pipeline. The flow veloiy ineses inle de o inesing of flow e onsn inle T nd. So inesing of flow veloiy leds o inesing of she sess nd s esl pesse dop ineses long he pipeline. As Regds hee is die elion beween flow e nd sqe oo of pesse dop (Shoede 00) inese of flow e leds o inesing in pesse dop nd bese inle pesse is onsn ole pesse will deese. Fige 0: Gs pesse viions vess flow e long he pipeline. Fige : Gs pesse dop viions vess flow e long he pipeline. Fige : Gs empee viions e shown fo nd 90 MMSMD flow es long he pipeline. Regding o edion of boh gs flow e nd gs veloiy esidene ime will inese fo gs moleles nd hey hve moe ime fo he nsfe wih pipe wll heefoe gs ehes mbien empee in 78
10 Noobehesh 0 less long of pipe. In fige 3 densiy viions nd in fige gs veloiy viions e shown fo nd 90 MMSMD flow es long he pipeline. Fige 3: Gs densiy viions long he pipeline fo vios flow es Fige : Gs veloiy viions long he pipeline fo vios flow es Flow e edion leds o ely ppoh of gs empee o mbien empee. This mens h empee gdien inese b pesse gdien deeses oding o fige 0. Ths seepe slope is fomed in gowh e of densiy nd mximm poin hppens in shoe disne. The effe of inle empee on he flow pmees Aoding o fige 5 empee viion gdien inese while inle gs empee ineses nd he vege gs flow empee inese onseqenly ill he eneed gs ehes o he empee of mbien nd gs flow empee emins onsn. Fige 5: Gs flow empee viions vess inle empee long he pipeline 78
11 J. Bsi. Appl. Si. Res. (8) If he inle gs flow empee is less hn mbien empee he gs flow eeives he fom he wll long he pipe nd is empee ineses. This leds o densiy edion nd gs veloiy inese long he pipeline (eliminion of pek poins in he figes). Gs densiy nd veloiy viions long he pipeline fo he vios inle empees e shown in fige 6 nd 7 seqenilly. Fige 6: Gs flow densiy viions in vios inle empees long he pipeline Fige 7: Gs flow veloiy viions in vios inle empees long he pipeline Aoding o fige 6 in he enne seion of pipe densiy deeses de o inesing of he empee in onsn inle pesse (eq. ). This end is onined lmos o he end of he pipeline. I mens h he end of pipeline he vle of densiy is deesed (howbei vey few) wih inesing of inle empee. In fige 7 inle veloiy is inesed de o inle empee inese. Bese moion of gs moleles ineses by empee nd lso densiy deese s empee ineses. Inese of veloiy is onining o he end of he line. As shown in fige 8 deese of inle empee leds o deese in gs veloiy heefoe he ole pesse ineses nd pesse dop deeses long he pipeline. 783
12 Noobehesh 0 Fige 8: Gs ole pesse viions vess inle empee. The effe of mbien empee on he flow pmees In his seion he effe of mbien empee o in ohe wods envionmenl ondiion on he flow pmees e exmined fo gs flow wih inle pesse of 7. bs nd inle empee of 6.86 C nd flow e of 7.6 MMSMD. Gs nsmission line nsfes nl gs fom sohen egion o nohen egion of In. Regding o viey of lime ondiions invesigion on he effe of mosphei fo on he sysem s pmees is impon. The gs empee viions fo diffeen mbien empees e ploed in fige 9. I is obvios h he empee viion gdien ineses when he mbien empee edes. The vege empee of gs flow lso deeses. Fige 9: Gs empee viion long he pipeline fo vios mbien empees. Densiy nd gs veloiy viions wih vios mbien empees e espeively shown in fige 0 nd. Densiy ineses wih gee slope he fis of he pipeline. I n be sid h genelly densiy ineses when mbien empee deeses 78
13 J. Bsi. Appl. Si. Res. (8) Fige 0: Gs densiy viions long he pipeline fo vios mbien empees Gs flow veloiy deeses de o deese of vege empee so h moleles moions deese nd densiy ineses. Fige : Gs veloiy viions long he pipeline fo vios mbien empees Fige : shows he ole pesse viions long he pipeline fo vios mbien empees. 785
14 Noobehesh 0 As i is obvios he ole pesse is inesed by deese of mbien empee o in ohe wods pesse dop of pipe delines (in onsn flow e) by deese of mbien empee. I n be eled o veloiy edion de o deese of mbien empee nd hen deese of vege empee of he flid. Using fige 3 deqe flow e vle in diffeen mosphei ondiions n be obined fo onsn pesse io (o speified ole pesse). Fige 3: Gs flow e viions fo vios mbien empees in speified pesse io (0.78) As obseved fom his fige in hoe lime egions fo ehing o speified pesse dop less flow e is neessied. Conlsion In his ppe nl gs flow in nsmission line ws modelled. Coniniy momenm enegy nd se eqions wee sed in wo-dimensionl ylindil lie. Tblene effes wee pplied o eqions. Then fo solving se of six DE opled eqions CFD (Compionl Flid Dynmi) nd finie volme mehods wee pplied fo diseiion of eqions. Resls whih obined by his mehod geed vey well o expeimenl d whih poded by Nionl Inin Gs Compny his shows y of he mehod. Afewd hoghogh nlysis ws ied o o show he effe of flow e inflow gs empee nd mbien empee on he ohe impon pmees sh s pesse dop in he pipeline. The esls showed h inese in he inflow empee o pipe-wll he flx ineses he pesse dop o deeses he mss flow e in pipe fo speified pesse io. Theefoe in ode o deese of pesse loss nd inese of gs flow e in he pipeline beween wo ompesso sions one shold ede he nsfe in ompison o he gs flow. Howeve his vle of hnge is moe onsideble when he inflow empee is deesed. In ohe wods fo inesing of mss flow e in speifi inflow pesse he inflow gs empee shold be deesed. REFERENCES Axelsson O. (960). Ieive solion mehods Cmbidge Univesiy ess UK. Bid R. B. Sew W. E. nd Lighfoo E. N. (00). Tnspo phenomen nd Ed John Wiley & Sons In New Yok. Blek J. (00). Compionl flid dynmis: piniples nd ppliions Elsevie Siene Ld London. Bnd A. (977). Mlilevel dpive solions o bondy vle poblems Mh. Comp Ceo L. S. Gosmn A. D. nk S. V. nd splding D. B. (97). Tow llion poedes fo sedy hee dimensionl flows wih eilion o. 3d In. Conf. Nm. Mehods Flid dyn. is. Chng T.J. (00). Compionl flid dynmis Cmbidge Univesiy ess UK. Feige J. H. nd ei M. (00). Compionl mehods fo flid dynmis 3d Ed Spinge Belin. Ibheem S.O. nd Adewmi M.A. (999). On ol viion diminishing shemes fo pesse nsiens J. Enegy Res. Teh.-Tns. ASME 30. ISO 3-3 (997). Nl gs llion of ompession fo. 3: Cllion sing physil popeies. Inenionl Ogniion fo Sndiion Tehnil Commiee ISO/TC 93 Nl gs Sb ommiee SC Anlysis of nl gs Geneve. Klik J. Siegle. Vosy Z. nd Zvok J. (988). Dynmi Modelling of Lge Sle Newoks Wih Appliion o Gs Disibion Elsevie New Yok. 786
15 J. Bsi. Appl. Si. Res. (8) Lnde B. E. nd Splding D. B. (97). The nmeil ompion of blen flows Comp. Mehods Appl. Meh.Eng Osid A.J. (987). Simlion nd nlysis of gs newoks Glf blishing Compny Hoson. Osid A.J. (00). Compison of isoheml nd non-isoheml pipeline gs flow models Chemil Engineeing Jonl 8-5. nk S. V. nd Splding D. B. (97). A llion poede fo he mss nd momenm nsfe in hee dimensionl pboli flows In. J. He Mss Tnsfe nk S. V. (980). Nmeil he nsfe nd flid flow Hemisphee blishing Copoion Tylo & Fnis Gop New Yok. Rd M. Z. nd Bojedi A. N. (008). Nmeil simlion of blen nsedy ompessible pipe flow wih he nsfe in he enne egion hys. S Shoede D.W. (00). A oil on pipe flow eqions poeedings SIG 33d Annl Meeing Sl Lke CiyUT. Shpio A.H. (95). The dynmis nd hemodynmis of ompessible flid flow The Ronld ess Compny New Yok. Tylo T.D. Wood N.E. nd owes J.E. (96). A ompe simlion of gs flow in long pipelines Soiey of eolem enginees Jonl Veseeg H.K. nd Mllseke W. (995). An inodion o ompionl flid dynmis: he finie volme mehod Longmn Gop Ld Hlow Englnd. Wylie E.B. Sone M.A. nd Seee V.L. (97). Newok sysem nsien llion by implii mehods Soiey of eolem enginees Jonl Wylie E.B. nd Seee V.L. (978). Flid Tnsiens MGw-Hill New Yok. Zho J. nd Adewmi M.A. (995). Simlion of nsien flow in nl gs pipelines oeedings SIG 7h Annl Meeing Albqeqe NM. 787
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