Modeling and Simulation of Real Gas Flow in a Pipeline

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1 Jornl of ppled Mhems nd Physs, 6, 4, Pblshed Onlne gs 6 n SRes. hp:// hp://d.do.org/.436/jmp Modelng nd Smlon of Rel Gs Flow n Ppelne gegneh en, lhn Mhe Deprmen of Mhems, Snnh Se Unersy, Snnh, US Reeed Jly 6; eped 7 gs 6; pblshed 3 gs 6 Copyrgh 6 by hors nd Senf Reserh Pblshng In. hs work s lensed nder he Cree Commons rbon Inernonl Lense (CC BY). hp://reeommons.org/lenses/by/4./ bsr In hs pper, mheml model h desrbes he flow of gs n ppe s formled. he model s smplfed by mkng some ssmpons. I s onsdered h he nrl gs flowng n long horzonl ppe, no he sore ors nsde he olme, rnsfer of he de o he ondon s domned by he ehnge wh he srrondng. he flow eqons re opled wh eqon of se. Dfferen ypes of eqons of se, rngng from he smple Idel gs lw o he more omple eqon of se Bened Webb Rbn Srlng (BWRS), re onsdered. he flow eqons re soled nmerlly sng he Godno sheme wh Roe soler. Some nmerl resls re lso presened. Keywords Gs Flow, Eqon of Se, Godno Sheme, Roe Soler, Ppe. Inrodon he prpose of hs pper s o desrbe he flow of nrl gs n ppelne by employng he fll se of dfferenl eqons long wh dfferen ypes of eqons of ses(eos), rngng from he smple Idel gs lw o he more omple eqon of se, Bened Webb Rbn Srlng (BWRS). he flow eqons re dered from he physl prnples of onseron of mss, momenm, nd energy. More deled dssson of onseron lws n be fond n []-[4]. he nrl gs s nsd nd ompressble. he gs flows n long horzonl ppe, nd hen n be onsdered s one-dmensonl flow. I s ssmed no he sore ors nsde he ppe nd rnsfer of he de o he he ondon s mh less hn he he ehnge wh he srrondng. In hs pper, he resls obned by solng he flow eqons long wh dfferen ypes of EOS re ompred How o e hs pper: en,. nd Mhe,. (6) Modelng nd Smlon of Rel Gs Flow n Ppelne. Jornl of ppled Mhems nd Physs, 4, hp://d.do.org/.436/jmp

2 . en,. Mhe [5]. he del gs eqon works resonbly well oer lmed emperre nd pressre rnges for mny sbsnes. Howeer, ppelnes ommonly opere osde hese rnges nd my moe sbsnes h re no del nder ny ondons. he more ompled EOS wll pprome he rel gs behor for wde rnge of pressre nd emperre ondons. he Godno sheme wh Roe soler [3] s sed o sole he Eler eqons nmerlly. he Godno sheme for onseron lws s known for s shok-prng pbly. he res of he rle s orgnzed s follows. In Seon () we reew he se of prl dfferenl eqons whh desrbe he flow of gs n ppe. Seerl eqons of ses re dsssed n hs seon. In Seon (3) hermodynml relonshps mong he physl qnes re presened. One n refer [6] for more hermodynml relonshps. Seon (4) onns he dssson of he nmerl mehod sed o sole he flow eqons ogeher wh dfferen ypes eqon of ses. Some nmerl resls re gen n hs seon. Conlsons re deferred o Seon (5).. Goernng Eqons of Rel Gs Flow n Ppe Le s onsder gs opyng sb domn Ω me Φ,, Ω desrbes he pos- Ω Φ Ω. he e-. Le ( ) on of he prle me. hen me he gs opes he domn { (, ), } loy of he gs poson nd me s gen by (, ) Φ (, )... rnspor heorem Le f : Ω [, ] be some physl qny rnspored by he fld. he ol mon ( ) qny f onned n Ω me s gen by F( ) f(, ) d. Ω Noon: J(, ) de Φ (, ) he re of hnge of F( ) s gen by: df( ) d f(, ) d d d Ω d f ( (, ) ) J(, ) d d Φ Ω d d f ( (, ) ) J(, ) f ( (, ), ) J(, ) d sng prod rle Ω Φ Φ d d f ( (, ), ) ( ) f ( (, ), ) J(, ) f ( (, ), ) J d Ω Φ Φ Φ f (, ) ( ) f(, ) f(, ) d Ω f (, ) f Ω ( )(, ) d d Ω s sefl n he deron of he goernng eqons. F of he hen we ge he rnspor heorem: f(, ) d (, ) ( f)(, ) d. he rnspor heorem d Ω f.. Conseron of Mss (he Conny Eqon) Ω s gen by ( ) he ol mss m n olme, d. Mss s onsered drng he deformon of Ω Ω Ω.e. d m d, (, ) d d d Ω ( ) d Ω d (By rnspor heorem) ( ) () d 653

3 . en,. Mhe Sne he boe negrl holds re for rbrry regon Ω.3. Conseron of Momenm (Eqon of he Moon) he ol momenm M of prles onned n Ω s gen by M (, ) (, ) d ordng o Newon s seond lw: he re of hnge of momenm eqls he on of ll he fores F ppled on Ω dm F d d (, ) (, ) d F d Ω ( ) d F Ω We he wo ypes of fores ng on Ω : ) Volme fores f, for emple gron, whh s gen by ( ) ( ) f Ω, g, d g s he gronl eleron. ) Srfe fores f s ng on Ω hrogh he bondry fores. Srfe fores re gen by σ s he sress ensor defned s: ( ) f σ, nds s Ω Ω of Ω Ω, sh s pressre nd nner fron σ σ σ3 σ σ σ σ3 σ3 σ3 σ 33 d nd n s he oer norml. he ol fore F f fs. hen, we he M f f s d ( ) d (, ) g(, ) d σ (, ) ds Ω Ω Ω n. By pplyng dergene heorem, he seond erm on he rgh sde of he boe eqon n be rnsformed o negrl oer he domn Ω nd hen we ge: ( ) d (, ) g(, ) d σ (, ) d Ω Ω Ω ( ) g σ or ( )( ) ( ) g σ For Newonn fld, he sress ensor depends lnerly on he deformon eloy,,.e. ( ) σ pi τ p λ I µ D τ s he sos pr of σ, p s pressre, I s he deny mr, λ nd µ re fron oeffens, nd D s he srn ensor gen by ( ( ) ) D For nsd fld, fron s negleed nd hen σ pi herefore, he eqon of moon for nsd fld beomes 654

4 . en,. Mhe ( ) ( ) g p ().4. Conseron of Energy Conseron of energy ons for effes of emperre rons on he flow or he rnsfer of he wh n he flow. he s Lw of hermodynms ses h: he ol energy of sysem nd s srrondngs remns onsn. Le be he ol energy of he fld n Ω nd Q be he mon of he rnsfered o Ω. he re of hnge of he ol energy of he fld opyng Ω s he sm of powers of he olme fore ng on he olme Ω, powers of he srfe fore ng on he srfe Ω, nd he mon of he rnsmed o Ω,.e. d (, ) g(, ) (, ) d (, ) (, ) ds Q d σ Ω n Ω ( ) E( ) densy, nd Ω,, d nd s he densy of kne energy. E e s he densy of energy (per n mss), e s nernl energy ( ) ( ) ( ) n( ) ( ) Q, q, d q, d s q, d Ω Ω Ω q s he densy of he sores (per n mss), nd q s he he fl (rnsfer of he by ondon). he rnsfer of he by ondon s gen by Forer s lw: q κ s he bsole emperre nd κ s he oeffen of herml ondy of he fld. q s he densy of he rnsfered from he srrondng nd s gen by: q kl srfe re ( o ) k L s he ol he rnsfer oeffen nd of he srrondng. hen he energy eqon for nsd gs flow beomes: d (, ) E(, ) d (, ) g(, ) (, ) d ( pi) (, ) ds d Ω Ω n Ω, q, d q, nd s q, d ( ) ( ) ( ) ( ) Ω Ω Ω o s he emperre By pplyng he rnspor nd dergene heorems o he boe eqon we obn he followng eqon: ( ) E (, ) ( E )(, ) d (, ) g (, ) (, ) d ( p ) d Ω Ω Ω, q, d qd q, d ( ) ( ) ( ) Ω Ω Ω E srfe re ( E) g ( p) q q kl ( o ) olme. E srfe re ( E) g ( p) q ( κ ) kl ( o ) (3) d olme here fore, from he eqons (), (), (3) we ge he followng sysem of eqons. ( ) ( ) ( ) g p E srfe re L o olme ( E) g ( p) q ( κ ) k ( ) (4) 655

5 . en,. Mhe.5. Smplfons In pre he form of mheml model res wh he ssmpons mde s regrds of operon ondons of he ppelne. Smplfed models re obned by negleng some erms n he bs eqons. In or se, we onsder nrl gs (Mehne) flowng n long horzonl ppelne. Hene we n onsder he flow s one dmensonl flow. By ssmng he ppe s horzonl, we n negle he onrbon of he gronl fore. srfe re 4 ssme lso no he sore ors nsde he olme. For ylndrl ppe, D s he olme D dmeer of he ppe. By pplyng he ssmpons we mde, (4) s reded o p (5) E ( E p) 4k L ( o ) D Frhermore, Mehne gs hs he followng properes. he spef he py p 65[ J kg K], herml ondy κ.3[ W m K], dynm sosy µ.e 5[ kg m s]. ypl les for he oerll he rnsfer oeffen k L re.6 W m K for.5 m dmeer nsled nd bred n sol. If he ppe s eposed on he r k L s 9 W m K. p µ Prndl nmber (Pr), defned s Pr, desrbes he rele srengh of sosy (he dffson of κ momenm) o h of he. I s enrely propery of he fld no he flow. In or se he le of Pr s bo.7, hs enbles s o regrd he flow s nsd flow. For gs flow ypl les of Pr re beween.7 nd. noher dmensonless onsn we n se o smplfy or sysem of eqons s he Nssel nmber (N). he kd L Nssel nmber s defned s N, D s hrers wdh of flow, for emple he dmeer κ of he ppe. he Nssel nmber ompres oneon he rnsfer o fld ondon he rnsfer. For Mehne gs flowng hrogh n nsled ppe of dmeer.5 m bred ndergrond, he le of N s ppromely. If he ppe s eposed o r, wll be rond 3. herefore, he erm nlded n he energy eqon de o he ondon n be negleed n for of he erm de o he ehnge wh he srrondng 4 k L ( ) D o. Inorporng hese ssmpons o Eqon (5) yelds: p (6) E ( E p) 4kL ( o ) D p p(, ) p p(, ) s n eqon of se sed o omplee he sysem of onseron lws. In he ne hper we wll sole Eqon (6) wh dfferen eqon of se nmerlly..6. Eqon of Se (EOS) n eqon of se s relonshp beween se rbles, sh h spefon of wo se rbles perms he llon of he oher se rbles. For n del gs, he eqon of se s he del gs lw. More ompled EOS he been formled by seerl workers o ry o model he behor of rel gses oer 656

6 . en,. Mhe rnge of pressres nd emperres. hs nldes Vn der Wls (VDW), Soe Redlh Kwong (SRK), Peng Robnson (PR), nd Bened Webb Rbn Srlng (BWRS)..6.. Idel Gs lw he del gs lw s gen by p R (7) p s he pressre, s he densy, R s he gs onsn, nd s he bsole emperre. he del gs lw s dered bsed on wo ssmpons: he gs moleles opy neglgble fron of he ol olme of he gs. he fore of ron beween gs moleles s zero. he del gs eqon works resonbly well oer lmed emperre nd pressre rnges for mny sbsnes. Howeer, ppelnes ommonly opere osde hese rnges nd my moe sbsnes h re no del nder ny ondons. Hene, we need o look for eqon of se wh wder ldy..6.. Vn der Wls (VDW) EOS I ws obsered h he del gs lw ddn qe work for hgher pressres nd emperres. he frs ssmpon works low pressres. B hs ssmpon s no ld s he gs s ompressed. Imgne for he momen h he moleles n gs were ll lsered n one orner of ylnder, s shown n he fgre below. norml pressres, he olme oped by hese prles s neglgbly smll fron of he ol olme of he gs. B hgh pressres (when he gs s ompressed), hs s no longer re. s resl, rel gses re no s ompressble hgh pressres s n del gs. he olme of rel gs s herefore lrger hn epeed from he del gs eqon hgh pressres. Vn der Wls proposed h we orre for he f h he olme of rel gs s oo lrge hgh pressres by sbrng erm from he olme of he rel gs before we sbse n o he del gs eqon. He herefore nroded onsn b n o he del gs eqon h ws eql o he olme lly oped by he gs prles. When he pressre s smll, nd he olme s resonbly lrge, he sbred erm s oo smll o mke ny dfferene n he llon. B hgh pressres, when he olme of he gs s smll, he sbred erm orres for he f h he olme of rel gs s lrger hn epeed from he del gs eqon. he ssmpon h here s no fore of ron beween he gs prles nno be re. If ws, gses wold neer ondense o form lqds. In rely, here s smll fore of ron beween gs moleles h ends o hold he moleles ogeher. hs fore of ron hs wo onseqenes: () gses ondense o form lqds low emperres nd () he pressre of rel gs s somemes smller hn epeed for n del gs. o orre for he f h he pressre of rel gs s smller hn epeed from he del gs eqon, Vn der Wls dded erm o he pressre n he del gs eqon. hs erm onns seond onsn. he omplee Vn der Wls eqon s wren s follows: R p (8) b Or n erms of molr olme p ( b) R 7R 64P R b 8P R s gs onsn, P rl pressre, nd rl emperre Noe h he les of he onsns nd b dffer from gs o gs. Een hogh, VDW EOS s beer hn Idel gs lw, sll s ndeqe o 657

7 . en,. Mhe desrbe rel gs behor. We wll onsder hree wdely sed eqons of se h do work resonbly well ner he dew pon: Soe-Redlh-Kwong (SRK), Peng-Robnson (PR), nd Bened-Webb-Rbn-Srlng (BWRS). In ddon o oerng wde rnge of ondons, hese eqons lso n be epressed n generlzed forms wh mng rles h perm he llon of he oeffens for dfferen omposons. SRK nd PR, long wh VDW re lled b eqon of se, bese epnson of he eqons no polynoml resls n he hghes order erms n densy (or spef olme) beng b. BWRS dds ffh nd sh power nd eponenl densy erms. he b eqon re ll of he form R p b B (9).6.3. he Soe-Redlh-Kwong (SRK) EOS he SRK EOS of se s gen by R p b b ( ( )) w r f.4748r P f w.76w w ().78664R b P w s he enr for whh s mesre of he gs moleles deon from he spherl symmery, R s gs onsn, P rl pressre, rl emperre, nd r s he reded emperre he Peng-Robnson (PR) EOS he PR EOS s defned s R p b b b () ( ( )) w r f.4574r P f w.699w w.778r b P.6.5. Bened-Webb-Rbn-Srlng (BWRS) EOS Probbly bese of s bly o oer boh lqds nd gses nd he lbly of oeffens nd mng 658

8 . en,. Mhe rles for mny hydrorbons n one ple, BWRS s he mos wdely sed eqon of se for smlon of ppelnes wh hgh densy hydrorbons, or wh ondenson. Smply s no mong he good qles of he BWRS eqon of se. he form of he eqon s: C D E d p 3 R BR br 3 4 d α γ ep γ ( ) ( ) 3 6 he eleen oeffens BCDEbdα,,,,,,,,,, nd γ re deermned from,, P, nd ω of he gs of neres nd he nersl onsns nd B s follows. Bω Bω B R C b Bω R Bω γ α D E B ω Bω R 4 B ω B ω R Bω R 8 8 d ( ω) B ωep B ω R 5 R B B B B B B B B B B B.43 BWRS n be dped for mres by he rles: B B k j j ( j ) ( ) 3 3 j j j γ ( γ ) 3 3 ( ) 3 3 b ( ) 3 ( 3) 3 3 α ( ) 3 C C C k b α ( ) 4 3 j j j d ( d ) 3 ( ) 5 j j j D D D k E E E k () 659

9 . en,. Mhe k j re he bnry neron oef- s he mole fron of he pre omponen of he mre, nd fens he Unersl Gs Lw he nersl gs lw s p Z R Z s lled he ompressbly for (Rel gs for). I s mesre of how fr he gs s from dely. mospher ondons, he le of Z s yplly rond.99. Under ppelne ondons, he le s yplly rond.9. good eqon of se n be seleed by s bly o pprome he ompressbly for rl ondons Z. For emple he epermenl le of Z for Mehne s.88. B s pprome le by VDW s.35, by SRK s.94, by PR s.894, nd by BWRS s hermodynml Relons In hs seon we wll brefly dsss hermodynml relons h es mong dfferen physl qnes. Frs lw of hermodynms ses h he spef ol enhlpy s defned s h e p whh mples de s d p d (3) dh s d p d (4) Dere relonshps: e e ssme e e( s, ), hen de ds d. Comprng he oeffens of hs eqon o h of Eqon (3) we ge s s Smlrly, ssmng h h( s, p) we ge e e, p s h h dh ds dp s p p nd omprng he oeffen of hs eqon wh h of Eqon (4) we ge h h, s p p Reprol relons nolng nernl energy e nd enropy s: Consder he nernl energy nd enropy o be fnon of emperre nd spef olme,.e, e e(, ) s(, ). s hen e e s s de d d, ds d d. Sbs- he oeffen of d, n he frs eqon, s by defnon he he py onsn olme, e hese wo eqons n (3) o ge nd e s e s p, Dfferenng he frs eqon of (8) wh respe o nd he seond wh respe o ges s e s s p s s s (5) (6), (7) (8) 66

10 . en,. Mhe Sbsng (9) n he frs eqon of (8) yelds e s s p e p p One sefl form nolng nernl energy s obned by sbsng he oeffen of d n he frs eqon of (7). (9) () for he oeffen of d n () for p de d p d Reprol relons nolng enhlpy h ssme h h( p, ), s s( p, ) hen h h dh dp d p p he oeffen of d s by defnon he he py onsn pressre, he nernl energy nd enropy se, boe we ge he followng relonshps. By doble dfferenng we do ge h s h s, p p p h p p p () () p. In smlr proedre s n (3) (4) dh pd dp p He pes By eqng he dfferene of (3) nd (4) o he dfferene of () nd (5) we ge Ddng by d nd holdng p onsn ges p ( p )d dp d p ( p ) p p (5) (6) (7) 4. Nmerl Mehods: Godno Sheme wh Roe Soler In hs seon we wll onsder nmerl sheme o sole homogeneos Eler eqon wh nl ondon by employng dfferen EOS. he Eler eqon n eor form: F( U) (,) ( ) U U U (8) 66

11 . en,. Mhe nd p p(, ) nd ( ) ( p) U F U p One of he mehods o sole D nonlner hyperbol sysems s he Godno sheme 4.. Godno Sheme Sppose we he sbdded or domn [, ] b. N U : U d. ssme Le s defne ( ) wh nl ondon on, b n o N sbnerls wh nd N b, so h n U me n s known nd h n s peewse onsn. hen we sole ely he lol Remnn problem for U F ( U ) n n on [ ] n (, ) U n U for < U for n n n Le s denoe he solon by w (, ). hen he solon w (, ) o defne he globl solon s hen he solon n U (, ) s defned by ( ) w, f nd w (, ) f nd n n n n n n n Conseron form: Sne s n e solon on,, we he n ( ) U, d n n n ( ) n ( ( )), dd F, dd n n,, of he lol Remnn problems re sed n n ( (, ) (, )) d F, F, d U, n n ( U U ) F U F U s onsn for n n 66

12 . en,. Mhe Wh he nmerl fl n n U U F U F U g( U, U) F U hs sheme s lled Godno sheme. Solng Remnn problem ely s no lwys n esy sk. hen we my need o onsder n pprome solon of he Remnn problem. 4.. Remnn Problem for Lner Sysem Sppose we he lner sysem U U wh nl ondon (,) U U U l r for < for Le λ < λ < λ3 re he egenles nd r, r, r 3 re he orrespondng egeneors. Defne α,,, 3 sh h U U α r. r l 3 hen he solon of he Remnn problem s gen by Ul for < λ U(, ) Uk for λk < λk, k, U r for λ3 U U α r k l rey of pprome Remnn solers he been proposed h n be ppled more esly hn he e Remnn soler. One of he mos poplr Remnn solers rrenly n se s de o Roe. Godno sheme wh Roe ppromon. he de s o reple he non-lner Remnn problem soled eh nerfe by n pprome one. U l nd ( l, r) (, ) ( ) k ( ) U U, U U l r U r re he lef nd rgh les nd U ( l, U r) ssfes F( U ) F( U ) ( U, U )( U U ) r l l r r l U U s dgonlzble wh rel egeneors. Ul Ur F U s Ul, Ur U Conseron form of he Roe sheme. he Roe sheme n be wren n onseron form s U U g U U g U U (, ) (, ) n n n n n n 663

13 . en,. Mhe 3 g( w, ) F( ) F( w) λα r 3 λ nd w, nd w α r. he mn sk n he Roe sheme s he deermnon of he mr of lnerzon. Now le s onsder or eqon (8) ogeher wh n eqon of se of he form p p(, ). hen we pprome hs non-lner sysem wh n pprome lner sysem s follows: U, U DF U r re he egenles nd egeneors of ( ) Defne ( l r) ( ) U nd l r h l l r r l h r l l r r l h r e p h s he spef enhlpy. hese erges re lled he Roe men les. U ( l, U r) ssfes he Roe ondons. o sole or problem wh he Roe sheme, we need o lle he egenles nd her egeneors of he Jobn mr DF ( U ) whh re needed o ompe he Roe fl. B for omple EOS he deermnon of hese egeneors my no be smple. One wy of deermnng he egeneors of hs Jobn s by epressng he Eler eqon n erms of prme rbles V (,, ). We hoose he emperre s one of prme rbles hn he pressre p, bese n mos eqon of se p s epressed n erms of. Le V BV be he Eler eqon n erms of he prme rbles V nd U F( U) be n onsere rbles. he pprome lner sysem s U DF ( U ) U M U V U U V DF U V V V ( ) ( ) V M DF U MV ( ) B M DF U M he mres B nd DF ( U ) he denl egeneors. Frher more, f B PΛ P hen DF( U) MP P M ( ) DF U Λ. hen R MP s he rgh egeneors of 4.3. Solng Eler Eqon Usng he Idel Gs Lw In hs seon we sole one dmensonl Eler eqon wh Idel gs EOS. Consder he Eler eqon (8) wh he del gs lw p R. Usng (), he hnge of nernl energy s gen by de d whh mples e, nd he ol energy 664

14 . en,. Mhe s gen by:. Now le s epress (8) n erms of he prme rbles V (,, ) sheme esly. Conny eqon: Momenm eqon: p p p R, nd R, he momenm eqon n erms of he prm- p p Now sng p, e rbles s Energy Eqon: Now, sng R nd he oeffen of p s. hen eqon (9) redes o ( ) ( ) ( p), so h we n pply he Roe R R (( p) ) p p ( ) ( ) ( p) ( ) p ( ) p ( ) ( p) p ( ) p p ( ) (9),, nd hen he Eler eqon n prme rbles s wren s, he oeffen of n Eqon (9) beomes R (3) R R R (3) 665

15 . en,. Mhe Or n eor form V BV Egenles nd egeneors of he oeffen mr B of (3) re omped s follows. he lol speed of sond s gen by U he mr of he orrespondng egeneors s: λ R λi B λ R R λ R R ( λ ) ( λ ) ( λ ) R λ or ( λ ) R λ, λ, nd λ 3 R R γ R P R R R o ompe he egeneors of he Jobn ( ) (,, ) nd V (,, ) Hene DF U we need o ompe he mr M he mr R of egeneors of DF ( U ) s gen by: M U V 666

16 . en,. Mhe R MP R ( ) R Sne he ol spef enhlpy h s gen by h R we n wre he egeneors n erms of h s R h ( ) h 4.4. Solng Eler Eqon Usng he Vn der Wls (VDW) EOS Here we sole one dmensonl Eler eqon wh VDW EOS. Consder gn he eler eqon (8) wh R 7R R VDW EOS p nd b, R s gs onsn, P rl pressre, b 64P 8P rl emperre, nd r s he reded emperre. gn sng (), he hnge of nernl energy s gen by: p de d p d p R p R p Here,,, nd p. b b Inegrng he boe dfferenl eqon ges he nernl energy e. he ol energy s gen by: Now le s epress (8) n erms of he prme rbles V (,, ) Conny eqon: ( ) Momenm eqon: ( ) ( p) p p p Here, p, p R ( b ) Hene he momenm eqon s reded o, nd p R. b 667

17 . en,. Mhe Energy Eqon: R R ( b) b (( p) ) p p ( ) ( ) ( p) ( ) p ( ) p ( ) ( p) p ( ) p p ( ) (3) Usng,, nd. he oeffen of n (3) beomes R b nd he oeffen of p s. hen (3) redes o he Eler eqon s wren s R ( b), 3 R ( b ) 3 3 R, nd b 3 R ( b ) Egenles nd egeneors of he oeffen mr B of (34) re omped s follows. λ λi B λ 3 3. λ (33) (34) he lol speed of sond s defned s ( λ ) ( λ ) ( λ )( ) 3 3 ( ) 3 3 λ or λ λ, λ nd λ 3 668

18 . en,. Mhe he mr of he orrespondng egeneors s: 33 P o ompe he egeneors of he Jobn ( ) (, e, ) nd V (,, ) U Hene DF U we need o ompe he mr M he mr R of egeneors of DF ( U ) s gen by: R MP m m m m m m m3 nd m33 Sne he ol spef enhlpy h s gen by s r3 m3 m h m m U M V we n wre he egeneors n erms of h R h r3 h 4.5. Solng Eler Eqon Usng he Soe-Redlh-Kwong (SRK) EOS Le s onsder (8) wh SRK EOS R p b b 669

19 . en,. Mhe ( ( )) fw r.4748r,, Fw w.76w, P s he enr for R s gs onsn, P rl pressre, rl emperre, nd emperre. he nernl energy s gen by: p de d p d ( w( r )) p R f w f b b ( ( )) p R fw r fw r b b ( ) ( ( )) p p ( f f f b b w r w r w r ( ( ))( ) f f f b b w r w wl.78664r b, w P s he reded r f wl fw. f w ( r ) fer negrng he dfferenl eqon of he nernl energy, we ge fwl e log b b ( ) he ol energy s gen by: fwl log ( b) b Conny eqon: Momenm eqon: p p sng p, momenm eqon s wren s Energy Eqon: ( b ) ( ) ( ) ( p) p ( b) ( b ) p R, nd p R fw f wl b b f ( b ) R fw fwl ( b ) b ( b ) ( fw) R ( b) (( p) ) ( ) ( ) w he 67

20 . en,. Mhe he oeffen of p e p ( ) ε ( ) ( p) ( ) p ( ) p ( ) ( p) p ( ) p p ( ) (35) fwl fwl log ( b ) b b f f w wl log b ( fw) n Eqon (35) beomes R f wl b b b ( b) nd he oeffen of p s. Noons: Le 3 denoe he oeffen of nd 3 denoe he oeffen of.e. hen Eqon (35) redes o he Eler eqon s wren s, fw fwl 3 log b b ( f ) w 3 3 ( ) (36) ( b ) ( b ) ( b ) R R fw fwl b b f ( ) ( ) Egenles nd egeneors of he oeffen mr of Eqon (37) re gen s follows w (37) 67

21 . en,. Mhe U λ λi B λ 3 3 λ ( λ ) ( λ ) ( λ )( ) 3 3 ( ) 3 3 λ or λ λ, λ nd λ 3 33 he mr of he orrespondng egeneors s: P o ompe he egeneors of he Jobn ( ) (, e, ) nd V (,, ) DF U we need o ompe he mr fwl log ( b) b Hene M m3 m 33 M U V f fwl 3 log b b wl m ( b ) f f w wl nd m33 log ( b) b ( fw) he mr R of egeneors of ( ) DF U s gen by: R MP 3 3 m3 m33 m3 m33 m3 m Sne he spef enhlpy h s gen by h m3 m33 we n wre he egeneors n erms of h s 67

22 . en,. Mhe r3 m3 m33. 3 R h r3 h 4.6. Solng Eler Eqon Usng he Peng-Robnson (PR) EOS Le s onsder (8) wh PR EOS. ( ( )) fw r, R p b b b.4574r, P F w.699w w,.778r b, w s he enr for R s gs onsn, P rl pressre, P r s reded emperre. he nernl energy s gen by: Here, f wl fw. f w p de d p d ( w( r )) p R f w f b b b ( ( )) p R f f b b b ( r ) w r w r ( ( )) p p fw r fw r b b ( w( r )) f b b ( w( r ))( w) f f b b f b b Inegrng he boe dfferenl eqon for nernl energy we ge he ol energy s gen by: fwl b e log 8b b fwl b log 8b b wl rl emperre, nd 673

23 . en,. Mhe Conny eqon: Momenm eqon: ( ) ( ) ( p) Usng he onny eqon, s reded o p p p Here, p, p R fw f wl b b b f ( b) p R ( b ) b b ( ) ( w) he momenm eqon s wren s ( )., nd R ( b ) R f f ( b) ( b b ) b ( b b ) ( fw) w wl Energy Eqon: (( p) ) p p ( ) e ( ) ( p) ( ) p ( ) p ( ) ( p) p ( ) Usng nd he oeffen of p p ( ) (38) f wl log b f wl 8b b b b n (38) beomes fw fwl b log, 8b f b ( ) w f wl R b b b b b 674

24 . en,. Mhe nd he oeffen of p s. Noons: Le 3 denoe he oeffen of nd 3 denoe he oeffen of hen (38) redes o he Eler eqon s wren s, 3 fw fwl b log 8b f b ( ) w (39) ( b ) R ( b) b b ( ) R f fwl b b b f w 3 ( ) ( w) λ λi B λ 3 3 λ ( λ ) ( λ ) ( λ )( ) 3 3 (4) ( ) 3 3 λ or λ λ, λ nd λ 3 U 33 he mr of he orrespondng egeneors s: P o ompe he egeneors of he Jobn ( ) (, e, ) nd V (,, ) DF U we need o ompe he mr M U V 675

25 . en,. Mhe Hene fwl b log 8b b M m3 m 33 fwl b m3 log f wl 8b b b b nd m fw fwl b log. 8b f b 33 ( w) he mr R of egeneors of ( ) DF U s gen by: R MP 3 3 m3 m33 m3 m33 m3 m Sne he spef enhlpy h s gen by h m3 m33 we n wre he egeneors n erms of h s R h r3 h r3 m3 m Solng Eler Eqon Usng he Bened-Webb-Rbn-Srlng (BWRS) EOS Le s onsder (8) wh BWRS EOS. C D E d d p R BR br α γ ep γ 3 4 he nernl energy s gen by: p de d p d ( ) ( ) p C 3D 4E d 3 αd 6 R ( BR ) ( br ) ( γ ) ep( γ ) p 3C 4D 5E d d 3 p α γ γ ( ) ep ( )

26 . en,. Mhe 5 3C 4D 5E d d 3 e α ep γ γ he ol energy s gen by: ( ) 3C 4D 5E d d 3 α ( γ ) 5 γ ep 3 4 Conny eqon: Momenm eqon: ( ) ( ) ( p) p p p p Le p C D E d R BR br d α ( 3 3γ γ ) ep ( γ ) p C 3D 4E d d R BR br α γ γ he momenm eqon s wren s Energy Eqon: ( ) ep ( ) p R C D E d d 4 4 6α ( 3 3γ γ ) ep ( γ ) BR 3 br 3 4 p C 3D 4E d 3 R BR br αd ( ) ep ( ) 5 γ γ 3 (( p) ) p p ( ) 677

27 . en,. Mhe he oeffen of ε ( ) ( p) ( ) p ( ) p ( ) ( p) p ( ) p p ( ) (4) 3C 4D 5E 3 d d 3 α γ ( γ ) 5 γ 5 4 ep 6C D E d αd 6 γ 5 γ ( ) ep n Eqon (4) beomes C 3D 4E d d R BR br α γ γ ( ) ep ( ) nd he oeffen of p s. Noons: Le 3 denoe he oeffen of nd 3 denoe he oeffen of.e, 3 hen (4) redes o he Eler eqon s wren s, (4) Egenles nd egeneors of he oeffen mr B of Eqon (43) re omped s follows. λ 3 3 λi B λ 3 3 λ ( λ ) ( λ ) ( λ )( ) 3 3 ( ) 3 3 λ or λ λ, λ nd λ 3 (43) 678

28 . en,. Mhe U he mr of he orrespondng egeneors s: 33 P o ompe he egeneors of he Jobn ( ) (, e, ) nd V (,, ) DF U we need o ompe he mr U M V C 4D 5E d d 3 α ep 3 4 ( γ ) 5 γ Hene M m3 m 33 3C 4D 5E 3 d m d 3 α γ ( γ ) 5 γ 5 4 ep 6 D E d αd 6 5 γ 3 nd m 3 6 ( 33 ep γ ) he mr R of egeneors of DF ( U ) s gen by: R MP 3 3 m3 m33 m3 m33 m3 m Sne he spef enhlpy h s gen by h m3 m33 we n wre he egeneors n erms of h s R h r3 h r3 m3 m

29 . en,. Mhe Fgre. he resls obned by solng he homogeneos Eler eqon by employng he del gs lw nd he oher for eqon of ses. Fgre. he resls obned by solng he Eler eqon (nldng he sore erm) by employng he PR nd BWRS EOS. 68

30 . en,. Mhe 4.8. pplon of he Roe soler Now o pply he Roe sheme on (8), on eh ell [, ] D ( FU ) n (, ) U U U n U for < U for >, we pprome he sysem by n nd U s deermned from he Roe erges. he solon s deermned s: U U g U U g U U (, ) (, ) n n n n n n 3 g( w, ) F( ) F( w) λα r 3 λ nd w, nd w α r. he ls eqon s sysem of smlneos lgebr eqons for he rbles α. he onsere rbles (,, ) re deermned by he sheme. he eloy s obned from nd. B o deermne he le of he emperre we se n eron mehod (espelly for he ses of omple EOS). hen he pressre P s omped from he EOS 4.9. Nmerl Resls r re he egenles nd egeneors of ( ) In hs seon we presen some nmerl resls. We onsder be of lengh, flled by Mehne gs, he nl dsonny s loed.5. In or smlon he followng nl d s sed. l 3, p l 3, l for.5 r, p r, r for >.5. In Fgre, we he ploed he densy, pressre, eloy, emperre, nd he rel gs ompressbly for omped by sng eh of EOS we dsssed. Fgre deps resls of (6),.e, he Eler eqon wh he sore erm nlded, obned by pplyng PR, nd BWRS EOS. 5. Conlson he model h desrbes he flow of gs n ppe s presened. Smplfons o he eqons re mde sng ppropre ssmpons. Seerl Eqons of ses h lose he sysem of eqons re emned nd he resls obned for eh eqon of se re ompred. Referenes [] Feser, M. (993) Mheml Mehods n Fld Dynms. Longmn Senf & ehnl, New York. [] Chorn,.J. nd Mrsden, J. E. (993) Mheml Inrodon o Fld Mehns. Sprnger, New York. hp://d.do.org/.7/ [3] LeVeqe, R.J. (99) Nmerl Mehods for Conseron Lws, Leres n Mhems. EH Zerh, Brkheser, Bsel. hp://d.do.org/.7/ [4] Kroener, D. (997) Nmerl Shemes for Conseron Lws/ John Wley & Sons Ld, Englnd nd B.G. ebner, Germny. [5] Modsee, J.L. () Eqons of Se orl. 3nd nnl Meeng PSIG (Ppelne smlon Ineres Grop), Snh, Georg. [6] Srlng, K.E. (973) Fld hermodynm Properes for Lgh Perolem Sysems. Glf Pbl., Hoson. 68

31 Sbm or reommend ne mnsrp o SCIRP nd we wll prode bes sere for yo: epng pre-sbmsson nqres hrogh Eml, Febook, LnkedIn, wer, e. wde seleon of jornls (nlse of 9 sbjes, more hn jornls) Prodng 4-hor hgh-qly sere User-frendly onlne sbmsson sysem Fr nd swf peer-reew sysem Effen ypeseng nd proofredng proedre Dsply of he resl of downlods nd ss, s well s he nmber of ed rles Mmm dssemnon of yor reserh work Sbm yor mnsrp : hp://ppersbmsson.srp.org/