PREDICTION OF THE TRANSIENT FORCE SUBSEQUENT TO A LIQUID MASS IMPACT ON AN ELBOW OF AN INITIALLY VOIDED LINE

Transcription

1 PEDICTION OF THE TNSIENT FOCE SUBSEQUENT TO IQUID MSS IMPCT ON N EBOW OF N INITIY OIDED INE THESIS SUBMITTED TO THE GDUTE SCHOO OF NTU ND PPIED SCIENCES OF MIDDE EST TECHNIC UNIESITY BY BÜENT BBS KYHN IN PTI FUFIMENT OF THE EQUIEMENTS FO THE DEGEE OF DOCTO OF PHIOSOPHY IN CII ENGINEEING FEBUY 9

2 ppoval of he hei: PEDICTION OF THE TNSIENT FOCE SUBSEQUENT TO IQUID MSS IMPCT ON N EBOW OF N INITIY OIDED INE bmie b BÜENT BBS KYHN in paial flfillmen of he eqiemen fo he egee of Doo of Philooph in Civil Engineeing Depamen, Mile Ea Tehnial Univei b, Pof. D. Canan Ögen Dean, Gaae Shool of Naal an pplie Siene Pof. D. Güne Öebe Hea of Depamen, Civil Engineeing o. Pof. D. Zafe Bokş Spevio, Civil Engineeing Dep., METU Examining Commiee Membe: Pof. D. Tg Tokemi Engineeing Siene Dep., METU o. Pof. D. Zafe Bokş Civil Engineeing Dep., METU Pof. D. Mein Ge Civil Engineeing Dep., METU i. Pof. D. Şahna Tiğek Civil Engineeing Dep., METU D. Uğaş Baan Havelan.Ş. Dae: Feba 7, 9

3 I heeb elae ha all infomaion in hi omen ha been obaine an peene in aoane wih aaemi le an ehial on. I alo elae ha, a eqie b hee le an on, I have fll ie an efeene all maeial an el ha ae no oiginal o hi wok. Name, a Name: BÜENT BBS KYHN Signae : iii

4 BSTCT PEDICTION OF THE TNSIENT FOCE SUBSEQUENT TO IQUID MSS IMPCT ON N EBOW OF N INITIY OIDED INE Kahan, Bülen bba Ph. D., Depamen of Civil Engineeing Spevio: o. Pof. D. Zafe Bokş Feba 9, 63 page The aim of he hei, i o fin he anien foe applie b an inivial anien liqi lg on an elbow a he en of a hoional line e o an impa. The liqi lg i iven b peie ai in a ank loae peam of he pipeline. The ime epenen pee iibion along he elbow an a veial exenion egmen afe he elbow wa olve, wih a -D nmeial appoah along a ve line meh. Fo hi ppoe; fil, a 3-D axial blen veloi pofile fnion wa ame fo he lg, wih i hape allowe o wa owa he onvex ie of he elbow along he ve meh wih he ai of a alibaion ool. Then, he pee vale wee allae b ing -D appliaion of enol Eqaion in linial pola an aeian ooinae fo he elbow an he veial exenion egmen, epeivel. The anien foe aing on he elbow iv

5 an he following veial exenion egmen wa fon b ing hee allae pee vale an appling onevaion of momenm piniple ove he volme elemen elee along he elbow an he veial exenion egmen. Fo he anali of he lg moion fom he peie ank o he elbow, a peviol wien ompe oe BOZKUŞ- wa ilie. Then, he elbow an he veial exenion egmen allaion in hi wee mae wih a new oe KYHN, whih i an impove veion of BOZKUŞ-. The allae anien foe an impa pee a he elbow wee alo ompae wih hoe fom pevio ie. Kewo: iqi Slg, Elbow, Tblen eloi Pofile, BOZKUŞ-, KYHN, Tanien Foe v

6 ÖZ BŞNGIÇT BOŞ ON Bİ BOU HTTININ DİSEĞİNE Bİ SII KÜTESİNİN ÇPMSI SONUCU OUŞN TNSİT KUETİN THMİNİ Kahan, Bülen bba Dokoa, İnşaa Müheniliği Bölümü Te Yöneiii: Doç. D. Zafe Bokş Feba 9, 63 afa B e çalışmaının amaı aa bi bo haının onnaki ieğe çapan ekil bi küleinin glaığı ani kvvein belilenmeii. S külei membaa blnan bi ankaki baınçlı hava aafınan iilmekei. Dieke ve ona bağlı, bonn bi üşe anı kımınaki baınç ağılımın amana bağlı eğişimi, eğiel bi ha şeklineki ağ üeine bol nümeik bi aklaşım ile çöümlenmişi. B amaçla, ilk olaak 3 bol bi ekenel, übülanlı akım hı pofili fonkion ahmin eilmişi ve b fonkionn şeklinin eçilen ha üeieki eğiel ağ bona, kalibaon paameei olaak kllanılan iğe bi fonkion aımı ile ieğin konvek aafına oğ efome olabilmei ağlanmışı. Baınç eğelei iliniik kpal ve kaeen kooinalaaki enol vi

7 Denklemlei nin, iek ve bonn üşe anı kıımlaına bol olaak glanmaı ile heaplanmışı. Dieğe ve ona bağlı, üşe anı kımına ekien ani kvve; heaplanan b baınç eğelei kllanılaak ve momenmn konm penibinin iek ve üşe anı kımı bona eçilen haimel elemanla üeine glanmaı ile blnmş. S küleinin baınç ankınan ieğe kaa olan haekeinin analii için aha öne aılmış bi bilgiaa ko olan BOZKUŞ- kllanılmışı. Daha ona b çalışmaaki iek ve üşe anı kımı için olan heaplamala, BOZKUŞ- nin gelişiilmiş bi veion olan KYHN iimli eni bi ko ile apılmışı. Tee, ieke heaplanan abeel baınçla ve blnan ani kvve eğeleinin aha öne apılan çalışmalaa blnan onçlala kaşılaşıılmaı a e almakaı. naha Kelimele: S külei, Diek, Tübülanlı kım Hı Pofili, BOZKUŞ-, KYHN, Tani Kvve vii

8 To M Paen viii

9 CKNOWEDGEMENTS I wol like o expe m eepe gaie o m pevio o. Pof. D. Zafe Bokş fo hi invalable giane an ggeion hogho hi. I wol like o expe m peial hank alo o m ooal ommiee membe, Pof. D. Mein Ge, Pof. D. Tg Tokemi, i. Pof. D. Şahna Tiğek an D. Uğaş Baan fo hei valable ggeion an ommen. M ve peial hank exen o i. Pof. D. Ögü Kç fo hi help in poviing he opponi o e METU Cenal Seve Sem a a oe of ompile ofwae. Speial hank alo go o i. Bak nıl, i. Kaan Öenç an i. İlke Küçükepe fo hei aiane in he poee of eing p he ompile ofwae on ompe an making i ea fo e. ix

10 TBE OF CONTENTS BSTCT.. iv ÖZ.. vi CKNOWEDGEMENTS... ix TBE OF CONTENTS x IST OF TBES.. xiii IST OF FIGUES xiv IST OF SYMBOS xvii CHPTES. INTODUCTION aionale fo he Peen S Speifi Objeive of he Peen S...3 Sope of he Peen S ITETUE EIEW MTHEMTIC DEEOPMENT FO THE HOIZONT PT Inoion Slg Dnami Eqaion Conevaion of Ma Eqaion Conevaion of Momenm Eqaion Slg Kinemai Eqaion Ga Dnami Eqaion Copling of Ga an Slg Dnami Eqaion Fnional Hol Up Coeffiien Cloe 6 4. MTHEMTIC DEEOPMENT FO THE EBOW PT 8 4. Inoion. 8 x

11 4. me xial eloi Pofile Fnion Callaion of he Impa Pee a he Elbow Simplifing mpion Pee Diibion Eqaion veage Pee Diibion along he eial Exenion Segmen veage Pee Diibion along he Elbow Callaion of Impa Pee a he Convex Sie of he Elbow Callaion fo ha Pa of he Slg in he Hoional Pipe The Iniial an he Bona Coniion Tanien Foe Callaion a he Elbow Pa Callaion a he Elbow Hoional Tanien Foe Diibion a he Elbow eial Tanien Foe Diibion a he Elbow Callaion a he eial Exenion Segmen Hoional Tanien Foe Diibion a he eial Exenion Segmen eial Tanien Foe Diibion a he eial Exenion Segmen Toal Hoional an he eial Tanien Foe Nmeial Inegaion Doble Inegaion in Pola Cooinae Evalaion of Nmeial ine Inegal Geneaion of -D Clee Meh ppliaion of 3-Poin Ga Qaae Meho.. 79 xi

12 4.6 Calibaion Fnion fo he eloi Pofile Non-Dimenionaliaion of he Eqaion Dieiaion of he Eqaion Evalaion of oal Peak Pee Inp Daa, Meh Sie an Eo nale Negleing of Some Tem.. 4. Inp Daa, Meh Sie an Eo nale Inp Daa fo Pogam KYHN Eo nali an Meh Sie fo ea Inegal Eo nali an Meh Sie fo ine Inegal Meh Sie an Eo nali fo he ine Meh along -Cve Eo nali an he Meh Sie fo Impa Pee Callaion ESUTS ND DISCUSSION Peak Pee a he Elbow Pee-Time Hio Plo Tanien Foe a he Elbow Impa Pee Diibion a he Elbow Diion on he Exiene of Waehamme Even 4 6. CONCUSIONS Smma Conling emak. 43 EFEENCES PPENDICES. MTHEMTIC DEITION OF THE EQUTIONS Fnional Hol p Coeffiien Deeminaion 5.. Fomlaion fo he veage Slg engh. 5.. Coelaion beween he Hol p an he Inananeo Slg engh... 5 xii

13 . Deivaion fo he me eloi Pofile Fnion Deivaion of Boom Cone Eqaion 5.. Deivaion of Uppe Cone Eqaion Callaion of Unknown Paamee. 6.3 Deivaion fo he Pee Diibion Eqaion Deivaion of veage Pee Diibion Eqaion a he eial Exenion Segmen Deivaion of veage Pee Diibion Eqaion a he Elbow Deivaion of he Eqaion fo he Pee a he Convex Sie of he Elbow 79.4 Deivaion fo he Tanien Foe Callaion a he Elbow Pa Deivaion fo he Hoional Tanien Foe Diibion a he Elbow Deivaion fo he eial Tanien Foe Diibion a he Elbow Deivaion fo he Hoional Tanien Foe Diibion a he eial Exenion Segmen Deivaion fo he eial Tanien Foe Diibion a he eial Exenion Segmen 94.5 Deivaion fo Nmeial Inegaion Fomla Deivaion fo Doble Inegal in Pola Cooinae Deivaion fo he ine Inegal Deivaion fo -D Clee Meh Geneaion Deivaion fo 3-Poin Ga Qaae Meho..6 Deivaion fo he Calibaion Fnion.. 5 xiii

14 B. COETION FUNCTION FO THE MXIMUM CIBTION NGE.. 6 C. COMPUTE POGMS... 3 C. Explanaion fo he Fnion of he Compe Pogam... 3 C. Flow-Cha fo he Compe Coe KYHN... 8 C.3 Compe Coe KYHN.. 4 xiv

15 IST OF TBES TBES Table 4. Iniial an he bona oniion fo he elbow an he veial exenion egmen allaion Table 4. ale fo he em in Eqaion (4.8).. Table 4.3 ale fo he em in Eqaion (4.85).. Table 4.4 aial an axial veloiie wih hei ime eivaive Table 4.5 Inp aa fo pogam KYHN.. 5 Table 5. Peak pee a he elbow (pig).. Table. Callaion ep fo / ( p m. ) p ave Table B.. Peak pee e fo oelaing max (all in pig) Table B. max vale v. p / in e fo oelaion poee... xv

16 IST OF FIGUES FIGUES Fige. Fenon expeimenal ep [] Fige. Sem ep e in Bokş [] Fige.3 Expeimenal ep e b Baan [7]... 9 Fige 3. Conol volme fo he liqi lg elee b Bokş [] along he hoional pipeline... 3 Fige 3. Compaional omain e b Bokş [] fo he olion of ga nami eqaion. Fige 3.3 aiaion of pee a he peam pee ank (Bokş []) Fige 4. eloi iibion in a ve pipe given b Panl [3], [4] (Shlihing []) Fige 4. Paamee fo he geneal obliqe one eqaion... 3 Fige 4.3 Co-eional hape of he ame blen veloi pofile fnion... 3 Fige 4.4 Paamee e fo he ame veloi pofile in a ingle ooinae em Fige 4.5 (a) Complee phial e p fo he em (Bokş []); Compaional omain fo (b) he elbow an he veial exenion pa, () he o-eion of he elbow, () he o-eion of veial exenion pa of he pipe Fige 4.6 Sample volmei pipe elemen on he -ve, in he veial exenion egmen.. 45 Fige 4.7 Sample volmei pipe elemen along he -ve, fo he elbow pa.. 48 Fige 4.8 -D aial meh a a o-eion of he elbow.. 5 Fige 4.9 -D aial meh eleion a elbow o-eion, wih he line mehe loae in he veial, mi x plane of he elbow xvi

17 Fige 4. Compaional omain wih he noal nmbeing along he -ve given Fige 4. Conol volme fo he anien foe allaion a he elbow.. 6 Fige 4. Seleion fo he loaion of he P vale along he -ve, fo he poee of inepolaion a he fae of he volme elemen Fige 4.3 Conol volme fo he anien foe allaion a he veial exenion egmen Fige 4.4 Cila meh ove he pipe o-eion. 7 Fige poin meh elemen elee fo oble inegaion in pola ooinae.. 74 Fige poin meh elemen fo -D Ga Qaae in aeian ooinae.. 75 Fige 4.7 Meh elemen fo he ene poin of he ila omain in pola ooinae. 76 Fige 4.8 ine meh fo a given poin K on he elbow o-eion.. 78 Fige 4.9 Paamee e fo he 3-poin Ga Qaae Meho.. 8 Fige 4. Shape of he alibaion fnion. 8 Fige 4. Geneal vaiaion of he hape of he ame axial veloi pofile along he ve.. 85 Fige 4. Peenage eo v. oal meh ie, fo m.. 7 Fige 4.3 Peenage eo v. oal meh ie, fo m Fige 4.4 Peenage eo v. oal meh ie, fo m Fige 4.5 Peenage eo v. oal meh ie, fo m Fige 4.6 Peenage eo v. oal meh ie, fo m.. 9 Fige 4.7 Peenage eo v. oal meh ie, fo m. 5. Fige 4.8 aiaion of peenage eo v. noe nmbe, Fige 4.9 aiaion of peenage eo v. noe nmbe, xvii M.. N lg.. 5

18 Fige 5. Nomalie peak pee v. / p in Fige 5. Pee-ime hio plo a he elbow fo 4 f an P pig Fige 5.3 Pee-ime hio plo a he elbow fo 4 f an P 3 pig Fige 5.4 Pee-ime hio plo a he elbow fo 4 f an P 4 pig Fige 5.5 Pee-ime hio plo a he elbow fo 5 f an P pig Fige 5.6 Pee-ime hio plo a he elbow fo 7 f an P pig... 5 Fige 5.7 Pee-ime hio plo a he elbow fo 7 f an P pig... 5 Fige 5.8 Pee-ime hio plo a he elbow fo 7 f an P 4 pig Fige 5.9 Pee-ime hio plo a he elbow fo 9 f an P pig Fige 5. Pee-ime hio plo a he elbow fo f an P pig Fige 5. Pee-ime hio plo a he elbow fo f an P pig Fige 5. Pee-ime hio plo a he elbow fo f an P 3 pig Fige 5.3 Pee-ime hio plo a he elbow fo f an P 4 pig Fige 5.4 Tanien foe omponen fo 4 f an P pig xviii in in in in in in in in in in in in in Fige 5.5 Tanien foe omponen fo 4 f an P pig in

19 Fige 5.6 Tanien foe omponen fo 7 f an P pig in Fige 5.7 Tanien foe omponen fo 7 f an P pig in Fige 5.8 Tanien foe omponen fo f an P pig in Fige 5.9 Tanien foe omponen fo f an P 4 pig in Fige 5. ale of F v. D (Bokş [] peak wee e fo nomaliaion) Fige 5. ale of F v. D (Bokş [] n peak wee e fo nomaliaion) Fige 5. Dnami pee iibion fo in 4 f, P pi,.539 an NEBW Fige 5.3 Dnami pee iibion fo in 4 f, P 4 pi,.375 an NEBW Fige 5.4 Dnami pee iibion fo in 7 f, P pi,.94 an NEBW Fige. Nomalie peak pee v. p in / fom pogam BOZKUŞ- an fom he expeimenal aa (Bokş []) Fige. aiaion of hol p oeffiien,, a a fnion of /.. 55 p Fige.3 Two ooinae em ombine ogehe fo he geneal obliqe an mmei one eqaion.. 56 Fige.4 Conol volme a he elbow wih he omponen of he foe an he veloiie hown Fige B. Nomalie peak pee a he elbow fo Fige B. Nomalie peak pee a he elbow fo Fige B.3 Nomalie peak pee a he elbow fo Fige B.4 Nomalie peak pee a he elbow fo Fige B.5 Nomalie peak pee a he elbow fo max max max max max xix

20 Fige B.6 Nomalie peak pee a he elbow fo Fige B.7 Coelaion fnion fo max v. p / in... max xx

21 IST OF SYMBOS egmen, a : wave pee, a : ooinae axi e a he anfomaion paamee, : o-eional aea of he pipeline, elbow an he veial exenion, : aea of he pa of he pipe o-eion, 3, 4 : oe fae aea of he infinieimal volme elemen pepenila o pipe axi an he a he pipe wall; epeivel, : lowe bona of he line inegal,,, B : poin on he ooinae em fo he alibaion fnion,, : iffeenial expeion, B : iffeenial expeion, C : wave pee, C : onol volme, CS : onol fae, C, C : paamee, : meh leing aio e fo he line meh, D : pipe iamee, D : ipeion iane, E : inegal expeion, E, E, E 3 : paamee, f : inegan fnion, f f : fiion fao, f fw : fiion fao fo wae, xxi

22 F : fiion foe pe ni lengh of he pipe aming no evee flow, F : foe veo aing on he onol volme, F m : peie foe a he elbow wih he nmeial moel, F p : expeimenall obaine peak foe a he elbow, F ve : veoial expeion, F x : foe aing on he volmei elemen in x -ieion, F x, F : x an omponen of he oal anien foe aing on he elbow an he veial exenion egmen, epeivel, F : foe aing on he volmei elemen in -ieion, F : nomalie foe, g : gaviaional aeleaion, g x, g, an ieion, g, ieion, elemen, g : omponen of gaviaional aeleaion epeivel in x, g : omponen of gaviaional aeleaion epeivel in an g : fnional paamee a inegan, g i, g i : paamee having iee vale a poin i, h, h, h 3 : paamee fo he geome of he veloi pofile fnion, h : an inegan fnion fo he line inegal, h, h, h 3 : vale of he fnion h ove he noe of he -D Ga i, j, k : inemenal gi loaion in, an ieion, epeivel, ip : inie fo he noe nmbe, I : aea inegal expeion, I m, n ; : expeion fo he aea inegal ove he meh elemen in he ila omain, wih he meh elemen ene poin a noe I m, n m, n ; ; : expeion fo he aea inegal ove he enal meh elemen in he ila omain, wih he meh elemen ene poin a noe m, n ; xxii

23 I : line inegal expeion, I : inegal expeion, I, q : expeion fo he line inegal ove he line meh elemen wih he meh elemen ene poin loae a noe q, I, I, I : inegal expeion, K m : mino lo oeffiien a he elbow, K : ppe bona of he line inegal, l : mixing lengh, l m : mixing lengh, : lg lengh emaining wihin he hoional pa of he pipeline, : aling fao fo lengh, ave : aveage lengh of he lg ha o ing he moion of he lg along he hoional pa of he pipeline, elb : enal a lengh of he elbow, ex : lengh of he veial exenion egmen afe he elbow, fin : final lg lengh of he hoional pipe allaion, final : final lengh of he lg emaining in he hoional pa of he pipeline obaine fom hoional pipe allaion, in : iniial lg lengh, p : lengh of he hoional pa of he pipeline, : inananeo lg lengh a an ime ing i avel along he hoional pa of he pipeline, imp : lg lengh a he inan ha he lg fon ene he elbow, : avel iane of he lg, o : oal axial lengh of he elbow an he veial exenion egmen, m : inie fo he aial loaion of he meh elemen in he ila omain, M an N : nmbe of meh elemen in he aial an he imfeenial ieion of he pipe o-eion, epeivel, xxiii

24 allaion, MPSS : nmbe of noe fo he line meh e fo he impa pee M : oal nmbe of lee line meh elemen in ae he omain of line inegal i he whole iamee of he o-eion of he elbow, in ieion, m : aio of he imfeenial meh ie o he aial one, a an oeion of he pipe, m, of maximm, epeivel, omain, m : lope of he alibaion fnion,, befoe an afe i poin n : inie fo he imfeenial loaion of he meh elemen ila NBOUND : poin on he -ve a he bona of he elbow an he veial exenion egmen, NEBW : nmbe of noe along he axi of he elbow, NFF : noal nmbe oeponing o he loaion of he lg fon fae along he -ve, N lg : nmbe of noe elee ove he lg lengh, imp. NSTOT : oal nmbe of noal poin along he -ve, N an k : nmbe of ank pee aa poin, n : oe ni nomal veo, O : ene of vae of he elbow, O: oigin of he loal ooinae em, O : oigin of he ppe loal ooinae em, O : oigin of he ooinae em fo he alibaion fnion,, p : enemble aveage loal pee, P : aveage pee ove he pipe o-eion, P al : allae aveage pee vale a he elbow o-eion, P D : iving ai pee aing on he lg peam fae, P D final : final iving ai pee vale a he peam fae of he lg fom hoional pipe allaion a he inan ha he lg fon fae eahe he enane eion of he elbow, xxiv

25 P E : aveage liqi pee eaing he moion of ha pa of he lg in he hoional pa of he pipeline, an aing a he enane eion of he elbow, P E al : allae aveage liqi pee eaing he moion of ha pa of he lg in he hoional pa of he pipeline, an aing a he enane eion of he elbow, P : iniial ank pee a he peie ank, P am : amophei abole pee, P : aveage pee aing a he owneam fae of he onol volme, P NFF : amophei pee a he fon fae of he liqi lg, P NBOUND : aveage pee a he onneion bona of he elbow an he veial exenion egmen, po : loal pee a he ene poin O of he elbow o-eion, P p, P m an P n : aveage pee a he ene poin of epeivel he pevio, en an nex volme elemen on he -ve, p peak : peak pee fom pogam KYHN a he loaion of ane #. P T : ank pee, P : final ank pee vale fom hoional pipe allaion a he T final inan ha he lg fon fae eahe he enane eion of he elbow, P : aveage pee aing a he peam fae of he onol volme, P w : iving ai abole pee aing a he peam fae of he lg, P we : wee peimee, PO : amophei pee, PO gage : amophei gage pee, q : an noe nmbe of he line meh, alo oiniing wih he ene poin of a -D Ga elemen, Q : ihage, QK : meh ie of he -D lee line meh, xxv

26 : aial axi of he linial ooinae em, : inenal ai of he pipe, : ai of he boom ile of he lowe one, : ai of he boom ile of he ppe one, e : enol nmbe, HS : igh-han ie of eqaion, g : ga onan, : ai of vae of he elbow, m : given aial ooinae in he o-eion of he elbow, wih epe o he linial ooinae em, m : iane of he poin of maximm veloi fom he oigin of he loal ooinae em, meae along he axi, q : aial loaion of he ene poin of he -D Ga elemen ove he lee meh, wih epe o he linial ooinae em, : vilinea ooinae, S : fae aea of he onol volme, S : owneam pipe o-eional aea fo he onol volme,, : loaion of epeivel he owneam an he peam fae of he en volme elemen a noe ip, on he -ve, egmen, elb : loaion on -ve fo he exi eion of he elbow, ex : loaion on -ve fo he exi eion of he veial exenion S m : aea of he mi-plane of he volme elemen pepenila o pipe axi, max : loaion on -ve whee he maximm vale of o, p, m, n : loaion of he ene poin of epeivel he pevio, en an nex volme elemen on he -ve, S : peam pipe o-eional aea fo he onol volme, n fol : lg fon fae loaion on he -ve a an ime ep n,, : aeian axe a paamee fo ooinae anfomaion, xxvi

27 : ime, T : empeae, : aling fao fo ime, in : iniial ime of lg moion, fin : final ime of lg moion, n : ime vale a an ime ep n, U : aveage lg veloi, U : aling fao fo veloi, U ave : aveage veloi of he lg aken ove he o-eion of he elbow, U : veloi of he bona of he onol fae, b U B : veloi of he bak fae of he lg, U final : final lg veloi obaine fom hoional pipe allaion, U : veloi of he fon fae of he lg, F U in : iniial lg veloi, U ix : fli veloi elaive o he global ooinae axi, on he fae i fo he onol volme, an in he x -ieion, U m : maximm veloi on he ame veloi pofile fnion, U pa : lg paage veloi hogh he elbow, U : liqi veloi elaive o he onol fae bona, U i : fli veloi elaive o he onol fae i, : veloi pofile fnion efine along he -ve,, : axial veloi pofile fnion a he peam an he owneam fae of he volme elemen, ve, : aveage vale of he ame axial veloi pofile fnion along he - U x : liqi veloi elaive o he fixe efeene fame, : axial veloi pofile fnion in he elbow, in ieion, : aveage vale of he veloi pofile fnion in he elbow, in xxvii

28 ieion,, : ame an alibae axial veloi pofile fnion evalae a epeivel he peam an owneam fae of he onol volme, ieion, : veloi wihin he ai volme, in he x -ieion, axial : axial veloi, : veloi veo a an poin of he flow,, : enemble aveage loal veloi omponen epeivel in an, : blene flaion of he veloi omponen epeivel in an ieion, : loal aial veloi a he owneam fae of he onol volme, : loal aial veloi a he peam fae of he onol volme, x, ieion, x,,, in x, an ieion, volme, : enemble aveage loal veloiie epeivel in x, an : blene flaion of he veloi omponen epeivel x : loal veloi in x -ieion a he owneam fae of he onol x : loal veloi in x -ieion a he peam fae of he onol volme, : aveage veloi in ieion,, : loal veloiie in -ieion a epeivel he peam an he owneam fae of he onol volme, : loal axial veloi a he owneam fae of he onol volme, : loal axial veloi a he peam fae of he onol volme, : aveage axial veloi in -ieion, w, w, w 3 : weighing oeffiien, x, : aeian ooinae axe, x,, : ooinae axe fo he loal ooinae em, xxviii

29 x,, : ooinae axe fo he ppe loal ooinae em, X in : iniial lg poiion, x,, : ooinae fo he apex poin of he geneal one eqaion, x,, : ooinae fo he apex poin of he ppe one wih epe o he ppe loal ooinae em, w : iane fom he pipe wall,, : paamee fo he geome of he veloi pofile fnion, : axi of he linial pola ooinae axe, n : given ooinae in he o-eion of he elbow, wih epe o he linial ooinae em. Geek mbol: : hol p oeffiien, : angla axi, : efeene angle fo he owneam fae of he onol volme, : efeene angle fo he peam fae of he onol volme, p : imfeenial ooinae fo he given o-eion of he elbow, wih epe o he linial ooinae em,, F : hoional an he veial omponen of he eaion foe F x applie b he pipe on he onol volme, epeivel, : whole hange in lg lengh ing he avel of he lg fom he peie ank p o he elbow, P WH : pee ie e o en valve loe, q : lengh of he lee line meh elemen a noe q, : axial lengh of he volme elemen,,, : ime inemen fo he hoional pipe allaion, pa : lg paage ime hogh he elbow, : ime inemen a an ime ep of he elbow an he veial exenion xxix

30 egmen allaion, : fi ime inemen vale e fo he elbow an he veial exenion egmen allaion, elbow, n : ime inemen a an ime ep, n, U : hange in lg veloi, W : weigh of he onol volme, : inemenal angle of vae ove he infinieimal elemen in he : inemenal angle of he meh in ieion (angla ieion), : inemenal lengh of he meh in ieion (aial ieion), : volme of he elee pipe elemen, : ivegene opeao, : oghne heigh of he pipe wall, : blen e vioi of he liqi, : aial ooinae axi ove he pipe o-eion, m : aial iane of he enal noe of he 9-poin meh elemen fom he ene-poin of he omain of he ila o-eion, : imfeenial ooinae axi ove he pipe o-eion, n : angla iane beween he line onneing he enal noe of he 9- poin meh elemen o he oigin of he ila omain, an he x axi, : nami vioi of wae, : pi nmbe, : kinemai vioi of he liqi, : eni of he liqi (wae), a : eni of ai, w : eni of wae, : imfeenial axi of he linial pola ooinae em, : alibaion fnion fo he angle a a elee poin of he veloi pofile fnion, xxx

31 en : vale a he enane eion of he elbow, ex : vale a he exi eion of he veial exenion egmen, max : maximm vale of along he -ve, f : angle fo he gaien of he veloi pofile a he pipe wall, : hea e vale a w iane awa fom he ila pipe wall, : wall hea e, : ivegene opeao, : volme of he onol volme, : volme of he volmei elemen, : volme of he boom one, : volme of he obliqe ppe one, : mbol iniaing enemble aveaging. Sbip: ip : vaiable a hi noe : loaion of he lef bona of he ga volme, : loaion of he igh bona of he ga volme, S : loaion of he igh a he mi-ime ep. Speip: n : ime level, : imenionle paamee. xxxi

32 CHPTE I INTODUCTION. aionale fo he Peen S S of he impa pee applie b high pee ai o eam iven liqi poke, alle lg, fome in voie line i an impoan phenomenon a fa a he eling poenial amage on he pipeline i onene. Thee ae ome iffeen mehanim ha ae fomaion of pee moion of liqi lg in nlea powe plan. One of hee mehanim i he eaion of a onenae in eam line of a powe plan ing i opeaion, an following moion of hi onenae in he fom of a liqi lg along a voie line (Kim [7]; Meilo an e al. [8]; Bokş an Wigge []). Inaeqae aining i one of he fao ha iel he fomaion of onenae in a eam line whih afewa ne he phing effe of high pee eam ma aeleae along he voie line, an n ino a lg a a whole liqi poke filling all he pipe o-eion ing i moion. Then, he liqi lg whih aain ve high pee along he pipeline ma hi elbow, paiall o fll open valve o jnion appling ve high impa pee on hee pa of he pipeline. nohe ae of lg eaion in voie line i he phing effe of high pee eam on wae, ollee in loop eal pa of he pipeline in powe plan (Bokş []; Wheele an Siegel [9]; Bokş an Wigge []). loop eal i ha pa of a peie wae eao whih i loae peam he peie afe an elief valve an i i e fo pevening exeive ga leakage hogh hee valve. When he peam pee in he eao oolan em exee a peviol efine limiing vale, he afe o elief valve enl open o elieve

33 he exe pee in he em. oon a he valve open, he high pee eam a o ph he wae in he loop eal aing he fomaion of a liqi lg whih aain ve high veloiie in he pipeline an applie high nami pee on h pipe omponen a elbow, ee an valve a a oe of poenial amage. The paamee o eibe he hali popeie of he lg moion an he following impa even oni of iniial ma an lengh of he lg, magnie of he iving pee aing on he peam fae of he lg, pipe maeial an pipe geome. Thee hali paamee ae all effeive in he vale ha he magnie of he nami pee on he pipe omponen will eah. Sine he magnie of he nami loa aing on he pipeline an i ppo i a meae of he poible amage ha he pipeline em will expeiene, a honami anali i eenial o pei an miigae an poenial ik of amage o he pipeline em. Thi fiel of honami ha anale he foe applie b popelle liqi lg in voie line i aall a bje on whih a limie nmbe of ie wee have been pefome o fa an hi make he bje an ie ha eqie moe aenion. lo onieing ha pipeline em an pipe ppo an eain onie qie a lage amon of he oal invemen o be mae fo nlea powe plan, peiion of he foe aing on hee ppo an eain e beome an impoan bje of eeah (Bokş []; Smih an an aan []).. Speifi Objeive of he Peen S In peviol pefome nmeial ie elae wih eimaion of impa pee an nami loa applie b liqi lg on elbow of iniiall voie line, iffeen -D mahemaial moel wee popoe fo he imlaion of lg moion along aigh pipeline an following impa even a elbow loae a he en of he pipeline. lhogh hee -D mahemaial moel wee aeqae enogh o eibe he aal flow oniion fo he lg moion along aigh pipeline, he have no inle a mh eaile moeling of elbow pa allaion p o now. all, iffeing fom ae fo he moion of a liqi lg along a aigh pa of a pipeline, he ammei axial veloi pofile of a lg in an elbow pa eqie peial aenion o onie ha ioe pofile in a nmeial moel.

34 The aim of he peen i o pei nmeiall he anien foe aing on a 9 egee elbow wih i axi exening in he veial plane, an on a following veial exenion egmen, beqen o a liqi ma impa on hoe pa. In he, boh of he elbow an he following veial exenion egmen wee aken o be loae a he en of a aigh an iniiall voie pipeline exening in he hoional plane. Fo he pa of he allaion ha aon fo he moion of he liqi lg in he hoional pipeline, a pevio -D pefome b Bokş an Wigge [, ] ha onain ga an lg nami eqaion wee ilie. n fo he elbow an he following veial exenion egmen allaion, a new -D anali wa pefome wih he inenion of making a moe eaile onieaion of he flow in hoe pa, an b aking ino aon he vaiaion of 3-D hape of he ammei axial veloi pofile along he elbow an he veial exenion egmen..3 Sope of he Peen S In he peen, he anien foe aing on a 9 elbow an he following veial exenion egmen wa allae b making a -D nmeial anali. Fo he anali, a alibaion ehniqe wa aope o eemine he nknown paamee fo he hange in he hape of an ame 3-D axial veloi pofile along he elbow an he veial exenion egmen. Fo he alibaion ppoe, ome expeimenal aa of peak pee obaine peviol b Bokş [] wee ilie a efeene vale. B ing he alibaion fnion obaine fom oelae aa, vaio ompe n wee mae wih iffeen iniial lg lengh an iniial ank pee o ge a e of plo fo he peak pee ime hioie a he elbow an he ime hioie fo he anien foe aing on he elbow an he veial exenion egmen. The obaine el fo he pee peak, an impa pee an anien foe ime hio plo wee alo ompae wih hoe fom peviol mae ie. 3

35 CHPTE II ITETUE EIEW Thee ae vaio ie pefome b iffeen eeahe, elae wih he peiion of foe ha aie e o impa of a liqi lg on an elbow of an iniiall voie pipeline, b he nmbe of hee wok i no qie high. Thoe of hem whih ae mo elevan o he peen wee aie o b Woo an Papaaki [3], Fenon [], Bokş [], Nemann an Giffih [4], Baan [7]. Woo an Papaaki [3] evelope a mahemaial moel fo he eeminaion of honami foe a pipe ben ha aie while an iniiall voie pipeline em wa being fille. In ha, a ombinaion of igi olmn heo an meho of haaeii wa ilie fo he mahemaial moeling. a el of he anali, ihage, an foe a he pipe an pipe ben allae wih ha moel wee eeme o be malle han hoe in a pevio mae moel b Papaaki an Hollinghea [37]. The eaon fo hi eion wa aibe o he fa ha in he peviol mae moel, mino hea loe a pipe jnion an pipe ben ha been neglee; an heefoe, i wa iniae ha he inlion of mino loe beame impoan if he veloiie in he em wee high [, 3]. Fenon [] pefome boh an analial an an expeimenal wih he ppoe of peiing he foe a a pipe ben, ine b he impa of a liqi lg whih wa iven b ai along a pipeline. The aim of he analial wa o evelop a imple meho fo he peiion of loa aing on he pipe ppo. n, an expeimenal wok wa pefome o hek he valii of he analial meho evelope in he ame. Fo hi ppoe, he e an expeimenal appaa a hown in Fige.. 4

36 Fige. Fenon expeimenal ep []. The expeimenal appaa wa ompoe of an ai peie veel o ppl high pee ai o he em an a -inh-iamee eel pipe wa onnee o hi veel. Thee wa ball valve a he enane of he eel pipe o onol ai flow o he em, an an oifie an a pee ane wee alo aahe o he eel pipe. he owneam ie, a -inh-iamee plexigla pipe whih wa inline pwa wa onnee o he eel pipe. The onneion hee wa mae b mean of a Tee eion whih wa alo ilie fo he ppoe of filling an aining wae o he lowe pa of he plexigla pipe. Thee wa an elbow loae a he owneam en of he plexigla pipe, whih wa open o he amophee. Fenon aie o hi expeimen wih iffeen iniial lg lengh, iniial ank pee, an alo b vaing he iane beween he lg fon fae an he elbow a he owneam ie. Fenon analial moel inle ome ampion, he mo eiive of whih wa ha he lg a a oheen ma. De o hi oheen ma ampion, 5

37 amon of liqi ha wa lef behin ing he lg moion wa neglee. ohe ampion, he ga fom he pee veel, iving he liqi lg wa aken o be ompee iohemall an a ieal, an he lg flow wa oniee a inompeible. Ding he flow of ga an he liqi lg, he hea eiane aing on he flow a he pipe wall wa aken ino onieaion. Fenon analial oveeimae he impa loa a he elbow epeiall when he nomalie lg avel iane wa eqal o o geae han 7 [, ]. Bokş [] pefome an expeimenal ogehe wih an analial moeling o inveigae he honami of a anien liqi ma iven b high pee ai in a voie line. Fo hi ppoe, he e an expeimenal ep whih onie of a peie an a hoional pipeline onnee o hi peie, an a he owneam en of he pipeline hee wa a 9 elbow aahe ogehe wih a veial exenion egmen, a hown in Fige.. Thee wa alo a lg geneao Pee Gage Peie i Tank -inh inie iamee PC 3 f. (9.45 m.)-long Ball alve Clea PC Pipe Elbow eleomee SGP Tane # Tane # 5 f. f. Fige. Sem ep e in Bokş []. pipe (SGP) loae beween he peie an a PC ball valve. B eleing a SGP of an eie lengh an filling i wih wae, i wa poible o geneae lg wih iffeen iniial lengh. Wih hi em ep, liqi lg wih vaio iniial 6

38 lengh an ne he popelling effe of iffeen iniial ank pee wee iven along he hoional an iniiall emp pipeline an he pee ime hioie ha fom e o impa of he liqi lg on he elbow wee eoe b he ane mone on he elbow a hown in Fige.. a el of he anali i wa onle ha long lg lo moe amon of hei iniial mae han i he ho lg ing hei moion along he hoional pipeline. In aiion, fo long an meim lg, wo iffeen peak pee wee obeve in he pee eoing wih epe o ime a he elbow. The eaon fo he oene of hi oble peak phenomenon wa aibe o he iinegaion of he lg ma ino wo piee following he en opening of he ball valve peam. nohe onlion awn in hi wa ha moe ai enainmen effe wa obeve in he ae of ho lg han ompae o long lg an hi iffeene beween he wo ae wa explaine wih he eaon ha ho lg ha a longe nomalie avel iane. I wa alo iniae in ha ha lage amon of ai enainmen wa a fao ha ae ignifian eion in he eniie of epeiall ho liqi lg. nalial of Bokş [] onie of wo iffeen mahemaial moel. The fi one of hee moel, alle Simple Moel [], inle wo iffeen onol volme anale; one fo he eeminaion of peak pee a he elbow an anohe fo obaining pee ime hio plo fo he vaiaion of impa pee a he elbow. The eon mahemaial moel wa alle he vane Moel [] an in hi moel, in aiion o he lg nami eqaion ha wa e in he Simple Moel, anohe e of eqaion ha alo ake ino aon he ga nami effe wee inle a a fhe impovemen. The peak pee a he elbow wee alo allae wih hi avane moel. Boh of he analial moel gave he peak pee a he elbow ine b he ho lg a being highe han he expeimenall obaine one. Thi oveeimaion of he analial moel wa iniae o be e o he fa ha he ai enainmen effe wa neglee in hee moel, whih wa a phenomenon ha ae he eni of he liqi lg o ee an h, he expeimenall obaine impa pee o beome malle a ompae o hoe fom he analial moel. The analiall obaine peak pee fo long an meim lg wee ae o mah loel wih he expeimenal el. eaile explanaion fo he vane Moel of Bokş 7

39 [] i given in Chape III ogehe wih he analial meho followe in wiing he ompe oe wih boh he Simple Moel an he vane Moel. Bokş, e. al [] pblihe he fining of he b ompaing hem o hoe fom Fenon an Giffih [6] an i wa ae ha he nomalie foe in Bokş expeimenal wee neal he ame a hoe given b Fenon an Giffih exep ha he el fom Bokş howe a vaiaion of aa ove a elaivel lage ange of nomalie foe vale. Thi iepan beween he wo el wa iniae o be pobabl e o he iffeene in he expeimenal ep e. Fo example, i wa poine o ha he pipe iamee in Bokş wa wie a lage a he one in he ae of Fenon an Giffih. In aiion, i wa expee ha he pipe in Bokş wa hoional ahe han he pipe inline pwa pipe owa owneam ha wa e in he of Fenon an Giffih. Fhemoe, i wa ae ha in Bokş, a ball valve wa e o onol he moion of he lg; howeve, Fenon an Giffih ilie a pe ik fo he ame ppoe. In onlion, Bokş an Wigge [] ggee ha hee iffeene beween he expeimenal ep wee likel o be he eaon fo highe ae of aa in Bokş a ompae o he vale obaine b Fenon an Giffih. Nemann an Giffih [4] ie how he foe aing on a 9 ben ae affee if he ben i loae owneam of a pipe expanion. a el of an anali b oning a e of expeimen i wa onle ha he peak foe a a ben loae afe an expanion wee ve loe o hoe ha oe in ae of a pipe wih a onan iamee. nohe onlion eahe in ha wa ha fo boh pe of he pipe oniee, he lg aveling iane eqie fo he foe a he pipe ben o ee o ve mall vale wee neal he ame. Baan alo [7] aie o an expeimenal an nmeial o inveigae he honami of an inivial anien liqi ma in voie line. Baan expeimenal ep onie of an ai ank an a pipeline whih wa inline pwa owa owneam, an hi pipeline wa onnee o an ai ank a i peam en hogh a ball valve a hown in Fige.3. Thee wa a 8

40 Tane Tane m Fill alve I TNK B E alve Dain alve mplifie Compe an D Sem COMPESSO Fige.3 Expeimenal ep e b Baan [7]. hap n elbow onnee o he owneam en of he inline pipeline, whih wa exening ownwa in he veial plane. ane wa loae on he elbow o be e wih he ppoe of meaing lg impa pee a ha poin of he elbow. I wa poible wih hi em o eae lg wih eie iniial lengh b filling wae fom he fill valve ino he pipeline, an hen ive hee lg ne he aion of high pee ai fom he ai ank along he inline pipeline pon en opening of he ball valve. Wih hi expeimenal ep, Baan one a e of e b popelling lg wih iffeen iniial lengh an ank pee an b meaing he impa pee a he elbow. The expeimenal el fom hee e ae wee ompae wih hoe fom Bokş expeimenal [] an i wa onle ha he el fom boh ie wee in aoane wih eah ohe alhogh hee wee ome iffeene beween he expeimenal ep e in hee wo ie a fa a he ie of he pipe iamee an he pipe inlinaion elee wee onene. Baan nmeial anali wa ompoe of he olion of Ele Eqaion wih Gonov Meho fo he ompeible ga pa in he em hown in Fige.3 an of he allaion of impa pee a he elbow b iliing a meho popoe b Daghe an e. al. [3]. In he mahemaial moeling, he ma lo 9

41 fom he lg bo ing he moion of he lg wa neglee in Baan. Fo he impa pee allaion a ggee b Daghe an e. al., a fiiio ge ank wa ame o exi a he elbow. Th, he impa pee wee allowe o ake vale omewhee beween hoe of en vale loe ae, an of a agnaion pee eimae ha wol have ompe wih he e eneg eqaion fo an elbow open o he amophei pee a one en. The impa pee a he elbow ompe wih hi meho neeimae he peak pee a he elbow wih epe o hoe fom he expeimenal fining obaine in he ame. a el of hi nmeial anali, i wa onle ha he moeling of he elbow o a o give el beween he ae of exiene of a ge ank an of a en valve loe wol have povie he impovemen of he el fo he peak pee obaine a he elbow. Baan el [7] fo he vaiaion of nomalie peak foe wih epe o nomalie ipeion iane wa ompae wih he vale in Bokş [] an Fenon [] ie in Bokş, e. al [8]. Fom hi ompaion, i wa onle ha Baan el omplie wih hoe of Bokş an Fenon alhogh he nomalie foe vale fom Baan ha ome enen o emain on he onevaive ie a Fenon aa. Making a ommen on he obaine el fom he hee ie pefome b Fenon [], Bokş [] an Baan [7], i wa ae ha hee wa a hap op in he nomalie foe when he nomalie lg avel iane wih epe o iniial lg lengh wa geae han 6. nohe onlion awn in hi wa ha he lane onfigaion of he lg fon hape wih epe o he inline pipeline ha an ineae effe in he oene of Talo- Inabili an heeb eling in moe ai enainmen ino he lg bo o ake plae epeiall fo ho lg ing hei moion.

42 CHPTE III MTHEMTIC DEEOPMENT FO THE HOIZONT PT 3. Inoion In he peen, he fomlaion fo he allaion of impa pee a he elbow of an iniiall voie hoional pipeline applie b an ai iven liqi lg ae mae in wo age. fi age, he allaion eqie fo he anali of he moion of he liqi lg along he hoional pa of he pipeline i mae b ing a peviol evelope fomlaion b Bokş []. Then, in he eon age, final vale fo he honami paamee obaine fom he fi pa ae e a iniial oniion fo he anali of he moion of he lg in he elbow an he following veial exenion egmen of he pipeline a he main ppoe of he hei In hi hape, a nmeial anali fo he honami of he moion of an ai iven liqi lg along an iniiall voie hoional pipeline, pefome peviol b Bokş [], i peene. The em ep e b Bokş fo hi ppoe i a hown in Fige. an he eipion of hi em wa given in Chape II. The mahemaial fomlaion given in hi hape whih wa name a vane Meho [] in Bokş ove he eipion of he aeleae moion of an inivial liqi lg fom eo iniial veloi pon a en valve opening, ne he aion of he iving effe of high pee ai aing a he peam ie of he lg. The ma lo fom he liqi lg e o i ineaion wih he pipe wall ing he lg moion along he hoional pipeline i alo oniee wih he e of a alibaion paamee. The flow of he high pee ga peam he liqi lg i anale b ing -D ga nami eqaion an h, he waehamme effe wihin he egion of lg iving ga volme i aken ino

43 aon. The olion of ga nami eqaion ae mae wih meho of haaeii an he mahemaial expeion obaine fom hi pa ae ople wih a e of lg nami eqaion obaine fom a onol volme anali, o loe he em. Ding he olion, he peam pee bona oniion fo he em i aken a he ank pee a he pipe inle an he owneam bona oniion i he eo amophei pee a he lg fon fae. The allaion ae pefome nil he ompaional poee el in he ae ha he lg fon fae of he liqi lg eahe he elbow a he owneam en of he hoional pipeline. The vale fo he lg iving pee an he lg veloi oeponing o he aival ime of he lg fon fae a he enane eion of he elbow wee hen, aken o be ilie a iniial oniion fo a eon age of allaion ha wee mae fo he anali of he lg moion wihin he elbow an he veial exenion egmen of he ame em hown in Fige.. In he following beion, he eivaion of lg nami eqaion wih a onol volme anali, an he fomlaion fo he flow of he ga iving he liqi lg b ing -D ga nami eqaion an meho of haaeii ae peene. n algoihm ha explain he e of he eling eqaion fom he above poee i alo given. In Seion 3.5 of hi hape, a new fomlaion fo he allaion of a fnional hol p oeffiien ha wa evelope in he peen a an impovemen of Bokş [] vane Meho i alo peene. 3. Slg Dnami Eqaion To anale he moion of a liqi lg popelle b peam high pee ai along a hoional pipeline, Bokş [] mae a onol volme anali b eleing a onol volme a hown in Fige 3.. Then, he applie he onevaion of ma an onevaion of momenm piniple ove hi onol volme. Fo he anali of he lg moion, a e of implifing ampion wee mae. Thee ampion ae ha: a) The lg flow i aepe a one-imenional, inompeible an plana. b) iqi eni i onan. ) Pipe maeial i igi an he pipe oe no vibae o move in an ieion ing he lg moion.

44 C 3 x n P U n P am= U U3 U = U3x = U F x U B F U x = Fige 3. Conol volme fo he liqi lg elee b Bokş [] along he hoional pipeline. ) Ga (ai) oe no enain ino he liqi lg (wae), in ohe wo, one phae ampion i vali ing he enie moion of he lg. e) qai-ea flow ampion i mae o fin he hea eiane o he lg flow. f) Some of he lg ma i lo ing he moion. The ma lo fom he liqi lg ing i moion e o ineaion of he lg wih he pipe wall b mean of heaing effe wa aken ino aon b ing a hol p oeffiien,. n vale fo he hol p oeffiien mean ha he lg loe i ma hogh a peenage of - of pipe o-eional aea, ing i moion. Fo example, if vale i eqal o.99, i mean ha lg loe i ma a a ae of % of he oal pipe o-eional aea. Th, ing he moion of he liqi lg b a iane of fom he lg eqal o he vale of x mbol ae a, : eni of wae x along he hoional pipeline, an amon of ma lo o. Hee, he meaning of he an : pipe o-eional aea. The hol p oeffiien vale,, an onl ake vale beween an. The 3

45 vale being eqal o iniae ha hee i no hol p o ma lo fom he lg, while eeaing vale of iniae ha moe amon of ma lo fom he lg i aking plae. In Bokş [], ing he moion of a liqi lg along he hoional pipeline onl ingle, elee fixe vale of hol p oeffiien wee e in he allaion. Howeve, in he peen, a fnion wa evelope whih aon fo he vaiaion of he vale of he hol p oeffiien,, along he hoional pipeline ing he lg moion, in em of he nomalie avel iane of he liqi lg wih epe o inananeo lg lengh. The mahemaial poee fo he evelopmen of hi fnion i given in Seion 3.5 of hi hape. Wih he above iniae implifing ampion, he appliaion of onevaion of ma an onevaion of momenm eqaion ae given in he following beion. 3.. Conevaion of Ma Eqaion Fo he anali of he lg moion, fil he onevaion of ma eqaion wa wien in i inegal fom fo a onol volme whih i moving ogehe wih he liqi lg along he hoional pipeline, a hown in Fige 3.: C.. C.S U n S U n S b C.S. (3.) In hi eqaion U b an U an fo he veloi of he bona of he onol fae, an he liqi veloi elaive o he onol fae bona, epeivel. lo, S i he fae aea an i he volme of he onol volme elee. Caneling fi an eon em of Eqaion (3.) an hen, appling emaining pa of Eqaion (3.) an he elaion (3.) (3.3) 4

46 an 3 (3.4) U U B U F (3.5) U U F (3.6) U 3 (3.7) o he onol volme in Fige 3.; Eqaion (3.8) below i obaine (Bokş []). U (3.8) F U B 3., In above eqaion, : o-eional aea of he pipe,,, 3 : aea of he pa of he pipe o-eion a hown in Fige U i : fli veloi elaive o he onol fae i, U B : veloi of he bak fae of he lg, U : veloi of he fon fae of he lg, F n : oe ni nomal veo. Now, apa fom he above onevaion of ma piniple, a iffeenial eqaion whih elae he hol p oeffiien,, o he lg lengh,, i ilie a given below. U F U B (3.9) Solving fo expeion i obaine: U B fom (3.9) an biing ino (3.8), he following ( ) U F (3.) 5

47 The eling eqaion given in (3.) i aall a fom of onini eqaion an given a vale of hol p oeffiien α, hi eqaion an be e a a elaion beween he lg lengh an he lg fon veloi U F. fom 3.. Conevaion of Momenm Eqaion The geneal fom of he onevaion of momenm eqaion ilie i of he Fx U x U x U C.. n S (3.) C.S. Hee, U x : liqi veloi elaive o he fixe efeene fame, U : liqi veloi elaive o he onol fae bona, F x : ne foe aing on he elee onol volme, in x -ieion. The lef ie of (3.) an be wien a: F x P D (3.) whee, an D : pipe iamee, f 8 f U (3.3) F P : ai gage pee iving he liqi lg aing a he lg peam fae, : pi nmbe, f f : he Moo fiion fao, : wall hea e. In hi fomlaion, he hange in he ieion of fiion foe wih ha of 6

48 fli veloi wa iae wih he ampion of no evee flow. In Bokş [], a fixe aveage vale of Moo fiion fao, f f, wa e in he allaion b in he peen, he fiion fao vale wa allae fom Swamee-Jain fomla (Walki []; Swamee an Jain []) given in (3.4) o impove he meho. f f.35 ln 3.7D e (3.4) In hi expeion, : oghne heigh of he pipe wall, e : eno nmbe. The enol nmbe an be allae fom U F D e (3.5) whee i he nami vioi of wae. Nex, biing, U (3.6) U x U B (3.7) U x (3.8) U 3 x U F (3.9) U B U F (3.) U (3.) U F U 3 (3.) in Eqaion (3.) aoing o he onol volme elee in Fige 3., an alo 7

49 ing Eqaion (3.8) ogehe wih he final fom of he onevaion of ma expeion given b (3.), he following eqaion i obaine (Bokş []): U F f f U D F P (3.3) In above eivaion, U ix an fo he fli veloi elaive o he global ooinae axi, on he fae i fo he onol volme, an in he x -ieion 3..3 Slg Kinemai Eqaion To omplee he lg nami eqaion, in aiion o above onevaion eqaion, whih ae Eqaion (3.) an (3.3), one moe eqaion inepenen of hee an be obaine fom he lg kinemai. Thi la eqaion i impl an expeion fo he ae of hange of he poiion of he fon fae of he lg wih ime, an an be given a: x U (3.4) F In all of he expeion (3.), (3.3) an (3.4), he lg fon veloi an be eplae wih he mean lg veloi U an hen, he final em of iffeenial lg nami eqaion beome U F U f f U D P (3.5) ( ) U (3.6) x U (3.7) 8

50 The aveage pee vale, P, in (3.5), i allae fom he anali of he flow of he high pee ga a he peam ie of he lg, an he fomlaion fo hi pa of he allaion ae given in he following eion whih inle he olion of ga nami eqaion. 3.3 Ga Dnami Eqaion Fo he anali of he flow of he high pee ga peam he liqi lg, he onevaion of ma, momenm an eneg eqaion in iffeenial fom given b Eqaion (3.8), (3.9), (3.3) below wee ilie b Bokş []; P P a C (3.8) x x x a P x F (3.9) a a x C P P x (3.3) whee F f f Pwe (3.3) 8 an C P a g T. (3.3) Hee, he meaning of he mbol ae a follow: a : eni of ai, : veloi wihin he ai volme, in he x -ieion, P : aveage pee a a o-eion of he pipeline, : ime, x : pae ooinae along he axe of he hoional pipe, F : fiion foe pe ni lengh of he pipe aming no evee flow, 9

51 P we : wee peimee of he o-eion, C : wave pee, g : ga onan, T : empeae. he ampion in ing he above ga nami eqaion given b (3.8), (3.9), (3.3); one-imenional, nea, nonnifom an a ompeible ga flow wa oniee. In aiion o hee, he flow wa alo aepe a aiabai, iohemal an blen. The em of hpeboli paial iffeenial eqaion given b (3.8), (3.9), (3.3) wee anfome o a e of oal oina iffeenial eqaion b ing meho haaeii. The anfomaion poee followe fo hi ppoe an be fon in efeene fom Moo [3], Chah [3] o Seee [33]. The eling oina iffeenial eqaion ae given b expeion (3.33) an (3.34). P C F fo x C (3.33) P C F fo x C (3.34) wih F f f CPwe (3.35) 8 Eqaion (3.33) an (3.34) ae vali ove he ompaional omain aing fom he peam peie p o he bak fae of he liqi lg hown in Fige 3.. he lg i moving owneam oninol in an aeleaing paen, he ompaional omain ge lage an lage b ime a hown in hi fige. Eqaion (3.33) hee i a igh avelling haaeii eqaion an i vali along he C line in he ompaional omain, an Eqaion (3.34) i a lef avelling haaeii eqaion whih i vali along he C line.

52 U/S D/S Slg Pah P T C + C C + - S C + x C - C, S -, S, S U Fige 3. Compaional omain e b Bokş [] fo he olion of ga nami eqaion. 3.4 Copling of Ga an Slg Dnami Eqaion To olve he ga nami an lg nami eqaion in a loe fom paen, he fi lg nami eqaion given b (3.5) an be ewien a in he following fom: U f fw U D P w P w am (3.36) The meaning of he mbol hee ae a: f fw : fiion fao fo wae, P w : iving ai abole pee aing a he peam fae of he lg, P am : amophei abole pee, w : eni of wae. Then, b inegaing he lg nami eqaion given in (3.6), (3.7) an (3.36) beween he gi poin along he lg pah ve hown in Fige 3.; an inegaing he ga nami eqaion given b (3.33) an (3.34) along he

53 oeponing haaeii line in he ompaional omain, a e of fomla fo he peam an he owneam bona ompaion of he peie ga egion in he hoional pipeline i obaine. While making hee allaion, he known bona oniion of he em ogehe wih Eqaion (3.3) an he ieal ga eqaion given in (3.37) ae alo e (Bokş []). P (3.37) T g The fomla fo he allaion of peam bona oniion ae a follow a obaine b Bokş []: P P T (3.38) P T (3.39) g C C T (3.4) g x x S S (3.4) ( C ) S S (3.4) P P f a C a S S a D S S C S (3.43) The bip an S of he vaiable in above eqaion iniae he loaion of lef igh bonaie whee he vaiable ae being evalae a hown in Fige 3.. The algoihm fo he allaion of he vale a he peam bona of he peie ga egion in he em i peene below. ) Callae P fom Eqaion (3.38) b ing he bona vale fo he

54 ga pee PT a he peie ank. ) Callae ga eni b ing (3.39). 3) Compe he wave pee C a he peam bona of he peie ga egion fom (3.4). 4) Callae fom Eqaion (3.4) b aking x, an ing he known vale of vaiable wih he bip S ha wee allae a he pevio ime ep (Fige 3.). 5) Compe fom (3.4). 6) Callae he ga veloi a he lef bona,, ing (3.43). The e of eqaion obaine b Bokş [] fo he allaion of owneam bona vale of he ompaional omain of he peie ga volme ae given below. x U ( C ) ( ) x U S S S S (3.44) C U S x ( C ) x x (3.45) C (3.46) x x S S (3.47) U S U S (3.48) C C T (3.49) g 3

55 C f a a C D C a (3.5) C f w U S U S D S (3.5) C Pw Pam C U P P a S (3.5) C w S Pw Pam U w S f w U U S D S S S (3.53) The owneam bona oniion iniae wih he bip, fo he omain of ga volme ae allae wih he following algoihm: ) Callae he poiion of he peam fae of he lg fom (3.44). ) Compe he ime inemen, an, vale fom Eqaion (3.45) an (3.47). 3) Callae vale b ing (3.46). 4) Compe lg lengh,, b ing he expeion given b (3.48). 5) Callae he wave pee, C, fom (3.49). 6) Compe he pee aing a he peam fae of he lg, P, b ing Eqaion (3.5). 7) Callae he veloi of he peam fae of he lg, U, fom Eqaion (3.53). Ding he above allaion, he ime epenen eeae of he ank pee, P T, i alo aken ino aon aoing o he aa given in Fige 3.3. In hi fige; P : iniial ank pee a he peie ank. 4