ACKNOWLEDGEMENTS. My husband, Mr Matthew Pegg, for bearing with my non-stop complaints and long hours of work.

Transcription

1 ABSTRACT This thesis describes the developmet, fudametal extesio ad extesive testig (validatio ad verificatio) of mathematical models for predictig outflow followig the failure of pressurised pipelies cotaiig icompressible liquids. The models, for the first time, accout for all the importat sequetial flow regimes takig place durig the discharge process. These iclude full pipe flow, bubble formatio ad propagatio, followed by ope chael flow. The system cofiguratios modelled iclude a draiig pipelie coected to a storage tak ad pipe with oe closed-ed. I the first part of this thesis, the developmet of outflow models to simulate the fullbore rupture of horizotal pipelies is preseted. I order to model the full pipe flow i a pipe fed from a upstream tak, the published model by Joye & Barrett (003) is employed i this study. Bubble propagatio ad ope chael flow for both cofiguratios (i the presece of upstream tak ad pipe with oe closed-ed) are modelled by assumig critical flow coditio throughout the pipe ad i the tak (where applicable). Bubble propagatio velocity is calculated based o Bejami s (1968) ad Bedikse s (1984) proposed equatios. The secod part of this study focuses o the extesio of the developed models to accout for pipe icliatio agle. Bubble propagatio ad ope chael flow are modelled by replacig the critical flow equatio with Darcy-Weisbach equatio, applicable to dowward-iclied pipes. The bubble propagatio patter i the pipe is determied based o the drift velocity method through the results obtaied from parametric studies. The developed models are validated by comparig the predicted values agaist experimetal measuremets recorded usig laboratory scale setups. Through sesitivity aalysis based o comparig the results of the models to case studies represetative of real evets, the importace of accoutig for post-full pipe flow o the total amout of ivetory discharged is demostrated. Abstract 1

2 ACKNOWLEDGEMENTS I wish to thak the followig people ad orgaisatios who have cotributed so much i may ways to facilitate the completio of this thesis. My husbad, Mr Matthew Pegg, for bearig with my o-stop complaits ad log hours of work. My parets, Mr ad Mrs Rafigh, my sister Shideh ad my brother Shahab for their ucoditioal love ad moral support. A big thak you to Dr Ja Stee from DNV Software for his ivaluable advice ad support throughout my PhD as well as the writig-up. Thak you ever so much. My advisors, Prof. Harou Mahgerefteh from UCL ad Mr Hek Witlox from DNV software, for their support throughout my PhD. Special thaks to Mr Alberto Barral for his help with the experimets. The techical ad admi. staff of the Departmet of Chemical egieerig, UCL.

3 تقدیم به پدر و مادر عسیس تر از جانم 3

4 TABLE OF CONTENTS ABSTRACT... 1 ACKNOWLEDGEMENTS... TABLE OF CONTENTS... 4 CHAPTER 1: INTRODUCTION... 8 CHAPTER : FUNDAMENTAL EQUATIONS AND BACKGROUND THEORY Itroductio Full Pipe Flow Coservatio of Mass (Cotiuity) (Coulso & Richardso, 1999) Coservatio of Eergy (Coulso & Richardso, 1999) Frictioal Loss Ope Chael Flow Coservatio of Mass (Cotiuity) (Sturm, 001) Specific Eergy (Aka, 006) Subcritical, Critical ad Supercritical Flow (Frech, 1994; Aka, 006) Uiform Flow Steep ad Mild/Horizotal Chaels Frictioal loss Coclusio CHAPTER 3: LITERATURE REVIEW Itroductio Full Pipe Flow Bubble Formatio ad Propagatio Bubble Propagatio i Stagat Liquid Bubble Propagatio i Co-curret Liquid Ope Chael Flow Horizotal Chaels Dowward-iclied Chaels Coclusio

5 CHAPTER 4: OUTFLOW SIMULATION UPON FULL-BORE RUPTURE IN HORIZONTAL PIPELINES Itroductio Key Models Assumptios Oe-dimesioal flow aywhere i the pipelie Icompressible flow, i.e. costat fluid desity Costat cross sectioal area of the pipe, much smaller tha tak cross sectioal area (where applicable) No frictio betwee the fluid ad tak Isothermal coditios i the pipe No ilet flow to the feed tak No hammer effect upo valve closure Negligible impact of surface tesio ad viscosity durig bubble formatio ad propagatio regime Trasiet Hydraulic Flow Modellig Followig Full-bore Rupture of a Horizotal Pipelie Fed from a Upstream Tak Itroductio Model Theory Full Pipe Flow Bubble Formatio ad Propagatio Ope Chael Flow Parametric Studies Discharge Velocity ad Wetted Area Normalised Cumulative Discharged Mass Experimets Upstream Bubble Propagatio Velocity u ub ad Liquid Depth d c Normalised Cumulative Discharged Mass ad Discharge Rate Trasiet Hydraulic Flow Modellig Followig Full-bore Rupture of a Isolated Horizotal Pipelie Itroductio Model Theory Bubble Formatio ad Propagatio Ope Chael Flow Parametric Studies

6 Discharge Velocity ad Wetted Area Released Mass durig Idividual Flow Regimes Error Aalysis ad Depedece of Covergece o c Experimets Coclusio CHAPTER 5: IMPACT OF PIPELINE INCLINATION ANGLE ON THE OUTFLOW FROM PIPELINES Itroductio Key Models Assumptios Trasiet Hydraulic Flow Modellig Followig Full-bore Rupture of Dowward-iclied Pipelies Itroductio Model Theory Full Pipe Flow Bubble Formatio & Propagatio Bubble Propagatio from Both Eds Bubble Propagatio Oly from the Upstream Ed of the Pipe Ope Chael Flow Parametric Studies Compariso of the Two Models for Bubble Propagatio Discharge Velocity ad Wetted Area Normalised Cumulative Discharged Mass Variatio of Liquid Depth Agle ( ) with Experimets Upstream Bubble Propagatio Velocity u ub ad Liquid Depth d Normalised Cumulative Discharged Mass ad Discharge Rate Trasiet Hydraulic Flow Modellig Followig Full-bore Rupture of a Isolated Dowward-iclied Pipelie Itroductio Model Theory Bubble Formatio ad Propagatio Ope Chael Flow Parametric Studies Discharge Velocity ad Wetted Area

7 Variatio of Liquid Depth Agle ( ) with Released Mass durig Idividual Regimes Error Aalysis ad Depedece of Covergece o Experimets Coclusio CHAPTER 6: CONCLUSION AND FUTURE WORK Coclusio Suggested Future Work NOMENCLATURE REFERENCES WEB REFERENCES

8 CHAPTER 1: INTRODUCTION Global oil demad is set to grow by 14% by 035 (The Ecoomic Times, 011) due to ecoomic growth ad expadig populatios i the world s developig coutries such as Idia ad Chia. The Iteratioal Eergy Agecy has estimated that such demad for oil will reach 99mbd i 035; 1mbd more tha i 010 (The Ecoomic Times, 011). This growig demad is expected to be met with icreased oil productio, resultig i a sigificat icrease i the use of pressurised pipelies, already by far the most widely used method for trasportig oil ad gas across the globe. Accordig to the research coducted by Lodo-based steel busiess cosultacy CRU, the demad for pipelies was said to go up by 78% i Easter Europe ad by over 100% i the Middle East ad Asia betwee 007 ad 011 (Eergy Global, 010). Give that such pipelies ca be several hudreds of kilometres log coveyig millios of toes of highly pressurised ivetory, their accidetal rupture may lead to catastrophic cosequeces, icludig ijuries, fatalities, sigificat evirometal damage ad fiacial loss. Accordig to data published by the US Departmet of Trasport ( ), short pipelies will have a reportable accidet durig a 0- year lifetime. Operators of log pipelie etworks (1000 km or over) ca expect a reportable accidet at a frequecy of oe per year. There are umerous examples of such icidets. Accordig to the US Departmet of Trasportatio Pipelie ad Hazardous Materials Safety Admiistratio (011), over 5000 hazardous liquid pipelie accidets were reported durig , represetig a total property damage of over US$ billio. The rupture of a Ebridge pipelie ear Cohasset, Miesota, USA i 00 resulted i the release of approximately 6,000 barrels of crude oil, represetig a fiacial loss of approximately US$5.6 millio (Pipelie Accidet Report, 00). The cost of the clea-up operatio was estimated to be sigificatly higher. I 010, a ruptured crude oil pipelie set at least 800,000 gallos of crude oil pourig ito the Kalamazoo Chapter 1: Itroductio 8

9 River, Michiga, USA. The spill is believed to be the largest i the history of the Midwest (Gree Techology, 010). The clea-up cost is estimated to exceed US$650 millio (Klug, 011). I Nigeria, the pucture ad rupture of a crude oil pipelie i Abule Egba ad Lagos i 006 ad 008 respectively resulted i more tha 60 (Iteratioal Busiess Times, 006) ad 100 (The Seattle Times, 008) fatalities. More recetly i Alberta, Caada, approximately 8,000 barrels of crude oil was released as a result of a crude oil pipelie rupture i 011. The accidet is believed to be the biggest spill from a crude oil pipelie i Alberta sice 1975 (CBC News, 011). I may developig coutries it is ow a statutory requiremet to evaluate the risks for all the major safety hazards associated with pressurised pipelies prior to their commissioig. I the Uited Kigdom, the Offshore Istallatios (Safety Case) Regulatios 005 (Health ad Safety Executive, 006) require quatitative assessmet of major accidet risks ad the measures employed to cotrol ad mitigate them. This is to esure that the relevat statutory provisios will be complied with. The above procedure idustry is ormally referred to as Quatitative Risk assessmet, or i short form QRA. By defiitio Risk is the likelihood (frequecy) of a specific cosequece of a specific accidet. The evet frequecy ca ormally be obtaied from the available historical data. O the other had, the cosequece of a specific accidet eeds to be determied before performig QRA. Followig a accidet ivolvig hazardous materials, first the ivetory is released to the atmosphere. This phase is ormally referred to as Discharge. I the absece of a immediate igitio of the released flow, depedig o the fluid phase there will be either Dispersio (cloud formatio) for gas/two phase flow or Pool Formatio for liquid releases. The resultig cosequeces ca the be fire, explosio, toxic release ad evirometal pollutio. The Discharge phase, which ivolves the calculatio of release rate, provides the source coditios for quatifyig all these major cosequeces. Chapter 1: Itroductio 9

10 O the other had, pipelie failure may be i the form of full-bore rupture where the pipe splits ito two, a simple pucture or logitudial tear. Amog these, full-bore rupture is cosidered to be the most catastrophic sceario. Numerous studies with various degrees of sophisticatio have bee coducted to model the trasiet outflow followig the failure of pressurised pipelies (see for example Bedikse et al. (1991), Richardso & Saville (1991, 1996a, 1996b) ad Mahgerefteh et al. (1997, 1999, 000)). Although importat, these studies are cofied to gas or flashig liquid pipelies. O the other had, the available models for failure of pipelies cotaiig liquids igore the draiage oce the pipelie pressure has reached the ambiet pressure. I other words, the pipe is assumed to remai full throughout the discharge process (see for example Loiacoo (1987), Schwarzhoff & Sommerfeld (1988), Sommerfeld & Stallybrass (199), Kossik (000), Joye & Barrett (003)). This regime, which is ormally referred to as full pipe flow, ca oly happe if the feed tak ever drais dry. Therefore, the subsequet flow regimes ivolvig bubble formatio followed by ope chael flow (Wallis et al., 1977) are ot cosidered. As a result, the remaiig mass i the pipe with ambiet pressure is igored. While this might be a valid assumptio for short/small diameter pipes, for log/large diameter pipelies, which are commoly used for trasportig petroleum products, may result i sigificat uderestimatio of released mass. I additio, if the pipe is isolated followig the closure of a ESD valve, it will reach the ambiet pressure istataeously due to the absece of feed tak. As the flow rate decreases over time, air igress will result i the formatio ad propagatio of a bubble i the pipe. Bubble formatio i the pipe is of sigificat iterest sice the formatio of air pockets ca reduce the effective pipe cross sectioal area thus reducig pipe capacity. Also the trasported air will be released at the discharge locatio, which ca raise evirometal cocers due to foamig (Lauchla et al., 005). Chapter 1: Itroductio 10

11 Based o a force balace, Bejami (1968) calculated the liquid velocity required to keep a bubble stagat at the dowstream ed of a horizotal pipe. The author determied that the calculated liquid velocity is the same as the velocity of the propagatig bubble ito a statioary liquid. Bedikse (1984) experimetally predicted the bubble propagatio velocity i o-statioary liquids i small diameter pipes with various icliatios. Iogamov & Opari (003) developed a aalytical model for the bubble propagatio velocity i a statioary liquid i dowwardiclied pipes. Their model for horizotal pipes produced close agreemet with Bejami s (1968) predictios. I all the above studies the pipe had either a closeded (statioary liquids) or the bubble was itroduced from the bottom of the pipe ito the flow (o-statioary liquids). The impact of a upstream storage tak o the flow regime ad bubble propagatio velocity was ot cosidered. Followig further decrease i the flow rate, the bubble will elogate formig a free surface o top of the liquid. This regime is called ope chael flow. Modellig such flows is of particular iterest i civil egieerig projects ivolvig water distributio etworks. The so called Califoria Pipe Method (Water measuremet maual, 001) for example predicts the discharge rate from a horizotal pipe based o the liquid depth at the free fall. The model is however limited to whe the pipe rus less tha half full. Sice the, a umber of authors have studied the relatio betwee the discharge rate ad liquid depth at the pipe exit (see for example Dey (001) ad Sterlig & Kight (001)). This thesis describes the developmet, verificatio ad validatio of aalytical models for simulatig the trasiet discharge rate followig the full-bore rupture of icompressible liquid trasportig pipelies with various icliatios. The models, for the first time, accout for all the importat flow regimes takig place durig discharge, focusig specifically o bubble formatio ad propagatio, ad ope chael flow. Two cofiguratios icludig pipe fed from a upstream storage tak ad with a closed-ed due to ESD valve closure are cosidered. For the pipe fed from a upstream tak the importace of accoutig for post-full pipe flow o the total amout of ivetory discharged is demostrated based o the simulatio data. I additio, the impact of upstream storage tak o the bubble propagatio velocity is Chapter 1: Itroductio 11

12 ivestigated. The developed models for pipes with closed-ed for the first time simulate the trasiet outflow throughout the draiage process. The relatio betwee the liquid depth ad pipelie characteristics durig ope chael flow is also determied for both horizotal ad dowward-iclied pipes. The ew equatios provide a alterative to the measured liquid depth data from the experimets, ot always available, required for the calculatio of the discharge rate. The developed models are validated agaist laboratory based experimets. This thesis is divided ito 6 chapters. I Chapter, the theoretical basis for the pipelie outflow model with its assumptios ad justificatios are preseted. The chapter presets the basic equatios goverig the flow of icompressible liquids i pipes, icludig the coservatio equatio for mass ad the Beroulli equatio. Chapter 3 presets a review of the mathematical models available i the ope literature for simulatig failures of pipelies cotaiig icompressible (oflashig) liquids. This icludes models for full pipe flow, bubble formatio ad propagatio, ad ope chael flow. Chapter 4 describes the developmet, verificatio ad validatio of hydraulic trasiet models to simulate the outflow followig full-bore rupture i horizotal pipes. First, the pipe is assumed to be fed from a upstream storage tak. The overall model for this cofiguratio icludes the published model for full pipe flow (Joye & Barret, 003) ad the reported models for bubble formatio ad propagatio, ad ope chael flow. The the pipe is assumed to be isolated istataeously upo rupture followig emergecy shut dow. Due to the absece of a upstream tak, the model developed for this cofiguratio oly icludes bubble formatio ad propagatio, ad ope chael flow. Ultimately the models are verified through sesitivity aalysis. Chapter 1: Itroductio 1

13 I additio, for the cofiguratio with upstream tak, a series of experimets ivestigates the applicability of Bedikse s (1984) proposed value for the empirical coefficiet C 0 for the upstream bubble propagatio velocity i the presece of upstream storage tak ad dowstream bubble. The accuracy of the two models i predictig the discharge rate is also examied through series of experimets. Chapter 5 focuses o the extesio of the developed models i Chapter 4 to accout for the pipe icliatio for both isolated pipes ad those fed from a upstream tak. The validity of Bedikse s (1984) proposed value for the empirical coefficiet C 0 for the upstream bubble propagatio velocity i the presece of upstream storage tak for dowward-iclied pipes is tested through experimets. Oce agai the efficacies of the models are tested based o their applicatio to the failure of a hypothetical pipelie system. The accuracy of the models predictios for the discharge rate is also assessed through series of experimets. Chapter 6 deals with geeral coclusios ad suggestios for future work. Chapter 1: Itroductio 13

14 CHAPTER : FUNDAMENTAL EQUATIONS AND BACKGROUND THEORY.1 Itroductio The developmet of a outflow fluid dyamics model etails three mai steps. The first ivolves formulatig the basic equatios goverig flow, thermodyamics ad pertiet boudary coditios. The ext stage is applyig a efficiet ad accurate method to resolve or simplify these equatios ito easily solvable forms. The fial step is cocered with the validatio of the model agaist field or experimetal data, ad/or evaluatio of its performace agaist case studies which are represetative of realistic scearios. A importat part of the first step metioed above is the formulatio of the coservatio equatios for mass, mometum ad eergy. The fial form of these equatios ca be obtaied through various assumptios ad simplificatios depedig o the type of the flow ad/or state of the fluid. The flow of a icompressible liquid i a coduit with a ope ed may be either ope chael or full pipe flow. The two flows are similar i may ways except i oe importat respect: i cotrast to full pipe flow, ope chael flow has a free surface (Chow, 1959) which is subjected to atmospheric pressure. Ope chaels may have a ope top; for example i rivers, streams ad estuaries. They also occur i coduits with a closed top. These iclude pipes ad culverts; provided that the coduit is partly full (Aka, 006). The trasitio regime from full pipe flow to ope chael flow is called bubble formatio ad propagatio where depedig o the pipe cofiguratio, air bubbles are formed ad propagate from the pipe ilet, outlet or both. The flow durig this regime is a combiatio of the two flows: ope chael flow where the bubbles exist, ad full pipe flow throughout the rest of the pipe. The full descriptio of this trasitio regime is preseted i Chapter 3. Chapter : Fudametal Equatios ad Backgroud Theory 14

15 This chapter covers the pertiet theory ad goverig equatios for full pipe flow ad ope chael flow. Descriptio of flows, rage of applicatios ad the pertaiig assumptios are also preseted.. Full Pipe Flow The flow is called full pipe flow as log as the pipe remais full of liquid at all times. Pressure differece ad gravitatioal forces are the key drivers for the liquid flow durig full pipe flow...1 Coservatio of Mass (Cotiuity) (Coulso & Richardso, 1999) Cosiderig oe-dimesioal flow i a straight pipe with a costat cross sectioal area, the cotiuity equatio is give by u t x 0 (.1) where, u, x, ad t represet the liquid desity, mea axial liquid velocity, positio alog the pipe ad time, respectively. The fluid is assumed to be icompressible; therefore 0. Cosequetly Equatio t (.1) becomes: u x 0 (.) or u Costat (.3) Chapter : Fudametal Equatios ad Backgroud Theory 15

16 .. Coservatio of Eergy (Coulso & Richardso, 1999) The total eergy of a fluid i motio is made up of umber of compoets icludig iteral, pressure, potetial ad kietic eergy. Therefore, the total eergy of uit mass of fluid, e, may be defied as: u e i P gz (.4) where i, P ad are iteral eergy, pressure ad volume per uit mass (1/) respectively. Furthermore, g, Z ad u are gravitatioal acceleratio, elevatio above the datum level ad the velocity of a fiite elemet of the fluid respectively. The above equatio may be applied to the fluid as it flows from poit 1 to assumig q ad W s represet the et heat absorbed from the surroudigs ad et work doe by the fluid o the surroudigs respectively: i u u1 P gz i1 P1 1 gz1 q W (.5) s or u i P gz q W (.6) s where deotes a fiite chage i the quatities. Also specific ethalpy, h, may be defied by: h i P (.7) Replacig i P i Equatio (.6) by h gives: Chapter : Fudametal Equatios ad Backgroud Theory 16

17 u h gz q W (.8) s Also for a irreversible process ethalpy may be defied by: dh q F dp (.9) where F represets the mechaical eergy coverted irreversibly ito heat. Replacig h from the above equatio ito Equatio (.8): u d u gdz dp Ws F 0 (.10) Whe the above equatio is applied over the whole cross sectioal area of the pipe, allowace must be made for the fact that the mea square velocity is ot equal to the square of the mea velocity. Therefore, a correctio factor,, is itroduced ito the kietic term with a value of 0.5 ad 1 for lamiar ad turbulet flow respectively. Thus, i the absece of exteral work ad assumig the flow to be turbulet, for fiite chages, Equatio (.10) is itegrated for flow from poit 1 to ad gives: u 0 g Z P F (.11)..3 Frictioal Loss For the flow of a liquid through the pipe, the total frictioal loss, F, ca be expressed as (Perry, 1997; Maa, 005): F F F F F (.1) c e ft f Chapter : Fudametal Equatios ad Backgroud Theory 17

18 where F c ad F e are the frictioal loss due to sudde cotractio ad expasio respectively. O the other had, F ft ad beds ad the fluid/pipe wall loss respectively. F f are the frictioal loss due to fittigs ad Loss due to sudde cotractio (F c ) For a sudde cotractio at a sharp-edged etrace to the pipe, the etry loss ca be approximated via the followig equatio (Crae, 1957; Evet et al., 1989): F c A u 0.5(1 ) (.13) A 1 A ad A 1 are the smaller ad larger areas of the pipe respectively. For the case of the pipe attached to a upstream vessel A 0. Therefore Equatio (.13) is simplified A to the followig equatio: 1 u F c 0.5 (.14) McCabe et al. (1956) proposed the followig equatio for the etry loss of a pipe attached to a upstream vessel: u F c 0.4 (.15) Loss due to sudde expasio (F e ) Frictio loss due to sudde expasio of ducts of ay cross sectio may be estimated by the Borda-Carot equatio (Perry, 1997): F e A1 u (1 ) (.16) A Chapter : Fudametal Equatios ad Backgroud Theory 18

19 A For the case of a pipe attached to a dowstream vessel, 1 0. Therefore Equatio A (.16) is simplified to the followig equatio: F e u (.17) Fittigs, beds ad valves frictio loss (F ft ) The two most commo methods for calculatig the fittigs, beds ad valves losses are (Perry, 1997): 1. equivalet legth method (L e ). velocity head method (K) I the equivalet legth method, the losses are reported as the legth of a straight pipe which has the same loss as the fittigs, beds or valves. For turbulet flow the equivalet legth is ormally reported as a umber of diameters of the pipe of the same size as the fittig coectio with fixed quatity for L e /D. For lamiar flow L e /D is depedet o Reyolds umber (Perry, 1997). I the velocity head method, the losses are reported as a umber of velocity heads, i.e. Ku /g. The values of K for differet fittigs, beds ad valves may be foud i refereces such as Crae (1957) ad Perry (1997). Fluid/Pipe Wall Frictioal loss (F f ) The fluid/pipe wall frictioal loss for lamiar flow may be calculated from the Hage-Poiseuille equatio (Bird et al., 007): 3Lu F f (.18) D ad i terms of head: Chapter : Fudametal Equatios ad Backgroud Theory 19

20 3Lu h l (.19) gd where h l, L, ad D are the head loss, pipe legth, dyamic viscosity ad pipe iteral diameter respectively. For turbulet flow, the fluid/pipe wall frictioal loss may be calculated by Darcy- Weisbach equatio (Frech, 1994): flu (.0) D F f ad i terms of head: flu (.1) gd h l where f is the Faig frictio factor. The Faig frictio factor should ot be cofused with the Darcy frictio factor which is 4 times greater. For turbulet flow i smooth pipes, the Blasius equatio gives f for Reyolds umber, Re, i the rage of (Perry, 1997): f (.) 0.5 Re where Re is defied via: Re ul (.3) The Colebrook formula is accepted as the most accurate for rough pipes as log as Re > 4000 (Perry, 1997): Chapter : Fudametal Equatios ad Backgroud Theory 0

21 1 f.51 log[ 3.7D Re f ] (.4) where is the pipe surface roughess. The disadvatage of the above equatio lies i expressig f implicitly, requirig iteratios for its evaluatio. Churchill (1977) (Perry, 1997) suggested the followig explicit equatio for f for both smooth ad rough pipes for Re > 4000: f log[ ( ) D Re ] (.5) All the above equatios are depedet o Reyolds umber ad cosequetly o velocity i the pipe which may be ukow. Coulso ad Richardso (1999) 0.5 proposed a simplistic expressio for f as log as Re f. 3 : D f [3..5l( / D)] (.6) Darcy-Weisbach equatio for turbulet flow (Equatio (.1)) is equivalet to Hage- Poiseuille equatio (Equatio (.19)) for lamiar flow with the exceptio of frictio factor f. Equatig the two equatios for head loss gives a expressio of f that allows the Darcy-Weisbach equatio to be applied to lamiar flow: 3Lu gd flu gd (.7) or f 16 Re (.8) Chapter : Fudametal Equatios ad Backgroud Theory 1

22 The above equatio is called Poiseuille's equatio (Rohseow et al., 1998)..3 Ope Chael Flow I ope chael flow, due to the presece of a free surface o top of the liquid, the pipe is ot pressurised. Despite the similarities betwee ope chael ad full pipe flow, it is much more difficult to solve problems for ope chael flow. This is due to variatio of the free surface positio with respect to time ad space (Chow, 1959). I fact, although the basic priciples of fluid mechaics are still applicable to ope chael flow, such flow is sigificatly more complex tha full pipe flow due to the presece of free surface o top of the liquid. I order to have a streamlie with atmospheric pressure at the free surface, the forces causig ad resistig the flow ad the iertia must form a balace. Cosequetly, ulike full pipe flow, the flow boudaries are o loger fixed by the coduit geometry. Here the free surface adjusts itself to accommodate the give flow coditios (Sturm, 001). Table.1 summarises the mai differeces betwee full pipe ad ope chael flow. Full pipe flow Ope chael flow Flow drive maily by Pressure Gravity Flow cross sectioal area Fixed (pipe cross sectio) Variable (reducig) Specific boudary coditios - Ambiet pressure at free surface Table.1:Compariso betwee full pipe ad ope chael flow As it ca be observed from the above table, gravity is the mai driver for the flow i ope chael flow. The mai dimesioless parameter for this type of flow is the ratio of iertia ad gravity forces, defied as the Froude umber: Fr u liquid A g T wet (.9) Chapter : Fudametal Equatios ad Backgroud Theory

23 where T, A wet ad u liquid are the top width of the chael, chael wetted area ad liquid film velocity respectively. The ratio A wet /T is also called hydraulic depth. Figure.1 presets the cross sectio of a circular chael with the various characteristics dimesios. Figure.1: Cross sectio of a circular chael represetig the various characteristic dimesios: d = liquid depth = liquid depth agle T = flow top width A wet = Wetted area.3.1 Coservatio of Mass (Cotiuity) (Sturm, 001) Oce agai cosiderig oe-dimesioal flow i a chael, the cotiuity equatio is give by: A Q wet liquid 0 (.30) t x where Q liquid is the liquid volumetric flow rate. Assumig A wet remais uchaged with time ad positio alog the chael, Equatio (.30) is simplified as: u liquid x 0 (.31) Chapter : Fudametal Equatios ad Backgroud Theory 3

24 or u Costat (.3) liquid.3. Specific Eergy (Aka, 006) The priciple for the eergy coservatio equatio i ope chael flow is the same as for full pipe flow. Assumig oe-dimesioal flow i a chael, the followig equatio describes eergy coservatio for ope chael flow: 1 g u liquid t u g u liquid x d x S e S 0 0 (.33) where d, S e ad S o are the liquid depth i the chael, eergy slope ad chael slope respectively. Eergy slope is related to the work doe by the frictio forces. I ope chael flow problems, it is ofte desirable to cosider the eergy cotet with respect to the chael base. This is called the specific eergy or specific head, E specific, ad is give by: E specific uliquid d (.34) g Aother commo term i ope chael flow is the eergy grade lie which is defied as sum of the specific head ad the elevatio of the chael base above a selected datum, z: E grade lie uliquid z d (.35) g Chapter : Fudametal Equatios ad Backgroud Theory 4

25 The slope (gradiet) of the eergy grade lie, S e i Equatio (.33), is the rate of eergy head loss due to frictio (h l ). Figure. shows the eergy grade lie ad specific eergy i a chael. Figure.: Total ad specific eergy lies i a chael.3.3 Subcritical, Critical ad Supercritical Flow (Frech, 1994; Aka, 006) Figure.3 shows the plot of d versus E specific for a fixed discharge ad chael sectio (Frech, 1994; Aka, 006). The depth at which the specific eergy is at its miimum is called critical depth, d c, ad the correspodig flow is referred to as critical flow. Flows with liquid depth higher ad lower tha critical depth are called subcritical ad supercritical flow respectively. The coditio of miimum specific eergy at critical depth implies that its derivative with respect to d must be zero: d E specific d( d) u 1 g liquid du liquid d( d) 0 (.36) Replacig u liquid i the above equatio by (Q critical / A wet ) produces: Chapter : Fudametal Equatios ad Backgroud Theory 5

26 Q g d critical wet 1 3 Awet A / d( d) 0 (.37) where Q critical is the critical liquid discharge rate. Notig T, the top width of the chael, is d(a wet ) / d(d), the above equatio may be rewritte as: u 1 g critical T A wet 0 (.38) or gawet ucritical (.39) T Based o the above equatio, the Froude umber (Equatio (.9)) is 1 for critical flow. Flows with Fr higher tha 1 are supercritical, while i subcritical flows, Fr is less tha Uiform Flow A chael is said to have uiform flow if the flow depth ad the velocity remai uchaged with time ad space. The correspodig flow depth is called ormal depth, d. This costitutes the fudametal type of the flow i a ope chael. It occurs whe gravity forces are i equilibrium with resistace forces. The defiitio of uiform flow implies that S e is the same as the chael slope, S o. This type of flow oly happes i prismatic chaels ad rarely occurs aturally. However, i very log chaels ad i the absece of flow cotrols such as hydraulic structures the flow becomes uiform (Aka, 006). Determiig the discharge rate for uiform flow for a give depth or vice versa has bee of particular iterest for may researchers. Historically, such formulas have bee preseted for the flow velocity as a fuctio of hydraulic radius ad slope. The Chapter : Fudametal Equatios ad Backgroud Theory 6

27 famous Chezy equatio empirically predicts the flow rate i a chael with uiform flow (White, 1999): u C ( R S ) 1/ liquid Chezy h 0 (.40) where R h ad C Chezy are the hydraulics radius ad Chezy coefficiet respectively. R h is defied as: R h 1 D 4 h A P wet wet (.41) Here D h ad P wet represet hydraulic diameter ad wetted perimeter of the chael respectively. Despite what the ame may suggest, the hydraulic diameter is ot twice the hydraulic radius, but four times. C chezy is defied as (Sturm, 001): 8g ( f 1/ C Chezy ) (.4) f is the Faig frictio factor which will be described i Sectio.3.6. A good approximatio of Equatio (.4) has bee proposed by Robert Maig (White, 1999): C Chezy 1/ 6 8g Rh ( ) 1/ c (.43) f Here is a roughess parameter which is preseted i Table. for various materials, ad c is a parameter = 1.0 m /6 s -1. is a dimesioless parameter. Chapter : Fudametal Equatios ad Backgroud Theory 7

28 Replacig C Chezy from Equatio (.43) ito Equatio (.40) gives: c / 3 1/ u liquid Rh S0 (.44) The use of Chezy coefficiet for ma-made chaels seems to be somewhat cotroversial. Chaso (004) stated that i ope chaels, the Darcy-Weisbach equatio is the oly soud method to estimate the frictio loss. Therefore, recallig Equatio (.1) for full pipe flow ad replacig D by D h gives: h l L fu liquid (.45) gd h For uiform flow the head loss per uit legth is the same as chael slope (S 0 ). Therefore, replacig the left side of the above equatio by S 0 produces: u liquid So gdh (.46) f For maily historical reasos empirical resistace coefficiets (e.g. Chezy coefficiet) are still used. Chaso (004) stated that their use is highly iaccurate for ma-made chaels. The Darcy-Weisbach equatio is the oly suitable method to predict the flow rate i ope chael flow..3.5 Steep ad Mild/Horizotal Chaels Depedig o the chael slope with a selected datum, a chael ca be classified as mild or steep. Whe the icliatio of the pipe is greater tha 6 o, the pipe is called steep (Aka, 006). Frech (1985) suggested a value of 10 o as the criteria for steep chaels. I mild/horizotal chaels, ormal depth (d ) is larger tha critical depth (d c ), while for steep pipes critical depth has a larger value. Figure.4 ad Figure.5 Chapter : Fudametal Equatios ad Backgroud Theory 8

29 respectively show schematic sketches for mild/horizotal ad steep chaels both edig with free fall. Figure.4: Flow i mild/horizotal chael edig with a free fall Figure.5: Flow i steep chael edig with a free fall Normally i mild/horizotal chaels due to low liquid velocity, the flow is subcritical. If the chael termiates at a free fall, the liquid depth at the free fall may be assumed to be the same as critical depth. I reality, whe the flow is subcritical, the critical depth occurs a short distace, about 4d c, upstream of the free falls. The depth at the ed of the pipe which is called brik depth, d e, is less tha critical depth. However, for sufficietly log chaels, the assumptio of critical depth is a reasoably good assumptio (Wallis et al., 1977; Aka, 006; Ti rek et al., 008). O the other had, i steep pipes due to high liquid velocity the flow is supercritical ad the free fall does ot affect the liquid depth. As such, the liquid depth at the pipe outlet ad alog the pipe is the same as ormal depth (d ) (Frech, 1985; Aka, 006). Chapter : Fudametal Equatios ad Backgroud Theory 9

30 .3.6 Frictioal loss I ope chaels the flow is ormally turbulet. Therefore the Faig frictio factor may be calculated from the Colebrook equatio by replacig D with D h : 1 f log[ 3.7D h.51 Re f ] (.47) Also sice typical chaels are rough, for fully turbulet flow, f may be defied by (White, 1999): 14.8R f log h (.48) Typically the equatios used to calculate the Faig frictio factor i full pipes are also applicable to ope chael flow, except that the pipe diameter eeds to be replaced by the hydraulic diameter for ope chael flow..4 Coclusio I this chapter, the two types of flows occurrig i icompressible liquid pipelies, i.e. full pipe flow ad ope chael flow, were described. The equatios describig mass ad eergy coservatio were preseted for both types of flow. The critical coditio at the pipe exit i ope chael flow was explaied ad the differece betwee subcritical ad supercritical flow was highlighted. Also mild/horizotal ad steep chaels were described ad the boudary coditios alog with the correspodig discharge equatios for each were explaied. This chapter cocluded that the mass coservatio ad the Beroulli equatio would be applicable to both types of flow. I ope chael flow, depedig o the pipe icliatio, the flow ca be either subcritical (mild/horizotal chaels) or supercritical (steep chaels). I additio, i horizotal chaels with free fall, critical coditio prevails at the chael exit, i.e. the liquid depth is the same as Chapter : Fudametal Equatios ad Backgroud Theory 30

31 critical depth. I steep chaels, o the other had, the flow is uiform across the chael with the liquid depth beig the same as the ormal depth. These coclusios form the basis for developig hydraulic-based outflow models to simulate the failure of horizotal ad dowward-iclied pipelies i the followig chapters. Chapter : Fudametal Equatios ad Backgroud Theory 31

32 CHAPTER 3: LITERATURE REVIEW 3.1 Itroductio Modellig of accidets ivolvig the failure of pressurised pipelies has bee the subject of sigificat iterest sice research carried out i the uclear power idustry (Offshore Techology Report, 1998) evaluatig the critical scearios of loss of coolat accidets (LOCAs) i Pressurised Water Reactors (PWRs). Sice the, umerous studies with various degrees of sophisticatio have bee coducted to model the trasiet outflow followig the failure of pressurised pipelies (see for example Bedikse et al.(1991); Richardso & Saville (1991, 1996a, 1996b) ad Mahgerefteh et al.(1997, 1999, 000)). However, these are cofied to gas or flashig liquid pipelies. Although equally importat, for pipelies cotaiig icompressible liquids there is o uified model to simulate the outflow throughout the complete draiage of the pipelie. Wallis et al. (1977), Motes (1997) ad Hager (1999) idetified three flow regimes which may be observed followig the rupture of horizotal, permaet liquid trasportig pipelies. These iclude full pipe flow, bubble formatio ad propagatio, ad fially ope chael flow. Full pipe flow occurs durig depressurisatio for as log as the pipe remais full. Bubble propagatio ad ope chael flow o the other had take place sequetially followig air igress to the pipe after full pipe flow. The three flow regimes occurrig i horizotal pipe have also bee observed i dowward-iclied pipes (Ye & Pasic, 1980; Joy & Barrett, 003; Pothof, 011). This chapter presets a review of the pertiet models for simulatig the trasiet or steady-state discharge followig the failure of pressurised pipelies cotaiig icompressible liquids. The trasiet modellig icludes the three regimes of full pipe flow, bubble formatio ad propagatio, ad ope chael flow. For the bubble formatio ad propagatio regime, researchers have maily focused o the bubble Chapter 3: Literature Review 3

33 propagatio velocity rather tha the liquid discharge rate. However, sice this forms the basis of the models developed i this study, the available equatios for bubble propagatio velocity are also preseted i this chapter. 3. Full Pipe Flow The most covetioal method to calculate the discharge rate followig icompressible liquid pipelie rupture is to assume the pipelie remais full throughout the discharge process. Loiacoo (1987) developed a full pipe flow model to calculate the required time, t f, to drai a vertical cylidrical tak through a vertical pipe coected to its base. First, Loiacoo (1987) applied the Beroulli equatio betwee poits 1 ad, preseted i Figure 3.1, which produced Equatio (3.1). The, Equatio (3.1) was combied with the trasiet flow mass balace, Equatio (3.), to produce Equatio (3.3) to calculate t f : ud fl gz D e u d 0 (3.1) dm dt Au d (3.) t f D D 4 fl g D ta k e 1 Z0 Z f (3.3) where: u d = discharge velocity Z = liquid level height Z 0 = iitial liquid level height at t = 0 Z f = fial liquid level height H = liquid head i the tak L e = equivalet legth of pipig ad fittig (see Sectio..3) A = pipe cross sectioal area Chapter 3: Literature Review 33

34 D tak = tak diameter D = Pipe ier diameter 1 H Z L u d Figure 3.1: Tak draiage through a vertical pipe Loiacoo (1987) oly cosidered taks with flat base. Shoaei & Sommerfeld (1989) modified Loiacoo s model to accout for taks with elliptical base which are more commo i idustry. The differece betwee a tak with flat base ad oe with elliptical base is the reductio i the tak cross sectioal area for the later case whe the liquid level approaches the pipe ilet. To accout for this, Shoaei & Sommerfeld (1989) suggested the followig equatio to describe A tak as a fuctio of liquid head i the tak: Dta k H A ta k H 4b b (3.4) where b is the depth of elliptical dished head. Defiig H = Z-L ad replacig A tak from the above equatio ito Equatio (3.3) yields: Chapter 3: Literature Review 34

35 Y BZ Z dz dt D D gz fl 1 D 4 ta k e (3.5) where B L b (3.6) ad Y L bl (3.7) Itegratig Equatio (3.5) betwee (0, Z 0 ) ad (t f, Z f ) yields: t f 4B 4B C Z0 Z0 Y Z0 Z f Z f Y Z f (3.8) where Dta C D k g fl D e (3.9) Joye & Barrett (003) modified Loiacoo s model (1987) by replacig the equivalet legth for the pipig ad fittigs (mior losses) with the resistace coefficiet due to wider availability of values for resistace coefficiet tha equivalet legth. Assumig the tak ever draied dry, they proposed the followig equatio for the trasiet discharge rate from pipes with ay icliatio: Chapter 3: Literature Review 35

36 t f D D ta k 4 fl K Z 0 g D Z f (3.10) where K represet the mior losses icludig the exit kietic loss ad the pipe etrace loss. Here the flow was assumed to be turbulet, thus f remaied costat. Joye & Barret (003) validated the above equatio through a series of experimets for taks with diameters of 0.075m ad 0.37m. The pipe fed from the small tak was smooth stailess steel with ier diameter ragig from 0.003m to 0.007m. The larger tak o the other had was coected to a smooth hard-draw copper pipe with the diameter of 0.019m. They also coducted some tests for a system of horizotal ad vertical pipe segmets. Figure 3. ad Figure 3.3 respectively L L preset their predicted efflux time (t f ) from Equatio (3.10) for 3 ad 3 D D for various pipe cofiguratios. From the graphs, they cocluded that Equatio (3.10) predicted the efflux time with good accuracy with maximum deviatio of about 15%. For lamiar flow, Bird et al. (007) combied Hage-Poiseuille equatio (Equatio.18) for steady state flow with trasiet flow mass balace (Equatio (3.)): dz dt 4 gd H L D ta k 18L 4 (3.11) or dz dt 3LDta k 4 gd H L (3.1) Although Hage-Poiseuille equatio was origially derived for steady-state flow, it could still be used here due to quasi-steady state ature of the flow (Bird et al., 007). Chapter 3: Literature Review 36

37 By rearragig the above equatio ad subsequetly itegratig betwee (0, H 0 ) ad (t f, H f ), Bird et al. (007) derived Equatio (3.14) for the efflux time for lamiar flow: dh H L gd 3LD 4 ta k dt (3.13) t f 3LD gd ta k 4 H l H 0 f L L (3.14) The above equatio assumes the exit kietic eergy ad other frictio losses are egligible. Joye & Barrett (003) tested this assumptio by coductig experimets for 98% glycerol/water. They compared the predicted efflux time from Equatio (3.14) with those obtaied from Equatio (3.10) where exit kietic eergy ad frictioal loss were also icluded. Faig frictio factor i Equatio (3.10) was calculated from Equatio (.8) for lamiar flow. The values for mior losses were assumed to be the same as those for the turbulet flow. Figure 3.4 ad Figure 3.5 L L show the results of their comparisos for 3 ad 3 respectively. Based o D D the graphs, Joye & Barrett (003) claimed that, overall, the predicted efflux time (t f ) from Equatio (3.10) was i slightly better agreemet with the experimetal results L tha Equatio (3.14). O the other had, for 1. 5 both equatios uderestimated D L t f, with the deviatio more tha 50% for 0. They suggested the better D agreemet from Equatio (3.10) for turbulet flow could be due to ucertaities for K values for lamiar flow. The cosequece of the ucertaity regardig K values was eve more detrimetal for shorter pipes where the mior losses domiated fluid/pipe wall loss. Chapter 3: Literature Review 37

38 The DISC model developed by DNV Software (005) predicts the steady-state discharge from a orifice i the tak wall or from a horizotal pipe attached to the tak for all fluid states. I the special case of icompressible liquids, the DISC model is simplified to a model very similar to Joye & Barett s model (003). Employig Equatio (3.15) istead of Equatio (.14) for pipe etry loss (F c ), the model calculates the discharge velocity via Equatio (3.16): F f e 1 (3.15) 0.5( 1) u d Cv where C v is velocity coefficiet ad is assumed to be 0.6. u d gz fl 0.89 D (3.16) The above model was validated agaist Uchida ad Nariai s experimetal data (1966) for water for the upstream pressure i the rage -8 bara. The pipe diameter was 0.004m ad the pipe legth varied i the rage 0-.5m. The maximum differece betwee the predicted ad experimetal discharge velocity was 8%. 3.3 Bubble Formatio ad Propagatio As the flow rate decreases i the pipe, air igress will result i the formatio ad propagatio of bubbles, either at the pipe upstream or dowstream or eve both, depedig o the pipe icliatio ad cofiguratio. Bubble formatio is of sigificat practical importace sice air pockets ca reduce the effective pipe cross sectioal area thus reducig pipe capacity. Also the trasported air may be released at the discharge locatio, which ca raise evirometal cocers due to foamig (Lauchla et al., 005). Chapter 3: Literature Review 38

39 3.3.1 Bubble Propagatio i Stagat Liquid The classical problem of determiig the drift velocity, i.e. propagatio velocity of a log bubble through a stagat liquid, i vertical pipes has bee of iterest to umerous authors i recet years. As log as surface tesio effects are egligible, the bubble velocity is proportioal to gd (Wiser et al., 1975; Falvey, 1980; Bedikse, 1984). Based o the assumptio of potetial flow, Dumitrescu (1943) obtaied a value of 0.35 for the proportioality coefficiet which was i excellet agreemet with experimetal measuremets. Zukoski (1966) experimetally ivestigated the impact of viscosity ad surface tesio o the bubble propagatio velocity for differet pipe icliatios. The coclusio was that for small pipe diameters with ay icliatios, the ifluece of surface tesio was to reduce the bubble propagatio velocity more tha gd. Bejami (1968) was the first to cosider bubble propagatio alog a closed-ed or isolated horizotal pipe. Igorig viscosity ad surface tesio ad based o a force balace betwee the approachig ad recedig sectios of the stream, Bejami (1968) calculated a value of 0.54 for the proportioality coefficiet for a horizotal pipe with the correspodig liquid depth below the bubble, d: u drift gd (3.17) d D (3.18) where u drift is the drift velocity. The calculated bubble velocity is the same as the miimum required liquid velocity to keep a bubble stagat (Bejami, 1968; Lauchla et al., 005). O the other had, for dowward-iclied pipes there has bee cosiderable debate o the miimum required liquid velocity to trasport air bubbles ad pockets alog the pipe. Majority of the available literature relate to the results of experimetal Chapter 3: Literature Review 39

40 ivestigatios (Lauchla, 005). Most researchers agree that the required velocity i a dowward-iclied pipelie is a fuctio of gd ad pipe slope S o (Kaliske & Bliss, 1943; Ket, 195). Ket (195) coducted detailed experimets for a 0.10m diameter, 5.5m log pipe with pipe agle varyig betwee 15 ad 60 relative to a horizotal level. Based o the curve preseted i Figure 3.6 showig si versus experimetal drift velocity, the author proposed the followig equatio for the miimum required velocity to trasport air bubbles ad pockets upstream of the pipe: u drift 1.3 gd si (3.19) is the pipe icliatio agle relative to a horizotal level. However Ket (196) made a crucial mistake by graphically fittig the obtaied data to the fuctioal relatio i Equatio (3.19) (Pothof, 011). The straight lie i Figure 3.6 is ot the same as what predicted by Equatio (3.19). By carefully studyig Figure 3.6 ad Equatio (3.19), it is clear that Equatio (3.19) icludes the origi, while the liear extrapolatio of the experimetal data does ot, ad the dotted parabolic curve is also icorrect (Pothof, 011). Mosvell (1976) suggested the followig equatio as a clearly better curve fit o Ket s data (195), which allows for a o-zero offset: u drift gd si (3.0) Bedikse (1984) suggested a empirical equatio based o the drift velocity for horizotal ad vertical pipes: Chapter 3: Literature Review 40

41 u drift 0.54 gd cos 0.35 gd si (3.1) The above equatio gives the same value for drift velocity as the predicted value by Bejami (1968) ad Dumitrescu (1943) for horizotal ad vertical pipes respectively. Iogamov & Opari (003) developed a equatio for bubble propagatio velocity i dowward-iclied pipes by applyig aalytical methods of potetial theory ad complex aalysis: cos / 6 u drift gd (3.) For horizotal ad vertical pipes, the above equatio predicts very similar results to what Bejami (1968) ad Dumitrescu (1943) predicted for horizotal ad vertical pipes respectively: 0 u drift gd (3.3) o gd (3.4) u drift 3.3. Bubble Propagatio i Co-curret Liquid Bubble propagatio i a co-curret movig liquid was first studied by Nickli et al. (196) for a 0.06m diameter, vertical tube for Reyolds umbers i the rage (8-50) l0 3. Their proposed model ivolves addig a extra term to the drift velocity to accout for the liquid movemet: u u C u (3.5) movig drift 0 Chapter 3: Literature Review 41

42 Here u movig, C 0 ad u are the bubble rise velocity i a movig liquid, a parameter = 1. ad the liquid velocity respectively. By coductig experimets with various diameter pipes, Bedikse (1984) showed that for horizotal pipe, C 0 varied i the rage of I additio, for dowwardiclied pipes with the icliatio agle less tha 30 o, C 0 was foud to be Ope Chael Flow Followig further decrease i the flow rate, the bubble will elogate formig a free surface o top of the liquid. This regime is called ope chael flow. Modellig such flows is of particular iterest i civil egieerig projects ivolvig water distributio etworks Horizotal Chaels The Califoria Pipe Method developed by Va Leer (19) (Water measuremet maual, 001) predicts the discharge rate of water from a horizotal pipe based o the liquid depth at the free fall, d e, via: Q d 1.88 de D (3.6) D where Q d, d e ad D are i ft 3 /s, ft ad ft respectively. The model is however limited to: whe the pipe rus less tha half full the liquid depth at the outlet (brik depth) is less tha 0.56D the pipe is loger tha 6D pipe diameter is i the rage of m Chapter 3: Literature Review 4

43 Whe the brik depth (d e ) is greater tha 0.56D, the more geeral Purdue pipe method developed at Purdue Uiversity by Greve (198) is used (Water measuremet maual, 001). This model is applicable equally well to both partially ad completely full water pipes. It cosists of measurig two coordiates of the upper surface of the jet, X ad Y, as preseted i Figure 3.7. Figure 3.7: Purdue pipe method for measurig flow from a horizotal pipe The model assumes that the horizotal velocity compoet of the flow is costat ad the oly force actig o the jet is gravity. I time t, a particle o the upper surface of the jet will travel a horizotal ad vertical distace X ad Y from the outlet of the pipe equal to: X u0t (3.7) Y 0.5gt (3.8) where u 0 is the velocity at X = 0. Combiig the two equatios ad itroducig a discharge coefficiet = 1.1, the discharge rate is calculated via: X Q d 0.864D g (3.9) Y Chapter 3: Literature Review 43

44 Sice the umerous authors carried out research to ivestigate the relatioship betwee the discharge rate ad the brik depth. A alterative approach to calculate the discharge rate from a horizotal pipe is based o the assumptio that the liquid depth at the exit to be the same as the critical depth, d c (Smith, 196; Wallis et al., 1977; Aka, 006; Ti rek et al., 008). Recallig Equatio (.39) for critical velocity: gawet ud (.39) T A wet ad T may be defied via: A D wet si (3.30) 8 T Dsi (3.31) where represets the liquid depth agle (radias) preseted i Figure.1. Replacig A wet ad T from Equatio (3.30) ad Equatio (3.31) ito Equatio (.39) produces: gd si u d (3.3) si( / ) Therefore, the discharge rate is calculated via: gd si D Qd Awetu d si (3.33) si( / ) 8 Chapter 3: Literature Review 44

45 or Q d 0.044D gd si si si (3.34) 3.4. Dowward-iclied Chaels For dowward-iclied chaels, there are two established methods to calculate the discharge rate ad velocity, as described i Sectio.3.4 i Chapter. The first is applyig the Maig equatio. Recallig Equatio (.44) for Maig equatio ad Equatio (.41) for hydraulic radius: k / 3 1/ u Rh S0 (.44) R h 1 D 4 h A P wet wet (.41) P wet may also be defied via: P wet D (3.35) Replacig A wet ad P wet from Equatios (3.30) ad (3.35) ito Equatio (.44) gives: R h D si 8 (3.36) D or Chapter 3: Literature Review 45

46 ad si R D h (3.37) 4 si D h D (3.38) Now by substitutig R h from the Equatio (3.37) ito the Maig equatio (Equatio (.44)), u d may be calculated via: u d c D 4 3 si / 1/ S 0 (3.39) or u d / 3 c / 3 si D S 1/ 0 (3.40) Fially the volumetric discharge rate is calculated by substitutig A wet ad u d from Equatio (3.30) ad Equatio (3.40) respectively: / 3 c / 3 si 1/ D Qd ud Awet D S0 si (3.41) 8 or / c 1/ Si 0.050D S0 si 1 Q d (3.4) However, as metioed i Sectio.3.4, the Maig equatio is a empirical formula ad is oly applicable for water pipes. A alterative approach to cover a Chapter 3: Literature Review 46

47 wider rage of liquids is usig the Darcy-Weisbach equatio (Equatio (.46)) described i Sectio.3.4. Substitutig D h from Equatio (3.38) ito Equatio (.46) produces: S0 gd si u d (3.43) f Therefore, the volumetric discharge rate is calculated via: S0gD si D Qd ud Awet si (3.44) f 8 or D 8 DS g 0 Q d (3.45) f si Coclusio Based o the above review, it is clear that sigificat work has bee doe to calculate both steady-state ad trasiet discharge durig full pipe flow for various upstream tak cofiguratios. O the other had, the focus of the previous research o the bubble formatio ad propagatio regime has bee o the bubble propagatio velocity rather tha discharge rate. Several studies have bee coducted to calculate the propagatio velocity for horizotal ad dowward-iclied pipes with stagat ad movig liquids. For the ope chael flow, umerous researches have studied the relatioship betwee the discharge rate ad brik depth, both theoretically ad experimetally, assumig the iformatio o the brik depth is available. The review also showed that the models dealig with pipelie failure have oly cosidered full pipe flow ad igored the subsequet regimes icludig bubble Chapter 3: Literature Review 47

48 formatio ad ope chael flow. Neglectig these regimes specifically for log pipelies may lead to sigificat uderestimatio of total released ivetory as will be show i Chapter 4 ad Chapter 5. Chapter 3: Literature Review 48

49 CHAPTER 4: OUTFLOW SIMULATION UPON FULL- BORE RUPTURE IN HORIZONTAL PIPELINES 4.1. Itroductio Crude oil ad other hydrocarbo products must be trasported from the storage taks at the productio site to refieries ad the cosumers. O-shore, this is doe by meas of pipelies. I order to stop the flow of a hazardous liquid followig detectio of a failure, Emergecy Shut Dow Valves (ESDV) are widely used i liquid trasportatio pipelies. These actuated valves esure the safety of the operatios by automatically closig i a emergecy. May authors have focused o the failure of pipelies cotaiig gas or two phase flow ad developed umerous mathematical models with various degrees of sophisticatio to simulate the outflow. However, despite its importace, little has bee doe for icompressible liquids. As discussed i Sectio 3., researchers who have worked o this field have oly cosidered full pipe flow regime where the feed tak ever drais dry, thus the pipe remais full (see for example Loiacoo (1987) ad Joye & Barrett (003)). I their studies, the subsequet regimes icludig bubble formatio ad propagatio ad ope chael flow are ot cosidered. O the other had, studies available o bubble formatio ad propagatio regime oly focus o predictig bubble propagatio velocity, rather tha the liquid discharge rate (see for example Bejami (1968) ad Bedikse (1984)). Fially, although for ope chael flow, methods such as Purdue pipe method (Water measuremet maual, 001) focus o the liquid discharge rate, they are based o kow liquid depth at the pipe exit. However, this data is ot always available durig the draiage process. This chapter presets the developmet ad testig of mathematical models to determie the discharge rate followig full-bore rupture at the ed of a horizotal pipelie cotaiig a icompressible liquid. Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 49

50 I Sectio 4. the key models assumptios are summarised. I Sectio 4.3, the pipelie is assumed to be attached to a upstream storage tak. The associated model accouts for all subsequet regimes icludig full pipe flow, bubble formatio ad propagatio, ad ope chael flow. Sectio 4.4 is cocered with pipes beig isolated istataeously usig emergecy shut dow upo rupture. Due to the absece of upstream tak, the associated model oly icludes bubble formatio ad ope chael flow. The efficacy ad accuracy of the developed models are tested through parametric studies ad series of experimets respectively. The results obtaied from the applicatio of the reported model for the system of pipe ad upstream tak highlight the sigificace of post-full pipe flow regime o the total ivetory loss. I Sectio 4.5 the mai coclusios are preseted. 4.. Key Models Assumptios The key assumptios made i derivig the flow models are: Oe-dimesioal flow aywhere i the pipelie I pipelies with large value of L/D the variatio of velocity alog the pipe radius is egligible. Therefore, for the liquid trasport pipelies the flow ca be easily assumed to be oe-dimesioal Icompressible flow, i.e. costat fluid desity The fluid cosidered i this study is liquid. As liquids are quite difficult to compress, they are ormally treated as icompressible. Cosequetly the desity ca well be assumed to remai uchaged. Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 50

51 4..3. Costat cross sectioal area of the pipe, much smaller tha tak cross sectioal area (where applicable) Pipelies trasportig liquids from a storage tak to the delivery poit have diameter much smaller tha the liquid storage tak diameter. Cosequetly, the pipe cross sectioal area is much smaller tha the tak cross sectioal area No frictio betwee the fluid ad tak Sice the pipelie legth is ormally much larger tha the liquid height i the tak, the frictioal loss alog the pipe is more sigificat tha that of tak wall. Therefore, the latter is igored i this study Isothermal coditios i the pipe The liquid is icompressible ad hece the desity is ivariat. Thus, the variatio of temperature for the duratio of the release is egligible No ilet flow to the feed tak I ormal operatig coditios storage taks are fed through a pump with costat iflow. However, i the evet of a accidet i a pipe charged from the tak the pump shuts dow. As a result there will be o iflow to the tak. Sice the time differece betwee the failure detectio ad emergecy shut dow is short (oly a few miutes), the ilet flow i this study is assumed to be zero No hammer effect upo valve closure Water hammer occurs whe a valve is closed suddely at the ed of a pipelie. This will produce a pressure wave which propagates alog the pipe. Due to the fiite compressibility of the liquid the iitial discharge as a result of this over pressure will occur i a very short period of time (more like a splash), comparig to the total release duratio. As such, the over pressure due to water hammer is igored here. Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 51

52 4..8. Negligible impact of surface tesio ad viscosity durig bubble formatio ad propagatio regime Pipelies trasportig liquids have ormally large diameter (more tha 50mm). Therefore, the impact of surface tesio ca be easily igored. I additio, the impact of viscosity o the bubble propagatio velocity for flows with the Reyolds Number higher tha 0 is egligible (Zukoski (1966). Due to the high Reyolds umber expected ormally for liquid trasport pipelies, i this study the impact of viscosity is also assumed to be egligible Trasiet Hydraulic Flow Modellig Followig Full-bore Rupture of a Horizotal Pipelie Fed from a Upstream Tak Itroductio The followig presets the formulatio of various flow equatios for predictig the discharge rate followig full-bore rupture i a horizotal pipe coected to the base of a storage tak cotaiig a icompressible fluid. The modellig ecompasses the three flow regimes amely full pipe flow, bubble formatio ad propagatio, ad ope chael flow. The overall model is the tested through parametric studies i Sectio to ivestigate the impact of pipelie ier diameter ad legth o the discharge velocity ad wetted area of the pipe. Normalised cumulative discharged mass ad the released mass durig idividual regimes are also calculated ad compared agaist the theoretical values. Based o these results, the importace of post-full pipe flow is discussed for a rage of pipelie ier diameters ad legths. Fially, the accuracy of the developed model is assessed through series of experimets, preseted i Sectio Model Theory Full Pipe Flow Followig a rupture i a horizotal pipe fed from a upstream tak, the pipe depressurises due to the reductio i the liquid head i the tak. Durig this phase, Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 5

53 the domiat regime is full pipe flow, i.e. the pipe remais full. Figure 4.1 presets a horizotal pipe with pipe attached to the base of a upstream storage tak. Recallig Equatio (3.10) developed by Joye & Barrett (003) for the calculatio of the required draiage time for a tak through a horizotal pipe ad rearragig it for a arbitrary time t produces: t f D D ta k 4 fl K Z 0 g D Z f (3. 10) Z Z 0 A A ta k g t 4 fl K D (4.1) Figure 4.1: Full pipe flow followig full-bore rupture i a horizotal pipe fed from a upstream storage tak The term ΣK icludes both kietic eergy ad the pipe etrace losses. Joye & Barrett (003) employed a value of 0.5 for K to accout for the pipe etrace loss (see Equatio.14): Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 53

54 u F c 0.5 (.14) By recallig Equatio (3.1) ad replacig L e with K = 1 for the kietic eergy loss ad 0.5 for pipe etrace loss, the discharge velocity may be calculated via: ud (1) ud (0.5) fl gz u D d 0 (4.) or gz u d fl 0.75 D (4.3) Replacig Z from (4.1) ito the above equatio gives: u d g ( 4 fl 1.5 D A A ta k ) t gz 0 fl 0.75 D (4.4) For a horizotal pipe, Z 0 = H 0 (see Figure 3.1). Therefore: u d g ( 4 fl 1.5 D A A ta k ) t gh 0 fl 0.75 D (4.5) ad i terms of mass discharge rate, m : d Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 54

55 m d g A gh 0 A ( ) t (4.6) 4 fl A fl 1.5 ta k 0.75 D D Equatio (4.6) is used to calculate the discharge rate util H = d c where d c is the critical depth of the flow as itroduced i Sectio.3.3. McCabe et al. (1956) proposed the lower value of 0.4 istead of 0.5 to take ito accout the loss at the pipe etrace. Followig Joye & Barrett (003), i this work due to various citatios the value of 0.5 is used to calculate the pipe etrace loss (see for example Crae (1957), Evett & Liu (1989) ad Joye & Barrett (003)). It should be oted that both values of 0.5 ad 0.4 are oly approximatio ad various publicatios are ot always i agreemet. I fact, some may eve differ by 5% (Joes et al, 008) Bubble Formatio ad Propagatio I practice, as soo as the liquid level i the tak falls below the pipe diameter, a bubble will form at the upstream ed of the pipe. This is followed by propagatio of bubbles from both eds of the pipe towards oe aother (Kadasamy, 1999). As described i Sectio.3.5, the liquid depth at the free fall below the dowstream bubble may be assumed to be equal to the critical depth (Ti rek et al., 008). Assumig a horizotal pipelie so that the upstream ad dowstream bubbles have the same height, the liquid depth below the upstream bubble ad iside the tak will also be equal to the critical depth. This depth ad cosequetly the discharge rate ad velocity are assumed to remai costat durig the bubble propagatio regime. Figure 4. shows a horizotal pipe attached to a tak with bubble formatio ad propagatio from both eds. Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 55

56 Figure 4.: Bubble propagatio from both eds of a horizotal pipe fed from a upstream storage tak I this study it is assumed that the bubble formatio at the pipe dowstream does ot occur durig full pipe flow. Istead, the two bubbles start propagatig towards each other simultaeously. As such, the discharge rate is calculated from Equatio (4.6) for as log as the liquid level i the tak is higher tha d c, the costat critical depth durig bubble propagatio regime. This is a coservative assumptio as the full pipe flow equatio produces higher discharge rate due to wetted area beig the same as pipe cross sectioal area. Based o the assumptio of egligible fluid/pipe wall frictio ad critical liquid depth below the bubbles, the critical discharge velocity (Equatio (3.3)) durig bubble formatio ad ope chael flow is give by: u d gd c si c (4.7) si( / ) c where c deotes the costat, time-ivariat, critical agle (see Figure.1) durig this regime. Thus, the correspodig critical liquid depth below the bubbles ( d c ) is give by: Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 56

57 d c D 1 c cos (4.8) Figure 4.3 is a schematic represetatio of bubble formatio ad propagatio regime showig the relevat flow parameters. For simplificatio, the bubble curvature is igored by assumig a rectagular cross-sectio for the bubbles alog the pipe axis. The applicatio of the cotiuity equatio betwee the rupture plae (sectio 1) ad full cross sectio of the pipe (sectio ) produces the followig formula for the volumetric liquid flow rate Q d : Q u A Au (4.9) d d wet or ud Awet u (4.10) A where u ad A wet are the liquid velocity i the full sectio of the pipe ad the costat wetted area durig bubble propagatio regime respectively. Figure 4.3: Schematic represetatio of bubble propagatio regime showig the relevat flow parameters Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 57

58 Also at ay give time the volumetric flow rate of the dischargig liquid is equal to that of air eterig the pipe. Therefore: Q d u A u A u A (4.11) d wet ub b db b u db ad u ub are the dowstream ad upstream bubble propagatio velocity respectively. Also Ab is the bubble cross sectioal area ad is give by: A A (4.1) b A wet Substitutig A b from the above equatio ito Equatio (4.11) produces: u d A A u u wet ub db (4.13) Awet Recallig Equatio (3.30), A may be calculated via: wet A wet D c sic (4.14) 8 Due to the absece of liquid head i the tak, the propagatio of the dowstream bubble is similar to the case of bubble propagatio i a isolated pipe. Therefore, recallig Equatio (3.17) developed by Bejami (1968) for bubble propagatio velocity i statioary liquid i the absece of fluid/pipe wall frictio: u db gd (3.17) Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 58

59 where u drift is replaced by u db. The above equatio is used i this study as it has bee referred to by various authors (see for example Bedikse (1984), Hager (1999), Lauchla et al. (005) ad Pothof (011)). For the upstream bubble Equatio (3.5) is recalled for the co-curret sigle bubble propagatio i movig liquids, iitially developed by Nickli et al. (196) for vertical pipes: u u C u (3.5) ub db 0 where u drift ad u movig are replaced by u db ad u ub respectively. By coductig experimets for various pipe diameters, Bedikse (1984) showed that with C 0 beig i the rage of , Equatio (3.5) may also be used for the case of sigle bubble propagatio i a horizotal pipe. I the preset study C 0 is take as the coservative value of 1. as it will produce the maximum u ub ad hece m d, thus represetig the worst case sceario. Bedikse (1984) proposed the above value for C 0 by oly cosiderig the upstream bubble. Cosequetly, the impact of dowstream bubble propagatio o C 0 is igored. The validity ad accuracy of the above value of C 0 for a system with bubble propagatio from both eds of the pipe is ivestigated experimetally i Sectio Replacig C 0 with 1. ad u db from Equatio (3.17) i Equatio (3.5) yields: u ub 0.54 gd 1. u (4.15) Therefore, replacig u from Equatio (4.10) ito Equatio (4.15) gives: Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 59

60 u u d Awet 0.54 gd 1. (4.16) A ub Fially the discharge velocity (u d ) may be calculated by replacig u ub ad u db from the above equatio ad Equatio (3.17) respectively ito Equatio (4.13) : u d A A wet u gd 1. A wet d A A wet (4.17) or Awet A A wet A wet 1. ud A gd (4.18) or u d gd Awet A 1. A Awet A wet (4.19) Substitutig produces: A from Equatio (4.14) ad A = wet D ito the above equatio 4 u d gd si ( si c ) 0.6 c c c ( si ) c c (4.0) Fially by equatig Equatios (4.7) ad (4.0) 16 o as solutio. Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 60 c is calculated iteratively producig

61 This is a importat fidig showig that durig bubble propagatio regime i a horizotal pipe where liquid/wall frictio may be igored (Pothof, 011), the agle of depth remais uiquely costat for all pipe legth ad diameters. The validity of this predictio is tested through experimets i Sectio By substitutig c =16 o ito Equatios (4.7), (4.8), (4.14), (4.10) ad (4.15), u d, d c, A wet, u ad u ub may be determied from: u d gd (4.1) d c D (4.) A wet 0.544D (4.3) u gd (4.4) u ub gd (4.5) Fially recallig Equatio (3.34) for critical volumetric discharge rate for mass discharge rate ( m d ) is calculated via: c, the m d 0.044D gd c si c c si c si c (4.6) Replacig c =16 o i the above equatio gives: Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 61

62 md 0.41D gd (4.7) d c, u ub ad u db are assumed to be time ivariat durig bubble formatio regime, thus producig a costat m d. As described i Sectio 3.3.1, the dowstream bubble velocity (u db ) is the same as the miimum required liquid velocity to keep a bubble stagat (Bejami, 1968; Lauchla et al., 005). I other words, the dowstream bubble propagates alog the pipe with a velocity equal to u db oly if u is less tha the calculated u db (see Equatio 3.17). Comparig Equatios (4.4) ad (3.17) for liquid velocity (u) i the full sectio of the pipe ad the dowstream bubble velocity (u db ), it may be observed that for the same pipe diameter, u is always smaller tha u db, cofirmig the propagatio of dowstream bubble upstream. I additio, the time lapsed for the dowstream ad upstream bubbles to merge, t bubble, is: t bubble L L (4.8) u u gd ub db Ope Chael Flow The mergig of the upstream ad dowstream bubbles marks the termiatio of bubble formatio regime ad start of ope chael flow. Figure 4.4 presets a horizotal pipe with ope chael flow attached to a upstream tak. Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 6

63 Figure 4.4: Ope chael flow i a horizotal pipe fed from a upstream storage tak Followig the discussio for bubble propagatio regime i Sectio 4.3.., the liquid depth at the pipe exit is equal to the critical depth. However, durig ope chael flow this depth ad cosequetly the correspodig c is ot costat ad decreases over time. The startig value of c for ope chael flow is c = 16 o, as calculated for the previous regime. For simplicity, it is assumed that the liquid depth is uiform across the pipe ad tak ad equal to the critical depth, d ad the pipe is defied as: c. Therefore M, the total mass of liquid i the tak M L L A eq wet (4.9) where L eq is the legth of a equivalet pipe of diameter D cotaiig the remaiig mass iside the tak at the commecemet of ope chael flow, M r, ad is give by: L eq M r 1 A wet (4.30) or Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 63

64 L eq M r D (4.31) Also assumig that full pipe flow termiates whe the liquid level i the tak reaches d c, M r is calculated via: M r d A (4.3) c ta k Replacig M r from the above equatio ito Equatio (4.31) ad Equatio (4.9) produces: d Ata L c k eq 0.544D (4.33) M d A c ta k L 0.544D A wet (4.34) I order to accout for the variatio of M with time, the critical discharge rate equatio (Equatio (3.34)) is combied with the trasiet flow mass balace represetig the rate of mass loss from the storage tak (Equatio (3.)): dm dt 0.044D gd si c c si c c si c (4.35) Substitutig M from Equatio (4.34) ito the above equatio gives: d c Ata d ( L 0.544D dt k ) A wet 0.044D gd c si c si c c si c (4.36) Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 64

65 or 0.044D gd da wet c sic c ta k si L c c 0.544D si c dt d A (4.37) At the iitial time t 0 = 0 at the oset of ope chael flow, A wet = A 0.544D wet from Equatio (4.3). Based o this, itegratig the above equatio betwee t = 0 ad a arbitrary time t produces: A wet 0.544D dawet t d A D gd (4.38) c si c c ta k si L c c 0.544D si c Equatio (4.38) is solved umerically by employig the trapezoidal rule where the regio uder the graph of the fuctio f(x) is approximated as a trapezoid ad its area is calculated via (Atkiso, 1989): a N 1 f ( x) dx xi 1 xi fi 1 fi b xi a (4.39) b i1 Comparig the left side of the above equatio with that of Equatio (4.38), f(x) ad x 1 represet m d ad Awet respectively. Figure 4.5 shows the flow diagram of the procedure used to estimate the above itegral. Here m d, i, A wet,i, c, i ad I i are the mass discharge rate, wetted area, agle Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 65

66 c ad the value of the itegral at time t i respectively. Oce I i is determied, the correspodig time t i is calculated via: d c Ata k t i I i L (4.40) 0.544D Figure 4.5: Calculatio flow diagram for determiig c, i at t i for a horizotal pipe attached to a upstream tak Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 66

67 Parametric Studies I the previous sectio a detailed mathematical model was developed to simulate the outflow followig a rupture i a horizotal pipe coected to a upstream storage tak cotaiig icompressible liquid. The model accouted for all the subsequet regimes icludig full pipe flow, bubble formatio ad propagatio, ad ope chael flow. This sectio presets the results of a series of parametric studies based o the applicatio of the above model. The study icludes ivestigatig chages i the pipelie legth ad ier diameter i order to ivestigate the sigificace of bubble propagatio ad ope chael flow regimes o the total ivetory loss. Table 4.1 gives the test coditios employed for the base case i the curret parameter studies. Tak cross sectioal area, A tak (m ) 5 Iitial liquid head i the tak, H o (m) 5 Ivetory Water Pipelie ier diameter, D (m) Pipelie legth, L (m) 100 Pipe roughess, (mm) 0.05 Table 4.1: Outflow simulatios test coditios for full-bore rupture sceario i a horizotal pipe fed from a upstream tak The cylidrical storage tak is assumed to have a cross sectioal area of 5m ad is coected to a 100m log, 0.356m diameter pipe. The tak is filled with water up to a depth of 5m prior to pipe failure. A pipe roughess of 0.05mm is assumed durig full pipe flow. The impact of frictio durig bubble propagatio ad ope chael flows are igored as it is assumed to be egligible. Uless otherwise specified, the characteristics give i the table are assumed to apply throughout the ivestigatios Discharge Velocity ad Wetted Area I this sectio the impact of pipelie characteristics o the discharge velocity ad wetted area (A wet ) is studied. The parametric studies are coducted for the pipelie Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 67

68 legth ad ier diameter i the rages 50-00m ad m respectively. Figure ad Figure respectively show the variatio of ormalised wetted area ad discharge velocity with time for various values of pipelie legth ad ier diameter. Normalised wetted area is defied as the ratio of the wetted area (A wet ) ad the pipe cross sectioal area (A). For discharge velocity all figures idicate three distict rages i behaviour. The first is a rapid ad sigificat liear drop correspodig to the discharge durig full pipe flow. This is the followed by a rapid ad relatively small recovery i the discharge velocity prior to reachig a costat discharge rate. Fially there is a slow o-liear declie i the discharge velocity which correspods to ope chael flow. It may be observed that the discharge velocity durig bubble formatio ad propagatio is the same for all pipelie legths. This is because the critical discharge velocity is idepedet of pipelie legth (see Equatio (4.7)). This is believed to be due to igorig fluid/wall frictio i the pipe durig this regime. The rapid icrease i discharge velocity as the flow regime chages from full pipe to bubble formatio ad propagatio is due to the impact of frictioal loss (fluid/pipe wall ad etry) durig full pipe flow. There is o etry loss durig bubble formatio regime due to the absece of liquid head i the tak. Also Pothof (011) claimed that water acceleratio alog the air bubble ose i a horizotal pipe is essetially frictioless. Thus, i the reported model the frictioal loss is ot cosidered durig this regime (see Sectio ). Cosequetly, the discharge velocity is lower at the ed of full pipe flow compared to that durig bubble formatio ad propagatio regime. Furthermore, the icrease i the magitude of the recovery i the discharge velocity with the pipelie legth is the result of the direct proportioality of fluid/pipe wall loss with pipelie legth (see Sectio..3). Therefore, the frictioal loss is more sigificat i loger pipes, resultig i greater reductio i discharge velocity durig full pipe flow. The same tred may also be observed for the impact of pipelie ier diameter o the discharge velocity. However, here the magitude of the recovery i the discharge Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 68

69 velocity decreases with pipe ier diameter. This is because the fluid/pipe wall loss is iversely proportioal to the pipe ier diameter ad thus, the impact of pipe wall loss o discharge velocity for large diameters is ot as sigificat as for small diameters. O the other had, the ormalised wetted area remais costat ad equal to 1 durig full pipe flow. This stems out from the defiitio of full pipe flow, i.e. A wet = A. The A wet drops to 0.693, show below, as the regime chages to bubble formatio ad propagatio. As assumed i the model, A wet remais uchaged throughout this regime. Upo iitiatio of ope chael flow alog the pipe, A wet decreases slowly util reachig zero, markig the termiatio of the draiage process. A wet A A wet A 0.544D D (4.41) 4 Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 69

70 Full pipe flow Bubble formatio ad propagatio Ope chael flow Figure 4.6: Impact of pipelie legth o the variatio of ormalised A wet with time followig full-bore rupture i a 0.356m diameter, horizotal pipe fed from a upstream tak Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 70

71 Full pipe flow Bubble formatio ad propagatio Ope chael flow Figure 4.7: Impact of pipelie legth o the variatio of the discharge velocity with time followig full-bore rupture i a 0.356m diameter, horizotal pipe fed from a upstream tak Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 71

72 Full pipe flow Bubble formatio ad propagatio Ope chael flow Figure 4.8: Impact of pipelie diameter o the variatio of ormalised A wet with time followig full-bore rupture i a 100m log, 0.305m diameter, horizotal pipe fed from a upstream tak Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 7

73 Full pipe flow Bubble formatio ad propagatio Ope chael flow Figure 4.9: Impact of pipelie diameter o the variatio of discharge velocity with time followig full-bore rupture i a 100m log, horizotal pipe fed from a upstream tak Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 73

74 Normalised Cumulative Discharged Mass Figure 4.10 shows the impact of the chage i the pipe legth i the rage m o the variatio of the ormalised cumulative discharged mass (ratio betwee cumulative discharged mass ad iitial ivetory i the system) with time. Figure 4.11 shows the correspodig data demostratig the impact of the chage i the pipe diameter i the rage m. I each figure, the trasitio from full pipe to bubble propagatio, ad from bubble propagatio to ope chael flow are marked by black circular ad triagular dots respectively. It may be observed from Figure 4.10 that the released mass durig post-full pipe flow icreases with the pipelie legth. For the logest pipelie (300m), more tha half of the iitial ivetory is released durig post-full pipe flow, whereas almost 70% of the mass is released durig full pipe flow for the shortest pipelie (100m). The same tred ca be see i Figure 4.11 for the impact of pipelie ier diameter. For the pipe with the largest ier diameter (0.508m), almost half of the ivetory is released durig bubble formatio ad ope chael flow. This study emphasises the sigificace of takig accout of such flow regimes i log distace or large diameter pipelies. Igorig these regimes ca result i sigificat uderestimatio of the released mass. Table 4. presets the compariso betwee the predicted ad theoretical discharged mass durig each regime. The theoretical mass released durig each flow regime is calculated as follow: Full pipe flow M full pipe flow Ata k H 0 c d (4.4) or M Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies A H 0. D (4.43) full pipe flow ta k

75 where 0.654D is the critical depth, d c from Equatio (4.). Bubble formatio ad propagatio M bubble propagatio 1 L D 4 L A A 0.544D wet 0.41LD (4.44) Ope chael flow 0.654DAta k 0.544D L M ope chael flow (4.45) 0.544D or M ope chael flow 1. 0Ata k 0.544D L (4.46) D where 0.544D ta ad DA k are A wet ad L eq from Equatios (4.3) ad 0.544D (4.33) respectively. O the other had, the predicted released mass durig each flow regime M predicted is calculated via: M predicted m t (4.47) d where Δt is the duratio of the flow regime. Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 75

76 It may be observed from Table 4. that for all scearios, durig each regime the predicted released mass is very close to the correspodig theoretical mass, with a maximum deviatio of 1%. Sceario Full pipe flow Bubble propagatio Ope chael flow Legth (m) Diameter (m) Theoretical (kg) Predicted (kg) Theoretical (kg) Predicted (kg) Theoretical (kg) Predicted (kg) Table 4.: Compariso of the theoretical ad predicted discharged mass durig idividual regimes followig full-bore i a horizotal pipe fed from a upstream tak Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 76

77 Trasitio from bubble formatio to ope chael flow Trasitio from full pipe flow to bubble formatio Figure 4.10: Impact of pipelie legth o the ormalised cumulative discharged mass followig full-bore rupture i a 0.356m diameter, horizotal pipe fed from a upstream tak Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 77

78 Trasitio from bubble formatio to ope chael flow Trasitio from full pipe flow to bubble formatio Figure 4.11: Impact of pipelie diameter o the ormalised cumulative discharged mass followig full-bore rupture i a 100m log, horizotal pipe fed from a upstream tak Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 78

79 Experimets The followig presets the results of a series of experimets coducted to validate the outflow model developed above. The first series of experimets are carried out to validate the developed equatios for the liquid depth below the bubble, d c (see Equatio (4.)) ad the upstream bubble propagatio velocity, u ub (see Equatio (4.5)) for water i a pipe upo a simulated full-bore rupture. The validity of the proposed value of 1. for the empirical coefficiet C 0 correspodig to bubble propagatio velocity by Bedikse (1984) for the case of bubble propagatio from both eds of a horizotal pipe attached to a upstream tak is ivestigated. Based o the secod series of experimets, the accuracy of the model developed for predictig the trasiet discharge rate followig full-bore rupture i a horizotal pipe fed from a storage tak cotaiig water is examied through measurig the cumulative discharged mass ad estimatig the correspodig discharge rate. The measured values are compared agaist those obtaied from the model Upstream Bubble Propagatio Velocity u ub ad Liquid Depth d c Figure 4.1 presets the experimetal setup costructed to validate Equatios (4.) ad (4.5) for the liquid depth below the bubble ( d propagatio velocity ( u ub ) respectively. c ) ad the upstream bubble Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 79

80 Figure 4.1: Experimetal setup to measure the bubble propagatio velocity (u ub ) ad the liquid depth below the bubble ( d c ) followig full-bore rupture i a horizotal pipe: 1) Stailess steel pipe ) Coectig clamps 3) Deaeratig orifice 4) Coectig flages 5) Acrylic pipe 6) Plug with a screw The acrylic pipe (5) is.5m log with 0.038m ier diameter. The fluid/wall frictio is ulikely to be sigificat as a result of the pipe material ad smooth ier wall. A stailess steel pipe (1), with the same ier diameter as the acrylic pipe represets the upstream tak. Whe fillig the pipe, the deaeratig orifice o top of the pipe is opeed ad closed frequetly to release ay trapped air bubbles i the system. Oce the pipe is full, the plug at the ed of the pipe (6) is rapidly uscrewed maually to simulate the full-bore rupture. I order to measure the bubble propagatio velocity, Kodak Ektapro high speed motio aalyser, model 4540 is used to record the bubble locatio at the speed of 500 frame/s. I order to reduce the impact of reactio error, the Motio Aalyser is switched o before the rubber plug is maually removed. Maually uscrewig the plug ca potetially cause some error as it takes some time for the screw to be ufasteed completely to simulate full-bore rupture. The mass released before the complete Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 80

81 removal of the plug ca potetially have a impact o the discharge rate ad cause a uderestimatio of the discharged mass. Figure 4.13 presets a selectio of the photographs take at differet time itervals followig the simulated pipe rupture usig the experimetal setup show i Figure 4.1. I order to calculate the upstream bubble velocity (u ub ), the distace travelled by the bubble from the photographs take at a give time iterval t is calibrated to obtai the actual distace approach adopted for the distace calibratio: xactual. The followig steps preset the 1. Measurig the pipe outer diameter from the photographs.. Fidig the correctio factor by dividig the real pipe outer diameter (i this case 0.056m) by the measured value from step Multiplyig the measured distace travelled by the bubble by the correctio factor from step to calculate. xactual Oce xactual is determied, the bubble velocity may be calculated via: u ub xactual (4.48) t I the above equatio is also calculated from the followig equatio: Frame Frsme1 t (4.49) 500 frame / s Here Frame1 ad Frame are the frame umbers reported by the Motio Aalyser for each photograph. Figure 4.14 shows the results i the form of a ratio betwee the measured ad predicted bubble velocity versus time. From the figure it may be observed that the predicted values for u ub from Equatio (4.5) are i relatively good agreemet with Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 81

82 experimetal data. The measured values rage betwee 75%-99% of the predicted values, which implies that the model is coservative. I additio, based o the photographs, the average measured liquid depth below the dowstream ad upstream bubbles ( d c ) is 0.04m. The correspodig liquid depth agle c = 10 o. This is i remarkably close agreemet with the predicted value of 16 o based o simultaeous solutio of Equatios (4.7) ad (4.0) (see Sectio 4.3..). These results cofirm the validity of Bedikse s (1984) suggested value for empirical coefficiet C 0 for a system with upstream storage tak ad two bubbles propagatig towards each other. Systematic ad Experimetal Errors Ucertaity i a experimet is due to either experimetal error or systematic error. Experimetal errors are statistical fluctuatios i the measured data as a result of the precisio limitatios of the measuremet device. Experimetal errors usually result from the experimeter's iability to take the same measuremet i exactly the same way to get exact the same umber. Systematic errors, o the other had, are reproducible iaccuracies that are cosistetly i the same directio. Systematic errors are ofte due to a problem which persists throughout the etire experimet. Based o the above defiitio, possible systematic ad experimetal error durig measurig the bubble propagatio velocity i horizotal pipes are: Systematic Errors 1. As described previously, here FBR is simulated by maually uscrewig ad removig the plug. As such, FBR does ot occur istataeously, resultig i releasig some mass prior to FBR. The liquid velocity durig this trasiet phase is lower tha that upo FBR. As a result, iitially bubble propagates slower tha the predicted value by the model (see Figure 4.14). The bubble propagatio velocity icreases oce the plug is fully removed. Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 8

83 . Maually uscrewig the plug also causes vibratios to the pipe ad cosequetly to the cotaied water. This will have a impact o the bubble propagatio velocity, producig scattered results for bubble propagatio velocity iitially. Oce the plug is fully removed ad the pipe is stable, the measured results for the bubble propagatio velocity become closer, evetually reachig the predicted value by the model (see Figure 4.14). Experimetal Errors 1. I order to miimise the umber of bubbles trapped alog the pipe while fillig the system with water, deaeratig orifice is opeed ad closed frequetly. However, sometimes there are still small bubbles trapped o top of the pipe. This will have a impact o the bubble propagatio velocity as the presece of these bubbles itroduces a ew frictioal loss betwee the bubbles ad the water.. Sice the process of removig the plus is ot automatic ad thus ot istataeous, the time take to uscrew the plug ca vary from oe experimet to aother. The loger it takes to maually uscrew the plug, the more mass is discharged prior to FBR, resultig i larger deviatio of the measured bubble propagatio velocity from the predicted value for FBR from the model. 3. I order to produce the upstream bubble, the liquid level i the stailess steel pipe represetig the upstream tak has to be aroud the critical depth. If the liquid head is much higher tha the critical depth, due to short legth of the pipe ad high iitial liquid mometum, the dowstream bubble will ot fully propagate towards the upstream bubble. O the other had, very small liquid head i the stailess still pipe causes the upstream bubble to form at the pipe ilet eve before the plug is removed. Both cases will produce differet results for bubble propagatio velocity for the cofiguratio cosidered i this study. Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 83

84 t = 4.78s t = 4.86s t = 4.94s t = 5.0s t = 5.10s t = 5.18s Figure 4.13: Sapshots take at the speed of 500frame/s from bubble propagatio i 0.038m ier diameter,.5m log, horizotal pipe Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 84

85 Figure 4.14: Variatio of (measured u ub / predicted u ub ) with time followig fullbore rupture i a 0.038m ier diameter,.5m log, horizotal pipe Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 85

86 Normalised Cumulative Discharged Mass ad Discharge Rate This sectio presets the results from a series of experimets coducted to determie the accuracy of the outflow model developed i Sectio 4.3. i predictig the trasiet discharge rate. Figure 4.15 shows the experimetal setup used to measure the cumulative discharged mass. Figure 4.15: Experimetal setup to measure the cumulative discharged mass followig full-bore rupture i a horizotal pipe fed from a plastic cotaier 1) Plastic cotaier ) Cable glad 3) Acrylic pipe 4) Rubber bug A 0.034m ier diameter, 3m log, acrylic pipe (3) is coected to a 0.05m 3, cylidrical plastic cotaier (1). Silicoe sealat is used at the coectio poit betwee the pipe ad the tak to avoid leakage. The ope ed of the pipe is iitially closed by a rubber bug (4). Oce the system is full, the rubber bug is removed to simulate the full-bore rupture. Similar to the setup used for measurig the bubble propagatio velocity, the bug is removed maually. Although more accurate, still this ca be a source of error as there will be some discharge before the complete removal of the ruer bug. The discharged water is collected i aother plastic cotaier placed o top of a Mettler PM30K idustrial balace whereby usig a LabView based software (Natioal Istrumets Compay, 011), the cumulative discharged mass is measured as a fuctio of time throughout the draiage of the Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 86

87 system. I order to mitigate the error i measurig the discharged mass, the plastic cotaier used here has a cross sectioal area almost the same size as the scale surface area. However, due to high liquid velocity the discharged water creates vibratios oce it hits the cotaier. This ca have a impact o the accuracy of the measured discharged mass. As such, the plastic cotaier has a relatively large height ot oly to miimise this impact, but also to prevet the icomig water from splashig outside the cotaier. The LabView-based software reports the discharged mass i the form a curve as a fuctio of time. I order to extract the data from the curve, Egauge Digitiser 4.1 (Mitch, 007) is used. The software trasfers the correspodig data from the curve, M i the cumulative discharged mass at t = t i, ito a Excel spreadsheet. The the discharge rate at t = t i, m d, i, is calculated via: M M i i1 m d, i (4.50) t i t i1 Figure 4.16 shows the compariso betwee the predicted ad measured ormalised cumulative discharged mass with time. Normalised cumulative discharged mass is defied as the ratio of the measured cumulative discharged mass ad system iitial ivetory. As metioed i Sectio (Equatio (.14)), the resistace coefficiet for the etry loss (K) is assumed to be 0.5 i the discharge model. The ed of full pipe flow for the model is idicated i the figure. It may be observed from the figure that durig full pipe flow, the model overestimates the cumulative discharged mass. As the regime chages from full pipe to bubble formatio, the deviatio of the predicted results from the measured values reduces to less tha 14%. The overestimatio of the cumulative discharged mass by the model durig full pipe flow ca be due to the additioal frictioal loss caused by the silicoe sealat used at Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 87

88 the pipe etrace iside the tak. I order to test the impact of this additioal frictioal loss o the discharge rate, the cumulative discharged mass is also calculated for K i the rage of The results are preseted i Figure 4.17 for K =.5 which produces the closest results to the measured value. With this ew value for K, the model predictios are very close to the measured data from the experimets. The maximum deviatio betwee the predicted ad measured values durig full pipe flow ad post-full pipe flow (bubble formatio ad ope chael flow) are 0% ad % respectively. Figure 4.18 shows the compariso betwee the measured discharge rate with the predicted values from the models with K = 0.5 ad K =.5. From the graph, it seems that upo iitiatio of full pipe flow the measured discharge rate is closer to the predicted values by the model with K = 0.5. However, towards the ed of this regime ad durig the subsequet regimes, bubble formatio ad ope chael flow, the model with K =.5 predicts more accurate results. Oe possible explaatio for the iitial agreemet betwee the measured ad predicted values for K = 0.5 could be the presece of large liquid head i the tak at the outset. As a result, there is egligible impact of additioal frictioal loss imposed by silicoe sealat upo iitiatio of full pipe flow. Cosequetly, usig K =.5 results i uderestimatio of discharge rate. Systematic ad Experimetal Errors Systematic Errors The measured total discharged mass at the ed of the draiage process is always slightly less tha the system iitial mass. Because: 1. Due to the maual removal of the rubber bug, there will be some release prior to FBR. Part of the released water durig this phase will ot be cotaied i the cotaier as it is ot a oe dimesioal flow.. Due to the pipe curvature, there will be some ivetory left at the base of the pipe eve after the draiage is complete. 3. Oce the discharged water hits the cotaier, it will partly bouce back ad splash outside the cotaier. Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 88

89 Experimetal Errors 1. Due to the high liquid discharge velocity there will be some vibratios imposed o the cotaier. I additio, part of discharge hits the cotaier wall first, rather tha the base. These two have a impact o the measured mass of the liquid by the scale, producig scattered results for the measured discharged mass. The LabView-based software takes the average of the measured results, producig a smooth curve as preseted i Figure Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 89

90 Ed of full pipe flow Figure 4.16: Compariso of the measured ormalised cumulative discharged mass with the predicted values from the model with K = 0.5 followig full-bore rupture i a horizotal pipe fed from a plastic cotaier Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 90

91 Figure 4.17: Compariso of the measured ormalised cumulative discharged mass with the predicted values from the model with K =.5 followig full-bore rupture i a horizotal pipe fed from a plastic cotaier Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 91

92 Figure 4.18: Compariso of the measured discharge rate with the predicted values from the models with K = 0.5 ad K =.5 followig full-bore rupture i a horizotal pipe fed from a plastic cotaier Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 9

93 4.4. Trasiet Hydraulic Flow Modellig Followig Full-bore Rupture of a Isolated Horizotal Pipelie Itroductio This sectio is cocered with the formulatio of various flow equatios for predictig the discharge rate followig full-bore rupture from the ed of a isolated horizotal pipe cotaiig a icompressible liquid. The pipe is assumed to be closed at oe ed due to emergecy shut dow, while the liquid releases from the rupture plae at the other ed of the pipe. Due to the absece of a upstream storage tak, the iitial full pipe flow regime is ot preset. Also i practice, a iitial discharge upo valve closure happes over a short period of time before the bubble formatio regime starts. Give the liquid fiite compressibility, the liquid mass released durig this rapidly trasiet regime is egligible ad thus igored i this study. Therefore, oly the flow regimes of bubble formatio ad ope chael flow are cosidered here. The associated theory is discussed i Sectio I Sectio the outflow model is tested through parametric studies by ivestigatig the impact of pipelie ier diameter ad legth o the discharge velocity ad wetted area. The released mass durig idividual regimes are also calculated ad compared agaist the correspodig theoretical values. Fially, the accuracy of the developed model is assessed through a series of experimets, preseted i Sectio Model Theory Bubble Formatio ad Propagatio Upo rupture at the ed of a isolated pipe, a bubble forms at the rupture plae ad propagates alog the pipe util it reaches the closed-ed. Figure 4.19 presets the bubble propagatio i a horizotal pipe with oe closed-ed. Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 93

94 Figure 4.19: Bubble propagatio i a isolated horizotal pipe Give the icompressibility of the liquid, the volume of the released liquid is the same as the volume of the air eterig the pipe. Therefore, the discharge velocity may be expressed as: u A u (4.51) d db b Awet or u d u A A db wet (4.5) Awet A wet may also be calculated from Equatio (4.14) : A wet D c si c (4.14) 8 u db is also determied by usig Equatio (3.17) proposed by Bejami (1968): u db gd (3.17) Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 94

95 O the other had, as described i Sectio.3.5, the liquid depth at the pipe exit ad below the bubble is the same as the critical depth. Therefore, Equatio (4.7) is recalled for the discharge velocity: u d gd c si c (4.7) si( / ) c Substitutig u db ad A wet the followig equatio for the discharge velocity: from Equatios (3.17) ad (4.14) ito Equatio (4.5) gives c si c ud gd (4.53) si c c Fially equatig Equatios (4.7) ad (4.53) ad solvig iteratively produces 175 o for. Similar to the pipe attached to a upstream tak, i the absece of frictioal c loss, the above value is idepedet of pipe characteristics. Isertig the value c = 175 o ito Equatios (4.6), (4.53) ad (4.14), m d, ud ad A wet ca be respectively determied via: md 0.5D gd (4.54) u d gd (4.55) A wet 0.371D (4.56) Cosequetly, the time required for the bubble to reach the closed-ed may be calculated from the followig equatio: Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 95

96 t bubble L L (4.57) u 0.54 gd db Ope Chael Flow As soo as the elogated bubble reaches the closed-ed of the pipe, ope chael flow prevails. Figure 4.0 shows a isolated horizotal pipelie with ope chael flow. Figure 4.0: Ope chael flow i a isolated horizotal pipe The model for this regime is similar to the described model i the presece of upstream storage tak i Sectio , except that the equivalet legth of the tak is ot applicable here. Therefore, recallig Equatio (4.37) with L eq = 0 gives: 0.044D da gd si c wet c c sic c si dt L (4.58) o Here at t = 0, c c 175 ad thus A wet = (4.56). Based o this, the above equatio is itegrated to produce: A wet which is calculated from Equatio A wet 0.371D dawet t L c si c 0.044D gd c si c (4.59) c si Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 96

97 Oce agai Equatio (4.59) must be solved umerically. Oce agai the procedure preseted i Figure 4.5 is used to evaluate the above itegral, except that here istead of o o,.oce the value of the itegral I i is determied, the c 16 c 175 correspodig time t i is calculated via: t i I L i (4.60) Parametric Studies I the previous sectio a detailed mathematical model was developed to simulate the outflow followig a rupture at the ed of a isolated horizotal pipe. Due to the absece of upstream storage tak, oly bubble propagatio ad ope chael flow were cosidered here. This sectio presets the results of a series of parametric studies based o the applicatio of the above model. Similar to the verificatio of the developed model with a upstream tak, the impact of the chages i the pipelie legth ad ier diameter o the discharge velocity ad the ormalised cumulative discharged mass durig idividual regimes are studied here. The characteristics give i Table 4.1 for the verificatio of the model with a upstream storage tak are also applicable here except the iitial liquid head. The impact of pipe frictio is igored here as it is assumed to be egligible durig bubble formatio ad ope chael flow Discharge Velocity ad Wetted Area I this sectio the impact of pipelie characteristics o the discharge velocity ad wetted area (A wet ) is studied. The parametric studies are coducted for the pipelie legth ad ier diameter i the rages 50-00m ad m respectively. Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 97

98 Figure 4.1 ad Figure 4. show the impact of pipelie legth ad diameter o the trasiet discharge velocity. O the other had, the impacts of the above characteristics o A wet are preseted i Figure 4.3 ad Figure 4.4. I all the graphs, the iitial horizotal lie shows the discharge velocity ad wetted area durig bubble formatio ad propagatio regime. This is the followed by a slow o-liear declie i the discharge velocity ad wetted area correspodig to ope chael flow. As it may be observed from the figures, the discharge velocity durig the bubble propagatio regime icreases with pipe ier diameter due to icrease i bubble propagatio velocity (Equatio (3.17)). However, the discharge velocity durig this regime is ot depedet o pipe legth. This ca be see from Figure 4.1 where for all pipe legths, the discharge velocity remais uchaged. The time required for the bubble to reach the closed-ed is show to be proportioal to the pipelie legth L, i lie with Equatio (4.57). O the other had, the ormalised wetted area durig bubble propagatio regime remais costat ad equal to 0.473, give below, regardless of pipelie legth ad ier diameter. This stems out from a costat c obtaied i Sectio based o the assumptio of egligible frictioal loss for all pipelie characteristics. A wet A A wet A 0.371D D (4.61) 4 Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 98

99 Figure 4.1: Impact of pipelie legth o the variatio of discharge velocity with time followig full-bore rupture i a 0.356m diameter, isolated, horizotal pipe Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 99

100 Figure 4.: Impact of pipelie diameter o the variatio of discharge velocity with time followig full-bore rupture i a 100m log, isolated, horizotal pipe Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 100

101 Figure 4.3: Impact of pipelie legth o the variatio of ormalised A wet with time followig full-bore rupture i a 0.356m diameter, isolated, horizotal pipe Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 101

102 Figure 4.4: Impact of pipelie diameter o the variatio of ormalised A wet with time followig full-bore rupture i a 100m log, isolated, horizotal pipe Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 10

103 Released Mass durig Idividual Flow Regimes Table 4.3 presets the compariso betwee the predicted ad theoretical discharged mass durig each regime for pipelie legth ad diameter i the rages m ad m respectively. The calculatio for the theoretical released mass durig each regime is similar to that described i Sectio except that oly bubble propagatio ad ope chael flow are relevat here. I additio, due to the absece of upstream tak, equivalet legth, L eq (see Equatio (4.31)) is also ot applicable. Bubble formatio ad propagatio M bubble propagatio 1 L D 4 L A A 0.371D wet 0.414LD (4.6) Ope chael flow M ope chael flow L 0.371D (4.63) where 0.371D is A wet from equatios (4.56). The predicted released mass durig each flow regime, M predicted, is calculated from Equatio (4.47): M predicted m t (4.47) d Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 103

104 Sceario Bubble propagatio Ope chael flow Legth (m) Diameter (m) Theoretical (kg) Predicted (kg) Theoretical (kg) Predicted (kg) Table 4.3: Compariso of the theoretical ad predicted discharged mass durig idividual regimes followig full-bore i a isolated horizotal pipe Oce agai it is clear from the table that the predicted ad the expected mass to release are very close for all scearios, less tha % deviatio Error Aalysis ad Depedece of Covergece o c This sectio presets the error aalysis for the draiage process icludig bubble formatio ad ope chael flow i a horizotal pipe. I additio, the impact of the icremet c o the covergece i trapezoidal rule used to estimate Equatio (4.59) is ivestigated. The default value is set to 1, as preseted i Figure 4.5. Figure 4. 5 shows the discharge rate versus time for various values of. The pipe legth ad diameter are 100m ad 0.356m respectively. As it may be observed from the graph, the covergece i the trapezoidal method used to estimate Equatio (4.58) is ot affected by the icremet c. c O the other had the impact of the icremet Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 104 c o the accuracy of the developed model i calculatig the total expelled mass is preseted i Figure Axis y shows total expelled mass where it is defied as predicted total expelled mass mius theoretical expelled mass. The theoretical values are calculated based o the approach described i Sectio It may be observed from the figure that as expected, reductio of mass from the theoretical values. c results i less deviatio of the predicted total expelled

105 Figure 4. 5: Impact of c o the covergece i trapezoidal method i a 100m log, 0.356m diameter, horizotal pipe Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 105

106 Figure 4. 6: Impact of c o the accuracy of the developed model for a 100m log, 0.356m diameter, horizotal pipe i predictig the total expelled mass Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 106

107 Experimets The followig presets the results from a series of experimets coducted to determie the accuracy of the above model by compariso of its predictios for the cumulative discharged mass ad trasiet discharge rate agaist measured data. The discharge rate from the experimets are calculated from Equatio (4.50) based o the measured cumulative discharged mass. Figure 4.7 shows the experimetal setup costructed for the model validatio. Here, the 0.034m ier diameter, 3m log, acrylic pipe () ilet is closed by usig a rubber bug (1). Oce the system is full, the rubber bug (3) is removed to simulate the fullbore rupture, while rubber bug (1) remais as the closed-ed of the pipe. The discharged water is collected i a plastic cotaier placed o top of a Mettler PM30K idustrial balace whereby usig a LabView based software (Natioal Istrumets Compay, 011), the cumulative discharged mass is measured as a fuctio of time throughout the draiage of the system. Figure 4.7: Experimetal setup to measure the cumulative discharged mass followig full-bore rupture i a isolated horizotal pipe 1 & 3) Rubber bug ) Acrylic pipe Figure 4.8, Figure 4.9 ad Figure 4.30 respectively show the compariso betwee the predicted ad measured discharge rate i liear ad logarithmic scale, ad measured ormalised cumulative discharged mass respectively. It may be observed from the figures that the measured discharge rate durig bubble propagatio regime margially varies, i lie with the assumptio of costat Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 107

108 discharge durig this regime i the model. Although the pipe is acrylic ad thus very smooth, there might still be some frictioal loss. The slight icrease i the discharge ca be due to the reductio of frictioal loss as the full sectio of the pipe decrease towards the ed of bubble propagatio regime. This ca potetially explai the over predictio of the discharge rate by the model as i developig the flow equatios, the frictioal loss is assumed to be egligible (see Sectio ). O the other had, durig ope chael flow the measured discharge rate is slightly higher tha the predicted values, showig the egligible impact of frictioal loss durig this regime. Based o the above, as expected, the predicted cumulative discharged mass is higher tha the measured values. The differece betwee the two reduces as the measured discharge rate durig ope chael flow becomes greater tha the predicted vales. Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 108

109 Figure 4.8: Compariso of the measured discharge rate with the predicted values from the model i liear scale followig full-bore rupture i a isolated horizotal pipe Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 109

110 Figure 4.9: Compariso of the measured discharge rate with the predicted values from the model i logarithmic scale followig full-bore rupture i a isolated horizotal pipe Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 110

111 Figure 4.30: Compariso of the measured ormalised cumulative discharged mass with the predicted values from the model followig full-bore rupture i a isolated horizotal pipe Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 111

112 4.5. Coclusio This chapter preseted the formulatio, verificatio ad validatio of mathematical models for simulatig the trasiet outflow followig full-bore rupture of a horizotal pipelie cotaiig a icompressible liquid. Two cofiguratios were cosidered for the system: Pipelie fed from a upstream storage tak Isolated pipelie where the pipe was closed at oe ed followig emergecy shut dow Historically, i order to model the trasiet discharge rate followig full-bore rupture i a pipe attached to a upstream tak, the pipe was assumed to remai full throughout the discharge process. Based o this assumptio, full pipe flow theory was the applied to simulate the outflow. I this chapter, two additioal flow regimes amely bubble formatio ad propagatio, ad ope chael flow were modelled to simulate the complete draiage of the system. I order to model the bubble formatio ad propagatio alog the pipe, Bejami s (1968) proposed equatio for bubble propagatio velocity was used to calculate the velocity of the bubble formed at the pipe dowstream. As suggested by Nickli et al. (196), the upstream bubble velocity was assumed to be a fuctio of the dowstream bubble velocity ad the liquid velocity i the full sectio of the pipe. A value of 1. was employed for the empirical coefficiet C o as proposed experimetally by Bedikse (1984) for the case of sigle, co-curret bubble propagatio i a horizotal pipe. By employig the above value for C o ad assumig critical flow throughout the pipe, a system of 6 equatios was produced ad solved simultaeously, givig 16 as solutio for the liquid depth agle c. Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 11

113 Based o the assumptio of egligible frictioal loss durig this regime, the calculated value for c was foud to be idepedet of pipe characteristics such as pipe ier diameter ad legth. Based o this value ad assumig a uiform, timeivariat liquid depth alog the pipe, the discharge velocity ad discharge rate durig this regime were determied. For the ope chael flow, assumig a uiform critical depth alog the pipe ad egligible frictioal loss, the trasiet flow mass balace equatio was coupled with the critical discharge equatio. The resultig itegral was the solved umerically by employig trapezoidal rule. The fial equatio predicted the variatio of c ad cosequetly the discharge rate ad velocity with time. The overall model for the cofiguratio with upstream tak was verified through parametric studies to ivestigate the impact of pipelie legth ad ier diameter o the discharge velocity, wetted area ad cumulative discharged mass. It was observed that for all rages of pipelie legth ad diameter, there was a small but rapid recovery i the discharge velocity at the trasitio from full pipe to bubble formatio regime. The magitude of this recovery icreased with pipelie legth, but decreased with pipe ier diameter due to impact of fluid/pipe wall frictioal loss. This was believed to be the model artefacts ad as a result of igorig frictioal loss durig post-full pipe flow. Further ivestigatios ca clarify this matter. I additio, it was show that for log or large-diameter pipelies, majority of the mass was released durig post-full pipe flow. Therefore, igorig bubble propagatio ad ope chael flow regimes would result i sigificat uderestimatio of released mass for such pipelies; sometimes by up to 50%. Through series of experimets ivolvig short pipes with low pipe roughess, the validity of the empirical coefficiet C 0 = 1. suggested by Bedikse (1984) for the case of sigle, co-curret bubble propagatio was tested for the case of upstream bubble propagatio i the presece of dowstream bubble ad upstream storage tak. It was observed that the predicted results from the model by usig the above value Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 113

114 for C 0 were i relatively good agreemet with the experimetal results, with less tha 0% deviatio o average. I additio, the agle c correspodig to the liquid depth below the bubble measured from the sapshots was 10 ; remarkably close to the calculated value of 16 from the model. I order to measure the accuracy of the model i predictig the discharge rate, the cumulative discharged mass was measured. The discharge rate was the calculated from the measured released mass. Based o the compariso with the experimetal results it was observed that the proposed model sigificatly overestimated the discharge rate ad cosequetly cumulative discharged mass durig full pipe flow. It was foud that the overestimatio was due to additioal etry loss (K) caused by the silicoe sealat used at the pipe etrace to avoid leakage. The best fit to the experimetal results was obtaied for K =.5 istead of the origial value of 0.5. With this value, apart from a short period of time at the begiig of discharge, the maximum deviatio betwee the predicted ad measured cumulative discharged mass reduced to 0% throughout the etire draiage. The other flow cofiguratio cosidered i this study was the full-bore rupture of a isolated horizotal pipelie with oe closed-ed. Give the absece of a upstream tak, oly bubble formatio ad ope chael flow were applicable here. The model based o the applicatio of Bejami s proposed bubble velocity equatio alog with the critical discharge equatio produced 175 for. c The model was also validated through a series of experimets by measurig the cumulative discharged mass ad calculatig the discharge rate. From the results it was observed that the discharge rate remaied almost uchaged durig the bubble propagatio regime, i lie with the assumptio made i the model. The slight icrease i the measured discharge rate towards the ed of this regime was believed to be the result of the reductio i the frictioal loss, although very small, as the bubble approached the closed-ed. Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 114

115 Based o the validatios results, it ca be cocluded that the developed models for a horizotal pipe with ad without a upstream storage tak provide reasoably accurate results for sufficietly short pipes with a small value of pipe roughess. The applicability of the model for loger pipes with larger values of pipe roughess requires additioal experimetal work. Chapter 4: Outflow Simulatio upo Full-Bore Rupture i Horizotal Pipelies 115

116 CHAPTER 5: IMPACT OF PIPELINE INCLINATION ANGLE ON THE OUTFLOW FROM PIPELINES 5.1. Itroductio Liquid trasport pipelies are ot always horizotal. A pipelie slope ca vary from rather flat to almost vertical depedig o the process specificatio or the pipe locatio terrai. Normally whe the pipelie icliatio is greater tha 6 o, the pipe is called steep (Aka, 006). The full pipe flow model developed by Joye & Barrett (003) is applicable to both horizotal ad dowward-iclied pipes ad accouts for frictioal losses for ay icliatio. However, as described i the previous chapter, i horizotal pipelies the fluid/pipe wall loss is egligible durig post-full pipe flow due to low liquid velocity. However, i the case of dowward-iclied pipelies, it eeds to be take ito accout. This chapter first focuses o developig a mathematical model for bubble formatio ad ope chael flow i dowward-iclied pipelies fed from a upstream tak followig full-bore rupture. The extesio ad modificatio of the model developed earlier for isolated horizotal pipelies to accout for pipe icliatio agle is preseted ext. For both cofiguratios, the Darcy-Weisbach equatio is employed istead of critical discharge formula to accout for gravity ad frictioal losses. The above is followed by the mai coclusios. The overall models icludig full pipe flow (for the cofiguratio with upstream tak), bubble formatio ad propagatio, ad ope chael flow are tested through parametric studies ad series of experimets. Chapter 5: Impact of Pipe Icliatio Agle o the Outflow from Pipelies 116

117 5.. Key Models Assumptios The key assumptios listed i Sectio 4. for horizotal pipelies are also employed to the model for dowward-iclied pipes Trasiet Hydraulic Flow Modellig Followig Full-bore Rupture of Dowward-iclied Pipelies Itroductio This sectio is cocered with the developmet ad testig of a mathematical model to simulate the outflow upo full-bore rupture i a dowward-iclied pipelie fed from a upstream tak. Similar to the horizotal pipe, the model icludes full pipe flow, bubble formatio ad propagatio, ad ope chael flow.the full pipe flow model developed by Joye & Barrett (003) is also used here to simulate the discharge durig this regime. Durig the bubble formatio ad propagatio regime, two distictive flow patters ca be assumed. The first patter is similar to the flow behaviour i the horizotal pipe, i.e. bubbles propagatig from both eds of the pipe. I the secod patter, due to the impact of pipe icliatio agle, oly the bubble formed at the pipe ilet will propagate alog the pipe. At the pipe outlet, istead of a bubble, there will be a stagat cavity which is washed away by the approachig upstream bubble at the ed of this regime. I order to determie the prevailig patter, the predicted drift velocity (u drift ), described i Sectio 3.3.1, is compared agaist the liquid velocity i the full sectio of the pipe (u). For pipes with u lower tha u drift the first patter is applicable, whereas the secod patter is domiat i pipes with equal or higher u tha u drift. Therefore, i this chapter first the previously developed model for the horizotal pipe is modified by itroducig the icliatio agle, based o the assumptio of two bubbles propagatig towards oe aother. This is the followed by a alterative model for the secod patter where the bubble oly propagates from the upstream of the pipe. The two models are the tested through a sesitivity aalysis by studyig Chapter 5: Impact of Pipe Icliatio Agle o the Outflow from Pipelies 117

118 the impact of pipelie legth, ier diameter ad pipe icliatio agle o the liquid velocity i the full sectio of the pipe (u). Based o the results ad by applyig the drift velocity method, the domiat patter is determied for various pipelie characteristics. The acuracy of the correspodig model is assessed through series of experimets. I additio, the impact of the above pipelie characteristics o the discharge velocity, wetted area ad cumulative discharged mass is studied. Ultimately, based o the results from the parametric studies, a approximate relatioship betwee the startig agle for ope chael flow,, ad the icliatio agle,, is established Model Theory Full Pipe Flow Similar to model for the case of a horizotal pipelie, Equatio (4.4) is used to calculate the trasiet discharge velocity for full pipe flow: u d g ( 4 fl 1.5 D A A ta k ) t gz 0 fl 0.75 D (4.4) ad i terms of discharge rate m : d m d g A g( H 0 Lsi ) A ( ) t (5.1) 4 fl A fl 1.5 ta k 0.75 D D where Z 0 is replaced by H 0 +Lsiθ with beig the agle of the pipe exit with a horizotal level. Figure 5.1 presets a dowward-iclied pipe with full pipe flow fed from a upstream tak. Chapter 5: Impact of Pipe Icliatio Agle o the Outflow from Pipelies 118

119 Figure 5.1: Full pipe flow i a dowward-iclied pipe fed from a upstream storage tak For the previously cosidered case of a horizotal pipe (see Sectio ), it was assumed that the domiat regime i the pipe was full pipe flow util the liquid level i the tak H reduced to the critical depth d c. For the curretly cosidered case of a dowward-iclied pipe, it is assumed that the domiat regime i the pipe is full pipe flow throughout the draiage of the tak, i.e. util H = Bubble Formatio & Propagatio Bubble formatio ad propagatio regime commeces after the tak drais dry. I a horizotal pipe, upo rupture there will be two bubbles propagatig towards each other from the pipe ilet ad outlet. I dowward-iclied pipes, there are two possibilities for the bubble propagatio patter alog the pipe: 1) Similar to the horizotal pipe, the upstream ad dowstream bubbles propagate towards each other alog the pipe. Chapter 5: Impact of Pipe Icliatio Agle o the Outflow from Pipelies 119