CHAPTER FIVE Network Hydraulics

Transcription

1 . ETWOR YDRAULICSE CATER IVE Network ydrauics The fudameta reatioships of coservatio of mass ad eergy mathematicay describe the fow ad pressure distributio withi a pipe etwork uder steady state coditios. To begi, parae ad series pipe systems are cosidered i Sectio.. The basic cocepts appied i these simpe systems to determie the pipe fow rates ad oda pressure heads are the exteded to fu etworks. our mathematica formuatios are discussed Sectios. ad.. uasi-dyamic fow for tak modeig uder time varyig coditios is the preseted. iay, fow coditios that chage i the short term are aayzed usig coservatio of mass with coservatio of mometum to icude the impact of dyamic chages. Notatio for this sectio ca become cofusig sice a umber of subscripts ad couters are used. To summarize, the subscripts i ad idetify odes ad pipes, respectivey. The pipe fows are typicay summed over the sets of pipes providig fow to or carryig fow from a ode. To avoid cofusio with pipe egths, L, the set of icomig ad outgoig pipes are defied as J i ad J out, respectivey. Depedig upo the coservatio of eergy formuatio, head osses i pipes are summed over the set of pipes i a oop, oop, or path, path. The otatio with the etter foowed by a idetifier is used for the umber of compoets such as ode, pipe, oop, ad poop for the umber of etwork odes, pipes, cosed oops, ad pseudo-oops, respectivey. The subscript j is used as a idetifier i summatios ad defied for the specific equatio icudig a idetifier for a ode at oe ed of a pipe. To compute the oda heads ad pipe fows a set of oiear equatios is soved by a iterative process. The couter m is used to defie the iteratio umber ad added to the ukow variabe as a superscript i paretheses, i.e., m. Lasty, variabes are show i itaics ad matrices ad vectors are deoted by bod characters.. SIMLE IING SYSTEMS Simpe pipig systems provide isight ito the uderstadig of pipe etworks. Variatio of tota head through a etwork is see i a set of pipes i series. Aaysis of pipes i parae is the first appicatio of the coservatio of mass at a juctio ad coservatio of eergy aroud a oop. I additio to the physica uderstadig that they offer, simpificatio of these systems to equivaet pipes reduces computatios durig aaysis. -

2 - CATER IVE.. ipes i Series As show i igure -, a series of pipes may have differet pipe diameters ad/or roughess parameters. The tota head oss is equa to the sum of the head oss i each pipe segmet or: h L path h L, path - where path is the umber of pipes i series, is the coefficiet for pipe cotaiig iformatio about the diameter, egth, ad pipe roughess, is the expoet from the head oss equatio, ad is the fow rate i pipe. If o withdrawas occur aog the pipe, each pipe carries the same fow rate but the rate of head oss i each pipe may be differet. If we use the same head oss equatio for a pipes i.e., the same, we ca take out of the summatio or: path path path h L s eq - s where eq is the equivaet coefficiet for series of pipes. If fow is s turbuet, the s are costat ad a sige equivaet eq ca be computed for a turbuet fows. A B C D D cm D cm D cm L m L m L m f. f. f. z A m, z B m, z C. m, z D 7. m igure -: ipes i series with data for Exampe.. Exampe. robem: or a fow rate of. m /s, determie the pressure ad tota heads at poits A, B, C, ad D for series pipes show i igure -. Assume fuy turbuet fow for a cases ad the pressure head at poit A is m.

3 NETWOR YDRAULICS - Soutio: The pressure ad tota heads are computed usig the eergy equatio aog the path begiig at poit A. Give the pressure head ad eevatio, the tota head at poit A is: p A A z A 6 m γ Note the veocity ad veocity head are: ad. V.7 m / s A π. V.7.7 m g 9.8 The veocity head is four orders of magitude ess tha the static head so it ca be egected. Negectig veocity head is a commo assumptio i pipe etwork aaysis. A eergy oss i the system is due to frictio. So foowig the path of fow the tota heads at A, B, C, ad D are: C D B B C h h A h BC f CD f AB f L V.7 6 f 6. D g m B. L. V π 7.6 f 7.6. D g C 7. m. L V π. 7. f 7.. D g D 6. 8 m The pressure heads are: at poit B pb z γ pb pb 7.6. γ γ B B 6 m

4 - CATER IVE at poit C ad at poit D pc pc pc C zc m γ γ γ pd pd pd D z D m γ γ γ Exampe. robem: or the series pipe system i Exampe., fid the equivaet roughess coefficiet ad the tota head at poit D for a fow rate of. m /s. Soutio: By Eq. -, the equivaet pipe oss coefficiet is equa to the sum of the pipe coefficiets or: or this probem: s eq 8 f L gπ D s 8 f L 8 f L 8 f L eq g π D g π D g π D π. 9.8π. 9.8π. s Note that pipe has the argest oss coefficiet sice it has the smaest diameter ad highest fow veocity. As see i Exampe., athough it has the shortest egth, most of the head oss occurs i this sectio. The head oss betwee odes A ad D for. m /s is the: AD s h f eq 89. h f 7. m eq AD We ca aso cofirm the resut i Exampe. by substitutig. m /s i which case: ad AD s h f eq m

5 NETWOR YDRAULICS - AD D A h f 6.6 D 6. 7 m that woud be equivaet to the earier resut if Exampe. was carried to decima paces... ipes i arae Whe oe or more pipes coect the same ocatios juctios, the hydrauics are much more iterestig. The reatioships i these sma etworks ead to the fudameta reatioships for fu etwork modeig. Locatios A ad B i igure - are described as odes or juctios of severa pipes. As i Exampe., coservatio of mass must be preserved at these ocatios. That is, i steady state the kow icomig fow at ode A must baace with the outgoig fows i pipes,, ad. Simiary, the icomig fows to ode B i the icomig pipes,, ad must equa the kow withdrawa at ode B. q A q B - where ad q j defie the fow rate i pipe ad the oda withdrawa/suppy at ode j, respectivey. The mass baace for ode B provides the same iformatio as the above ad is redudat. L ft, D i, C W, A q A cfs A 8 ft L ft, D 6 i, C W, L ft, D 8 i, C W, B q B cfs B? igure -: ipes i parae with data for Exampe.. The secod reatioship that must hod is that the head oss i pipes,, ad must be the same. Sice a begi at a sige ode A ad a ed at a sige ode B ad the differece i head betwee those two odes is uique, regardess of the pipe characteristics the head oss i the pipes is the same or: A B hl hl hl h -,,, L where A ad B are the tota heads at odes A ad B, respectivey, h L, is the head oss i pipe, ad h L is the sige vaue of head oss betwee odes A ad

6 -6 CATER IVE B. Eq. - is a statemet of coservatio of eergy for a pipe ad is used i severa formuatios for sovig for fows ad heads i a geera etwork. I other etwork soutio methods, we write coservatio of eergy for cosed oops. A cosed oop is a path of pipes that begis ad eds at the same ode. ipes ad form a cosed oop begiig ad edig at ode A. Startig at ode A aroud the path, eergy, h L,, is ost as water fows from A to B. As we foow the path back to ode A to cose the oop, we gai eergy, h L,, sice we are movig i the directio opposite to the fow. We ca write the path equatio aroud the oop ad maipuate it to show: h h h h h h - A L, L, A L, L, L, L, Now usig Eq. - ad either Eqs. - or -, we ca determie the head p oss ad fow for each pipe ad a equivaet pipe coefficiet, eq. I ay pipe etwork system the oda ifows ad outfows q A ad q B ad at east oe oda head s tota eergy A i this case must be kow to provide a datum for the pressure head. or steady fow coditios i the etwork i igure -, we have a tota of seve ukows, ode B s tota eergy B, three pipe fows,, ad ad three head osses h L,, h L,, ad h L,. Eq. - provides two idepedet equatios reatig the head osses h L, h L,, ad h L,, h L,. The third equatio is that the head oss i ay pipe equas the differece i head betwee odes A ad B the first part of Eq. -. Coservatio of mass at ode A Eq. - is the fourth reatioship. The fia three equatios are the head oss versus discharge equatios: h L, or h L, -6 We ca substitute Eq. -6 i the mass baace equatios Eq. - with h L equa to each pipe s head oss or: h L h L h L q A -7 I this equatio, a terms except for h L are kow. After sovig for h L, the ukow pipe fows ca be computed by Eq. -6 ad B ca be determied i Eq. -. Like pipes i series, a equivaet pipe coefficiet ca be computed for parae pipes. I Eq. -7, h L ca be pued from each term o the eft had side or for a geera discharge ad three parae pipes:

7 NETWOR YDRAULICS -7 q h L -8 The equivaet coefficiet is the: p p eq -9 where p eq is the equivaet pipe coefficiet for parae pipes. As show i the ast term, Eq. -9 ca be geeraized for p parae pipes. The head oss betwee the two ed odes is: tota p eq L h - Exampe. robem: Give the data for the three parae pipes i igure -, compute the equivaet parae pipe coefficiet, the head oss betwee odes A ad B, the fow rates i each pipe, ad the tota head at ode B. Soutio: The equivaet parae pipe coefficiet aows us to determie the head oss that ca the be used to disaggregate the fow betwee pipes. The oss coefficiet for the aze-wiiams equatio for pipe with Egish uits is:.9 / D C L Simiary, ad equa.86 ad.9, respectivey. The equivaet oss coefficiet is: p eq p eq ad The head oss betwee odes A ad B is the:

8 -8 CATER IVE p hl eq.67. ft tota So the head at ode B, B, is:.8 A B h L 8 B. ft B 79.7 ft The fow i each pipe ca be computed from the idividua pipe head oss equatios sice the head oss is kow for each pipe h L. ft... hl..6.9 cfs The fows i pipes ad ca be computed by the same equatio ad are. ad. cfs, respectivey. The sum of the three pipe fows equas cfs, which is same as ifow to ode A.. SYSTEM O EUATIONS OR STEADY LOW Coservatio of mass at a juctio ode Eq. - ad coservatio of eergy Eq. - ca be exteded from parae pipes to geera etworks for steady state hydrauic coditios. The resutig set of simutaeous quasi-iear equatios ca be soved for the pipe fows ad oda heads for steady state ad step-wise quasi dyamic kow as exteded period simuatio or ES aayses. ES aaysis requires a additioa reatioship describig chages i tak eves due to ifow/outfows ad is discussed i ater sectios. Oy steady state hydrauics is cosidered i Sectios. ad.... Coservatio of Mass As defied earier, a juctio ode is a coectio of two or more pipes. Athough demads are distributed aog pipes, these demads are umped at juctios ad defied as q ode. Coservatio of mass at a ode was preseted i Sectio... or a juctio ode i, coservatio of mass ca be writte as: Ji Jout q i - where q i is the extera demad withdrawa, J i,i ad J out,i are the set of pipes suppyig ad carryig fow from ode i, respectivey, ad J i deotes that is i the set of pipes i J i. This equatio ca be writte for every juctio ode i the system.

9 NETWOR YDRAULICS -9.. Coservatio of Eergy The secod goverig aw is that eergy must be coserved betwee ay two poits. Aog the path betwee odes A ad B that oy icudes pipes, coservatio of eergy is writte as: L i path h, - A B i path where A ad B are the tota eergy at odes A ad B, h L,,, are the head oss, oss equatio coefficiet ad fow rate i pipe ad is the expoet from the head oss equatio. The sig for the fow rate is defied usig the symbo ad this symbo does ot have its covetioa meaig. This symbo is iteded as a short form otatio ad remider of how the sigs of this reatioship shoud be iterpreted. The absoute vaue of is raised to the power of ad the sig of the pipe term is based o the fow directio. If fow is movig from ode A toward ode B the the sig shoud be take positive ad a egative sig is used if fow is away from B toward A. Eq. - ca be writte for a cosed or pseudo-oop or a sige pipe. path defies the set of pipes i the path. A cosed oop is oe that begis ad eds at the same ode. Sice each ocatio i the etwork has a uique eergy the et eergy oss aroud a cosed oop is zero. or a oop begiig ad edig at ode A: h, - A A L oop where oop is the set of pipes i the cosed oop. A pseudo-oop is a path of pipes betwee two poits of kow eergy such as two taks or reservoirs. Eq. - appies directy to pseudo-oops. seudooop equatios icude additioa iformatio regardig the fow distributio ad are eeded for some soutio methods. iay, Eq. - aso appies directy for idividua pipes with A ad B beig the tota heads at the two eds of the sige pipe or:.. Systems of Equatios A B - The ukows i a steady state hydrauic aaysis are the fows i each pipe,, ad the tota eergy head at each juctio ode,. I a system with ode

10 - CATER IVE odes ad pipe pipes, the tota umber of ukows is ode pipe. our equatio formuatios ca be deveoped to sove for these ukows. They ca be expressed i terms of ukow pipe fows or oda heads. A sets are oiear due to the eergy oss reatioships ad require iterative soutios. The Newto-Raphso method is the most widey used iterative soutio procedure i etwork aaysis. Its covergece properties have bee studied i detai by Atma ad Bouos 99. At east oe poit of kow eergy is required to provide a datum or root for the oda heads. The four soutio approaches are summarized beow ad mathematica detais are preseted i Sectio..... Loop Equatios The smaest set of equatios is the oop equatios that icude oe equatio for each cosed oop ad pseudo-oop for a tota of oop poop equatios where oop ad poop are the umber of cosed ad pseudo-oops, respectivey. The ukows i the oop equatios are s that are defied as the correctios to the fow rate aroud each oop. Begiig with a fow distributio that satisfies coservatio of mass, the correctios maitai those reatioships. Whe zero correctios are eeded i a oops, the fow rates i each oop ad each pipe has bee foud. After the fows have bee determied, Eq. - is appied begiig at a ocatio of kow tota eergy e.g., root to determie the oda heads. The ardy Cross method is oe approach to sove the oop equatios. This method first determies correctios for each oop idepedety the appies the correctios to compute the ew pipe fows. With the ew fow distributio, aother set of correctios is computed. ardy Cross itroduced this method i 96 ad, athough amedabe to had cacuatios, it is ot efficiet compared to methods that cosider the etire system simutaeousy. Epp ad ower 97 preseted a more efficiet method that simutaeousy soves for a oop correctios usig the Newto-Raphso method with the correctios as the ukows.... Node-Loop Equatios Wood ad Rayes 98 compared a umber of soutio agorithms with their modified iear theory fow adjustmet method ad showed that this approach was efficiet ad robust. Modified iear theory soves directy for the pipe fow rates rather tha the oop equatio approach of oop fow correctios. The oop poop oop equatios Eq. - icorporate the cocept of eergy coservatio ad ode ode equatios Eq. - itroduce coservatio of mass. The tota umber of idepedet equatios is ode oop poop that umber is equa to the umber of ukow pipe fow rates,

11 NETWOR YDRAULICS - pipe. A Newto s type method is used to sove for the s directy rather tha the oop fow rate correctios, s. As i the oop equatios, after the pipe fow rates are foud, they are substituted i Eq. - begiig at a poit of kow eergy to compute the oda heads.... Node Equatios The ode equatios ca be rewritte i terms of the oda heads by writig Eq. - for pipe that coects odes j ad i as: j i - These terms are substituted for the fow rate i Eq. - for each pipe ad oe equatio of the form of Eq. - is writte for each ode. This substitutio combies the coservatio of eergy ad mass reatioships resutig i ode equatios i terms of the ode ukow oda heads,. Shamir ad oward 968 soved these equatios usig the Newto-Raphso method. After the oda heads are computed, they ca be substituted i Eq. - to compute the pipe fow rates.... ipe Equatios The previous methods sove for the pipe fows,, or oda heads,, i a oiear soutio scheme the use coservatio of eergy to determie the other set of ukows. ama ad Brameer 97 ad Todii ad iati 987 devised a method to sove for ad simutaeousy. They wrote the ode equatios Eq. - with respect to the pipe fows ad Eq. - for each pipe icudig both the pipe fows ad the oda heads. Athough the umber of equatios ode pipe is arger tha the other methods, the soutio times ad the covergece to the true soutio are simiar or better. I additio, the agorithm does ot require defiig oops that may be a time cosumig task.. SOLUTION ALGORITMS OR STEADY LOW.. Soutio of the Loop Equatios... ardy Cross Method Sige Loop Adjustmet Agorithm The odest ad probaby best kow soutio method for pipe etworks is the ardy Cross method that is foud i most textbooks ad taught i udergraduate hydrauics courses Cross 96. As oted i Sectio..., the

12 - CATER IVE method soves the eergy equatios for oops ad pseudo-oops for a oop fow correctio. Athough a set of oop equatios must be soved for the system, this agorithm was deveoped for had cacuatios ad soves oe oop at a time. Oe cosed oop equatio is writte Eq. - for each oop. or cosed oops that cotai oy pipes, the oop equatio for oop L by Eq. - is: -6 L oop I this equatio, the sig of is appied to the term ad the absoute vaue is raised to the expoet. The sig is based o the fow directio reative to oop L ad discussed i more detai beow. Sice the fow rates that satisfy the set of oop equatios are ot kow, a oop equatio is expaded to a Tayor series trucated at the first order term or: L m oop L m m oop oop m m m m d d d d L L -7 where m- is the estimate of the fow at iteratio m- ad L / is the derivative of the L th oop equatio with respect to the th pipe fow,. Defiig m m- ad substitutig i Eq. -7: L m oop L m oop L m m -8 The deveopmet of Eq. -8 is equivaet to the Newto-Raphso method except that is computed rather tha the updated fow m. I additio, the ardy Cross method simpifies the determiatio of the correctio term by cosiderig each oop idepedety rather tha a oops simutaeousy. Sice a pipes i a oop wi have the same correctio, a sige is determied by Eq. -8. The umerator of Eq. -8 is computed from Eq. -6 with appropriate sigs for fow directios. Stadard covetio is to defie cockwise fow i each oop as positive. If equas zero, the equatio has bee satisfied. The deomiator is the sum of the absoute vaues of the derivative terms of Eq. -6 evauated at m-. The idividua gradiet terms are:

13 NETWOR YDRAULICS - m L / h -9 or oop L, Eq. -8 is the: L oop oop oop oop oop h / L, oop / - where the deomiator becomes a absoute vaue because the sigs of h L ad are the same. A simiar equatio ca be writte for each oop i the etwork. Sice a first-order Tayor series is used to approximate a oiear equatio, a sige set of correctios is ikey isufficiet to coverge to the true fows ad the process must be repeated uti a oops equatios are satisfied withi a desired toerace. I summary, the ardy Cross agorithm cosists of the foowig steps. Defie oops ad set m. Assume a iitia set of pipe fows that satisfy coservatio of mass at a odes. Note that the oop correctios wi maitai coservatio of mass after this iitia step. Update m m Compute the sum of head osses aroud a oop by sovig Eq. -6 for each oop substitutig m- for. This sum is the umerator of Eq. -. Compute the deomiator of Eq. - for each oop. Note that the deomiator is the sum of the absoute vaues of h L / over the set of pipes i the oop, oop. Compute the oop correctio, L, by sovig Eq Repeat steps - for each oop. 7 Appy correctio factors to a pipes or: m m ± p cp p where cp is the set of oe or two oops i the etwork that cotai pipe. 8 Check if a s are ess tha specified sma toerace. If so, stop. If ot, go to step.

14 - CATER IVE The equatios above accout for the fow directio chage without modificatio. I the umerator, the appropriate sig is take from the fow rate as defied by the iitiay assumed fow rate ad reative to the oop beig cosidered. Absoute vaues are aways take i the deomiator. A egative fow reative to the iitiay defied oops deotes that the fow is i the opposite directio of the origia assumptio with the computed magitude. I step 7, the correctios are appied with simiar ogic. A positive correctio impies a arger fow i the cockwise directio. Thus, is added to pipes with assumed positive fow directios. If the assumed directio is couter-cockwise, the correctio is subtracted from the m-. This covetio is aso appied if the actua fow directios is opposite of the assumed directio. I the approach described above, the oop correctios are appied after a correctios have bee computed. It is possibe to be more sophisticated by, for exampe, appyig the correctios as the method proceeds through a iteratio. owever, athough the ardy Cross method is acceptabe for had cacuatio, it is ot efficiet for or appied to arge systems so these improvemets are ot cosidered here. Exampe. robem: Determie the fow rates i the pipes i the three oop etwork i igure E-a ad the oda heads at a odes usig the ardy Cross method ad the aze-wiiams equatio. With the fow rates, compute the eergy at ode. Soutio: Step : m. oowig the procedure outied above, the first step is to defie a set of oops. our oops are idetified i igure E-b. The head oss aroud the cosed oops I, II ad III is zero sice the oops begi ad ed at the same ode ad the ode has a uique tota head as i the parae pipe aaysis i Sectio... A pseudo-oop is defied betwee the two reservoirs ad the differece i eergy betwee the ocatios is ft. The positive directio is defied i the cockwise directio for a oops. Assumig the fow directios show o igure E-b, the oop equatios are show beow. or Loop the pseudo-oop startig at reservoir ad cotiuig aog a path to reservoir : Loop : res. res We have added the sig covetio reative to the oop ad the iitiay assumed fow directios. ipe is assumed to fow i the couter-cockwise

15 NETWOR YDRAULICS - directio reative to oop ad its term is give a egative sig. A positive fow i pipe deotes that the fow directio is from the tak to ode. If fow is assumed icorrecty ad fow is actuay from ode toward the tak i pipe, the sig is switched to a egative. I this case, pipe s head oss term i Loop wi be positive the iitiay assumed egative sig times the egative sig from the fow term. I III 9 6 II 8 igure E-b: Exampe hydrauic aaysis pipe etwork with defied oops ad assumed fow directios. ipe 9 is assumed to fow from ode to ode that is positive reative to oop ad its head oss term is positive. The directio of fow through the pump is aso cockwise but the pump provides a head gai so it is give a egative sig. No sig covetio is appied to the pump sice it ca oy be o-egative, that is, a positive fow or zero. or oop III begiig ad edig at ode : Agai, positive sigs are give for fow movig cockwise reative to the oop. Thus, pipe 9 is positive reative to oop but egative i oop III. or oop II begiig ad edig at ode 6:

16 -6 CATER IVE res. ft,,,,,, [],,,,, 6, 7,, 6, [] 6 7 [],, 6, 6,,, 8,, 8, [] [] igure E-a: Exampe hydrauic aaysis pipe etwork. 9,, 6, res. ft ump, [6],, 6, Leged ipe umber, Legth ft, Diameter ich, C-vaue [Noda demad cfs]

17 NETWOR YDRAULICS -7 iay, oop I begis ad eds at ode : Ay pipe that appears i two oops has a egative sig i oe oop equatio ad a positive i the other equatio. This covetio must aso be cosidered whe updatig fows. The for each pipe is give i Tabe E-a usig:.7 D L.87.8 C W with i cfs, D ad L i ft. Tabe E-a aso gives a set of pipe fow rates that satisfy coservatio of mass. Iitia pipe fows were determied for the sequece of odes,,, 6, 7, ad. The ast ode is the checked to cofirm system mass baace. Sice the tota etwork demad is 8 cfs, the fows i pipe ad the pump must equa 8 cfs. To begi, the fow i pipe is assumed to be cfs. At a give ode, a but oe outfow pipe fows are assumed ad the ast vaue is computed by the ode s mass baace equatio. Step : Set m m Tabe E-a: Iitia data ad coefficiets for Exampe.. ipe Uode Tak 6 Dode h L h L / ipe Uode 7 6 Tak Dode h L h L / * Uode ad Dode are the iitiay assumed upstream ad dowstream odes, respectivey. Step : Compute sum of the head osses aroud each oop usig the vaues isted i Tabe E-a for the assumed fow coditios ad the oop equatios defied i Step. The sums are isted i Tabe E-b. Usig the iitia guess, the residua of oop s eergy equatio, p, is:

18 -8 CATER IVE Step : The computed sums of absoute vaues of the derivatives are aso isted i Tabe E-b. or oop : oop hl, hl, hl,9 h, / L Note that ast term is the derivative of the pump equatio with respect to. Step ad 6: Compute the correctio for each oop usig Eq. -. or oop : h L.8 / Tabe E-b gives the vaues for a oops. Tabe E-b: Loop correctios for iteratio of ardy Cross method exampe. Loop I II III Σ h L Σ h L / Step 7: The oop correctios are appied to each pipe as foows. or pipe, the pseudo-oop s correctio is appied with a egative sig sice the correctio is i the cockwise directio ad pipe is assumed to fow i the couter-cockwise directio or: Correctios for oops ad I are appied to pipe sice it is ocated i both oops. Loop s correctio is egative sice pipe fow is assumed to fow i the couter-cockwise directio for that oop egative ad the oop I correctio is positive sice pipe s fow is cockwise reative to that oop.

19 NETWOR YDRAULICS -9 I Simiary for the other pipes: I..879 III I II II III I II II III III Noda fow baaces cotiue to be coserved ad ca be verified. It is worthwhie to compute those baaces to check if a computatioa or sig error has bee itroduced. I this case, pipe is rouded dow to preserve the mass baace for ode. Step 8: The maximum correctio is.76 cfs so iteratios cotiue. Go to step. To provide some isight ito the sig covetio, cosider pipe durig the first iteratio. If we had assumed that pipe s fow was from ode to the tak ad the fow was exitig the tak, the sig o woud be egative i.e., -. I cacuatig the correctio, the deomiator of woud be the same sice the absoute vaue of the idividua terms are summed. The resutig umerator woud aso be the same because a differet sig woud be appied to the pipe term or: ipe woud be positive sice the assumed fow from ode to the tak is a cockwise fow reative to oop. owever, the actua fow woud carry a egative sig show i parethesis sice fow was opposite of the assumed directio. Thus, the umerator ad correctio term woud ot chage. Based o the fow directio assumptio, the fow is i the positive cockwise directio for oop so the correctio woud be added to or:

20 - CATER IVE Thus, the magitude for the ext iteratio woud be same as above ad the egative sig woud deote that the fow was i the opposite directio of the assumptio ode to tak. A umber of iteratios are required for the ardy Cross method to coverge to the soutio. ow vaues ad the oop correctios are give i Tabe E-c ad d, respectivey, for additioa iteratios uti the argest oop correctio is ess tha. cfs. Vaues i the tabes are computed without roudig i the spreadsheet that performed the cacuatios. The tota head at ode ca be computed by begiig a path at either reservoir or. Startig at reservoir with a head of ft, a path to ode is pipes,, ad. Sigs o the eergy oss terms are based o fow directios i the path. Sice fow goes from the tak to ode, eergy is ost as the water traves through pipe. Simiary, eergy is ost as fow moves from ode to 6 i pipe ad from ode 6 to i pipe. The overa eergy equatio is the:.8 res ft Tabe E-c: ipe fow vaues for ardy Cross iteratios. ipe ow cfs m Node s head ca aso be computed startig at reservoir. I this path, eergy is gaied as water is ifted by the pump ad ost as water moves from ode to i pipe. The eergy at ode is greater tha at ode sice water is fowig from ode to i pipe 8. The head oss i pipe 8 therefore must be added i the path equatio from reservoir to ode or: res ft

21 NETWOR YDRAULICS - Sight differeces resut from the two paths sice the ardy Cross iteratios were stopped before fu covergece. Tabe E-d: Loop correctios for ardy Cross iteratios. Iteratio Loop Loop I Loop II Loop III Simutaeous Loop Equatio Soutio Simutaeous Loop ow Adjustmet Agorithm I the ardy Cross method, each oop correctio is determied idepedety of the other oops. As see i igure E-b, severa oops may have commo pipes so correctios to those oops wi impact eergy osses aroud more tha oe oop. Epp ad ower 97 deveoped a more efficiet approach by simutaeousy computig correctios for a oops. As i the ardy Cross method, a iitia soutio that satisfies cotiuity at a odes is required. or a simutaeous oop equatio soutio, Eq. -6 for oop L becomes: m L p - oop p cp where cp is the set of oe or two oops i the etwork that cotai pipe e.g., oops ad I for pipe i Exampe.. The sig covetio o the pipe fow reative to the oop is the same as for the ardy Cross method. The Newto-Raphso method is the used to sove Eq. - for the s. A system of iear equatios must ow be iterativey soved rather tha the sige equatios of the ardy Cross method. A first order Tayor series approximatio of Eq. - for oop L is: oop L p L L p c L m p m p L m -

22 - CATER IVE where cl is the set of oops that have a commo pipe with oop L e.g., oops I ad III for oop II i Exampe.. I vector form for a oops simutaeousy Eq. - ca be writte as: J L - m- - where m- is the vector of pipe pipe fow, is the [ x ooppoop] vector of oop fow correctios ad m- is the [ x ooppoop] vector of residuas of the oop coservatio of eergy equatios Eq. - evauated at m-. Residuas are the vaues of the right had side at the tria vaues of. The objective is for a of those terms to equa zero such that a oop equatios are satisfied. J L equas / ad is the Jacobia matrix of first derivatives of the oop equatios evauated at m-. J L is square [ooppoop x ooppoop], symmetric ad positive defiite. The rows i J L correspod to the oop equatios ad the coums are reated to the oop correctios. The L th diagoa term of the Jacobia is the sum of the first derivatives of the oop pipes i oop L or the summatio i the first term i Eq. -. This term is aso equivaet to the deomiator of Eq. -. The differece betwee the simutaeous oop method ad the ardy Cross method is that some of the off-diagoa terms are o-zero. The Jacobia term i row L oop L s coservatio of eergy equatio ad coum p commo oop correspods to the gradiet term for the pipe that is commo to oops L ad p. These terms are equa to zero if the oops do ot have commo pipes. If the oops have commo pipes, these terms are the egative of the sum of the absoute vaues of hl / for pipes that appear i oop L ad p. The egative sig resuts because the fow directio i oop p is opposite to oop L. A exampe of formig Eq. - is give i Exampe.. Oce the matrices are formed, Eq. - ca be soved by ay iear equatio sover for. The pipe fows are updated by the oop correctios as i the ardy Cross method i.e., m m- /-. The soutio agorithm is the same as the ardy Cross method except steps ad 6 are reduced to a sige step ad a correctios are computed simutaeousy. Sice the equatios are oiear, severa iteratios may be ecessary to coverge to the soutio, ike the ardy Cross method. To ed the agorithm, oe of severa stoppig criteria ca be appied: umber of iteratios aowed or the magitude of the chage i oop fow rates,. As i the ardy Cross method, if a pipe s fow directio chages from the assumed vaue, the sigs for that pipe head oss terms are switched for a oops cotaiig the pipe durig the ext iteratio i the oop equatios usig the sig covetio oted above. The directio chage does ot ater the coefficiet matrix, J L, or the sigs o the idividua terms. The diagoa terms are the sum

23 NETWOR YDRAULICS - of the absoute vaues of the gradiets ad the off-diagoa terms are aways egative. Exampe. robem: Determie the fow rates i the pipes i the three oop etwork i Exampe. ad the oda heads at a odes usig the simutaeous oop method ad the aze-wiiams equatio. Soutio: The Exampe. startig poit is used agai i this exampe. Equatio - is soved to provide the simutaeous oop correctios. or the exampe etwork, the four oop equatios were deveoped i Exampe. for oops, I, II, ad III as: res res.... : : I : II : III.... The coefficiet matrix terms are the gradiets of the oop equatios with respect to each oop fow correctio or: III III II III I III III III II II II I II II III I II I I I I III II I L J The diagoa terms are idetica to the deomiator of the ardy Cross correctio: oop L L L h,

24 - CATER IVE where oop is the umber of pipes i oop L. rom the vaues i Tabe E-a ad b, a symmetric coefficiet matrix ca be fied. The rows - correspod to oops, I, II, ad III. As shows i step of Exampe.: hl, hl, hl, hl,9.976 oop L L 9 The off-diagoa terms are the gradiets for pipes that appear i oop L ad aother oop, p or: L p cpipe L, p h L, p L where cpipe L,p is the set of pipes that are commo to oops L ad p e.g., pipe for oops I ad. The vaue i row correspodig to oop equatio ad coum correspodig to oop I is the gradiet of the oy commo pipe with respect to the fow correctio i oop I or: h, h L L I cpipe, I, As oted above, this gradiet is aso I /. The remaiig diagoa terms are give i Tabe E-b ad the offdiagoas correspod to pipes commo to two oops Tabe E-a. ipe appears i oops ad I, pipe 9 appears i ad II, pipe is ocated i I ad II ad pipe 6 is i oops II ad III. No pipe is commo to oops ad II so the coefficiets are zero i those ocatios coum -row ad coum -row. The J L matrix is the: J L The right had side of the equatio is the computed vaue of the equatio with the curret fow estimates that were computed for the ardy Cross method ad isted i the secod row of Tabe E-b. T [ ,.6, -.97]

25 NETWOR YDRAULICS - or exampe, row correspods to oop II: II The system of iear equatios, J L -, is soved for the ukow fow correctios..8 J L I II III Resutig i: T [, I, II, III ] T [-.9, -., -.66, -.] T The correctios are appied usig the equatios show i Exampe., step 7. or exampe, the fow rates i pipes ad become: I The oop fow correctios are sti arge so additioa iteratios are eeded to coverge to the soutio. Sice fow correctios are made o a oops simutaeousy, this method coverges i four iteratios to a absoute chage i of. cfs compared to for the ardy Cross method with a arger toerace. The resuts are summarized i Tabes E-a ad b. Sice the fow rates from the ardy Cross ad simutaeous oop methods are the same the oda heads wi be the same for both resuts... Soutio of the Node-Loop Equatios ow Adjustmet Agorithm Usig the oop equatios to represet coservatio of eergy, Wood ad Chares 97 deveoped the iear theory ow Adjustmet method by

26 -6 CATER IVE coupig the oop equatios with the ode equatios. Wood ad Rayes 98 ater showed that a modified iear theory preseted here exhibits superior covergece characteristics compared with the origia iear theory method. Iter. Tabe E-a: ipe fow vaues for the simutaeous oop correctio method. ipe ow cfs m Tabe E-b: Loop correctios for simutaeous oop correctio iteratios. Iteratio Loop Loop I Loop II Loop III I the modified method, rather tha sove for oop correctios ad be required to provide a feasibe iitia soutio, coservatio of eergy aroud a oop Eq. -6 is writte directy i terms of the pipe fow rates or for a cosed oop: - oop owever, the umber of ukow pipe fows is equa to the umber of pipes p but oy oop poop equatios of the form of Eq. -6 are avaiabe. Therefore, these equatios are couped with the oda coservatio of mass equatios Eq. -: Ji J out With coservatio of mass, the umber of equatios is ode ode equatios pus oop cosed oop equatios ad poop pseudo-oop equatios or a tota of p equatios writte i terms of the p ukow pipe fow rates. These oiear equatios are aso soved iterativey by appyig the Newto-Raphso method. Takig a Tayor series expasio of a oop equatio resuts i: q

27 NETWOR YDRAULICS -7 oop d d L m m m L m where is the oop equatio Eq. -6 ad m- are the kow pipe fows for the previous iteratio ad m is the ukow fow rates at iteratio m. This equatio ca be rearraged with the kow terms o the right had side as: oop d d m m m oop d d m m - A simiar reatioship ca be writte for coservatio of mass. It is iear with respect to the ukows, m sice the gradiet terms ad the fuctios ca be evauated at m-. Eq. - ca be writte i matrix form as: J NL m NL - J NL m- -6 where J NL is the Jacobia of the ode-oop equatios ad NL is the vector of fuctios of kow vaues from the previous iteratio. J NL ad NL vary betwee iteratios as moves toward the soutio. is the vector of residuas computed by substitutig m- i the ode-oop equatios. The rows i J NL correspod to the coservatio of mass ad eergy equatios ad the coums reate to the ukow pipe fow rates. or the coservatio of mass equatio for ode i, the terms i the correspodig row i J NL wi be if the pipe is ot coected to ode i, if the pipe is carryig fow to the ode i.e., the pipe is i the set J i,i, e.g., pipes 6 ad 8 for ode i Exampe. ad if the pipe is i set J out,i ad carries water from ode i e.g., pipes ad for ode i Ex... or the coservatio of eergy equatios, the gradiet terms are the same as the ardy Cross terms Eq. - i.e., h L / -, if the pipe appears i the oop, ad zero, otherwise. The fu term becomes more compex whe a pump appears i the oop Exampe.6. J NL ad NL are evauated at m- ad Eqs. -6 are soved for the ew pipe fows, m. This iterative process cotiues uti a defied stoppig criteria is met, such as whe the absoute or percetage differece betwee two iteratios fows, m ad m-, is ess tha a toerace for a pipes or a imitig umber iteratios are competed. Sice coservatio of mass is soved as part of Eq. -6, the iitia soutio does ot have to satisfy this coditio. Exampe.6 robem: Determie the fow rates i the pipes i the three oop etwork i Exampe. ad the oda heads at a odes usig the modified iear theory

28 -8 CATER IVE method ad the aze-wiiams equatio. Use the same startig poit as Exampe.. Soutio: The ode-oop equatios cosist of the ode equatios writte with respect to the pipe fow Eq. - ad the oop equatios Eq. -6. Loop : Loop I Loop II Node: Node : Node : Node : Node Node 6 : Node 7 : q 9 q 9 q q : 8 q 7 q6 7 6 q res. res : : 7 6 Loop III : These equatios ca be soved for the pipe fows ad pump fow. A arbitrary positive fow directio has bee assiged to each pipe that is cosistety appied i the coservatio of mass ad eergy equatios. or exampe, pipe is positive whe water fow from ode to ode. Thus, it is a outfow from ode ad is give a egative sig i that coservatio of mass equatio. It is a ifow to ode ad give a positive sig i that ode s mass baace equatio. I additio, choosig cockwise as positive i a coservatio of eergy equatios, fow from ode to is coutercockwise egative i oop ad cockwise positive i oop I. Usig our covetio for takig the sig of the fow rate, fows that are opposite of the assumed directio become egative ad chage the sigs o the terms. or exampe, if became with.8 the: Thus, egative fows are possibe ad distiguish the proper magitude ad that the assumed directio was icorrect.

29 NETWOR YDRAULICS -9 As oted above, the coums i the coefficiet matrix, J NL, for these equatios correspod to pipes ad the rows correspod to the odes pus oop equatios. The first seve ode rows of the right had side vector are oda demads Eq. -. or ode 6, the o-zero Jacobia terms correspod to pipes, ad 7. or that ode, pipe is i ifow pipe ad ad 7 outfow pipes so the gradiets for these pipes are: ; ; 7,6,6 6, N N N where N,6 deotes the ode equatio for ode 6. The sixth row i J NL is: [ ] 6 row NL, J The oop equatios are iearized by a Tayor series expasio ad the terms i the gradiet matrix, L, are the derivatives of oop equatio for oop L with respect to fow i pipe or h L, / for the pipes i the oop ad zero otherwise. The sig for the term reates if the assumed fow is cockwise or couter-cockwise - reative to the oop. Therefore, vaues of the ast four rows of J NL correspodig to the oop equatios come directy from the ast row of Tabe E-a. Based o the assumed fow directios, the sigs are: Loop [-, -, 9, -ump], Loop I [, -,, -7], Loop II [-, 6, 7, -8], ad Loop III [-, -6, -9, ]. or the iitia poit used i the ardy Cross method, J NL is: J NL The right had side of Eq. - for the ode equatios is equa to the oda demad, q, demostratig that the iearizatio of the ode resuts does ot ater the equatio. or ode 6 with the assumed fows the RS is:

30 - CATER IVE 7 N,6 q 6 N Ji, Jout,6 N,6 N,6 N,6.... The reader ca cofirm that a ubaaced assumptio of pipe fows gives the same resuts e.g.,, -. ad The ast four rows are more tha the deviatios i the eergy baaces i.e., - as i the simutaeous oop soutio. The additioa terms are the gradiets of the oop equatios the ast four rows of J NL times the preset fow estimates m-. or oop I that cotais oy pipes, the right had side is: m h L h m L m h L h L h L / The resutig reatioship is differet for oop that cotais a pump. h L h E m h L / The fu oop equatio is icuded i the first term icudig the eergy differece for pseudo-oops. The gradiet of the eergy differece is zero so it does ot appear i the secod term. or the first iteratio, the resutig right had side is: NL m m

31 NETWOR YDRAULICS - The system of iear equatios J NL NL, Eq. - are the soved for the ukow : T [.9, 9., 9.,.,.88, 6.7,.,.6,.7,.6,.6, 7.6] Updatig J NL ad the RS vaues, the method coverges to the soutio i iteratios as show i Tabe E-6. Tabe E-6: ipe fows for iteratios of the modified iear theory method for the pipe equatios. ipe m Soutio of the Node Equatios Simutaeous Node Adjustmet Agorithm The pipe head oss equatio for pipe that coects odes i ad j: L, j i -7 h ca be trasformed to oda head equatio as: / j i -8 Shamir ad oward 968 soved used this trasformatio to form ode ode equatios for the oda heads usig the Newto-Raphso method. Substitutig Eq. -8 i Eq. - for a geera ode i gives: / j i N, i : qi J J -9, i out where the summatio is over the pipes eterig or eavig the ode. I the head differece, the ode for which the mass baace is writte is aways the secod, provides the fow directio. If i is greater term. The sig otatio, [ ] j i

32 - CATER IVE tha j, fow is from ode i towards j a outfow ad the sig is egative. Whe j exceeds i, the sig is positive ad fow is suppied to ode i from ode j a ifow. or ode i, appicatio of the Newto-Raphso method yieds:, m i code i i i i N - where m- are the oda heads for the previous ad preset iteratios ad codei is the set of odes that are coected by pipes to ode i ad ode i e.g., odes,, 6, ad 7 for ode 7. I matrix form for a equatios ad odes: J N - N - where J N is the Jacobia matrix of the ode equatios with respect to the chages i oda heads ad N is the residuas of the ode equatios. Both J N ad N are evauated at equa to zero or the preset iteratio s head estimates. It shoud be oted that the square Jacobia matrix ode x ode is symmetric ad positive defiite. or pipe that coects odes i ad j, the Jacobia terms gradiets for the coservatio of mass at ode i are:, i j i j j i N m -a, i j i cp i j i cp i i N m -b Regardess of the fow directio, the gradiet sig for the fow baace at ode i are a positive for i terms whie the terms for the coectig odes j are a egative. After Eq. - are soved for, the heads are updated by subtractig the oda correctios or: i m i m i -

33 NETWOR YDRAULICS - The process is competed iterativey uti the chages i oda heads for a odes are ess tha a toerace or a desired umber of iteratios are competed. The overa process is: Iitiaize m ad defie startig set of oda heads,. Set m m Compute oda baaces usig Eq. -9 ad gradiets usig Eq. - Sove the system of equatios - for Update oda heads usig Eq. - 6 Check stoppig criteria. If satisfied, stop. If ot satisfied, go to step. Exampe.7 robem: Determie the oda heads at a odes i the three oop etwork i Exampe. usig the modified iear theory method ad the aze-wiiams equatio. Assume a iitia head vector, Compute the fow rate i pipe.,t [98, 9, 9, 7, 88, 9, 8] Soutio: Give the iitia head distributio step, we ca ow update the vaues at m step. Step : As isted i Exampe.6, the oda mass baace equatios are: Node: Node : Node : Node : 6 Node : Node 6 : Node 7 : q 6 q q q q q 6 q 7 6 or the ode equatio soutio, these seve equatios are writte i terms of the seve oda heads as:... res. 6, N q... 7 N, q 9

34 - CATER IVE ,.... 9, q q N res N , , , q q q N N N Sice this form is oiear, a ode equatios are iearized about the ode correctios. To set up the iear equatios, the pipe fows are computed usig the previousy computed s Tabe E-a ad the iitia oda heads ad are isted i Tabe E-7a. or pipe ad ode : cfs res The gradiet of the fow rate i pipe with respect to a chage i head at ode oe is give by Eq. -b: res res res res The gradiet term for each pipe is isted i the ast row of Tabe E-7a. Tabe E-7a: ipe fows computed with the defied iitia oda heads. ipe i - j / i - j

35 NETWOR YDRAULICS - Tabe E-7a cot.: ipe fows computed with the defied iitia oda heads. ipe i - j / i - j or the first iteratio, the ode x ode coefficiet matrix, J N, is: J N The diagoa terms are the sum of the gradiets for that ode. or ode, where the idividua terms were take from Tabe E-7a. The off-diagoa terms are equa to the gradiets of the ode equatio with respect to the adjacet ode ad are aways egative. or exampe, the chage i ode s mass baace due to a chage i head at ode is: 9.6 This term is paced i row two for mass baace at ode -coum for coectig ode. A chage i the head at ode causes a equa chage i the mass baace at ode. So the term aso appears i row ode equatio three coum coectig ode two. The right had side of the system of equatios is N, the residuas of the oda baace equatios. They are computed by substitutig the summed head

36 -6 CATER IVE vaues i the mass baace equatios isted above. or exampe for ode 6 at iteratio the residua is: N, q Their vaues are show i first row of Tabe E-7b. The sigs of these terms are the chaged i the soutio of Eq. -. The set of equatios - is the: J N N Step : The first iteratio soutio for T [-., -.9, -.,.68,.78, -.7, -.]. Step : The updated oda heads are isted i Tabe E-7c where -. or ode : Step 6: A arge chage i oda heads was appied so retur to step m. The oda mass baace equatios are evauated agai ad mass baace

37 NETWOR YDRAULICS -7 has ot bee achieved Tabe E-7b. The oda heads are updated two more times ad the vaues coverge as see i Tabe E-7c. After the fows have coverged after iteratio, the fow rate i pipe ca be computed by:. fia cfs.9 Tabe E-7b: Residuas of the oda baace equatios, N, computed usig the oda heads at the begiig of iteratio. To sove the set of equatios these residuas are mutipied by i Equatio -. Node Iteratio Tabe E-7c: Noda heads for the three iteratios. Node Iteratio Soutio of the ipe Equatios I the oop equatio formuatio, head osses were baaced aroud a series of pipes betwee poits with a kow differece i eergy. amam ad Brameer 97 for gas etworks ad Todii ad iati 987 for water etworks wrote coservatio of eergy for each pipe Eq. - resutig i a set of pipe equatios with the pipe pipe fows ad the ode oda heads as ukow. They couped these equatios with the ode equatios writte i terms of the pipe fows Eq. - to form a set of pipe pus ode equatios for a equa umber of ukows. The method is aso kow as the hybrid or gradiet approach. or additioa backgroud o the method, see Osiadacz 99.

38 -8 CATER IVE or the four-pipe etwork show i igure -, the pipe equatios icude oe equatio for each ode ad each pipe. With the assumed set of pipe fow directios, the ode equatios are the coservatio of mass reatioships or: : q {ode } : q {ode } : q {ode } The pipe equatios are writte for each pipe with the oda head ad pipe fow o the eft-had side of the equatio. The frictio oss equatio is give a positive sig, so the upstream source ode head has a egative sig ad the dowstream head takes a positive sig. The equatios for pipes to i the four-pipe etwork are: where ad a s are kow., : - {pipe }, : - {pipe }, : - {pipe }, : - {pipe } q. q. igure -: our pipe exampe etwork for pipe equatios formuatio.

39 NETWOR YDRAULICS -9 The otatio,, represets that the absoute vaue of the pipe fow is raised to the power ad the sig of the pipe fow is appied to the head oss equatio term. The fow s sig aso is appied i the ode equatios. Thus, a egative fow is acceptabe ad defies a fow that is i the opposite directio from the iitia assumptio. I our exampe, coservatio of mass ad eergy comprise seve equatios writte with respect to four pipe fows ad three oda heads. Appyig the Newto-Raphso method but sovig for the chages i fow ad head for pipe that coects odes i ad j gives: Ji, Jout m m - i m i j m j m m, m - The derivatives of the mass baace equatios are outfow pipe, - ifow pipe or ot coected to ode. The right-had side is cacuated by substitutig i the preset estimates of the fow rates ad defied as dq. or ode, Eq. - becomes: m m m m m- - m- - m- - q - dq {, : ode } -6a Simiary for odes ad : - - m- - m- - q - dq {, : ode } -6b - m- m- -q - dq {, : ode } -6c or the eergy baaces, the derivatives with respect to the oda heads are sik ode, - source ode, ad ot coected to pipe. The derivatives with respect to pipe fow are. The right-had side is computed usig the preset iteratios fows ad oda heads ad defied as de. or pipe, Eq. - is: