Optimization model of water supply networks design

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1 Opimizaion model of wae upply newok deign Pof. d. eng. IOAN SÂRBU Depamen of Building Sevice Poliehnica Univeiy of imioaa Piaa Bieicii, no. 4A, 333 imioaa ROMANIA Abac: he pape appoache he opimizaion of wae diibuion newok upplied fom one o moe node ouce, accoding o demand vaiaion. adiionally, in pipe opimizaion, he obecive funcion i alway focued on he co cieia of newok componen. In hi udy an impoved linea model i developed, which ha he advanage of uing no only co cieia, bu alo enegy conumpion, conumpion of cace eouce, opeaing expene ec. he pape ea looped newok wich have concenaed ouflow o unifom ouflow along he lengh of each pipe. An impoved model i developed fo opimal deing of new o paially exended wae diibuion newok, which opeae eihe by mean of gaviy o a pump yem.he model i baed on he mehod of linea pogamming and allow he deeminaion of an opimal diibuion of commecial diamee fo each pipe in he newok and he lengh of he pipe which coepond o hee diamee. Alo, i i poible o ake ino accoun he vaiou funcional iuaion chaaceiic found duing opeaion. hi pape compae linea opimizaion model o he ome ohe, uch a he claic model of aveage economical velociie and Mohnin opimizaion model. hi how he good pefomance of he new model. Fo diffeen analyzed newok, he aving of elecical enegy, due o diminihing peue loe and opeaion co when applying he developed model, epeen abou 35 %. Key-Wod: Wae upply, Diibuion, Looped newok, Opimal deign, Linea opimizaion model. Inoducion Diibuion newok ae an eenial pa of all wae upply yem. he eliabiliy of upply i much geae in he cae of looped newok. Diibuion yem co wihin any wae upply cheme may be equal o o geae han 6 % of he enie co of he poec. Alo, he enegy conumed in a diibuion newok upplied by pumping may exceed 6 % of he oal enegy conumpion of he yem [6]. Aemp hould be made o educe he co and enegy conumpion of he diibuion yem hough opimizaion in analyi and deign. A wae diibuion newok ha include pump mouned in he pipe, peue educing valve, and check-valve can be analyzed by eveal common mehod uch a Hady Co, linea heoy, and Newon Raphon []. adiionally, pipe diamee ae choen accoding o he aveage economical velociie (Hady Co mehod) [3]. hi pocedue i cumbeome, uneconomical, and equie ial, eldom leading o an economical and echnical opimum. hi pape develop a linea model fo opimal deign of new and paially exended diibuion yem upplied by pumping o gaviaion. I i baed on linea pogamming and allow fo he deeminaion of opimal diibuion of commecial diamee along he lengh of each pipe and he lengh of pipe eco coeponding o hee diamee. I i poible o ake ino accoun vaiou funcional iuaion chaaceiic found duing opeaion and unifom ouflow along he lengh of each pipe. Bai of hydaulic calculaion A diibuion newok may be epeened by oienaion compiing a finie numbe of ac (pipe, pump, fiing) and a e of node a well a eevoi and pump o pipe ineecion. In he cae of a complex opology, wih eevoi and pump a he node, he numbe of open loop (peudoloop) N RP i added o he numbe of cloed loop, o ha he oal numbe M of independen loop i deemined fom he equaion: M N + N RP () whee: i numbe of pipe; N numbe of node; N RP numbe of peue geneaing faciliie. he hydaulic calculaion of a diibuion newok involve in deemining he diamee, dichage and head loe in pipe, in ode o guaanee a each node he neceay dichage and peue. When pefoming he hydaulic calculaion of a diibuion newo he law of wae flow in all he pipe mu be epeced: dichage coninuiy a node: ISSN: ISBN:

2 f N i i i + q (,..., N N ) () in which: f i he eidual dichage a he node ; i dichage hough pipe i, wih he ign (+) when eneing node and ( ) when leaving i; q conumpion dichage (demand) a node wih he ign (+) fo node inflow and ( ) fo node ouflow. enegy conevaion in loop: h m i m i RP ε h f ( m,..., M ) (3) i i m in which: h m i he eidual head lo (divegence) in he loop m; h i head lo of he pipe i; ε i oienaion of flow hough he pipe, having he value (+) o ( ) a he wae flow ene i he ame o opoie o he pah ene of he loop m, and () value if i m; f m peue head inoduced by he poenial elemen of he loop m, given by: imple cloed loop: f m (4) cloed loop conaining boe pmp inalled in he pipe: f m ε H (5) i m i i p, i open loop wih pump and/o eevoi a node: f m Z I Z E (6) whee: Z I, Z E ae piezomeic head a peue device a he enance o exi fom he loop; H p,i pump head of he pump inegaed on he pipe i, fo he dichage i, appoximaed by paabolic inepolaion on he pump cuve given by poin [6]. 3 Newok deign opimizaion cieia Opimizaion of diibuion newok diamee conide a mono o mulicieial obecive funcion. Co o enegy cieia may be ued, imple o complex, which conide he newok co, pumping enegy co, opeaing expene, included enegy, conumed enegy, oal expene ec. hee cieia can be expeed in a complex obecive mulicieial funcion [7], wih he geneal fom: F ξ c in which: NP α ( a + bdi ) Li + ψp, ( hi + H ) i a ( + β ) (8) β ( + β ) a p + ; ξ a p (7) ξ + (9) ψ 9.8 ( f σξ 73 τ ) η + Φ a e () k whee: i he numbe of pipe in a newok; a, b, α co/enegy paamee depending on pipe maeial [9]; D i, L i diamee and lengh of pipe i; NP numbe of pump aion; p, pumped dichage of pump aion ; Σh i um of head loe along a pah beween he pump aion and he ciical node; H geodezic and uilizaion componen of he pumping oal dynamic head; β / amoizaion pa fo he opeaion peiod ; p, p epai, mainenance and peiodic eing pa fo newok pipe and pump aion, epecively; peiod fo which he opimizaion cieion expeed by he obecive funcion i applied, having he value o ; η efficiency of pump aion; f inallaion co of uni powe; σ a faco geae han one which ake ino accoun he inalled eeve powe; e co of elecical enegy; τ p /876 pumping coefficien, which ake ino accoun he effecive numbe p of pumping hou pe yea; Φ κ aio beween he aveage monhly dichage and he pumped dichage [6]. Fo newok upplied by pumping, he lieaue [], [], [] ugge he ue of minimum annual oal expene cieion (CAN), bu chooing he opimal diamee obained in hi way, he newok become uneconomical a ome ime afe conucion, due o inflaion. heefoe, i i ecommended he foe menioned cieion be ubec o dynamizaion by uing he cieion of oal updaed minimum expene (CA), he fome being in fac a pecific cae of he lae when he invemen i ealied wihin a yea; he opeaing expene ae he ame fom one yea o anohe and he expeced life-ime of he diibuion yem i high. In paicula, he ue of enegeical cieia diffeen fom co cieia i ecommendable. hu, anohe way o appoach he poblem, wih ha a bee validiy in ime and he homogenizaion of he obecive funcion i newok deign accoding o minimum enegeic conumpion (W). he geneal funcion (7) enable u o obain a paicula obecive funcion by paiculaizaion of he ime paamee and of he ohe economic and enegeic paamee, chaaceiic of he diibuion yem. Fo example, fom, a, e, f he minimum enegeic conumpion cieion i obained. 4 Developmen of opimizaion model Uing following nondimenional paamee: ISSN: ISBN:

3 fo ouflow: fo pipe: θ c i () ω i α Di, () λ L i he expeion of head lo h i, fo pipe wih unifom ouflow along hei lengh, ake he fom: Ω i h i hi Θ i (3) Li whee: 4 θi 3θi + 3 Θ i (4) 3 ( θ ) i i 4α Dk, i Ω i (5) λ ( θ ) i in which: h i i he head lo fo pipe i wih concenaed ouflow; c ouflow along he lengh of he pipe i; inflow in he iniial ecion of he pipe i; α nonunifomiy coefficien of velociy diibuion in he co ecion of he pipe; λ i ficion faco of pipe i. he oal lengh of a pipe i, wih he dichage i, may be divided ino i paial lengh (eco), of D i diamee and X i lengh. aking ino accoun he Dacy Weibach funcional equaion, he ficion lope J i fo each eco k of he pipe i can be calculaed, in he hypohei of concenaed ouflow, wih he equaion: hk, i 8 i J i λ i (6) X π g D i i i whee: i an exponen having he value 5.; g gaviaional acceleaion; λ i ficion faco of eco k in pipe i, can be calculaed uing he Colebook Whie fomula. Since in eal condiion he dichage deceae fom one co ecion o anohe in he ene of ouflow, an inceae of peue i accomplihed a he oule of he pipe, by a phenomenon imila o ebound, which ha a he effec of diminihing he head lo. hu, he expeion of ficion lope in each eco k of he pipe i, fo he unifom ouflow along he lengh of he pipe, i ewien a: h Ω i i J Θ i J i i (7) X i X i Specific conumpion of enegy fo diibuion of wae w d, in kwh/m 3, i obained by efeing he hydaulic powe diipaed in pipe o he um of dichage: w d i R q< β +.7 (8), whee q i he ouflow a he node. Compuaion of he opimal deign of looped newok mu be pefomed in wo age: eablihmen of opimal diibuion fo dichage hough pipe, i, accoding o he minimum bulk anpo cieion [8], which ake ino accoun he newok eliabiliy; compuaion of opimal pipe diamee, D i, aking ino accoun he opimized dichage. he eie of commecial diamee which can be ued D i [D max,i, D min,i ] fo each pipe i ae eablihed uing he limi value of opimal diamee D max,i and D min,i, compued by opimizaion elaion (9) fo pumping opeaion newok o elaion () fo gaviy newok: in which: D D max min), i N i α+ max(min) q i, α+ p β α+ i ( E (9) max( min), i 4i () πv min( max), i.33 n' ψ E () abξ whee: i i he dichage of he pipe i; p Σ p, pumped dichage; V min, V max limi of he economic velociie; n Manning oughne coefficien of he pipe; E economy-enegy faco of he pipe, which ha a maximum value and a minimum value [9], coeponding o he limi value of he vaiaion of economy-enegy paamee (p, p, η, f, σ, e, τ, Φ k ) fo he diibuion yem, included in ψ and ξ. A penaly coefficien p i i ued when opimizing diamee in he cae of exending a newo which ha he value equal o he value of coeponding impoed diamee, fo pipe wih fixed diamee, euling in D i p i. Admiing ha a pipe i of lengh L i of a pumping opeaion newok made up of pipe, can be divided ino i eco k of diamee D i and lengh X i and aking ino accoun he noaion: α c i ξ( a + bdk, i ) () Z IPP, ( hi + H ) (3) he obecive funcion (7) ake he fom: ISSN: ISBN:

4 F c i i k k i NP c, X, + ψ, Z, min (4) k i p IPP he unknown of he obecive funcion ae vaiable X i and Z IPP,. When he peue device i compied of one o moe eevoi (ψ ), he expeion (4) of he obecive funcion become: i F c X min, (5) c i k i i minimizing he included enegy o he newok co and having a unknown he vaiable X i. Hence, he value of he vaiable mu be deemined in ode o minimize he obecive funcion F c, povided he following conain ae aified: conucive conain: i k X k, i Li ( i,..., ) (6) funcional conain which ae wien fo each opeaing iuaion, and which mu povide he neceay peue HN o a he ciical node, aing on diffeen pah fom he peue device IPP (fig. ): Z IPP N i, ε i Θi J i X i Zo + HN i k N i ( ε i Ω k i J i + H p, i k, i ) o (7) whee: N i he pipe numbe of a pah IPP O; Z o elevaion head a he ciical node O; Z IPP, available piezomeic head a he peue device ; H p,i pumping head of he boe pump mouned in he pipe i. Fig. Scheme of a pah IPP ciical node O hydaulic conain chaaceiic only fo looped newok, expeing he enegy conevaion in loop: ε i Θi J k i k i i k i k i m i m i i m m, X, ε Ω, J, + f ( m,..., M ) (8) in which he peue head f m i given by he elaion (4), (5), (6) and ε i i he oienaion of he pipe. In he cae ha he available piezomeic head Z IPP, ae known, and i being unneceay o deemine hem by opimizaion, he obecive funcion (4) ake he fom (5), while value Z IPP, ae conained in he fee em of conain (7) and (8). A he obecive funcion (4) o (5) and conain (6), (7), (8) ae linea wih epec o he unknown of yem he opimal oluion i deemined accoding o he linea pogamming mehod, uing he Simplex algoyhm. Compuing he unknown Z IPP, by opimizaion, fo pumping opeaion newok eul in he coeponding pumping head: Z Z (9) H p, IPP, SP, whee Z SP, i he wae level in he ucion bain of IPP. aking ino accoun head lo SH n on he pah IPP n: SH n N i N i Θi J i X i ( Ω i J i + H p, i ) i k i k (3) he piezomeic head Z n and he eidual peue head H n a he node n ae deemined fom he elaion: Z n Z IPP, SH n (3) H n Z n Zn (3) whee Z n i he elevaion head a he node n. Fo an opimal deign, he piezomeic line of a pah of N pipe, iuaed in he ame peue zone, mu epeen a polygonal line which eemble a cloely a poible he opimal fom expeed by he equaion: βα + α+ N d Z Z h (33) IPP, - N n i i Li i in which: Z n i he piezomeic head a he node n; d diance beween node n and he peue device. he compue pogam OPLIRA ha been elaboaed baed on he linea opimizaion model, in he FORRAN pogamming language fo IBM PC compaible micoyem. 5 Numeical applicaion he looped diibuion newok wih he opology fom Figue i conideed. I i made of ca ion and i upplied by pumping wih a dichage of.3 m 3 /. he following daa i known: pipe lengh L i, in m, elevaion head Z, in m, and neceay peue HN 4 m H O. ISSN: ISBN:

5 A compaaive udy of newok dimenioning i pefomed uing he claic model of aveage economical velociie (MVE), Mohnin opimizaion model (MOM) [] and he linea opimizaion model (MOL) developed above, he la being applied in he hypohei of concenaed ouflow (MOL N), a well a of unifom ouflow along he lengh of he pipe (MOL D). able. Hydaulic chaaceiic of he pipe deemined wih he model MOL N and MOL D Fig. Scheme of he deigned diibuion newok Calculu wa pefomed conideing a anioy ubulence egime of wae flow and he opimizaion cieion ued wa ha of minimum enegeic conumpion. Reul of he numeical oluion pefomed by mean of an IBM PC compue, efeing o he hydaulic chaaceiic of he pipe ae peened in able and. able. Hydaulic chaaceiic of he pipe deemined wih he model MVE and MOM Pipe L MVE MOM i D i h i V i i D i h i V i [m 3 /] [mm] [m] [m/] [m 3 /] [mm] [m] [m/] MOL-N MOL-D Pipe i - i k X i D i h i V i i k X i D i h i V i [m 3 /] [m] [mm] [m] [m/] [m 3 /] [m] [mm] [m] [m/] , he ignificance of ( ) ign of dichage and head loe in able and i he change of flow ene in he epecive pipe wih epec o he iniial ene conideed in he Figue. In Figue 3 hee i a gaphic epeenaion, aing fom he node ouce 8 o he ciical node, on he pah 8 5, he piezomeic line being obained uing he hee menioned model of compuaion, evidencing hei deviaion fom he opimal heoeical fom. Figue 3 alo include he coeponding value of he obecive funcion F c, he newok included enegy W c, pumping enegy W e, a well a pecific enegy conumpion fo wae diibuion w d. Accoding o he pefomed udy i wa eablihed ha: all he pipe of he newok ae opeaing in a anioy ubulence egime of wae flow; hee i a geneal inceae of pipe diamee obained by opimizaion model (MOM, MOL) wih epec o MVE, becaue he claical model doe no ake ino accoun he minimum conumpion of enegy and he diveiy of economical paamee; ISSN: ISBN:

6 Fig. 3 Piezomeic line along he pah 8 5 in compaion wih he eul obained by MVE, he one obained by opimizaion model ae moe economical, a ubanial educion of pecific enegy conumpion fo wae diibuion i achieved (MOM.3 %, MOL-N 4.3 %, MOL-D 45.3 %) a well a a educion of pumping enegy (MOM 6,4%, MOL-N.3%, MOL-D.6%), a he ame ime he obecive funcion ha alo malle value (MOM.3 %, MOL-N 4.5 %, MOL-D 4.8 %); he opimal eul obained uing MOL ae upeio enegeically o hoe offeed by MOM, leading o pumping enegy aving of 5 %; alo, he applicaion of MOL fo unifom ouflow along he lengh of he pipe, ha led o he minimum deviaion fom he opimal fom of he piezomeic line, epecially o a moe unifom diibuion of he pumping enegy, by eliminaion of a high level of available peue a ome node even a maximum conumpion. he malle value of he pecific enegeic conumpion, namely ha of.4 kwh/m 3, alo uppo hi aeion; educion of he peue in he diibuion newok achieved in hi way, i of mao pacical impo, conibuing o he diminihing of wae loe fom he yem. 6. Concluion he popoed opimizaion model, a vey geneal and pacical one, offe he poibiliy of opimal deign of wae upply newok uing muliple opimizaion cieia and conide he anioy o quadaic ubulence flow. I ha he advanage of uing no only co cieia, bu alo enegy conumpion, conumpion of cace eouce, and ohe cieia can be expeed by imple opion in he obecive funcion (7). he model of linea opimizaion could be applied o any looped o ee haped newo eihe when piezomeic head a peue device (pump aion o ank) mu be deemined o when hee head ae given. I pemi he deeminaion of an opimal diibuion of comecial diamee along he lengh of each pipe of he newok and he lengh of pipe eco coeponding o hee diamee. Alo, hi faciliae he conideaion of unifom ouflow along he lengh of he pipe newok. A moe unifom diibuion of pumping enegy i achieved o ha head loe and paamee of pump aion can be deemined moe peciely. Fo diffeen analized newok, he aving of elecical enegy due o diminihing peue loe and opeaion co when applying he model of linea opimizaion epeen abou...35%, which i of gea impoance, conideing he geneal e- negy iue. Refeence: [] Abamov, N.N. eoia i meodica acea iem podaci i apedelenia vodî, Soizda, Mokva, 97. [] Cenedee, A. Mele, P. Opimal deign of wae diibuion newok, Jounal of he Hydaulic Diviion, ASCE, no. HY, 978. [3] Co, H. Analyi of flow in newok of condui o conduco, Bullein no. 86, Univ. of Illinoi Engg. Expeimen Saion, III, 936. [4] Jame M. e al. Opimal oluion fo diic heaing pipe newok, Poc. of he 5 h IASME/ WSEAS In. Conf. on Enegy & Envionmen, Cambidge, Febuay 3-5,, pp [5] Kope, J. Dobee D. Goicanec, D. Vaiaion of pipe diamee and i influence on he flowpeue condiion, WSEAS anacion on Fluid Mechanic, vol., no., 6, pp [6] Sâbu, I. Enegeical opimizaion of wae diibuion yem, Publihing Houe of he Romanian Academy, Buchae, 997. [7] Sâbu, I. Kalmà, F. Opimizaion of looped wae upply newok, Peiodica Polyechnica, Budape, vol 46, no.,, pp [8] Sâbu, I. Opimizaion of he dichage diibuion in wae upply newok, Hydoechnic, no. 6, 5, pp [9] Sâbu, I. Kalma, F. Compue aided deign of building equipmen, Publihing Houe Mion, imioaa,. [] Sephenon, D. Pipe flow analyi, Elevie Science Publihe B.V., 984. [] hawa, W. Lea co deign of wae diibuion yem, Jounal of he Hydaulic Diviion, ASCE, no. HY9, 973. ISSN: ISBN:

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