Comparison of Newton Raphson and Hard Darcy methods for gravity main nonlinear water network

Transcription

1 IOSR Journal of Mecancal and Cvl Engneerng (IOSRJMCE) eissn:,pissn: 0X, Volume, Issue Ver. III (May. June. 0), PP 0 Comparson of Newton Rapson and Hard Darcy metods for gravty man nonlnear water network AbdulamdSaad, HasanAweed, HusamELDnAbdulkaleq, Alwaleedaleel, Amed Abdulsalam. a aculty of Engneerng, Tobruk Unversty, Tobruk, Lbya. Abstract: A water network of ppes dependng on manly gravty and covers an area of. square klometers was taken an as a case study to test and compare te analyss. Te governng equaton of ts network are nternal flow n ppe equatons, wc consst of te contnuty equaton, te modfed Bernoull's equaton, and te ead loss due to te lengt of te ppe. Te tree equatons are nonlnear algebrac equatons because of te square power of te dscarge n te ead loss equatons, wc need to be solved numercally. Hard Darcy metod and Newton Rapson metod are used to solve te system of nonlnear equatons, and to compare te soluton.so, twenty four nonlnear equatons (nne Bernoull's equatons and ffteen contnuty equatons) n twenty four unknowns dscarges were got by tese two metod by usng MATLAB code. Tere are not dfferences n te resulted dscarges between Hard Darcy and Newton Rapson metods. Also, t was found tat Newton Rapson was faster tan Hard Darcy Metod wen tey compared by te number of teraton. Te fnal soluton of te dscarges ave tested by te basc of flud mecancs tat says te summaton of ead losses nsde a loop must be equal zero wc can be seen clearly n te plots of te two metods. eywords: comparson, duscarge, ppe, HardDrcy, Newton Rapson I. Introducton Water ppe network systems are desgned and operated to supply fres water from te source (or treatment faclty) to customers (HundDer & YuCang, 00). Nearly 0% to % of te cost of a total water supply system s contrbuted toward water transmsson and te water dstrbuton network (Abdulamd, 0). In ts project, te dstrbuton network of ppes wt nne looped network and gravty man s consdered. Analyss wll take place bysettng up a system of a nonlnear equaton as results of nternal flow n ppe suc as, te contnutyequaton, Bernoull equaton, and te major losses equatons. Ts system cannot be solved analytcally. Terefore, numercal metod by usng MATLAB software s used to solve te nonlnear systems of te network. Nonlnear equatons set can be formulated to descrbe te relatonsp between te nodal ead and ppe flow rate. Hard Darcy metod and Newton Rapson metod was commonly used to solve te nonlnear equaton set for obtanng te soluton of te network (HundDer & YuCang,00). Te ydraulc and optmzaton analyss are lnked troug an teratve procedure. Te analyss of te ppe network s to estmate te dscarge n eac ppe, veloctes, and te total cost of te system. Also, proof of te soluton n eac metod and te comparson between te two wll be consdered.. Te modfed Bernoull equaton Te Bernoull equaton s a relaton between pressure, velocty,and elevaton n steady, ncompressble flow(yunus A & Jon M,00) as sown n te next equaton. V V gz V Were s te flow energy, gz L s knetc energy, gz s potental energy and L s ead losses. []. Te major losses n ppe Te ead loss due to vscous effects n te stragt ppes, termed te major loss and denoted (Munson et al., 00). L V L f D g L major DOI: 0.0/000 Page []

2 . Te mnor losses n ppe Te flud n a typcal ppng system passes troug varous fttngs, valves,bends, elbows, tees, nlets, exts, enlargements, and contractons n addtonto te ppes. Tese components nterrupt te smoot flow of te flud andcause addtonal losses because of te flow separaton and mxng teynduce. In a typcal system wt long ppes, tese losses are mnor comparedto te total ead loss n te ppes (te major losses) and are called mnor losses (Yunus A & Jon M,00). L mn or V L g. Volumetrc flow rate (dscarge) Te volume of te flud flowng troug a cross secton per unt tme s: [] VA C. Seres and parallel network [] or ppes n seres, te flow rate s te same n eac ppe, and te total ead loss s te sum of te ead losses n ndvdual ppes. LT L L L Snce te same dscarge passes troug all te ppes, te contnutyequaton s... n [] or ppes n parallel, te ead loss s te same n eac ppe, and te total flow rate s te sum of te flow rates n ndvdual ppes. A B [] L= L II. Te Problem Water supply networks consstof a of sources, ppe loops (M. Tabes,00) n ts case study desgn a water network from node No wc s te upstream to node No wc s te downstream by gravty man as sown fgure (). Te dmensons of te network are lsted ntables and. Te network covers an area of. klometers square, conssted of nne loops ( ppes, man lnes and mnor lnes) wat's more, te outsde border of te network consdered as te man lnes, and te nner lnes consdered as mnor lne of te network. urtermore, ts network ncluded of nodes, te frst node consdered te upstream (wt neglectedmnor losses) (Swamee&Sarma,00). [] [] g : Gravty man looped network of ppes DOI: 0.0/000 Page

3 Table : Te dmensons of te network No. of ppe Dameter Lengt No. of ppe Dameter Lengt No. of ppe Dameter Lengt Table : Te elevaton of eac node No. of node Hegt No. of node Hegt III. Numercal Soluton Of Nonlnear System Of Equatons One of te most common mportant steps n water resources engneerng s ppe network analyss, te key metods for ts analyss are Hard Darcy and NewtonRapson (I.A. Oke;00). IV. Te Soluton By Usng Newton Rapson. Te assumpton All te dscarges can be assumed for one value or dfferent values as sown n table (Moosavan& Jaefarzade, 0).. Terefore, n Newton Rapson not necessary to assume an ntal guesses tat satsfes te contnuty equatons as sown n table.. Table : ntal guesses of Newton Rapson metod Ppe dscarges Te assumed values Ppe dscarges Te assumed values Ppe dscarges Te assumed values Te equatons of Newton Rapson metod Te dscarge equaton of eac node We know te summaton of nflow and out flow at node sould be equal zero, terefore: [] [0] [] [] [] [] [] [] [0] [] [] [] [] DOI: 0.0/000 Page

4 0 Te ead losses equaton of eac loop By usng te basc of flud mecancs, te sum of losses nsde eac loop sould be equal zero, terefore: L L L L L L L L L L L0 L L L L L 0 L L L L L0 L L L L L L L L fl gd Were L L L L0. ndng te jacoban Te equaton () as te jacoban matrx wc can be found as follow. f ( J L L0 L L * f (,) (,,... ) ( f f... f f... 0.) f f [] [] [] [] [] [] [] [] [0] [] [] [] [] J f (,) ( L L L L ). Te fnal matrx Te next matrx sows te calculaton of te frst teraton of eac loop DOI: 0.0/ Page

5 .Te result of te ppe dscarges of te frst teraton Table: Te ppe dscarge for te frst teraton Ppe dscarge Dscarges Ppe dscarge Te ppe dscarges and veloctes of te last teraton Te correct dscarges and veloctes can be got after teraton (MATLAB code by usng Newton Rapson metod see App A), sowed n table and. Table : Te ppe dscargesfor te last teraton Ppe dscarges Ppe dscarges Ppe dscarges In addton, by apply te equaton [] vac we get te followng veloctes: Table : Te ppe veloctesfor te last teraton Ppe veloctes Ppe veloctes Ppe veloctes ` Te accuracy of frst teraton soluton In flud mecancs bascs, te algebrac sum of te ead losses around a loop must be zero wc s not sown n tables and. L.00 L. L. Table : Te losses of eac ppe for te frst teraton L. L0.0 L 0. L. L. L. L. L. L0. L. L. L.00 L.0 L.0 L.00 L 0. L.. L. L. L 0. Table : Te summaton of losses n eac loop for te frst teraton Loop number (loop) ( loop) (loop) (loop) Summaton of ead Loop number (loop) (loop) (loop) (loop) Summaton of ead (loop) 0.0 DOI: 0.0/000 Page

6 . Te accuracy of last teraton soluton By usng newton Rapson metods and usng MATLAB code we got te sum of ead loss around eac loop s zero as sown n tables and 0. L.0 L. L. L.00 L0. L 0.0 Table : Te losses of eac ppe for te last teraton L.0 L. L.0 L.0 L.0 L0.0 L.0 L.00 L. L.0 L.0 L. L 0. L..00 L. L. L 0. Loop number Summaton of ead Loop number Summaton of ead Table 0: Te summaton of losses n eac loop for te last teraton (loop) ( loop) (loop) (loop) 0 (loop) 0 (loop) (loop) 0 (loop) 0 0 (loop) V. Te Soluton By Usng Hard Darcy Te overall procedure for te looped network analyss can be summarzed n te followng steps:. Number all te node and ppe lnks, Also number te loops, for clarty, ppe numbers are crcled and te loop numbers are put n square brackets.. Adopt a sgn conventon tat a ppe dscarge s postve f t flows from a lower node number te ger node number, oterwse negatve.. Apply nodal contnuty equaton at all nodes to obtan ppe dscarge.startng from nodes avng two ppes wt unknown dscarge, assume an arbtrary dscarge (say 0. ) n one of te ppes and apply contnutyto obtan dscarge n te oter ppe. Repeat te procedure untl all te ppe flows are known.f tere exst more tan two ppes avng unknown dscarges, assume arbtrary dscarges n all te ppe except one and apply contnuty equaton to get dscarge n te oter ppe. Te total number of prmary loops n te network.. Assume frcton factors f 0. 0 n all ppes lnks and compute correspondng. Assume loop ppe flow sgn conventon to apply loop dscarge correctons; generally, clockwse flows postve and counterclockwse flows negatve are consdered.. Calculate k for te exstng ppe flows and apply ppe correctons algebracally.. Apply te smlar procedure n all te loops of a ppe network. Repeat steps and untl te dscarge correctons n all te loops are relatvely very small (Swamee& Sarma,00).. Te assumpton Te ntal dscarges sould satsfy contnuty equaton at eac node as table (Moosavan& Jaefarzade,0). Also, te number of assumed dscarge sould be equaled to te number of loops wc s nne. Table : Te assumed ntal guesses for te frst teraton Ppe dscarge Te assumed value Ten te rest of te dscarge of te frst teraton are lsted n table. Table.: Te dscarge obtaned from contnuty equaton Ppe dscarge Te assumed values Ppe dscarge Te assumed values DOI: 0.0/000 Page

7 . Te equaton of Hard Darcy metod Te dscarge equaton of eac node We know te summaton of nflow and out flow at node sould be equal zero, terefore: Te loss equaton [] Te algebrac sum of te ead loss n a loop must be equal to zero k loop, k Were k,,,...,k 0 for all loops L fl gd. Te frst teraton of Hard Darcy metod Table to table sow te calculaton of te frst teraton of eac loop. Were: k loop, k loop, k [] [] [] [] [] [] [0] [] [] [] [] [] [] [] [] [0] Ppe Dscarge )m /s) )s /m ) Table : loop )s/m ) Corrected low (m /s) DOI: 0.0/000 Page

8 Ppe Dscarge )m /s) )s /m ) Table : loop )s/m ) Corrected low (m /s) Ppe Dscarge )m /s) )s /m ) Table : loop )s/m ) Corrected low (m /s) Ppe Dscarge )m /s) )s /m ) Table : loop )s/m ) Corrected low (m /s) Ppe Dscarge )m /s) )s /m ) Table : loop )s/m ) Corrected low (m /s) Ppe Dscarge )m /s) )s /m ) Table : loop )s/m ) Corrected low (m /s) DOI: 0.0/000 Page

9 Ppe 0 Dscarge )m /s) )s /m ) Table : loop )s/m ) Corrected low (m /s) Ppe 0 Dscarge )m /s) )s /m ) Table 0: loop )s/m ) Corrected low (m /s) Ppe Dscarge )m /s) )s /m ) Table : loop )s/m ) Corrected low (m /s). Te ppe dscarges for te frst teraton Te dscarges of te frst teraton are sown n table by (MATLAB code). Table : Te ppe dscarge of te frst teraton Ppe dscarge Ppe dscarge Ppe dscarge Te ppe dscarges and veloctes of te last teraton Te correct dscarges and veloctes can be got after many number of teraton (MATLAB code by usng Hard Darcy metod), sowed n tables and. Table : Te ppe dscarges for te last teraton Ppe dscarge Ppe dscarge Ppe dscarge DOI: 0.0/000 Page

10 In addton, by apply te equaton [] vac we get te followng veloctes: Table : Te ppe veloctes for te last teraton Veloctes Veloctes ` Veloctes Te accuracy of frst teraton soluton Te soluton tat sowed above only for te frst teraton, wc s not correct. Te next test, sows tat are not equal to zero wc s not correct as sown n table No. loop Table : Te correcton factor n eac loop of te frst teraton. Te accuracy of last teraton soluton Hard Darcy metod by usng MATLAB code was run to get te next results as a proof of te accuracy of te soluton of te dscarges as sown n table No. loop Table : Te correcton factor n eac loop of te last teraton VI. low Rate Comparson Te dfferences between te dscarges obtaned by Newton Rapson and Hard Darcy metod are approxmately zero as sown n fg () g : low rates obtaned by Newton Rapson and Hard Darcy metods wt number p VII. Te Number Of Iteraton Wt Te Summaton Of Head Losses In Eac Loop or Newton Rapson Metod Te correct flow rates by Newton Rapson metod were got after teraton as sown fg (). g : Te relatonsp between te numbers of teraton wt te summaton of ead losses equatons n eac loop for Newton Rapson metod DOI: 0.0/000 Page

11 VIII. Te Number Of Iteraton Wt Te Summaton Of Head Losses In Eac Loop or Hard Darcy Metod Te correct flow rates by Hard Darcy metod were got after 0 teraton as sown fg (). g : Te relatonsp between te numbers of teraton wt te summaton of ead losses equatons n eac loop for Hard Darcy metod IX. Comparson Between Te Summaton Of Head Losses Equatons By Newton Rapson And Hard Darcy Te next table sows te summaton of te ead loss equaton n eac loop tat must be approxmately zero, wc can be seen tat newton Rapson s faster tan ard Darcy to converge to te soluton. Table : Tesummaton ead losses equatons by Newton Rapson and Hard Darcy metods Loop number Te summaton of ead losses (Newton Rapson) after teraton. Te summaton of ead losses (Hard Darcy) after teraton..e0.0e0 0.0e0.e0 0.e0.e0.e0.e0 0.e0 0.0e0 0.e0.e00.0e0 X. Concluson A nonlnear systems network were smulated by Newton Rapson and Hard Darcy metods usng MATLAB software. Te nonlnearty s sowed n te square power of te dscarge n ead losses equatons. Te dscarges resulted of eac ppe were found te same n eac metod. Also, te fnal soluton was valdated by usng te basc of flud mecancs wc tat te summaton of losses nsde a loop must be equal to zero. Tus numercally, n Newton Rapson, wc summaton as a g accuracy and approxmately zero compared to Hard Darcy metod. Also, te soluton n Newton Rapson metod can be got at less number of teratons (faster) compared to Hard Darcy metod. In addton, ntal guesses (te assumpton) s more complcated n Hard Darcy because te value of eac dscarge must satsfy te contnuty equatons wc need more calculatons. However, te ntal guesses can be cosen randomly n Newton Rapson metod wtout satsfyng te contnuty equatons. References []. HundDer Ye& YuCang Ln.ppe network system analyss usng smulated annealng. journal of water supplyresearc and tecnology AUA;aug00Vol () :. []. Abdulamd El IdrsSaad. Desgn of Nonlnear Ppng Water System Network by Usng Newton Rapson Metod. IOSR Journal of Mecancal and Cvl Engneerng (IOSRJMCE); MarApr 0Vol () :. []. Yunus A. Cengel, Jon M. Cmbala. flud mecancs fundamental and applcaton, Hger Educaton;00. []. Bruce R. Munson, Donald. Young, Teodore H. Oks, Wade W. Huebsc, fundamental of flud mecancs, Sxt Edton, Jon Wley&Sons.INC;00. []. M. Tabes, T. T. Tanymbo, R. Burrows. Headdrven smulaton of water supply networks. J SID;00. []. N. Moosavan*,M. R. Jaefarzade. Hydraulc Analyss of water supply networks usng a modfed ard Darcy metod. Internatonal Journal of Engneerng;September0Vol () :. []. I.A.Oke, Relabllty and statstcal assessment of metods for ppe network analyss, envronmental engneerng scence, December 00 Vol (0) :0. []. Prabata. Swamee, Asok. Sarma. Desgnof water supplu ppe network, Jon Wley & Sons, Inc; 00 DOI: 0.0/000 Page