DESIGN OF OPTIMAL WATER DISTRIBUTION SYSTEMS

Transcription

1 DESIGN OF OPIMAL WAER DISRIBUION SYSEMS IOAN SÂRBU Politehnica Univesity of imişoaa. Intoduction Distibution netwoks ae an essential pat of all wate supply systems. he eliability of supply is much geate in the case of looped netwoks. Distibution system costs within any wate supply scheme may be equal to o geate than 60 % of the entie cost of the poect. Also, the enegy consumed in a distibution netwok supplied by pumping may exceed 60 % of the total enegy consumption of the system [9]. Attempts should be made to educe the cost and enegy consumption of the distibution system though optimization in analysis and design. A wate distibution netwok that includes pumps mounted in the pipes, pessue educing valves, and checkvalves can be analyzed by seveal common methods such as Hady-Coss, linea theoy, and Newton-Raphson [0]. aditionally, pipe diametes ae chosen accoding to the aveage economical velocities (Hady-Coss method) [3]. his pocedue is cumbesome, uneconomical, and equies tials, seldom leading to an economical and technical optimum. his pape develops a nonlinea model fo optimal design of looped netwoks supplied by diect pumping fom one o moe node souces, which has the advantage of using not only cost citeia, but also enegy consumption, consumption of scace esouces, opeating expenses etc. Also, this new model consides the tansitoy o quadatic tubulence egime of wate flow. he dischage continuity at nodes, enegy consevation in loops, and enegy consevation along some paths between the pump stations and the adequate citical nodes ae consideed as containts. he nonlinea optimization model consides head losses o dischages though pipes as vaiables to be optimized in ode to establish the optimal diametes of pipes and is coupled with a hydaulic analysis. his model can seve as guidelines to supplement existing pocedues of netwok design. 2. Netwoks design optimization citeia Optimization of distibution netwok diametes consides a mono- o multiciteial obective function. Cost o enegy citeia may be used, simple o complex, which consides the netwok cost, pumping enegy cost, opeating expenses, included enegy, 2002 CIB W62 INERNAIONAL SYMPOSIUM WAER SUPPLY AND DRAINAGE FOR BUILDINGS C0-

2 consumed enegy etc. hese citeia can be expessed in a complex obective multiciteial function [8], with the geneal fom: α F = ξ ( a+ bd ) L + ψ Q, ( h + H ) () c p o = = in which: t ( + β o ) t t a = (2) ξ t = a p + ; ξ 2 = a p2 + (3) β o( + β o) ψ = ( fσξ a eτ Φ k) (4) η whee: is the numbe of pipes in a netwok; a, b, α cost paametes depending on pipe mateial [9]; D, L diamete and length of pipe ; NP numbe of pump stations; Q p, pumped dischage of pump station ; Σh sum of head losses along a path between the pump station and the citical node; H o geodezic and utilization component of the pumping total dynamic head; β o = / amotization pat fo the opeation peiod ;. p, p 2 epai, maintenance and peiodic testing pat fo netwok pipes and pumping stations, espectively; t peiod fo which the optimization citeion expessed by the obective function is applied, having the value o ; η efficiency of pumping station; f installation cost of unit powe; σ a facto geate than one which takes into account the installed eseve powe; e cost of electical enegy; τ = p /8760 pumping coefficient, which takes into account the effective numbe p of pumping hous pe yea; Φ k atio between the aveage monthly dischage and the pumped dischage [8]. he geneal function () enables us to obtain a paticula obective function by paticulaization of the time paamete t and of the othe economic and enegetic paametes, chaacteistic of the distibution system. Fo example, fom t =, a =, e =, f =0 the minimum enegetic con-sumption citeion is obtained. Fo netwoks supplied by pumping, the liteatue [], [2], [4], [5], [] suggests the use of minimum annual total expenses citeion (CAN), but choosing the optimal diametes obtained in this way, the netwoks become uneconomical at some time afte constuction, due to inflation. heefoe, it is ecommended the foe-mentioned citeion be subect to dynamization by using the citeion of total updated minimum expenses (CA), the fome being in fact a specific case of the latte when the investment is ealised within a yea; the opeating expenses ae the same fom one yea to anothe and the expected life-time of the distibution system is high. In paticula, the use of enegetical citeia diffeent fom cost citeia is ecommendable. hus, anothe way to appoach the poblem, with has a bette validity in time and the homogenization of the obective function is netwok dimensioning accoding to minimum enegetic consumption (W). NP 3. Nonlinea model of optimization in designing distibution netwoks he head loss is given by the Dacy-Weisbach functional elation: L h g D Q = 8 2 λ 2 (5) π 2002 CIB W62 INERNAIONAL SYMPOSIUM WAER SUPPLY AND DRAINAGE FOR BUILDINGS C0-2

3 whee: is an exponent having the value 5.0; g acceleation of gavity; λ fiction facto of pipe which can be calculated using the Colebook-White fomula; Q dischage of the pipe. Equation (5) is difficult to use in the case of pipe netwoks and theefoe it is convenient to wite it simila to the Chézy-Manning fomula: β h = R Q (6) whee: R = KL / D is the hydaulic esistance of pipe ; β exponent which has values between.85 and 2. Specific consumption of enegy fo distibution of wate w sd, in kwh/m 3, is obtained by efeing the hydaulic powe dissipated in pipes to the sum of node dischages: w sd = = R N = q< 0 Q q ' β + ' whee q is the outflow at the node. he nonlinea optimization model (MON) allows the optimal designing of looped netwoks by using one of the CAN, CA o W optimization citeia, expessed by the obective function (). If the diamete D is expessed in elation (7) though the dischage and head losses: β D K Q = h L (8) and in the obective function () is eplaced the esulting expession, we have: α βα α α Fc a bk Q NP = ξ + h L L + ψ Qp, ( h + Ho ) (9) = = which is limited by the following constaints: dischage continuity at nodes : N Q + q = 0 ( =,..., N NP) (0) i= i enegy consevation in loops: ε h f m = 0 ( m=,..., M) () m = enegy consevation along some paths between the pump stations and adequate citical nodes : N ZSP, ε ( h Hp, ) Zo, = 0 ( =,..., NP) (2) = in which: Q is the dischage though pipe, with the sign (+) when enteing node and ( ) when leaving it; q concentated dischage at node with the sign (+) fo node inflow and ( ) fo node outflow; h head loss of the pipe ; ε oientation of flow tough the pipe, having the values (+) o (-) as the wate flow sense is the same o opposite to of the path sense of the loop m and (0) value if m; f m pessue head intoduced by the 2002 CIB W62 INERNAIONAL SYMPOSIUM WAER SUPPLY AND DRAINAGE FOR BUILDINGS C0-3 (7)

4 potential elements of the loop m [8]; M numbe of independent loops (closed-loops and pseudoloops); Z SP, available piezometic head at the pump station SP ; H p, pumping head of the pump mounted in the pipe, fo the dischage Q, appoximated by paabolic intepolation on the pump cuve given by points [9]; Z o, piezometic head at the citical node O ; N pipe numbe of a path SP - O. he optimization model (9) (2) epesent a nonlinea pogamming poblem, which esults in a system of non-linea equations by applying the Lagange nondetemined coefficients method. his system will have 2+NP equations with unknown vaiables Q, h, Z SP,, fomed by: a) N NP nodal equations (0); b) M loop equations (); c) NP functional equations (2); d) N NP enegy-economy equations fo nodes: N Q A Q p NP = ψ, ; fo pumping nodes ( =,..., ) (3) i NP N NP i= 0; fo othe nodes ( = +,..., ) in which: βα α+ α + Q = Q L h (4) A = α ξ bk (5) e) M enegy-economy equations fo loops: H = 0 ( m=,..., M) (6) = m α α+ βα in which: H = h L Q (7) Equations (3) can be expessed similaly with the dischage continuity equations, by giving Q * the same sign as Q. Equations (6) ae simila to the enegy consevation equations in the loops, by giving H * the same sign as h. In ode to establish an exteme of the obective function, we should specify a set of vaiables (Q o h ). hus, if the flow dischages in pipes ae known, the values h ae to be detemined by minimizing the obective function F c. If only the head losses ae the given values, the vaiables Q ae to be detemined by maximizing the obective function F c. Consideing vaiables h to be unknown, pipes dischages could be calculated in an vaiety of ways fo equations (0) to be satisfied; this howeve affects the eliability and technical and economic-enegetical conditions of the system. hat is why optimization of the flow dischages in pipes must be pefomed acco-ding to the minimum bulk tanspot citeion [7], which takes into account the netwok eliability. In this case, computation of the optimal design of looped netwoks must be pefomed in the following stages: Establishment of optimal distibution fo dischages though pipes, Q [8]. Detemination of head losses though pipes and piezometic heads at the supply nodes, by solving the nonlinea equation system (), (2), (3) using the gadients method [6]. Computation of optimal pipes diametes D using expession (8) and thei appoximation to the closest commecial values. α 2002 CIB W62 INERNAIONAL SYMPOSIUM WAER SUPPLY AND DRAINAGE FOR BUILDINGS C0-4

5 A new computation of the head losses using elation (5) o (6) and the hydaulic equilibium fo pipes netwok using Hady-Coss method. If the head losses ae the given values, the unknown vaiables Q ae to be detemined by solving the equation system (0), (2), (6), and used to calculate the optimal diametes in elation (8). he piezometic heads Z n can be detemined stating fom a node of known piezometic head. he esidual pessue head H n at the node n is calculated fom the elation: Hn = Zn Zn (8) whee Z n is the elevation head at the node n. Fo an optimal design, the piezometic line of a path of N pipes, situated in the same pessue zone, must epesent a polygonal line which esemble as closely as possible the optimal fom expessed by the equation: βα N d Zn = ZS h N P, - (9) L = = in which: Z n is piezometic head at the node n; d distance between the node n and the pump station SP. he compute pogam OPNELIRA was designed [9] based on the nonlinea optimization model. It was ealized in the FORRAN 5. pogamming language, fo IBM-PC compatible computes. α Numeical application he looped distibution netwok with the topology fom figue is consideed. It is made of cast ion and is supplied with a dischage of 0.23 m 3 /s. he following data ae known: pipe length L, in m, elevation head Z, in m, and necessay pessue HN = 24 m. A compaative study of netwok dimensioning is pefomed using the classic model of aveage economical velocities (MVE), Moshnin optimization model (MOM) [] and the nonlinea optimization model (MON) developed above. Calculus was pefomed consideing a tansitoy tubulence egime of wate flow and the optimization citeion used was that of minimum enegetic consumption. Results of the numeical solution pefomed by means of an IBM-PC PENIUM III compute, efeing to the hydaulic chaacteistics of the pipes (dischage, diamete, head loss, velocity) ae pesented in table. he significance of the ( ) sign of dischages and head losses in table is the change of flow sense in the espective pipes with espect to the initial sense consideed in figue CIB W62 INERNAIONAL SYMPOSIUM WAER SUPPLY AND DRAINAGE FOR BUILDINGS C0-5

6 Fig. Scheme of the designed distibution netwok In figue 2 thee is a gaphic epesentation, stating fom the node souce 8 to the citical node, on the path 8 5 2, the piezometic lines being obtained by using the thee mentioned models of computa-tion, and highlighting thei deviation fom Fig. 2 Repesentation of piezometic lines along the path the optimal theoetical fom. Figue 2 also includes the coesponding values of the obective function F c, the netwok included enegy W c, pumping enegy W e, as well as specific enegy consumption fo wate distibution w sd. able. Hydaulic chaacteistics of the pipes Pipe L Classic model (MVE) Moshnin model (MOM) Non-linea model (MON) i - [m] Q D h V Q D h V Q D h V [m 3 /s] [mm] [m] [m/s] [m 3 /s] [mm] [m] [m/s] [m 3 /s] [mm] [m] [m/s] Accoding to the pefomed study it was established that: thee is a geneal incease of pipe diametes obtained by optimization models (MOM, MON) with espect to MVE, because the classical model does not take into account the minimum consumption of enegy and the divesity of economical paametes; in compaison with the esults obtained by MVE, those obtained by optimization models ae moe economical, a substantial eduction of specific enegy consumption fo wate disti-bution is achieved (MOM %, MON - 64 %), as well as a eduction of pumping enegy (MOM %, MON - 2 %); at the same time the obective function has also smalle values (MOM %, MON %); 2002 CIB W62 INERNAIONAL SYMPOSIUM WAER SUPPLY AND DRAINAGE FOR BUILDINGS C0-6

7 the optimal esults obtained using MON ae supeio enegetically to those offeed by MOM, leading to pumping enegy savings of 6.2 %; also, the application of MON has led to the minimum deviation fom the optimal fom of the piezometic line, especially to a moe unifom distibution of the pumping enegy, by elimination of a high level of available pessue at some nodes. he smallest value of the specific enegetic consumption, namely that of kwh/m 3, also suppots this assetion; eduction of the pessue in the distibution netwok achieved in this way, is of mao pactical impot, contibuting to the diminishing of wate losses fom the system. 5. Conclusions he compute pogam developed in this study, a vey geneal and pactical one, offes the possibility of optimal design of wate supply netwoks using multiple citeia of optimization and consides the tansitoy o quadatic tubulence egime of wate flow. It has the advantage of using not only cost citeia, but also enegy consumption, consumption of scace esouces, and othe citeia can be expessed by simple options in the obective function (). he optimization appoach used in this study does not equie calculation of deivatives. his makes the method moe efficient and consequently helps the designe to get the best design of wate distibution systems with fewe effots. he nonlinea optimization model could be applied to any looped netwok, when piezometic heads at pump stations must be detemined. A moe unifom distibution of pumping enegy is achieved so that head losses and paametes of pump stations can be detemined moe pecisely. Fo diffeent analyzed netwoks, the saving of electical enegy due to diminishing pessue losses and opeation costs when applying this new optimization model, epesents about %, which is of geat impotance, consideing the geneal enegy issues. Refeences [] ABRAMOV, N. N.: eoia i metodica asceta sistem podaci i aspedelenia vodî, Stoizdat, Moskva, 972. [2] CENEDESE, A. MELE, P.: Optimal design of wate distibution netwoks, Jounal of the Hydaulics Division, ASCE, no. HY2, 978. [3] CROSS, H.: Analysis of flow in netwok of conduits o conductos, Bulletin no. 286, Univ. of Illinois Engg. Expeiment Station, III, 936. [4] DIXI, M. RAO, B. V.: A simple method in design of wate distibution netwoks, Afo-Asian Confeence on Integated Wate Management in Uban Aeas, Bombay, 987. [5] ORMSBEE, L.E. WOOD, D.J.: Hydaulic design algoithms fo pipe netwoks, Jounal of Hydaulic Engineeing, ASCE, no. HY2, 986. [6] PCHÉNICHNY, B. DANILINE Y.: Méthodes numéiques dans les poblèmes d extemum, Edition Mi, Moscou, 977. [7] SÂRBU, I.: L optimisation de la épatition des débits dans les éseaux maillés de distibution d eau, Buletinul Ştiinţific al U.P. imişoaa, om 36, 99. [8] SÂRBU, I. BORZA I.: Optimal design of wate distibution netwoks, Jounal of Hydaulic Reseach, no., CIB W62 INERNAIONAL SYMPOSIUM WAER SUPPLY AND DRAINAGE FOR BUILDINGS C0-7

8 [9] SÂRBU, I.: Enegetical optimization of wate distibution systems, Editua Academiei, Bucueşti, 997. [0] SEPHENSON, D.: Pipe flow analysis, Elsevie Science Publishes B.V., 984. [] HAWA, W.: Least-cost design of wate distibution systems, Jounal of the Hydaulics Division, ASCE, no. HY9, 973. Main autho pesentation Phd. eng. Ioan Sâbu is pofesso at POLIEHNICA Univesity of imişoaa, Romania CIB W62 INERNAIONAL SYMPOSIUM WAER SUPPLY AND DRAINAGE FOR BUILDINGS C0-8