analysis of water distribution systems using a

ELSEVIER

Analysis of water distribution systems using a perturbation method

H. A. Basha and B. G. Kassab

Faculty of Engineering and Architecture, American University of Beirut, Beirut, Lebanon

The analysis of a water distribution network requires the solution of a set of nonlinear equations. The current methods are all iterative and require a good initial estimate to reach the solution quickly without any convergence problems. In this study a perturbation expansion is applied to the set of nonlinear equations to obtain a series of linear equations that can be solved easily using matrix methods. The advantage of the proposed approach is that the solution is obtained directly without iterations, initial estimates, and issues of convergence. The method of solution is simple and straightforward to implement because it requires only one matrix inversion and four matrix multiplications. Hence the solution process is fast and efficient, which could prove useful in the optimization of water distribution systems wherein the network is solved for every trial set of design parameters. The solution is expressed in an explicit fashion which might be of use for further mathematical manipulation and implementation in an optimization algorithm. The method has been tested on various networks and the results obtained show a relatively high degree of accuracy.

Keywords: pipe networks, water distribution systems, perturbation solution, analytical method

1. Introduction

Water distribution network analysis is an important problem in civil engineering. It has gained more importance in recent years since the optimization of water distribution networks has become a focus of current research. The design of an optimal water distribution system involves an extensive simulation of the flow in the system for every trial set of design parameters. An additional concern has been the modeling of water quality in a distribution system whose solution also requires an efficient method of pipe network analysis. Most of the current methods used in the analysis of water distribution networks are iterative since the system of equations is nonlinear. An explicit and efficient method is therefore beneficial.

The basic hydraulic equations have been expressed in two principal fashions: either in terms of the unknown flow rates or in terms of the unknown nodal heads. The flow equations are expressed in terms of the flow rates in the links and consist of nonlinear energy equations and linear continuity equations. The head equations are formed of nonlinear continuity equations expressing the flow rates in the links as a function of the nodal heads. Each of them is a set of nonlinear algebraic equations that cannot be

Address reprint requests to Dr. H. A. Basha at the Faculty of Engineering

and Architecture, American University of Beirut, Beirut, Lebanon.

Received 31 January 1995; revised 6 July 1995; accepted 16 August 1995

solved directly. The current approaches are mostly iterative and the convergence of the solution depends significantly on the initial estimate.

There are basically four methods of solution: Hardy Cross, Newton-Raphson, linearization, and numerical min- imization. The Hardy Cross method’ consists of adjusting the flows along each loop in a consecutive manner such that the energy equations are satisfied within a certain error limit. The method depends on the initial estimate of flows and it suffers from slow convergence. The Newton- Raphson method2x3 consists of adjusting the flows or heads simultaneously along all the loops; it uses simultaneous corrections instead of local or individual corrections as in the Hardy Cross technique. Although the simultaneous corrections have improved convergence in simple networks, the Newton-Raphson techniques depend highly on the initial guess of the solution. The further the initial estimate is from the exact solution the more the convergence problems are experienced. Additional convergence problems have been encountered in the nodal subdomain (head equations) if a pipe in the network has a low value of frictional resistance resulting from a large diameter and/or a short length. In such a case the head loss between the two junctions is small and the nodal heads are close to each other which results in an ill-conditioned matrix. Recently a hybrid element formulation4 has been proposed to overcome such a problem.

The linear theory method5 has been applied to the flow equations and consists of transforming the nonlinear term in the energy equation into a linear one. The method

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Analysis of water distribution systems: H. A. Basha and B. G. Kassab

converges in a small number of iterations as compared with the Newton-Raphson method, but it uses a larger matrix and sometimes suffers from oscillation around the exact solution.6 The linear theory method was also devel- oped for the head equations’ but it has proven to be unreliable in certain cases.’

Todini and Pilati’ used the concept of minimizing an objective function to formulate a system of equations in terms of the flow rates and the nodal heads. The system formed of partly linear and partly nonlinear equations is solved using the Newton-Raphson method to find simultaneously the unknown flow rates in the pipes and the heads at the nodes. There are also other methods of solution that have been published in related fields.lO~”

Most of these methods depend on an initial estimate of the flows or the heads and some suffer from slow convergence or failure as has been tabulated by Wood and Rayes.‘* The proposed method follows an analytical approach rather than a numerical one. It consists of solving the original nonlinear equation in an approximative fashion using a perturbation expansion. The advantages of the method are that the solution is obtained in a straightforward manner without any iterations, initial estimate, and convergence problems. It just involves the determination and summation of the terms of the perturbation series.

2. Perturbation solution

The basic hydraulic equations describing the flow in a water distribution system are derived from the principles of mass and energy conservation. The mass continuity equation can be written for each node, and the energy conservation equation can be written for any path or loop. A network has n, links, nj junction nodes, nr fixed-grade nodes, and 1 independent closed loops. At the nj junction nodes, the sum of the flows in each connected pipe must be equal to the consumption

C(Qi”-Q,“,)=Cj .i=l,2,...,nj (1)

where Cj is the consumption or demand at junction node j, positive for outflow and negative for inflow. The energy equation states that the sum of the head losses and gains along a flow path must be equal to the difference of the end nodal heads AH. In particular, the sum of the head losses and gains around a closed loop must be equal to zero since AH = 0:

;h=AH k=l, 2,...,1+n,-1 (2)

The discharge is related to the head loss through

Q=cxhx (3)

where x varies between 0.5 and 0.54 depending on the head loss equation used, and cy is function of the length L, the diameter D, and the roughness in the pipe. There are

mainly three relationships that are expressed in S.I. units as Hazen-Williams

0.8491r C HW D2.h3 1

“=F LO.54 x=-

1.85 (4)

Darcy-Weisbach

1 x=-

2 (5)

Manning

T D8/3 1 a=---

45’3 n6 xc-

2 (6)

Substituting equation (3) into equation (1) and expressing equation (1) for every node, we obtain a set of nonlinear equations in terms of the head losses which are coupled with the linear equations derived from equation (2) for every loop. An approximate solution of the system of nonlinear equations can be obtained using a perturbation technique called the delta expansion.” The technique requires the replacement of the exponent x by S + 1 where 6 is the perturbation parameter. The perturbation approach then consists of expanding the head loss in powers of 6 and determining analytically the terms of the series.

Expressing the head loss in the form of a perturbation series in powers of 6

h=ho+h,S+h262+h3S3+0(S4)

6=x- 1 (7)

we obtain

hX=hhs=hexp(6 In h)=h,+(h,+ho In ho)6

i

ho + h, + h, + h, In ho + yln2 h, a2

I

+ h3+h2+$+(hI+h2) In ho

I 0

+:ln2 ho+ :ln’ ho S3+0(S4) 1 (8) where, in deriving equation (8), the following expansions were used

2 z3 z4

el=l+r+s+31+z+...

z2 z3 z4 ln(l+z)=z-1+3_Qf - ... I.2 <I

(10)

The flow in the pipe is therefore, from equations (3) and (81,

Q=aho+a[hI+hO In h,]S

h0 h, + h, + h, In ho + Tln2 ho a2

I

Appl. Math. Modelling, 1996, Vol. 20, April 291


I h: +a h,+h2+2h+(h,+h2) In ho

0

+ ;ln’ ho + ;Ln3 h, a3 + O( a4) 1 (11) Equation (11) expresses the discharge in the pipe in terms of the various orders of the head losses.

Pipe fittings can also be included in the above analysis by expressing the minor losses as

Q, = a,h;

(12)

where (Y, is a function of the minor loss coefficient k, which depends on the type of fittings such as valves or bends. Expanding equation (12) as in equation (8), we calculate that the discharge through the fitting is

Q, = q,$,o + ~,h,,~ +h,o ln k,ol~+ s[hrn~

+ h,, + h,, In h,, + OSh,, ln2 h,,] 6’

+ . . . (13)

For a system that includes a pump, an additional equation that relates the head change across the pump to the discharge can be expressed as a second-order pump-characteristic curve12”4

h, = pQ2 + wQ + w2 (14)

where p, q, and r are the coefficients of the quadratic curve representing the actual operation for a pump operating at full speed and 77 is the ratio of the rotational speed at any time to the rotational speed used to determine the coefficients (v= 1 for constant-speed operation). Equation (14) can be fitted to actual operating data using at least three points from a pump test relating the discharge to the head differential across the pump. It can be written in a more suitable form for perturbation analysis using the following transformationr4

G=Q+$ (15)

and substituting into equation (14) to get

q2v2 h, =pG2 + rq2 - -

4P (16)

Using equations (15) and (16), the pump discharge can be expressed as

Q = Qr, = u,( h, + cP), - b, 1

xc- 2 (17)

where up = I/ 6, b, = OSvq/p, and cp =pb,f - rq2. Expressing h, as a perturbation series and using equation

(8) wherein the cp parameter in the expansion is combined with the zeroth-order term h,,, we obtain

Qp = -b, + a,( $0 + c,)

+ a,[ $1 + (h,o + c,) ln(h,o + c,)] 8

+ 4 $2 + $1 + h,, ln(h,o + c,>

+OS(h,,+c,) ln2(h,o+c,)]62+ . . .

(18)

Equation (18) expresses the pump discharge as a function of the different perturbation orders of the pump head h,,, h,,, . . . etc.

Substituting equations (ll), (13), and (18) into the continuity equation (1) and collecting same powers of S we obtain a sequence of systems of linear algebraic equations. Each system consists of nj continuity equations and 1 + nf - 1 energy equations obtained from equation (2) Zeroth order:

E‘Mhol= +E,l

kl[hol= +[Affl (19) First order:

[A,l[hrl= -[oh, ln hoI

Mhl = + WI (20) Second order:

[ A,][ h2] = - [ cr(h, + h, In h, + OSh, ln2 h,,)]

L4Jh21 = + PI (21)

Third order:

[( h: [A,][h,]= - cx hZ+~+hlpnho+h2 In ho

0

4 h0 + yin’ ho + dln3 ho

MM = + WI (22) where the coefficient matrix [A,] is a sparse matrix that corresponds to the continuity equations and whose elements depend on the network topology and the characteristics of the links [ aI, rx2,. . . , an,], while [A,] is a matrix whose elements are zeros and ones representing the energy equations. These two matrices are coupled together to solve for the vector [ho] = [ho 1 ho 2.. . ho n ] which consists of all the n, unknown zerdth-order head losses in the network links. The systems (19)-(22) are solved recur- sively whereby at each step the right-hand side is known from previous steps; the first-order solution is a function of the zeroth-order one, and the second-order solution depends on [ho] and [h,]. The zeroth-order solution [ho1 is the linear solution and the higher order terms can be considered as refinements to the solution. Note that in the above set of systems (19)-(22), only the right-hand side vector is different for every order while the coefficient matrix remains unaltered. Therefore, once the inverse matrix is determined, the solutions for the different orders can be evaluated by a simple matrix product thereby reducing the computational time.

292 Appl. Math. Modelling, 1996, Vol. 20, April


Since quadratic functions are used in the case of pumps and minor losses, a similar exponent must be used for the pipe friction formula because the perturbation parameter must be the same for all the nonlinear terms. Therefore either the Darcy-Weisbach’s equation or the Manning’s equation can be used since it has a similar exponent. This is not a severe limitation since all three formulas (4)-(6) are commonly used. Switching from one head loss equation to another does not compromise the accuracy because the three formulations are equivalent especially since each equation resorts to a roughness coefficient that must be calibrated and whose value is often uncertain. Note that in case only minor losses are present, the Hazen-Williams’ equation can still be used by implementing the equivalent pipe length approach which transforms the minor loss associated with any valve into an equivalent length of a fictitious pipe to be added to the system.

The procedure for including pressure regulating or sus- taining valves can be mathematically accommodated by the perturbation method since they generally follow the same discharge head relationship as equation (12). The modeling approach follows the one presented by Jeppson and Davis.” The analysis of the pipe network must first determine whether the pressure reducing valve is in the normal mode of operation which consists of maintaining a constant downstream pressure or whether the valve is inoperative because the upstream pressure is below the valve setting or whether the valve acts as a check valve when there is a reverse flow. The linear solution can be used to determine which case applies. In case the upstream pressure is below the valve setting, the valve has no effect and the analysis proceeds as if no valve were present. In case of normal operation of the valve, the procedure consists of replacing the pressure regulating valve by an artificial reservoir with a constant head equal to the pressure setting of the valve and the analysis proceeds as if an extra reservoir were present. In the case of reverse flow, the pipe is closed and a zero flow rate is obtained by assigning a zero head loss to the closed pipe. The reservoir head then becomes a variable and it is set equal to the head at the downstream node.

In order to ensure the validity of the logarithmic expansion (10) where I z 1 = 1 Sh,/h, + 62h,/h, + . . . 1 must be less than one, the head loss and the flow rate are normalized with respect to some characteristic values. Choosing Q, and h * as the normalizing variables, we can define the following dimensionless variables

Q+ &7;h_ (23)

* *

The discharge then becomes

(24)

and the perturbation solution is as-given by equations (1_9)-(22) but with h replaced by h, (Y by (Y, [Cl by [Cl = [C/Q, I and AH by AH/h,. The normalizing variables Q * and h, can be taken as the total demand of the system and the maximum head difference among the various reservoirs, respectively. In case there is only one

reservoir or none, Q * and h * can be set to one. Equations (23) and (24) have sometimes proven to be instrumental in assuring the accuracy of the perturbation series in the case of more than one reservoir in the network system (see Example II).

The above perturbation solution can be further simpli- fied by expressing the head loss in every link in terms of the end nodal heads. The resulting system is formed only of the continuity equations and it results in a matrix that is smaller in size and nearly symmetric since the system of head loss equations exceeds the system of head equations by the number of energy equations If n, - 1. The solution of such a system gives the various orders of the nodal heads. Note that the right-hand side remains the same as in the head loss formulation. The flow obtained with the head equations is exactly the same as with the head loss formulation since both are derived from the same perturbation expansion. However, the head formulation requires the specification of at least one nodal head in order for the solution not to be ill-conditioned. In this work, the head loss formulation has been chosen for ease of presentation and programming, noting that the extra computational effort due to the added energy equations was deemed negligible. It should be mentioned that, unlike the Newton-Raphson approach for the head equations, small values of the head loss do not cause problems in the proposed head loss formulation as the head loss terms do not appear in the coefficient matrix.

In summary, the perturbation method reduces the solution procedure to a series of matrix equations (19)--(22) that can be solved directly using efficient matrix methods for sparse matrices. Having obtained the various orders of the head losses, the discharges in the pipes, fittings, and pumps are then obtained through equations (11) (13), and (18), respectively. Starting from a reference head, the nodal heads can also be determined by adding or subtract- ing the head loss h as obtained from h = (Q/cx)‘/“. The solution process is therefore direct and efficient without any difficulties of poor initial estimates and convergence- related problems. It is simple and straightforward to implement as it does not require sophisticated algorithms and numerical techniques besides matrix inversion and multi- plication. The method is also fast since it necessitates only one matrix inversion and four matrix multiplications; however, it may not necessarily be the fastest method since some iterative methods can reach the solution in a very small number of iterations. Iterative methods involve a number of matrix operations that are roughly equal to the number of iterations since every iteration requires the solution of the nonlinear set of equations. An iterative method may reach an acceptable solution in three iterations which makes it faster than the perturbation solution or in five or more iterations which makes it slower. Obviously the number of iterations depends on the choice of the initial estimates.

3. Application and results

The perturbation solution has been tested on many networks of various sizes, some of which have been presented



.l”“ction nurnbar - n k

Figure 1. Hydraulic network of Example I.

in the literature, and two are discussed here. The first example is presented in detail in order to illustrate the application of the method, while the second example has been used as a benchmark network in a previous study.” The calculation procedure consists of solving equations (19)-(22) in a stepwise fashion. The various orders of perturbation solutions are then compared with the numerical solution as obtained by the Environmental Protection Agency Network (EPANET) software which is based on the algorithm by Todini and Pilati.’ The differences between the analytical and numerical solutions, the average difference and the maximum difference are also tabulated. The percent average and percent maximum difference is determined by dividing the difference by the average flow rate in the system since such a relative measure is more useful for comparison purposes. The performance of the method is also assessed by calculating the residual of the head losses which in principle must be equal to zero for closed loops. The ratio of the residual head losses to the absolute ones gives another relative indication of the degree of error in the complete solution. Using the definition of Wood and Rayes, I2 failure of the method is recorded when the percent average difference is more than 10% or when the percent maximum difference is more than 30%.

3.1 Example I

The first example consists of a simple five-pipe network taken from Boulos and Wood16 to illustrate the application of the method. Table 1 lists the network parameters, and

Table 1. System characteristics data of sample network I

the schematic diagram of the system is shown in Figure 1. The network is supplied from two reservoirs and includes a pump with specifications shown in Table 1. The steady- state head loss equations include the nonlinear continuity equations at the junction nodes

‘huh; - a2h; - a3h; = 0 (25)

a2h,” - qh; + ash; = 0.1 (26)

qh; + qh; = 0.15 (27)

the nonlinear pump equation

aIht = a,( h, + cP)’ - bP (28)

and the linear energy equations

h, + h, - h, = 0 (29)

H,+h,=H,+h,+h,-h, (30)

Upon perturbation expansion, we obtain from equation (19) the zeroth-order head loss h,

cqh, 1 - qh, 2 - cz3h, 3 = 0 (31)

(Y2h02-(Yqh04+t(Ygh05=0.1

%h0,3 + czq h, ‘, = 0.15

czIh,, , - a,h, p = apcp -b,

ho,, + ho,, - ho,, = 0

h 0-P - ho,, - ho,, + ho,, = 40

where the first subscript on h denotes the perturbation order of the head loss term and the second subscript denotes the link number. The first-order head loss solution h, is obtained from equation (20):

a&,, - azh, z - ‘Y$,,,

= - ( a@,,; ln ho,, - a~ho,, ln ho,, -

a&o,3 ln ho,,) (32)

Pipe no. Length Diameter (m) (mm)

Manning’s Node Demand Head Pump data

n (I/s) (m) h,(m) Q”(l/S)

1 300 250 0.010 A 10 40 0 2 200 200 0.010 1 0 30 100 3 250 200 0.010 2 100 22.8 150 4 150 200 0.010 3 150 5 200 250 0.010 B 50

Table 2. Flow rates in I/s and absolute error for sample network I

Pipe no. Q, Q, Q, Q, - Q, Q, - Q,

1 85.57 86.26 85.73 - 0.69 -0.16 2 18.88 20.27 18.77 - 1.39 0.11 3 66.70 65.98 66.96 0.72 -0.26 4 83.30 84.02 83.04 - 0.72 0.26 5 164.43 163.74 164.27 0.69 0.16 Average difference 0.82 0.19 Maximum difference 1.39 0.26 % average difference 0.97 0.22 % maximum difference 1.66 0.31

0, = 1 I/s; h, = 1 m: Qavg = 84.08 I/s.



Table3. System characteristicsdata of sample network11

Link no. Length Diameter n or km Nodeno. Demand Head Pump data

(m) (mm) (I/s) (m) h,(m) a, (I/s)

1 610 356 0.012 1 0 170.0 0 2 944 356 0.012 2 0 165.6 200 3 762 356 0.012 3 113 131.9 600 4 610 356 0.012 4 113 5 610 356 0.012 5 57 6 457 305 0.012 6 0 7 610 356 0.012 7 57 8 914 356 0.012 8 57 9 762 356 0.012 9 0 10 610 356 0.012 10 0 11 30 203 0.012 11 0 12 61 152 0.012 12 0 13 1524 406 0.012 13 0 14 975 305 0.012 A 3.05 15 valve 356 5 B 33.53 16 valve 356 8 C 30.48 17 valve 152 10 18 valve 356 10

19 pump

azh,,, - adh,,, + Q,,, = - ( Q0,2 ln ho,, - Go,, ln ho,,

+ Qo,s In ho,,) ‘Y34.3 + qh, 4 = - ( a3h0,3 In ho,, + a4ho,4 In ho,,)

a,h,,, - a,h,,, = - [ alho, 11, ho,, - a,(h,,, + c,)

x 14h”., + 41

h, 2 + hI.4

h ’ - hl,l

-h,,, = 0

I,P - ht.* + h,,S = 0

and so on for the higher orders. The flow rates in the pipes are calculated for the different orders from equation (ll), whereby the zeroth-order is Q, = cxho, the first order is Q, = ahO + c_x(hl + h, In h,)S, and so on for Q, and Qa.

Table 2 shows the second and third-order flow rates alongside their differences with the solution as obtained from the EPANET solver. We notice that in general the approximate perturbative method gives reasonably accurate solutions. The maximum difference in the second-order solution is 1.39 l/s which is further reduced to 0.26 l/s with the third order. The percent average difference, as

Table4. Flow rates in I/s and absoluteerrorsforsample network II

Link no. Q, Q2 Q, Q,- Q2 Q,- Qs

1 271.42 284.71 275.60 -13.29 - 4.18 2 147.50 156.15 150.15 - 8.65 - 2.65 3 - 34.50 - 43.15 - 37.15 8.65 2.65 4 - 0.65 12.30 9.43 -12.95 -10.08 5 57.65 44.70 47.57 12.95 10.08 6 95.87 98.76 97.24 - 2.89 - 1.37 7 - 140.47 -147.42 -140.70 6.95 0.23 8 -112.66 -117.61 -112.49 4.95 -0.17 9 258.80 265.12 261.01 - 6.32 - 2.21 10 123.92 128.56 125.44 - 4.64 - 1.52 11 - 62.86 - 65.27 - 55.42 2.41 -7.44 12 18.78 2.94 7.33 15.84 11.45 13 530.23 549.84 536.61 -19.61 - 6.38 14 - 89.14 - 90.51 - 91.52 1.37 2.38 15 530.23 549.84 536.61 -19.61 - 6.38 16 271.42 284.71 275.60 -13.29 - 4.18 17 18.78 2.94 7.33 15.84 11.45 18 57.65 44.70 47.57 12.95 10.08 19 530.23 549.84 536.61 -19.61 - 6.38 Averagedifference 10.67 5.33 Maximum difference 19.61 11.45 % averagedifference 6.05 3.02 % maximum difference 11.11 6.49

0, = 400 I/s; h, = 30 m; Qava =176.5 I/s.



Junctlsn nunh.r - n

Figure 2. Hydraulic network of Example II.

calculated with respect to the average flow rate in the system, is about 1% for Qa and 0.2% for Qa which is quite a small error. The percent maximum difference for the third order is only 0.3% of the average flow rate. The sum of the residual head losses in the pipes around the loop in the clockwise direction is Z;h = -0.089 m which is close to zero. In fact, it forms only 0.73% of the sum of absolute losses _Z 1 h 1 around the loop. The head difference between the two reservoirs A and B is 40.045 m as obtained using Q, and has an error of 0.045 m. Hence, the third-order solution produces a result with a high degree of accuracy for this simple network.

3.2 Example II

The second example is the benchmark network used by Wood and Rayes’* to compare the various iterative methods. It is composed of 14 pipes, three fixed-head reservoirs, four valves, and one pump. The network is shown in Figure 2, and the system characteristics data are tabulated in Table 3.

Table 4 compares the second-and third-order perturbation solutions to the EPANET results. The normalizing variable proved instrumental in assuring the accuracy of the solution as the solution of the original problem with Q .+ = h * = 1 was in error because the condition of equation (10) was not satisfied. The normalizing variables were chosen as the total demand of the system Q, = 0.4 m3/s and the maximum head difference among the three reservoirs h, = 30 m. Other values for Q * and h * could have been used and would have given slightly different flow rates but with an error of the same order of magnitude.

We see that the largest difference occurs in pipe 12 and it is, for order 3,6.5% of the average flow rate; whereas the percent average error is only 3%. The error can be further reduced by including additional perturbation terms up to the fifth order. However, the accuracy gained does not justify the computational effort especially in that the number of terms on the right-hand side of equation (22) increases significantly with increasing order. It reaches 11 terms for order 4, 18 terms for order 5, and 30 terms for order 6. The ratios of Zh, to 2 I h, 1 for the different loops show that the energy equations are reasonably satisfied in most of the loops using Q3, the maximum error 5.6% being in loop III which includes pipe 12 which also has the largest error.

4. Concluding remarks

The networks tested show that the perturbation method for the head loss equations gives solutions of sufficient accuracy with a third-order perturbation expansion, The head loss equations turn out to give better results than the flow equations when similarly expanded in a perturbation series.17 Few terms are needed here since the perturbation parameter is small (- .50 I 6 I -0.46), whereas 6 is higher for the flow equations ranging between 0.85 and 1.0 since x in that case ranges from 1.85 to 2. The smaller perturbation parameter causes the perturbation series to reach an accurate solution with fewer terms. In principle, the perturbation solution can be further improved using PadC approximants,i’ however they have not proven useful here because the accuracy gained was not worth the extra effort.

In summary, the third-order solution Q3 can be considered satisfactory for a preliminary design or as part of an optimization program whereby hundreds of trial runs are executed to get the optimum set of design parameters. The perturbation solution is useful by itself or as a very good initial estimate for the iterative methods if an exact solution is so desired. The proposed method is simple and straightforward to implement in that it does not require sophisticated algorithms and numerical techniques and it has a comparable computational time to the current iterative methods. Its main advantage is in its simplicity and ease of programming while most current numerical methods require fairly sophisticated algorithms and programming skills. This may prove advantageous when a water distribution network solver is needed as part of a larger model or program as in optimization or water quality modelling.

Acknowledgments

This work was supported by the University Research Board at the American University of Beirut.

Nomenclature

nl nj nf

$i” O”f

‘j L

HA D c HW

h

kin

h, P 4 r

number of links number of junctions number of fixed-grade nodes number of closed loops flow in flow out consumption at junction node j length of pipe head loss for reservoir A diameter Hazen- Williams’ roughness coefficient head loss minor loss coefficient the head change across the pump coefficient of the quadratic curve coefficient of the quadratic curve coefficient of the quadratic curve



[A,1

Q,> h, AH

‘j (Y

; n 6

expansion parameters of the pump discharge expansion parameters of the pump discharge expansion parameters of the pump discharge coefficient matrix corresponding to the continuity equation coefficient matrix corresponding to the energy equation normalizing variables difference in nodal heads consumption at node j coefficient of head loss equation exponent of head loss equation Darcy-Weisbach’s friction factor Manning’s roughness coefficient perturbation parameter

subscript O,I@ler of perturbation or link number coefficient of minor loss equation flow rate as obtained from EPANET

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