Pipe Network Components

Computers and Electronics in Agriculture; 7 (1992) 301-321 Elsevier Science Publishers B.V., Amsterdam

301

A linear formulation model for pipe network components

K a m y a r Haghighi a, Rab i M oh t a r b, Vincent F. Bralts b and Larry J. Segerlind b aAgricultural Engineering Department, Purdue University, West Lafayette, IN 47907, USA

bAgricultural Engineering Department, Michigan State University, East Lansing, 3H 48824, USA

(Accepted 15 December 1991 )

ABSTRACT

Haghighi, K., Mohtar, R., Bralts, V.F. and Segerlind, L.J., 1992. A linear formulation model for pipe network components. Comput. Electron. Agric., 7: 301-321.

A new model and efficient formulaiion for the incorporation of various essential network components such as bends, tees, expansions and contractions, diffusers, valves, and booster pumps into the anal- ysis of pipe distribution networks is presented. The solution technique is based on the linearized nodal flow and head equations. The technique has a unique feature of formulating a general scheme for handling all network components. The linear formulation model results were compared with the re- suits of a linear theory model. The correlation in terms of values of junction pressures and element flow rates was high. The advantages of the technique include reduced computer space requirement, implementation through existing matrix inversion or finite element codes, simplicity of application to large networks, the ability to model and incorporate all essential network components into the analysis, and ease of use with large and small computers. The technique has potential applications to municipal water supplies, irrigation pipe networks,food and material handling and transportation, complicated hydraulic control systems, fuel distribution networks, and engine cooling systems.

INTRODUCTION

T h e p r o b l e m o f f low in p ipe ne t work s is genera l ly so lved b y o b t a i n i n g a f low d i s t r i bu t ion wh ich sat isf ies f low c o n t i n u i t y at each j u n c t i o n o r n o d e a n d c o n s e r v a t i o n o f ene rgy a r o u n d all c losed loops in the ne twork . T h e c o n t i n u i t y r e l a t ionsh ips are l inea r a lgebra ic e q u a t i o n s whi le the c o n s e r v a t i o n o f ene rgy a r o u n d a c losed l oop genera l ly yie lds n o n l i n e a r equa t ions . N o d i rec t m e t h o d for the s i m u l t a n e o u s so lu t ion o f these e q u a t i o n s is known . I t e r a t i ve so lu t ions for p ipe f low ra te or n o d a l p re s su re are o b t a i n e d us ing the H a r d y - C r o s s , N e w t o n - R a p h s o n , a n d L i n e a r T h e o r y ( L T ) m e t h o d s . A d v a n t a g e s a n d l imi-

Correspondence to: K. Haghighi, Agricultural Engineering Department, Purdue University, West Lafayette, IN 47907, USA.

Journal Paper 12241, Agricultural Experiment Station, Purdue University, West Lafayette, IN.

0168-1699/92/$05.00 © 1992 Elsevier Science Publishers B.V. All fights reserved.

302 K. HAGHIGHI ETAL.

tations of each technique have been discussed by Bralts (1983), Wood and Rayes ( 1981 ) and Jeppson (1982).

In addition to pipes, a distribution system contains a variety of compo- nents that include bends, tees, pumps, valves, tanks and other types of control and measuring devices. Network components alter the flow pattern in the system usually creating a turbulence that results in head loss in excess to the normal frictional loss occurring in the pipes (Jeppson, 1982). It is essential to incorporate the special characteristics of these components into the.analy- sis, and to solve the system as it physically exists.

It has been emphasized by Villemonte (1977) that the use of the name 'minor loss' is often a misnomer. In many pipe networks, distribution and collection systems, the energy losses associated with these components are 'major' since the average L/D ratio (pipe length over pipe diameter) between fittings is less than 1000. For example, 'minor' losses are truly 'major' in many pumping stations, industrial plants, and domiciles.

Numerous research papers have been published on the steady-state analysis of hydraulic networks. Most however, deal primarily with simple system components such as pipes, reservoirs, pressure reducing and check valves, and constant head pumps (Pitachi, 1966; Shamir and Howard, 1968; Lem- ieux, 1972; Wood and Charles, 1973; Jeppson and Tavallaee, 1975; Jeppson and Davis, 1976; Chandrashekar, 1980; Isaacs and Mills, 1980; Gofman and Rodeh, 1981; Kamand et al., 1987). Valves and pumps are pressure (or head) controlling devices. Since the node method solves directly for such pressures (or heads), inclusion of such devices into a numerical algorithm is relatively straight forward. In the case of a loop oriented method one solves for flow rates, and pressures become available only after a second step.

Most of the present computer algorithms have the capability to deal with pressure reducing and check valves, pressure sustaining valves, and booster pumps in addition to reservoirs and constant head pumps. Many of these programs use a fraction of the velocity head or an equivalent pipe length to evaluate some of these components. To the authors' knowledge none of the present widely used algorithms incorporate such essential network compo- nents as bends, tees, wyes, contractions, expansions, and diffusers into the solution as a separate component. Very rarely can the flow disturbances caused by the division and combination of flow at a junction be defined in a precise, mathematical and yet simple form.

Bralts and Segerlind (1985) developed a linear formulation for the analysis of pipes and emitters in micro irrigation systems. The method is based on the iterative solution of a set of linearized flow equations. The laminar flow for- mulation of Norrie and De Vries (1978) was extended to include turbulent flow conditions in drip irrigation hydraulic ne.tworks. Results obtained when comparing the model to actual data showed a high statistical correlation. This

PIPE NETWORK COMPONENTS

a. Pipe

b ' -" q

b. Bend c. Tee

d • k j

303

d. C o n t r a c t i o n e. D i f f use r f. Valve g. Pump

i,, ~ , °J 2

Fig. I. Network model components.

study did not include other important components such as bends, expansions, diffusers, wyes, tees, valves and pumps into the analysis.

Modeling of network components as separate elements in the whole system has the potential of becoming a break-through in the analysis of distribution networks. There are numerous significant potential applications such as to municipal water supplies (Shamir and Howard, 1968 ), electrical load distri- bution and circuit analysis (Norrie and De Vries, 1978; Allaire, 1985), irri- gation network analysis (Bralts, 1983), food and material handling systems (Segerlind et al., 1983), compressed air distribution networks, engine and cooling systems, fuel distribution systems (Ohtmer, 1983), and many other fluid distribution networks. Neglecting the effects ofthese components could lead to poor design, calibration and operation of such systems.

The objective of this research was to develop a new model and formula- tions for the tee, bend, contraction, expansion, valve and booster pump com- ponents in the analysis of hydraulic and distribution networks. Pipe network components of Fig. 1 will be handled by a general all-purpose element shown in Fig. 2. The element matrices can be used to obtain the junction pressures and flow rates in a pipe network.

T H E O R E T I C A L D E V E L O P M E N T

The detailed pressure distribution in any component can be calculated by taking mechanical energy balances through the junctions along appropriate stream tubes (Villemonte, 1977). The following analysis is based on the en- ergy balance equation and the contribution of a specific network component to the nodal equations. All network components can be represented by the general component shown in Fig. 2. The only component that has a defined length is the pipe element which has already been modeled by Bralts and Se-

304 K. HAGHIGHI ET AL

D i

Oii. K ii l" . . . . . . . . .

Fig. 2. General component element showing flow directions, loss coefficients K U and Kik, and neighboring elements. (Di> Dk, Di> Dj and dividing flow moves to ij and ik directions).

gerlind ( 1985 ). The numb ering o f the no des and elements (components ) fol- lows standard finite element procedure (Segerlind, 1984).

The mechanical energy balance per unit of incompressible fluid mass be- tween the two nodes i a n d j of an element (e) is:

gZ, + ~-~ + Ei "b W= gZj + ~-~ + Ej W EL (1)

where g is the acceleration due to gravity; Z is the elevation above a reference plane; P is the pressure; p is the density; E is the average kinetic energy per unit mass; W is the work input to the network; and EL is the energy loss per unit mass due to friction or minor losses. For a turbulent flow, the kinetic energy correction factor approaches one and the velocity profile is nearly con- stant and kinetic energy is:

W 2 E= (2)

2

where Vis the average velocity. Writing the mechanical energy balance equation ( 1 ) in terms of energy per

unit weight and incorporating equation (2) yields:

Zi AI- Hi -~-~g .~- hm = Zjdt- I-Ij ~- W2 -~- hL 2g

(3)

where hm= Wig is the head gain to the network due to external mechanical energy supply such as a pump; H=P/7 is the pressure head, and hL=EL/g is the total head loss due to pipe friction or network components.

The pipe element has been treated before by Bralts and Segerlind (1985) and will not be considered here. For the network components considered here,

PIPE NE'Ia, VORK COMPONENTS 305

other than the booster pump that is considered later in this paper, the expres- sion for the head loss used is:

V 2 hL=hc =K2g (4)

where hc is the head loss due to component of the network and K is the com- ponent loss coefficient. Equations (3) and (4) can now be used to develop new equations which describe flow through various network components shown in Fig. 1.

Linear form ulation model (LFM)

The proposed linear matrix formulation method can be used for the solu- tion of discrete element problems such as occur in structural analysis. The following development is based on a physical analysis of what a single com- ponent (element) contributes to the continuity equation.

1. General element. The general component of Fig. 2 has three legs and three nodes associated with them, namely i, j and k. Crosses can be modeled as two tees separated by an infinitesimal pipe section. This component is considered as a separate element whose equations contribute to the global system of equations. In the following derivations it is assumed that flow into a node is negative and flow away from a node is positive.

The mechanical energy balance equations for the flow in the ij and ik direc- tions of the general element, respectively, are:

- +g+•O 2g (5)

o

Z, + H, +E, = Zk + Hk . Ek . .~. V ]k g * g * A i k ' ~ - ~ (6)

where Vii, Ku and V~k, Kik are velocities and loss coefficients (friction factors) in ij and ik directions, respectively. Equations (5) and (6) assume that there is no energy input from any mechanical device and can be rearranged as:

- " V2 ~ ( E , - E j ) (7) ( Z i - Z j ) +

V i2k 1 ( Z i - - Zk ) "~- ( H, -- Hk ) = K,k-~g - g ( E, -- Ek ) (8)

The evaluation of (E i -E j ) and (E i -Ek ) in eqtiations (7) and (8) yields results which have the general form E=BQ, where B is a coefficient depen-

306 K. HAGHIGHI El" AL.

dent on the fluid properties and the element dimensions, and Q is the fl0w rate through a component. Using Qu= (nD2/4) V/j and Ei-E. i= ½ [ ( V~ - V] )], the mechanical energy balance equations (7) and (8)become:

/ 8 Kij Qo ~ ( Z i _ Z j ) ~ I . ( H i _ H j ) : i ~ ) Q i j _ ( D 4 1 ~(8Qij~Q, tg. s

1 (9) (Ko+ 1 )-D,.ltgrc Sj.

1 =-~6Qu

and

. . ['8K~kQ,k'~.-, { 1 I "~{8Q,k'~,., ( Zi -- Zk) dr (H i - .1-1 k ) = t ~ ' ) ~ . i k - - t - ' D ~ i - - ' D ~ k ) t ~ ) ~ik

-l(8Q,,)ye,,< j •

1 = C.<

where

(1o)

2 4 4 grc DiDj (11)

CiJ=8QoID~(Ku+ 1 ) - D ] ]

and 2 4 4 gr~ DiDk (12)

Cil, = 8Q,k[Da(Ki k + 1 ) - D 4 ]

are coefficients for the general element for flow from node i to nodes j and k, respectively. It is evident that C coefficients in ( 11 ) and (12) are dependent on the flow rate and involve the diameter of the upstream and downstream flows.

In the above derivations it is assumed that the element in Fig. 2 has Di> Dk, Di>Dj, and flow is from i to k and from i t o j (dividing flow). It is always necessary that node i matches with the larger diameter leg or maximum flow and this is the main criteria for node identification by the computer program. For complicated systems where flow direction is not known a simple run not including components might be needed as will be explored later. For the case that flow is from the smaller to the larger diameter legs (i.e. flow f romj to i and k to i), which is the case for combining flow, the same procedure is fol- lowed and the resulting coefficients would be:

PIPE NETWORK COMPONENTS 307

2 4 4 gx D; Dj Cjl = 8Qji[D] (Kj ' _ 1 )+D 4) (13)

gx2D~ D~" (14) Cki = 8Qki[ D~( K ,u - 1)+D 4 ]

where subscripts j i and ki for parameters K, C, and Q correspond t their re- spective values for combining flow from nodesj and k to node i. Loss coeffi- cients Ko, K~k, Kji, and Kki are available from experimental data charts (Miller, 1971, 1978 ) for any particular type of component and for various configura- tions of dividing and combining flows (Haghighi et al., 1988).

Equations (9) and (10) may now be written as:

aq = cq( n , - Hj) + Cij( Zi - Zj) (15)

Qik = Ctk(H, --Hk) + C~k(Z~ -Zk) (16)

Figure 2 shows the general element (e) which is connected to neighboring elements ( e - 1), (e+ 1 ), and (e+2) of the pipe network. Element (e) is bounded by nodes i , j and k, therefore, its contribution to the final system of equations is limited to the equations for these three nodes.

Assuming flow into a node is negative and flow away from a node is posi- tive, the nodal continuity equations for the general element of Fig. 2 are:

-Q( , e - l )+Q}e)=o (17) -QJe)+Q}e+')=O (18) _ Q~e) + a~e+2) = 0 (19)

where _Q}e-I) is the total flow into node i from element ( e - 1 ), +Q~e+2) is the flow out of node k to element (e+ 2), ere .... The contribution of element (e) to equations ( 17)- (19) is simply Q}e), _ Qje) and - Q~e). Applying this to ( 15 ) and (16) separately, results in:

Qu = Cu(H, - t l j )+Co ' (Z , -Z . i ) (20) Q,.i = - Co( Ht - HJ) - Co( Z, - Z.i) (21) Qik = 0 (22)

and

{ atk=Ctk(Ht --Kk) + Cik( Zi --Zk) (23) Qo =0 (24) Qik = Cik ( Hi - H a ) - Cik ( Zi -- Zk ) (25)

Arranging (20)- (25 ) in a matrix form yields:

04 7 = 1 --Cij Cij 0 (l ~ -- CijAZij ~ (26) QkJ L - C , , ¢ o c,,, (Hk) ( C;kAZ,k )

308 K. HAGHIGHI ET AL

where AZo.= Z~- Z j and AZ~= Z~- Zk

Equation (26) has the weighted residual finite element form (Segerlind, 1984):

{R (e)} = [k(e) ] {Hie)}_ {f.(~)) (27)

where - the general element residual vector is"

{R (e)} = ( 2 8 )

- the general element stiffness matrix is:

FCu+C~" -Co - O ~k] (29) [k(~>] =/ - G G k -c,~ o c,~ A

- the unknown nodal head vector is:

, fz, {H( )}=~Hj ;, (30)

L//~J

-and the element force vector is:

f -- gu -- gik ] ~f('>}=] go ( (31)

L gik ) in which

gu = Co AZu and gik = C~k AZ, k

Equation (27) is a general equation that describes flow through different components of a hydraulic network and for various configurations of combin- ing or dividing flow.

The formulation presented above is for a general network component. Ap- plication to any particular network component reqiaires substitution of spe- cific conditions into the general equation (27). Generally, hm or W are zero for all the components of the network, other than the booster pump. When the cross sectional area at nodes i and j or i and k of the general element remains constant (Ai=Aj and/or A~=Ak), the kinetic energy at those nodes may be assumed constant (Ei=Ej and/or Ei=Ek) or V/= Vj, V/= Vk. This ease is equivalent to a tee or wye component [Fig. 1 (c) ] with equal diameters in any of its two legs or even in all three legs. In this case, D~=Dj=Dk and

PIPE NETWORK COMPONENTS 309

element coefficients C o and C;k of ( 11 ) and (12) reduce to terms that are listed in Table 1. Note that the case of combining flow in a tee or wye element is not shown in Table I since its coefficients are represented in ( 13 ) and (14) and its element matrices are the same as the ones listed in Table 1 for dividing flow.

For a two noded element, the leg associated with node k does not exist, and all the terms associated with k will disappear in element matrices of equation (26). When D~¢Dj, this is equivalent to a contraction/expansion [Fig. 1 (d) ] and diffuser [Fig. 1 (e) ] elements whose coefficients and element matrices are also listed in Table 1.

Other network components, namely bend [Fig. 1 (b)] , and valve [Fig. 1 (f) ] elements require two conditions to be satisfied; node k does not exist, and D~=Dj. As is evident from ( 11 ) and (12), proper loss coefficient or fric- tion factors K o. and Kik a r e needed to evaluate C o and Cik. These are specific to particular components and various configurations of combining or divid- ing flow. Evaluation and use of friction factors as they apply to tee and bend components has been demonstrated in detail by Haghighi et al. (1988).

2. Booster pump element. Among all network components, the booster pump has unique features that can not be represented by the general element model of Fig. 2. Only a few investigators have dealt with booster pumps (Chan- drashekar, 1980; Ohtmer, 1983) but in a different way that is outlined here. Many pipe network solution programs dealt with the pump as a constant pump head.

For the booster pump of Fig. 1 (e), the energy balance equation (3) is:

Hbr' + Z' + ~ =Zj +Pj7 (32)

n o

"-r-

I _ o

E5

COH L

oO b

Hbp

Pump Flow Rote

Fig. 3. A typical p u m p character is t ic curve.

TA

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312 K. HAGHIGHI ETAL.

where Hbp is the total net head supplied by the booster pump. From a typical pump characteristic curve (Fig. 3), a relation for determining Hbp could be written as:

nbp = COH--aQ b (33)

where COH is the Cut Off Head supplied by the pump at zero-flow condition, usually the peak head for most pumps; and a and b are the regression coeffi- cient and the exponent in pump flow friction term, respectively. Substituting for Hbp in (32) yields:

COH--aQb+ Z,+Hi=Zj+tly (34)

Equation (34) can be rearranged as:

(Zi+coH)+H~=Zj+Hj+aQ b (35)

and linearized with respect to flow to give:

a l/ba_. [ (Zi q- COH +Hi) -- (Zj +t-Ij) ]l/b (36)

or

Q=Cbp( Hi-- l-lj) + Cbp( Zi + COH-- Z/) (37)

where the linearized coefficient for the booster pump is:

Cbp= [ (Zi"]- COH'bni) -- (ZjarHj)[ C'-b)/b al/b (38)

Application of the nodal continuity equations ( 17 ) - (19) to (37) results in:

Q(i e) = Cbp(H i - H i ) "lr Cbp( Z i +COH-- Zj) (39)

Q)e) = -Cbp(Hi-IIj) - Cbp (Zi +COH-- Zj) (40)

Equations 39 and 40 can be written in matrix form as:

{a (ie)~__~ Cbp - C b p ~ n , ~ ~-- Cbp (AZd~ COH)~ Q t e ) J - [ - C b p CbpJ[HyJ-- [ Cbp (AZ+ COH)J (41)

Therefore the booster pump element matrices are:

and

f f ) A z + ) ], Cbp (z~Zq- COH)J

(42)

PIPE NETWORK COMPONENTS 313

Sohttion procedure

The element matrices for all network components are assembled similar to the direct stiffness method in structural analysis and to the nodal method in electrical circuit analysis. Direct stiffness algorithm yields a system of equa- tions which have the general matrix form:

[K] [ H ] - [F] = [0] (43)

where [K] and IF] are the global stiffness matrix and force vector, respec- tively, and the vector [H] contains the nodal pressure head values for the network. The assembly of the stiffness matrix, [K], is done by calculating individual coefficients in [k (e) ] for each component of the network in turn, and adding this value to appropriate locations in [K]. The system of (43) is solved for the unknown nodal pressure heads using the direct method. The flow in each element or component is calculated once the pressure head val- ues are known. Since the coefficients such as Cb for a bend element and Cbp for the booster pump component, are functions of the flow rate, Q, the com- ponent geometry, and the component loss coefficients, Kc's, new values of the component coefficients must be calculated and the solution process is re- peated. The iterations are continued until the nodal pressure head values con- verge to the desired accuracy.

The loss coefficients of pressure heads for different components, Kc's are often given in diagrams (Mille r, 1971, 1978 ). Therefore, these diagrams have to be digitized and stored for every component of the network, as was de- scribed by Haghighi et al. (1988). During each iteration step the loss coeffi- cients within different network components are computed for a given flow rate and then C coefficients such as Cb and Cv are obtained from Table 1 for every iteration. Special double interpolation functions were developed to analyze the diagrams.

Generally, the solution of large sparse linear system of equations, such as (43), is done very efficiently today by subroutines in finite element program packages. The matrix [K] is symmetrical, diagonally dominant, and the non- zero coefficients occur within a bandwidth determined by the maximum dif- ference between the node (junction) numbers for any one element (compo- nent). The bandwidth can be reduced by proper renumbering of the nodes and elements. Only the upper half of the bandwidth needs to be stored for the solution of equations. A large distribution network problem can be stored and solved in a relatively small computer.

E X A M P L E N E T W O R K

A program for pipe network analysis, ANALYZER, has been developed (Mohtar et al., 1991 ) which utilizes the formulation discussed in this paper

314

• (i) (3)

(14) (13) ( ~ (9) 17|6--x._.)--'15 1," '13 (12)

Fig. 4. Example network.

(7

K. HAGHIGHI EL" AL.

for incorporating the various network components and the booster pump. The program can be used to assist in the design of complex pipe networks with branches or mult i looped layout.

The example network shown in Fig. 4 and whose physical parameters are in Table 2 was solved using ANALYZER. The nodes were numbered to min- imize the band width. The pressure head at node 17 is specified at 39.9 m and serves as input. There are 16 unknown pressure values in this example. The boundary condition was incorporated into the solution as a known nodal pressure. The computer storage requirement for this example has been re- duced by 81.25% as a result of the symmetrical and banded matrix.

A comparison of the linear formulation model (LFM) results and the lin- ear theory (LT) model results of Wood (1979) was made. The same initial conditions shown in Table 2 were used for both models. The calculated nodal pressure heads and element flow rates for the two methods are given in Tables 3 and 4 respectively. A graphical comparison and linear regression of the lin- ear formulation model (LFM) and the linear theory (LT) model for the ex- ample network of Fig. 4 are given in Figs. 5 and 6. Note that points A and B in Fig. 6 are each three data points as is shown in Table 4.

The accuracy and confidence in the computer models was checked using a series of verification, calibration and validation tests. The model was verified using a simple data set checked with hand calculations. A calibration and val- idation of model was conducted in comparison with the linear theory pro- gram of Wood (1979). The results of the comparison are presented in Tables 3 and 4 and in Figs. 5 and 6 and show that nodal pressure heads and element flow rates are very close. The slight differences are statistically insignificant and the two models show high correlation. The slight variation arises due to

PIPE NETWORK COMPONENTS

TABLE 2

Physical parameters and equations for the example network

315

Parameters Elements (I) (2)

Pipe length= 30.48 m Pipe length= 15.24 m Pipe size = 25.4 mm Pipe size= 50.8 mm H-W: C= 150 Tee Bow Expansion Contraction Valve Pump Bhnd width=5

( I ) , (3), (6), (7), (9) (13), (15) (6) (1), (3), (7), (9), (13), (15)

(2), (12) (4), (8) (5) (11) (1o) (14)

Equations used in ANALYZER:

Hazen Williams (H-W), for pipe elements, 4.73 0 1"852

hf= Cl.~s2 D4.sT/L

Orifice flow for node 1 V=feo ~n

the differences in handling the pump element. The pump used in'the example was a 2UI centrifugal 2900 RPM, 771/74 pump (Worthington-Simpson, 1984). The linear formulation approach models the pump as a power curve (equation 33). Ten data points were used in a regression calculation of the exponent and coefficients for (33). The point of this curve that matched with the system curve was taken as the operating head, as seen in Fig. 3. The LT model, on the other hand, deals with the pump as an exponential curve with only.three data points used for regression. Having different regression proce- dures and different degrees of freedom will result in different pump curves. Since both models are iterative and the solution procedure is simultaneous, shifting the system curve will shift the pump operating point. This causes a slight variation in flow and pressure results of the two models.

It should also be noted that the linear theory (LT) program can not account for the components as separate units. Therefore, components were treated as single nodes with their head losses included in the head loss of the preceding pipe element. As a result LT has less nodes and the nodal pressure in the LT program of Wood (1979) was compared with upstream nodal pressure of the component element in the LFM where head losses for that particular element were not yet deducted from the pressure head. In addition, special attention

316

TABLE 3

Results of nodal pressure heads for the example network

K. HAGHIGHI ET AL.

Node LFM (m) LT (m) (1) (2) (3)

1 1.63 2 28.24 3 29.0 4 28.28 5 45.76 6 30.48 7 77.84 8 46.60 9 78.41

10 81.77 11 63.37 12 64.21 13 80.98 14 82.05 15 95.35 16 26.62 17 39.93

30.02

47.65

65.31

83.10

26.81

TABLE 4

Results of element flow rates for the example network

Element LFM ( L / s ) (1)

Flow 1 (2)

Flow 2 (3)

LT (L/s) (4)

1 15.3 2 11.9 3.4 3 11.9 4 11.9 5 3.4 6 3.4 7 11.9 8 11.9 9 11.9

10 3.4 11 3.4 12 11.9 3.4 13 15.3 14 15.3 15 15.3

15.3

12.18

3.11 12.18

12.18

15.3

15.3

1 0 0 . 0 - -

317

E v

3

n

LL .J

80.0 -

Linear Regression

Y = 0.577+0.977X = ,

6 0 . 0 -

4 0 . 0 -

P I P E N E T W O R K C O M P O N E N T S

2 0 0 ~ I ' I ' I , I 20.0 40.0 60.0 80.0 100.0

LT pressure (m)

Fig. 5. Comparison of nodal pressure heads and linear regression of the LFM vs. LT for the example network.

1 6 . 0 - - _ Linear Regression /

1 4 . 0 - - Y = 0.21+0,9B X [3 - - = .

1 2 . 0 - -

~ 1 o . o - -

o

r r 8 . 0

{ - 6.0

4 .0

2 . o - , i , i , i i i i i i i i i 2.0 4.0 6.0 8.0 10.0 12.0 14.0 15.0

LT Flow (L/s)

Fig. 6. Comparison of element flow rates and linear regression of the LFM vs. LT for the ex- ample network. Points A and B are each three data points as is shown in Table 4.

should be paid to the selection o f the proper head loss coefficients in accord- ance with flow direction, quantity and element diameters. In more complex looped pipe networks the direction o f flow may not be known. In this ease a redundant computer run is essential to decide on flow direction and correct

318 r.. HAGHIGHI ETAL.

for Kvalues accordingly. Such a run will have a different pressure distribution but flow direction will be the same.

FEATURES OF THE TECHNIQUE

The technique is very simple to apply in new or existing computer pro- grams. The formulation would allow for an efficient use of computer storage space with few band width restrictions. All numerical methods for pipe net- work analysis use iterative procedures and the solution accuracy depends on the tolerance adopted and on the precision of the numerical computations. In this technique the final solution satisfies continuity and any error may be interpreted in terms of pipe friction losses or losses due to components of the network.

Using the formulation outlined here one can define flow disturbances and characteristics (volume flow rates and pressure heads) at junctions of the network components in a precise, mathematical from where the frictional losses are divided into the direction of flow. This has not been used in earlier techniques. All essential components of a network such as tees, bends, diffus- ers, expansions, contractions, valves, and pumps are easily incorporated into the system equations. The known analysis techniques account for component losses as a part of the pipe losses either by the equivalent pipe length method (Saldivia et al., 1990) or by adding the product of velocity and the friction coefficient for that component to the head losses of the connecting pipe (Wood, 1979). Both of these techniques treat the total head losses across a component, in particular a tee, as a single unit head loss. An additional fea- ture of this formulation is the generalized scheme for the derivation of ele- ment matrices which eliminates the need for individual component deriva- tions. The general scheme could be extended to all pipe network components that satisfy equation (4). No other technique is available with such impor- tant features.

Another feature of this formulation is the ease of data preparation, pro- gramming and computer implementation. The data required for all compo- nents are node numbers and loss coefficients. The data for the nodes are es- timates of pressure heads, except for the source node where the known head value is used. The program did not have any problem of convergence. The number of iterations were dependent on number o f elements in the network and to a lesser extent on the initial estimates of nodal pressure heads. Sixteen iterations were needed for the network presented in Fig. 4 to converge to ac- curacy of0.0017 m total accumulated error for all nodal pressures. The initial head estimates were 21.33 m of fon the average. In addition to the ease of use, the method is easy to understand and program using existing finite element subroutines or matrix solvers. For people who do not have access to large computers, the technique is as well suited for micro-computers. In addition

PIPE NETWORK COMPONENTS 319

to the application to field design problems the technique also helps under- stand component interaction in the construction of the stiffness matrix.

CONCLUSIONS

A linear formulation and a general element scheme for the incorporation of various network components and booster pumps into the analysis of pipe networks was presented. The technique was highly correlated with the linear theory model of Wood (1979). The technique's advantages are ease of use and data preparation, ability of modeling and incorporating all the essential network components into the analysis, cost effectiveness, ease of implemen- tation on large and small computers, in addition to the fact that many of the existing finite element programs can easily be modified for use.

APPENDIX: NOTATION

The following symbols are used in this paper:

A = cross-section area a = regression coefficient in pump flow friction term for the pump characteristic curve B = coefficient related to the fluid properties and the element dimensions b = exponent in pump flow friction term for the pump characteristic curve C = linearized coefficient for a specific component C U = coefficient for the general element for flow from node i to node j C~k = coefficient for the general element for flow from node i to node k COH = cut off head supplied by the pump at zero flow condition, usually the peak head for most

pumps and constant for any specific pump D = inlet diameter of the component E = average kinetic energy per unit mass EL = energy loss per unit mass due to friction or minor loss IF] = global force vector

f = orifice flow coefficient g = acceleration due to gravity H = pressure head (P/y) [HI = global nodal head vector {H (e)} = element unknown nodal head vector Hbp = total net head supplied by the booster pump hc = head loss due to components of the network hr. = total head loss due to pipe friction or network components (EL/g) hm = head gain to the network due to external mechanical energy supply such as a pump ( IV~

g) K = loss coefficient for a specific component [K] = global stiffness matrix [k (ol = element stiffness matrix P = pressure Q = flow rate through a component (1rD'V/4) {R (e)} = element residual vector V = mass average velocity

320

IV = work input to the network Z = elevation above a reference plane

Greek letters ~, = specific weight p = density

Subscripts b = bend element bp = booster pump c = contraction element d = diffuser element e = expansion element ec = expansion/contraction element o = orifice v = valve element

I~ HAGHIGHI El" AL.

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