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Water Resour Manage (2011) 25:2951–2987DOI 10.1007/s11269-011-9784-3

Iterative Methods for Looped Network Pipeline

Calculation

Dejan Brki´ c

Received: 14 December 2009 / Accepted: 3 February 2011 /

Published online: 17 March 2011© Springer Science+Business Media B.V. 2011

Abstract Since the value of the hydraulic resistance depends on flow rate, problemof flow distribution per pipes in a gas or water distributive looped pipelines has tobe solved using iterative procedure. A number of iterative methods for determiningof hydraulic solution of pipeline networks, such as, Hardy Cross, Modified HardyCross, Node-Loop method, Modified Node method and M.M. Andrijašev methodare shown in this paper. Convergence properties are compared and discussed using

a simple network with three loops. In a municipal gas pipeline, natural gas can betreated as incompressible fluid. Even under this circumstance, calculation of waterpipelines cannot be literary copied and applied for calculation of gas pipelines. Somediferences in calculations of networks for distribution of these two fluids, i.e. waterapropos natural gas are also noted.

Keywords Pipeline networks · Water distribution system · Natural gas distributionsystem · Calculation methods · Flow · Hydraulic pipeline systems

1 Introduction

A number of iterative methods for determining the hydraulic solution of waterand natural gas pipeline networks which take ring-like form, such as, Hardy Cross,Modified Hardy Cross (including Andrijašev method), Node-Loop method andModified Node method are compared in this paper. All presented methods assumeequilibrium among pressure and friction forces in steady and incompressible flow.As a result, they cannot be successfully used in unsteady and compressible flowcalculations with large pressure drops where inertia force is important. Problem of

flow in pipes and open conduits was one which had been of considerable interest toengineers for nearly 250 years. Even today, this problem is not solved definitively.

D. Brki´ c (B)Ministry of Science and Technological Development, Strumi ˇ cka 88, 11050 Beograd, Serbiae-mail: [email protected]


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2952 D. Brki´ c

The dificulty to solve the turbulent flow problems lies in the fact that the frictionfactor is a complex function of relative surface roughness and Reynolds number.Precisely, hydraulic resistance depends on flow rate and hence flow problem inhydraulic networks has to be solved iteratively. Similar situation is with electric

current when diode is in circuit. With common resistors in electrical circuits wherethe electric resistances do not depend on the value of electric current in the conduit,problem is linear and no iterative procedure has to be used. So problem of flowthrough a single tube is already complex. Despite of it, very efficient procedures areavailable for solution of flow problem in a complex pipeline such as looped pipelinelike waterworks or natural gas distribution network is. Most of the shown methodsare based on solution of the loop equations while the node equations are used onlyas control of accuracy. Node-Loop method is also based on solution of the loopequations but the node equations are also used in calculation and not only for controlpurposes. Node method and here shown Modified Node method is based on solutionof the node equations while the loop equations is used for control purposes.

In this paper, speed of convergence of presented methods are compared anddiscussed. This is done for one simple network with three loops, both, for waterand gas distribution networks. Similar example can be done for the air ventilationsystems in the buildings or mines. For ventilation problem readers can consult paperof Aynsley (1997), Mathews and Köhler (1995), Wang and Hartman (1967), etc.While the disposal of water or gas distribution system is in single plane with certainelevation of pipes, ventilation network is almost always spatial (Brki ´ c 2009).

2 Hydraulics Frictions and Flow Rates in Pipes

Each pipe is connected to two nodes at its ends. In a pipe network system, pipes arethe channels used to convey fluid from one location to another. Pipes are sometimesreferred to as tubes or conduits, and as a part of network to as lines, edges, arcs orbranches (in the graph theory, special kinds of branches are referred to as trees). Thephysical characteristics of a pipe include the length, inside diameter, roughness, andminor loss coefficient. When fluid is conveyed through the pipe, hydraulic energy

is lost due to the friction between the moving fluid and the stationary pipe surface.This friction loss is a major energy loss in pipe flow (Farshad et al. 2001; Kumar et al.2010).

Various equations were proposed to determinate the head losses due to fric-tion, including the Darcy-Weisbach, Fanning, Chezy, Manning, Hazen-Williams andScobey formulas. These equations relate the friction losses to physical characteristicsof the pipe and various flow parameters. The Darcy-Weisbach formula for calculatingof friction loss is more accurate than the Hazen-Williams. Beside this, the Hazen-Williams relation is only for water flow.

2.1 Gas Flow Rates and Pressure Drops in Pipes

When a gas is forced to flow through pipes it expands to a lower pressure andchanges its density. Flow-rate, i.e. pressure drop equations for condition in gasdistribution networks assumes a constant density of a fluid within the pipes. Thisassumption applies only to incompressible, i.e. for liquids flows such as in water


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Iterative Methods for Looped Network Pipeline Calculation 2953

distribution systems for municipalities (or any other liquid, like crude oil, etc.). Forthe small pressure drops in typical gas distribution networks, gas density can betreated as constant, which means that gas can be treated as incompressible fluid(Pretorius et al. 2008), but not as liquid flow. Assumption of gas incompressibility

means that it is compressed and forced to convey through pipes, but inside thepipeline system pressure drop of already compressed gas is small and hence furtherchanges in gas density can be neglected. Fact is that gas is actually compressed andhence that volume of gas is decreased and then such compressed volume of gas isconveying with constant density through gas distribution pipeline. So, mass of gas isconstant, but volume is decreased while gas density is according to this, increased.Operate pressure for distribution gas network is 4·105 Pa abs i.e. 3·105 Pa gauge andaccordingly volume of gas is decreased four times compared to volume of gas atnormal conditions.

Inner surface of polyethylene pipes which are almost always used in gas distribu-tion networks are practically smooth and hence flow regime in the typical network ishydraulically ‘smooth’ (Sukharev et al. 2005). For this regime is suitable Renouard’sequation adjusted for natural gas flow (1):

F g = C = p21 − p22 =4810 · Q1.82n · L · ρr

D4.82in(1)

Regarding to Renouard’s formula has to be careful since it does not relate pressuredrop but actually diference of the quadratic pressure at the input and the outputof pipe. This means that

√ C is not actually pressure drop in spite of the same

unit of measurement, i.e. same unit is used for the pressure (Pa). For gas pipelinecalculation is very useful fact that when

√ C → 0 this consecutive means that also

C→0. Parameter√

C can be noted as pseudo-pressure drop.

2.2 Liquid Flow in Pipes

In Renoard’s equation adjusted for gas pipelines (1) friction factor is rearrangedin the way to be expressed using other flow parameters and also using somethermodynamic properties of natural gas. Using formulation for Darcy friction factor

in hydraulically smooth region Renouard suggests his equation for liquid flow (2):

λ = 0.172Re0.18

(2)

Then pressure drop (Ekinci and Konak 2009) can be found very easily using Darcy-Weisbach Eq. 3:

F w = p = p1 − p2 = λ ·L

D5in· 8 · Q

2

π 2 · ρ (3)

Renouard’s Eq. 2 is based on power law. Liquid flow is better fit using some kind of logarithmic formula like Colebrook’s (4):

1√ λ= −2 · log

2.51

Re ·√

λ+ ε

3.71 · Din

(4)

Colebrook’s equation is suitable especially for flow through steel pipes (Colebrook1939). Some researchers adopt a modification of the Colebrook equation, using the


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2954 D. Brki´ c

2.825 constant instead of 2.51 especially for gas flow calculation (Haaland 1983;Coelho and Pinho 2007).

In the case of waterworks, pressures will also be expressed in Pa, not in meter(m) equivalents. In that way, clear comparison with gas distribution network can be

done. Pressures expressed in Pa can be very easily recalculated in heads expressed inmeters (m) for water networks knowing water density. Example network is located ina single-flat area with no variation in elevation (otherwise the correction term mustbe used which is significant for water network).

3 Looped Pipeline Networks for Distribution of Fluids

A pipeline network is a collection of elements such as pipes, compressors, pumps,valves, regulators, tanks, and reservoirs interconnected in a specific way. In thisarticle focus is on pipes. The behavior of the network is governed by the specificcharacteristics of the elements and how the elements are connected together. Ourassumption is that pipes are connected in a smooth way, i.e. so called minor hydraulicloses are neglected. Including other elements diferent than pipes is a subject of sufficient diversity and complexity to merit a separate review.

The analysis of looped pipeline systems by formal algebraic procedures is verydificult if the systems are complex. Electrical models had been used in study of thisproblem in the time before advanced computer became available as backgroundto support demandable numerical procedures (Mah and Shacham 1978). Here

presented methods use successive corrections or better to say these methods requireiterative procedure. The convergence properties of presented methods are the mainsubject of this article. All of such methods can be divided into two groups (1)Methods based on solution of the loop equations, and (2) Methods based on solutionof the node equations.

Most of the methods used commonly in engineering practice belong to thegroup based on solution of the loop equations. Such method presented in thispaper is Hardy Cross method (Cross 1936). Method of Andrijašev is variation of Hardy Cross method (Andrijašev 1964).1 Contemporary with Hardy Cross, sovietauthor V.G. Loba ˇ cev (Latišenkov and Loba ˇ cev 1956) developed very similar method

compared to original Hardy Cross method. Modified Hardy Cross method proposedEpp and Fowler (1970) which considers entire system simultaneously is also sortof loop method. Node-Loop method proposed by Wood and Charles (1972) andlater improved by Wood and Rayes (1981) is combination of the loop and nodeoriented methods, but despite of its name is essentially belong to the group of loopmethods. Node-Loop method is based on solution of the loop equations while thenode equations are also involved in calculation. Only Node method proposed byShamir and Howard (1968) is real representative of node oriented methods. Nodemethod uses idea of Hardy Cross but to solve node equations instead of loops ones.

Improved Node method uses similar idea as Epp and Fowler (1970) suggested for

1There are some dificulties with citation of Russian methods. Both methods, Loba ˇ cev andAndrijašev, are actually from 1930s but original papers from Soviet time are problem to be found.Many books in Russian language explain methods of Loba ˇ cev and Andrijašev but in these booksreference list is not available.


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the improvement of original Hardy Cross method, but of course here applied to theNode method.

Example network with three loops is shown in the Fig. 1. Before calculation,maximal consumption for each node including one of more inlet nodes has to be

determined. Pipe diameters and node inputs and outputs cannot be changed duringthe iterative procedure. Goal is to find final flow distribution for this pipeline system(simulation problem).

Pipe lengths and pipes diameters are listed in Fig. 1. Final flows do not dependon first assumed fluid flows per pipes in the case of loop oriented methods or onfirst assumed pressure drops or pseudo-pressure drops in the case of node orientedmethods. To have a better appreciation of the utility of these representations, firstwill be considered the laws that govern flow rates and pressure drops in a pipelinenetwork. These are the counterparts to Kirchhoff’s laws for electrical circuits andaccordingly for hydraulic networks, namely, (1) the algebraic sum of flows at eachnode must be zero; and (2) the algebraic sum of pressure drops for water network(i.e. pseudo-pressure drop for gas network) around any cyclic path (loop) mustbe approximately zero. For a hydraulic network, fact that both laws are satisfiedsimultaneously means that calculation for steady state for simulation problem isfinished.

3.1 Methods Based on Solution of the Loop Equations

Loops are sometimes referred also to as contours or paths. Note that contours and

loops are not synonyms in the M.M. Andrijašev method.For the loop oriented methods first Kirchhoff’s law must be satisfied for all

nodes in all iterations. Second Kirchhoff’s law for each loop must be satisfied with

Fig. 1 Example of pipeline network


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2956 D. Brki´ c

acceptable tolerance at the end of the calculation. Or, in other words, final flows arethese ones which values are not changed between two successive iterations (must besatisfied for flow in each pipe).

In a loop oriented methods, first initial flow pattern must be chosen to satisfied

first Kirchhoff’s law. Endless number of flow combinations can satisfy this condition.Someone can conclude that it is crucial to start with good initial guesses. But, howdoes one obtain good initial guesses? And how sensitive are the methods to the initialguesses? Answer is that initial flow pattern does not have significant influence onconvergence properties of observed method. These indicate that the choice of initialflows is not critical and need to be chosen only to satisfied first Kirchhoff’s law forall nodes (Gay and Middleton 1971). It would be equally satisfactory to generate theinitial distribution within the computer program. Initial flow pattern for our examplenetwork is shown in the Fig. 2. Loops are also defined in the Fig. 2.

Results of calculation using original Hardy Cross, Modified Hardy Cross, M.M.Andrijašev method are not actually flows Q but correction of flows Q calculated foreach loop. These corrections have to be added algebraically to flows from previousiteration for each pipe according to specific rules. A pipe common to two loopsreceives two corrections. The upper plus or minus sign (shown in Tables 1, 2, 3, 4,5, 6, 7 and 8) indicates direction of flow in that conduit in these two loops and isobtained from Q for previous iteration. The upper sign is the same as the sign in frontof Q if the flow direction in each loop coincides with the assumed flow direction inthe particular loop under consideration, and opposite if it does not. The lower signis copied from the primary loop for this correction (sign from the loop where this

Fig. 2 Example network initial parameters prepared for loop oriented calculation


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T a b l e 1

T h e H a r d y

C r o s s c a l c u l a t i o n f o r e x a m p l e g

a s n e t w o r k

L o o p

P i p e

I t e r a t i o n 1

Q a

C =

p 2 1

− p

2 2 b

F =

∂ C ( Q )

∂ Q

Q

1

=

F F

c

Q

2 d

Q 1

= Q e

I

3

− 1 9 , 4 4 4 · 1

0 −

5

− 1 , 2 6 4 , 9 3 3 , 3 3 9

1 1 , 8 3 9

, 7 7 6 , 0 5 5

+ 1 3 , 2 6 8 · 1 0 −

5

–

− 6 , 1 7 7 · 1 0 −

5

4

+ 2 , 7 7 8 · 1 0 −

5

+ 2 0 , 3 5 7 , 1 3 7

1 , 3 3 3

, 7 9 9 , 6 2 2

+ 1 3 , 2 6 8 · 1 0 −

5

+ 9 , 7 2 2 · 1 0 −

5 ∓

+ 2 5 , 7 6 7 · 1 0 −

5

7

− 3 0 , 5 5 6 · 1 0 −

5

− 2 , 3 9 9 , 6 2 0 , 9 6 3

1 4 , 2 9 3

, 0 1 5 , 0 4 7

+ 1 3 , 2 6 8 · 1 0 −

5

+ 8 8 1 · 1 0 −

5 ‡

− 1 6 , 4 0 7 · 1 0 −

5

F I = − 3 , 6 4 4 , 1 9 7 , 1 6 5

2 7 , 4 6 6

, 5 9 0 , 7 2 5

I I

1

+ 2 7 , 7 7 8 · 1 0 −

5

+ 1 , 3 4 4 , 9 8 2 , 7 0 9

8 , 8 1 2

, 3 2 6 , 7 1 3

− 9 , 7 2 2 · 1 0 −

5

–

+ 1 8 , 0 5 6 · 1 0 −

5

2

− 2 7 , 7 7 8 · 1 0 −

5

− 2 0 0 , 6 1 5 , 4 7 6

1 , 3 1 4

, 4 3 2 , 6 0 1

− 9 , 7 2 2 · 1 0 −

5

–

− 3 7 , 5 0 0 · 1 0 −

5

4

− 2 , 7 7 8 · 1 0 −

5

− 2 0 , 3 5 7 , 1 3 7

1 , 3 3 3

, 7 9 9 , 6 2 2

− 9 , 7 2 2 · 1 0 −

5

− 1 3 , 2 6 8 · 1 0 −

5 ±

− 2 5 , 7 6 7 · 1 0 −

5

5

+ 2 , 7 7 8 · 1 0 −

5

+ 1 , 8 2 8 , 4 2 5

1 1 9

, 7 9 8 , 4 5 2

− 9 , 7 2 2 · 1 0 −

5

+ 8 8 1 · 1 0 −

5 ∓

− 6 , 0 6 3 · 1 0 −

5 f

F I I = + 1 , 1 2 5 , 8 3 8 , 5 2 1

1 1 , 5 8 0

, 3 5 7 , 3 8 8

I I I

5

− 2 , 7 7 8 · 1 0 −

5

− 1 , 8 2 8 , 4 2 5

1 1 9

, 7 9 8 , 4 5 2

− 8 8 1 · 1 0 −

5

+ 9 , 7 2 2 · 1 0 −

5 ‡

+ 6 , 0 6 3 · 1 0 −

5 f

6

+ 2 , 7 7 8 · 1 0 −

5

+ 6 5 , 6 0 4 , 9 4 0

4 , 2 9 8

, 4 3 5 , 7 3 0

− 8 8 1 · 1 0 −

5

–

+ 1 , 8 9 7 · 1 0 −

5

7

+ 3 0 , 5 5 6 · 1 0 −

5

+ 2 , 3 9 9 , 6 2 0 , 9 6 3

1 4 , 2 9 3

, 0 1 5 , 0 4 7

− 8 8 1 · 1 0 −

5

− 1 3 , 2 6 8 · 1 0 −

5

=

+ 1 6 , 4 0 7 · 1 0 −

5

8

− 1 6 , 6 6 7 · 1 0 −

5

− 2 , 0 9 6 , 8 6 4 , 1 0 5

2 2 , 8 9 7

, 7 5 6 , 0 3 5

− 8 8 1 · 1 0 −

5

–

− 1 7 , 5 4 8 · 1 0 −

5

F I I I = + 3 6 6 , 5 3 3 , 3 7 2

4 1 , 6 0 9

, 0 0 5 , 2 6 4


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2958 D. Brki´ c

T a b l e 1

( c o n t i n u e d

)

L o o p

P i p e

I t e r a t i o n 2

Q 1

= Q e

C = p

2 1

− p

2 2 b

F =

∂ C ( Q )

∂ Q

Q

1

=

F F c

Q

2 d

Q 2

= Q

I

3

− 6 , 1 7 7 · 1 0 −

5

− 1

5 6 , 9 0 4 , 9 1 7

4 , 6 2

3 , 2 9 3 , 4 6 7

− 1 , 1 2 8 · 1 0 −

5

–

− 7 , 3 0 5 · 1 0 −

5

4

+ 2 5 , 7 6 7 · 1 0 −

5

+ 1

, 1 7 3 , 1 0 9 , 3 3 5

8 , 2 8

5 , 8 6 3 , 4 1 4

− 1 , 1 2 8 · 1 0 −

5

− 5 , 5 7 2 · 1 0 −

5

=

+ 1 9 , 0 6 7 · 1 0 −

5

7

− 1 6 , 4 0 7 · 1 0 −

5

− 7

7 3 , 7 9 7 , 5 6 1

8 , 5 8

3 , 6 4 8 , 1 4 3

− 1 , 1 2 8 · 1 0 −

5

− 4 , 1 5 4 · 1 0 −

5 ±

− 2 1 , 6 8 9 · 1 0 −

5

F I = + 2 4 2 , 4 0 6 , 8 5 5

2 1 , 4 9

2 , 8 0 5 , 0 2 4

I I

1

+ 1 8 , 0 5 6 · 1 0 −

5

+ 6

1 4 , 0 8 7 , 3 9 6

6 , 1 8 9

, 9 1 3 , 4 5 2

+ 5 , 5 7 2 · 1 0 −

5

–

+ 2 3 , 6 2 8 · 1 0 −

5

2

− 3 7 , 5 0 0 · 1 0 −

5

− 3

4 6 , 3 9 0 , 9 3 0

1 , 6 8

1 , 1 6 2 , 0 9 1

+ 5 , 5 7 2 · 1 0 −

5

–

− 3 1 , 9 2 7 · 1 0 −

5

4

− 2 5 , 7 6 7 · 1 0 −

5

− 1

, 1 7 3 , 1 0 9 , 3 3 5

8 , 2 8

5 , 8 6 3 , 4 1 4

+ 5 , 5 7 2 · 1 0 −

5

+ 1 , 1 2 8 · 1 0 −

5 ‡

− 1 9 , 0 6 7 · 1 0 −

5

5

− 6 , 0 6 3 · 1 0 −

5 f

− 7

, 5 6 9 , 6 7 1

2 2

7 , 2 1 6 , 6 2 2 . 4

+ 5 , 5 7 2 · 1 0 −

5

− 4 , 1 5 4 · 1 0 −

5 ±

− 4 , 6 4 5 · 1 0 −

5

F I I = − 9 1 2 , 9 8 2 , 5 4 1

1 6 , 3 8

4 , 1 5 5 , 5 7 9

I I I

5

+ 6 , 0 6 3 · 1 0 −

5 f

+ 7

, 5 6 9 , 6 7 1

2 2

7 , 2 1 6 , 6 2 2

+ 4 , 1 5 4 · 1 0 −

5

− 5 , 5 7 2 · 1 0 −

5

=

+ 4 , 6 4 5 · 1 0 −

5

6

+ 1 , 8 9 7 · 1 0 −

5

+ 3

2 , 7 6 7 , 1 8 0

3 , 1 4

3 , 9 1 5 , 8 5 7

+ 4 , 1 5 4 · 1 0 −

5

–

+ 6 , 0 5 1 · 1 0 −

5

7

+ 1 6 , 4 0 7 · 1 0 −

5

+ 7

7 3 , 7 9 7 , 5 6 1

8 , 5 8

3 , 6 4 8 , 1 4 3

+ 4 , 1 5 4 · 1 0 −

5

+ 1 , 1 2 8 · 1 0 −

5 ∓

+ 2 1 , 6 8 9 · 1 0 −

5

8

− 1 7 , 5 4 8 · 1 0 −

5

− 2

, 3 0 2 , 9 2 7 , 5 8 9

2 3 , 8 8

5 , 5 2 4 , 9 6 1

+ 4 , 1 5 4 · 1 0 −

5

–

− 1 3 , 3 9 4 · 1 0 −

5

F I I I = − 1 , 4 8 8 , 7 9 3 , 1 7 5

3 5 , 8 4

0 , 3 0 5 , 5 8 3

a P i p e l e n g t h s a n d d i a m e t e r s a r e s h o w n i n F i g . 1 a n d

i n i t i a l f l o w p a t t e r n i n F i g . 2 ; s i x

i t e r a t i o n s a r e e n o u g h t o a c h i e v e r e s u l t s s h o w n i n T a b l e 9 ( t h i s i s f o r F → 0 )

b T h i s i s F c a l c u l a t e d

u s i n g ( 1 )

c A l s o u s i n g ( 9 )

d Q

2

i s Q

1

f r o m a d j a c e n t l o o p

e F i n a l c a l c u l a t e d f l o

w i n t h e f i r s t i t e r a t i o n i s u s e d f o

r t h e c a l c u l a t i o n i n t h e s e c o n d i t e r a t i o n

f O p p o s i t e f l o w d i r e c t i o n t h a n i n p r e v i o u s i t e r a t i o n


9/37


T a b l e 2

T h e H a r d y

C r o s s c a l c u l a t i o n f o r e x a m p l e w

a t e r n e t w o r k

L o o p

P i p e

I t e r a t i o n 1

Q a

p

= p

1

− p 2 b

F =

∂ p ( Q )

∂ Q

Q

1

=

F F

c

Q

2 d

Q 1

= Q e

I

3

− 1 9 , 4 4 4 · 1 0 −

5

− 5

, 2 7 9 , 0 9 5

5 4 , 2 9 9 , 2 7 2

+ 1 2 , 3 0 6 · 1 0 −

5

–

− 7 , 1 3 8 · 1 0 −

5

4

+ 2 , 7 7 8 · 1 0 −

5

+ 6

9 , 1 4 6

4 , 9 7 8 , 5 7 3

+ 1 2 , 3 0 6 · 1 0 −

5

+ 9 , 3 4 9 · 1 0 −

5 ∓

+ 2 4 , 4 3 3 · 1 0 −

5

7

− 3 0 , 5 5 6 · 1 0 −

5

− 1

0 , 7 1 8 , 5 4 9

7 0 , 1 5 7 , 7 8 0

+ 1 2 , 3 0 6 · 1 0 −

5

+ 1 , 0 6 8 · 1 0 −

5 ‡

− 1 7 , 1 8 1 · 1 0 −

5

F I = − 1 5 , 9 2 8 , 4 9 8

1 2 9 , 4 3 5 , 6 2 5

I I

1

+ 2 7 , 7 7 8 · 1 0 −

5

+ 5

, 9 1 9 , 8 5 0

4 2 , 6 2 2 , 9 2 4

− 9 , 3 4 9 · 1 0 −

5

–

+ 1 8 , 4 2 9 · 1 0 −

5

2

− 2 7 , 7 7 8 · 1 0 −

5

− 8

1 6 , 5 8 5

5 , 8 7 9 , 4 1 3

− 9 , 3 4 9 · 1 0 −

5

–

− 3 7 , 1 2 7 · 1 0 −

5

4

− 2 , 7 7 8 · 1 0 −

5

− 6

9 , 1 4 6

4 , 9 7 8 , 5 7 3

− 9 , 3 4 9 · 1 0 −

5

− 1 2 , 3 0 6 · 1 0 −

5 ±

− 2 4 , 4 3 3 · 1 0 −

5

5

+ 2 , 7 7 8 · 1 0 −

5

+ 5

, 9 6 7

4 2 9 , 6 7 7

− 9 , 3 4 9 · 1 0 −

5

+ 1 , 0 6 8 · 1 0 −

5 ∓

− 5 , 5 0 3 · 1 0 −

5 f

F I I = + 5 , 0 4 0 , 0 8 6

5 3 , 9 1 0 , 5 8 7

I I I

5

− 2 , 7 7 8 · 1 0 −

5

− 5

, 9 6 7

4 2 9 , 6 7 7

− 1 , 0 6 8 · 1 0 −

5

+ 9 , 3 4 9 · 1 0 −

5 ‡

+ 5 , 5 0 3 · 1 0 −

5 f

6

+ 2 , 7 7 8 · 1 0 −

5

+ 2

3 2 , 1 8 6

1 6 , 7 1 7 , 4 1 5

− 1 , 0 6 8 · 1 0 −

5

–

+ 1 , 7 0 9 · 1 0 −

5

7

+ 3 0 , 5 5 6 · 1 0 −

5

+ 1

0 , 7 1 8 , 5 4 9

7 0 , 1 5 7 , 7 8 0

− 1 , 0 6 8 · 1 0 −

5

− 1 2 , 3 0 6 · 1 0 −

5

=

+ 1 7 , 1 8 1 · 1 0 −

5

8

− 1 6 , 6 6 7 · 1 0 −

5

− 8

, 8 7 4 , 2 5 7

1 0 6 , 4 9 1 , 0 8 9

− 1 , 0 6 8 · 1 0 −

5

–

− 1 7 , 7 3 5 · 1 0 −

5

F I I I = + 2 , 0 7 0 , 5 1 0

1 9 3 , 7 9 5 , 9 6 3


10/37

2960 D. Brki´ c

T a b l e 2

( c o n t i n u e d

)

L o o p

P i p e

I t e r a t i o n 2

Q 1

= Q e

p = p

1

− p b 2

F =

∂ p ( Q )

∂ Q

Q

1

=

F F c

Q

2 d

Q 2

= Q

I

3

− 7 , 1 3 8 · 1 0 −

5

−

7 5 0 , 3 7 8

2 1 , 0 2 3 , 9 3 6

− 4 0 0 · 1 0 −

5

–

− 7 , 5 3 8 · 1 0 −

5

4

+ 2 4 , 4 3 3 · 1 0 −

5

+

4 , 5 9 6 , 4 4 9

3 7 , 6 2 5 , 1 3 1

− 4 0 0 · 1 0 −

5

− 4 , 5 7 3 · 1 0 −

5

=

+ 1 9 , 4 5 9 · 1 0 −

5

7

− 1 7 , 1 8 1 · 1 0 −

5

−

3 , 4 5 0 , 6 4 0

4 0 , 1 6 7 , 9 9 8

− 4 0 0 · 1 0 −

5

− 3 , 9 2 0 · 1 0 −

5 ±

− 2 1 , 5 0 1 · 1 0 −

5

F

I = + 3 9 5 , 4 3 0

9 8 , 8 1 7 , 0 6 6

I I

1

+ 1 8 , 4 2 9 · 1 0 −

5

+

2 , 6 3 9 , 6 0 3

2 8 , 6 4 6 , 4 9 5

+ 4 , 5 7 3 · 1 0 −

5

–

+ 2 3 , 0 0 2 · 1 0 −

5

2

− 3 7 , 1 2 7 · 1 0 −

5

−

1 , 4 4 3 , 7 8 4

7 , 7 7 7 , 5 9 7

+ 4 , 5 7 3 · 1 0 −

5

–

− 3 2 , 5 5 3 · 1 0 −

5

4

− 2 4 , 4 3 3 · 1 0 −

5

−

4 , 5 9 6 , 4 4 9

3 7 , 6 2 5 , 1 3 1

+ 4 , 5 7 3 · 1 0 −

5

+ 4 0 0 · 1 0 −

5 ‡

− 1 9 , 4 5 9 · 1 0 −

5

5

− 5 , 5 0 3 · 1 0 −

5 f

−

2 1 , 3 4 6

7 7 5 , 8 4 2

+ 4 , 5 7 3 · 1 0 −

5

− 0 . 0 3 9 2 0 ±

− 4 , 8 5 0 · 1 0 −

5

F

I I = − 3 , 4 2 1 , 9 7 7

7 4 , 8 2 5 , 0 6 7

I I I

5

+ 5 , 5 0 3 · 1 0 −

5 f

+

2 1 , 3 4 6

7 7 5 , 8 4 2

+ 3 , 9 2 0 · 1 0 −

5

− 4 , 5 7 3 · 1 0 −

5

=

+ 4 , 8 5 0 · 1 0 −

5

6

+ 1 , 7 0 9 · 1 0 −

5

+

9 2 , 7 1 8

1 0 , 8 4 8 , 2 0 5

+ 3 , 9 2 0 · 1 0 −

5

–

+ 5 , 6 2 9 · 1 0 −

5

7

+ 1 7 , 1 8 1 · 1 0 −

5

+

3 , 4 5 0 , 6 4 0

4 0 , 1 6 7 , 9 9 8

+ 3 , 9 2 0 · 1 0 −

5

+ 4 0 0 · 1 0 −

5 ∓

+ 2 1 , 5 0 1 · 1 0 −

5

8

− 1 7 , 7 3 5 · 1 0 −

5

−

1 0 , 0 2 8 , 1 2 9

1 1 3 , 0 8 8 , 1 6 8

+ 3 , 9 2 0 · 1 0 −

5

–

− 1 3 , 8 1 5 · 1 0 −

5

F

I I I = − 6 , 4 6 3 , 4 2 4

1 6 4 , 8 8 0 , 2 1 4


i n i t i a l f l o w p a t t e r n i n F i g . 2 ; s i x

i t e r a t i o n s a r e e n o u g h t o a c h i e v e r e s u l t s s h o w n i n T a b l e 9 ( t h i s i s f o r F → 0 )


u s i n g ( 3 )

c A l s o u s i n g ( 9 ) o r ( 1

0 )

d Q

2

i s Q

1







11/37


12/37

2962 D. Brki´ c

T a b l e 3

( c o n t i n u e d

)

L o o p

P i p e

I t e r a t i o n 2

Q 1

= Q e

C

= p

2 1

− p

2 2 b

F =

∂ C ( Q )

∂ Q

Q c 1

Q

2 d

Q 2

= Q

I

3

− 4 , 3 3 1 · 1 0 −

5

− 8 2 , 2 2 6 , 3 6 9

3 , 4 5 5 , 5 3 9 , 1 8 1

+ 2 , 0 7 8 · 1 0 −

5

–

− 2 , 2 5 3 · 1 0 −

5

4

+ 2 5 , 8 2 8 · 1 0 −

5

+ 1 , 1 7 8 , 1 5 1 , 8 8 4

8 , 3 0 1 , 8 9 1 , 4 3 0

+ 2 , 0 7 8 · 1 0 −

5

− 5 , 5 7 2 · 1 0 −

5

=

+ 2 2 , 2 6 3 · 1 0 −

5

7

− 1 9 , 7 3 0 · 1 0 −

5

− 1 , 0 8 2 , 4 3 5 , 4 0 3

9 , 9 8 5 , 0 5 7 , 8 8 7

+ 2 , 0 7 8 · 1 0 −

5

− 4 , 1 5 4 · 1 0 −

5 ‡

− 1 7 , 6 2 0 · 1 0 −

5

F I = + 1 3 , 4 9 0 , 1 1 2

2 1 , 7 4 2 , 4 8 8 , 4 9 9

I I

1

+ 1 9 , 8 4 1 · 1 0 −

5

+ 7 2 9 , 0 3 7 , 7 1 6

6 , 6 8 7 , 4 3 2 , 8 9 9

+ 5 , 6 4 3 · 1 0 −

5

–

+ 2 5 , 4 8 4 · 1 0 −

5

2

− 3 5 , 7 1 5 · 1 0 −

5

− 3 1 6 , 9 6 7 , 6 7 0

1 , 6 1 5 , 2 5 1 , 7 1 6

+ 5 , 6 4 3 · 1 0 −

5

–

− 3 0 , 0 7 2 · 1 0 −

5

4

− 2 5 , 8 2 8 · 1 0 −

5

− 1 , 1 7 8 , 1 5 1 , 8 8 4

8 , 3 0 1 , 8 9 1 , 4 3 0

+ 5 , 6 4 3 · 1 0 −

5

+ 1 , 1 2 8 · 1 0 −

5 ±

− 2 2 , 2 6 3 · 1 0 −

5

5

− 9 , 4 4 7 · 1 0 −

5 f

− 1 6 , 9 6 6 , 0 7 0

3 2 6 , 8 5 8 , 3 4 6 . 3

+ 5 , 6 4 3 · 1 0 −

5

− 4 , 1 5 4 · 1 0 −

5 ‡

− 3 , 7 7 2 · 1 0 −

5

F I I = − 7 8 3 , 0 4 7 , 9 0 8

1 6 , 9 3 1 , 4 3 4 , 3 9 2

I I I

5

+ 9 , 4 4 7 · 1 0 −

5 f

+ 1 6 , 9 6 6 , 0 7 0

3 2 6 , 8 5 8 , 3 4 6 . 3

− 3 2 · 1 0 −

5

− 5 , 5 7 2 · 1 0 −

5

=

+ 3 , 7 7 2 · 1 0 −

5

6

+ 7 , 0 6 6 · 1 0 −

5

+ 3 5 8 , 8 1 2 , 8 4 6

9 , 2 4 2 , 4 0 5 , 9 4 9

− 3 2 · 1 0 −

5

–

+ 7 , 0 3 4 · 1 0 −

5

7

+ 1 9 , 7 3 0 · 1 0 −

5

+ 1 , 0 8 2 , 4 3 5 , 4 0 3

9 , 9 8 5 , 0 5 7 , 8 8 7

− 3 2 · 1 0 −

5

+ 1 , 1 2 8 · 1 0 −

5

=

+ 1 7 , 6 2 0 · 1 0 −

5

8

− 1 2 , 3 7 9 · 1 0 −

5

− 1 , 2 2 0 , 3 3 1 , 6 1 9

1 7 , 9 4 2 , 0 5 4 , 1 3 0

− 3 2 · 1 0 −

5

–

− 1 2 , 4 1 1 · 1 0 −

5

F I I I = + 2 3 7 , 8 8 2 , 7 0 1

3 7 , 4 9 6 , 3 7 6 , 3 1 2


i n i t i a l f l o w p a t t e r n i n F i g . 2 ; t h r

e e i t e r a t i o n s a r e e n o u g h t o a c h i e v e r e s u l t s s h o w n i n T a b l e 9 ( t h i s i s f o r F → 0 )


u s i n g ( 1 )

c U s i n g ( 1 2 ) w h e r e ( p ) i s r e p l a c e d w i t h ( C )

d Q

2

i s Q

1







13/37


T a b l e 4

C a l c u l a t i o n

a f t e r t h e m o d i f i e d H a r d y C r o s

s m e t h o d f o r e x a m p l e w a t e r n e t

w o r k

L o o p

P i p e

I t e r a t i o n 1

Q a

p

= p 1

− p 2 b

F =

∂ p ( Q )

∂ Q

Q

1 c

Q

2 d

Q 1

= Q e

I

3

− 1 9 , 4 4 4 · 1 0 −

5

− 5

, 2 7 9 , 0 9 5

5 4 , 2 9 9 , 2 7 2

+ 1 4 , 1 9 5 · 1 0 −

5

–

− 5 , 2 5 0 · 1 0 −

5

4

+ 2 , 7 7 8 · 1 0 −

5

+ 6

9 , 1 4 6

4 , 9 7 8 , 5 7 3

+ 1 4 , 1 9 5 · 1 0 −

5

+ 8 , 0 0 6 · 1 0 −

5 ∓

+ 2 4 , 9 7 8 · 1 0 −

5

7

− 3 0 , 5 5 6 · 1 0 −

5

− 1

0 , 7 1 8 , 5 4 9

7 0 , 1 5 7 , 7 8 0

+ 1 4 , 1 9 5 · 1 0 −

5

− 4 , 0 5 3 · 1 0 −

5 ±

− 2 0 , 4 1 3 · 1 0 −

5

F I = − 1 5 , 9 2 8 , 4 9 8

1 2 9 , 4 3 5 , 6 2 5

I I

1

+ 2 7 , 7 7 8 · 1 0 −

5

+ 5

, 9 1 9 , 8 5 0

4 2 , 6 2 2 , 9 2 4

− 8 , 0 0 6 · 1 0 −

5

–

+ 1 9 , 7 7 2 · 1 0 −

5

2

− 2 7 , 7 7 8 · 1 0 −

5

− 8

1 6 , 5 8 5

5 , 8 7 9 , 4 1 3

− 8 , 0 0 6 · 1 0 −

5

–

− 3 5 , 7 8 4 · 1 0 −

5

4

− 2 , 7 7 8 · 1 0 −

5

− 6

9 , 1 4 6

4 , 9 7 8 , 5 7 3

− 8 , 0 0 6 · 1 0 −

5

− 1 4 , 1 9 5 · 1 0 −

5 ±

− 2 4 , 9 7 8 · 1 0 −

5

5

+ 2 , 7 7 8 · 1 0 −

5

+ 5

, 9 6 7

4 2 9 , 6 7 7

− 8 , 0 0 6 · 1 0 −

5

− 4 , 0 5 3 · 1 0 −

5

=

− 9 , 2 8 1 · 1 0 −

5

F I I = + 5 , 0 4 0 , 0 8 6

5 3 , 9 1 0 , 5 8 7

I I I

5

− 2 , 7 7 8 · 1 0 −

5

− 5

, 9 6 7

4 2 9 , 6 7 7

+ 4 , 0 5 3 · 1 0 −

5

+ 8 , 0 0 6 · 1 0 −

5 ±

+ 9 , 2 8 1 · 1 0 − 5 f

6

+ 2 , 7 7 8 · 1 0 −

5

+ 2

3 2 , 1 8 6

1 6 , 7 1 7 , 4 1 5

+ 4 , 0 5 3 · 1 0 −

5

–

+ 6 , 8 3 0 · 1 0 −

5

7

+ 3 0 , 5 5 6 · 1 0 −

5

+ 1

0 , 7 1 8 , 5 4 9

7 0 , 1 5 7 , 7 8 0

+ 4 , 0 5 3 · 1 0 −

5

− 1 4 , 1 9 5 · 1 0 −

5 ±

+ 2 0 , 4 1 3 · 1 0 −

5

8

− 1 6 , 6 6 7 · 1 0 −

5

− 8

, 8 7 4 , 2 5 7

1 0 6 , 4 9 1 , 0 8 9

+ 4 0 5 3 · 1 0 −

5

–

− 1 2 , 6 1 4 · 1 0 −

5

F I I I = + 2 , 0 7 0 , 5 1 0

1 9 3 , 7 9 5 , 9 6 3


14/37

2964 D. Brki´ c

T a b l e 4

( c o n t i n u e d

)

L o o p

P i p e

I t e r a t i o n 2

Q 1

= Q e

p = p 1

− p 2 b

F =

∂ p ( Q )

∂ Q

Q

1 c

Q

2 d

Q 2

= Q

I

3

− 5 , 2 5 0 · 1 0 −

5

−

4 1 6 , 0 2 4

1 5 , 8 4 9 , 7 0 0

+ 2 , 5 0 5 · 1 0 −

5

−

− 2 , 7 4 4 · 1 0 −

5

4

+ 2 4 , 9 7 8 · 1 0 −

5

+

4 , 8 0 0 , 9 0 2

3 8 , 4 4 0 , 4 1 4

+ 2 , 5 0 5 · 1 0 −

5

− 5 , 3 1 2 · 1 0 −

5

=

+ 2 2 , 1 7 2 · 1 0 −

5

7

− 2 0 , 4 1 3 · 1 0 −

5

−

4 , 8 4 0 , 3 7 3

4 7 , 4 2 3 , 5 5 6

+ 2 , 5 0 5 · 1 0 −

5

− 1 0 7 · 1 0 −

5 ±

− 1 8 , 0 1 5 · 1 0 −

5

F

I = − 4 5 5 , 4 9 4

1 0 1 , 7 1 3 , 6 7 0

I I

1

+ 1 9 , 7 7 2 · 1 0 −

5

+

3 , 0 3 0 , 6 7 6

3 0 , 6 5 6 , 2 7 5

+ 5 , 3 1 2 · 1 0 −

5

–

+ 2 5 , 0 8 4 · 1 0 −

5

2

− 3 5 , 7 8 4 · 1 0 −

5

−

1 , 3 4 2 , 7 9 0

7 , 5 0 5 , 0 6 3

+ 5 , 3 1 2 · 1 0 −

5

–

– 3 0 , 4 7 2 · 1 0 −

5

4

− 2 4 , 9 7 8 · 1 0 −

5

−

4 , 8 0 0 , 9 0 2

3 8 , 4 4 0 , 4 1 4

+ 5 , 3 1 2 · 1 0 −

5

− 2 , 5 0 5 · 1 0 −

5 ±

− 2 2 , 1 7 2 · 1 0 −

5

5

− 9 , 2 8 1 · 1 0 −

5

−

5 7 , 4 3 0

1 , 2 3 7 , 6 3 2

+ 5 , 3 1 2 · 1 0 −

5

− 1 0 7 · 1 0 −

5 ±

− 4 , 0 7 6 · 1 0 −

5

F

I I = − 3 , 1 7 0 , 4 4 6

7 7 , 8 3 9 , 3 8 4

I I I

5

+ 9 , 2 8 1 · 1 0 − 5 f

+

5 7 , 4 3 0

1 , 2 3 7 , 6 3 2

+ 1 0 7 · 1 0 −

5

− 5 , 3 1 2 · 1 0 −

5

=

+ 4 , 0 7 6 · 1 0 −

5

6

+ 6 , 8 3 0 · 1 0 −

5

+

1 , 3 1 2 , 3 1 1

3 8 , 4 2 5 , 4 4 7

+ 1 0 7 · 1 0 −

5

–

+ 6 , 9 3 7 · 1 0 −

5

7

+ 2 0 , 4 1 3 · 1 0 −

5

+

4 , 8 4 0 , 3 7 3

4 7 , 4 2 3 , 5 5 6

+ 1 0 7 · 1 0 −

5

− 2 , 5 0 5 · 1 0 −

5

=

+ 1 8 , 0 1 5 · 1 0 −

5

8

− 1 2 , 6 1 4 · 1 0 −

5

−

5 , 1 3 6 , 6 8 1

8 1 , 4 4 4 , 0 1 0

+ 1 0 7 · 1 0 −

5

–

− 1 2 , 5 0 7 · 1 0 −

5

F

I I I = + 1 , 0 7 3 , 4 3 4

1 6 8 , 5 3 0 , 6 4 5


i n i t i a l f l o w p a t t e r n i n F i g . 2 ; t h r

e e i t e r a t i o n s a r e e n o u g h t o a c h i e v e r e s u l t s s h o w n i n T a b l e 9 ( t h i s i s f o r F → 0 )


u s i n g ( 3 )

c U s i n g ( 1 2 )

d Q

2

i s Q

1







15/37


T a b l e 5

C a l c u l a t i o n

a f t e r M o d i f i e d M . M . A n d r i j a š e v m e t h o d f o r e x a m p l e g a s n e t w

o r k

C o n t o u r O

P i p e

I t e r a t i o n 1

Q a

C = p

2 1

− p

2 2 b

F =

∂ C ( Q )

∂ Q

Q o 1 c

Q o 2 d

Q 1

= Q e

I O

6

+ 2 , 7 7 8 · 1 0 −

5

6 5 , 6 0 4 , 9 4 1

4 , 2 9 8 , 4 3 5 , 7 3 0

+ 1 3 , 6 6 9 · 1 0 −

5

− 9 , 3 8 1 · 1 0 −

5 ‡

+ 7 , 0 6 6 · 1 0 −

5

8

− 1 6 , 6 6 7 · 1 0 −

5

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