Reference:  Analysis of Flow in Pipe Networks, 1976, Publisher Ann Arbor Science, Rowland W. Jeppson, Example problem 6 page 98 - 99


Pipe Flow Expert File: Case_48_Water_Twenty_Seven_Pipe_Network.pfe


Problem Description: 

Water is supplied from two reservoirs to a twenty seven pipe network. 

The pipes are connected at sixteen node points. 

The network contains 3 pumps.

Out flows from the network occur at several node points.

The pipe network is based on the twenty eight pipe network in case 47. 

Pipe P28 has been removed from the network.


Find the new flow rate and head loss for each pipe.

Find the pressure and Hydraulic Grade Line at each node point.


A screenshot of a computer??Description automatically generated


Fluid Data: Water at 59° F (assumed).


Result Comparison: 


● = Pipe Flow Expert reversed pipe direction and reported flow as positive


Pipe

Published

Flow (ft³/s)

Pipe Flow Expert (ft³/s)


Pipe

Published 

Head Loss (ft)

Pipe Flow Expert 

Head  Loss (ft)

P 1 

2.92

2.8138


P 1 

6.35

6.17

P 2 

-1.90

● 1.8259


P 2 

13.76

14.34

P 3

-0.19

● 0.1496


P 3 

0.09

0.05

P 4 

1.60

1.5885


P 4 

7.83

6.89

P 5 

1.43

1.4052


P 5 

26.30

24.01

P 6

-1.18

● 1.1447


P 6 

7.38

6.32

P 7

-1.90

● 1.8358


P 7

15.89

14.49

P 8

1.60

1.6794


P 8

28.79

30.41

P 9

2.62

2.8491


P 9

32.88

27.98

P 10

-0.31

● 0.1671


P 10

4.06

0.81

P 11

-0.25

● 0.1200


P 11

2.09

0.33

P 12

0.43

0.3953


P 12

3.95

3.52

P 13

0.78

0.7010


P 13

10.60

9.74

P 14

0.96

0.9408


P 14

32.85

31.22

P 15

1.02

0.9879


P 15

22.47

21.00

P 16

0.72

0.6912


P 16

19.04

17.06

P 17

-2.09

● 1.9755


P 17

6.63

6.33

P 18

-0.65

● 0.6306


P 18

15.71

14.27

P 19

0.30

0.3249


P 19

2.77

2.85

P 20

1.83

1.7613


P 20

12.73

13.36

P 21

1.52

1.5807


P 21

12.57

12.96

P 22

0.54

0.5202


P 22

1.72

1.37

P 23

0.57

0.5938


P 23

1.89

1.77

P 24

-0.06

● 0.0958


P 24

0.16

0.40

P 25

4.45

4.4560


P 25

13.84

15.21

P 26

5.55

5.5629


P 26

0.78

0.58

P 27

3.81

3.6947


P 27

0.39

0.26

P 28

closed

Closed


P 28

closed

Closed



Node

Published 

Press. (ft hd)

Pipe Flow 

Expert (psig)


Node

Published 

HGL. (ft)

Pipe Flow 

Expert HGL (ft)

N1

163.00

162.5977


N1

1365

1365.43

N2

146.00

146.0675


N2

1359

1359.26

N3

133.00

132.9273


N3

1345

1344.92

N4

125.00

124.7194


N4

1345

1344.97

N5

118.00

117.7421


N5

1353

1351.86

N6

121.00

121.6356


N6

1327

1327.85

N7

128.00

128.4283


N7

1319

1321.53

N8

127.00

128.2147


N8

1303

1307.04

N9

150.00

151.7808


N9

1332

1337.45

N10

139.00

140.0037


N10

1336

1338.26

N11

134.00

133.6521


N11

1338

1338.60

N12

126.00

126.0820


N12

1342

1342.12

N13

160.00

161.0361


N13

1470

1471.82

N14

161.00

161.7472


N14

1457

1458.47

N15

166.00

166.2500


N15

1458

1458.86

N16

150.00

150.3247


N16

1455

1457.07


Commentary:  

The published node pressure and HGL data compare well with the calculated results.


The reference text uses the Hazen Williams method with a linear approximation to calculate head losses.

Pipe Flow Expert uses the more accurate Darcy-Weisbach equation and converges to within a pressure balance tolerance of 0.000145 psi.


The reference text indicates that the flow direction in a pipe is opposite to the direction shown on the schematic diagram by reporting the flow as negative value.

Pipe Flow Expert reversed the flow direction of the pipes indicated with ‘●’ automatically, and reported the flow rates in these pipes as a positive value.


There are differences in the flow rate and head loss calculations for each pipe.

Although the flow and head loss results may not agree to the normal expected accuracy, it will be noted that the results are around the same order of magnitude for the vast majority of the pipes.


The flow and head loss differences are due errors produced by the Hazen Williams empirical approximation formula, used to produce the reference text results. 


The Hazen Williams formula uses an arbitrary factor ‘C’ to estimate the head loss based on a particular flow rate.

The same ‘C’ factor is used for pipes P1 and P2, but when comparing the published results from these pipes it can

that the fluid velocities are 3.718 ft/sec and 5.443 ft/sec respectively.

The relative roughness factors are 0.0010 and 0.0015 respectively.

Given these comparisons it is obvious that these pipes cannot have the same friction factor.

Hence the use of the same ‘C’ factor will produce a degree of error in the flow and head loss calculations.