Reference:  Analysis of Flow in Pipe Networks, 1976, Publisher Ann Arbor Science, Rowland W. Jeppson, Example problem 6 page 95 - 98


Pipe Flow Expert File: Case_47_Water_Twenty_Eight_Pipe_Network.pfe


Problem Description: 

Water is supplied from two reservoirs to a twenty eight 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.


Find the flow rate and head loss for each pipe.

Find the 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 

2.94

2.7772


P 1 

6.41

6.01

P 2 

-1.76

● 1.6707


P 2 

11.85

12.04

P 3

-0.54

● 0.5778


P 3 

0.62

0.59

P 4 

1.74

1.7883


P 4 

9.10

8.68

P 5 

0.88

0.8904


P 5 

10.76

9.86

P 6

-2.55

● 2.4480


P 6 

30.20

28.11

P 7

-3.35

● 3.2081


P 7

45.50

43.63

P 8

2.17

2.2590


P 8

51.10

54.53

P 9

3.07

3.2297


P 9

43.90

35.86

P 10

-0.44

● 0.3661


P 10

7.92

3.59

P 11

-0.58

● 0.5495


P 11

9.90

6.06

P 12

0.64

0.6207


P 12

8.20

8.45

P 13

0.73

0.6720


P 13

9.38

8.97

P 14

1.32

1.2899


P 14

59.00

58.12

P 15

1.18

1.1065


P 15

29.60

26.25

P 16

0.80

0.7601


P 16

23.40

20.55

P 17

-2.29

● 2.2485


P 17

7.86

8.15

P 18

-0.17

● 0.1478


P 18

1.37

0.89

P 19

0.09

0.0965


P 19

0.28

0.29

P 20

3.27

3.0903


P 20

37.60

40.51

P 21

2.45

2.5527


P 21

30.70

33.32

P 22

-0.04

● 0.1040


P 22

0.01

0.07

P 23

1.15

1.2180


P 23

6.90

7.13

P 24

-0.41

● 0.4435


P 24

6.91

7.20

P 25

6.84

6.7570


P 25

30.60

34.60

P 26

6.01

6.0069


P 26

0.91

0.65

P 27

3.35

3.3507


P 27

0.31

0.21

P 28

-2.39

● 2.3010


P 28

8.43

6.94



Node

Published 

Press. (ft hd)

Pipe Flow 

Expert (psig)


Node

Published 

HGL. (ft)

Pipe Flow 

Expert HGL (ft)

N1

not published

162.6073


N1

1365

1365.45

N2

not published

146.1446


N2

1359

1359.44

N3

not published

133.9999


N3

1347

1347.40

N4

not published

126.0271


N4

1348

1347.99

N5

not published

119.8254


N5

1357

1356.67

N6

not published

119.8479


N6

1346

1346.81

N7

not published

127.2001


N7

1316

1318.70

N8

not published

114.3679


N8

1270

1275.07

N9

not published

148.3774


N9

1321

1329.60

N10

not published

137.8059


N10

1329

1333.19

N11

not published

133.9343


N11

1339

1339.25

N12

not published

128.4999


N12

1347

1347.70

N13

not published

127.4462


N13

1392

1394.27

N14

not published

116.3962


N14

1354

1353.75

N15

not published

123.8450


N15

1361

1360.95

N16

not published

105.5984


N16

1354

1353.82


Commentary:  

The published HGL data and the calculated results compare well.


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.


The correlation of the HGL values is good, indicating that the calculated pressures at the nodes must be similar to un-published pressures obtained in the reference calculation.


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 to 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 be seen that the fluid velocities are 3.743 ft/sec and 5.042 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 published flow and head loss calculations.