21
Pipe Network Analysis
12’’- 1500’
10’’- 3500’
8’’- 1000’6’’- 1000’
12’’- 3000’
3.34 cfs
1.11 cfs
4.45 cfs
Junction 3 Junction 4
Junction 2 Junction 1
1
2
43
5
Figure 1: A Small Pipe Network
12’’- 1500’
10’’- 3500’
8’’- 1000’6’’- 1000’
12’’- 3000’
3.34 cfs
1.11 cfs
4.45 cfs
Junction 3 Junction 4
Junction 2 Junction 11
2
43
5
Figure 1: A Small Pipe Network
F1
F2
F3
F4
1 3 4.45 0Q Q
1 2 4 1.11 0Q Q Q
54 3.34 0Q Q
53 2 0Q Q Q
10’’- 3500’
12’’- 1500’
8’’- 1000’6’’- 1000’
12’’- 3000’
3.34 cfs
1.11 cfs
4.45 cfs
Junction 3 Junction 4
Junction 2 Junction 11
2
43
5
Figure 2: A Small Pipe Network Loops
Loop 1
Loop 2
dimensional constantpipe length
The friction loss function
The friction loss functio
the , , , depends on
and the , .
the
Hazen-Williams , depen roughds on nesn s
KCL
K directly
K inversely , , and th
coefficient pipe diametere , .
HWCD
0ni iK Q
1.852 4.87 The friction loss K
HW
C LK
C D
Dimensional coefficients
D (pipe diameter) L (pipe length) CK (dimensional constant)
Feet Feet 4.73
Inches Feet 8.56 X 105
Meters Meters 10.67
Some typical values of roughness coefficient CHW
Material CHW
PVC 150
Very Smooth Pipe 140
Cement-Lined Ductile Iron 140
New Cast Iron or Welded Steel 130
Wood, Concrete 120
Clay or New Riveted Steel 110
Old Cast Iron, Brick 100
Badly Corroded Cast Iron 80
10’’- 3500’
12’’- 1500’
8’’- 1000’6’’- 1000’
12’’- 3000’
3.34 cfs
1.11 cfs
4.45 cfs
Junction 3 Junction 4
Junction 2 Junction 1
1
2
43
5
Loop 1
Loop 2
1.852 1.852 1.8521 2
1 1 2 2 3 3
32.004 5.68 1
0
9.53 0
n n nK Q
Q Q Q
K Q K Q
1.852 1.852 1.8525 54 4 2 2
54 2
0
4.811 1.002 5.68 0
n n nK Q
Q Q Q
K Q K Q
Initial guess values Q1 2.0 Q2 2.0 Q3 1.0 Q4 1.0 Q5 1.0
Given Beginning of MathCAD's solve block
Q1 Q3 4.45 0
Q1 Q2 Q4 1.11 0
Q4 Q5 3.34 0
K1 Q1n K2 Q2
n K3 Q3n 0
K4 Q4n K5 Q5
n K2 Q2n 0
Qvec Find Q1 Q2 Q3 Q4 Q5 Qvec
3.261
1.286
1.189
0.865
2.475
10’’- 3500’
12’’- 1500’
8’’- 1000’6’’- 1000’
12’’- 3000’
3.34 cfs
1.11 cfs
4.45 cfs
Junction 3 Junction 4
Junction 2 Junction 1
1
2
43
5
Loop 1
Loop 2
,n nx f x
mmm FDxx }{}{}{}{ 1)()1(
Root sought
0 xn
x
y
1 , ' 0'
nn n n
n
f xx x f x slope
f x
y f x
Newton’s Method
1. Guess a first approximation to a root of the equation
2. Use the first approximation to get a second, the second to get a third, and so on, using the formula
0f x
F1
F2
F3
F4
F5
1 3 4.45 0Q Q
1 2 4 1.11 0Q Q Q
54 3.34 0Q Q
1.852 1.852 1.8521 2 32.004 5.68 19.53 0Q Q Q 1.852 1.852 1.852
54 24.811 1.002 5.68 0Q Q Q
mmm FDxx }{}{}{}{ 1)()1(
2
2
2
1 1
2
1 2
1 2
1
1. . .
. . .
. . . . . .
. . . . . .
. . .
n
n n n
n
n
F F FQ Q
F F F
F F
Q
Q
Q
D
Q
FQ
0.852 0.852 0.8521 2 3
0.852 0.852 0.8522 4 5
1 0 1 0 01 1 0 1 0
0 0 0 1 13.71 10.52 36.17 0 0
0 10.52 0 8.91 1.86
Q Q Q
Q Q Q
Initial guess values Q1 2.0 Q2 2.0 Q3 1.0 Q4 1.0 Q5 1.0
Beginning of iteration 1
D
1
1
0
2.0041.852 Q10.852
0
0
1
0
5.681.852 Q20.852
5.68 1.852 Q20.852
1
0
0
19.53 1.852 Q30.852
0
0
1
1
0
4.811 1.852 Q40.852
0
0
1
0
1.002 1.852 Q50.852
D
1
1
0
6.699
0
0
1
0
18.987
18.987
1
0
0
36.17
0
0
1
1
0
8.91
0
0
1
0
1.856
F
Q1 Q3 4.45
Q1 Q2 Q4 1.11
Q4 Q5 3.34
K1 Q1n K2 Q2
n K3 Q3n
K4 Q4n K5 Q5
n K2 Q2n
F
1.45
2.11
1.34
8.209
14.693
z rref augment D F( )( ) 5
Extraction of the last column of the augmented rref matrix z
1.401
0.834
0.049
0.125
1.215
Q1
Q2
Q3
Q4
Q5
Q1
Q2
Q3
Q4
Q5
z
Q1
Q2
Q3
Q4
Q5
3.401
1.166
1.049
1.125
2.215
Q 1
Q 2
Q 3
Q 4
Q 5
3.401
1.166
1.049
1.125
2.215
Q vec
3.261
1.286
1.189
0.865
2.475
First Iteration MathCAD
Second Iteration MathCADQ 1
Q 2
Q 3
Q 4
Q 5
3.275
1.209
1.175
0.956
2.384
Q vec
3.261
1.286
1.189
0.865
2.475
Third Iteration MathCADQ 1
Q 2
Q 3
Q 4
Q 5
3.267
1.26
1.183
0.897
2.443
Q vec
3.261
1.286
1.189
0.865
2.475
Fourth Iteration MathCAD
Q 1
Q 2
Q 3
Q 4
Q 5
3.263
1.276
1.187
0.877
2.463
Q vec
3.261
1.286
1.189
0.865
2.475
Fifth Iteration MathCAD
Q vec
3.261
1.286
1.189
0.865
2.475
Q 1
Q 2
Q 3
Q 4
Q 5
3.261
1.282
1.189
0.869
2.471
Sixth Iteration MathCAD
Q vec
3.261
1.286
1.189
0.865
2.475
Q 1
Q 2
Q 3
Q 4
Q 5
3.261
1.284
1.189
0.867
2.473
Seventh Iteration MathCAD
Q vec
3.261
1.286
1.189
0.865
2.475
Q 1
Q 2
Q 3
Q 4
Q 5
3.261
1.285
1.189
0.866
2.474
Q1
Q2
Q3
Q4
Q5
3.267
1.26
1.183
0.897
2.443
Q1
Q2
Q3
Q4
Q5
Q1
Q2
Q3
Q4
Q5
zz
7.996 10 3
0.051
7.996 10 3
0.059
0.059
Extraction of the last column of the augmented rref matrix
z rref augment D F( )( ) 5
F
0
0
0
0.212
1.356
F
Q1 Q3 4.45
Q1 Q2 Q4 1.11
Q4 Q5 3.34
K1 Q1n K2 Q2
n K3 Q3n
K4 Q4n K5 Q5
n K2 Q2n
D
1
1
0
10.198
0
0
1
0
12.368
12.368
1
0
0
41.491
0
0
1
1
0
8.574
0
0
1
0
3.89
D
1
1
0
2.0041.852 Q10.852
0
0
1
0
5.681.852 Q20.852
5.68 1.852 Q20.852
1
0
0
19.53 1.852 Q30.852
0
0
1
1
0
4.811 1.852 Q40.852
0
0
1
0
1.002 1.852 Q50.852
Beginning of iteration 3
Q1
Q2
Q3
Q4
Q5
3.275
1.209
1.175
0.956
2.384
Q1
Q2
Q3
Q4
Q5
Q1
Q2
Q3
Q4
Q5
zz
0.126
0.043
0.126
0.169
0.169
Extraction of the last column of the augmented rref matrix
z rref augment D F( )( ) 5
F
0
0
0
5.559
2.803
F
Q1 Q3 4.45
Q1 Q2 Q4 1.11
Q4 Q5 3.34
K1 Q1n K2 Q2
n K3 Q3n
K4 Q4n K5 Q5
n K2 Q2n
D
1
1
0
10.532
0
0
1
0
11.991
11.991
1
0
0
37.666
0
0
1
1
0
9.851
0
0
1
0
3.654
D
1
1
0
2.0041.852 Q10.852
0
0
1
0
5.681.852 Q20.852
5.68 1.852 Q20.852
1
0
0
19.53 1.852 Q30.852
0
0
1
1
0
4.811 1.852 Q40.852
0
0
1
0
1.002 1.852 Q50.852
Beginning of iteration 2
Q1
Q2
Q3
Q4
Q5
3.261
1.282
1.189
0.869
2.471
Q1
Q2
Q3
Q4
Q5
Q1
Q2
Q3
Q4
Q5
zz
1.487 10 3
5.903 10 3
1.487 10 3
7.391 10 3
7.391 10 3
Extraction of the last column of the augmented rref matrix
z rref augment D F( )( ) 5
F
0
0
0
9.367 10 4
0.165
F
Q1 Q3 4.45
Q1 Q2 Q4 1.11
Q4 Q5 3.34
K1 Q1n K2 Q2
n K3 Q3n
K4 Q4n K5 Q5
n K2 Q2n
D
1
1
0
10.166
0
0
1
0
12.95
12.95
1
0
0
41.86
0
0
1
1
0
7.964
0
0
1
0
4
D
1
1
0
2.0041.852 Q10.852
0
0
1
0
5.681.852 Q20.852
5.68 1.852 Q20.852
1
0
0
19.53 1.852 Q30.852
0
0
1
1
0
4.811 1.852 Q40.852
0
0
1
0
1.002 1.852 Q50.852
Beginning of iteration 5
Q1
Q2
Q3
Q4
Q5
3.263
1.276
1.187
0.877
2.463
Q1
Q2
Q3
Q4
Q5
Q1
Q2
Q3
Q4
Q5
zz
4.249 10 3
0.016
4.249 10 3
0.021
0.021
Extraction of the last column of the augmented rref matrix
z rref augment D F( )( ) 5
F
0
0
0
0.01
0.46
F
Q1 Q3 4.45
Q1 Q2 Q4 1.11
Q4 Q5 3.34
K1 Q1n K2 Q2
n K3 Q3n
K4 Q4n K5 Q5
n K2 Q2n
D
1
1
0
10.177
0
0
1
0
12.808
12.808
1
0
0
41.732
0
0
1
1
0
8.124
0
0
1
0
3.972
D
1
1
0
2.0041.852 Q10.852
0
0
1
0
5.681.852 Q20.852
5.68 1.852 Q20.852
1
0
0
19.53 1.852 Q30.852
0
0
1
1
0
4.811 1.852 Q40.852
0
0
1
0
1.002 1.852 Q50.852
Beginning of iteration 4
3.261
1.286
1.189
0.865
2.475
Q1
Q2
Q3
Q4
Q5
3.261
1.285
1.189
0.866
2.474
Q1
Q2
Q3
Q4
Q5
Q1
Q2
Q3
Q4
Q5
zz
1.912 10 4
7.633 10 4
1.912 10 4
9.545 10 4
9.545 10 4
Extraction of the last column of the augmented rref matrix
z rref augment D F( )( ) 5
Comparison with MathCAD
F
0
0
0
1.787 10 5
0.021
F
Q1 Q3 4.45
Q1 Q2 Q4 1.11
Q4 Q5 3.34
K1 Q1n K2 Q2
n K3 Q3n
K4 Q4n K5 Q5
n K2 Q2n
D
1
1
0
2.0041.852 Q10.852
0
0
1
0
5.681.852 Q20.852
5.68 1.852 Q20.852
1
0
0
19.53 1.852 Q30.852
0
0
1
1
0
4.811 1.852 Q40.852
0
0
1
0
1.002 1.852 Q50.852
Beginning of iteration 7
Q1
Q2
Q3
Q4
Q5
3.261
1.284
1.189
0.867
2.473
Q1
Q2
Q3
Q4
Q5
Q1
Q2
Q3
Q4
Q5
zz
5.323 10 4
2.122 10 3
5.323 10 4
2.654 10 3
2.654 10 3
Extraction of the last column of the augmented rref matrix
z rref augment D F( )( ) 5
F
0
0
0
1.267 10 4
0.059
F
Q1 Q3 4.45
Q1 Q2 Q4 1.11
Q4 Q5 3.34
K1 Q1n K2 Q2
n K3 Q3n
K4 Q4n K5 Q5
n K2 Q2n
D
1
1
0
2.0041.852 Q10.852
0
0
1
0
5.681.852 Q20.852
5.68 1.852 Q20.852
1
0
0
19.53 1.852 Q30.852
0
0
1
1
0
4.811 1.852 Q40.852
0
0
1
0
1.002 1.852 Q50.852
Beginning of iteration 6
