Case 46: Water – Sixty Three Pipe Network - 30 Loops – 5 Pumps
Reference: Analysis of Flow in Pipe Networks, 1976, Publisher Ann Arbor Science, Rowland W. Jeppson, Example problem 10 page 105 -109
Pipe Flow Expert File: Case_46_Water_Sixty_Three_Pipe_Network.pfe
Problem Description:
Water is supplied from five reservoirs to a sixty three pipe network.
The pipes are connected at thirty three node points.
The network contains 5 pumps.
Out flows from the network occur at each node point.
Find the flow rate and head loss for each pipe.
Find the pressure and Hydraulic Grade Line at each node point.
Fluid Data: Water at 68° F (assumed).
Result Comparison:
● = Pipe Flow Expert reversed pipe direction and reported flow as positive
|
Pipe |
Published Flow (m³/s) |
Pipe Flow Expert (m³/s) |
|
Pipe |
Published Head Loss (m) |
Pipe Flow Expert Head Loss (m) |
|
P 1 |
0.0869 |
0.0873 |
|
P 1 |
201.1 |
202.536 |
|
P 2 |
0.0208 |
0.0209 |
|
P 2 |
53.6 |
54.586 |
|
P 3 |
0.0132 |
0.0133 |
|
P 3 |
19.1 |
19.216 |
|
P 4 |
0.0153 |
0.0153 |
|
P 4 |
27.4 |
27.304 |
|
P 5 |
0.0092 |
0.0091 |
|
P 5 |
7.2 |
7.066 |
|
P 6 |
0.0274 |
0.0276 |
|
P 6 |
27.5 |
27.657 |
|
P 7 |
0.0254 |
0.0256 |
|
P 7 |
27.3 |
27.439 |
|
P 8 |
0.0148 |
0.0148 |
|
P 8 |
8.3 |
8.223 |
|
P 9 |
-0.0027 |
● 0.0027 |
|
P 9 |
0.2 |
0.218 |
|
P 10 |
0.0299 |
0.0297 |
|
P 10 |
68.4 |
67.431 |
|
P 11 |
0.0212 |
0.0211 |
|
P 11 |
63.1 |
62.284 |
|
P 12 |
0.0081 |
0.0080 |
|
P 12 |
5.1 |
4.930 |
|
P 13 |
0.0089 |
0.0091 |
|
P 13 |
10.1 |
10.616 |
|
P 14 |
0.0164 |
0.0163 |
|
P 14 |
38.1 |
37.446 |
|
P 15 |
0.0273 |
0.0270 |
|
P 15 |
91.0 |
89.114 |
|
P 16 |
0.0107 |
0.0107 |
|
P 16 |
78.9 |
79.465 |
|
P 17 |
0.0118 |
0.0118 |
|
P 17 |
165.5 |
163.557 |
|
P 18 |
0.0078 |
0.0077 |
|
P 18 |
86.5 |
84.091 |
|
P 19 |
0.0243 |
0.0239 |
|
P 19 |
48.4 |
46.645 |
|
P 20 |
0.0791 |
0.0783 |
|
P 20 |
166.9 |
163.139 |
|
P 21 |
0.0351 |
0.0349 |
|
P 21 |
150.0 |
147.546 |
|
P 22 |
-0.0035 |
● 0.0034 |
|
P 22 |
16.3 |
15.072 |
|
P 23 |
0.0106 |
0.0106 |
|
P 23 |
117.7 |
115.973 |
|
P 24 |
0.0166 |
0.0165 |
|
P 24 |
45.8 |
44.811 |
|
P 25 |
0.0119 |
0.0119 |
|
P 25 |
29.7 |
29.739 |
|
P 26 |
0.0277 |
0.0280 |
|
P 26 |
25.3 |
25.636 |
|
P 27 |
0.0805 |
0.0809 |
|
P 27 |
287.6 |
289.787 |
|
P 28 |
-0.0043 |
● 0.0041 |
|
P 28 |
1.4 |
1.223 |
|
P 29 |
0.0061 |
0.0061 |
|
P 29 |
12.5 |
12.317 |
|
P 30 |
-0.0065 |
● 0.0065 |
|
P 30 |
3.7 |
3.706 |
|
P 31 |
0.0148 |
0.0148 |
|
P 31 |
11.2 |
11.094 |
|
P 32 |
0.0071 |
0.0071 |
|
P 32 |
6.6 |
6.602 |
|
P 33 |
0.0046 |
0.0046 |
|
P 33 |
0.8 |
0.785 |
|
P 34 |
0.0274 |
0.0274 |
|
P 34 |
27.6 |
27.378 |
|
P 35 |
0.0094 |
0.0094 |
|
P 35 |
3.5 |
3.462 |
|
P 36 |
0.0219 |
0.0219 |
|
P 36 |
31.8 |
31.625 |
|
P 37 |
0.1306 |
0.1305 |
|
P 37 |
297.8 |
300.250 |
|
P 38 |
-0.0214 |
● 0.0213 |
|
P 38 |
12.7 |
12.553 |
|
P 39 |
-0.0299 |
● 0.0299 |
|
P 39 |
24.4 |
24.324 |
|
P 40 |
-0.0238 |
● 0.0238 |
|
P 40 |
15.7 |
15.533 |
|
P 41 |
0.0239 |
0.0239 |
|
P 41 |
37.8 |
37.593 |
|
P 42 |
0.0287 |
0.0287 |
|
P 42 |
45.2 |
44.894 |
|
P 43 |
0.0450 |
0.0450 |
|
P 43 |
54.6 |
54.469 |
|
P 44 |
0.0139 |
0.0140 |
|
P 44 |
11.1 |
11.084 |
|
P 45 |
-0.0002 |
● 0.0001 |
|
P 45 |
0.1 |
0.010 |
|
P 46 |
0.0056 |
0.0056 |
|
P 46 |
55.0 |
55.375 |
|
P 47 |
0.0051 |
0.0052 |
|
P 47 |
54.9 |
55.365 |
|
P 48 |
0.0167 |
0.0167 |
|
P 48 |
267.5 |
266.671 |
|
P 49 |
0.0137 |
0.0137 |
|
P 49 |
322.4 |
322.209 |
|
P 50 |
-0.0078 |
● 0.0078 |
|
P 50 |
14.3 |
14.173 |
|
P 51 |
0.0222 |
0.0222 |
|
P 51 |
336.7 |
336.209 |
|
P 52 |
0.0547 |
0.0550 |
|
P 52 |
240.7 |
242.802 |
|
P 53 |
0.0210 |
0.0210 |
|
P 53 |
94.2 |
93.916 |
|
P 54 |
0.0165 |
0.0165 |
|
P 54 |
58.1 |
57.898 |
|
P 55 |
0.0135 |
0.0135 |
|
P 55 |
39.1 |
38.952 |
|
P 56 |
-0.0020 |
● 0.0020 |
|
P 56 |
3.0 |
2.934 |
|
P 57 |
0.0116 |
0.0116 |
|
P 57 |
93.0 |
92.689 |
|
P 58 |
0.0109 |
0.0109 |
|
P 58 |
82.9 |
82.732 |
|
P 59 |
0.0031 |
0.0031 |
|
P 59 |
7.1 |
7.023 |
|
P 60 |
0.0072 |
0.0072 |
|
P 60 |
36.2 |
36.054 |
|
P 61 |
0.0078 |
0.0078 |
|
P 61 |
43.2 |
42.963 |
|
P 62 |
0.0003 |
0.0003 |
|
P 62 |
0.1 |
0.114 |
|
P 63 |
0.1276 |
0.1276 |
|
P 63 |
284.5 |
286.981 |
|
Node (text ref) |
Published Press. (m hd) |
Pipe Flow Expert (m hd) |
|
Node (text ref) |
Published HGL. (m) |
Pipe Flow Expert HGL (m) |
|
N1 (n/a) |
n/a |
n/a |
|
N1 (n/a) |
Fluid Surface |
500.00 |
|
N2 (1) |
8.60 |
6.41 |
|
N2 (1) |
308.60 |
306.41 |
|
N3 (2) |
5.00 |
2.83 |
|
N3 (2) |
255.00 |
252.83 |
|
N4 (4) |
29.60 |
27.20 |
|
N4 (4) |
289.60 |
287.20 |
|
N5 (3) |
7.20 |
4.89 |
|
N5 (3) |
262.20 |
259.89 |
|
N6 (5) |
1.12 |
-1.24 |
|
N6 (5) |
281.10 |
278.76 |
|
N7 (6) |
1.34 |
78.97 |
|
N7 (6) |
281.30 |
278.97 |
|
N8 (7) |
9.50 |
6.19 |
|
N8 (7) |
349.50 |
346.19 |
|
N9 (8) |
16.40 |
13.90 |
|
N9 (8) |
286.40 |
283.90 |
|
N10 (9) |
3.40 |
5.57 |
|
N10 (9) |
339.40 |
335.57 |
|
N11 (10) |
17.50 |
13.02 |
|
N11 (10) |
377.50 |
373.02 |
|
N12 (11) |
0.45 |
-3.89 |
|
N12 (11) |
260.40 |
256.11 |
|
N13 (12) |
5.93 |
-0.34 |
|
N13 (12) |
425.90 |
419.66 |
|
N14 (n/a) |
n/a |
n/a |
|
N14 (n/a) |
Fluid Surface |
540.00 |
|
N15 (13) |
5.88 |
2.12 |
|
N15 (13) |
275.90 |
272.12 |
|
N16 (14) |
15.60 |
17.05 |
|
N16 (14) |
259.60 |
257.05 |
|
N17 (31) |
30.10 |
27.32 |
|
N17 (31) |
230.10 |
227.31 |
|
N18 (15) |
24.90 |
22.68 |
|
N18 (15) |
284.90 |
282.68 |
|
N19 (n/a) |
n/a |
n/a |
|
N19 (n/a) |
Fluid Surface |
530.00 |
|
N20 (16) |
13.80 |
11.59 |
|
N20 (16) |
273.80 |
271.59 |
|
N21 (17) |
14.60 |
12.37 |
|
N21 (17) |
274.60 |
272.37 |
|
N22 (18) |
22.20 |
19.75 |
|
N22 (18) |
302.20 |
299.75 |
|
N23 (21) |
20.30 |
18.13 |
|
N23 (21) |
270.30 |
268.13 |
|
N24 (n/a) |
n/a |
n/a |
|
N24 (n/a) |
Fluid Surface |
600.00 |
|
N25(19) |
7.70 |
5.43 |
|
N25(19) |
277.70 |
275.43 |
|
N26 (20) |
15.50 |
13.02 |
|
N26 (20) |
315.50 |
313.02 |
|
N27 (22) |
8.40 |
6.05 |
|
N27 (22) |
268.40 |
266.05 |
|
N28 (30) |
24.80 |
22.67 |
|
N28 (30) |
284.80 |
282.67 |
|
N29 (32) |
12.50 |
10.64 |
|
N29 (32) |
-37.50 |
-39.37 |
|
N30 (33) |
8.20 |
6.46 |
|
N30 (33) |
-51.80 |
-53.54 |
|
N31 (n/a) |
n/a |
n/a |
|
N31 (n/a) |
Fluid Surface |
480.00 |
|
N32 (24) |
14.20 |
12.13 |
|
N32 (24) |
174.20 |
172.13 |
|
N33 (23) |
10.40 |
8.15 |
|
N33 (23) |
210.40 |
208.15 |
|
N34 (25) |
11.30 |
9.20 |
|
N34 (25) |
171.30 |
169.20 |
|
N36 (26) |
11.30 |
9.44 |
|
N36 (26) |
81.30 |
79.45 |
|
N37 (28) |
18.30 |
16.47 |
|
N37 (28) |
88.30 |
86.47 |
|
N38 (29) |
5.10 |
3.39 |
|
N38 (29) |
45.10 |
43.39 |
|
N39 (n/a) |
25.20 |
23.51 |
|
N39 (n/a) |
45.20 |
43.51 |
Commentary:
The published data and the calculated results compare well.
The reference text uses 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 0.00001 bar.
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 reference text indicates that the out flow from node N14 is 0.045 cfs (should be cms), and this value appears to be incorrect due a printing error.
The sum of flows in the pipes entering and leaving node N14 indicate that a flow rate of 0.0045 m³/s leaves this node.
The close correlation of the calculation results indicates that the reference text calculations were based on 0.0045 m³/s leaving node N14.
The correlation of the HGL values is very good.
The reference text lists the HGL at node N6 as 281.3 m and pressure at node N6 as 1.34 m hd, indicating an elevation of 280 m at this node, however the reference lists the elevation of this node at 200 m.
Pipe Flow Expert results are based on an elevation of 200 m for Node 6, and hence the results for pressure at this node are affected by the change in elevation.
The pressure at the node is calculated as 78.97 m hd.
Using an elevation of 200 m, the HGL is calculated as 278.97 m, which closely agrees with the published HGL of 281.3 m for this node.
As this node is completely surrounded by pipes which do not connect to atmosphere the change in elevation does not affect the flow rate through other pipes in the system.