0ed59715-5444-4288-8757-ea07a93bb9a7 83 4 5 false false 0 decimal

&[DATE] &[TIME] - &[FILENAME]

Internet: Example 3

Chemical Engineer's Guide :

Solve for water piping network as shown in diagram. Hazen-Williams coefficient, C for each pipe is provided in table below.

Note: This is an example given in the link below, solved by Hardy Cross method using EXCEL.
Essentially it is an overdetermined nonlinear pipe network with 22 variables {pipe flows) and 23 equations (16 Nodes + 7 Loops).
Hence, the solution using SMath is based on a snippet by Viacheslav N. Mezentsev (uni) in SMath Forum.

https://cheguide.com/pipe_network.html

t.0 0 time :

Head loss is calculated using Hazen-Williams equation (SI units).

H.L 10.671 L * Q abs 0.85^* C 1.85^D 4.87^* / Q * ≡ (K Q abs (0.85^* Q * ≡

where,

K 10.671 L * C 1.85^D 4.87^* / ≡

Based on method discussed above 7 loop equations are formed and it takes 5 iterations to converge to final flow values. Refer attached excel spreadsheet for solution.

Define K (Pipe Constant) as given below

D 10007507507505005005005007501000100010007501000750750750500500500500500221 mat mm * :

LL 10009251000925350671400650100010009258007636504001254001258001258001000221 mat m * :

C 100100100100100100100100120120120120120120120120120120120120120120221 mat :

K 10.671 LL m / * C 1.85^D m / (4.87^* / vectorize : 2.1291 7.9946 8.6428 7.9946 21.7916 41.7776 24.9047 40.4701 6.1684 1.5196 1.4056 1.2157 4.7065 0.9877 2.4674 0.771 2.4674 5.5545 35.5489 5.5545 35.5489 44.4361 221 mat

D LL C K 41 mat rows vectorize 2222222241 mat

Q.A Q.B Q.C Q.D Q.E Q.F Q.G Q.H Q.I Q.J Q.K Q.L Q.M Q.N Q.O Q.P 161 mat 21 - 121 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 161 mat :

Q.issues Q.A Q.B Q.C Q.D Q.E Q.F Q.G Q.H Q.I Q.J Q.K Q.L Q.M Q.N Q.O Q.P 161 mat :

Q.issues sum 0

Sanity check

Number of Issue points Points Q.issues rows : 16

Number of Loops Loops 7 :

Number of Pipes Pipes 22 :

Tot.eqns Points Loops + : 23

Tot.eqns Pipes > 1

guess 1 Pipes range el 0.1 :

Hence, overdetermined system

guess 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 221 mat

AlgLib Solver for a system of nonlinear equations(Uni) result k 1 Q f rows range u k el Q k el : for Q Jac u f u Jacobian : StepMax 0 : Eps 1014 -^: u guess StepMax Eps Q f Q Jac al_nleqsolve : 51 line :

result guess 1014 -^Q f al_nleqsolve :

Define the system of 23 equations (16 Nodes + 7 Loops) ; 22 Variables (pipe flows) : Hence, Overdetermimed Nonlinear System

Q f Q.A Q 1 el - Q 4 el - Q.B Q 1 el + Q 10 el + Q 2 el - Q.C Q 10 el - Q 11 el - Q.D Q 4 el + Q 3 el + Q 5 el - Q.E Q 2 el + Q 9 el - Q 8 el - Q 3 el - Q.F Q 11 el + Q 12 el - Q 9 el + Q.G Q 12 el + Q 13 el - Q 14 el - Q.H Q 5 el + Q 6 el - Q.I Q 6 el + Q 7 el - Q.J Q 8 el + Q 7 el + Q 22 el - Q.K Q 22 el + Q 21 el + Q 20 el - Q.L Q 21 el - Q 14 el + Q 15 el - Q 18 el - Q.M Q 13 el + Q 15 el + Q 16 el - Q.N Q 20 el + Q 19 el - Q.O Q 18 el + Q 19 el + Q 17 el - Q.P Q 16 el + Q 17 el + K 1 el Q 1 el abs (0.85^* Q 1 el * K 2 el Q 2 el abs (0.85^* Q 2 el * + K 3 el Q 3 el abs (0.85^* Q 3 el * + K 4 el Q 4 el abs (0.85^* Q 4 el * - K 10 el Q 10 el abs (0.85^* Q 10 el * - K 11 el Q 11 el abs (0.85^* Q 11 el * + K 9 el Q 9 el abs (0.85^* Q 9 el * - K 2 el Q 2 el abs (0.85^* Q 2 el * - K 3 el Q 3 el abs (0.85^* Q 3 el * - K 8 el Q 8 el abs (0.85^* Q 8 el * + K 7 el Q 7 el abs (0.85^* Q 7 el * - K 6 el Q 6 el abs (0.85^* Q 6 el * - K 5 el Q 5 el abs (0.85^* Q 5 el * - K 9 el Q 9 el abs (0.85^* Q 9 el * K 12 el Q 12 el abs (0.85^* Q 12 el * + K 14 el Q 14 el abs (0.85^* Q 14 el * + K 21 el Q 21 el abs (0.85^* Q 21 el * + K 22 el Q 22 el abs (0.85^* Q 22 el * - K 8 el Q 8 el abs (0.85^* Q 8 el * - K 13 el Q 13 el abs (0.85^* Q 13 el * K 15 el Q 15 el abs (0.85^* Q 15 el * - K 14 el Q 14 el abs (0.85^* Q 14 el * - K 21 el Q 21 el abs (0.85^* Q 21 el * - K 18 el Q 18 el abs (0.85^* Q 18 el * + K 19 el Q 19 el abs (0.85^* Q 19 el * - K 20 el Q 20 el abs (0.85^* Q 20 el * - K 15 el Q 15 el abs (0.85^* Q 15 el * K 16 el Q 16 el abs (0.85^* Q 16 el * + K 17 el Q 17 el abs (0.85^* Q 17 el * - K 18 el Q 18 el abs (0.85^* Q 18 el * - 231 mat :

Q f rows 23

Results

Internet Example Result using in EXCEL

Q.final result : 0.53 - 2.535 0.211 2.53 1.741 0.741 0.259 - 1.477 0.153 - 4.066 7.934 6.782 1.848 3.934 0.943 1.791 0.791 - 1.05 0.841 - 0.159 0.94 0.218 221 mat

Q.final f 1.7764 1015 -^* 2.2204 1015 -^* - 0 6.8834 1015 -^* - 1.9984 1015 -^* - 7.2164 1016 -^* - 1.0214 1014 -^* 5.9952 1015 -^* 9.992 1016 -^* - 0 9.992 1016 -^* 2.2204 1015 -^* - 1.1102 1015 -^* 0 1.1102 1016 -^* - 5.107 1015 -^* - 8.218 1015 -^* 1.5081 1013 -^* 2.6136 1013 -^* - 7.4561 1014 -^* 3.952 1014 -^* - 3.9784 1014 -^* 1.3478 1014 -^* - 231 mat

Q.final f 2.6645 1015 -^* 3.1086 1015 -^* - 0 3.9968 1015 -^* 9.992 1016 -^* - 2.72 1015 -^* - 1.0214 1014 -^* 6.9944 1015 -^* - 3.9968 1015 -^* 1.9984 1015 -^* - 9.992 1016 -^* 2.2204 1015 -^* - 9.1038 1015 -^* - 0 1.1102 1016 -^* - 5.107 1015 -^* 8.218 1015 -^* 1.5968 1013 -^* 2.2052 1013 -^* 4.1084 1014 -^* 3.952 1014 -^* - 3.9784 1014 -^* 1.0406 1014 -^* 231 mat

0 time t.0 - 0.9 s *