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4 appVersion 1.0.8151.26392

https://cheguide.com/pipe_network.html

Example

A water supply distribution system is shown in the figure below. All pipes are cast iron with lengths and diameters as provided in table below. Perform pipe network analysis and calculate water flow in all branches .

Relative roughness

Nodal flows at A,B,C,D

Length

Diameter

LL 10002000100020001000750220020005002200101 mat m * :

D 0.4 0.45 0.30 0.30 0.40 0.30 0.30 0.30 0.25 0.25 101 mat m * :

Ks 0.00087 0.00104 0.00130 0.00130 0.00130 0.00130 0.00130 0.00173 0.00130 0.00173 101 mat :

Q.A Q.B Q.C Q.D 41 mat 300150 - 50 - 100 - 41 mat L s / * :

t.0 0 time :

Darcy-Weisbach 'n' n 2 :

Kinematic Viscosity of water ν 1106 -^* m 2^s / * :

Sanity check

Q.A Q.B Q.C Q.D 41 mat sum 0

Define

cumec m 3^s / :

nr 1 D rows range :

n_loops 3 :

PROGRAM 1: Find 'head loss' in a pipe q k d l Calc_HL v q abs π 4 / d 2^* / : Re v d * ν m 2^s / / / abs : f 64 Re / Re 2100 ≤ 0.25 k 3.7 / 5.74 Re 0.9^/ + log10 (2 -^* cases : K# 8 f * l * g.e m s 2^/ / (d 5^* π 2^* / : hL K# q * q abs (n 1 -^* : 51 line :

Note:
Formula for f as given in the spreadsheet example.
(See link above)

Head Loss (HL)

guess 110 range el 0.1 :

For testing intermediate claculationls only Q.init 10020030307080201702020101 mat L s / * :

guess 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 101 mat

Testing only

Head Loss in all pipes Q HL Q Ks D m / LL m / Calc_HL vectorize :

Q.init cumec / HL 1.6249 7.3303 0.7014 1.4028 0.8774 3.5615 0.7083 45.0635 0.3944 1.8302 101 mat

Flows at Nodes Q eq.nodes Q.A cumec / Q 1 el - Q 2 el - Q.B cumec / Q 8 el + Q 9 el - Q.C cumec / Q 5 el + Q 7 el - Q.D cumec / Q 6 el + Q 7 el + Q 1 el Q 4 el - Q 5 el - Q 2 el Q 3 el - Q 8 el - Q 9 el Q 10 el - Q 4 el Q 3 el + Q 10 el + Q 6 el - 81 mat :

Clockwise flow + ve
Anticlockwise flow - ve

Head loss in the 3 loops Q eq.Loops Q HL 1 el - Q HL 2 el + Q HL 3 el + Q HL 4 el - Q HL 3 el - Q HL 8 el + Q HL 9 el + Q HL 10 el + Q HL 4 el Q HL 5 el - Q HL 6 el + Q 7 el HL 7 el - 31 mat :

Testing only

Q.init cumec / eq.Loops 5.004 46.5867 3.3786 31 mat

Define the system of 11 equations ( 8 Nodes + 3 Loops ) ; 10 Variables (pipe flows) Hence, Overdetermimed Nonlinear System

Q f Q eq.nodes Q eq.Loops stack 11 line :

result guess 104 -^Q f al_nleqsolve 11 line :

Final head loss in each pipe

Q.final result :

Excel Result

Q.final HL 3.326 4.4834 2.3583 3.5157 1.6287 1.6483 3.5354 15.6995 2.3468 - 10.9945 - 101 mat

Q.final 0.1442 0.1558 0.056 0.0482 0.096 0.054 0.046 0.0999 0.0501 - 0.0501 - 101 mat

In Method 1, "f" for each pipe is directly calculated using the formula shown in Program 1. Each pipe has a diffrent "f" value.

0 time t.0 - 7.3 s *

Method 2: Using Colebrook-White equation

ε.f 1010 -^:

Define

t1 0 time :

Numerical solution of Colebrook-White equation using SMath Studio 'NewtonRaphson' function

1 f sqrt / 2 k 3.7 D * / 2.51 Re f sqrt * / + log10 * - ≡

x 1 f sqrt / ≡

x 2 k 3.7 d * / 2.51 Re / x * + log10 * + 0 ≡

Substitute

PROGRAM 2: 'NewtonRaphson' function to find 'f' of a pipe k d Re NR_SOLVE x# F x# 2 k 3.7 d * / 2.51 Re / x# * + log10 * + : x.init 1 0.01 sqrt / : x.nr x# F x.init ε.f NewtonRaphson : f.nr 1 x.nr 2^/ : 41 line :

Colebrook_White Eqn

Start with min 'f' value in Moody chart

Use "Newton Raphson" method to solve for 'x
Note: "LevenbergMarquadt" method is slow.

Friction factor of pipe

PROGRAM 3: Find 'head loss' in a pipe using PROGRAM 2 q k d l Calc_HL_NR v q abs π 4 / d 2^* / : Re v d * ν m 2^s / / / : f 64 Re / Re 2100 ≤ k d Re NR_SOLVE cases : K# 8 f * l * g.e m s 2^/ / d (5^* π 2^* / : hL K# q * q abs (n 1 -^* : 51 line :

Velocity in pipe

Kinamatic viscosity

Reynold number

ν m 2^s / 1106 -^*

friction factor (if laminar flow)

friction factor (call Program 2 if Re>2100)

n 2

Darcy-Weisbach

Head loss

Head loss in all pipes Q HL_NR Q Ks D m / LL m / Calc_HL_NR vectorize 11 line :

Testing only

Q.init cumec / HL_NR 1.9768 8.8302 0.9157 1.8313 1.0814 4.8014 0.9069 62.6084 0.5354 2.5575 101 mat

Clockwise flow + ve
Anticlockwise flow - ve

Head loss in the 3 loops Q eq_NR.Loops Q HL_NR 1 el - Q HL_NR 2 el + Q HL_NR 3 el + Q HL_NR 4 el - Q HL_NR 3 el - Q HL_NR 8 el + Q HL_NR 9 el + Q HL_NR 10 el + Q HL_NR 4 el Q HL_NR 5 el - Q HL_NR 6 el + Q 7 el HL_NR 7 el - 31 mat 11 line :

Testing only

Q.init cumec / eq_NR.Loops 5.9378 64.7857 4.6444 31 mat

Define the system of 11 equations ( 8 Nodes + 3 Loops ) ; 10 Variables (pipe flows). Hence, Overdetermimed Nonlinear System

Q f_NR Q eq.nodes Q eq_NR.Loops stack 11 line :

In Method 2, finding "f" for each pipe involves the numerical solution of Colebrook-White equation as per Program 2. Hence, it takes a longer time than Method 1.

result_NR guess 104 -^Q f_NR al_nleqsolve 11 line :

Q_NR.final result_NR :

0 time t1 - 14.8 s *

Summary

Method 1

Final head loss in each pipe

Method 2

Final head loss in each pipe

Q.final 0.1442 0.1558 0.056 0.0482 0.096 0.054 0.046 0.0999 0.0501 - 0.0501 - 101 mat

Q.final HL 3.326 4.4834 2.3583 3.5157 1.6287 1.6483 3.5354 15.6995 2.3468 - 10.9945 - 101 mat

Q_NR.final 0.1435 0.1565 0.0561 0.0475 0.0961 0.0539 0.0461 0.1004 0.0496 - 0.0496 - 101 mat

Q_NR.final HL 3.2966 4.5202 2.3646 3.4147 1.6309 1.6444 3.5451 15.8781 2.2946 - 10.7492 - 101 mat

Q.final f normi 6.794 106 -^*

Q_NR.final f_NR normi 1.8377 106 -^*

0 time t.0 - 22.5 s *

Q.final f 2.9447 107 -^* 4.8125 108 -^* - 1.9665 107 -^* - 2.2789 107 -^* 3.8278 108 -^* - 1.8542 108 -^* 2.6455 107 -^* - 6.6995 109 -^* 7.0598 107 -^* 6.794 106 -^* - 2.463 107 -^* 111 mat

Q_NR.final f_NR 1.5181 108 -^* - 2.3648 108 -^* - 4.4888 109 -^* - 9.2153 108 -^* 1.5986 108 -^* 6.3011 108 -^* - 1.9469 109 -^* - 1.366 1010 -^* 4.2784 107 -^* 1.8377 106 -^* - 1.3972 107 -^* 111 mat