787e83ae-806a-431c-8b74-1e580b86cf10 520 4 5 decimal

PIPE NETWORK - Ex 3 Improved

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This document is courtesy from Smath Collab NDTMA. I have sharpened/reworked a bit the diagrams, clarified the failure of the vectorise operator [jmG].

The system above contains 4 pipes, and therefore, we need 4 independent equations to solve.

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1. DATA:

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Nodal Supplies & demands QJ 0.32 0.28 0.14 0.10 41 mat m 3^s / * :

Pipe diameters D 0.25 0.25 0.25 0.25 41 mat m * :

Pipe lengths L 20010020010041 mat m * :

Friction factor f 0.02 :

Exponent for friction head loss. 
"n" may be 2.0 or 1.85 or ... n 2.0 :

Elevation at Node 1 WS.1 100 m * :

Density of Water γ 1000 kg m 3^/ * :

n=2 in Darcy Eqn

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Assume pipe flow directions as shown in the diagram -------------------------------------------------------->

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Flow at node:1

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QJ.1 Q.1 - Q.4 - 0 ≡

Sign Convention

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Flow into + ve

Flow out - ve

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at node:2

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Q.1 Q.2 + QJ.2 - 0 ≡

Linear Eqns

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at node:3

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Q.3 Q.2 - QJ.3 - 0 ≡

Friction loss

in real loop-1

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K.1 Q.1 n^* K.2 Q.2 n^* - K.3 Q.3 n^* - K.4 Q.4 n^* - 0 ≡

Clockwise + ve

Anticlockwise - ve

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Non-Linear Eqn

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One real loop : Pipes 1-2-3-4. Use the convention that positive is clockwise, in the direction of decreasing head.

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2. CALCULATIONS:

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h.f f L * V 2^* 2 g.e * D * / ≡

V 2^1 π 4 / D 2^* / (2^Q 2^* ≡

h.f f L * 2 g.e * D * / 16 π 2^D 4^* / * Q 2^* ≡ 8 f * L * g.e D 5^* π 2^* / Q 2^* ≡

Hence K 8 f * L * g.e D 5^* n 2^* / ≡

------->

K is simply a lumped constant for a specific pipe

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Alternatively, using 
"Vectorize" operator K 8 f * L * g.e D (5^* π 2^* / vectorize : s 2^m 5^/ 338.5552 169.2776 338.5552 169.2776 41 mat

j 1 D rows range K j el 8 f * L j el * g.e D j el (5^* π 2^* / : for

K s 2^m 5^/ 338.5552 169.2776 338.5552 169.2776 41 mat

----------->

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Note: if n = 2, Smath computes correct units with no problem. However if n is not equal to 2, (say if n = 1.85), then we have to balance the units in the equations for h_f as shown below.

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General eqn for head loss hf K Q n^* ≡ K Q * Q abs (n 1 -^* ≡

Generates an error in "FindRoot"

Q comes out not indexed

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Pipe friction losses, 
with units adjusted. r 1 K rows range hf r el K r el s 2^m 5^/ / (Q r el m 3^s / / (* Q r el m 3^s / / abs (n 1 -^* m * : 11 line for

Using vectorize operator hf K s 2^m 5^/ / (Q m 3^s / / (* Q m 3^s / / abs (n 1 -^* vectorize 11 line :

-- 1A

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-- 1B

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Symbolic Optimization hf 655360000 s * Q 1 el * s Q 1 el * m 3^/ abs * 196133 m 2^* π 2^* / 327680000 s * Q 2 el * s Q 2 el * m 3^/ abs * 196133 m 2^* π 2^* / 655360000 s * Q 3 el * s Q 3 el * m 3^/ abs * 196133 m 2^* π 2^* / 327680000 s * Q 4 el * s Q 4 el * m 3^/ abs * 196133 m 2^* π 2^* / 41 mat

Symbolic optimization hf 655360000 Q * s * Q s * m 3^/ abs * 196133 m 3^* π 2^* / 327680000 Q * s * Q s * m 3^/ abs * 196133 m 3^* π 2^* / 655360000 Q * s * Q s * m 3^/ abs * 196133 m 3^* π 2^* / 327680000 Q * s * Q s * m 3^/ abs * 196133 m 3^* π 2^* / 41 mat

-- 2A

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-- 2B

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Compare

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Governing Equations required for solution

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F QJ 1 el Q 1 el - Q 4 el - 0 ≡ Q 1 el Q 2 el + QJ 2 el - 0 ≡ Q 3 el Q 2 el - QJ 3 el - 0 ≡ hf 1 el hf 2 el - hf 3 el - hf 4 el - 0 ≡ 41 mat :

Guess values g_val 111141 mat m 3^s / * :

Explicit the algo solver u v GivenFind u v FindRoot :

-ve value (if any). indicates that flow is opposite to the assumed direction.

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Q F g_val GivenFind : m 3^s / 0.218 0.062 0.202 0.102 41 mat

hf 16.149 m * 0.6423 m * 13.7594 m * 1.7473 m * 41 mat

Assumed 'n' n 2

Sanity Check

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QJ 1 el Q 1 el - Q 4 el - Q 1 el Q 2 el + QJ 2 el - Q 3 el Q 2 el - QJ 3 el - QJ 4 el Q 4 el + Q 3 el - hf 1 el hf 2 el - hf 3 el - hf 4 el - 51 mat 00000 m * 51 mat