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Calculate the head losses and the corrected fows in the various pipes of a

Flow to Node: +ve
Flow out of Node: -ve

Flows at Nodes QJ 7522 - 15 - 10 - 31 - 361 mat L s / * :

Lengths LOOP 1 L1 50030050030041 mat m * :

Dia. LOOP 1 D1 30020020020041 mat mm * :

Lengths LOOP 2 L2 50030050030041 mat m * :

Dia. LOOP 2 D2 20015015015041 mat mm * :

Continuity of the whole network QJ sum 0

For Darcy-Weisbach: n = 2

Assumed Flows forContinuity in LOOP 1 Q1 452315 - 30 - 41 mat L s / * :

Assumed Flows forContinuity in LOOP 2 Q2 15238 - 5 - 41 mat L s / * :

Hazen William 'n' n 1.85 :

Friction Factor f 0.02 :

Flows in LOOPS Sign Convention
1. Clockwise: +ve
2. Anticlockwise: -ve

Following 3 Methods will be used in this excercise, using Hardy Cross Method

Successive Steps – starting with assumed flow in the pipes after balancing at each node. 3 steps in this example.
One step for any number of Iterations. More Iterations, better accuracy. 5 Iterations in this example.
One step using a "While Loop" with a pre-defined degree of accuracy. Flow accuracy 0.001 L/s.

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METHOD 1: Hardy Cross Method using Successive Steps

PROGRAM 1 starts with assumed flow for both LOOPS

Program 1: Using Vectorize Function. Find 'Corrected Flow' for both LOOPS Q# L# D# Calc_ΔQ# k# 8 f * L# * g.e D# (5^* π 2^* / vectorize : HL# k# Q# * Q# abs (n 1 -^* vectorize : δHL# HL# Q# / abs vectorize : ΔQ# HL# sum n δHL# sum (* / - : 41 line :

<---------- K values for all pipes

<---------- Head Loss for all pipes

<---------- Net Head Loss

<---------- Corrections for both LOOPS . To be minimized with successive steps

1st correction (Starts with assumed flows in both LOOPS)

Apply PROGRAM 1 to get 1st correction ΔQ.cor1 Q1 Q2 21 mat L1 L2 21 mat D1 D2 21 mat Calc_ΔQ# vectorize : L s / 2.06 4.93 - 21 mat

ΔQ.cor1 1 el L s / 2.06

ΔQ.cor1 2 el L s / 4.93 -

1st Corrected flows in LOOP 1 Q1.cor1 Q1 ΔQ.cor1 1 el + :

1st Corrected flows in LOOP 2 Q2.cor1 Q2 ΔQ.cor1 2 el + : L s / 10.07 18.07 12.93 - 9.93 - 41 mat

Now Apply 1st Correction for Pipe CD common to both loops

Adjusted flow for Pipe CD in LOOP 1 ΔCD1 Q1.cor1 3 el ΔQ.cor1 2 el - : L s / 8 -

Adjusted flow for Pipe DC in LOOP 2 Q2.cor1 1 el ΔCD1 - :

Q1.cor1 3 el ΔCD1 :

Q1.cor1 L s / 47.06 25.06 8 - 27.94 - 41 mat

Q2.cor1 L s / 8 18.1 12.9 - 9.9 - 41 mat

Q1.cor1 3 el 0.008 m 3^s / * -

2nd correction (Starts with flows in correction 1)

Apply PROGRAM 1 to get 2nd correction ΔQ.cor2 Q1.cor1 Q2.cor1 21 mat L1 L2 21 mat D1 D2 21 mat Calc_ΔQ# vectorize : L s / 1.22 - 0.45 21 mat

ΔQ.cor2 1 el L s / 1.22 -

ΔQ.cor2 2 el L s / 0.45

2nd Corrected flows in LOOP 1 Q1.cor2 Q1.cor1 ΔQ.cor2 1 el + : L s / 45.8 23.8 9.2 - 29.2 - 41 mat

2nd Corrected flows in LOOP 2 Q2.cor2 Q2.cor1 ΔQ.cor2 2 el + : L s / 8.5 18.5 12.5 - 9.5 - 41 mat

Now Apply 2nd Correction for Pipe CD common to both loops

Adjusted flow for Pipe CD in LOOP 1 ΔCD2 Q1.cor2 3 el ΔQ.cor2 2 el - : L s / 9.68 -

Adjusted flow for Pipe DC in LOOP 2 Q2.cor2 1 el ΔCD2 - :

Q1.cor2 3 el ΔCD2 :

Q2.cor2 1 el L s / 9.7

Q1.cor2 3 el L s / 9.7 -

Corrected flows in LOOP 1 Q1.cor2 L s / 45.8 23.8 9.7 - 29.2 - 41 mat

Corrected flows in LOOP 2 Q2.cor2 L s / 9.7 18.5 12.5 - 9.5 - 41 mat

3rd correction (Starts with flows in correction 2)

Apply PROGRAM 2 to get 3rd correction ΔQ.cor3 Q1.cor2 Q2.cor2 21 mat L1 L2 21 mat D1 D2 21 mat Calc_ΔQ# vectorize : L s / 0.12 0.09 - 21 mat

ΔQ.cor3 1 el L s / 0.12

ΔQ.cor3 2 el L s / 0.09 -

3rd Corrected flows in LOOP 1 Q1.cor3 Q1.cor2 ΔQ.cor3 1 el + : L s / 4624 9.6 - 29 - 41 mat

3rd Corrected flows in LOOP 2 Q2.cor3 Q2.cor2 ΔQ.cor3 2 el + : L s / 9.6 18.4 12.6 - 9.6 - 41 mat

Q1.cor3 3 el L s / 9.56 -

Q2.cor3 1 el L s / 9.59

Now Apply 3rd Correction for Pipe CD common to both loops

Adjusted flow for Pipe CD in LOOP 1 ΔCD3 Q1.cor3 3 el ΔQ.cor3 2 el - : L s / 9.47 -

Adjusted flow for Pipe DC in LOOP 2 Q2.cor3 1 el ΔCD3 - :

Q1.cor3 3 el ΔCD3 :

Q2.cor3 1 el L s / 9.47

Q1.cor3 3 el L s / 9.47 -

Corrected flows in LOOP 1 Q1.cor3 L s / 45.96 23.96 9.47 - 29.04 - 41 mat

Diff of flow in CD/DC in LOOP1 and LOOP 2 Q1.cor3 3 el Q2.cor3 1 el + abs L s / 0

Corrected flows in LOOP 2 Q2.cor3 L s / 9.47 18.43 12.57 - 9.57 - 41 mat

Calculated Flows - After 3 Corrections

From Book Example - After 2 Corrections

METHOD 3: Hardy Cross Method: Using Any Given Number of Iterations

Index of common pipe in LOOP 1 n1 3 :

Index of common pipe in LOOP 2 n2 1 :

PROGRAM 2: Calculates 'Corrected Flows' & 'Corrections' CALLS PROGRAM 1 q1 q2 Calc_All_Δ Δq1 q1 q2 21 mat L1 L2 21 mat D1 D2 21 mat Calc_ΔQ# vectorize : q1.cor q1 Δq1 1 el + : q2.cor q2 Δq1 2 el + : Δcd1 q1.cor n1 el Δq1 2 el - : q1.cor n1 el Δcd1 : q2.cor n2 el Δcd1 - : q1.cor q2.cor 21 mat Δq1 1 el Δq1 2 el 21 mat 12 mat 71 line :

1st Corrections example C1 Q1 Q2 Calc_All_Δ :

C1 L s / 47.06 25.06 8 - 27.94 - 41 mat 8 18.07 12.93 - 9.93 - 41 mat 21 mat 2.06 4.93 - 21 mat 12 mat

Solution by Repeated Iteration Method using PROGRAM 2

PROGRAM 3: Repeated Iteration Method. CALLS PROGRAM 2 q1 q2 iter Q j 1 iter range j 1 ≡ P j el q1 q2 Calc_All_Δ : P j el P j 1 - el 1 el 1 el P j 1 - el 1 el 2 el Calc_All_Δ : if 11 line for P iter el 21 line :

1st Corrections example C_Iter Q1 Q2 1 Q :

C_Iter L s / 47.06 25.06 8 - 27.94 - 41 mat 8 18.07 12.93 - 9.93 - 41 mat 21 mat 2.06 4.93 - 21 mat 12 mat

1st column shows FINAL flows in LOOPS 1 & 2. The 2 nd column shows FINAL Corrections after 3 Iterations

With 5 Iterations: Q1 Q2 5 Q L s / 45.94 23.94 9.5 - 29.06 - 41 mat 9.5 18.44 12.56 - 9.56 - 41 mat 21 mat 0021 mat 12 mat

METHOD 4: Hardy Cross Method: Using While Loop

PROGRAM 4: Using 'While' loop : CALLS PROHRAM 2 q1 q2 Z A q1 q2 Calc_All_Δ : ΔQ A 2 el : qq1 A 1 el 1 el : qq2 A 1 el 2 el : Δ 0.001 L s / * : ΔQ 1 el abs Δ ≥ (ΔQ 2 el abs Δ ≥ (& B qq1 qq2 Calc_All_Δ : ΔQ B 2 el : qq1 B 1 el 1 el : qq2 B 1 el 2 el : ΔQ B 2 el : 51 line while B 71 line :

Q1 Q2 Z L s / 45.94 23.94 9.5 - 29.06 - 41 mat 9.5 18.44 12.56 - 9.56 - 41 mat 21 mat 0021 mat 12 mat

Ans2 Q1 Q2 Z 1 el : L s / 45.94 23.94 9.5 - 29.06 - 41 mat 9.5 18.44 12.56 - 9.56 - 41 mat 21 mat

Results from Different Methods

'While' Loop Method using PROGRAM 4 : Both LOOPS Q1 Q2 Z L s / 45.94 23.94 9.5 - 29.06 - 41 mat 9.5 18.44 12.56 - 9.56 - 41 mat 21 mat 0021 mat 12 mat

Repeated Iterations Method using PROGRAM 3 : Both LOOPS Q1 Q2 5 Q L s / 45.94 23.94 9.5 - 29.06 - 41 mat 9.5 18.44 12.56 - 9.56 - 41 mat 21 mat 0021 mat 12 mat

45.94 23.94 9.5 - 29.06 - 41 mat 9.5 18.44 12.56 - 9.56 - 41 mat 21 mat 0021 mat 12 mat L s / *

Successive Step Method Using PROGRAM 1. LOOP 1 Q1.cor3 L s / 45.96 23.96 9.47 - 29.04 - 41 mat

Successive Step Method Using PROGRAM 1. LOOP 2 Q2.cor3 L s / 9.47 18.43 12.57 - 9.57 - 41 mat

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