<content> <p>Numerical Solution</p> </content>

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4 appVersion 0.99.7774.1835

Numerical Solution of Colenrook-White Equation
using SMath Studio 'NewtonRaphson' function

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

t.0 0 time :

<content> <p>Numerical Solution</p> </content>

'NewtonRaphson' Method : To find friction factor 'f' of Colebrook-White Eqn

ζ 1010 -^:

Define

x 1 f sqrt / ≡

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

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

Substitute

PROGRAM 1: Use 'NewtonRaphson' function to find 'f' for 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 x F x.init ζ NewtonRaphson : f 1 x 2^/ : 41 line :

Linear Eqn for single pipe

Start with min 'f' value in Moody chart

Solve for 'x'

Friction factor

PROGRAM 2: Find 'f' for ALL PIPES in a LOOP Calls PROGRAM 1 q k d Calc_LOOP_f v q abs π 4 / d 2^* / vectorize : Re v d * ν / abs vectorize : ff k d Re NR_SOLVE vectorize : 31 line :

PROGRAM 3 : 'f' values for ALL LOOPS Calls PROGRAM 2 with Vectorize Function for Nested Array q k d Calc_All_LOOP_f q k d Calc_LOOP_f vectorize 11 line :

1. DATA

Assumed Flows forContinuity in LOOP 1 Q1 103.56 - 20040 53.56 - 41 mat L s / * :

Assumed Flows forContinuity in LOOP 2 Q2 40 - 160 7.22 7.22 41 mat L s / * :

Assumed Flows forContinuity in LOOP 3 Q3 53.56 50 - 100.78 13.8 - 41 mat L s / * :

Kinematic viscosity ν 1.005 106 -^* m 2^s / * :

Lengths LOOP 1 L1 100020001000200041 mat m * :

Lengths LOOP 2 L2 10002000500220041 mat m * :

Lengths LOOP 3 L3 20001000750220041 mat m * :

Dia. LOOP 1 D1 0.4 0.45 0.30 0.30 41 mat m * :

Dia. LOOP 2 D2 0.3 0.3 0.25 0.25 41 mat m * :

Dia. LOOP 3 D3 0.3 0.4 0.3 0.3 41 mat m * :

Ks: LOOP 1 Ks1 0.00005 0.00005 0.00005 0.00005 41 mat m * :

Ks: LOOP 2 Ks2 0.00005 0.00003 0.00003 0.00003 41 mat m * :

Ks: LOOP 3 Ks3 0.00005 0.00003 0.00003 0.00003 41 mat m * :

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2. CALCULATIONS

'f' values for Each LOOP using Program 2

Define Nested Arrays for All LOOPS

LOOP 1 f1 Q1 Ks1 D1 Calc_LOOP_f :

QQ Q1 Q2 Q3 31 mat :

DD D1 D2 D3 31 mat :

KK Ks1 Ks2 Ks3 31 mat :

LOOP 2 f2 Q2 Ks2 D2 Calc_LOOP_f :

LOOP 3 f3 Q3 Ks3 D3 Calc_LOOP_f :

3. RESULTS: 'f' values for ALL LOOPS

Call PROGRAM 3 : for all LOOPS ff QQ KK DD Calc_All_LOOP_f : 0.0155 0.0144 0.0173 0.0166 41 mat 0.0173 0.014 0.0228 0.0228 41 mat 0.0166 0.0169 0.0147 0.0206 41 mat 31 mat

Compare

f1 0.0155 0.0144 0.0173 0.0166 41 mat

f2 0.0173 0.014 0.0228 0.0228 41 mat

f3 0.0166 0.0169 0.0147 0.0206 41 mat

0 time t.0 - 0.4 s *