A pipe network with two loops is shown in Fig. 11.9. Determine the flow in each pipe for an inflow of 5 units at the junction A and outflows of 2.0 units and 3.0 units at junctions D and C respectively. The resistance R for different pipes are shown in the figure.

Question

Accepted Answer

Flow direction is assumed positive clockwise for both the loops ABD and BCD. The iterative solutions based on Hardy-Cross method has been made.

The five trials have been made and the results of each trial is shown in Fig. 11.10; for each trial, dQ is calculated from Eq. (11.31). After fifth trial, the error dQ is so small that it changes the flow only in the third place of decimal. Hence the calculation has not been continued beyond the fifth trial.

[latex]d Q=\frac{\Sigma R|Q| Q}{\Sigma 2 R|Q|}[/latex] (11.31)

First trial

Loop ABD		Loop BCD
R\|Q\|Q	2R\|Q\|	R\|Q\|Q	2R\|Q\|
[latex]120 imes 2^{2}=480[/latex]	2 × 120 × 2 = 480	[latex]300 imes(1.2)^{2}=432[/latex]	2 × 300 × 1.2 = 720
[latex]400 imes(0.8)^{2}=256[/latex]	2 × 400 × 0.8 = 640	[latex]-150 imes(1.8)^{2}=-486[/latex]	2 × 150 × 1.8 = 540
[latex]-200 imes 3^{2}=-1800[/latex]	2 × 400 × 0.8 = 640	[latex]-400 imes(0.8)^{2}=-256[/latex]	2 × 400 × 0.8 = 640
[latex]\Sigma R\|Q\| Q=-1064[/latex]	[latex]2 \Sigma R\|Q\|=2320[/latex]	[latex]\Sigma R\|Q\| Q=-310[/latex]	[latex]2 \Sigma R\|Q\|=1900[/latex]
[latex]d Q=\frac{\sum R\|Q\| Q}{\sum 2 R\|Q\|}[/latex]		[latex]d Q=\frac{\sum R\|Q\| Q}{\sum 2 R\|Q\|}[/latex]
[latex]=\frac{-1064}{2320}[/latex]		[latex]=\frac{-300}{1900}[/latex]
[latex]=-0.46[/latex]		[latex]=-0.16[/latex]

Second trial:

Loop ABD		Loop BCD
R\|Q\|Q	2R\|Q\|	R\|Q\|Q	2R\|Q\|
[latex]120 imes(2.46)^{2}=726.19[/latex]	[latex]2 imes 120 imes 2.46=590.40[/latex]	[latex]300 imes(1.36)^{2}=554.88[/latex]	[latex]2 imes 300 imes 1.36=816[/latex]
[latex]400 imes(1.10)^{2}=484.00[/latex]	[latex]2 imes 400 imes 1.10=880.00[/latex]	[latex]-150 imes(1.64)^{2}=-403.44[/latex]	[latex]2 imes 150 imes 1.64=492[/latex]
[latex]-1200 imes(2.54)^{2}=-1290.32[/latex]	[latex]2 imes 200 imes 2.54=1016.00[/latex]	[latex]-400 imes(1.10)^{2}=-484.00[/latex]	[latex]2 imes 400 imes 1.10=880[/latex]
[latex]\Sigma R\|Q\| Q=-50.13[/latex]	[latex]2 \Sigma R\|Q\|=2486.40[/latex]	[latex]\Sigma R\|Q\| Q=-332.56[/latex]	[latex]2 \Sigma R\|Q\|=2188[/latex]
[latex]d Q=\frac{\sum R\|Q\| Q}{\sum 2 R\|Q\|}[/latex]		[latex]d Q=\frac{\sum R\|Q\| Q}{\sum 2 R\|Q\|}[/latex]
[latex]=\frac{-50.13}{2486.40}[/latex]		[latex]=\frac{-332.56}{2188}[/latex]
[latex]=-0.02[/latex]		[latex]=-0.15[/latex]

Third trial:

Loop ABD		Loop BCD
R\|Q\|Q	2R\|Q\|	R\|Q\|Q	2R\|Q\|
[latex]120 imes(2.48)^{2}=738.05[/latex]	[latex]2 imes 120 imes 2.48=595.20[/latex]	[latex]300 imes(1.51)^{2}=684.03[/latex]	[latex]2 imes 300 imes 1.51=906.00[/latex]
[latex]400 imes(0.97)^{2}=376.36[/latex]	[latex]2 imes 400 imes 0.97=776.00[/latex]	[latex]-150 imes(1.49)^{2}=-333.01[/latex]	[latex]2 imes 150 imes 1.49=447.00[/latex]
[latex]-200 imes(2.52)^{2}=-1270.08[/latex]	[latex]2 imes 200 imes 2.52=1008.00[/latex]	[latex]-400 imes(0.97)^{2}=-376.36[/latex]	[latex]2 imes 400 imes 0.97=776.00[/latex]
[latex]\Sigma R\|Q\| Q=-155.67[/latex]	[latex]2 \Sigma R\|Q\|=2379.20[/latex]	[latex]\Sigma R\|Q\| Q=-25.34[/latex]	[latex]2 \sum R\|Q\|=2129[/latex]
[latex]d Q=\frac{\sum R\|Q\| Q}{\sum 2 R\|Q\|}[/latex]		[latex]d Q=\frac{\sum R\|Q\| Q}{\sum 2 R\|Q\|}[/latex]
[latex]=\frac{-155.67}{2379.67}[/latex]		[latex]=\frac{-25.34}{2129}[/latex]
[latex]=-0.06[/latex]		[latex]=-0.01[/latex]

Fourth trial:

Loop ABD		Loop BCD
R\|Q\|Q	2R\|Q\|	R\|Q\|Q	2R\|Q\|
[latex]120 imes(2.54)^{2}=774.20[/latex]	[latex]2 imes 120 imes 2.54=609.60[/latex]	[latex]300 imes(1.52)^{2}=693.12[/latex]	[latex]2 imes 300 imes 1.52=912.00[/latex]
[latex]400 imes(1.02)^{2}=416.16[/latex]	[latex]2 imes 400 imes 1.02=816.00[/latex]	[latex]-150 imes(1.48)^{2}=-328.56[/latex]	[latex]2 imes 150 imes 1.48=444.00[/latex]
[latex]-200 imes(2.46)^{2}=-1210.32[/latex]	[latex]2 imes 200 imes 2.46=984.00[/latex]	[latex]-400 imes(1.02)^{2}=-416.16[/latex]	[latex]2 imes 400 imes 1.02=816.00[/latex]
[latex]\Sigma R\|Q\| Q=-19.96[/latex]	[latex]2 \Sigma R\|Q\|=2409.60[/latex]	[latex]\Sigma R\|Q\| Q=-51.6[/latex]	[latex]2 \Sigma R\|Q\|=2172[/latex]
[latex]d Q=\frac{\sum R\|Q\| Q}{\sum 2 R\|Q\|}[/latex]		[latex]d Q=\frac{\sum R\|Q\| Q}{\sum 2 R\|Q\|}[/latex]
[latex]=\frac{-19.96}{2409.60}[/latex]		[latex]=\frac{-51.6}{2172}[/latex]
[latex]=-0.008[/latex]		[latex]=-0.02[/latex]

Fifth trial:

Loop ABD		Loop BCD
R\|Q\|Q	2R\|Q\|	R\|Q\|Q	2R\|Q\|
[latex]120 imes(2.58)^{2}=779.08[/latex]	[latex]2 imes 120 imes 2.58=619.20[/latex]	[latex]300 imes(1.54)^{2}=711.48[/latex]	[latex]2 imes 300 imes 1.54=924.00[/latex]
[latex]400 imes(1.008)^{2}=406.42[/latex]	[latex]2 imes 400 imes 1.008=806.40[/latex]	[latex]-150 imes(1.46)^{2}=-319.74[/latex]	[latex]2 imes 150 imes 1.46=438.00[/latex]
[latex]-200 imes(2.452)^{2}=-1202.46[/latex]	[latex]2 imes 200 imes 2.452=980.80[/latex]	[latex]-400 imes(1.08)^{2}=-406.42[/latex]	[latex]2 imes 400 imes 1.008=806.40[/latex]
[latex]\Sigma R\|Q\| Q=-16.96[/latex]	[latex]2 \Sigma R\|Q\|=2406.40[/latex]	[latex]\Sigma R\|Q\| Q=-14.68[/latex]	[latex]2 \Sigma R\|Q\|=2168.40[/latex]
[latex]d Q=\frac{\sum R\|Q\| Q}{\sum 2 R\|Q\|}[/latex]		[latex]d Q=\frac{\sum R\|Q\| Q}{\sum 2 R\|Q\|}[/latex]
[latex]=\frac{-16.96}{2406.40}[/latex]		[latex]=\frac{-14.68}{2168.40}[/latex]
[latex]=-0.007[/latex]		[latex]=-0.007[/latex]

Loop ABD		Loop BCD
R\|Q\|Q	2R\|Q\|	R\|Q\|Q	2R\|Q\|
$120 \times 2^{2}=480$	2 × 120 × 2 = 480	$300 \times(1.2)^{2}=432$	2 × 300 × 1.2 = 720
$400 \times(0.8)^{2}=256$	2 × 400 × 0.8 = 640	$-150 \times(1.8)^{2}=-486$	2 × 150 × 1.8 = 540
$-200 \times 3^{2}=-1800$	2 × 400 × 0.8 = 640	$-400 \times(0.8)^{2}=-256$	2 × 400 × 0.8 = 640
$\Sigma R\|Q\| Q=-1064$	$2 \Sigma R\|Q\|=2320$	$\Sigma R\|Q\| Q=-310$	$2 \Sigma R\|Q\|=1900$
$d Q=\frac{\sum R\|Q\| Q}{\sum 2 R\|Q\|}$		$d Q=\frac{\sum R\|Q\| Q}{\sum 2 R\|Q\|}$
$=\frac{-1064}{2320}$		$=\frac{-300}{1900}$
$=-0.46$		$=-0.16$

Loop ABD		Loop BCD
R\|Q\|Q	2R\|Q\|	R\|Q\|Q	2R\|Q\|
$120 \times(2.46)^{2}=726.19$	$2 \times 120 \times 2.46=590.40$	$300 \times(1.36)^{2}=554.88$	$2 \times 300 \times 1.36=816$
$400 \times(1.10)^{2}=484.00$	$2 \times 400 \times 1.10=880.00$	$-150 \times(1.64)^{2}=-403.44$	$2 \times 150 \times 1.64=492$
$-1200 \times(2.54)^{2}=-1290.32$	$2 \times 200 \times 2.54=1016.00$	$-400 \times(1.10)^{2}=-484.00$	$2 \times 400 \times 1.10=880$
$\Sigma R\|Q\| Q=-50.13$	$2 \Sigma R\|Q\|=2486.40$	$\Sigma R\|Q\| Q=-332.56$	$2 \Sigma R\|Q\|=2188$
$d Q=\frac{\sum R\|Q\| Q}{\sum 2 R\|Q\|}$		$d Q=\frac{\sum R\|Q\| Q}{\sum 2 R\|Q\|}$
$=\frac{-50.13}{2486.40}$		$=\frac{-332.56}{2188}$
$=-0.02$		$=-0.15$

Loop ABD		Loop BCD
R\|Q\|Q	2R\|Q\|	R\|Q\|Q	2R\|Q\|
$120 \times(2.48)^{2}=738.05$	$2 \times 120 \times 2.48=595.20$	$300 \times(1.51)^{2}=684.03$	$2 \times 300 \times 1.51=906.00$
$400 \times(0.97)^{2}=376.36$	$2 \times 400 \times 0.97=776.00$	$-150 \times(1.49)^{2}=-333.01$	$2 \times 150 \times 1.49=447.00$
$-200 \times(2.52)^{2}=-1270.08$	$2 \times 200 \times 2.52=1008.00$	$-400 \times(0.97)^{2}=-376.36$	$2 \times 400 \times 0.97=776.00$
$\Sigma R\|Q\| Q=-155.67$	$2 \Sigma R\|Q\|=2379.20$	$\Sigma R\|Q\| Q=-25.34$	$2 \sum R\|Q\|=2129$
$d Q=\frac{\sum R\|Q\| Q}{\sum 2 R\|Q\|}$		$d Q=\frac{\sum R\|Q\| Q}{\sum 2 R\|Q\|}$
$=\frac{-155.67}{2379.67}$		$=\frac{-25.34}{2129}$
$=-0.06$		$=-0.01$

Loop ABD		Loop BCD
R\|Q\|Q	2R\|Q\|	R\|Q\|Q	2R\|Q\|
$120 \times(2.54)^{2}=774.20$	$2 \times 120 \times 2.54=609.60$	$300 \times(1.52)^{2}=693.12$	$2 \times 300 \times 1.52=912.00$
$400 \times(1.02)^{2}=416.16$	$2 \times 400 \times 1.02=816.00$	$-150 \times(1.48)^{2}=-328.56$	$2 \times 150 \times 1.48=444.00$
$-200 \times(2.46)^{2}=-1210.32$	$2 \times 200 \times 2.46=984.00$	$-400 \times(1.02)^{2}=-416.16$	$2 \times 400 \times 1.02=816.00$
$\Sigma R\|Q\| Q=-19.96$	$2 \Sigma R\|Q\|=2409.60$	$\Sigma R\|Q\| Q=-51.6$	$2 \Sigma R\|Q\|=2172$
$d Q=\frac{\sum R\|Q\| Q}{\sum 2 R\|Q\|}$		$d Q=\frac{\sum R\|Q\| Q}{\sum 2 R\|Q\|}$
$=\frac{-19.96}{2409.60}$		$=\frac{-51.6}{2172}$
$=-0.008$		$=-0.02$

Loop ABD		Loop BCD
R\|Q\|Q	2R\|Q\|	R\|Q\|Q	2R\|Q\|
$120 \times(2.58)^{2}=779.08$	$2 \times 120 \times 2.58=619.20$	$300 \times(1.54)^{2}=711.48$	$2 \times 300 \times 1.54=924.00$
$400 \times(1.008)^{2}=406.42$	$2 \times 400 \times 1.008=806.40$	$-150 \times(1.46)^{2}=-319.74$	$2 \times 150 \times 1.46=438.00$
$-200 \times(2.452)^{2}=-1202.46$	$2 \times 200 \times 2.452=980.80$	$-400 \times(1.08)^{2}=-406.42$	$2 \times 400 \times 1.008=806.40$
$\Sigma R\|Q\| Q=-16.96$	$2 \Sigma R\|Q\|=2406.40$	$\Sigma R\|Q\| Q=-14.68$	$2 \Sigma R\|Q\|=2168.40$
$d Q=\frac{\sum R\|Q\| Q}{\sum 2 R\|Q\|}$		$d Q=\frac{\sum R\|Q\| Q}{\sum 2 R\|Q\|}$
$=\frac{-16.96}{2406.40}$		$=\frac{-14.68}{2168.40}$
$=-0.007$		$=-0.007$

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