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Derive an expression for the equivalent size of the pipe to replace the pipes in series.

Derive an expression for the equivalent size of the pipe to replace the pipes in series.

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Equivalent pipe : The pipe of uniform diameter having loss of head and discharge equal to the loss of head and discharge of a compound pipe, consisting of several pipe of different length and diameters.

Now, Let $L_1$ = Length of pipe 1 and $d_1$ = dia.

$L_2$ = Length of pipe 2 and $d_2$ = dia.

$L_3$ Length of pipe 3 and $d_3$ = dia.

H = total head loss

L = length of equivalent pipe

d = dia. of equivalent pipe.

$\therefore L = L_1 + L_2 + L_3$

neglect minor losses, the head loss in compound pipe,

$H = \frac{4f_1L_1v_1^2}{d\times2g} + \frac{4f_2L_2V_2^2}{d\times2g} + \frac{4f_3L_3V_3^2}{d\times2g} ----- (1)$

(Assume $(f_1 = f_2 = f_3 = f)$

Discharge $Q = Av_1 = A_2 V_2 = A_3 V_3 = \frac{\pi}{4} d_1^2v_1 = \frac{\pi}{4} d_2^2v_2 = \frac{\pi}{4} d_3^2 v_3^2$

Substituting these values in the form of

$v = (\frac{4Q}{\pi d^2})$ in (1)

$H = 4fL_1 \times \frac{ \bigg( \frac{4Q}{\pi d_1^2 } \bigg)^2 }{d_1 \times 2g} + 4fL_2 \times \frac{ \bigg( \frac{4Q}{\pi d_2^2 } \bigg)^2 }{d_2 \times 2g} + 4fL_3 \times \frac{ \bigg( \frac{4Q}{\pi d_3^2 } \bigg)^2 }{d_3 \times 2g} $

$H = \frac{4\times1bfQ}{\pi ^2 \times 2g} (\frac{L_1}{d1^5} + \frac{L_2}{d2^5} + \frac{L_3}{d3^5}) $

$\therefore \text{Head loss in equivalent pipe}, H = \frac{4fLv^2}{d\times2g}$

where, v = $\frac{4Q}{\pi d^2}$

$\therefore$ H = 4fL x $(\frac{4Q}{\pi d^2})^2$ = $\frac{4\times16Q^2f}{\pi ^2 \times 2g}$ $(\frac{L}{d^5})$ ----- (2)

Head loss in compound pipe and in equivalent pipe is same hence equating equation (2) and (1)

$\therefore \frac{4\times16fQ^2}{\pi ^2\times2g}$ $(\frac{L_1}{d1^5} + \frac{L_2}{d2^5} + \frac{L_3}{d3^5}) = \frac{4\times16Q^2}{\pi ^2 \times 2g} (\frac{L}{d^5})$

$\therefore$ $\frac{L_1}{d1^5} + \frac{L_2}{d2^5} + \frac{L_3}{d3^5} = \frac{L}{d^5}$

Equation (3) is known as Dupuit's Equation where L = $L_1 + L_2 + L_3$ and $d_1, d_2, d_3$ are known. Hence equivalent size of pipe (d) can be found out.

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