Head lost
$HL_{1-2} = 0$
$HL_{2-3} = 0.6 \, \text{ m}$
$HL_{3-4} = 9 \, \text{ m}$
$HL_{4-5} = 3 \, \text{ m}$
$HL_{1-5} = HL_{1-2} + HL_{2-3} + HL_{3-4} + HL_{4-5}$
$HL_{1-5} = 0 + 0.6 + 9 + 3$
$HL_{1-5} = 12.6 \, \text{ m}$
Energy equation between 1 and 5
$E_1 - HL_{1-5} = E_5$
$\dfrac{{v_1}^2}{2g} + \dfrac{p_1}{\gamma} + z_1 - HL_{1-5} = \dfrac{{v_5}^2}{2g} + \dfrac{p_5}{\gamma} + z_5$
$(0 + 0 + 30) - 12.6 = \dfrac{8Q^2}{\pi^2g{D_5}^4} + 0 + 0$
$17.4 = \dfrac{8Q^2}{\pi^2(9.81)(0.05^4)}$
$Q = 0.0363 \, \text{ m}^3\text{/s} = 36.3 \, \text{ L/s}$ answer
Velocity heads
$\dfrac{v^2}{2g} = \dfrac{8Q^2}{\pi^2gD^4}$
$\dfrac{{v_3}^2}{2g} = \dfrac{{v_4}^2}{2g} = \dfrac{8(0.0363^2)}{\pi^2(9.81)(0.15^4)} = 0.2151 \, \text{ m}$
$\dfrac{{v_5}^2}{2g} = \dfrac{8(0.0363^2)}{\pi^2(9.81)(0.05^4)} = 17.4202 \, \text{ m}$
Pressure heads
$\dfrac{p_2}{\gamma} = \text{El } 1 - \text{El } 2 = 30 - 25$
$\dfrac{p_2}{\gamma} = 5 \, \text{ m}$
Energy equation between 2 and 3
$E_2 - HL_{2-3} = E_3$
$\dfrac{{v_2}^2}{2g} + \dfrac{p_2}{\gamma} + z_2 - HL_{2-3} = \dfrac{{v_3}^2}{2g} + \dfrac{p_3}{\gamma} + z_3$
$(0 + 5 + 25) - 0.6 = 0.2151 + \dfrac{p_3}{\gamma} + 25$
$\dfrac{p_3}{\gamma} = 4.1849 \, \text{ m}$
Energy equation between 3 and 4
$E_3 - HL_{3-4} = E_4$
$\dfrac{{v_3}^2}{2g} + \dfrac{p_3}{\gamma} + z_3 - HL_{3-4} = \dfrac{{v_4}^2}{2g} + \dfrac{p_4}{\gamma} + z_4$
$(0.2151 + 4.1849 + 25) - 9 = 0.2151 + \dfrac{p_4}{\gamma} + 0$
$\dfrac{p_4}{\gamma} = 20.1849 \, \text{ m}$
Tabulated result
| Point |
Elevation head (m) |
Velocity head (m) |
Pressure head (m) |
Total head (m) |
| 1 |
30 |
0 |
0 |
30.0 |
| 2 |
25 |
0 |
5 |
30.0 |
| 3 |
25 |
0.2151 |
4.1849 |
29.4 |
| 4 |
0 |
0.2151 |
20.1849 |
20.4 |
| 5 |
0 |
17.4202 |
0 |
17.4 |
