# Problem 17 - Bernoulli's Energy Theorem

**Problem 17**

In Figure 4-08 is shown a siphon discharging water from reservoir A into the air at B. Distance 'a' is 1.8 m, 'b' is 6 m, and the diameter is 150 mm throughout. If there is a frictional loss of 1.5 m between A and the summit, and 1.5 m between the summit and B, what is the absolute pressure at the summit in kiloPascal? Also determine the rate of discharge in cubic meter per second and in gallons per minute.

**Solution 17**

$HL_{A-C} = 1.5 \, \text{ m}$

$HL_{C-B} = 1.5 \, \text{ m}$

Velocity head at B and C in terms of Q

$\dfrac{v^2}{2g} = \dfrac{8Q^2}{\pi^2gD^4} = \dfrac{8Q^2}{\pi^2(9.81)(0.15^4)}$

$\dfrac{v^2}{2g} = 163.21Q^2$

Energy Equation between A and B

$E_A - HL_{A-C} - HL_{C-B} = E_B$

$(0 + 0 + 0) - 1.5 - 1.5 = 163.21Q^2 + 0 - 4.2$

$163.21Q^2 = 1.2$

$Q = 0.0857 \, \text{ m}^3\text{/s}$ *answer*

$Q = 85.7 \dfrac{\text{Liters}}{\text{sec}} \times \dfrac{1 \text{gallon}}{3.78 \text{Liters}} \times \dfrac{60 \text{sec}}{1 \text{min}}$

$Q = 1360.32 \, \text{ gallons/min}$ *answer*

Thus, the velocity head at B and C is

$\dfrac{v^2}{2g} = 163.21(0.0857^2) = 1.2 \, \text{ m}$

Energy equation between A and C

$E_A - HL_{A-C} = E_C$

$(0 + 0 + 0) - 1.5 = 1.2 + \dfrac{p_C}{\gamma} - 4.2$

$\dfrac{p_C}{\gamma} = 1.5 \, \text{ m}$

$p_C = 1.5\gamma = 1.5(9.81)$

$p_C = 14.715 \, \text{ kPa}$ *answer*