# Problem 07 - Bernoulli's Energy Theorem

**Problem 7**

Compute the velocity head of the jet in Figure 4-03 if D_{1} = 75 mm, D_{2} = 25 mm, the pressure head at 1 is 30 m of the liquid flowing, and the lost head between points 1 and 2 is 5 percent of the velocity head at point 2.

**Solution 7**

$\dfrac{v^2}{2g} = \dfrac{8Q^2}{\pi^2gD^4}$

$\dfrac{{v_1}^2}{2g} = \dfrac{8Q^2}{\pi^2(9.81)(0.075^4)} = 2611.42Q^2$

$\dfrac{{v_2}^2}{2g} = \dfrac{8Q^2}{\pi^2(9.81)(0.025^4)} = 211~524.75Q^2$

Head lost

$HL = 0.05\left( \dfrac{{v_2}^2}{2g} \right) = 0.05(211~524.75Q^2)$

$HL = 10576.24Q^2$

Energy equation between 1 and 2

$E_1 - HL = E_2$

$\dfrac{{v_1}^2}{2g} + \dfrac{p_1}{\gamma_w} + z_1 - HL = \dfrac{{v_2}^2}{2g} + \dfrac{p_2}{\gamma_w} + z_2$

$2611.42Q^2 + 30 + 0 - 10576.24Q^2 = 211~524.75Q^2 + 0 + 0$

$219~489.57Q^2 = 30$

$Q = 0.01169 \, \text{ m}^3\text{/s}$

Velocity head at point 2

$\dfrac{{v_2}^2}{2g} = 211~524.75(0.01169^2)$

$\dfrac{{v_2}^2}{2g} = 28.91 \, \text{ m}$ *answer*