# Problem 10 - Bernoulli's Energy Theorem

**Problem 10**

The diameter of a pipe carrying water changes gradually from 150 mm at A to 450 mm at B. A is 4.5 m lower than B. If the pressure at A is 70 kPa and that B is 50 kPa, when 140 L/s is flowing.

(a) Determine the direction of flow.

(b) Find the frictional loss between the two points.

**Solution 10**

Velocity heads

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

$\dfrac{{v_A}^2}{2g} = \dfrac{8(0.14^2)}{\pi^2(9.81)(0.15^4)} = 3.1990 \, \text{ m}$

$\dfrac{{v_B}^2}{2g} = \dfrac{8(0.14^2)}{\pi^2(9.81)(0.45^4)} = 0.0395 \, \text{ m}$

Pressure heads

$\dfrac{p_A}{\gamma} = \dfrac{70}{9.81} = 7.1356 \, \text{ m}$

$\dfrac{p_B}{\gamma} = \dfrac{50}{9.81} = 5.0968 \, \text{ m}$

Total head

$E = \dfrac{v^2}{2g} + \dfrac{p}{\gamma} + z$

$E_A = 3.1990 + 7.1356 + 0 = 10.3346 \, \text{ m}$

$E_B = 0.0395 + 5.0968 + 4.5 = 9.6363 \, \text{ m}$

The flow is always from higher energy to lower energy. E_{A} > E_{B}, thus, the flow will be from A to B. *answer*

Energy equation between A and B

$E_A - HL = E_B$

$10.3346 - HL = 9.6363$

$HL = 0.6983 \, \text{ m}$ *answer*

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