Hydropower - Environmental Impacts of Renewable Energy - page 122

Solution: Using the equation A 1 V 1 = A 2 V 2 , we need to determine the area of each pipe:

Area ( A ) = π × ( D 2 /4)

12-in. pipe area = 3.14 × (1 ft 2 /4) = 0.785 ft 2

6-in. pipe area = 3.14 × (0.5 ft 2 /4) = 0.196 ft 2

The continuity equation now becomes:

0.785 ft 2 × 3 ft/s = 0.196 ft 2 × V 2

Solving for V 2 :

2

0.785 ft 3ft/s
0.196 ft

×

V 2 = = 12 ft/s (fps)

2

Pressure and Velocity
In a closed pipe flowing full (under pressure), the pressure is indirectly related to the
velocity of the liquid.

Velocity 1 × Pressure 1 = Velocity 2 × Pressure 2 (4.14)

or

V 1 P 1 = V 2 P 2

Piezometric Surface and Bernoulli's Theorem
The volume of water flowing past any given point in the pipe or channel per unit
time is called the flow rate or discharge —or just flow . The continuity of flow and the
continuity equation have been discussed (see Equation 4.13). Along with the conti-
nuity of flow principle and continuity equation, the law of conservation of energy,
piezometric surface, and Bernoulli's theorem (or principle) are also important to our
study of water hydraulics.

Conservation of Energy
Many of the principles of physics are important to the study of hydraulics. When
applied to problems involving the flow of water, few of the principles of physical
science are more important and useful to us than the law of conservation of energy .
Simply, the law of conservation of energy states that energy can be neither created
nor destroyed, but it can be converted from one form to another. In a given closed
system, the total energy is constant.

Energy Head
Hydraulic systems have two types of energy—kinetic and potential—and they
have three forms of mechanical energy—potential energy due to elevation, poten-
tial energy due to pressure, and kinetic energy due to velocity. Energy has the