Basic Hydraulics: Energy and Momentum concepts

Energy of flow Three kinds of energy gradients cause flow Elevation (called potential energy) Pressure (another kind of potential) Kinetic (related to how fast water is moving) 1 2 p 1, v 1 p 2, v 2 Elevation 1 Elevation 2

Pressure Pressure at point = p = h For US customary units,  = 62.4 lb/ft 3 Example: At point 1, p 1 = h 1 At bottom of tank, p bottom = h bottom Pressure energy = h h bottom 1 h1h1

Potential and Kinetic Energy Potential energy is the sum of the elevation head and the pressure head Sometimes called the static head Kinetic energy is the energy of motion Proportional to the square of the mean section velocity The sum of potential and kinetic energy is the total energy (head).

Total energy Express energy in consistent units. Elevation (h) has units of ft. Pressure has units of lb/ft 2. If we divide p by  (62.4 lb/ft 3 ), we get units of L for the pressure term. Velocity has units of ft/sec. Energy related (velocity) 2. Measure of velocity energy consistent with other energy units is v 2 /2g where g = gravitational acceleration. These energy terms referred to as “head”. Total energy (head) = h + p/ + v 2 /2g

Bernoulli Equation If friction losses are neglected and no energy is added to, or taken from a piping system, the total head, H, which is the sum of the elevation head, the pressure head and the velocity head will be constant for any point on a fluid streamline. This expression head conservation of head in a conduit or streamtube is known as the Bernoulli equation: where is: Z 1,2 - elevation above reference level; p 1,2 - absolute pressure; v 1,2 - velocity; ρ 1,2 - density; g - acceleration of gravity

Energy losses Due to Boundary resistance (friction losses) Effects of changes in flow geometry (local losses) Local losses often expressed as h L = K v 2 /2g in which K = the head loss coefficient Friction losses commonly computed using empirical equation, such as Manning’s equation, Chezy equation, Darcy-Weisbach equation or Hazen- Williams (water only!)

Conservation of energy Total energy at a point = E = h + p/ + V 2 /2g Between any two points in the flow E 1 = E 2 + h L1-2 where h L1-2 = energy “loss” between locations So h 1 + p 1 / + V 1 2 /2g = h 2 + p 2 / + V 2 2 /2g + h L1-2

Energy Equation If friction losses are included, the equation is called the energy equation Turbine extraction is probably uncommon for transportation infrastructure, but the other two (pumps and friction) are common Added head (pump) Extracted head (turbine) Frictional Loss

Momentum Concept Momentum is defined as mass of object multiplied by velocity of object Dealing with momentum is more difficult than dealing with mass and energy because momentum is vector quantity, having both magnitude and direction.

Momentum Concept Show momentum as 3 component equations

Momentum Concept Thrust block example

Momentum Concept Force on a pier