Summary
Highlights
The video introduces the final topic for the prelims in Hydraulics: energy and head in fluid flow, specifically focusing on Bernoulli's equation. Bernoulli's equation is defined as an energy equation relating to work done at a specific distance, composed of velocity head, pressure head, and elevation head, all measured in meters.
Kinetic energy is the ability of a fluid mass to do work by virtue of its velocity (KE = 0.5 * m * v^2). Potential energy is the energy possessed by a fluid due to its position or elevation (PE = mass * gravity * height) or the pressure it experiences. Both are present in moving fluids.
Head is defined as the amount of energy per unit weight (pound or Newton) of fluid. This explains why head units are in meters (Energy in Joules / Weight in Newtons = Newton-meters / Newtons = meters). The video quantifies kinetic head (velocity head) as V^2 / 2G, potential head (elevation head) as Z, and pressure head as P / γ (pressure over unit weight).
Bernoulli's energy theorem states that in steady, frictionless flow, total energy is constant at every point. This means energy at point 1 equals energy at point 2 (V1^2/2G + P1/γ + Z1 = V2^2/2G + P2/γ + Z2). When considering real-world scenarios with friction, head losses (HL) are introduced. These dissipate energy, hence being subtracted from the upstream energy or added to the downstream energy.
An example problem demonstrates calculating head loss, energy at point B, and energy at point A in a water pipeline with varying diameters, pressures, and elevations. The continuity equation (Q1 = Q2) is used to find velocities, and the generalized Bernoulli equation (including head loss) is applied. The solution details calculation of velocities, substitution into the Bernoulli equation, and solving for head loss and total energies at respective points.
Another example calculates the discharge rate from a hose attached to a hydrant, considering a friction factor and horizontal orientation. The Bernoulli equation is used, simplifying due to equal velocities and elevations, with head loss derived from the equation. The Darcy-Weisbach equation (HL = 0.0826 * f * L * Q^2 / D^5) is then employed to find the discharge (Q).
The previous problem is extended by attaching a 25mm nozzle. The calculation now includes head loss from both the hose (friction) and the nozzle itself (minor loss). The continuity equation, along with nozzle head loss formula (Vn^2/2G * (1/Cv^2 - 1)), helps determine the new head loss. This leads to calculating the velocity and consequent discharge if a nozzle is present.
The video introduces pumps and turbines in the energy equation. A pump increases the head (energy added, denoted as Ha), while a turbine extracts head (energy extracted, denoted as He). The impact of pumps (Ha is positive) and turbines (He is negative in Bernoulli's equation on the upstream side, or positive on the downstream side) on the energy equation is discussed. Power associated with pumps and turbines (Power = γ * Q * Energy) and their efficiencies are also covered.
An example with a pump details calculating discharge (Q) and input horsepower (HP). A pump adds 16 kW of power, and head losses are neglected. The generalized Bernoulli equation, including head added (Ha), is used. The power formula (Power = γ * Q * Ha) links the pump's power to the head added, allowing for the determination of Q. Input HP is then calculated considering the pump's efficiency.
The final example involves a turbine between two reservoirs. The elevation difference and a Hazen-Williams coefficient for head loss are given. The velocity and pressure heads are considered zero for reservoirs. Head loss is calculated using the Hazen-Williams formula. The Bernoulli equation (simplified for reservoirs) then determines the head extracted (He) by the turbine. Finally, the power generated by the turbine is calculated, considering its efficiency (P = γ * Q * He).