ENGG2500 Chap.4 Bernoulli and the General Energy Equation
Bernoulli and the General Energy Equation
Bernoulli's equation is a restricted mechanical-energy relation for steady, incompressible flow with negligible viscous dissipation and no shaft work between chosen points. The general energy equation retains pump head, turbine head and losses. Hydraulic and energy grade lines turn the terms into a system picture, while cavitation checks require absolute pressure and comparison with vapour pressure.
Point selection is part of the modelling: reservoir surfaces, free jets and pipe sections expose different known pressure, velocity and elevation terms, and the chosen datum must remain consistent through the calculation.
What this chapter covers
- 01
Bernoulli assumptions
- 02
Pressure, velocity and elevation heads
- 03
Hydraulic and energy grade lines
- 04
Pump and turbine head
- 05
Major and local energy losses
- 06
Pitot tubes, flowmeters and cavitation
Worked example · free
Pumping between reservoirs
- 1Choose both reservoir surfaces as energy points. This weighting is not an official assessment scheme.
- 1Cancel equal gauge pressures and negligible surface speeds.
- 1Pump head is 18 + 6.5 = 24.5 m.
- 1Hydraulic power is 4.80 kW.
Key terms
- Velocity head
- Velocity head is kinetic energy per unit weight, V squared over two g.
- Pump head
- Pump head is mechanical energy added to fluid per unit weight.
- Head loss
- Head loss is mechanical energy dissipated per unit weight.
- Cavitation
- Cavitation is vapour formation when local absolute pressure approaches vapour pressure.
Bernoulli and the General Energy Equation FAQ
When is Bernoulli's equation valid?
Use it along an appropriate streamline when the flow is steady, density is effectively constant, viscous loss is negligible and no pump or turbine transfers shaft work between the selected points. If those conditions fail, use the general energy equation or add the missing terms explicitly.
What is the difference between the grade lines?
The hydraulic grade line contains elevation head plus pressure head. The energy grade line adds velocity head, so their separation represents speed. Both usually fall with loss in the flow direction, while a pump produces an upward jump and a turbine produces a downward jump.
Why must cavitation use absolute pressure?
Vapour pressure is measured from a true vacuum, so it is an absolute quantity. A gauge pressure can be negative while the absolute pressure remains positive. Convert to a common absolute reference before deciding whether local pressure is low enough for vapour bubbles to form.
How do I choose the two energy points?
Choose locations where pressure, velocity or elevation is known or requested and where the assumptions apply along the connecting path. Large open reservoir surfaces often remove pressure and velocity unknowns, while pipe taps retain them. The choice must represent the actual pump, turbine and loss elements between the points.
What sign should head loss have?
Define head loss as a positive amount of mechanical energy dissipated per unit weight. In a balance written from point one to point two, subtract that positive loss on the left or add it to the required upstream head. A negative solved loss usually signals a reversed flow assumption or missing machine.
How is hydraulic power different from shaft power?
Hydraulic power is density times gravity times flow rate times head transferred to or from the fluid. A pump needs more shaft input than the fluid receives, while a turbine delivers less shaft output than the fluid loses. Apply efficiency only after identifying which of those powers is requested.
Why can pressure rise in an enlarging pipe?
An area increase reduces average speed for steady incompressible flow, so some velocity head can convert into pressure head. Total mechanical head still falls when losses are present. A pressure recovery therefore does not violate energy conservation, and the energy grade line should not rise through a passive enlargement.
How do I detect a wrong energy-balance sign?
Sketch the energy and hydraulic grade lines after solving. A passive pipe should lose total head in the flow direction, a pump should raise it and a turbine should lower it. If the computed picture does the opposite, recheck flow direction, pressure reference and the signs assigned to machines and losses.
Exam move
Build an energy ledger before entering numbers. Draw the system, select two sections, set one elevation datum and list pressure, velocity and elevation head at both. Mark every pump, turbine and loss element between them. Write the general energy equation first, then cancel equal atmospheric pressures, negligible reservoir speeds or absent machines only with a stated reason.
Keep pump head positive into the fluid, turbine head positive out of it and head loss positive as dissipation. Use continuity separately when a diameter change makes a section velocity unknown. After solving, sketch qualitative hydraulic and energy grade lines: their separation must equal velocity head, passive lines fall in the flow direction, and machines create discrete jumps.
Convert head to pressure or power only once the head balance closes. For cavitation, restore absolute pressure and compare the lowest local value with vapour pressure. This sequence prevents reference, sign and efficiency errors from being hidden inside one substitution line.