Bernoulli, Energy Equation, Pipe Losses, and Pumps
Key Takeaways
- Use simple Bernoulli only when flow is steady, incompressible, along a streamline, and losses or shaft devices are negligible.
- Use the mechanical energy equation when a problem includes pump head, turbine head, head loss, pressure change, velocity change, or elevation change.
- Pipe-loss problems require the Darcy-Weisbach model, a Darcy friction factor, Reynolds-number judgment, and careful length-to-diameter units.
- Minor losses scale with velocity head and can dominate short systems with valves, elbows, entrances, and exits.
- Pump hydraulic power is tied to flow rate and head; motor input power must account for efficiency in the correct direction.
- NPSH questions are cavitation-avoidance questions, not ordinary pump-power questions.
Bernoulli is a special case
Bernoulli's equation is powerful because it converts pressure, velocity, and elevation into comparable energy-per-weight terms. The usual head form is pressure head plus velocity head plus elevation head. It is valid as a clean shortcut only for steady, incompressible, frictionless flow along a streamline with no shaft work between the two points. FE questions often describe a horizontal, frictionless nozzle or pipe section to invite this model.
If the problem mentions a pump, turbine, pipe friction, valve, long line, roughness, or loss coefficient, move to the mechanical energy equation. The structure is the same, but now pump head adds energy, turbine head removes energy, and losses remove energy. Do not force a loss problem into ideal Bernoulli just because pressure and velocity are given.
Head terms and signs
Head is length. Pressure head is p/gamma, velocity head is V^2/(2g), and elevation head is z. A pump adds h_p; a turbine extracts h_t; head loss h_L is always a positive loss term. The safest setup is to write the equation symbolically from point 1 to point 2 before substituting numbers.
| Term | Meaning | Common mistake |
|---|---|---|
p/gamma | pressure energy per unit weight | Mixing psi with ft of water without conversion |
V^2/(2g) | kinetic energy head | Forgetting area changes velocity through continuity |
z | elevation datum | Reversing uphill and downhill signs |
h_p | pump head added | Multiplying by efficiency in the wrong direction |
h_L | pipe and fitting losses | Treating losses as negative values inside a positive loss term |
Continuity is usually paired with energy. For incompressible flow, Q = AV, so a smaller pipe means larger velocity and larger velocity head. In many FE items, pressure change is not directly from area change; it comes from the energy balance after velocity and elevation are included.
Major and minor losses
Major loss in a round pipe is commonly modeled by Darcy-Weisbach: h_f = f (L/D) V^2/(2g). The friction factor in the FE Reference Handbook is the Darcy friction factor unless a problem explicitly says otherwise. Laminar pipe flow uses f = 64/Re. Turbulent flow needs a Moody chart or a correlation based on Reynolds number and relative roughness.
Minor losses use h_m = K V^2/(2g) for valves, elbows, entrances, exits, contractions, and expansions. Minor does not mean negligible. A compact system with several fittings can have more fitting loss than straight-pipe loss. Use the velocity associated with the fitting location unless the problem states a different convention.
Pumps, efficiency, and NPSH
Pump hydraulic power is rho g Q H in SI, where H is pump head. If pump efficiency is eta, shaft input power is hydraulic power divided by eta. For a turbine, useful output is hydraulic power times efficiency. This direction is a common trap: machines that consume power need more input than the fluid receives; machines that produce power deliver less output than the fluid loses.
Net positive suction head (NPSH) protects against cavitation at the pump inlet. NPSH available depends on suction pressure, vapor pressure, elevation, suction velocity, and suction-side losses. NPSH required comes from the pump manufacturer. The FE-style conclusion is usually qualitative: cavitation risk increases when suction pressure drops, temperature rises, vapor pressure rises, or suction losses increase.
Exam workflow
Draw two points, pick a datum, label pressure type, compute velocities from flow rate, then add losses and machinery. Only after the model is complete should you calculate. If the answer is off by a factor of 144, convert psi to psf or Pa correctly. If pump power seems too small, check whether flow is in gallons per minute but head and density were treated as SI.
A pipeline problem includes elevation change, pressure change, velocity change, a pump, and friction loss. Which model should be used?
Water is pumped at 0.020 m^3/s through a system requiring 25 m of pump head. If pump efficiency is 80%, what motor shaft power is closest?
In a short pipe system, four elbows and a partially open valve are present. What is the best treatment of these fittings in an FE pipe-loss calculation?