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The engineering energy balance adds pump input, turbine extraction and irreversible head loss between two flow sections.
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Formula
All head terms in m; p in Pa; ρ in kg/m3; v in m/s; g in m/s2
| Symbol | Meaning | Unit |
|---|---|---|
| Static pressure at a section | Pa | |
| Kinetic-energy correction factor | dimensionless | |
| Mean flow speed | m/s | |
| Elevation above a datum | m | |
| Pump head added to the fluid | m | |
| Turbine head extracted | m | |
| Irreversible head loss | m |
\frac{p_1}{\rho g}+\alpha_1\frac{v_1^2}{2g}+z_1+h_p=\frac{p_2}{\rho g}+\alpha_2\frac{v_2^2}{2g}+z_2+h_t+h_LThis is a steady, incompressible control-volume energy balance between two sections. A pump adds energy, a turbine removes it, and friction or fittings contribute positive loss hL. The factor α accounts for a nonuniform velocity profile. With no machinery or losses and α≈1, it reduces to ideal Bernoulli along a streamline. Compressible, strongly unsteady or strongly three-dimensional flows require a different or extended model.
Steady incompressible flow between sections; model machinery, losses and velocity profiles appropriately.
Balance mechanical energy per unit weight between two sections.
Head loss
H includes pressure, velocity and elevation heads.
A 40 kPa pressure drop in water with other heads equal: find hL.
hL=40000/(1000×9.81)≈4.08 m.
Applying the ideal equation to a real pipe system.
Include machine and loss terms; check assumptions.
Formula cluster
Reynolds characterizes regime; the balance describes energy heads.
If velocity and elevation heads are equal at both sections, p1=300 kPa, p2=260 kPa, ρ=1000 kg/m3 and g=9.81 m/s2, the loss head is about 4.08 m.
Analyze pipe networks, pump sizing, turbines and pressure losses through fittings and pipe runs.
Here is how to work through a typical Extended Bernoulli energy equation with head loss (p₁/ρg + α₁v₁²/2g + z₁ + hp = p₂/ρg + α₂v₂²/2g + z₂ + ht + hL) task step by step:
Task
A 40 kPa pressure drop in water with other heads equal: find hL.
Solution path
hL=40000/(1000×9.81)≈4.08 m.
Ideal Bernoulli balances pressure, velocity and elevation heads without friction loss or machinery. The engineering energy equation adds pump head, turbine extraction and head loss between sections. Without those terms and with a suitable velocity-profile factor, it reduces to the ideal relation. The extension is usually essential for real pipe systems. Every term in the engineering equation is an energy head in metres, so pressure is divided by ρg. hL is defined as positive irreversible loss. Apply the form to suitable sections and steady incompressible cases. Friction and local-loss models may need pipe, fitting and flow data.
hL is defined as the magnitude of mechanical energy irreversibly lost per unit weight. Friction and fittings convert organized flow energy into internal energy; they do not create useful mechanical energy in a passive pipe system. Thus hL appears as a loss in the balance. A negative calculated value often signals a sign or model problem. Every term in the engineering equation is an energy head in metres, so pressure is divided by ρg. hL is defined as positive irreversible loss. Apply the form to suitable sections and steady incompressible cases. Friction and local-loss models may need pipe, fitting and flow data.
A pump transfers mechanical energy to the fluid and adds head hp. A turbine extracts energy, so head ht appears as an output. Check the sign convention of the source being used; signs must represent energy direction. Pump efficiency may be needed when converting shaft power into fluid head. Every term in the engineering equation is an energy head in metres, so pressure is divided by ρg. hL is defined as positive irreversible loss. Apply the form to suitable sections and steady incompressible cases. Friction and local-loss models may need pipe, fitting and flow data.
The ideal form typically assumes steady, incompressible flow, negligible friction and application along a streamline. It can be a first approximation for low-loss regions without pumps or turbines. Long pipes, valves or strong profile changes need loss and correction terms. Check that pressure, velocity and elevation are known at the selected sections. Every term in the engineering equation is an energy head in metres, so pressure is divided by ρg. hL is defined as positive irreversible loss. Apply the form to suitable sections and steady incompressible cases. Friction and local-loss models may need pipe, fitting and flow data.
If elevation and velocity heads are equal and no machine lies between sections, pressure drop corresponds to hL=Δp/(ρg). For water and 40 kPa, this is about 4.08 m. The shortcut fails when elevation or velocity changes; then include every term in the energy balance. Every term in the engineering equation is an energy head in metres, so pressure is divided by ρg. hL is defined as positive irreversible loss. Apply the form to suitable sections and steady incompressible cases. Friction and local-loss models may need pipe, fitting and flow data.
Quanta exam set
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Which formula describes Extended Bernoulli energy equation with head loss?
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How do you rearrange Extended Bernoulli energy equation with head loss for Head loss?
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Which common mistake happens with Extended Bernoulli energy equation with head loss?
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Create a curated FSRS exam set for p₁/ρg + α₁v₁²/2g + z₁ + hp = p₂/ρg + α₂v₂²/2g + z₂ + ht + hL: formula recall, variables, derivation, rearrangement, worked example, common mistakes and exam context.
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