How to use it
- Choose the unknown in the Find list.
- Enter the fluid density: 1000 kg/m³ for water, about 1.225 kg/m³ for air at sea level.
- Enter the other values: pressures in kPa, speeds in m/s, heights in metres from any common reference level.
- Read the result and the dynamic pressures and head below it.
Formula
P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂, with g = 9.80665 m/s². Solved for each unknown:
- P₂ = P₁ + ½ρ(v₁² − v₂²) + ρg(h₁ − h₂)
- v₂ = √(v₁² + 2(P₁ − P₂) ÷ ρ + 2g(h₁ − h₂))
- h₂ = h₁ + (P₁ − P₂) ÷ (ρg) + (v₁² − v₂²) ÷ (2g)
and the same with 1 and 2 swapped. The dynamic pressure is ½ρv², the total head is the total energy per volume divided by ρg: the height of fluid column it is worth.
Worked example
Water flows at 2 m/s with 200 kPa through a pipe at height 0. Further on the pipe narrows so the water moves at 4 m/s, and it is 1 m higher. Pressure there:
P₂ = 200,000 + ½ × 1000 × (2² − 4²) + 1000 × 9.80665 × (0 − 1) = 200,000 − 6000 − 9806.65 = 184.193 kPa.
The dynamic pressure grows from 2 kPa to 8 kPa; the total stays at 202 kPa, a head of 20.5983 m.
Tips
- Speed from pipe sizes: if you know the speed in one section, continuity gives the other, v₂ = v₁ × A₁ ÷ A₂.
- Heights are relative. Only the difference h₁ − h₂ matters, so measure both from the same level.
- Watch low pressures. With absolute pressures a negative result is impossible; in reality a liquid starts to boil (cavitate) once its pressure falls to the vapour pressure.