Data Source and Methodology
Authoritative Source: Munson, Young and Okiishi’s Fundamentals of Fluid Mechanics, 8th Edition (2016), Wiley. See publisher page: Wiley — Fundamentals of Fluid Mechanics.
Tutti i calcoli si basano rigorosamente sulle formule e sui dati forniti da questa fonte.
This tool implements the steady, incompressible Bernoulli energy equation between two points, with an optional head loss term, and performs unit-consistent calculations in double precision.
The Formula Explained
Core energy balance with head loss:
$$ \frac{P_1}{\rho g} + \frac{v_1^2}{2g} + z_1 \;=\; \frac{P_2}{\rho g} + \frac{v_2^2}{2g} + z_2 \;+\; h_L $$
Rearranged to solve for commonly needed unknowns:
$$ P_2 = P_1 + \tfrac{1}{2}\rho\left(v_1^2 - v_2^2\right) + \rho g \left(z_1 - z_2 - h_L\right) $$
$$ v_2 = \sqrt{\,v_1^2 + 2g\!\left( \frac{P_1 - P_2}{\rho g} + z_1 - z_2 - h_L \right)} $$
$$ z_2 = \frac{P_1 - P_2}{\rho g} + \frac{v_1^2 - v_2^2}{2g} + z_1 - h_L $$
Glossary of Variables
- P1, P2 — Pressure at point 1 and 2 (Pa, kPa, MPa, bar, or psi)
- v1, v2 — Average velocity at point 1 and 2 (m/s or ft/s)
- z1, z2 — Elevation head at point 1 and 2 (m or ft)
- ρ — Fluid density (kg/m³ or slug/ft³)
- g — Gravitational acceleration (m/s² or ft/s²)
- hL — Head loss between points (m or ft)
- Heads — Pressure head P/(ρg), velocity head v²/(2g), elevation head z
Worked Example
How it works: a step-by-step example
Given water (ρ = 1000 kg/m³), g = 9.80665 m/s², P1 = 200 kPa, v1 = 2 m/s, z1 = 2 m, z2 = 0 m, v2 = 5 m/s, and head loss hL = 1 m. Solve for P2.
Use: $$P_2 = P_1 + \tfrac{1}{2}\rho\left(v_1^2 - v_2^2\right) + \rho g (z_1 - z_2 - h_L)$$
Compute terms: 0.5·ρ·(v1² − v2²) = 0.5·1000·(4 − 25) = −10500 Pa; ρg(z1 − z2 − hL) = 1000·9.80665·(2 − 0 − 1) = 9806.65 Pa.
Therefore P2 = 200000 − 10500 + 9806.65 ≈ 199306.65 Pa ≈ 199.31 kPa.
Frequently Asked Questions (FAQ)
When is Bernoulli applicable?
For steady, incompressible, inviscid flow along a streamline with no pumps/turbines between the two points. Losses can be lumped into hL.
How do I include frictional losses?
Estimate hL using Darcy–Weisbach with friction factor and minor loss coefficients, then input that value in the head loss field.
Are the results gauge or absolute?
They match your input reference. Use the same reference for P1 and P2 to ensure consistency.
What if the computed velocity is not real?
If the square-root argument is negative, the inputs violate energy balance. Re-check pressures, elevations, and head loss.
Does the calculator support US units?
Yes. Switch the unit system or choose units per field. All calculations are performed with consistent base units internally.
Can I solve for upstream variables too?
Yes. Choose P1, v1, or z1 in “Solve for” to compute the corresponding upstream unknown.
What precision is used?
Double precision internally, with results displayed to a practical number of significant digits. You can change units to see different scales.