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Bernoulli's equation relates the pressure P, flow velocity v, and elevation h at two points along a streamline in an ideal incompressible fluid of density ρ. Misapplication of this principle causes systematic errors in pipe sizing, ventilation design, and pitot-tube airspeed readings. The equation assumes steady, inviscid, irrotational flow along a single streamline with constant density. It does not account for viscous losses, turbulence, or compressibility effects above Mach 0.3. For real engineering problems, friction losses require the extended Darcy-Weisbach correction.

This calculator solves for any one unknown among the seven variables (P₁, v₁, h₁, ρ, P₂, v₂, h₂) given the other six. Gravitational acceleration is fixed at g = 9.80665 m/s² per ISO 80000-3. All inputs use SI units: pressure in Pa, velocity in m/s, height in m, density in kg/m³. Note: negative pressure results indicate gauge pressure below atmospheric reference and may signal a physically unrealizable condition in your setup.

Formulas

Bernoulli's equation for steady, incompressible, inviscid flow along a streamline:

P₁ + 12ρv₁² + ρgh₁ = P₂ + 12ρv₂² + ρgh₂

Each term has dimensions of pressure (Pa = N/m²). The static pressure term P represents the thermodynamic pressure of the fluid. The dynamic pressure term 12ρv² captures kinetic energy per unit volume. The hydrostatic pressure term ρgh accounts for gravitational potential energy per unit volume.

Solving for velocity at point 2:

v₂ = √2(P₁ − P₂ + 12ρv₁² + ρg(h₁ − h₂))ρ

Where P₁, P₂ = static pressure at points 1 and 2 (Pa); v₁, v₂ = flow velocity (m/s); h₁, h₂ = elevation above datum (m); ρ = fluid density (kg/m³); g = 9.80665 m/s² (standard gravitational acceleration per ISO 80000-3).

Reference Data

Fluid	Density ρ (kg/m³)	Condition	Typical Application
Air (dry)	1.225	Sea level, 15°C	HVAC, aerodynamics
Air (dry)	0.4135	Altitude 10000m	Aviation
Fresh Water	998.2	20°C, 1atm	Plumbing, hydraulics
Seawater	1025	15°C	Marine engineering
Mercury	13546	20°C	Manometers, barometers
Ethanol	789	20°C	Chemical processing
Gasoline	720	15°C	Fuel systems
Diesel	850	15°C	Fuel injection
Glycerin	1261	25°C	Pharmaceutical, cosmetics
Crude Oil	870	Average, 15°C	Pipeline transport
Kerosene (Jet A)	804	15°C	Aviation fuel
Acetone	784	25°C	Solvent processing
Olive Oil	913	25°C	Food processing
Helium	0.164	Sea level, 20°C	Leak testing, balloons
Carbon Dioxide	1.842	Sea level, 20°C	Carbonation, fire suppression
Nitrogen	1.165	Sea level, 20°C	Inerting, purging
Propane (liquid)	493	25°C	LPG systems
Sulfuric Acid	1840	98%, 25°C	Chemical industry
Blood (human)	1060	37°C	Biomedical engineering
Honey	1420	25°C	Food processing

Frequently Asked Questions

A negative value under the square root when solving for velocity means the specified pressure and elevation conditions are physically impossible for the given fluid. The total energy at one point would need to exceed what is available at the other point. This typically indicates an input error, or that external work (a pump) is required. The calculator will report this condition explicitly rather than returning an imaginary number.

No. Bernoulli's equation applies to ideal, inviscid flow. For real pipe systems with friction, you must add the Darcy-Weisbach head loss term: h_f = f ⋅ LD ⋅ v²2g, where f is the Darcy friction factor, L is pipe length, and D is pipe diameter. For turbulent flows use the Moody chart or Colebrook equation.

The incompressible form of Bernoulli's equation is valid for Mach numbers below approximately 0.3. For air at sea level and 15°C, the speed of sound is about 340 m/s, so the limit is roughly 102 m/s (367 km/h). Above this threshold, density changes become significant and you must use the compressible Bernoulli equation or isentropic flow relations.

No. Bernoulli's equation requires a single, homogeneous fluid with uniform density ρ along the entire streamline between the two points. If the fluid changes (e.g., oil flowing into water), the equation does not apply. Each fluid segment must be analyzed separately with appropriate boundary conditions at the interface.

Elevation h is measured upward from an arbitrary horizontal datum. A point above the datum has positive h; below the datum, negative h. The absolute datum does not matter because only the difference h₁ − h₂ enters the equation. For horizontal flow at a constant elevation, set both to 0.

When solving for ρ, the equation becomes quadratic because density appears in both the kinetic term (12ρv²) and the hydrostatic term (ρgh). The calculator solves the resulting quadratic and selects only the physically meaningful positive root. If both roots are negative or complex, the input conditions are unrealizable.