What equation does the tool solve?

It solves p/(ρg) + v²/(2g) + z = constant using g = 9.81 m/s².

Which downstream value can I calculate?

You can calculate downstream velocity v₂, pressure p₂, or elevation z₂.

What values are required?

Provide p₁, v₁, z₁, fluid density, and two of p₂, v₂, and z₂.

Which units are supported?

Density supports kg/m³ and g/cm³; pressure supports Pa, kPa, bar, atm, and psi; elevation supports m, cm, and ft; velocity uses m/s.

What happens if the velocity calculation has a negative radicand?

The tool rejects it because a negative radicand is physically impossible for the velocity formula.

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Bernoulli Equation Solver (Velocity / Pressure / Elevation)

Solve the Bernoulli equation p/(ρg) + v²/(2g) + z = const for steady, ideal, incompressible flow between two points on a streamline. Supply all three upstream quantities (p₁, v₁, z₁) and any two of the three downstream quantities; the tool solves for the third. Solve velocity: v₂ = √[v₁² + 2(p₁-p₂)/ρ + 2g(z₁-z₂)] — a negative radicand is physically impossible and is rejected. Solve pressure: p₂ = p₁ + ½ρ(v₁²-v₂²) + ρg(z₁-z₂). Solve elevation: z₂ = z₁ + (p₁-p₂)/(ρg) + (v₁²-v₂²)/(2g). g = 9.81 m/s². Density in kg/m³/g/cm³, pressure in Pa/kPa/bar/atm/psi, velocity in m/s, elevation in m/cm/ft. The upstream pressure/velocity/elevation heads and the total head are also reported.

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2 examples

Venturi-style: solve v₂, water ρ=1000, p₁=200000 Pa, v₁=2 m/s, z₁=0, p₂=100000 Pa, z₂=0 -> v₂≈14.2829 m/s

Solve velocity. Water ρ=1000 kg/m³. p₁=200000 Pa, v₁=2 m/s, z₁=0, p₂=100000 Pa, z₂=0. v₂=√[2²+2(200000-100000)/1000+2g(0-0)]=√[4+200]=√204≈14.2829 m/s (per code).

{"result":"Bernoulli equation (velocity solved). ρ=1000 kg/m³. p₁=200000 Pa, v₁=2 m/s, z₁=0 m. v₂ = 14.2829 m/s. Upstream heads: pressure=20.3874 m, velocity=0.2039 m, elevation=0 m, total=20.5912 m.","metadata":{"input":{"unknownVar":"velocity","density":1000,"densityUnit":"kg/m3","p1":200000,"v1":2,"z1":0,"pressureUnit":"Pa","velocityUnit":"m/s","lengthUnit":"m","p2":100000,"z2":0},"result":{"unknownVar":"velocity","solved":{"value_SI":14.2829,"unit_SI":"m/s","value_input":14.2829,"inputUnit":"m/s"},"p1_Pa":200000,"v1_m_s":2,"z1_m":0,"p2_Pa":100000,"v2_m_s":14.2829,"z2_m":0,"density_kg_m3":1000,"gravity_m_s2":9.81,"upstreamHeads_m":{"pressureHead":20.3874,"velocityHead":0.2039,"elevationHead":0,"totalHead":20.5912}}}}

Solve elevation: water ρ=1000, p₁=0 Pa, v₁=0 m/s, z₁=10 m, p₂=0 Pa, v₂=0 m/s -> z₂=10 m (equal head)

Solve elevation. Water ρ=1000 kg/m³. p₁=0 Pa, v₁=0 m/s, z₁=10 m, p₂=0 Pa, v₂=0 m/s. z₂=z₁+(p₁-p₂)/(ρg)+(v₁²-v₂²)/(2g)=10+0+0=10 m.

{"result":"Bernoulli equation (elevation solved). ρ=1000 kg/m³. p₁=0 Pa, v₁=0 m/s, z₁=10 m. z₂ = 10 m (10 m). Upstream heads: pressure=0 m, velocity=0 m, elevation=10 m, total=10 m.","metadata":{"input":{"unknownVar":"elevation","density":1000,"densityUnit":"kg/m3","p1":0,"v1":0,"z1":10,"pressureUnit":"Pa","velocityUnit":"m/s","lengthUnit":"m","p2":0,"v2":0},"result":{"unknownVar":"elevation","solved":{"value_SI":10,"unit_SI":"m","value_input":10,"inputUnit":"m"},"p1_Pa":0,"v1_m_s":0,"z1_m":10,"p2_Pa":0,"v2_m_s":0,"z2_m":10,"density_kg_m3":1000,"gravity_m_s2":9.81,"upstreamHeads_m":{"pressureHead":0,"velocityHead":0,"elevationHead":10,"totalHead":10}}}}

Inputs

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13 options

SettingsAdjust formats, ranges, numbers, and modes.13

Unknown to SolveselectOptionalWhich downstream quantity to solve for: velocity v₂, pressure p₂, or elevation z₂. Provide the other two downstream quantities.Fluid Density ρnumberRequiredFluid density ρ. Enter in the selected Density Unit.Density UnitselectOptionalUpstream Pressure p₁numberRequiredUpstream (point 1) pressure p₁. Enter in the selected Pressure Unit.Upstream Velocity v₁numberRequiredUpstream (point 1) velocity v₁ in m/s.Upstream Elevation z₁numberRequiredUpstream (point 1) elevation z₁. Enter in the selected Length Unit.Pressure UnitselectOptionalVelocity UnitselectOptionalLength UnitselectOptionalDownstream Pressure p₂ (leave blank if solving p₂)numberOptionalDownstream (point 2) pressure p₂. Enter in the selected Pressure Unit. Required unless solving for p₂.Downstream Velocity v₂ (leave blank if solving v₂)numberOptionalDownstream (point 2) velocity v₂ in m/s. Required unless solving for v₂.Downstream Elevation z₂ (leave blank if solving z₂)numberOptionalDownstream (point 2) elevation z₂. Enter in the selected Length Unit. Required unless solving for z₂.Decimal PlacesnumberOptional

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Bernoulli Equation Solver (Velocity / Pressure / Elevation)

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Examples

1. Venturi-style downstream velocity

2. Equal-head elevation check

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3. Downstream pressure from velocity increase