What is Bernoulli's equation?

For steady, incompressible, frictionless flow along a streamline, P + ½ρv² + ρgz is constant. Pressure, kinetic energy, and potential energy per unit volume can trade off, but their sum does not change. With friction, a head loss term is subtracted from the downstream side.

How do I find the pressure after a pipe reducer?

Use continuity to find the new velocity, v₂ = v₁(D₁/D₂)², then apply Bernoulli. Water at 2 m/s and 200 kPa going from 100 mm to 50 mm reaches 8 m/s, and the pressure falls to about 170 kPa.

Does Bernoulli's equation work with friction?

Not by itself. The basic form assumes no losses. Add a head loss term hL, from Darcy-Weisbach and minor loss coefficients, to account for friction in a real pipe. The calculator takes hL as an optional input.

Should I use gauge or absolute pressure?

Either, as long as you use the same one at both points. The calculator warns when a pressure falls below a perfect vacuum, which signals a mix of bases or a flow that is not physically possible.

Can I use Bernoulli's equation across a pump?

Not directly. A pump adds energy, so the equation needs the pump head added to the upstream side. The pump power calculator relates that head to the power required.

Bernoulli Equation Calculator

Solve Bernoulli's equation between two points for an unknown pressure, velocity, or elevation, with continuity for diameter changes and optional head loss.

Scratchpad (not saved)

Save to create a persistent, shareable workspace.

Solve For

Fluid (Density ρ)

Custom Density ρ

Used only when the fluid is set to Custom.

Custom Density Unit

Unit for the custom density.

Pressure at Point 1, P1

Gauge or absolute, matching point 2. Ignored when solving for P1.

Pressure at Point 2, P2

Used when solving for v2, z2, or P1. Ignored when solving for P2.

Pressure Unit

Unit for both pressures.

Velocity at Point 1, v1

Mean velocity at point 1.

Velocity at Point 2, v2

Used when no pipe diameters are given and you are not solving for v2.

Velocity Unit

Unit for both velocities.

Pipe Diameter at Point 1, D1 (optional)

Enter both diameters to get v2 from continuity. Use 0 to skip.

Pipe Diameter at Point 2, D2 (optional)

Enter both diameters to get v2 from continuity. Use 0 to skip.

Diameter Unit

Unit for both diameters.

Elevation at Point 1, z1

Height above any common datum.

Elevation at Point 2, z2

Used when not solving for z2.

Elevation Unit

Unit for elevations and head loss.

Head Loss hL (optional)

Friction and fitting losses between the points, as a head of fluid. Use 0 to ignore.

What This Calculator Does

Bernoulli's equation is an energy balance for a flowing fluid: along a streamline, the sum of pressure, kinetic, and potential energy per unit volume stays constant, less any losses. This calculator applies it between two points. Choose the unknown (the pressure at point 2, the velocity at point 2, the elevation at point 2, or the pressure at point 1), and enter the rest. If you give both pipe diameters, the velocity at point 2 comes from continuity, and an optional head loss $h_L$ accounts for friction between the points. Use gauge pressure at both points, or absolute at both, but not a mix.

It combines Solve For, Fluid (Density ρ), Custom Density ρ, Custom Density Unit to estimate Pressure at point 2, P2, Velocity at point 2, v2, Pressure change P2 − P1.

Formula & Method

Bernoulli equation with head loss:

P_1 + \tfrac{1}{2}\rho v_1^2 + \rho g z_1 = P_2 + \tfrac{1}{2}\rho v_2^2 + \rho g z_2 + \rho g h_L

Continuity for incompressible flow:

v_2 = v_1\left(\frac{D_1}{D_2}\right)^2

Solved for the pressure at point 2:

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)

Solved for the velocity at point 2:

v_2 = \sqrt{v_1^2 + \frac{2\left(P_1 - P_2\right)}{\rho} + 2g\left(z_1 - z_2 - h_L\right)}

where

P

is pressure,

v

is mean velocity,

z

is elevation,

\rho

is density,

g = 9.80665

m/s²,

D

is pipe inside diameter, and

h_L

is the head loss between the points, in metres of fluid.

Notation used in the formulas: $R$ = Pressure at point 2, P2; $x_{1}$ = Solve For; $x_{2}$ = Fluid (Density ρ); $x_{3}$ = Custom Density ρ; $x_{4}$ = Custom Density Unit; $x_{5}$ = Pressure at Point 1, P1; $x_{6}$ = Pressure at Point 2, P2.

Method summary: inputs are normalized to consistent units, core equations are evaluated, then secondary values are derived and rounded for display.

Use it for steady, incompressible flow between two points on the same streamline or in the same pipe: pressure change through a reducer or nozzle, discharge speed from a tank, or the height a jet can reach. When a pump or turbine sits between the points, add its head to the equation or use the pump power calculator.

Worked Examples

Pressure after a reducer, 100 mm to 50 mm (metric)

Given: water ρ = 998 kg/m³, P1 = 200 kPa, v1 = 2 m/s, D1 = 100 mm, D2 = 50 mm, z1 = z2, hL = 0 Method: continuity gives v2, then Bernoulli gives P2. v2 = 2 × (100/50)² = 8 m/s P2 = 200,000 + ½ × 998 × (2² − 8²) + 0 = 200,000 − 29,940 = 170,060 Pa P2 = 170.06 kPa. The velocity gain of 6 m/s costs about 30 kPa of pressure.

Reducer that also climbs 20 ft (imperial)

Given: water, P1 = 30 psi (gauge), v1 = 6 ft/s in a 4 in pipe, 2 in pipe at a point 20 ft higher, hL = 0 Method: continuity, then Bernoulli with the elevation term in SI units. v2 = 6 × (4/2)² = 24 ft/s = 7.315 m/s; v1 = 1.829 m/s; Δz = 6.096 m P2 = 206,843 + ½ × 998 × (1.829² − 7.315²) − 998 × 9.80665 × 6.096 P2 = 206,843 − 25,034 − 59,662 = 122,147 Pa ≈ 17.7 psi (gauge) Roughly 12.3 psi of the 30 psi is converted to velocity and elevation.

Velocity Head and Dynamic Pressure for Water

Each term of Bernoulli's equation can be expressed as a pressure or as a head. The values below are for water at 998 kg/m³.

Velocity (m/s)	Dynamic pressure ½ρv² (kPa)	Velocity head v²/2g (m)
1	0.499	0.051
2	1.996	0.204
3	4.491	0.459
5	12.48	1.275
8	31.94	3.263
10	49.90	5.099

Both columns grow with the square of the velocity, so doubling the speed quadruples the pressure needed to produce it.

Reference

The energy equation for steady incompressible flow is derived in Çengel and Cimbala, Fluid Mechanics, and White, Fluid Mechanics. Munson, Young, and Okiishi, Fundamentals of Fluid Mechanics, give the extended form with head loss and shaft work.

Common Mistakes

Mixing gauge and absolute pressure. Use the same basis at both points.
Applying Bernoulli across a pump, turbine, or fan without adding the shaft work. The simple equation assumes no energy is added or removed.
Ignoring friction in a long pipe. Over real lengths the head loss can exceed the other terms, so include $h_L$ from the pipe head loss calculator.
Using it for compressible gas flow at high speed. It holds for gases only when the density change is small, roughly below Mach 0.3.
Entering a diameter at only one of the two points. Continuity needs both; otherwise enter $v_2$ directly.
Forgetting that $h_L$ is a head in metres, not a pressure. A pressure loss must be divided by $\rho g$ first.

When Not To Use This Calculator

Compressible flow at high Mach number. Use isentropic flow relations.
Unsteady flow, such as water hammer or a draining tank at the moment of a valve change.
Flow across a pump or turbine, unless you add the machine's head explicitly.
Flow across a shock, a hydraulic jump, or a sudden expansion where mechanical energy is lost in mixing. Use momentum or the loss coefficient.
Points that are not on the same streamline in a rotational flow.

Frequently Asked Questions

What is Bernoulli's equation?: For steady, incompressible, frictionless flow along a streamline, P + ½ρv² + ρgz is constant. Pressure, kinetic energy, and potential energy per unit volume can trade off, but their sum does not change. With friction, a head loss term is subtracted from the downstream side.
How do I find the pressure after a pipe reducer?: Use continuity to find the new velocity, v₂ = v₁(D₁/D₂)², then apply Bernoulli. Water at 2 m/s and 200 kPa going from 100 mm to 50 mm reaches 8 m/s, and the pressure falls to about 170 kPa.
Does Bernoulli's equation work with friction?: Not by itself. The basic form assumes no losses. Add a head loss term hL, from Darcy-Weisbach and minor loss coefficients, to account for friction in a real pipe. The calculator takes hL as an optional input.
Should I use gauge or absolute pressure?: Either, as long as you use the same one at both points. The calculator warns when a pressure falls below a perfect vacuum, which signals a mix of bases or a flow that is not physically possible.
Can I use Bernoulli's equation across a pump?: Not directly. A pump adds energy, so the equation needs the pump head added to the upstream side. The pump power calculator relates that head to the power required.

Inputs Used

Solve For: Used directly in the calculation.
Fluid (Density ρ): Used directly in the calculation.
Custom Density ρ: Used only when the fluid is set to Custom.
Custom Density Unit: Unit for the custom density.
Pressure at Point 1, P1: Gauge or absolute, matching point 2. Ignored when solving for P1.
Pressure at Point 2, P2: Used when solving for v2, z2, or P1. Ignored when solving for P2.
Pressure Unit: Unit for both pressures.
Velocity at Point 1, v1: Mean velocity at point 1.
Velocity at Point 2, v2: Used when no pipe diameters are given and you are not solving for v2.
Velocity Unit: Unit for both velocities.
Pipe Diameter at Point 1, D1 (optional): Enter both diameters to get v2 from continuity. Use 0 to skip.
Pipe Diameter at Point 2, D2 (optional): Enter both diameters to get v2 from continuity. Use 0 to skip.
Diameter Unit: Unit for both diameters.
Elevation at Point 1, z1: Height above any common datum.
Elevation at Point 2, z2: Used when not solving for z2.
Elevation Unit: Unit for elevations and head loss.
Head Loss hL (optional): Friction and fitting losses between the points, as a head of fluid. Use 0 to ignore.

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