Bernoulli Equation Calculator
Solve Bernoulli's equation for p, v or z at either point, with head loss and a pipe mode that uses diameters and continuity to find downstream pressure or flow.
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About this toolWhat is the Bernoulli Equation Calculator?
Bernoulli's equation is an energy balance between two points in a moving fluid: pressure, kinetic energy and height trade against each other, and whatever is lost to friction comes out as head loss. This calculator holds all six variables, p₁, v₁, z₁ at the upstream point and p₂, v₂, z₂ downstream, and solves for whichever one you choose. With the defaults, water entering at 200 kPa and 2 m/s and leaving 5 m higher at 4 m/s arrives at 145.0 kPa.
A second mode is built for pipes. Give the two diameters and the flow, and continuity (A₁v₁ = A₂v₂) supplies both velocities so you only need the upstream pressure to get the downstream one. Or give both pressures and it solves for the flow, which is how a Venturi meter works: 30 kPa across a 100 to 50 mm contraction passes 15.7 L/s of water in the ideal case.
Pressure, velocity and height each have a unit picker (kPa, Pa, bar, MPa or psi; m/s or ft/s; m or ft), and a table breaks the answer into pressure head, velocity head and elevation head at each point so you can see where the energy went.
- Solves for any of p₁, v₁, z₁, p₂, v₂ or z₂
- Optional head loss for real, frictional flow
- Pipe mode with continuity: downstream pressure from flow, or flow from two pressures (Venturi)
- Independent units for pressure, velocity and height with in-place conversion
- Energy table of pressure, velocity and elevation heads at both points
How to use the Bernoulli Equation Calculator
- Choose six-variable mode for any two points, or pipe mode when you know the diameters.
- Set the units for pressure, velocity and height. Changing a unit converts all the matching fields.
- Pick the unknown. In pipe mode choose downstream pressure (given the flow) or flow (given both pressures).
- Fill in the remaining values for point 1 and point 2, the head loss between them (0 for an ideal flow) and the fluid density.
- Read the solved value, the pressure change and the energy table. If no real answer exists the tool says why.
Worked examples
Pipe rising to an upper floor
Input
p₁ = 200 kPa, v₁ = 2 m/s, z₁ = 0; v₂ = 4 m/s, z₂ = 5 m; water, no loss
Output
p₂ = 144.97 kPa
Climbing 5 m costs 49.0 kPa and speeding up from 2 to 4 m/s costs 6.0 kPa. Add a realistic head loss and p₂ falls further.
Venturi meter
Input
100 mm to 50 mm, p₁ = 150 kPa, p₂ = 120 kPa, water
Output
Q = 15.71 L/s, v₁ = 2.0 m/s, v₂ = 8.0 m/s
A real Venturi multiplies this by a discharge coefficient of about 0.98 to 0.995.
Draining a tank
Input
Surface: p = 0, v = 0, z = 3 m; outlet: p = 0, z = 0
Output
v₂ = 7.67 m/s
This is Torricelli's law, v = √(2gh). Real outlets deliver less because of the contraction and losses; see the orifice flow calculator.
Background and insight
Daniel Bernoulli described the pressure and velocity trade-off in Hydrodynamica in 1738; Leonhard Euler later wrote it in its modern form. Along a stream line in steady, incompressible, frictionless flow, p + ½ρv² + ρgz is constant. Divide by ρg and each term becomes a height: pressure head, velocity head and elevation head, whose sum is the total head that hydraulic grade line drawings show.
Real flow loses energy to friction and turbulence. The engineering form adds a head loss term, h_L, on the downstream side. You can take h_L from the pipe pressure drop calculator, then use Bernoulli to find the pressure that is left. Across a pump the equation also needs the pump head added, and across a turbine the extracted head subtracted.
Pressures must share one reference. Use gauge at both points or absolute at both. A negative gauge result below about −100 kPa is physically impossible, and long before that the liquid reaches its vapour pressure and cavitates, so the calculator flags those results.
Common use cases
Pressure at a fixture or nozzle
Find the pressure left at a higher or faster point after rises, changes in diameter and losses.
Venturi and flow meters
Get the ideal flow through a contraction from a measured differential pressure.
Siphons and tank outlets
Estimate outlet velocity, or check whether the crest of a siphon drops toward vapour pressure.
Teaching energy grade lines
Show pressure, velocity and elevation heads trading off between two points.
Tips and limitations
- Bernoulli applies along one stream line between two points with no pump or turbine between them. Add pump head separately.
- Use average velocities from continuity for pipe flow. The kinetic energy correction factor is about 1.05 for turbulent flow and 2 for laminar flow, which matters in laminar cases.
- For gases, Bernoulli in this form only holds when density barely changes, roughly below Mach 0.3 (about 100 m/s in air).
- Do not apply it across a sudden expansion or a valve without a loss term; those convert kinetic energy into heat, not pressure.
- A velocity solve with a negative value under the square root means the flow cannot go from 1 to 2 with the energy you gave it.
Frequently asked questions
What is Bernoulli's equation?
For steady, incompressible, frictionless flow along a stream line, p + ½ρv² + ρgz is the same at every point. Between two points: p₁ + ½ρv₁² + ρgz₁ = p₂ + ½ρv₂² + ρgz₂. Engineers add a head loss term for real flows.
How do you solve Bernoulli's equation for velocity?
Rearrange for the unknown v: v₂ = √[v₁² + 2(p₁ − p₂)/ρ + 2g(z₁ − z₂) − 2g·h_L]. If the number under the root is negative there is no real solution with those inputs.
Why does pressure drop when velocity increases?
The total energy per unit volume is fixed in ideal flow, so if kinetic energy (½ρv²) rises, pressure energy must fall by the same amount. That is why the throat of a Venturi has the lowest pressure.
Should I use gauge or absolute pressure in Bernoulli?
Either, as long as both points use the same one. Gauge is convenient for liquids open to the air. Use absolute when checking for cavitation against vapour pressure.
What are the assumptions of the Bernoulli equation?
Steady flow, incompressible fluid, no friction (unless you add head loss), no shaft work between the points, and points on the same stream line or in irrotational flow.
How is Torricelli's law related to Bernoulli?
It is Bernoulli between a tank's free surface and a small outlet, both at atmospheric pressure, with the surface velocity near zero: v = √(2gh). A 3 m head gives 7.67 m/s before losses.
Is there an in-depth guide to the Bernoulli Equation Calculator?
Yes. “Bernoulli's Equation With Worked Examples: Heads, Venturis and Tanks” covers the background, how it works, common pitfalls and worked examples. Read the full guide.
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