Bernoulli Equation Calculator - Pressure, Velocity, Elevation Solver

A bernoulli equation calculator is a fluid-mechanics tool that solves Bernoulli's equation for the unknown term between two points along a streamline in steady, incompressible flow.

• • Pipe flow problems: Compute pressure drop or velocity change between an upstream and a downstream section of a pipe.
• • Tank draining: Estimate exit height, exit speed, or surface pressure from the energy balance between the free surface and the outlet.
• • Venturi and nozzle analysis: Find the throat velocity or pressure drop for a constriction in a duct or pipe.
• • Pitot-static and airspeed work: Use Bernoulli's equation with density and pressure readings to back out flow speeds.

The two-point form of Bernoulli's equation is P1 + 1/2 rho v1^2 + rho g h1 = P2 + 1/2 rho v2^2 + rho g h2. Each term has units of energy per unit volume, so the equation is conservation of mechanical energy along a streamline in an ideal fluid.

This calculator lets you supply five of the six Bernoulli variables and fluid density, then pick which unknown to solve for. The result returns the chosen variable in its natural unit (Pa, m/s, or m) plus the per-point dynamic and elevation head contributions so you can audit the energy balance.

If you are studying velocity and acceleration in non-fluid motion, the kinematics motion calculator walks through a similar solver pattern for 1D problems.

The calculator evaluates Bernoulli's equation in pressure-energy form and rearranges it for the variable you selected. Standard gravity g = 9.80665 m/s^2 is used for the elevation head term.

• P1, P2: Static pressure at points 1 and 2, in pascals (Pa).
• v1, v2: Flow speed at points 1 and 2, in metres per second (m/s).
• h1, h2: Elevation of points 1 and 2 above a shared reference datum, in metres (m).
• rho: Fluid mass density, in kilograms per cubic metre (kg/m^3).
• g: Gravitational acceleration, fixed at 9.80665 m/s^2 (standard gravity).

The dynamic pressure 1/2 rho v^2 is the kinetic energy per unit volume of the moving fluid, and the term rho g h is the gravitational potential energy per unit volume relative to the chosen datum. According to Wikipedia, Bernoulli's equation for incompressible, steady flow along a streamline is exactly this conservation of static, dynamic, and hydrostatic pressure energy.

As published by the Engineering Toolbox, the two-point form is the version most often used in pipe-flow and tank-draining problems, so the calculator reports the Bernoulli constant so you can confirm both sides agree to the precision of your inputs.

According to Wikipedia, Bernoulli's equation for incompressible, steady flow along a streamline states that static pressure, dynamic pressure, and hydrostatic pressure sum to a constant.

As published by Engineering Toolbox, The two-point form P1 + 1/2 rho v1^2 + rho g h1 = P2 + 1/2 rho v2^2 + rho g h2 is the form most often used in pipe-flow and tank-draining problems.

If you need the friction-loss extension, the Reynolds number calculator tells you whether the flow is laminar or turbulent before you add a Darcy-Weisbach term on top of the energy balance returned here.

Four ideas behind the bernoulli equation calculator that are worth understanding before you trust the numbers.

These four ideas reappear throughout fluid-mechanics homework. The straight-line energy bookkeeping used in mechanics problems complements Bernoulli's streamline energy balance.

If you also work on momentum-style problems, the work energy power calculator covers the straight-line energy bookkeeping that complements Bernoulli's streamline energy balance.

Use the bernoulli equation calculator in five steps.

1. 1 Pick the unknown: Open the Solve For menu and choose the variable you want the calculator to return: P1, P2, v1, v2, h1, or h2. If you are not sure, leave the default (v2) for an exit-velocity solve.
2. 2 Set the fluid density: Enter the density rho in kg/m^3. Use 1000 for water, 1.225 for sea-level air, or the actual value for the fluid in your problem.
3. 3 Enter the upstream point: Type the static pressure, flow speed, and elevation at point 1 in the P1, v1, h1 inputs.
4. 4 Enter the downstream point: Type the static pressure, flow speed, and elevation at point 2 in the P2, v2, h2 inputs. The input for the variable you picked in step 1 can be left at zero; the calculator will overwrite it.
5. 5 Read the solved value and energy balance: The primary output shows the solved variable in its natural unit, and the secondary outputs break out the dynamic and elevation heads and the conserved Bernoulli constant so you can audit the energy balance.

For a draining water tank with P1 = 100000 Pa, v1 = 0 m/s, h1 = 10 m, P2 = 100000 Pa, and h2 = 0 m, leave Solve For on the default v2 and read the result. The calculator returns about 14 m/s, which is the classic Torricelli exit velocity sqrt(2 g h) for a 10 m head of water.

If you only know the fluid in pounds per gallon, the density calculator can convert to kg per cubic metre before you fill the fluid density field.

Practical reasons to use this bernoulli equation calculator instead of solving Bernoulli's equation by hand.

• • One tool for all six unknowns: Switch the Solve For menu to rearrange for P1, P2, v1, v2, h1, or h2 without rewriting the energy balance each time.
• • Auditable energy balance: The Bernoulli constant, dynamic head, and elevation head are all reported in pascals, so you can confirm both sides of the equation agree.
• • Works for any incompressible fluid: Set rho to the actual density (water, oil, air, glycerol) instead of hard-coding water at 1000 kg/m^3.
• • Hand-check friendly precision: Four-decimal output matches the precision expected in a fluid-mechanics class.
• • Quick textbook sanity check: Recreate worked examples in seconds and compare your handwritten answer to the calculator before turning in homework.
• • Connects to related motion problems: The same physical intuition (kinetic energy traded for pressure or height) shows up in free-fall and projectile problems, so related motion calculators share inputs and units.

The calculator is intentionally narrow: it does one ideal-fluid energy balance well. For viscous losses in real pipes, you will need a Darcy-Weisbach extension on top of the Bernoulli constant reported here.

If your Bernoulli problem turns out to also involve free fall or a projectile, the same density and gravity values feed the projectile motion calculator.

What changes the answer the bernoulli equation calculator returns, and what it cannot capture.

• • The Bernoulli equation assumes steady, incompressible, inviscid flow along a single streamline, so it does not capture pipe friction, turbulent dissipation, shock waves, or large density changes in compressible flow.
• • Solving for a velocity requires the expression under the square root to be non-negative. If the inputs force v^2 below zero, the calculator returns null and an error message so you can check the sign or units of your pressures.
• • The calculator uses a single shared density across both points, so temperature- or pressure-driven density changes within the same flow are not modelled. The result is an idealised energy balance, not a viscous or compressible CFD answer.

According to NASA Glenn Research Center, Bernoulli's equation is the conservation of energy statement for an ideal fluid moving along a streamline, which is exactly the scope of this calculator. The Engineering Toolbox notes that adding head losses from fittings, friction, and pumps is the next step when you need a realistic pipe-flow answer.

According to NASA Glenn Research Center, Bernoulli's equation P + 1/2 rho v^2 + rho g h = constant expresses conservation of energy for an ideal fluid moving along a streamline.

When the elevation change in the factor list sets the dominant term, the free fall time calculator helps estimate the fall time of a particle dropped through the same height for a quick physical cross-check.