Bernoulli Equation Calculator

Fluid Density (kg/m³)

Solve For (Point 2)

Point 1 (Inlet)

Point 2 (Outlet)

Pressure P₁ (Pa)

Pressure P₂ (Pa)

Velocity v₁ (m/s)

Velocity v₂ (m/s)

Elevation h₁ (m)

Elevation h₂ (m)

System Energy & Results

Calculated Value 0.00

This calculator applies the principle of conservation of energy to a steady, incompressible, and frictionless fluid along a streamline. It uses the standard gravity constant g = 9.81 m/s². Losses due to friction (head loss) or pump/turbine work are not included in this ideal state calculation.

What Is a Bernoulli Equation Calculator?

A Bernoulli Equation Calculator applies the conservation of mechanical energy to fluid moving along a streamline. It relates pressure energy, kinetic energy from velocity, and potential energy from elevation. This calculator can solve for pressure P₂, velocity v₂, or elevation h₂ at Point 2.

The Bernoulli equation calculator uses known pressure, velocity, elevation, fluid density, and gravity values to find one unknown condition at Point 2. It assumes steady, incompressible, frictionless flow without pumps, turbines, or energy losses. The result is an ideal estimate rather than a complete model of every real fluid system.

The tool also displays the total fluid energy per unit volume at Point 1. This total is expressed in joules per cubic meter, which is numerically equivalent to pascals. It breaks the total into pressure, kinetic, and potential energy components so you can see how each term contributes.

How the Bernoulli Equation Formula Works

The calculator uses the standard Bernoulli equation between Point 1 and Point 2:

P_1 + \frac{1}{2}\rho v_1^2 + \rho g h_1 = P_2 + \frac{1}{2}\rho v_2^2 + \rho g h_2

P₁ and P₂ are the fluid pressures at Points 1 and 2, measured in pascals.
ρ is the fluid density in kilograms per cubic meter.
v₁ and v₂ are the fluid velocities in meters per second.
g is gravitational acceleration. The calculator uses 9.81 m/s².
h₁ and h₂ are elevations in meters relative to the same reference level.

The calculator first computes the total energy per unit volume at Point 1:

E_1 = P_1 + \frac{1}{2}\rho v_1^2 + \rho g h_1

It then rearranges the equation according to the selected Point 2 unknown.

To calculate pressure at Point 2:

P_2 = E_1 – \frac{1}{2}\rho v_2^2 – \rho g h_2

To calculate velocity at Point 2:

v_2 = \sqrt{\frac{2}{\rho}\left(E_1-P_2-\rho g h_2\right)}

To calculate elevation at Point 2:

h_2 = \frac{E_1-P_2-\frac{1}{2}\rho v_2^2}{\rho g}

Worked Example: Calculating Pressure at Point 2

Assume water density is 1,000 kg/m³. At Point 1, pressure is 150,000 Pa, velocity is 2.5 m/s, and elevation is 0 m. At Point 2, velocity is 4 m/s and elevation is 5 m.

First, calculate Point 1 total energy:

E_1 = 150000 + \frac{1}{2}(1000)(2.5^2) + (1000)(9.81)(0) = 153125\ \text{J/m}^3

Next, subtract the Point 2 kinetic and potential energy terms:

P_2 = 153125 – \frac{1}{2}(1000)(4^2) – (1000)(9.81)(5) = 96075\ \text{Pa}

The calculated pressure at Point 2 is 96,075 Pa. The calculator displays numerical results using U.S. number formatting and up to two decimal places.

How to Use the Bernoulli Equation Calculator: Step by Step

Enter the fluid density in kilograms per cubic meter. Density must be greater than zero.
Use the “Solve For” menu to select pressure P₂, velocity v₂, or elevation h₂ at Point 2. The selected field becomes disabled because the calculator will determine it.
Enter the pressure P₁ at Point 1 in pascals.
Enter the velocity v₁ at Point 1 in meters per second.
Enter the elevation h₁ at Point 1 in meters.
Complete the two enabled Point 2 fields. For example, calculating pressure P₂ requires velocity v₂ and elevation h₂.
Select “Calculate” to display the requested result, total fluid energy, and Point 1 energy breakdown.
Select “Reset” to restore the default values and return the calculator to solving for pressure P₂.

The main result shows the calculated Point 2 pressure, velocity, or elevation with its unit. The detail area shows Point 1 pressure energy, kinetic energy, potential energy, and their combined total. These values help explain how energy is distributed within the ideal fluid system.

What to Check Before Using Your Result

Use Consistent SI Units

The calculator expects density in kg/m³, pressure in Pa, velocity in m/s, and elevation in meters. Mixing units can produce a result that looks reasonable but is incorrect. Convert values before entering them. For example, convert kilopascals to pascals by multiplying by 1,000.

Use the Same Elevation Reference

Both elevation values must use the same reference level. The reference can be the floor, pipe centerline, sea level, or another chosen datum. A negative elevation is allowed, but it must still be measured from the same reference used for the other point.

Understand the Ideal-Flow Assumptions

Calculator assumption	What it means
Steady flow	Conditions do not change with time.
Incompressible fluid	Fluid density remains constant.
Frictionless flow	Energy loss from pipe friction and fittings is excluded.
Single streamline	The two points are treated as lying along the same flow path.
No pump or turbine work	No mechanical energy is added or removed by equipment.
Constant gravity	Gravitational acceleration is fixed at 9.81 m/s².

Real systems may include pipe friction, valves, bends, pumps, turbines, changing density, or unsteady flow. Those effects can change the actual pressure, velocity, or elevation. This calculator does not include head loss, pump head, turbine extraction, pipe diameter, or flow rate.

Watch for an Impossible Velocity State

When solving for velocity, the value inside the square root must be zero or positive. If the entered Point 2 pressure and elevation require more energy than Point 1 provides, the calculator reports an impossible state instead of returning a real velocity.

This result is an ideal engineering estimate. It should not replace detailed fluid-system analysis, equipment data, safety review, or advice from a qualified engineer when designing or operating a real system.

Frequently Asked Questions

What does a Bernoulli equation calculator calculate?

A Bernoulli equation calculator finds pressure, velocity, or elevation at one point in an ideal fluid system. This tool solves only for the selected Point 2 variable. It also calculates total energy per unit volume and shows the pressure, kinetic, and potential energy terms at Point 1.

How do I calculate pressure at Point 2?

Select pressure P₂ from the “Solve For” menu. Enter fluid density, pressure P₁, velocity v₁, elevation h₁, velocity v₂, and elevation h₂. The calculator subtracts Point 2 kinetic and potential energy from the total energy calculated at Point 1.

How do I calculate fluid velocity with Bernoulli’s equation?

Select velocity v₂ and enter the other required values. The calculator isolates the Point 2 kinetic energy term and takes its square root. If the available energy would require a negative value under the square root, the tool reports that the entered state is physically impossible.

Why does the calculator show an impossible state?

An impossible state appears only when solving for velocity and the calculated value of v₂ squared is negative. This means the entered Point 2 pressure and elevation require more energy than the Point 1 conditions provide under the calculator’s frictionless, steady-flow assumptions.

Is joules per cubic meter the same as pascals?

Yes. One joule per cubic meter is dimensionally equal to one pascal. Bernoulli’s equation in this calculator expresses every term as energy per unit volume. Pressure, kinetic energy density, potential energy density, and total energy can therefore be compared using equivalent units.

Does this Bernoulli equation calculator include friction loss?

No. The calculator assumes frictionless flow and does not include head loss from pipe walls, valves, bends, fittings, or other restrictions. It also excludes energy added by pumps and energy removed by turbines. Real fluid systems may require an extended Bernoulli equation.

How accurate is the Bernoulli equation calculator?

The arithmetic follows the standard ideal Bernoulli equation using gravity of 9.81 m/s². Practical accuracy depends on the entered values and how closely the real system matches the assumptions. Friction, turbulence, equipment, compressibility, and measurement error can make actual results different.

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