Bernoulli Equation Calculator

TL;DR: To calculate velocity at point 2 using the Bernoulli equation, solve for v₂ with the formula v₂ = √(v₁² + 2(P₁ − P₂)/ρ + 2g(h₁ − h₂)), which directly accounts for changes in pressure, height, and fluid density.

What Is the Bernoulli Equation Calculator?

The Bernoulli Equation Calculator is a free online tool that solves for an unknown fluid velocity when you provide pressure, height, and density values at two points along a streamline. It applies the principle of conservation of mechanical energy to incompressible, inviscid fluid flow. Engineers, physics students, and HVAC or piping system designers use this calculator to determine how fast a fluid moves after passing through a pressure change or elevation difference.

Real-world applications include sizing pipe diameters for water supply systems, predicting flow rates in venturi meters, analyzing aircraft wing lift, and evaluating blood flow in medical devices. The calculator eliminates tedious algebra by automatically computing each energy term and solving for the target variable. You input pressure 1, velocity 1, height 1, pressure 2, height 2, and fluid density; the calculator outputs the velocity at point 2 instantly. This makes it ideal for quick design checks, homework verification, and scenario testing without manual rearrangement of the governing equation.

Because the tool uses standard SI units (Pascals for pressure, meters per second for velocity, meters for height, and kg/m³ for density), it ensures that results are physically meaningful and directly comparable to textbook or industry values. By handling the potential energy term automatically, it prevents the common error of assuming horizontal flow when elevation changes are significant.

How to Use the Calculator

1. Enter Pressure 1 (P₁): Input the static pressure at the first location, measured in Pascals (Pa). For example, 150,000 Pa for 1.5 bar.
2. Enter Velocity 1 (v₁): Provide the fluid speed at point 1 in meters per second (m/s). This must be a known value to use as the baseline for solving v₂.
3. Enter Height 1 (h₁): Specify the elevation of point 1 above your reference datum in meters (m).
4. Enter Pressure 2 (P₂): Input the static pressure at the second point, also in Pascals. Ensure it is the absolute or gauge pressure consistent with P₁.
5. Enter Height 2 (h₂): Give the elevation of point 2 in meters.
6. Enter Fluid Density (ρ): Input the density of the working fluid in kg/m³. For water at 20°C, use 998 kg/m³; for air at standard conditions, use 1.225 kg/m³.
7. Click Calculate: The tool computes the total energy at point 1, applies Bernoulli's equation, and instantly displays Velocity 2 (v₂) in m/s.

Formula and Calculation Method

The Bernoulli equation states that the sum of pressure energy, kinetic energy, and potential energy along a streamline remains constant for steady, incompressible, frictionless flow. In plain language: as a fluid speeds up, its pressure drops, or as it moves uphill, its pressure must compensate. The mathematical formulation is:

P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂

Here, P is static pressure (Pa), ρ is density (kg/m³), v is velocity (m/s), g is gravitational acceleration (9.81 m/s²), and h is elevation (m). The term ½ρv² is called dynamic pressure (kinetic energy per unit volume), and ρgh is the hydrostatic pressure (potential energy per unit volume).

Worked Example: Suppose water flows through a pipe that narrows and rises. We have P₁ = 150,000 Pa, v₁ = 3.5 m/s, h₁ = 2 m, P₂ = 120,000 Pa, h₂ = 5 m, and ρ = 1000 kg/m³. First, calculate the energy at point 1:
Pressure term: 150,000 J/m³
Kinetic term: ½ × 1000 × (3.5)² = 6,125 J/m³
Potential term: 1000 × 9.81 × 2 = 19,620 J/m³
Total energy at point 1 = 175,745 J/m³

Now apply Bernoulli's equality. At point 2, the sum must equal 175,745 J/m³. Write the equation with v₂ unknown:
150,000 + 6,125 + 19,620 = 120,000 + ½(1000)v₂² + 1000(9.81)(5)
175,745 = 120,000 + 500v₂² + 49,050
175,745 = 169,050 + 500v₂²
6,695 = 500v₂²
v₂² = 13.39
v₂ ≈ 3.66 m/s

The calculator performs these algebraic steps automatically, starting from the rearranged general solution:

v₂ = √(v₁² + 2(P₁ − P₂)/ρ + 2g(h₁ − h₂))

Plugging the example numbers yields v₂ = √(12.25 + 60 + 2×9.81×(−3)) = √(72.25 − 58.86) = √13.39 = 3.66 m/s. Note that if the term under the square root becomes negative, the input pressures or heights violate Bernoulli's principle for the given fluid, which could indicate an error in input or a location where flow separation occurs.

Practical Examples

In the first scenario, the velocity increases from 3.5 to 3.66 m/s because the pressure drop (30,000 Pa) outweighs the energy lost to the 3 m elevation gain. The second example shows a classic venturi effect: a modest pressure drop in air creates a dramatic six-fold speed increase because of the low density. In the third, a tank open to the atmosphere with water starting from rest at 10 m height produces an outlet velocity of about 14 m/s, closely matching Torricelli's law (√(2gh) = 14.0 m/s). Each result tells you whether the fluid accelerates or decelerates based on the competing pressure and gravity effects.

Tips for Accurate Results

• Use consistent units: The calculator expects pressure in Pascals, velocity in meters per second, height in meters, and density in kg/m³. Convert bar to Pa (×100,000), cm/s to m/s (÷100), and g/cm³ to kg/m³ (×1000) before entering values.
• Check the density term carefully: Forgetting to include ρ in the potential energy term ρgh is a common mistake. For water, this term is roughly 9,810 Pa per meter of height; for air it is only about 12 Pa per meter, so ignore it only when heights and densities are negligible.
• Never assume horizontal flow: If heights differ by more than 1 meter for liquids, the potential energy term substantially affects the result. The calculator includes h₁ and h₂ specifically to handle this — only set them equal if the pipe is truly level.
• Use gauge or absolute pressure consistently: Mixing gauge pressure at point 1 with absolute pressure at point 2 will introduce errors. Both pressures must refer to the same reference level.
• Verify the radicand is positive: If you compute v₂² as negative, the inputs are physically inconsistent for a single streamline. Reduce the pressure difference, increase the height drop, or check whether the fluid could cavitate.
• Know the limitations: Bernoulli's equation ignores viscous friction, turbulence, and energy losses from bends or fittings. For long pipes with significant friction, use the Darcy-Weisbach equation instead.

Frequently Asked Questions

Can I use this calculator for compressible fluids like air at high speeds?

No. The Bernoulli equation as implemented in this calculator assumes incompressible flow (constant density). For air or gases moving at speeds above roughly Mach 0.3 (about 100 m/s at sea level), density changes become significant and the incompressible assumption fails. For low-speed air applications like HVAC ducting, venturi meters, or wind tunnel analysis below 30 m/s, using ρ = 1.225 kg/m³ gives reliable results. If you need to analyze high-speed gas flow, you must use the compressible Bernoulli equation with the isentropic flow relations instead.

What does it mean if the calculator returns a very high or zero velocity?

A zero velocity at point 2 indicates that the total energy at point 1 exactly equals the pressure and potential energy at point 2, leaving no kinetic energy. This happens when a pump or elevation head exactly counteracts the pressure difference. Conversely, a very high velocity (above 50 m/s for water) usually indicates an unreasonably large pressure drop or elevation drop — check whether cavitation would occur. For water, the maximum possible velocity before vaporization occurs is about 14 m/s at 20°C when the pressure drops to vapor pressure (2,337 Pa). If you get values above that, the pressure input at point 2 may be below vapor pressure, and liquid would boil, invalidating the calculation.

What is the difference between using gauge pressure and absolute pressure?

The Bernoulli equation works with either gauge or absolute pressure as long as you use the same convention at both points, because only pressure differences appear in the formula v₂ = √(v₁² + 2(P₁ − P₂)/ρ + ...). Using gauge pressure simply shifts both values by atmospheric pressure (~101,325 Pa), and the difference P₁ − P₂ remains unchanged. However, if you are comparing results with tabulated data or checking for cavitation, you must use absolute pressure because vapor pressure is absolute. The calculator does not convert between the two — it uses exactly what you type — so be careful to input either two gauge values or two absolute values, never a mix.

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