What is Bernoulli Equation Calculator?
Bernoulli's equation is the energy-conservation law for fluids in motion. It states that along a streamline, the sum of pressure energy, kinetic energy per unit volume (½ρv²), and gravitational potential energy per unit volume (ρgh) stays constant. In simple terms, when a fluid speeds up through a narrow section of pipe, its pressure drops — the effect that lifts airplane wings, shapes baseball curves, and powers carburetor and atomizer designs.
When to Use This Calculator
- Solving fluid dynamics problems for physics or engineering coursework and exams.
- Designing or troubleshooting piping systems where pressure drops must be predicted.
- Estimating flow velocities in venturi meters, orifices, and pitot tubes.
- Explaining everyday phenomena — lift, spray bottles, curveballs — to students or curious learners.
- Analyzing how elevation changes along a pipeline affect downstream pressure.
- Quickly checking tank-drain times using Torricelli's law before detailed simulation.
Steps:
- Choose the two points on the same streamline where you know enough values — label them point 1 and point 2.
- Enter the fluid density in kg/m³. Use 1000 for water, 1.225 for air at sea level, or look up the density of your specific fluid.
- Input pressure, velocity, and height at point 1, and at least two of the three values at point 2. The unknown value is computed automatically.
- Check the units — pressure must be in pascals, velocity in meters per second, height in meters, and density in kg/m³.
- Read the result — the missing value (pressure, velocity, or height) appears with the full step-by-step working shown beneath it.
- Validate with conservation — compare the total energy terms (P + ½ρv² + ρgh) at both points; they should be equal when your inputs are consistent.
Formula
Bernoulli's Equation (per unit volume):
P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂
Where:
P = Fluid pressure (pascals, Pa)
ρ = Fluid density (kg/m³)
v = Flow velocity (m/s)
g = Gravitational acceleration (9.81 m/s²)
h = Height above a reference level (m)
For horizontal flow (h₁ = h₂):
P₁ + ½ρv₁² = P₂ + ½ρv₂²
Torricelli's law (tank draining):
v = √(2gh)
Use Cases
- Pipe and duct design — predicting pressure changes when a pipeline or air duct narrows, widens, or changes elevation.
- Aviation — understanding how differences in air speed and pressure across a wing generate lift.
- Orifice and venturi meters — calculating flow rate from the pressure difference across a constriction in a pipe.
- Tank and reservoir draining — using Torricelli's law to estimate the efflux velocity of water leaving a tank.
- Medical and biological flows — analyzing blood flow in narrowed arteries and airflow in the lungs.
Key Benefits
- Instant results — solve for pressure, velocity, or height without rearranging the equation by hand.
- Full step-by-step working — every calculation is shown so students and engineers can follow the logic.
- Unit-safe — consistent SI inputs give correct answers in pascals, meters per second, and meters every time.
- Educational — interactive examples make the energy-conservation concept behind fluid flow tangible.
- Free and unlimited — no sign-up, no download, and no usage limits for classroom or professional work.
Pro Tips
- Start with the horizontal-flow form (P₁ + ½ρv₁² = P₂ + ½ρv₂²) when no height change is involved — it has one less term to track.
- Use the continuity equation A₁v₁ = A₂v₂ to find the missing velocity whenever a pipe changes cross-section.
- Remember that where velocity is highest, pressure is lowest — a handy mental check before trusting any computed result.
- When using Torricelli's law, measure the depth from the free surface of the liquid to the hole, not from the bottom of the tank.
- For air at room temperature, 1.225 kg/m³ is a reliable density value at sea level; at high altitude use a smaller value.
Common Mistakes to Avoid
- Mixing units — using bar or psi for pressure instead of pascals produces answers off by orders of magnitude. Convert to Pa first.
- Ignoring height differences — in vertical flow the ρgh term is often significant; only drop it for horizontal pipes.
- Applying the equation across different streamlines — Bernoulli's equation is valid along a single streamline, not between arbitrary points in the flow.
- Forgetting the continuity equation — you cannot compute the velocity at point 2 from geometry alone; use A₁v₁ = A₂v₂.
- Using it for viscous or turbulent flow — friction and turbulence dissipate energy, so the ideal Bernoulli equation overestimates real pressure changes in long pipes.
Key Terms Explained
- <strong>Streamline:</strong> A path traced by a fluid particle in steady flow, along which Bernoulli's equation is applied.
- <strong>Static pressure (P):</strong> The pressure exerted by the fluid on a surface moving with the flow.
- <strong>Dynamic pressure (½ρv²):</strong> The pressure equivalent of the fluid's kinetic energy per unit volume.
- <strong>Continuity equation:</strong> A₁v₁ = A₂v₂, expressing conservation of mass for incompressible flow.
- <strong>Torricelli's law:</strong> v = √(2gh), the efflux speed of a fluid draining through an orifice under gravity.
Related Concepts
- Continuity Equation: A₁v₁ = A₂v₂ — the companion law that ties pipe cross-section to flow velocity, used together with Bernoulli's equation.
- Fluid Density: The mass per unit volume (ρ) that scales the kinetic and potential energy terms. Our density calculator helps you find it.
- Torricelli's Law: The special case of Bernoulli's equation for a tank draining through an orifice under gravity.
- Hydrostatic Pressure: The pressure at a depth in a static fluid (P = ρgh) — Bernoulli's equation reduces to this when velocity is zero.
- Kinetic Energy: The energy of motion, KE = ½mv², which Bernoulli's equation tracks on a per-unit-volume basis. Our kinetic energy calculator explores the concept.
Example
Pipe narrowing example. Water (ρ = 1000 kg/m³) flows through a horizontal pipe that narrows from a 4 cm radius at point 1 to a 2 cm radius at point 2. The velocity at point 1 is 2 m/s and the pressure there is 100 kPa.
By the continuity equation, A₁v₁ = A₂v₂, so v₂ = (A₁/A₂) × v₁ = (16π/4π) × 2 = 8 m/s.
Using the horizontal-flow form: P₂ = P₁ + ½ρv₁² − ½ρv₂²
P₂ = 100,000 + ½(1000)(4) − ½(1000)(64) = 100,000 + 2,000 − 32,000 = 70,000 Pa (70 kPa).
The pressure drops by 30 kPa as the water speeds up — a direct demonstration of Bernoulli's principle.
Interpreting Your Results
If you solved for pressure, a lower result at a faster-flowing section is expected — it confirms Bernoulli's principle. When you solve for velocity, a larger velocity at a narrower section satisfies continuity. Always compare the total energy (P + ½ρv² + ρgh) at both points: the values must match for a consistent, physically valid solution. If they differ, revisit your inputs for unit or streamline errors.