Loading calculator…

What is Bernoulli's Principle?

Bernoulli's principle is a fundamental concept in fluid mechanics stating that in a steady fluid flow, an increase in fluid velocity is accompanied by a decrease in pressure or potential energy. Mathematically expressed as: P + ½ρv² + ρgh = constant, which is a statement of the conservation of energy for flowing fluids. Kalkulab's Bernoulli's Law Calculator is designed to help high school students (grades 11-12), civil and mechanical engineering students, and piping professionals understand fluid dynamics. This tool covers four main calculations: (1) flow rate from pipe cross-section area and velocity, (2) continuity equation for pipe diameter changes, (3) Bernoulli's equation to find pressure or velocity at two points, and (4) Torricelli's theorem for tank discharge velocity.

Bernoulli's Equation & Continuity

P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂Formula: Q = A × v ; A₁v₁ = A₂v₂ ; v = √(2gh) for Torricelli

Variables:

  • PPressure (Pascal/Pa)
    Fluid pressure at a point(e.g.: 100 kPa)
    💡 Water pressure in a pipe
  • ρDensity (kg/m³)
    Fluid density(e.g.: 1000 kg/m³ (water))
    💡 Density of water vs oil
  • vVelocity (m/s)
    Fluid flow velocity(e.g.: 2 m/s)
    💡 Water velocity in a hose
  • gGravity (m/s²)
    Earth's gravitational acceleration(e.g.: 9.8 m/s²)
    💡 Earth's gravitational constant
  • hHeight (meters)
    Height from reference point(e.g.: 10 m)
    💡 Water tank height
  • QFlow Rate (m³/s)
    Volume of fluid per unit time(e.g.: 0.02 m³/s)
    💡 Household faucet flow rate
  • ACross-Sectional Area (m²)
    Flow area(e.g.: 0.00785 m² (10cm pipe))
    💡 Water pipe area

Conservation of Energy for Fluids

Bernoulli's equation states that the sum of static pressure (P), kinetic energy per volume (½ρv²), and potential energy per volume (ρgh) is constant along a streamline in steady, incompressible, inviscid flow.

  1. 1Static Pressure (P): Energy per volume from fluid pressure
  2. 2Kinetic Energy (½ρv²): Energy per volume from fluid motion
  3. 3Potential Energy (ρgh): Energy per volume from height
  4. 4Continuity: A₁v₁ = A₂v₂ - Constant flow rate (Q = Av)

Categories:

Flow RateQ = A × v
ContinuityA₁v₁ = A₂v₂
Bernoulli's LawP + ½ρv² + ρgh = const
Torricelliv = √(2gh)

How to Use the KalkuLab Bernoulli's Law Calculator

This calculator has 4 main modes. Follow the steps for your calculation:

  1. 1

    Select Calculation Mode

    Choose: (1) Flow rate (Q = A×v), (2) Continuity (A₁v₁ = A₂v₂), (3) Bernoulli pressure (P + ½ρv² + ρgh = constant), or (4) Torricelli (v = √(2gh)).

  2. 2

    Enter Known Values

    Enter cross-section area (A), velocity (v), pressure (P), density (ρ), and height (h). Use SI units (m, m/s, Pa, kg/m³).

  3. 3

    Identify Points 1 and 2

    For Bernoulli's equation, mark two points in the fluid system—e.g., wide pipe vs. narrow pipe. Assume steady, incompressible flow.

  4. 4

    Click Calculate

    Get instant results in m³/s or L/s for flow rate, Pa or kPa for pressure.

  5. 5

    Analyze Results

    Pipe constriction increases velocity but decreases pressure (Venturi effect). Torricelli speed depends only on height, not fluid density.

💡 Tip:

  • •Water density = 1000 kg/m³, oil ≈ 800–900, air ≈ 1.2
  • •1 m³/s = 1000 L/s; household faucet ≈ 0.1–0.2 L/s
  • •Smaller area (A₂ < A₁) means higher velocity (v₂ > v₁)
  • •On a horizontal pipe (h₁ = h₂): P₁ + ½ρv₁² = P₂ + ½ρv₂²
  • •Venturi effect: constriction lowers pressure—used in carburetors and sprayers

Examples

Example 1: Household Faucet Flow Rate

Problem:

Faucet diameter 2 cm, water speed 1.5 m/s. Find flow rate.

Solution:
  1. 1.d = 0.02 m, v = 1.5 m/s
  2. 2.A = π × 0.01² = 0.000314 m²
  3. 3.Q = 0.000471 m³/s = 0.471 L/s
Result:0.471 L/s

About 28 liters per minute flows from the faucet.

Example 2: Venturi Effect in a Pipe

Problem:

Water at 2 m/s, 150 kPa narrows to 6 m/s on a horizontal pipe (ρ = 1000 kg/m³). Find pressure in the narrow section.

Solution:
  1. 1.P₂ = P₁ + ½ρ(v₁² − v₂²)
  2. 2.P₂ = 150000 + 500×(4−36) = 134000 Pa
Result:134 kPa

Pressure drops as velocity rises—the classic Venturi effect.

Example 3: Continuity in a Hose

Problem:

Hose diameter shrinks from 2 cm to 0.5 cm. Speed in main hose 1 m/s. Speed at tip?

Solution:
  1. 1.v₂ = (A₁/A₂) × v₁ = 16 m/s
Result:16 m/s

Pinching the hose makes water exit 16× faster.

Example 4: Torricelli Tank Drain

Problem:

Tank height 5 m above outlet. Exit speed? (g = 9.8 m/s²)

Solution:
  1. 1.v = √(2 × 9.8 × 5) = √98 ≈ 9.9 m/s
Result:9.9 m/s

Exit speed equals free-fall from 5 m— independent of fluid type.

Example 5: Pressure at Two Heights

Problem:

Point 1: h=10 m, P=200 kPa, v=1 m/s. Point 2: h=2 m, v=3 m/s. Find P₂ (ρ=1000, g=9.8).

Solution:
  1. 1.P₂ = P₁ + ½ρ(v₁²−v₂²) + ρg(h₁−h₂)
  2. 2.P₂ ≈ 274.4 kPa
Result:274.4 kPa

Lower elevation increases pressure despite higher velocity.

Frequently Asked Questions

What is Bernoulli's principle and why does it matter?
Bernoulli's principle states that in steady, incompressible flow, the sum of static pressure, kinetic energy per volume, and potential energy per volume is constant along a streamline: P + ½ρv² + ρgh = constant. It explains airplane lift, Venturi tubes, and carburetor operation.
What is the Venturi effect?
Fluid pressure drops when it passes through a constriction (velocity increases). Applications include carburetors, medical suction, pesticide sprayers, MAF sensors, and ventilators.
What is the continuity equation?
Flow rate Q = A × v is constant in a pipe with no storage: A₁v₁ = A₂v₂. Smaller area means higher velocity—why pinching a hose makes water spray faster.
What is Torricelli's law?
Exit speed from a small hole in a tank equals free-fall speed from the fluid surface to the hole: v = √(2gh). Speed depends on height and gravity, not fluid density.
Does Bernoulli's law apply to all fluids?
It assumes steady, incompressible, inviscid flow along streamlines. High-speed compressible air needs corrections; highly viscous fluids need friction factors.
How do I calculate flow rate in L/s?
Q = A × v. For a circular pipe: A = πr². Result in m³/s; multiply by 1000 for L/s.
How does Bernoulli's principle explain airplane lift?
Curved upper wing surface forces air to travel farther, increasing speed above the wing. Lower pressure above (P_top < P_bottom) creates upward lift.
Is this calculator accurate for engineering?
Accurate for ideal conditions. Real piping needs friction losses (Darcy-Weisbach), local losses at bends/valves, and turbulence corrections.

Related Calculators

Pressure Calculator

Hydrostatic and Pascal pressure

Density Calculator

Fluid and material density

Speed Calculator

Calculate velocity

Work & Power Calculator

Mechanical energy and work

References