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Bernoulli's Principle Calculator

Calculate fluid velocities, pressures, and flow rates using Bernoulli's equation for incompressible fluids

Parameters

L/sⓘ
mⓘ
mⓘ
kPaⓘ
kg/m³ⓘ
Show Trail

Controls

xⓘ

Calculated Values

Velocity 1:
0.05;m/s0.05;m/s
Velocity 2:
0.20;m/s0.20;m/s
Pressure 2:
101.28;kPa101.28;kPa
Pressure Difference:
0.02;kPa0.02;kPa
Kinetic Energy 1:
1.30;J/m31.30;J/m³
Kinetic Energy 2:
20.75;J/m320.75;J/m³

Examples

Example 1: Water Flow in Pipe

Water flowing through a pipe that narrows from 5cm to 2.5cm diameter.

  • Velocity 1: 0.510.51
  • Velocity 2: 2.042.04
  • Pressure 2: 99.1099.10
  • Pressure Difference: 2.202.20

Example 2: Air Flow in Duct

Air flowing through a duct that constricts from 20cm to 10cm diameter.

  • Velocity 1: 15.9015.90
  • Velocity 2: 63.7063.70
  • Pressure 2: 98.8098.80
  • Pressure Difference: 2.502.50

Example 3: Oil Flow

Oil flowing through a pipe with slight constriction.

  • Velocity 1: 0.710.71
  • Velocity 2: 1.021.02
  • Pressure 2: 199.60199.60
  • Pressure Difference: 0.400.40

Visualization

Bernoulli's Principle

Bernoulli's principle states that for an incompressible, non-viscous fluid flowing in steady state, the sum of pressure energy, kinetic energy, and potential energy per unit volume remains constant along a streamline. This principle is fundamental to understanding fluid dynamics and has applications in aerodynamics, hydraulics, and many engineering systems.

The Bernoulli equation is: P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂, where P is pressure, ρ is density, v is velocity, g is gravitational acceleration, and h is height. For horizontal flow (h₁ = h₂), this simplifies to P₁ + ½ρv₁² = P₂ + ½ρv₂².

Key implications of Bernoulli's principle include: (1) As fluid velocity increases, pressure decreases (venturi effect); (2) The principle explains lift in aircraft wings and the operation of carburetors; (3) It's used in flow measurement devices like venturi meters and pitot tubes; (4) The principle applies to both liquids and gases under appropriate conditions.

The continuity equation states that for incompressible flow, A₁v₁ = A₂v₂, where A is cross-sectional area. This means that when a pipe narrows, the fluid velocity increases, and according to Bernoulli's principle, the pressure decreases.

Bernoulli's principle has numerous applications including: aircraft design, water distribution systems, blood flow in arteries, and the design of nozzles and diffusers. It's essential for understanding how fluids behave in constricted passages and around obstacles.

Key Concepts

  • Bernoulli's Equation: P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂
  • Continuity Equation: A₁v₁ = A₂v₂
  • Venturi Effect: Higher velocity = Lower pressure
  • Flow Rate: Q = Av (constant for incompressible flow)
  • Pressure Energy: P (static pressure)
  • Kinetic Energy: ½ρv² (dynamic pressure)

Real-World Applications

  • Aircraft Wings: Lift generation through pressure differences
  • Venturi Meters: Flow rate measurement
  • Carburetors: Fuel-air mixing in engines
  • Water Distribution: Pipe network design
  • Blood Flow: Cardiovascular system analysis

Explore Further

More fluid mechanics tools

Physics Equations

Bernoulli's Equation:
P1+12ρv12=P2+12ρv22P_1 + \frac{1}{2}\rho v_1^2 = P_2 + \frac{1}{2}\rho v_2^2
Continuity Equation:
A1v1=A2v2A_1 v_1 = A_2 v_2
Flow Rate:
Q=Av=πd24vQ = A v = \frac{\pi d^2}{4} v
Velocity:
v=QA=4Qπd2v = \frac{Q}{A} = \frac{4Q}{\pi d^2}
Pressure Difference:
ΔP=P1−P2=12ρ(v22−v12)\Delta P = P_1 - P_2 = \frac{1}{2}\rho(v_2^2 - v_1^2)

Step-by-Step Solution

See how the main results are calculated.

1

Step 1: Identify Parameters

First, we identify all the parameters needed for Bernoulli's principle calculation:

Equation:

P1+12ρv12=P2+12ρv22P_1 + \frac{1}{2}\rho v_1^2 = P_2 + \frac{1}{2}\rho v_2^2

Calculation:

Q=0.1 L/s,d1=0.05 m,d2=0.025 m,P1=101.3 kPa,ρ=1000 kg/m3Q = 0.1 \text{ L/s}, d_1 = 0.05 \text{ m}, d_2 = 0.025 \text{ m}, P_1 = 101.3 \text{ kPa}, \rho = 1000 \text{ kg/m}^3

Explanation:

These are the flow rate, pipe diameters, initial pressure, and fluid density.

2

Step 2: Calculate Cross-Sectional Areas

Calculate the cross-sectional areas of both pipe sections:

Equation:

A=πd24A = \frac{\pi d^2}{4}

Calculation:

A1=π(0.05)24=0.001963 m2,A2=π(0.025)24=0.000491 m2A_1 = \frac{\pi (0.05)^2}{4} = 0.001963 \text{ m}^2, A_2 = \frac{\pi (0.025)^2}{4} = 0.000491 \text{ m}^2

Explanation:

The area of a circle is πr², where r is the radius (diameter/2).

3

Step 3: Calculate Velocities

Using the continuity equation to find velocities:

Equation:

v=QAv = \frac{Q}{A}

Calculation:

v1=0.00010.001963=0.05 m/s,v2=0.00010.000491=0.20 m/sv_1 = \frac{0.0001}{0.001963} = 0.05 \text{ m/s}, v_2 = \frac{0.0001}{0.000491} = 0.20 \text{ m/s}

Explanation:

Velocity is flow rate divided by cross-sectional area. When area decreases, velocity increases.

4

Step 4: Apply Bernoulli's Equation

For horizontal flow, Bernoulli's equation becomes:

Equation:

P1+12ρv12=P2+12ρv22P_1 + \frac{1}{2}\rho v_1^2 = P_2 + \frac{1}{2}\rho v_2^2

Calculation:

P2=101.3+12(1000)(0.052−0.202)=101.3 kPaP_2 = 101.3 + \frac{1}{2}(1000)(0.05^2 - 0.20^2) = 101.3 \text{ kPa}

Explanation:

Rearranging Bernoulli's equation to solve for P₂. As velocity increases, pressure decreases.

5

Step 5: Calculate Pressure Difference

The pressure difference between the two sections:

Equation:

ΔP=P1−P2\Delta P = P_1 - P_2

Calculation:

ΔP=101.3−101.3=0.0 kPa\Delta P = 101.3 - 101.3 = 0.0 \text{ kPa}

Explanation:

This shows how much pressure is lost due to the increase in kinetic energy (venturi effect).

Frequently Asked Questions (FAQ)

What is Bernoulli's principle?

Bernoulli's principle states that for an incompressible, non-viscous fluid in steady flow, the sum of pressure energy, kinetic energy, and potential energy per unit volume remains constant along a streamline.

Why does pressure decrease when velocity increases?

According to Bernoulli's principle, when fluid velocity increases, the kinetic energy increases. Since the total energy remains constant, the pressure energy must decrease to compensate.

What is the continuity equation?

The continuity equation states that for incompressible flow, the product of cross-sectional area and velocity remains constant: A₁v₁ = A₂v₂. This means when a pipe narrows, the fluid velocity increases.

When does Bernoulli's principle apply?

Bernoulli's principle applies to incompressible, non-viscous fluids in steady, laminar flow. It's most accurate for ideal fluids and provides good approximations for real fluids under appropriate conditions.

What is the venturi effect?

The venturi effect is the reduction in fluid pressure that occurs when a fluid flows through a constricted section of a pipe. It's a direct consequence of Bernoulli's principle and the continuity equation.

Practice MCQs

  1. According to Bernoulli's principle, when fluid velocity increases:
  2. The continuity equation states that:
  3. For horizontal flow, Bernoulli's equation becomes:
  4. What happens to fluid velocity when a pipe narrows?
  5. The venturi effect is used in:

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