Bernoulli's Equation
Understand Bernoulli's equation: P + ½ρv² + ρgh = constant.
Essential for fluid dynamics, aerodynamics, and hydraulic engineering.
The Formula
Bernoulli's equation relates pressure, velocity, and elevation in a flowing fluid.
It states that the total mechanical energy along a streamline remains constant.
This applies to steady, incompressible, non-viscous (ideal) fluid flow.
Variables
| Symbol | Meaning |
|---|---|
| P | Static pressure of the fluid (Pascals, Pa) |
| ρ | Fluid density (kg/m³) |
| v | Flow velocity (m/s) |
| g | Acceleration due to gravity (9.81 m/s²) |
| h | Height above a reference point (metres, m) |
| ½ρv² | Dynamic pressure (the kinetic energy per unit volume) |
| ρgh | Hydrostatic pressure (the potential energy per unit volume) |
Example 1
Water (ρ = 1000 kg/m³) flows through a horizontal pipe. At point 1, the pressure is 200,000 Pa and velocity is 2 m/s. At point 2, the velocity increases to 5 m/s. Find the pressure at point 2.
Since the pipe is horizontal, h₁ = h₂, so the ρgh terms cancel.
P₁ + ½ρv₁² = P₂ + ½ρv₂²
200,000 + ½(1000)(2²) = P₂ + ½(1000)(5²)
200,000 + 2,000 = P₂ + 12,500
P₂ = 202,000 - 12,500
P₂ = 189,500 Pa ≈ 189.5 kPa
Example 2
Water flows from a tank through a hole 3 m below the surface. Find the exit velocity (assume the tank surface velocity is negligible).
Apply Bernoulli's equation between the surface and the hole.
At both points, pressure equals atmospheric pressure (open tank and open hole), so P terms cancel.
½ρv₁² + ρgh₁ = ½ρv₂² + ρgh₂
Since v₁ ≈ 0 and setting h₂ = 0: ρg(3) = ½ρv₂²
v₂ = √(2 × 9.81 × 3) = √(58.86)
v₂ ≈ 7.67 m/s (this is Torricelli's theorem)
When to Use It
Use Bernoulli's equation when you need to:
- Analyse flow through pipes that change diameter
- Calculate the speed of fluid exiting a tank or nozzle
- Understand lift on aircraft wings (pressure difference)
- Design venturi meters, pitot tubes, and carburettors
Remember: this equation assumes ideal conditions.
Real fluids have viscosity and friction losses, which require correction factors.