Understanding Bernoulli's Equation & Flow Measurement
Learn about energy conservation in fluid flow, Pitot tubes, and Venturi meters
Bernoulli's Equation
For steady, incompressible flow along a streamline, the total mechanical energy per unit weight remains constant. Energy can convert between pressure, velocity, and elevation forms.
p/(ρg) + v²/(2g) + z = constant
Total Head H = Pressure Head + Velocity Head + Elevation Head
- • p/(ρg): Pressure head (m)
- • v²/(2g): Velocity head (m)
- • z: Elevation head (m)
Continuity Equation
For incompressible flow, mass is conserved. The product of cross-sectional area and velocity is constant throughout the flow—when area decreases, velocity must increase.
A₁v₁ = A₂v₂ = Q (constant)
Q = volumetric flow rate (m³/s)
This explains why water speeds up in a narrow nozzle and why the Venturi effect creates low pressure in constrictions.
Pitot Tube / Stagnation Probe
A Pitot tube measures flow velocity by comparing stagnation pressure (where flow stops) to static pressure. The kinetic energy converts entirely to pressure at the stagnation point.
v = √(2Δp/ρ)
Δp = p_stagnation - p_static = ½ρv²
Used in aircraft airspeed indicators, wind tunnels, and industrial flow measurement.
Venturi & Orifice Meters
These devices measure flow rate using a constriction. The velocity increase in the throat creates a pressure drop proportional to flow. A discharge coefficient (C_d) accounts for real-world losses.
v₂ = C_d × √(2Δp / (ρ(1 - β⁴)))
β = D₂/D₁ (throat to pipe diameter ratio)
- • Venturi: C_d ≈ 0.95–0.98 (low losses)
- • Orifice: C_d ≈ 0.60–0.65 (vena contracta)
Extended Bernoulli Equation
Real systems often include pumps (adding energy), turbines (extracting energy), and head loss due to friction and fittings.
H₁ + h_pump = H₂ + h_turbine + h_loss
h_pump (m)
Head added by a pump—energy input to the system
h_turbine (m)
Head extracted by a turbine—energy output from the system
h_loss (m)
Head lost to friction and minor losses—always positive
Common Applications
| Application | Principle Used | Key Formula |
|---|---|---|
| Aircraft Airspeed | Pitot-static tube | v = √(2Δp/ρ) |
| Pipe Flow Rate | Venturi meter | Q = C_d × A₂ × v₂ |
| Pump Sizing | Head calculation | h_pump = ΔH + h_loss |
| Carburetors | Venturi effect | Low pressure draws fuel |
| Wing Lift | Pressure difference | Faster flow = lower pressure |
Frequently Asked Questions
Common questions about Bernoulli's equation, flow measurement, and fluid dynamics
Related Calculators
Explore more physics tools for your calculations
Fluid Pressure & Hydrostatic Force
Calculate pressure at depth and forces on submerged surfaces
Force, Work & Power
Calculate force, work, and power relationships
Energy Calculator
Calculate kinetic, potential, and mechanical energy
Thermodynamics Calculator
Heat transfer, ideal gas law, and thermal energy
Heat Exchanger LMTD Helper
Calculate log mean temperature difference for heat exchanger design