Bernoulli's Equation Calculator
Bernoulli's equation describes the conservation of mechanical energy in fluid flow. It states that for an inviscid, incompressible fluid flowing steadily along a streamline, the sum of pressure energy, kinetic energy, and potential energy per unit volume remains constant. Daniel Bernoulli derived this principle in 1738, and it remains fundamental to fluid dynamics.
Bernoulli's principle explains lift generation in aircraft, flow measurement devices, carburetor operation, and many hydraulic systems. It's essential for engineering design in aerospace, civil, mechanical, and biomedical fields. The equation connects pressure, velocity, and elevation in fluid systems.
Key concepts in Bernoulli's equation:
- Conservation of energy: Total mechanical energy constant along streamline
- Pressure-velocity tradeoff: Higher velocity → lower pressure (and vice versa)
- Static pressure: Pressure exerted by fluid at rest
- Dynamic pressure: ½ρv² - pressure due to fluid motion
- Hydrostatic pressure: ρgh - pressure due to fluid weight
- Total pressure: Sum of static + dynamic + hydrostatic pressures
This calculator solves for any unknown variable in Bernoulli's equation when you know the other parameters at two points along a streamline:
- Find Pressure at Point 2: Enter P₁, v₁, h₁, v₂, h₂ → Get P₂
- Find Velocity at Point 2: Enter P₁, v₁, h₁, P₂, h₂ → Get v₂
- Find Height at Point 2: Enter P₁, v₁, h₁, P₂, v₂ → Get h₂
The calculator provides:
- Complete Bernoulli equation solution: All energy terms calculated
- Multiple unit systems: SI (Pa, m/s, m) and Imperial (psi, ft/s, ft)
- Common fluid presets: Water, air, oil, mercury densities
- Energy conservation check: Verifies total head remains constant
- Visual Bernoulli principle: Shows pressure-velocity relationship
- Detailed results: Pressure change, velocity change, height change
Practical examples showing Bernoulli's principle in action:
| Application | Point 1 | Point 2 | Bernoulli Effect | Result |
|---|---|---|---|---|
| Airplane Wing | Below wing: Slow air, high pressure | Above wing: Fast air, low pressure | Pressure difference creates lift | Airplane flies |
| Venturi Tube | Wide section: Slow flow, high P | Narrow section: Fast flow, low P | Pressure drop measures flow rate | Flow measurement |
| Carburetor | Air intake: Atmospheric pressure | Venturi throat: Low pressure | Sucks fuel into airstream | Fuel-air mixing |
| Chimney Draft | Inside: Hot air, low density | Outside: Cool air, high density | Pressure difference creates upward flow | Smoke drawn up |
| Atomizer | Air over tube: Fast, low P | Liquid in tube: Atmospheric P | Liquid drawn up and sprayed | Fine mist |
| Baseball Curve | One side: Smooth airflow | Other side: Turbulent airflow | Pressure difference deflects ball | Curveball |
| Roof Lift | Under roof: Still air | Over roof: Fast wind | Low pressure above lifts roof | Hurricane damage |
| Blood Flow | Wide artery: Slow flow | Narrow artery: Fast flow | Pressure drop in constriction | Atherosclerosis effect |
Static Pressure (P): Measured pressure in moving fluid
Dynamic Pressure (½ρv²): Pressure due to motion, converts to static when stopped
Hydrostatic Pressure (ρgh): Pressure due to weight of fluid above
Total Pressure: P + ½ρv² + ρgh = constant along streamline
Head: Pressure expressed as equivalent height of fluid
Below are answers to frequently asked questions about Bernoulli's equation:
Combine Bernoulli's equation with continuity equation for pipes of varying diameter:
Given: Pipe diameter reduces from 0.1m to 0.05m, P₁ = 200 kPa, v₁ = 2 m/s, ρ = 1000 kg/m³
Continuity: A₁v₁ = A₂v₂ → (π×0.05²)×2 = (π×0.025²)×v₂
v₂ = v₁ × (A₁/A₂) = 2 × (0.05²/0.025²) = 2 × 4 = 8 m/s
Bernoulli (horizontal, h constant): P₁ + ½ρv₁² = P₂ + ½ρv₂²
200,000 + ½×1000×2² = P₂ + ½×1000×8²
200,000 + 2,000 = P₂ + 32,000 → P₂ = 170,000 Pa = 170 kPa
30 kPa pressure drop due to velocity increase from 2 to 8 m/s.
Key insight: Pipe narrowing → velocity increase → pressure decrease. This principle powers carburetors, atomizers, venturi meters.
Real fluids have viscosity causing energy loss. Modified Bernoulli equation includes head loss (hₗ):
P₁/ρg + v₁²/2g + h₁ = P₂/ρg + v₂²/2g + h₂ + hₗ
Where hₗ = head loss due to friction (major losses) and fittings (minor losses)
Darcy-Weisbach equation: hₗ = f × (L/D) × (v²/2g)
f = friction factor (depends on Reynolds number and pipe roughness)
L = pipe length, D = pipe diameter, v = average velocity
Minor losses: hₗ = K × (v²/2g) where K = loss coefficient
Practical calculation: For water in smooth pipe: hₗ ≈ 0.02 × (L/D) × (v²/2g). Our calculator assumes ideal (frictionless) flow unless you manually subtract estimated losses.
Airplane lift combines Bernoulli principle and Newton's third law:
| Mechanism | Bernoulli Contribution | Newton Contribution | Total Lift |
|---|---|---|---|
| Wing Shape (Airfoil) | Air travels faster over curved top → lower pressure above (67%) | Air deflected downward by wing bottom (33%) | 100% lift force |
| Angle of Attack | Increased angle → more curvature effect | Increased angle → more air deflected down | Greatly increased lift |
| Wing Area | Larger area → larger pressure difference area | Larger area → more air deflected | Proportional increase |
| Air Speed | Lift ∝ v² (Bernoulli: P ∝ v²) | Lift ∝ v² (momentum change ∝ v) | Lift ∝ v² |
| Air Density | Lift ∝ ρ (Bernoulli: P ∝ ρ) | Lift ∝ ρ (more mass deflected) | Lift ∝ ρ |
Lift equation: L = ½ × ρ × v² × A × Cₗ
Where Cₗ = lift coefficient (depends on wing shape and angle of attack), A = wing area.
Example: Boeing 747: ρ = 1.225 kg/m³ (sea level), v = 250 m/s (takeoff), A = 511 m², Cₗ ≈ 1.5 → L ≈ 30 million Newtons (enough to lift 3000 metric tons).
Venturi meters use Bernoulli principle to measure flow without moving parts:
- Converging section: Pipe narrows, velocity increases, pressure decreases
- Throat: Minimum diameter, maximum velocity, minimum pressure
- Diverging section: Pipe widens, velocity decreases, pressure recovers
- Measurement: Pressure difference (P₁ - P₂) measured by manometer
- Calculation: Q = A₂ × √[2(P₁ - P₂)/ρ(1 - (A₂/A₁)²)]
- Calibration: Discharge coefficient C_d accounts for friction (~0.98 for smooth venturi)
Example: Water flow in 0.1m pipe with 0.05m throat, ΔP = 10 kPa, ρ = 1000 kg/m³
A₁ = 0.00785 m², A₂ = 0.00196 m², A₂/A₁ = 0.25, 1-(A₂/A₁)² = 0.9375
Q = 0.00196 × √[2×10000/(1000×0.9375)] = 0.00196 × √[21.33] = 0.00905 m³/s = 9.05 L/s
Bernoulli's equation applies only under specific ideal conditions:
| Assumption | Real-World Deviation | Correction Method | When Critical |
|---|---|---|---|
| Steady flow | Pulsating or transient flow | Use unsteady Bernoulli or computational methods | Heart valves, engine cycles |
| Incompressible fluid | Gases at high speed (Mach > 0.3) | Use compressible flow equations | Aircraft at high speed, gas pipelines |
| Frictionless (inviscid) | All real fluids have viscosity | Add head loss term (Darcy-Weisbach) | Long pipes, small diameters |
| Along streamline | Flow rotation or turbulence | Use different streamlines or average values | Bends, obstacles, mixing |
| No energy added/removed | Pumps, turbines, heat transfer | Add pump head or turbine work term | Piping systems with pumps |
| Constant density | Temperature variations, mixing | Use variable density or Boussinesq approx. | Hot water systems, stratified flows |
Modified Bernoulli (with pump/turbine):
P₁/ρg + v₁²/2g + h₁ + h_pump = P₂/ρg + v₂²/2g + h₂ + h_turbine + h_loss
Where h_pump = pump head added, h_turbine = turbine head extracted.
For gases at high velocity (Mach > 0.3), compressibility effects become significant:
For isentropic (adiabatic, reversible) flow of ideal gas:
v²/2 + γ/(γ-1) × P/ρ = constant along streamline
Where γ = c_p/c_v = specific heat ratio (1.4 for air)
Stagnation properties: When fluid is brought to rest isentropically:
P₀/P = [1 + (γ-1)/2 × M²]^(γ/(γ-1))
T₀/T = 1 + (γ-1)/2 × M²
ρ₀/ρ = [1 + (γ-1)/2 × M²]^(1/(γ-1))
Where M = Mach number = v/c, c = speed of sound = √(γRT)
Example - aircraft at Mach 0.8:
M = 0.8, γ = 1.4, P = 30 kPa (at altitude), T = 220 K
P₀/P = [1 + 0.2 × 0.8²]^(3.5) = [1.128]^(3.5) = 1.524
Stagnation pressure P₀ = 30 × 1.524 = 45.7 kPa
Our calculator assumes incompressible flow (valid for liquids and gases at low speed).