Bernoulli's Equation & Fluid Pressure Head Conservation Calculator

Fluid Mechanics: Bernoulli's Principle & The Venturi Effect

Formulated by Daniel Bernoulli in Hydrodynamica (1738), Bernoulli's principle states that for an incompressible, inviscid fluid in steady streamline flow, the sum of static pressure, dynamic pressure (kinetic energy), and hydrostatic pressure (potential energy) remains constant along any streamline:

• The Three Energy Forms:

• Static Pressure (P): The actual thermodynamic pressure exerted by the fluid on its container walls.

• Dynamic Pressure ((1) / (2)ρ v^2): Kinetic energy per unit volume representing the fluid's directed motion.

• Hydrostatic Elevation (ρ g h): Gravitational potential energy per unit volume.

• The Venturi Effect: When a fluid flows through a constricted throat, conservation of mass) forces velocity to increase. By Bernoulli's equation, an increase in velocity must be accompanied by a simultaneous drop in static pressure (the physical mechanism powering carburetors, aircraft wing lift, atomizers, and Venturi flowmeters).

• Cavitation Danger: If flow velocity becomes excessively high, static pressure can drop below the fluid's saturation vapor pressure for water at 20°C), causing the liquid to boil spontaneously and form vapor bubbles that violently collapse against pipe walls, causing severe pitting damage.

How to Use This Calculator

Enter your Variable to Solve for — choose from Solve for Downstream Pressure (P₂) [Venturi / Elevation Change], Solve for Downstream Velocity (v₂) [Nozzle / Efflux], Fluid Density (ρ in kg/m³) (e.g. 1000 (Water: 1000, Air: 1.225, Oil: 850)), Upstream Static Pressure (P₁ in kPa) (e.g. 200 (200 kPa = 2 bar)), Upstream Flow Velocity (v₁ in m/s) (e.g. 2.0), Upstream Height / Elevation (h₁ in meters) (e.g. 0), Downstream Flow Velocity (v₂ in m/s [when solving P₂]) (e.g. 5.0), Downstream Static Pressure (P₂ in kPa [when solving v₂]) (e.g. 160), and Downstream Height / Elevation (h₂ in meters) (e.g. 3.0). Results update instantly as you type — adjust any value to compare scenarios.

The Formula

Bernoulli's Principle (Daniel Bernoulli, 1738) \& Energy Conservation equations:

P_1 + (1) / (2)ρ v_1^2 + ρ g h_1 = P_2 + (1) / (2)ρ v_2^2 + ρ g h_2 = Constant (Total Stagnation Pressure)

Total Hydraulic Head (meters): H = (P) / (ρ g) + (v^2) / (2g) + h

Downstream Static Pressure: P_2 = P_1 + (1) / (2)ρ (v_1^2 - v_2^2) + ρ g (h_1 - h_2)

Downstream Velocity: v_2 = √(v_1^2 + (2(P_1 - P_2)) / (ρ) + 2g(h_1 - h_2))

Torricelli's Law (Tank Drain): v = √(2g h)

Worked Example

For example, water) flowing through a pipe at and \ enters a constricted nozzle at an elevation \ where velocity accelerates to Initial total energy is 2^2) + 0 = 200,000 + 2,000 = 202.00 kPa. Downstream dynamic pressure is (1) / (2)(1000)(5^2) = 12,500 Pa (12.5 kPa) and elevation head is \ (29.42 kPa). Downstream static pressure is .

Understanding Your Result

Evaluate your result in the context of your specific use case and relevant industry standards or reference values. A single calculated result is a starting point — test two to three scenarios to understand how sensitive the outcome is to your key input assumptions. For high-stakes decisions, verify your result against a secondary source or subject-matter expert before acting.