Bernoulli's Equation Calculator
Apply Bernoulli's principle to find the total mechanical energy per unit volume at a point in a flowing fluid. Used in aerodynamics, pipe flow analysis, and hydraulic engineering to relate pressure, velocity, and elevation.
About this calculator
Bernoulli's equation expresses the conservation of mechanical energy along a streamline for an incompressible, inviscid fluid in steady flow. The total energy per unit volume at any point equals: P + ½ρv² + ρgh = constant, where P is the static pressure (Pa), ρ is the fluid density (kg/m³), v is the flow velocity (m/s), g is gravitational acceleration (9.81 m/s²), and h is the elevation above a reference point (m). This calculator computes the total energy at Point 1 using: E₁ = P₁ + 0.5 × ρ × v₁² + ρ × 9.81 × h₁. In a frictionless system, this value equals the total energy at any other point along the streamline. The equation reveals the trade-off between pressure, kinetic energy, and potential energy — as velocity increases, pressure decreases, explaining how airplane wings generate lift and how venturi meters measure flow.
How to use
Suppose water (density = 1,000 kg/m³) flows through a pipe where Point 1 has a pressure of 200,000 Pa, a velocity of 3 m/s, and an elevation of 5 m. Compute each term: static pressure = 200,000 Pa; kinetic pressure = 0.5 × 1,000 × 3² = 4,500 Pa; potential pressure = 1,000 × 9.81 × 5 = 49,050 Pa. Total at Point 1: E₁ = 200,000 + 4,500 + 49,050 = 253,550 Pa. If Point 2 is at elevation 2 m with velocity 5 m/s, you can solve for P₂ = 253,550 − 12,500 − 19,620 = 221,430 Pa.
Frequently asked questions
What assumptions must be true for Bernoulli's equation to be valid?
Bernoulli's equation is derived under four key assumptions: the fluid must be incompressible (constant density), inviscid (no viscosity or friction losses), the flow must be steady (not changing with time), and the equation applies along a single streamline. In real-world applications, these assumptions are approximations. Viscous losses are ignored, which means the equation overestimates actual pressure at downstream points. For engineering systems with significant friction, the extended Bernoulli equation — which adds a head loss term — should be used instead.
How is Bernoulli's equation used to measure flow velocity in a pipe?
A venturi meter or pitot tube exploits Bernoulli's equation to measure flow velocity by comparing pressures at two points with different cross-sectional areas or stagnation conditions. In a venturi meter, the pipe narrows at the throat, increasing velocity and decreasing pressure. By measuring the pressure difference between the wide section and the throat, and knowing the geometry, you can solve Bernoulli's equation for the velocity. Pitot tubes measure the difference between stagnation pressure (where flow is brought to rest) and static pressure to directly calculate airspeed or water speed.
What is the difference between static pressure, dynamic pressure, and total pressure in Bernoulli's equation?
In Bernoulli's equation, static pressure (P) is the actual thermodynamic pressure exerted by the fluid on surfaces parallel to the flow — it is what a pressure gauge measures when flush-mounted to a pipe wall. Dynamic pressure (½ρv²) represents the kinetic energy per unit volume and is the additional pressure that would result if the flow were brought to rest. Total pressure (stagnation pressure) is the sum of static and dynamic pressure and represents the maximum pressure achievable by decelerating the fluid. When elevation is included, the total mechanical energy per unit volume equals P + ½ρv² + ρgh, and Bernoulli's principle states this sum is constant along a frictionless streamline.