The Bernoulli Equation Calculator solves the fundamental fluid mechanics equation relating pressure, velocity, and elevation at two points along a streamline in ideal flow. Used for pipe flow analysis, venturi meter design, Pitot tube calculations, and any application requiring fluid energy balance.
112,858.25
21.60692
m
10.332275
m
1.274645
m
10
m
211,891.5
Pa
123.0665
m2/s2
112,858.25
21.60692
m
10.332275
m
1.274645
m
10
m
211,891.5
Pa
123.0665
m2/s2
A fluid moving faster has lower pressure — this counterintuitive relationship is the essence of Bernoulli's principle, and it underlies aircraft wing lift, venturi carburetors, the Pitot tube on every airplane, and the pressure drop in any pipe constriction. The calculator for the Bernoulli equation solves for any unknown variable (pressure, velocity, or elevation at either point) given the other five, enabling complete energy accounting along any streamline in ideal fluid flow.
For steady, incompressible, inviscid (frictionless) flow along a streamline:
P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂ = constant
where P is static pressure (Pa), ρ is fluid density (kg/m³), v is flow velocity (m/s), g = 9.81 m/s², and h is elevation above a datum (m). The three terms represent pressure energy, kinetic energy, and potential energy per unit volume respectively — Bernoulli's equation is simply the statement that their sum is conserved along a streamline. For a horizontal flow (h₁ = h₂): P₁ + ½ρv₁² = P₂ + ½ρv₂² — pressure decreases wherever velocity increases. For a 1 m/s water flow in a 100 mm pipe constricting to a 50 mm pipe: v₂ = v₁ × (D₁/D₂)² = 1 × (100/50)² = 4 m/s; ΔP = ½ × 1000 × (4² − 1²) = 7,500 Pa. Use this online calculator for any two-point comparison. The flow rate calculator provides the continuity equation complement to Bernoulli's pressure analysis.
Bernoulli's equation alone cannot determine both velocity and pressure at the constriction — it requires the continuity equation (conservation of mass) as a second constraint:
A₁ × v₁ = A₂ × v₂ (for incompressible flow)
Together, Bernoulli and continuity form the basis of venturi meter design. In a venturi meter, the measured pressure drop between the inlet and throat sections directly gives the flow rate: Q = A₂ × √(2ΔP / (ρ(1 − (A₂/A₁)²))). This is the principle behind fuel injectors, carburetors, and industrial flow measurement devices across chemical processing plants worldwide.
The Pitot tube converts a velocity measurement problem into a pressure measurement problem using Bernoulli's equation. At the stagnation point (where the flow is brought to rest), all kinetic energy converts to pressure: P_stagnation = P_static + ½ρv². Measuring the difference between stagnation pressure and static pressure gives dynamic pressure q = ½ρv², from which: v = √(2q/ρ). Aircraft airspeed indicators measure this pressure difference (in practice across the Pitot-static system) and display the result as indicated airspeed. The Reynolds number calculator and fluid dynamics calculators provide the complete fluid mechanics toolkit.
Bernoulli's equation assumes ideal conditions that are never perfectly met in real flows. The primary failure modes:
Bernoulli's equation states that for steady, incompressible, inviscid flow along a streamline:
$$P_1 + \frac{1}{2}\rho v_1^2 + \rho g h_1 = P_2 + \frac{1}{2}\rho v_2^2 + \rho g h_2$$
where:
Each term represents an energy per unit volume:
The sum of these three terms is constant along a streamline, forming the Bernoulli constant:
$$P + \frac{1}{2}\rho v^2 + \rho g h = \text{const}$$
Solving for unknown variables:
Bernoulli's equation assumes no energy losses from friction or turbulence. In real systems, the modified Bernoulli equation adds head loss terms to account for viscous dissipation.
The Bernoulli constant represents the total mechanical energy per unit volume of the fluid. When velocity increases (such as in a constriction), pressure must decrease — this is the Bernoulli effect. The calculator shows both the Bernoulli constant and the solved unknown value, allowing you to verify energy conservation along the streamline.
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Water draining from a tank with 5 m head exits at ~9.9 m/s (Torricelli's theorem: v = √(2gh)).
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When pipe flow accelerates from 2 m/s to 8 m/s, pressure drops by 30 kPa — the Bernoulli effect.
Bernoulli's equation assumes: (1) steady flow, (2) incompressible fluid, (3) inviscid (no friction), and (4) flow along a single streamline. Violations of these assumptions require corrections or different equations.
Yes, for low-speed gas flows (Mach < 0.3) where compressibility effects are negligible. At higher speeds, compressible flow equations must be used instead.
The Bernoulli effect states that an increase in fluid velocity occurs simultaneously with a decrease in pressure. This principle explains how airplane wings generate lift and how atomizers work.
Torricelli's theorem is a special case of Bernoulli's equation where both points are at atmospheric pressure and the tank surface velocity is approximately zero. This gives v = √(2gh) for the exit velocity.
A zero result for velocity means the expression under the square root is negative, which indicates the specified conditions are physically impossible — the pressure and height differences cannot sustain flow. Check your input values.
Use the extended Bernoulli equation that includes a head loss term: P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂ + h_loss, where h_loss is calculated using the Darcy-Weisbach equation.
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