Bernoulli Equation Calculator

Bernoulli Equation Solver

Quick Application Presets

Load pre-configured scenarios to see Bernoulli's equation applied to real-world situations. Each preset fills in known values and solves for the appropriate unknown.

Understanding Bernoulli's Equation

Bernoulli's equation is a statement of energy conservation for flowing fluids. Published by Daniel Bernoulli in 1738 in his work Hydrodynamica, it connects three forms of mechanical energy along a simplify: pressure energy, kinetic energy, and potential energy.

For a steady, incompressible, frictionless flow along a single simplify, the equation states that the sum of static pressure, adaptable pressure, and hydrostatic pressure remains constant between any two points.

This relationship is written as:

When comparing two points along the same simplify, this becomes the working form used in this calculator:

Each variable represents a measurable physical quantity: P is the static pressure in Pascals, ρ is the fluid density in kg/m³, v is the flow velocity in m/s, g is gravitational acceleration (9.81 m/s²), and h is the elevation above a chosen reference datum in meters.

The equation can also be expressed in "head" form by dividing every term by ρg. This converts all terms to units of length (meters of fluid column), which is common in hydraulic engineering and pump specifications.

The Three Pressure Terms

Each term in Bernoulli's equation has units of pressure (Pascals in SI) and represents a distinct form of fluid energy per unit volume.

Static Pressure (P)

The actual thermodynamic pressure of the fluid at a given point. This is what a pressure gauge moving with the flow would measure. It acts equally in all directions and is the pressure that would push on a surface immersed in the fluid. In pipe flow, it is read by a wall tap perpendicular to the flow direction.

adaptable Pressure (½ρv²)

The kinetic energy per unit volume of fluid. This term accounts for the energy carried by the bulk motion of the fluid. When fluid speeds up, adaptable pressure increases; when it slows down, adaptable pressure decreases. The sum of static and adaptable pressure is called the total pressure or stagnation pressure. At a stagnation point (v = 0), all adaptable pressure converts to static pressure.

Hydrostatic Pressure (ρgh)

The gravitational potential energy per unit volume. This term matters whenever there is an elevation change between the two points being analyzed. For horizontal flows where h₁ equals h₂, this term cancels out and the equation simplifies to P₁ + ½ρv₁² = P₂ + ½ρv₂². For liquids, the hydrostatic term is significant even over small height differences due to the high density. For gases, it is often negligible unless the height difference is large.

Venturi Effect Applications

The Venturi effect is one of the most practical demonstrations of Bernoulli's principle. When a fluid flows through a section of pipe that narrows (the throat), the continuity equation (A₁v₁ = A₂v₂) dictates that velocity must increase in the narrower section. Bernoulli's equation then requires that static pressure must decrease correspondingly.

This pressure drop in a constriction powers many engineering devices:

• Venturi flow meters measure flow rate by reading the pressure difference between the full pipe section and the throat section. Accuracy is typically within 1 to 2 percent with proper installation.
• Carburetors draw fuel into an airstream by creating a low-pressure zone at the throat of a Venturi tube. The fuel-air ratio depends on the throat geometry.
• Aspirators and ejectors create suction by passing a high-speed jet past an opening connected to a chamber that needs evacuation.
• Spray nozzles atomize liquid by accelerating air past a liquid feed tube, pulling liquid upward into the airstream.

The Venturi flow rate can be calculated as Q = C_d · A₂ · √(2ΔP / (ρ(1 - (A₂/A₁)²))), where C_d is the discharge coefficient, typically 0.95 to 0.99 for a well-designed Venturi tube with gradual convergence and divergence angles.

Pitot Tube Measurements

A Pitot tube (named after Henri Pitot, 1732) determines fluid velocity from pressure readings. The device has two ports: one facing directly into the flow (measuring stagnation pressure) and one perpendicular to the flow (measuring static pressure). Modern Pitot-static tubes combine both measurements in a single probe.

At the stagnation point, the flow velocity is zero. Applying Bernoulli's equation between the free stream (point 1) and the stagnation point (point 2, where v₂ = 0, same height):

In aircraft, the Pitot-static system feeds differential pressure to the airspeed indicator. Standard atmosphere corrections are applied for altitude since air density decreases with elevation. The indicated airspeed (IAS) differs from true airspeed (TAS) at altitude, requiring density correction: TAS = IAS · √(ρ₀/ρ_actual).

For precise Pitot tube readings, the tube axis must be aligned within about 15 degrees of the flow direction. Larger misalignment requires correction factors or multi-hole pressure probes. Heated Pitot tubes prevent ice blockage on aircraft, a safety-critical consideration.

Airplane Lift and Aerodynamics

Bernoulli's principle contributes to explaining how wings generate lift. An airfoil shape causes air passing over the upper surface to accelerate. According to Bernoulli's equation, this higher velocity results in lower pressure on the upper surface compared to the lower surface.

The pressure difference integrated over the wing area produces a net upward force. The total lift force is given by:

Where A is the wing planform area and C_L is the lift coefficient. Typical C_L values range from 0.2 during cruise to 1.5 or higher with flaps deployed. The angle of attack (the angle between the wing chord line and the oncoming airflow) directly affects C_L up to the stall angle.

The full explanation of lift also involves the reaction force from deflecting the airstream downward (Newton's third law). Both the pressure-difference mechanism (Bernoulli) and the flow-deflection mechanism (Newton) act together. Neither explanation alone is complete.

Torricelli's Theorem

Torricelli's theorem is a direct simplification of Bernoulli's equation for the case of a large tank draining through a small orifice. Named after Evangelista Torricelli (1608 to 1647), it states that the velocity of fluid exiting an orifice at depth h below the fluid surface is:

This result follows from setting P₁ = P₂ (both at atmospheric pressure), v₁ = 0 (the tank surface barely moves if the tank is much larger than the orifice), and h₁ - h₂ = h (the height difference between the surface and the orifice).

The resulting exit velocity is exactly the same as the speed a freely falling object would reach after dropping from height h. This is not a coincidence. The fluid accelerates under gravity, converting potential energy to kinetic energy, just as a falling object does.

For real orifices, a discharge coefficient C_d accounts for friction and contraction effects. Typical values: sharp-edged orifice C_d = 0.61, short tube C_d = 0.82, well-rounded nozzle C_d = 0.97. The actual flow rate becomes Q = C_d · A_orifice · √(2gh).

Applications include tank drainage time calculations, dam spillway design, fire hydrant flow estimation, and sizing drain valves on process vessels.

Common Fluid Properties

precise density values are needed for Bernoulli calculations. The table below lists density at standard conditions (20°C and 1 atm unless noted). Temperature and pressure changes affect gas density significantly and liquid density slightly.

Use the fluid preset dropdown in the calculator above to automatically load the correct density for common fluids. For gases at non-standard conditions, calculate density using the ideal gas law: ρ = PM / (RT), where M is molar mass, R is the universal gas constant, and T is absolute temperature.

Limitations and Assumptions

Bernoulli's equation is a effective analysis tool, but it has clear boundaries where it produces inaccurate results. Knowing when not to apply it is as important as knowing how to use it.

• Steady flow only: The equation applies to flows that do not change with time. For flows that vary (such as pulsating pump output or water hammer), the unsteady Bernoulli equation adds a time-dependent integral term.
• Incompressible fluids: Density must remain constant. For gases above Mach 0.3 (roughly 100 m/s in air), compressibility effects become significant and the standard equation gives wrong answers.
• Along a simplify: The equation connects two points on the same simplify. Comparing points on different simplifies requires the flow to be irrotational (zero vorticity), a condition met in many practical external flows.
• No viscous losses: Real fluids lose energy to friction. For pipe flows, the modified Bernoulli equation (the energy equation) adds a head loss term h_L to the downstream side. Friction losses are computed using the Darcy-Weisbach equation and Moody chart.
• No work interactions: Pumps add energy and turbines remove energy. When these are present, pump head h_p or turbine head h_t terms must be included.

Worked Examples

Example 1: Horizontal Pipe Constriction

Water flows through a horizontal pipe that narrows from 10 cm diameter to 5 cm diameter. The upstream pressure is 200 kPa and the upstream velocity is 1.5 m/s. Find the downstream pressure.

First, find v₂ using continuity: v₂ = v₁ · (A₁/A₂) = v₁ · (d₁/d₂)² = 1.5 × (10/5)² = 6.0 m/s.

Apply Bernoulli (horizontal, so h terms cancel):

P₂ = P₁ + ½ρ(v₁² − v₂²) = 200,000 + 0.5 × 998.2 × (2.25 − 36) = 200,000 − 16,847 = 183,153 Pa ≈ 183.2 kPa.

The pressure dropped by about 16.8 kPa due to the velocity increase in the constriction.

Example 2: Water Tank Drain (Torricelli)

A large open tank has water filled to 4 meters above an orifice at the bottom. Find the ideal exit velocity.

By Torricelli's theorem: v = √(2 × 9.81 × 4) = √78.48 = 8.86 m/s.

For a sharp-edged orifice (C_d = 0.61) with 5 cm diameter: Q = 0.61 × π(0.025)² × 8.86 = 0.01062 m³/s = 10.6 L/s.

Example 3: Airspeed from Pitot Tube

A Pitot tube on an aircraft at sea level reads a stagnation pressure of 102,500 Pa and a static pressure of 101,325 Pa. Find the airspeed.

v = √(2 × (102,500 − 101,325) / 1.204) = √(2 × 1,175 / 1.204) = √1,952 = 44.2 m/s = 159 km/h.

Engineering Applications

Bernoulli's equation is used across nearly every engineering discipline that involves fluid motion:

• Chemical engineering: Sizing pipes and valves, designing reactor feed systems, analyzing flow distribution in heat exchangers and distillation columns.
• Civil engineering: Open-channel flow analysis, spillway design, water distribution network sizing, and stormwater management.
• Mechanical engineering: Pump selection and system curve analysis, HVAC duct sizing, nozzle design, and flow measurement instrumentation.
• Aerospace engineering: Airfoil pressure distribution, wind tunnel data analysis, and Pitot-static airspeed instrumentation.
• Biomedical engineering: Blood flow analysis in arteries and heart valves (with pulsatile flow corrections), and medical device design for dialysis and oxygenation equipment.
• Environmental engineering: Groundwater flow modeling, leachate collection system design, and atmospheric dispersion analysis.

For pipe system design that accounts for friction losses, engineers typically pair Bernoulli's equation with the Darcy-Weisbach equation. The Moody diagram provides friction factors for different pipe roughness values and Reynolds numbers. Try our Reynolds number calculator to determine the flow regime for your system.

References and Standards

• NASA Glenn Research Center - Bernoulli's Equation (educational explanation with interactive demonstrations)
• MIT OpenCourseWare - 2.06 Fluid Dynamics, Spring 2013 (full lecture notes and problem sets covering Bernoulli's equation derivations)
• Munson, Young, Okiishi - Fundamentals of Fluid Mechanics, 8th Edition, Wiley (textbook treatment of Bernoulli's equation in Chapter 3)
• ASME MFC-3M - Measurement of Fluid Flow in Pipes Using Orifice, Nozzle, and Venturi (industry standard for differential pressure flow measurement)

Frequently Asked Questions

What is Bernoulli's equation?

Bernoulli's equation states that for an ideal, incompressible fluid flowing along a simplify, the total mechanical energy per unit volume remains constant. It is written as P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂, relating static pressure, adaptable pressure, and hydrostatic pressure between two points.

What assumptions does Bernoulli's equation require?

The equation requires steady-state flow, incompressible fluid (constant density), inviscid flow (no viscous friction losses), and analysis along a single simplify. For real applications, correction factors and additional loss terms handle deviations from these ideal conditions.

Can I use this calculator for air flow?

Yes, for air flowing below about Mach 0.3 (approximately 100 m/s or 225 mph at sea level). At these speeds, air behaves as an effectively incompressible fluid and Bernoulli's equation gives precise results. Select "Air (sea level)" from the fluid preset dropdown to load the correct density.

How does the Venturi effect work?

When fluid flows through a constriction, the velocity increases (by conservation of mass) and the pressure decreases (by Bernoulli's principle). This pressure drop in a narrowed section is the Venturi effect, used in carburetors, flow meters, and aspirators.

What is Torricelli's theorem?

A special case of Bernoulli's equation for fluid draining from a tank. The exit velocity at an orifice with fluid depth h above it equals v = √(2gh). This is the same speed a freely falling object would reach after falling from height h.

How precise is this calculator?

This calculator gives exact results for ideal (inviscid, incompressible, steady) flow conditions. For real pipe flows, actual pressures and velocities differ by 2 to 20 percent due to friction losses, depending on pipe length, roughness, and Reynolds number. Apply discharge coefficients for orifice and nozzle calculations.

What is adaptable pressure?

adaptable pressure (½ρv²) represents the kinetic energy per unit volume of a moving fluid. It quantifies the pressure increase when a flow is brought to rest (as at the tip of a Pitot tube). adaptable pressure is always positive and increases with the square of velocity.

Why does faster air have lower pressure?

In a flowing system where total energy is conserved, an increase in kinetic energy (velocity) must be offset by a decrease in pressure energy. The pressure gradient is what accelerates the fluid in the first place, and the resulting higher velocity corresponds to lower static pressure. It is a consequence of energy conservation, not a standalone cause-and-effect rule.