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Bernoulli Equation Calculator
Calculate fluid pressure, velocity, and height using Bernoulli's principle
Bernoulli Equation Calculator
Calculate fluid pressure, velocity, and height using Bernoulli's principle
Density of the fluid (water: 1000 kg/m³, air: 1.225 kg/m³)
Point 1 (Reference)
Pressure at point 1 (1 atm = 101,325 Pa)
Flow velocity at point 1
Elevation height at point 1
Point 2 (Target)
Flow velocity at point 2
Elevation height at point 2
Enter fluid parameters and click Calculate to see results
About Bernoulli Equation Calculator
Bernoulli Equation Calculator
What is the Bernoulli Principle?
The Bernoulli equation, derived from Daniel Bernoulli's work in the 18th century, is a fundamental principle in fluid dynamics that describes the conservation of energy in flowing fluids. It states that for an inviscid (frictionless), incompressible fluid in steady flow, the sum of pressure energy, kinetic energy, and potential energy remains constant along a streamline.
The Bernoulli Equation Formula
P + ½ρv² + ρgh = constantVariables Explained
- • P = Static pressure (Pa)
- • ρ = Fluid density (kg/m³)
- • v = Flow velocity (m/s)
- • g = Gravity (9.81 m/s²)
- • h = Elevation height (m)
Energy Components
- • P = Pressure energy
- • ½ρv² = Kinetic energy
- • ρgh = Potential energy
Common Units
- • Pressure: Pascals (Pa), psi, bar
- • Velocity: m/s, ft/s, knots
- • Density: kg/m³, slugs/ft³
- • Height: meters, feet
Key Assumptions of Bernoulli's Equation
Required Conditions
- • Steady flow (no time dependence)
- • Incompressible fluid (constant density)
- • Inviscid flow (no friction losses)
- • Flow along a streamline
- • No energy added/removed
When It Doesn't Apply
- • Turbulent or highly viscous flows
- • Compressible gases at high speeds
- • Flows with significant heat transfer
- • Rotating reference frames
- • Across shock waves
Practical Applications
Aerospace Engineering
- • Airfoil lift calculations
- • Venturi meter designs
- • Pitot tube airspeed measurement
- • Wind tunnel analysis
Civil Engineering
- • Water distribution systems
- • Dam and spillway design
- • Pipe network analysis
- • Pump station design
Medical Applications
- • Blood flow in arteries
- • Respiratory airflow
- • Venturi masks for oxygen
- • Aneurysm risk assessment
Extended Bernoulli Equation
For real-world applications, engineers often use the extended Bernoulli equation that accounts for energy losses and additions:
P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂ + ρghL - ρghPAdditional Terms
- • hL = Head loss (friction)
- • hP = Pump head added
- • 1,2 = Two points in system
Head Loss Calculations
- • Darcy-Weisbach equation
- • Hazen-Williams formula
- • Minor losses (fittings, valves)
- • Turbulent vs laminar flow
Common Problem Types
Venturi Effect Problems
A horizontal pipe narrows from 0.5m to 0.2m diameter. If velocity at wide section is 2m/s and pressure is 50kPa, find narrow section pressure.
Solution approach: Use continuity equation first to find v₂, then apply Bernoulli.
Tank Drainage Problems
A 10m tall tank with 2m diameter has a 5cm hole at bottom. Calculate exit velocity and flow rate.
Solution approach: Apply Torricelli's law (special Bernoulli case).
Pump Power Calculations
A pump moves water 20m uphill through 100m of 15cm pipe at 0.03m³/s. Calculate required pump power.
Solution approach: Calculate head loss first, then power = ρgQhP.
Aircraft Lift Problems
Air moves 15% faster over a wing's top surface than bottom. If ambient pressure is 101kPa and vbottom=90m/s, find lift pressure.
Solution approach: Assume same elevation, solve for ΔP.
Limitations and Modifications
Compressible Flow
For gases at high speeds (Mach > 0.3), use the compressible Bernoulli equation with density as a function of pressure. The energy equation must account for temperature changes and enthalpy.
Viscous Effects
Real fluids experience viscous losses described by the Navier-Stokes equations. Engineers use Moody charts and friction factors to estimate these losses in the extended Bernoulli equation.
Unsteady Flow
For transient conditions like water hammer or pulsatile flow, additional terms accounting for local acceleration must be included, leading to the unsteady Bernoulli equation.
Rotational Effects
In rotating systems like turbomachinery, centrifugal forces introduce additional terms. The Euler turbine equation extends Bernoulli principles for these applications.
Historical Context
Daniel Bernoulli published his principle in 1738 in "Hydrodynamica," establishing the relationship between pressure and velocity. His work built on earlier concepts from Leonardo da Vinci and Evangelista Torricelli. The equation gained practical importance in the 19th century with the development of hydraulic engineering and aerodynamics.
Key Milestones
- • 1738: Original Bernoulli equation published
- • 1843: Saint-Venant and Wantzel extend to viscous flow
- • 1904: Prandtl develops boundary layer theory
- • 1930s: Widespread use in aeronautics
Modern Applications
- • CFD software validation
- • Microfluidics device design
- • Cardiovascular flow modeling
- • Renewable energy systems
Using Our Bernoulli Equation Calculator
Our calculator simplifies Bernoulli equation calculations for both students and professionals. Enter known values for any two points in a system, and the calculator will solve for the remaining variables. The tool handles:
- Unit conversions between metric and imperial systems
- Solutions for pressure, velocity, or elevation changes
- Basic head loss estimations
- Common fluid properties database
- Multiple calculation modes (simple, extended, compressible)
For complex systems with significant energy losses or compressibility effects, consider using our advanced fluid dynamics calculator.
Frequently Asked Questions
What is the Bernoulli equation?
The Bernoulli equation states that for an ideal fluid in steady flow, the sum of pressure, kinetic energy per unit volume, and potential energy per unit volume remains constant along a streamline.
What is the mathematical form of the Bernoulli equation?
P + ½ρv² + ρgh = constant, where P is pressure, ρ is density, v is velocity, g is gravity, and h is height.
What are the assumptions of the Bernoulli equation?
The equation assumes steady flow, incompressible fluid, no viscosity, no heat transfer, and flow along a streamline.
How does pressure change with velocity in the Bernoulli equation?
According to Bernoulli's principle, as fluid velocity increases, pressure decreases, and vice versa.
What is the relationship between pressure and height in the Bernoulli equation?
Pressure increases with depth according to the hydrostatic pressure term ρgh in the Bernoulli equation.
How do I calculate pressure difference using the Bernoulli equation?
Pressure difference equals the change in kinetic and potential energy terms: ΔP = ½ρ(v₁² - v₂²) + ρg(h₁ - h₂).
What is the significance of the constant in the Bernoulli equation?
The constant represents the total mechanical energy per unit volume, which is conserved along a streamline in ideal flow.
How does the Bernoulli equation apply to pipe flow?
In pipe flow, the Bernoulli equation relates pressure, velocity, and elevation changes between different points in the pipe.
What is the effect of pipe diameter on pressure in the Bernoulli equation?
When pipe diameter decreases, velocity increases and pressure decreases according to the continuity equation and Bernoulli's principle.
How do I use the Bernoulli equation for venturi meters?
Venturi meters use the Bernoulli equation to measure flow rate by creating a pressure difference through a constriction.
What is the relationship between the Bernoulli equation and lift?
The Bernoulli equation explains how pressure differences around airfoils create lift forces in aerodynamics.
How does the Bernoulli equation apply to water flow from a tank?
For water flowing from a tank, the Bernoulli equation relates the height of water to the exit velocity: v = √(2gh).
What is the effect of viscosity on the Bernoulli equation?
Viscosity causes energy losses, making the Bernoulli equation less accurate for real fluids with significant viscous effects.
How do I calculate velocity from pressure difference using Bernoulli?
Velocity can be calculated from pressure difference: v = √(2ΔP/ρ), assuming no height change.
What is the relationship between the Bernoulli equation and energy conservation?
The Bernoulli equation is a statement of energy conservation for flowing fluids, equating total mechanical energy at different points.
How does the Bernoulli equation apply to blood flow?
The Bernoulli equation helps explain pressure changes in blood vessels, though blood's viscosity requires modifications to the ideal equation.
What is the effect of compressibility on the Bernoulli equation?
For compressible fluids like gases, the Bernoulli equation must be modified to account for density changes with pressure.
How do I use the Bernoulli equation for flow measurement?
Flow measurement devices like pitot tubes and orifice meters use the Bernoulli equation to relate pressure differences to flow rates.
What is the relationship between the Bernoulli equation and the continuity equation?
The continuity equation (A₁v₁ = A₂v₂) is often used with the Bernoulli equation to solve for unknown flow parameters.
How does the Bernoulli equation apply to wind flow around buildings?
The Bernoulli equation explains pressure differences around buildings that can affect ventilation and structural loads.
What is the effect of turbulence on the Bernoulli equation?
Turbulence creates energy losses and pressure fluctuations that make the Bernoulli equation less accurate for turbulent flow.
How do I calculate pressure at different heights using Bernoulli?
Pressure at different heights can be calculated using the hydrostatic term: P₂ = P₁ + ρg(h₁ - h₂).
What is the relationship between the Bernoulli equation and the energy equation?
The Bernoulli equation is a simplified form of the energy equation for incompressible, inviscid flow.
How does the Bernoulli equation apply to aircraft wings?
The Bernoulli equation explains how pressure differences between the upper and lower surfaces of wings create lift.
What is the effect of flow direction on the Bernoulli equation?
The Bernoulli equation applies along streamlines, so flow direction affects how the equation is applied between different points.
How do I use the Bernoulli equation for pump calculations?
The Bernoulli equation helps calculate the work required by pumps to move fluids between different elevations and pressures.
What is the relationship between the Bernoulli equation and the momentum equation?
The Bernoulli equation is derived from the momentum equation for steady, inviscid flow along streamlines.
How does the Bernoulli equation apply to water jets?
Water jets use the Bernoulli equation to relate nozzle pressure to exit velocity and flow rate.
What is the effect of surface tension on the Bernoulli equation?
Surface tension can create additional pressure effects that may need to be considered in addition to the Bernoulli equation.
How do I calculate flow rate using the Bernoulli equation?
Flow rate can be calculated by combining the Bernoulli equation with the continuity equation to solve for velocity and area.
What is the relationship between the Bernoulli equation and the Navier-Stokes equation?
The Bernoulli equation is a simplified form of the Navier-Stokes equation for inviscid, incompressible flow.
How does the Bernoulli equation apply to siphon flow?
Siphon flow uses the Bernoulli equation to explain how fluid can flow uphill due to pressure differences.
What is the effect of temperature on the Bernoulli equation?
Temperature affects fluid density and viscosity, which can influence the accuracy of the Bernoulli equation.
How do I use the Bernoulli equation for pressure drop calculations?
Pressure drop can be calculated using the Bernoulli equation by comparing total energy at two points in a flow system.
What is the relationship between the Bernoulli equation and the Euler equation?
The Bernoulli equation is derived from the Euler equation for steady, inviscid flow along streamlines.
How does the Bernoulli equation apply to wind turbines?
The Bernoulli equation helps explain pressure changes around wind turbine blades that generate power.
What is the effect of flow separation on the Bernoulli equation?
Flow separation creates regions where the Bernoulli equation doesn't apply due to recirculation and energy losses.
How do I calculate velocity head using the Bernoulli equation?
Velocity head equals v²/(2g) and represents the kinetic energy per unit weight of the fluid.
What is the relationship between the Bernoulli equation and the energy grade line?
The energy grade line represents the total energy per unit weight and is constant for ideal flow described by the Bernoulli equation.
How does the Bernoulli equation apply to rocket propulsion?
The Bernoulli equation helps explain pressure changes in rocket nozzles that create thrust.
What is the effect of cavitation on the Bernoulli equation?
Cavitation occurs when pressure drops below vapor pressure, creating conditions where the Bernoulli equation may not apply.
How do I use the Bernoulli equation for orifice flow?
Orifice flow uses the Bernoulli equation to relate the pressure difference across an orifice to the flow rate.
What is the relationship between the Bernoulli equation and the hydraulic grade line?
The hydraulic grade line represents pressure head plus elevation head and changes with velocity according to the Bernoulli equation.
How does the Bernoulli equation apply to water hammer?
Water hammer creates pressure waves that temporarily violate the steady flow assumption of the Bernoulli equation.
What is the effect of flow curvature on the Bernoulli equation?
Flow curvature can create additional pressure gradients that may need to be considered beyond the Bernoulli equation.
How do I calculate pressure recovery using the Bernoulli equation?
Pressure recovery can be calculated by comparing pressure at the entrance and exit of a diffuser using the Bernoulli equation.
What is the relationship between the Bernoulli equation and the Froude number?
The Froude number indicates the relative importance of inertial to gravitational forces in open channel flow.
How does the Bernoulli equation apply to wave motion?
Wave motion involves unsteady flow, so the Bernoulli equation must be modified to include time-dependent terms.
What is the effect of boundary layers on the Bernoulli equation?
Boundary layers create viscous effects that make the Bernoulli equation less accurate near solid surfaces.
How do I use the Bernoulli equation for flow through contractions?
Flow through contractions uses the Bernoulli equation to relate pressure and velocity changes in converging sections.
What is the relationship between the Bernoulli equation and the Reynolds number?
The Reynolds number indicates the relative importance of inertial to viscous forces in determining flow behavior.
How does the Bernoulli equation apply to atmospheric pressure changes?
Atmospheric pressure changes with altitude can be described using a form of the Bernoulli equation for static fluids.
What is the effect of flow unsteadiness on the Bernoulli equation?
Unsteady flow requires additional terms in the Bernoulli equation to account for time-dependent effects.
How do I calculate energy losses using the Bernoulli equation?
Energy losses can be calculated as the difference between the ideal Bernoulli equation prediction and actual measurements.
What is the relationship between the Bernoulli equation and the Mach number?
The Mach number indicates compressibility effects that require modifications to the Bernoulli equation for high-speed gas flow.
How does the Bernoulli equation apply to flow through expansions?
Flow through expansions uses the Bernoulli equation to relate pressure recovery and velocity changes in diverging sections.
What is the effect of flow rotation on the Bernoulli equation?
Flow rotation can create additional pressure gradients that may need to be considered beyond the Bernoulli equation.
How do I use the Bernoulli equation for flow measurement in open channels?
Open channel flow measurement uses modified forms of the Bernoulli equation that account for free surface effects.
What is the relationship between the Bernoulli equation and the Weber number?
The Weber number indicates the relative importance of inertial to surface tension forces in fluid flow.
How does the Bernoulli equation apply to flow through porous media?
Flow through porous media requires modifications to the Bernoulli equation to account for resistance and tortuosity effects.
What is the effect of flow stratification on the Bernoulli equation?
Flow stratification can create density variations that affect the application of the Bernoulli equation.
How do I calculate pressure distribution using the Bernoulli equation?
Pressure distribution can be calculated by applying the Bernoulli equation between different points in a flow field.
What is the relationship between the Bernoulli equation and the Strouhal number?
The Strouhal number characterizes unsteady flow phenomena that may require modifications to the Bernoulli equation.
How does the Bernoulli equation apply to flow through valves?
Flow through valves uses the Bernoulli equation to relate pressure drops to flow rates and valve characteristics.
What is the effect of flow compressibility on the Bernoulli equation?
Compressible flow requires modifications to the Bernoulli equation to account for density changes with pressure.
How do I use the Bernoulli equation for flow visualization?
Flow visualization techniques often rely on pressure differences predicted by the Bernoulli equation to show flow patterns.
What is the relationship between the Bernoulli equation and the Knudsen number?
The Knudsen number indicates when molecular effects become important, requiring modifications to continuum flow equations.
How does the Bernoulli equation apply to flow through filters?
Flow through filters uses the Bernoulli equation to relate pressure drops across filter media to flow rates.
What is the effect of flow pulsation on the Bernoulli equation?
Flow pulsation creates unsteady effects that require modifications to the Bernoulli equation for accurate predictions.
How do I calculate flow efficiency using the Bernoulli equation?
Flow efficiency can be calculated by comparing actual energy losses to ideal predictions from the Bernoulli equation.
What is the relationship between the Bernoulli equation and the Peclet number?
The Peclet number indicates the relative importance of convective to diffusive transport in fluid flow.
How does the Bernoulli equation apply to flow through heat exchangers?
Flow through heat exchangers uses the Bernoulli equation to relate pressure drops to flow rates and heat transfer performance.
What is the effect of flow mixing on the Bernoulli equation?
Flow mixing can create additional energy losses and pressure changes that affect the application of the Bernoulli equation.
How do I use the Bernoulli equation for flow control?
Flow control systems use the Bernoulli equation to predict how pressure and velocity changes affect flow rates and system performance.
What is the relationship between the Bernoulli equation and the Biot number?
The Biot number indicates the relative importance of convective to conductive heat transfer in fluid-solid systems.
How does the Bernoulli equation apply to flow through pumps?
Flow through pumps uses the Bernoulli equation to relate the work input to pressure and velocity changes in the fluid.
What is the effect of flow separation on pressure recovery?
Flow separation can reduce pressure recovery compared to ideal predictions from the Bernoulli equation.
How do I calculate flow stability using the Bernoulli equation?
Flow stability can be analyzed by examining how pressure and velocity changes predicted by the Bernoulli equation affect system behavior.
What is the relationship between the Bernoulli equation and the Grashof number?
The Grashof number indicates the relative importance of buoyancy to viscous forces in natural convection flows.
How does the Bernoulli equation apply to flow through turbines?
Flow through turbines uses the Bernoulli equation to relate pressure drops to power extraction and efficiency.
What is the effect of flow compressibility on pressure measurements?
Flow compressibility can affect pressure measurements and the accuracy of the Bernoulli equation for gas flows.
How do I use the Bernoulli equation for flow optimization?
Flow optimization uses the Bernoulli equation to identify conditions that minimize energy losses and maximize system performance.
What is the relationship between the Bernoulli equation and the Nusselt number?
The Nusselt number characterizes convective heat transfer, which can affect fluid properties and Bernoulli equation applications.
How does the Bernoulli equation apply to flow through compressors?
Flow through compressors uses the Bernoulli equation to relate work input to pressure and velocity changes in compressible flow.
What is the effect of flow turbulence on pressure fluctuations?
Flow turbulence creates pressure fluctuations that can affect the accuracy of steady-state Bernoulli equation predictions.
How do I calculate flow resistance using the Bernoulli equation?
Flow resistance can be calculated by comparing actual pressure drops to ideal predictions from the Bernoulli equation.
What is the relationship between the Bernoulli equation and the Prandtl number?
The Prandtl number indicates the relative importance of momentum to thermal diffusivity in fluid flow.
How does the Bernoulli equation apply to flow through nozzles?
Flow through nozzles uses the Bernoulli equation to relate pressure changes to velocity increases and flow acceleration.
What is the effect of flow curvature on pressure distribution?
Flow curvature can create pressure gradients that affect the application of the Bernoulli equation in curved flow paths.
How do I use the Bernoulli equation for flow analysis in complex systems?
Complex flow systems can be analyzed by applying the Bernoulli equation between different points while accounting for energy losses and system interactions.
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