Engineering ToolBox - Resources, Tools and Basic Information for Engineering and Design of Technical Applications!

Bernoulli Equation

Conservation of energy - non-viscous, incompressible fluid in steady flow

Sponsored Links

The statement of conservation of energy is useful when solving problems involving fluids. For a non-viscous, incompressible fluid in a steady flow, the sum of pressure, potential and kinetic energies per unit volume is constant at any point.

A special form of the Euler’s equation derived along a fluid flow streamline is often called the Bernoulli Equation:

Bernoulli equation

For steady state in-compressible flow the Euler equation becomes (1). If we integrate (1) along the streamline it becomes (2). (2) can further be modified to (3) by dividing by gravity.

Head of Flow

Equation (3) is often referred to the "head" because all elements has the unit of length.

Dynamic Pressure

(2) and (3) are two forms of the Bernoulli Equation for steady state in-compressible flow. If we assume that the gravitational body force is negligible, (3) can be written as (4). Both elements in the equation have the unit of pressure and it's common to refer the flow velocity component as the dynamic pressure of the fluid flow (5).

Since energy is conserved along the streamline, (4) can be expressed as (6). Using the equation we see that increasing the velocity of the flow will reduce the pressure, decreasing the velocity will increase the pressure.

This phenomena can be observed in a venturi meter where the pressure is reduced in the constriction area and regained after. It can also be observed in a pitot tube where the stagnation pressure is measured. The stagnation pressure is where the velocity component is zero.

Example - Bernoulli Equation and Flow from a Tank through a small Orifice

Liquid flows from a tank through a orifice close to the bottom. The Bernoulli equation can be adapted to a streamline from the surface (1) to the orifice (2) as (e1):

Bernoulli equation - flow out of tank example

Since (1) and (2)'s heights from a common reference is related as (e2), and the equation of continuity can be expressed as (e3), it's possible to transform (e1) to (e4).

Vented tank

A special case of interest for equation (e4) is when the orifice area is much lesser than the surface area and when the pressure inside and outside the tank is the same - when the tank has an open surface or "vented" to the atmosphere. At this situation the (e4) can be transformed to (e5).

"The velocity out from the tank is equal to speed of a freely body falling the distance h." - also known as Torricelli's Theorem.

Example - outlet velocity from a vented tank

The outlet velocity of a tank with height 10 m can be calculated as

V2 = (2 (9.81 m/s2) (10 m))1/2

    = 14 (m/s)

Pressurized Tank

If the tanks is pressurized so that product of gravity and height (g h) is much lesser than the pressure difference divided by the density, (e4) can be transformed to (e6). The velocity out of the tank depends mostly on the pressure difference.

Example - Outlet Velocity from a Pressurized Tank

The outlet velocity of a pressurized tank where

p1 = 0.2 (MN/m2)

p2 = 0.1 (MN/m2)

A2 / A1 = 0.01

h = 10 (m)

can be calculated as

V2 = ( (2 / (1 - (0.01)2) ((0.2 106 N/m2) - (0.1 106 N/m2)) / (1000 kg/m3) + (9.81 m/s2) (10 m)))1/2

        = 19.9 (m/s)

Coefficient of Discharge - Friction Coefficient

Due to friction the real velocity will be somewhat lower than this theoretic examples. If we introduce a friction coefficient c (coefficient of discharge), (e5) can be expressed as (e5b).

The coefficient of discharge can be determined experimentally. For a sharp edged opening it may bee as low as 0.6. For smooth orifices it may bee between 0.95 and 1.

Energy Loss through a Reduction Valve

When fluid flows through a reduction valve and pressure is reduced - there is a energy loss. By neglecting the change in elevation (h1 = h2) and the change in pipe velocity (v1 = v2) the pressure energy before the valve and the pressure energy after the valve including the energy loss through the valve - is constant, and the Bernouilli equation can be modified to

p1 / ρ = p2 / ρ + dE                                       (7)


dE = energy loss through valve (J)

(7) can be transformed to:

dE = (p1 - p2) / ρ

Sponsored Links

Related Topics

Related Documents

Tag Search

  • en: bernoulli euler flow height pressure
Sponsored Links

Search the Engineering ToolBox

Engineering ToolBox - SketchUp Extension - Online 3D modeling!

3D Engineering ToolBox Extension to SketchUp - add parametric components to your SketchUp model

Add standard and customized parametric components - like flange beams, lumbers, piping, stairs and more - to your Sketchup model with the Engineering ToolBox - SketchUp Extension - enabled for use with the amazing, fun and free SketchUp Make and SketchUp Pro .Add the Engineering ToolBox extension to your SketchUp from the SketchUp Pro Sketchup Extension Warehouse!

Translate this page to
About the Engineering ToolBox!


This page can be cited as

  • Engineering ToolBox, (2003). Bernoulli Equation. [online] Available at: [Accessed Day Mo. Year].

Modify access date.

Customize Ads in the ToolBox

Make ads more useful in Google Ad Settings .

. .


3D Engineering ToolBox - draw and model technical applications! 2D Engineering ToolBox - create and share online diagram drawing templates! Engineering ToolBox Apps - mobile online and offline engineering applications!

Scientific Online Calculator

Scientific Calculator

6 22

Sponsored Links