Bernoulli Equation ( Head of Flow)

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 incompressible 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)



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 from 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