Aula Teórica 12
Equação de Bernoulli
Bernoulli’s Equation
• Let us consider a Stream pipe such as indicated in
the figure and an ideal fluid
(without viscosity) .
• Using the mass and
momentum conservation
principles,
• obtain an equation relating
the energy in two sections.
Mass conservation
• Being a stream pipe there is flow across the
tops only.
Bernoulli Equation Requirements
•
•
•
•
Ideal fluid (no viscosity)
Incompressible flow ( constant)
Permanent flow (partial time derivative null)
Along a streamline.
Performing a mass balance

t

t
 out  m
 in  0;

dV

m

CV
 dV
  0
 dm
CV

Ads   dm  0
t
If A is very small dA is even smaller and we
are on a streamline
Below we will use:
 

V   Ads  dm   0
 t

Momentum Balance
Forces
 

V   Ads  dm   0
 t

Bernoulli’s Equation
Exercise
• In a domestic water pipe the pressure is
typically 6 kg/cm2.
– If the velocity is 1m/s, how much does the kinetic
energy account for the total energy?
– If whole the pressure energy was transformed into
kinetic energy, how much would the velocity be?
Where do you expect the energy to be dissipated?
Is the Bernoulli applicable in this flow?
1
1




2
2
 p  U  gz    p  U  gz 
2
2

1 
2
 p

 p

U2
U2


 z   

 z 
 g 2g
1  g 2g
2
• Computing the pressure and the kinetic
energy: p  6 kg  6 9.8N  6 * 10 Nm  6 * 10 Pa
cm2
10
2
m

5
2
p
6 * 105 Nm 2

 60m
g 103 * 9.8kgm 3ms  2
U2
12

 0.05m
2g
2 * 9.8
U2
0.05
2g
* 100% 
* 100%  0.1%
p
60
g
2
5
Ppiezo  P  gz
Ptotal
1
 P  V 2
2
Considerations
• The Mechanical Energy remains constant along a streamline
in steady, incompressible, frictionless flow.
• Pressure is a form of energy: is the energy (work) necessary
for moving a unit of volume from a region with null pressure
into a region of pressure P.
• Inside pipes (pressurised flows) pressure is usually the main
form of energy.
• In liquids the potential energy can be very important. Inside
pipes, discharging liquids pressure and kinetic energy are
usually the important forms of energy.
• In external flows pressure and kinetic energy are usually the
most important forms of energy and determine the shape of
the flow around a body.
Applications
• Consider the flow in a Ventura pipe with entrance area
5 cm2 and contraction area 2 cm2. If the fluid is air and
h is 10cm of water, compute the flow in the pipe
• Considere o escoamento num tubo de Ventouri cuja
área de entrada (e saída) é de 5 cm2 e na garganta é 2
cm. Se o fluido que circula no Ventouri for ar e h for 10
cm de água, determine o caudal que circula no
Ventouri.
h
Nozzle: compute the force knowing
the discharge.
Chimney
• Consider a chimney discharging gas with 1.1
kgm-3 . Make a relation between the outlet
velocity and the height h and exterior air
density.
Bernoulli equation can be applied to relate
energy in two points on the same
streamline only if fluid properties remains
constant between the two points. For this
reason one cannot apply the equation
between a point inside the chimney and
another located outside. One has to do it
in two steps.
Chimney: resolution
S
E