Aula Teórica 8

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

0

0

0

mdAdst

mddVt

;mmdVt

CV

inoutCV

0

mdAdst

V Below we will use:

If A is very small dA is even smaller and we are on a streamline

Momentum Balance

Forces

0

mdAdst

V

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?

• Computing the pressure and the kinetic energy:

2

2

1

2

2

2

1

2

22

2

1

2

1

zgU

gp

zgU

gp

gzUpgzUp

%.%*.

%*

gpgU

m..*g

U

mmskgm.*

Nm*gp

Pa*Nm*m

N.cmkg

p

1010060

050100

2

050892

1

2

608910

106

10610610

8966

2

22

233

25

525222

2

2

1VPPtotal

gzPPpiezo

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