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4 (Bernoulli Equation)

Daniel Bernoulli (1700 1782)

Born 8 February 1700:

Broningen, Netherlands

Died 8 March 1782 (aged 82):

Basel, Switzerland

Residence Trizzy

Known for Bernoulli's

Principle, early Kinetic theory

of gases, Thermodynamics

Intro. Fluid Mechanics 2


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First Published Bernoullis principle Bernoulli's principle

is named after the

Dutch-Swiss

mathematician

Daniel Bernoulli

who published his

principle in his

bookHyd odynam ca in

1738.



(Bernoulli Equation)

(frictionless flow, inviscid flow)

2

constant (J/k2

g)p V

gz


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valid


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valid



net net system

cv cs

Q W e d e V dAt

4

1. (shaft work) CS

2. (normal stress work)

3. (viscouse stress work)

4. (other work)


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steady state, incompressible flow, frictionless flow inviscid flow,,- 1 intensive properties


2 2

2 2 1 1, 2 2 1 12 2

net shaft net

p V p Vq w u gz u gz

net shaft shear other net

cv cs

pQ W W W e d e V dA

t

2 2

1 1 2 2, 1 2 2 1

2 2shaft net net

p V p Vw gz gz u u q



2 2

1 1 2 2, 1 2 2 1

2 2shaft net net

p V p Vw gz gz u u q

4

2 2

1 1 2 21 2

2 2

p V p Vgz gz

2constant

2

p Vgz


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2

( kgJ/ )2

p Vgz const


(mechanical energy per unit mass of working fluid)

2 2

1 1 2 21 2

(J/ )2 2

kgp V p V

gz gz

2

2

V

gz

(flow energy)

(kinetic energy)

(potential energy)



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(mechanical energyper unit weight)


2 2

1 1 2 21 2

= (J/ m)2 2

Np V p V

z z Hg g g g

2

2

V

g

z

(pressure head)

(velocity head)

(elevation head)

H (total head)



2 2

1 1 2 2 = (J/ m)2 2

Np V p V

Hg g g g


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2 221 2

1 1 2 2 (N/m )2 2

V Vp gz p gz

2

2

V

z

(static pressure)

(dynamic pressure)

(hydrostatic pressure)


: 4.1

0.1 0.02 m2 50 m/s


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4.1


2 21 21 2

( ) ( )

2 2

atm g atm g P P P Pv v

g g g g

2 21 21 2

( ) ( )

2 2

g gP Pv v

g g g g

23

2 2 2 2 31 2 1 2

1.0

50 10 1.2 102 2

g

kg

g mmP v v Pag s

2

3 2 2 2 2

.

. .

kg m kg kg m

m s m s m s

1 2m m

1 1 2 2A v A v 2

2 2 21 2

1 1

A v Dv v

A D

1

0.02 5010 /

0.1v m s


: 4.2

1 m 7 m


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4.2

(1) (Steady flow)

(2) (Incompressible flow)

(3) (Inviscid fluid)

(4) (5) (V

1= 0)


2

2

221

2

11

22gz

vPgz

vP

4.2() 1 2

1 2

P1

= P2

V1 V

1= 0


smms

mzzgv

gzv

gz

gz

vP

gz

vP

/7.117807.92)(2

2

22

2212

2

2

21

2

2

22

1

2

11


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4.2() A 1

Continuity equation VA

= V2

V1 V

1= 0

Intro. Fluid Mechanics 21}5.78{8.22

.

.)1(807.91000

.

.

2

7.111000)10013.1(

)(2

)(2

22

0;

22

)(

2

23

2

2

22

32

5

1

2

21

1

2

21

1

2

11

2

2

12

1

2

11

2

gageabs

AA

AA

AA

A

AAA

kPakPa

mkg

sNm

s

m

m

kg

mkg

sN

s

m

m

kg

m

N

zzgv

PP

zzgvPP

gzvP

gzvP

vvv

gzvP

gzvP


: 4.3

0.45 m 50mm (uniform flow)

2


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: 4.4

2 2

1 1 2 21 2

(J/ )2 2

kgp V p V

gz gz

1 1atmp gD 2 2atmp p gD

2 2

1 21 2

2 2

atm atmp gD p gDV V

2 2

1 21 2

2 2

V VD gD


: 4.4

2 2

1 21 2

2 2

V VD gD

2 1 22

2 9.81 0.45 0.005

2.8 m s

V g D D


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: 4.4

540 mm2 10 kW

3 m


:

Wall pressure tab Static pressure probe


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: (stagnation pressure)

22 (N/m )

2stag

Vp p

2 2

2 2

stag

stag

V Vp p


pitot tube Pitot

: (stagnation pressure)

Henry de Pitot (1695-1771)


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A flow of air into a venturi meter. The

kinetic energy increases at the

expense of the fluid pressure, as

shown by the difference in height of

the two columns of water.

This article is about Bernoulli's principle and Bernoulli's equation in fluid dynamics.

For an unrelated topic in ordinary differential equations, see Bernoulli differential

equation.

A flow of air into a venturi meter. The kinetic energy increases at the expense of the

fluid pressure, as shown by the difference in height of the two columns of water.In

fluid dynamics, Bernoulli's principle states that for an inviscid flow, an increase in

the speed of the fluid occurs simultaneously with a decrease in pressure or a

decrease in the fluid's potential energy.[1][2]Intro. Fluid Mechanics 29


3

:

22 (N/m )

2stag

Vp p


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: Pitot-Static Probe


: 4.5

pitottube static tube

30 mm

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