Macroscopic energy balance 2: Bernoulli equation

CHEE 3363Spring 2014Handout 12

Reading: Fox 4.4 (Bernoulli section)

Learning objectives for lecture

1. State the Bernoulli equation and give the conditions under which it can be used.

2. Apply the Bernoulli equation to solve problems.

2

Particularly simple case: streamline CVFluid is steady, incompressible, frictionless077;-;8-+1)4+7674=5-*7=6,-,*AE7?;

t

CV

dV +

CS

v dA = 0

VsA+ ((Vs + ds)(A+ dA)) = 0

(Vs + ds)(A+ dA) = VsA

VsdA+AdVs + dAdVs = 0

VsdA+AdVs = 0

Continuity equation

dsVs + dVs

A+ dA

Vs

p

p+ dp

x

y

z

g

streamline

!64AE7?1;

Momentum equation along s 1

dsVs + dVs

A+ dA

Vs

p

p+ dp

x

y

z

g

streamline

Equation:

Surface force:

5

Fsurf,s + Fbody,s =

t

CV

us dV +

CS

usv dA

Fsurf,s = pA (p+ dp)(A+ dA) +

(p+

dp

2

)dA

= Adp1

2dpdA

pressure forces on end faces

pressure force acting in s direction on surface

Momentum equation along s 2Body force:

75-6

Momentum equation along s 3

7

Divide by A and neglect 2nd order terms:

dp

g dz = Vs dVs = d

(V

2s

2

)

d

(V

2s

2

)+

dp

+ g dz = 0

integrate

Adp1

2dp dA gAdz

1

2g dAdz = VsAdVs

Put terms together:

v2

2+

p

+ gz = const

Bernoulli equation

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