Student Handout 12 2014

Macroscopic energy balance 2: Bernoulli equation CHEE 3363 Spring 2014 Handout 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.

Particularly simple case: streamline CV Fluid is steady, incompressible, frictionless                 ρ

ds 

 p + dp

ds

g

V  s + dV  s

z 

θ

 p

V  s y   x 

Assumptions: 1. 2. 3. 4.

ρ

A + dA

streamline

Continuity equation ρ

 p + dp

ds

V  s + dV  s

g

streamline

V  s

z 

 

 

       ∂  ρ dV   + ρv · dA = 0 Equation: ∂ t CV CS y 

 x 

A + dA

θ

 p

−ρV   A + (ρ(V    +  ds )(A +  dA )) = 0 s

neglect

2nd order

terms

s

ρ(V  s  +  ds )(A +  dA )

= ρV   A s

V  s dA + AdV  s + dAdV  s = 0 V  s d

+

dV  s = 0

Momentum equation along s 1  p + dp

ρ

ds

V  s + dV  s

g

A + dA

θ

 p

streamline

V  s

z 

y 

F surf ,s + F body,s =

Equation:  x 

  ∂ 

∂ t

us ρ dV   +

CV

Surface force: F surf  s = pA − ( p + dp)(A +  dA) +  p +

pressure forces on end faces =

1

−Adp − dpdA 2

us ρv

· dA

CS



,

 

dp

2



dA

pressure force acting in s direction on surface

Momentum equation along s 2 Body force:



dA

F body,s = ρgs dV   = ρ(−g sin θ) A +

sin θ ds = dz



F body s = −ρg A + ,

dA 2



2



ds

dz

        

 

us ρ v · dA

= V   (−ρV   A) + (V    + dV   ) [ρ(V    + dV   )(A + dA)] s

s

s

s

s

s

Cs =

ρV  s

by continuity

= V   (−ρV   A) + (V    + dV   )(ρV   A) = ρV   AdV   s

s

s

s

s

s

s

Momentum equation along s 3 Put terms together: 1

1

2

2

−Adp − dpdA − ρgAdz −

ρg

dA dz

= ρV   s A dV   s

Divide by  ρA and neglect 2 nd order terms: dp

−

d

V  2

  s

2

+

dp ρ

ρ

− g dz

+ g dz = 0

=

V    dV   s

integrate

s

=

d

V  2

  s

2

Bernoulli equation v2 2

+

 p

+ gz  = const

ρ

1.   2. Flow along streamline 3.    ρ = constant 4. No frictional terms                                

  1 Problem 

                            Calculate :

minimum h     plate in place. Assumptions:

Bernoulli’s equation:

From water surface to jet:

Fluid statics:

CV 2 1

  2 Conservation of momentum:

Use Bernoulli:

  1 Problem 

    air with diameter d        D.   v       Calculate :             Assumptions:

1

Bernoulli’s equation: 2

CS

      

  2 Fluid statics:

1

x  momentum:

2

Force of jet on plate is in opposite direction

CS

    1 Problem : M  

           from below by vertical jet of water. Water jet speed V 0 = 10 m/s, diameter d  = 25 mm. Calculate :          function of height h .          stationary. Assumptions:

Bernoulli’s equation:

    2 Problem : M  

              vertical jet of water. Water jet speed V 0 = 10 m/  s, diameter d  = 25 mm. Calculate :          function of height h .          stationary.    z  direction:

    3 Problem : M  

              vertical jet of water. Water jet speed V 0 = 10 m/  s, diameter d  = 25 mm. Calculate :          function of height h .          stationary. Apply continuity: