Applications of Bernoulli’s Equation

2 21 1 2 2

1 2 2 2

p V p Vz z

g g

2 21 1 1 2 2 2

1 1

2 2p V z p V z

22 1 0

1

2p p V

What happens if the bicyclist is accelerating or decelerating?

Figure E3.2 (p. 101)

Static and Dynamic Pressures

• Along a streamline:2

constant2

Vp gz

g

p= static pressure, thermodynamic pressure

=dynamic pressure, it represents the pressure

rise when the fluid In motion is brought to a stop sentropically.

ρg z= hydrostatic pressure, not actually a pressure, but represent a change in pressure due to potential energy variations of the fluid as a result of elevation changes

The sum of these is called total pressure

2

2

V

g

Stagnation Pressure

2

2stag

Vp p

g

Pitot Static Probe

Connected to a pressure transducer or a manometer

to measure the dynamic pressure

Cross-section of a directional finding pitot-tube

For symmetric object, stagnation is clearly at the tip or front

For non-symmetric object, stagnation point is not very clear

Figure 3.5 (p. 108)Stagnation points on bodies in flowing fluids.

Example 1. An airplane flies 100 mi/hr at an elevation of 10,000 ft in a standard atmosphere. Determine the pressure at point (1) far ahead of the airplane, the pressure at the stagnation point on the nose of the airplane (point 2)

Figure E3.6a (p. 110)

Figure E3.6b (p. 110)

Free Jets

Applying Bernoulli Equation between points (1) and (2),and

using p1=0=p2; z1=h; z2= 0, V1=02

2

Vh

2 2 at point 2h

V gh

V= 2 ( ) at point 5g h HFigure 3.11 (p. 112)

Vertical flow from a tank.

An object falling a distance h from rest, velocity is given as:

Page 113

Use the centerline velocity as the average velocity as d<<h

Figure 3.12 (p. 113)Horizontal flow from a tank.

Vena contracta is because of the inability of the fluid to turn a sharp

corner

Figure 3.13 (p. 113)Vena contracta effect for a sharp-edged orifice.

Figure 3.14 (p. 114)

Typical flow patterns and contraction coefficients for various

round exit configurations.