Ways to express Bernoulli equation Energy per unit volume: Energy per unit mass: Energy per unit weight: - conservation of energy (no friction loss)

Ways to express Bernoulli equation

Energy per unit volume:

Energy per unit mass:

Energy per unit weight:

)streamline (alongconstant 2

1 2 Vzp

)streamline (alongconstant 2

1 2 Vgzp

)streamline (alongconstant 2

2

g

Vz

p

- conservation of energy (no friction loss)

Civil Engineers often use the “energy per unit weight” form:

)streamline (alongconstant 2

2

g

Vz

p

2

2

g

Vz

p

is often referred to as total head

z is often referred to as elevation (or potential) head

p

is often referred to as pressure head

2

2

g

Vis often referred to as velocity head

Mechanical engineers often use the “energy per unit volume” form:

2

1 2Vzp is often referred to as total pressure

z is often referred to as hydrostatic pressure

p is often referred to as static pressure

2

1 2V is often referred to as dynamic pressure

)streamline (alongconstant 2

1 2 Vzp

Pressure measurements (static, dynamic and stagnation pressure)

V

Consider the following closed channel flow (neglect friction):

Uniform velocity profile

z1 2

Hh

4

5

openopen

Point 2 is at the entrance of the pitot tube where velocity is zero

piezometertube

pitot tube

Velocity at point 1 is the velocity of the flow: VV 1

Pressure measurements (static pressure)

V

z1 2

Hh

4

5

To measure static pressure say at point 1 we use piezometer tube along withp + γz = constant across straight streamlines between pts. 1 and 4:

piezometertube

pitot tube

hpp 41

hpp atm 1

hp gage 0)( 1

Pressure measurements (dynamic pressure)

V

z1 2

Hh

4

5

To measure dynamic pressure say at point 1 we use pitot tube along withBernoulli equation from point 1 to point 5:

piezometertube

pitot tube

2555

2111 2

1

2

1VzpVzp

112

5552

1 2

1

2

1 1 pt.at pressure dynamic pzVzpV

Pressure measurements (dynamic pressure)

V

z1 2

Hh

4

5piezometertube

pitot tube

112

5552

1 2

1

2

1 1 pt.at pressure dynamic pzVzpV

Pressure measurements (dynamic pressure)

V

z1 2

Hh

4

5piezometertube

pitot tube

112

5552

1 2

1

2

1 1 pt.at pressure dynamic pzVzpV

0atmp 0 h

Pressure measurements (dynamic pressure)

V

z1 2

Hh

4

5piezometertube

pitot tube

112

5552

1 2

1

2

1 1 pt.at pressure dynamic pzVzpV

0atmp 0 h

)()(2

115

21 hHhzzV

H

1 VV

Pressure measurements (stagnation pressure (pressure at pt. 2))

V

z1 2

Hh

4

5piezometertube

pitot tube

Stagnation pressure is pressure where velocity is zero (at entrance of pitot tube (pt. 2))

Pressure measurements (stagnation pressure (pressure at pt. 2))

V

z1 2

Hh

4

5piezometertube

pitot tube

Stagnation pressure is pressure where velocity is zero (at entrance of pitot tube (pt. 2))

Bernoulli from pt. 1 to pt. 2: 212

2222

111 ; 2

1

2

1zzVzpVzp

Pressure measurements (stagnation pressure (pressure at pt. 2))

V

z1 2

Hh

4

5piezometertube

pitot tube

Stagnation pressure is pressure where velocity is zero (at entrance of pitot tube (pt. 2))

Bernoulli from pt. 1 to pt. 2: 212

2222

111 ; 2

1

2

1zzVzpVzp

0

Pressure measurements (stagnation pressure (pressure at pt. 2))

V

z1 2

Hh

4

5piezometertube

pitot tube

Stagnation pressure is pressure where velocity is zero (at entrance of pitot tube (pt. 2))

Bernoulli from pt. 1 to pt. 2: 212

2222

111 ; 2

1

2

1zzVzpVzp

0

Stagnation pressure at pt. 2 is: 2112 2

1Vpp 1 VV

Pressure measurements

Stagnation pressure

212 2

1Vpp

Static pressure

Note that

)(2 12 pp

V

Airplanes use pitot-static tubes (a combination of piezometer and pitot tubes) to measure p2 and p1 and compute airplane speed using

previous equation

Energy Grade Line (EGL) and Hydraulic Grade Line (HGL)

Graphical interpretations of the energy along a pipeline may be obtained through the EGL and HGL:

E G Lp V

gz

2

2

H G Lp

z

EGL and HGL may be obtained via a pitot tube and a piezometer tube,respectively

In our discussion we will be taking atmospheric pressure equal to zero, thuswe will be working with gage pressures

Energy Grade Line (EGL) and Hydraulic Grade Line (HGL)

E G Lp V

gz

2

2H G L

pz

h hL f - head loss, say, due to friction

piezometertube

V

g22

2

z 2z1

pitot tube

( )z0

EGL

HGL h L

p 2 /

Datum

Energy Grade Line (EGL) and Hydraulic Grade Line (HGL)

Large V2/2g becausesmaller pipe here

Steeper EGL and HGLbecause greater hL per length of pipe

Head loss atsubmerged discharge

EGL

HGL

p /

z

z0

H G Lp

z

E G Lp V

gz

2

2h hL f

Positive

Energy Grade Line (EGL) and Hydraulic Grade Line (HGL)

If then and cavitation may be possible P

0H G L z

HGL

EGLp /

Positive

V

g

2

2

p

Negative

p

z

z0

E G Lp V

gz

2

2

H G Lp

z

h hL f

Energy Grade Line (EGL) and Hydraulic Grade Line (HGL)

Helpful hints when drawing HGL and EGL:

1. EGL = HGL + V2/2g, EGL = HGL for V=0

2. If p=0, then HGL=z

3. A change in pipe diameter leads to a change in V (V2/2g) due to continuity and thus a change in distance between HGL and EGL

4. A change in head loss (hL) leads to a change in slope of EGL and HGL

5. If then and cavitation may be possible H G L zP

0

6. A sudden head loss due to a turbine leads to a sudden drop in EGL and HGL

7. A sudden head gain due to a pump leads to a sudden rise in EGL and HGL

Helpful hints when drawing HGL and EGL (cont.):

8. A sudden head loss due to a submerged discharge leads to a sudden drop in EGL

Hydrostatic Paradox

At Lake Mudd and Lake Mead, the depth is ~600 ft.

At Lake Mead, the horizontal thrust near the base of the dam is ~18 tons per square foot.

Here is the paradox: in both cases, the horizontal thrust on the dam is the SAME

Hydrostatic Paradox

The reason for this paradox is that the pressure depends only on the depth of the water, not on its horizontal extent:

constzp

Hydrostatic Paradox

See movie at: http://www.ac.wwu.edu/~vawter/PhysicsNet/QTMovies/PressureFluids/HydrostaticParadoxMain.html