Ways to express Bernoulli equation Energy per unit volume: Energy per unit mass: Energy per unit...
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Transcript of Ways to express Bernoulli equation Energy per unit volume: Energy per unit mass: Energy per unit...
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