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TRANSCRIPT
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Fluid MechanicsFluid Mechanics
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FluidFluid
Liquid GasGas
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DensityDensitym massVm
=densitymass
volumedensity SI kg/m3
or
m = Vm = V
density of water
w = 1.00 x 103 kg/m3 = 1.00 g/cm3 = 1.00 g/c.c. = 1.00 kg/Ldensity
1 c.c. = 1 mL = 1 cm31 L = 103 cm31 m3 = 103 L = 106 cm3
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Volume Volume 1 cm3 (c.c. or mL) water = 1 g water
1 L water = 1 kg water
( ) g
1000
1 L = 103 cm3 (c.c. or mL)
1 cm3 (c.c. or mL) = 10-3 L
m = 10-3
x 103 =
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1 L water = 1 kg water
1 m
1 m3 = 103 L1 Lg
1 m3 water = 103 kg water
1000
1 m 1 m1 m3 = 103 L
1 kg water
103 kg water103 kg water
1 m
1 m1 m
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Specific GravitySpecific Gravity
w
grsp
=..
specific gravity
= (w )(sp. gr.)
sp. gr.
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density density density kg/m3 sp. gr.
water 1.00 x 103 1.00sea water 1.03 x 103 1.03ice 0.917 x 103 0.917mercury 13.6 x 103 13.6gold 19 3 x 103 19 3gold 19.3 x 103 19.3copper 8.92 x 103 8.92iron 7.86 x 103 7.86 air 1.29 1.29 x 10-3
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pressure buoyant force buoyant force fluid dynamics
continuityflow rate
Bernoullis Equation pressure
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pressurepressure
AFP =pressure
force
pressure pressure SI N/m2Pa
F = PA
or P
FF = PA F
AF = PA
A
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P0 pressure P0 pressure
h
P pressuredepth
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P0A
P0A + W = PA
m = V V = Ah P0A + Ahg = PAA
h
W = mg = Vg = Ahg
P = P0 + gh
PA
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A B pressure
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Atmospheric PressureAtmospheric Pressure
Patm = 1.013 x 105 Pa (N/m2)
pressure= pressure
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pressure= pressurep
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1 29 k / 3 a = 1.29 kg/m3 1 atm = 1.013 x 105 Pa P0 = 0
P = agh => h = P/(ag) = 1.013 x 105/1.29/9.8 =
8000 m
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Pressure Measurement Pressure Measurement P0 = 0
barometerP
Patm + gh
manometer
P
pressure
pressure
P = Patm + gh
0 + gh = Patm density
P - Patm = gh
gauge pressure
gauge pressure
density
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Barometer Barometer
1 atm = 760 mmHgBarometer 760 mm 0.76 m Hg g
Patm = 1 atm = (13.6)(103)(9.8)(0.76) = 1.013 x 105 N/m2(Pa) g105 N/m2(Pa)
Hg sp. gr. w
g
Hg
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barometer barometer 21 atm = 1.013 x 105 N/m2 = wghw
1.013 x 105 N/m2 = (103)(9.8)hw( )( ) w
( )( ) m31010013.1 5
=
=h ( )( ) m3.108.9103wh pressure P0 pressure P 0
13.6 0.76 x 13.6 P0 = 0 > 10.3 m
Patm = P0 + gh
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10.3 m
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ManometerManometer
pressure P2 = P1 + ghp 2 1 g
P2 P1
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b l t absolute pressure 0 P
gauge pressureg g p = -1 atm gauge pressure :
> > 0 < < 0
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density
pressure
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pressure
pressure
p
P0 + oilgL
Oil 0 750
P0 + wg(L h)
Oil = 0.750 L = 5.00 cm h =
oilgL = wg(L h)h =
( )( ) Oil
( )( ) cm25.1cm00.575.011 oiloil ==
== LLh
ww
w
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Pascals LawPascal s Law
P1 = P2
2
2
1
1
AF
AF
=2
1
2
1
AA
FF
=pressure
pressure
P2 = P1 + gh ghAF
AF +=
1
1
2
2
p
AA 1221
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96 96
U A = 0.01 m2
il = 500 kg/m3 t = 1000 kg/m3oil 500 kg/m water 1000 kg/m F =
WWWWF airwaterairoil +=++
AA
wateroil WWF =+ h
AhgAhgF woil =+
( ) ( )( )( )( ) N9489100105001000 === AhgF ( ) ( )( )( )( ) N9.48.91.001.05001000 === AhgF oilw
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Buoyant Force Buoyant Force P1 pressure P1 pressure
h
P2 pressuredepth
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P1A
P A P A
Ah buoyant force B =
dispalcement volume
W
P2A P1A = (P1 + gh)A P1A= ghA V = Ah
d sp ce e vo u e
g= Vg= mg
m = V
P2A
P2 = P1 + gh Archemedes principlebuoyant force =
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= B = V g
densitydisplacement volume = object volume
B Vog W = oVog
density
F= B W = ( o)Vog > > F >0
> o > F >0
< o < F
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= < B = V'g displacement volume = volume d i B V g W = oVog
volume
density
densityVo
F = B W = 0
density
V'
B = W => V' = oVo
1
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= buoyant forcebuoyant force
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buoyant force
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Fluid DynamicsFluid Dynamics incomprissible
compressible compressible nonviscous
viscous steady
unsteady laminar
turbulent flowirrotational irrotational
rotational
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streamlinestreamline
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t = 0
t = t
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AAAv = const
A2x2 = A1x1A2v2t = A1v1t
2211 vAvA =
or
21
AA
vv
=12 Av
x2 = v2tx1 = v1t
A2 > A1v2 < v1 2 1 2 1 A2 < A1v2 > v1
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or flow rate
or flow rate t = 0 t V
V xAxA 2211
v1 v2
t
VQ
= txA
txAQ
=
= 2211
m3/s
V = Qt 2211 vAvAQ ==
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V = vAt
A
v
vt
Q = V/t = Av
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v
Q = Av A = Q/vQ = Av
A Qv = Q/A
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Q1
Q = Q + QQ3 Q1 + Q2
QQ3 = Q1 + Q2
Q
Q1
Q2Q2
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pressure Bernoullis Equation
Pressure Pressure
y2 y1
y 0y 0
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2
t = 0 t V V
2
Kinetic energy K2 = 22
22 2
121 Vvmv =
Potential energy Ug2 = 22 Vgymgy =
pressure work W2= VPxAPxF 222222 ==
force 1
Kinetic energy K1 =
Potential energy U = Vgymgy =
21
21 2
121 Vvmv =
Potential energy Ug1 = 11 Vgymgy =
pressure work W1= VPxAPxF 111111 ==
force
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11
volume
work kinetic energy potential energy
K + Ug = W 22221
211 2
121 gyvPgyvP ++=++
pressure V or
constant1 2 =++ gyvP
pressure V
Bernoullis equation
2gy
Patm
22221
211 2
121 gyvPPgyvPP atmatm ++=++
atm
22
1 gauge pressure 2 gauge pressure
Bernoullis equation P absolute pressure gauge pressure
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Bernoullis Equation Bernoulli s Equation
22221
211 2
121 gyvPgyvP ++=++ 222111 22
gygy
g
222
211 11 vPvP
222
111
21
21 y
gv
gPy
gv
gP
++=++ y
= = w
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v = v = 0 v1 = v2 = 0
2211 gyPgyP +=+ ( )2112 yygPP += P = P2P = PP = P0 + gh
P0 = P1h = y1- y2
h y
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22 11 PP y1 = y2 222211 2
121 vPvP +=+
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94 94
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DE DE
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pressure -> kinetic energy
kinetic energy -> pressure
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Pitote tube
pressurePA = P0
P0 pressure
B = 0
Ba PvP =+2
0 21
PAP P = ghBa0 2
21 AB PP 2 AB PPv = 2
a
PB PA = gh density pressure
ABa PPv =2
21
a
AB PPv
= 22a
v
= 2
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v
1 y1 = 0 m
v1
y = 0P = Patm
g = 10 0 m/s2
25.00 m
g 10.0 m/s
2
34
5A2 = 0.02 m2
A = 0 04 m23.00 m 4A3 = 0.04 m2
A4 = 0.01 m2
1.00 m
gauge pressure
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y y1 = 0 m y2 = -5.00 m y3 = -8.00 my3 y4 = -9.00 m y5 = -5 00 my5 5.00 m
gauge pressure P 0 P1 = 0
P4 = 0
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00 0 0
42441
211 2
121 gyvPgyvP wwww ++=++
v1
v1
( ) ( )( )( ) m/s4.1318000.91022 44 ==== ygvh = 9.00hh
Bernoullis Equation continuity
( )( ) /sm134.04.1301.0 3444 === vAQBernoullis Equation pressure
/sm134.0 3432 === QQQ
m/s71.602.0
134.0
2
22 === A
Qv m/s35.304.0134.0
3
33 === A
Qv
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22 11 PP
0 0 v2 = 6.71 y2 = -5.000
22221
211 2
121 gyvPgyvP wwww ++=++
P 0 ga ge press re
( )( ) ( )( )( ) Pa107520051010002/71610001 422222 === gyvP
w = 1000 kg/m3 P1 = 0 gauge pressure gauge pressure
( )( ) ( )( )( ) Pa1075.200.51010002/71.610002 222
gyvP ww
P2 = wgh2 = (1000)(10)(5.00) = 5.00 x 104 Pa2 wg 2 ( )( )( )
0
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22 11 PP
0 0 v3 = 3.35 y3 = -8.000
32331
211 2
121 gyvPgyvP wwww ++=++
P 0 ga ge press re
( )( ) ( )( )( ) Pa104470081010002/35310001 423233 === gyvP
P1 = 0 gauge pressure gauge pressurew = 1000 kg/m3
( )( ) ( )( )( ) Pa1044.700.81010002/35.310002 333
gyvP ww
P3 = wgh3 = (1000)(10)(8.00) = 8.00 x 104 Pa3 wg 3 ( )( )( )
0
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22 11 PP
0 0 0 0 y5 = -5.00
52551
211 2
121 gyvPgyvP wwww ++=++
( )( )( ) Pa1000.500.5101000 455 === gyP w
P5 = wgh5 = (1000)(10)(5.00) = 5.00 x 104 Pa
2 pressure
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0 hP 21
y2 = h
v2
0
v = 0 ghvP wwatm ++222
01 2 ++ vP
ghv 21 =v1 = (2gh)1/2
02 1
++ vP watm y1 = 0
v = 0
(2 h)1/2
v 0
v1 = (2gh)1/2
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continuity
A2211 vAvA =
continuity
Bernoullis equation
21
21 vA
Av =
021
21 2
122 ++=++ vPghvP watmwwatm
Bernoulli s equation
22
22 11 Ah
( )[ ] 22212 1/2 vAAgh =2
21
222 2
121 v
AAghv
=+
( )[ ] 212g
O !
A1
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continuity Bernoullis Equation A1 = A2 ti it1 2v1 = v2P1 = P2
continuityBernoullis Equation