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Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, Donald Young, and Theodore Okiishi
Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved.
Lecture 2
Fluid Statics and Pressure Measurements
Unit Operations 2 CHE3012
Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, Donald Young, and Theodore Okiishi
Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved.
Overview
•Pressure at a point
•Basic Equation for Pressure Field
•Pressure variation in a fluid at rest
•Standard Atmosphere
•Measurement of Pressure and Pressure
Measuring Devices
Unit Operations 2 CHE3012
Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, Donald Young, and Theodore Okiishi
Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved.
Figure 2.1 (p. 39) Forces on an arbitrary wedge-shaped element of fluid.
Pressure at a Point
Free-body Diagram of a Fluid wedge
Unit Operations 2 CHE3012
Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, Donald Young, and Theodore Okiishi
Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved.
Assumptions
•zero shearing stress (fluid element moves as a rigid body)
•the only forces are due to pressure and weight
•Neglect forces in the x-direction (for simplicity)
ysyya
zyxsxpzxpF
maF
2sin
From Newton’s 2nd Law of motion
zszz azyxzyx
sxpyxpF22
cos
sin,cos szsy
0,,limit
2
2
zyxthetake
zapp
yapp
zsz
ysy
szsy pppp ,
Conclusion
Pressure at a point in a fluid at rest or in motion, is independent of direction as
long as there are no shearing stress. Pascal’s Law
Unit Operations 2 CHE3012
Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, Donald Young, and Theodore Okiishi
Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved.
Figure 2.2 (p. 40) Surface and body forces acting on
small fluid element.
Basic Equation for Pressure Field
Assumptions
•zero shearing stress (fluid element
moves as a rigid body)
•only surface force due to pressure and
body force equal to the weight of the
element are acting on the element
•Neglect other types of body forces
•Neglect forces in the x-direction (for
simplicity)
Unit Operations 2 CHE3012
Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, Donald Young, and Theodore Okiishi
Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved.
in the y-direction
zyxy
pF
zxy
y
ppzx
y
y
ppF
y
y
22
zyxz
pF
zyxx
pF
z
x
in the x- and z-directions
zyxkz
pj
y
pi
x
pF
or
kFjFiFF
s
zyxs
ˆˆˆ
ˆˆˆ
pkz
pj
y
pi
x
p
ˆˆˆpressure gradient
Resultant surface force
Unit Operations 2 CHE3012
Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, Donald Young, and Theodore Okiishi
Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved.
kz
jy
ix
ˆˆˆ
where
thus, the resultant surface force pzyx
Fs
the weight in the z-direction
kzyxkW ˆˆ
akp
therefore
azyxkzyxzyxp
or
amkWFF s
ˆ
ˆ
ˆ
Unit Operations 2 CHE3012
Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, Donald Young, and Theodore Okiishi
Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved.
For a fluid at rest a = 0,
0
x
p
0ˆ kp
z
p0
y
p
Thus as we move from point to point on the horizontal plane the pressure does not
change.
dz
dp
Pressure Variation in a Fluid at a Rest Unit Operations 2 CHE3012
Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, Donald Young, and Theodore Okiishi
Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved.
Figure 2.3 (p. 43)
Notation for pressure variation in a fluid at
rest with a free surface.
dz
dp
1212
2
1
2
1
zzppdzdpp
p
z
z
Incompressible Fluid (Hydrostatic Pressure)
1221 zzpp
hpp 21
21 pp
h
(Pressure head)
ophp
P2 represent your datum P0
Unit Operations 2 CHE3012
Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, Donald Young, and Theodore Okiishi
Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved.
Figure 2.4 (p. 44) Fluid equilibrium in a container of arbitrary shape.
Unit Operations 2 CHE3012
Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, Donald Young, and Theodore Okiishi
Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved.
Figure E2.1
Question 1
Because of a leak in a buried
gasoline storage tank, water
has seeped in to the depth
shown in the figure. If the
specific gravity of gasoline is
SG = 0.68, determine the
pressure at the gasoline water
interface and at the bottom of
the tank. Express the answer
as absolute pressure, gauge
pressures and pressure head.
m
m
Unit Operations 2 CHE3012
Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, Donald Young, and Theodore Okiishi
Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved.
Figure 2.5 Transmission of fluid pressure.
The pressure in a closed system is transmitted equally throughout.
1
212 A
AFF
Unit Operations 2 CHE3012
Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, Donald Young, and Theodore Okiishi
Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved.
dz
dp
RTp
Compressible Fluid
RT
gp
dz
dp
Gases are considered compressible fluids, since the density can change
considerably with changes in pressure and temperature.
Specific weight for gases are relatively small when compared to liquids, thus it
follows that from the equation the pressure gradient in the vertical direction is
correspondingly small, and even over distance of several hundred feet the
pressure will remain essentially constant for a gas.
For large variation in height we recall:
These relationships can be combined to give
Unit Operations 2 CHE3012
Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, Donald Young, and Theodore Okiishi
Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved.
Compressible Fluid
2
1
2
1
z
z
p
p T
dz
R
g
p
dp
By separating the variables we get
To complete this integration the relationship between temperature and
elevation must be specified. If we assume constant temperature T0
over the range z1 to z2 (isothermal condition), then
0
1212 exp
RT
zzgpp
Unit Operations 2 CHE3012
Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, Donald Young, and Theodore Okiishi
Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved.
Table 2.1 (p. 47) Properties of U.S. Standard Atmosphere at Sea Level
Standard Atmosphere Unit Operations 2 CHE3012
Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, Donald Young, and Theodore Okiishi
Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved.
Figure 2.6 (p. 47) Variation of temperature with altitude in the U.S. standard atmosphere.
Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, Donald Young, and Theodore Okiishi
Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved.
Measurement of Pressure and Pressure
Measuring Devices
Monometers
Mechanical and Electronic Pressure measuring devices
Unit Operations 2 CHE3012
Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, Donald Young, and Theodore Okiishi
Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved.
0php 02211 hhpA
Piezometer Tube
Fundamental equation describing their use
U-Tube Manometer
1122 hhpA 11hpA
22hpA
If fluid A was a gas then
Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, Donald Young, and Theodore Okiishi
Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved.
02211 hhpA
113322BA
B332211A
hγhγhγpp
phγhγhγp
Differential U-tube manometer. Incline-Tube manometer.
113322BA
B332211A
hγhγsinθlγpp
phγsinθlγhγp
sinθγ
ppl
or
sinθlγpp
2
BA2
22BA
If pipes A and B contain a gas