15 Fluid Statics
1. PRESSURE-MEASURING DEVICES
...
15 Fluid Statics
1. Pressure-Measuring Devices . . . 15-1 2. Manometers . . . 15-3 3. Hydrostatic Pressure . . . 15-4 4. Fluid Height Equivalent to Pressure . . . 15-5 5. Multifluid Barometers . . . 15-5 6. Pressure on a Horizontal Plane Surface . . . . 15-6 7. Pressure on a Rectangular Vertical Plane
Surface . . . 15-7 8. Pressure on a Rectangular Inclined Plane
Surface . . . 15-7 9. Pressure on a General Plane Surface . . . 15-9 10. Special Cases: Vertical Surfaces . . . 15-10 11. Forces on Curved and Compound
Surfaces . . . 15-10 12. Torque on a Gate . . . 15-11 13. Hydrostatic Forces on a Dam . . . 15-11 14. Pressure Due to Several Immiscible
Liquids . . . 15-12 15. Pressure from Compressible Fluids . . . 15-12 16. Externally Pressurized Liquids . . . 15-14 17. Hydraulic Ram . . . 15-14 18. Buoyancy . . . 15-15 19. Buoyancy of Submerged Pipelines . . . 15-16 20. Intact Stability: Stability of Floating
Objects . . . 15-17 21. Fluid Masses Under External
Acceleration . . . 15-19 Nomenclature
a acceleration ft/sec2 m/s2
A area ft2 m2
b base length ft m
d diameter ft m
e eccentricity ft m
F force lbf N
FS factor of safety – –
g gravitational acceleration, 32.2 (9.81)
ft/sec2 m/s2 gc gravitational constant,
32.2
lbm-ft/lbf-sec2 n.a.
h height ft m
I moment of inertia ft4 m4
k radius of gyration ft m
k ratio of specific heats – –
L length ft m
m mass lbm kg
M mechanical advantage – –
M moment ft-lbf Nm
n polytropic exponent – –
N normal force lbf N
p pressure lbf/ft2 Pa
r radius ft m
R resultant force lbf N
R specific gas constant ft-lbf/lbm-R J/kgK
SG specific gravity – –
T temperature R K
v velocity ft/sec m/s
V volume ft3 m3
w width ft m
W weight lbf n.a.
x distance ft m
x fraction – –
y distance ft m
Symbols
specific weight lbf/ft3 n.a.
efficiency – –
angle deg deg
coefficient of friction – –
density lbm/ft3 kg/m3
specific volume ft3/lbm m3/kg
! angular velocity rad/sec rad/s
Subscripts a atmospheric
b buoyant
bg between CB and CG c centroidal
f frictional
F force
l lever or longitudinal
m manometer fluid, mercury, or metacentric
p plunger
r ram
R resultant
t tank
v vapor or vertical
w water
1. PRESSURE-MEASURING DEVICES
There are many devices for measuring and indicating fluid pressure. Some devices measure gage pressure;
others measure absolute pressure. The effects of nonstandard atmospheric pressure and nonstandard gravitational acceleration must be determined, partic-ularly for devices relying on columns of liquid to indicate pressure. Table 15.1 lists the common types of devices and the ranges of pressure appropriate for each.
WaterResources
The Bourdon pressure gauge is the most common pressure-indicating device. (See Fig. 15.1.) This mechan-ical device consists of a C-shaped or helmechan-ical hollow tube that tends to straighten out (i.e., unwind) when the tube is subjected to an internal pressure. The gauge is referred to as a C-Bourdon gauge because of the shape of the hollow tube. The degree to which the coiled tube unwinds depends on the difference between the internal and external pressures. A Bourdon gauge directly indi-cates gage pressure. Extreme accuracy is generally not a characteristic of Bourdon gauges.
In non-SI installations, gauges are always calibrated in psi, unless the dial is marked “altitude” (measuring in feet of water) or“vacuum” (measuring in inches of mer-cury) on its face. To avoid confusion, the gauge dial will be clearly marked if other units are indicated.
The barometer is a common device for measuring the absolute pressure of the atmosphere.1It is constructed by filling a long tube open at one end with mercury (or alcohol, or some other liquid) and inverting the tube so that the open end is below the level of a mercury-filled container. If the vapor pressure of the mercury in the
tube is neglected, the fluid column will be supported only by the atmospheric pressure transmitted through the container fluid at the lower, open end.
Strain gauges, diaphragm gauges, quartz-crystal trans-ducers, and other devices using the piezoelectric effect are also used to measure stress and pressure, partic-ularly when pressure fluctuates quickly (e.g., as in a rocket combustion chamber). With these devices, cali-bration is required to interpret pressure from voltage generation or changes in resistance, capacitance, or inductance. These devices are generally unaffected by atmospheric pressure or gravitational acceleration.
Manometers (U-tube manometers) can also be used to indicate small pressure differences, and for this purpose they provide great accuracy. (Manometers are not suit-able for measuring pressures much larger than 10 psi (70 kPa), however.) A difference in manometer fluid surface heights is converted into a pressure difference.
If one end of a manometer is open to the atmosphere, the manometer indicates gage pressure. It is theoreti-cally possible, but impractical, to have a manometer indicate absolute pressure, since one end of the manom-eter would have to be exposed to a perfect vacuum.
A static pressure tube (piezometer tube) is a variation of the manometer. (See Fig. 15.2.) It is a simple method of determining the static pressure in a pipe or other vessel, regardless of fluid motion in the pipe. A vertical trans-parent tube is connected to a hole in the pipe wall.2 (None of the tube projects into the pipe.) The static pressure will force the contents of the pipe up into the tube. The height of the contents will be an indication of gage pressure in the pipe.
The device used to measure the pressure should not be confused with the method used to obtain exposure to the pressure. For example, a static pressure tap in a pipe is merely a hole in the pipe wall. A Bourdon gauge,
Table 15.1 Common Pressure-Measuring Devices device
approximate range (in atm)
water manometer 0–0.1
mercury barometer 0–1
mercury manometer 0.001–1
metallic diaphragm 0.01–200
transducer 0.001–15,000
Bourdon pressure gauge 1–3000
Bourdon vacuum gauge 0.1–1
Figure 15.1 C-Bourdon Pressure Gauge
connection to pressure source gear and pinion
link closed end pointer
Bourdon tube
1A barometer can be used to measure the pressure inside any vessel.
However, the barometer must be completely enclosed in the vessel, which may not be possible. Also, it is difficult to read a barometer enclosed within a tank.
2Where greater accuracy is required, multiple holes may be drilled around the circumference of the pipe and connected through a mani-fold (piezometer ring) to the pressure-measuring device.
Figure 15.2 Static Pressure Tube
WaterResources
...
manometer, or transducer can then be used with the tap to indicate pressure.
Tap holes are generally1=8–1=4 in (3–6 mm) in diameter, drilled at right angles to the wall, and smooth and flush with the pipe wall. No part of the gauge or connection projects into the pipe. The tap holes should be at least 5 to 10 pipe diameters downstream from any source of turbulence (e.g., a bend, fitting, or valve).
2. MANOMETERS
Figure 15.3 illustrates a simple U-tube manometer used to measure the difference in pressure between two ves-sels. When both ends of the manometer are connected to pressure sources, the name differential manometer is used. If one end of the manometer is open to the atmo-sphere, the name open manometer is used.3 The open manometer implicitly measures gage pressures.
Since the pressure at point B in Fig. 15.3 is the same as at point C, the pressure differential produces the verti-cal fluid column of height h. In Eq. 15.2, A is the area of the tube. Equation 15.2 and Eq. 15.3 assume consistent density units. In the absence of any capillary action, the inside diameters of the manometer tubes are irrelevant.
Fnet¼ weight of fluid column 15:1
ðp2 p1ÞA ¼ mghA 15:2
p2 p1¼ mgh ½SI 15:3
In countries that do not use SI units, densities are commonly quoted in pounds per cubic foot. In that case, Eq. 15.3 can be written as
p2 p1¼ mh g
gc¼ mh ½U:S: 15:4
The quantity g/gc has a value of 1.0 lbf/lbm in almost all cases, somis numerically equal tom, with units of lbf/ft3.
Equation 15.3 and Eq. 15.4 assume that the manometer fluid height is small, or that only low-density gases fill the tubes above the manometer fluid. If a high-density fluid (such as water) is present above the measuring fluid, or if the columns h1or h2are very long, corrections will be necessary. (See Fig. 15.4.)
Fluid column h2 “sits on top” of the manometer fluid, forcing the manometer fluid to the left. This increase must be subtracted out. Similarly, the column h1 restricts the movement of the manometer fluid. The observed measurement must be increased to correct for this restriction.
p2 p1¼ gðmhþ 1h1 2h2Þ ½SI 15:5ðaÞ
p2 p1¼ ð mhþ 1h1 2h2Þ g gc
¼ mhþ 1h1 2h2 ½U:S: 15:5ðbÞ
When a manometer is used to measure the pressure difference across an orifice or other fitting where the same liquid exists in both manometer sides (shown in Fig. 15.5), it is not necessary to correct the manometer reading for all of the liquid present above the manome-ter fluid. This is because parts of the correction for both sides of the manometer are the same. Therefore, the distance y in Fig. 15.5 is an irrelevant distance.
Manometer tubes are generally large enough in diameter to avoid significant capillary effects. Corrections for capillarity are seldom necessary.
3If one of the manometer legs is inclined, the term inclined manometer or draft gauge is used. Although only the vertical distance between the manometer fluid surfaces should be used to calculate the pressure difference, with small pressure differences it may be more accurate to read the inclined distance (which is larger than the vertical distance) and compute the vertical distance from the angle of inclination.
Figure 15.3 Simple U-Tube Manometer
A
B C
h
p2 p1
Figure 15.4 Manometer Requiring Corrections
h1
1
m
2
p2 p1
h2 h
WaterResources
...
Example 15.1
The pressure at the bottom of a water tank ( = 62.4 lbf/ft3; = 998 kg/m3) is measured with a mercury manometer located below the tank bottom, as shown.
(The density of mercury is 848 lbm/ft3; 13 575 kg/m3.) What is the gage pressure at the bottom of the water tank?
hw ⫽ 120 in (3.0 m)
hm⫽ 17 in (0.43 m) water
mercury
SI Solution From Eq. 15.5(a),
Dp ¼ gðmhm whwÞ
¼ 9:81 m s2
13 575 kg
m3
ð0:43 mÞ
998 kg m3
ð3:0 mÞ 0
BB B@
1 CC CA
¼ 27 892 Pa ð27:9 kPa gageÞ
Customary U.S. Solution From Eq. 15.5(b),
Dp ¼ mhm whw
¼
848lbf ft3
ð17 inÞ 62:4lbf ft3
ð120 inÞ 12in
ft