3.3-Pressure Measurements, fluid mech

3.3-Pressure Measurements

3.3-Pressure Measurements, fluid mech

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Pressure Measurements
3.3 Pressure Measurements
This section describes five scientific instruments for measuring pressure: the barometer, Bourdontube gage,
piezometer, manometer, and transducer. This information is used for experimental work, for equipment testing
and process monitoring.
Barometer
An instrument that is used to measure atmospheric pressure is called a
barometer
. The most common types are
the mercury barometer and the aneroid barometer. A mercury barometer is made by inverting a mercuryfilled
tube in a container of mercury as shown in Fig. 3.8. The pressure at the top of the mercury barometer will be the
vapor pressure of mercury, which is very small:
p
v
= 2.4 × 10
6
atm at 20°C. Thus, atmospheric pressure will
push the mercury up the tube to a height
h
. The mercury barometer is analyzed by applying the hydrostatic
equation:
(3.17)
Thus, by measuring
h
, local atmospheric pressure can be determined using Eq. (3.17).
Figure 3.8
A mercury barometer.
An aneroid barometer works mechanically. An aneroid is an elastic bellows that has been tightly sealed after
some air was removed. When atmospheric pressure changes, this causes the aneroid to change size, and this
mechanical change can be used to deflect a needle to indicate local atmospheric pressure on a scale. An aneroid
barometer has some advantages over a mercury barometer because it is smaller and allows data recording over
time.
BourdonTube Gage
A
Bourdon-tube
gage, Fig. 3.9, measures pressure by sensing the deflection of a coiled tube. The tube has an
elliptical cross section and is bent into a circular arc, as shown in Fig. 3.9
b
. When atmospheric pressure (zero
gage pressure) prevails, the tube is undeflected, and for this condition the gage pointer is calibrated to read zero
pressure. When pressure is applied to the gage, the curved tube tends to straighten (much like blowing into a
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Pressure Measurements
party favor to straighten it out), thereby actuating the pointer to read a positive gage pressure. The Bourdontube
gage is common because it is low cost, reliable, easy to install, and available in many different pressure ranges.
There are disadvantages: dynamic pressures are difficult to read accurately; accuracy of the gage can be lower
than other instruments; and the gage can be damaged by excessive pressure pulsations.
Figure 3.9
Bourdon-tube gage.
(a) View of typical gage.
(b) Internal mechanism (schematic).
Piezometer
A
piezometer
is a vertical tube, usually transparent, in which a liquid rises in response to a positive gage
pressure. For example, Fig. 3.10 shows a piezometer attached to a pipe. Pressure in the pipe pushes the water
column to a height
h,
and the gage pressure at the center of the pipe is
p
= γ
h
, which follows directly from the
hydrostatic equation (3.7c). The piezometer has several advantages: simplicity, direct measurement (no need for
calibration), and accuracy. However, a piezometer cannot easily be used for measuring pressure in a gas, and a
piezometer is limited to low pressures because the column height becomes too large at high pressures.
Figure 3.10
Piexometer attached to a pipe.
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Manometer
A
manometer
, often shaped like the letter “U,” is a device for measuring pressure by raising or lowering a
column of liquid. For example, Fig. 3.11 shows a Utube manometer that is being used to measure pressure in a
flowing fluid. In the case shown, positive gage pressure in the pipe pushes the manometer liquid up a height
h
.
To use a manometer, engineers relate the height of the liquid in the manometer to pressure as illustrated in
Example 3.6.
Figure 3.11
U-tube manometer.
EXAMPLE 3.6 PRESSURE MEASUREMET (UTUBE
MAOMETER)
Water at 10°C is the fluid in the pipe of Fig. 3.11, and mercury is the manometer fluid. If the
deflection
h
is 60 cm and ℓ is 180 cm, what is the gage pressure at the center of the pipe?
PROBLEM DEFINITION
Situation:
Pressure in a pipe is being measured using a Utube manometer.
Find:
Gage pressure (kPa) in the center of the pipe.
Properties:
1.
Water (10°C), Table A.5, γ = 9810 N/m
3
.
2.
Mercury, Table A.4: γ = 133, 000 N/m
3
.
PLAN
Start at point 1 and work to point 4 using ideas from Eq. (3.7c). When fluid depth increases, add a
pressure change. When fluid depth decreases, subtract a pressure change.
SOLUTION
1. Calculate the pressure at point 2 using the hydrostatic equation (3.7c).
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Pressure Measurements
2. Find the pressure at point 3.
·
The hydrostatic equation with
z
3
=
z
2
gives
·
When a fluidfluid interface is flat, pressure is constant across the interface. Thus, at the
oilwater interface
3. Find the pressure at point 4 using the hydrostatic equation given in Eq. (3.7c).
Once one is familiar with the basic principle of manometry, it is straightforward to write a single equation rather
than separate equations as was done in Example 3.6. The single equation for evaluation of the pressure in the
pipe of Fig 3.11 is
One can read the equation in this way: Zero pressure at the open end, plus the change in pressure from point 1 to
2, minus the change in pressure from point 3 to 4, equals the pressure in the pipe. The main concept to
remember is that pressure increases as depth increases and decreases as depth decreases.
The general equation for the pressure difference measured by the manometer is:
(3.18)
where γ
i
and
h
i
are the specific weight and deflection in each leg of the manometer. It does not matter where one
starts; that is, where one defines the initial point 1 and final point 2. When liquids and gases are both involved in
a manometer problem, it is well within engineering accuracy to neglect the pressure changes due to the columns
of gas. This is because γ
liquid
» γ
gas
. Example 3.7 shows how to apply Eq. (3.18) to perform an analysis of a
manometer that uses multiple fluids.
Interactive Application: Multiple Liquid Manometer
EXAMPLE 3.7 MAOMETER AALYSIS
Sketch:
What is the pressure of the air in the tank if ℓ
1
= 40 cm, ℓ
2
= 100 cm, and ℓ
3
= 80 cm?
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PROBLEM DEFINITION
Situation:
A tank is pressurized with air.
Find:
Pressure (kPa gage) in the air.
Assumptions:
Neglect the pressure change in the air column.
Properties:
1. Oil:
2.
Mercury, Table A.4: γ = 133, 000 N/m
3
.
PLAN
Apply the manometer equation (3.18) from elevation 1 to elevation 2.
SOLUTION
Manometer equation
Because the manometer configuration shown in Fig. 3.12 is common, it is useful to derive an equation specific
to this application. To begin, apply the manometer equation (3.18) between points 1 and 2:
Simplifying gives
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: 2011 Feynman-Kac-Type Theorems and Gibbs Measures on Path Space, Richard Feynman, Books

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