This document discusses the measurement of pressure, velocity, and flow. It begins with an overview of measuring pressure and velocity, as pressure measurements can be used to obtain velocity. Key relationships between pressure and velocity are explained, such as Bernoulli's equation. Common pressure measurement devices are then described, including manometers, which use liquid columns, and electromechanical transducers like Bourdon tubes. The response of these devices and considerations for measurement ranges are also covered. Finally, the document discusses the differences between measuring flow rate versus velocity and different units used to express these quantities.

1. Pressure, Velocity & Flow
Measurements
ME 310 – Instrumentation
Prof. Jamey D. Jacob
Spring, 2004

2. Overview
• Measurement of Pressure & Velocity
– Related, usually measure pressure to obtain velocity
• Review of Fluid Properties
• Relationship between pressure and velocity
• Pressure variation in the atmosphere and ocean
• Pressure Measurement (Chap. 9)
• Velocity Measurement (Chap. 9)
• Flow Rate Measurement (Chap. 10)

3. Why measure pressure?
• From pressure variation in a moving fluid, one
can obtain velocity
• Measure losses due to friction in ducts and
diffusers
• Use pressure variation to determine
temperature,
density, etc.
• Rate processes in reactions
• Changes in atmospheric conditions
• Safety (max./min. pressure)

4. Measures of Weight
• Specific Weight
– Weight per unit volume
– Units of N/m3 or lb/ft3
• Specific gravity
– Ratio of density of the fluid to water at 4ºC
– Dimensionless

5. Some Numbers of Interest
• The “Common” Fluids
– Air and water (in SI except for ν in cgs) at STP

6. The Continuum Hypothesis
Number of
molecules in
1 cm3 of air =
2.7 ·1019 at STP;
Number of
particles in the
solar wind =
~10-100/cm3

7. Continuum versus Non-continuum Gases
• All liquids can be viewed as a continuum
• Gases at low pressures and densities may be noncontinuum
in nature if the Knudson # is not <<1 - special
devices are required to measure these very low pressures

8. What Causes Motion?
• Balance of forces
– Equilibrium between pressure, viscous, and inertial
forces.
– F=ma ⇒ Navier-Stokes equation
• True for steady and unsteady flows.
– A steady flow reaches an equilibrium, fluid is moving
but flow field does not changes in time.
– An unsteady flow is constantly changing in time.
• Nature abhors a vacuum?
• Nature abhors a pressure gradient!

9. Relation of Pressure to Velocity
• Conservation of momentum (Navier-Stokes)
• Bernoulli’s equation (derived from N-S eq’n)

10. Pressure Measurement
• Pressure = force per area

11. Pressure: gage versus absolute
• Pressure is always positive (negative pressures do
exist, but not in practicality) – can be measured on
an absolute scale or respect to atmospheric

12. Units of Pressure
• Pressure
– Multitudes of units.
– 1 atmosphere is equal to . . . (these are just a few!)

13. Pressure Ranges
• Some typical values of pressure
– Atmospheric
• Hurricane, 700 mm Hg
– Biological
• 120/70 mm Hg ⇒ 1.15 atm to 1.09 atm absolute pressure
(about 2 psi gage)
– Vacuum chambers
• 29 inches Hg (25 mm Hg absolute) to 10-6 Torr (“a lot of
nothingness”)
– ASME Pressure Vessels
• Very small (0 psia) to very big 100 psig to 10,000 psig

14. Atmosphere
• Pressure variation
– Decreases logarithmically
with height, P = 10^(-0.06H,
where P is pressure in
atmospheres and H is
height in km, density
behaves similarly.
• Temperature variation
– Highly complex with
height, due to solar heating
in the troposphere, and
chemical absorption, reradiation
in the
stratosphere, mesosphere,
and thermosphere.

15. Ocean

16. What pressure are you measuring?
• Static pressure and dynamic pressure are related
to each other by the stagnation or total pressure
– at equal elevations,
static pressure + dynamic pressure = total pressure
• The total or stagnation pressure is that pressure
that results if the flow is ideally brought to rest
(stagnated)

17. Static and stagnation pressure probes

18. Pressure Measuring Devices
• Basic pressure measuring devices fall into two
basic categories
– Liquid-Column Manometers: pressure is measured
by
using hydrostatic behavior of a column of liquid
– Electro-Mechanical Transducers: pressure is
measured
by (essentially) direct measurement of force or
deflection due to pressure on a given area
• It is common to find multiple devices in a system,
particularly when different magnitudes of
pressure are being measured

19. Device Breakdown
• A more rigorous categorization reveals the
following
– Gravitational devices
• Liquid columns
• Pneumatic or hydraulic pistons
– Direct-acting elastic
• Loaded tubes, symmetric or asymmetric
• Elastic diaphragms
• Bellows devices
– Other devices

20. Direct versus Differential
• A pressure measuring device (manometer or
transducer) can measure pressure of a single
input or the difference between two pressure
inputs
• Differential is useful when measuring fluid
velocity or losses in a duct, for example

21. The Liquid Column Manometer
• The basis of the liquid column manometer is the
hydrostatic pressure relation
which can be integrated to
obtain
showing the pressure variation from one height
to another in a liquid column density ρ
• Pressure is a function of density and height alone

22. The Piezometer Tube
• Uses a single tube to relate height of liquid
column to pressure of a liquid in a pipe or vessel
– Accurate, but requires high
columns
of liquid for large pressures (for a
pressure of 14.7 psig, a column of
water must be 34 feet!)
– Can be used with liquids only
– Cannot measure vacuum pressure
(air is sucked into the liquid)

23. The Inclined Piezometer Tube
• Since pressure is a function of height only, the
tube can be inclined to increase accuracy
Greater accuracy can be attained
by decreasing the value of the
inclination angle since
– For example, if θ =10o

24. Well Manometer
• Useful for barometric pressure measurements
(pressure variation due to atmospheric pressure)
– Pressure in closed tube can be
known at time of construction so
– or air in closed tube can
withdrawn so that it is nearly
vacuum (p=0), thus

25. U-tube Manometer
• Most common type of liquid column manometer
– Measures pressure difference
between two inputs
– Inputs can be both static, both
total, or a mix
– Can be inclined as well to achieve
better accuracy
– Can use two different fluids to
obtain greater accuracy

26. U-tube Manometers
• Can be easily modified to provide large or small
pressure ranges

27. Example
• Take a U-tube manometer that using water that
has a height difference of 1 inch. What is p?
• This difference in pressures is approximately
0.2% above atmospheric pressure - i.e., a very
small change.

28. Single and Two-Fluid Manometers
• Two Fluid Manometer
• Single Fluid Manometer

29. Liquid Column Manometer Samples
Examples of liquid
column manometers
from Dwyer, Inc

30. Electro-Mechanical Pressure
Gages
• Mechanical
– Dead-weight tester
– Bourdon-tube
– Bellows Gage
– Diaphragm (with mechanical displacement sensor)
• Mechanical-Electrical
– Diaphragm (with strain gages)
– Bridgman Gage
– Other

31. Dead Weight Tester
• Balances a fluid pressure against a known weight
• Used for pressure gage calibration

32. Pressure ⇒ Displacement
• Consider the spring system
• Now use this relation to determine pressure
• This is a Bellows gage

33. Bourdon Tube Gage
• Most common gage – widely used in all field from
engineering to medical
• Gages available over pressure ranges of <1 to
>106 mm Hg (.001 atm to 1000 atm)

34. Diaphragm Gages
• Uses the deflection of a diaphragm to measure
pressure – the greater the pressure applied on
the diaphragm, the larger the deflection
• Can be flat or corrugated
Top
Side
From omega.com

35. Strain-Gage Based Transducer
• Strain-gage placed on diaphragm
• Can be used for small spans and differential
pressures
From omega.com

36. Capacitance Transducer
• Deflection in diaphragm
registers a change in
capacitance that is picked
up by a bridge circuit
• Can be used from high
vacuums (<< 1 Torr) to 70
MPa (10,000 psig)

37. Pontentiometric Transducer
• Bourdon or Bellows tube connected to a wiper
arm that changes variable resistor in a
Wheatstone Bridge circuit
• Ranges from 35 kPa to 70 MPa (5 to 10,000 psig)

38. Other Types (Just a Few!)
Resonant Wire
Optical
Piezoelectric

39. Gage Ranges
• Need to consider gage span, accuracy, and drift
when selecting a gage

40. Low Pressure Gages
• Manometers
– McLeod Gage
• Electrical
– Pirani thermal-conductivity gage
– Knudsen gage
– Ionization gage
– Alphatron

41. Gage Ranges: Low Pressure & Vacuum

42. Gage Connections
• Since transducers require
periodic calibration and
replacement, the gage is
placed in a test circuit is
shown
• Process pressure is
measured by opening valve P
while D is used to drain the
fluid from the sensor; T is
used for calibration
• Other items, such as
condensors, filters and
snubbers may also be needed

43. Transient Response
• Fluctuations in pressure common in many
systems (take the human body for example; or
the pressure in a internal combustion engine)
• Thus, transient response is important
• The mass of fluid in a sensor vibrates under the
influence of fluid friction which tends to dampen
the oscillatory motion
Sensor

44. Transient Response
• Since the tube is small, assume the flow is laminar; the
resulting expression for pressure-amplitude ratio is
• Where f is the frequency of the pressure signal. The natural
frequency fn is
• While the damping ratio is
• Where L is the tube length, r is the tube radius, V is the
sensor volume, and c the speed of sound

45. Transient Response Example
• A tube with d=0.5 mm diameter and L=7.5 cm
length is connected to a pressure transducer with
a volume V=3.5 cm3; air at STP is the working
fluid – how much does 100 Hz signal attenuate?

46. Transient Response Example
• For the given conditions, the pressure
attenuates
as a function of the input signal
frequency as
shown below
p/po

47. Fluid Flow Measurement:
Velocity and Flow Rate

48. Flow Rate versus Velocity
• Flow rate is an integral quantity
– Flux of flow through a given area
• Velocity is a differential quantity
– Velocity of flow at a given point
• Velocity over an area can be used to
obtain flow
rate; flow rate can be used to determine an
average velocity only

49. Flow Rate
• Integral quantity
• Measures total flow rate in a duct or
enclosure;
no account for variation in local flow rate
• Volumetric flow rate: [L3/t]
– US: ft3/s, GPM (gallons/min), CFM (cubic
feet/min)
– SI: m3/s
• Mass flow rate: [M/t]
– US: lbm/s, slug/s
– SI: kg/s

50. Flow Rate
• From conservation of mass

51. Velocity
• Velocity is a differential quantity;
function of
space (and possibly time)
• Thus, any measure of velocity is a point
measurement; i.e., only the value of velocity at
that measurement location is known
€ • Continuity equation in differential form is

52. Relation between Q (or dm/dt) and V
• Volumetric flow rate, Q, and mass flow rate,
dm/dt, can be related to velocity by continuity
• Only average velocity can be obtained if flow rate
is known

53. U versus Q
• Average velocity may or may not be close to
maximum velocity; highly Re dependent

54. Flow Rate Meters
• Obstruction meters
– Venturi
– Orifice meter (plate and nozzle)
• Variable-area meters
• Turbine meters
• Vortex shedding meters
• Magnetic meters
• Ultrasonic meters

55. Flow through a Constriction
• In incompressible lossless flow through a
constriction, mass conservation dictates that the
velocity must increase while Bernoulli’s equation
predicts a subsequent pressure drop

56. Obstruction Meters
• This effects leads to the development and use of
obstruction meters; the flow is restricted and the
resulting pressure drop can be used to determine
flow rate in a pipe
• Type of obstruction meter depends on geometry
of restriction
– Venturi: smooth restriction
– Orifice plate: plate with hole (sudden contraction
and
expansion)
– Nozzle: plate with nozzle (gradual contraction,
sudden
expansion)

57. Venturi meter
• Tube installed into pipeline
• Pressure drop from 1 to 2 determines flow rate
from theory (Bernoulli) with corrections for
losses (K)
• Pressure drop varies as square of flow rate;
requires sensitive gage

58. Orifice plate
• Similar in concept to venturi meter, but simpler to
manufacture
• Simpler geometry increases flow losses and
accuracy
• Vena contracta determines discharge coefficient

59. Nozzle Meter
• Improved orifice meter; lower losses but harder
to construct

60. Problem 10.6
• Find the discharge coefficient for a 5 cm diameter
square edged orifice plate using flange taps in a
15 cm pipe if Re=250,000.
• From Figure 10.6, K=0.6=CE

61. Variable-area meters
• Also called rotameters;
flow is diverted in a
tube of variable area
with a float
• Buoyancy and weight
of the float are
balanced by pressure
and viscous fluid
forces
• Linear with flow rate

62. Turbine Meter
• Rate of rotor spin
related to flow rate
• Counter devices
measures RPMs

63. Vortex Shedding Meters
• Vortex shedding frequency can be nondimensionalized
and shown to be constant for a
given geometry over a range of Re
• Thus, flow rate (average velocity) can be
determined by measuring shedding frequency

64. Electromagnetic
• Operates off of Faraday’s law; a voltage is
induced when a conductor (the liquid) moves
through a magnetic field
• Voltage is proportional to flow rate
• Measures forward and reverse flow

65. Ultrasonic
• Doppler type and time-of-travel meters
• Doppler uses a single trasmitter and
measures
the time it takes for a signal to bounce off;
Doppler shift is related to the flow
velocity
• Time of travel meters use two elements

66. Vibrating Flow Meters
• Also called Coriolis meters
• Deflection in a u-tube is directly
related to flow
rate due to Coriolis forces from turning
fluid

67. Velocity probes
• Pressure probes
– Pitot and pitot-static probes
– Multi-hole probes (direction sensing)
• Vane-meters
• Hot-wire/hot-film probes
• Laser-based techniques
– LDV
– PIV
• Ultrasonic

68. Velocity Measurement Using Pressure

69. U-tube Manometer
• Most common type of liquid column manometer
• Can be used to obtain velocity
by using static and stagnation
pressure as the inputs
• Usually denoted on manometers
by high and low pressure inputs
(stagnation and static,
respectively)

70. The Liquid Column Manometer
• When using a liquid column manometer,
combine
the hydrostatic equation with Bernoulli’s
equation

71. The Inclined Manometer
• Since pressure is a function of height only, the
tube can be inclined to increase accuracy

72. Problem 9.5
• An inclined manometer measures 5.6 cm H20 with
an inclination of 30o; what is the actual pressure
change?

73. Pitot-Static Tube
• Accurate to within 15-20o of
orientation with
respect to free-stream
• Measures velocity magnitude
only; not direction

74. Turbometer
• Same principle as turbine-meters, except
turbine
rotation is correlated with velocity not flow rate

75. Direction-Sensing Probes
• Multi-hole Pitot probes

76. Hot-Wire Anemometry
• Consider a thin wire mounted to supports and exposed to a
velocity U. When a current is passed through wire, heat is
generated (I2Rw). In equilibrium, this must be balanced by
heat loss (primarily convective) to the surroundings.
If velocity changes,
convective heat
transfer coefficient
will change, wire
temperature will
change and
eventually reach a
new equilibrium.

77. HWA
• Constant-Temperature Anemometry

78. Laser-Doppler Velocimetry
• LDV
– Focused laser beams intersect and form the
measurement
volume
– Plane wave fronts: beam waist in the plane
of intersection
– Interference in the plane of intersection
– Pattern of bright and dark stripes/planes

79. LDV
• Velocity=distance/time

80. LDV

81. PIV
• Particle Image Velocimetry
– Field technique – measures 2D (u and v) velocity in a
plane instantaneously
– Essentially tracks seeded flow from one digital
image to
another

82. PIV Layout

83. Stereo PIV

84. Quiz!
• Quiz on Thursday
• Be familiar with
– Pitot-static tubes
– Pressure measuring
devices
• Manometers
• Transducers