This document provides an overview of pressure and velocity measurement techniques for fluid flow applications. It discusses pressure concepts like absolute, gauge, and vacuum pressure. It also covers the Bernoulli equation and how it relates pressure, velocity, and elevation. Measurement devices like manometers and Pitot-static tubes are explained. A sample problem demonstrates how to calculate the uncertainty in a pressure measurement using an inclined manometer.
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Fluid flow, the seemingly effortless movement of liquids and gases, plays a crucial role in various scientific and engineering fields. From blood circulation to airplane design, understanding fluid mechanics is essential. This note explores the basics of fluid flow, keeping it under 3000 words.
Understanding Fluids:
What is a fluid? Any substance that readily adapts to its container's shape, like liquids and gases.
Flow types: Laminar (ordered layers) vs. Turbulent (chaotic swirls), internal (in pipes) vs. external (around objects), steady (unchanging) vs. unsteady (variable).
Governing Principles:
Conservation of mass, momentum, and energy: Fundamental principles ensure mass, momentum, and energy are conserved within a system.
The Core Mechanism: Navier-Stokes Equations
These complex equations describe viscous fluid motion, incorporating the above principles.
Analytical solutions are often challenging, leading to the use of numerical methods like CFD.
Key Concepts:
Reynolds Number (Re): Ratio of inertial to viscous forces, predicting laminar-turbulent transition.
Boundary Layer Theory: Analyzes the thin region near solid boundaries where viscosity dominates.
Drag and Lift Forces: Forces exerted by flowing fluids on objects, important in aerodynamics.
Fluid Properties: Density, viscosity, and compressibility significantly impact flow behavior.
Applications and Importance:
Civil Engineering: Design of pipelines, dams, and water distribution systems.
Aerospace Engineering: Designing airplanes, rockets, and understanding airfoils.
Chemical Engineering: Designing reactors, pumps, and separation processes.
Biomedical Engineering: Understanding blood flow and designing medical devices.
1. Measurements Course 4600:483-001
Lecture 7 – Pressure and Velocity Measurements
Information about this course can be found at the home page of the instructor:
http://gozips.uakron.edu/~dorfi/
Introduction
This lecture focuses on standard pressure and velocity measurement techniques in fluid flow applications
and the instrumentation necessary to perform the measurements. Fluid flow is governed by thee
conservation equations:
• Conservation of Mass
• Conservation of Momentum
• Conservation of Energy
These equations govern the fluid flow and its field variables: fluid pressure, velocity and temperature.
Pressure Concepts
Pressure is defined as contact force per unit
area. It always acts inward and is therefore
always positive (for absolute pressure). Since
Gage
pressure is the macroscopic manifestation of Pressure
the kinetic energy of fluid molecules, zero Reference
pressure is found in absolute vacuum or at Pressure Absolute
absolute zero temperature. In fluid Standard System
Atmospheric Pressure
applications, pressure is a local scalar quantity.
Pressure
101.3 KPa abs
The pressure relative to a reference pressure 14.696 psia
such as the current atmospheric pressure is
Perfect
called gage pressure. Gage pressure can be
Vacuum
both positive and negative, since it is simply a Pressure Scales
pressure difference.
Bernoulli Equation
The Bernoulli Equation is the conservation of momentum equation for steady-state, inviscid (no wall
friction) flow for an incompressible fluid
p1 v12 p v2
+ + gh1 = 2 + 2 + gh2 = const
ρ 2 ρ 2
where p is the absolute pressure, v is the fluid velocity, g gravity and h the elevation at arbitrary locations 1
and 2 on the stream line. The Bernoulli equation states that the kinetic and potential energy of the fluid
remains constant along a stream line.
It is important to remember the assumptions made in deriving this equation:
• steady flow
• incompressible flow
• frictionless flow
• flow along a streamline
Dr. Hans R. Dorfi 11/03/03 Page 1of 4
2. Measurements Course 4600:483-001
Manometer
The manometer is an instrument, which converts p1 p1 p2
pressure differences into elevation differences (flow
velocity is zero). If we define the specific gravity of a Specific x
fluid as γ=ρ/g and set the velocity to zero, the Gravity of
Bernoulli equation simplifies to Gas: γ
p1 + γ h1 = p2 + γ h2
θ
L=H /sinθ H
Applying the Bernoulli equation to the U-tube
Specific Gravity
manometer as shown on the right gives the equation of Measurement
p1 + γ ( x + H ) = p2 + γ x + γ m H Fluid: γm
Solving for the pressure difference yields
p1 − p2 = (γ m − γ ) H
The pressure difference is directly proportional to the
height difference H. The difference in specific weight θ
of the measurement fluid and the gas is the
proportionality factor. Usually the specific weight of U-tube Manometer (optionally with inclined tube)
the measurement fluid is several orders of magnitude
larger than the specific weight of the gas.
In order to increase the sensitivity of the instrument, one of the arms of the manometers can be inclined by
an angle θ. The pressure difference for an inclined manometer is then found from
p1 − p2 = p = (γ m − γ ) L sin θ
Uncertainty in inclined manometer measurements
The above equation shows that the measurement of pressure differentials p with inclined manometers is
affected by the uncertainty in the fluid specific weight (density/gravity), the ability to read the manometer
(uncertainty in L) and the uncertainty for the inclination angle θ. The normalized uncertainty equation for
the manometer is thus given by
e p = eγ2m + eγ2 + eL + eθ2
2
Example: An inclined manometer with indicating leg at 30° is used at 20°C to measure a gas pressure of
nominal magnitude of 100N/m2 relative to ambient. The specific weight of the measurement fluid is 9770
N/m3. The manometer resolution is 1mm and the specific gravity of the gas is 11.5 N/m3. The relative
uncertainty in the specific gravities is assumed to be 1%. We can measure the angle of the inclined tube to
within 1° resolution.
What is the design state uncertainty for the pressure measurement with this manometer?
From the uncertainty analysis we know that the relative uncertainty ei for parameter i is
1 dp
ei = Ui
p dxi
Substitution of the relative uncertainty equation for the specific gravities yields
1 dp γ m Uγ m 9770
eγ m = Uγ m = = * .01 ≈ .01
p dγ m γ m − γ γ m 9770 − 11.5
1 dp Uγ γ Uγ 11.5
eγ = Uγ = = = * .01 ≈ 10 −5
p dγ γm −γ γm −γ γ 9770 − 11.5
Notice how the uncertainty due to the specific gravity of the gas is negligibly small. This is because the
specific gravity of the measurement fluid is about 1000 times larger than the gas.
Dr. Hans R. Dorfi 11/03/03 Page 2of 4
3. Measurements Course 4600:483-001
The relative uncertainty in the length is
1 dp U
eL = UL = L
p dL L
In order to calculate the relative uncertainty, we need to calculate the Length L of the fluid column the
manometer at the nominal pressure of 100N/m2.
p 100 N / m2
L= = = 0.0205m = 20.5mm
(γ m − γ ) sin θ (9770 − 11.5) sin 30o N / m3
The manometer resolution is 1mm, therefore the zero order uncertainty is 0.5mm. The relative uncertainty
in length L is therefore
UL 0.5mm
eL = = = 0.024
L 20.5mm
The resolution in the angle measurement is 1°. Therefore the zero order uncertainty is 0.5° or 0.0087
radians. The relative uncertainty in the angle is found from
1 dp Uθ 0.5o 0.0087 rad
eθ = Uθ = = = = 0.015
p dθ tan θ tan 30 o
0.5774
Putting all uncertainties together yields
e p = 10 −2 * 12 + 10 −3*2 + 2.4 2 + 1.52 = 10 −2 * 1 + 10 −6 + 5.76 + 2.25 = 0.03 = 3%
The uncertainty in the length measurement L is the primary contributor to the uncertainty in the pressure
measurement. For the nominal pressure of 100N/m2 the design stage uncertainty is 3% or 3N/m2.
Pitot-Static Tubes
The Bernoulli equation can also be used to
measure the fluid velocity. Recall the equation :
for zero elevation change p1 is sensed normal to
p1v12 p2 v2
2
Point 1: flow direction
+ = +
ρ 2 ρ 2 p1,v1
If we could slow down the fluid flow to zero at
point 2, we can solve for the fluid velocity at
point 1.
= Point 2
2( p2 − p1 ) 2∆p p2,v2=0
v1 = =
ρ ρ
This is the basic idea of the Pitot-Static tube. The
Pitot-Static (P-S) tube consists basically of two
concentric tubes, with the end turned through a
right angle so that the tip can be faced into the
airstream after insertion through the duct wall.
The modified ellipsoidal nose form has a single
forward facing hole for sensing Total Pressure
and a ring of side holes for sensing the Static
Pressure. Both these inlets are individually
connected to tapping outlets at the tail of the unit.
p2 is called the stagnation or total pressure at
the forward facing inlet to the Pitot-static probe
where the velocity becomes zero, p1 or pstatic is Schematic of Pitot-Static Tube
the static pressure along the sides of the Pitot-
Dr. Hans R. Dorfi 11/03/03 Page 3of 4
4. Measurements Course 4600:483-001
static probe where the velocity is unchanged from the upstream duct velocity v. The pressure difference,
∆P, is called the dynamic pressure because it is related to the change in fluid velocity. We can calculate
the duct velocity from the dynamic pressure as
2∆P
v=
ρ fluid
Note that this expression is only accurate if the P-S tube points directly into v1 such that all of v1 is
stagnated. If the P-S tube is misaligned, the measured velocity will be too low. Remember that this
equation was derived based on the Bernoulli equation, which assumes laminar and incompressible fluid
flow. For flow velocities greater than 30% of the sonic velocity, the fluid must be treated as compressible.
Then the simplified equation for the P-S tube no longer holds.
Textbook descriptions of Pitot-static probes usually describe their use in a laminar flow. What happens
when Pitot-static probes are used in time-varying turbulent flows? The pressure difference associated with
the fluctuation velocity must move a mass in the pressure sensor to measure the pressure change associated
with a given velocity change. The measurement devices are thus second-order mechanical systems with
their own natural frequency and damping ratio. If the frequency of the velocity fluctuation is much faster
than the natural frequency of the measuring system, then it will display the average value of the fluctuating
signal. This will only hold true for moderately turbulent flows (less than 10% turbulence intensity) because
the velocity vector must remain approximately parallel to the Pitot-static probe. Duct flows typically have
low enough turbulence intensities that the effect of turbulence can be neglected, but disturbed regions of
flow near sharp edges or area changes can prevent good readings.
Uncertainty in Pitot-Static Tubes
See the example in the lecture on uncertainty.
Dr. Hans R. Dorfi 11/03/03 Page 4of 4