MECHANICS of FLUIDS LABORATORY - Mechanical Engineering

A Manual for the
MECHANICS of FLUIDS LABORATORY
William S. Janna
Department of Mechanical Engineering
Memphis State University

©1997 William S. Janna
All Rights Reserved.
No part of this manual may be reproduced, stored in a retrieval
system, or transcribed in any form or by any means—electronic, magnetic,
mechanical, photocopying, recording, or otherwise—
without the prior written consent of William S. Janna
2

TABLE OF CONTENTS
Item
Page
Report Writing.................................................................................................................4
Cleanliness and Safety ....................................................................................................6
Experiment 1 Density and Surface Tension.....................................................7
Experiment 2 Viscosity.........................................................................................9
Experiment 3 Center of Pressure on a Submerged Plane Surface.............10
Experiment 4 Measurement of Differential Pressure..................................12
Experiment 5 Impact of a Jet of Water ............................................................14
Experiment 6 Critical Reynolds Number in Pipe Flow...............................16
Experiment 7 Fluid Meters................................................................................18
Experiment 8 Pipe Flow .....................................................................................22
Experiment 9 Pressure Distribution About a Circular Cylinder................24
Experiment 10 Drag Force Determination .......................................................27
Experiment 11 Analysis of an Airfoil................................................................28
Experiment 12 Open Channel Flow—Sluice Gate .........................................30
Experiment 13 Open Channel Flow Over a Weir ..........................................32
Experiment 14 Open Channel Flow—Hydraulic Jump ................................34
Experiment 15 Open Channel Flow Over a Hump........................................36
Experiment 16 Measurement of Velocity and Calibration of
a Meter for Compressible Flow.............................39
Experiment 17 Measurement of Fan Horsepower .........................................44
Experiment 18 Measurement of Pump Performance....................................46
Appendix .........................................................................................................................50
3

REPORT WRITING
All reports in the Fluid Mechanics
Laboratory require a formal laboratory report
unless specified otherwise. The report should be
written in such a way that anyone can duplicate
the performed experiment and find the same
results as the originator. The reports should be
simple and clearly written. Reports are due one
week after the experiment was performed, unless
specified otherwise.
The report should communicate several ideas
to the reader. First the report should be neatly
done. The experimenter is in effect trying to
convince the reader that the experiment was
performed in a straightforward manner with
great care and with full attention to detail. A
poorly written report might instead lead the
reader to think that just as little care went into
performing the experiment. Second, the report
should be well organized. The reader should be
able to easily follow each step discussed in the
text. Third, the report should contain accurate
results. This will require checking and rechecking
the calculations until accuracy can be guaranteed.
Fourth, the report should be free of spelling and
grammatical errors. The following format, shown
in Figure R.1, is to be used for formal Laboratory
Reports:
Title Page–The title page should show the title
and number of the experiment, the date the
experiment was performed, experimenter's
name and experimenter's partners' names.
Table of Contents –Each page of the report must
be numbered for this section.
Object –The object is a clear concise statement
explaining the purpose of the experiment.
This is one of the most important parts of the
laboratory report because everything
included in the report must somehow relate to
the stated object. The object can be as short as
one sentence and it is usually written in the
past tense.
Theory –The theory section should contain a
complete analytical development of all
important equations pertinent to the
experiment, and how these equations are used
in the reduction of data. The theory section
should be written textbook-style.
Procedure – The procedure section should contain
a schematic drawing of the experimental
setup including all equipment used in a parts
list with manufacturer serial numbers, if any.
Show the function of each part when
necessary for clarity. Outline exactly step-
Bibliography
Calibration Curves
Original Data Sheet
(Sample Calculation)
Appendix
Title Page
Discussion & Conclusion
(Interpretation)
Results (Tables
and Graphs)
Procedure (Drawings
and Instructions)
Theory
(Textbook Style)
Object
(Past Tense)
Table of Contents
Each page numbered
Experiment Number
Experiment Title
Your Name
Due Date
Partners’ Names
FIGURE R.1. Format for formal reports.
by-step how the experiment was performed in
case someone desires to duplicate it. If it
cannot be duplicated, the experiment shows
nothing.
Results – The results section should contain a
formal analysis of the data with tables,
graphs, etc. Any presentation of data which
serves the purpose of clearly showing the
outcome of the experiment is sufficient.
Discussion and Conclusion – This section should
give an interpretation of the results
explaining how the object of the experiment
was accomplished. If any analytical
expression is to be verified, calculate % error †
and account for the sources. Discuss this
experiment with respect to its faults as well
† % error–An analysis expressing how favorably the
empirical data approximate theoretical information.
There are many ways to find % error, but one method is
introduced here for consistency. Take the difference
between the empirical and theoretical results and divide
by the theoretical result. Multiplying by 100% gives the
% error. You may compose your own error analysis as
long as your method is clearly defined.
4

as its strong points. Suggest extensions of the
experiment and improvements. Also
recommend any changes necessary to better
accomplish the object.
Each experiment write-up contains a
number of questions. These are to be answered
or discussed in the Discussion and Conclusions
section.
Appendix
(1) Original data sheet.
(2) Show how data were used by a sample
calculation.
(3) Calibration curves of instrument which
were used in the performance of the
experiment. Include manufacturer of the
instrument, model and serial numbers.
Calibration curves will usually be supplied
by the instructor.
(4) Bibliography listing all references used.
Short Form Report Format
Often the experiment requires not a formal
report but an informal report. An informal report
includes the Title Page, Object, Procedure,
Results, and Conclusions. Other portions may be
added at the discretion of the instructor or the
writer. Another alternative report form consists
of a Title Page, an Introduction (made up of
shortened versions of Object, Theory, and
Procedure) Results, and Conclusion and
Discussion. This form might be used when a
detailed theory section would be too long.
Graphs
In many instances, it is necessary to compose a
plot in order to graphically present the results.
Graphs must be drawn neatly following a specific
format. Figure R.2 shows an acceptable graph
prepared using a computer. There are many
computer programs that have graphing
capabilities. Nevertheless an acceptably drawn
graph has several features of note. These features
are summarized next to Figure R.2.
Features of note
• Border is drawn about the entire graph.
• Axis labels defined with symbols and
units.
• Grid drawn using major axis divisions.
• Each line is identified using a legend.
• Data points are identified with a
symbol: “ ´” on the Q ac line to denote
data points obtained by experiment.
• The line representing the theoretical
results has no data points represented.
• Nothing is drawn freehand.
• Title is descriptive, rather than
something like Q vs ∆h.
flow rate Q in m 3 /s
0.2
0.15
0.1
0.05
0
Q th
Q ac
0 0.2 0.4 0.6 0.8 1
head loss ∆ h in m
FIGURE R.2. Theoretical and actual volume flow rate
through a venturi meter as a function of head loss.
5

CLEANLINESS AND SAFETY
Cleanliness
There are “housekeeping” rules that the user
of the laboratory should be aware of and abide
by. Equipment in the lab is delicate and each
piece is used extensively for 2 or 3 weeks per
semester. During the remaining time, each
apparatus just sits there, literally collecting dust.
University housekeeping staff are not required to
clean and maintain the equipment. Instead, there
are college technicians who will work on the
equipment when it needs repair, and when they
are notified that a piece of equipment needs
attention. It is important, however, that the
equipment stay clean, so that dust will not
accumulate too badly.
The Fluid Mechanics Laboratory contains
equipment that uses water or air as the working
fluid. In some cases, performing an experiment
will inevitably allow water to get on the
equipment and/or the floor. If no one cleaned up
their working area after performing an
experiment, the lab would not be a comfortable or
safe place to work in. No student appreciates
walking up to and working with a piece of
equipment that another student or group of
students has left in a mess.
Consequently, students are required to clean
up their area at the conclusion of the performance
of an experiment. Cleanup will include removal
of spilled water (or any liquid), and wiping the
table top on which the equipment is mounted (if
appropriate). The lab should always be as clean
or cleaner than it was when you entered. Cleaning
the lab is your responsibility as a user of the
equipment. This is an act of courtesy that students
who follow you will appreciate, and that you
will appreciate when you work with the
equipment.
Safety
The layout of the equipment and storage
cabinets in the Fluid Mechanics Lab involves
resolving a variety of conflicting problems. These
include traffic flow, emergency facilities,
environmental safeguards, exit door locations,
etc. The goal is to implement safety requirements
without impeding egress, but still allowing
adequate work space and necessary informal
communication opportunities.
Distance between adjacent pieces of
equipment is determined by locations of floor
drains, and by the need to allow enough space
around the apparatus of interest. Immediate
access to the Safety Cabinet is also considered.
Emergency facilities such as showers, eye wash
fountains, spill kits, fire blankets and the like
are not found in the lab. We do not work with
hazardous materials and such safety facilities
are not necessary. However, waste materials are
generated and they should be disposed of
properly.
Every effort has been made to create a
positive, clean, safety conscious atmosphere.
Students are encouraged to handle equipment
safely and to be aware of, and avoid being
victims of, hazardous situations.
6

EXPERIMENT 1
FLUID PROPERTIES: DENSITY AND SURFACE TENSION
There are several properties simple
Newtonian fluids have. They are basic
properties which cannot be calculated for every
fluid, and therefore they must be measured.
These properties are important in making
calculations regarding fluid systems. Measuring
fluid properties, density and viscosity, is the
object of this experiment.
W 1
W 2
Part I: Density Measurement.
Equipment
Graduated cylinder or beaker
Liquid whose properties are to be
measured
Hydrometer cylinder
Scale
The density of the test fluid is to be found by
weighing a known volume of the liquid using the
graduated cylinder or beaker and the scale. The
beaker is weighed empty. The beaker is then
filled to a certain volume according to the
graduations on it and weighed again. The
difference in weight divided by the volume gives
the weight per unit volume of the liquid. By
appropriate conversion, the liquid density is
calculated. The mass per unit volume, or the
density, is thus measured in a direct way.
A second method of finding density involves
measuring buoyant force exerted on a submerged
object. The difference between the weight of an
object in air and the weight of the object in liquid
is known as the buoyant force (see Figure 1.1).
FIGURE 1.1. Measuring the buoyant force on an
object with a hanging weight.
Referring to Figure 1.1, the buoyant force B is
found as
B = W 1
- W 2
The buoyant force is equal to the difference
between the weight of the object in air and the
weight of the object while submerged. Dividing
this difference by the volume displaced gives the
weight per unit volume from which density can be
calculated.
Questions
1. Are the results of all the density
measurements in agreement?
2. How does the buoyant force vary with
depth of the submerged object? Why?
Part II: Surface Tension Measurement
Equipment
Surface tension meter
Beaker
Test fluid
Surface tension is defined as the energy
required to pull molecules of liquid from beneath
the surface to the surface to form a new area. It is
therefore an energy per unit area (F⋅L/L 2 = F/L).
A surface tension meter is used to measure this
energy per unit area and give its value directly. A
schematic of the surface tension meter is given in
Figure 1.2.
The platinum-iridium ring is attached to a
balance rod (lever arm) which in turn is attached
to a stainless steel torsion wire. One end of this
wire is fixed and the other is rotated. As the wire
is placed under torsion, the rod lifts the ring
slowly out of the liquid. The proper technique is
to lower the test fluid container as the ring is
lifted so that the ring remains horizontal. The
force required to break the ring free from the
liquid surface is related to the surface tension of
the liquid. As the ring breaks free, the gage at
the front of the meter reads directly in the units
indicated (dynes/cm) for the given ring. This
reading is called the apparent surface tension and
must be corrected for the ring used in order to
obtain the actual surface tension for the liquid.
The correction factor F can be calculated with the
following equation
7

FIGURE 1.2. A schematic of the
surface tension meter.
balance rod
platinum
iridium ring
test liquid
clamp
torsion wire
F = 0.725 + √⎺⎺⎺⎺⎺⎺⎺⎺⎺⎺⎺⎺⎺⎺⎺⎺⎺⎺⎺⎺⎺⎺⎺
0.000 403 3(σ a
/ρ) + 0.045 34 - 1.679(r/R)
where F is the correction factor, σ a is the
apparent surface tension read from the dial
(dyne/cm), ρ is the density of the liquid (g/cm 3 ),
and (r/R) for the ring is found on the ring
container. The actual surface tension for the
liquid is given by
σ = Fσ a
8

EXPERIMENT 2
FLUID PROPERTIES: VISCOSITY
One of the properties of homogeneous liquids
is their resistance to motion. A measure of this
resistance is known as viscosity. It can be
measured in different, standardized methods or
tests. In this experiment, viscosity will be
measured with a falling sphere viscometer.
The Falling Sphere Viscometer
When an object falls through a fluid medium,
the object reaches a constant final speed or
terminal velocity. If this terminal velocity is
sufficiently low, then the various forces acting on
the object can be described with exact expressions.
The forces acting on a sphere, for example, that is
falling at terminal velocity through a liquid are:
Weight - Buoyancy - Drag = 0
ρ s g 4 3 πR3 - ρg 4 3 πR3 - 6πµVR = 0
where ρ s and ρ are density of the sphere and
liquid respectively, V is the sphere’s terminal
velocity, R is the radius of the sphere and µ is
the viscosity of the liquid. In solving the
preceding equation, the viscosity of the liquid can
be determined. The above expression for drag is
valid only if the following equation is valid:
average the results. With the terminal velocity
of this and of other spheres measured and known,
the absolute and kinematic viscosity of the liquid
can be calculated. The temperature of the test
liquid should also be recorded. Use at least three
different spheres. (Note that if the density of
the liquid is unknown, it can be obtained from any
group who has completed or is taking data on
Experiment 1.)
Questions
1. Should the terminal velocity of two
different size spheres be the same?
2. Does a larger sphere have a higher
terminal velocity?
3. Should the viscosity found for two different
size spheres be the same? Why or why not?
4. If different size spheres give different
results for the viscosity, what are the error
sources? Calculate the % error and account
for all known error sources.
5. What are the shortcomings of this method?
6. Why should temperature be recorded.
7. Can this method be used for gases?
8. Can this method be used for opaque liquids?
9. Can this method be used for something like
peanut butter, or grease or flour dough?
Why or why not?
ρVD
µ < 1
where D is the sphere diameter. Once the
viscosity of the liquid is found, the above ratio
should be calculated to be certain that the
mathematical model gives an accurate
description of a sphere falling through the
liquid.
Equipment
Hydrometer cylinder
Scale
Stopwatch
Several small spheres with weight and
diameter to be measured
Test liquid
FIGURE 2.1. Terminal velocity measurement (V =
d/time).
V
d
Drop a sphere into the cylinder liquid and
record the time it takes for the sphere to fall a
certain measured distance. The distance divided
by the measured time gives the terminal velocity
of the sphere. Repeat the measurement and
9

EXPERIMENT 3
CENTER OF PRESSURE ON A SUBMERGED
PLANE SURFACE
Submerged surfaces are found in many
engineering applications. Dams, weirs and water
gates are familiar examples of submerged
surfaces used to control the flow of water. From
the design viewpoint, it is important to have a
working knowledge of the forces that act on
submerged surfaces.
A plane surface located beneath the surface
of a liquid is subjected to a pressure due to the
height of liquid above it, as shown in Figure 3.1.
Increasing pressure varies linearly with
increasing depth resulting in a pressure
distribution that acts on the submerged surface.
The analysis of this situation involves
determining a force which is equivalent to the
pressure, and finding the location of this force.
FIGURE 3.1. Pressure distribution on a submerged
plane surface and the equivalent force.
For this case, it can be shown that the
equivalent force is:
F = ρgy c
A (3.1)
in which ρ is the liquid density, y c is the distance
from the free surface of the liquid to the centroid
of the plane, and A is the area of the plane in
contact with liquid. Further, the location of this
force y F below the free surface is
y F = I xx
y c
A + y c
(3.2)
in which I xx is the second area moment of the
plane about its centroid. The experimental
F
y F
verification of these equations for force and
distance is the subject of this experiment.
Center of Pressure Measurement
Equipment
Center of Pressure Apparatus
Weights
Figure 3.2 gives a schematic of the apparatus
used in this experiment. The torus and balance
arm are placed on top of the tank. Note that the
pivot point for the balance arm is the point of
contact between the rod and the top of the tank.
The zeroing weight is adjusted to level the
balance arm. Water is then added to a
predetermined depth. Weights are placed on the
weight hanger to re-level the balance arm. The
amount of needed weight and depth of water are
then recorded. The procedure is then repeated for
four other depths. (Remember to record the
distance from the pivot point to the free surface
for each case.)
From the depth measurement, the equivalent
force and its location are calculated using
Equations 3.1 and 3.2. Summing moments about the
pivot allows for a comparison between the
theoretical and actual force exerted. Referring to
Figure 3.2, we have
WL
F =
(y + y F )
(3.3)
where y is the distance from the pivot point to
the free surface, y F is the distance from the free
surface to the line of action of the force F, and L is
the distance from the pivot point to the line of
action of the weight W. Note that both curved
surfaces of the torus are circular with centers at
the pivot point. For the report, compare the force
obtained with Equation 3.1 to that obtained with
Equation 3.3. When using Equation 3.3, it will be
necessary to use Equation 3.2 for y F .
Questions
1. In summing moments, why isn't the buoyant
force taken into account?
2. Why isn’t the weight of the torus and the
balance arm taken into account?
10

L
level
R i
y
zeroing weight
torus
pivot point
(point of contact)
weight
hanger
R o
y F
h
F
w
FIGURE 3.2. A schematic of the center of pressure apparatus.
11

EXPERIMENT 4
MEASUREMENT OF DIFFERENTIAL PRESSURE
Pressure can be measured in several ways.
Bourdon tube gages, manometers, and transducers
are a few of the devices available. Each of these
instruments actually measures a difference in
pressure; that is, measures a difference between
the desired reading and some reference pressure,
usually atmospheric. The measurement of
differential pressure with manometers is the
subject of this experiment.
Manometry
A manometer is a device used to measure a
pressure difference and display the reading in
terms of height of a column of liquid. The height
is related to the pressure difference by the
hydrostatic equation.
Figure 4.1 shows a U-tube manometer
connected to two pressure vessels. The manometer
reading is ∆h and the manometer fluid has
density ρ m . One pressure vessel contains a fluid of
density ρ 1 while the other vessel contains a fluid
of density ρ 2 . The pressure difference can be found
by applying the hydrostatic equation to each
limb of the manometer. For the left leg,
p 1
p 2
1
p A
z 1
FIGURE 4.1. A U-tube manometer connected to
two pressure vessels.
p 1
+ ρ 1
gz 1
= p A
z 2
Likewise for the right leg,
p 2
+ ρ 2
gz 2
+ ρ m
g∆h = p A
Equating these expressions and solving for the
pressure difference gives
h
p A
m
2
p 1
- p 2
= ρ 2
gz 2
+ ρ 1
gz 1
+ ρ m
g∆h
If the fluids above the manometer liquid are both
gases, then ρ 1 and ρ 2 are small compared to ρ µ .
The above equation then becomes
p 1
- p 2
= ρ m
g∆h
Figure 4.2 is a schematic of the apparatus
used in this experiment. It consists of three U-tube
manometers, a well-type manometer, a U-
tube/inclined manometer and a differential
pressure gage. There are two tanks (actually, two
capped pieces of pipe) to which each manometer
and the gage are connected. The tanks have bleed
valves attached and the tanks are connected
with plastic tubing to a squeeze bulb. The bulb
lines also contain valves. With both bleed valves
closed and with both bulb line valves open, the
bulb is squeezed to pump air from the low pressure
tank to the high pressure tank. The bulb is
squeezed until any of the manometers reaches its
maximum reading. Now both valves are closed
and the liquid levels are allowed to settle in
each manometer. The ∆h readings are all
recorded. Next, one or both bleed valves are
opened slightly to release some air into or out of a
tank. The liquid levels are again allowed to
settle and the ∆h readings are recorded. The
procedure is to be repeated until 5 different sets of
readings are obtained. For each set of readings,
convert all readings into psi or Pa units, calculate
the average value and the standard deviation.
Before beginning, be sure to zero each manometer
and the gage.
Questions
1. Manometers 1, 2 and 3 are U-tube types and
each contains a different liquid. Manometer
4 is a well-type manometer. Is there an
advantage to using this one over a U-tube
type?
2. Manometer 5 is a combined U/tube/inclined
manometer. What is the advantage of this
type?
3. Note that some of the manometers use a
liquid which has a specific gravity
different from 1.00, yet the reading is in
inches of water. Explain how this is
possible.
4. What advantages or disadvantages does
the gage have over the manometers?
12

5. Is a low value of the standard deviation
expected? Why?
6. What does a low standard deviation
imply?
7. In your opinion, which device gives the
most accurate reading. What led you to this
conclusion?
High pressure tank
Low pressure tank
Bleed valves
Gage
U-tube manometers
Well-type
manometer
U-tube/inclined
manometer
FIGURE 4.2. A schematic of the apparatus used in this experiment.
13

EXPERIMENT 5
IMPACT OF A JET OF WATER
A jet of fluid striking a stationary object
exerts a force on that object. This force can be
measured when the object is connected to a spring
balance or scale. The force can then be related to
the velocity of the jet of fluid and in turn to the
rate of flow. The force developed by a jet stream
of water is the subject of this experiment.
Impact of a Jet of Liquid
Equipment
Jet Impact Apparatus
Object plates
Figure 5.1 is a schematic of the device used in
this experiment. The device consists of a tank
within a tank. The interior tank is supported on a
pivot and has a lever arm attached to it. As
water enters this inner tank, the lever arm will
reach a balance point. At this time, a stopwatch
is started and a weight is placed on the weight
hanger (e.g., 10 lbf). When enough water has
entered the tank (10 lbf), the lever arm will
again balance. The stopwatch is stopped. The
elapsed time divided into the weight of water
collected gives the weight or mass flow rate of
water through the system (lbf/sec, for example).
The outer tank acts as a support for the table
top as well as a sump tank. Water is pumped from
the outer tank to the apparatus resting on the
table top. As shown in Figure 5.1, the impact
apparatus contains a nozzle that produces a high
velocity jet of water. The jet is aimed at an object
(such as a flat plate or hemisphere). The force
exerted on the plate causes the balance arm to
which the plate is attached to deflect. A weight
is moved on the arm until the arm balances. A
summation of moments about the pivot point of
the arm allows for calculating the force exerted
by the jet.
Water is fed through the nozzle by means of
a centrifugal pump. The nozzle emits the water in
a jet stream whose diameter is constant. After the
water strikes the object, the water is channeled to
the weighing tank inside to obtain the weight or
mass flow rate.
The variables involved in this experiment
are listed and their measurements are described
below:
1. Mass rate of flow–measured with the
weighing tank inside the sump tank. The
volume flow rate is obtained by dividing
mass flow rate by density: Q = m/ρ.
2. Velocity of jet–obtained by dividing volume
flow rate by jet area: V = Q/A. The jet is
cylindrical in shape with a diameter of 0.375
in.
3. Resultant force—found experimentally by
summation of moments about the pivot point
of the balance arm. The theoretical resultant
force is found by use of an equation derived by
applying the momentum equation to a control
volume about the plate.
Impact Force Analysis
The total force exerted by the jet equals the
rate of momentum loss experienced by the jet after
it impacts the object. For a flat plate, the force
equation is:
F = ρQ2
A
For a hemisphere,
F = 2ρQ2
A
(flat plate)
(hemisphere)
For a cone whose included half angle is α,
F = ρQ2
(1 + cos α) (cone)
A
For your report, derive the appropriate
equation for each object you use. Compose a graph
with volume flow rate on the horizontal axis,
and on the vertical axis, plot the actual and
theoretical force. Use care in choosing the
increments for each axis.
14

pivot
balancing
weight
lever arm with
flat plate attached
water
jet
flat plate
nozzle
drain
flow control
valve
weigh tank
tank pivot
plug
weight hanger
sump tank
motor
pump
FIGURE 5.1. A schematic of the jet impact apparatus.
15

EXPERIMENT 6
CRITICAL REYNOLDS NUMBER IN PIPE FLOW
The Reynolds number is a dimensionless ratio
of inertia forces to viscous forces and is used in
identifying certain characteristics of fluid flow.
The Reynolds number is extremely important in
modeling pipe flow. It can be used to determine
the type of flow occurring: laminar or turbulent.
Under laminar conditions the velocity
distribution of the fluid within the pipe is
essentially parabolic and can be derived from the
equation of motion. When turbulent flow exists,
the velocity profile is “flatter” than in the
laminar case because the mixing effect which is
characteristic of turbulent flow helps to more
evenly distribute the kinetic energy of the fluid
over most of the cross section.
In most engineering texts, a Reynolds number
of 2 100 is usually accepted as the value at
transition; that is, the value of the Reynolds
number between laminar and turbulent flow
regimes. This is done for the sake of convenience.
In this experiment, however, we will see that
transition exists over a range of Reynolds numbers
and not at an individual point.
The Reynolds number that exists anywhere in
the transition region is called the critical
Reynolds number. Finding the critical Reynolds
number for the transition range that exists in pipe
flow is the subject of this experiment.
Critical Reynolds Number Measurement
Equipment
Critical Reynolds Number Determination
Apparatus
Figure 6.1 is a schematic of the apparatus
used in this experiment. The constant head tank
provides a controllable, constant flow through
the transparent tube. The flow valve in the tube
itself is an on/off valve, not used to control the
flow rate. Instead, the flow rate through the tube
is varied with the rotameter valve at A. The
head tank is filled with water and the overflow
tube maintains a constant head of water. The
liquid is then allowed to flow through one of the
transparent tubes at a very low flow rate. The
valve at B controls the flow of dye; it is opened
and dye is then injected into the pipe with the
water. The dye injector tube is not to be placed in
the pipe entrance as it could affect the results.
Establish laminar flow by starting with a very
low flow rate of water and of dye. The injected
dye will flow downstream in a threadlike
pattern for very low flow rates. Once steady state
is achieved, the rotameter valve is opened
slightly to increase the water flow rate. The
valve at B is opened further if necessary to allow
more dye to enter the tube. This procedure of
increasing flow rate of water and of dye (if
necessary) is repeated throughout the
experiment.
Establish laminar flow in one of the tubes.
Then slowly increase the flow rate and observe
what happens to the dye. Its pattern may
change, yet the flow might still appear to be
laminar. This is the beginning of transition.
Continue increasing the flow rate and again
observe the behavior of the dye. Eventually, the
dye will mix with the water in a way that will
be recognized as turbulent flow. This point is the
end of transition. Transition thus will exist over a
range of flow rates. Record the flow rates at key
points in the experiment. Also record the
temperature of the water.
The object of this procedure is to determine
the range of Reynolds numbers over which
transition occurs. Given the tube size, the
Reynolds number can be calculated with:
Re = VD
ν
where V (= Q/A) is the average velocity of
liquid in the pipe, D is the hydraulic diameter of
the pipe, and ν is the kinematic viscosity of the
liquid.
The hydraulic diameter is calculated from
its definition:
D =
4 x Area
Wetted Perimeter
For a circular pipe flowing full, the hydraulic
diameter equals the inside diameter of the pipe.
For a square section, the hydraulic diameter will
equal the length of one side (show that this is
the case). The experiment is to be performed for
both round tubes and the square tube. With good
technique and great care, it is possible for the
transition Reynolds number to encompass the
traditionally accepted value of 2 100.
16

Questions
1. Can a similar procedure be followed for
gases?
2. Is the Reynolds number obtained at
transition dependent on tube size or shape?
3. Can this method work for opaque liquids?
dye reservoir
drilled partitions
B
transparent tube
on/off valve
rotameter
inlet to
tank
overflow
to drain
FIGURE 6.1. The critical Reynolds number determination apparatus.
A
to drain
17

EXPERIMENT 7
FLUID METERS IN INCOMPRESSIBLE FLOW
There are many different meters used in pipe
flow: the turbine type meter, the rotameter, the
orifice meter, the venturi meter, the elbow meter
and the nozzle meter are only a few. Each meter
works by its ability to alter a certain physical
characteristic of the flowing fluid and then
allows this alteration to be measured. The
measured alteration is then related to the flow
rate. A procedure of analyzing meters to
determine their useful features is the subject of
this experiment.
The Venturi Meter
The venturi meter is constructed as shown in
Figure 7.1. It contains a constriction known as the
throat. When fluid flows through the
constriction, it must experience an increase in
velocity over the upstream value. The velocity
increase is accompanied by a decrease in static
pressure at the throat. The difference between
upstream and throat static pressures is then
measured and related to the flow rate. The
greater the flow rate, the greater the pressure
drop ∆p. So the pressure difference ∆h (= ∆p/ρg)
can be found as a function of the flow rate.
1
h
2
FIGURE 7.1. A schematic of the Venturi meter.
Using the hydrostatic equation applied to
the air-over-liquid manometer of Figure 7.1, the
pressure drop and the head loss are related by
(after simplification):
p 1
- p 2
ρg
= ∆h
By combining the continuity equation,
Q = A 1
V 1
= A 2
V 2
with the Bernoulli equation,
p 1
ρ + V 1 2
2 = p 2
ρ + V 2 2
2
and substituting from the hydrostatic equation, it
can be shown after simplification that the
volume flow rate through the venturi meter is
given by
Q = A th 2
√⎺⎺⎺⎺
2g∆h
1 - (D 24
/D 14
)
(7.1)
The preceding equation represents the theoretical
volume flow rate through the venturi meter.
Notice that is was derived from the Bernoulli
equation which does not take frictional effects
into account.
In the venturi meter, there exists small
pressure losses due to viscous (or frictional)
effects. Thus for any pressure difference, the
actual flow rate will be somewhat less than the
theoretical value obtained with Equation 7.1
above. For any ∆h, it is possible to define a
coefficient of discharge C v as
C v
= Q ac
Q th
For each and every measured actual flow rate
through the venturi meter, it is possible to
calculate a theoretical volume flow rate, a
Reynolds number, and a discharge coefficient.
The Reynolds number is given by
Re = V 2 D 2
(7.2)
ν
where V 2
is the velocity at the throat of the
meter (= Q ac
/A 2
).
The Orifice Meter and
Nozzle-Type Meter
The orifice and nozzle-type meters consist of
a throttling device (an orifice plate or bushing,
respectively) placed into the flow. (See Figures
7.2 and 7.3). The throttling device creates a
measurable pressure difference from its upstream
to its downstream side. The measured pressure
difference is then related to the flow rate. Like
the venturi meter, the pressure difference varies
with flow rate. Applying Bernoulli’s equation to
points 1 and 2 of either meter (Figure 7.2 or Figure
7.3) yields the same theoretical equation as that
for the venturi meter, namely, Equation 7.1. For
any pressure difference, there will be two
associated flow rates for these meters: the
theoretical flow rate (Equation 7.1), and the
18

actual flow rate (measured in the laboratory).
The ratio of actual to theoretical flow rate leads
to the definition of a discharge coefficient: C o
for
the orifice meter and C n
for the nozzle.
rotor supported
on bearings
(not shown)
to receiver
h
1 2
flow
straighteners
turbine rotor
rotational speed
proportional to
flow rate
FIGURE 7.4. A schematic of a turbine-type flow
meter.
FIGURE 7.2. Cross sectional view of the orifice
meter.
h
1 2
FIGURE 7.3. Cross sectional view of the nozzletype
meter, and a typical nozzle.
For each and every measured actual flow
rate through the orifice or nozzle-type meters, it
is possible to calculate a theoretical volume flow
rate, a Reynolds number and a discharge
coefficient. The Reynolds number is given by
Equation 7.2.
The Turbine-Type Meter
The turbine-type flow meter consists of a
section of pipe into which a small “turbine” has
been placed. As the fluid travels through the
pipe, the turbine spins at an angular velocity
that is proportional to the flow rate. After a
certain number of revolutions, a magnetic pickup
sends an electrical pulse to a preamplifier which
in turn sends the pulse to a digital totalizer. The
totalizer totals the pulses and translates them
into a digital readout which gives the total
volume of liquid that travels through the pipe
and/or the instantaneous volume flow rate.
Figure 7.4 is a schematic of the turbine type flow
meter.
The Rotameter (Variable Area Meter)
The variable area meter consists of a tapered
metering tube and a float which is free to move
inside. The tube is mounted vertically with the
inlet at the bottom. Fluid entering the bottom
raises the float until the forces of buoyancy, drag
and gravity are balanced. As the float rises the
annular flow area around the float increases.
Flow rate is indicated by the float position read
against the graduated scale which is etched on
the metering tube. The reading is made usually at
the widest part of the float. Figure 7.5 is a sketch
of a rotameter.
freely
suspended
float
outlet
tapered, graduated
transparent tube
inlet
FIGURE 7.5. A schematic of the rotameter and its
operation.
Rotameters are usually manufactured with
one of three types of graduated scales:
1. % of maximum flow–a factor to convert scale
reading to flow rate is given or determined for
the meter. A variety of fluids can be used
with the meter and the only variable
19

encountered in using it is the scale factor. The
scale factor will vary from fluid to fluid.
2. Diameter-ratio type–the ratio of cross
sectional diameter of the tube to the
diameter of the float is etched at various
locations on the tube itself. Such a scale
requires a calibration curve to use the meter.
3. Direct reading–the scale reading shows the
actual flow rate for a specific fluid in the
units indicated on the meter itself. If this
type of meter is used for another kind of fluid,
then a scale factor must be applied to the
readings.
Experimental Procedure
Equipment
Fluid Meters Apparatus
Stopwatch
The fluid meters apparatus is shown
schematically in Figure 7.6. It consists of a
centrifugal pump, which draws water from a
sump tank, and delivers the water to the circuit
containing the flow meters. For nine valve
positions (the valve downstream of the pump),
record the pressure differences in each
manometer. For each valve position, measure the
actual flow rate by diverting the flow to the
volumetric measuring tank and recording the time
required to fill the tank to a predetermined
volume. Use the readings on the side of the tank
itself. For the rotameter, record the position of
the float and/or the reading of flow rate given
directly on the meter. For the turbine meter,
record the flow reading on the output device.
Note that the venturi meter has two
manometers attached to it. The “inner”
manometer is used to calibrate the meter; that is,
to obtain ∆h readings used in Equation 7.1. The
“outer” manometer is placed such that it reads
the overall pressure drop in the line due to the
presence of the meter and its attachment fittings.
We refer to this pressure loss as ∆H (distinctly
different from ∆h). This loss is also a function of
flow rate. The manometers on the turbine-type
and variable area meters also give the incurred
loss for each respective meter. Thus readings of
∆H vs Q ac are obtainable. In order to use these
parameters to give dimensionless ratios, pressure
coefficient and Reynolds number are used. The
Reynolds number is given in Equation 7.2. The
pressure coefficient is defined as
C p
= g∆H
V 2 /2
(7.3)
All velocities are based on actual flow rate and
pipe diameter.
The amount of work associated with the
laboratory report is great; therefore an informal
group report is required rather than individual
reports. The write-up should consist of an
Introduction (to include a procedure and a
derivation of Equation 7.1), a Discussion and
Conclusions section, and the following graphs:
1. On the same set of axes, plot Q ac vs ∆h and
Q th vs ∆h with flow rate on the vertical
axis for the venturi meter.
2. On the same set of axes, plot Q ac vs ∆h and
Q th vs ∆h with flow rate on the vertical
axis for the orifice meter.
3. Plot Q ac
vs Q th
for the turbine type meter.
4. Plot Q ac
vs Q th
for the rotameter.
5. Plot C v vs Re on a log-log grid for the
venturi meter.
6. Plot C o vs Re on a log-log grid for the orifice
meter.
7. Plot ∆H vs Q ac for all meters on the same set
of axes with flow rate on the vertical axis.
8. Plot C p vs Re for all meters on the same set
of axes (log-log grid) with C p vertical axis.
Questions
1. Referring to Figure 7.2, recall that
Bernoulli's equation was applied to points 1
and 2 where the pressure difference
measurement is made. The theoretical
equation, however, refers to the throat area
for point 2 (the orifice hole diameter)
which is not where the pressure
measurement was made. Explain this
discrepancy and how it is accounted for in
the equation formulation.
2. Which meter in your opinion is the best one
to use?
3. Which meter incurs the smallest pressure
loss? Is this necessarily the one that should
always be used?
4. Which is the most accurate meter?
5. What is the difference between precision
and accuracy?
20

manometer
orifice meter
venturi meter
volumetric
measuring
tank
rotameter
return
sump tank
turbine-type meter
motor pump valve
FIGURE 7.6. A schematic of the Fluid Meters Apparatus. (Orifice and Venturi meters: upstream
diameter is 1.025 inches; throat diameter is 0.625 inches.)
21

EXPERIMENT 8
PIPE FLOW
Experiments in pipe flow where the presence
of frictional forces must be taken into account are
useful aids in studying the behavior of traveling
fluids. Fluids are usually transported through
pipes from location to location by pumps. The
frictional losses within the pipes cause pressure
drops. These pressure drops must be known to
determine pump requirements. Thus a study of
pressure losses due to friction has a useful
application. The study of pressure losses in pipe
flow is the subject of this experiment.
Pipe Flow
Equipment
Pipe Flow Test Rig
Figure 8.1 is a schematic of the pipe flow test
rig. The rig contains a sump tank which is used as
a water reservoir from which a centrifugal pump
discharges water to the pipe circuit. The circuit
itself consists of four different diameter lines and
a return line all made of drawn copper tubing. The
circuit contains valves for directing and
regulating the flow to make up various series and
parallel piping combinations. The circuit has
provision for measuring pressure loss through the
use of static pressure taps (manometer board not
shown in schematic). Finally, because the circuit
also contains a rotameter, the measured pressure
losses can be obtained as a function of flow rate.
As functions of the flow rate, measure the
pressure losses in inches of water for (as specified
by the instructor):
1. 1 in. copper tube 5. 1 in. 90 T-joint
2. 3 /4-in. copper tube 6. 1 in. 90 elbow (ell)
3. 1 /2-in copper tube 7. 1 in. gate valve
4. 3 /8 in copper tube 8. 3 /4-in gate valve
• The instructor will specify which of the
pressure loss measurements are to be taken.
• Open and close the appropriate valves on the
apparatus to obtain the desired flow path.
• Use the valve closest to the pump on its
downstream side to vary the volume flow
rate.
• With the pump on, record the assigned
pressure drops and the actual volume flow
rate from the rotameter.
• Using the valve closest to the pump, change
the volume flow rate and again record the
pressure drops and the new flow rate value.
• Repeat this procedure until 9 different
volume flow rates and corresponding pressure
drop data have been recorded.
With pressure loss data in terms of ∆h, the
friction factor can be calculated with
2g∆h
f =
V 2 (L/D)
It is customary to graph the friction factor as a
function of the Reynolds number:
Re = VD
ν
The f vs Re graph, called a Moody Diagram is
traditionally drawn on a log-log grid. The graph
also contains a third variable known as the
roughness coefficient ε/D. For this experiment
the roughness factor ε is that for drawn tubing.
Where fittings are concerned, the loss
incurred by the fluid is expressed in terms of a loss
coefficient K. The loss coefficient for any fitting
can be calculated with
K =
∆ h
V 2 /2g
where ∆h is the pressure (or head) loss across the
fitting. Values of K as a function of Q ac are to be
obtained in this experiment.
For the report, calculate friction factor f and
graph it as a function of Reynolds number Re for
items 1 through 4 above as appropriate. Compare
to a Moody diagram. Also calculate the loss
coefficient for items 5 through 8 above as
appropriate, and determine if the loss coefficient
K varies with flow rate or Reynolds number.
Compare your K values to published ones.
Note that gate valves can have a number of
open positions. For purposes of comparison it is
often convenient to use full, half or one-quarter
open.
22

otameter
tank
valve
motor
static pressure tap
pump
FIGURE 8.1. Schematic of the pipe friction apparatus.
23

EXPERIMENT 9
PRESSURE DISTRIBUTION ABOUT A CIRCULAR CYLINDER
In many engineering applications, it may be
necessary to examine the phenomena occurring
when an object is inserted into a flow of fluid. The
wings of an airplane in flight, for example, may
be analyzed by considering the wings stationary
with air moving past them. Certain forces are
exerted on the wing by the flowing fluid that
tend to lift the wing (called the lift force) and to
push the wing in the direction of the flow (drag
force). Objects other than wings that are
symmetrical with respect to the fluid approach
direction, such as a circular cylinder, will
experience no lift, only drag.
Drag and lift forces are caused by the
pressure differences exerted on the stationary
object by the flowing fluid. Skin friction between
the fluid and the object contributes to the drag
force but in many cases can be neglected. The
measurement of the pressure distribution existing
around a stationary cylinder in an air stream to
find the drag force is the object of this
experiment.
Consider a circular cylinder immersed in a
uniform flow. The streamlines about the cylinder
are shown in Figure 9.1. The fluid exerts pressure
on the front half of the cylinder in an amount
that is greater than that exerted on the rear
half. The difference in pressure multiplied by the
projected frontal area of the cylinder gives the
drag force due to pressure (also known as form
drag). Because this drag is due primarily to a
pressure difference, measurement of the pressure
distribution about the cylinder allows for finding
the drag force experimentally. A typical pressure
distribution is given in Figure 9.2. Shown in
Figure 9.2a is the cylinder with lines and
arrowheads. The length of the line at any point
on the cylinder surface is proportional to the
pressure at that point. The direction of the
arrowhead indicates that the pressure at the
respective point is greater than the free stream
pressure (pointing toward the center of the
cylinder) or less than the free stream pressure
(pointing away). Note the existence of a
separation point and a separation region (or
wake). The pressure in the back flow region is
nearly the same as the pressure at the point of
separation. The general result is a net drag force
equal to the sum of the forces due to pressure
acting on the front half (+) and on the rear half
(-) of the cylinder. To find the drag force, it is
necessary to sum the components of pressure at
each point in the flow direction. Figure 9.2b is a
graph of the same data as that in Figure 9.2a
except that 9.2b is on a linear grid.
Freestream
Velocity V
Stagnation
Streamline
Wake
FIGURE 9.1. Streamlines of flow about a circular
cylinder.
separation
point
p
0 30 60 90 120 150 180
separation
point
(a) Polar Coordinate Graph
(b) Linear Graph
FIGURE 9.2. Pressure distribution around a circular cylinder placed in a uniform flow.
24

Pressure Measurement
Equipment
A Wind Tunnel
A Right Circular Cylinder with Pressure
Taps
Figure 9.3 is a schematic of a wind tunnel. It
consists of a nozzle, a test section, a diffuser and a
fan. Flow enters the nozzle and passes through
flow straighteners and screens. The flow is
directed through a test section whose walls are
made of a transparent material, usually
Plexiglas or glass. An object is placed in the test
section for observation. Downstream of the test
section is the diffuser followed by the fan. In the
tunnel that is used in this experiment, the test
section is rectangular and the fan housing is
circular. Thus one function of the diffuser is to
gradually lead the flow from a rectangular
section to a circular one.
Figure 9.4 is a schematic of the side view of
the circular cylinder. The cylinder is placed in
the test section of the wind tunnel which is
operated at a preselected velocity. The pressure
tap labeled as #1 is placed at 0° directly facing
the approach flow. The pressure taps are
attached to a manometer board. Only the first 18
taps are connected because the expected profile is
symmetric about the 0° line. The manometers will
provide readings of pressure at 10° intervals
about half the cylinder. For two different
approach velocities, measure and record the
pressure distribution about the circular cylinder.
Plot the pressure distribution on polar coordinate
graph paper for both cases. Also graph pressure
difference (pressure at the point of interest minus
the free stream pressure) as a function of angle θ
on linear graph paper. Next, graph ∆p cosθ vs θ
(horizontal axis) on linear paper and determine
the area under the curve by any convenient
method (counting squares or a numerical
technique).
The drag force can be calculated by
integrating the flow-direction-component of each
pressure over the area of the cylinder:
π
D f
= 2RL
0
∫ ∆p cosθdθ
The above expression states that the drag force is
twice the cylinder radius (2R) times the cylinder
length (L) times the area under the curve of ∆p
cosθ vs θ.
Drag data are usually expressed as drag
coefficient C D
vs Reynolds number Re. The drag
coefficient is defined as
D f
C D
=
ρV 2 A/2
The Reynolds number is
Re = ρVD
µ
inlet flow
straighteners
nozzle
test section
diffuser
fan
FIGURE 9.3. A schematic of the wind tunnel used in this experiment.
25

where V is the free stream velocity (upstream of
the cylinder), A is the projected frontal area of
the cylinder (2RL), D is the cylinder diameter, ρ
is the air density and µ is the air viscosity.
Compare the results to those found in texts.
60
90
120
0
30
static pressure
taps attach to
manometers
150
180
FIGURE 9.4. Schematic of the experimental
apparatus used in this experiment.
26

EXPERIMENT 10
DRAG FORCE DETERMINATION
An object placed in a uniform flow is acted
upon by various forces. The resultant of these
forces can be resolved into two force components,
parallel and perpendicular to the main flow
direction. The component acting parallel to the
flow is known as the drag force. It is a function of
a skin friction effect and an adverse pressure
gradient. The component perpendicular to the
flow direction is the lift force and is caused by a
pressure distribution which results in a lower
pressure acting over the top surface of the object
than at the bottom. If the object is symmetric
with respect to the flow direction, then the lift
force will be zero and only a drag force will exist.
Measurement of the drag force acting on an object
immersed in the uniform flow of a fluid is the
subject of this experiment.
Equipment
Subsonic Wind Tunnel
Objects
A description of a subsonic wind tunnel is
given in Experiment 9 and is shown schematically
in Figure 9.3. The fan at the end of the tunnel
draws in air at the inlet. An object is mounted on a
stand that is pre calibrated to read lift and drag
forces exerted by the fluid on the object. A
schematic of the test section is shown in Figure
10.1. The velocity of the flow at the test section is
also pre calibrated. The air velocity past the
object can be controlled by changing the angle of
the inlet vanes located within the fan housing.
Thus air velocity, lift force and drag force are
read directly from the tunnel instrumentation.
There are a number of objects that are
available for use in the wind tunnel. These
include a disk, a smooth surfaced sphere, a rough
surface sphere, a hemisphere facing upstream,
and a hemisphere facing downstream. For
whichever is assigned, measure drag on the object
as a function of velocity.
Data on drag vs velocity are usually graphed
in dimensionless terms. The drag force D f is
customarily expressed in terms of the drag
coefficient C D (a ratio of drag force to kinetic
energy):
in which ρ is the fluid density, V is the free
stream velocity, and A is the projected frontal
area of the object. Traditionally, the drag
coefficient is graphed as a function of the
Reynolds number, which is defined as
Re = VD
ν
where D is a characteristic length of the object
and ν is the kinematic viscosity of the fluid. For
each object assigned, graph drag coefficient vs
Reynolds number and compare your results to
those published in texts. Use log-log paper if
appropriate.
Questions
1. How does the mounting piece affect the
readings?
2. How do you plan to correct for its effect, if
necessary?
uniform flow
lift force
measurement
object
mounting stand
drag force
measurement
FIGURE 10.1. Schematic of an object mounted in
the test section of the wind tunnel.
D f
C D
=
ρV 2 A/2
27

EXPERIMENT 11
ANALYSIS OF AN AIRFOIL
A wing placed in the uniform flow of an
airstream will experience lift and drag forces.
Each of these forces is due to a pressure
difference. The lift force is due to the pressure
difference that exists between the lower and
upper surfaces. This phenomena is illustrated in
Figure 11.1. As indicated the airfoil is immersed
in a uniform flow. If pressure could be measured at
selected locations on the surface of the wing and
the results graphed, the profile in Figure 11.1
would result. Each pressure measurement is
represented by a line with an arrowhead. The
length of each line is proportional to the
magnitude of the pressure at the point. The
direction of the arrow (toward the horizontal
axis or away from it) represents whether the
pressure at the point is less than or greater than
the free stream pressure measured far upstream of
the wing.
Experiment I
Mount the wing with pressure taps in the
tunnel and attach the tube ends to manometers.
Select a wind speed and record the pressure
distribution for a selected angle of attack (as
assigned by the instructor). Plot pressure vs chord
length as in Figure 11.1, showing the vertical
component of each pressure acting on the upper
surface and on the lower surface. Determine
where separation occurs for each case.
Mount the second wing on the lift and drag
balance (Figure 11.2). For the same wind speed
and angle of attack, measure lift and drag exerted
on the wing.
lift
c
drag
c
uniform flow
mounting stand
stagnation
point
C p
pressure
coefficient
negative pressure
gradient on upper
surface
lift force
measurement
drag force
measurement
stagnation
point
chord, c
positive pressure
on lower surface
FIGURE 11.2. Schematic of lift and drag
measurement in a test section.
FIGURE 11.1. Streamlines of flow about a wing
and the resultant pressure distribution.
Lift and Drag Measurements for a Wing
Equipment
Wind Tunnel (See Figure 9.3)
Wing with Pressure Taps
Wing for Attachment to Lift & Drag
Instruments (See Figure 11.2)
The wing with pressure taps provided
pressure at selected points on the surface of the
wing. Use the data obtained and sum the
horizontal component of each pressure to obtain
the drag force. Compare to the results obtained
with the other wing. Use the data obtained and
sum the vertical component of each pressure to
obtain the lift force. Compare the results
obtained with the other wing. Calculate %
errors.
28

Experiment II
For a number of wings, lift and drag data
vary only slightly with Reynolds number and
therefore if lift and drag coefficients are graphed
as a function of Reynolds number, the results are
not that meaningful. A more significant
representation of the results is given in what is
known as a polar diagram for the wing. A polar
diagram is a graph on a linear grid of lift
coefficient (vertical axis) as a function of drag
coefficient. Each data point on the graph
corresponds to a different angle of attack, all
measured at one velocity (Reynolds number).
Referring to Figure 11.2 (which is the
experimental setup here), the angle of attack α is
measured from a line parallel to the chord c to a
line that is parallel to the free stream velocity.
If so instructed, obtain lift force, drag force and
angle of attack data using a pre selected velocity.
Allow the angle of attack to vary from a negative
angle to the stall point and beyond. Obtain data
at no less than 9 angles of attack. Use the data to
produce a polar diagram.
Analysis
Lift and drag data are usually expressed in
dimensionless terms using lift coefficient and drag
coefficient. The lift coefficient is defined as
L f
C L
=
ρV 2 A/2
where L f is the lift force, ρ is the fluid density, V
is the free stream velocity far upstream of the
wing, and A is the area of the wing when seen
from a top view perpendicular to the chord
length c. The drag coefficient is defined as
D f
C D
=
ρV 2 A/2
in which D f
is the drag force.
29

EXPERIMENT 12
OPEN CHANNEL FLOW—SLUICE GATE
Liquid motion in a duct where a surface of the
fluid is exposed to the atmosphere is called open
channel flow. In the laboratory, open channel
flow experiments can be used to simulate flow in a
river, in a spillway, in a drainage canal or in a
sewer. Such modeled flows can include flow over
bumps or through dams, flow through a venturi
flume or under a partially raised gate (a sluice
gate). The last example, flow under a sluice gate,
is the subject of this experiment.
Flow Through a Sluice Gate
Equipment
Open Channel Flow Apparatus
Sluice Gate Model
Figure 12.1 shows a schematic of the side
view of the sluice gate. Flow upstream of the gate
has a depth h o while downstream the depth is h.
The objective of the analysis is to formulate an
equation to relate the volume flow rate through
(or under) the gate to the upstream and
downstream depths.
sluice gate
p atm
h o
hand crank
direction of
movement
h
p atm
FIGURE 12.1. Schematic of flow under a sluice
gate.
The flow rate through the gate is maintained at
nearly a constant value. For various raised
positions of the sluice gate, different liquid
heights h o and h will result. Applying the
Bernoulli equation to flow about the gate gives
p 0
ρg + V 0 2
2g + h 0 = p ρg + V2
2g + h
Pressures at the free surface are both equal to
atmospheric pressure, so they cancel. Rearranging
gives
h 0 = V2
2g - V 0 2
2g +h
In terms of flow rate, the velocities are written as
V 0 = Q A = Q
bh 0
V = Q bh
where b is the channel width at the gate.
Substituting into the Bernoulli Equation and
simplifying gives
h 0 = Q2
2gb 2 ⎝ ⎛ 1
h ⎠ ⎞ 2 - 1
h
2 + h
0
Dividing by h 0 ,
Q
1 =
2
2gb 2 h 0 ⎝ ⎛ 1
h ⎠ ⎞
2 - 1
h
2 + h 0 h 0
Rearranging further,
Q 2
⎛1 - h ⎞ =
⎝ h 0 ⎠ 2gb 2 h 2 h 0 ⎝ ⎛ 1 - h 2
h ⎠ ⎞ 2 0
Multiplying both sides by h 2 /h 0 2 , and continuing
to simplify, we finally obtain
h 2 /h
2 0 Q
=
2
1 + h/h 0 2gb 2 h
3 0
here Q is the theoretical volume flow rate. The
right hand side of this equation is recognized as
1/2 of the upstream Froude number. So by
measuring the depth of liquid before and after
the sluice gate, the theoretical flow rate can be
calculated with the above equation. The
theoretical flow rate can then be compared to the
actual flow rate obtained by measurements using
the orifice meters.
For 9 different raised positions of the sluice
gate, measure the upstream and downstream
depths and calculate the actual flow rate. In
addition, calculate the upstream Froude number
for each case and determine its value for
maximum flow conditions. Graph h/h 0 (vertical
30

axis) versus (Q 2 /b 2 h 0 3 g). Determine h/h 0
corresponding to maximum flow. Note that h/h 0
varies from 0 to 1.
Figure 12.2 is a sketch of the open channel
flow apparatus. It consists of a sump tank with a
pump/motor combination on each side. Each pump
draws in water from the sump tank and
discharges it through the discharge line to
calibrated orifice meters and then to the head
tank. Each orifice meter is connected to its own
manometer. Use of the calibration curve
(provided by the instructor) allows for finding
the actual flow rate into the channel. The head
tank and flow channel have sides made of
Plexiglas. Water flows downstream in the
channel past the object of interest (in this case a
sluice gate) and then is routed back to the sump
tank.
Questions
1. For the required report, derive the sluice
gate equation in detail.
2. What if it was assumed that V 0

EXPERIMENT 13
OPEN CHANNEL FLOW OVER A WEIR
Flow meters used in pipes introduce an
obstruction into the flow which results in a
measurable pressure drop that in turn is related to
the volume flow rate. In an open channel, flow
rate can be measured similarly by introducing an
obstruction into the flow. A simple obstruction,
called a weir, consists of a vertical plate
extending the entire width of the channel. The
plate may have an opening, usually rectangular,
trapezoidal, or triangular. Other configurations
exist and all are about equally effective. The use
of a weir to measure flow rate in an open channel
is the subject of this experiment.
Flow Over a Weir
Equipment
Open Channel Flow Apparatus (See
Figure 12.1)
Several Weirs
The open channel flow apparatus allows for
the insertion of a weir and measurement of liquid
depths. The channel is fed by two centrifugal
pumps. Each pump has a discharge line which
contains an orifice meter attached to a
manometer. The pressure drop reading from the
manometers and a calibration curve provide the
means for determining the actual flow rate into
the channel.
Figure 13.1 is a sketch of the side and
upstream view of a 90 degree (included angle) V-
notch weir. Analysis of this weir is presented
here for illustrative purposes. Note that
upstream depth measurements are made from the
lowest point of the weir over which liquid flows.
This is the case for the analysis of all
conventional weirs. A coordinate system is
imposed whose origin is at the intersection of the
free surface and a vertical line extending upward
from the vertex of the V-notch. We select an
element that is dy thick and extends the entire
width of the flow cross section. The velocity of
the liquid through this element is found by
applying Bernoulli's equation:
p a
ρ + V o 2
2 + gh = p a
ρ + V2
+ g(h - y)
2
Note that in pipe flow, pressure remained in the
equation when analyzing any of the differential
pressure meters (orifice or venturi meters). In open
channel flows, the pressure terms represents
atmospheric pressure and cancel from the
Bernoulli equation. The liquid height is
therefore the only measurement required here.
From the above equation, assuming V o
negligible:
V = √⎺⎺⎺2gy (13.1)
Equation 13.1 is the starting point in the analysis
of all weirs. The incremental flow rate of liquid
through layer dy is:
dQ = 2Vxdy = √⎺⎺⎺2gy(2x)dy
From the geometry of the V-notch and with
respect to the coordinate axes, we have y = h - x.
V o
p a
h
V
p a
y axis
y
dy
x
FIGURE 13.1. Side and upstream views of a 90° V-notch weir.
x axis
32

Therefore,
Q = ∫
0
Integration gives
h
(2√⎺⎺2g)y 1/2 (h - y)dy
Q th = 8 15 √⎺⎺ 2g h 5/2 =Ch 5/2 (13.2)
where C is a constant. The above equation
represents the ideal or theoretical flow rate of
liquid over the V-notch weir. The actual
discharge rate is somewhat less due to frictional
and other dissipative effects. As with pipe
meters, we introduce a discharge coefficient
defined as:
C' = Q ac
Q th
The equation that relates the actual volume flow
rate to the upstream height then is
Q ac
= C'Ch 5/2
It is convenient to combine the effects of the
constant C and the coefficient C’ into a single
coefficient C vn for the V-notch weir. Thus we
reformulate the previous two equations to obtain:
C vn ≈ Q ac
Q th
(13.3)
Q ac
= C vn h 5/2 (13.4)
Each type of weir will have its own coefficient.
Calibrate each of the weirs assigned by the
instructor for 7 different upstream height
measurements. Use the flow rate chart provided
with the open channel flow apparatus to obtain
the actual flow rate. Derive an appropriate
equation for each weir used (similar to Equation
13.4) above. Determine the coefficient applicable
for each weir tested. List the assumptions made
in each derivation. Discuss the validity of each
assumption, pointing out where they break down.
Graph upstream height vs actual and theoretical
volume flow rates. Plot the coefficient of
discharge (as defined in Equation 13.3) as a
function of the upstream Froude number.
FIGURE 13.2. Other types of weirs–semicircular, contracted and suppressed.
33

EXPERIMENT 14
OPEN CHANNEL FLOW—HYDRAULIC JUMP
When spillways or other similar open
channels are opened by the lifting of a gate,
liquid passing below the gate has a high velocity
and an associated high kinetic energy. Due to the
erosive properties of a high velocity fluid, it
may be desirable to convert the high kinetic
energy (e.g. high velocity) to a high potential
energy (e.g., a deeper stream). The problem then
becomes one of rapidly varying the liquid depth
over a short channel length. Rapidly varied flow
of this type produces what is known as a
hydraulic jump.
Consider a horizontal, rectangular open
channel of width b, in which a hydraulic jump
has developed. Figure 14.1 shows a side view of a
hydraulic jump. Figure 14.1 also shows the depth
of liquid upstream of the jump to be h 1
, and a
downstream depth of h 2
. Pressure distributions
upstream and downstream of the jump are drawn
in as well. Because the jump occurs over a very
short distance, frictional effects can be neglected.
A force balance would therefore include only
pressure forces. Applying the momentum equation
in the flow direction gives:
p 1
A 1
- p 2
A 2
= ρQ(V 2
- V 1
)
Pressure in the above equation represents the
pressure that exists at the centroid of the cross
section. Thus p = ρg(h/2). With a rectangular
cross section of width b (A = bh), the above
equation becomes
h 1
g
2 (h 1 b) - h 2 g
2 (h 2 b) = Q(V 2 - V 1 )
From continuity, A 1
V 1
= A 2
V 2
= Q. Combining and
rearranging,
h
2
1
- h
2
2
= Q 2
2 gb 2 ⎝ ⎛ 1
- 1 h ⎠ ⎞
2
h 1
Simplifying,
h 2
2
+ h 2
h 1
- 2
Q 2
gb 2 h 1
= 0
Solving for the downstream height yields one
physically (nonnegative) possible solution:
h 2
= - h 1
2 √⎺⎺⎺⎺
+ 2Q 2
4
gb 2 h 1
+ h 1 2
from which the downstream height can be found.
By applying Bernoulli’s Equation along the free
surface, the energy lost irreversibly can be
calculated as
Lost Energy = E =
g(h 2
- h 1
) 3
4h 2
h 1
and the rate of energy loss is
dW
dt = ρQE
The above equations are adequate to properly
describe a hydraulic jump.
Hydraulic Jump Measurements
Equipment
Open Channel Flow Apparatus (Figure 12.1)
The channel can be used in either a
horizontal or a sloping configuration. The device
contains two pumps which discharge water
through calibrated orifice meters connected to
manometers. The device also contains on the
channel bottom two forward facing brass tubes.
Each tube is connected to a vertical Plexiglas
tube. The height of the water in either of these
tubes represents the energy level at the
respective tube location. The difference in height
is the actual lost energy (E) for the jump of
interest.
FIGURE 14.1. Schematic of a
hydraulic jump in an open
channel.
h 1
V 2
V 1
p 1
p 2
h 2
34

Develop a hydraulic jump in the channel;
record upstream and downstream heights,
manometer readings (from which the actual
volume flow rate is obtained) and the lost energy
E. By varying the flow rate, upstream height,
downstream height and/or the channel slope,
record measurements on different jumps. Derive
the applicable equations in detail and substitute
appropriate values to verify the predicted
downstream height and lost energy. In other
words, the downstream height of each jump is to
be measured and compared to the downstream
height calculated with Equation 14.1. The same
is to be done for the rate of energy loss (Equation
14.2).
Analysis
Data on a hydraulic jump is usually specified
in two ways both of which will be required for
the report. Select any of the jumps you have
measurements for and construct a momentum
diagram . A momentum diagram is a graph of
liquid depth on the vertical axis vs momentum on
the horizontal axis. The momentum of the flow is
given by:
M = 2Q2
gb 2 h + h2
4
Another significant graph of hydraulic jump
data is of depth ratio h 2 /h 1 (vertical axis) as a
function of the upstream Froude number, Fr 1 (=
Q 2 /gb 2 h 1
3
). Construct such a graph for any of the
jumps for which you have taken measurements.
35

EXPERIMENT 15
OPEN CHANNEL FLOW OVER A HUMP
Flow over a hump in an open channel is a
problem that can be successfully modeled in order
to make predictions about the behavior of the
fluid. This experiment involves making
appropriate measurements for such a system, and
relating flow rate to critical depth. The flow
rate, critical depth, and specific energy are
determined theoretically and experimentally.
Theory
Flow in a channel is modeled in terms of a
parameter called the specific energy head (or just
specific energy) of the flow, E. The specific
energy head is defined as
Q 2
E = h +
2gh 2 b 2 (15.1)
where h is the depth of the flow, Q is the volume
flow rate, g is gravity, and b is the channel
width. The dimension of the specific energy head
is L (ft or m).
Figure 15.1 is a sketch of flow over a hump,
with flow from left to right. Shown is the channel
bed and the hump. Upstream of the hump
(subscript 1 notation), the flow is subcritical;
downstream (subscript 2) the flow is supercritical.
Just at the highest point of the hump,
the flow is critical (subscript c). Also shown in
the figure is the total energy line, which we
assume is parallel to the flow channel bed; i.e.,
the total energy remains a constant in the flow.
Upstream of the hump, the total specific
energy head of the flow is denoted as E 1 , and the
depth of the liquid is h 1 , as shown graphically in
Figure 15.1. At any location z on the hump before
z c , the energy head is E, and the depth is h. At
this same height z downstream of z c , the liquid
depth is h’, but the energy head is still E. At the
highest point of the hump z c , the energy head is
E c and the liquid depth is h c . The total specific
energy head and the liquid depth anywhere are
related according to Equation 15.1.
total energy line
flow
direction
h
E
h c
h'
h 2
E 1
h 1
z
E c
z c
hump
channel bed
FIGURE 15.1. Flow over a
hump in an open
channel.
We can illustrate the relationship between
these parameters graphically by drawing a
specific energy head diagram, as illustrated in
Figure 15.2. This graph has flow depth on the
vertical axis and specific energy head on the
horizontal axis. The condition of the flow is
represented by the solid line with arrows
showing how the flow changes from subcritical to
supercritical. At the location on the hump where
the height is z, the energy head is E. We draw a
vertical line at this value of the specific energy
head; it will intersect the line at h (upstream)
and h’ (downstream).
depth h
h 1
h
z
z c
h c
h'
h 2
E c E E 1 , E 2
specific energy head E
FIGURE 15.2. Specific energy diagram.
subcritical
supercritical
36

At any upstream (of the hump) location, say
h 1 , we see that the corresponding specific energy
head is E 1 . The vertical line that locates E 1 also
locates the energy E 2 which is downstream of the
hump. A vertical line drawn at E 1 intersects the
line at h 1 and h 2 , which are the upstream and
downstream liquid heights, respectively. Note
that the minimum specific energy head is at the
highest point of the hump z c , and the energy
head there is E c .
As water flows over the hump, the initial
specific energy head E 1 is reduced to a value E by
an amount equal to the height of the hump. So at
any location along the hump, the specific energy
head is E 1 - z, where z is the elevation above the
channel bed. At the point where the flow is
critical, the critical depth h c is given by
Q 2
1/3 2E
h c = ⎛ ⎞ = c
⎝ b 2 g⎠
3
Flow Over a Hump
(15.2)
Equipment
Open Channel Flow Apparatus (Figure
12.1)
Installed hump
The open channel flow apparatus is described
in Experiment 12 and illustrated schematically in
Figure 12.1. Adjust the channel so that it is
horizontal. Make every effort to minimize
leakage of water past the sides of the hump.
Start both pumps and adjust the valves to give a
smooth water surface profile over the hump. For
one set of conditions, take readings from the
manometers to determine the volume flow rate
over the hump.
The open channel flow apparatus has a
depth gage attached. It will be necessary to
measure the water depth at certain specific
locations on or about the hump. These locations
are shown in Figure 15.3 (dimensions are in feet).
There are 8 water depths to be measured. So for
one flow rate, two manometer readings and 8
water depths will be recorded. Gather data for
the assigned number of flow rates.
Results
Although the data taken in this experiment
seem simple, the calculations required to reduce
the data appropriately can occupy much time.
With the data obtained:
Determine the flow rate using the manometer
readings. This value will be referred to as the
actual flow rate Q AC (subscript AC will refer
to an actual value, while TH refers to a
theoretical value).
Calculate the flow rate using a rearranged
form of Equation 15.2. This value will be
referred to as the theoretical flow rate Q TH .
Compare the two flow rates and find % error.
Use Equation 15.2 to find the value of the
critical depth using Q AC. Compare this value
to the measured value, and find % error.
Calculate the theoretical and actual values
of the minimum energy E c using Equation 15.2.
Compare the results.
Calculate the actual specific energy head E AC
at each measurement station using Equation
15.1. Determine also the total energy head
H AC (= E AC + z) for all readings.
Compose a chart using the column and row
headings shown in Table 15.1.
1 2 3 4 5 6 7 8
flow
direction
hump
2
0.276
0.313
0.313 0.313 0.313
1
FIGURE 15.3. Water depth
measurement locations
for flow over a hump.
(Dimensions in feet.)
37

TABLE 15.1. Data reduction table for flow over a hump.
Station 1 2 3 4 5 6 7 8
Depth of flow h AC in ft
Specific energy head E AC in ft
Height of hump above channel bed z
in ft
Total energy head H AC in ft
Construct a graph of the flow configuration.
On the horizontal axis, plot distance
downstream, and plot depth on the vertical
axis. On this set of axes, plot (a) the total
energy line (H AC ); (b) the water surface
profile; and, (c) the elevation z. Show data
points on the graph.
Construct a specific energy head diagram
similar to that of Figure 15.2. Show the
theoretical results (based on Q TH ), and show
the actual data points.
Derive Equation 2.
Questions
1. What is the value of the Froude number (a)
upstream of the hump, (b) at the highest
point of the hump, and (c) downstream of the
hump?
2. Is the Froude number used in finding the
critical depth in Equation 15.2?
3. What equations is used to develop the
expression for specific energy head (Equation
15.1)?
4. How is the second term in Equation 15.1 (i.e.;
Q 2 /2gh 2 b 2 ) related to the Froude number?
5. Is the total energy line (H AC ) a constant as we
assumed with reference to Figure 15.1, or does
it change?
38

EXPERIMENT 16
MEASUREMENT OF VELOCITY
AND
CALIBRATION OF A METER FOR COMPRESSIBLE FLOW
The objective of this experiment is to
determine a calibration curve for a meter placed
in a pipe that is conveying air. The meters of
interest are an orifice meter and a venturi meter.
These meters are calibrated in this experiment by
using a pitot-static tube to measure the velocity,
from which the flow rate is calculated.
Pitot Static Tube
When a fluid flows through a pipe, it exerts
pressure that is made up of static and dynamic
components. The static pressure is indicated by a
measuring device moving with the flow or that
causes no velocity change in the flow. Usually, to
measure static pressure, a small hole
perpendicular to the flow is drilled through the
container wall and connected to a manometer (or
pressure gage) as indicated in Figure 16.1.
The dynamic pressure is due to the movement
of the fluid. The dynamic pressure and the static
pressure together make up the total or stagnation
pressure. The stagnation pressure can be measured
in the flow with a pitot tube. The pitot tube is an
open ended tube facing the flow directly. Figure
16.1 gives a sketch of the measurement of
stagnation pressure.
flow
stagnation pressure
measurement
static pressure
measurement
pitot tube
h
FIGURE 16.1. Measurement of static and
stagnation pressures.
The pitot-static tube combines the effects of
static and stagnation pressure measurement into
one device. Figure 16.2 is a schematic of the pitotstatic
tube. It consists of a tube within a tube
which is placed in the duct facing upstream. The
pressure tap that faces the flow directly gives a
measurement of the stagnation pressure, while
h
the tap that is perpendicular to the flow gives
the static pressure.
When the pitot-static tube is immersed in the
flow of a fluid, the pressure difference
(stagnation minus static) can be read directly
using a manometer and connecting the pressure
taps to each leg. Applying the Bernoulli equation
between the two pressure taps yields:
four to eight holes
equally spaced
flow direction
section A-A
enlarged
A
A
manometer
connections
FIGURE 16.2. Schematic of a pitot-static tube.
p 1
ρg + V 1 2
2g + z 1 = p 2
ρg + V 2 2
2g + z 2
where state “1” as the stagnation state (which
will be changed to subscript “t”), and state “2” as
the static state (no subscript). Elevation
differences are negligible, and at the point where
stagnation pressure is measured, the velocity is
zero. The Bernoulli equation thus reduces to:
p t
ρg = p ρg + V2
2g
Next, we rearrange the preceding equation and
solve for velocity
V =
√⎺⎺⎺
2(p t - p)
ρ
A manometer connected to the pitot-static tube
would provide head loss readings ∆h given by
39

∆h = p t - p
ρg
where density is that of the flowing fluid. So
velocity in terms of head loss is
V = √⎺⎺⎺⎺⎺ 2g∆h
Note that this equation applies only to
incompressible flows. Compressibility effects are
not accounted for. Furthermore, ∆h is the head
loss in terms of the flowing fluid and not in terms
of the reading on the manometer.
For flow in a duct, manometer readings are to
be taken at a number of locations within the cross
section of the flow. The velocity profile is then
plotted using the results. Velocities at specific
points are then determined from these profiles.
The objective here is to obtain data, graph a
velocity profile and then determine the average
velocity.
Average Velocity
The average velocity is related to the flow
rate through a duct as
V = Q A
where Q is the volume flow rate and A is the
cross sectional area of the duct. We can divide
the flow area into five equal areas, as shown in
Figure 16.3. The velocity is to be obtained at
those locations labeled in the figure. The chosen
positions divide the cross section into five equal
concentric areas. The flow rate through each area
labeled from 1 to 5 is found as
Q 1 = A 1 V 1 Q 2 = A 2 V 2
Q 3 = A 3 V 3 Q 4 = A 4 V 4
Q 5 = A 5 V 5
0.548 R
0.316 R
R
0.837 R
0.707 R 0.949 R
FIGURE 16.3. Five positions within the cross
section where velocity is to be determined.
The total flow rate through the entire cross
section is the sum of these:
5
Q total = ∑Q i = A 1 V 1 + A 2 V 2 + A 3 V 3 + A 4 V 4
1
+ A 5 V 5
or Q total = A 1 (V 1 + V 2 + V 3 + V 4 + V 5 )
The total area A total is 5A 1 and so
V = Q total
A total
= (A total/5)(V 1 + V 2 + V 3 + V 4 + V 5 )
A total
The average velocity then becomes
V = (V 1 + V 2 + V 3 + V 4 + V 5 )
5
The importance of the five chosen radial
positions for measuring V 1 through V 5 is now
evident.
Velocity Measurements
Equipment
Axial flow fan apparatus
Pitot-static tube
Manometer
The fan of the apparatus is used to move air
through the system at a rate that is small enough
to allow the air to be considered incompressible.
While the fan is on, make velocity profile
measurements at a selected location within the
duct at a cross section that is several diameters
downstream of the fan. Repeat these
measurements at different fan speed settings so
that 9 velocity profiles will result. Use the
velocity profiles to determine the average
velocity and the flow rate.
Questions
1. Why is it appropriate to take velocity
measurements at several diameters
downstream of the fan?
2. Suppose the duct were divided into 6 equal
areas and measurements taken at select
positions in the cross section. Should the
average velocity using 6 equal areas be the
same as the average velocity using 5 or 4
equal areas?
40

Incompressible Flow Through a Meter
Incompressible flow through a venturi and an
orifice meter was discussed in Experiment 9. For
our purposes here, we merely re-state the
equations for convenience. For an air over liquid
manometer, the theoretical equation for both
meters is
Q th = A √⎺⎺⎺⎺⎺
2g∆h
2
(1 - D 4 2 /D 4 1 )
Now for any pressure drop ∆h i , there are two
corresponding flow rates: Q ac and Q th . The ratio of
these flow rates is the venturi discharge
coefficient C v , defined as
C v = Q ac
Q th
= 0.985
for turbulent flow. The orifice discharge
coefficient can be expressed in terms of the Stolz
equation:
C o = 0.595 9 + 0.031 2β 2.1 - 0.184β 8 +
+ 0.002 9β 2.5 10
⎛
6 0.75
⎞
⎝ Re β⎠
where Re = ρV oD o
µ
L 1 = 0
L 1 = 1/D 1
L 1 = 1
+ 0.09L 1
⎝ ⎛ β 4
1 - β ⎠ ⎞ 4 - L 2 (0.003 37β 3 )
= 4ρQ ac
πD o µ
for corner taps
for flange taps
for 1D & 1 2 D taps
β = D o
D 1
β 4
and if L 1 ≥ 0.433 3, the coefficient of the ⎛ ⎞
⎝ 1 - β 4 ⎠
term becomes 0.039.
L 2 = 0 for corner taps
L 2 = 1/D 1 for flange taps
L 2 = 0.5 - E/D 1 for 1D & 1 2 D taps
E = orifice plate thickness
Compressible Flow Through a Meter
When a compressible fluid (vapor or gas)
flows through a meter, compressibility effects
must be accounted for. This is done by introduction
of a compressibility factor which can be
determined analytically for some meters
(venturi). For an orifice meter, on the other hand,
the compressibility factor must be measured.
The equations and formulation developed
thus far were for incompressible flow through a
meter. For compressible flows, the derivation is
somewhat different. When the fluid flows
through a meter and encounters a change in area,
the velocity changes as does the pressure. When
pressure changes, the density of the fluid changes
and this effect must be accounted for in order to
obtain accurate results. To account for
compressibility, we will rewrite the descriptive
equations.
Venturi Meter
Consider isentropic, subsonic, steady flow of
an ideal gas through a venturi meter. The
continuity equation is
ρ 1 A 1 V 1 = ρ 2 A 2 V 2 = ·m isentropic = ·m s
where section 1 is upstream of the meter, and
section 2 is at the throat. Neglecting changes in
potential energy (negligible compared to changes
in enthalpy), the energy equation is
h 1 + V 1 2
2 = h 2 + V 2 2
2
The enthalpy change can be found by assuming
that the compressible fluid is ideal:
h 1 - h 2 = C p (T 1 - T 2 )
Combining these equations and rearranging gives
or
C p T 1 +
·
m s
2
2ρ 1 2 A 1
2 = C pT 2 +
m
· 2 s
2ρ 2 2 A
2 2
m
· 1
2 s
⎛
⎞
⎝ ρ 2 2 A
2 - 1
2 ρ 2 1 A
2 1 ⎠
= 2C p(T 1 - T 2 )
= 2C p T 1
⎝ ⎛ 1 - T 2
T ⎠ ⎞
1
If we assume an isentropic compression process
through the meter, then we can write
p 2 T
= ⎛ 2
⎞
p 1 ⎝ T 1 ⎠
γ
γ - 1
where γ is the ratio of specific heats (γ = C p /C v ).
Also, recall that for an ideal gas,
C p = R γ
γ - 1
Substituting, rearranging and simplifying, we get
41

m
· 2 s
ρ 2 2 A
2 2 ⎝ ⎛ 1 - ρ 2 2 A
2 2
ρ ⎠ ⎞ 2 1 A
2 = 2 R γ
1 γ - 1 T 1
⎢ ⎡ p
1 - ⎛ 2
⎞
⎣ ⎝ ⎠
γ - 1
γ
p 1
For an ideal gas, we write ρ = p/RT. Substituting
for the RT 1 term in the preceding equation yields
·
m s
2
A 2
2 = 2ρ 2 2
γ
γ - 1 ⎝ ⎛
p 1
ρ 1
(γ - 1)/γ
⎠ ⎞ 1 - (p 2 /p 1 )
1 - (ρ 2 2 A 2 2 /ρ 2 1 A 2 1 )
For an isentropic process, we can also write
or
p 1
ρ 1
γ = p 2
ρ 2
γ
p
ρ 2 = ⎛ 2
⎞
⎝ p 1 ⎠
1/γ
ρ1
from which we obtain
ρ 2 p
2 = ⎛ 2
⎞
⎝ ⎠
p 1
2/γ
ρ1
2
Substituting into the mass flow equation, we get
after considerable manipulation Equation 16.1 of
Table 16.1, which summarizes the results.
Thus for compressible flow through a venturi
meter, the measurements needed are p 1 , p 2 , T 1 ,
the venturi dimensions, and the fluid properties.
By introducing the venturi discharge coefficient
C v , the actual flow rate through the meter is
determined to be
·
m ac = C v
·ms
Combining this result with Equation 16.1 gives
Equation 16.2 of Table 16.1.
It would be convenient if we could re-write
Equation 16.2 in such a way that the
compressibility effects could be consolidated into
one term. We attempt this by using the flow rate
equation for the incompressible case multiplied
by another coefficient called the compressibility
factor Y; we therefore write
m
·
2(p
ac = C v Yρ 1 A 2
√⎺⎺⎺⎺⎺⎺
1 - p 2 )
ρ 1 (1 - D 4 2 /D 4 1 )
We now set the preceding equation equal to
Equation 16.2 and solve for Y. We obtain Equation
16.3 of the table.
The ratio of specific heats γ will be known for
a given compressible fluid, and so Equation 16.3
⎦ ⎥⎤
could be plotted as compressibility factor Y versus
pressure ratio p 2 /p 1 for various values of D 2 /D 1 .
The advantage of using this approach is that a
pressure drop term appears just as with the
incompressible case, which is convenient if a
manometer is used to measure pressure. Moreover,
the compressibility effect has been isolated into
one factor Y.
Orifice Meter
The equations and formulation of an analysis
for an orifice meter is the same as that for the
venturi meter. The difference is in the evaluation
of the compressibility factor. For an orifice meter
the compressibility factor is much lower than
that for a venturi meter. The compressibility
factor for an orifice meter cannot be derived, but
instead must be measured. Results of such tests
have yielded the Buckingham equation, Equation
16.4 of Table 16.1, which is valid for most
manometer connection systems.
Calibration of a Meter
Figures 16.4 and 16.5 show how the apparatus
is set up. An axial flow fan is attached to the
shaft of a DC motor. The rotational speed of the
motor, and hence the volume flow rate of air, is
controllable. The fan moves air through a duct
into which a pitot-static tube is attached. The
pitot static tube is movable so that the velocity
at any radial location can be measured. An orifice
or a venturi meter can be placed in the duct
system.
The pitot static tube has pressure taps which
are to be connected to a manometer. Likewise each
meter also has pressure taps, and these will be
connected to a separate manometer.
A meter for calibration will be assigned by
the instructor. For the experiment, make
measurements of velocity using the pitot-static
tube to obtain a velocity profile. Draw the
velocity profile to scale. Obtain data from the
velocity profile and determine a volume flow
rate.
For one velocity profile, measure the pressure
drop associated with the meter. Graph volume
flow rate as a function of head loss ∆h obtained
from the meter, with ∆h on the horizontal axis.
Determine the value of the compressibility factor
experimentally and again using the appropriate
equation (Equation 16.3 or 16.4) for each data
point. A minimum of 9 data points should be
obtained. Compare the results of both
calculations for Y.
42

TABLE 16.1. Summary of equations for compressible flow through a venturi or an orifice meter.
m
·
s = A 2 ⎨ ⎧
⎩
2p 1 ρ 1 (p 2 /p 1 ) 2/γ [γ/(γ - 1)] [1 - (p 2 /p 1 ) (γ - 1)/γ 1/2
]
1 - (p ⎭ ⎬⎫
2 /p 1 ) 2/γ (D 4 2 /D 4 1 )
(16.1)
m
·
ac = C v A 2 ⎨ ⎧
⎩
2p 1 ρ 1 (p 2 /p 1 ) 2/γ [γ/(γ - 1)] [1 - (p 2 /p 1 ) (γ - 1)/γ 1/2
]
1 - (p ⎭ ⎬⎫
2 /p 1 ) 2/γ (D 4 2 /D 4 1 )
(16.2)
Y = √⎺⎺⎺⎺⎺⎺⎺⎺⎺⎺⎺⎺⎺⎺⎺⎺
γ [(p 2 /p 1 ) 2/γ - (p 2 /p 1 ) (γ + 1)/γ ](1 - D 4 2 /D 4 1 )
γ - 1 [1 - (D 4 2 /D 4 1 )(p 2 /p 1 ) 2/γ ](1 - p 2 /p 1 )
Y = 1 - (0.41 + 0.35β 4 ) (1 - p 2/p 1 )
γ
(venturi meter) (16.3)
(orifice meter) (16.4)
rounded
inlet
manometer
connections
motor
pitot-static
tube
axial flow outlet duct
fan
FIGURE 16.4. Experimental setup for calibrating a venturi meter.
venturi meter
manometer
connections
rounded
inlet
motor
axial flow
fan
outlet duct
pitot-static
tube
orifice plate
FIGURE 16.5. Experimental setup for calibrating an orifice meter.
43

EXPERIMENT 17
MEASUREMENT OF FAN HORSEPOWER
The objective of this experiment is to measure
performance characteristics of an axial flow fan,
and display the results graphically.
Figure 17.1 shows a schematic of the
apparatus used in this experiment. A DC motor
rotates an axial flow fan which moves air
through a duct. The sketch shows a venturi meter
used in the outlet duct to measure flow rate.
However, an orifice meter or a pitot-static tube
can be used instead. (See Experiment 16.) The
control volume from section 1 to 2 includes all the
fluid inside. The inlet is labeled as section 1, and
has an area (indicated by the dotted line) so huge
that the velocity at 1 is negligible compared to
the velocity at 2. The pressure at 1 equals
atmospheric pressure. The fan thus accelerates
the flow from a velocity of 0 to a velocity we
identify as V 2 . The continuity equation is
m· 1 = m· 2
The energy equation is
0 = - dW
dt + m· ⎡
1 h ⎣
1 + V 1 2 ⎤ - m· ⎡
2 h
2 ⎦ ⎣
2 + V 2 2 ⎤
2 ⎦
where dW/dt is the power input from the fan to
the air, which is what we are solving for. By
substituting the enthalpy terms according to the
definition (h = u + pv), the preceding equation
becomes
dW
dt = m·
(u 1 - u 2 )
+ m·
⎧ p 1
⎨
⎩
p 2
⎡ ⎤
⎣ ρ + V 1 2
- ⎡ ⎤
2 ⎦ ⎣ ρ + V 2 2
2 ⎦
Assuming ideal gas behavior, we have
⎫
⎬
⎭
u 1 - u 2 = C v (T 1 - T 2 )
With a fan, however, we assume an isothermal
process, so that T 1 ≈ T 2 and ρ 1 ≈ ρ 2 = ρ. With m·
=
ρAV (evaluated at the outlet, section 2), the
equation for power becomes
dW
dt
= A 2 V 2 ⎨ ⎧ ⎡p ⎩ ⎣
1 + ρV 1 2 ⎤ - ⎡p 2 ⎦ ⎣
2 + ρV 2 2 ⎤
2 ⎦ ⎭ ⎬⎫
Recall that in this analysis, we set up our control
volume so that the inlet velocity V 1 = 0; actually
V 1

placed on the torque arm to reposition the motor
to its balanced position. The product of weight
and torque arm length gives the torque input from
motor to fan.
A tachometer is used to measure the
rotational speed of the motor. The product of
torque and rotational speed gives the power input
to the fan:
dW a
dt
= Tω (17.2)
This is the power delivered to the fan from the
motor.
The efficiency of the fan can now be
calculated using Equations 1 and 2:
η = dW/dt
dW a /dt
(17.3)
Thus for one setting of the motor controller, the
following readings should be obtained:
1. An appropriate reading for the flow meter.
2. Weight needed to balance the motor, and its
position on the torque arm.
3. Rotational speed of the fan and motor.
4. The static pressure at section 2.
With these data, the following parameters
can be calculated, again for each setting of the
motor controller:
1. Outlet velocity at section 2: V 2 = Q/A 2 .
2. The power using Equation 17.1.
3. The input power using Equation 17.2.
4. The efficiency using Equation 17.3.
Presentation of Results
On the horizontal axis, plot volume flow
rate. On the vertical axis, graph the power using
Equation 1, and Equation 2, both on the same set of
axes. Also, again on the same set of axes, graph
total pressure ∆p t as a function of flow rate. On a
separate graph, plot efficiency versus flow rate
(horizontal axis).
45

EXPERIMENT 18
MEASUREMENT OF PUMP PERFORMANCE
The objective of this experiment is to perform
a test of a centrifugal pump and display the
results in the form of what is known as a
performance map.
Figure 18.1 is a schematic of the pump and
piping system used in this experiment. The pump
contains an impeller within its housing. The
impeller is attached to the shaft of the motor
and the motor is mounted so that it is free to
rotate, within limits. As the motor rotates and
the impeller moves liquid through the pump, the
motor housing tends to rotate in the opposite
direction from that of the impeller. A calibrated
measurement system gives a readout of the torque
exerted by the motor on the impeller.
The rotational speed of the motor is obtained
with a tachometer. The product of rotational
speed and torque is the input power to the
impeller from the motor.
Gages in the inlet and outlet lines about the
pump give the corresponding pressures in gage
pressure units. The gages are located at known
heights from a reference plane.
After moving through the system, the water
is discharged into an open channel containing a
V-notch weir. The weir is calibrated to provide
the volume flow rate through the system.
The valve in the outlet line is used to control
the volume flow rate. As far as the pump is
concerned, the resistance offered by the valve
simulates a piping system with a controllable
friction loss. Thus for any valve position, the
following data can be obtained: torque, rotational
speed, inlet pressure, outlet pressure, and volume
flow rate. These parameters are summarized in
Table 18.1.
TABLE 18.1. Pump testing parameters.
Raw Data
Parameter Symbol Dimensions
torque T F·L
rotational speed ω 1/T
inlet pressure p 1 F/L 2
outlet pressure p 2 F/L 2
volume flow rate Q L 3 /T
The parameters used to characterize the
pump are calculated with the raw data obtained
from the test (listed above) and are as follows:
input power to the pump, the total head
difference as outlet minus inlet, the power
imparted to the liquid, and the efficiency. These
parameters are summarized in Table 18.2. These
parameters must be expressed in a consistent set of
units.
TABLE 18.2. Pump characterization parameters.
Reduced Data
Parameter Symbol Dimensions
input power dW a /dt F·L/T
total head diff ∆H L
power to liquid dW/dt F·L/T
efficiency η —
The raw data are manipulated to obtain the
reduced data which in turn are used to
characterize the performance of the pump. The
input power to the pump from the motor is the
product of torque and rotational speed:
- dW a
dt
= Tω (18.1)
where the negative sign is added as a matter of
convention. The total head at section 1, where
the inlet pressure is measured (see Figure 18.1), is
defined as
H 1
= p 1
ρg + V 1 2
2g + z 1
where ρ is the liquid density and V 1
(= Q/A) is
the velocity in the inlet line. Similarly, the
total head at position 2 where the outlet pressure
is measured is
H 2
= p 2
ρg + V 2 2
2g + z 2
The total head difference is given by
46

∆H = H 2
- H 1
= p 2
ρg + V 2 2
2g + z 2
p
- ⎛ 1
⎞
⎝ ρg + V 1 2
2g + z 1⎠
The dimension of the head H is L (ft or m). The
power imparted to the liquid is calculated with
the steady flow energy equation applied from
section 1 to 2:
- dW
dt
p 2
= m· g
⎡⎛
⎞
⎣⎝ ρg + V 2 2
2g + z 2
⎠
-
⎛
⎝
p 1
ρg + V 1 2
In terms of total head H, we have
⎞⎤
2g + z 1⎠
⎦
- dW
dt = m· g (H 2
- H 1
) = m· g ∆H (18.2)
The efficiency is determined with
η = dW/dt
dW a
/dt
(18.3)
Experimental Method
The experimental technique used in obtaining
data depends on the desired method of expressing
performance characteristics. For this experiment,
data are taken on only one impeller-casing-motor
combination. One data point is first taken at a
certain valve setting and at a preselected
rotational speed. The valve setting would then be
changed and the speed control on the motor (not
shown in Figure 18.1) is adjusted if necessary so
that the rotational speed remains constant, and
the next set of data are obtained. This procedure
is continued until 6 data points are obtained for
one rotational speed.
Next, the rotational speed is changed and
the procedure is repeated. Four rotational speeds
should be used, and at least 6 data points per
rotational speed should be obtained.
v-notch weir
return
valve
control panel
and gages
pressure
tap
1 nominal
schedule 40
PVC pipe
•
sump tank
inlet
z 2
pressure
tap
•
valve
motor
motor
shaft
pump
z 1
1-1/2 nominal
schedule 40
PVC pipe
FIGURE 18.1. Centrifugal pump testing setup.
47

Performance Map
A performance map is to be drawn to
summarize the performance of the pump over its
operating range. The performance map is a graph
if the total head ∆H versus flow rate Q
(horizontal axis). Four lines, corresponding to the
four pre-selected rotational speeds, would be
drawn. Each line has 6 data points, and the
efficiency at each point is calculated. Lines of
equal efficiency are then drawn, and the resulting
graph is known as a performance map. Figure 18.2
is an example of a performance map.
Total head in ft
40
30
20
10
3600 rpm
2700
1760
900
65%
Efficiency in %
75%
80%
85%
75%
0
0 200 400 600 800
Volume flow rate in gallons per minute
65%
FIGURE 18.2. Example of a performance map of
one impeller-casing-motor combination
obtained at four different rotational speeds.
Dimensionless Graphs
To illustrate the importance of
dimensionless parameters, it is prudent to use the
data obtained in this experiment and produce a
dimensionless graph.
A dimensional analysis can be performed for
pumps to determine which dimensionless groups
are important. With regard to the flow of an
incompressible fluid through a pump, we wish to
relate three variables introduced thus far to the
flow parameters. The three variables of interest
here are the efficiency η, the energy transfer rate
g∆H, and the power dW/dt. These three
parameters are assumed to be functions of fluid
properties density ρ and viscosity µ, volume flow
rate through the machine Q, rotational speed ω,
and a characteristic dimension (usually impeller
diameter) D. We therefore write three functional
dependencies:
η = f 1
(ρ, µ, Q, ω, D )
dW
dt
= f 3
(ρ, µ, Q, ω, D)
Performing a dimensional analysis gives the
following results:
where
ρωD
η = f 1
⎛
2
⎞
⎝ µ , Q
ωD 3 ⎠
g∆H
ω 2 D 2 = f 2
⎝ ⎛ ρωD 2
µ , Q
ωD ⎠ ⎞ 3
dW/dt
ρω 3 D 5 = f 3
⎝ ⎛ ρωD 2
µ , Q
ωD ⎠ ⎞ 3
g∆H
ω 2 D2 = energy transfer coefficient
Q
ωD 3
ρωD 2
µ
dW/dt
ρω 3 D 5
= volumetric flow coefficient
= rotational Reynolds number
= power coefficient
Experiments conducted with pumps show that the
rotational Reynolds number (ρωD 2 /µ) has a
smaller effect on the dependent variables than
does the flow coefficient. So for incompressible
flow through pumps, the preceding equations
reduce to
Q
η ≈ f 1
⎛ ⎞
⎝ ωD 3 (18.4)
⎠
g∆H
ω 2 D 2 ≈ f 2
⎝ ⎛ Q
ωD ⎠ ⎞ 3 (18.5)
dW/dt
ρω 3 D 5
Q
≈ f 3
⎛ ⎞
⎝ ωD 3 (18.6)
⎠
For this experiment, construct a graph of
efficiency, energy transfer coefficient, and power
coefficient all as functions of the volumetric flow
coefficient. Three different graphs can be drawn,
or all lines can be placed on the same set of axes.
g∆H = f 2
(ρ, µ, Q, ω, D)
48

Specific Speed
A dimensionless group known as specific
speed can also be derived. Specific speed is found
by combining head coefficient and flow
coefficient in order to eliminate characteristic
length D:
ω ss
= ⎝
⎛
or ω ss
=
Q
⎞
⎠
ωD 3 1/2
ωQ 1/2
(g∆H) 3/4
ω
⎛
2 D 2
⎞
⎝ g∆H ⎠
3/4
[dimensionless]
Exponents other than 1/2 and 3/4 could be used (to
eliminate D), but 1/2 and 3/4 are customarily
selected for modeling pumps. Another definition
for specific speed is given by
ω s
= ωQ1/2
∆H 3/4 ⎣ ⎡ rpm = rpm(gpm)1/2
⎦ ⎤
ft 3/4
in which the rotational speed ω is expressed in
rpm, volume flow rate Q is in gpm, total head ∆H
is in ft of liquid, and specific speed ω s
is
arbitrarily assigned the unit of rpm. The equation
for specific speed ω ss
is dimensionless whereas
ω s
is not.
The specific speed of a pump can be
calculated at any operating point, but
customarily specific speed for a pump is
determined only at its maximum efficiency. For
the pump of this experiment, calculate its
specific speed using both equations.
49

Appendix
Calibration Curves
Orifice plates—open channel flow apparatus ....................................51
V-notch weir—turbomachinery experiments...................................... 52
50

1
large orifice
volume flow rate in ft 3 /s
0.1
small orifice
0.01
0.01 0.1 1 10
manometer deflection in ft of water
FIGURE A.1. Calibration curve for the open channel flow device.
51

120
100
80
height reading in mm
60
40
20
0
0 50 100 150 200 250 300 350
volume flow rate in liters/min
FIGURE A.2. Calibration curve for the V-notch weir, turbomachinery experiments.
52