1
Statics: Pressure Measurement
data and converts the voltage data to the measured
physical property using a conversion of the form
Objectives
In this laboratory, you will learn how to measure
pressure using a computerized data acquisition
system. You will build and test a bubbler system to
measure the depth of water in a tank based on the
relationship between pressure and depth of water.
You will also take pressure measurements to
determine the elevation of the distilled water storage
tanks in Hollister Hall.
Theory
Pressure Transducers
Pressure measurements were first made almost
exclusively with manometers and mercury and water
were both commonly used as manometer fluids. The
risk of mercury spills precludes the use of mercury.
Pressure gages that create a deflection of a needle as
pressure increases are also commonly used. With the
advent of computerized data acquisition, sensors that
can produce a voltage output that is related to a
physical property are preferred.
Pressure transducers produce a voltage output that
is proportional to the applied pressure. Pressure
transducers are available in gage, absolute, and
differential configurations. The pressure transducers
used in this experiment are differential and thus can
be used as gage pressure transducers by connecting
only one of the two ports.
Pressure transducers contain a pressure sensitive
diaphragm with strain gages bonded to it. The strain
gage converts the deflection of the diaphragm into a
measurable voltage. The strain gage output is
affected by temperature changes and the zero value
(no applied pressure) would normally vary from
sensor to sensor. Pressure transducers contain
circuitry to compensate for temperature and to
correctly zero the output.
Data Acquisition System
Pressure transducers produce a voltage output that
is proportional to the pressure applied. The output
voltages are all monitored by a dedicated "data
server" computer with a multi-channel dataacquisition system. The data server sends the
digitized voltage data to client computers on demand
across the Internet. Signal Monitor software logs
onto the data server, receives the digitized voltage
Y  a V  V0 
b
1.1
where V is the measured voltage, V0 is a voltage
offset, the coefficients a and b are user defined, and
Y has the desired physical units. The voltage offset
may be measured at a reference pressure or at a
depth datum.
The Signal Monitor software can monitor any
voltage data being acquired using the data server and
can monitor up to 64 channels simultaneously.
However, the graphical display is limited to 16
channels of data (and is rather cluttered with 16
plots!). Different conversion equations can be
applied to each channel. The channels are numbered
and correspond to labels on the ports located on the
bench tops.
The Signal Monitor software can be used to
average the voltage signal and to log the data to disk.
In this lab, it will not be necessary to save the data to
disk. It will be easier to simply record the pressure
measurements by reading the values on the computer
display.
Pressure Transducer Calibration
The pressure transducer used for this experiment
measures the differential pressure between two
ports. The pressure transducer has a range of 0 to 6.8
kPa with a corresponding output voltage range of 0
to 16.7 mV. The pressure transducer can be
calibrated to determine the actual relationship
between volts (the measured signal) and pressure
differential by connecting a pressure transducer to a
static column of water. The relationship between
pressure and voltage should be expressed in the form
of equation 1.1 where Y has dimensions of
centimeters of water for entry into the Signal
Monitor program. Alternately, the relationship
between pressure and voltage can be obtained from
the
pressure
transducer
specifications
(http://www.omega.com/Pressure/pdf/PX26.pdf).
Error Analysis
Both the pressure transducers and the data
acquisition system contribute to the measurement
errors. The pressure transducers have an accuracy of
1% FS (FS is their full-scale measurement) and a
hysteresis and repeatability of 0.2% FS. The
Statics: Pressure Measurement
2
pressure transducers are rated to measure 1 psi (6.8
kPa) and produce an output voltage of 20 mV at that
pressure. The data acquisition system is set to
measure a range of ±25 mV. The data acquisition
system is 12 bit meaning that the measured voltage
range is digitized into 212 (4096) intervals. The
smallest difference that the data acquisition system
can measure is 50 mV/4096 or 0.012 mV.
the end of the tube. The small radius of curvature of
the bubbles can result in a significant pressure
increase in the gas line. As the bubbles are formed
the radius of curvature will vary from close to
infinite to the radius of the released bubbles and the
pressure in the line will vary.
p
2
r
1.3
Statics
Pressure variation with depth in a constant density
fluid is linear.
p h
1.2
The simple relationship between pressure and depth
suggests that pressure transducers can be used to
measure either pressure or depth by simply applying
an appropriate calibration constant.
Bubbler system
Bubbler systems are used by United States
Geological Survey (USGS) to measure stage (depth)
of streams and rivers. Stations that use a bubbler
system can be located hundreds of feet from the
stream. In a bubbler system, an orifice is attached
securely below the water surface and connected to
the instrumentation by a length of tubing.
Pressurized gas (usually nitrogen or air) is forced
through the tubing and out the orifice. Because the
pressure in the tubing is a function of the depth of
water over the orifice, a change in the stage of the
river produces a corresponding change in pressure in
the tubing. Changes in the pressure in the tubing are
recorded and are converted to a record of the river
stage.
The accuracy of bubbler system is affected by the
head loss of the gas flowing through the tubing that
connects the pressure transducer to the river. If the
gas flow rate is variable, the head loss through the
tubing will also be variable. If the head loss in the
tubing is small relative to the desired accuracy of
water depth measurement then small changes in flow
rate will be insignificant. Alternately accuracy can
be maintained by carefully regulating the flow of gas
in the tubing. A final source of error is the pressure
variation due to the formation of small air bubbles at
Experimental Apparatus
A 10 cm diameter tube that can be filled with up
to 50 cm of water is used to model a small reservoir.
A pressure transducer connected to a stainless steel
tube is used to measure the relationship between
depth and pressure. The pressure transducer is wired
to be connected to a port installed on the lab bench.
The ports on the lab bench are numbered and the
numbers correspond to channels in the Signal
Monitor software.
Experimental Methods
Calibrate Pressure Transducer
Start the Signal Monitor software (available on
the desktop of the computer). A description of the
software
is
available
at
http://ceeserver.cee.cornell.edu/mw24/software/sign
al_monitor.htm.
The software control palette is set up so that it is
logical to proceed from top to bottom. The first step
is to enter (or retrieve
from a file) the
calibration
matrices
that are entered in the
form of equation 1.1
by clicking on the
"Select
Channels"
button (Figure 1). A
dialog box similar to
Figure 2 will appear.
Select the channel that
you want to monitor
and then select “from
Figure 1. Palette of
options in Signal Monitor
software.
CEE331 Summer . Fluid Mechanics. Cornell University.
3
(maximum of 50 Hz) and for averaging the
data. If desired you can log the data to a file by
selecting “Enable Logging Data.” After setting
the method you can “Monitor Signal.”
Build and Test a Bubbler System
Use the plumbing supplies at your station to
build a bubbler system. You will be using the
bubbler system to measure the pressure as a
function of depth in a 10-cm diameter column
Figure 2. Dialog box for selecting channels and calibrating
sensors.
file” under “calibration”. If you need to monitor
multiple sensors you can select as many channels as
needed. Each of the channels can then be assigned
an appropriate conversion. The sensor calibration
makes it possible to enter a series of types of sensors
with different physical units as long as the desired
unit can be obtained by a transformation based on
equation 1.1. Enter a descriptive name, the units (m),
the coefficients (ai) and the exponents (bi). If desired
the calibration matrix can be saved for later use.
of
wat
er.
U
se
the
peri
stalt
ic
pum
Figure 4. Dialog box showing data rate.
p as your air supply. The 6 mm diameter stainless
steel tube can be submerged to variable depths in a
tank of water to test your bubbler system. Have the
TA check your bubbler system after you've built it.
Set the peristaltic pump rate so that a bubble is
formed every 2 to 5 seconds. Figure out a
reasonable method to zero the bubbler so that the
Signal Monitor will display 0 cm of water when the
end of the bubbler rod is at zero depth. Use the "Set
Offset" button (see Figure 3) to measure the offset
voltage. Only one sensor can be zeroed at a time.
Figure 3. Dialog box for calibrating sensors. Note that
the scale type needs to be polynomial so the coefficient
and exponent arrays are used for converting volts into the
sensor physical units.
Connect a 7-kPa pressure transducer to one of the
numbered ports on the top row on the bench top.
Instruct the signal monitor software to monitor that
channel and to apply an appropriate calibration to
the voltage measurements from that channel. Select
"Select Channels" to activate and apply the correct
calibration to the channel that your sensor (or
sensors) is connected to. The "Sample Rate" dialog
box has options for setting the data acquisition rate
1) Submerge the end of the bubbler rod in the
4-L volumetric detector and measure the
depth of submergence using a ruler. Record
the depth of submergence measured using
the ruler and that obtained using the
bubbler system. Record values at 5
different depths (repeatable depths can be
created by measuring out 500 mL volumes
of water). Repeat the 5 measurements to
check repeatability.
2) Use the bubbler system to determine if
pressure is a function of the diameter of the
reservoir. Explain your test method and your
results.
Statics: Pressure Measurement
4
3) Use the bubbler system to determine if the
pressure at a point is a function of direction.
Explain briefly how you tested your
hypothesis and report the results obtained.
4) What happens if you don’t pump air through
the bubbler?
5) Explain why it is necessary to continually
pump air through the bubbler.
6) Observe the response of the bubbler system
to rapid changes in submergence. Rapidly
plunge the bubbler tube to the bottom of the
reservoir and observe the response of the
pressure transducer using the Signal
Monitor software. What happens to the rate
of bubble formation?
Lab Prep Notes
Table 2. Equipment list.
Description
Pressure
transducer
Pressure
transducer
4 L volumetric
detector
4 L reservoir
45 cm ruler
Peristaltic pump
Supplier
Omega
Omega
Catalog number #/group
PX26-001DV
1
PX26-030DV
1
CEE shop
1
Fisher
1
1
1
ColeParmer
7) Explain why the bubbler system responds
slowly to changes in depth.
8) What could you do to decrease the response
time?
9) What equations would you use to determine
the location of the air-water interface inside
the bubbler tube if you weren’t pumping air?
Measure the Elevation of a Reservoir Surface
10) The distilled water system is fed by a
reservoir located somewhere in Hollister
Hall. Use a 200-kPa pressure transducer to
determine on which floor the reservoir is
located. According to the manufacturer, the
pressure transducer output is 100 mV/206.8
kPa. Note that distilled water taps have
white handles. Connect the pressure
transducer to the distilled water tap using
appropriate tubing. You shouldn't use the
bubbler for this! Determine the
elevation difference between
the lab bench tops and the water Table 1. Recommended measurements.
surface of the distilled water
Added Depth Bubbler Bubbler
reservoir.
Volume Using
Depth,
Depth,
Lab Report
Submit a brief group report at the end
of lab containing your responses to the
questions.
(mL)
500
1000
1500
2000
2500
CEE331 Summer . Fluid Mechanics. Cornell University.
Ruler
(cm)
Trial #1
(cm)
Trial #2
(cm)
Difference Difference
between
between
Ruler &
Ruler &
Bubbler
Bubbler
(cm)
(cm)
5
Bernoulli and the Free Jet
Objectives
In this laboratory you will measure fluid
velocities in a free jet using a stagnation tube. You
will confirm that the Bernoulli equation can be used
to measure fluid velocities using a simple stagnation
tube.
pstagnation  
V jet2

2g
SS Tubing with
7.25 mm ID
V jet2
2
2.4
7 kPa Pressure
Sensor z
Theory
In addition, mass must be conserved and thus
Q1=Q2. The depth relationship and mass
conservation will be verified using the Bernoulli
equation.
V2

z C
 2g
p
Free Jet
Stagnation
Point
2.1
A stagnation tube will be connected to a pressure
sensor. The stagnation tube will be filled with water
prior to connecting to the pressure sensor and the
pressure sensor output will be zeroed with the
stagnation tube held vertically (in the same
orientation used for taking measurements.) Thus the
pressure sensor will measure the pressure at point 3
(Figure 1). From the Bernoulli equation we can
obtain the following relationship.
V12 p2
V22
 z1 

 z2 

2g 
2g
p1
2.2
where point 1 is on a horizontal line in the jet away
from the stagnation point and point 2 is the
stagnation point. The stagnation pressure,
p2

, will
be measured using a pressure transducer. Equation
2.2 requires the assumption that the streamlines are
straight and parallel (allowing us to cross
p1 V22
streamlines from point 1 to point 2). Since
,
 2g
are zero and z1  z2 we can obtain
2
1
V
p
 2
2g 
Stagnation
Tube filled
with water
Centrifugal
Pump
Figure 1. Stagnation tube and pressure sensor used
to measure velocity in a jet.
Experimental Methods and Analysis
Set up a small jet powered by a centrifugal pump.
Fill the 10 cm diameter column with 15 cm of water.
Fill the stagnation tube completely with water before
connecting the pressure sensor. Monitor the pressure
sensor with the Signal monitor software and apply
scaling so the output is measured in Pascals.
Make the following measurements and
calculations.
1) What is the stagnation pressure at z = 0?
2) What is the velocity at z = 0?
2.3
3) What is the jet flow rate?
Solving for the pressure head at the tip of the
stagnation tube
Connect a 7 kPa pressure sensor to the volumetric
detector and monitor the sensor with the Signal
Bernoulli and the Free Jet
6
monitor software and apply scaling so the output is
measured in mL. Log the data to file and with the
pump running turn the jet so it discharges into a
different container. You will use the initial slope of
the resulting data to determine the flow rate out of
the volumetric detector.
4) Why should you use the initial slope to measure
the flow rate?
5) The data will be close to linear, but a parabolic
fit is a better approximation. What is the
equation of the parabolic fit?
6) How can you use the parabolic fit to determine
the initial slope?
7) What is the flow rate that you obtain from the
initial slope and how does it compare with the
flow rate calculated using the stagnation tube?
8) Fill the volumetric detector to 15 cm again and
measure the stagnation tube pressure at various
elevations in the jet. Use Excel to plot the
velocity in the jet as a function of elevation. On
the same graph plot the prediction based on
Bernoulli’s equation.
V12
V22
z1 
 z2 
2g
2g
2.5
where points one and two are any two points in a
free
jet.
Format
the
graph
correctly
(http://ceeserver.cee.cornell.edu/mw24/cee331/lab_r
eport.htm#graphs ) and email the Excel file to the
TA and to the Instructor.
CEE331 Summer . Fluid Mechanics. Cornell University.
7
Mass Conservation in Unsteady Flow
O  I  
Objectives
To demonstrate mass conservation in a finite
control volume in unsteady flow and to illustrate the
effects of reservoir storage on flood flows.
According to the control volume (cv) equations
the mass leaving - mass entering = - rate of increase
of mass in cv

cs  v  dA   t cv  d 
3.1
Density can often be considered constant and thus
equation 3.1 simplifies to

 v  dA   t
3.2
cs
The integral of v·dA is simply the sum of the
volumetric flow rates through each control surface.
The sign of the dot product is positive when the flow
is out of the control volume.
d
dt
3.3
Equation 3.3 can be written in finite difference form
to be used for experimental data over short time
intervals.

 Qin  Qout
t
3.4
The change in volume of water in the control
volume is equal to the inflow minus the outflow.
Alternately, equation 3.4 can be read as "the net rate
of flow into the control volume is equal to the inflow
minus the outflow."
Equation 3.3 can be integrated to obtain a form of
the mass balance equation that is in terms of total
mass rather than in terms of rates.
t
Q
out
t0
Finally, we can define the change in volume in the
control volume as change in storage to obtain a form
of the mass conservation equation that is well suited
to describe reservoir operation.
S  I  O
Theory
Qout  Qin  
3.6
t

t0
0
dt   Qin dt    d 
3.5
3.7
Both equation 3.7 and equation 3.4 can be used to
analyze a reservoir during unsteady flow.
Experimental Apparatus
The experimental apparatus consists of a water
source and a reservoir with an overflow. The flow
rate of the tap water is measured with a 1.6-mm
diameter orifice, the depth of water in the reservoir
is measured with a sensor attached at the base, and
the outflow from the reservoir is measured in a
volumetric detector. The Signal Monitor data
acquisition software will be used to monitor the
pressure drop across the orifice, the hydrostatic
pressure at the bottom of the reservoir, and the
hydrostatic pressure at the bottom of the volumetric
detector. A calibration matrix will be used to convert
the output as shown in Table 1.
Table 1. Sensor outputs.
Location
orifice
reservoir
volumetric detector
Pressure
transducer
206 kPa
7 kPa
7 kPa
output
mL/s
mL
mL
The flow through an orifice causes a pressure
drop. The flow is proportional to the square root of
the pressure drop
Q  K orifice
d2
2
4

p
3.8
The pressure drop will be measured by a pressure
transducer that will have a voltage output
proportional to the pressure drop. The voltage offset
of the pressure transducer, V0, will be measured
under conditions of no flow.
If we define O as the cumulative outflow from t0 to t
and define I similarly,
Mass Conservation in Unsteady Flow
p  k pt V  V0 
3.9
8

 d2
Q   Korifice
4

0.5
 2k pt  
0.5

  V  V0 
   
3.10
therefore our calibration coefficients are:

 d 2  2k pt  
a   Korifice

 ,
4    


b  0.5,
0.5
3.11
and kpt is the calibration coefficient that converts the
pressure transducer output from Volts to Pascals.
The orifice diameter is 1.6 mm and Korifice has a
value near 0.82.
The volume of water in the reservoir will be
measured based on the pressure at the bottom of the
reservoir. The reservoir walls slope outward, but to
simplify the calibration the average cross sectional
area will be used when calculating volume. The
reservoir bottom has dimensions of 17.5 cm x 19.0
cm, and the top has dimensions 19.5 cm x 21 cm.
The volume in the reservoir is:
  mny
3.12
where m is average width, n is the average length,
and the water depth, y, is defined as:
y
k pt

(V  V0 )
3.13
The volume of the water in the volumetric
detector is calculated in a similar manner. The
volumetric detector internal diameter is 10 cm. The
relationship between volume and depth is thus

d2
4
y
3.14
and the relationship between voltage and volume is
given by
 d 2 k pt

V  V0 
4
3.15
Experimental Methods
1) Enter the calibration coefficients for the three
sensors into the Signal Monitor software (done
by TA).
CEE331 Summer . Fluid Mechanics. Cornell University.
2) Use the Select Channels command to instruct the
computer to monitor the three sensors (for the
orifice, reservoir, and volumetric detector) with
the appropriate conversions.
3) Set sampling rate to measure 1 scan per sample
and 250-sample averaging for a final data rate of
0.2 Hz.
4) Set the offset for all three sensors under zero
flow and zero volume conditions (the zero
volume condition for the reservoir is when it is
full to the 3-L mark).
5) Verify that the three sensors are working
correctly by measuring a flow rate (use
stopwatch and graduated cylinder) and volumes
in the reservoir and volumetric detector.
Compare with values given by the Signal
Monitor software.
6) Empty the volumetric detector and fill the
reservoir to the level of the overflow (3-L mark),
and again set the offsets for the three channels.
7) Create a new file to log data by clicking the
Enable Logging Data button. This file can be
imported into Excel for the data analysis.
8) Instruct the computer to begin Monitoring the
Signal and make sure data is being logged to file
(if the button says “Stop Logging”, then data is
being written to file). Slowly open the brass
valve until the flow rate is approximately 20-25
mL/s (you may take a minute to open the valve
if desired).
9) When the reservoir level reaches the top of the
overflow tube (~5-L mark) close the brass valve
but continue data acquisition for about 15
minutes or until the flow from the reservoir
stops.
Pre-lab Questions
Pre-lab questions are to be handed in by the
team at the start of the lab period.
Calculate the coefficients of the calibration matrix
that will be needed to convert the voltage readings
into the desired dimensions. The 206 kPa pressure
transducer has a kpt of 2.068 x 106 Pa/V and the 7
kPa pressure transducer has a kpt of 412.8 x 103
Pa/V. Remember that we will be using mL as our
volume measurement and mL/s as our flow rate
measurement. Note that the calibration coefficients
‘a’ and ‘b’ (equation 3.11) apply to the orifice only!
9
Table 2. Calibration coefficients.
Description
Orifice
Reservoir
Volumetric
detector
a
Lab Prep Notes
b
Table 3. Equipment list.
Data Analysis
1) Draw a control volume around the reservoir and
indicate the surfaces where mass is entering or
leaving the control volume. Compute and plot
the inflow rate, the outflow rate and the change
in storage with respect to time. (Plot all three
plots on the same graph.) Remember that flow
rate is the change in volume with respect to
time, i.e. flowI = (Voli+1-VolI-1)/(2*time step),
where i is the current time index and the time
step for this experiment is 5 seconds. Note that
the orifice transducer may give a small flow rate
after it is shut off; these values should be
manually set to zero when performing the
spreadsheet analysis. Show graphically that the
inflow rate is equal to the sum of the outflow
rate and the change in storage with respect to
time.
2) Compute and plot the total volume inflow, the
total volume outflow, and the total volume
storage as a function of time for the duration of
your measurements. (Put all three plots on the
same graph.) Remember that volume is the
integral, or accumulation, of flow with respect to
time, i.e. Volumei = Volumei-1 + (flowi+ flowi1)/2*time step. Show graphically by summing
the outflow and storage curves that mass is
conserved at all times (i.e. inflow = outflow +
storage).
Description
Supplier
Pressure
transducer
Pressure
transducer
4 L volumetric
detector
Nupro angled
3/8 swage valve
Omega
Omega
CEE shop
Rochester
Valve & Fitting
Co., INC.
orifice holder
CEE shop
3/8" OD tubing
Cole-Parmer
1/4" OD tubing
Cole-Parmer
Pressure reducer
ID Booth
Reservoir
RubberMaid
3) When does the maximum outflow occur?
Lab Report
Submit a group report containing the answers to the
“Data Analysis” questions. The report should
follow the lab report guidelines given on the web at
http://ceeserver.cee.cornell.edu/mw24/cee331/lab_re
port.htm#graphs.
Mass Conservation in Unsteady Flow
Catalog
number
PX26001DV
PX26030DV
#/group
2
1
1
B-6JNA
1
1
H-06490-15
H-06490-15
FB-38
1
1
10
Momentum and Energy Conservation Experiment:
Expansion/Contraction
Objectives
To demonstrate energy conservation and the
hydraulic grade line for steady pipe flow.
a sudden contraction are actually due to the flow
expansion that occurs after the section of greatest
contraction of the jet (see page 500 in Munson, et
al.),
hc  K L
Theory
V22
2g
4.5
The losses due to a sudden expansion in a
pipeline can be calculated using conservation of
momentum and the conservation of energy
equations. The head loss due to a sudden expansion
is
where K L is a function of the area ratio of the
downstream and upstream pipes.
V1  V2 
The test piece consists of a 25 cm long 3 mm ID
brass tube connected to a 50 cm long 8 mm ID brass
tube. Pressure ports are installed on the tubes 10
pipe diameters from the ends of each tube. The water
source is cold tap water that passes through a
pressure regulator and a needle control valve. The
flow of water can be reversed to change it from an
expansion to a contraction.
he 
2
4.1
2g
2
A
 V2
he   2  1 2
 A1
 2g
4.2
Equation 4.2 can be rewritten in terms of the
volumetric flow rate (using conservation of mass) to
obtain
8  1
1 
he  Q 2
 2
2 
2
g  D1 D2 
Q  2 ghe
2
4.3

4.4
 1
1 
4 2  2 
 D1 D2 
The losses due to a sudden contraction are less
than losses due to a sudden expansion. The losses in
200 kPa
7 kPa
Experimental Apparatus
Experimental Methods
1) Measure the distances between connected
pressure ports as well as the exact location of the
expansion/contraction. Note that the pressure
ports are installed on the tubes 10 pipe diameters
from the ends of the tubes and from the change
in pipe diameter.
2) Plug all of the sensors into the top row of
channel ports (maximum voltage measured will
be less than 25 mV even for the 200 kPa sensor).
3) Connect the pressure sensors so that the higher
7 kPa
Figure 2. Experimental apparatus showing pressure transducers connected to brass tubing test
section.
CEE331 Summer . Fluid Mechanics. Cornell University.
11
pressure is where the cable leaves the sensor.
4) Connect a 7 kPa pressure sensor to the
volumetric detector.
5) Use the Select Channels command to instruct the
computer to monitor the 3 sensors with the
appropriate conversions.
4) Compare the head loss due to the expansion and
contraction with the values obtained from
equations 4.3 and 4.5.
5) Use the data acquired to determine the
contraction loss coefficient (equation 4.5).
Lab Prep Notes
6) Set the “samples to average” to 50 (data rate
will be 1 Hz).
7) Set the offset for all sensors under zero flow
conditions.
8) Open the brass valve until the flow rate is
approximately 20-30 mL/s. Use the volumetric
detector to measure flow rate. (Note that the first
derivative of the volumetric detector output is in
mL/s.)
9) Enable logging the data to file and begin
monitoring the signal.
10) Record the pressure change in the thin tube, the
expansion/contraction, and the thick tube.
11) Reverse the flow direction through the test
section and repeat steps 3 to 8. Note that you
must use a 200-kPa sensor across the contraction
instead of the 7-kPa sensor that you used for the
expansion.
Table 2. Equipment list.
Description
Supplier
#/group
Omega
Catalog
number
PX26-001DV
Pressure
transducer
Pressure
transducer
Nupro angled
3/8 swage valve
Omega
PX26-030DV
1
B-6JNA
1
H-06490-15
FB-38
1
Rochester
Valve &
Fitting Co.,
INC.
3/8" OD tubing Cole-Parmer
Pressure reducer
ID Booth
2
Pre-lab Questions
1) Calculate the Reynolds number for both sections
of tubing for a flow of 30 mL/s.
2) What should the contraction loss coefficient KL
(equation 4.5) be for the experimental setup?
3) Why will the pressure drop between the ports
closest to the expansion/contraction be greater
than that predicted by equations 4.3 and 4.5?
Data Analysis
1) Plot the hydraulic grade line for the expansion
and contraction on separate graphs.
2) Plot the energy grade lines on the corresponding
graphs,
keeping
consistent
scaling.
3) Measure the head loss due to the Table 1. Recommended measurements.
expansion and the contraction
Desired Flow
Actual
p
based on the drop in the energy
Rate
Flow
3 mm
grade line. (This can be done
(mL/s)
(mL/s)
tube
using simple equations or by
accurately measuring on the
(Pa)
graph.)
30 (expansion)
p
Expansion/
Contraction
(Pa)
30 (contraction)
Momentum and Energy Conservation Experiment: Expansion/Contraction
p
8 mm
tube
(Pa)
12
Determination of the Friction Factor in Small Pipes
Turbulent flow
Objective
To study the variation in friction factor, f, used in
the Darcy Formula with the Reynolds number in
both laminar and turbulent flow. The friction factor
will be measured as a function of Reynolds number
and the roughness will be calculated using the
Swamee-Jain equation.
Theory
The loss of head resulting from the flow of a fluid
through a pipeline is expressed by the Darcy
Formula
hf  f
L V2
D 2g
5.1
where hf is the loss of head (units of length) and the
average velocity is V. The friction factor, f, varies
with Reynolds number and a roughness factor.
When the flow is turbulent the relationship
becomes more complex and is best shown by means
of a graph since the friction factor is a function of
both Reynolds number and roughness. Nikuradse
showed the dependence on roughness by using pipes
artificially roughened by fixing a coating of uniform
sand grains to the pipe walls. The degree of
roughness was designated as the ratio of the sand
grain diameter to the pipe diameter (/D).
The relationship between the friction factor and
Reynolds number can be determined for every
relative roughness. From these relationships, it is
apparent that for rough pipes the roughness is more
important than the Reynolds number in determining
the magnitude of the friction factor. At high
Reynolds numbers (complete turbulence, rough
pipes) the friction factor depends entirely on
roughness and the friction factor can be obtained
from the rough pipe law.
Laminar flow
The Hagen-Poiseuille equation for laminar flow
indicates that the head loss is independent of surface
roughness.
32LV
hf 
 gD 2
5.2
64
Re
or
f 
64
VD 
5.3
indicating that the friction factor is proportional to
viscosity and inversely proportional to the velocity,
pipe diameter, and fluid density under laminar flow
conditions. The friction factor is independent of pipe
roughness in laminar flow because the disturbances
caused by surface roughness are quickly damped by
viscosity.
Equation 5.2 can be solved for the pressure drop
as a function of total discharge to obtain
p 
128LQ
D 4
CEE331 Summer . Fluid Mechanics. Cornell University.
5.5
For smooth pipes the friction factor is independent
of roughness and is given by the smooth pipe law.
 Re f
 2log 
 2.51
f

1
Thus in laminar flow the head loss varies as V and
inversely as D2. Comparing equation 0 and equation
1 it can be shown that
f 
 3.7 D 
 2 log 

f
  
1



5.6
The smooth and the rough pipe laws were developed
by von Karman in 1930.
Many pipe flow problems are in the regime
designated “transition zone” that is between the
smooth and rough pipe laws. In the transition zone
head loss is a function of both Reynolds number and
roughness. Colebrook developed an empirical
transition function for commercial pipes. The
Moody diagram is based on the Colebrook equation
in the turbulent regime.
  D 2.523
 2 log 

 3.7 Re f
f

1



5.7
5.4
The Colebrook equation can be used to determine
the absolute roughness, , by experimentally
measuring the friction factor and Reynolds number.
13



  3.7 D  e
1.151
f

2.523 

Re f 
5.8
Alternatively the explicit equation for the friction
factor derived by Swamee and Jain can be solved for
the absolute roughness.
f 
0.25
  
5.74  
log  3.7 D  Re0.9  

 
2
5.9
Experimental Apparatus
The experimental apparatus consists of a pressure
regulating valve, a flow control valve, a test section
of tubing with pressure taps and a pressure
transducer. The 10-cm-diameter volumetric detector
will be used to measure flow rates.
Experimental Methods
The experiment consists of measuring the head
loss in a length of tubing as a function of discharge.
Head loss will be measured in small diameter brass
pipe using pressure transducers. Discharge will be
obtained by measuring the volume of discharge over
a time interval using two volumetric sensors. An 85
cm section of tubing with an inside diameter of 3.4
mm will be used.
1) Record the distance between pressure ports.
2) Use Setup Channels to load the calibration
coefficients for the two pressure transducers
(output in cm) that will be used to measure
pressure drop, and for volumetric detector and to
set the offsets for both sensors under zero flow
conditions.
3) Use “Sample Rate” to set the samples to average
to 50 so the data frequency is 1 Hz.
4) Log the data to a file. Either keep a record of
what you are doing using the current time
displayed on the on the Signal Monitor graph so
you can decode the data log later or create a
separate file for each trial.
5) Instruct the computer to begin Monitoring the
Signal.
6) Measure the pressure drop in the tubing and the
flow rate.
7) Record the values in Table 1.
8) Repeat steps 3-8 for the other flow rates listed in
Table 1, remembering to change pressure
transducer as needed.
Table 1. Recommended measurements.
Pressure
transducer
(for
headloss)
7 kPa
7 kPa
7 kPa
200 kPa
200 kPa
200 kPa
200 kPa
200 kPa
200 kPa
200 kPa
200 kPa
200 kPa
200 kPa
Desired
Flow rate
(mL/s)
Actual
flow
(mL/s)
hf
(cm)
1.5
3.0
4.0
5
6
7
10
12
14
17
20
25
max
Prelab Questions
1) What is the expected pressure drop in a 85 cm
section of 3 mm ID brass tubing when the flow
rate is 10 mL/s?
2) Why are two different pressure transducers used
to measure the head loss? (The answer isn't
explicitly in the lab manual!)
Data Analysis
1) What is the advantage of expressing the friction
factor as a function of the Reynolds number
rather than as a function of the flow rate?
2) Determine the absolute roughness, , for the
brass tubing using equation 8.
3) Create a diagram similar to the one created by
Moody showing the friction factor as a function
of Reynolds number (log-log plot). Clearly
indicate the laminar and turbulent regions. In
addition to your data, plot the results obtained by
Hagen-Poiseuille in the laminar region and the
Swamee-Jain equation in the turbulent region
using your best estimate of the roughness of the
brass tubing.
Determination of the Friction Factor in Small Pipes
14
Lab Prep Notes
Table 2. Equipment list.
Description
Supplier
Pressure
Omega
transducer
Pressure
Omega
transducer
Nupro angled Rochester Valve
3/8 swage valve
& Fitting Co.,
INC.
3/8" OD tubing
Cole-Parmer
Pressure reducer
ID Booth
Volumetric
CEE shop
detector
Catalog
number
PX26-001DV
PX26-030DV
B-6JNA
H-06490-15
FB-38
CEE331 Summer . Fluid Mechanics. Cornell University.
15
Hydraulic Systems: Pumps and Valves
Objectives
The objective of this experiment is to familiarize
students with centrifugal pumps as well as several
flow control devices.
Experimental Apparatus
The apparatus consists of a low-pressure water
source, a small centrifugal pump, a needle valve, a
ball valve, pressure transducers, an orifice, an
elevated reservoir, and miscellaneous tubing and
connectors.
Experimental Methods
Students will design and build a simple hydraulic
system to deliver water to an elevated reservoir from
a low-pressure water source. The flow rate to the
reservoir must be monitored with an orifice. The
system must also be able to compare the
performance of a needle valve and a ball valve to
determine which valve is most appropriate for a
system that requires flow control. You must also be
able to monitor pressure changes across the pump.
1) Draw a schematic on paper and show your plans
to the TA to get approval to build your system.
2) Assembly the hydraulic system.
3) Get approval from the TA to open source valve
and plug in pump. The counter should be wiped
dry before plugging in pump. (Make sure you
don't run the pump with the pump full of air.
The pump rotor will be permanently damaged in
less than 1 minute!)
4) Set the Signal Monitor software to monitor the
orifice and the pressure transducer.
5) Test your system performance so you can
answer all the questions in the analysis section.
Analysis
1) How would you design the hydraulic system so
that the flow rate would be unaffected by the
level of water in the elevated reservoir?
2) How would you design the connection to the
elevated reservoir to obtain the maximum flow
rate from the pump (and lowest energy usage!)?
Hydraulic Systems: Pumps and Valves
16
Bernoulli and the Hydraulic Jump
p1
Objectives
In this laboratory you will measure fluid
velocities in open channel flow using a stagnation
tube. You will confirm that the Bernoulli equation
describes the flow through a sluice gate and that the
Bernoulli equation can be used to measure fluid
velocities using a simple stagnation tube.
Theory
Flow through a sluice gate can be reasonably
modeled using the Bernoulli equation. The potential
energy of the water behind the sluice gate is
converted into kinetic energy as the water passes
under the gate. Thus the velocity of the water can be
calculated directly from the height of the water
behind the sluice gate. Hydraulic jumps occur in
open channel flow when the flow transitions from
supercritical to subcritical flow. A description of the
phenomena can be found in Munson, et al. page 671.
The upstream (y1) and downstream (y2) depths are
related
by
equation
Error! Reference source not found..

y2 1
 1  1  8Fr12
y1 2

Fr1 
7.2
gy1
In addition, mass must be conserved and thus
Q1=Q2. The depth relationship and mass
conservation will be verified using the Bernoulli
equation.
p


V2
z C
2g
p3

7.4
will be measured using a pressure
transducer. Equation 2.2 requires the assumption that
the streamlines are straight and parallel (allowing us
to cross streamlines from point 1 to point 2). Since
p1 V32
,
are both zero we can obtain
 2g
V12 p3
z1 

 z3
2g 
7.5
Solving for the pressure head at the tip of the
stagnation tube
V12
 z1  z3 

2g
p3
7.6
Thus the stagnation pressure head includes both the
static head based on the submergence of the
stagnation tube tip as well as the velocity head.
Pressure sensor
Stagnation tube
1
2
7.3
A stagnation tube will be connected to a pressure
sensor. The stagnation tube will be filled with water
prior to connecting to the pressure sensor and the
pressure sensor output will be zeroed with the
stagnation tube held vertically (in the same
orientation used for taking measurements.) Thus the
pressure sensor will measure the pressure at point 3
(Figure 1). From the Bernoulli equation we can
obtain the following relationship.
CEE331 Summer . Fluid Mechanics. Cornell University.
where
V2
V12 p2
V2 p

 z2  2  3  z3  3
2g 
2g 
2g
7.1
where the upstream Froude number (Fr1) is
defined as
V1

 z1 
3
z
Figure 1. Stagnation tube and pressure sensor used
to measure velocity in open channel flow.
Experimental Methods
A small flume will be set up with a stable
hydraulic jump. Your goal is to measure the flume
dimensions and fluid velocity upstream and
downstream from the hydraulic jump.
17
Make the following measurements using the
bottom of the channel as your elevation datum.
Height of water in the reservoir (cm)
Stagnation pressure head at the
opening of the sluice gate (cm)
Stagnation pressure head just upstream
of the hydraulic jump (cm)
Depth of submergence of the
stagnation
tube
for
previous
measurement (cm)
Depth of water just upstream of the
hydraulic jump (cm)
Depth of water downstream of the
hydraulic jump (cm)
In addition to these measurements you should play
with the stagnation tube and the hydraulic jump so
you can answer the questions for the lab report.
Lab Prep Notes
Table 1. Equipment list.
Description
Supplier
Pressure
Omega
transducer
Pressure
Omega
transducer
Nupro angled Rochester Valve
3/8 swage valve
& Fitting Co.,
INC.
3/8" OD tubing
Cole-Parmer
Pressure reducer
ID Booth
Centrifugal
McMaster
pump
Ball valve
McMaster
Lab Report
1) If you make a small disturbance in the water
surface upstream from the hydraulic jump which
way does the disturbance travel?
2) If you make a small disturbance in the water
downstream from the hydraulic jump do any of
the waves from the disturbance travel upstream?
3) Calculate the upstream Froude number.
4) Calculate the downstream water depth based on
Equation Error! Reference source not found..
Is this estimate close to your measurement? If
not, recall the difficulty of measuring the
upstream water depth.
5) What was the water velocity immediately
downstream from the sluice gate? Compare the
velocity head with the elevation of the reservoir
surface.
6) What happens if you measure the stagnation
pressure at the very bottom of the channel?
Explain based on the properties of real fluids.
3)
4) Which type of valve made it easiest to set the
flow rate to a desired value?
5) Which type of valve is easiest to shut off
quickly?
6) What is the maximum pressure increase that the
pump produces?
7) What is the maximum flow rate that the pump
produces?
Bernoulli and the Hydraulic Jump
Catalog
number
PX26-001DV
PX26-030DV
B-6JNA
H-06490-15
FB-38
99335k16
4912K47