Energy and Hydraulic Grade Lines through an Expansion
Objectives
To demonstrate measurement of mechanical energy losses and the concepts of energy and hydraulic
grade lines for steady pipe flow through an expansion.
Theory
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
he 
Vin  Vout 
2
1.1
2g
V2 
A 
he  in 1  in 
2g 
Aout 
2
1.2
Equation 1.2 can be rewritten in terms of the volumetric flow rate (using conservation of mass) to obtain
8  1
1 
he  Q



g 2  D12 D22 
2
2
Q  2 ghe

 1
1 
4 2  2 
 D1 D2 
1.3
1.4
The head loss through the expansion occurs over several pipe diameters as much of the kinetic energy
available upstream of the expansion is converted to thermal energy. A small amount of the kinetic energy is
converted into potential energy and will be manifest as an increase in pressure. Although the conversion from
mechanical to thermal energy occurs over several pipe diameters it can be represented as a drop in the energy
grade line (EGL) that occurs exactly at the expansion. The drop in the EGL can be measured experimentally
by measuring the pressures at several locations upstream and downstream from the expansion. The hydraulic
grade line (HGL) can then be plotted. The EGL can be obtained by adding the velocity head (based on the
flow rate) to the HGL. In straight sections of tubing the EGL is expected to decreases linearly with distance
and thus the EGL and HGL can be extrapolated to the expansion. The difference between the values of the
EGL obtained from extrapolating downstream to the expansion and upstream to the expansion is the head loss
due to the expansion.
Experimental Apparatus
The test piece (Figure 1-1) 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 needle control valve. The effluent from the test section is
diverted to a 10-cm diameter volumetric detector for flow measurement.
The pressures at the 4 ports along the test piece are monitored with 200-kPa pressure sensors. If necessary
7-kPa sensors can be used at the ports connected to the 8 mm diameter tubing. The pressure sensors will be
configured to measure head in cm. The volumetric detector is monitored with a 7-kPa pressure sensors with a
10-cm diameter volumetric detector
Needle flow control valve
200 kPa pressure sensors
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.75
distance along pipe (m)
0.67
0.33
-5.00
0.22
0.25
15.00
0
0.03
conversion
to
volume in mL
based on the area
of the volumetric
detector.
The
derivative of the
volume
with
respect to time is
the flow rate. If
the flow rate is
relatively constant
the volume vs.
time data can be
fit with a straight
line.
1000
500
0
7 kPa pressure
sensor
Figure 1-1.
Experimental apparatus showing pressure transducers connected
to brass tubing test section.
Experimental Methods
1) Plug the sensors connected to the test section into the top row of channel ports (100 mV max).
2) Plug the sensor connected to the volumetric detector into the middle row of channel ports (20 mV max).
3) Verify that the pressure sensors are connected so that the higher pressure is where the cable leaves the
sensor.
4) Open Easy Data.
5) Open the needle valve approximately ½ turn until water (and no air) comes out of the test section.
6) Close the needle valve.
7) Verify that the tubes connecting the pressure sensors to the ports contain water and if necessary purge the
air by carefully removing the pressure sensor while clamping the tubing between your finger and thumb.
After the pressure sensor is removed allow a small amount of water to discharge into a sponge and then
reclamp and reconnect the pressure sensor. Be very careful to not get the outside of the pressure sensor
wet!
8) Open the needle valve again to ensure that the test section is full of water.
9) Close the needle valve.
10) Set the output of the 5 sensors to zero by clicking on
at the top of the Easy Data window.
11) Enable logging data
and create a new file in the cee 331 folder.
12) Open the brass valve ½ turn. Note that this can be done quickly and doesn’t have to be precise. Observe
the Easy Data graph. After 10 seconds of consistent data open the brass valve an additional ½ turn. Again
wait for 10 seconds of consistent data. Repeat until the valve is completely open.
13) Stop acquiring data and open your data file with Excel to begin analysis.
Data Analysis
To make the analysis easy convert all units to the SI system of meters, kilograms and seconds. If you
don’t know how to use names in a spreadsheet, ask the TA! Use names for the pipe diameters and
any other constants that you use in your spreadsheet. Make sure you set up your spreadsheet so you
can answer problem 11! Before you start constructing your spreadsheet spend a couple of minutes
designing your spreadsheet so it will be neat and easy to create and understand.
1) Plot the output of the sensor connected at 0.22 m vs. time. Use this graph to determine where the changes
in valve setting occurred. Select the times corresponding to the flat sections of the graph and copy the
data (time and 5 pressure sensor values) and paste it into separate worksheets. Each new worksheet will
contain the data corresponding to one flow rate. Make sure each new worksheet is setup identically to
make further analysis easier. Name the sheets exp1..expn where n is the number of different flow rates.
2) Create an analysis worksheet that you will use to analyze all of the data.
3) Enter and name constants that you need (, diameters, g…)
4) In the analysis worksheet calculate the flow rate for each case using the slope function.
5) Calculate the pressures at the 4 ports by taking the averages of the data collected during each valve
setting.
6) Calculate any other parameters you need to plot the HGL and EGL
7) Use the slope and intercept functions (Don’t use linear fit of graphed values!!!) to determine the values
of the HGL and EGL at the location of the expansion.
8) Plot the HGL and EGL for the lowest and highest flows (one graph per flow rate). Verify that the EGL
always slopes down in the direction of flow. (If it doesn’t, immediately obtain a patent for your system.)
9) Calculate the head loss due to the expansion based on the drop in the energy grade line at the expansion
for each case and compare with equation 1.2 or equation 1.3.
10) What is the percentage error (theory-measured)/theory for the head loss for each case?
11) Suppose that the error at the highest flow rate is due to an inaccurate measure of the diameter of the
smaller tube. Use Excel to determine what diameter the smaller tube could be to make the measured head
loss agree with theory.
12) Calculate the loss coefficient, kexp
Vin2
from he  kexp
based on the measured values of head loss and
2g
velocity. Do you agree that kexp is independent of flow rate?
13) Why is the energy grade line decrease (i.e. head loss) between the port immediately upstream from the
expansion and the port immediately downstream from the expansion greater than that predicted by
equation 1.3?
Email your spreadsheet with well formatted graphs and clearly labeled answers to questions 9-13 to
the TA and to the Instructor.
Lab Prep Notes
Table 1-1.
Equipment list.
Description
Supplier
#/group
Omega
Catalog
number
PX26-001DV
Pressure
transducer
Pressure
transducer
Nupro angled
3/8 swage valve
Omega
PX26-030DV
4
B-6JNA
1
H-06490-15
FB-38
1
Rochester
Valve &
Fitting Co.,
INC.
3/8" OD tubing Cole-Parmer
Pressure reducer
ID Booth
1
1) Configure the top row of ports to have a maximum voltage of 100 mV. The middle row of ports should
have a maximum voltage of 20 mV.
2) Set up the physical apparatus and create configuration files for the Easy data software.
3) The volume should be measured in L and the pressures should be in cm of water. The volume
measurement units can be changed by editing the calibration file.
4) Use 3/8” tubing to connect the tap at the sink to the needle valves. Pressure regulators aren’t needed since
the duration of the experiment is so short that it is unlikely that groups will significantly interfere with
each other.