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FALL 2015
MIDDLE EAST TECHNICAL UNIVERSITY DEPARTMENT OF MECHANICAL ENGINEERING
ME 305 FLUID MECHANICS I GROUP 01
EXPERIMENT 2 DETERMINATION OF LINEAR MOMENTUM RATE OF AIR
FLOW
PREPARATION: In this course, you will conduct the experiments at the Fluid Mechanics
Laboratory, by yourself, with little help or instruction from the teaching assistants. You must read
the lab sheet thoroughly and understand what you are expected to do (and why) for each
experiment, before coming to the lab. You must use a pen (not a pencil) when recording your
data. Although you are going to perform the experiment as a group, each student will submit a
separate report using the data recorded during the experiment. The report of the experiment is
attached to this manual. You will complete the report and submit it at the end of the lab period.
You will complete the report in 1 hour following the experiment and submit it before leaving the
lab. There cannot be a “group study” in writing the reports – everyone will prepare his/her report
individually using the data recorded during the experiment.
1. OBJECTIVE
The purpose of this experiment is the calculation of the linear momentum rate of air
flowing out of a duct, using the integral formulation of the momentum equation. In the first part
of the experiment, the force with which the air flowing out from the duct impinges on a plate is to
be measured using a measurement plate (scale). In the second part, the velocity distribution at the
exit plane of the duct is to be obtained through Pitot tube measurements and the use of Bernoulli
equation. This velocity distribution can be used to obtain the momentum rate of air. From the
results of this portion of the experiment, the momentum coefficient will also be calculated.
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2. THEORETICAL BACKGROUND
2.1 Momentum Equation in Integral Form
According to Newton’s second law of motion, the net external force acting on a control
volume is equal to the total rate of change of its linear momentum. This can be stated in integral
form with the below equation
d ( . )ˆ dA
At
F V V V n
(1)
In Equation (1), first term on the right hand side represents the local rate of change of linear
momentum in the control volume. The second term on the right hand side represents the net
linear momentum rate of flow crossing the surfaces of the control volume. Under steady state
conditions, the first term on the right hand side of Equation (1) cancels out as
( . )ˆ dA
A F V V n
(2)
2.2 Fluid Jet Impinging Normally on a Flat Plate
Suppose that a jet of air (leaving a duct) strikes a planar surface (a plate) steadily as
illustrated in Figure 1the below figure. Observing Figure 1, Equation (2) can be expressed in
horizontal and vertical directions, for the indicated control volume (which encloses the plate).
The air pressure is atmospheric everywhere therefore there is no net pressure force acting on the
control volume. The fluid weight is negligible but weight of the plate, Wpl, is not. The horizontal
components of the momentum equation cancel out, yielding a zero net force in the horizontal
direction since the air leaves the control volume from all sides at equal linear momentum rates,
exitM (supposing there is perfect symmetry in the flow). Considering the vertical momentum
balance, the air linear momentum rate at the top boundary, AM and the plate weight, Wpl, is
balanced by the reaction force, R’, applied to the planar surface to keep it in place. A portion of
the reaction force R’opposes the plate weight, Wpl. Then, the remaining reaction force
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R = R’ - Wpl balances the air linear momentum rate at the top boundary of the control volume, i.e.
AR M .
Figure 1 Air stream hitting a planar surface
2.3 Measurement of Flow Velocity at a Flow Section Using Pitot Tubes
In the second part of the experiment, the velocity distribution at the duct exit section
(section A in Figure 1) is to be obtained using a series of Pitot tubes.
Consider a thin tube with a right angle bend, as shown in Figure 2, for flow velocity
measurements. Air is coming out of a duct steadily and the tube is positioned to face the air flow.
This tube device is known as a simple Pitot tube. When this Pitot tube is inserted into the flow
with the tube opening directed upstream as shown, the air is pushed into and comes to a complete
stop at the nose of the Pitot tube (a “stagnation” point). The compressed air in the Pitot tube
pushes the manometer fluid and a manometer deflection of h is obtained as the pressure within
the Pitot tube increases. The fluid in front of the tube opening (point O) remains stagnant (at rest).
Suppose the streamline leading to the point O (the stagnation point), passes through a point X at
the exit of the duct.
The Bernoulli equation for the steady, incompressible flow of air with a density of a may
be applied between points X and O along this streamline to yield
planar surface
control volume
section A
R’
Wpl
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Figure 2 Velocity measurement by using a Pitot tube
2
0 02
2 2 x x
a a
p V p V
(3)
where px and p0 are the static pressures at points x and 0, and Vx and V0 are the corresponding
velocities at these two sections. In the above equation, the change in the elevation is not
considered since points x and 0 are close to each other. Likewise, frictional effects are negligible,
too. Observing that point x is exposed to the atmosphere (pressure is atmospheric at x) and point
0 is a stagnation point, then V0 = 0 and px = patm. The velocity at point x is found as
02( )atmxa
p pV
(4)
Neglecting the pressure changes in an air column, for a manometer fluid density of m,
0 y atm mp p p g h (5)
where py is the pressure at point y (shown in Figure 2), h’ = hsin, h is the reading from the
inclined manometer, inclined at an angle of , and m is the manometer fluid density. Thus,
x
air flow
0
streamline
plastic tube
Pitot tube
inclined manometer
h
y
duct
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2 sinmxa
g hV
(6)
If a series of Pitot tubes, connected to individual manometers, are positioned at a flow section
(such as the exit plane of the duct in Figure 2, i.e. section A in Figure 1), the velocity distribution
at that flow section can be obtained using Equation (6). The average velocity V at this flow
section (duct exit) can be obtained by mass-averaging the velocity distribution Vi as
i a i i i ia a a
m V A V AmVA A A A
(7)
where m is the total mass flow rate through the duct across the duct area (total flow area) A and
im is the mass flow rate across the Ai portion of the total flow area, with Ai = A. Vi (the velocity
measured by the Pitot tube) is assumed to be the average velocity on Ai and the density is
constant.
2.4. Momentum Coefficient
The momentum rate at a flow section may change depending on the velocity profile even
if the mass flow rate does not change. Consider Figure 3 where two different velocity profiles
(one uniform, one parabolic) are shown for the same flow section. The mass flow rates are
identical in both profiles. Even though the average velocities are the same in both profiles, the
momentum rates of the flows are not the same.
(a) Nonuniform velocity profile (b) Uniform velocity profile
Figure 3 Nonuniform and uniform velocity profiles of the same mass flow rate at a section
A
V(r) A
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The momentum coefficient is defined for a specific flow profile at a flow section. It
indicates the deviation of the momentum rate associated with this velocity profile, from the
momentum rate of a uniform flow of the same mass flow rate, and is defined as
2
2A
V dA
V A
(8)
Note that = 1 for uniform flow and > 1 for nonuniform flow.
3. EXPERIMENTAL SET-UP
The sketch of the experimental set-up is shown in Figure 4. There are three major
components in this set-up. The first is the air duct that discharges air created by the operation of a
fan (air blower). The fan is not shown in the sketch; however, the duct is connected to the outlet
of the fan and by starting the fan an air flow is created in the duct. The second component of the
experimental set-up is the measurement plate (scale) and the table on which it lies (Figure 4.a).
This component will be used to measure the force with which the air jet (leaving the duct)
impinges on a horizontal plate. Air is directed through the duct onto the top plate, so that it
impinges on the measurement plate. The load cell under the measurement plate (not visible in the
figure) reads the “force” on the plate and displays it on the reading screen. The momentum rate of
this air flow is to be obtained during this experiment. The third component is the Pitot tube-
manometer apparatus that will be used to obtain the air velocity profile at the exit of the duct, in
order to obtain the momentum rate (Figure 4.b). In the first part of the experiment, the
measurement plate will be positioned below the duct (Figure 4.a). Once data is taken, the table
will be moved aside and the Pitot tube apparatus will be brought and positioned below (Figure
4.b) to obtain the air velocity profile at the duct exit. Note that Figure 4 is a schematic and does
not show the actual number of Pitot tubes/manometers or their positions.
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(a) (b)
Figure 4 Sketch of the experimental setup
4 EXPERIMENTAL PROCEDURE
Part 1:
The photographs of the set-up in the laboratory for this part of the experiment are shown
in Figure 5.
a) With the help of your teaching assistant, run the air blower (fan) and obtain a steady
air flow. The fan is behind the wall against which the duct is mounted. Note that you will need to
wait for a while for the flow to reach steady-state. The space below the duct must be clear when
you start the fan.
measurement plate (scale)
air
air source
reading screen
top plate
bottom plate
duct
Pitot tube rack
open to atmosphere
air source
manometer fluid supply
inclined manometer
rack
inclination angle,
duct
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Figure 5 Experimental set-up for creating air flow in a duct and for force measurement
b) Plug the electric cord of the measurement plate to the wall outlet. (there are several
behind the duct) Numbers will appear on the reading screen. Wait until there is no longer a
change in the numbers. Then, press the “ O ” button to reset the value on the screen to zero.
This way, the weight of the measurement plate is zeroed so that the measurements to be taken
during the flow will only reflect the effect of air momentum rate.
c) Move the table that carries the measurement plate under the duct air outlet. Air exit
area must be centered with respect to the opening on the top plate. To help you to position the
table properly, markers have been placed on the floor for the four wheels of the table. Study the
positioning of the wheels in Figure 5. Your lab supervisor will help you with the positioning.
Behind the wall…
duct
air coming into the duct from behind
air flow
(to the duct)
fan
air flow
air flow
Reading screen
duct
measurement plate
top plate
bottom plate
marker
reading screen
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d) Read and note the value in the reading screen. Don’t forget that this value is in [kg].
There might be fluctuations in the reading. Record an average value as best as you can, on the
data sheet.
e) Unplug the electric cord and move the table aside, clearing the space below the exit
of the duct for the next portion of the experiment. Do not turn off the air blower.
Part 2:
The photographs of the set-up in the laboratory for this part of the experiment are shown
in Figure 6.
Figure 6. Experimental set-up for obtaining the air velocity profile at the exit of the duct
1 2 3 4 5 6 7 8 9 10 11 12
manometer fluid supply
Pitot tubes
protractor
manometer rack
duct
Pitot tube rake
manometer rack
markers to position the Pitot tube rack
plastic tubing manometers open
to atmosphere
manometers connected to the 12
Pitot tubes
to Pitot tubes
manometers open to atmosphere
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a) Record the ambient temperature and pressure on the data sheet. Your teaching
assistant will direct you to the thermometer & barometer. You will use these values to calculate
the density of air using ideal gas law.
b) Move the Pitot tube rake under the duct by using markers. Your lab supervisor will
help you with the positioning of the rake.
c) Read the inclination angle, of the manometer rack from the protractor and record
on the data sheet.
d) Record the deflection, h in each manometer connected to the respective Pitot tube
facing the exit flow, on the data sheet. Note that there are a total of 12 Pitot tubes, numbered as
shown in Figure 6. These Pitot tubes are connected to the respective manometers, also numbered
as shown. The manometer rack has several manometers open to atmosphere as shown (not
connected to Pitot tubes). The deflection you read for the Pitot tubes should be the relative to the
deflection in a manometer open to atmosphere.
e) Once you have finished recording the deflections, stop the air blower with the help of
your teaching assistant. Move aside the Pitot tube rake so that you clear the space below the duct
exit.
5. CALCULATIONS
Part 1
Convert the measurement plate reading into [N]. State what this reading means
physically, in the discussion part of your report.
Part 2
a) Assuming that the air density remains constant under the experimental conditions
with a gas constant of 287 J/kg K, calculate the air density.
b) Calculate the velocity, Vei at each Pitot tube measurement point using equation (6).
The manometer fluid is alcohol and its density is given in the data sheet.
c) Determine the linear momentum rate at the plane under the exit of the duct, using the
calculated velocities, the air density and the area segments. Assume that each calculated velocity
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is uniform on its corresponding area segment. In order to find the momentum rate across the duct
exit, first determine the momentum rates across each area segment and then sum them.
d) Determine the average flow velocity on the exit plane of the duct from Equation (7).
e) Determine the momentum coefficient for this flow from Equation (8).
6. DISCUSSION
At the end of your report, you will add a Discussion section (label it “3. Discussion of
Results”) on a separate sheet of paper in which you will discuss your results. Specifically you
should answer the below questions.
(i) Explain the physical significance of the force obtained from the measurement plate
in the first part of the experiment.
(ii) Compare the force measured on the measurement plate in the first part of the
experiment and the linear momentum rate at the exit plane of the duct obtained in the second
part of the experiment. Are they the same? Should they be the same? If they are/are not the
same, why/why not?
(iii) Comment on the significance of the momentum coefficient value found in this
experiment
(iv) Why do you think inclined manometers were used for velocity measurements
(rather than vertical manometers)?
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NAME : LABORATORY GROUP:
ID NUMBER: DATE:
ME 305 FLUID MECHANICS I
EXPERIMENT 2: DETERMINATION OF LINEAR MOMENTUM RATE OF AIR FLOW
EXPERIMENT REPORT
1. DATA & RESULTS
Part 2 Ambient temperature (οC)
Ambient pressure (Pa) Gas constant of air 287 J/kgK Air density (kg/m3)
Density of alcohol (manometer fluid) 810 kg/m3 Manometer inclination, (º)
h (cm of alcohol) Vei (m / s) Aei (m2) eiM (linear momentum rate) 1 0.00105 2 0.00105 3 0.00105 4 0.00105 5 0.00105 6 0.00105 7 0.00105 8 0.00105 9 0.00105 10 0.00105 11 0.00105 12 0.00105
:
Average duct exit velocity (m/s)
Momentum Coefficient
Part 1
Measurement Plate Reading (kg)
Force on the plate (N)
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2. CALCULATIONS 2.1. Force on the measurement plate 2.2. Air density 2.3. Sample calculation of the velocity measured by Pitot tube 5 2.4. Sample calculation of the linear momentum rate across the area segment 5 on the duct exit section 2.5. Average velocity of air flow at the duct exit 2.6. Momentum coefficient