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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  

 

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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