EAS361 LabManual Fall2004 r3

7/28/2019 EAS361 LabManual Fall2004 r3

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Laboratory Manual for EAS 361Engineering Fluid Mechanics

Fall 2004

Department of Mechanical EngineeringPortland State UniversityP.O. Box 751Portland, Oregon 97207www.me.pdx.edu

November 16, 2004


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PrefaceThe Laboratory for EAS 361, Engineering Fluid Mechanics complements the learning experience of the lecture. Laboratory exercises provide opportunities for direct study of uid behavior. All of thelaboratory experiments reinforce material presented during lecture. Some of the experiments willalso expose material that is not presented during lecture. You are responsible for the union of thelaboratory and lecture experience, not their intersection.

Use the laboratory as a chance to enhance your understanding of uid statics and dynamics.The following Learning Objectives for the laboratory will guide you in taking an active role in youreducation.

1. Gain familiarity with physical manifestations of uid mechanics.You will perform experiments dealing with

the basic uid properties: viscosity and pressure; static uid forces; dynamic uid forces; the relation between pressure and velocity in a owing uid.

These experiments will give you rst-hand experience with uid behavior. As a result of performing these experiments you should be able to recognize the effects of uid pressure andviscosity, to relate measurements of pressure to hydrostatic force in a stationary uid, and torelate measurements of pressure to velocity in a moving uid.In addition to learning about uid behavior, you should be able to recognize the physicalequipment in the laboratory and explain the basic operating principles of the equipment. Youshould learn how to operate the equipment properly and safely.

2. Develop and reinforce measurement skills.You should know how to read gages, manometers, owmeters, spring scales, and balance scales.You should be able to time events with a stopwatch. You should strive to measure quantitieswith the maximum precision of the instruments provided in the laboratory.

3. Develop and reinforce skills in documenting observations.You should develop good habits in the organization and recording of raw data in a notebook,and take care to document the data such that it can be analyzed at a later time. You shouldsketch the physical apparatus used in each experiment. In doing so, pay special attentionto the specic mechanical and operational details that enable the apparatus to achieve thepurpose for which it was designed. You should be able to list and describe the steps used toobtain the desired measurements. You should be able to identify whether any actions weretaken to improve the outcome of the experiment. Likewise, you should be able to identify anyactions that may have contributed to undesirable outcomes

4. Develop skills at writing laboratory reports.You will create reports to document your measurements in the laboratory. You will usea writing style and format that is common to technical documentation used in Civil andMechanical Engineering. Your reports should be complete, yet concise. By writing the report,you should develop a clear understanding of the laboratory exercise, and communicate thatunderstanding in your written words.


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ii


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Contents

1 Viscosity Measurement 1

2 Calibration of Pressure Gages 53 Hydrostatic Force on a Submerged Surface 9

4 Impact of a Jet 15

5 Bernoulli Equation 21

6 Flow Meters 27

A Report Writing 31

iii


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Experiment 1

Viscosity Measurement

Purpose

The purpose of this experiment is to measure the viscosity of a glycerin-water mixture with aThomas-Stormer viscometer.

Apparatus

Figure 1.1 is a schematic of the viscometer. A weight, W , is used to drive a rotor that is partiallysubmerged in a sample of liquid. The torque exerted by viscous shear on the rotor is balanced bythe work input of the falling weight. The experiment involves measurement of the time it takes fora known number of revolutions of the rotor.

W

Rotor

Fixedcylinder

Spindle2 r s

r r

V L

Figure 1.1: Thomas-Stormer viscometer.

1


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2 EXPERIMENT 1. VISCOSITY MEASUREMENT

Theory

The operation of the viscometer relies on a linear velocity prole in the gap between the rotor andthe xed cylinder. If the velocity prole is linear the viscous shear stress on the surface of the rotorcan be written

= k (1.1)

where is the viscous shear stress, is the uid viscosity, k is a constant that depends only on thegeometry of the viscometer, and is the angular velocity of the rotor. Given the shear stress fromEquation (1.1), the torque exerted by the rotor on the uid is

T f = ( A)r r (1.2)

where A is the wetted surface area of the rotor, and r r is the radius of the rotor. The area, A,accounts for the inner and outer surfaces of the rotor. Since the uid is in contact with both surfaces

r r is an effective radius.Neglecting any friction in the pulleys and bearings, the power dissipated by viscous stresses in

the uid, P f , is equal to the power input of the falling weight, P w .

P f = P w = T f = WV (1.3)

where W is the magnitude of the weight, and V is the velocity of the falling weight. CombiningEquations (1.1) through (1.3) gives

k 2 A r r = WV (1.4)

The velocity of the weight falling a distance L in time t is

V =L

t=

2 r s n s

t(1.5)

where n s is the number of revolutions of the spindle, and r s is the radius of the spindle. The angularvelocity of the rotor is

=2 n r

t(1.6)

where n r is the number of revolutions of the rotor in time t.Substitution of Equations (1.5) and (1.6) into Equation (1.4) and rearranging yields

=1

k 2 n r

tA r r

W 2 r s n s

t=

r s k A r r

W

(1.7)

where = n s /n r is the overall gear ratio between the spindle and the rotor. Dening the viscometerconstant as

C =r s

k A r r(1.8)

Equation (1.7) can be written

= C W

(1.9)

Assuming that the model of viscous shear in Equation (1.1) is valid, the constant C depends onlyon the geometry of the device.


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3

Procedure

Use of the Thomas-Stormer viscometer requires determination of C in Equation (1.9) by calibratingthe instrument with a uid having a known viscosity. With C known from the calibration step,measurements of W and can be used with Equation (1.9) to compute the viscosity of an unknownuid. The overall procedure may be divided into three phases: (1) setting up the viscometer, (2)adjusting the weight in preparation for the tests, and (3) running the tests.

Set Up the Viscometer

1. Fill the test cup with test uid to 0.25 in (0.6 cm) below the side vanes. Make sure that thedepth of the uid is the same for all tests.

2. Replace the test cup in the viscometer

3. Raise the platform that supports the bath and test cup until the uid is 0.6 cm (0.25 inch)below the top of the rotor. Make sure that the platform is raised so that it touches the stop.

4. Place the thermometer in the holder. Allow the system to come into thermal equilibrium andrecord the temperature.

Adjust Weight in Preparation for Tests

The viscosity measurement is based on the assumption that the ow on the surface of the rotor islaminar. After placing a new sample of liquid in the test cup, and lowering the rotor into position,adjust the weight until 100 revolutions on the counter takes at least 20 seconds. Shorter run timeswill cause turbulent ow, and result in erroneously high viscosity values. The preliminary weightadjustment will also allow you to become familiar with the measurement procedure.

All data runs should be taken over a range of weights no greater than the weight that gives 100

revolutions in no less than 20 seconds. You will also need to allow about 20 revolutions for the rotorto attain steady state velocity. Make sure, therefore, that the weight can fall far enough to causeat least 120 revolutions of the rotor.

Running Tests

Once the instrument is set up, the calibration runs and the viscosity measurement runs use thefollowing procedure.

1. Turn the brake on and raise the driving weight by turning the handle of the rewinding drumcounter-clockwise until the weight nearly touches the pulley.

2. Release the brake one quarter turn and allow the weight to slowly descend until the pointeron the dial is located between 80 and 90. Reset the brake. The time for the weight to fall 100revolutions will be measured. The starting time is when the revolution counter crosses 0, andthe stopping time is when the revolution counter crosses 100. Setting the starting position sothat the revolution counter is between 80 and 90 allows several revolutions of the rotor to becompleted before the beginning of the measured time interval. This guarantees that the rotoris rotating at steady angular velocity during the interval of the time measurement.

3. Reset the stopwatch.

4. Fully release the brake. Start the stopwatch when the counter passes 0 and stop it when thecounter passes 100. Record the time and mass of the weight box.


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4 EXPERIMENT 1. VISCOSITY MEASUREMENT

5. Repeat the preceding steps for several weights (at least 5) by adding or removing shot fromthe weight box.

Data Reduction

Equation (1.9) can be rearranged as

= C W

(1.10)

Data from the calibration runs allows C to be determined from a least squares curve t of versusW/ . Note that many software packages provide the capability for nding curve ts of the form

y = a 0 + a1 x

but Equation (1.10) requires a 0 = 0. If the curve t does not pass through ( W/, ) = (0 , 0) the

value of the slope (and hence C ) will be in error.A simple formula for the least squares t to

y = a 1 x (1.11)

is not hard to derive. Given a set of measured ( x, y ) data pairs the least squares t to Equation (1.11)is

a 1 =x i yix2i

Report

1. Plot the data for the calibration run and report the value of C . Be sure to include the point

(W/, ) = (0 , 0) to verify that the t is reasonable.2. Report the average value for the viscosity of the unknown substance, and compare this to

published values.

3. Discuss discrepancies and anomalies in your data.

4. Which measurements (data points) are most reliable? Which measurements are most limitedby your ability to measure time?

5. How well does the versus W/ data t the model of a line with zero intercept?

6. What other methods are available for measuring the viscosity of liquids?


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Experiment 2

Calibration of Pressure Gages

Purpose

The objective of this experiment is to assess the accuracy of a pressure gage with a dead weightgage tester.

Apparatus

Figure 2.1 is a schematic of a dead weight tester. There are three primary components of thisdevice: a uid that transmits the pressure, a weight and piston used to apply the pressure, andan attachment point for the gage to be calibrated. The weight applies a force over a preciselyknown area, thereby applying a known pressure to the uid. The uid is an oil that is essentiallyincompressible. Since a dead weight tester is relatively compact the effect of elevation changes onthe pressure are negligible. The pressure at the piston face, therefore, is equal to the pressurethroughout the oil in the tester.

Secondary components of the dead weight tester are a reservoir and an adjusting piston. Thereservoir accumulates oil displaced by the the vertical piston during tests when a large range of weights are used for a given gage. The adjusting piston is used to make sure that the vertical pistonis freely oating on the oil.

Procedure

1. Attach the gage to the stem, B.

2. Select a weight and place it on the vertical piston, A.

3. Move the handle of the adjusting piston C to insure that the weight and piston are supportedby oil, not the bottom stop.

4. Spin the vertical piston to insure it is oating freely.

5. Record the gage reading and the weight.

6. Repeat steps 2 through 5 for increasing and decreasing weights for each gage. Be sure to coveras much of the range of the gage that can be achieved with available weights.

5


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6 EXPERIMENT 2. CALIBRATION OF PRESSURE GAGES

reservoir test gage

weight

vertical piston

adjusting piston

A

B

C

valves

Figure 2.1: Dead weight gage tester.

Data Reduction

For each gage tested, draw two curves like those in Figure 2.2 and Figure 2.3. Figure 2.2 is a plot of the pressure indicated on the gage versus the pressure of the oil in the dead weight tester. Figure 2.3is the discrepancy between the pressure gage reading and the pressure applied by the weight on thedead weight tester.

A difference plot like that in Figure 2.3 is a good way to compare two quantities that havenearly the same value. The data in Figure 2.2 suggest that the calibration is quite good, but thereis no indication of the magnitude of the discrepancy. Figure 2.3 clearly shows the magnitude of the discrepancy between the indicated reading of the pressure gage and the dead weight tester.Furthermore, by plotting the calibration data as in Figure 2.3 one sees that the indicated pressuretends to be lower than the calibration standard (more points fall below the line pindicated pdwt = 0).

Report

Briey explain the principle involved in the deadweight gage tester. How is the pressuregenerated? How is it transmitted to the gage? How is the pressure level controlled?

What is the maximum error to be expected when this gage is used to measure pressure? Isthis error more likely to happen at low or high pressures? Is there a range of pressures forwhich the gage gives signicantly more (or less) accurate readings?

Is there any difference in the calibration errors between the data taken in order of increasingpressure, and the data taken in order of decreasing pressure? If so, give a plausible explanationfor this error.

Could you use this apparatus to calibrate a vacuum gage? How?

The dead weight tester is just a standard to which the pressure gages are compared? How doyou imagine the dead weight tester was calibrated?


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7

0 500 1000 1500 20000

200

400

600

800

1000

1200

1400

1600

1800

pdwt

(psi)

p i n d i c

a t e d

( p s

i )

Increasing pressureDecreasing pressure

Figure 2.2: Calibration results for model XYZ gage.

0 500 1000 150010

8

6

4

2

0

2

4

6

8

10

pdwt

(psi)

p i n d i c a

t e d

p d w

t ( p s

i )

Increasing pressureDecreasing pressure

Figure 2.3: Discrepancy between the gage and dead weight tester for model XYZ gage.


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8 EXPERIMENT 2. CALIBRATION OF PRESSURE GAGES


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Experiment 3

Hydrostatic Forceon a Submerged Surface

Purpose

The purpose of this experiment is to experimentally locate the center of pressure of a vertical,submerged, plane surface. The experimental measurement is compared with a theoretical prediction.

Apparatus

Figure 3.1 is a sketch of the device used to measure the center of pressure on a submerged verticalsurface. It consists of an annular sector of solid material attached to a balance beam. When the

device is properly balanced the face of the sector that is not attached to the beam is directly below(coplanar) with the pivot axis. The solid sector and the balance beam is supported above a tank of water.

h B

AF W

L

H

Balanceadjustment

Balance beam

CG

r 1r 2

P Q

T S

O

Figure 3.1: Apparatus for measuring the location of the center of pressure.

9


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10 EXPERIMENT 3. HYDROSTATIC FORCE ON A SUBMERGED SURFACE

h

x

y

y R

Figure 3.2: Detailed nomenclature for locating the center of pressure.

Theory

Figure 3.2 shows the submerged surface viewed from the left side of the tank in Figure 3.1. Thedepth of the centroid below the surface of the water is h. The x-y coordinate system has its originat the centroid. The y-direction position of the center of pressure, yR , is (Munson et al., 2.8)

yR = yc +I xcyc A

(3.1)

where I xc is the moment of inertia of the surface about the x-axis, and A is the surface area.The location of the center of pressure can be measured using the apparatus sketched in Figure 3.1.

The counterweight is adjusted so that the beam is horizontal when there is no water in the tank

and no weight in the pan. When the tank is lled with water the unbalanced hydrostatic forcecauses the beam to tilt. Adding weight W to the pan at a distance L from the pivot O exertsa moment W L that counterbalances the resultant moment due to the hydrostatic forces on thequarter-annulus-shaped body ABPQ .

When the water level is as shown in the gure, there are hydrostatic forces on surfaces AB ,BS and AT . Since BS and AT are concentric cylindrical surfaces with the common axis passingthrough O, the hydrostatic forces on BS and AT do not exert any moment about O. As a resultWL is equal to the moment due to the hydrostatic force F acting on the vertical plane surface AB .

In this experiment the force F is not measured. Instead the theoretical value F = ghA isassumed, where h is the depth of the centroid of the surface. The moment due to F is measuredand the theoretical value of F is used to compute the location of the center of pressure.

Balancing the moments about O gives

W L = F (H + yR )

Substituting F = ghA and solving for yR yields

yR =W L

ghA H (3.2)


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11

Laboratory Procedure

1. Adjust the counterweight so that the balance the beam is horizontal with no water in thetank.

2. Add water up to some level. During the lab you will use at least four water levels. Make suresome water levels leave part of the vertical face exposed.

3. Add weights to the pan to restore the beam to a horizontal position. Record the weight.Measure H .

4. Measure and record h.

5. Return to step 2 and repeat the measurements using at least three other water levels.

Analysis1. Calculate yR from equation (3.1). Call this the theoretical value yR, th .

2. For each water depth, calculate yR from equation (3.2). Call this value the measured valueyR,m .

3. Plot yR, th versus h and yR,m versus h on the same axes.

4. Plot yR, th yR,m versus h.

5. Plot yR, th yc versus h.

The plots created in step 3 and step 4 allow a comparison of the theoretical and measured values

of yR . The plot from step 4 shows the difference between the measured and theoretical values.A difference plot (like that required in step 4) is a good way to compare two quantities that have

nearly the same value. For example, Figure 2.2 and Figure 2.3 in the lab manual for Experiment 2are two plots of the calibration data for a pressure gage. The data in Figure 2.2 suggest that thecalibration is quite good, but there is no indication of the magnitude of the errors. Figure 2.3 clearlyshows the magnitude of the discrepancy between the indicated reading of the pressure gage and thedead weight tester. Furthermore, by plotting the calibration data as in Figure 2.3 one sees that theindicated pressure tends to be lower than the calibration standard (more points fall below the line pindicated pdwt = 0).

Report

1. How does the design of the apparatus enable the resultant force on the vertical surface to bemeasured? Are any signicant forces being neglected? Does the section of the vertical surfacethat is above the water surface contribute any error to the measurement?

2. Compare the experimental and theoretical values of yR and explain any discrepancy. Pick onepoint in the middle of the range of measurements. For that data point, how much of a changein the measured yR would be caused by an error of 10 grams in the weight measurement?

3. What is the primary trend in yR yc versus water depth? Is this consistent with the theorypresented in lecture and in the textbook?


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Reference

B.R. Munson, D.F. Young, and T.H. Okiishi, Fundamentals of Fluid Mechanics , 4th ed., 2002,Wiley and Sons, New York.

Appendix: What About Buoyancy?

Is the buoyancy force being neglected in the analysis of the experimental data? The answer is no.To understand why, we will consider two ways to analyze the experiment. The rst analysis involvesa moment balance that causes the buoyancy force to appear. The second analysis is the same asthat presented in the preceding sections, and no buoyancy force appears.

How can the buoyancy force be made to disappear? Remember that the buoyancy force is dened as the net pressure force acting on a submerged body. If we consider the pressure force components

acting in the horizontal and vertical directions, then the buoyancy force contributes to the momentabout the device pivot.

If instead we consider the pressure forces acting normal to the surface of the acrylic arc, then thebuoyancy force does not appear because the normal forces on the curved surface do not contributea moment about the pivot of the device. This result is due to the design of the experiment. Inother words, the person who designed this device chose the circular arc shape because it allows usto measure the hydrostatic pressure forces without accounting for the buoyancy effect.

Initial Balance

First consider the force balance on device when the apparatus is dry (the tank is empty), and thebalance weight has been properly adjusted. This situation is depicted in Figure 3.3. The balanceweight W c is moved left or right until the moment W c Lc is equal and opposite to the momentW a La . When the tank is lled with water, pressure forces on the surface of the curved acrylic causean additional moment. The moment due to the pressure forces is balanced by adding weights tothe pan shown on the right side of Figure 3.3. Adding water does not affect the moment balanceW c Lc = W a La because the water does not change the weight of the device.

O

W a

L a

W c

L c

Figure 3.3: Moments acting while balance weight is being adjusted and the tank is empty.


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W

L

O

W a

F b

L b

L a

W c

L c

Figure 3.4: Horizontal and vertical forces that create moments.

Moments due to Horizontal and Vertical Forces

Figure 3.4 shows the horizontal and vertical force components acting the acrylic after water is addedto the tank. The horizontal forces are depicted as acting on vertical planes that are the projectionof the curved surface. The contributions to the horizontal pressure forces on the left and rightsides cancel exactly. Thus, by considering the horizontal pressure forces separately from the verticalpressure forces, we see that the net horizontal force must be zero. The horizontal force on the atface of the acrylic does not appear separately because it is already included the balance of horizontal

pressure forces.The vertical forces acting on the top and bottom of the curved surface create a buoyancy forceF b, which acts through the center of buoyancy. The center of buoyancy is the centroid of thatportion of the acrylic that is submerged. The weight of the acrylic W a acts through the center of gravity of the solid material. When the system is analyzed with the forces identied in Figure 3.4,the weight W creates a moment W L that balances the buoyancy force F bLb . The moment W a Lacaused by the weight of the acrylic is still cancelled exactly by the moment from the balance weightW c Lc .

Moments due to Normal Forces

Now consider the moment balance depicted in Figure 3.5. In this view only the force componentsnormal to the surface are identied. No forces are neglected because the pressure force acts normalto the surface. In this particular apparatus it is easier to analyze the normal forces directly than toseparate the forces into horizontal and vertical components.

The local pressure force on the curved surface of the acrylic is not zero. However, the pressureforces on the curved surface do not contribute to moments about O because these forces have linesof action that pass directly through O. In other words, the device is cleverly designed to eliminatethe contributions of all surface forces except the force acting on the vertical surface.

A moment balance about point O shows that the moment F (H + h) is balanced by WL. Thebuoyancy force is not neglected.


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W

L

O

F

H + h

Figure 3.5: Forces normal to curved surface.

Summary of Buoyancy Affect

The analysis of Figure 3.4 givesWL = F bLb (3.3)

and the analysis of Figure 3.5 givesWL = F (H + h). (3.4)

Thus, the weight can be used to measure either the magnitude of the buoyancy force or the magni-tude of the net pressure force on the vertical face of the acrylic. To use Equation (3.3) we need tocompute Lb , which requires locating the center of buoyancy. This is not trivial because the centroidof the submerged region of the acrylic is not a regular geometric shape.


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Experiment 4

Impact of a Jet

Purpose

The purpose of this experiment is to demonstrate and verify the integral momentum equation. Theforce generated by a jet of water deected by an impact surface is measured and compared to themomentum change of the jet.

Apparatus

The experimental apparatus consists of a water nozzle, a set of impact surfaces, a spring scaleconnected to a balance beam, a ow meter, and plumbing for recirculating the water. Figure 4.1is a schematic of these components. The pump draws water from the collection tank and provides

sufficient head for the water to ow through the nozzle and the ow meter. The jet of water fromthe nozzle impinges on the impact surface. The balance beam attached to impact surface allowsmeasurement of the force necessary to deect the water jet.

Theory

A theoretical model for the force necessary to hold the impact surface stationary is obtained byapplying the integral forms of the continuity and momentum equations. The details of the modeldepend on whether or not the uid stream leaving the impact surface is symmetric relative to thevertical axis of the surface.

Symmetric Jet

The geometric and uid parameters for this experiment are identied in the sketch in Figure 4.2. Astream of water with average velocity V ows upward from the nozzle. It impinges on the impactsurface and turns to ow radially outward from the axis of the impact surface.

The control volume, bounded by the dashed lines, is chosen so that it crosses the jet streams atright angles. To proceed with the analysis make the following assumptions

friction between the impact surface and the water jet is negligible

the magnitude of the jet velocity does not change as the jet is turned


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16 EXPERIMENT 4. IMPACT OF A JET

F

L 3

Balanceadjustment

Scaleadjustment

L 2

Pump

Flow meter

Flow control

valveCollection tank

Nozzle

Impact surface

L 1

Figure 4.1: Apparatus used in the jet impact experiment.

velocity proles are uniform where the ow crosses the control surface

the jet exit is circumferentially symmetrical

If any of the impact surfaces used in the experiment cause ows that violate these assumptions, theformulas for reaction forces given below will not match the measured reaction forces.

Applying the conservation of mass to the jet streams gives

V 1 A1 V 2 A2 = 0 (4.1)

where V is the average velocity at a given cross-section, and A is the cross-sectional area normalto the direction of the average velocity. The subscripts 1 and 2 refer to the inlet and outlet of the control volume, respectively. Since the magnitude of the velocity is assumed to not change,Equation(4.1) simplies to

A1 = A2 = A (4.2)

The integral equation for momentum conservation in the x-direction is

F x = CS V x V n dA= Rh = V 2 cos V 2 A2 + ( V 2 )cos V 2 A2 = 0 (4.3)

where Rh is the reaction force in the x-direction necessary to hold the impact surface stationary,and is the angle between the horizontal and the velocity vector of the uid leaving the controlvolume. Equation (4.3) shows that F h = 0 if the ow leaving the impact surface is symmetric aboutthe vertical axis of the impact surface. If there is any disruption to the symmetry, e.g., variationsin V 2 or around the periphery of the exit, Rh will not be zero.


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17

Rv

Rh

V 1

V 2

q

Impact surface

x

y

V 2

Figure 4.2: Nomenclature for control volume analysis of the jet. Ideally, the apparatus and jet aresymmetric about the centerline of the jet.

Applying the y-direction integral momentum equation gives

F y = CS V y V n dA= Rv = V 1 ( V 1 ) A1 + ( V 2 sin )V 2 A2 (4.4)

where F v is the reaction force in the y-direction. Using the simplications A1 = A2 = A andV 1 = V 2 = V , Equation(4.4) reduces to

Rv = mV (1 + sin ) (4.5)

where m = V 1 A1 = V 2 A2 .Equation (4.5) is the theoretical model for predicting the vertical force on the impact surface.

The experimental apparatus is designed to measure F v . A moment balance about the point O inFigure 4.3 yields

F v L2 + F h L1 F s L3 = 0 (4.6)

where F v and F h are the vertical and horizontal forces transmitted from the impact surface to itssupport, and F s is the force measured by the spring balance. Solving this equation for F v allows a

F s

F v

F h

L 1

L 2 L 3

O

Figure 4.3: Moments arising from forces in the jet experiment.


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Rv

Rh

V 1

V 2q

Impact surface

x

y

Figure 4.4: Forces in the jet experiment when the impact surface is not symmetric.

comparison between the theoretical and measured reaction forces caused by the uid jet. Note thatF v = |Rv | and F h = |Rh | .

Asymmetric Jet

Now consider the case depicted in Figure 4.4 where the impact surface is not symmetric about itsvertical axis. The uid stream leaving the surface will cause a nonzero horizontal reaction force.Applying the momentum integral equation in the x direction yields

Rh = ( V 2 cos )V 2 A2

which simplies toRh = mV 2 cos (4.7)

where m = V 2 A2 . Applying the momentum integral equation in the y direction gives

Rv = V 1 ( V 1 )A1 + (V 2 sin )V 2 A2

orRv = m(V 1 V 2 sin ) (4.8)

where m = V 1 A1 = V 2 A2 has been used to simplify the expression.Equations (4.7) and (4.8) can be simplied further if we know the cross sectional area of the

jets entering and leaving the control volume. Without approximation we can use the incompressiblemass conservation relationshipV 1 A1 = V 2 A2 . (4.9)

Using Equation (4.9) to eliminate V 2 from Equations (4.7) and (4.8) gives

Rh = mV 1A1A2

cos (4.10)

Rv = mV 1 1 A1A2

sin (4.11)


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19

Unfortunately, there is no easy way to measure A1 /A 2 . In the absence of additional information,assume A1 /A 2 = 1. Under this assumption, the formulas for the horizontal and vertical reactionforces simplify to

Rh = mV 1 cos (4.12)Rv = mV 1 (1 sin ) (4.13)

Rearranging Equation (4.13) gives

mV 1 =Rh

1 sin and substituting this result into Equation 4.12 and simplifying gives

Rh = Rvcos

1 sin (4.14)

Finally, substituting Equation (4.13) and (4.14) into Equation (4.6) and solving for F v gives

F v = F sL3 /L 2

1 +L1L2

cos 1 sin

(4.15)

This equation is the theoretical model for predicting the vertical force on the impact surface whenthe impact surface is not symmetric about its vertical axis. Note that Equation (4.15) is based onthe assumption that A1 /A 2 = 1. This is consistent with an assumption that the uid velocity doesnot decrease in magnitude as the uid impinges on and leaves the impact surface.

Procedure

In the theoretical calculation of the force on the impact surface it is assumed that the jet exit issymmetric around the impact surface. For the ow to be symmetric the balance beam must behorizontal. Two adjustments are necessary to keep the balance beam horizontal: one on the balancebeam and one on the scale.

Balance adjustment : For each impact surface, adjust the knurled knob on the balance beamso that with no ow and with no load on the scale the balance beam is horizontal. Do not makefurther adjustments to this knob unless the impact surface is changed.

Scale adjustment : The mechanism inside the spring scale stretches as the force on it is in-creased. This causes the balance beam to tip as the ow rate is changed. The new equilibriumorientation of the balance beam is established when the reaction moment from the scale balancesthe moment exerted by the water on the impact nozzle. To get a proper force reading you have toadjust the scale so the balance beam is restored to horizontal. This is achieved by turning the nut

on the rod that passes through the support for the scale.

Step-by-step Instructions

To perform the experiment:

1. Measure the length of the lever arms of the balance beam.

2. Install an impact surface and adjust the balance knob as described above.

3. Set the desired ow rate with the ow control valve.


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4. Adjust the scale as described above so that the balance beam is horizontal.

5. Record the ow rate and the force on the scale.6. Repeat steps 4 through 6 for a total of six ow rates

7. Repeat steps 2 through 7 for at least two different impact surfaces.

Analysis

Analysis of the data for the symmetric impact surface is almost identical to the analysis to theanalysis for the asymmetric impact surface. The only differences are in the formula used to computethe theoretical reaction force and the formula used to compute the reaction force from the measuredspring force. In the following steps, the references to the formulas for the asymmetric impact surfaceare given in parenthesis.

1. For each setting of the control value:

Convert the ow rate to average velocities for the jet. Use Equation (4.5) (or Equation (4.13)) to compute the theoretical reaction force given

V computed from the ow rate measurement. Use Equation (4.6) (or Equation (4.15)) to compute F v from F s and that apparatus

dimensions.

2. On the same axes, plot Rv and F v versus V for each impact surface. (Create a separate plotfor each surface, but compare Rv and F v on each plot.)

3. Plot the discrepancy Rv F v versus V . (Create a separate plot for each surface.)

Report

1. Discuss the trends in reaction forces versus jet velocity. Is the trend consistent with thetheory? Does it make sense?

2. How well does the theoretical model predict the measured force for the symmetric and asym-metric impact surfaces?

3. Do any of the measured values point to weaknesses in the theoretical models?


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Experiment 5

Bernoulli Equation

Purpose

The objective of this experiment is to measure the variation in air velocity along the axis of aduct with variable cross sectional area. The measurements enable experimental verication of theBernoulli Equation.

Apparatus

Figure 5.1 depicts the apparatus used in the experiment. It consists of a blower connected to a ductwith a Venturi. At the exit of the duct is a support for a Pitot probe. The support allows the probeto be positioned at different axial locations in the duct.

The Pitot tube (see Munson et al., 3.5) is a device that enables simultaneous measurement of the stagnation and static pressure of a moving uid. Figure 5.2 is a schematic of the measurementtip of the Pitot tube used in the experiment.

A Pitot tube has an opening that faces upstream. Fluid approaching this opening is assumedto be brought to rest isentropically. Thus, the upstream-facing opening measures the stagnationpressure. The stagnation pressure port is the open end of a continuous tube that extends to apressure tap outside of the ow stream. The stagnation tap is attached to a pressure measuringdevice.

Downstream a short distance from the stagnation port are a series of openings around thecircumference of the probe. These ports sense the static pressure. Inside the probe body, the staticpressure ports are connected to a continuous tube that extends to a pressure tap outside of the owstream. By connecting multiple static pressure ports to a single pressure transmitting duct, theeffects of misalignment of the probe with the local ow direction are minimized. The static pressuretap is attached to a pressure measuring device.

Although the static pressure ports are downstream from the stagnation port, one normallyassumes that the stagnation port and the static ports measure their respective pressures at thesame axial location. More precisely, the assumption is that the static pressure does not changesignicantly as the uid moves the short distance from the stagnation port to the static port(s).For the Pitot probe used in this laboratory exercise, the distance between the stagnation and staticpressure ports is 11/16 inch.

In the laboratory exercise, a Pitot tube is used to measure air velocity in a duct with variablecross sectional area in the ow direction. Because the duct area is not constant, the average uidvelocity varies in the ow direction. The Bernoulli equation shows that changes in velocity along

21


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22 EXPERIMENT 5. BERNOULLI EQUATION

Pitot probe

Venturi

Flowstraightener

Blower

Figure 5.1: Apparatus for verifying the Bernoulli Equation.

a horizontal streamline cause a change in static pressure. Thus, in regions of the duct where thearea is rapidly changing in the ow direction, it is not safe to assume that the static pressure is thesame at the stagnation and static pressure ports. The measurement procedure and data reductionsteps in the laboratory exercise must compensate for the variation in static pressure due to changesin duct area.

On the opposite end of the Pitot probe from the measuring tip are taps that allow the stagnationand total pressure ports to be connected to a pressure measuring device. Not shown in Figure 5.1are three U-tube manometers that are used to measure the static pressure, dynamic pressure andstagnation pressure from the Pitot probe. The manometer measuring dynamic pressure measuresthe difference between the stagnation and static pressure ports. This measurement will indicatethe true dynamic pressure only in sections of the duct where the area does not change in the owdirection.

Theory

The Bernoulli equation applies to steady, incompressible ow along a streamline with no heat orwork interaction. One form of the Bernoulli equation is

p0 = p1 +12

V 21 + z 1 = p2 +12

V 22 + z 2

where p is the pressure, is the density, V is the uid velocity, g is the acceleration of gravity, andz is the elevation measured from an arbitrary datum. The subscripts 1 and 2 denote two positionsalong the streamline. The total or stagnation pressure, p0 , is a measure of total energy of the owingstream. When the Bernoulli equation applies the stagnation pressure is constant along a streamline.

Figure 5.2 is a sketch of the ow eld near the tip of the Pitot probe. The streamline thatterminates at point A is called the stagnation streamline because as the uid approaches A itdecelerates until it has zero velocity. The uid velocity is V on the stagnation streamline far

upstream of A. If we assume that the deceleration is reversible, the pressure tap at point A measuresthe total pressure in the vicinity of the tip. Note that there is no ow through the Pitot tube.The Pitot probe is a small aerodynamic body that does not signicantly disturb the ow eld

except for the stagnation streamline. Although the streamlines curve as the uid passes around thetip there is negligible change in velocity for uid that follow streamlines near the probe. Along thestreamline through point C , for example, the uid velocity is assumed to be constant. In addition,over distances on the order of the Pitot tube diameter, elevation changes are negligible. All thestreamlines in Figure 5.2, therefore, have the same stagnation pressure

p0 ,A = p0 ,C


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23

C

Static pressure taps Stagnation point

Uniform flowat velocity V

B

A

0.688 inch

D

Figure 5.2: Flow eld near the tip of a Pitot probe.

There is a different relationship between points B and C . Since there is no velocity in the directionbetween B and C the static pressure is the same at these points,

pB = pC

In other words the pressure tap at point B measures the static pressure in the vicinity of the probetip. Combining the foregoing equations and assumptions, the pressure difference between point Aand point C is the dynamic pressure

pA pB =12

V 2 (5.1)

Equation (5.1) applies if the uid velocity does not change between points D and C .

Procedure

The Pitot tube is used to measure the variation of stagnation, static, and dynamic pressure alongthe length of the variable area duct. The basic measurement procedure is to record the differentialheights of the uid columns for the three manometers attached to the Pitot tube. The manometerreadings are recorded at a series of probe positions along the duct. Care must be taken in selectingthe probe locations so that the correct dynamic pressure can be obtained from measurements of stagnation and static pressure at two separate probe locations.

Figure 5.3 shows the location of the Pitot probe at two measurement stations located x =11/ 16 = 0 .688 inch apart. The term station refers to the position of the probe in the apparatus. Ateach station, the static ports are 0.688 inch downstream from the stagnation port. If the probe ismoved so that subsequent stations are 0.688 inches apart, then the stagnation port at station i + 1is at the same location as the static pressure port for station i. Therefore, by moving the probe inincrements of 0.688 inch, the correct dynamic pressure can be obtained by subtracting the staticpressure at station i from the stagnation pressure at station i + 1.

Table 5.1 is a suggested layout of a table to record raw data during the experiment. The rstcolumn is a station number. The second column is the axial position of the probe measured from anarbitrary reference point. It makes sense to dene x as the position of the probe tip. The locationof x = 0 is up to you. The last three columns in the table are for recording the height differentialof the three U-tube manometers on the apparatus. Record all three manometer readings for eachposition of the probe.


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ab

V ( x)

c

0.688 0.688

ab

V ( x)

c Station i+1

Station i

Figure 5.3: Using a Pitot probe to measure velocity in a duct with variable area. If the probe ismoved a distance equal to the spacing between the stagnation port and the static port, the dynamicpressure can be measured without introducing an area due to the change in duct area.

Table 5.1: Layout of table to record pressure measurements

Indicated Pressures (inch H 2 O)Station x (in.) static dynamic stagnation

123...


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25

Analysis

The raw data must rst be converted from manometer heights to pressures. The indicated dynamicpressure is the difference between the stagnation and static pressure taps at each probe position.In other words, the indicated dynamic pressure is obtained by converting the readings in the fourthcolumn of Table 5.1 to pressure. The indicated dynamic pressure is then used to compute theindicated velocity.

The corrected dynamic pressure is obtained by subtracting the static pressure at location x (inthe duct) from the stagnation pressure at location x. The corrected dynamic pressure can only becalculated if the probe was moved in increments of 0.688 inch. The corrected dynamic pressure isused to computed the corrected velocity.

Conversion of the raw data yields the variation of station pressure p0 (x), the static pressure p(x), the indicated velocity V i (x), and corrected velocity V c (x) along the centerline of the duct.

1. Plot p0 (x) and p(x) on the same axes.

2. Plot V i (x) and V c (x) on the same axes.

3. Plot V i (x) V c (x).

4. Estimate the ow rate through the duct.

Report

Present and discuss the plots listed in the Analysis section. Answer the following questions.

1. When using a Pitot tube in a duct of constant cross-sectional area the total and static pressureare measured at two streamwise locations. What assumption is being made when the distancebetween these locations is neglected?

2. Suppose pressure gages with a range of 0 to 10 psi were used instead of the manometers. Howwould this affect the accuracy of the measurements?

3. Suppose mercury instead of water was used as the manometer uid. How would this affectthe accuracy of the measurements?

4. Suppose the experiment was performed with an apparatus that was scaled up by a factor of two. Furthermore, suppose that the fan was adjusted so that the pressure readings for all of themanometers in the large apparatus was identical to the pressure readings for the manometersin the experiment you performed. Would the velocity values for the two experiments be thesame or different? What other ow properties would be affected by the change in scale?

ReferenceB.R. Munson, D.F. Young, and T.H. Okiishi, Fundamentals of Fluid Mechanics , 4th ed., 2002,Wiley and Sons, New York.


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Experiment 6

Flow Meters

Purpose

The objective of this experiment is to show how pressure measurements can be used to measureow rates in pipes.

Apparatus

Figure 6.1 is a schematic of the apparatus used to demonstrate obstruction-type ow meters. Theapparatus is a ow loop with a Venturi meter, a sharp-edged orice, a paddle- type ow meter,a pump, and a collection tank. The pressure taps around the ow loop have quick-disconnectconnectors. By switching taps it is possible to measure pressure differentials at different points

around the loop with a single U-tube manometers.

Theory

Obstruction type ow meters work on the principle that changes ow area result in changes in uidvelocity as required by continuity, and the changes in velocity result in changes in uid pressure.The theory of ow meters is discussed in 3.6.3 and 8.6 of Munson et al. The volumetric owrate through an obstruction type ow meter is

Q = CA 2 p(1 4 )where = d/D , d is the diameter of the orice (or throat of the Venturi), and D is the diameter of the pipe.

Procedure

1. Turn on the pump and choose a ow rate.

2. Move the pressure lines connected to the U-tube manometer to the taps on either side of oneof the ow meter (orice or Venturi).

27


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28 EXPERIMENT 6. FLOW METERS

Venturimeter

Sharp-edgedorifice meter

Paddle wheelflow meter

Flow controlvalve

pump

Collectiontank

Pressure taps

Figure 6.1: Flow loop for testing obstruction-type ow meters.

3. Record the pressure drop across the ow meter (orice or Venturi).

4. Repeat steps 2 and 3 for a total of six different ow rates.

5. Repeat steps 2 through 4 for the other ow meter

Analysis

1. Calculate the discharge coefficient, C , for each ow meter at each ow rate.

2. Calculate the Reynolds number for each ow rate.

3. Make a plot of C versus Re on semi-log paper for the orice and the Venturi.

Report

1. Compare the physical appearance of the owmeters with the diagrams of these devices in youruids textbook.

2. Discuss any discrepancies with your results and the discharge coefficients from your uidstextbook.

3. Which ow measurement device in this apparatus is most accurate? Why?


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29

Reference

B.R. Munson, D.F. Young, and T.H. Okiishi, Fundamentals of Fluid Mechanics , 4th ed., 2002,Wiley and Sons, New York.


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30 EXPERIMENT 6. FLOW METERS


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Appendix A

Report Writing

The primary objective of an engineering report is to transmit technical information to individualshaving training comparable to that of the author. The information in the report should be presentedas clearly and concisely as possible, but always with sufficient detail that the methods and data canbe well understood by the reader.

Imagine that you are writing a report that you would want to read. Imagine that you will haveto make an important decision based on the information in the report. This does not mean thatthe report has to be long or elaborate. It does require you to explain the equipment used, theprocedure followed, and to identify and explain the signicant results. In addition to presentingessential information, a good report is well organized and uses a conventional style.

In many situations the reader of a report will not have seen the apparatus or performed theexperiment. The reader needs a description of the experimental apparatus and the procedure usedto make the measurements. The use of line drawings (schematics) is an essential aid to a textdescription of the apparatus.

While preparing your report, ask these questions:

Could someone with your education reproduce your results with the same apparatus? In otherwords, is the description of the apparatus, procedure, and theory complete?

Could someone with your education make make a judgment on the quality and usefulness of the results without having to reproduce the experiment?

Content

EAS 361 Laboratory reports should consist of the following sections:

1. Cover page . The cover page allows quick identication of the report. It should containthe number and title of the experiment, your name and names of lab partners, the date theexperiment was performed, and the lab section (identied by day and group).

2. Introduction . The introduction is where you explain the purpose of the experiment. Give anoverview of the methods used and the expected results. In this class the introduction shouldbe brief. One paragraph should be sufficient.

3. Apparatus . Sketch the equipment used in the experiment and describe the key componentsin words. Use of computer generated drawings are strongly encouraged. All drawings shouldbe labeled according to the Style Conventions discussed below. Additional drawings may

31


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32 APPENDIX A. REPORT WRITING

be necessary to help explain pieces of the apparatus that are referred to in later sections of the report. Specialized equipment should be identied by brand and model number.

4. Procedure . Briey describe the procedure used in the experiment. Do not copy verbatimfrom the instruction sheets. Be sure to describe any special steps needed to achieve goodresults.

5. Theory . Provide a concise listing of equations used to obtain your results from the measure-ments. Equations should appear as in the following page excerpt.

The line of action of the hydrostatic force is through thecenter of pressure. For a vertical plate, the the center of pressure is at yR . The theoretical formula for yR is

yR = yc +I xcyc A

(A.5)

where yc is the depth of the centroid of the plate, I xc isthe moment of inertia about the horizontal axis throughthe centroid, and A is the surface area of the plate.

The equation is centered between the margins. The equation number (which is (A.5) in thiscase) is aligned ush with the right margin. These effects can be obtained by placing acentering tab stop in the center of the page, and a right-aligned tab stop on the right margin.

More important than the appearance of the equation is the documentation of the equation

content. Each symbol used in the equation must be dened in the text of the report. In thepreceding excerpt, notice that yR , yc , I xc , and A are all identied in the sentences precedingor following the equation. The reader should not be left to guess about the meaning of asymbol. The one exception is that universal mathematical constants like or functions likesin() need not be explained (though should be dened).

6. Results . The Results section should contain the reduced data in either graphical or tabularformat. If possible, list known values for comparison. See Style Conventions below for adiscussion of the proper format for graphs and tables. Raw data should appear as part of the Appendix. The Results section should contain a brief narrative that describes what iscontained in each graph and table. A few sentences are usually sufficient. Results sectionslacking a narrative will be ignored.

7. Discussion . State what has been learned from the test and the signicance of the results.Discuss the accuracy of the results. Explain any sources of error. Do the limits of accuracyexplain discrepancies in the data, or is there something missing or wrong with the experiment?If possible give alternate procedures to obtain the same or better results. Answer any questionsgiven on instruction sheets. Did the experiment achieve its objective?

8. References . Always give complete citations for material on other sources. A proper referenceinvolves two components: the citation in the text and the complete bibliographic entry in theReferences section. Consider the following excerpt.


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33

The viscosity of water at 22 C is 9.61 10 4 kg/m/s.

This value was obtained by linear interpolation of thedata in Table B.2 in the book by Munson et. al [2]....

References

1. Department of Mechanical Engineering, PortlandState University, Laboratory Manual for EAS 361Engineering Fluid Mechanics: Fall 2002 , Portland,OR.

2. Munson, B.R., Young, D.F., and Okiishi, T.H.,Fundamentals of Fluid Mechanics , 4th ed., Wiley,New York, 2002.

Notice that the citation in the text uses a number [2] in square brackets. This tells the readerto look at the second entry in the reference section for the complete bibliographic citation.

When in doubt, emulate publications that you have. For example, look at the way citationsare made in your course textbooks. Refer to the ASME style guide at http://www.asme.org/pubs/MS4.html for additional examples.

9. Appendices .

A. Sample Calculation . Give an example of how the reduced data was obtained from theraw data.

B. Raw Data . All the data collected during the experiment should be presented in a neatand clearly readable format.

C. Additional calculations or information supporting arguments made in the report. Oc-casionally it is necessary to make a lengthy justication or mathematical proof of anargument made in the body of the report. For example, you may want to show that thevariation of uid viscosity with temperature could not account for the scatter in the data.That conclusion could be stated in the body of the report, while the quantitative justica-tion, especially if it involves detailed calculations, should be in the appendix. In general,for this class, if the proof takes more than half a page of algebra or computations, itshould be relegated to the appendix.

Style Conventions

Engineering technical reports are structured documents with the contents described in the precedingsection. The style of the lab report should conform to standards for professional communications,and good usage of the English language. The visual appearance should be clean and rather plain.Do not distract the reader with fancy fonts, borders, and cute graphics.

For information on the style suitable for ASME journal publications, see http://www.asme.org/pubs/MS4.html . For questions on usage of English consult a style manual such as Strunk, W.,Jr., and White, E.B., The Elements of Style , 3rd ed., Macmillan, New York, 1979.


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Format

The report should be typed. Use of a word-processor is preferred. All text, Figures and Tablesshould appear on only one side of each sheet of paper. All pages other than the cover sheet shouldhave page numbers that begin with 1 on the rst page after the title page, and should continuethrough the last page of the Appendix.

Handwritten reports, or reports that contain pages of scrap paper or notebook pages with raggededges are unacceptable, and will be returned without grading.

Text

The text of the report should be written in complete sentences. The style should be formal. Thisdoes not mean that you should use vocabulary words that are not part of your speaking vocabulary.Rather, formal style means to avoid slang, cliches, abbreviations that are common in spoken Englishor advertising copy. In short, imitate a engineering textbook, not the writing that appears in popularmagazines.

It is convention that formal reports are written in the third person.

First Person: We repeated the test ve times and computed an average fromthis data.

Third Person: The test was repeated ve times and an average was computedfrom this data.

Use clear, exact prose. Be specic.

Bad: The numbers were close enough considering all the data.

Good: The viscosity values were within 15 percent of the publishedvalues listed in Table 3.

Clear thinking and understanding of the material is a necessary but not sufficient conditionfor good report writing. Consider the quality of your report as evidence that you understood theexperiment.

Figures and Tables

A Figure is any drawing, photograph, or data plot. All Figures and Tables should have a number and a caption . When your laboratory reports are graded, Figures and Tables without captions willbe ignored. All Figures and Tables should lie within the margins of the text of the report.

Table A.1 is an example of a properly formatted table. The caption includes the number of thetable, i.e., A.1, and a brief caption identifying the contents of the table. Units of data in the tableare usually be placed in the column headings. The exception is when an individual column containsquantities with different dimensions. Never put dimensional data in a table without somehowindicating the units. Of course, percent is dimensionless.

Computer-generated graphs are suggested, but not required. Hand drawn graphs should beon graph paper, and to scale. Graphical display of quantitative data must be precise. Never usefreehand sketches to present quantitative data. Axes should be labelled with a symbol or word,and the units of the scale. Multiple curves should be identied with a legend. If colors are used torepresent different data on a graph, make sure that the nal printout is in color. Figure A.1 is anexample of a properly formatted graph.


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35

Table A.1: Summary of viscosity measurements at 20 C . Percent deviation is relative to the

published values.

Viscosity values, kg/m s

Fluid Test Ball Experimental Published PercentDeviation

Glycerin small metal 1 .85 1.50 +23Glycerin large metal 1 .72 1.50 +15Water large metal 0 .75 10 3 1.00 10 3 25Water glass 1 .13 10 3 1.00 10 3 +13

0 50 100 150 200 25010

-5

10-4

10-3

10 -2

T (

C)

( P a

s )

waterair

Figure A.1: Variation of the dynamic viscosity of air and water with temperature.

Word-Processing

The purpose of using a word processor is to increase your productivity, not to produce elaboratereports. Properly used a word processor allows you to (1) type your rough draft directly into thecomputer, (2) easily make editorial revisions, (3) present your report in a neat, easily readableformat and (4) check your spelling. In short, a word-processor should help you write better reports.A word processor should not be used to (1) copy the report of another student, (2) waste timeplaying with multiple fonts and formats, (3) waste paper by printing your report after making smallchanges, (4) write poorly. The GIGO phenomenon, Garbage In, Garbage Out is as common withthe use of word-processors as any computer tool.


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Free Advice

Writing well has nothing to do with word processors, fancy font selection, or plastic report covers.Writing well is hard work. It is about clear communication with a reader who is not able to directlyask you for clarication.

In order to write well you will need to revise your manuscript. In other words, do not simplytype out a report and print it. Rather, sit down with the printed draft and mark it up with a pencil.After you have edited the draft, go back and revise the document in your word processor. Repeatthis write-edit-revise loop at least twice.

There are many paths to a completed manuscript. It is usually a bad idea to start by writing atthe beginning of the report. Instead, you might want to try the following procedure

1. Complete the data reduction. Perform all necessary computations and make all plots andtables of nal results. This information forms the core of your Results section.

2. Write the text of the Results section. Explain the content of each plot and table in words.Guide the readers attention to the most important information. Explain any unexpectedresults. Use equations to support quantitative arguments.

3. Write the Theory section. Include all of the equations necessary to convert your raw data tothe reduced data in the Results section. Because you wrote the Results section rst, you willknow exactly what equations are necessary. Include any additional background equations thathelp to explain the results.

4. Write the Apparatus and Procedure sections. Be sure to include diagrams of the equipment thatare useful in explaining the results. Close up diagrams or alternative views of the equipmentmay be necessary.

5. Write the Introduction section. Because you have already completed the Results , Theory , andApparatus sections, you should have a clear idea of the entire experiment. Now you are in aposition to prepare the reader for what is to follow in the report.

6. Write the Conclusion section.

7. Write the Abstract (if required).

Policy on Collaboration

Writing a laboratory report is an educational experience. Copying the lab report of another studentmeans loosing out on that experience. It also constitutes a misrepresentation of your achievements.

Students working in laboratory groups are expected to communicate with each other aboutperforming the experiments and analyzing the results. Students are expected to turn in reportsthat are substantially their own, independent work.

It is unacceptable to turn in photocopies of any part of another students work.

It is unacceptable to exchange any part of word-processing documents used forlab reports.

It is unacceptable to exchange spreadsheets or computer programs used to analyzedata or prepare graphs for any part of a lab report.


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Visual similarity of the nal report constitutes proof of unacceptable collaboration. Other evidencesuch as duplicate les on computer disks also constitute proof of unacceptable collaboration. Sinceit is usually impossible to determine who is the source and who is the recipient of such unacceptablecopying, grades of zero will be given for lab reports for any and all students who turn in duplicatedwork.

EAS361 LabManual Fall2004 r3

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