A Manual for the

M E C H A N I C So f

FLUIDS LABORATORY

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William S. JannaDepartment of Mechanical Engineering

The University of Memphis

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©2012 William S. Janna

All Rights Reserved.No part of this manual may be reproduced, stored in a retrieval

system, or transcribed in any form or by any means—electronic, magnetic,mechanical, photocopying, recording, or otherwise—

without the prior written consent of William S. Janna.

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TABLE OF CONTENTS

Item Page

Course Learning Outcomes, Cleanliness and Safety ................................................ 4Code of Student Conduct ............................................................................................... 5Statistical Treatment of Experimental......................................................................... 6Report Writing ............................................................................................................... 16Experiment 1 Density and Surface Tension................................................... 18Experiment 2 Viscosity ....................................................................................... 20Experiment 3 Center of Pressure on a Submerged Plane Surface ............. 21Experiment 4 Impact of a Jet of Water ............................................................ 23Experiment 5 Critical Reynolds Number in Pipe Flow............................... 26Experiment 6 Fluid Meters ................................................................................ 28Experiment 7 Pipe Flow ..................................................................................... 32Experiment 8 Pressure Distribution About a Circular Cylinder................ 34Experiment 9 Drag Force Determination ....................................................... 37Experiment 10 Analysis of an Airfoil................................................................ 38Experiment 11 Open Channel Flow—Sluice Gate ......................................... 40Experiment 12 Open Channel Flow Over a Weir .......................................... 42Experiment 13 Open Channel Flow—Hydraulic Jump ................................ 44Experiment 14 Measurement of Pump Performance .................................... 46Experiment 15 Measurement of Velocity and Calibration of

a Meter for Compressible Flow ............................. 50Experiment 16 Measurement of Fan Horsepower ......................................... 55Experiment 17 External Laminar Flows Over Immersed Bodies................ 57Experiment 18 Series-Parallel Pump Performance ........................................ 59Experiment 19 Design of Experiments: Calibration of an Elbow Meter..... 63Experiment 20 Design of Experiments: Measurement of Force on a

Conical Object ........................................................... 65Appendix ......................................................................................................................... 67

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Course Learning OutcomesThe Fluid Mechanics Laboratory experiments areset up so that experiments can be performed tocomplement the theoretical information taughtin the fluid mechanics lecture course. Thus topicalareas have been identified and labeled as CourseLearning Outcomes (CLOs). The CLOs in theMECH 3335 Laboratory are as follows:

TABLE 1. Course Learning Outcomes

1. Identify safe operating practices andrequirements for laboratory experiments

2. Measure fluid properties3. Measure hydrostatic forces on a submerged

body4. Use flow meters to measure flow rate in a

pipe5. Measure pressure loss due to friction for pipe

flow6. Measure drag/lift forces on objects in a flow,

or measure flow rate over a weir7. Design and conduct an experiment, as well as

analyze and interpret data8. Function effectively as a member of a team

CleanlinessThere are “housekeeping” rules that the user

of the laboratory should be aware of and abideby. Equipment in the lab is delicate and eachpiece is used extensively for 2 or 3 weeks persemester. During the remaining time, eachapparatus just sits there, literally collecting dust.University housekeeping staff are not required toclean and maintain the equipment. Instead, thereare college technicians who will work on theequipment when it needs repair, and when theyare notified that a piece of equipment needsattention. It is important, however, that theequipment stay clean, so that dust will notaccumulate too heavily.

The Fluid Mechanics Laboratory containsequipment that uses water or air as the workingfluid. In some cases, performing an experimentwill inevitably allow water to get on theequipment and/or the floor. If no one cleaned uptheir working area after performing anexperiment, the lab would not be a comfortable orsafe place to work in. No student appreciateswalking up to and working with a piece ofequipment that another student or group ofstudents has left in a mess.

Consequently, students are required to cleanup their area at the conclusion of the performanceof an experiment. Cleanup will include removal

of spilled water (or any liquid), and wiping thetable top on which the equipment is mounted (ifappropriate). The lab should always be as cleanor cleaner than it was when you entered. Cleaningthe lab is your responsibility as a user of theequipment. This is an act of courtesy that studentswho follow you will appreciate, and that youwill appreciate when you work with theequipment.

SafetyThe layout of the equipment and storage

cabinets in the Fluid Mechanics Lab involvesresolving a variety of conflicting problems. Theseinclude traffic flow, emergency facilities,environmental safeguards, exit door locations,unused equipment stored in the lab, etc. The goalis to implement safety requirements withoutimpeding egress, but still allowing adequate workspace and necessary informal communicationopportunities.

Distance between adjacent pieces ofequipment is determined by locations of watersupply valves, floor drains, electrical outlets,and by the need to allow enough space around theapparatus of interest. Immediate access to theSafety Cabinet and the Fire Extinguisher is alsoconsidered. We do not work with hazardousmaterials and safety facilities such as showers,eye wash fountains, spill kits, fire blankets, etc.,are not necessary.

Safety Procedures. There are five exit doors inthis lab, two of which lead to other labs. Oneexit has a double door and leads directly to thehallway on the first floor of the EngineeringBuilding. Another exit is a single door that alsoleads to the hallway. The fifth exit leadsdirectly outside to the parking lot. In case of fire,the doors to the hallway should be closed, andthe lab should be exited to the parking lot.

There is a safety cabinet attached to thewall of the lab adjacent to the double doors. Incase of personal injury, the appropriate itemshould be taken from the supply cabinet and usedin the recommended fashion. If the injury isserious enough to require professional medicalattention, the student(s) should contact the CivilEngineering Department in EN 104, Extension2746.

Every effort has been made to create apositive, clean, safety conscious atmosphere.Students are encouraged to handle equipmentsafely and to be aware of, and avoid beingvictims of, hazardous situations.

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THE CODE OF STUDENT CONDUCTTaken from The University of Memphis

1998–1999 Student Handbook

Institution Policy StatementThe University of Memphis students are citizensof the state, local, and national governments, andof the academic community. They are, therefore,expected to conduct themselves as law abidingmembers of each community at all times.Admission to the University carries with itspecial privileges and imposes specialresponsibilities apart from those rights andduties enjoyed by non-students. In recognition ofthis special relationship that exists between theinstitution and the academic community which itseeks to serve, the Tennessee Board of Regentshas, as a matter of public record, instructed “thepresidents of the universities and colleges underits jurisdiction to take such action as may benecessary to maintain campus conditions…and topreserve the integrity of the institution and itseducational environment.”

The following regulations (known as the Codeof Student Conduct) have been developed by acommittee made up of faculty, students, and staffutilizing input from all facets of the UniversityCommunity in order to provide a secure andstimulating atmosphere in which individual andacademic pursuits may flourish. Students are,however, subject to all national, state and locallaws and ordinances. If a student’s violation ofsuch laws or ordinances also adversely affects theUniversity’s pursuit of its educational objectives,the University may enforce its own regulationsregardless of any proceeding instituted by otherauthorities. Additionally, violations of anysection of the Code may subject a student todisciplinary measures by the University whetheror not such conduct is simultaneously violative ofstate, local or national laws.

The term “academic misconduct” includes, butis not limited to, all acts of cheating andplagiarism.

The term “cheating” includes, but is not limitedto:a. use of any unauthorized assistance in taking

quizzes, tests, or examinations;

b. dependence upon the aid of sources beyondthose authorized by the instructor in writingpapers, preparing reports, solving problems,or carrying out other assignments;

c. the acquisition, without permission, of testsor other academic material before such

material is revealed or distributed by theinstructor;

d. the misrepresentation of papers, reports,assignments or other materials as the productof a student’s sole independent effort, for thepurpose of affecting the student’s grade,credit, or status in the University;

e. failing to abide by the instructions of theproctor concerning test-taking procedures;examples include, but are not limited to,talking, laughing, failure to take a seatassignment, failing to adhere to starting andstopping times, or other disruptive activity;

f . influencing, or attempting to influence, anyUniversity official, faculty member,graduate student or employee possessingacademic grading and/or evaluationauthority or responsibility for maintenance ofacademic records, through the use of bribery,threats, or any other means or coercion inorder to affect a student’s grade orevaluation;

g. any forgery, alteration, unauthorizedpossession, or misuse of University documentspertaining to academic records, including, butnot limited to, late or retroactive change ofcourse application forms (otherwise known as“drop slips”) and late or retroactivewithdrawal application forms. Alteration ormisuse of University documents pertaining toacademic records by means of computerresources or other equipment is also includedwithin this definition of “cheating.”

The term “plagiarism” includes, but is not limitedto, the use, by paraphrase or direct quotation, ofthe published or unpublished work of anotherperson without full or clear acknowledgment. Italso includes the unacknowledged use ofmaterials prepared by another person or agencyengaged in the selling of term papers or otheracademic materials.

Course PolicyAcademic misconduct (acts of cheating and ofplagiarism) will not be tolerated. The StudentHandbook is quite specific regarding the course ofaction to be taken by an instructor in cases whereacademic misconduct may be an issue.

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Statistical Treatment of Experimental Data

IntroductionThis laboratory course concerns making

measurements in various fluid situations andgeometries, and relating results of thosemeasurements to derived equations. The objectiveis to determine how well the derived equationsdescribe the physical phenomena we aremodeling. In doing so, we will need to makephysical measurements, and it is essential thatwe learn how to practice good techniques inmaking scientific observations and in obtainingmeasurements. We are making quantitativeestimates of physical phenomena undercontrolled conditions.

MeasurementsThere are certain primary desirable

characteristics involved when making thesephysical measurements. We wish that ourmeasurements would be:

a ) Observer-independent,b) Consistent, andc) Quantitative

So when reporting a measurements, we will bestating a number. Furthermore, we will have toadd a dimension because a physical valuewithout a unit has no significance. In reportingmeasurements, a question arises as to how shouldwe report data; i.e., how many significant digitsshould we include? Which physical quantity isassociated with the measurement, and howprecise should it or could it be? It is prudent toscrutinize the claimed or implied accuracy of ameasurement.

Performing experimentsIn the course of performing an experiment, we

first would develop a set of questions or ahypothesis, or put forth the theory. We thenidentify the system variables to be measured orcontrolled. The apparatus would have to bedeveloped and the equipment set up in aparticular way. An experimental protocol, orprocedure, is established and data are taken.

Several features of this process areimportant. We want accuracy in ourmeasurements, but increased accuracy generallycorresponds to an increase in cost. We want theexperiments to be reproducible, and we seek tominimize errors. Of course we want to address allsafety issues and regulations.

After we run the experiment, and obtain data,we would analyze the results, draw conclusions,and report the results.

Comments on Performing Experiments• Keep in mind the fundamental state of

questions or hypotheses.• Make sure the experiment design will answer

the right questions.• Use estimation as a reality check, but do not

let it affect objectivity.• Consider all possible safety issues.• Design for repeatability and the appropriate

level of accuracy.

Error & Uncertainty—DefinitionsThe fluid mechanics laboratory is designed to

provide the students with experiments thatverify the descriptive equations we derive tomodel physical phenomena. The laboratoryexperience involves making measurements ofdepth, area, and flow rate among other things. Inthe following paragraphs, we will examine ourmeasurement methods and define terms thatapply. These terms include error, uncertainty,accuracy, and precision.

Error. The error E is the difference between aTRUE value, x, and a MEASURED value, xi:

E x xi= − (1)

There is no error-free measurement. Allmeasurements contain some error. How error isdefined and used is important. The significance ofa measurement cannot be judged unless theassociated error has been reliably estimated. InEquation 1, because the true value of x is unknown,the error E is unknown as well. This is always thecase.

The best we can hope for is to obtain theestimate of a likely error, which is called anuncertainty. For multiple measurements of thesame quantity, a mean value, x , (also called anominal value) can be calculated. Hence, theerror becomes:

E x x= −

However, because x is unknown, E is stillunknown.

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Uncertainty. The uncertainty, ∆x, is an estimateof E as a possible range of errors:

∆x E≈ (2)

For example, suppose we measure a velocity andreport the result as

V = 110 m/s ± 5 m/s

The value of ± 5 m/s is defined as the uncertainty.Alternatively, suppose we report the results as

V = 110 m/s ± 4.5%

The value of ± 4.5% is defined as the relativeuncertainty. It is common to hear someone speakof “experimental errors,” when the correctterminology should be “uncertainty.” Both termsare used in everyday language, but it should beremembered that the uncertainty is defined as anestimate of errors.

Accuracy. Accuracy is a measure (or an estimate)of the maximum deviation of measured values, xi,from the TRUE value, x:

accuracy estimate of x xi= −max (3)

Again, because the true value x is unknown, thenthe value of the maximum deviation is unknown.The accuracy, then, is only an estimate of theworst error. It is usually expressed as apercentage; e.g., “accurate to within 5%.”

Accuracy and Precision. As mentioned, accuracy isa measure (or an estimate) of the maximumdeviation of measured values from the true value.So a question like:

“Are the measured values accurate?”

can be reformulated as

“Are the measured values close to the truevalue?”

Accuracy was defined in Equation 3 as

accuracy estimate of x xi= −max (3)

Precision, on the other hand, is a measure (or anestimate) of the consistency (or repeatability).Thus it is the maximum deviation of a reading(measurement), xi, from its mean value, x :

precision estimate of x xi= −max (4)

Note the difference between accuracy andprecision.

Regarding the definition of precision, there isno true value identified, only the mean value (oraverage) of a number of repeated measurements ofthe same quantity. Precision is a characteristic ofthe measurement. In everyday language we oftenconclude that “accuracy” and “precision” are thesame, but in error analysis there is a difference.So a question like:

“Are the measured values precise?”

can be reformulated as

“Are the measured values close to eachother?”

As an illustration of the concepts of accuracy andprecision, consider the dart board shown in theaccompanying figures. Let us assume that the bluedarts show the measurements taken, and that thebullseye represents the value to be measured.When all measurements are clustered about thebullseye, then we have very accurate and,therefore, precise results (Figure 1a).

When all measurements are clusteredtogether but not near the bullseye, then we havevery precise but not accurate results (Figure 1b).

When all measurements are not clusteredtogether and not near the bullseye, but theirnominal value or average is the bullseye, then wehave accurate (on average) but not precise results(Figure 1c).

When all measurements are not clusteredtogether and not near the bullseye, and theiraverage is the not at the bullseye, then we haveneither accurate nor precise results (Figure 1d).

We conclude that accuracy refers to thecorrectness of the measurements, while precisionrefers to their consistency.

Classification of ErrorsRandom error. A random error is one that arisesfrom a random source. Suppose for example that ameasurement is made many thousands of timesusing different instruments and/or observersand/or samples. We would expect to have randomerrors affecting the measurement in eitherdirection (±) roughly the same number of times.Such errors can occur in any scenario:• Electrical noise in a circuit generally produces

a voltage error that may be positive ornegative by a small amount.

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2173

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FIGURE 1a. Accurate and Precise

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FIGURE 1b. Precise but not Accurate.

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FIGURE 1c. Precise but not Accurate.

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FIGURE 1d. Neither Precise nor Accurate.

• By counting the total number of pennies in alarge container, one may occasionally pick uptwo and count only one (or vice versa).

The question arises as to how can we reducerandom errors? There are no random error freemeasurements. So random errors cannot beeliminated, but their magnitude can be reduced.On average, random errors tend to cancel out.

Systematic Error. A systematic error is one that isconsistent; that is, it happens systematically.Typically, human components of measurementsystems are often responsible for systematicerrors. For example, systematic errors are commonin reading of a pressure indicated by an inclinedmanometer.

Consider an experiment involving dropping aball from a given height. We wish to measure thetime it takes for the ball to move from where it isdropped to when it hits the ground. We mightrepeat this experiment several times. However,the person using the stopwatch may consistentlyhave a tendency to wait until the ball bouncesbefore the watch is stopped. As a result, the timemeasurement might be systematically too long.

Systematic measurements can be anticipatedand/or measured, and then corrected. This can bedone even after the measurements are made.

The question arises as to how can we reducesystematic errors? This can be done in severalways:

1. Calibrate the instruments being used bychecking with a known standard. Thestandard can be what is referred to as:

a) a primary standard obtained from the“National Institute of standards andtechnology” (NIST— formerly the NationalBureau of Standards); or

b) a secondary standard (with a higheraccuracy instrument); or

c) A known input source.

2. Make several measurements of a certainquantity under varying test conditions, suchas different observers and/or samples and/orinstruments.

3. Check the apparatus.

4. Check the effects of external conditions

5. Check the coherence of results.

A repeatability test using the same instrument isone way of gaining confidence, but a far more

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reliable way is to use an entirely differentmethod to measure the desired quantity.

Uncertainty AnalysisDetermining Uncertainty. When we state ameasurement that we have taken, we should alsostate an estimate of the error, or the uncertainty.As a rule of thumb, we use a 95% relativeuncertainty, or stated otherwise, we use a 95%confidence interval.

Suppose for example, that we report theheight of a desk to be 38 inches ± 1 inch. Thissuggests that we are 95% sure that the desk isbetween 37 and 39 inches tall.

When reporting relative uncertainty, wegenerally restrict the result to having one or twosignificant figures. When reporting uncertainty ina measurement using units, we use the samenumber of significant figures as the measuredvalue. Examples are shown in Table 1:

TABLE 1. Examples of relative and absoluteuncertainty.

Relative uncertainty Uncertainty in units3.45 cm ± 8.5% 5.23 cm ± 0.143 cm6.4 N ± 2.0% 2.5 m/s ± 0.082 m/s

2.3 psi ± 0.1900% 9.25 in ± 0.2 in9.2 m/s ± 8.598% 3.2 N ± 0.1873 N

The previous tables shows uncertainty inmeasurements, but to determine uncertainty isusually difficult. However, because we are usinga 95% confidence interval, we can obtain anestimage. The estimate of uncertainty depends onthe measurement type: single samplemeasurements, measurements of dependentvariables, or multi variable measurements.

Single-sample measurements. Single-samplemeasurements are those in which theuncertainties cannot be reduced by repetition. Aslong as the test conditions are the same (i.e., samesample, same instrument and same observer), themeasurements (for fixed variables) are single-sample measurements, regardless of how manytimes the reading is repeated.

Single-sample uncertainty. It is often simple toidentify the uncertainty of an individualmeasurement. It is necessary to consider the limitof the “scale readability,” and the limitassociated with applying the measurement toolto the case of interest.

Measurement Of Function Of More Than OneIndependent Variables. In many cases, several

different quantities are measured in order tocalculate another quantity—a dependentvariable. For example, the measurement of thesurface area of a rectangle is calculated usingboth its measured length and its measured width.Such a situation involves a propagation ofuncertainties.

Consider some measuring device that has asits smallest scale division δx. The smallest scaledivision limits our ability to measure somethingwith any more accuracy than δx/2. The ruler ofFigure 2a, as an example, has 1/4 inch as itssmallest scale division. The diameter of thecircle is between 4 and 4 1/4 inches. So we wouldcorrectly report that

D = 41/8 ± 1/8 in.

This is the correct reported measurement forFigure 2a and Figure 2b, even though the circlesare of different diameters. We can “guesstimate”the correct measurement, but we cannot reportsomething more accurately than our measuringapparatus will display. This does not mean thatthe two circles have the same diameter, merelythat we cannot measure the diameters with agreater accuracy than the ruler we use will allow.

0 1 2 3 4 5 6

(a )

0 1 2 3 4 5 6

(b)

FIGURE 2. A ruler used to measure the diameterof a circle.

The ruler depicted in the figure could be anyarbitrary instrument with finite resolution. Theuncertainty due to the resolution of anyinstrument is one half of the smallest increment

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displayed. This is the most likely single sampleuncertainty. It is also the most optimistic becausereporting this values assumes that all othersources of uncertainty have been removed.

Multi-Sample Measurements. Multi-samplemeasurements involve a significant number ofdata points collected from enough experiments sothat the reliability of the results can be assuredby a statistical analysis.

In other words, the measurement of asignificant number of data points of the samequantity (for fixed system variables) undervarying test conditions (i.e., different samplesand/or different instruments) will allow theuncertainties to be reduced by the sheer number ofobservations.

Uncertainty In Measurement of a Function ofIndependent Variables. The concern in thismeasurement is in the propagation ofuncertainties. In most experiments, severalquantities are measured in order to calculate adesired quantity. For example, to estimate thegravitational constant by dropping a ball from aknown height, the approximate equation wouldbe:

g

Lt

= 22

(5)

Now suppose we measured: L = 50.00 ± 0.01 m andt = 3.1 ± 0.5 s. Based on the equation, we have:

g

Lt

= = × =2 2 50 003 1

10 42 2

2..

. m/s

We now wish to estimate the uncertainty ∆g inour calculation of g. Obviously, the uncertainty∆g will depend on the uncertainties in themeasurements of L and t. Let us examine the“worst cases.” These may be calculated as:

gmin

..

.= × =2 49 993 6

7 72

2m/s

and

gmax

..

.= × =2 50 012 6

14 82

2m/s

The confidence interval around g then is:

7 7 14 82 2. .m/s m/s≤ ≤g (6)

Now it is unlikely for all single-sampleuncertainties in a system to simultaneously be the

worst possible. Some average or “norm” of theuncertainties must instead be used in estimating acombined uncertainty for the calculation of g.

Uncertainty In Multi-Sample Measurements.When a set of readings is taken in which thevalues vary slightly from each other, theexperimenter is usually concerned with the meanof all readings. If each reading is denoted by xiand there are n readings, then the arithmeticmean value is given by:

x

x

n

ii

n

=∑=1 (7)

Deviation. The deviation of each reading isdefined by:

d x xi i= − (8)

The arithmetic mean deviation is defined as:

d

ndi

i

n= ∑ =

=

10

1

Note that the arithmetic mean deviation is zero:

Standard Deviation. The standard deviation isgiven by:

σ =

−∑

−=

( )x x

n

ii

n2

1

1(9)

Due to random errors, experimental data isdispersed in what is referred to as a belldistribution, known also as a Gaussian or NormalDistribution, and depicted in Figure 3.

xi

f(xi )

FIGURE 3. Gaussian or Normal Distribution.

The Gaussian or Normal Distribution is whatwe use to describe the distribution followed byrandom errors. A graph of this distribution is

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often referred to as the “bell” curve as it lookslike the outline of a bell. The peak of thedistribution occurs at the mean of the randomvariable, and the standard deviation is a commonmeasure for how “fat” this bell curve is. Equation10 is called the Probability Density Function forany continuous random variable x.

f x e

x x

( )( )

=− −1

2

2

22

σ πσ (10)

The mean and the standard deviation are allthe information necessary to completely describeany normally-distributed random variable.

Integrating under the curve of Figure 3 overvarious limits gives some interesting results.

• Integrating under the curve of the normaldistribution from negative to positiveinfinity, the area is 1.0 (i.e., 100 %). Thus theprobability for a reading to fall in the rangeof ±∞ is 100%.

• Integrating over a range within ± σ from themean value, the resulting value is 0.6826. Theprobability for a reading to fall in the rangeof ± σ is about 68%.

• Integrating over a range within ± 2σ from themean value, the resulting value is 0.954. Theprobability for a reading to fall in the rangeof ± 2σ is about 95%.

• Integrating over a range within ± 3σ from themean value, the resulting value is 0.997. Theprobability for a reading to fall in the rangeof ± 3σ is about 99%.

TABLE 2. Probability for Gaussian Distribution(tabulated in any statistics book)

Probability ± value of the mean50% 0.6754σ

68.3% σ86.6% 1.5σ95.4% 2σ99.7% 3σ

Estimating Uncertainty. We can now use theprobability function to help in determining theaccuracy of data obtained in an experiment. Weuse the uncertainty level of 95%, which meansthat we have a 95% confidence interval. In otherwords, if we state that the uncertainty is ∆x, wesuggest that we are 95% sure that any reading xi

will be within the range of ± ∆x of the mean.Thus, the probability of a sample chosen atrandom of being within the range ± 2σ of themean is about 95%. Uncertainty then is defined astwice the standard deviation:

∆x ≈ 2σ

Example 1. The manufacturer of a particularalloy claims a modulus of elasticity of 40 ± 2 kPa.How is that to be interpreted?

Solution: The general rule of thumb is that ± 2kPa would represent a 95% confidence interval.That is, if we randomly select many samples ofthis manufacturer’s alloy we should find that95% of the samples meet the stated limit of 40 ± 2kPa.

Now it is possible that we can find a samplethat has a modulus of elasticity of 37 kPa;however, it means that it is very unlikely.

Example 2 If we assume that variations in theproduct follow a normal distribution, and thatthe modulus of elasticity is within the range 40 ±2 kPa, then what is the standard deviation, σ?

Solution: The uncertainty ≈ 95% of confidenceinterval ≈ 2σ. Thus

± 2 kPa = ± 2σSo

σ = 1 kPa

Example 3. Assuming that the modulus ofelasticity is 40 ± 2 kPa, estimate the probabilityof finding a sample from this population with amodulus of elasticity less than or equal to 37 kPa.

Solution: With σ = 1 kPa, we are seeking thevalue of the integral under the bell shaped curve,over the range of -∞ to – 3σ. Thus, the probabilitythat the modulus of elasticity is less than 37 kPais:

P(E < 37 kPa) = 100 - 99.7

2 = 0.15%

Statistically Based Rejection of “Bad” Data–Chauvenet’s Criterion Occasionally, when a sample of nmeasurements of a variable is obtained, theremay be one or more that appear to differmarkedly from the others. If some extraneous

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influence or mistake in experimental techniquecan be identified, these “bad data” or “wildpoints” can simply be discarded. More difficult isthe common situation in which no explanation isreadily available. In such situations, theexperimenter may be tempted to discard thevalues on the basis that something must surelyhave gone wrong. However, this temptation mustbe resisted, since such data may be significanteither in terms of the phenomena being studied orin detecting flaws in the experimental technique.On the other hand, one does not want an erroneousvalue to bias the results. In this case, a statisticalcriterion must be used to identify points that canbe considered for rejection. There is no otherjustifiable method to “throw away” data points. One method that has gained wide acceptanceis Chauvenet’s criterion; this technique definesan acceptable scatter, in a statistical sense,around the mean value from a given sample of nmeasurements. The criterion states that all datapoints should be retained that fall within a bandaround the mean that corresponds to aprobability of 1-1/(2n). In other words, datapoints can be considered for rejection only if theprobability of obtaining their deviation from themean is less than 1/(2n). This is illustrated inFigure 4.

xi

f(xi )Probability1 - 1/(2n)

Rejectdata

Rejectdata

FIGURE 4. Rejection of “bad” data.

The probability 1-1/(2n) for retention of datadistributed about the mean can be related to amaximum deviation dmax away from the mean byusing a Gaussian probability table. For the givenprobability, the non dimensional maximumdeviation τmax can be determined from the table,where

τmax = |(xi – –x )|max

sx = dmax

sx

and sx is the precision index of the sample. All measurements that deviate from the

mean by more than dmax/sx can be rejected. A newmean value and a new precision index can then becalculated from the remaining measurements. Nofurther application of the criterion to the sampleis allowed.

Using Chauvenet’s criterion, we say that thevalues xi which are outside of the range

x C± σ (11)

are clearly errors and should be discarded for theanalysis. Such values are called outliers. Theconstant C may be obtained from Table 3. Notethat Chauvenet’s criterion may be applied onlyonce to a given sample of readings.

The methodology for identifying anddiscarding outlier(s) is a follows:

1. After running an experiment, sort theoutcomes from lowest to highest value. Thesuspect outliers will then be at the top and/orthe bottom of the list.

2. Calculate the mean value and the standarddeviation.

3. Using Chauvenet’s criterion, discard outliers.

4. Recalculate the mean value and the standarddeviation of the smaller sample and stop. Donot repeat the process; Chauvenet’s criterionmay be applied only once.

TABLE 3. Constants to use in Chauvenet’scriterion, Equation 11.

Number,n

dmax

sx = C

3 1.384 1.545 1.656 1.737 1.808 1.879 1.9110 1.9615 2.1320 2.2425 2.3350 2.57

100 2.81300 3.14500 3.29

1,000 3.48

Example 4. Consider an experiment in which wemeasure the mass of ten individual “identical”objects. The scale readings (in grams) are asshown in Table 4.

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By visual examination of the results, wemight conclude that the 4.85 g reading is too highcompared to the others, and so it represents anerror in the measurement. We might tend todisregard it. However, what if the reading was2.50 or 2.51 g? We use Chauvenet’s criterion todetermine if any of the readings can be discarded.�

TABLE 4. Data obtained in a series ofexperiments.

Number, n reading in g1 2.412 2.423 2.434 2.435 2.446 2.447 2.458 2.469 2.4710 4.85

We apply the methodology described earlier.The results of the calculations are shown in Table5:

1. Values in the table are already sorted.Column 1 shows the reading number, andthere are 10 readings of mass, as indicated incolumn 2.

2. We calculate the mean and standarddeviation. The data in column 2 are added toobtain a total of 26.8. Dividing this value by10 readings gives 2.68, which is the meanvalue of all the readings:

m– = 2.68 g

In column 3, we show the square of thedifference between each reading and themean value. Thus in row 1, we calculate

(x– – x1)2 = (2.68 – 2.41)2 = 0.0729

We repeat this calculation for every datapoint. We then add these to obtain the value5.235 shown in the second to last row ofcolumn 3. This value is then divided by (n –1)= 9 data points, and the square root is taken.The result is 0.763, which is the standarddeviation, as defined earlier in Equation 9:

σ =

−∑

−=

( )x x

n

ii

n2

1

1 = 0.763 (9)

3. Next, we apply Chauvenet’s criterion; for 10data points, n = 10 and Table 3 reads C = 1.96.We calculate Cσ = 1.96(0.763) = 1.50. Therange of “acceptable” values then is 2.68 ±1.50, or:

m– – Cσ ≤ mi ≤ m– + Cσ

1.18 g ≤ m– ≤ 4.18 g

Any values outside the range of 1.18 and 4.18are outliers and should be discarded.

4. Thus for the data of the example, the 4.85value is an outlier and may be discarded. Allother points are valid. The last two columnsshow the results of calculations madewithout data point #10. The mean becomes2.44, and the standard deviation is 0.019(compare to 2.68, and 0.763, respectively).

14

TABLE 5. Calculations summary for the data of Table 4.

Number, n reading in g (x– – xi)2remove #10 (x– – xi)2

1 2.41 0.0729 2.41 0.0008352 2.42 0.0676 2.42 0.0003573 2.43 0.0625 2.43 0.0000794 2.43 0.0625 2.43 0.0000795 2.44 0.0576 2.44 0.0000016 2.44 0.0576 2.44 0.0000017 2.45 0.0529 2.45 0.0001238 2.46 0.0484 2.46 0.0004469 2.47 0.0441 2.47 0.00096810 4.85 4.7089∑ = 26.8 5.235 21.95 0.002889

2.68 0.763 2.44 0.019

\f(∂T,∂x

16

REPORT WRITING

All reports in the Fluid MechanicsLaboratory require a formal laboratory reportunless specified otherwise. The report should bewritten in such a way that anyone can duplicatethe performed experiment and find the sameresults as the originator. The reports should besimple and clearly written. Reports are due oneweek after the experiment was performed, unlessspecified otherwise.

The report should communicate several ideasto the reader. First the report should be neatlydone. The experimenter is in effect trying toconvince the reader that the experiment wasperformed in a straightforward manner withgreat care and with full attention to detail. Apoorly written report might instead lead thereader to think that just as little care went intoperforming the experiment. Second, the reportshould be well organized. The reader should beable to easily follow each step discussed in thetext. Third, the report should contain accurateresults. This will require checking and recheckingthe calculations until accuracy can be guaranteed.Fourth, the report should be free of spelling andgrammatical errors. The following format, shownin Figure R.1, is to be used for formal LaboratoryReports:

Title Page–The title page should show the titleand number of the experiment, the date theexperiment was performed, experimenter'sname and experimenter's partners' names, allspelled correctly.

Table of Contents –Each page of the report mustbe numbered for this section.

Object –The object is a clear concise statementexplaining the purpose of the experiment.This is one of the most important parts of thelaboratory report because everythingincluded in the report must somehow relate tothe stated object. The object can be as short asone sentence.

Theory –The theory section should contain acomplete analytical development of allimportant equations pertinent to theexperiment, and how these equations are usedin the reduction of data. The theory sectionshould be written textbook-style.

Procedure – The procedure section should containa schematic drawing of the experimentalsetup including all equipment used in a partslist with manufacturer serial numbers, if any.Show the function of each part whennecessary for clarity. Outline exactly step-

Bibliography

Calibration Curves

Original Data Sheet(Sample Calculation)

Appendix Title Page

Discussion & Conclusion(Interpretation)

Results (Tablesand Graphs)

Procedure (Drawingsand Instructions)

Theory(Textbook Style)

Object(Past Tense)

Table of ContentsEach page numbered

Experiment NumberExperiment Title

Your Name

Due Date

Partners’ Names

FIGURE R.1. Format for formal reports.

by-step how the experiment was performed incase someone desires to duplicate it. If itcannot be duplicated, the experiment showsnothing.

Results – The results section should contain aformal analysis of the data with tables,graphs, etc. Any presentation of data whichserves the purpose of clearly showing theoutcome of the experiment is sufficient.

Discussion and Conclusion – This section shouldgive an interpretation of the resultsexplaining how the object of the experimentwas accomplished. If any analyticalexpression is to be verified, calculate % error†

and account for the sources. Discuss this

†% error–An analysis expressing how favorably theempirical data approximate theoretical information.There are many ways to find % error, but one method isintroduced here for consistency. Take the differencebetween the empirical and theoretical results and divideby the theoretical result. Multiplying by 100% gives the% error. You may compose your own error analysis aslong as your method is clearly defined.

16

experiment with respect to its faults as wellas its strong points. Suggest extensions of theexperiment and improvements. Alsorecommend any changes necessary to betteraccomplish the object. Each experiment write-up contains anumber of questions. These are to be answeredor discussed in the Discussion and Conclusionssection.

Appendix(1) Original data sheet.(2) Show how data were used by a samplecalculation.(3) Calibration curves of instrument whichwere used in the performance of theexperiment. Include manufacturer of theinstrument, model and serial numbers.Calibration curves will usually be suppliedby the instructor.(4) Bibliography listing all references used.

Short Form Report FormatOften the experiment requires not a formal

report but an informal report. An informal reportincludes the Title Page, Object, Procedure,Results, and Conclusions. Other portions may beadded at the discretion of the instructor or thewriter. Another alternative report form consistsof a Title Page, an Introduction (made up ofshortened versions of Object, Theory, andProcedure) Results, and Conclusion andDiscussion. This form might be used when adetailed theory section would be too long.

GraphsIn many instances, it is necessary to compose a

plot in order to graphically present the results.Graphs must be drawn neatly following a specificformat. Figure R.2 shows an acceptable graphprepared using a computer. There are manycomputer programs that have graphingcapabilities. Nevertheless an acceptably drawngraph has several features of note. These featuresare summarized next to Figure R.2.

FEATURES OF NOTE

• Border is drawn about the entire graph.• Axis labels defined with symbols and

units.• Grid drawn using major axis divisions.• Each line is identified using a legend.• Data points are identified with a

symbol: “ ´” on the Qac line to denotedata points obtained by experiment.

• The line representing the theoreticalresults has no data points represented.

• Nothing is drawn freehand.• Title is descriptive, rather than

something like Q vs ∆h.

0

0.05

0.1

0.15

0.2

0 0.2 0.4 0.6 0.8 1

Qth

Qac

Q

∆ hhead loss in m

flow

rat

e

in m

3 /s

FIGURE R.2. Theoretical and actual volume flow ratethrough a venturi meter as a function of head loss.

17

EXPERIMENT 1

FLUID PROPERTIES: DENSITY AND SURFACE TENSION

There are several properties simpleNewtonian fluids have. They are basicproperties which cannot be calculated for everyfluid, and therefore they must be measured.These properties are important in makingcalculations regarding fluid systems. Measuringfluid properties, density and surface tension, isthe object of this experiment.

Part I: Density Measurement.

EquipmentGraduated cylinder or beakerLiquid whose properties are to be

measuredHydrometer cylinderScale

Method 1. The density of the test fluid is to befound by weighing a known volume of the liquidusing the graduated cylinder or beaker and thescale. The beaker is weighed empty. The beakeris then filled to a certain volume according to thegraduations on it and weighed again. Thedifference in weight divided by the volume givesthe weight per unit volume of the liquid. Byappropriate conversion, the liquid density iscalculated. The mass per unit volume, or thedensity, is thus measured in a direct way.

Method 2. A second method of finding densityinvolves measuring buoyant force exerted on asubmerged object. The difference between theweight of an object in air and the weight of theobject in liquid is known as the buoyant force (seeFigure 1.1).

W1

W2

FIGURE 1.1. Measuring the buoyant force on anobject with a hanging weight.

Referring to Figure 1.1, the buoyant force B isfound as

B = W1 - W2

The buoyant force is equal to the differencebetween the weight of the object in air and theweight of the object while submerged. Dividingthis difference by the volume displaced gives theweight per unit volume from which density can becalculated.

Method 3. A third method of making a densitymeasurement involves the use of a calibratedhydrometer cylinder. The cylinder is submergedin the liquid and the density is read directly onthe calibrated portion of the cylinder itself.

ExperimentMeasure density using the methods assigned bythe instructor. Compare results of allmeasurements.

Questions1. Are the results of all the density

measurements in agreement?

2. How does the buoyant force vary withdepth of the submerged object? Why?

3. In your opinion, which method yielded the“most accurate” results?

4. Are the results precise?

5. What is the mean of the values youobtained?

6. What is the standard deviation of theresults?

7. Using Chauvenent’s rule, can any of themeasurements be discarded?

18

Part II: Surface Tension Measurement

EquipmentSurface tension meterBeakerTest fluid

Surface tension is defined as the energyrequired to pull molecules of liquid from beneaththe surface to the surface to form a new area. It istherefore an energy per unit area (F⋅L/L2 = F/L).A surface tension meter is used to measure thisenergy per unit area and give its value directly. Aschematic of the surface tension meter is given inFigure 1.2.

The platinum-iridium ring is attached to abalance rod (lever arm) which in turn is attachedto a stainless steel torsion wire. One end of thiswire is fixed and the other is rotated. As the wireis placed under torsion, the rod lifts the ringslowly out of the liquid. The proper technique isto lower the test fluid container as the ring islifted so that the ring remains horizontal. Theforce required to break the ring free from theliquid surface is related to the surface tension ofthe liquid. As the ring breaks free, the gage atthe front of the meter reads directly in the units

indicated (dynes/cm) for the given ring. Thisreading is called the apparent surface tension andmust be corrected for the ring used in order toobtain the actual surface tension for the liquid.The correction factor F can be calculated with thefollowing equation

F = 0.725 + √0.000 403 3(σa/ρ) + 0.045 34 - 1.679(r/R)

where F is the correction factor, σa is theapparent surface tension read from the dial(dyne/cm), ρ is the density of the liquid (g/cm3),and (r/R) for the ring is found on the ringcontainer. The actual surface tension for theliquid is given by

σ = Fσa

ExperimentMeasure the surface tension of the liquidassigned. Each member of your group should makea measurement to become familiar with theapparatus. Are all measurements in agreement?

FIGURE 1.2. A schematic of thesurface tension meter.

torsion wire

test liquid

platinumiridium ring

clampbalance rod

19

EXPERIMENT 2

FLUID PROPERTIES: VISCOSITY

One of the properties of homogeneous liquidsis their resistance to motion. A measure of thisresistance is known as viscosity. It can bemeasured in different, standardized methods ortests. In this experiment, viscosity will bemeasured with a falling sphere viscometer.

The Falling Sphere Viscometer When an object falls through a fluid medium,

the object reaches a constant final speed orterminal velocity. If this terminal velocity issufficiently low, then the various forces acting onthe object can be described with exact expressions.The forces acting on a sphere, for example, that isfalling at terminal velocity through a liquid are:

Weight - Buoyancy - Drag = 0

ρsg 43 πR3 - ρg

43 πR3 - 6πµVR = 0

where ρs and ρ are density of the sphere andliquid respectively, V is the sphere’s terminalvelocity, R is the radius of the sphere and µ isthe viscosity of the liquid. In solving thepreceding equation, the viscosity of the liquid canbe determined. The above expression for drag isvalid only if the following equation is valid:

ρVD

µ < 1

where D is the sphere diameter. Once theviscosity of the liquid is found, the above ratioshould be calculated to be certain that themathematical model gives an accuratedescription of a sphere falling through theliquid.

EquipmentCylinder filled with test liquidScaleStopwatchSeveral small spheres with weight and

diameter to be measured

Drop a sphere into the cylinder liquid andrecord the time it takes for the sphere to fall acertain measured distance. The distance dividedby the measured time gives the terminal velocityof the sphere. Repeat the measurement andaverage the results. With the terminal velocityof this and of other spheres measured and known,the absolute and kinematic viscosity of the liquid

can be calculated. The temperature of the testliquid should also be recorded. Use at least threedifferent spheres. (Note that if the density ofthe liquid is unknown, it can be obtained from anygroup who has completed or is taking data onExperiment 1.)

d

V

FIGURE 2.1. Terminal velocity measurement (V =d/t ime) .

Questions1. Should the terminal velocity of two

different size spheres be the same?

2. Does a larger sphere have a higherterminal velocity?

3. Should the viscosity found for two differentsize spheres be the same? Why or why not?

4. What are the shortcomings of this method?

5. Why should temperature be recorded?

6. Can this method be used for gases?

7. Can this method be used for opaque liquids?

8. Can this method be used for something likepeanut butter, or grease or flour dough?Why or why not?

9. Perform an error analysis for one of the datapoints. That is, determine the errorassociated with all the measurements, andprovide an error band about the mean value.

20

EXPERIMENT 3

CENTER OF PRESSURE ON A SUBMERGEDPLANE SURFACE

Submerged surfaces are found in manyengineering applications. Dams, weirs and watergates are familiar examples of submerged planesurfaces. It is important to have a workingknowledge of the forces that act on submergedsurfaces.

A plane surface located beneath the surfaceof a liquid is subjected to a pressure due to theheight of liquid above it, as shown in Figure 3.1.Pressure increases linearly with increasing depthresulting in a pressure distribution that acts onthe submerged surface. The analysis of thissituation involves determining a force which isequivalent to the pressure, and finding the line ofaction of this force.

F

yF

FIGURE 3.1. Pressure distribution on a submergedplane surface and the equivalent force.

For this case, it can be shown that theequivalent force is:

F = ρgycA (3.1)

in which ρ is the liquid density, yc is the distancefrom the free surface of the liquid to the centroidof the plane, and A is the area of the plane incontact with liquid. Further, the location of thisforce yF below the free surface is

yF = Ix x

ycA + yc (3.2)

in which Ixx is the second area moment of theplane about its centroid. The experimentalverification of these equations for force anddistance is the subject of this experiment.

Figure 3.2a is a sketch of an apparatus thatwe use to illustrate the concepts behind thisexperiment. The apparatus consists of one-fourthof a torus, consisting of a solid piece of material.The torus is attached to a lever arm, which is

free to rotate (within limits) about a pivot point.The torus has inside and outside radii, Ri and Rorespectively, and it is constructed such that thecenter of these radii is at the pivot point of thelever arm. The torus is now submerged in a liquid,and there will exist an unbalanced force F that isexerted on the plane of dimensions h x w. In orderto bring the torus and lever arm back to theirbalanced position, a weight Wmust be added tothe weight hanger. The force and its line ofaction can be found with Equations 3.1 and 3.2.

Consider next the apparatus sketched inFigure 3.2b. It is quite similar to that in Figure3.2a, in that it consists of a torus attached to alever arm. In this case, however, the torus ishollow, and can be filled with liquid. If thedepth of the liquid is equal to that in Figure 3.2a,(as measured from the bottom of the torus), thenthe forces in both cases will be equal in magnitudebut opposite in direction. Moreover, the distancefrom the free surface of the liquid to the line ofaction of both forces will also be equal. Thus,there is an equivalence between the two devices.

Center of Pressure Measurement

Equipment Center of Pressure Apparatus

(Figure 3.2b)Weights

The torus and balance arm are located on a pivotrod. Note that the pivot point for the balancearm is the point of contact between the rod andthe torus. Place the weight hanger on theapparatus, and add water into the trim tank (notshown in the figure) to bring the submerged planeback to the vertical position.

Start by adding 20 g to the weight hanger.Then pour water into the torus until thesubmerged plan is brought back to the verticalposition. Record the weight and the liquiddepth. Repeat this procedure for 4 more weights.(Remember to record the distance from the pivotpoint to the free surface for each case.)

From the depth measurement, the equivalentforce and its location are calculated usingEquations 3.1 and 3.2. Summing moments about thepivot allows for a comparison between thetheoretical and actual force exerted. Referring toFigure 3.2b, we have

21

F = W L

(y + yF) (3.3)

where y is the distance from the pivot point tothe free surface, yF is the distance from the freesurface to the line of action of the force F, and L isthe distance from the pivot point to the line ofaction of the weight W. Recalling that bothcurved surfaces of the torus are circular withcenters at the pivot point, we see that the forcesacting on the curved surfaces have a zero momentarm.

For the report, compare the force obtainedwith Equation 3.1 to that obtained with Equation3.3. When using Equation 3.3, it will be necessaryto use Equation 3.2 for yF.

Questions1. In summing moments, why isn't the buoyant

force taken into account in Figure 3.2a?2. Why isn’t the weight of the torus and the

balance arm taken into account?

weighthanger

L

Ri

F

y

h

w

yFRo

torus

FIGURE 3.2a

weighthanger

L

F

y

h

w

yF

Ri

Ro

torus

FIGURE 3.2b. A schematic of the center of pressure apparatus.

22

EXPERIMENT 4

IMPACT OF A JET OF WATER

A jet of fluid striking a stationary objectexerts a force on that object. This force can bemeasured when the object is connected to a springbalance or scale. The force can then be related tothe velocity of the jet of fluid and in turn to therate of flow. The force developed by a jet streamof water is the subject of this experiment.

Impact of a Jet of Liquid

EquipmentJet Impact ApparatusObject plates

Figure 4.1 is a schematic of the device used inthis experiment. The device consists of a catchbasin within a sump tank. A pump moves waterfrom the sump tank to the impact apparatus,after which the water drains to the catch basin.The plug is used to allow water to accumulate inthe catch basin. On the side of the sump tank is asight glass (not shown in Figure 4.1) showing thewater depth in the catch basin.

When flow rate is to be measured, water isallowed to accumulate in the catch basin, and astopwatch is used to measure the time requiredfor the water volume to reach a pre-determinedvolume, using the sight glass as an indicator. Inother words, we use the stopwatch to measure thetime required for a certain volume of water toaccumulate in the catch basin. The sump tank acts as a support for the tabletop which supports the impact apparatus. Asshown in Figure 4.1, the impact apparatuscontains a nozzle that produces a high velocityjet of water. The jet is aimed at an object (such asa flat plate or hemisphere). The force exerted onthe plate causes the balance arm to which theplate is attached to deflect. A weight is movedon the arm until the arm balances. A summationof moments about the pivot point of the armallows for calculating the force exerted by the jet.

Water is fed through the nozzle by means ofa pump. The nozzle emits the water in a jetstream whose diameter is constant. After the

water strikes the object, the water is channeled tothe catch basin to obtain the volume flow rate.

The variables involved in this experimentare listed and their measurements are describedbelow:1. Volume rate of flow–measured with the

catch basin (to obtain volume) and astopwatch (to obtain time). The volume flowrate is obtained by dividing volume by time:Q = V/t.

2. Velocity of jet–obtained by dividing volumeflow rate by jet area: V = Q/A. The jet iscylindrical in shape.

3. Resultant force—found experimentally bysummation of moments about the pivot pointof the balance arm. The theoretical resultantforce is found by use of an equation derived byapplying the momentum equation to a controlvolume about the plate.

Impact Force Analysis(Theoretical Force)

The total force exerted by the jet equals therate of momentum loss experienced by the jet afterit impacts the object. For a flat plate, the forceequation is:

F = ρQ2

A(flat plate)

For a hemisphere,

F = 2ρQ2

A(hemisphere)

For a cone whose included half angle is α,

F = ρQ2

A (1 + cos α) (cone)

These equations are easily derivable from themomentum equation applied to a control volumeabout the object.

23

flat platepivot

balancing weight lever arm withflat plate attached

water jet nozzle

drain

sump tank

flow controlvalve

motorpump

plug

catch basin

FIGURE 4.1. A schematic of the jet impact apparatus.

ProcedureI . Figure 4.2 shows a sketch of the lever arm

in the impact experiment. The impact objectshould be in place and the thumbscrew onthe spring should be used to zero the leverarm. This is done without any water flow.(Units of the scales in the figures arearbitrary.)

II. The pump is now turned on and a water jethits the impact object, which will deflectthe lever arm causing it to rotate slightly

counterclockwise. The balancing weight ismoved from the zero position to theposition required to re-balance the leverarm (in this case identified as “3” in Figure4.3). The spring is left untouched. Only thebalancing weight is moved in order to re-balance the lever arm.

III. During the time that the water jet impactsthe object, the time required to calculatevolume flow rate is measured.

24

40 1 2 3 5

Fs

Fo

Fw

dw1

do

ds

O

FIGURE 4.2. Lever arm in zero position withoutany water flow.

40 1 2 3 5

Fs

Fo

Fw

F

dw2

do

ds

O

water jet

FIGURE 4.3. Lever arm in zero position when thewater jet is on.

Nomenclature

SYMBOL FORCE DISTANCEFs spring force ds

Fw balancing weight dw

Fo impact object do

F exerted by water jet do

Analysis (Actual Force as Measured)Summing moments about point O in Figure 4.2gives the following equation for the lever arm:

Fsds + Fodo + Fwdw1 = 0 (4.1)

Summing moments about point O gives thefollowing equation for the lever arm in Figure 4.3:

Fsds + Fodo – Fdo + Fwdw2 = 0 (4.2)

Now we compare Equations 4.1 and 4.2. We canidentify parameters that appear in bothequations that are constants. These are Fsds andFodo. We rearrange Equation 4.1 to solve for thesum of these force-distance products:

Fsds + Fodo = – Fwdw1 (4.3)

Likewise, Equation 4.2 gives

Fsds + Fodo = + Fdo – Fwdw2 (4.4)

Subtracting Equation 4.4 from 4.3, we get

0 = – Fwdw1 – Fdo + Fwdw2

The force we are seeking is that exerted by thewater jet F; rearranging gives

Fdo = – Fwdw1 + Fwdw2 = Fw(dw2 – dw1)

or

F = Fw(dw2 – dw1)

do(4.5)

Thus, the force exerted by the water equals theweight of what we have called the balancingweight times a ratio of distances. The distance(dw2 – dw1) is just the difference in readings of theposition of the balancing weight. The distance dois the distance from the pivot to the location ofthe impact object.

For your report, derive the appropriateequation for each object you are assigned to use.Compose a graph with volume flow rate on thehorizontal axis, and on the vertical axis, plot theactual and theoretical force. Use care in choosingthe increments for each axis.

25

EXPERIMENT 5

CRITICAL REYNOLDS NUMBER IN PIPE FLOW

The Reynolds number is a dimensionless ratioof inertia forces to viscous forces and is used inidentifying certain characteristics of fluid flow.The Reynolds number is extremely important inmodeling pipe flow. It can be used to determinethe type of flow occurring: laminar or turbulent.Under laminar conditions the velocitydistribution of the fluid within the pipe isessentially parabolic and can be derived from theequation of motion. When turbulent flow exists,the velocity profile is “flatter” than in thelaminar case because the mixing effect which ischaracteristic of turbulent flow helps to moreevenly distribute the kinetic energy of the fluidover most of the cross section.

In most engineering texts, a Reynolds numberof 2 100 is usually accepted as the value attransition; that is, the value of the Reynoldsnumber between laminar and turbulent flowregimes. This is done for the sake of convenience.In this experiment, however, we will see thattransition exists over a range of Reynolds numbersand not at an individual point.

The Reynolds number that exists anywhere inthe transition region is called the criticalReynolds number. Finding the critical Reynoldsnumber for the transition range that exists in pipeflow is the subject of this experiment.

Critical Reynolds Number Measurement

EquipmentCritical Reynolds Number Determination

Apparatus

Figure 5.1 is a schematic of the apparatusused in this experiment. The constant head tankprovides a controllable, constant flow throughthe transparent tube. The flow valve in the tubeitself is an on/off valve, not used to control theflow rate. Instead, the flow rate through the tubeis varied with the rotameter valve at A. Thehead tank is filled with water and the overflowtube maintains a constant head of water. Theliquid is then allowed to flow through one of thetransparent tubes at a very low flow rate. Thevalve at B controls the flow of dye; it is openedand dye is then injected into the pipe with thewater. The dye injector tube is not to be placed inthe pipe entrance as it could affect the results.Establish laminar flow by starting with a verylow flow rate of water and of dye. The injected

dye will flow downstream in a threadlikepattern for very low flow rates. Once steady stateis achieved, the rotameter valve is openedslightly to increase the water flow rate. Thevalve at B is opened further if necessary to allowmore dye to enter the tube. This procedure ofincreasing flow rate of water and of dye (ifnecessary) is repeated throughout theexperiment.

Establish laminar flow in one of the tubes.Then slowly increase the flow rate and observewhat happens to the dye. Its pattern maychange, yet the flow might still appear to belaminar. This is the beginning of transition.Continue increasing the flow rate and againobserve the behavior of the dye. Eventually, thedye will mix with the water in a way that willbe recognized as turbulent flow. This point is theend of transition. Transition thus will exist over arange of flow rates. Record the flow rates at keypoints in the experiment. Also record thetemperature of the water.

The object of this procedure is to determinethe range of Reynolds numbers over whichtransition occurs. Given the tube size, theReynolds number can be calculated with:

Re = VDν

where V (= Q/A) is the average velocity ofliquid in the pipe, D is the hydraulic diameter ofthe pipe, and ν is the kinematic viscosity of theliquid.

The hydraulic diameter is calculated fromits definition:

D = 4 x Area

Wetted Perimeter

For a circular pipe flowing full, the hydraulicdiameter equals the inside diameter of the pipe.For a square section, the hydraulic diameter willequal the length of one side (show that this isthe case). The experiment is to be performed forboth round tubes and the square tube. With goodtechnique and great care, it is possible for thetransition Reynolds number to encompass thetraditionally accepted value of 2 100.

26

Questions1. Can a similar procedure be followed for

gases?2. Is the Reynolds number obtained at

transition dependent on tube size or shape?3. Can this method work for opaque liquids?

drilled partitions

dye reservoir

on/off valverotameter

A

to drain

inlet totank

overflowto drain

B

transparent tube

FIGURE 5.1. The critical Reynolds number determination apparatus.

27

EXPERIMENT 6

FLUID METERS IN INCOMPRESSIBLE FLOW

There are many different meters used in pipeflow: the turbine type meter, the rotameter, theorifice meter, the venturi meter, the elbow meterand the nozzle meter are only a few. Each meterworks by its ability to alter a certain physicalcharacteristic of the flowing fluid and thenallows this alteration to be measured. Themeasured alteration is then related to the flowrate. A procedure of analyzing meters todetermine their useful features is the subject ofthis experiment.

The Venturi Meter The venturi meter is constructed as shown in

Figure 6.1. It contains a constriction known as thethroat. When fluid flows through theconstriction, it must experience an increase invelocity over the upstream value. The velocityincrease is accompanied by a decrease in staticpressure at the throat. The difference betweenupstream and throat static pressures is thenmeasured and related to the flow rate. Thegreater the flow rate, the greater the pressuredrop ∆p. So the pressure difference ∆h (= ∆p/ρg)can be found as a function of the flow rate.

12

h

FIGURE 6.1. A schematic of the Venturi meter.

Using the hydrostatic equation applied tothe air-over-liquid manometer of Figure 6.1, thepressure drop and the head loss are related by(after simplification):

p1 - p2

ρg = ∆h

By combining the continuity equation,

Q = A1V1 = A2V2

with the Bernoulli equation,

p1

ρ + V1

2

2 = p2

ρ + V2

2

2

and substituting from the hydrostatic equation, itcan be shown after simplification that thevolume flow rate through the venturi meter isgiven by

Qth = A2 √2g∆h1 - (D2

4/D14) (6.1)

The preceding equation represents the theoreticalvolume flow rate through the venturi meter.Notice that is was derived from the Bernoulliequation which does not take frictional effectsinto account.

In the venturi meter, there exists smallpressure losses due to viscous (or frictional)effects. Thus for any pressure difference, theactual flow rate will be somewhat less than thetheoretical value obtained with Equation 6.1above. For any ∆h, it is possible to define acoefficient of discharge Cv as

Cv = Qac

Qth

For each and every measured actual flow ratethrough the venturi meter, it is possible tocalculate a theoretical volume flow rate, aReynolds number, and a discharge coefficient.The Reynolds number is given by

Re = V2D2

ν (6.2)

where V2 is the velocity at the throat of themeter (= Qac/A2).

The Orifice Meter andNozzle-Type Meter

The orifice and nozzle-type meters consist ofa throttling device (an orifice plate or bushing,respectively) placed into the flow. (See Figures6.2 and 6.3). The throttling device creates ameasurable pressure difference from its upstreamto its downstream side. The measured pressuredifference is then related to the flow rate. Likethe venturi meter, the pressure difference varieswith flow rate. Applying Bernoulli’s equation topoints 1 and 2 of either meter (Figure 6.2 or Figure6.3) yields the same theoretical equation as thatfor the venturi meter, namely, Equation 6.1. Forany pressure difference, there will be twoassociated flow rates for these meters: thetheoretical flow rate (Equation 6.1), and the

28

actual flow rate (measured in the laboratory).The ratio of actual to theoretical flow rate leadsto the definition of a discharge coefficient: Co forthe orifice meter and Cn for the nozzle.

1 2

h

FIGURE 6.2. Cross sectional view of the orificemeter.

1 2

h

FIGURE 6.3. Cross sectional view of the nozzle-type meter, and a typical nozzle.

For each and every measured actual flowrate through the orifice or nozzle-type meters, itis possible to calculate a theoretical volume flowrate, a Reynolds number and a dischargecoefficient. The Reynolds number is given byEquation 6.2.

The Turbine-Type Meter The turbine-type flow meter consists of a

section of pipe into which a small “turbine” hasbeen placed. As the fluid travels through thepipe, the turbine spins at an angular velocitythat is proportional to the flow rate. After acertain number of revolutions, a magnetic pickupsends an electrical pulse to a preamplifier whichin turn sends the pulse to a digital totalizer. Thetotalizer totals the pulses and translates theminto a digital readout which gives the totalvolume of liquid that travels through the pipeand/or the instantaneous volume flow rate.Figure 6.4 is a schematic of the turbine type flowmeter.

rotor supportedon bearings(not shown)

turbine rotorrotational speedproportional to

flow rate

to receiver

flowstraighteners

FIGURE 6.4. A schematic of a turbine-type flowmeter.

The Rotameter (Variable Area Meter) The variable area meter consists of a tapered

metering tube and a float which is free to moveinside. The tube is mounted vertically with theinlet at the bottom. Fluid entering the bottomraises the float until the forces of buoyancy, dragand gravity are balanced. As the float rises theannular flow area around the float increases.Flow rate is indicated by the float position readagainst the graduated scale which is etched onthe metering tube. The reading is made usually atthe widest part of the float. Figure 6.5 is a sketchof a rotameter.

tapered, graduatedtransparent tube

freelysuspendedfloat

inlet

outlet

FIGURE 6.5. A schematic of the rotameter and itsoperation.

Rotameters are usually manufactured withone of three types of graduated scales:1. % of maximum flow–a factor to convert scale

reading to flow rate is given or determined forthe meter. A variety of fluids can be usedwith the meter and the only variable

29

encountered in using it is the scale factor. Thescale factor will vary from fluid to fluid.

2. Diameter-ratio type–the ratio of crosssectional diameter of the tube to thediameter of the float is etched at variouslocations on the tube itself. Such a scalerequires a calibration curve to use the meter.

3. Direct reading–the scale reading shows theactual flow rate for a specific fluid in theunits indicated on the meter itself. If thistype of meter is used for another kind of fluid,then a scale factor must be applied to thereadings.

Experimental Procedure

EquipmentFluid Meters ApparatusStopwatch

The fluid meters apparatus is shownschematically in Figure 6.6. It consists of acentrifugal pump, which draws water from asump tank, and delivers the water to the circuitcontaining the flow meters. For nine valvepositions (the valve downstream of the pump),record the pressure differences in eachmanometer. For each valve position, measure theactual flow rate by diverting the flow to thevolumetric measuring tank and recording the timerequired to fill the tank to a predeterminedvolume. Use the readings on the side of the tankitself. For the rotameter, record the position ofthe float and/or the reading of flow rate givendirectly on the meter. For the turbine meter,record the flow reading on the output device.

Note that the venturi meter has twomanometers attached to it. The “inner”manometer is used to calibrate the meter; that is,to obtain ∆h readings used in Equation 6.1. The“outer” manometer is placed such that it readsthe overall pressure drop in the line due to thepresence of the meter and its attachment fittings.We refer to this pressure loss as ∆H (distinctlydifferent from ∆h). This loss is also a function offlow rate. The manometers on the turbine-typeand variable area meters also give the incurredloss for each respective meter. Thus readings of∆H vs Qac are obtainable. In order to use theseparameters to give dimensionless ratios, pressurecoefficient and Reynolds number are used. TheReynolds number is given in Equation 6.2. Thepressure coefficient is defined as

Cp = g∆HV2/2 (6.3)

All velocities are based on actual flow rate andpipe diameter.

The amount of work associated with thelaboratory report is great; therefore an informalgroup report is required rather than individualreports. The write-up should consist of anIntroduction (to include a procedure and aderivation of Equation 6.1), a Discussion andConclusions section, and the following graphs:

1. On the same set of axes, plot Qac vs ∆h andQth vs ∆h with flow rate on the verticalaxis for the venturi meter.

2. On the same set of axes, plot Qac vs ∆h andQth vs ∆h with flow rate on the verticalaxis for the orifice meter.

3. Plot Qac vs Qth for the turbine type meter.4. Plot Qac vs Qth for the rotameter.5. Plot Cv vs Re on a log-log grid for the

venturi meter.6. Plot Co vs Re on a log-log grid for the orifice

meter.7. Plot ∆H vs Qac for all meters on the same set

of axes with flow rate on the vertical axis.8. Plot Cp vs Re for all meters on the same set

of axes (log-log grid) with Cp vertical axis.

Questions1. Referring to Figure 6.2, recall that

Bernoulli's equation was applied to points 1and 2 where the pressure differencemeasurement is made. The theoreticalequation, however, refers to the throat areafor point 2 (the orifice hole diameter)which is not where the pressuremeasurement was made. Explain thisdiscrepancy and how it is accounted for inthe equation formulation.

2. Which meter in your opinion is the best oneto use?

3. Which meter incurs the smallest pressureloss? Is this necessarily the one that shouldalways be used?

4. Which is the most accurate meter?5. What is the difference between precision

and accuracy?

Air Over Liquid ManometryEach corresponding pair of pressure taps on

the apparatus is attached to an air over liquid(water, in this case), inverted U-tube manometer.Use of the manometers can lead to somedifficulties that may need attention.

Figure 6.7 is a sketch of one manometer. Theleft and right limbs are attached to pressure taps,

30

denoted as p1 and p2. Accordingly, when thesystem is operated, the liquid will rise in eachlimb and reach an equilibrium point. The pressuredifference will appear as a difference in heightof the water columns. That is, the pressuredifference is given by:

p1 – p2 = ρg∆h

where ρ is that of the liquid, and ∆h is readdirectly on the manometer.

In some cases, the liquid levels are at placesbeyond where we would like them to be. Toalleviate this problem, the air release valvemay be opened (slowly) to let air out or in. Whenthis occurs, the two levels will still have thesame ∆h reading, but located at a different placeon the manometer.

Sometimes, air bubbles will appear withinthe liquid. The apparatus used has water with asmall amount of liquid soap dissolved to reducethe surface tension of the water. However, if thepresence of bubbles persists, the pump should be

cycled on and off several times, and this shouldsolve the problem.

air

liquid

air releasevalve

p1 p2

∆h

FIGURE 6.7. Air over liquid manometer.

orifice meter

venturi meter

manometer

valve

turbine-type meter

rotameter sump tank

volumetricmeasuring

tank

return

pump

motor

FIGURE 6.6. A schematic of the Fluid Meters Apparatus. (Orifice and Venturi meters: upstreamdiameter is 1.025 inches; throat diameter is 0.625 inches.)

31

EXPERIMENT 7

PIPE FLOW

Experiments in pipe flow where thepresence of frictional forces must be taken intoaccount are useful aids in studying the behaviorof traveling fluids. Fluids are usually transportedthrough pipes from location to location by pumps.The frictional losses within the pipes causepressure drops. These pressure drops must beknown to determine pump requirements. Thus astudy of pressure losses due to friction has a usefulapplication. The study of pressure losses in pipeflow is the subject of this experiment.

Pipe Flow

EquipmentPipe Flow Test Rig

Figure 7.1 is a schematic of the pipe flow testrig. The rig contains a sump tank which is used asa water reservoir from which one or twocentrifugal pumps discharge water to the pipecircuit. The circuit itself consists of six differentdiameter lines and a return line all made of PVCpipe. The circuit contains ball valves fordirecting and regulating the flow, and can be usedto make up various series and parallel pipingcombinations. The circuit has provision formeasuring pressure loss through the use of staticpressure taps (manometer board and pressure tapsare not shown in the schematic).

The six lines are 1/2, 3/4, 1, 11/4, 11/2, and 2inch schedule 80 pipe. The topmost line is a returnline, and it is made of 2 inch pipe as well. Theapparatus contains two flow meters. The 1/2 linecontains a flow meter which is used only for thatpipe. The other flow meter in the return line is forall other flows through the system. Because thecircuit contains flow meters, the measuredpressure losses can be obtained as a function offlow rate.

As functions of the flow rate, measure thepressure losses in inches of water for whatevercombination of flows and minor losses specified bythe instructor

• The instructor will specify which of thepressure loss measurements are to be taken.

• Open and close the appropriate valves on theapparatus to obtain the desired flow path.

• Use the valve closest to the pump(s) on itsdownstream side to vary the volume flowrate.

• With the pump on, record the assignedpressure drops and the actual volume flowrate from the flow meter.

• Using the valve closest to the pump, changethe volume flow rate and again record thepressure drops and the new flow rate.

• Repeat this procedure until 9 differentvolume flow rates and corresponding pressuredrop data have been recorded.

With pressure loss data in terms of ∆h, thefriction factor can be calculated with

f = 2g∆h

V 2(L/D)

It is customary to graph the friction factor as afunction of the Reynolds number:

Re = VDν

The f vs Re graph, called a Moody Diagram, istraditionally drawn on a log-log grid. The graphalso contains a third variable known as theroughness coefficient ε/D. For this experimentthe roughness factor ε is that for smooth walledtubing.

Where fittings are concerned, the lossincurred by the fluid is expressed in terms of a losscoefficient K. The loss coefficient for any fittingcan be calculated with

K = ∆ h

V2/2g

where ∆h is the pressure (or head) loss across thefitting. Values of K as a function of the flow rateare to be obtained in this experiment.

For the report, calculate friction factor f andgraph it as a function of Reynolds number Re forthe assigned pipe(s). Compare to a Moodydiagram. Also calculate the loss coefficient forthe fitting(s) assigned, and determine if the losscoefficient K varies with flow rate or Reynoldsnumber. Compare your K values to published ones.

32

flow meter

Q2

open ball valve

closed ball valve

union fitting

flow direction

pumps

Q1

Q2

1/2 nominal

3/4

1

1 1/4

1 1/2

2

FIGURE 7.1. Schematic of the pipe friction apparatus (not to scale).

33

EXPERIMENT 8

PRESSURE DISTRIBUTION ABOUT A CIRCULAR CYLINDER

In many engineering applications, it may benecessary to examine the phenomena occurringwhen an object is inserted into a flow of fluid. Thewings of an airplane in flight, for example, maybe analyzed by considering the wings stationarywith air moving past them. Certain forces areexerted on the wing by the flowing fluid thattend to lift the wing (called the lift force) and topush the wing in the direction of the flow (dragforce). Objects other than wings that aresymmetrical with respect to the fluid approachdirection, such as a circular cylinder, willexperience no lift, only drag.

Drag and lift forces are caused by thepressure differences exerted on the stationaryobject by the flowing fluid. Skin friction betweenthe fluid and the object contributes to the dragforce but in many cases can be neglected. Themeasurement of the pressure distribution existingaround a stationary cylinder in an air stream tofind the drag force is the object of thisexperiment.

Consider a circular cylinder immersed in auniform flow. The streamlines about the cylinderare shown in Figure 8.1. The fluid exerts pressureon the front half of the cylinder in an amountthat is greater than that exerted on the rearhalf. The difference in pressure multiplied by theprojected frontal area of the cylinder gives thedrag force due to pressure (also known as formdrag). Because this drag is due primarily to apressure difference, measurement of the pressuredistribution about the cylinder allows for findingthe drag force experimentally. A typical pressuredistribution is given in Figure 8.2. Shown in

Figure 8.2a is the cylinder with lines andarrowheads. The length of the line at any pointon the cylinder surface is proportional to thepressure at that point. The direction of thearrowhead indicates that the pressure at therespective point is greater than the free streampressure (pointing toward the center of thecylinder) or less than the free stream pressure(pointing away). Note the existence of aseparation point and a separation region (orwake). The pressure in the back flow region isnearly the same as the pressure at the point ofseparation. The general result is a net drag forceequal to the sum of the forces due to pressureacting on the front half (+) and on the rear half (-) of the cylinder. To find the drag force, it isnecessary to sum the components of pressure ateach point in the flow direction. Figure 8.2b is agraph of the same data as that in Figure 8.2aexcept that 8.2b is on a linear grid.

FreestreamVelocity V

StagnationStreamline Wake

FIGURE 8.1. Streamlines of flow about a circularcylinder.

separationpoint

separationpoint

0 30 60 90 120 150 180

p

(a) Polar Coordinate Graph (b) Linear Graph

FIGURE 8.2. Pressure distribution around a circular cylinder placed in a uniform flow.

34

Pressure Measurement

EquipmentA Wind TunnelA Right Circular Cylinder with Pressure

Taps

Figure 8.3 is a schematic of a wind tunnel. Itconsists of a nozzle, a test section, a diffuser and afan. Flow enters the nozzle and passes throughflow straighteners and screens. The flow isdirected through a test section whose walls aremade of a transparent material, usuallyPlexiglas or glass. An object is placed in the testsection for observation. Downstream of the testsection is the diffuser followed by the fan. In thetunnel that is used in this experiment, the testsection is rectangular and the fan housing iscircular. Thus one function of the diffuser is togradually lead the flow from a rectangularsection to a circular one.

Figure 8.4 is a schematic of the side view ofthe circular cylinder. The cylinder is placed inthe test section of the wind tunnel which isoperated at a preselected velocity. The pressuretap labeled as #1 is placed at 0° directly facingthe approach flow. The pressure taps areattached to a manometer board. Only the first 18taps are connected because the expected profile issymmetric about the 0° line. The manometers willprovide readings of pressure at 10° intervalsabout half the cylinder. For two differentapproach velocities, measure and record thepressure distribution about the circular cylinder.

Plot the pressure distribution on polar coordinategraph paper for both cases. Also graph pressuredifference (pressure at the point of interest minusthe free stream pressure) as a function of angle θon linear graph paper. Next, graph ∆p cosθ vs θ(horizontal axis) on linear paper and determinethe area under the curve by any convenientmethod (counting squares or a numericaltechnique).

The drag force can be calculated byintegrating the flow-direction-component of eachpressure over the area of the cylinder:

Df = 2RL ∫0

π

∆p cosθdθ

The above expression states that the drag force istwice the cylinder radius (2R) times the cylinderlength (L) times the area under the curve of∆p cosθ vs θ.

Drag data are usually expressed as dragcoefficient CD vs Reynolds number Re. The dragcoefficient is defined as

CD = Df

ρV2A/2

The Reynolds number is

Re = ρVD

µ

inlet flowstraighteners

nozzle

test sectiondiffuser fan

FIGURE 8.3. A schematic of the wind tunnel used in this experiment.

35

where V is the free stream velocity (upstream ofthe cylinder), A is the projected frontal area ofthe cylinder (2RL), D is the cylinder diameter, ρis the air density and µ is the air viscosity.Compare the results to those found in texts.

static pressuretaps attach tomanometers

60

0

30

90120

150

180

FIGURE 8.4. Schematic of the experimentalapparatus used in this experiment.

36

EXPERIMENT 9

DRAG FORCE DETERMINATION

An object placed in a uniform flow is actedupon by various forces. The resultant of theseforces can be resolved into two force components,parallel and perpendicular to the main flowdirection. The component acting parallel to theflow is known as the drag force. It is a function ofa skin friction effect and an adverse pressuregradient. The component perpendicular to theflow direction is the lift force and is caused by apressure distribution which results in a lowerpressure acting over the top surface of the objectthan at the bottom. If the object is symmetricwith respect to the flow direction, then the liftforce will be zero and only a drag force will exist.Measurement of the drag force acting on an objectimmersed in the uniform flow of a fluid is thesubject of this experiment.

EquipmentSubsonic Wind TunnelObjects

A description of a subsonic wind tunnel isgiven in Experiment 8 and is shown schematicallyin Figure 8.3. The fan at the end of the tunneldraws in air at the inlet. An object is mounted on astand that is pre calibrated to read lift and dragforces exerted by the fluid on the object. Aschematic of the test section is shown in Figure9.1. The velocity of the flow at the test section isalso pre calibrated. The air velocity past theobject can be controlled by changing the rotationalspeed of the fan. Thus air velocity, lift force anddrag force are read directly from the tunnelinstrumentation.

There are a number of objects that areavailable for use in the wind tunnel. Theseinclude a disk, a smooth surfaced sphere, a roughsurface sphere, a hemisphere facing upstream,and a hemisphere facing downstream. Forwhichever is assigned, measure drag on the objectas a function of velocity.

Data on drag vs velocity are usually graphedin dimensionless terms. The drag force Df iscustomarily expressed in terms of the dragcoefficient CD (a ratio of drag force to kineticenergy):

CD = Df

ρV2A/2

in which ρ is the fluid density, V is the freestream velocity, and A is the projected frontalarea of the object. Traditionally, the dragcoefficient is graphed as a function of theReynolds number, which is defined as

Re = VDν

where D is a characteristic length of the objectand ν is the kinematic viscosity of the fluid. Foreach object assigned, graph drag coefficient vsReynolds number and compare your results tothose published in texts. Use log-log paper ifappropriate.

Questions1. How does the mounting piece affect the

readings?2. How do you plan to correct for its effect, if

necessary?

drag forcemeasurement

lift forcemeasurement

uniform flow mounting stand

object

FIGURE 9.1. Schematic of an object mounted inthe test section of the wind tunnel.

37

EXPERIMENT 10

ANALYSIS OF AN AIRFOIL

A wing placed in the uniform flow of anairstream will experience lift and drag forces.Each of these forces is due to a pressuredifference. The lift force is due to the pressuredifference that exists between the lower andupper surfaces. This phenomena is illustrated inFigure 10.1. As indicated the airfoil is immersedin a uniform flow. If pressure could be measured atselected locations on the surface of the wing andthe results graphed, the profile in Figure 10.1would result. Each pressure measurement isrepresented by a line with an arrowhead. Thelength of each line is proportional to themagnitude of the pressure at the point. Thedirection of the arrow (toward the horizontalaxis or away from it) represents whether thepressure at the point is less than or greater thanthe free stream pressure measured far upstream ofthe wing.

stagnationpoint

negative pressuregradient on uppersurface

positive pressureon lower surface

Cp

pressurecoefficient

stagnationpoint

c

chord, c

FIGURE 10.1. Streamlines of flow about a wingand the resultant pressure distribution.

Lift and Drag Measurements for a Wing

EquipmentWind Tunnel (See Figure 8.3)Wing with Pressure TapsWing for Attachment to Lift & Drag Instruments (See Figure 10.2)

Experiment For a number of wings, lift and drag data

vary only slightly with Reynolds number andtherefore if lift and drag coefficients are graphedas a function of Reynolds number, the results arenot that meaningful. A more significantrepresentation of the results is given in what isknown as a polar diagram for the wing. A polardiagram is a graph on a linear grid of liftcoefficient (vertical axis) as a function of dragcoefficient. Each data point on the graphcorresponds to a different angle of attack, allmeasured at one velocity (Reynolds number).Referring to Figure 10.2 (which is theexperimental setup here), the angle of attack α ismeasured from a line parallel to the chord c to aline that is parallel to the free stream velocity.

Obtain lift force, drag force and angle ofattack data using a pre selected velocity. Allowthe angle of attack to vary from a negative angleto the stall point and beyond. Obtain data at noless than 9 angles of attack. Use the data toproduce a polar diagram.

Analysis Lift and drag data are usually expressed in

dimensionless terms using lift coefficient and dragcoefficient. The lift coefficient is defined as

CL = Lf

ρV2A/2

where Lf is the lift force, ρ is the fluid density, Vis the free stream velocity far upstream of thewing, and A is the area of the wing when seenfrom a top view perpendicular to the chordlength c. The drag coefficient is defined as

CD = Df

ρV2A/2

in which Df is the drag force.

38

drag forcemeasurement

lift forcemeasurement

uniform flow mounting stand

cdrag

lift

FIGURE 10.2. Schematic of lift and drag measurement in a test section.

39

EXPERIMENT 11

OPEN CHANNEL FLOW—SLUICE GATE

Liquid motion in a duct where a surface of thefluid is exposed to the atmosphere is called openchannel flow. In the laboratory, open channelflow experiments can be used to simulate flow in ariver, in a spillway, in a drainage canal or in asewer. Such modeled flows can include flow overbumps or through dams, flow through a venturiflume or under a partially raised gate (a sluicegate). The last example, flow under a sluice gate,is the subject of this experiment.

Flow Through a Sluice GateEquipment

Open Channel Flow ApparatusSluice Gate Model

Figure 11.1 is a sketch of the flow patternunder a sluice gate. Upstream of the gate, thevelocity is V0, and the liquid height is h0. Thegate is a distance h1 above the bottom of thechannel, and downstream, the liquid height ish2. The channel width is b.

The objective of this experiment is to makemeasurements for a number of gate positions andflow rates, and to determine whether theequations we derive for a sluice gate are accuratein their description of the resulting flows.

h0

h1

h2

V0

sluicegate

FIGURE 11.1. Schematic of flow under a sluicegate .

TheoryThe continuity equation applied about the

sluice gate is

V0h0b = V2h2b (11.1)

Under real conditions, h2 is somewhat less thanh1. We therefore introduce a contractioncoefficient Cc defined as

Cc = h2

h 1

Substituting into Equation 11.1, canceling thechannel width b, and solving for V0, we get

V0 = V2 Cch1

h 0(11.2)

The Bernoulli Equation applied about the gate is

V02

2g + h0 = V2

2

2g + h2 (11.3)

Substituting for V0 from Equation 11.2, we obtain

V22Cc

2h12/h0

2

2g + h0 = V2

2

2g + Cch1

Rearranging and solving for V2, we have

V2 =

2g

h0 – C ch1

1 – Cc2h1

2/h02

1/2

Factoring h0 and noting the relationship betweenthe numerator and denominator, the precedingequation becomes

V2 =

2gh0

1 + Cch1/h0

1/2(11.4)

The flow rate is the product of area and velocity.At section 2, the flow rate may be written as

Q = V2h2b = V2Cch1b

Substituting from Equation 11.4 yields

Q = Cch1b

2gh0

1 + Cch1/h0

1/2(11.5)

Introducing a discharge coefficient Cs, we write

Q = Csb √2gh0 (11.6)

Comparison with Equation 11.5 gives

Cs = Cc

√1 + Cch1/h0

(11.7)

40

We see that the contraction coefficient and thedischarge coefficient depend only on theupstream height and the gate height.

ProcedureSet up the open channel flow apparatus

(Figure 11.2) to obtain flow under a sluice gate.For nine (if possible) different flow rate/gateposition combinations, record upstream height,gate height, downstream height, and volumeflow rate.

CalculationsCalculate the contraction coefficient, the

discharge coefficient (Equation 11.7), and the

expected volume flow rate (Equation 11.6).Compare the calculated flow rate (theoretical)with the measured (actual) value.

Include in Your Report• Detailed derivation of the equations• Graph of contraction coefficient as a function

of the ratio h1/h0• Graph of discharge coefficient as a function of

the ratio h1/h0• Graph of flow rates (actual and theoretical)

as a function of the ratio h1/h0• Discussion of your results

sump tankpump/motor

pump dischargepipe

valve

head tanksluice gate

turbinemeter

flow channel

FIGURE 11.2. Schematic of the open channel flow apparatus.

41

EXPERIMENT 12

OPEN CHANNEL FLOW OVER A WEIR

Flow meters used in pipes introduce anobstruction into the flow which results in ameasurable pressure drop that in turn is related tothe volume flow rate. In an open channel, flowrate can be measured similarly by introducing anobstruction into the flow. A simple obstruction,called a weir, consists of a vertical plateextending the entire width of the channel. Theplate may have an opening, usually rectangular,trapezoidal, or triangular. Other configurationsexist and all are about equally effective. The useof a weir to measure flow rate in an open channelis the subject of this experiment.

Flow Over a Weir

EquipmentOpen Channel Flow Apparatus (See Figure 11.2)Several Weirs

The open channel flow apparatus allows forthe insertion of a weir and measurement of liquiddepths. The channel is fed by two centrifugalpumps. Each pump has a discharge line whichcontains an turbine meter with digital readout,which provide the means of determining theactual flow rate into the channel.

Figure 12.1 is a sketch of the side andupstream view of a 90 degree (included angle) V-notch weir. Analysis of this weir is presentedhere for illustrative purposes. Note thatupstream depth measurements are made from thelowest point of the weir over which liquid flows.This is the case for the analysis of all

conventional weirs. A coordinate system isimposed whose origin is at the intersection of thefree surface and a vertical line extending upwardfrom the vertex of the V-notch. We select anelement that is dy thick and extends the entirewidth of the flow cross section. The velocity ofthe liquid through this element is found byapplying Bernoulli's equation:

pa

ρ + Vo

2

2 + gh = pa

ρ + V2

2 + g(h - y)

Note that in pipe flow, pressure remained in theequation when analyzing any of the differentialpressure meters (orifice or venturi meters). In openchannel flows, the pressure terms representsatmospheric pressure and cancel from theBernoulli equation. The liquid height istherefore the only measurement required here.From the above equation, assuming Vo negligible:

V = √2gy (12.1)

Equation 12.1 is the starting point in the analysisof all weirs. The incremental flow rate of liquidthrough layer dy is:

dQ = 2Vxdy = √2gy(2x)dy

From the geometry of the V-notch and withrespect to the coordinate axes, we have y = h - x.

pa

pa

Vo

Vh

y

dy

x

x axis

y axis

FIGURE 12.1. Side and upstream views of a 90° V-notch weir.

42

Therefore,

Q = ∫ 0

h

(2√2g)y1/2(h - y)d y

Integration gives

Qth = 8

15 √2g h5/2 =Ch5/2 (12.2)

where C is a constant. The above equationrepresents the ideal or theoretical flow rate ofliquid over the V-notch weir. The actualdischarge rate is somewhat less due to frictionaland other dissipative effects. As with pipemeters, we introduce a discharge coefficientdefined as:

C' = Qac

Qth

The equation that relates the actual volume flowrate to the upstream height then is

Qac = C'Ch5/2

It is convenient to combine the effects of theconstant C and the coefficient C’ into a singlecoefficient Cvn for the V-notch weir. Thus wereformulate the previous two equations to obtain:

Cvn ≈ Qac

Qth(12.3)

Qac = Cvnh5/2 (12.4)

Each type of weir will have its own coefficient. Calibrate each of the weirs assigned by the

instructor for 7 different upstream heightmeasurements. Derive an appropriate equationfor each weir used (similar to Equation 12.4)above. Determine the coefficient applicable foreach weir tested. List the assumptions made ineach derivation. Discuss the validity of eachassumption, pointing out where they break down.Graph upstream height vs actual and theoreticalvolume flow rates. Plot the coefficient ofdischarge (as defined in Equation 12.3) as afunction of the upstream Froude number.

FIGURE 12.2. Other types of weirs–semicircular, contracted and suppressed, respectively.

43

EXPERIMENT 13

OPEN CHANNEL FLOW—HYDRAULIC JUMP

When spillways or other similar openchannels are opened by the lifting of a gate,liquid passing below the gate has a high velocityand an associated high kinetic energy. Due to theerosive properties of a high velocity fluid, itmay be desirable to convert the high kineticenergy (e.g. high velocity) to a high potentialenergy (e.g., a deeper stream). The problem thenbecomes one of rapidly varying the liquid depthover a short channel length. Rapidly varied flowof this type produces what is known as ahydraulic jump.

Consider a horizontal, rectangular openchannel of width b, in which a hydraulic jumphas developed. Figure 13.1 shows a side view of ahydraulic jump. Figure 13.1 also shows the depthof liquid upstream of the jump to be h1, and adownstream depth of h2. Pressure distributionsupstream and downstream of the jump are drawnin as well. Because the jump occurs over a veryshort distance, frictional effects can be neglected.A force balance would therefore include onlypressure forces. Applying the momentum equationin the flow direction gives:

p1A1 - p2A2 = ρQ(V2 - V1)

Pressure in the above equation represents thepressure that exists at the centroid of the crosssection. Thus p = ρg(h/2). With a rectangularcross section of width b (A = bh), the aboveequation becomes

h1g

2 (h1b) - h2g

2 (h2b) = Q(V2 - V1)

From continuity, A1V1 = A2V2 = Q. Combining andrearranging,

h1

2 - h22

2 = Q 2

gb2

1

h 2 -

1 h 1

Simplifying,

h22 + h2h1 - 2

Q2

gb2h1 = 0

Solving for the downstream height yields onephysically (nonnegative) possible solution:

h2 = - h1

2 + √2Q 2

gb2h1 +

h12

4

from which the downstream height can be found.By applying Bernoulli’s Equation along the freesurface, the energy lost irreversibly can becalculated as

Lost Energy = E = g(h2 - h1)3

4h 2h 1

and the rate of energy loss is

d Wd t

= ρQE

The above equations are adequate to properlydescribe a hydraulic jump.

Hydraulic Jump Measurements

EquipmentOpen Channel Flow Apparatus (Figure 11.2)

The channel can be used in either ahorizontal or a sloping configuration. The devicecontains two pumps which discharge waterthrough calibrated turbine meters connected todigital readouts. The device also contains on thechannel bottom two forward facing brass tubes.Each tube is connected to a vertical Plexiglastube. The height of the water in either of thesetubes represents the energy level at therespective tube location. The difference in heightis the actual lost energy (E) for the jump ofinterest.

FIGURE 13.1. Schematic of ahydraulic jump in an openchannel.

h1

V2

V1

p1

p2

h2

44

Develop a hydraulic jump in the channel;record upstream and downstream heights,manometer readings (from which the actualvolume flow rate is obtained) and the lost energyE. By varying the flow rate, upstream height,downstream height and/or the channel slope,record measurements on different jumps. Derivethe applicable equations in detail and substituteappropriate values to verify the predicteddownstream height and lost energy. In otherwords, the downstream height of each jump is tobe measured and compared to the downstreamheight calculated with Equation 13.1. The sameis to be done for the rate of energy loss (Equation13.2).

Analysis Data on a hydraulic jump is usually specified

in two ways both of which will be required forthe report. Select any of the jumps you havemeasurements for and construct a momentumdiagram . A momentum diagram is a graph of

liquid depth on the vertical axis vs momentum onthe horizontal axis, an example of which isshown in Figure 13.2. As indicated in the figure,the momentum of the flow before the jump equalsthe momentum of the flow after the jump.

The momentum of the flow is given by:

M = 2Q2

gbh +

h2b4

If momentum before and after the jump isunchanged, then the he diagram shows merelythat kinetic and potential energy changes equalone another.

Another significant graph of hydraulic jumpdata is of depth ratio h2/h1 (vertical axis) as afunction of the upstream Froude number, Fr1 (=Q2/gb2h1

3 ). Construct such a graph for any of thejumps for which you have taken measurements.

FIGURE 13.2. Momentumdiagram for ahydraulic jump.

h1

h

M

Fr1 < 1

Fr2 > 1

zcr

Mmin

h2

45

EXPERIMENT 14

MEASUREMENT OF PUMP PERFORMANCE

The objective of this experiment is to performa test of a centrifugal pump and display theresults in the form of what is known as aperformance map.

Figure 14.1 is a schematic of the pump andpiping system used in this experiment. The pumpcontains an impeller within its housing. Theimpeller is attached to the shaft of the motorand the motor is mounted so that it is free torotate, within limits. As the motor rotates andthe impeller moves liquid through the pump, themotor housing tends to rotate in the oppositedirection from that of the impeller. A calibratedmeasurement system gives a readout of the torqueexerted by the motor on the impeller.

The rotational speed of the motor is obtainedwith a tachometer. The product of rotationalspeed and torque is the input power to theimpeller from the motor.

Gages in the inlet and outlet lines about thepump give the corresponding pressures in gagepressure units. The gages are located at knownheights from a reference plane.

After moving through the system, the wateris discharged into an open channel containing aV-notch weir. The weir is calibrated to providethe volume flow rate through the system.

The valve in the outlet line is used to controlthe volume flow rate. As far as the pump isconcerned, the resistance offered by the valvesimulates a piping system with a controllablefriction loss. Thus for any valve position, thefollowing data can be obtained: torque, rotationalspeed, inlet pressure, outlet pressure, and volumeflow rate. These parameters are summarized inTable 14.1.

TABLE 14.1. Pump testing parameters.

Raw Data

Parameter Symbol Dimensions

torque T F·L

rotational speed ω 1/T

inlet pressure p1 F/L2

outlet pressure p2 F/L2

volume flow rate Q L3/T

The parameters used to characterize thepump are calculated with the raw data obtainedfrom the test (listed above) and are as follows:input power to the pump, the total headdifference as outlet minus inlet, the powerimparted to the liquid, and the efficiency. Theseparameters are summarized in Table 14.2. Theseparameters must be expressed in a consistent set ofunits.

TABLE 14.2. Pump characterization parameters.

Reduced Data

Parameter Symbol Dimensions

input power dWa/dt F·L/T

total head diff ∆H L

power to liquid dW/dt F·L/T

efficiency η —

The raw data are manipulated to obtain thereduced data which in turn are used tocharacterize the performance of the pump. Theinput power to the pump from the motor is theproduct of torque and rotational speed:

- dWa

d t = Tω (14.1)

where the negative sign is added as a matter ofconvention. The total head at section 1, wherethe inlet pressure is measured (see Figure 14.1), isdefined as

H1 = p1

ρg +

V12

2g + z1

where ρ is the liquid density and V1 (= Q/A) isthe velocity in the inlet line. Similarly, thetotal head at position 2 where the outlet pressureis measured is

H2 = p2

ρg +

V22

2g + z2

The total head difference is given by

46

∆H = H2 - H1 = p2

ρg +

V22

2g + z2

-

p1

ρg +

V12

2g + z1

The dimension of the head H is L (ft or m). Thepower imparted to the liquid is calculated withthe steady flow energy equation applied fromsection 1 to 2:

- d Wd t

= m· g

p2

ρg +

V22

2g + z2

-

p1

ρg +

V12

2g + z1

In terms of total head H, we have

- d Wd t

= m· g (H2 - H1) = m· g ∆H (14.2)

The efficiency is determined with

η = d W / d td W a/d t

(14.3)

Experimental MethodThe experimental technique used in obtaining

data depends on the desired method of expressingperformance characteristics. For this experiment,data are taken on only one impeller-casing-motorcombination. One data point is first taken at acertain valve setting and at a preselectedrotational speed. The valve setting would then bechanged and the speed control on the motor (notshown in Figure 14.1) is adjusted if necessary sothat the rotational speed remains constant, andthe next set of data are obtained. This procedureis continued until 6 data points are obtained forone rotational speed.

Next, the rotational speed is changed andthe procedure is repeated. Four rotational speedsshould be used, and at least 6 data points perrotational speed should be obtained.

•sump tank

inlet

valve

valve

pumpmotor

control paneland gages

v-notch weir

return

1-1/2 nominalschedule 40PVC pipe

1 nominalschedule 40PVC pipe

pressuretap

pressuretap

•

z1

z2

motorshaft

FIGURE 14.1. Centrifugal pump testing setup.

47

Performance MapA performance map is to be drawn to

summarize the performance of the pump over itsoperating range. The performance map is a graphif the total head ∆H versus flow rate Q(horizontal axis). Four lines, corresponding to thefour pre-selected rotational speeds, would bedrawn. Each line has 6 data points, and theefficiency at each point is calculated. Lines ofequal efficiency are then drawn, and the resultinggraph is known as a performance map. Figure 14.2is an example of a performance map.

0 200 400 600 8000

10

20

30

40

Volume flow rate in gallons per minute

Tot

al h

ead

in ft

3600 rpm

1760

2700

900

85%

80%

75%

75%

65%

65%

Efficiency in %

FIGURE 14.2. Example of a performance map ofone impeller-casing-motor combinationobtained at four different rotational speeds.

Dimensionless GraphsTo illustrate the importance of

dimensionless parameters, it is prudent to use thedata obtained in this experiment and produce adimensionless graph.

A dimensional analysis can be performed forpumps to determine which dimensionless groupsare important. With regard to the flow of anincompressible fluid through a pump, we wish torelate three variables introduced thus far to theflow parameters. The three variables of interesthere are the efficiency η, the energy transfer rateg∆H, and the power dW/dt. These threeparameters are assumed to be functions of fluidproperties density ρ and viscosity µ, volume flowrate through the machine Q, rotational speed ω,and a characteristic dimension (usually impellerdiameter) D. We therefore write three functionaldependencies:

η = f1(ρ, µ, Q, ω, D )

g∆H = f2(ρ, µ, Q, ω, D)

d Wd t

= f3(ρ, µ, Q, ω, D)

Performing a dimensional analysis gives thefollowing results:

η = f1

ρωD2

µ , Q

ωD3

g∆Hω2D2 = f2

ρωD2

µ , Q

ωD3

d W / d t ρω3D5 = f3

ρωD2

µ , Q

ωD3

where

g∆Hω2D2 = energy transfer coefficient

QωD3 = volumetric flow coefficient

ρωD2

µ = rotational Reynolds number

d W / d t ρω3D5 = power coefficient

Experiments conducted with pumps show that therotational Reynolds number (ρωD2/µ) has asmaller effect on the dependent variables thandoes the flow coefficient. So for incompressibleflow through pumps, the preceding equationsreduce to

η ≈ f1

Q

ωD3 (14.4)

g∆Hω2D2 ≈ f2

Q

ωD3 (14.5)

d W / d t ρω3D5 ≈ f3

Q

ωD3 (14.6)

For this experiment, construct a graph ofefficiency, energy transfer coefficient, and powercoefficient all as functions of the volumetric flowcoefficient. Three different graphs can be drawn,or all lines can be placed on the same set of axes.

48

Specific SpeedA dimensionless group known as specific

speed can also be derived. Specific speed is foundby combining head coefficient and flowcoefficient in order to eliminate characteristiclength D:

ωss =

Q

ωD3

1/2

ω2D2

g∆H

3/4

or ωss = ωQ1/2

(g∆H)3/4 [dimensionless]

Exponents other than 1/2 and 3/4 could be used (toeliminate D), but 1/2 and 3/4 are customarilyselected for modeling pumps. Another definitionfor specific speed is given by

ωs = ωQ1/2

∆H3/4

rpm =

rpm(gpm)1/2

ft3/4

in which the rotational speed ω is expressed inrpm, volume flow rate Q is in gpm, total head ∆His in ft of liquid, and specific speed ωs isarbitrarily assigned the unit of rpm. The equationfor specific speed ωss is dimensionless whereasωs is not.

The specific speed of a pump can becalculated at any operating point, butcustomarily specific speed for a pump isdetermined only at its maximum efficiency. Forthe pump of this experiment, calculate itsspecific speed using both equations.

49

EXPERIMENT 15

MEASUREMENT OF VELOCITYAND

CALIBRATION OF A METER FOR COMPRESSIBLE FLOW

The objective of this experiment is todetermine a calibration curve for a meter placedin a pipe that is conveying air. The meters ofinterest are an orifice meter and a venturi meter.These meters are calibrated in this experiment byusing a pitot-static tube to measure the velocity,from which the flow rate is calculated.

Pitot Static TubeWhen a fluid flows through a pipe, it exerts

pressure that is made up of static and dynamiccomponents. The static pressure is indicated by ameasuring device moving with the flow or thatcauses no velocity change in the flow. Usually, tomeasure static pressure, a small holeperpendicular to the flow is drilled through thecontainer wall and connected to a manometer (orpressure gage) as indicated in Figure 15.1.

The dynamic pressure is due to the movementof the fluid. The dynamic pressure and the staticpressure together make up the total or stagnationpressure. The stagnation pressure can be measuredin the flow with a pitot tube. The pitot tube is anopen ended tube facing the flow directly. Figure15.1 gives a sketch of the measurement ofstagnation pressure.

static pressuremeasurement

stagnation pressuremeasurement

pitot tube

flow

hh

FIGURE 15.1. Measurement of static andstagnation pressures.

The pitot-static tube combines the effects ofstatic and stagnation pressure measurement intoone device. Figure 15.2 is a schematic of the pitot-static tube. It consists of a tube within a tubewhich is placed in the duct facing upstream. Thepressure tap that faces the flow directly gives ameasurement of the stagnation pressure, while

the tap that is perpendicular to the flow givesthe static pressure.

When the pitot-static tube is immersed in theflow of a fluid, the pressure difference(stagnation minus static) can be read directlyusing a manometer and connecting the pressuretaps to each leg. Applying the Bernoulli equationbetween the two pressure taps yields:

A

A

section A-Aenlarged

four to eight holesequally spaced

manometerconnections

flow direction

FIGURE 15.2. Schematic of a pitot-static tube.

p1

ρg +

V12

2g + z1 =

p2

ρg +

V22

2g + z2

where state “1” as the stagnation state (whichwill be changed to subscript “t”), and state “2” asthe static state (no subscript). Elevationdifferences are negligible, and at the point wherestagnation pressure is measured, the velocity iszero. The Bernoulli equation thus reduces to:

ptρg

= pρg

+ V2

2g

Next, we rearrange the preceding equation andsolve for velocity

V = √2(p t - p)ρ

A manometer connected to the pitot-static tubewould provide head loss readings ∆h given by

50

∆h = pt - p

ρg

where density is that of the flowing fluid. Sovelocity in terms of head loss is

V = √2g∆h

Note that this equation applies only toincompressible flows. Compressibility effects arenot accounted for. Furthermore, ∆h is the headloss in terms of the flowing fluid and not in termsof the reading on the manometer.

For flow in a duct, manometer readings are tobe taken at a number of locations within the crosssection of the flow. The velocity profile is thenplotted using the results. Velocities at specificpoints are then determined from these profiles.The objective here is to obtain data, graph avelocity profile and then determine the averagevelocity.

Average VelocityThe average velocity is related to the flow

rate through a duct as

V = QA

where Q is the volume flow rate and A is thecross sectional area of the duct. We can dividethe flow area into five equal areas, as shown inFigure 15.3. The velocity is to be obtained atthose locations labeled in the figure. The chosenpositions divide the cross section into five equalconcentric areas. The flow rate through each arealabeled from 1 to 5 is found as

Q1 = A1V1 Q2 = A2V2

Q3 = A3V3 Q4 = A4V4

Q5 = A5V5

0.316 R

R

0.837 R0.949 R0.707 R0.548 R

FIGURE 15.3. Five positions within the crosssection where velocity is to be determined.

The total flow rate through the entire crosssection is the sum of these:

Qtotal = ∑1

5

Qi = A1V1 + A2V2 + A3V3 + A4V4

+ A5V5

or Qtotal = A1 (V1 + V2 + V3 + V4 + V5)

The total area Atotal is 5A1 and so

V = Qtotal

Atotal =

(Atotal/5)(V1 + V2 + V3 + V4 + V5)Atotal

The average velocity then becomes

V = (V1 + V2 + V3 + V4 + V5)

5

The importance of the five chosen radialpositions for measuring V1 through V5 is nowevident.

Velocity MeasurementsEquipment

Axial flow fan apparatusPitot-static tubeManometer

The fan of the apparatus is used to move airthrough the system at a rate that is small enoughto allow the air to be considered incompressible.While the fan is on, make velocity profilemeasurements at a selected location within theduct at a cross section that is several diametersdownstream of the fan. Repeat thesemeasurements at different fan speed settings sothat 9 velocity profiles will result. Use thevelocity profiles to determine the averagevelocity and the flow rate.

Questions1. Why is it appropriate to take velocity

measurements at several diametersdownstream of the fan?

2. Suppose the duct were divided into 6 equalareas and measurements taken at selectpositions in the cross section. Should theaverage velocity using 6 equal areas be thesame as the average velocity using 5 or 4equal areas?

51

Incompressible Flow Through a MeterIncompressible flow through a venturi and an

orifice meter was discussed in Experiment 9. Forour purposes here, we merely re-state theequations for convenience. For an air over liquidmanometer, the theoretical equation for bothmeters is

Qth = A2 √2g∆h(1 - D2

4/D14)

Now for any pressure drop ∆hi, there are twocorresponding flow rates: Qac and Qth. The ratio ofthese flow rates is the venturi dischargecoefficient Cv, defined as

Cv = Qac

Qth = 0.985

for turbulent flow. The orifice dischargecoefficient can be expressed in terms of the Stolzequation:

Co = 0.595 9 + 0.031 2β 2.1 - 0.184β 8 +

+ 0.002 9β 2.5

106

Re β 0.75

+ 0.09L1

β 4

1 - β 4 - L2 (0.003 37β 3)

where Re = ρVoDo

µ =

4ρQac

πDoµβ =

Do

D1

L1 = 0 for corner tapsL1 = 1/D1 for flange tapsL1 = 1 for 1D & 12D taps

and if L1 ≥ 0.433 3, the coefficient of the

β 4

1 - β 4

term becomes 0.039.

L2 = 0 for corner tapsL2 = 1/D1 for flange tapsL2 = 0.5 - E/D1 for 1D & 12D taps

E = orifice plate thickness

Compressible Flow Through a MeterWhen a compressible fluid (vapor or gas)

flows through a meter, compressibility effectsmust be accounted for. This is done by introductionof a compressibility factor which can bedetermined analytically for some meters(venturi). For an orifice meter, on the other hand,the compressibility factor must be measured.

The equations and formulation developedthus far were for incompressible flow through ameter. For compressible flows, the derivation is

somewhat different. When the fluid flowsthrough a meter and encounters a change in area,the velocity changes as does the pressure. Whenpressure changes, the density of the fluid changesand this effect must be accounted for in order toobtain accurate results. To account forcompressibility, we will rewrite the descriptiveequations.

Venturi MeterConsider isentropic, subsonic, steady flow of

an ideal gas through a venturi meter. Thecontinuity equation is

ρ1A1V1 = ρ2A2V2 = ·misentropic = ·ms

where section 1 is upstream of the meter, andsection 2 is at the throat. Neglecting changes inpotential energy (negligible compared to changesin enthalpy), the energy equation is

h1 + V1

2

2 = h2 + V2

2

2

The enthalpy change can be found by assumingthat the compressible fluid is ideal:

h1 - h2 = Cp(T1 - T2)

Combining these equations and rearranging gives

CpT1 + ·ms

2

2ρ12A1

2 = CpT2 + ·ms

2

2ρ22A2

2

or

·ms2

1

ρ22A2

2 - 1

ρ12A1

2 = 2Cp(T1 - T2)

= 2CpT1

1 -

T2

T1

If we assume an isentropic compression processthrough the meter, then we can write

p2

p1 =

T2

T1

γγ - 1

where γ is the ratio of specific heats (γ = Cp/Cv).Also, recall that for an ideal gas,

Cp = R γ

γ - 1

Substituting, rearranging and simplifying, we get

52

·ms2

ρ22A2

2

1 -

ρ22A2

2

ρ12A1

2 = 2 R γ

γ - 1 T1

1 -

p2

p1

γ - 1γ

For an ideal gas, we write ρ = p/RT. Substitutingfor the RT1 term in the preceding equation yields

·ms2

A22 = 2ρ2

2 γγ - 1

p1

ρ1

1 - (p2/p1) (γ - 1)/γ 1 - (ρ2

2A22/ρ1

2A12)

For an isentropic process, we can also write

p1

ρ1γ =

p2

ρ2γ

or ρ2 =

p2

p1

1/γ ρ1

from which we obtain

ρ22 =

p2

p1

2/γ ρ1

2

Substituting into the mass flow equation, we getafter considerable manipulation Equation 15.1 ofTable 15.1, which summarizes the results.

Thus for compressible flow through a venturimeter, the measurements needed are p1, p2, T1,the venturi dimensions, and the fluid properties.By introducing the venturi discharge coefficientCv, the actual flow rate through the meter isdetermined to be

·mac = Cv ·ms

Combining this result with Equation 15.1 givesEquation 15.2 of Table 15.1.

It would be convenient if we could re-writeEquation 15.2 in such a way that thecompressibility effects could be consolidated intoone term. We attempt this by using the flow rateequation for the incompressible case multipliedby another coefficient called the compressibilityfactor Y; we therefore write

·mac = CvYρ1A2 √2(p1 - p2)ρ1(1 - D2

4/D14)

We now set the preceding equation equal toEquation 15.2 and solve for Y. We obtain Equation15.3 of the table.

The ratio of specific heats γ will be known fora given compressible fluid, and so Equation 15.3

could be plotted as compressibility factor Y versuspressure ratio p2/p1 for various values of D2/D1.The advantage of using this approach is that apressure drop term appears just as with theincompressible case, which is convenient if amanometer is used to measure pressure. Moreover,the compressibility effect has been isolated intoone factor Y.

Orifice MeterThe equations and formulation of an analysis

for an orifice meter is the same as that for theventuri meter. The difference is in the evaluationof the compressibility factor. For an orifice meterthe compressibility factor is much lower thanthat for a venturi meter. The compressibilityfactor for an orifice meter cannot be derived, butinstead must be measured. Results of such testshave yielded the Buckingham equation, Equation15.4 of Table 15.1, which is valid for mostmanometer connection systems.

Calibration of a MeterFigures 15.4 and 15.5 show how the apparatus

is set up. An axial flow fan is attached to theshaft of a DC motor. The rotational speed of themotor, and hence the volume flow rate of air, iscontrollable. The fan moves air through a ductinto which a pitot-static tube is attached. Thepitot static tube is movable so that the velocityat any radial location can be measured. An orificeor a venturi meter can be placed in the ductsystem.

The pitot static tube has pressure taps whichare to be connected to a manometer. Likewise eachmeter also has pressure taps, and these will beconnected to a separate manometer.

A meter for calibration will be assigned bythe instructor. For the experiment, makemeasurements of velocity using the pitot-statictube to obtain a velocity profile. Draw thevelocity profile to scale. Obtain data from thevelocity profile and determine a volume flowrate.

For one velocity profile, measure the pressuredrop associated with the meter. Graph volumeflow rate as a function of head loss ∆h obtainedfrom the meter, with ∆h on the horizontal axis.Determine the value of the compressibility factorexperimentally and again using the appropriateequation (Equation 15.3 or 15.4) for each datapoint. A minimum of 9 data points should beobtained. Compare the results of bothcalculations for Y.

53

TABLE 15.1. Summary of equations for compressible flow through a venturi or an orifice meter.

·ms = A2

2p1ρ1 (p2/p1)2/γ [γ/(γ - 1)] [1 - (p2/p1)(γ - 1)/γ ]

1 - (p2/p1)2/γ (D24/D1

4)

1/2

(15.1)

·mac = Cv A2

2p1ρ1 (p2/p1)2/γ [γ/(γ - 1)] [1 - (p2/p1)(γ - 1)/γ ]

1 - (p2/p1)2/γ (D24/D1

4)

1/2

(15.2)

Y = √γγ - 1

[(p2/p1)2/γ - (p2/p1)(γ + 1)/γ](1 - D24/D1

4)[1 - (D2

4/D14)(p2/p1)2/γ](1 - p2/p1) (venturi meter) (15.3)

Y = 1 - (0.41 + 0.35β 4) (1 - p2/p1)

γ (orifice meter) (15.4)

roundedinlet

outlet ductaxial flowfan

motor

manometerconnections

pitot-statictube

venturi meter

FIGURE 15.4. Experimental setup for calibrating a venturi meter.

roundedinlet

outlet ductaxial flowfan

motor

orifice plate

manometerconnections

pitot-statictube

FIGURE 15.5. Experimental setup for calibrating an orifice meter.

54

EXPERIMENT 16

MEASUREMENT OF FAN HORSEPOWER

The objective of this experiment is to measureperformance characteristics of an axial flow fan,and display the results graphically.

Figure 16.1 shows a schematic of theapparatus used in this experiment. A DC motorrotates an axial flow fan which moves airthrough a duct. The sketch shows a venturi meterused in the outlet duct to measure flow rate.However, an orifice meter or a pitot-static tubecan be used instead. (See Experiment 16.) Thecontrol volume from section 1 to 2 includes all thefluid inside. The inlet is labeled as section 1, andhas an area (indicated by the dotted line) so hugethat the velocity at 1 is negligible compared tothe velocity at 2. The pressure at 1 equalsatmospheric pressure. The fan thus acceleratesthe flow from a velocity of 0 to a velocity weidentify as V2. The continuity equation is

m· 1 = m· 2

The energy equation is

0 = - dWd t

+ m· 1

h1 +

V 12

2 - m· 2

h2 +

V22

2

where dW/dt is the power input from the fan tothe air, which is what we are solving for. Bysubstituting the enthalpy terms according to thedefinition (h = u + pv), the preceding equationbecomes

d Wd t

= m· (u1 - u2)

+ m·

p1

ρ + V1

2

2 -

p2

ρ + V2

2

2

Assuming ideal gas behavior, we have

u1 - u2 = Cv(T1 - T2)

With a fan, however, we assume an isothermal

process, so that T1 ≈ T2 and ρ1 ≈ ρ2 = ρ. With m· =ρA V (evaluated at the outlet, section 2), theequation for power becomes

d Wd t

= A2V2

p1 +

ρV12

2 -

p2 +

ρV22

2

Recall that in this analysis, we set up our controlvolume so that the inlet velocity V1 = 0; actuallyV1 << V2. Thus

p1 +

ρV12

2 -

p2 +

ρV22

2 ≈

p1 - p2 -

ρV22

2

in which p1 is atmospheric pressure, and p2 ismeasured at section 2. The quantity in brackets inthe previous equation is the change in totalpressure ∆pt. Thus, the power is

d Wd t

= ∆ptQ (16.1)

This is the power imparted to the air from thefan.

Data AcquisitionThe motor controller is used to set the

rotational speed of the fan, which in turn controlsthe volume flow rate of air through the duct. Onthe side of the motor is a shaft (a torque arm)that extends outward normal to the axis ofrotation. When the fan rotates, the motor tends torotate in the opposite direction. A weight can be

roundedinlet

outlet ductaxial flowfan

motor

manometerconnections

venturi meter

1 2

FIGURE 16.1. Schematic of setup of fan horsepower experiment.

55

placed on the torque arm to reposition the motorto its balanced position. The product of weightand torque arm length gives the torque input frommotor to fan.

A tachometer is used to measure therotational speed of the motor. The product oftorque and rotational speed gives the power inputto the fan:

dWa

d t = Tω (16.2)

This is the power delivered to the fan from themotor.

The efficiency of the fan can now becalculated using Equations 1 and 2:

η = d W / d td W a/d t

(16.3)

Thus for one setting of the motor controller, thefollowing readings should be obtained:

1. An appropriate reading for the flow meter.2. Weight needed to balance the motor, and its

position on the torque arm.3. Rotational speed of the fan and motor.4. The static pressure at section 2.

With these data, the following parameterscan be calculated, again for each setting of themotor controller:1. Outlet velocity at section 2: V2 = Q/A2.2. The power using Equation 16.1.3. The input power using Equation 16.2.4. The efficiency using Equation 16.3.

Presentation of ResultsOn the horizontal axis, plot volume flow

rate. On the vertical axis, graph the power usingEquation 1, and Equation 2, both on the same set ofaxes. Also, again on the same set of axes, graphtotal pressure ∆pt as a function of flow rate. On aseparate graph, plot efficiency versus flow rate(horizontal axis).

56

���EXPERIMENT 17

EXTERNAL LAMINAR FLOWS OVER IMMERSED BODIES

The flow of fluid over an immersed body iscalled an external flow. There are qualitativeexplanations for the behavior of flows overbodies. Purely analytic solutions to flow fieldsare limited to only a few simple geometries.External flows are characterized by two regionsin the flow field—the viscous boundary layernear the solid surface of the body, and theinviscid free stream. For flow past some immersedbodies, the boundary layer can separate when theflow near the body develops an adverse pressuregradient, resulting in a region of flow reversal.Locating the point of separation is importantwhen analyzing external flows, because thelocation will have a significant impact on thepressure distribution and on the correspondingforces acting on the body.

Hele-Shaw flows are a special class ofexternal flows where a body is positionedbetween two parallel flat plates and the Re < 1.The resulting pattern of streamlines for thesecreeping laminar flows is identical to thepotential or inviscid flow solutions obtainedanalytically.

EquipmentLaminar Flow Table ApparatusDigital Camera

Available Shapes(use those assigned by the instructor)Cylinder (disk)RectangleAirfoilVenturi

Figure 17.1 is a schematic of the laminar flowtable used in this experiment. The apparatusconsists of a one piece base made of fiberglas withwater reservoirs at each end. Water enters theupstream reservoir and flows over the lowersurface of the device to the downstream tank.Water leaves this tank and goes to the drain.

A glass cover can be placed over the table andbody to model what is referred to as inviscidHele-Shaw flow behavior. Alternatively, theglass cover is removed to observe the influence ofthe viscous boundary layer on the flow. The lowersurface is marked with a grid to aid inobservation and orientation of the bodies. Eightsmall holes in the table surface (not shown in the

sketch) allow for the introduction of source andsink flows. Control valves for each of the eightsource and sink holes are also included in theapparatus.

Mounted at the back of the table is the dyereservoir. A valve at the base of this reservoircontrols the flow of the dye, used forvisualization of the flow, to the injection needlespositioned at the entrance of the table surface.

ProcedureBegin by filling the source reservoir with

water. Water will fill the reservoir until itreaches the level of the marked table surface. Asa result of the incline of the apparatus, the watershould flow down in a thin sheet along the tablesurface. If the glass cover is in place, then thewater should be flowing between the tablesurface and the glass cover. Any air pocketsshould be evacuated before injecting the dye.Place the desired body on the table surface andbegin injecting the dye. Data collection, in theform of digital photos, can begin once the flowhas reached steady-state. Source and sink flowsmay be added to obtain other classic flows, suchas flow past a Rankine Oval.

Questions1. How do the observed flow patterns for the

Hele-Shaw flows compare to publishedinviscid (potential) flow analyticsolutions? Discuss your results and includeyour sources in the discussion.

2. How do the observed flow patterns for theboundary layer flows compare to otherpublished experimental results or analyticsolutions? Discuss your results and includeyour sources in the discussion.

3. In observing (non Hele-Shaw) flow past acylinder, would you expect the separationpoints on the cylinder to shift location ifthe flow rate were increased? Why or whynot?

4. Are the dye lines observed on the LaminarFlow Table Apparatus streamlines,streaklines, or both? Explain your answer.

AcknowledgementsTo Dr. Jeffrey Marchetta for implementing thisexperiment, and to Darrin Bevard for getting theapparatus installed and working.

57

removable glass cover

dye reservoirvalve

manifold with injectors

dye water flowwater inlet

to drain

planview

profileview

grid on surfacebeneath glass

manifold with injectors

grid on surfacebeneath glass

circulardisk

circulardisk

FIGURE 17.1. Plan and profile views of the laminar flow table. The setup is for Hele-Shaw flow past acylinder.

58

EXPERIMENT 18

SERIES-PARALLEL PUMP PERFORMANCE

The objective of this experiment is to makemeasurements on an apparatus that containstwo identical pumps, and determine: (a) theperformance of one of the pumps; (b) theperformance of both pumps while operating inseries; and, (c) the performance of both pumpswhile operating in parallel.

Figure 18.1 is a sketch of the apparatus. Awater tank acts as a supply reservior fromwhich water is pumped. For individual pumpperformance, we use pump (1). Water is pumpedfrom the tank through the valve at B. It isdischarged through the valve at A, with valveD closed. Valve E is used as a flow controlvalve, while the others are used for on-offoperation.

For series pump operation, pump (1) takesin water with valve B fully open. Water isdischarged with the valve at A closed, thevalve at D fully open, and the valve at Cclosed. Thus, flow that is discharged frompump (1) enters pump (2). The water from pump(2) is discharged through valve E which is theflow control valve.

For parallel operation, the valves at B andC are both open, and the valve at D is closed.Valves A and E are both open with the valve

at E used to control the flow. Thus both pumpsoperate simultaneously, with the dischargefrom both combined through the flow controlvalve.

Measurement and Control SystemsBoth motors are controlled by external

rotary rheostats, and for series or paralleloperation, should be run at the same powersetting. Rotational speed sensors are mounted oneach motor housing. There are flow meters atthe outlet of each pump and in the pump justdownstream of valve E. Attached to the flowmeters are differential pressure sensors.Upstream of pump (1) is a temperature sensor.Output from all sensors is transmitted to anelectronics circuit, which transmits signals to acomputer. Measurements taken are compiledand displayed by a computer, which removesthe need to take measurements manually, andto analyze the results “by hand.” The sensorsand the electronics systems are alreadyconnected to the computer, and the programsneeded for analysis are loaded. The computerwill produce performance graphs and tables forany of the experiments.

centrifugalpump (1)

centrifugalpump (2)

motor

motor

water tank

A

B

C

DE

differentialpressuresensor

differentialpressuresensor

rotationalspeedsensor

temperaturesensor

FIGURE 18.1. Schematic of the series-parallel pump apparatus.

59

Single Pump PerformanceThe objective here is to obtain a head flow

curve (∆H vs Q) for a centrifugal pumpoperating at one speed.

Procedure• Launch the program FM21SNGL on the

computer.• Use pump (1) and be sure that the valves

are set appropriately: valves A, B, and Eare open. Valves C and D are closed.

• Start pump (1); pump (2) should remain off.Decide on a power setting for pump (1) [suchas 50% or 75% or 100%] and set its controlleraccordingly. This setting is used for thisand the subsequent experiments.

• Select “Diagrm” and note the value of thevolume flow rate displayed by thecomputer.

• Decide on suitable increments to use for flowrate so that typically 15 data points areobtained between zero and maximum flowrate.

• Close valve E for the condition of no flow(Q = 0). When the readings on the screenbecome sufficiently steady, select “TakeSample.” This is the first data point. DONOT allow the pump to operate at zeroflow for any longer than necessary.

• Open valve E slightly to the first incrementin flow rate decided upon earlier. Whenthe readings are sufficiently steady, select“Take Sample.”

• Repeat the previous step for other settingsof valve E, corresponding to increasingvalues of flow rate. The last sample pointcorresponds to valve E being fully open.

• The recorded data set can now be examinedvia any of the selectable options:“Graphs,” “Tables,” or downloaded into aspreadsheet. Select the “Graphs” optionand obtain a head versus flow rate curve.(See software help screens if necessary.)

Series Pump PerformanceThe objective here is to obtain a head flow

curve (∆H vs Q) for a two identical centrifugalpumps operating at the same speed, andoperated such the flow leaving pump (1) entersthat of pump (2). When two pumps operate inseries, the combined head versus flow ratecurves is found by adding the heads of thesingle pump curves at the same flow rates.

Procedure• Launch the program FM21SERS on the

computer.

• Use both pumps and be sure that the valvesare set appropriately: valve A is closed.All other valves are open.

• Start both pumps, and set them on the samepower setting that was used in the singlepump experiment. This setting is used forthis and the subsequent experiment.

• Select “Diagrm” and note the value of thevolume flow rate displayed by thecomputer.

• Decide on suitable increments to use for flowrate so that typically 15 data points areobtained between zero and maximum flowrate.

• Close valve E for the condition of no flow(Q = 0). When the readings on the screenbecome sufficiently steady, select “TakeSample.” This is the first data point. DONOT allow the pump to operate at zeroflow for any longer than necessary.

• Open valve E slightly to the first incrementin flow rate decided upon earlier. Whenthe readings are sufficiently steady, select“Take Sample.”

• Repeat the previous step for other settingsof valve E, corresponding to increasingvalues of flow rate. The last sample pointcorresponds to valve E being fully open.

• The recorded data set can now be examinedvia any of the selectable options:“Graphs,” “Tables,” or downloaded into aspreadsheet. Select the “Graphs” optionand obtain a head versus flow rate curve.(See software help screens if necessary.)

Parallel Pump PerformanceThe objective here is to obtain a head flow

curve (∆H vs Q) for a two identical centrifugalpumps operating at the same speed, andoperated in parallel. Both pumps take in waterfrom the tank, and discharge the water into acommon line containing the valve at E. Whentwo pumps operate in parallel, the combinedhead versus flow rate curves is found by addingthe flow rates of the single pump curves at thesame head.

Procedure• Launch the program FM21PARA on the

computer.• Use both pumps and be sure that the valves

are set appropriately: valve D is closed.All other valves are open. .

• Start both pumps, and set them on the samepower setting that was used in the singlepump experiment.

60

• Click on the button “single/parallel” on the“Diagrm” screen. Note that the computercan analyze data for single or for parallelpump operation. (Single pump operationwas already completed, but if it was not, itcan be performed while the pumps areoperating in parallel.)

• Note the value of the volume flow ratedisplayed by the computer.

• Decide on suitable increments to use for flowrate so that typically 15 data points areobtained between zero and maximum flowrate.

• Close valve E for the condition of no flow(Q = 0). When the readings on the screenbecome sufficiently steady, select “TakeSample.” This is the first data point. DONOT allow the pump to operate at zeroflow for any longer than necessary.

• Open valve E slightly to the first incrementin flow rate decided upon earlier. Whenthe readings are sufficiently steady, select“Take Sample.”

• Repeat the previous step for other settingsof valve E, corresponding to increasingvalues of flow rate. The last sample pointcorresponds to valve E being fully open.

• The recorded data set can now be examinedvia any of the selectable options:“Graphs,” “Tables,” or downloaded into aspreadsheet. Select the “Graphs” optionand obtain a head versus flow rate curve.(See software help screens if necessary.)

ResultsCompare the results obtained from this

experiment. Do the head versus flow ratecurves come out as expected; i.e., as illustratedin Figure 18.2? Select several data points andmake calculations to verify the performancegraphs. What conclusions can be drawnregarding series and parallel pump operation?Is the use of the computer more convenient thanmaking calculations and producing graphs byhand (or with a spreadsheet)?

61

two pumps

one pump

H

H2

H1

Q

any Q

H

two pumps

one pump

Q

any H

Q1 Q2

series pump performancefor any Q

∆H2 = 2∆H1

parallel pump performancefor any ∆HQ2 = 2Q1

FIGURE 18.2. Performance graphs for series and parallel pump configurations.

62

EXPERIMENT 19

DESIGN OF EXPERIMENTS:CALIBRATION OF AN ELBOW METER

There are many types of meters that can beinstalled in a pipeline—venturi, orifice,rotameter, and turbine-type. These meters can allbe calibrated to provide a reading of the volumeflow rate of fluid through the pipe.

An alternative, less expensive flow meter—known as an elbow meter—can also be used. For anexisting pipeline containing elbows, an elbowmeter is perhaps the easiest meter to set up. Allthat would be required is to drill and tap a coupleof holes in the elbow, and attach them to adevice for measuring pressure drop. Informationon an elbow meter is available from [1]. A sketchof an elbow meter is shown in Figure 19.1

flow

D

R

45o

pressuretap

FIGURE 19.1. Schematic of an elbow metershowing pressure tap locations.

Installation. It is necessary to select an elbow in aline where flow rate is to be measured. Theinternal dimensions of R and D must be known forthe elbow of interest. Holes are drilled in specificlocations as shown to accept 1/8th nominal pipethreads or something different if desired.

Pressure measurements are made to determinethe pressure drop that exists as well as thecorresponding flow rate.

Theory. The pressure measurement devices areattached to the connectors and flow through thepipe is initiated. The difference in pressurebetween the tap locations is to be determined.

As given in [1], the flow rate through themeter can be calculated with:

Q = A K √RD ∆ pρ

where

A = cross sectional area = πD2/4 in ft2

K= 1 - 6.5

√Re = correction factor (19.1)

Re = Reynolds number = ρVD/µ

104 ≤ Re ≤ 106

ρ = fluid density in slug/ft3

µ = fluid viscosity in lbf·s/ft2

∆p = pressure drop in lbf/ft2

D = diameter in ft

R = radius of elbow in ft

Sample Calculation. Show how flow rate iscalculated using a 4 nominal schedule 80 shortradius elbow meter. Pressure is measured with amanometer. Water is the working fluid.

Solution: For a 4 nominal, short radius elbow wehave [2]:

R = 4 in. = 0.333 ft D = 0.3198 ft

We calculate

A = πD2

4 = π(0.3198)2

4 = 0.0803 ft2

For water,

ρ = 1.94 slug/ft3 µ = 1.9 x 10-5 lbf·s/ft2

A calibration curve is one that relates the volumeflow rate through the meter to the pressure drop,using whatever units are convenient. We arecalculating the calibration results for only onedata point in such a curve. In this example, wemeasure flow rate in gpm, and because we areusing a manometer, we express pressure drop in

63

terms of inches of water. A sample calculation fora flow rate of, say, 150 gpm is as follows. Wehave

Q = 150 gpm (2.229 x 10-3) = 0.334 ft3/s

The flow velocity then is

V = QA

= 0.3340.0803 = 4.16 ft/s

To find ∆p, we must first find K which in turndepends on Reynolds number:

Re= ρVD

µ =

1.94(4.16)(0.3198)1.9 x 10-5 = 1.36 x 105

Then

K = 1 - 6.5

√Re = 1 -

6.5

√1.36 x 105 = 0.982

For an elbow meter,

Q = A K √RD ∆ pρ

Rearranging and solving for ∆p gives

∆p = ρ

Q

A K

2 DR

(19.2)

Substituting

∆p = (1.94)

0.334

0.0803(0.982)2 0.31980.333

or ∆p = 33.4 lbf/ft2

Now in terms of a column of water,

∆h = ∆pρg

= 33.4

1.94(32.2) = 0.534 ft of water

or ∆h = 6.41 in. of water

For calculations of this type on an elbow that hasnot been tested in the laboratory, the result isaccurate to within ± 4%. That is, for a reading ∆hof 6.41 in. of water, the flow rate can be as highas 156 gpm or as low as 144 gpm. Note that amanometer is not readable to the nearesthundredth of an inch. Typically a reading will beto the nearest tenth of an inch.

So based on the results here, one data point onthe calibration curve is:

∆h = 6.4 in of water Q = 150 gpm

Calibration. In the preceding calculation, notethe dependence of the results on having anequation for the correction factor K (Equation19.1). The equation for K was derived fromtheoretical considerations [1], but it is desirableto have experimental results to determine a“better” relationship for it. We could useEquation 19.2 with experimental data todetermine K, which is the subject of this exercise.

Thus, if we had an apparatus that is set upand ready to use, we could measure flow rate Q,determine R and D from a handbook [2], calculateA, and reduce Equation 19.2 to:

∆p = (a known constant) x

Q

K

2(19.3)

Then with the apparatus, we could obtain ∆p vsQ for a number of data points. Using Equation19.3, we could then calculate K.

Experiment Design1. Design an apparatus for making measurementson an elbow meter. It is desired to have thecapability of making measurements on 3/4, 1 and1 1/2 inch line sizes/elbows. A sketch of theapparatus is required, showing:

a) what fluid is to be usedb) the prime mover for pumping the fluidc) where the fluid is to be storedd) an accurate method for determining actual

flow rate

2. Write a procedure for the operator(s) to followin order to obtain the desired data.

3. Write a theory section that leads the readerthrough a sample calculation and showsspecifically how the correction factor K isdetermined.

4. How many data points should be obtained sothat K can be determined with 95% confidence?

References[1] Fluid Meters: Their Theory and Application,6th edition, 1971, ASME, New York, page 75.

[2] (Perry’s Chemical Engineering Handbook, pg.6-57.

64

EXPERIMENT 20

DESIGN OF EXPERIMENTS:MEASUREMENT OF FORCE ON A CONICAL OBJECT

A jet of fluid striking a stationary objectexerts a force on that object. This force can bemeasured when the object is connected to a springbalance or scale. The force can then be related tothe velocity of the jet of fluid and in turn to therate of flow. This was the subject of the Impact ofa Jet of Water experiment, in which the forceexerted by a water jet on a flat plate or ahemisphere was measured. Calculations weremade to theoretically predict this force, andwere compared to the measured results.

The force developed by a jet stream of waterdirected at a concave cone is the subject of thisexperiment.

Figure 20.1 is a schematic of the device usedin this experiment. The device consists of a tankwithin a tank. The interior tank is supported on apivot and has a lever arm attached to it. Aswater enters this inner tank, the lever arm willreach a balance point. At this time, a stopwatchis started and a weight is placed on the weighthanger (e.g., 10 lbf). When enough water hasentered the tank (10 lbf), the lever arm willagain balance. The stopwatch is stopped. Theelapsed time divided into the weight of watercollected gives the weight or mass flow rate ofwater through the system (lbf/sec, for example).

The outer tank acts as a support for the tabletop as well as a sump tank. Water is pumped fromthe outer tank to the apparatus resting on thetable top. As shown in Figure 20.1, the impactapparatus contains a nozzle that produces a highvelocity jet of water. The jet is aimed at an object(such as a flat plate or hemisphere). The forceexerted on the plate causes the balance arm towhich the plate is attached to deflect. A weightis moved on the arm until the arm balances. Asummation of moments about the pivot point ofthe arm allows for calculating the force exertedby the jet.

Water is fed through the nozzle by means ofa centrifugal pump. The nozzle emits the water ina jet stream whose diameter is constant. After thewater strikes the object, the water is channeled tothe weighing tank inside to obtain the weight ormass flow rate.

The variables involved in this experimentare listed and their measurements are describedbelow:1. Mass rate of flow–measured with the

weighing tank inside the sump tank. The

volume flow rate is obtained by dividingmass flow rate by density: Q = m/ρ.

2. Velocity of jet–obtained by dividing volumeflow rate by jet area: V = Q/A. The jet iscylindrical in shape with a diameter of 0.375in.

3. Resultant force—found experimentally bysummation of moments about the pivot pointof the balance arm. The theoretical resultantforce is found by use of an equation derived byapplying the momentum equation to a controlvolume about the plate.

Impact Force Analysis The total force exerted by the jet equals the

rate of momentum loss experienced by the jet afterit impacts the object. For a cone, as shown inFigure 20.2, whose included half angle is α,

F = ρQ2

A (1 + cos α) (20.1)

A

A

FIGURE 20.2. A schematic of a conical impactobject.

Experiment Design1. Design a conical object to be used with theapparatus in the lab; i.e., the apparatus used inExperiment 5. The cone dimensions must beselected as well as a method for mounting thecone in the apparatus. Prepare a dimensioneddrawing of your design.

65

2. The method of making flow rate measurementsis believed to be antiquated, and should bereplaced with another method. Determine analternative method of measuring flow rate thatis compatible with the apparatus used inExperiment 4. The instrument(s) selected shouldhave a digital readout, and should provide datain appropriate units.

3. Write a procedure for the operator(s) to followin order to obtain the desired data.

4. Write a theory section that leads the readerthrough a sample calculation and showsspecifically how Equation 20.1 is derived.

flat platepivot

balancing weight lever arm withflat plate attached

water jet nozzle

drain

weigh tank

plug

sump tank

motor pump

weight hanger

flow controlvalve

tankpivot

FIGURE 20.1. A schematic of the jet impact apparatus.

66

Appendix

Calibration CurvesV-notch weir—turbomachinery experiments

67

0 50 100 150 200 250 300 3500

20

40

60

80

100

120

volume flow rate in liters/min

heig

ht r

ead

ing

in m

m

FIGURE A.2. Calibration curve for the V-notch weir, turbomachinery experiments.