Heat Tr Manual

8/3/2019 Heat Tr Manual

1/89

A Manual for

MECH 4335THERMODYNAMICS

ANDHEAT TRANSFER

LABORATORY

William S. JannaDepartment of Mechanical Engineering

University of Memphis


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2008 William S. Janna

All Rights Reserved.No part of this manual may be reproduced, stored in a retrievalsystem, or transcribed in any form or by any meanselectronic, magnetic,

mechanical, photocopying, recording, or otherwisewithout the prior written consent of William S. Janna


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TABLE OF CONTENTS

Item Page

Course Learning Outcomes, Cleanliness, and Safety .................................................................4

Code of Student Conduct .........................................................................................................5Statistical Treatment of Experimental Data ...........................................................................6

Report Writing ......................................... ........................................... ................................ 16

Experiment 1 Thermocouples and Instrumentation .............................. .................... 21

Experiment 2 Measurement of Thermal Conductivity of a Metal ...................... ....... 24

Experiment 3 Effect of Area on One Dimensional Conduction................................... 26

Experiment 4 Measurement of Thermal Conductivity of an Insulator ....................... 28

Experiment 5 Determination of Contact Resistance................................................. 30

Experiment 6 Radial One Dimensional Conduction ............................ .................... 32

Experiment 7 Heat Transfer from a Fin ............................... ................................... 33

Experiment 8 Determination of Conduction Shape Factor .......................... ............. 37

Experiment 9 Graphical Solutions to Unsteady Heat Transfer Problems.................. 40

Experiment 10 Transient Conduction with Convection ................................ .............. 43

Experiment 11 Natural Convection Heat Transfer: Flat, Finned, and Pinned Plates... 47

Experiment 12 Forced Convection Flat Plate to Air .............................. .................... 49

Experiment 13 Radiation Heat Transfer I .................................... ............................ 51

Experiment 14 Radiation Heat Transfer II............................................................... 54

Experiment 15 Emissivity of Black and Gray Surfaces.............................................. 55Experiment 16 Radiation View Factor..................................................................... 59

Experiment 17 Analysis of a Double Pipe Heat Exchanger........................................ 64

Experiment 18 Analysis of a Plate and Frame Heat Exchanger ......................... ........ 70

Experiment 19 Analysis of a Shell and Tube Condenser .............................. .............. 76

Experiment 20 The Vapor Compression Refrigeration Cycle ........................ ............. 80

Appendix ..................................... ......................................... .......................... 83

Experiment TBA Analysis of a Cross Flow Heat Exchanger..............................................


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Course Learning OutcomesThe Heat Transfer Laboratory experiments areset up so that experiments can be performed tocomplement the theoretical information taughtin the heat transfer lecture course. Thus topicalareas have been identified and labeled as CourseLearning Outcomes (CLOs). The CLOs in theMECH 4335 Laboratory are as follows:

TABLE 1. Course Learning Outcomes

1. Identify safe operating practices andrequirements for laboratory experiments

2. Measure temperature with a thermocouple.3. Perform one dimensional conduction

experiment(s).4. Perform multidimensional conduction

experiment.5. Perform transient conduction experiment.6. Perform forced convection experiment.7. Perform natural convection experiment.8. Perform radiation experiment.9. 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, there

are 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 Heat Transfer 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 removalof spilled water (or any liquid), and wiping thetable top on which the equipment is mounted (ifappropriate). The lab should always be as clean

or 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 storagecabinets in the Heat Transfer Lab involvesresolving a variety of conflicting problems. Theseinclude traffic flow, emergency facilities,environmental safeguards, exit door locations,etc. The goal is to implement safety requirementswithout impeding egress, but still allowingadequate work space and necessary informalcommunication opportunities.

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. However, waste materials aregenerated periodically and they should bedisposed of properly.

Not infrequently, specimens under study areheated by the use of a heat source. The student isadvised to use caution when conductingexperiments that involve heated surfaces,because usually there is no visual indication thata specimen is hot.

Safety Procedures. There is unmistakablyonly one, clearly marked exit in this laboratory.It has a single door and leads directly to thehallway on the third floor of the EngineeringBuilding. In case of fire, exit the lab to thehallway. After closing the door, take the stairsdown to first floor, and leave the building.

There is a safety cabinet attached to thewall of lab adjacent to the door. In case ofpersonal injury, the appropriate item should be

taken from the supply cabinet and used in therecommended fashion. If the injury is seriousenough to require professional medical attention,the student(s) should contact the MechanicalEngineering Department in EN 312, Extension2173.

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

19981999 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 be

necessary to maintain campus conditionsand 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 students violation ofsuch laws or ordinances also adversely affects the

Universitys 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 violatesstate, local or national laws.

The term academic misconduct includes, butis not limited to, all acts ofcheating andplagiarism.

The term cheating includes, but is not limited

to:a. use of any unauthorized assistance in takingquizzes, 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 students sole independent effort, for thepurpose of affecting the students 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, any

University 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 students 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 asdrop 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 of

materials 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 heat transfer geometriesand relating those measurements to derived

equations. The objective is to determine how wellthe derived equations describe the physicalphenomena we are modeling. In doing so, we willneed to make physical measurements, and it isessential that we learn how to practice goodtechniques in making scientific observations andin obtaining measurements. We are makingquantitative estimates of physical phenomenaunder controlled conditions.

MeasurementsThere are certain primary desirable

characteristics involved when making these

physical measurements. We wish that ourmeasurements would be:

a ) Observer-independent,b) Consistent, andc) Quanti tative

So when reporting a measurements, we will bestating a number, but whats in a number? A singlenumber, in isolation, has almost no significance,but the implied question is, Is it large or small?

Is 26 a large number? Is 6 x 105 a large number?The answer requires another number for reference

purposes. Is 26 a large number compared to 6 x 105?Furthermore, we will have to add a dimensionand this leads to another question: Are wetalking about a number or a dimensional physicalquantity? We know from experience that aphysical value without a unit has nosignificance.

In reporting measurements, another questionarises as to how should we report data; i.e., howmany significant digits should we include?Which physical quantity is associated with themeasurement, and how precise should it or could

it be.For example, does 2.54 cm = 2.54001 cm? It isimpossible to answer this question without somemeasure of the expected natural variation in themeasurement. So it is prudent to scrutinize theclaimed or implied accuracy of a measurement.

Performing experimentsIn the course of performing an experiment, we

first would develop a set of questions or ahypothesis, or put forth the theory. We then

identify 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.

EstimationIn some situations, there is no time to run

formal experiments to answer a question or verifyan equation. In such cases, it is often useful tomake careful estimates. These can help todetermine the ranges of parameters to investigatein the experiments. Also, estimates are necessaryfor partial validation of experimental results.

Consider, for example, that we must obtain aquick estimate of the density of a rock. We

observe that it sinks in water, so it must be moredense than water, 1 000 kg/m3. As an upper bound,we might suggest that it is less dense than steelat 3 000 kg/m3.

So if we conduct an experiment and obtain avalue outside this range, we would be suspiciousand check the equipment and the experimentalapproach.

Comments on Performing Experiments Keep in mind the fundamental state of

questions or hypotheses.

Make sure the experiment design will answerthe 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.


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Error & UncertaintyDefinitionsThe laboratory in heat transfer is designed to

provide the students with experiments thatverify the descriptive equations we derive tomodel physical phenomena. The laboratoryexperience involves making measurements of heatflux and temperature, among other things.However, we have to ask if the measurements we

make are accurate and/or precise. In thefollowing paragraphs, we will examine ourmeasurement methods and define terms thatapply. These terms include error, uncertainty,accuracy, and precision.

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

E x xi= (1 )

There is no error-free measurement. All

measurements 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 ofx is unknown,then the error E is unknown as well. This isalways the case.

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 anominalvalue) can be calculated. Hence, theerror becomes:

E x x=

However, because x is unknown, E is stillunknown.

Uncertainty. The uncertainty, x, is an estimateofE 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%.

Example 1. Consider a measurement that isreported as:

p = 50 psi 5 psi

What is the accuracy of the pressure probe usedfor making this measurement?

Solution: The relative uncertainty is calculatedto be

pp

5

50 0 1 10. %

Thus the accuracy may be estimated to be(around) 10%.

Example 2. A sensor is claimed to be accurate to5%. What will be the uncertainty (in psi) in themeasurement of a pressure of 50 psi?

Solution: The accuracy ( relative uncertainty)is 5%, so

pp

5%

p p 0 05 0 05 50. .

or p psi 2 5.


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The uncertainty in p in psi is 2.5 psi, so themeasurement should be reported as follows:

p = 50 psi 2.5 psi.

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 2 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 blue

darts 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.

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 systematic

errors. 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,


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1

18

4

13

6

10

15

2

173

7

16

8

11

14

9

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5

FIGURE 1a.Accurate and Precise

1

18

4

13

6

10

15

2

173

7

16

8

11

14

9

12

5

FIGURE 1b.Precise but not Accurate.

1

18

4

13

6

10

15

2

173

7

16

8

11

14

9

12

5

FIGURE 1c.Precise but not Accurate.

1

18

4

13

6

10

15

2

173

7

16

8

11

14

9

12

5

FIGURE 1d.Neither Precise nor Accurate.

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 anticipated

and/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 theNational Institute of standards andtechnology (NIST formerly the National

Bureau of Standards); orb) 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 morereliable 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:


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TABLE 1.Examples of relative and absoluteuncertainty.

Relative uncertainty Uncertainty in units

3.45 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. So as a rule of thumb, we use a95% confidence interval which gives us anestimate. Now the estimate of uncertaintydepends on the measurement type: single samplemeasurements, measurements of dependentvariables, or multi variable measurements.

Single-sample measurements. Single-samplemeasurements are those in which the

uncertainties cannot be reduced by repetition. Aslong as the test conditions are the same (i.e.,same sample, same instrument and sameobserver), the measurements (for fixed variables)are single-sample measurements, regardless ofhow many times the reading is repeated.

Measurement Of Function Of More Than OneIndependent Variables. In many cases, severaldifferent quantities are measured in order tocalculate another quantitya dependentvariable. For example, the measurement of thesurface area of a rectangle is calculated using

both its measured length and its measured width.Such a situation involves a propagation ofuncertainties.

Multi-Sample Measurements. Multi-samplemeasurements involve a significant number ofdata points collected from enough experiments sothat the reliability of the results can be assured by 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 samples

and/or different instruments) will allow theuncertainties to be reduced by the sheer number ofobservations.

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.

Consider some measuring device that has asits smallest scale division x. The smallest scaledivision limits our ability to measure somethingwith any more accuracy than x/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 guesstimatethe 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 incrementdisplayed. 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.


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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 a

known height, the approximate equation wouldbe:

g

L

t=

22

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

g

L

t= =

=

2 2 50 00

3 110 4

2 22.

.. m/s

We now wish to estimate the uncertainty g inour calculation ofg. Obviously, the uncertaintyg will depend on the uncertainties in themeasurements ofL and t. Let us examine theworst cases. These may be calculated as:

gmin

.

..=

=

2 49 99

3 67 7

22m/s

and

gmax

.

..=

=

2 50 01

2 614 8

22m/s

The confidence interval aroundg then is:

7 7 14 82 2. .m/s m/s g

Now it is unlikely for all single-sampleuncertainties in a system to simultaneously be theworst possible. Some average or norm of theuncertainties must instead be used in estimating acombined uncertainty for the calculation ofg.

Definition of the Euclidean Norm. In general, ifthe quantityf is determined by an equation

involving n independent variables xi:

f x x xn( , , ..., )1 2 (5 )

and the uncertainty in each independentmeasurement variable xi is called xi then theuncertainty inf is given by:

ff

xx

ii

i

n=

=

2

1(6 )

We will need a process or an algorithm forcalculating f. The procedure is as follows:1. Write expression forf in terms of its

independent variables, xi

2. Evaluate eachf

xiterm separately

3. Calculate

ff

xx

ii

i

n=

=

2

1

4. Calculate the relative uncertainty f/f.

As an example of this procedure, we calculatethe Euclidean Norm in the example for

determining the gravitational constant:

1. Write expression for g in terms of itsindependent variable(s):

g

L

t= 2

2

2. Evaluate each partial derivative termseparately:

=g

L t

22

=g

t t

43

3. Calculate the Euclidean norm

g

tL

L

tt=

+

2 42

2

3

2

g =

+

2

3 10 01

4 50 00

3 10 5

2

2

3

2

..

.

..

g = 3 42. m/s

4. Alternatively, calculate the relativeuncertainty g/g:

gg

L

L

t

t=

+

2 22


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gg

32%

Note that the expression for g/g is simpler thanthat for g. Also, in the g/g expression, theindividual terms are dimensionless. This isconvenient if quantities are originally given in %,

or if the units are incompatible.Now, in calculatingg and g/g, we found:

g = 10 42. m/s

and

gg

32%

So, the measurement should be reported as:

g = 10 322m/s %

This is an example of bad experiment or poorresults.

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:

dn

dii

n= =

=

10

1

Note that the arithmetic mean deviation is zero:

Standard Deviation. The standard deviation isgiven by:

=

=

( )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

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

(10)

The mean and the standard deviation are all

the 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 from themean value, the resulting value is 0.6826. The

probability for a reading to fall in the rangeof is about 68%.

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

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


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

Probability value of the mean

50% 0.675468.3%

86.6% 1.595.4% 299.7% 3

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 xiwill be within the range of x of the mean.

Thus, the probability of a sample chosen atrandom of being within the range 2of themean is about 95%. Uncertainty then is defined astwice the standard deviation:

x 2

Example 3. 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 2

kPa would represent a 95% confidence interval.That is, if we randomly select many samples ofthis manufacturers 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 4. 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, ?

Solution: The uncertainty 95% of confidenceinterval 2. Thus

2 kPa = 2So

= 1 kPa

Example 5. 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 = 1 kPa, we are seeking thevalue of the integral under the bell shaped curve,over the range of - to 3. 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 DataChauvenets 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

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, astatistical criterion must be used to identifypoints that can be considered for rejection. Thereis no other justifiable method to throw awaydata points.

One method that has gained wide acceptanceis Chauvenets 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, data

points can be considered for rejection only if theprobability of obtaining their deviation from themean is less than 1/(2n). This is illustrated inFigure 4.


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14

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 max can be determined from the table,

where

max =|(xi

x )|maxsx

=dmaxsx

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

mean by more than dmax/sxcan 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 Chauvenets criterion, we say that thevalues xiwhich are outside of the range

x C (11)

are clearly errors and should be discarded for theanalysis. Such values are called outliers. Theconstant C may be obtained from Table 3. Notethat Chauvenets 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 Chauvenets criterion, discard outliers.

4. Recalculate the mean value and thestandard deviation of the smaller sampleand stop. Do not repeat the process;Chauvenets criterion may be applied onlyonce.

TABLE 3. Constants to use in Chauvenetscriterion, Equation 11.

Number,n

dmaxsx

= C

3 1.384 1.545 1.656 1.737 1.808 1.879 1.9110 1.9615 2.1320 2.2425 2.33

50 2.57100 2.81300 3.14500 3.29

1,000 3.48

Example 5. Consider an experiment in which wemeasure the mass of ten individual identicalobjects. The scale readings (in grams) are asshown in Table 4.

By visual examination of the results, wemight conclude that the 4.85 g reading is too high

compared 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 Chauvenets 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 2.42

3 2.434 2.435 2.446 2.447 2.458 2.469 2.4710 4.85


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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

= 0.0729We 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 in Equation 9:

=

=

( )x x

n

ii

n2

1

1= 0.763 (9)

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

m Cmi m + C

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).

Exercise. Define the following terms:1. Error2 . Uncertainty3. Accuracy4. Precision5. Random Error6. Systematic Error7. Confidence Interval8. Outlier

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

Number, n reading in g (x xi)2 remove #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 0.000079

5 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


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REPORT WRITING

All reports in the Heat Transfer Laboratoryrequire a formal laboratory report unlessspecified otherwise. The report should be writtenin such a way that anyone can duplicate theexperiment and obtain the same results as theoriginator. The reports should be simple andclearly written. Reports are due one week afterthe experiment was performed, unless specifiedotherwise.

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 with

great 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. Following is a summary ofthe key elements in a well written, formallaboratory report. Details regarding each ofthese elements are also provided.

SUMMARY

TITLE PAGEExperiment Number, Title of the Experiment, Name of the Author, Name ofPartners, Date the Experiment was Performed, Due date for this Report, TheUniversity of Memphis, Department of Mechanical Engineering, proofreaderssignature.

ABSTRACTBrief summary of the objective of the experiment, the procedures, the results,conclusions and recommendations. Present or future tense.

INTRODUCTIONDescription of the problem, references, objective. Past or present tense.

THEORY ANDANALYSIS

Theory associated with this experiment, equation derivation. Past or present

tense.

PROCEDUREDescribes the equipment used, equipment setup, model and/or serial numbers,experimental procedure. Past tense.

RESULTS AND

DISCUSSION

Summarize your outcome, graphs and tables, sample calculation. Past tense.

CONCLUSIONSConclusions, observations, trends, and recommendations. Past or present tense;

recommendations in future tense.

APPENDICESTitle page, references, original data sheet, calibration curves.


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Sample Title Page

Experiment Number

TITLE OF THE EXPERIMENT

Name of the Author

Name of Partner #1Name of Partner #2

etc.

Date the Experiment was Performed

Due date for this Report

The University of MemphisDepartment of Mechanical Engineering

ABSTRACTThis report was designed to contain the instructions on how to write a report, and to serve as an

example of the format and style expected in all reports. It was based on the style and format ofengineering reports used in the writing of professional engineering publications. The Title Page andAbstract are the first two components of the report.

The Abstract summarizes the information in the report. It provides a brief summary of the objective

of the experiment, the procedures, the results, conclusions and recommendations. It should not referenceany tables, figures or appendices. A short abstract may appear on the title page as in this example. Alonger abstract would appear on the sheet following the Title Page.

The Abstract allows the reader to determine whether to read the report. It is written in the pasttense, except for the recommendations, which may be written in the present or future tense.


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INTRODUCTIONThis section tells the reader what the experimentis about. It begins with a description of theproblem that is being investigated. It includesthe background and provides the reader with aclear concise statement explaining the purpose ofthe experiment. This is one of the most importantparts of the laboratory report because everything

included in the report must somehow relate to thestated object. The object can be as short as onesentence and it is usually written in the pasttense.

Subheadings. The beginning of this sectionshowed how a major heading (e.g., Introduction)should appear in a report: all capital letters,boldface type, and left justified. It may benecessary to use subheadings, and the format forthese is shown at the beginning of thisparagraph. Note also, that the beginning of a

paragraph is not indented, but instead ispreceded by a blank line.

Write for the Reader. Consider that the report iswritten for a technically competent person who isunfamiliar with the specific subject matter, butwill be after he/she reads the report. Alsoconsider that the reader is not as closelyassociated with the test as you are. Checkgrammar and spelling. Check continuity of page,figure, and table numbers. Have an associate whodid not perform the experiment with you, but whohas technical competency, proofread your report.

Report Preparation. Reports must be composed ona word processor. Use white paper and black text.Use only one side of a page. All margins should be1 in. Do not right justify the text. Each sectiondoes not need to begin on a new page.

Each page is to be numbered with an Arabicnumeral centered at the bottom of the page. Donot number the title page. Begin numbering withpage 2.

Figures should be numbered sequentially usingArabic numbers. Each figure is to have adescriptive title. Figures should be drawn using acomputer and a drawing program, or use thefigures available with the lab manual. Figuresare to be located near the place in the text wherethey are first referred to. Figures should becentered left-to-right either on the page (singlecolumn) or within the column (two or morecolumns). The figure number and title shouldappear centered just below the figure itself.

Tables should contain as much information aspossible. They are to be enclosed in a border. They

can be placed in the text or at the end of thesection where they are first referred to. Tablesare to be numbered consecutively with Arabicnumerals. An acceptable table format is asfollows:

TABLE 1. Reduced data for heat transferred pasta flat plate.

TrialVelocityVin m/s

Heattransferred

q in WTemperature T

in C1 0.5 13.6 74.82 1.0 16.2 75.53 1.5 17.5 75.34 2.0 18.9 74.5

Note carefully the following features regardingthis table: The first column is trial or run. Each column heading is of a parameter,

followed by the symbol and the unit. Each column heading is centered within the

column. The table is centered left to right within the

page or column. A border has been placed around the table,

and around each cell. The font size is smaller than that used for the

text in the report. TABLE 1 is in all capital letters, and the

actual title is in italics.These features are referred to collectively as thestyle of the report.

Graphs. In many instances, it is necessary tocompose a plot in order to graphically present theresults. Graphs must be drawn neatly following aspecific format. Figure R.1 shows an acceptablegraph prepared using a computer. There are manycomputer programs that have graphingcapabilities. An acceptably drawn graph hasseveral features of note. These features aresummarized next to Figure R.1.

Graphs especially should have descriptivetitles. A graph of temperature versus time, forexample, should not have a title of:

FIGURE 1. Temperature versus time.The reader can see by looking at the graph thatthis is so. A better title would be:

FIGURE 1. Temperature variation with time for abrass sphere cooling in air.

Note that FIGURE 1 is in all capital letters,and the actual title is in italics.


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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: oon the Qac line to denote datapoints 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

rate

inm

3/s

FIGURE R.1. Theoretical and actual volume flow ratethrough a venturi meter as a function of head loss.

Writing Style. Use simple words with exactmeanings. Use technical terms to express a precisetechnical meaning. Do not use a large and unusualterm to add false importance to the report or toyourself. Do not use slang words or expressions.

Use simple sentences that have a subject anda predicate. Add adjectives as required. Avoidextra long sentences. Never use I regardless ofwhat you were taught in any previous courses.We is acceptable. Insert only one space betweena period and the beginning of the next sentence.

Do not bind the report in a folder. Staple thepages together in the upper left-hand corner.

The introduction section should conclude witha brief statement of what the objective of theexperiment is.

The Introduction is written in the past orpresent tense.

The report can be written using 1 column ortwo columns per page.

THEORY AND ANALYSISThis section explains the theory associated withthis experiment. The theory should be discussedin great detail and it should contain anexplanation of the theoretical model. Forexample, if an experiment was performed with apendulum, then include a brief derivation of themathematical model of a pendulum. Put thesignificant portions of the derivation in thissection. Cite references if appropriate. Includesimple sketches or diagrams to help the readervisualize the physical phenomenon beingstudied.

If there is little or no theory involved in thisexperiment, include the theory with theIntroduction section.

All equations in the report should be indentedand numbered consecutively with Arabicnumerals. Each symbol in the equations should benamed and its dimensional unit given. Anexample:

Newtons Second Law of Motion can be writtenas :

F = ma (1 )

where F is the unbalanced external force in N, mis the mass of the block in kg, and a is theacceleration in m/s2.

There are several inconspicuous but extremelyimportant details associated with this example,specifically in the way equations and units arewritten. Note that the letters used in theequation are in italics. Every reference to force,for instance, is in italics. The units used for eachvariable are in normal type (e.g., non italics).Numerical subscripts and superscripts are innormal type as well. However, subscripts andsuperscripts that are variables are italicized.

When a number is written with a unit, a spaceshould separate the two. For example, 5 N, or17.3 kPa. Numbers are written in normal type.

These features are very important in reportwriting. It is these features that will make a


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well done technical report appear professional inevery way.

The theory section in its entirety is written inthe past or present tense.

PROCEDUREThis section describes the equipment used in theexperiment and the test procedures. The

equipment setup should be shown in a figure. Thetest equipment and instrumentation used shouldbe listed with model and/or serial numbers, andthe expected instrument precision. Figures ofspecific components should be provided ifnecessary to help the reader to better understandthe test procedure.

Briefly describe the steps of theexperimental procedure in the order in whichthey were conducted. Include sufficient detail inthis section such that the reader could repeat theexperiment.

The procedure is written in the past tense.

RESULTS AND DISCUSSIONThe section should discuss the results. Summarizeyour outcome in the topic sentence, then supportthat summary with the results. Use graphs andtables to concisely present the results. Do notdraw conclusions in this section; only list anddiscuss results. This is also the section where acomparison of results with referenced valuesshould be presented.

A sample calculation should also beprovided. Start with raw data obtained while

performing the experiment, and show thecalculations involved in finding one of thenumbers in this section.

The Results and Discussion section should bewritten in the past tense.

CONCLUSIONSThis section is a clear and concise qualitative andquantitative summary of the experiment andresults. It includes conclusions, observations,trends, and recommendations. Recommendationsare especially valuable if the experiment failedor was impaired. Do not refer to tables or figures

in this section. Coordinate the material in thissection with the Introduction section. If there wasa clear objective in this experiment, statewhether the objective was reached. Makerecommendations regarding the experiment.

Do not use sentences such as We learned a lotin this experiment. Remember that your

perspective is that of an engineer writing atechnical report to others who are technicallyminded. It is not that of a student writing to a labreport grader.

The conclusions should be written in the pastor present tense, except for the recommendationswhich are in the future tense.

APPENDICESThe Appendix section contains its own title page,with a list of what the reader will find inside.

References. This portion of the appendix listsreferences used in the preparation of the report.You must cite the source publication for the workof all others which you include. This gives themdue credit for their work, and shows the researcheffort you put into your report. Do not list the labmanual as a reference. An example of analphabetical Reference list follows:

Bannister, L. (1991). University Style Manual,Fourth edition, Los Angeles: LoyolaMarymount University, 3639 and 5861.

Kovarik, M. (1989). Optimal Heat Exchangers, Journal of Heat Transfer, 111 , 287293.

Main, B. W. and A. C. Ward. (1992). What doDesign Engineers Really Know AboutSafety? Mechanical Engineer, 114 , 8, 4451.

Resnick, R. and D. Halliday. (1966). Physics.

New York: John Wiley.

Original Data Sheet. The data sheet completedwhen the experiment was conducted is includedhere.

Calibration Curves. If provided by the instructoror the manufacturer of the lab equipment,calibration curves for each meter used should beincluded in this section.

SHORT FORM REPORT FORMAT

Once in a while the experiment requires not aformal report but an informal report. An informalreport includes the Title Page, ExperimentObjective, Procedure, Results, and Conclusions.Other portions may be added at the discretion ofthe instructor or the writer.


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EXPERIMENT 1

THERMOCOUPLES & INSTRUMENTATION

Thermocouples are used extensively in thiscourse, and so it is prudent to learn how they aremade, how they work, and how to use them to

make measurements of temperature.A thermocouple is described as a device that

uses an electrical potential to obtain ameasurement of temperature. Using what isknown as a thermoelectric circuit, it is possible tomeasure temperature by measuring the voltageacross an open circuit. However, it should beremembered that we are actually measuring atemperature difference with the open circuitvoltage.

Temperature sensors that work on the opencircuit voltage principle are calledthermocouples. A thermocouple consists of two

wires made of two dissimilar metals and meltedor welded together at one end, called the junction.

Consider the simple circuit sketched in Figure1.1. Shown are two dissimilar metals, A and B,that are melted together at both ends to form twojunctions. If one of the junctions is heated, atemperature difference T1 - T2 will exist, and thiswill cause a current to flow through the wires.This phenomena is referred to as the Seebeckeffect. If the thermocouple is connected to anexternal circuit and a current passes through thewires, then the voltage generated will be alteredslightly. This is known as the Peltier effect. If atemperature gradient exists along either or bothwires, the voltage again will be altered slightly.This is called the Thomson effect.

i

i

T1

T2

q

metal A

metal B

FIGURE 1.1. Induced current in a circuit made oftwo dissimilar metals.

If the circuit of Figure 1.1 is broken, as inFigure 1.2, then there will exist an open circuitvoltage EAB , the magnitude of which is a functionof the junction temperature T2. This voltage iscalled the Seebeck voltage and is given by

EAB(T2) = ABT2

where AB is called the Seebeck coefficient,whose value depends on the two metals. Thenotation indicates that the voltage EAB is a

function of the junction temperature T2. Thevariation of the Seebeck coefficient withtemperature is given in Figure 1.3.

T2

q

metal A

metal B

EAB

+

FIGURE 1.2. Open circuit voltage in a

thermocouple.

Linear region

Temperature in C

SeebeckCo

efficientinV/C

FIGURE 1.3. Seebeck coefficient variation withtemperature for different types ofthermocouples. (From Omega.com.)

The objective is to measure the voltage EAB,and relate it to the junction temperature T2. So weconnect the thermocouple to a voltmeter, as inFigure 1.4. We see that Metal A is now connectedto another metal (copper, for example, as mightbe found in the meter), and so is Metal B. Whenconnected to the voltmeter, two newthermocouples have been created: A-Cu, and B-Cu. Although our desire is to measure EAB, thevoltmeter will read a value E which is given by:


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22

T2

T3

T3

T1

T1

q

metal A

metal B

EAB

+

voltmeter

E

Cu

Cu

FIGURE 1.4. Attempt at reading the Seebeckvoltage with a voltmeter.

E = ECuA(T1) + EAB(T2) + EBCu(T1)

E = CuAT1 + ABT2 + BCuT1

Note that these equations reflect a clockwisedirection in writing the subscripts of the voltages.Also, remember that we want only the voltagethat corresponds to the temperature of thejunction T2, which is EAB.

We can solve this problem by the addition ofa second thermocouple that we insert into anenvironment of known temperature which werefer to as the reference temperature. Thisreference temperature can be that of an ice bath,as illustrated in Figure 1.5. Now when writingthe equation for voltage and paying attention tothe + to direction, we get

T2

T3

T3

T1

T1

q

metal A

metal A

metal B

EAB+

voltmeter

E

Cu

Cu

reference junction

flask

corkthermometer

ice bath(water/crushed

ice mix)

FIGURE 1.5. Measurement of temperature T2 withrespect to a reference temperature.

E = ECuA(T1) + EAB(T2) + EBA(Tref) + EACu(T1)

E = CuAT1 + ABT2 BATref CuAT1

or

E = AB(T2 Tref)

If the reference temperature is known fromcalibration results, then T2 can be measuredknowing E. The reference temperature can be that

of an ice bath, or of the ambient.The effect of the reference temperature

environment can be replaced with either softwareor hardware compensation. Thus the need for aphysical device like an ice bath can beeliminated, and we are able to use the setup ofFigure 1.2 to obtain a measurement oftemperature. The only difference is that thevoltmeter is replaced with a digitalthermometer, which is internally compensatedfor a reference temperature.

There are a number of types of thermocouples

manufactured to close tolerances to give accurate,reliable and repeatable results. The types areusually designated with a letter which refers tothe two dissimilar metals used. The standardsare established and recorded by ANSI. Some ofthe common types are listed in Table 1.1. Thevariation of temperature with voltage fordifferent types of thermocouples is shown inFigure 1.6.

TABLE 1.1. Common thermocouple types andmetals used.

ANSIType Metals+ E Chromel ConstantanJ Iron ConstantanK Chromel AlumelR Platinum Platinum/13% RhodiumS Platinum Platinum/10% RhodiumT Copper Constantan

Temperature in C

Millivolts

FIGURE 1.6. Variation of voltage reading withtemperature for different types ofthermocouples. (From Omega.com.)


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The different types of thermocouples aremanufactured using a color code on the insulationfor the wires. The color code used for Types J, K,and T appear in Table 1.2. Note that the redinsulated wire is always connected to thenegative terminal of the digital thermometer.

Summary

Thermocouples produce a voltage that isproportional to the difference in temperaturebetween the hot junction and the referencejunction.

Because thermocouples measure relativetemperature differences, cold junctioncompensation is required if the system is toreport absolute temperatures.

Thermocouples have a small Seebeck voltagecoefficient, typically on the order of fifty

microvolts per degree Celsius over the range0C < Tabs < 150C

Thermocouples are non-linear across theirtemperature range. Linearization, if needed,is best done in software.

ExperimentThe instructor will demonstrate the making of athermocouple, and its use in measuring

temperature.The instructor will use a portable welding

device to make a thermocouple. Carefullyobserve the procedure, and note the appearance ofthe junction.

1. Using a digital thermometer as a readoutdevice, connect a thermocouple to the posts as

in Figure 1.1 and measure room temperature.2. Measure the temperature of someone in the

room.3. Reverse the leads and determine how the

output is affected by again measuring roomtemperature and the temperature of the sameindividual.

4. Compare the results.

QuestionsWhat metals are used in the alloys Constantan,Chromel, and Alumel? Why are these metals

used? How accurate are the temperaturemeasurements using thermocouples? What is athermistor?

For Your ReportDescribe in your own words what a thermocoupleconsists of, and how a thermocouple is made.

Describe the procedure demonstrated by theinstructor, and the results that were obtained.

TABLE 1.2. Thermocouple Color Code

ThermocoupleANSI

DesignationPositive

WireNegative

Wire OutsideTemperature

Range FIron-

Constantan J White Re d Brown 32 to 1382

Copper-Constantan T Blue Re d Brown 32 to 662

Chromel-Alumel K Yellow Re d Brown 32 to 2282


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EXPERIMENT 2

MEASUREMENT OF THERMAL CONDUCTIVITY OF A METAL

In order to make accurate predictions of heattransfer rates through materials, it is necessaryto first know the value of the thermal

conductivity of the material itself. Thermalconductivity can be measured using standardmethods, devices and techniques. In thisexperiment, we will measure thermalconductivity of a metal, and in addition,calculate an overall heat transfer coefficient forthree metals in series.

Figure 2.1 is a sketch of the apparatus used inthis experiment. It consists of three separablesections. The center section is removable. The leftend section contains a brass rod, and an electricalheater. The heat input to the heater can becontrolled and measured. The right end section is

also made of brass, and contains a hollowed outcavity with water tubes attached. Thus heatflows through from the heater through the leftend section, then through the center section, andfinally through the right end section to thewater.

The entire apparatus is insulated so that onedimensional heat conduction is wellapproximated. The end sections containinstrumentation for measuring temperature. Therods in the end sections have a diameter of 25 mmwhile the distance between adjacent temperaturemeasurements is 10 mm. The center section is 30

mm long.Temperature versus length readings can be

obtained with this apparatus. Severalexperiments can be performed depending on whatis used in the center section.

ProcedureInstall the center section that is not

instrumented for temperature measurement. The

heater control is turned on to some value on therheostat. Water is circulated through the cavity.Once steady state is reached, record temperatureversus distance. The steady state data can be usedto verify the accuracy of Fouriers Law, and tocalculate the thermal conductivity of thematerial.

AnalysisFouriers Law of Heat Conduction is most

easily verified (or tested) in the one dimensionalconfiguration of this experiment. In equationform, Fouriers Law is

q = kATx

(2.1)

where q is the heat flowing through the rod ofdimensions FL/T (BTU/hr or W), k is thethermal conductivity of the material ofdimensions FL/(TLt) [BTU/(hrftR) orW/(mK)], A is the cross sectional area ofdimensions L2 (ft2 or m2 ), and T/x is thetemperature gradient of dimensions t/L (R/ft orK/m). Because temperature decreases withincreasing distance, the gradient T/x isnegative, and T/x is actually positive.

Temperature is measured at discrete pointsalong the rod in this experiment. It is thereforeappropriate to rewrite Equation 2.1 in a differentform:

brass rod brass rod

t

temperaturemeasurement

water coolingchamber

center sectionelectrical heaterwater inwater out

A

A

insulation

25 mmdiameter

section A-A

FIGURE 2.1. A schematic of the apparatus used to verify Fouriers Law of Conduction.


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q = kATx

(2.2)

where Tis the temperature difference betweenany two thermocouples (adjacent or not) and x isthe distance between the two thermocouples ofinterest.

For one dimensional heat flow, we can writethe following:

qA

= kLT1 TIL

xL= k

TIL TIRx

= kRTIR T6

xR(2.3)

where T1 is the temperature at the warmest pointon the rod at the left end, TILis the interfacetemperature between the left end rod and thecenter rod, TIR is the interface temperaturebetween the center rod and the right end rod, T6 is

the temperature at the coolest point of the rod onthe right, and the xs correspond to theappropriate distances. The interfacetemperatures are sketched in Figure 2.2. Thermalconductivity k values correspond to theappropriate materials. The first and third

materials are brass. The center section is stainlesssteel.

By manipulation of Equation 2.2 and 2.3, it ispossible to express the heat transferred along therods in terms of a heat transfer coefficient, U, as

qA

= U(T1 T6) (2.4)

where

1U

=xLkL

+xk

+xRkR

(2.5)

The preceding equations do not include the effectof contact resistance.

ResultsPlot temperature versus distance along the

rods. Determine from your plot the interface

temperatures (see figure below). Using publishedthermal conductivity values for brass, andEquation 2.3, find the thermal conductivity ofstainless steel and compare your results topublished values. Do not include the effects ofcontact resistance. Also calculate the overallheat transfer coefficient.

brass rod brass rod


T

z

TIL

TIR

interface

x

stainlesssteel

T1

T6

FIGURE 2.2. Determination of interface temperatures from the measured temperature values.


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EXPERIMENT 3

EFFECT OF AREA ON ONE DIMENSIONAL CONDUCTION

In some geometries, one dimensional heatconduction occurs through a material with anarea that is not constant. In such cases, it is still

necessary to be able to make accurate predictionsof heat transfer rates through the material. Wewill investigate the effect of variable area on onedimensional conduction in this experiment.Thermal conductivity of the material will beknown.

Figure 3.1 is a sketch of the apparatus used inthis experiment. It consists of three separablesections. The center section is removable. The leftend section contains a brass rod, and an electricalheater. The heat input to the heater can becontrolled and measured. The right end section isalso made of brass, and contains a hollowed out

cavity with water tubes attached. Thus heatflows through from the heater through the leftend section, then through the center section, andfinally through the right end section to thewater.

The entire apparatus is insulated so that onedimensional heat conduction is wellapproximated. The end sections containinstrumentation for measuring temperature. Therods in the end sections have a diameter of 25 mmwhile the distance between adjacent temperaturemeasurements is 10 mm. The center section is 30mm long.

Temperature versus length readings can beobtained with this apparatus. Severalexperiments can be performed depending on whatis used in the center section.

ProcedureInstall the center section which has a smaller

cross sectional diameter than the end sections.

This center section is not instrumented fortemperature measurement. The heater control isturned on to some value on the rheostat. Water iscirculated through the cavity. Once steady stateis reached, record temperature versus distance.

AnalysisFouriers Law of Heat Conduction is easily

verified (or tested) in the one dimensionalconfiguration of this experiment. In equationform, Fouriers Law is

q = kA

T

x (3.1)



brass rod brass rod

t




A

A

insulation

25 mmdiameter

section A-A

FIGURE 3.1. A schematic of the apparatus used to measure the effect of area on one dimensionalconduction.


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q = kATx

(3.2)


For one dimensional heat flow, we can writethe following:

qA

= kLT1 TIL

xL= k

TIL TIRx

= kRTIR T6

xR(3.3)

where T1 is the temperature at the warmest pointon the rod at the left end, TILis the interfacetemperature between the left end rod and thecenter rod, TIR is the interface temperaturebetween the center rod and the right end rod, T6 isthe temperature at the coolest point of the rod on

the right, and the xs correspond to theappropriate distances. Thermal conductivity k

values correspond to the appropriate materials.The first and third materials are brass. Thecenter section is also brass, with dimensions of 13mm diameter by 30 mm long.

By manipulation of Equation 3.2 and 3.3, it ispossible to write an equation in terms of area andtemperature gradient:

AL

T

xL

= A

T

x= AR

T

xR

(3.4)

Note that the preceding equations do not includethe effects of contact resistance.

ResultsPlot temperature versus distance along the rods.Using published thermal conductivity values forbrass (if you need them), verify whetherEquation 3.4 is correct. Do not include the effectsof thermal contact resistance in your calculations.


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EXPERIMENT 4

MEASUREMENT OF THERMAL CONDUCTIVITY OF AN INSULATOR

In order to make accurate predictions of heattransfer rates through materials, it is necessaryto first know the value of the thermal

conductivity of the material itself. Thermalconductivity can be measured using standardmethods, devices and techniques. In thisexperiment, we will measure thermalconductivity of an insulator.

Figure 4.1 is a sketch of the apparatus used inthis experiment. It consists of two separablesections, between which is placed the insulator ofinterest. The left end section contains a brass rod,and an electrical heater. The heat input to theheater can be controlled and measured. The rightend section is also made of brass, and contains ahollowed out cavity with water tubes attached.

Thus heat flows through from the heater throughthe left end section, then through the centersection, and finally through the right end sectionto the water.

The entire apparatus is insulated so that onedimensional heat conduction is wellapproximated. The end sections containinstrumentation for measuring temperature. Therods in the end sections have a diameter of 25 mmwhile the distance between adjacent temperaturemeasurements is 10 mm.

Temperature versus length readings can beobtained with this apparatus. Several

experiments can be performed depending on whatis used in the center section.

ProcedureInstall an insulating material between the

two end sections. The heater control is turned on to

some value on the rheostat (ensure that thetemperature at the hottest point does not exceed100C). Water is circulated through the cavity.Once steady state is reached, record temperatureversus distance. The steady state data can be usedto calculate the thermal conductivity of thematerial.



q = kATx

(4.1)

where q is the heat flowing through the rod ofdimensions FL/T (BTU/hr or W), k is thethermal conductivity of the material ofdimensions FL/(TLt) [BTU/(hrftR) orW/(mK)], A is the cross sectional area ofdimensions L2 (ft2 or m2 ), and T/x is thetemperature gradient of dimensions t/L (R/ft orK/m). Because temperature decreases withincreasing distance, the gradient T/x is

negative, and T/x is actually positive.Temperature is measured at discrete points

along the rod in this experiment. It is thereforeappropriate to rewrite Equation 4.1 in a differentform:

brass rod brass rod



insulatingmaterial

electrical heaterwater inwater out

A

A

insulation

25 mmdiameter

section A-A

t

FIGURE 4.1. A schematic of the apparatus used to measure thermal conductivity.


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q = kATx

(4.2)


We can apply Equation 4.2 across theinsulating material to obtain:

k =qA

x

T(4.3)

Thus, x is the thickness of the insulatingmaterial while it is clamped in position, and Tis the difference between the interfacetemperatures; i.e., the temperature differenceacross the insulating material.

Thermal conductivity should be expressed in

the appropriate units.


rods. Determine from your plot the interfacetemperatures. Use them with Equation 4.3 todetermine thermal conductivity. If publishedvalues of the thermal conductivity are availablecompare your results to them. Do not include theeffects of thermal contact resistance.


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EXPERIMENT 5

DETERMINATION OF CONTACT RESISTANCE

Energy in the form of heat travels from aregion of high temperature to a region of lowtemperature. When one dimensional flow exists,

the heat transfer rate can be described byFouriers Law of Heat Conduction. Fouriers Lawis an experimentally observed law whichpredicts a linear temperature distribution in onedimension for a constant heat flux. Itsexperimental verification is the subject of thisexperiment.

Figure 5.1 is a sketch of the apparatus used inthis experiment. It consists of three separablesections. The center section is removable. The leftend section contains a brass rod, and an electricalheater. The heat input to the heater can becontrolled and measured. The right end section is

also made of brass, and contains a hollowed outcavity with water tubes attached. Thus heatflows through from the heater through the leftend section, then through the center section, andfinally through the right end section to thewater.

The entire apparatus is insulated so that onedimensional heat conduction is wellapproximated. The end sections containinstrumentation for measuring temperature. Therods in the end sections have a diameter of 25 mmwhile the distance between adjacent temperaturemeasurements is 10 mm. The center section is 30

mm long.Temperature versus length readings can be

obtained with this apparatus. Severalexperiments can be performed depending on whatis used in the center section.

ProcedureInstall the center section that is instrumented

for temperature measurement. The heater control

is turned on to some value on the rheostat. Wateris circulated through the cavity. Once steadystate is reached, record temperature versusdistance. The steady state data can be used toverify the accuracy of Fouriers Law, and tocalculate the thermal conductivity of thematerial.



q = kATx

(5.1)



brass rod brass rod

t


water cooling

chamber


A

A

insulation

25 mmdiameter

section A-A

FIGURE 5.1. A schematic of the apparatus used to verify Fouriers Law of Conduction.


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q = kATx

(5.2)

where Tis the temperature difference betweenany two thermocouples (adjacent or not) and x isthe distance between the two thermocouples ofinterest. If in fact Fouriers Law is accurate, thenT/x is a constant for any number of pairs ofthermocouples within the same material.

Referring to Equation 5.2, the cross sectionalarea A is calculated knowing the diameter, andthe gradient T/x is calculated from thetemperature versus distance data. With theseparameters known, the thermal conductivity k iseasily calculated using Equation 5.2.

AnalysisFigure 5.2 is a sketch of temperature versus

distance for two materials touching each other at

a location labeled interface. Data points aregraphed, and a best line is drawn through themfor both materials. The lines are extrapolated tothe interface. Data for the material on the leftindicates that the interface temperature is Ti.,while data for the other material indicates aninterface temperature ofTir. The difference inthese two temperatures is due to what is calledthe contact resistance to heat transfer.

The contact resistance Rtc is found from theone dimensional conduction equation:

q

A=

Til - Tir

Rtc(5.3)

ResultsThe apparatus used in this experiment has 3materials in series, and so there are twointerfaces. Plot temperature versus distance alongthe rods. Determine the associated temperaturesfor both interfaces, and calculate the contactresistances. Compare to values found in a

textbook.

T

z

data point extrapolatedprofiles

TilTir

interface

FIGURE 5.2. A sketch of temperature versusdistance.

QuestionsDoes thermal conductivity vary with

temperature? If so, should T/x vary instead ofbeing constant? If the rod itself is tapered, thenwhat should be constant in addition to or insteadofT/x?

Are the values for both contact resistancesequal? Should they be?


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EXPERIMENT 6

RADIAL ONE DIMENSIONAL CONDUCTION

Conduction in the radial direction is made tooccur in a number of geometries. In such cases, it isimportant to be able to make accurate predictions

of the heat transfer rate. In this experiment, wewill investigate the temperature profile andheat transfer rate for a radial system.

Figure 6.1 is a sketch of the apparatus used inthis experiment. It consists of a 55 mm diameterdisk 3 mm in thickness, which has a hole in itscenter that is 4 mm in diameter. The disk has anelectrical heater attached to its center, andcontains a circumferential water channel near itsouter edge. With the heater on and cooling waterflowing through the channel, heat flowsradially through the disk. The disk and heaterare well insulated to prevent heat losses in other

directions. Provision is made to measuretemperature at selected, evenly spaced locationsalong the disk.


disk

insulation

electricalheater

circumferentialwater channel

55 mm 4 mm

FIGURE 6.1. A schematic of the apparatus used toinvestigate radial, one dimensional conduction.

ProcedureTurn the heater and the cooling water on and

allow sufficient time to elapse for steady state tobe reached. Take readings of temperature at allradial locations. Measure the distance betweenadjacent thermocouples.

AnalysisFouriers Law of Heat Conduction is verified

(or tested) in the radial, one dimensional

configuration of this experiment. In equationform, Fouriers Law is

q = kATr

(6.1)

where q is the heat flowing through the disk, ofdimensions FL/T (BTU/hr or W), k is thethermal conductivity of the material ofdimensions FL/(TLt) [BTU/(hrftR) orW/(mK)], A is the cross sectional area normal tothe heat flow, of dimensions L2 (ft2 or m2 ), andT/x is the temperature gradient of dimensions

t/L (R/ft or K/m). Because temperaturedecreases with increasing distance from the heatsource, the gradient T/r is negative, and(T/r) is actually positive.

The cross-sectional area is given by

A = 2rL

where r is the radial coordinate (R1 < r < R2), andL is the disk thickness. Substituting into Equation6.1 and separating variables for integration gives

T1

T2

dT=

R1

R2

12kL

qrdrr

where the subscripts 1 and 2 are at any twodifferent points in the disk. Integrating gives

T2 T1 = qr

2k L ln

R2R1

or T1 T2 =qr

2k L ln

R2R1

(6.2)


disk. Use the data with Equation 6.2 to calculatethe rate of radial heat conduction. Compare theresults with the heat input read from the wattmeter. Are you expecting to get a straight linegraph?


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EXPERIMENT 7

HEAT TRANSFER FROM A FIN

Conduction is the primary mode of heattransfer through solids. When a heated solid isexposed to a lower temperature fluid, heat is

transferred from the solid to the fluid byconvection. The amount of heat transferred at thesurface is calculated by means of a convectioncoefficient h. Measuring the convectioncoefficient for a fin is the subject of thisexperiment.

Figure 7.1 is a sketch of a fin, also known asan extended surface. Fins are used to increase thesurface area of a solid. When the surface areaAsis increased, the rate of heat transfer q isincreased because q is directly proportional to As.

In the apparatus of this experiment, thereare three fins attached to a chamber into which

steam is admitted. The steam will heat the endof each fin. Heat will be conducted axially alongeach fin and will be transferred by naturalconvection to the air. Thermocouples areembedded at intervals along each fin so thattemperature is known at selected points. It is atthese points where the convection coefficient willbe determined.

IntroductionFins are usually characterized with a ratio of

parameters denoted as m which is defined as

m =hP k Awhere h is the convection coefficient [W/(m2K)or BTU/(hrft2R)], P is the perim

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