Embed Size (px)
DESCRIPTION
Mechanical Engineering Laboratory II
Citation preview
MEAM 347Mechanical Engineering Laboratory II
Laboratory Manual
Do Not Print this on CETS or SEAS Printers
Department of Mechanical EngineeringUniversity of Pennsylvania
September, 1998
2
TABLE OF CONTENTS
Introduction ................................................................................................................................... 3
Objectives............................................................................................................................ 3Course Content ................................................................................................................... 3
Writing a Lab Report ...................................................................................................................... 8
Some General Comments..................................................................................................... 8Types of Reports ................................................................................................................. 9Contents of a Report ......................................................................................................... 11Graphical Presentations ..................................................................................................... 18Miscellaneous Helpful Hints............................................................................................... 19
Presenting Experimental Data........................................................................................................ 20
Introduction....................................................................................................................... 20Causes and Types of Experimental Errors .......................................................................... 20Error Analysis on a Commonsense Basis............................................................................ 22Uncertainty Analysis .......................................................................................................... 23Evaluation of Uncertainties for complicated Data Reduction .............................................. 28Graphical Analysis and Curve Fitting ................................................................................. 30General Considerations in Data Analysis ............................................................................ 31Summary ........................................................................................................................... 31Review Questions.............................................................................................................. 35Problems ........................................................................................................................... 35
Experiment #1............................................................................................................................... 43Lift and Drag on 1/48 Scale Model Aircraft
Experiment #2............................................................................................................................... 48Fluid Friction Flow
Experiment #3............................................................................................................................... 51Pressure Distribution Over an Airfoil
Experiment #4............................................................................................................................... 57Cooling Tower Experiment
3
1. INTRODUCTION
1.1 OBJECTIVES
The objectives of the MEAM 247 and 347 laboratory courses are:
(1) To teach you how to do the experimental work expected from modern engineeringprofessionals.
(2) To demonstrate to you physical concepts that you are learning in various engineeringcourses. The experiments are not necessarily limited to material you have learned in class.Some of the experiments are designed to make you think and introduce new concepts andideas that are beyond traditional classroom instruction.
(3) To stimulate your curiosity and imagination. Hopefully, the course will give you a sense ofmeasuring and discovering nature, and will introduce you to the excitement of research.
(4) To give you experience with modern instrumentation, and to acquaint you with computer-aided data acquisition, control, real time data processing and graphical display of results.
(5) To train you in team work.(6) To enhance your communication and writing skills. Great emphasis is put on the quality and
appearance of the laboratory reports.
1.2 COURSE CONTENT
The material covered and the experiments in the laboratory course are classified into four categories:i. Experimental methodsii. Computer aided data acquisition, processing and analysisiii. Experiments in mechanics, thermal and fluid sciences and mechanical systemsiv. Computer integrated design and manufacturing
We briefly discuss each of these categories next.
Experimental methods
A good experimentalist must be equipped with analytical tools that are required to model physicalsystems (including sensors), design experiments and analyze data. In addition, it is very important tobe able to write a coherent and concise report. The basics of system dynamics, signal processing,statistical analysis and technical writing will be taught in this section. Some of this material will becovered in the classroom.
Computer aided data acquisition, processing and analysis
The primary advantage of computer-aided data acquisition and control of experiments is that the dataare acquired, reduced and analyzed, almost in real time, results are graphically displayed on thecomputers' CRTs, and neat, formatted output which you can prepare to be ready for final submission,can be almost immediately printed out. The experimental data can be compared either with theoretical
4
predictions (if available) or with known engineering correlations. Should discrepancies betweenexperimental observations and the expected results arise, you can either repeat the experiment or tryto establish the source of the discrepancy. (Contrast this with the more conventional mode ofoperation in which you record the data manually, reduce and analyze it later at home. Perhaps youdiscover, in the final stages of your analysis, that there is something wrong with the data, and the onlysolution is to repeat the experiment another day. Of course, the graphs and tables must bepainstakingly generated manually!) In this section of the course, you will learn what is involved ininterfacing sensors and controllers to computers, how to set up computers to take data or to outputdata to control the external world, and how to process, interpret and analyze experimental data that isoften corrupted by errors that are either predictable (explained by a suitable model) or random(noise). You will use personal computers, data acquisition cards, and appropriate software to collectdata and then analyze it. You will learn to interface sensors and analyze their response characteristics.
Experiments in mechanics, thermal and fluid sciences and mechanical systems
In this section of the course, you will perform experiments that will enhance the materials you arelearning in various engineering courses.
In these experiments, the computer integrated environment with microcomputers and associatedinstrumentation is typical of what engineers may expect to find in modern, automated industry. Themicrocomputers will substantially enhance your learning environment since you can delegate tedious,repetitive tasks to the machines and concentrate on observing the fundamental phenomenademonstrated by the experiments. At the same time, the experiments have been designed to avoid theuse of "black boxes" (as much as possible) and "canned programs".
Computer integrated design and manufacturing
This section of the course is designed to train you in computer integrated engineering analysis, designand manufacturing techniques. The emphasis is on the intellectual component in "integration" and itsimportance in engineering. The experiments will give you a flavor of the automation that is typical inthe manufacturing industry. You will be able to design and draft a product, build a 3-D model of theproduct, verify its manufacturability, and manufacture a prototype in the same environment. Thisallows you to concurrently design and manufacture with databases integrated into one environment.(Contrast this with the conventional approach in which a designer sketches out the design, has itdrawn by the draftsman, and then sends it to the shop floor to get it fabricated. Very often, thedesigner does not take into consideration the actual manufacturing process and specifies parts thatcannot be machined. Or, the draftsman misinterprets the sketch. At every stage there is delay and theprocess of transforming a conceptual idea into a prototype is stretched out to a point where it is afinancial disaster.) This idea of electronic prototyping is fast becoming a reality in today's industry.You will use state of the art software and numerically controlled milling machines and lathes to buildsmall machine components out of machinable wax.
Course structure
5
Both MEAM 247 and 347 courses will be continuous through the academic year. Each courseconsists of 8-10 laboratory sessions. You will work in groups, typically of 3, and prepare reports forlaboratory experiments.
Time and Place
The lab period for each group will be set during the organizational meeting. The main laboratory isthe GM Lab which is located in room 193 on the west side, ground floor of the Towne Building (onthe same floor as the SEAS machine shop and copy center). Experiments will be conducted in otherlabs as well including the wind tunnel facility in 175 Towne, which is behind the machine shop, themechanical testing lab in the basement of LRSM, and the Chemical Engineering undergraduate lab in116 Towne. The lab for each experiment will be noted on the master schedule.
You will be able to perform the experiments only during the scheduled times. However, students willhave additional access to the laboratory to use the computers and prepare for their next experiment.This access is only during normal work hours - Monday through Friday, 9:00 a.m. to 4:30 p.m.
Books and Manuals
The primary text for this course is this manual to be purchased from the copy center (room 143,Towne Building). The material specific to each experiment is also kept near the experimental set-up,consult with the TA if you need it.
Groups
The laboratory exercises and report writing are to be done by an individual student or in teams ofthree. You should select partners to form a laboratory team. Select your team members wisely, sinceyou will work together for the year. Team work is an important aspect of this course and it willenhance your performance. All students who cannot identify partners should see the instructor or theTA during the first week the lab meets.
Laboratory reports
You will submit a report for each experiment that you perform. All reports will be graded on thetechnical content as well as the writing and presentation. The deadline for submitting rough and finaldrafts are shown on the syllabus for your individual group. These deadlines are not changeable.
Technical report submission will be an important factor in your future career. It is an avenue forcommunicating your ideas to your superiors and colleagues. The opinion that your superiors form ofyou will depend heavily on the quality of such reports. You should make a considerable effort tomaster the technique of report writing.
The laboratory exercise grade will be based on:
6
(a) The quality of preparation for the laboratory session. This may include a short verbal orwritten quiz before or during each experiment.
(b) Attendance and performance in the laboratory.(c) Quality of the report: This will be based on the technical content as well as the quality of
presentation and technical writing. Late assignments will result in lower grades.
Preparation for the laboratory session
Before coming to the laboratory:
1. Read carefully and understand the description of the experiment in the lab manual. You maygo to the lab at an earlier date to look at the experimental facility and understand it better.Consult the appropriate references to be completely familiar with the concepts and hardware.At the beginning of the class, if the TA or the instructor finds that a student is not adequatelyprepared, they will be graded zero for that experiment and not be allowed to take it.
2. If software is required for the experiment, it must be prepared and brought on floppy disketteto the lab. You will not be allowed (or have time) to write the software during the laboratoryperiod.
3. Bring necessary material needed to perform the required preliminary analysis. It is a goodidea to do sample calculations and as much of the analysis as possible during the session. TAhelp will be available. Errors in the procedure may thus be easily detected and rectified.
4. Check with one of the TAs one week before the experiment to make sure that you have thehandout for that experiment and all the apparatus.
After the laboratory session
1. Clean up your work area.2. check with the TA or technician before you leave.3. Make sure you understand what kind of report is to be prepared and when it is due.4. Do sample calculations and some preliminary work to verify that the experiment was
successful.
Make-ups and late work
Students must participate in all laboratory exercises as scheduled. They must obtain permission fromthe instructor for absence, which would be granted only under justifiable circumstances. In such anevent, a student must make arrangements for a make-up laboratory, which will be scheduled whenlaboratory time is available.
Laboratory policies
1. Food and beverages are not allowed in the laboratory at any time.2. Do not sit or place anything on instrument benches.3. The teaching assistants and staff are concerned about your safety and for the equipment in the
laboratory. Follow their instructions. If you have any doubt about the way to operate some
7
equipment, do not guess. Read the instructions carefully and if still in doubt, ask the staff forassistance.
4. Organizing laboratory experiments requires the help of laboratory technicians and staff. Bepunctual. If you come late to the laboratory, you may be asked to leave. Remember thatsome of the sessions include short tests before the experiment and no make-ups will be given.
Notices, Electronic Mail
The entire class will meet only once during the two semesters. Therefore, changes in schedule anddeadlines in class will be announced in the following ways:
1. Announcements will be posted on the notice board outside the GM Lab and on the eniacnewsgroup. It is your responsibility to check this board every week.
2. Messages will be posted on the electronic bulletin board on eniac upenn.meam.meam247 or347. Read your mail at least twice a week and check the bulletin board.
The easiest way to get in touch with a TA, instructor or staff person is via electronic mail. In theclass notes, there are two pages which tell you how to use it. If you want to meet with an instructor,send mail to the TA requesting an appointment with preferred times.
8
2. WRITING A LAB REPORT
2.1 SOME GENERAL COMMENTS
The importance of good report writing and data presentation cannot be overemphasized. No matterhow good an experiment or how brilliant a discovery, it is worthless unless the information is properlycommunicated to other people.
A man who uses a great many words to express his meaning is like a bad marksman whoinstead of aiming a single stone at an object takes up a handful and throws at it in hopes hemay hit. -Samuel Johnson
Succinct and precise expression are essential for a good technical report. When graphs or tables willpresent the idea clearly, use them, but do not also include a wordy explanation which tells the readerwhat can be plainly seen by careful inspection of the graph.
Third-person past tense is generally accepted as the most formal grammatical style for technicalreports, and it is seldom incorrect to use such a style. In some instances, first person may beemployed in order to emphasize a point or to stress the fact that a statement is primarily the opinionof the writer. The usual scientific writing style is also in the passive mode. Examples of the twostyles are:
Third person Equation (5) is recommended for the correlation in accordance with thelimitations of the data as discussed above.First person We (I) recommended Eq. (5) for the final correlation in accordance with thelimitations of the data presented in our discussion above.
In the first-person statement, the writer is making the recommendation on a much more personal basisthan in the third-person statement. The selection of the proper statement is a matter of idiom, whichdepends on many factors, including consideration of the person(s) who will read the report. For aformal paper in a scientific journal, the third-person statement might be preferable, whereas the first-person usage might be desirable in an engineering report to an individual.
A writer should make sure that the strong points of a report are clearly set forth and not buried inexcess words. Be specific if you have something to be specific about. Consider the following twostatements:
1. An analysis of the experimental data showed that the average deviation from the theoreticalvalues was less than 1 percent. Uncertainties in the primary data were shown to account for adeviation of 0.5 percent. In view of this excellent agreement between Eq. (42) and theexperimental data, this relation is believed to be an adequate representation of the physicalphenomena and is recommended for calculation purposes.
2. The experimental data are in good agreement with the theoretical development. In view ofthis favorable comparison the assumptions pertaining to the derivation are verified.
9
Note the difference between the two statements. The first statement is quite specific and leaves thereader with a feeling of confidence in the experimental data and theoretical analysis. Upon examiningthe second statement the reader will immediately ask: How good is "good"; how favorable is"favorable"? The author has seen such statements applied to experimental data that differed fromtheoretical values by such a large factor that the veracity of the writer might be questioned by acareful reader.
A brief, general procedure for report writing might take the following form:
1. Make a written outline of the report with as much detail as possible.2. Let the outline "cool" for a period of time while you direct your thoughts to other matters.3. Go back to the outline and make whatever changes you feel are necessary.4. Write the report in rough draft form as quickly as possible.5. Let the report cool for a period of time.6. Go back and make corrections on your draft. You will probably find that you did not say
things quite the way you would like to, did not include as much information as you wanted to,or made several stupid mistakes in some of the data analysis.
7. If possible, have a colleague scrutinize the report carefully. This person should be one whosecompetence you respect.
8. Consider your colleague's comments very carefully, and rewrite the report in its final form.
The above remarks on report writing can be considered as general ones which may be applicable tothe broad range of reports, papers, or monographs the engineer will be called upon to write. Ofcourse, a carefully constructed outline is always the best starting point and will help the writer tomake sure that all pertinent information is included.
2.2 TYPES OF REPORTS
Informational Reports
Informational reports usually follow a letter or interoffice memo format and are brief and to the point.Two examples are given below.
MEMO FORMATTO: J. J. BrownFROM: B. R. SmithDATE:SUBJECT: Test of TV Satellite Uplink Quality
In accordance with your request of _______, we conducted tests with the TV satellite uplink on______. Signals were monitored and videotaped at locations in Boulder, Colorado; Andover,Massachusetts; and Dayton, Ohio. All three sites reported satisfactory video quality at transmission
10
levels below 125 watts. This value is well below the maximum power level of 300 watts for theuplink. Difficulties were experienced with the audio signal caused by 60-Hz interference from the on-desk monitor. The interference was eliminated by better location of the desk microphone or usingand alternate lapel mike.
The three receiving sites are returning videotapes of all the tests to my office where they will beavailable for your inspection. Based on our verbal communications with the sites, we believe that theentire uplink operation is now delivering satisfactory video and audio signals.
LETTER FORMATMr. J. R. MarshallAcme Development Corp.501 Main StreetDallas, Texas 75201
Dear Mr. Marshall:Following out meeting of ______ we conducted preliminary test of the HVAC system in your
two-story building at 10123 North Road. The purpose of these tests was to determine if the mainunit was performing in accordance with specifications. We made the following measurements:
1. Dry-bulb and wet-bulb temperatures for all inlet and exit airstreams of the main unit.2. Air velocities for these airstreams.3. Leakage from the outside air damper.4. Power consumption of the main unit.
Calculations based on these measurements indicate that the main unit is capable of achievingits rated cooling capacity; however, measurements on the outside air damper indicate a leakage of 15percent when the damper is fully closed. This leakage places an additional load on the cooling unit sothat some deterioration in performance is experienced in very hot and humid weather.
We recommend that the outside air damper be replaced; therefore, we would expect entirelysatisfactory operation of the system.
Please let us know if we can be of further service.
Sincerely,.R.W. Smith, P.E.Consulting Engineer
Formal Reports
Formal reports are usually organized to include several of the elements described in the followingsection. Lengths obviously vary with the complexity of the report. The format for the report may be
11
specified by the person or organization to receive the document without much discretion on the partof the writer.
Papers and Journal Articles
Papers and journal articles written for the professional engineering or scientific community maycertainly be classified as "formal" reports. In many cases the journal or literature of the sponsoringorganization will include a section entitles "Information of Authors" which specifies the format to befollowed. The writer should pay close attention to such information because non-adherence to therequired format may result in rejection of the paper even if it is judged by the referees to have goodtechnical merit.
Tutorial Reports
A tutorial report may also be classified as "formal," but - in contrast to papers or journal articles - it isnot directed to a knowledgeable professional in the field. The purpose of the tutorial report is toteach someone who is not well informed in the subject matter. For example, a manufacturer offlowmeters might ask a number of its engineering staff to prepare a report which describes how theflowmeters work, the difficulties to be encountered in installation, proper selection techniques, andaccuracies which may be expected. Such a report would probably be issued as a company brochuredirected at possible users of the product. The report must be factual and helpful to the reader, but itdoes not convey "new" information as would a paper or a journal article.
Books and Monographs
We mention books and monographs mainly for the sake of completeness. Obviously, some largereports are long enough to be called "books," and indeed, some are eventually published as books.Book publishers frequently have their own standards for format and style.
2.3 CONTENTS OF A REPORT
We have mentioned that a good outline is always the starting point for a good report. To aid in thisoutline construction we now give a brief discussion of different report elements. Not all theseelements will be included in every report, and the writer must choose those appropriate to theaudience.
Front Matter
Front matter includes the title page, with author affiliation(s), sponsor of report activity (if any), tableof contents, list of nomenclature, list of figures, preface (if any), and a letter of transmittal if required.
In many cases the letter of transmittal will be a simple interoffice memo for the company. For a majorreport to a project sponsor the letter may be more elaborate.
12
A variety of schemes are used for organizing the table of contents. Many authors like to numberevery paragraph, thereby creating many subheadings. A technical paper for a journal may or may notuse numbered headings and would not include a table of contents.
Nomenclature lists, identifying the symbols naming the used variables, are normally required for bothjournal articles and comprehensive technical reports. They may not be necessary in brief reports for alimited audience.
Common practice is to list terms or variables in alphabetical order, followed by Greek or foreignsymbols in order, and then followed by subscripts and superscripts. The units of all terms should begiven with the nomenclature list. If more than one set of units may be used, as with English and SI,both should be stated. Standard abbreviations should be employed for stating the units. Someexamples of nomenclature listings are as follows:
a Local velocity of sound, m/s or ft/sAc Flow cross-sectional area, m2 or ft2
k Thermal conductivity, W / m ⋅°C or Btu/h⋅ft⋅°FRe=pud/µ Reynolds number, dimensionlessµ Dynamic viscosity, kg/m⋅s or lbm/h⋅ftp Density, kg/m3 or lbm/ft3
13
Subscriptsi Insideo Outside8 Free-stream conditions
When using built-up units, as with thermal conductivity k above, several choices of style are available,such as:
1.Btu
(h)(ft)(°F)2. Btu/(h)(ft)(°F)3. Btu/h ft °F4. Btu/h-ft-°F5. Btu/h/ft/°F
The first four styles all mean the same thing. Styles 1 and 2 are more work for the typist, style 34 iseasiest for the typist, and is common in typeset publications. Style 5 does appear from time to time,but the multiple slash marks can be confused for multiple divisions unless the reader knows thesubject. In general, style 5 should be avoided. Inclusion of units in the nomenclature list is veryimportant. A reader unfamiliar with the terms finds the absence of units very annoying.
Lists of tables and figures are frequently included in large reports but are never used in papers forjournal publication. Inclusion of a preface is optional in reports: it is normally used only inmonographs (books).
Abstracts
Someone stated an old rule for army methods of communications: Tell them what you are going totell them; tell them; and then tell them what you told them. The abstract should attempt toaccomplish the first objective in a very short, succinct format, without mathematical formulations. Itshould tell, specifically, but very briefly, what was done, and the conclusions which resulted from thework. Keep in mind that there are many people who will read only the abstract of a report because ofheavy demands on their time. This is especially true of managers who must review a large number ofreports on a broad range of subjects. A well-written abstract becomes particularly useful for thesepeople, and may arouse their interest in examining the report in more detail.
Some writers may choose to use the terms "summary" or "executive summary" instead of abstract,but the purpose of the section is the same.
Introduction
The introduction section is used to clearly state the motivation for performing the work, i.e., to definethe problem, and sometimes a review of previous work is given too.
Background and Previous Work
14
A survey of the literature may be appropriate and can be included under this title. It could also beincluded in an introduction. In most cases the survey should not be simply chronological, but writtento present the current knowledge of the subject and the specific deficiencies in that knowledge.
This section can vary widely in length, depending on what one assumes for the knowledge level of theintended reader. The length also depends on what the reader needs to know in order to understandand appreciate the remainder of the report. It is rather easy to go off the deep end in this sectionbecause with most technical subjects there are great volumes of background and previous work. Thewriter should keep this section focused so that it both acknowledges previous work and clearly pointsto the need for the current study.
Because of space limitations, most technical papers and journal articles try to minimize the length ofthis section.
Theoretical Presentation(s)
In some reports a large section will be devoted to development of theoretical information applicableto the subject. This section enables the reader to understand the implications of the experimentalwork and aids in proper interpretation of the data.
Of course, some reports are strictly theoretical in nature, and thus this section forms the main bodyfor those reports. To encourage more efficient use of the reader's time, some writers may use thissection as a vehicle to summarize theoretical presentations and relegate long detailed derivations toappendices. In this way, the reader may get to the heart of the presentation more quickly, whileretaining the option of examining details later.
In the theoretical presentations(s) one has the option of defining units for the various terms as theyare introduced, or specifying the units only in the nomenclature list. The latter practice is preferablefor papers because it conserves space. For unusual variables the units might be defined when they areintroduced.
Display of mathematical formulas should follow standard practice in the field.
Experimental Apparatus and Procedure
Sufficient information must be supplied on the apparatus and experimental procedures for the readerto understand what was done. If the report is concerned with research and new knowledge, a ratherdetailed discussion of the apparatus may be necessary. If test results are being reported in accordancewith standard procedures (ASME, ASTM, etc.), then the appropriate procedures can be cited withoutfurnishing details in the report. An error analysis and calibration procedures for the experiments mustbe presented, as explained in Chapter 3.
15
Great variations in length of this section can be used. An extensive report might include all details ofthe apparatus and instrumentation, while a technical paper or journal article would only give a briefsummary.
Results of Experiments
One normally will include a separate section in the report to give results of the experiments in a formwhich is consistent with the needs of the intended audience. Clear tabular and graphical presentationsshould be used as much as possible to conserve words in the report. Of course, some verbaldiscussions of the graphs and tables must be given, but such discussion should focus the reader'sattention on salient features of the data, not just recite numbers or parameters which are obvious uponinspection. In other words the written and graphical presentations should be complementary so thatthe reader's time is conserved.
Section 2.5, on graphical presentations, gives suggestions for good practice.
Interpretation of Results
Once the experimental results have been presented in a clear form, the author has a responsibility tointerpret the results in the light of the theoretical presentation and the work of others in the samegeneral subject area. In this section the background, theoretical presentation, and experimentalresults are all brought together to lead the reader to the conclusions of the study. It is very importantto ensure that the results can be justified based on lows of nature, even if only on a qualitative level.
Of course, there are many instances where one is concerned only with presentation of results and nointerpretation is required. For instance, a calibration test for a thermocouple calls only for acalibration curve, table, or polynominal relation. The results speak for themselves.
Conclusions and Recommendations
By the time the reader has reached this section, most of the conclusions of the work should alreadyhave been drawn. The object of the conclusion section is to collect all the important results andinterpretations in clear summary form. This is the section which tells the reader what was covered inthe body of the report. There will be many readers who will read only the abstract or conclusionsections of a report, so it is especially important that they be carefully written.
Some writers like to include recommendations for action or further study in this section. A typicalaction statement might be:
Calibration of the model 802 flowmeter has now been completed and production runs maybegin immediately.
While a further-study recommendation could be:
16
The calibration procedure for the Model 802 flowmeter has now been established and reliabledata obtained for the temperature range of 50 to 100°F. It is recommended that further databe obtained up to a temperature of 300°F. Once these data are obtained, production runs maybegin.
Acknowledgments
Many people may contribute to a project who are not listed as formal authors of the report. Anacknowledgment section may be used to recognize these contributions, as well as the sponsors of thework (if applicable). For technical papers it is normally placed after the conclusions section, while forother reports it may appear as a part of front matter.
Appendix Materials
Wide latitude is available to the writer in the types of material which may be placed in appendices. Ofcourse, the writer may choose to have no appendix at all and include all information in the body of thereport. Otherwise, one or more of the following might appear in the appendix:
1. Detailed mathematical derivations which are summarized in the body of the report, asdescribed under "Theoretical Presentation(s)" above.
2. Tabulations of raw experimental data which are summarized or correlated in the body of thereport.
3. Calibration information for instruments or sensors employed in the experiments.4. Uncertainty analysis of the experiments, the results of which may be stated or summarized in
the body of the report.5. Tabulations or graphs of material properties used in the report.6. In some cases, calculation charts or materials obtained from other sources.7. Detailed computer programs used in the work, which may be referenced in the report.
We can see that the general purpose of the appendix is to remove detailed clutter from the body ofthe report so that the reader may appreciate the work and conclusions in a shorter period of time. Forthose interested, the details are still available for study. Most technical papers and journal articles donot have much appendix material (if any at all) because the very object of these papers is to presentjust the essence of the work.
References and Bibliographies
Most technical reports require references to the work of others. References should be cited when awork was used in writing the report. For example, the statement:
Cheesewright [1] discusses turbulent natural convection...
with a citation:
17
1. Cheesewright, R.: Turbulent Natural Convection from a Vertical Plane Surface, J. Heat Trans.,vol. 90, p. 1, 1968.
is a valid reference citation. On the other hand, a standard textbook in heat transfer might havegeneral information on free convection and yet not be employed as a specific reference for the report.It is common practice to list such sources under the title of Bibliography. Normally, only extensiveworks like textbooks or monographs have separate bibliographies. Because of the extra spaceconsumed, bibliographies are almost never used in papers for journal publication.
Some latitude is available in the manner of literature citation. The above reference could be listedwithout its title, i.e.,
1. Cheesewright, R., J. Heat Trans., vol. 90, p. 1, 1968.
Although such citations may be accepted by various journals, and do conserve space, they should beavoided whenever possible because the absence of the article title is very annoying to a reader. Theobject of a report is to communicate. A procedure which impedes communication is to be shunned.
An alternate citation technique is to list references in alphabetical order of the first author, thensecond author, and then year. No specific reference number is given and citation in the body of thereport refers to the year of publication. When the same author has more than one citation in one year,the year is followed by a, b, c, and so on. For one or two authors both names are cited; for more thantwo authors only the first author followed by "et. al." is cited. The following examples illustrate thistechnique.
1. Lee, Y., Korpela, S. A., and Horne, R. N., 1982, "Structure of Multi-Cellular Natural Convectionin a Tall Vertical Annulus," Proceedings, 7th International Heat Transfer Conference, U. Grigullet al., eds., Hemisphere Publishing Corp., Washington, D.C., vol. 2, pp. 221-226.
2. Kwon, O.K., and Pletcher, R.H., 1981, "Prediction of the Incompressible Flow Over a Rearward-Facing Step," Technical Report HTL-26, CFD-4, Iowa State Univ., Ames, Iowa.
3. Sparrow, E.M., 1980a, "Fluid-to-Fluid Conjugate Heat Transfer for a Vertical Pipe-InternalForced Convection and External Natural Convection," ASME Journal of Heat Transfer, vol. 102,pp. 402-407.
4. Sparrow, E.M., 1980b, "Forced-Convection Heat Transfer in a Duct Having Spanwise-PeriodicRectangular Protuberances," Numerical Heat Transfer, vol. 3, pp. 149-167.
The corresponding citations in the report would appear as:
1. Lee et al. (1982) discovered...2. Kwon and Pletcher (1981) predicted...3. Sparrow (1980a) studied...4. Sparrow (1980b) observed...
18
The reader will notice that some of the above citations list the journal title in full (Journal of HeatTransfer) while only an abbreviation (J. Heat Trans) is used in others. Either is correct, but somejournals and/or publishers follow their own standard format.
2.4 GRAPHICAL PRESENTATIONS
Most engineering reports include some form of graphical presentation: a simple plot of raw data, astatistical distribution of data, correlation of data, and perhaps a comparison of experimental datawith analytical predictions. Some of the types of graphs used are:
1. General curves or "trends"2. Detailed plots of experimental data3. Accurate graphs which are used for calculation purposes4. Schematic diagrams5. Scale diagrams for experimental apparatus6. Graphs in a format to be used for presentation
Style
Regardless of the purpose of the graph, there are certain elements of good style which should becommon to all.
1. Label the graph. Make sure all coordinates are labeled consistently, i.e., don't use all uppercaseon one coordinate and mixed upper- and lowercase on the other.
2. If numerical values of variables are graphed, make sure that each coordinate has scale markingsand that each label for a variable includes the units for that variable. Try to maintain the same unitsystem for all variables, i.e., do not mix English and SI units.
3. Recognize that some graphs may be reduced in size later because of space limitations. This isparticularly true for those to be published in technical papers or symposia volumes. If anyreduction is anticipated, the size of the labels and unit specifications on the graph must beincreased so that they will be legible in their eventual format. For graphs prepared on computersthis will be an easy matter to arrange.
4. The purpose of graphs is to convey information. Sometimes this is best accomplished by plottingseveral curves on one graph; sometimes not. For example, suppose that six pressure traverses aretaken along the length of a wind tunnel for six different flow rates. In this case it would probablybe best to include all six sets of data on one graph with each curve clearly labeled. A use of sixseparate graphs would not communicate as well and would certainly consume more space. Onthe other hand, suppose a heated plate is placed in the tunnel and heat transfer data collected forthe same six flow rates. It would not be prudent to present these data on the same graph as thepressure data because the display would be too cluttered. A second graph would be in order.There is no hard rule to apply in these circumstances. Common sense usually works fine.
19
5. There is a class of presentations called G and A graphs (for Generals and Admirals). They areusually broad-brush and never loaded with a lot of data points or complicated mathematicalexpressions. The purpose of these graphs is to present general trends without precise data points,and they are normally employed for readers or viewers not very familiar with the subject.
6. Careful thought should be given to the selection of coordinates for a graph. Even with plots ofraw data one may wish to choose a logarithmic plot over a linear system if the data spans are avery wide range and is expected to have an exponential trend.
Legends for Graphs
Although a pertinent part of the legend for a graph may be placed directly on the graph itself, thecurves are labeled typically with numerals (1, 2, 3, etc.) or letters (a, b, c, etc.), or by different linetexture or symbols, which are then defined or explained in the figure title placed under the figure.
Audiovisual presentations expose the audience more briefly to the graphs, and legends have to bemore clear than those used in a written report.
2.5 MISCELLANEOUS HELPFUL HINTS
As a further aid to the writer we give the following bits of advice which do not fall in the categoriesof other sections on report writing.
1. Always give great attention to the audience for which the report is written.2. Set up a list of symbols to be used in the report before starting to write. Then be sure to use these
symbols consistently as the report is written. Be particularly careful about switching betweenuppercase and lowercase letters for the same variable.
3. In choosing symbols to be used in a report try to follow the practice used in standard textbooks ortechnical journals in the field. Don't invent new symbols which will make the report hard for thereader to understand.
4. Once writing is started, write the report as if you had to make it correct on the first draft. Takeyour time. This will pay off later. Of course, unless you are very good, or very lucky, changeswill have to be made, but they usually can be minimized with this approach.
5. If you are not word-processing or typing the report yourself, consider the typist or word-processor operator. Be careful with handwriting, and avoid confusion between English and Greeksymbols.
Acknowledgement: The material, in Sections 2 and 3, is reproduced (with some changes) from"Experimental Methods for Engineers", J.P. Holman and W.J. Gajda Jr. Fifth Ed. (1989) McGraw-Hill, Inc. with permission from the publisher.
20
3. PRESENTING EXPERIMENTAL DATA
3.1 INTRODUCTION
This relatively brief summary should be augmented by readings from the reference list, especiallyHolman [1].
Some form of analysis must be performed on all experimental data. The analysis may be a simpleverbal appraisal of the test results, or it may take the form of a complex theoretical analysis of theerrors involved in the experiment and matching of the data with fundamental physical principles.Even new principles may be developed in order to explain some unusual phenomenon. Ourdiscussion in this chapter will consider the analysis of data to determine errors, precision, and generalvalidity of experimental measurements. The correspondence of the measurements with physicalprinciples is another matter, quite beyond the scope of our discussion. Some methods of graphicaldata presentation will also be discussed. The interested reader should consult the monograph byWilson [2] for many interesting observations concerning correspondence of physical theory andexperiment.
The experimentalist should always know the validity of data. The automobile test engineer mustknow the accuracy of the speedometer and gas gage in order to express the fuel-economyperformance with confidence. A nuclear engineer must know the accuracy and precision of manyinstruments just to make some simple radioactivity measurements with confidence. In order tospecify the performance of an amplifier, an electrical engineer must know the accuracy with which theappropriate measurements of voltage, distortion, etc., have been conducted. Many considerationsenter into a final determination of the validity of the results of experimental data, and we wish topresent some of these considerations in this chapter.
Errors will creep into all experiments regardless of the care which is exerted. Some of these errorsare of a random nature, and some will be due to gross blunders on the part of the experimenter. Baddata due to obvious blunders may be discarded immediately. But what of the data points that just"look" bad? We cannot throw out data because they do not conform with our hopes and expectationsunless we see something obviously wrong. If such "bad" points fall outside the range of normallyexpected random deviations, they may be discarded on this basis of some consistent statistical dataanalysis. The key here is "consistent". The elimination of data points must be consistent and shouldnot be dependent on human whims and bias based on what "ought to be". In many instances it is verydifficult for the individual to be consistent and unbiased. The pressure of a deadline, disgust withprevious experimental failures, and normal impatience all can influence rational thinking processes.However, the competent experimentalist will strive to maintain consistency in the primary dataanalysis. Our objective in this chapter is to show how one may go about maintaining this consistency.
3.2 CAUSES AND TYPES OF EXPERIMENTAL ERRORS
In this section we present a discussion of some of the types of errors that may be present inexperimental data and begin to indicate the way these data may be handled. First, let us distinguishbetween single-sample and multisample data.
21
Single-sample data are those in which some uncertainties may not be discovered by repetition.Multisample data are obtained in those instances where enough experiments are performed so that thereliability of the results can be assured by statistics. Frequently, cost will prohibit the collection ofmultisample data, and the experimentor must be content with single-sample data and prepared toextract as much information as possible from experiments. The reader should consult Refs [1-3] forfurther discussion on this subject, but we state a simple example at this time. If one measurespressure with a pressure gage and a single instrument is the only one used for the entire set ofobservations, then some of the error that is present in the measurement will be sampled only once nomatter how many times the reading is repeated. Consequently, such as experiment is a single-sampleexperiment. On the other hand, if more than one pressure gage is used for the same total set ofobservations, then we might say that a multisample experiment has been performed. The number ofobservations will then determine the success of this multisample experiment in accordance withaccepted statistical principles.
The magnitude of an experimental error is ultimately unknown. If the experimenter know what theerror was, he or she would correct it and it would no longer be an error. In other words, the realerrors in experimental data are those factors that are always vague to some extent and carry someamount of uncertainly. Our task is to determine just how uncertain a particular observation may beand to devise a consistent way of specifying the uncertainty in analytical form. A reasonabledefinition may be taken as the possible range that error may have. This uncertainty may vary a greatdeal depending upon the circumstances of the experiment. Perhaps it is better to speak ofexperimental uncertainty instead of experimental error because of magnitude of an error is alwaysuncertain. Both terms are used in practice, however, so the reader should be familiar with themeaning attached to the terms and the ways that they relate to each other.
At this point, we may mention some of the types of errors that may cause uncertainty in anexperimental measurement. First, there can always be those gross blunders in apparatus or instrumentconstruction which may invalidate the data. Hopefully, the careful experimenter will be able toeliminate most of these errors. Second, there may be certain fixed errors which will cause repeatedreadings to be in error by roughly the same amount but for some unknown reason. These fixed errorsare sometimes called systematic errors. Third, there are the random errors, which may be caused bypersonal fluctuations, random mechanical and electronic fluctuations in the apparatus or instruments,various influences of friction, etc. These random errors usually follow a certain statistical distribution,but not always. In many instances it is very difficult to distinguish between fixed errors and randomerrors.
The experimentalist may sometimes use theoretical methods to estimate the magnitude of a fixederror. For example, consider the measurement of the temperature of a hot gas stream flowing in aduct with a mercury-in-glass thermometer. It is well known that heat may be conducted from thestem of the thermometer into the surrounds. In other words, the fact that part of the thermometer isexposed to the surroundings at a temperature different from the gas temperature to be measured mayinfluence the temperature of the stem of the thermometer. Therefore, the temperature we read on thethermometer is not the true temperature of the gas, and it will not make any difference how manyreadings are taken we shall always have an error resulting from the heat-transfer condition of the stem
22
of the thermometer. This is fixed error, and its magnitude may be estimated with theoreticalcalculations based upon known heat transfer processes and thermal properties of the gas and the glassthermometer.
3.3 ERROR ANALYSIS ON A COMMONSENSE BASIS
We have already noted that it is somewhat more explicit to speak of experimental uncertainty ratherthan experimental error. Suppose that we have satisfied ourselves with the uncertainty in some basicexperimental measurements, taking into consideration such factors as instrument accuracy,competence of the using the instruments, etc. Often, the primary measurements must be combined tocalculate a particular result that is desired. We shall be interested in knowing the uncertainty in thefinal result due to the uncertainties in the primary measurements. This may be done by acommonsense analysis of the data which may take many forms. To find the worst-case error, all theerrors in the primary measurements are combined in the most detrimental way . Consider thecalculation of electric power from
P = EI
where E (voltage) and I (current) are measured as
E = 100 V ± 2 VI = 10 A ± 0.2 A
The nominal value of the power is 100 x 10 =1000 W. By taking the worst possible variations involtage and current, we could calculate.
Pmax = (100 + 2)(10 + 0.2) = 1040.4 WPmin = (100 - 2)(10 - 0.2) = 960.4 W
Thus, using this method of calculation, the uncertainty in the power is +4.04 percent, -3.96 percent.It is quite unlikely that the power would be in error by these amounts because the voltmeter variationswould probably not correspond with the ammeter variations. When the voltmeter reads an extreme"high," there is no reason why the ammeter must also read an extreme "high" at that particular instant;indeed, this combination is most unlikely.
The simple calculation applied to the electric-power equation above is a useful way of inspectingexperimental data to determine what errors could result in a final calculation; however, the test is toosevere and should be used only for rough inspections of data. It is significant to note, however, that ifthe results of the experiments appear to be in error by more than the amounts indicated by the abovecalculation, then the experimenter had better examine the data more closely. In particular, theexperimenter should look for certain fixed errors in the instrumentation (such as the reading on thebathroom scale when no weight is on it), which may be eliminated by applying either theoretical orempirical corrections.
23
As another example we might conduct an experiment where heat is added to a container of water. Ifour temperature instrumentation should indicate a drop in temperature of the water, our good sense(i.e., knowledge of the laws of nature) would tell us that something is wrong and the data point(s)should be thrown out. No sophisticated analysis procedures are necessary to discover this kind oferror.
3.4 UNCERTAINTY ANALYSIS
A method of estimating uncertainty in experimental results has been presented by Kline andMcClintock [3]. The method is based on a careful specification of the uncertainties in the variousprimary experimental measurements. For example, a certain pressure reading might be expressed as
p = 100 kN/m2 ± 1 kN/m2
When the plus or minus notation is used to designate the uncertainty, the person making thisdesignation is stating the degree of accuracy with which he or she believes the measurement has beenmade. We may note that this specification is in itself uncertain because the experimenter is naturallyuncertain about the accuracy of these measurements.
If a very careful calibration of an instrument has been performed recently, with standards of very highprecision, then the experimentalist will be justified in assigning a much lower uncertainty tomeasurements than if they were performed with a gage or instrument of unknown calibration history.
To add a further specification of the uncertainty of a particular measurement, Kline and McClintockpropose that the experimenter specify certain odds for the uncertainty. The above equation forpressure might thus be written
p = 100 kN/m2 ± 1 kN/m2 (20 to 1)
In other words, the experimenter is willing to bet with 20 to 1 odds that the pressure measurement iswithin ± 1 kN/m2. It is important to note that the specification of such odds can only be made bythe experimenter based on the total laboratory experience.
Suppose a set of measurements is made and the uncertainty in each measurement may be expressedwith the same odds. These measurements are then used to calculate some desired result of theexperiments. We wish to estimate the uncertainty in the calculated result on the basis of theuncertainties in the primary measurements. The result R is a given function of the independentvariables x1, x2, x3, ..., xn. Thus,
R = R(x1, x2, x3,....,xn) (3.1)
Let wr be the uncertainty in this result and w1, w2, ..., wn be the uncertainties in the independentvariables. If the uncertainties in the independent variables are all given with same odds, then theuncertainty in the result having these odds is given in Ref. [1] as
24
w R =∂R∂x1
w1
2
+∂R∂x2
w2
2
+ L +∂R∂x n
wn
2
1/2
(3.2)
If this relation is applied to the electric power relation of the previous section, the expecteduncertainty (also called the root-mean-square error) is 2.83 percent instead of 4.04 percent.
Example 3.1 The resistance of a certain size of copper wire is given as
R = Ro [1 + α (T-20)]
where Ro = 6? ±0.3 percent is the resistance at 20°C, α = 0.004°C-1±1 percent is the temperaturecoefficient of resistance, and the temperature of the wire is T = 30±1°C. Calculate the resistance ofthe wire and its uncertainty.
Solution. The nominal resistance is
R = (6)[1 + (0.004)(30 - 20)] = 6.24?
The uncertainty in this value is calculated by applying Eq. (3.2). The various terms are:
∂R∂Ro
= 1 + α T - 20( ) = 1 + (0.004)(30- 20) = 1.04
∂R∂α
= Ro T - 20( ) = (6)(30 - 20) = 60
∂R∂T
= Roα = (6)(0.004) = 0.024
w ro = (6)(0.003) = 0.018Ωwα = (0.004)(0.01) = 4 x 10-5 °C-1
wT = 1°C
Thus, the uncertainty in the resistance iswR = [(1.04)2(0.018)2 + (60)2(4x10-5)2 + (0.024)2(1)2]1/2
= 0.0305? or 0.49%
Particular notice should be given to the fact that the uncertainty propagation in the result wRpredicted by Eq. 3.2 depends on the squares of the uncertainties in the independent variables wn.This means that if the uncertainty in one variable is significantly larger than the uncertainties in theother variables, say, by a factor of 5 or 10, then it is the largest uncertainty that predominates and theothers may probably be neglected.
25
To illustrate, suppose there are three variables with a product of sensitivity and uncertainty[(?R/?x)wx] of magnitude 1, and one variable with a magnitude of 5. The uncertainty in the resultwould be
52 + 12 + 12 + 12( )1/2 = 28 = 5.29
The importance of this brief remark concerning the relative magnitude of uncertainties is evidentwhen one considers the design of an experiment procurement of instrumentation, etc. Very little isgained by trying to reduce the "small" uncertainties. Because of the square propagation it is the largeones that predominate, and any improvement in the overall experimental result must be achieved byimproving the instrumentation or technique connected with these relatively large uncertainties. In theexamples and problems that follow, both in this chapter and throughout the book, the reader shouldalways note the relative effect of uncertainties in primary measurements on the final result.
The reader is cautioned to examine possible experimental errors before the experiment is designedand conducted. Equation (3.2) may be used very effectively for such analysis, as we shall see in thesections and chapters that follow. A further word of caution may be added here. It is equally asunfortunate to overestimate uncertainty as to underestimate it. An underestimate gives false security,while an overestimate may make one discard important results, miss a real effect, or buy much tooexpensive instruments. The purpose of this chapter is to indicate some of the methods for obtainingreasonable estimates of experimental uncertainty.
In the previous discussion of experimental planning we noted that an uncertainty analysis may aid theinvestigator in selecting alternative methods to measure a particular experimental variable. It mayalso indicate how one may improve the overall accuracy of a measurement by attacking certain criticalvariables in the measurement process. The next three examples illustrate these points.
Example 3.2 Selection of measurement method . A resistor has a nominal stated value of10? ±1 percent. A voltage is impressed on the resistor, and the power dissipation is to be calculatedin two different ways: (1) from P = E2/R and (2) from P = EI. In (1) only a voltage measurement willbe made, while both current and voltage will be measured in (2). Calculate the uncertainty in thepower determination in each case when the measured values of E and I are:
E = 100 V ± 1%I = 10 A ± 1%
FIGURE EXAMPLE 3.2Power measurement across a resistor.
Solution. The schematic is shown in the accompanying figure. For the first case we have
26
∂P∂E
= 2ER
∂P∂R
= -E2
R2
and we apply Eq. (3.2) to give
w P = 2ER
2
wE2 + -
E 2
R2
2
w R2
1/2
(a)
Dividing by P = E2/R givesw P
P = 4
wE
E
2
+ w R
R
2
1/2
(b)
Inserting the numerical values for uncertainty,
w P
P = 4 0.01( )2 + 0.01( )2[ ]1 / 2
= 2.236%
For the second case we have∂P∂E
= I ∂P∂I
= E
and after similar algebraic manipulation, we obtainw P
P =
wE
E
2
+ w I
I
2
1/2
(c)
Inserting the numerical values of uncertainty,
w P
P = 0.01( )2 + 0.01( )2[ ]1/2
= 1.414%
Thus, the second method of power determination provides considerably less uncertainty than the firstmethod, even though the primary uncertainties in each quantity are the same. In this example theutility of the uncertainty analysis is that it affords the individual a basis for selection of a measurementmethod to produce a result with less uncertainty.
Example 3.3 Instrument selection. The power measurement in Example 3.2 is to be conducted bymeasuring voltage and current across the resistor with the circuit shown in the accompanying figure.The voltmeter has an internal resistance Rm, and the value of R is known only approximately.Calculate the nominal value of the power dissipated in R and the uncertainty for the followingconditions:
R = 100? (not known exactly)Rm = 1000? ± 5%
27
I = 5 A ±1%E = 500 V ± 1%
FIGURE EXAMPLE 3.3Effect of meter impedance on measurement.
Solution. A current balance on the circuit yields
I1 + I2 = I
ER
+ ERm
= I
and
I1 = I - ERm
(a)
The power dissipated in the resistor is
P = EI1 = EI - E2
Rm
(b)
The nominal value of the power is thus calculated as
P = (500)(5) - 5002
1000 = 2250 W
In terms of known quantities the power has the functional form P = f(E,I,Rm), and so we form thederivatives
∂P∂E
= I - 2ERm
∂P∂I
= E
∂P∂Rm
= E2
Rm2
The uncertainty for the power is now written as
w P = I -2ERm
2
w E2 + E2w1
2 + E2
Rm2
2
wRm
2
1/2
(c)
28
Inserting the appropriate numerical values gives
w P = 5 - 10001000
2
52 + 25 x 104( )25 x 10 -4( ) + 25 x 104
106
2
2500( )
1/2
= 16 + 25 + 6.25[ ]1/2 5( ) = 34.4 W
orw P
P =
34.42250
= 1.53%
In order of influence on the final uncertainty in the power we have
1. Uncertainty of current determination2. Uncertainty of voltage measurement3. Uncertainty of knowledge of internal resistance of voltmeter
Comment. There are other conclusions we can draw from this example. The relative influence ofthe experimental quantities on the overall power determination is noted above. But this listing may bea bit misleading in that it implies that the uncertainty of the meter impedance does not have a largeeffect on the final uncertainty in the power determination. This results from the fact that Rm>>R(Rm= 10R). If the meter impedance were lower, say, 200? , we would find that it was a dominant factorin the overall uncertainty. For a very high meter impedance there would be little influence, even witha very inaccurate knowledge of the exact value of Rm. Thus, we are led to the simple conclusion thatwe need not worry too much about the precise value of the internal impedance of the meter as long asit is very large compared with the resistance we are measuring the voltage across. This fact shouldinfluence instrument selection for a particular application.
3.5 EVALUATION OF UNCERTAINTIES FOR COMPLICATED DATA REDUCTION
We have seen in the preceding discussion and examples how uncertainty analysis can be a useful toolto examine experimental data. In many cases data reduction is a rather complicated affair and is oftenperformed with a computer routine written specifically for the task. A small adaptation of the routinecan provide for direct calculation of uncertainties when analytical determination of the partialderivatives in Eq. (3.2) is difficult. We still assume that this equation applies, although it couldinvolve several computational steps. We also assume that we are able to obtain estimates by somemeans of the uncertainties in the primary measurements, i.e., w1, w2, etc.
Suppose a set of data is collected in the variables x1, x2,....,xn and a result calculated. At thesame time one may perturb the variables by ∆x1, ∆x2, and so on, and calculate new results. Wewould have
R(x1) = R(x1, x2,....,xn)
R(x1 + ∆x1) = R(x1, + ∆x1, x2, ..., xn)
29
R(x2) = R(x1, x2,...,xn)
R(x2 + ∆x2) = R(x1, x2 + ∆x2,...,xn)
For small enough values of ∆x the partial derivatives can be well approximated by the finite differenceexpressions
∂R∂x1
≈ R x1 + ∆x1( ) - R x1( )
∆x1
∂R∂x2
≈ R x 2 + ∆x 2( ) - R x2( )
∆x 2
and these values could be inserted in Eq. (3.2) to calculate the uncertainty in the result.
At this point we must again alert the reader to the ways uncertainties or errors of instruments arenormally specified. Suppose a pressure gage is available and the manufacturer states that it isaccurate within ±1.0 percent. This statement normally refers to percent of full scale. So a gage witha range of 0 to 100 kPa would have an uncertainty of ±10 percent when reading a pressure of only10 kPa. Of course, this means that the uncertainty in the calculated result, either as an absolute valueor percentage, can vary widely depending on the range of operation of instruments used to make theprimary measurements. The above procedure can be used to advantage in complicated data-reductionschemes.
A very full description of this technique and many other considerations of uncertainty analysis aregiven by Moffat [4]. An example of an industry standard on uncertainty analysis is given in Ref. [5].
Example 3.5. Calculate the uncertainty of the wire resistance in Example 3.1 using the techniquedescribed in this section.
Solution. In Example 3.1 we have already calculated the nominal resistance at 6.24? . We nowperturb the three variables Ro, α, and T by small amounts to evaluate the partial derivatives. We shalltake
∆Ro = 0.01 ∆α = 1 x 10-5 ∆T = 0.1Then
R(Ro + ∆Ro) = (6.01)[1 + (0.004)(30-20)] = 6.2504
and the derivative is approximated as
∂R∂Ro
≈ R Ro + ∆Ro( )- R
∆Ro
= 6.2504 - 6.24
0.01 = 1.04
30
or the same result as in Example 3.1. Similarly,
R(α + ∆α) = (6.01)[1 + (0.00401)(30-20)] = 6.2406
∂R∂α
≈ R α + ∆α( )- R
∆α =
6.2406 - 6.241 x 10-5 = 60
R(T + ∆T) = (6)[1 + (0.004)(30.1-20)] = 6.2424
∂R∂T
≈ R T + ∆T( )- R
∆T =
6.2424 - 6.240.1
= 0.24
All the derivatives are the same as in Example 3.1 so the uncertainty in R would be the same, or0.0305? .
3.6 GRAPHICAL ANALYSIS AND CURVE FITTING
Successful analysis of experimental data requires good understanding of the physical processes behindthe data. Unless thought through carefully, curve-plotting and cross-plotting usually generate anexcess of displays, which are confusing not only to the management or supervisory personnel whomust pass on the experiments, but sometimes even to the experimenter.
Assuming that the engineer knows what is to be examined with graphical presentations, the plots maybe carefully prepared and checked against appropriate theories. Frequently, a correlation of theexperimental data is desired in terms of analytical expression between variables that were measured inthe experiment; the easiest to plot and understand is a linear relationship. It is most convenient, then,to try to plot the data in such a linear form, which could sometimes be accomplished by a coordinatetransformation.
Table 3.1 summarizes several different types of functions and transformations that may be used toproduce straight lines on graph paper. The graphical measurements, which may be made to determinethe various constants, are also shown. It may be remarked that the method of least squares may beapplied to all these relations to obtain the best straight line to fit the experimental data. A number ofcomputer software packages are available to accomplish the functional plots illustrated in Table 3.1.See, for example, Refs.. [6], [7], and [8].
Note that when using logarithmic or semilog graph paper is is unnecessary to make log calculations;the scaling of the paper automatically accomplishes this.
3.7 GENERAL CONSIDERATIONS IN DATA ANALYSIS
Our discussions in this chapter have considered a variety of topics: statistical analysis, uncertaintyanalysis, curve plotting, least squares, etc. With these tools the reader is equipped to handle a varietyof circumstances that may occur in experimental investigations. As a summary to this chapter let us
31
now give an approximate outline of the manner in which one would go about analyzing a set ofexperimental data.
1. Examine the data for consistency. No matter how hard one tries, there will always be somedata points that appear to be grossly in error. If we add heat to a container of water, thetemperature must rise, and so if a particular data point indicates a drop in temperature for aheat input, that point might be eliminated. In other words, the data should follow consistencywith laws of nature, and points that do not appear proper in that way should be eliminated. Ifvery many data points fall in the category of "inconsistent," perhaps the entire experimentalprocedure should be investigated for gross mistakes or miscalculation.
2. Perform a statistical analysis of data where appropriate. A statistical analysis is onlyappropriate when measurements are repeated several times. If this is the case, make estimatesof such parameters as standard deviation, etc.
3. Estimate the uncertainties in the results. We have discussed uncertainties at length.Hopefully, these calculations will have been performed in advance and the investigator willalready know the influence of different variables by the time the final results are obtained.
4. Anticipate the results from theory. Before trying to obtain correlations of the experimentaldata, the investigator should carefully review the theory appropriate to the subject and try toglean some information that will indicate the trends the results may take. Importantdimensionless groups, pertinent functional relations, and other information may lead to afruitful interpretation of the data.
5. Validate the data. The experimental investigator should make sense of the data in terms ofphysical theories or on the basis of previous experimental work in the field. Certainly, theresults of the experiments should be analyzed to show how they conform to or differ fromprevious investigations or standards that may be employed for such measurements.
6. Correlate the data. Develop the mathematical relationship between the parameter of interestand the independently measured variables which define it. For example, the equation Nu =cRe0.8Pr0.4 is the mathematical relationship between the Nusselt number and the Reynolds andPrandth numbers which are the independent variables sufficient to determine it.
3.8 SUMMARY
By now the reader will have sensed the central theme of this chapter as that of uncertainty analysisand the use of this analysis to influence experiment design, instrument selection, and evaluation of theresults of experiments. At this point we must reiterate statements we have made before. We stillmust recognize that uncertainty is not the same as error, even though some people interchange theterms. The determination of "error" is eventually related to a comparison with a standard. Even then,there is still "uncertainty" in the error because the "standard" has its own uncertainty.
32
Table 3.1Methods of plotting various functions to obtain straight lines
33
Table 3.1Methods of plotting various functions to obtain straight lines (Continued)
34
In the chapters which follow we shall examine a large number of instruments and measurementdevices and will see how the concepts of error, uncertainty, and calibration apply to each.
REVIEW QUESTIONS (Answers require also the reading of the references)
3.1 How does an error differ from an uncertainty?3.2 What is a fixed error; random error?3.3 Define standard deviation and variance.3.4 In the normal error distribution, what does P(x) represent?3.5 What is meant by measure of precision?3.6 What is Chauvenet's criterion and how is it applied?3.7 What are some purposes of uncertainty analyses?3.8 Why is an uncertainty analysis important in the preliminary stages of experiment planning?3.9 How can an uncertainty analysis help to reduce overall experimental uncertainty?3.10 What is meant by standard deviation of the mean?3.11 What is a least-squares analysis?3.12 What is the correlation coefficient?3.13 What is meant by a regression analysis?
PROBLEMS
3.1 The resistance of a resistor is measured 10 times, and the values determined are 100.0, 100.9,99.3, 99.9, 100.1, 100.2, 99.9, 100.1, 100.0, and 100.5. Calculate the uncertainty in theresistance.
3.2 A certain resistor draws 110.2 V and 5.3 A. The uncertainties in the measurements are ±0.2V and ±0.06 A, respectively. Calculate the power dissipated in the resistor and theuncertainty in the power.
3.3 A small plot of land has measured dimensions of 50.0 by 150.0 ft. The uncertainty in the 50-ftdimension is ±0.01 ft. Calculate the uncertainty with which the 150-ft dimension must bemeasured to ensure that the total uncertainty in the area is not greater than 150 percent of thatvalue it would have if the 150-ft dimension were exact.
3.4 Two resistors R1 and R2 are connected in series and parallel. The values of the resistance are
R1 = 100.0 ± 0.1?R2 = 50.0 ± 0.03?
Calculate the uncertainty in the combined resistance for both the series and the parallelarrangements.
3.5 A resistance arrangement of 50? is desired. Two resistances of 100.0±0.1? and tworesistances of 25.0±0.02? are available. Which should be used, a series arrangement with the25? resistors or a parallel arrangement with the 100-? resistors? Calculate the uncertaintyfor each arrangement.
3.6 The following data are taken from a certain heat-transfer test. The expected correlationequation is y=axb. Plot the data in an appropriate manner, and use the method of leastsquares to obtain the best correlation.
35
x 2040 2580 2980 3220 3870 1690 2130 2420 2900 3310 1020
Calculate the mean deviation of these data from the best correlation.3.7 A horseshoes player stands 30 ft from the target. The results of the tosses are
Deviation from Deviation fromToss target, ft Toss target, ft1 0 6 +2.42 +3 7 -2.63 -4.2 8 +3.54 0 9 +2.75 +1.5 10 0
On the basis of these data would you say that this is a good player or a poor player? Whatadvice would you give this player in regard to improving at the game?
3.8 Calculate the probability of drawing a full house (three of a kind and two of a kind) in the first5 cards from a 52-card deck.
3.9 Calculate the probability of filling an inside straight with one draw from the remaining 48cards of a 52-card deck.
3.10 A voltmeter is used to measure a known voltage of 100 V. Forty percent of the readings arewithin 0.5V of the true value. Estimate the standard deviation for the meter. What is theprobability of an error of 0.75V?
3.11 In a certain mathematics course the instructor informs the class that grades will be distributedaccording to the following scale provided that the average class score is 75:
Grade A B C D FScore 90-100 80-90 70-80 60-70 below 60
Estimate the percentage distribution of grades for 5, 10, and 15 percent failing. Assume thatthere are just as many A's as F's.
3.12 For the following data points y is expected to be a quadratic function of x. Obtain thisquadratic function by means of a graphical plot and also by the method of least squares.
x 1 2 3 4 5y 1.9 9.3 21.5 42.0 115.7
3.13 It is suspected that the rejection rate for a plastic-cup-molding machine is dependent on thetemperature at which the cups are molded. A series of short tests is conducted to examinethis hypothesis with the following results:
Temperature Total production Number rejectedT1 150 12
36
T2 75 8T3 120 10T4 200 13
On the basis of these data do you agree with the hypothesis?3.14 A capacitor discharges through a resistor according to the relation E/Eo=e-t/RC where
Eo=voltage at time zero; R=resistance; C=capacitance. The value of the capacitance is to bemeasured by recording the time necessary for the voltage to drop to a value E1. Assumingthat the resistance is known accurately, derive an expression for the percent uncertainty in thecapacitance as a function of the uncertainty in the measurements of E1 and t.
3.15 In heat-exchanger applications, a log mean temperature is defined by
∆Tm = Th1
- Tc 1( )- Th2 - Tc 2( )
ln Th1 - Tc1( )/ Th2
- Tc 2( )[ ]where the four temperatures are measured at appropriate inlet and outlet conditions for theheat-exchanger fluids. Assuming that all four temperatures are measured with the sameabsolute uncertainty wT, derive an expression for the percentage uncertainty in ∆Tm in termsof the four temperatures and the value of wT. Recall that the percentage uncertainty is
w∆Tm
∆Tm
x 100
3.16 A certain length measurement is made with the following results:
Reading 1 2 3 4 5 6 7 8 9 10x, in 49.36 50.12 48.98 49.24 49.26 50.56 49.18 49.89 49.33 49.39
Calculate the standard deviation, the mean reading, and the uncertainty. Apply Chauvenet'scriterion as needed.
3.17 A citizen's traffic committee decides to conduct its own survey and analysis of the influence ofdrinking on car accidents. By some judicious estimates the committee determines that in theircommunity 30 percent of the drivers on a Saturday evening between 10 p.m. and 2 a.m. haveconsumed some alcohol. During this same period there were 50 accidents, varying fromminor scratched fenders to fatalities. In these 50 accidents 50 of the drivers had hadsomething to drink (there are 100 drivers for 50 accidents). From these data what conclusionsdo you draw about the influence of drinking on car accidents? Can you devise a better way toperform this analysis?
3.18 The grades for a certain class fall in the following ranges:
Number 10 30 50 40 10 8Score 90-100 80-90 70-80 60-70 50-60 Below 50
37
The arithmetic mean grade is 68. Devise your own grade distribution for this class. Be sureto establish the criteria for the distribution.
3.19 A certain length measurement is performed 100 times. The arithmetic mean reading is 6.823ft, and the standard deviation is 0.01 ft. How many readings fall within (a) ±0.005 ft, (b)±0.02 ft, (c) ±0.05 ft, and (d) 0.001 ft of the mean value?
3.20 A series of calibration tests is conducted on a pressure gage. At a known pressure of 1000psia, it is found that 30 percent of the readings are within 1 psia of the true value. At a knownpressure of 500 psia, 40 percent of the readings are within 1 psia. At a pressure of 200 psia,45 percent of the readings are within 1 psia. What conclusions do you draw from thesereadings? Can you estimate a standard deviation for the pressure gage?
3.21 Two resistors are connected in series and have the following values:
R1 = 10,000 ? ± 5% R2 = 1 M? ± 10%
Calculate the percent uncertainty for the series total resistance.3.22 Two groups of secretaries operate under the same manager. Both groups have the same
number of people, use the same equipment, and turn out about the same amount of work.During one maintenance period, group A had 10 service calls on the equipment while group Bhad only 6 calls. From these data would you conclude that group A was harder on theequipment?
3.23 A laboratory experiment is conducted to measure the viscosity of a certain oil. A series oftests gives the values as 0.040, 0.041, 0.041, 0.042, 0.039, 0.040, 0.043, 0.041, and 0.039ft2/s. Calculate the mean reading, the variance, and the standard deviation. Eliminate anydata points as necessary.
3.24 The following data are expected to follow a linear relation of the form y = ax + b. Obtain thebest linear relation in accordance with a least-squares analysis. Calculate the standarddeviation of the data from the predicted straight-line relation.
x 0.9 2.3 3.3 4.5 5.7 6.7y 1.1 1.6 2.6 3.2 4.0 5.0
3.25 The following data points are expected to follow a functional variation of y = axb. Obtain thevalues of a and b from a graphical analysis.
x 1.21 1.35 2.40 2.75 4.50 5.1 7.1 8.1y 1.20 1.82 5.0 8.80 19.5 32.5 55.0 80.0
3.26 The following data points are expected to follow a functional variation of y = aebx. Obtainthe values of a and b from a graphical analysis.
x 0 0.43 1.25 1.40 2.60 2.9 4.3y 9.4 7.1 5.35 4.20 2.60 1.95 1.15
38
3.27 The following heat-transfer data points are expected to follow a functional form of N = aRb.Obtain the values of a and b from a graphical analysis and also by the method of least squares.
R 12 20 30 40 100 300 400 1000 3000N 2 2.5 3 3.3 5.3 10 11 17 30
What is the average deviation of the points from the correlating relationship?3.28 In a student laboratory experiment a measurement is made of certain resistance by different
students. The values obtained were
Calculate the standard deviation, the mean reading, and the uncertainty.3.29 In a certain decade resistance box resistors are arranged so that four resistances may be
connected in series to obtain a desired result. The first selector uses 10 resistances of 1000,2000,...,9000, the second uses 10 of 100, 200,...,900, the third uses 10 of 10, 20,...,90, andthe fourth, 1,2,...,9? . Thus the overall range is 0 to 9999 ? . If all the resistors have anuncertainty of 1.0 percent, calculate the percent uncertainties for total resistances of 9, 56,148, 1252, and 9999 ? .
3.30 Calculate the chances and probabilities that data following a normal-distribution curve will fallwithin 0.2, 1.2, and 2.2 standard deviations of the mean value.
3.31 Suggest improvements in the measurement uncertainties for Example 3.4 which will result inreduction in the overall uncertainty of flow measurement to 1.0 percent.
3.32 What uncertainty in the resistance for the first part of Example 3.2 is necessary to produce thesame uncertainty in power determination as results from the current and voltagemeasurements?
3.33 Use the technique of Sec. 3.5 with Example 3.4.3.34 Use the technique of Sec. 3.5 with Examples 3.3 and 3.2.3.35 Obtain the correlation coefficient for Prob. 3.24.3.36 Obtain the correlation coefficient for Prob. 3.25.3.37 Obtain the correlation coefficient for Probs. 3.26 and 3.27.3.38 Obtain the correlation coefficient for Probs. 3.6 and 3.12.3.39 For the heat exchanger of Prob. 3.15 the temperatures are measured as Th1
= 100°C, Th2=
80°C, Tc1 = 75°C, and Tc2
= 55°C. All temperatures have an uncertainty of ±1°C. Calculatethe uncertainty in ∆Tm using the technique of Sec. 3.5.
3.40 A radar speed-measurement device for state police is said to have an uncertainty of ±4percent when directed straight at an oncoming vehicle. When directed at some angle θ fromthe straight-on position, the device measures a component of the vehicle speed. The policeofficer can only obtain a value for the angle θ through a visual observation having anuncertainty of ±10°. Calculate the uncertainty of the speed measurement for θ values of 0,10, 20, 30, and 45°. Use the techniques of both Secs. 3.4 and 3.5.
39
3.41 An automobile is to be tested for its acceleration performance and fuel economy. Plan thisproject taking into account the measurements which must be performed and expecteduncertainties in these measurements. Assume that three different drivers will be used for thetests. Make plans for the number of runs which will be used to reduce the data. Also preparea detailed outline with regard to form and content of the report which will be used to presentthe results.
3.42 A thermocouple is used to measure the temperature of a known standard maintained at 100°C.After converting the electrical signal to temperature the readings are: 101.1, 99.8, 99.9, 100.2,100.5, 99.6, 100.9, 99.7, 100.1, and 100.3. Using whatever criteria seem appropriate, makesome statements about the calibration of the thermocouple.
3.43 Seven students are asked to make a measurement of the thickness of a steel block with amicrometer. The actual thickness of the block is known very accurately as 2.000 cm. Theseven measurements are: 2.002, 2.001, 1.999, 1.997, 1.998, 2.003, and 2.003 cm. Commenton these measurements using whatever criteria you think appropriate.
3.44 A collection of 120 rock aggregate samples is taken and the volumes measured for each. Themean volume is 6.8cm3 and the standard deviation is 0.7cm3. How many rocks would youexpect to have volumes ranging from 6.5 to 7.2cm3?
3.45 Plot the equation y=5e1.2x on semilog paper. Arbitrarily assign fictitious data points on bothsides of the line so that the line appears by eye as a reasonable representation. Then, usingthese points, perform a least-squares analysis to obtain the best fit to the points. What do youconclude from this comparison?
3.46 The following data are presumed to follow the relation y=axb. Plot the values of x and y onlog-log graph paper and draw a straight line through the points. Subsequently, obtain thevalues of a and b. Then determine the values of a and b by the method of least squares.Compute the standard deviation for both cases. If a packaged computer routine for the least-squares analysis is available, use it.
x y4 1055.3 155
11 32021 58030 105050 1900
3.47 The variables x and y are related by the quadratic equationy = 2 - 0.3x + 0.01x2
for 0<x<2. Compute the percentage uncertainty in y for uncertainties in x of ±1, 2, and 3percent. Use both an analytical technique and the numerical technique discussed in Sec. 3.5.
3.48 Reynolds numbers for pipe flow may be expressed as
µdm
•
= 4Re
40
where •m is the mass flow in kg/s, d is pipe diameter in m, and µ is viscosity in kg/m ⋅s . In a
certain system the flow rate is 12 lbm/min, ±0.5 percent, through a 0.5-in diameter (±0.005-in) pipe. The viscosity is 4.64 x 10-4 lbm/hr ⋅ft , ±1 percent. Calculate the value of theReynolds number and its uncertainty. Use both the analytical and numerical techniques.
3.49 The specific heat of a gas at constant volume is measured by determining the temperature riseresulting from a known electrical heat input to a fixed mass and volume. Then
P = EI = mcν∆T = mcν(T2-T1)where the mass is calculated from the ideal gas law and the volume, that is,
m = p1VRT1
Suppose the gas is air with R = 287 J/ kg⋅K and cν=0.714 kJ/ kg⋅ °C, and the measurementsare to be performed on a 1-liter volume (known accurately) starting at p1 = 150kPa and T1 =30°C. Determine suitable power and temperature requirements, assign some uncertainties tothe measured variables, and estimate the uncertainty in the value of specific heat determined.
3.50 A model race car is placed on a tethered circular track having a diameter of 10 m ±1cm. Thespeed of the car is determined by measuring the time required for traveling each lap. A hand-held stopwatch is used for the measurement, and the estimated uncertainty in both starting andstopping the watch is ±0.2 sec. For a nominal speed of 100 mi/hr calculate the uncertainty inthe speed measurement when made over 1, 2, 3, and 4 laps.
REFERENCES
1. Holman, J.P. and Gajada, W.J. Jr., "Experimental Methods for Engineers", 5th ed., McGrawHill, N.Y. 1989.
2. Wilson, E.B.: "An Introduction to Scientific Research," McGraw-Hill Book Company, NewYork, 1952.
3. Kline, S.J., and F.A. McClintock: Describing Uncertainties in Single-Sample Experiments,Mech. Eng., p. 3, January 1953.
4. Moffat, R.J.: Contributions to the Theory of Single-Sample Uncertainty Analysis, J. FluidEngr., vol. 104, p. 250, June 1982.
5. "Measurement Uncertainty," ANSI/ASME Power Test Code 19.1-1985, American SocietyMechanical Engineers, New York, 1986.
6. Johnson, A.T.: Microcomputer Programs for Instrumentation and Data Analysis, Int. Appl.Engineering Education, vol. 3, no. 2, p. 149, 1987.
7. Canale, R.P., and S.C. Chapra: "Electronic Tool Kit," McGraw-Hill Book Company, 1988.8. Chapra, S.C., and R.P. Canale: "Numerical Methods for Engineers," 2d Ed., McGraw-Hill
Book Company, 1988.
41
EXPERIMENT No. 1.LIFT AND DRAG ON 1/48 SCALE MODEL AIRCRAFT
APPARATUS
WIND TUNNEL: The AEROLAB Educational Wind Tunnel is of the Eiffel or Open Circuit typewith a 12”x12” test section. The airspeed is infinitely variable from 0 to 145 mph with a 10 hp driveoperated by a variable frequency system. The tunnel achieves a high energy ratio through its largecontraction ratio of 9.5:1, its small-angle diffuser, efficient fan system and low turbulence level of0.13%. The fan operates in an acoustically treated section to reduce the noise level. The tunnel isequipped with an honeycomb and two anti-turbulence screens. An orifice ring encircles the upstreamend of the test section to provide the average static pressure of this section. This provides the staticpressure of the undisturbed flow which is used as the reference pressure for pressures measured onmodels when the pressures are non-dimensionalized as pressure coefficients. This static pressure isalso used as the throat Venturi pressure in the measurement of airspeed.
START UP PROCEDURE: Before operating the tunnel make sure that the model to be tested issecurely fastened and all loose objects have been removed from the test section.1. Keep the FAN switch OFF and turn the fan speed control fully counterclockwise.2. Turn on panel power by deflecting METER switch ON. All meters should light.3. Turn on the FAN switch. Slowly turn fan speed control clockwise until motor begins to turn.4. Check to see that the direction of rotation is correct. Air should be blowing out the exhaust and
the fan blades should be turning clockwise looking through the exhaust port. If the motor isrunning backwards, contact your TA or the lab technician.
ANGLE OF ATTACT: Angle of Attack is sensed by a potentiometer connected to the lower armshaft of the parallelogram linkage. The zero adjustment is a second potentiometer on the front panel.The calibration can be checked by leveling and zeroing the sting and then adjusting the sting a knownnumber of degrees and reading the new indication. DO NOT EVER position sting at an angle ofattack greater than +30°. This places great pressure on the cable and will usually shear thevery fine gage wires inside the cable.
ZERO AIRSPEED: Setting the airspeed reading to zero before starting a run is complicated by theinternal square root circuit which cannot take the square root of negative numbers. This requires thatzero be adjusted from an initially positive reading until it just reaches zero.
MODEL INSTALLATION: A socket in the model is provided to mount the model on the sting.Before mounting a model make sure that the set screw is retracted. Push the model gently to the stopand tighten set screw with moderate torque.
As in most large subsonic tunnels, the model is sting mounted for balance readings. The modelpositioning system is of the parallelogram type which rotates the model in yaw or pitch about a pointnear the end of the sting, keeping the model substantially in the center of the test section. A turntablein the floor of the test section permits rotation in yaw up to the limits imposed by the side walls(±30°).
For airplane models the airplane center of gravity (C.G.) is used as a reference center. Themeasured moment is referred to the balance moment center. The moment must be transferred to a
42
parallel axis through the C.G. Usually the C.G. of the model is in the vicinity of the sting balancecenter so the moment transfer is small.
STING BALANCE: The sting balance is designed with a 3/8” diameter hardened stainless steel stingequipped with separate strain gauges for Normal Force and Pitching Moment. Axial Force ismeasured by parallel flexures which are designed for both strength and sensitivity. Normal Force isdifferentially wired. Pitching Moment is measured by a completely separate Wheatstone bridge.Because of this wiring, each component is direct reading without resorting to arithmeticalcalculations. The maximum load ranges are:
Normal Force: 25 lbs.Axial Force: 10 lbs.Pitching Moment: 50 inch-lbs.
To protect the balance, a stop is provided to counteract abnormally large forces in the axialdirection. The outer end of the balance is cylindrical to accommodate models with a straight hole anda transverse set screw. The balance is designed to rotate 90 in its socket so that Normal Forcebecomes Side Force and Pitch Moment becomes Yawing Moment. This feature is important formeasurements on a model in yaw.
To convert the Normal Force and Axial Force to Lift and Drag components the followingformulas should be used:
Drag = N sinα + AcosαLift = Ncosα − Asin α
where α = angle of attackN = Normal ForceA = Axial Force
BALANCE CALIBRATION: The strain gauge balance is both the most fragile and the mostexpansive component of the system. The wires leading from the balance are only a few thousandths ofan inch in diameter and therefore cannot stand much tension or rough handling.
For checking the calibration of the balance, the procedure is as follows:1. Install the 3/4” diameter aluminum alloy calibration barrel on the sting balance by sliding it back to
the stop and tightening the set screw.2. Level the calibration barrel. The angle of attack readout should be adjusted to 0°.3. Adjust the sting to an angle of attack at 30°.4. Zero the balance panel meters: Normal Forces, Axial Forces and Pitching Moment. Make sure the
roll-pin at the end of the sting balance support bears against the near end of the slot marked NF.5. Hang the weight holder onto the calibration barrel, and record the reading for the Normal and
Axial Forces.6. Add a known weight, one pound at a time, onto the weight holder, and record the Normal and
Axial Forces. From the recorded data obtain a calibration factor relates the applied forces to thereadouts.
7. For Pitching Moment calibration, move the known weight forward two grooves (i.e. 2” forwardof the balance moment center). The Normal Force readout should remain approximately constant.(Not used in this experiment.)
PRESSURE SYSTEM: Tunnel pressures are connected to the 24-input selector valve to the left ofthe control panel. The control knob rotates an internal valve, which successively connects each of theports to a Sen-Sym transducer that offers the optimum combination of high accuracy, stability andfreedom from hysteresis. The transducer is of the differential type so pressures may be sensed relative
43
to any desired pressure. The most useful reference pressure is the static pressure of the approachingor undisturbed flow which is averaged by the static pressure ring at the upstream end of the testsection. The reference port is therefore ordinarily connected to the static ring. The pressure range is±20” of water.
PRESSURE CALIBRATION: This wind tunnel offers a uniform low-turbulence airstream at speedsup to 145 mph. Accurate data can be obtained provided that the airspeed, or more appropriately thedynamic pressure, is measured correctly along with forces, moments and pressures on models.Starting out with the Airspeed calibration, the procedure is as follows:1. Zero Airspeed, see the instruction.2. Disconnect tube running from test section static ring.3. Connect one of the lines from the tee (connected to the manometer) to the line just disconnect
from the static ring.4. Apply suction to the tube connected to tee and hold the pressure when 5” to 6” of vacuum is
registered.5. Read Airspeed on display.6. Obtain the correction factor. The reading is considered “correct” when the speed (in miles per
hour) corresponds to the vacuum reading on the manometer. The Airspeed is based on standardair density of 0.002378 slugs/ft 3 . The dynamic pressure q can therefore be calculated as 1
2 ρairV2
where V is in ft/sec. The density of water in the manometer is taken at 62.3 lbs/ft3 correspondingto a temperature of 20°C centigrade.
The pressure in the static pressure ring relative to atmospheric tunnel inlet pressure is usually termed∆h . For a typical tunnel, q/∆h = 1.02 over the range of 100 to 150 mph. This value is factored intothe airspeed versus Inches of Water Curve.
The pressure readout system can be calibrated by temporary disconnecting the tubing connectedto the center of the 24-inlet selection valve and connecting it to one leg of the U-tube manometersupplied. Positive or negative pressure can be applied. The procedure is the same as the airspeedcalibration.
INTRODUCTION
FORCE BALANCE: In straight and level steady flight, the forces on an aircraft are in balance. Tomaintain flight at a fixed altitude, the lift must balance the weight of aircraft. To maintain a fixedflight speed, the thrust must balance the drag, as indicated in the figure 1.
Figure 1. Force balance on an aircraft in flight.We will measure here Lift and Drag as a function of angle of attack α. Then we will calculate lift
and drag coefficients
44
CL = L12 ρU 2S
, CD = D12 ρU 2S
and plot CL and CD as functions of α. Here ρ = air density = .002327slug ft3 , U = Flight speed, S =wing area. Corrections in a wind tunnel must be made as described below.
WIND TUNNEL CORRECTIONS: An airplane flies in practically a limitless volume of air (exceptwhen near the ground) whereas an airplane model “flies” in a wind tunnel test section in a volume ofair which is more confined. This difference gives rise to a number of corrections which are applied towind tunnel data. Many are quite small and in this experiment we will consider only the correctionarising from the streamlines being forced straight by the flat walls of the wind tunnel whereas they arecurved at corresponding distances from the full scale airplane, the so-called wind tunnel wallcorrection. The correction depends on the shape and sweep of the wing as well as its span relative tothe test section width, but as an approximation we suggest the following additive corrections to angleof attack and to the drag coefficient for any given value of CL:
( )202.
degreesin 5.1
LD
L
CC
C
=∆=∆α
FORCE RESOLUTION: The force balance measures forces along the sting axis (axial force A) andperpendicular to the sting axis (normal force N). These are resolved into Lift (perpendicular to theflight path U) and Drag (parallel to the flight path U) through the angle of attack α, the angle betweenthe flight path and the sting axis, as indicated in figure 2,
Drag = N sinα + AcosαLift = Ncosα − Asin α
where α = angle of attackN = Normal ForceA = Axial Force
Figure 2. Force resolution.
PROCEDURE
1. Get familiar with the software HP VEE, especially the function on the data acquisition.2. Calibrate the angle of attack according to the manual. Zero the angle of attack. You may run the
calibration software and follow the instruction on screen. The software will calibrate Angle ofAttack, Normal and Axial Forces, and Pressure.
3. Calibrate the strain gauge balance according to the instruction listed in the APPARATUS orfollow the instruction of the calibration software.
4. Install the aircraft model according to the instruction.
45
5. Zero the airspeed. Set the normal and axial force components equal to zero. At the six values ofangle of attack, note axial and normal force components. These are due to the model weight andmust be subtracted from the aerodynamic forces measured when the wind is on.
6. Measure normal and axial forces for two windspeeds U=100mi/hr, U=130mi/hr for each of the sixvalues of angle of attack: oooooo 16,12,8,4,0,4−=α . Please note that at each angle of attack, theNormal and Axial forces should be zeroed.
7. Load the measured data into Excel (automatically done in HP-VEE), graph and check your datain Excel.
8. Calculate Lift and Drag (including the wind tunnel corrections) for each of the 12 cases.Calculate CL, CD using the density of air given, the airspeed, and the measured wing surface areaS=12.87in2 for the model.
9. Plot CL and CD vs. α for the two airspeeds. Characterize the two airspeeds by their Reynoldsnumbers Re = ρUc µ where c= wing root chord = 8.15 cm and µ=shear viscosity of air= 3.78 ×10-7 slug/ft ⋅sec . Note that CL, CD, Re are completely dimensionless so unit conversionsmust be made as necessary. Forces A and N are measured in lbs. and airspeed U in mi/hr.
QUESTIONSince the plots are dimensionless, we may apply the results to low speed flight of the real aircraft.With a gross weight of 22,500 lb., calculate the minimum take off and landing speeds of this aircraftbased on the wind tunnel measurements.
46
EXPERIMENT NO. 2.FLUID FRICTION FLOW
OBJECTIVE:To investigate the frictional losses across pipes.
DISCUSSION:In this experiment, you will be measuring frictional losses across pipes of different diameters andsmoothness as a function of flow rate, looking at the effect of valves and bends in the flow stream,and examining the performance of several types of flow meters, including a Venturi meter, an orificemeter, and a Pitot tube. The apparatus allows one to measure the pressure drop across the variousparts of the flow stream using either a mercury or water manometer and to determine the flow rate.Tabulated data for the viscosity and density of water at given temperature allow friction factors to bedetermine and compared to standard charts.A brief description of how to do the measurements is attached. While the actual experiments arestraightforward, you should plan your time carefully so that you can study one aspect of frictionlosses or flow metering fully, rather than taking a collection of unrelated results. Examples ofexperiments which you could perform include the following:
1) Determine the relationship between head losses due to friction (h) and flow rate (Q) in smoothtubes. What relationship do you expect to see in the laminar and turbulent flow regimes? Plotlog(h) versus log(Q) to see if this is observed. Calculate friction factors as a function ofReynolds number from your data and see if it agrees with tabulated data. A roughened pipe isalso available for determining the effect of roughness on head losses.
2) Measure the head losses due to friction for each of the various fittings and vales as a functionof flow rate. The fittings include a sudden contraction and expansion, a 45° elbow, a 90° elbowand bend, a T-junction, a Y-junction and a strainer. For valves, losses can be measured as afunction of how "open" the valves is for a gate valve, a ball valve, and a globe valve.
3) Determine the performance of the differential head devices for measurement of flow rate andwater velocity in a pipe. Examine the use of the orifice meter, the Venturi meter, and the Pitottube. (The theory of theses devices is described in most fluid mechanics texts.)
47
NOTES ON OPERATION OF THE APPARATUS:
The schematic of the experiment is shown in Fig.1.
Figure 1. General Schematic of the Fluid Friction Experiment
Flow rates through the apparatus may be adjusted by operation of outlet control valve (V6).Simultaneous operation of inlet flow control valve (V2) will permit adjustment of the static pressurein the apparatus together with the flow rate. Operation of by-pass flow control (V1) will permitreduction in static pressure when operating at low flow rates. Fine outlet control valve (V5) willpermit accurate control at very low flow rates. Suitable selection and operation of these controlvalves will enable tests to be performed at different, independent combinations of flow rate andsystem static pressure.
Measurement of Flow Rates using the Volumetric TankThe service module incorporates a molded volumetric measuring tank (22) which is stepped toaccommodate low or high flow rates. A stilling baffle is incorporated to reduce turbulence. A remotesight gauge, consisting of a sight tube and scale, is connected to a tapping in the base of the tank andgives an instantaneous indication of water lever. The scale is divided into zones corresponding to thevolume above and below the step in the tank. A dump valve in the base of the volumetric tank is
48
operated by a remote actuator. In operation, the volumetric tank is emptied by lifting the dump valve,allowing the entrained water to return the sump (23). When test conditions have stabilized, the dumpvalve is lowed, entraining the water in the tank.Timing are taken as the water level rises in the tank. Low flow rates are monitored on the lowerportion of the scale corresponding to the small volume beneath the step. Larger flow rates aremonitored on the upper scale corresponding to the main tank. Before operation, the position of thescale relative to the tank should be adjusted.When extremely small volumetric flow rates are to be measured, the measuring cylinder should beused rather than volumetric tank. When using the measuring cylinder, division of the flow to and fromthe cylinder should be synchronized as closely as possible with the starting and stopping of a watch.Do not attempt to use a definite time of a definite volume.
Operation of the Self-Bleeding ManometersBoth the mercury and pressured water manometers installed on the apparatus are fitted with quickconnection test probes and self-bleeding pipework.Each test point on the apparatus is fitted with a self-sealing connection. In operation, the dust cover isunscrewed from the test point selected and the required test probe inserted into the exposedreceptacle. Screwing the collar of the test probe onto the test point opens the integral valvepermitting measurement of the system pressure via the pipe connected to the probe. Each test probeis connected to the limb of a manometer via a vented ball valve which is situated over the volumetrictank. In operation, the connecting valves are set to the 90° position and the test probes screwed ontothe required test points. Pressure in the test pipe, drives fluid along the flexible connecting pipepushing air bubbles to the valve where the mixture of air and water is ejected into the volumetric tankvia the vent in the body. In this condition the valve connection to the manometer remains sealedkeeping the manometer fully primed.When all air bubbles have been expelled at the vent, the valve is turned through 90° to the liveposition connecting the test point directly to the manometer.Prior to removal of the test probe, the valve is returned to 90° position to prevent loss of water fromthe manometer. Using this procedure, the manometers once primed will remain free from air bubblesensured accuracy in readings.The pressured water manometer incorporated a Schrader valve which is connected to the topmanifold. This permits the level at high static pressures.
49
EXPERIMENT No. 3.PRESSURE DISTRIBUTION OVER AN AIRFOIL
OBJECTIVE:To measure the pressure distribution over a Clark Y-14 (CY-14) Airfoil atvarious angles of attack.
INTRODUCTION:An airfoil develops Lift at a positive angle of attack through generally lower pressures above the
wing and higher below with respect to the pressure of the approaching air. The lift on the airfoilincreases as the angle of attack increases until a flow separation occurs. Then the lift decreasessuddenly. This phenomenon is called stall.
The overall pressure distribution can be measured with small tubes embedded in the wing leadingto a suitable pressure transducer. The airfoil model is equipped with 18 pressure openings. Theopenings are located 0, 5, 10, 20, 30, 40, 50, 60 and 70% chord on both upper and lower surfacesand there is an additional opening at 80% chord on the upper surface, see the attached data sheet.
Calculation of the Lift From the Measured Pressure DistributionAssume the pressure p is only a function of s (or x), please refer to the Figure 1 below. The force
on a strip of width b and length ds is
dF = p ⋅b ⋅ds . (1)This force acts normal to the surface of the airfoil.
The force dF makes an angle θ with the normal to the center line of the cross-section (see Fig.2).
Let the vertical component of this force be dV. Then
dV = cos α − θ( )⋅dF . (2)Note that θ is a function of s but it is small over most of the airfoil. Also, dx = cosθ ⋅ds. . Thus, weapproximate
x
y
s
b
dF
α
V•
Figure 1. Force dF on an airfoil with angle of attack α.
50
x
s
α
dFdV
αθ
Figure 2. Decomposition of force dF into a vertical component dV.
dx ≅dsdV ≅cosα ⋅dF = p ⋅b ⋅cosα ⋅ds
. (3)
Thus, the resultant vertical force is approximately
V ≅b ⋅cosα ⋅ p x( )⋅dx0
C
∫ . (4)
Let pl be the pressure distribution on the lower surface and let pu be the pressure distribution on theupper surface. The pressure coefficients are defined by
Cp,l = pl − p∞
q∞ and Cp ,u = pu − p∞
q∞(5)
where p∞ is the freestream pressure and q∞ = 12ρoV∞
2 is the dynamic pressure. The coefficient of lift isdefined by
CL = Vq∞ ⋅b⋅c
. (6)
Using these definitions, the coefficient of lift CL can be expressed
CL = cosα ⋅ Cp ,lxc
− Cp ,u
xc
⋅d
xc
0
1
∫ . (7)
The resultant moment of the pressure distribution about the leading edge of the airfoil at small anglesof attack is given by
M = b ⋅ pl − pu( )⋅xdx0
C
∫ . (8)
The coefficient of moment is defined by
CM = Mq∞ ⋅b ⋅c2 = Cp,l − Cp ,u[ ]⋅ x
c
d
xc
0
1
∫ . (9)
The location of the center of pressure is
xp = CM
CL
. (9)
PROCEDURE:
51
1. Calibrate the pressure reading of the wind tunnel system, following the instruction listed in theExperiment No.1 or in the calibration software.
2. Install the pressure wing vertically in the wind tunnel and connect the pressure tubes, in order, tothe inlet nipples of the tunnel pressure system. Make sure that the static pressure of the testsection (the upstream orifice ring) is attached to the reference connection of the pressuretransducer. Any unconnected input nipples will register the difference between atmospheric andthe test section pressure which can be taken as q∞ , the dynamic pressure of the airstream.Dividing the pressure measured at any point on the airfoil by q∞ provides the pressure coefficientat that point, p − pref( )q
∞= Cp .
3. Operate the wind tunnel at two airspeeds of 100 and 130 mph and make pressure measurementsat angles of attacks of 0°, 3°, 6°, 9°, 12°, 15° and with small increment up to as far as the windtunnel allows. Check to see if you have observed the stall of the airfoil at large angle of attack.
4. Load the measured data into Excel (automatically done in HP-VEE), graph and check your datain Excel.
5. Plot the pressure coefficients along the chord line on both the lower and upper surface of theairfoil.
6. Integrate the pressure coefficients to determine Lift and Drag coefficients. Note that this methoddoes not measure viscous drag forces.
52
Data Sheet for Experiment 3
Date
Test Section Airspeed
PRESSURE DATAStationNumber
% ofChord
α1= α2= α3= α4= α5= α6= α7=
0 01 52 103 204 305 406 50
7 60
8 70
9 80
10 5
11 10
12 20
13 30
14 40
15 50
16 60
17 70
53
Ordinates of Airfoils
54
Section Characteristics for Miscellaneous Airfoils
55
COOLING TOWER EXPERIMENTAlthough the laboratory experiment will be carried out by a team of three students, each
student must familiarize him/herself with the laboratory instructions and submit an assignment prior tothe beginning of the lab session.
This laboratory experiment consists of three parts:1. Preparation before the lab:
(i) Read carefully the laboratory notes.(ii) Answer the preliminary questions in writing and submit the answers to the TA before thestart of the lab session. The answers should be worked out and submitted by each student.Failure to submit these homeworks may result in banning the student from the lab session.
2. Laboratory Experiment:(i) Make sure that you understand what you are doing. If in doubt, ask.(ii) Make sure to collect all the needed data.
3. After the Laboratory:Prepare a laboratory report. The report writing is a team effort. Only one report will be
submitted by each team. The report should be written clearly since, in addition to technical content, itwill be graded for the quality of the writing.
1. INTRODUCTIONCooling towers are widely used to dissipate heat to the environment. Typically, a condenser
of a power plant or of an air-conditioning/chilling system is cooled by water. The warmwater is transmitted to the cooling tower,where it is sprayed from the top of the tower.The falling water passes through a series ofbaffles intended to keep the water as a thin filmto increase its surface area and to promoteevaporation. Atmospheric air is supplied at thetower's bottom. The air rises counter to thedirection of the falling water. The air flow canbe driven by either buoyancy (natural draft) orby means of a fan (forced draft). As the twostreams come into direct contact, a smallfraction of the liquid water evaporates into themoist air which exits the tower at a greaterhumidity ratio than that of the incoming air.The energy needed for the latent heat ofevaporation is mostly provided by the liquidwater, with the net result that the liquid exitingthe tower is at a lower temperature than thewater entering the tower. Since some of thewater evaporates in the process, a smallamount of makeup water needs to be added tothe liquid stream.
Fig. 1: A bank of cooling towers
56
OBJECTIVES:The objectives of this experiment are:1. To familiarize you with a piece of frequently used thermodynamic equipment.2. To use the first law of thermodynamics in the analysis of thermodynamic hardware.
Note:This experiment involves a few new concepts that are not taught in MEAM 203. These new conceptsare essential for carrying out the experiment and they are described, as much as needed, in section 2of this laboratory manual.
2. BACKGROUND MATERIAL
2.1 SPECIFIC HUMIDITYThe amount of water vapor carried by the air is specified conveniently by the specific humidity
(w). The specific humidity is the ratio between the mass of the water-vapor (mv) and that of the dry-air (ma):
w = mv
ma
= mv
•
ma
• . (1)
The specific humidity is a dimensionless quantity. Although in most thermodynamic calculations onedefines the fractional mass as the ratio between the mass of a particular species and the total mass ofthe mixture, in air - vapor systems, it is more convenient to use the dry-air mass rather then the totalmass (ma+mv) as the normalizing quantity since the mass of the dry-air typically does not change
during the process.
2.2 STATE EQUATIONS FOR THE AIR-VAPOR MIXTUREThe dry- air (air without any water vapor content) is a mixture of a number of components.
Since in our application the air's composition does not change, we can think of the air as if it were apure substance with representative molecules having an average molecular mass of Ma=28.97
kg/kmol. This molecular mass was calculated as the weighted average of the molecular masses of thevarious components of air such as O2, N2 and Ar.
A simple way of thinking of an air - vapor mixture is to consider the air and the vapor as ifthey were existing all by themselves in a volume, V, at a uniform temperature, T, each at its own(partial) pressure. The total pressure of the mixture (or the atmospheric pressure in our case), p0, isthe sum of the partial pressures of the air, pa, and the vapor, pv.
p0 = pa + pv . (2)
57
This model is known as the Dalton Model and it is valid when the gasses are ideal and there are nointermolecular forces. When the Dalton Model is valid, we say that the mixture is ideal. Not allmixtures are ideal. The ideal mixture approximation provides a reasonably accurate description of thebehavior of air-vapor mixtures.
p a pv p
Dry Air at volume V & partial pressure, p
a
Vapor at volume V & partial pressure, p
v
Air -Vapor mixture at volume, V, and total pressure, p = p + p
a vFig. 2: The Dalton Model: the dry-air and the vapor can be thought of as each of them existed all by itself at its own
partial pressure and occupied the entire volume of the mixture.
To the first approximation, we assume that both the dry-air and the water-vapor behave likeideal gases. When we studied the properties of water, we emphasized over and over again thatsuperheated vapor (or steam) cannot be assumed to behave like an ideal gas. In this particularexample, however, such an assumption can be made because the partial pressure of the vapor, pv, is
very small.We write the law of ideal gases separately for the dry-air and for the vapor:
pa = na RTV
and pv = nvRTV
, (3)
where na and nv are, respectively, the number of moles of dry-air and vapor; R=8.314 kJ/kmol-K is
the universal constant of ideal gases; and T is the mixture's temperature expressed in degrees Kelvin(K). Witness that the mixture as a whole also obeys the law of ideal gases. By adding the stateequations for the dry-air and the water-vapor, we get the state equation for the mixture:
p = pa + pv = na RTV
+ nvRTV
= na + nv( )RTV
. (4)
The ideal gas law also provides us with the means of determining the masses of the dry-air andthe water-vapor in a given volume, V, and temperature, T. For example, the mass of the air,
ma = na Ma . (5)
Consequently,
ma = Ma
paVRT
and mv = Mv
pvVRT
. (6)
Furthermore, we can express the specific humidity, w, in terms of the partial pressures,
w = mv
ma
= Mv
Ma
pv
pa
= 0.622pv
pa
= 0.622pv
p − pv
. (7)
58
The air cannot carry infinite amounts of water-vapor. Once the partial pressure of the water-vapor, pv, exceeds the water saturation pressure, psat, water will start to condense. The partialpressure of the vapor-water cannot be larger than the saturation pressure, pv<psat. The saturationpressure is a function of the temperature. Since psat increases as the temperature increases, the
capacity of the air to carry water-vapor increases as the temperature increases.
2.3 RELATIVE HUMIDITYThe quantity that determines our level of comfort and is reported daily by the weathermen is
the relative humidity (φ) which represents the ratio between the amount of vapor the air actuallycarries, mv, and the maximum amount of vapor the air could possibly carry, mv,max, at the same
temperature, T. According to equation (6),
mv, max = Mv
psatVRT
and mv = Mv
pvVRT
. (7)
Thus, the relative humidity,
φ= mv
mv, max
= pv
psat
. (8)
When the air carries water-vapor to its full capacity, we say that the air is saturated, and φ=1.
2.4 THE INTERNAL ENERGY AND ENTHALPY OF AIR-VAPOR MIXTURESThe internal energy and enthalpy of air vapor mixtures are obtained simply by summing up the
corresponding properties of the individual constituents. Since both the dry-air and the water-vaporare assumed to behave like ideal gases, the internal energy and the enthalpy are functions of thetemperature alone,
Hmix = Ha + Hv = maha + mvhv (kJ ). (9)
It is convenient to define the specific enthalpy of the mixture,hmix = ha + whv (kJ / kg dry − air). (10)
Witness that the normalization is done with respect to the mass of the dry-air rather than the totalmass of the mixture.
In processes when the temperature changes are not large, the change in the dry-air's enthalpycan be calculated using the air's specific heat at a constant pressure, i.e.,
ha ,2 − ha ,1 = Cpa T2 − T1( ), (11)where Cpa=1.0035kJ/kg-K. The enthalpy of the water-vapor can be read from the steam tables usingthe saturation data at the given temperature, i.e., hv(T)=hv,sat(T) and hl(T)=hl,sat(T).
2.5 THE DETERMINATION OF THE HUMIDITY AND DRY & WET BULBTEMPERATURES
59
The air's humidity is measured by an instrument known as a hygrometer. One common way ofmeasuring the specific humidity is through the use of an adiabatic saturation process. Consider, forexample, the apparatus depicted in the Fig. 3.
Air with an unknown specific humidity, w
Saturated air
liquid water
1 2
3
Fig. 3: An adiabatic saturation process
Air containing an unknown amount of water-vapor is brought into contact with a body ofliquid water until the air becomes fully saturated. The apparatus is assumed to be well insulated sothat there is no thermal interaction with the environment and the process is adiabatic. Thetemperatures at locations 1, 2, and 3 are monitored. The temperature at the inlet, T1, is known as thedry-bulb temperature. The temperature of the fully saturated air at the exit, T2, is the wet-bulb
temperature.Once initial transients die out (i.e., the temperatures T1 and T2 no longer change as functions
of time), the process is a steady-state-steady-flow process. Mass conservation statements can bewritten separately for the air and the water. The mass flow rate of the air is the same at the inlet andexit,
m•
a ,1 = m•
a , 2 = m•
a . (12)When writing mass conservation for the water, one needs to account for the fact that the
amount of water-vapor may vary during the process. For example, in the adiabatic saturationprocess, liquid water evaporates. The mass conservation statement for the water is:
m•
v,2 = m•
v ,1 + m•
l ,3 , (13)
where m•
l ,3 is the amount of make-up water one needs to add to maintain a fixed level of water in theliquid reservoir. It is convenient to normalize this last equation by the mass flow rate of dry air,
w2 = w1 + m•
l ,3
m•
a
. (14)
Since the process is adiabatic and there is no shaft work, the first law of thermodynamicsstates that
m•
a ha ,1 + w1hv,1( )+ m•
l ,3 hl,3 = m•
a ha,2 + w2hv ,2( ). (15)
Using equations (14) and (15), we obtain a formula for w1,
60
w1 =ha ,2 − ha,1( )+ w2 hv,2 − hl ,3( )
hv ,1 − hl,3( ) . (16)
Thus, by measuring the temperatures at locations 1, 2, and 3, one can compute the specific humidityw1. Usually, the temperature of the make-up water (point 3) is not measured and it is assumed toequal the temperature T2. This introduces only a negligibly small error.
In practice, the wet-bulb temperature is measured by a thermometer whose bulb is enclosed bya wick moistened with water. The dry-bulb temperature is measured by simply placing a thermometeris the air-vapor mixture.
Example: The dry and wet bulb temperatures of air at 101.325kPa are 25C and 15C, respectively.Determine the specific and relative humidities of the air.
In order to carry out the calculations, I'll use my Maple-based thermodynamic tables. You canalso obtain the data from the tables at the end of your thermodynamics book. First, I read thethermodynamic tables into my worksheet.> read `Macintosh HD:Thermo Tool Box:ASME_SteamTables.m`;
Second, I insert into Maple the given information:> p_ambient:=0.101325*MPa: T1:=(25+273.15)*K: T2:=(15+273.15)*K:
Cp_air:=1.0035*kJ/kg/K:Third, I will evaluate all the needed thermodynamic properties. For the purpose of the Maple
calculations, I do not really need to view explicitly the values of the various properties; but I'll do sofor the benefit of those of you who work with hard copy thermodynamic tables.> psat1:=SaturationPressureWater(T1); psat1 := .003165990008 MPa> psat2:=SaturationPressureWater(T2); psat2 := .001703925815 MPa
> hv1:=EnthalpySaturatedVaporWater(T1); kJ hv1 := 2547.284813 ---- kg
> hv2:=EnthalpySaturatedVaporWater(T2);
kJ hv2 := 2529.054737 ----
61
kg
> hl3:=EnthalpySaturatedLiquidWater(T2);
kJ hl3 := 62.94098261 ---- kg
Given the fact that the exiting air (point 2 in Fig. 3) is saturated (φ2=1), I can readily calculate
the specific humidity of the exiting air (equation 7).w2:=.622*psat2/(p_ambient-psat2); w2 := .01063873147
Finally, I have all the information needed for the evaluation of the specific humidity of the air,w1 (equation 16):> w1:=(Cp_air*(T2-T1)+w2*(hv2-hl3))/(hv1-hl3); w1 := .006521368663and the relative humidity (φ1).
> ph1:=pv1/psat1; ph1 := .3320670449
Since in the course of the experiment you will need to evaluate the specific humidity a largenumber of times, it is convenient to define a Maple function for the evaluation of specific humidity.
> ws:=proc(T_DB,T_WB,p_ambient)> local Cp_a, psat, w_out, hv2, hv1, hl;> Cp_a:= 1.0035*kJ/kg/K;> psat:=SaturationPressureWater(T_WB);> w_out:=0.622*psat/(p_ambient-psat);> hv2:= EnthalpySturatedVaporWater(T_WB);> hv1:= EnthalpySturatedVaporWater(T_DB);> hl:= EnthalpySturatedLiquidWater(T_WB);> (Cp_a*(T_WB-T_DB)+w_out*(hv2-hl))/(hv1-hl);> end:
Example:
62
> ws(298*K, 288*K,0.1*MPa);.006559
Similarly, we can write a Maple procedure that would allow us to evaluate the relativehumidity.
> relative_humidity:=proc(specific_humidity,temp, p0)> local pv;> pv:=p0*specific_humidity/(specific_humidity+0.622);> pv/SaturationPressureWater(temp);> end:
3. THE COOLING TOWER EXPERIMENT
3.1 THE EXPERIMENTAL SET-UPThe experimental apparatus is depicted schematically in Fig. 4. The cooling tower
accommodates counterflow of air and water. The air flows upwards while the water dripsdownwards and spreads on the baffle plates. In the process, some of the water evaporates. Primarilydue to the latent heat of evaporation, the water cools down.
Prior to its entering the cooling tower, the water is heated (to simulate the "heating load") inthe "load tank". Subsequently, the water is pumped through a control valve that allows for theadjustment of the flow rate and through a flow meter to the top of the column. After its temperatureis measured, the water is distributed over the top packing deck. The water spreads over the bafflesand flows downwards as a thin film whose surface is exposed to the air.
The cooled water leaves the cooling tower and collects into the "load tank." Before re-entering into the tank, the temperature of the exiting water is measured. Due to evaporation, the levelof the water in the "load tank" tends to fall. This causes the float-operated needle valve to open andtransfer water from the "make-up tank" into the load tank. Under steady-state conditions, the rate atwhich the water leaves the "make-up tank" is approximately equal to the rate of evaporation of thewater in the tower.
The pump's power consumption is approximately W•
= − 100W , where W•
represents thepump work and one can assume that all this power is consumed by the water. The water's flow rate ismeasured by a rotometer.
63
64
Atmospheric air enters the fan at a rate that is controlled by the intake damper setting. The fandischarges into a distribution chamber and the air passes dry and wet bulb thermometers before itenters the column. The air flows upwards in the column. On leaving the top of the column, the airpasses through a droplet arrester which traps most of the entrained droplets and returns them to thecolumn. The air is then discharged to the atmosphere via an air flow-rate measuring orifice and wetand dry bulb thermometers.
The air mass flow rate can be obtained from the formula:
ma
•= 0.0137
∆pvb
, (17)
where ∆p is the pressure difference (expressed in mm of H2O) between the atmosphere and the airstream; and vb (m3/kg) is the specific volume of the water-air mixture leaving the top of the column
(cooling tower).
3.2 ANALYSISWe assume that the cooling tower is adiabatic (i.e., it does not interact thermally with its
environment). When analyzing the cooling tower, one can consider two different control volumes. Inthe first case, the control volume includes the water re-circulation conduits, the load tank, the pump,and the heater (Fig. 5a) . In the second case, the control volume excludes the water recirculation unit(Fig. 5b). The boundaries of the control volumes are denoted by dashed lines in Figs. 5a and 5b.Points (A) and (B) denote, respectively, the inlet and exit of the air-vapor mixture. Points (C) and(D) denote, respectively, the inlet and exit of the water. Point (E) denotes the inlet of the make-upwater.
65
ma
•
ma
•
Make-up Water Tank
pump
A
BC
D
E
water
ma
•
ma
•
Make-up Water Tank
pump
A
BC
D
E
Fig 5.a: One possibility of a control volume thatincludes the load tank, the heater, and the pump.
Fig 5.b: Another possibility of a control volume thatexcludes the load tank, the heater, and the pump.
We analyze first the control volume shown in Fig. 5a. Under steady-state conditions, massconservation of air and water requires that:
ma
•
A= ma
•
B= ma
•(18)
and
mv
•
B− mv
•
A= ml
•
E, (19)
where subscripts a, v, and l denote, respectively, air, vapor, and liquid.The water mass-conservation can be rewritten in terms of the specific humidity as:
w B − wA =ml, E
•
ma
• . (20)
Next, we use the first law of thermodynamics to write the energy balance.
( ) ( )( ) Ell
mixturevaporairthebygainedEnergy
AvaBvaaworkpumploadHeat
hmwhhwhhmWQ ,
•
−
•••−+−+=− 44444 344444 21 , (21)
66
For the control volume shown in Fig. 5b, the water mass conservation statement is:
mv
•
B− mv
•
A= ml
•
C− ml
•
D, (22)
and the energy conservation (the first law) is:
( ) ( )( )4444 34444 21
44444 344444 21
waterthebylEnergy
Dll
Cll
mixturevaporairthebygainedEnergy
AvaBvaa hmhmwhhwhhm
ost
0
−
−+−+=••
−
•
, (23)
In the experiment, we will measure the temperatures at locations A, B, C, D, and E. Knowledge ofthese temperatures will allow us to obtain the enthalpies needed to compute the energy balances.
4. PREPARATORY WORKThe requirements of this section must be completed before the laboratory session and handed to theTA at the start of the lab. The preparatory work is designed to assure that you are well familiar withthe processes occurring in the cooling tower. This will minimize the possibility of you collectinginsufficient data during the experiment.
67
4.1 Read carefully the laboratory handout and make surethat you understand the material well.
DATAThe numerical subscripts correspondto the numerical designations in theexperimental set-up.
4.2 Visit the experimental set-up, and identify the variouscomponents of the cooling tower.
Air Outlet, Dry BulbtemperatureTB,DB=T0 (C)
21.2
4.3 Suppose that the data given in the table on the right wascollected in an experiment. Determine:
Air Outlet, WetBulb temperatureTB,WB=T1 (C)
19.9
4.3.1 the specific and relative humidity at the air inlet(point A);
Air Inlet, Dry BulbtemperatureTB,DB=T2 (C)
22.7
4.3.2 the specific and relative humidity at the air exit(point B);
Air Inlet, Wet BulbtemperatureTB,WB=T3 (C)
13.5
4.3.3 the mass flow rate of the dry-air; Water OutlettemperatureTD=T4 (C)
19.9
4.3.4 the flow rate of the make-up water; and Water Make-uptemperatureTE=T5 (C)
20.2
4.3.5 the heating load of the system. Water InlettemperatureTC=T7 (C)
27.9
Air OrificeDifferential Pressurein mm H20
15
5. EXPERIMENTAL PROCEDURENecessary equipment: Graduated Cylinder, Stop Watch, Syringe, Distilled Water
In this section, some of the experimental procedures are described. The description is notexhaustive and you will need to exercise common sense.5.1 Make sure that the upper side of the water manometer is connected to the outlet located beneaththe orifice and the other side is exposed to the atmosphere.5.2 Turn on the fan and set the damper so that the air pressure drop will be around 15 mm H20.
5.3 Make sure that the wicks of the wet-bulb thermometers are saturated with distilled water. Thesewicks tend to dry-up during the experiment. Make sure to re-check them from time to time and refillthe wells with distilled water as needed.5.4 Measure the dry and wet bulb temperatures at the air inlet and exit.
68
5.5 Turn on the pump and, using the Flow Control Valve, adjust the water flow rate to 25-30 g/s.
Warning: Do not turn on the heaters in the absence of water circulation!5.6 The first set of experiments will be conducted in the absence of any heat load.5.7 Once you have turned on the water pump, monitor the wet-bulb temperature at the cooling towerexit as a function of time, i.e., measure this wet-bulb temperature every 30s and record the time andtemperature readings.5.8 Once the temperature does not change as a function of time and steady state conditions areestablished (this should take about 10-15 minutes) , fill the make-up water container up to the markand start the stop-watch.5.9 Periodically, re-examine the water flow rate and, if needed, re-adjust it so as to keep the waterflow rate as constant as possible throughout each experiment. You may change the water flow ratewhen you change the heat-load.5.10 Read the seven temperatures (set the temperature switch to 0, 1, 2, 3, 4, 5, and 7) anddocument the readings in the enclosed worksheets together with all the other relevant data. Repeatthese measurements three times. Note: switch setting 6 is inactive.5.11 Wait a sufficient amount of time until at least 100 ml of water have drained from the make-uptank. This may take as long as 10 minutes (this time interval decreases as the heating load increases).5.12 Use the graduated cylinder to refill the make-up tank up to the mark. Take note of the waterlevel in the graduated cylinder before and after the re-fill operation so that you can obtain the amountof make-up water added. As soon as you have added water up to the mark, stop the stopwatch.Monitor the amount of water added and the amount of time that have elapsed.5.13 Set the heating load to 0.5kW.5.14 Repeat steps 5.7-5.12.5.15 Set the heating load to 1.0kW.5.16 Repeat steps 5.7-5.12.5.17 Set the heating load to 1.5kW.5.18 Repeat steps 5.7-5.12.5.19 At the conclusion of the experiment, make sure to first turn off the heating load. Then turn offthe liquid pump, the air fan, and the power to the entire experiment.
6. REPORTThe requirements listed here represent the bare minimum that should be included in the report.
You are encouraged to further elaborate upon and find new ways to analyze the data. You are alsowelcome to conduct additional experiments.
69
6.1 In the absence of water flow, calculate the specific and relative humidities of the air entering andexiting the cooling tower. Would you expect the relative and specific humidities at the inlet and exitto be the same? Explain.6.2 Depict the exit-air wet-bulb temperature as a function of time for every change in the heatingload. Determine the time-constant of the system.6.3 For all the heating loads, calculate the specific humidity and the relative humidity at the air inletand exit. Using the specific humidities obtained from the dry and wet bulb thermometers, calculatethe amount of make-up water and compare it to the amount of make-up water which you measureddirectly. Determine the percentage difference between the calculated and measured make-up rates.Comment about the possible reasons for any discrepancies. Plot the mass flow rate of the make-up
water normalized with the mass flow rate of dry air as a function of Q•− W
•
ma
• .
6.4 For all the heating loads, calculate the amount of energy gained by the air-vapor mixture andcompare it to the amount of energy lost by the water and to the amount of energy supplied by theheater. Comment on any possible sources of measurement errors.
REFERENCES1. Bench Top Cooling Tower H890, P.A. Hilton Ltd, Experimental Operating & MaintenanceManual. A copy of the manual is available in the MEAM laboratory. You may borrow the manualbefore the experiment.2. Moran, M., J., and Shapiro, H., N., Fundamentals of Engineering Thermodynamics, Chapter 12.3. Sonntag, R. E., and Van Wylen, G., J., 1991, Introduction to Thermodynamics, Classical andStatistical, John Wiley, Chapter 11.3. COOLING TOWERS ON THE NET, See, for example,http://www.texasonline.net/us/history.htm.
70
COOLING TOWER OBSERVATION SHEET #1
Date: __________ Atmospheric Pressure: _____________
Heating Load: ____________
Test # 1 2 3 4Air Outlet, Dry BulbTemperature, TB,DB=T0 (C)
Air Outlet, Wet BulbTemperature, TB,WB=T1 (C)
Air Inlet, Dry BulbTemperature, TB,DB=T2 (C)
Air Inlet, Wet BulbTemperature, TB,WB=T3 (C)
Water Outlet TemperatureTD=T4 (C)
Water Make-up TemperatureTE=T5 (C)
Water Inlet TemperatureTC=T7 (C)
Air Orifice DifferentialPressure in mm H20Water Flow Rate, ml,C (g/s)
Make-up Quantity (ml)
Time Interval to MeasureMake-up water (s)
Comments: ______________________________________________________________
________________________________________________________________________
________________________________________________________________________
71
COOLING TOWER OBSERVATION SHEET #2
Date: __________ Atmospheric Pressure: _____________
Heating Load: ____________
Test # 1 2 3 4Air Outlet, Dry BulbTemperature, TB,DB=T0 (C)
Air Outlet, Wet BulbTemperature, TB,WB=T1 (C)
Air Inlet, Dry BulbTemperature, TB,DB=T2 (C)
Air Inlet, Wet BulbTemperature, TB,WB=T3 (C)
Water Outlet TemperatureTD=T4 (C)
Water Make-up TemperatureTE=T5 (C)
Water Inlet TemperatureTC=T7 (C)
Air Orifice DifferentialPressure in mm H20Water Flow Rate, ml,C (g/s)
Make-up Quantity (ml)
Time Interval to MeasureMake-up water (s)
Comments: ______________________________________________________________
________________________________________________________________________
________________________________________________________________________
72
COOLING TOWER OBSERVATION SHEET #3
Date: __________ Atmospheric Pressure: _____________
Heating Load: ____________
Test # 1 2 3 4Air Outlet, Dry BulbTemperature, TB,DB=T0 (C)
Air Outlet, Wet BulbTemperature, TB,WB=T1 (C)
Air Inlet, Dry BulbTemperature, TB,DB=T2 (C)
Air Inlet, Wet BulbTemperature, TB,WB=T3 (C)
Water Outlet TemperatureTD=T4 (C)
Water Make-up TemperatureTE=T5 (C)
Water Inlet TemperatureTC=T7 (C)
Air Orifice DifferentialPressure in mm H20Water Flow Rate, ml,C (g/s)
Make-up Quantity (ml)
Time Interval to MeasureMake-up water (s)
Comments: ______________________________________________________________
________________________________________________________________________
________________________________________________________________________
73
COOLING TOWER OBSERVATION SHEET #4
Date: __________ Atmospheric Pressure: _____________
Heating Load: ____________
Test # 1 2 3 4Air Outlet, Dry BulbTemperature, TB,DB=T0 (C)
Air Outlet, Wet BulbTemperature, TB,WB=T1 (C)
Air Inlet, Dry BulbTemperature, TB,DB=T2 (C)
Air Inlet, Wet BulbTemperature, TB,WB=T3 (C)
Water Outlet TemperatureTD=T4 (C)
Water Make-up TemperatureTE=T5 (C)
Water Inlet TemperatureTC=T7 (C)
Air Orifice DifferentialPressure in mm H20Water Flow Rate, ml,C (g/s)
Make-up Quantity (ml)
Time Interval to MeasureMake-up water (s)
Comments: ______________________________________________________________
________________________________________________________________________
________________________________________________________________________
74
COOLING TOWER EXPERIMENT - EVALUATION SHEETAt the conclusion of the experiment, please fill this evaluation sheet and submit it to Professor Haim H. Bau's mail-boxin the MEAM department. Verbal comments will also be appreciated. The objective of this evaluation sheet is toenable us to improve this laboratory experiment for the benefit of future generations of mechanical engineeringstudents. If needed, please add comments on additional sheets. Thank you.
About Yourself (Please circle the appropriate)
1. Sophomore / Junior / Senior
2. Did you take? MEAM203 yes / no MEAM338 yes / no
3. Name (optional): _____________________ E-mail address (optional): __________________
Laboratory Handout:
4. Was the laboratory handout clear? _______________________________________________
5. Did you detect any errors? where? _______________________________________________
6. Was the background material sufficient? ___________________________________________
Preparatory Assignment
7. Did the preparatory assignment help clarify relevant concepts? __________________________
8. Did the assignment help you utilize the laboratory time more effectively? __________________
9. Should any additional material be included in the preparatory assignment?
______________________________________________________________________________
Laboratory Session
10. Did the laboratory experiment run without glitches? _________________________________
11. Could these glitches be avoided by appropriate instructions that were missing from the handout?
_____________________________________________________________________
12. Should changes/additions be made in the laboratory instructions? ______________________
______________________________________________________________________________
Report
13. Did you use Maple? ________________________
General
14. Was this laboratory experiment useful? __________________________________________
