AME341b 2010 Lecture&Lab Notes Jerry

AME341BMECHOPTRONICS IILecture & Lab Notes

Cheng-Yuan Jerry Chen & Geo¤ Spedding

Spring 2011

Contents

1 Course Information 11.1 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Textbooks . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Laboratory materials . . . . . . . . . . . . . . . . . . . 1

1.2 Grading and Conduct . . . . . . . . . . . . . . . . . . . . . . . 21.2.1 Predicted/Approximate Grade Distribution . . . . . . 21.2.2 Computing the Grade . . . . . . . . . . . . . . . . . . 31.2.3 Timing, Scheduling and Make-up . . . . . . . . . . . . 31.2.4 Academic Integrity . . . . . . . . . . . . . . . . . . . . 31.2.5 How to Get an A . . . . . . . . . . . . . . . . . . . . . 5

1.3 General Lab Rules . . . . . . . . . . . . . . . . . . . . . . . . 51.3.1 Time Management . . . . . . . . . . . . . . . . . . . . 51.3.2 Space Management . . . . . . . . . . . . . . . . . . . . 61.3.3 Computer Usage & Printing Rules . . . . . . . . . . . 61.3.4 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . 71.3.5 Academic Integrity . . . . . . . . . . . . . . . . . . . . 7

2 Vibration Analysis 92.1 Second Order Systems . . . . . . . . . . . . . . . . . . . . . . 9

2.1.1 Governing Equations . . . . . . . . . . . . . . . . . . . 92.1.2 Free Vibration (Unforced Dynamic Response) . . . . . 102.1.3 Solutions of the Underdamped System . . . . . . . . . 11

2.2 Strain Gauges . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2.1 Strain Sensitivity and Gauge Factor . . . . . . . . . . . 142.2.2 Practical Gauge Design and Materials . . . . . . . . . 162.2.3 The Wheatstone Bridge . . . . . . . . . . . . . . . . . 18

2.3 Flexure Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 212.4 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

v

vi CONTENTS

2.4.1 Measurement of Microstrain in a Second Order Me-chanical System . . . . . . . . . . . . . . . . . . . . . . 23

2.4.2 An Engineering Spreadsheet Report . . . . . . . . . . . 272.4.3 Important Reminder . . . . . . . . . . . . . . . . . . . 28

3 Heat Transfer and Thermocouples 313.1 De�nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2 Useful Properties . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.2.1 Law of Intermediate Metals (LIM) . . . . . . . . . . . 323.2.2 Practical Consequences of LIM . . . . . . . . . . . . . 323.2.3 Law of Intermediate Temperatures (LIT) . . . . . . . . 33

3.3 Static Calibration . . . . . . . . . . . . . . . . . . . . . . . . . 343.4 Dynamic Response . . . . . . . . . . . . . . . . . . . . . . . . 34

3.4.1 Conservation of Energy . . . . . . . . . . . . . . . . . . 353.4.2 Solution of First Order System for Step-Function Forcing 373.4.3 Meanwhile, Back in the Lab... . . . . . . . . . . . . . . 39

3.5 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.5.1 Thermocouples and Heat Transfer . . . . . . . . . . . . 413.5.2 Sometime Later, Furiously Report-Writing..... . . . . . 503.5.3 Important Reminder . . . . . . . . . . . . . . . . . . . 51

4 Fluid Flow and Turbulence 554.1 Fluid Turbulence and Jet Flows . . . . . . . . . . . . . . . . . 56

4.1.1 Jets: Applications and de�nitions . . . . . . . . . . . . 574.1.2 Turbulence . . . . . . . . . . . . . . . . . . . . . . . . 594.1.3 Time-Averaged Quantities and Universal Statistical States 594.1.4 Dimensional Analysis and the Reynolds Number . . . . 614.1.5 Practical Examples . . . . . . . . . . . . . . . . . . . . 634.1.6 Dynamic Pressure - How to Measure u from p? . . . . 65

4.2 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.2.1 Pressure/Velocity Measurements in a Turbulent Jet . . 664.2.2 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . 704.2.3 Testing Velocity Pro�les in Turbulent Jets for Self-

Similarity �A Practical Guide . . . . . . . . . . . . . . 724.2.4 Assignment: 3 Plots + Minitalk . . . . . . . . . . . . . 774.2.5 Important Reminder . . . . . . . . . . . . . . . . . . . 79

CONTENTS vii

5 LabVIEW Programming 815.1 Lab 1: Basic of LabVIEW Programming . . . . . . . . . . . . 81

5.1.1 Acquire a Voltage from Channel 0 . . . . . . . . . . . . 815.1.2 Construct a Digital Thermometer using LabVIEW . . 845.1.3 Homework Assignment: . . . . . . . . . . . . . . . . . 865.1.4 Important Reminder . . . . . . . . . . . . . . . . . . . 91

5.2 Lab 2: Further Adventures in LabVIEW . . . . . . . . . . . . 915.2.1 Stepper Motor Control and Simple Data Acquisition . 915.2.2 Homework Assignment: . . . . . . . . . . . . . . . . . . 1005.2.3 Important Reminder . . . . . . . . . . . . . . . . . . . 100

5.3 Lab 3: Automated Sampling of Turbulent Jet Flows . . . . . . 1015.3.1 Part 1: Check Calibration Constants . . . . . . . . . . 1025.3.2 Part 2: Make a Standalone Flow Meter . . . . . . . . . 1035.3.3 Part 3: Modifying and Testing your Customized Jets.vi 1045.3.4 Part 4: Obtain and Analyze U(y) Pro�les Across a

Turbulent Jet . . . . . . . . . . . . . . . . . . . . . . . 1065.3.5 Part 5: Prepare for the Talk . . . . . . . . . . . . . . . 106

5.4 Lab 4: Turbulent Jets II . . . . . . . . . . . . . . . . . . . . . 1075.4.1 Additional Notes for Measuring Turbulent Jets . . . . . 1075.4.2 Special Lab Rules for Turbulent Jets II Lab . . . . . . 108

Chapter 1

Course Information

1.1 Materials

1.1.1 Textbooks

There is currently no required textbook for this course, because of the uniquemix of materials. There are two recommended textbooks that have beenordered from the bookstore:

� Alciatore DG & Histand MB 2007 �Introduction to Mechatronics andMeasurement Systems, 3rd Edition�. McGraw-Hill. ISBN 0072963050

� Figliola RS & Beasley DE 2011 �Theory and Design for MechanicalMeasurements, Fifth Edition�. Wiley. ISBN 0471445932

Both of the books are strongly recommended. Reference will be madeto both of these texts throughout the semester, so that background materialcan be found in either one.

1.1.2 Laboratory materials

1. Notebook. Each student must have a lab notebook where experi-mental details are recorded. Remember, together with any computer-generated �les, the information collected here forms the basis for yourreport. Frequently, this is where you will note the signi�cance and con-text of the data �les. Rough translation: no good notebook = no good

1

2 CHAPTER 1 COURSE INFORMATION

report = no good grade. The notebooks will be collected and gradedat the end of the semester.

2. USB �ash drive. (provided last semester) For data �les, electroniccopies of lab and lecture notes, and submission of some assignments.

3. Lecture notes/Lab Notes. (provided) Always bring lecture notes/LabNotes to the lab. This way the connection between the more theoreticalmaterial and the practical exercises can be clari�ed.

4. Calculator. It is often convenient and/or necessary to calculate andcheck quantities on the �y. Do your sanity checks in the lab, so thatgross errors can be detected before it is too late.

5. Your brain. The laboratory exercises will be pointless and boringunless you bring some active component into the room.

1.2 Grading and Conduct

1.2.1 Predicted/Approximate Grade Distribution

The grading will be based primarily on evaluation of your written lab reportsand spreadsheets, together with two quizzes. There will also be a number oftalks during which you will be presenting some of the results obtained duringthe labs. These make up the balance of the grade. The following table isgiven as an approximate guide only.

% of grade

Report and Spreadsheets 58

Talks 16

Quiz 17

Lab Performance 9

Total 100

Clearly, the most important component is the written reports. We at-tempt to arrange the semester so that you always get feedback from oneattempt before another of similar style is graded.

1.2 GRADING AND CONDUCT 3

The Lab Performance contribution is assigned by sta¤ and TA�s through-out the semester, and is a measure of each student�s e¤ort and contributionduring lab hours.A detailed breakdown is given in class schedule. Assignments and their

relative weights are allowed to vary during the semester, as we adapt toconditions in the lab.

1.2.2 Computing the Grade

The grade is calculated from a weighted sum of normalized grades for allassignments usingBlackboard system�sGrade Center. Roughly speaking,students with grades higher than half a standard deviation above the meanare likely to receive �A� grades; Students with grades lower than half astandard deviation below the mean are likely to receive �C�grade or lower;Students with grades in between half a standard deviation above and belowthe mean will be falling into �B�grade region. This grade calculation is forapproximate guides only, the actual calculation formula may vary slightlyas we adapt to conditions in the lab.

1.2.3 Timing, Scheduling and Make-up

Each assignment is due within one or two weeks of the lab, as speci�ed at labtime, or in the Web announcements. Physical documents must be handed inat the lab in BHE 301. They must be handed in on time. In dealing withlarge numbers of complex assignments such as this, it is necessary to enforcestrict rules concerning the production and grading of homework assignmentsand reports. Consequently, a late report will be docked 10% per day, orfraction of a day, overdue. One (1) minute late counts as 1 day late and thereare no exceptions. Late work will not be accepted more than 1 week afterthe due date. For similar reasons, there are no make-up labs. All labs willcount towards the total grade (i.e. none are dropped). Absences for medicalreasons must be justi�ed with some reasonable evidence. It is not possibleto pass the course if you are missing two or more assignments.

1.2.4 Academic Integrity

Each student must write their own report, do their own homework, and �llin their own worksheet. This includes the generation of graphs, any data


analysis and preparation. You are encouraged to work together during thelab, and also to discuss your work with your partner or other colleagues, butyou must do your own work on the write-up.Some aspects bear repeating in detail. Part of the challenge of a class like

AME341 is in the design and construction of reports and analysis strategiesof lab data. Your grade is based on how well you are able to do this. Youmust decide for yourself how to present the material and in what order andin what format. You alone decide what goes in and what stays out. Youalone decide whether to show a result as a table, a graph or an equation.Collaboration on these points is not OK.Similarly, you may not show the results of your report to anyone else in

the class until it is graded. If you do, then you share the penalty if thatmaterial is copied or is used as inspiration for work that is judged to becopied or not original.There are three main ways in which students most commonly attempt to

take illegal shortcuts in this class. They are:

1. Collaborating with one or more classmates in report writing.

2. Copying text or graphics directly from the published class notes or theWeb.

3. Taking material from previous years.

To ensure fair play, it is our duty as faculty and sta¤to be actively vigilantfor these transgressions and to follow through on advertised rules of imposingpenalties.If you are in any doubt about this, ask us, because we take plagiarism and

cheating very seriously. It is usually very clear when plagiarism has occurred,and the subject is often very surprised to learn this. Do not let this be you.Refer to the documents on the 341 memory stick, which come from the USCStudent Judicial A¤airs and Community Standards Web pages.As a rule of thumb, do not ever copy and paste. Do not copy mater-

ial from previous years, because the material changes. It is a simple andstraightforward matter of fairness and respect for your fellow students thatyou all abide by the same rules.All copied reports or reports that contain copied material receive an im-

mediate grade of zero. These cases will then be referred by written reportto the O¢ ce for Student Judicial A¤airs. The recommended action will bea grade of �F�for the course.

1.3 GENERAL LAB RULES 5

1.2.5 How to Get an A

Let us rather end on an upbeat note here! Here are some tips to maximizethe chances of doing well.

1. Read the notes. All the absolute minimum required notes, based on thelectures themselves are in the course reader, and are on the Web pages.In addition, there are background reading sections in the two optionaltextbooks. To get the most out of the course, read the backgroundmaterial in either one of these books, and make sure that it makessense along with the lecture notes.

2. Budget your time carefully. Writing a reasonable report takes manyhours of sometimes hard, and sometimes straightforward, work. Fig-ure on 10 hours, at least, for a full-length report. Do not make thisimpossible by leaving insu¢ cient time. Do not leave it until the lastmoment.

3. Make the connection between the lecture/Web materials and the labwork. The tremendous potential of this course for real learning is madeby those who realize this connection. It is the job and responsibility ofthe individual student to make this connection, by applying thought toboth components.

4. Get help. Use the o¢ ce hours, discussion boards and the TA�s withintelligence. We can prod the most mediocre student to great �OK �better things!

5. Come to lectures. To a large extent, you can get all handouts andassignments without physically coming to the lecture room. However,with all the newfangled techniques, asking questions, and writing thingsdown on the board, and listening to explanations still work really well.

That�s it! Have a great time, keep that brain switched on, and good luck!

1.3 General Lab Rules

1.3.1 Time Management

1. Lab hours are from 2pm till 5pm with 30 minutes extension if needed.Doors will close at 5:30pm.


2. Always come to lab on time. Penalties will be applied to lab perfor-mance score if late for any lab.

3. Always turn in your assignments on time. 10% penalties per daylate will be applied to each assignment score if late. No assignmentwill be accepted if late for more than 7 days.

4. Study lecture notes and lab notes are requiredBEFORE entering yourlab to ensure that you �nish your lab on time.

1.3.2 Space Management

1. Leave your personal belongings (backpacks, skateboards,...etc.) in thepresentation room during lab hours.

2. Keep the lab clean. No food or drinks allow in the lab area. (OKin presentation room).

3. Our sta¤will ensure the lab devices and equipment are organized beforeyou enter lab. Make sure that you put away your lab components anddevices(cables, beakers, measuring devices, tapes,...etc.) back to wherethey belong after you�ve done your lab. It is your responsibility tomake the lab devices and equipment organized after you�vedone your lab.

4. Library books are free to use ONLY in our library area. Do not takeany books away from the lab. Do not make any mark to any librarybooks in the lab.

1.3.3 Computer Usage & Printing Rules

1. Login name for all lab stations is JStude and the password is AME-lab.

2. Do not customize any lab stations. Do not install any software ontoany lab stations.

3. Save your lab data �leONLY under the following directoryE:nhomenJStuden.You may create personal sub-folders under above directory. All �les inabove directory will be kept for the entire semester; all personal �les inother locations will be removed after each lab.

1.3 GENERAL LAB RULES 7

4. Save your lab data �rst in lab stations�hard drives under above direc-tory then transfer your data �les onto your USB drive to prevent anydata loss.

5. No internet activities are allow during lab hours. You maytransfer your lab data via email system only AFTER you�ve doneyour lab.

6. Lab printers are for lab reports ONLY. No lecture notes or other non-341 related �les are allowed to print from our lab printers. Be sure thatyou prepare at least 30 minutes to print out your lab papers to avoidany printer problems. Remember all lab assignments are strictly dueat 2:00pm. No excuse of late reports due to printer failurewill be accepted.

7. Log o¤ your lab station (Do not turn o¤) after your lab.

1.3.4 Miscellaneous

1. Switch lab partner for each lab. No repeated lab partners allowed inthe same semester.

2. Lab notebook needs to be signed and checked by a sta¤ member BE-FORE you leave lab.

3. Concentrate during lab hours. Turn o¤your cellular phone during yourlab.

1.3.5 Academic Integrity

1. Do your own work for all the assignments.

2. Do not copy-paste any materials from others (your classmates, formerstudents, internet contents, Wikipedia,...etc.).

3. No plagiarism will be tolerated.

Chapter 2

Vibration Analysis

2.1 Second Order Systems

2.1.1 Governing Equations

All mechanical systems are governed by Newton�s laws of motion, the mostfamiliar of which is X

F (t) =Md2y(t)

dt2(2.1)

whereM is the mass, y is the displacement, and F is the sum of forces actingon the system. Since the inertial force is usually proportional to d2y=dt2,Eq.(2.1) shows that such mechanical systems are always second order. Theseare called linear second order systems as long as no nonlinear forces aredisturbing the system.This turns out to be of great practical signi�cance, since almost all dy-

namic mechanical systems are indeed linear second order, so knowledge ac-quired about one system can be readily transferred to another problem. [Brieftranslation: Learn this stu¤, and you will �nd it useful for more than justthis lab/report/class/degree. Really!]The simplest mechanical system is the mass-spring-damper problem shown

in Figure 2.1. When displaced from its equilibrium at y = 0, the restoringforce due to the spring is

Fsp(t) = �ky(t); (2.2)

where k is the spring constant. The force due to wall friction can be modeledas a viscous damper which is linearly proportional to the velocity of the

9

10 CHAPTER 2 VIBRATION ANALYSIS

Figure 2.1: (a)Mass-spring-damper system. (b)Free-body diagram.

mass

Fd(t) = �bdy(t)

dt; (2.3)

where b is the damping coe¢ cient of the system. Thus, the governing equa-tion of such system is:

Md2y(t)

dt2+ b

dy(t)

dt+ ky(t) = r(t); (2.4)

where r(t) represents external forcing to the system.

2.1.2 Free Vibration (Unforced Dynamic Response)

If we consider the free vibration of a second order system, the forcing functionis zero and the general solution of Eq.(2.4) is of the form

y(t) = k1 exp(s1t) + k2 exp(s2t); (2.5)

where the coe¢ cients k1 and k2 are called residues of the system. The rootsof the characteristic equation are

s1;2 = ��!n � !nq�2 � 1; (2.6)

where !n =pk=M is the natural frequency, � = b=(2

pkM) is the dimen-

sionless damping ratio of the system. When � > 1, the roots are real and

2.1 SECOND ORDER SYSTEMS 11

the system is overdamped; when 0 � � < 1, the roots are complex conju-gates and the system is underdamped; when � = 1, the roots are repeatedand real, and the condition is called critical damping.

2.1.3 Solutions of the Underdamped System

Vibration occurs when the system is underdamped, i.e. � < 1. The rootsof characteristic equation in Eq.(2.6) can then be re-written as

s1;2 = ��!n � j!nq1� �2; (2.7)

where j =p�1. The characteristic roots are plotted in the complex plane

in Figure 2.2. As � varies with !n constant, the complex conjugate roots

Figure 2.2: A complex plane plot of mass-spring-damper system under freevibration.

follow a circular locus, as shown in Figure 2.3. the transient response isincreasingly oscillatory as the roots approach the imaginary axis when � ap-proaches zero.The unforced dynamic response y(t) of the spring-mass-dampermechanical system having initial conditions y(0) = y0 and dy(t)=dt = 0 canbe obtained from evaluating the residues k1 and k2. Through Laplace trans-formation and Taylor series expansion techniques, k1 and k2 can be evaluated


Figure 2.3: The locus of roots as � varies with !n constant.

ask1 =

y0

2p1� �2

exp�j(�2� �); (2.8)

andk2 =

y0

2p1� �2

exp j(�

2� �); (2.9)

where � = cos�1 � represents phase angle in the complex plane as shown inFigure 2.2. Substitute Eq.(2.8) and Eq.(2.9) into Eq.(2.5), the solution ofEq.(2.4) becomes

y(t) =y0p1� �2

exp(��!nt) sin(!nq1� �2t+ �): (2.10)

The transient responses of the overdamped (� > 1) and underdamped (� < 1)cases are shown in Figure 2.4. Let�s focus on the underdamped case where� < 1. The decay rate is �!nt. When �!nt = 1, the amplitude has decayedto 1

etimes its original value; if current time is t1 then the damping ratio is

� =1

!nt1: (2.11)

2.1 SECOND ORDER SYSTEMS 13

When damping ratio � is small, then

!n

q1� �2 �= !n; (2.12)

and one can calculate � from the approximate relation,

� =1

2�(n� 1) ln(y1yn); (2.13)

where y1 and yn are the peak amplitudes measured n� 1 cycles apart. Now,given an estimate for �, it is possible to estimate a value for the undampednatural frequency, !n :

!n =2�

Tp1� �2

: (2.14)

Figure 2.4: Transient Response of the mass-spring-damper system.


2.2 Strain Gauges

2.2.1 Strain Sensitivity and Gauge Factor

For most common materials under reasonable loading conditions, there is aconstant ratio between stress and strain. The relationship is then expressedby Hooke�s law :

E =�

"; (2.15)

where:E = the modulus of elasticity or Young�s modulus� = stress" = strainAll electrically conductive materials possess a strain sensitivity, which

is de�ned as the ratio of the relative change in electrical resistance of aconductor to the applied relative change in conductor length, or

Fs =�R=R0�L=L0

: (2.16)

where:Fs = strain sensitivity factor�R = resistance change ()R0 = initial conductor resistance ()�L = change in length (mm)L0 = initial conductor length (mm)

Note that Fs is a dimensionless quantity, as are both the numerator andthe denominator. The term �L=L0 = " is called the strain. The electricalresistance of a conductor of uniform cross section behaves according to theequation:

R = �(L=A): (2.17)

where:R = resistance ()L = length of conductorA = cross-sectional area of conductor� = resistivity constant, a property of the speci�c conductor material

2.2 STRAIN GAUGES 15

If a straight wire is stretched elastically, the length will increase and thecross-section will decrease by Poisson�s ratio, v = (�D=D)=(�L=L), whereD is the diameter of the wire, which is about 30% for most metals usedin electrical resistance wire. From Eq.(2.17), one can see that, provided �remains constant, the two e¤ects are additive in causing the resistance toincrease. Also notice that

�R

R=�L

L+�A

A=�L

L+ 2

�D

D: (2.18)

The overall e¤ect is that the percentage change in the electrical resistanceof the conductor will be 1:6 times greater than the applied strain percentage.This also means that Fs will be about 1:6 for an elastically stretched wire.For most alloys, the speci�c resistivity �, is not constant. It is a¤ected by

the applied strain. More precisely, resistivity changes occur when a conductoris strained because of an elastic distortion of the lattice structure whichin�uences electron �ow through the conductor. It can be said then thatthe resistance change in a conductor is made up of a geometric e¤ect plus aresistivity change due to the internal state of stress of the conductor. In orderfor the overall resistance change of a conductor to be a linear function of theapplied strain, the resistivity change must be proportional to the internalstress level. This requirement is met by most, but not all metals. Nickel forexample, is very non-linear.As the conductor is stressed beyond its elastic limit, two e¤ects become

important. First, additional resistivity change due to internal stress willapproach zero. Second, Poisson�s ratio will approach a value of 50%, whichapplies to a constant volume or purely plastic deformation. As a result ofthese factors, further changes in resistance of the conductor will be due togeometric changes only, and Fs will approach 2.As a practical consequence, only alloys that exhibit a strain sensitivity

of 2:0 in the elastic region can exhibit essentially linear behavior over verylarge strain ranges.Strain sensitivity is a basic property of the alloy used in a strain gauge.

When this metal is formed into a grid, and provided with attachment pointsfor leadwires, the gauge will exhibit a somewhat di¤erent relationship be-tween resistance change and applied strain. The term gauge factor (Fg) isused to quantify this relationship. It is de�ned as


Fg =�R=R0�L=L0

=�R=R0"

: (2.19)

where :�R = resistance change in gauge (ohms)R0 = original gauge resistance (ohms)" = strain in the specimen surface under the gauge

Notice that the expressions for Fs and Fg are identical except that�R=R0is measured on a speci�c gauge/grid design in the case of Fg. In most cases,because of the factors mentioned above, Fg = 2:0.Actually, it is not practical to use any alloy under continuous repetitive

cycling into its plastic range because the strain sensitive alloy will fail infatigue. Normal strain-based gauge transducers therefore always operate atstrain levels within the elastic range of the strain gauge alloy.

2.2.2 Practical Gauge Design and Materials

Typical wire strain gauge designs consist of a grid, rather than a single �la-ment as shown in Figure 2.5. This is necessary to achieve the desired gaugeresistance while maintaining a practical �lament diameter and overall gaugelength. The grid is attached to a backing material with a bonding adhesive.The backing material in turn is attached with epoxy to the specimen. Such agrid behaves as several straight �laments connected in series and is thereforethe electrical equivalent of a long, straight, single wire.There are two undesirable e¤ects of forming this grid. First, each end loop

acts like the end of a single, short bonded �lament. This is a disadvantagebecause for a single bonded �lament, the shear stress is largest at the endsand is practically zero elsewhere. Because of the additional shear stress, thereis a lack of strain transmission at the end points of a �lament which thereforeis a source of error. Secondly, the end loop represents a short but signi�cantlength of bonded wire at right angles to the desired measuring axis. This willmake the grid somewhat responsive to transverse strains. The degradationin overall performance that results by forming a grid out of a single longconductor is most signi�cant for very short grids which contain a greaternumber of end loops.Wire strain gauges went out of style with the advent of printed circuit

technology. They were replaced by foil strain gauges as shown in Figure 2.6.


Figure 2.5: Single �lament strain gauge.

The foil strain gauge is essentially a small printed circuit. Instead of wireloops, the gauge pattern is etched into a thin sheet of foil of the desired alloy.The resulting gauge grid cross-section is rectangular instead of circular. Thebasic di¤erences in performance between foil and wire gauges derive from thefollowing:

1. For the same cross-sectional area, the exposed surface of the foil conduc-tor is much greater. Correspondingly, less unit shear stress is requiredin the backing and adhesive to strain the conductor. As a result, straintransmission is more complete for foil gauges.

2. A much better thermal path exists between the foil conductor and thesubstrate. Foil gauges can therefore operate at considerably higherpower levels.

3. The width of the foil conductor is usually large compared to the thick-ness of the backing and adhesive layer; therefore, unlike wire gauges,transverse strains are transmitted to some extent into the �active�partof the conductor. Additionally, end loop resistance can be minimizedby using large end tabs on foil gauges.

The most signi�cant advantage of a foil gauge is that it is much easierto manufacture. The photo-etching processes used for making foil gauges


Figure 2.6: Construction of a typical metallic foil strain gauge.

permits many identical gauges of exact size and geometry to be formed atthe same time. In contrast, wire gauges must be manufactured by hand,placing limitations on cost, performance and reproducibility from gauge togauge. A wide variety of foil gauge patterns are available as shown in Figure2.7.

2.2.3 The Wheatstone Bridge

The output of a strain gauge is a resistance change as a function of appliedstrain level. At strain sensitivities on the order of 2:0, the resistance changeswill be in the order of hundreds to a few thousand parts per million for strainlevels normally encountered in experimental analysis. Resistance changes ofthis magnitude are generally much too low for direct indication by standardlaboratory ohmmeters. It is therefore desirable to employ so called bridgecircuitry to measure small changes in resistance, usually referred to as theWheatstone bridge as shown in Figure 2.8.For readout of a single active strain gauge, it would be conventional prac-

tice for arms RB and RC to consist of precision resistors of equal value.RD would then be a precision potentiometer approximately equal to the un-


Figure 2.7: Di¤ersnt types of foil strain gauges.

strained resistance of strain gauge RA. Apply the excitation voltage V totop and bottom nodes in Figure 2.8, the output voltage eo is measured fromea and eb nodes, where eo = ea � eb.The currents through ea and eb are:

ia =V

RC +RD(2.20)

and

ib =V

RB +RA: (2.21)

ia can also be written as,

ia =V � eaRC

: (2.22)

Combine Eq.(2:20) and Eq.(2:22) yield

V � eaRC

=V

RC +RD; (2.23)


Figure 2.8: Basic Wheatstone bridge con�guration.

and ea can be expressed in terms of V , RC , and RD only:

ea = V

�1� RC

RC +RD

�: (2.24)

Similarly,

eb = V

�1� RB

RA +RB

�: (2.25)

So, the output, eo, is

eo = ea � eb = V�

RDRC +RD

� RARA +RB

�; (2.26)

and since RB = RC ,

eo =V RB (RD �RA)

(RB +RD) (RB +RA): (2.27)

Note that eo = 0 when the potentiometer, RD is set equal to the straingauge resistance, RA. Under this condition, the bridge is said to be balanced.Now set RD = RA and observe the response when RA changes by a small

amount �R,

eo =V RB (RA �RA + �R)

(RB +RA) (RB +RA + �R): (2.28)

This can be written in terms of �R=RA,

2.3 FLEXURE ANALYSIS 21

eo =�VbRBRA(RB +RA)

2

��R

RA

�; (2.29)

and from Eq.(2.19) we get

eo =�VbRBRA(RB +RA)

2 (Fg") : (2.30)

As the resistance of RA changes due to applied strain, eo will vary in bothpolarity and magnitude in accordance with �R. This is sometimes referredto as the unbalanced, or direct output, bridge and provides a fairly linearrelationship between eo and RA as long as the resistance of RA does notchange more than about 1%.In practice, RD may be another strain gauge used as a dummy, or com-

pensating gauge; or may be another strain gauge which experiences equalstrain levels of opposite polarity. Strain gauge based transducers are usuallyof the full bridge type, in which all four arms are active gauges. The use oftwo or four active arms increases the output signal available from the bridge,and reduces the nonlinearity which exists when only a single arm is active.

2.3 Flexure Analysis

Consider a simple cantilevered beam in bending as shown in Figure 2.9 (i.e.steal the equations from your structures book): The bending stress � is

� =MdyI; (2.31)

where M is the applied moment, dy is the distance measured from neutralaxis, and I is the moment of inertia of the beam. The applied moment, M;is the weight or applied force at the tip of the beam, W , multiplied by theunsupported length of the beam, L,

M = WL: (2.32)

The moment of inertia of the beam is I = bh3

12, where b is the width of the

beam measured from z direction and h is the height of the beam measuredfrom y direction. The bending stress also can be described as

� = "E; (2.33)


Figure 2.9: Simple cantilevered beam with point load at its tip.

where " is the strain and E is the Young�s modulus which is the same asdescribed in Eq.(2.15).Combine equations above to get :

" =6WL

Ebh2; (2.34)

which is perhaps more commonly written as

" =MdyEI

: (2.35)

Eq.(2.35) can be thought about in terms of two components that determinehow the strain, ", depends on the applied moment, M . Here I

dyis the section

modulus which is a geometric parameter, and EI is the �exural sti¤ness.So, using a strain gauge to monitor the strain, which is directly propor-

tional to the applied moment as in Eq.(2.35), we can measure the staticresponse of the beam to an applied load. For the purposes of measuringthe frequency response, one can model the vibration of a cantilevered beamas a second order system. A damped second order system response can becharacterized by two parameters, the undamped natural frequency, !n, andthe damping ratio, �, as shown in the previous section. By measuring thefrequency and amplitude decay in free vibration, both parameters can beestimated by experiment.

2.4 EXPERIMENT 23

BUT ....... enquiring minds might want to know: Why? What doesthe damping ratio correspond to, physically? Why should the damping beproportional to the velocity? Why is it like viscous damping? Think.

2.4 Experiment

2.4.1 Measurement of Microstrain in a Second OrderMechanical System

General Background and Procedure

Many di¤erent mechanical systems can be set up to demonstrate damped,second order systems. A cantilevered beam with attached strain gauges isone of the simplest. Two strain gauges have been mounted on the beam afew centimeters from one end. Since the resistances of the gauges changeonly slightly, they must be incorporated in a Wheatstone bridge circuit, asshown in Figure 2.10, so that small changes in the resistance are convertedto measurable changes in the voltage. A Sensotec SA-B analog ampli�er, orits digital successor SC-1000, will be used to provide a 10 VDC excitationto the Wheatstone bridge as well as amplify the output of the bridge. Leadsto and from the bridge should be connected to the SA-B unit as shown inFigure 2.11.The cantilevered beam should be attached to the work bench as illustrated

on the whiteboard. Using known weights, the strain gauge can be calibratedfor forces applied to the end of the beam. Then the dynamic response to anon-zero initial displacement can be measured for two cases: the un-loadedbeam and beam with a mass at the end. With su¢ cient care, measurementsof the beam geometry and mass allow theoretical predictions to be made forthe undamped natural frequencies of the system, and these can be comparedwith measurements.

Detailed Experimental Procedure

1. Measure the resistance of each strain gauge. They should all be ap-proximately either 120 or 350.

2. Pick up proper resistors for yourWheatstone bridge circuit and measuretheir values.


Figure 2.10: Wheatstone Bridge setup.

3. Using the C-clamp and the small piece of aluminum, clamp the beamto the counter as shown on the whiteboard. Be sure not to clampdown on either the strain gauges or the electrical connectingwires. Measure the un-supported beam length after the clamp is se-cured. Don�t forget to re-measure it if you change the clamp locationat any time.

4. Build the Wheatstone bridge circuit with proper resistors, connectingthe strain gauges as shown in Figure 2.10. Make very sure that allconnectors are stable and tight.

5. Measure the excitation voltage from the SA-B�s power supply. Blackis ground and red is nominal +10 VDC. Connect the power supply tothe bridge.

6. Connect the bridge output to the DMM. Check if increasing loads out-put increasing voltages. If not, switch the output polarity. Balancethe bridge by adjusting the 10 potentiometer so that the output is0:000mV!!!

7. Now, re-connect the bridge output to the ampli�er input.

8. Connect the ampli�er output to the DMM and zero the amp outputby adjusting the coarse and �ne zero adjustment screws on the front

2.4 EXPERIMENT 25

Figure 2.11: Sensotec SA-B power supply/ampli�er.

panel of the analog box or press the �TARE�button on the front panelof the digital box. Do not touch the gain screw. The resting out-put will be drifting by some amount due primarily to uncompensatedtemperature �uctuations. Observe these on VScope. Do not panic.Make sure the �uctuations are approximately around a mean value of0V, and then simply design your procedures for the remainder of thelab accordingly.

9. Calibrate the strain gauge using the platform and the weights provided(bolts). Add the weights, one by one, and record the static responseas a function of the applied force. Do not exceed a mass of ap-proximately 400 g as you might bend the beam permanently.Add the weights carefully, one by one and make the voltage readingsquickly. Then do an unloading curve, by measuring the voltage outputwhen removing the weights, one-by-one. With practice, the entire cali-bration procedure can be done in a matter of minutes. So, if somethinggoes wrong, then do it again until the results are consistent.

10. De�ect the beam at its tip up or down about 1 to 2 centimeters andrelease. Observe the output of the ampli�er on Vscope. You should seea damped sine wave. Digitize, save and plot the time response so youcan determine � and !n. Add a known mass (clamp) to the end of the


beam and repeat. You will then be able to determine the new � and!n.

11. Check your saved data �les very carefully using Excel BEFORE leav-ing lab. Also, don�t forget to have one of the lab sta¤ members signyour lab notebook BEFORE leaving lab.

Beam Bending Theory

The natural frequency of vibration of a cantilevered beam can be written as:For the beam only1,

!n = 3:516

rEI

ml4: (2.36)

and for the beam and clamp2,

!n =

r3EI

Ml3: (2.37)

where!n= natural frequencyE = Young�s modulus (N=m2)I = moment of inertia (m4)l = unsupported length of beam (m)m = mass of beam per unit length ( kg=m)M =Mclamp + (Mbeam=3) for case with clamp ( kg)

I can be calculated from

I =bh3

12(2.38)

where b is the width, and h, the height, of the beam. If the beam is composedsolely of aluminum, then E is 6:9� 1010 N=m2.Thus, from careful measurements of the dimensions of the beam and

the masses of the beam and clamp, two estimates of !n can be made fromEq.(2.36) and Eq.(2.37). How do these values compare with the values ob-served directly from the dynamic response? Remember, you need measure-ments of the uncertainties in order to estimate this. How do the measure-ments of � compare with some reference value? Are they the same, or di¤er-ent when !n changes?

1p.420. Steidel, R.F. (1989), An Introduction to Mechanical Vibrations, Wiley.2p.92. Steidel, R.F. (1989), An Introduction to Mechanical Vibrations, Wiley.

2.4 EXPERIMENT 27

2.4.2 An Engineering Spreadsheet Report

The report from the strain gauge experiment will be in the form of a singlespreadsheet �le. The purpose of the spreadsheet will be to present the quan-titative results of the experiment, stating simple conclusions deriving fromthis comparison. The spreadsheet will be graded from paper and electroniccopies according to the following criteria:

1. Logic and clarity of the layout.

� Clear presentation of raw data� Clear calculations of derived quantities� Clear, labelled and legible plots of time response, static calibra-tion, referred to when used in calculations.

� Clear statements of fact or conclusions from experiment and the-ory

2. Completeness of data/calculations

� Su¢ cient number of data points?� Derived quantities identi�ed and calculated, together with usefulintermediate steps?

3. Flexibility of spreadsheet calculations

� If a variable is changed, does the change propagate through thecalculations?

� Are variable quantities identi�ed?

4. Uncertainty analysis

� Technically correct?� Used to compare numbers and make statements of fact?

5. Control of numbers

� Appropriate precision.


� Units correct and identi�ed.

6. Graphics

� Technically correct (lines, symbols, labels, units)� Control of format �decisions rather than defaults.

7. On time

The goal of the spreadsheet is much the same as that of a formal report�to tell a story based on experimental �ndings, and to make comparisonswith appropriate theoretical models. Note that both paper and electroniccopies will be used for grading �pay attention to the layout and appearanceof both.Remember some of the techniques introduced last semester. Here it will

quite useful to name cell variables, and to use formatting to outline and iden-tify di¤erent parts of the spreadsheet. Name the worksheets appropriatelyand remove unused ones.When you�re done, print out all relevant pages of the spreadsheet (maybe

2-3 of these, formatted deliberately and carefully for printing and reading)+ graphs. Staple them together and hand the paper copies in together withthe electronic version. All materials will be returned in one week.

2.4.3 Important Reminder

You must do your own work. It is very important that every part of thisexercise comes from your e¤orts alone, from design of the write-up, to designof the plots to choice of topics to write about. The only data you share withyour lab partner are the columns of numbers that were written in your labbook or saved in Excel. All plots, tables and �gures for the report must begenerated by yourself, from scratch.For those who are used to working together on problems, or �nd them-

selves side-by-side at their computers, then you must enforce an arti�cialseparation of your e¤orts. It will seem a bit unfriendly, and as though youare now competing against each other, but you are not, because an absolutenormalized scale is not imposed in the �nal grades. We just want to see whateach of you has learned, individually, to make sure that you have all learnedit.

2.4 EXPERIMENT 29

It is not so relevant to this lab in particular, but we will follow this generalrule in 341: do not ever copy-paste material unless you are copyingit from your own work. Copy-pasting from your own Excel spreadsheetto your own Word document will be common, but will be the only form ofthis we allow.The recommended penalty from Student Judicial A¤airs and Community

Standards for breach of such conduct codes is F for the course. This willbe imposed and paperwork will be sent to SJACS after an initial meetingbetween the student(s) and professor. The same penalty will be given tothose from whom work is copied as for those who copy it.

Chapter 3

Heat Transfer andThermocouples

3.1 De�nitions

A thermocouple is a simple device for measuring temperature. Two metals(wires) of di¤erent composition are joined (welded or soldered together) atone end and that�s it! They produce a small voltage between the open endsthat is proportional to the temperature that the joint is exposed to.The operation of a thermocouple can be understood by remembering two

basic principles:

1. When two dissimilar metals are in contact, their di¤erent a¢ nities forfree electrons produces a small current across the junction as shown inFigure 3.1.

2. The magnitude of the di¤erence varies with temperature as shown inFigure 3.2.

� The voltage produced is small, on the order of a few millivolts, so itrequires ampli�cation.

� The metals used are alloys that are specially formulated to maximizethe emf output, i.e., maximize sensitivity.

� The standard con�guration is often denoted as in Figure 3.3.

31

32 CHAPTER 3 HEAT TRANSFER AND THERMOCOUPLES

Figure 3.1: Two dissimilar metals in contact produce a small electric currentacross the junction under temperature T1.

� Note that if T1 = T2, both thermocouples will produce the same emfbut with opposite sign, so the total emf will be zero.

� If the circuit is opened, and T1 6= T2, there will be an emf across theterminals as in Figure 3.4.

� The second thermocouple is not strictly necessary, but is useful as areference junction. It is usually kept at 0 �C.

3.2 Useful Properties

3.2.1 Law of Intermediate Metals (LIM)

The introduction of a third metal into the circuit does not a¤ect the emf,provided T is uniform across all the junctions.

3.2.2 Practical Consequences of LIM

� Consider how the thermocouple is constructed. Solder sits between thetwo metals. Essentially, this creates 2 thermocouples, one of Metal A

3.2 USEFUL PROPERTIES 33

Figure 3.2: Two dissimilar metals are in contact produce a larger elcectriccurrent across the junction under higher temperature T2.

and the solder, and another of Metal B and the solder as shown inFigure 3.5. The output emf then is EAS + ESB = EAB.

� Another pair of thermocouple junctions are created when the thermo-couple wires are attached to electrical wires (Cu). It is important thatthose junctions are kept at the same temperature so that the net emfthey produce is zero as shown in Figure 3.6.

� It is also possible to use one single thermocouple junction, with themeasuring device acting as the reference junction as in Figure 3.7. Pro-vided the temperature is uniform across the connecting terminals, andprovided this temperature does not change, then the single thermocou-ple can be calibrated for use in measuring an arbitrary temperature.

3.2.3 Law of Intermediate Temperatures (LIT)

� LIT does not require the thermocouple response to be linear with tem-perature.

� Furthermore, it allows the use of reference tables, based on standardtemperatures.


Figure 3.3: Standard con�guration of a thermocouple.

3.3 Static Calibration

� Determine the thermocouple composition by �nding its sensitivity (mV= �C)and comparing with published values of common types of thermocou-ples.

� Vary temperature of water in a beaker from (close to) 0 �C to 100 �C.Keep track of temperature with a thermometer and record thermocou-ple output at 5 �C increments.

� Do the same while decreasing temperature from 100 �C to about 30 �C.

3.4 Dynamic Response

� How quickly can the thermocouple respond to changes in temperature?

� Find time constant by measuring the response to stepwise change intemperature. The larger the temperature change, the better the esti-mate will be. So dunk the bead from one extreme to another. Water

3.4 DYNAMIC RESPONSE 35

Figure 3.4: Open circuit con�guration of a thermocouple.

is convenient (more convenient than air, for example; why?), so switchfrom 0 �C� 100 �C.

� Consider an ideal thermocouple bead as in Figure 3.12:

3.4.1 Conservation of Energy

Conservation of energy: Rate of increase of energy in a volume = �ux of heatacross its boundaries.

1. If we assume that, within the boundary, the material behaves as a singleentity �i.e. there are no gradients in T (t), then the energy balance canbe written:

d(energy)dt

=d

dt(mCpT (t)) = mCp

dT (t)

dt; (3.1)

where Cp is the mass-speci�c, constant pressure heat coe¢ cient in unit[ J= ( kg �C)].

2. Further, assume negligible heat �ux along the wires.


Figure 3.5: A thermocouple is constructed by two metals soldering togetherat their tips.

3. Assume that all heat exchange is through convective heat transfer withsurrounding �uid (ignore radiative heat transfer).

The heat �ux will depend on the exposed surface area, S, the temper-ature di¤erence at the surface, and how e¢ cient is the exchange process(h =convective heat transfer coe¢ cient):

qout(t) = h[T (t)� T1(t)]S; (3.2)

qin(t) = �h[T (t)� T1(t)]S; (3.3)

where ~q; ~n > 0 for net �ux out. Therefore,

mCpdT (t)

dt= �h[T (t)� T1(t)]S: (3.4)

Now, substitute for the geometry of a sphere with diameter d,

�th4

3

�d3

8CpdT (t)

dt+ h�d2T (t) = h�d2T1(t): (3.5)

Eq.(3.5) divide by h�d2, yields��thdCp6h

�dT (t)

dt+ T (t) = T1(t): (3.6)

This is a linear �rst order di¤erential equation with characteristic time con-stant � = �thdCp

6h. [Does this have units of time? Do a sanity check, just in

case. h has SI units in Watts per meter squared-kelvin [W=(m2K)].


Figure 3.6: A pair of thermocouples attach to copper electrical wires toconnect to measuring devices.

3.4.2 Solution of First Order System for Step-FunctionForcing

Recall the general solution for �rst order systems,

�dT (t)

dt+ T (t) = T1(t); (3.7)

which was composed of two parts, the complementary function and the par-ticular integral. The step function can be written as

t � 0 ! T1(t) = T1;

t > 0 ! T1(t) = T2; (3.8)

where T1 and T2 are constants. The output is

t � 0 ! T (t) = T1;

t > 0 ! T (t) =?: (3.9)


Figure 3.7: Single thermocouple attaches to copper electrical wires to connectto measuring devices.

T1 is the starting temperature and the �rst order system can be rewritten interms of temperature di¤erences:

t � 0 ! �d (T (t)� T1)

dt+ (T (t)� T1) = 0;

t > 0 ! �d (T (t)� T1)

dt+ (T (t)� T1) = (T2 � T1) : (3.10)

The solution to the homogeneous equation (recall 1st Order System notes)has the form,

(T (t)� T1) = A exp(�t

�): (3.11)

Now, �nd the particular solution for RHS = (T2 � T1). It is a constant,

(T (t)� T1) = (T2 � T1) : (3.12)

Therefore, T (t) = T2. The temperature di¤erence at any time t is T (t)�T1,and can be written as a function of the complementary and particular partsof the general solution,


Figure 3.8: Diagrams showing law of intermediate temperatures.

(T (t)� T1) = A exp(�t

�) + (T2 � T1) : (3.13)

Now apply initial condition T (0) = T1 to Eq.(3.13),

0 = A+ (T2 � T1) :

The value of A can be solved as

A = � (T2 � T1) : (3.14)

Substitute Eq.(3.14) into Eq.(3.13), yields

(T (t)� T1) = (T2 � T1)�1� exp(� t

�)

�: (3.15)

The step response of Eq.(3.10) has solution described by Eq.(3.15) asshown in Figure 3.13.

3.4.3 Meanwhile, Back in the Lab...

The time constant, � , can be measured, and is related to the geometry andproperties of the thermocouple bead as described in Eq.(3.4) and Eq.(3.6)by:


Figure 3.9: A pair of thermocouple junctions output E = 4:277mV whenone end has temperature 0 �C and the other end has temperature 100 �C:

� = �th1

6dCph=mCphS

: (3.16)

The static calibration allows the composition of the thermocouple bead tobe estimated, and �th and Cp can then be looked up in tables provided in labnotes. So, we can measure d (rather approximately) and calculate h as,

h =�thdCp6�

: (3.17)

The step response can be measured going from cold!hot, or hot!cold.That gives us two measurements of the time constant, � . If we do both:

1. Do we get the same answer?

2. Would we expect to?

3. Presumably, �th, Cp and d do not change, so what might?

4. Just exactly what is this h-thing anyway? Figure 3.18 shows thath = h(Re).

5. Why would Re change with temperature?

6. If we can estimate Re, then do our estimates of h agree with Figure3.18?

3.5 EXPERIMENT 41

Figure 3.10: A pair of thermocouple junctions output E = 3:488mV whenone end has temperature 20 �C and the other end has temperature 100 �C:

3.5 Experiment

3.5.1 Thermocouples and Heat Transfer

Introduction

One of the simplest devices for measuring temperature is a thermocouple,which consists only of a junction between two di¤erent metals. As the tem-perature of the junction changes, the attraction of the di¤erent metals for thefree electrons in the junction varies. As one metal attracts more electrons,an electromotive force is produced. This force is of the order of several milli-volts for typical metals used in thermocouples and the emf is approximatelylinear over a broad range of temperatures. Because of its simplicity, near-indestructability (this is not a challenge), and low cost, the thermocouple isvery popular for measuring temperature in many environments.

Static Calibration

Whenever the sensitivity of an instrument is unknown, it must be determinedeither by estimation or by calibration. The accuracy of any experiment isimproved if a reliable calibration can be obtained. The thermocouple can becalibrated by varying the junction through a range of known temperaturesand plotting this against the observed emf.


Figure 3.11: Thermocouples emf output due to temperature changes in in-creasing temperature(heating) and decreasing temperature(cooling).

Procedure

1. Place a beaker of ice water on the ring stand and immerse the ther-mocouple bead in it. Note that the emf produced can be ampli�edand read on the VScope as shown in Figure 3.14. The ice water nowprovides the �rst data point on the static calibration curve as shownin Figure 3.15:

2. Light the Bunsen burner and place it under the beaker of ice watercontaining the thermocouple. Read the temperature of the water witha thermometer while continually stirring with the stirring stick (do notuse the thermometer to stir as it breaks easily). As the water is heated,record the temperature from the thermometer and the DC voltage onVScope at approximately 5 �C intervals between 0 and 100 �C. Whenplotted as in Figure 3.15, a reasonably straight line should result.

3. After reaching the boiling point, add small amounts of ice to the waterand decrease the temperature while recording the emf. Cool the waterback down to 25 �C or so. Do the data points agree with your previouscalibration or is there some hysteresis?

3.5 EXPERIMENT 43

Figure 3.12: A sphere thermocouple bead connects two metals.

Figure 3.13: Step response of a 1st order system.

Dynamic Response

In the static calibration, it was inherently assumed that the thermocouplealways had the same temperature as the water. That is, there was never anytime lag between the actual temperature of the water and the emf voltage.This approximation is only valid when the temperature of the environmentchanges slowly, because the voltage due to the thermocouple does not re-spond instantaneously to the applied temperature. The heat lost by thethermocouple is


Figure 3.14: Small voltage output from thermocouple is ampli�ed beforeacquisition and analysis in VScope.

q(t) = �mCpdT (t)

dt; (3.18)

wherem is the mass of the thermocouple, Cp is the constant-pressure speci�cheat, and T is the temperature. Neglecting the conduction through thethermocouple wires, the heat is primarily lost to the environment accordingto

q(t) = hS(T (t)� T1(t)); (3.19)

where h is the convective heat transfer coe¢ cient, S is the surface area ofthe thermocouple and T1(t) is the environment temperature. CombiningEq.(3.18) and Eq.(3.19) yields

�dT (t)

dt+ T (t) = T1(t); (3.20)

3.5 EXPERIMENT 45

Figure 3.15: Construction of a static calibration curve for thermocouple.

where � = mCphS

is the time constant. Eq.(3.20) is identical to Eq.(3.7),when starting from initial environment temperature T1(t) = T1 and sud-denly jumping to another environment temperature T1(t) = T2, the outputtemperature T (t) becomes

T (t) = T2 � (T2 � T1) exp(�t

�): (3.21)

If the output voltage of the thermocouple is linearly related to the tempera-ture by the sensitivity s, i.e., e(t) = sT (t), then Eq.(3.21) becomes

e(t) = sT (t) = sT2 � s (T2 � T1) exp(�t

�): (3.22)

Therefore,

e(t) = e2 � (e2 � e1) exp(�t

�); (3.23)

where e1 is the voltage output when the thermocouple measures the initialenvironment temperature T1(t) = T1, and e2 is the voltage output when thethermocouple measures the new environment temperature T1(t) = T2. Inother words, if T1(t) is a step function, then the thermocouple emf followsit exponentially as in Eq.(3.23) and shown in Figure 3.16.As seen in equations (3:20) ; (3:21) and (3:23), the thermocouple behaves

as a �rst order system. As with all such systems, the dynamic response


Figure 3.16: Step response of the thermocouple.

is completely determined by the time constant, � , and once this quantityis measured, some interesting characteristics of the physics of heat transferin moving �uids can be inferred. The step response can be measured byrapidly transferring the thermocouple bead between containers of water heldat di¤erent, and �xed, temperatures.

Detailed Experimental Procedure

1. Set up the equipment as shown in Figure 3.14.

2. (Re-)Familiarize yourself with the use of the Op-Amps ampli�er andVScope.

3. Verify that the gain of the ampli�er is � 100 using a signal from thefunction generator and measure its exact value from the DMM.

4. Connect the thermocouple to the op-amp input. Heat up a beakerabout 1=3 full of water and have another, also about 1=3 full, with ice

3.5 EXPERIMENT 47

water. Check the response of the thermocouple and set the VScopecontrols to useful values. The temperature of the ice-water mix will bevery nearly 0 �C, that of boiling water will be 100 �C. Take 5 pairs ofdata points, recording the voltage output of the thermocouple at thesetwo reference points. This helps to debug the set-up and later to checkand verify the calibration curve data.

5. Do the static calibration described in earlier section. Start with theice-water used above. Take care over the physical set-up, making surethat various instruments are not touching the glass beaker, for example,and making sure that you are not setting your partners hair on �re,for another example. Do both loading and unloading curves, i.e. forpositive and negative temperature changes.

6. Prepare a second beaker of water, and observe and record the transientresponse as the thermocouple is transferred from boiling water to icewater and vice versa.

7. When all data has been collected, measure the bead diameter, d, to-gether with a realistic estimate of the uncertainty in this measurement.Also make sure that some measure of the bead speed, �b, can be made,either from the time traces themselves, or by timing a typical experi-ment and measuring the �ight path length. Once again, make a realisticnumerical estimate of the uncertainty.

Later on..

1. The slope of the calibration curve allows you to deduce the compositionof the thermocouple by comparing with reference curves in Figure 3.17.The thermocouple will be one of Type E, T , J or K. Remember totake into account the gain of the ampli�er, as the reference curves arefor un-ampli�ed output.

2. Assume a 50 : 50 mix of the two alloys, and calculate Cp for the ther-mocouple bead.

3. Calculate the time constant, � , from both sets of transient responses(hot to cold, and cold to hot). Are they the same?


4. From � , and the bead diameter, d, it is now possible to calculate theheat transfer coe¢ cient h. So do it. If there are two values of � , thenthere will also be two values of h. That means it is not a constant.Why not?

Given estimates of �b, then the curve-�t of empirical data in the litera-ture in Figure 3.18 allows independent estimates for h to be made for theappropriate Re for the ice- and boiling water cases. Does h depend on thetemperature T? Are these values of h consistent with those derived from thestep responses?

Useful Tables

Nomenclature

� = time constant

m = mass of bead

Cp = constant-pressure speci�c heat

S = surface area of bead

h = convective heat transfer coe¢ cient

�b = speed of bead entering water

d = diameter of thermocouple bead

k = thermal conductivity of water

� = ��= kinematic viscosity of water

Re =�bd�

= Ratio of inertial to viscous forces in �ow over bead

Nu =hdk

= E¢ ciency of heat exchange by convection vs. conduction

Some useful numbers

T �C � ( cm2= s) k ( J= cm � s � �C)

0 1:787� 10�2 5:6� 10�3

100 0:295� 10�2 6:7� 10�3

3.5 EXPERIMENT 49

Cp values of di¤erent materials

Material Cp ( J= g � �C)

Al :900

Cr :460

Cu :385

Fe :452

Mn :482

Ni :440


Density of common thermocouple metals, alloys, and their compositions

Stu¤ Density( g= cm3)

Pure Metals

Iron (Fe) 7:9

Nickel (Ni) 8:9

Manganese (Mn) 7:47

Molybdenum (Mo) 10:2

Aluminium (Al) 2:71

Copper (Cu) 8:93

Silver (Ag) 10:5

Gold (Au) 19:3

Tungsten (W) 19:25

Platinum (Pt) 21:45

Rhodium (Rh) 12:42

Chromium (Cr) 7:14

Platinum Alloys

Pt�Rh(10%) 19:95

Nickel Alloys

Constantan (Cu(55%)�Ni(45%)) 8:86

Chromel(Ni(90%)�Cr(10%)) 8:73

Alumel(Ni(95%)�Mn(2%)�Al(2%)) 8:6

Note: Missing entries can be looked up at http://www.webelements.com/

3.5.2 Sometime Later, Furiously Report-Writing.....

� This experiment, and accompanying report, can be complex. Startwith an outline.

3.5 EXPERIMENT 51

� The objective of the experiment is to investigate and characterize theresponse of a thermocouple bead.

� Keep details of procedure to the minimum required. What must thereader know to repeat the same experiment?

� Keep results section free of interpretation. Just put what you got.

� The questions in the notes above are prompting you for physical ex-planations of the phenomena you observe. How can we understand thethermocouple results in terms of the physics of heat transfer, and �uidmotion?



selves side-by-side at their computers, then you must enforce an arti�cialseparation of your e¤orts. It will seem a bit unfriendly, and as though youare now competing against each other, but you are not, because an absolutenormalized scale is not imposed in the �nal grades. We just want to see whateach of you has learned, individually, to make sure that you have all learnedit.It is not so relevant to this lab in particular, but we will follow this general

rule in 341: do not ever copy-paste material unless you are copyingit from your own work. Copy-pasting from your own Excel spreadsheetto your own Word document will be common, but will be the only form ofthis we allow.The recommended penalty from Student Judicial A¤airs and Community



Figure 3.17: Calibration curves of di¤erent type thermocouples.

3.5 EXPERIMENT 53

Figure 3.18: Least-squares �t to experimental data for heat transfer from asphere over a large range of Reynolds numbers.

56 CHAPTER 4 FLUID FLOW AND TURBULENCE

Chapter 4

Fluid Flow and Turbulence

4.1 Fluid Turbulence and Jet Flows

4.1 FLUID TURBULENCE AND JET FLOWS 57

Opening Ceremonies, Olympic Games Sydney 2000. (LA times)

4.1.1 Jets: Applications and de�nitions

De�nition 1 A jet is a �ow produced by a pressure drop across an ori�ce.

Examples:

� Aircraft propulsion

� Rocket propulsion

� Fuel injectors in ICE

� Chimneys, smoke stacks, car exhaust

� Air-conditioning vents

� Sprays - perfume, hair, medical, �re extinguishers, terrorist activities

Two major areas of concern:

1. Propulsion

2. Mixing (may be related to #1)

Jets are one type of a very widespread class of �uid �ows known asboundary-free shear �ows, which evolve in space and time due either to ex-ternal gradients (in temperature or pressure), or may evolve themselves.


Three types of free-shear �ows: a wake, a jet, and a shear layer. (Tennekes& Lumley, 1972)

Free-shear �ows are:

� Evolving in space and time.

� Very important in many technological applications.


� Very hard to study.

� Very poorly understood.

Free-shear �ows are almost always turbulent. (Non-turbulent �ows canbe regarded as artifacts of theoreticians and wind-tunnels.)

4.1.2 Turbulence

Characteristics of turbulence:

� Complex, irregular �ow.

� Large, continuous range of scales.

� Highly dissipative - viscous shear stresses dissipate kinetic energy.

� Highly di¤usive - good for mixing.

� Re is large.

� Mathematically intractable.

Note: Mix analytical tools with physical reasoning/assumptions, basedon experimental evidence.

4.1.3 Time-Averaged Quantities and Universal Statis-tical States

Postulate: Turbulence evolves with complex motions over many time andlength scales, which interact with each other locally to produce a dynamicallyself-similar state, where only local scales of length and velocity matter. Sucha state is called fully developed turbulence. It has reached a state of dynamicequilibrium, where local forces and time scales only are important. It isuniversal. One patch of turbulence is very much like another, and they canbe re-scaled so their properties (energy dissipation, mixing,...) are equivalent.Note that all of the above is really a postulate, and not a proven fact. It

is also correct only in a statistical sense. The individual detailed variationsin turbulent �ows defy any such simpli�cations. It is possible to demonstratethat self-similar solutions to simpli�ed equations of motion exist. It is not


Figure 4.1: Evolution of a plane turbulent jet. (Tennekes & Lumley, 1972)

possible to show that they must exist in real life. Instead, one looks forevidence of the existence of such solutions in experiment.In practice, the circular jet is governed by a single length scale, its diam-

eter, D, and the average strength is related to some time-averaged velocity,U .If the �ow is self-similar, then it should be possible to rescale velocity

pro�les by using the characteristic local scales shown in Figure 4.3.The y

12 point de�nes a characteristic point on the bell-shaped velocity

pro�le, proportional to its width. If the pro�le were Gaussian, then onecould also use the Gaussian half-width. The magnitude, Umax, is simply thehighest point of the curve.Now, the argument goes: if the physical mechanism, the �ow dynamics,

are self-similar, then self-similar �ow pro�les should be observed. One shape�ts all. The pro�le shape does not change, only its relative scales in y andU as x increases. Umax will decrease and y

12 will increase. This being the

case, if each pro�le is rescaled by equivalent measures of local length and


Figure 4.2: Coordinate system for turbulent jet �ow.

velocity, then the resulting normalized pro�le should be universal, describingall pro�les at all locations in the jet. Figure 4.4 shows an example, takenfrom some lab data, where the y

12 point was used to de�ne a new x coor-

dinate, x�, as described in further detail in the lab handout. The pro�lescollapse, within experimental uncertainty, when rescaled this way. They allhave the same shape. Thus, the self-similarity hypothesis is supported in thiscase. Moreover, if another application showed similar data (a smokestack,for example), then the lab results could be rescaled to �t and/or predictthat data too. This could be very important if it proves di¢ cult to makesuch detailed measurements in the real smokestack, or at large numbers of xlocations (both are likely).

4.1.4 Dimensional Analysis and the Reynolds Number

� Solutions to systems of equations that describe possible physical sys-tems are dimensionally homogeneous.

� Solutions are therefore invariant to change in units.

� A solution is similar if the dimensionless groups are equal in magnitude.(Buckingham-Pi theorem)


Figure 4.3: De�ning a local measure of the jet width in U(y).

� For a given set of boundary and initial conditions, the only dimension-less group for incompressible �ow in a homogeneous �uid is Reynoldsnumber, Re.

The Navier-Stokes equation relate the physical variables: x, u, �, and �,where x and u are the physical coordinate and velocity respectively of a �uidelement with density (mass per unit volume) �, and viscosity �. The physicalunits are M , L, and T . Consequently, there are 4 � 3 = 1 dimensionlessgroups made up of these quantities. This is the Reynolds number,

Re =�ul

�; (4.1)

or, commonly written as:

Re =ul

�; (4.2)

where� =

�

�; (4.3)

is called the kinematic viscosity. If two �ows have the same Re, then theyare dynamically similar. The balance of forces is identical. This is the onlyrequirement for geometrically similar problems in �uids.


Figure 4.4: Self-similarity in turbulent jet pro�les.

4.1.5 Practical Examples

Cigarettes

l � 2 cm 2 cm

u � 10 cm= s 1m= s

Re � 10�20:15�= 140 100�2

0:15�= 1400

Smoke-Stack

l � 10m

u � 1m= s

Re � 103�1020:15

�= 106

Flow is very turbulent. (meaning?)


Figure 4.5: Thermal plume caused by heating of earth surface. Estimate Re.

Rules of Thumb

�air = 0:15 cm2= s

�water = 0:01 cm2= s

! �air�water

� 15:

Example

Check the Reynolds numbers, Re, of two di¤erent jet conditions in the fol-lowing table:


d u Re

Jet in air 10 cm 1:0m= s ?

Jet in water 7 cm 10 cm= s ?

:

You may apply the following rule of thumb,

Reair = 7� u( cm= s)� l( cm)

Rewater = 100� u( cm= s)� l( cm):

What can you conclude from above calculations?

Interesting Corollaries

� The �ows are (dynamically) identical (not just kinda sorta the same-ish).

� full-scale model/tests are not always required.

� We can do experiments on hair dryers!

� Aerospace engineers sometimes use water channels.

4.1.6 Dynamic Pressure - How to Measure u from p?

Principles of the Pitot Tube

The dividing streamline arrives at B in Figure 4.6, where the �ow comesto rest, and so u = u0 = 0. The total pressure is the sum of the staticand dynamic pressures, and must be constant along a streamline. Hence, atpoints A and B in Figure 4.6 along the streamline,

pA +1

2�u2A = pB +

1

2�u2B: (4.4)

At B in Figure 4.6, uB = u0 = 0: Denote the pressure at B as p0. At Cin Figure 4.6, the measured pressure is the static pressure (only), and sinceps �= p1,

1

2�u21 = p0 � ps; (4.5)


and so,

u21 =2 (p0 � ps)

�: (4.6)

Assumptions are:

Figure 4.6: Dividing streamline �ow over a pitot tube.

� Direction is straight.

� Flow is attached, and parallel.

� Flow is incompressible.

4.2 Experiment

4.2.1 Pressure/Velocity Measurements in a TurbulentJet

Jets

A jet is formed by �ow issuing from a nozzle into ambient �uid, which canbe either moving or at rest. A jet is a fundamental �ow con�guration withmany practical applications, e.g. propulsion, combustion, various chemically-reacting �ows, mixing of temperature and pollutants, and chemical lasers.The velocity at the exit of the nozzle of a typical laboratory jet has

a smooth pro�le and a low turbulence level, about 0:1~0:5% of the meancenterline velocity. In practical cases, the turbulence can be as much as 20%,

4.2 EXPERIMENT 67

but the basic �ow properties observed in a clean laboratory jet can still begeneralized and applied to real world problems.Due to the velocity di¤erence between the jet and the ambient �uid, a

thin shear layer is created. Fluid from the jet and from the ambient mix in-side the shear layer. In the central portion of the �ow, there is a region withalmost uniform mean speed, called the potential core. Due to entrainmentof the outside �uid, the jet spreads in the streamwise direction as shown inFigure 4.7. Eventually, the potential core disappears at a distance of about�ve diameters downstream of the nozzle.The entrainment process continues

Figure 4.7: Regions of a jet.

further downstream, but the rate of spreading is di¤erent from that observedupstream of the potential core. The velocity distributions in the region down-stream of the potential core have a bell-shaped pro�le as shown in Figure4.8.Y 12is de�ned as the distance between the jet axis and the location where

the velocity equals half of the maximum velocity, U0. The cross-stream ylocation of this characteristic point can be used to measure the thicknessof the jet pro�le. The bell-shaped velocity pro�les can be thought of asif they originate from a single point, called the virtual origin. The virtualorigin is determined by joining the Y 1

2points and extrapolating to the X

axis. At some distance from the jet exit, the turbulent jet pro�les collapseonto a single curve when the y axis is normalized with X as shown in Figure4.9. At this point, the jet is said to have reached the self-similar region;this usually occurs about 6 to 8 jet diameters, D, downstream from thenozzle. The fact that the pro�les are self-similar and collapse when scaledwith local U0 and X, implies that only local length and velocity scales are


Figure 4.8: Velocity pro�les of a jet.

important in determining the �uid dynamics. The fully-developed, turbulent�ow thus has certain universal scaling characteristics that do not dependon the initial generating conditions. When true, the practical consequenceis that reasonable predictions can be made concerning the mean propertiesand evolution of an extremely complicated and poorly understood �ow (�uidturbulence was identi�ed, �rst in 1948, and then again in 1995, as one of thegreat unsolved applied physics problems).

Pitot-Static Tubes

The Pitot-static tube (also called a Prandtl tube) is one of the most commoninstruments for measuring the mean velocity of a �ow �eld. The deviceconsists of two coaxial tubes as shown in Figure 4.10, one is open at thetip and measures the total pressure or Pitot pressure.Another is open at thesurface and measures the static pressure. The basic principle used in thePitot tube is given by Bernoulli�s equation,

pt � ps =1

2�airU

2; (4.7)

4.2 EXPERIMENT 69

Figure 4.9: Self-similar region velocity pro�le after normalization.

where�air = density of air,

ps = static pressure,

pt = total pressure,

U = freestream velocity.

The �ow stagnates as it reaches the tip, and the velocity can be measuredfrom the pressure di¤erence,

U =

s2 (pt � ps)

�: (4.8)

Manometers

A manometer is an instrument used for measuring pressure as shown inFigure 4.11. The pressure di¤erence, pt � ps, is equal to the weight of theliquid column between point B and point A,

pt � ps = �0gh; (4.9)

where �0 is the density of the manometer �uid.


Figure 4.10: Schematic of pitot-static tube.

In the case of a small pressure di¤erence, an inclined manometer is oftenused to increase the accuracy of the reading of h as shown in Figure 4.12.Thepressure di¤erence is still represented by �0gh. However,

h = L sin �; (4.10)

from Eq.(4.9),pt � ps = �0gL sin �: (4.11)

The resolution is thus much improved by measuring L instead of h.

4.2.2 Procedure

NOTE: Despite the crudity of the available measuring instruments, it ispossible to measure reasonably accurate velocity pro�les of the jet �ow. Inthis lab, you will be left more or less to your own devices as to how to makesatisfactory measurements. Proceed carefully and thoughtfully. How shouldthe coordinate system be set up? Is the Pitot-static tube aligned with mean�ow? How many data points are required for each pro�le? How far in yshould measurements be made? Think.

1. Level and adjust the manometer to zero reading while the Pitot tube isin still air. Note that the scale on the manometer is already in vertical

4.2 EXPERIMENT 71

Figure 4.11: Schematic of manometer.

inches (shudder) of water so Eq.(4.11) is not necessary. Convert thesemeasurements to cm for subsequent calculations.

2. Place the Pitot tube on the centerline of the exit of the nozzle, onejet diameter (1D) downstream. Turn on the blower (high speed withno heating). Note the manometer displacement. Then, calculate andmark the point on the manometer scale corresponding to 63% of thetotal displacement.

3. Now estimate the time constant for the system by observing the re-sponse to a step function in U . With a piece of paper, block the Pitot-static tube openings so that the manometer reading is zero. Removethe paper quickly. Watch the liquid movement in the manometer andestimate the time constant using a stopwatch. Repeat �ve times andcompute the average. Move the Pitot tube to x = 8D. Repeat the testagain. Is there a di¤erence? Do you expect one? Report the results ofthis test to a sta¤ member during lab.

4. Move the Pitot tube back to x = 1D, on the centerline of the jet, andtake a pressure reading (is it the same as in step 2?). Now measurethe response of the Pitot tube as a function of angle of incidence, �, tothe mean �ow by rotating it about this point (the tip should remain


Figure 4.12: An inclined manometer.

at x = 1D, on the centerline). Take enough data so that you can plotU(�) for 0 � � � � 60 �. Think about how you would interpret thedata for both small and large �.

5. Measure and plot the velocity pro�les at x = 1D, 6D, 7D and 8D. Thevelocity pro�les will be plots of the mean streamwise velocity, U , vs. y,the cross-stream location. Make sure you take enough points to clearlymeasure the shape of the U(y) pro�les. Do not assume symmetry aboutthe centerline.

6. Re-plot the velocity pro�les measured at 6D, 7D and 8D (not 1D(why not?)) by using the self-similar normalization method explainedin lecture note. Plot them all on the same graph. Are they self-similar?i.e. do they collapse?. What is the physical signi�cance of this?

4.2.3 Testing Velocity Pro�les in Turbulent Jets forSelf-Similarity �A Practical Guide

De�ning a Coordinate System

First, let us clearly de�ne a sensible coordinate system for this �ow as shownin Figure 4.13. This is the usual way of doing it, and it is worth noting whythis is so. First, in this model of the �ow, only two spatial dimensions areconsidered. This includes both planar jets and axisymmetric jets. Whenever

4.2 EXPERIMENT 73


there is a mean �ow, or most dominant �ow component (in this case, the �owalong the jet axis), it is common to assign the x coordinate, and correspondingu velocity component parallel to that direction. The �ow measured is a time-averaged component (owing to the relatively large time constant of the pitottube/manometer system), and so it is denoted U , rather than u. x is de�nedparallel to the mean �ow and the long axis of the jet exit nozzle, beginningat the nozzle exit. Positive x moves in the direction of the �ow, with positiveU . The remaining coordinate, y, should be perpendicular to x. It has itsorigin at the centerline, which is the line of symmetry of the jet. Note howthe coordinate system is �xed with respect to the jet �ow itself, and not tosome arbitrary laboratory reference.

The Mean Velocity Pro�le in a Turbulent Jet

The evolution of the jet �ow at x > 5D~6D is thought to occur in a self-similar fashion because turbulent velocity �uctuations have had su¢ cienttime to re-arrange the initial �ow that further �ow evolution occurs with onlypre-existing turbulence as its precursor. If this is true, then it ought to bepossible to renormalize any equivalent pro�le by local dimensional variablesso that they collapse onto one curve. In the coordinate system de�ned inFigure 4.13, a velocity pro�le at x > 5D looks like Figure 4.14.Note howthe symbols and units are de�ned clearly, and self-consistently on both axes,


Figure 4.14: Standard velocity pro�le for a turbulent jet.

how y = 0 at the center, and how the data points are clearly shown. It is notnecessary to join up the dots with a line, since the data are not believed to becompletely free of measurement error and noise. In fact, it is not necessaryat this stage to make any attempt to �t the pro�le, and points alone are just�ne. In a more sophisticated investigation, a model predicting some kind ofGaussian �t might be predicted, in which case a curve of this functional formmight be �tted. For now, lacking any such detailed theory, the data pointscan be left alone.Figure 4.15 shows an example of two such U(y) velocity pro�les. The

x = 7D data look similar to the x = 6D data, except the maximum valueof U is lower, and the pro�le is a little broader. A moments thought showsthat conservation of momentum requires that the �rst be accompanied bythe second.

Normalizing the Pro�les

In principle, the pro�les of Figure 4.15 should be rescaleable provided suitablelength and velocity scales can be found. A convenient measure of the local

4.2 EXPERIMENT 75

Figure 4.15: U(y) for x = 6D (circles) and 7D (squares).

velocity magnitude is simply the local maximum value, Umax. Umax is foundat the jet centerline for a �ow with this symmetry. In fact the y = 0 pointcan be most accurately determined from Umax. Umax decays with increasingdownstream distance, x, and so the pro�le width increases with increasingx. The y coordinate might therefore be expected to rescale with x. But xis somewhat arbitrarily de�ned by the end of the jet nozzle, which may notreally correspond to the beginning of the jet. Perhaps one can make a moreprecise de�nition of the jet origin by extrapolating back from a measuredtrend in the jet. Conceptually, imagine continuing the dotted lines in Figure4.13, marking the edge of the jet, back to where they meet and the jethas zero width. Now, how can one de�ne the edge of the jet? When thepro�les have a shape such as Figure 4.14 and Figure 4.15, the edge is veryhard to de�ne precisely because U very gradually approaches zero, fallingeventually below the resolution of the measuring instrument. This cannotbe used as a criterion, because the resolution is �xed (in m= s) while theabsolute magnitude of the pro�le steadily decreases with increasing x. It isbetter, rather, to take some well-de�ned point, such as the so-called y

12 point,

where U = Umax=2, as shown in Figure 4.16.


Figure 4.16: De�ning a local measure of the jet width in U(y).

This de�nes a length scale in y, with units in cm, characterizing the localjet width at each downstream location in x. Now the y

12 length scale can

be plotted at each downstream location (in our experiment, there are three),and the straight-line extrapolation back to y

12 = 0 marks the point at which,

according to this measure, the jet has zero width.Note how the x coordinate here has been normalized by the jet diameter.

This is unessential, but at the very least, takes care of the unit conversionsin a tidy way. Here, the virtual origin, x0, is at negative values of x as shownin Figure 4.17, implying that the physical jet evolution began before the �owleaves the nozzle. You might expect x0 to be in the range [�10;+10 cm].Values outside this range should be treated with suspicion; they are takenfrom extrapolating only 3 points, after all.Having identi�ed the virtual origin, the x coordinate can be re-expressed

in terms of downstream distance from this point,

x� = x� x0:

Hence, at each downstream location (x = 6D, 7D, 8D), a value of x� canbe calculated. Now, each of the three pro�les at these locations can be

4.2 EXPERIMENT 77

Figure 4.17: Location of the virtual origin, x0.

renormalized so that U is divided by Umax and y is divided by x�. Thepredicted result of all this is shown in Figure 4.18. Note how the axes areclearly labelled, in symbols, as de�ned in the text. Also, both axes are nowdimensionless. U=Umax has a maximum value of 1, by de�nition, at y=x� = 0.Make sure the units of y and x� are the same before dividing. The �ow evolvesslowly in x, and characteristic x� values will be larger then y, so y=x� shouldbe a small number.

Finally. . .

This completes the procedure for making the normalized pro�les. They cannow be compared with each other, and with similar pro�les in the literature.Examples will be given in class. Now, see if you can explain why any of thismatters. If someone says "So what?", what do you say? What is the physicalsigni�cance?

4.2.4 Assignment: 3 Plots + Minitalk

The format for this week�s write-up is less formal than usual, and rests pri-marily on the correct plotting of the data from the experiment. These plots


Figure 4.18: Self-similarity in turbulent jet pro�les.

will then be used as the basis for a minitalk. The minitalk will be a 5 minuteinformal presentation given to a sta¤ member. The grade will be based on(i) the plots, (ii) the presentation, and (iii) a one paragraph abstract.

Plots

There are 3 pages of graphs that can be imagined to come from this ex-periment. The �rst is the response of the pitot tube to changes in angle ofincidence. The second would include raw velocity pro�les at all downstreamlocations, where mean �ow speed in m= s is plotted as a function of cross-stream distance in cm or m. Finally, the last page would have normalizedplots for the far-downstream pro�les.Make the plots carefully and think about each component, rather than

blindly accepting Excel (or equivalent) defaults. These will be the raw ma-terials for your minitalk next week. Make sure that di¤erent symbols can bereadily identi�ed, and that axes are labelled correctly, units are given, andso on. Unlike in a report, where graph headings are super�uous to the �gurecaption, there is no caption in a graph used for a talk (one does not stand

4.2 EXPERIMENT 79

there and say to the audience, �See Fig. 2�) and so graph headings can beuseful.

Abstract

Write a one-paragraph abstract describing what you did and what the resultswere. Include your name on the sheet of paper and staple it as a front pageto the graphs, which you will hand in after your talk.

Talk

You will have 5 minutes to give a one-on-one presentation of your results to asta¤member. The talk will be based only on the results. The raw materialsshould be only the recommended plots discussed above. Since time is short,it will be important to have a story line worked out in advance, and to stickto it when the time comes. After 5 minutes your talk will be interrupted.The sta¤ member may ask questions during or after the talk. Time will beadded on to account for questions.

Grading

The total grade will be based on the graphs, abstract and talk, with a per-centage breakdown of roughly 40 : 20 : 40. The key to success will be inmaking clear and correct plots based on the data. Then the talk can beclear, and so can the abstract.



selves side-by-side at their computers, then you must enforce an arti�cialseparation of your e¤orts. It will seem a bit unfriendly, and as though youare now competing against each other, but you are not, because an absolutenormalized scale is not imposed in the �nal grades. We just want to see what


each of you has learned, individually, to make sure that you have all learnedit.It is not so relevant to this lab in particular, but we will follow this general



Chapter 5

LabVIEW Programming

5.1 Lab 1: Basic of LabVIEW Programming

5.1.1 Acquire a Voltage from Channel 0

LabVIEW DAQ Setup

1. Launch LabVIEW 2009 ! Select Blank VI ! save this VI underE:/home/JStude/Your_own_folder and give a meaningful name(not Untitled 1.vi).

2. Right-click in the Block Diagram ! underMeasurement I/O se-lect DAQmx-Data Acquisition! select DAQ Assistant ! placein your Block Diagram.

3. Select Acquire Signals ! Analog Input ! Voltage ! select ai0! click Finish.

4. CheckTiming Settings!Acquisition Mode! selectN Samples! select OK.

5. On DAQ Assistant create control inputs for sampling rate andnumber of samples.

6. On DAQ Assistant create a Graph Indicator on the output ofdata.

81

82 CHAPTER 5 LABVIEW PROGRAMMING

7. On the Front Panel right-click the control for rate and number ofsamples and Replace with a Dial or Knob found under Express! place in proper location on the Front Panel.

Acquire Real-World Signal

1. ConnectWaveformGenerator to both theDMM andADC�sChan-nel 0 input AI0.

2. Set reasonable input signal onWaveform Generator. Click on RunContinuously button of LabVIEW Front Panel and observe theoutput graph. This is a sanity check to ensure your hardware workproperly.

3. Click on Abort Execution to stop the program.

Analyze the Real-World Signal

1. Right-click in the Block Diagram ! under Express select SignalAnalysis ! select Statistics ! place in your Block Diagram.

2. Under Statistical Calculations ! select Arithmetic mean andStandard deviation ! select OK.

3. Connect data output fromDAQ Assistant to Signals input on Sta-tistics and createNumeric Indicator on the outputs forArithmeticmean and Standard deviation.

4. On the Front Panel right-click the indicator for Arithmetic meanand Replace with aMeter found under Express ! place in properlocation on the Front Panel.

5. Repeat step 4 for the Standard deviation indicator.

6. Create a user friendly Front Panel as shown in the example in Figure5.1. Remember no good decision is left to default. Note: Don�t worryif you don�t know how to make as good cosmetics work as shown inFigure 5.1 in lab, but do practice at home to get familiar with it.

7. Don�t forget to save your �nal functional VI.

5.1 LAB 1: BASIC OF LABVIEW PROGRAMMING 83

Figure 5.1: Sample front panel of LabVIEW data acquisition.

Make Engineering Measurements

1. Click on theRun Continuously button and observe theFront Panel.

2. Make sysmatic measurements to get mean and standard deviation volt-ages from varying the waveform generator�s input parameters and chang-ing the ADC�s sampling methods.

3. Make a table in your notebook to show input parameters, ADC sam-pling parameters, and output voltages. Answer the following questions:

� What does DC voltage depends on? Do you always get steadyvalue or when does it �uctuate?

� What is the relationship between standard deviation voltages andAC voltages? What does AC voltage depends on? Do you alwaysget steady value or when does it �uctuate?


� What is your choice of input frequency? Why? What is yourchoice of sampling frequency? Why?

� What is the model number of our ADC card? What is its maxi-mum sampling rate? How about its resolution?

Write down all your answers in your notebook, demonstrate the opera-tional use of your VI and show the values of your measurements includinginterpretation of di¤erent input /output responses to a sta¤ and get a checkmark.

5.1.2 Construct a Digital Thermometer using LabVIEW

Construct an Algorithm for Digital Thermometer

1. Your goal is to make use of your knowledge from �Heat Transferand Thermocouples� lab to construct a digital thermometer usingLabVIEW. Since you already knew that thermocouples output emfis proportional to the temperature surrounding the thermocouplesbead, you may use two known temperature readings (convenient to use0 �C and 100 �C) to calibrate your digital thermometer disregard eitherthe sensitivity (or type) of thermocouples or the ampli�cation fromOP-Amp circuit. This step is similar to Static Calibration in yourthermocouples lab, the di¤erence is you need only two data points todo it. Therefore, this step is called �Two-point Calibration�.

2. Before you wire up any circuit or making any LabVIEW programmingyou should formulate an algorithm to convert thermocouple outputvoltages, eo, to temperature readings, T , in unit of �C: In short, you areconstructing a formula to describe T (eo). Do this on your notebook.First draw a diagram of expected eo(T ) vs T linear curve and thenmake formula to show T as a function of eo with two known constantvalues eo(0 �C) (voltage output measured at 0 �C) and eo(100 �C) (voltageoutput measured at 100 �C). Show your work to a sta¤ and get a checkmark.

Construct Thermocouples Setup with OP-Amp Ampli�cation

1. Follow the same steps as in �Heat Transfer and Thermocouples�lab to construct thermocouples setup with OP-Amp ampli�cation as


shown in Figure 3.14.

2. Test with DMM to see if your output voltages are in the reasonableranges when thermocouples bead submerged into 0 �C water and 100 �Cwater.

3. Keep monitoring with DMM. Add a new connection from the thermo-couples output to ADC�s Channel 0 input AI0.

Make a New VI to Acquire Thermocouples Output Voltage

1. Open a new Blank VI from LabVIEW 2009 ! save this VI underE:/home/JStude/Your_own_folder and give a meaningful name(not Untitled 2.vi).

2. Right-click in the Block Diagram ! underMeasurement I/O se-lect DAQmx-Data Acquisition! select DAQ Assistant ! placein your Block Diagram.

3. Select Acquire Signals ! Analog Input ! Voltage ! select ai0! click Finish.

4. CheckTiming Settings!Acquisition Mode! selectN Samples! select OK.

5. On DAQ Assistant create control inputs for sampling rate andnumber of samples.

6. On DAQ Assistant create a Numeric Indicator on the output ofdata.

7. Give proper input values to sampling rate and number of samples.

8. Click on theRun Continuously button and observe theFront Panel.

9. Measure two output voltages eo(0 �C) (voltage output measured at 0 �C)and eo(100 �C) (voltage output measured at 100 �C) then write down theirvalues on your notebook.

10. Click on Abort Execution to stop the program.


Implement the Digital Thermometer Algorithm into LabVIEW

1. Implement your algorithm into your previous LabVIEW VI. Nowyou may use the values of eo(0 �C) and eo(100 �C) as your Two-pointCalibration input constants and wire up DAQ Assistant�s outputas the input variable in your algorithm.

2. Construct your Front Panel as shown in Figure 5.2, which allows usersto enter two input constants eo(0 �C) and eo(100 �C) and make temperaturemeasurements from thermocouples then display the temperature read-ings on the Front Panel. Also, add a warning light to your FrontPanel to indicate danger situations (like fever when a human bodytemperature exceeds 37:5 �C).

3. Click on theRun Continuously button and observe the Front Panelto test your new device. Do the sanity check by measuring water tem-perature at 0 �C and 100 �C.

4. Do some cosmetic work to yourFront Panel, make sure all informationthat you want to show is clear, sensible, organized and pretty.


6. Make a table in your notebook to show following measurements:

� What is your ice water temperature?� What is your boiling water temperature?� What is our room temperature?

� What is your body temperature? Does your Warning light goon when temperature exceeds 37:5 �C?

Write down all your answers in your notebook, demonstrate your func-tional VI to a sta¤ and get a check mark.

5.1.3 Homework Assignment:

Part 1: LabVIEW DAQ Setup and Measurements

1. Make a table in Excel to show your lab measurements and make gooduse of that table to write a brief summary paragraphs that make sen-sible comments to answer the following questions:


Figure 5.2: Sample LabVIEW digital thermometer Front Panel.

� What does DC voltage depends on? Do you always get steadyvalue or when does it �uctuate?

� What is the relationship between standard deviation voltages andAC voltages? What does AC voltage depends on? Do you alwaysget steady value or when does it �uctuate?

� What is your choice of input frequency? Why? What is yourchoice of sampling frequency? Why?

� What is the model number of our ADC card? What is its maxi-mum sampling rate? How about its resolution?

Submit a hard copy of your work to lab next week.


Part 2: Digital Thermometer LabVIEW VI

At home, replace your LabVIEW DAQ with Dummy_DAQ_Assist.viprovided under LabVIEW Docs on Blackboard, making sure that yourdigital thermometer VI is still functional.. After you �nish your cosmeticwork submit your �nal version of digital thermometer LabVIEW VI to theAssignment Drop Box on Blackboard as well as a hard copy of yourFront Panel and Block Diagram to your lab next week. Don�t forget toinclude your name as part of the �le name. Again, make sure all informationthat you want to show is clear, sensible, organized and pretty. Thise¤ort should still be personal. So I expect your cosmetic work is di¤erentthan your lab partner�s or anyone else. Creativity is highly encouraged!

Part 3: Programming a Transducer Conversion Sub-VI

At the core of the automated jet program will be a part of a sequence thatsamples a voltage at one analog input channel, and then converts it to avelocity in units of m= s. This conversion problem can conveniently be iso-lated and solved in a small LabVIEW VI. The debugged and functionalVI can then be converted into a sub-VI, by de�ning the connections on aconnector panel.It is worth noting that the sub-VI can still be run as a standalone pro-

gram, with all its fancy Front Panel controls. This is a tremendous advan-tage of LabVIEW over any other programming language.The freestream velocity is directly related to the dynamic pressure, or the

di¤erence between total pressure, Pt, and static pressure, Ps,

u2 =2 (Pt � Ps)

�air; (5.1)

for a given air density, �air.The pressure transducer will give an output voltage that can be related

through the manufacturer�s calibrating curve to vertical inches of water. Themodel used in our lab is made by Honeywell and its model number isDRAL5(5 in H2O). You may search online to �nd detail speci�cations. Thispressure transducer has full range of �5 in H2O which outputs a voltage dif-ference of 1:0V to 6:0V from �5 in H2O to +5 in H2O, respectively. At zeropressure di¤erence, there is an o¤set voltage of 3:5V output from transducer.


Step 1: Calculate a Conversion Factor(s) to go from Volts to Cen-timeter of Water Let h be the vertical water height in cm. Then thepressure di¤erence is,

Pt � Ps = �wgh; (5.2)

where g is the acceleration due to Earth gravity, and �w is the density ofwater. Combining Eq.(5:1) and Eq.(5:2),

u2 =2�wgh

�air; (5.3)

so h is simply multiplied by a constant, whose value can be calculated for agiven set of environmental conditions. Use the following table:

Physical Quantity Value Units

�air 1:18 kg=m3

�w 998 kg=m3

g 9:81 m= s2

Note: STP at 20 �C.

Step 2: Write a VI to Convert Transducer Volts into Velocity Con-struct a LabVIEW VI to convert DRAL5 pressure transducer voltage into�ow velocity in m= s as shown in Figure 5.3.

Step 3: Convert LabVIEW VI into a Sub-VI

1. Right-click on the standard LabVIEW icon-rectangle in the top rightcorner and select Show Connector.

2. The icon panel changes to a grid pattern with sub-rectangles, or panes,for every input (Control) and every output (Indicator). By default,inputs are on the left, outputs are on the right. Youi have one of eachtype, and so there ought to be two panes in the connector. To beginwith, both panes will be clear, or white in color.

3. Note that the cursor changes to the wiring tool. Click on the left pane,move to the input control on either Front Panel or Block Diagram


Figure 5.3: Sample �owmeter.

and click again. The pane is �lled with a dark reddish brownish colorwhen a successful connection has been made. Do the same thing forthe right side pane, wiring it up to the output.

4. Right-click again on the connector pane and select Edit Icon to makea custom icon for your new sub-VI.

5. Again, make sure all information that you want to show is clear, sen-sible, organized and pretty. This e¤ort should still be personal. So Iexpect your cosmetic work is di¤erent than your lab partner�s or anyoneelse. Creativity is highly encouraged!


7. After you �nish your cosmetic work submit your �nal version of digital�owmeter LabVIEW VI to the Assignment Drop Box on Black-

5.2 LAB 2: FURTHER ADVENTURES IN LABVIEW 91

board as well as a hard copy of your Front Panel and Block Dia-gram to your lab next week



selves side-by-side at their computers, then you must enforce an arti�cialseparation of your e¤orts. It will seem a bit unfriendly, and as though youare now competing against each other, but you are not, because an absolutenormalized scale is not imposed in the �nal grades. We just want to see whateach of you has learned, individually, to make sure that you have all learnedit.It is not so relevant to this lab in particular, but we will follow this general



5.2 Lab 2: Further Adventures in LabVIEW

5.2.1 Stepper Motor Control and Simple Data Acqui-sition

This afternoon will be an exercise in LabVIEW programming that will againrequire full concentration and willingness to experiment and learn. The proofof success at each stage will be in a demo of a functioning program to a sta¤


member. Motors will run and lights will �ash, so get the checkmarks as youproceed.

Part 1: motor.vi: Basic Stepper Motor Control

The stepper motors in the 341 lab are controlled directly by mysterious blackboxes produced by Anaheim Automation. The function of these controllerboxes is to perform the elementary step control timing so that simple com-mand strings can be used to determine complete event sequences. This re-lieves the main computer CPU from the tedious and unimaginative businessof basically going [up..count..down..count..] many thousands of times (PWMsignals). The command strings are ASCII coded sequences, with short lettercodes, followed by numbers. The �rst example LabVIEW program, mo-tor.vi, generates these control codes and sends them to the motor controllerthrough the serial port.

1. Start up LabVIEW and open motor.vi [in User Libraries], it�sFront Panel is shown in Figure 5.4. Make sure the Anaheim controllerbox is turned on, and that the serial cable is connected. Run theprogram (in single-shot mode [Run], NOT (Never, ever) continuousloop [Run Continuously]), just as it is, with default values for theinputs. If the motor does not turn smoothly, alert a sta¤member. If itdoes, then �ddle with the front panel controls and see what happens.

2. Open up the Block Diagram of motor.vi. Take a deep breath.Note how the serial port output is done by a single VI (Serial PortWrite.vi), you may double-click the VI icon to view the details insideif you are curious about how it works.

3. Switch back to the Front Panel, and Run motor again. Examine theoutput string that is reported. This is the string that is sent to themotor controller. It actually does 8 things. Look up these 8 thingsin the Anaheim Program Reference Manual provided, and write themdown in the checklist table below. Copy the checklist table to your


Figure 5.4: A simple stepper motor controller.

notebook and get your �rst checkmark of the week from a sta¤member.

Command Substring Function

1

2

3

4

5

6

7

8


Part 2: Calibrated Stepper Motor Control with Kill Switch

When moving devices by stepper motor, there are several basic considerationsthat are common to almost all practical set-ups. We will consider two here.The �rst concern is in making simple conversions from number of steps in themotor to some physical variable, such as distance traveled (linear bearing), orangle swept (telescope). The second consideration is some kind of emergencycut-o¤ system when the device is reaching the end of its physical range.Failure to stop a device from powering through these limits could be verycostly. The objective of this part of the lab is to make a VI that moves adevice through a pre-determined distance, and that also stops when somecuto¤ condition is encountered.

1. Simply from the Front Panel of motor.vi, you can see that the motoroperation is controlled by 5 inputs �three slider bars, a numeric paneland one Boolean logic switch. This motor.vi can either be used as astand-alone program or be converted into a sub-VI. Your task is to usemotor.vi as a sub-VI (just like a subroutine or function in regular-programming) to construct your own custom VI that moves a device acertain distance (either forwards, or backwards) through stepper motorcontrol. If a kill switch is closed, then stop. The Front Panel maylook something like Figure 5.5. Note that we keep the axis selectionswitch on the Front Panel, even though it will always be set equal tozero for this lab (single axis).

2. First, think about the program logic a little. The kill switch valuewill be sampled at channel 0 of the LabVIEW DAQ. If the voltageexceeds some threshold value, then . . . well, do nothing, except maybelight up a warning light. Otherwise, move a certain number of steps bysending the appropriate control string to the motor.

3. Open up a new VI, and save it immediately in E:/home/JStude/.This makes subsequent save operations fast, which makes them morelikely to happen, which makes it less likely for big disasters will happen.

4. Start o¤ by laying out the main control function. The [if x [do this]else [do that]] kind of construction is handled quite elegantly by theCase Structure in LabVIEW. It works pretty much the same wayas the CASE/case statements in Fortran & C. Drag it into the Block


Figure 5.5: Motor control with kill switch.

Diagram. Here�s a simple example of how it works as in Figure 4. Notethat the expression on the left is evaluated. According to the result,which is wired into the [?] box, one of several frames is evaluated.Here, since 0 > 1 is false, the [do this] code is executed. The otherpossibility is shown in Figure 4.Here, since 2 > 1, so the contents of the[True] frame are executed. Switch between the two alternatives withthe little sideways arrows. Now you should be able to make a versionthat does what you need. It will depend on a voltage read operationfrom channel 0 of the LabVIEW DAQ.

5. Test that it works, by reading a simple voltage value (arrange this


yourself, using the equipment available in lab) and lighting up a lighton the front panel if it exceeds some threshold value. Note that therewill be two case possibilities [true/false], while there will probably beonly one button to light on the front panel. Which frame should thelight go on? Neither. Put it outside the frame, on the right. To connectvalues from inside the frame to outside now requires you to cross theboundary. This is ok. LabVIEWmakes a so-called "Tunnel" for you.You may also use True Constant or False Constant in Booleanoperator to control the warning light.

6. Now, in the appropriate frame, insert the motor.vi, and appropriatecontrols, including magic constants and simple arithmetic to convertcm into motor number of steps. An input in units of cm will beconverted to number of motor steps, most likely by multiplication bysome magic constant (What units would this constant have? How doyou �nd it?). For initial testing, set this constant equal to one, andwhen all the VI logic is correct, put in the correct value (after havingsuccessfully worked out how to �nd that correct value, of course).

7. Test your VI, and measure the traverse�s true moving distances attwo pre-determined distances. Write down your measurements in yournotebook.

8. Well, that seems to be that then. . . or is it? What�s wrong with thisdesign? How would you �x it? Demonstrate your functional VI as wellas your measurements and answers the above questions to a sta¤ andget a check mark.

Kill switch ________________ (check)


Forward/backward ___________ (check)Design weakness ___________________Possible solution ____________________

Part 3: Moving and Sampling: Principles of Automation

Now we wish to write a control program that combines stepper motor controlwith data acquisition, in preparation for Jets II lab �the automated versionof the turbulent jets experiment. The objective is to write aVI whose FrontPanel might look something like Figure 5.6.

Figure 5.6: Example Front Panel for combined data acquisition (graph onright) and motor control (top left box).

In Figure 5.6, the top left part is clearly just a re-arranged copy of theprevious part. The right part is a display of a waveform of some kind, andthe bottom left box controls the overall loop. The middle box displays thetotal distance moved, based on input values for distance/step and number ofiterations.


1. Use [Save as..] to make a new VI name, and work on this.

2. Think about the program control again. We want a VI that will movea device to a certain location, take some data, and then move on again,all the while checking the kill switch. Note that the operations move-sample-move-sample and so on... must be in a speci�c sequence (out-of-order data�ow control will not be convenient). The outer loop control,where the basic operation is repeated n times can most conveniently bedone with a For Loop. It is probably most convenient to leave this steptill last, but remember, after checking that step#5 in this list de�nitelypositively always works for sure, then you will want to enclose the wholelot in a giant for loop. By contrast, enforcing sequential operation ofthe move-sample operations requires a new Structure, the Sequence,which has no simple equivalent in programming languages that aresequential by default.

3. Enlarge the Block Diagram window, and the Case Structure in-side, so there is some room round the borders for new stu¤. Selecta Sequence Structure and lay it out so that it encloses the motorcontrol commands, inside the existing case frame. Right click on theSequence border and [Add frame after]. Operations that are put inthis second frame in the Sequence will execute only after the �rst asin Figure 5.7.

Figure 5.7: Sequence frames of a Structure in LabVIEW Block Dia-gram.

4. Use another standard LabVIEW DAQ to sample a time sequence tochannel 1. Put this VI in the second frame of the sequence. For now,


send signals from Waveform Generator to the ADC box. Use aWaveform Graph in the Front Panel to display the result.

5. We�re done! Or are we? Remember that the stepper motor controlworks by sending a command string out to the controller box, andthen immediately returning to the main program. The stepper motormay well still be moving when control returns and we move on to thenext program step. Consequently, a time delay must be insertedimmediately after the motor control commands i.e. in Frame 0 ofthe Sequence. Figure 5.8 shows a schematic of the required timing.Clearly the wait time depends on how long it takes the motors to run.Select and wire up the [Wait (ms)] function to do this. The waitperiod required is proportional to the number of steps and the steps/secsetting. Calculate the correct value (in ms) and use this to control thetimer. Add a 0:5 second increment for safety. Check the functioning ofyour program carefully at this point, before moving on.

Figure 5.8: Schematic of the required timing on di¤erent steps.

6. Demonstrate your new gizmo by sampling a sequence of di¤erent, recog-nizable functions from the function generator. You should be able to


move the motor n times to a new location, sample a voltage on channel1 and display the result. Collect the checkmark from a sta¤ member.

7. Finally, don�t forget to save your �nal version VI.

5.2.2 Homework Assignment:

At home, replace yourLabVIEWDAQwithDummy_KS.vi andDummy_PT.viprovided under LabVIEW Docs on Blackboard, making sure that yourMove-N-Sample VI is still functional.Now that your VI can perform move and sample tasks, it�s often useful

to record your samples of data to a spreadsheet. Make use of the Lab-VIEW Search andHelp features to �nd appropriate sub-VIs to build andsave your spreadsheet. Before running your VI, remember to choose properSampling rate andNumber of Samples. The output spreadsheet shouldrecord a table of the following data: Kill Switch On/O¤, Current Dis-tance Travelled in cm, Analyzed Results from Dummy_PT.vi. Setyour VI to move a total of 10 cm with 1 cm per step and Run your VI.After you �nish your cosmetic work submit your �nal version of move-n-

sample LabVIEW VI to the Assignment Drop Box on Blackboard aswell as a hard copy of your Front Panel, Block Diagram, and Spread-sheet to your lab next week. Don�t forget to include your name as part ofthe �le name. Again, make sure all information that you want to show isclear, sensible, organized and pretty. This e¤ort should still be personal.So I expect your cosmetic work is di¤erent than your lab partner�s or anyoneelse. Creativity is highly encouraged!



selves side-by-side at their computers, then you must enforce an arti�cialseparation of your e¤orts. It will seem a bit unfriendly, and as though you

5.3 LAB 3: AUTOMATED SAMPLINGOFTURBULENT JET FLOWS101

are now competing against each other, but you are not, because an absolutenormalized scale is not imposed in the �nal grades. We just want to see whateach of you has learned, individually, to make sure that you have all learnedit.It is not so relevant to this lab in particular, but we will follow this general



5.3 Lab 3: Automated Sampling of Turbu-lent Jet Flows

Now that we have accumulated some LabVIEW programming experience,it can be put to work by doing something useful. Next lab, the TurbulentJets experiment will be revisited, but with two twists: it is your own projectdesign and it is an automated version, with superior instrumentation thatyou know how to modify and calibrate.This afternoon you will be making the �nal modi�cations to yourMove&Sample.vi

from last week to adapt to your Turbulent Jets II lab next week. They in-clude:

1. Checking the distance calibration for traverse motion.

2. Writing and checking a standalone �owmeter that uses your pressuretransducer conversion VI to report �ow speeds in m= s.

3. Incorporating the Flowmeter intoMove&Sample.vi.

4. Include the Standard Deviation output fromFlowmeter.vi, to mea-sure �ow velocity �uctuations.

5. Record your samples of data to a spreadsheet and save a output �le.


The objective is to make velocity measurements in some kind of a turbu-lent jet experiment, and it is important to realize that all these programmingtricks and niceties are means to an end. The end result should be good qual-ity velocity pro�les that can be compared with each other, or with referencematerial in the research literature.

5.3.1 Part 1: Check Calibration Constants

1. Make use of your knowledge from previous LabVIEW labs to create anew standalone Traverse.vi as in Figure 5.9. Traverse.vi is a genericprogram to move a linear traverse through a calibrated distance.

Figure 5.9: Front panel for generic Traverse.vi

2. Note the kill switch voltage threshold for your traverse setup and setits value appropriately to ensure your Traverse.vi is fully functional.

3. Each setup needs to be calibrated, in both x and y directions. Calculatethe steps= cm constants, Cx and Cy, then use it to demonstrate motionalong each axis. Fill in the �rst table in the checklist below and copyit onto your notebook. Are the constants di¤erent for the di¤erent


directions? Take several measurements of each one, calculate the meanand standard deviation to answer this question.

4. After you �nish your new VI construction, run Traverse.vi to checkthe traverse motion in each direction, taking extreme care not to exceedtravel limits. Alert a sta¤ member if there are set-up problems.

5. Checkllist for your Traverse.vi,

Value �� Unit

Distance selected in x

Cx =

x Distance moved

Distance selected in y

Cy =

y Distance moved

Cx = Cy?

Conclusion

5.3.2 Part 2: Make a Standalone Flow Meter

1. Make use of your knowledge from previous LabVIEW labs to createa new standalone �ow meter, Flowmeter.vi, and test it qualitativelyin the jet �ow with the real instrument.

2. Display the converted result on a meter as shown in Figure 5.10.

3. Check that Flowmeter.vi works and use it to read the jet exit velocity,Uexit(m= s). Use this value to calculate a �ow Reynolds number.

4. Copy the checklist below for �ow meter onto your notebook and get a


Figure 5.10: Front Panel of a generic �ow meter.

check mark from a sta¤ member.

High velocity setting Low velocity setting

Value �� Unit Value �� Unit

UH UL

u0H u0L

ReH ReL

5.3.3 Part 3: Modifying and Testing your CustomizedJets.vi

Some simple modi�cations to your Move&Sample.vi program from lastweek can be made to run this experiment. Here�s the Front Panel view ofwhat a generic Jet.vi program might look like in Figure 5.11.

1. Open your existingVIs and use [Save as..] to start the modi�ed version.


Figure 5.11: Front Panel of a generic traverse-mover and pressure trans-ducer data-gatherer.

2. Add the voltage-velocity conversion sub-VI so that mean and standarddeviation are both converted to some velocity component in m= s. Thestandard deviation value will need to be adjusted by the resting voltageo¤set for the result to make sense in the conversion routine.

3. At each step through the pro�le (iteration round the For-Loop), com-pile an array that will probably have 3 elements: the y location, U andu0. Compile a cumulative 2D array by sending the 1D array valuesacross the right edge of the For-Loop. (Recall the general rule: inputson left, outputs on right. Send inputs to loops through the left margin,and take outputs from the right.)


4. Add a VI to output the data to a spreadsheet �le. Wire up the 2Dterminal to the data.

5. The Jet program should now be complete.

6. Checklist for Jet.vi

Traverse motion ____________(check)Kill switch ________________ (check)Data acquisition ____________ (check)File I/O __________________ (check)

5.3.4 Part 4: Obtain and Analyze U(y) Pro�les Acrossa Turbulent Jet

1. Design and implement an experimental investigation of some aspect ofthe turbulent jet.

2. Decide on the locations in x where you wish to make U(y) pro�les.Think ahead.

3. Use your versions of Traverse.vi and Flowmeter.vi to control themotion and check the �ow in real-time. Then make selected pro�lesusing Jet.vi.

4. Plot the pro�les in Excel as you get them, checking both mean andstandard deviation values. Name the output �les as

some_name_that_make_sense:xlsx

so it indicates the function of program, the creator of program, andExcel is the default format. Make sure the data is su¢ cient for yourparticular requirements.

5.3.5 Part 5: Prepare for the Talk

1. Your data should be quite good, certainly compared with manual mea-surements, and probably even better when compared with literaturedata. Make sure that, whatever your project, there is some compara-tive aspect of your work, either with existing literature or with previousdata, within your own experiment.

5.4 LAB 4: TURBULENT JETS II 107

2. Try to put this laboratory work in a broad context by comparison withother applications. Credit will be given to those who can locate andinclude appropriate information from speci�c real-world applications.This does not mean vague references to sewage out�ows at Hyperion,but should have some numerical details of a speci�c �ow.

3. Remember, the focus of your talk is on turbulent jets, and not Lab-VIEW programming. Include only those experimental details that arenecessary to specify the physical experiment and the data acquisitionstrategy. Do not show LabVIEW block diagrams in the talk.

4. De�ne your coordinate system and variable names explicitly and care-fully. Avoid all references to vague concepts such as �position� or�velocity�.

5. Each talk will be 10 minutes in length. The talk will be given on alab station or your laptop computer using PowerPoint which will beprojected onto a big screen in lab.

5.4 Lab 4: Turbulent Jets II

5.4.1 Additional Notes for Measuring Turbulent Jets

� Design and conduct an experiment involving �ow measurement in tur-bulent jets.

� Compare results with something: Jets I, literature values, . . .

� Evaluate and discuss in broad context, with speci�c example applica-tions.

As shown in Figure 5.12, the velocity components in fx; yg are fu; vg,and

u (x; y; t) = U (x; y) + u0 (x; y; t) ; (5.4)

where U is mean, time-averaged value of the �ow velocity, and u0 is the�uctuation of the �ow. If we can get multiple velocity measurements at thesame location, the mean �ow velocity can be calculated from the averagevalue of all measurements while �ow �uctuation can be calculated from thestandard deviation value of the same measurements.Possible topics:



� Compare with paper from S.C. Crow, P.H. Champagne (x=D = 0:025; 2; 4; 6; 8)

� Compare with paper from G. Xu, R.A. Antonia (various x=D including0)

� Low speed vs. high speed

� With, and without heat (correction of �)

� U and u0 vs. x=D at �xed y, or at cones of constant U=Umax

� Jet �ow, near, or around an obstacle.

� Thermocouple measurements.

� Twin turbulent jets

5.4.2 Special Lab Rules for Turbulent Jets II Lab

The basic philosophy behind the Turbulent Jets II lab is very di¤erent fromthe standard setup. Each group assumes the ENTIRE responsibility fordesigning, running and checking the experiment during the afternoon. Thereare no instructions to follow, except your own.

5.4 LAB 4: TURBULENT JETS II 109

Because the time is so unregulated, here are some simple rules that wewill abide by during this lab. They will help the lab run smoothly, will helpmake it fair for all and may prevent some kind of predictable catastrophe.

1. The lab will open at 1:00pm and will close down at 5:30pm. No excep-tions. There is no late penalty for showing up after 1:00pm; there isno penalty for leaving early, either. Do what you have to do to makethe experiment and check the results. There is no possibility to o¤eradditional lab hours to anyone if any group cannot �nish their dataacquisition during lab or lost their data at home. You must be done by5:30pm in your own lab day.

2. The purpose of the sta¤ is NOT to help you build your experimentor LabVIEW VI�s. They are on hand to provide assistance andadvice. Each group is completely responsible for making plans to getthe experiment up and running, and doing it.

3. You may wish to run your pitot tube long distances, say 20 cm, betweentaking pro�les composed of multiple short steps. Do not smash the killswitches by entering a large distance into your Jet.vi. It is quite usefulto have a program like Traverse.vi sitting around, so you can movethe apparatus independent of your main fancy program.

4. For similar reasons, it is quite handy to have Flowmeter.vi sittingaround. You may get an instantaneous �ow velocity reading of themeasured jet whenever you need it.

5. Do not run multiple LabVIEWVI�s simultaneously. Remember, theycompete for the same hardware, and results may be unpredictable.Similarly, you may run VBench, if you wish, but not the same timeas LabVIEW itself.

6. Check the results BEFORE leaving lab. Check them carefully. Norepeat experiments will be permitted!

7. At the end of lab, your workstations must be as you found it. with allcustom items removed and dismantled. The station should be readyfor the next day�s operations after you�ve done your clean-up work.

AME341b 2010 Lecture&Lab Notes Jerry

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