heat transfer lab

AER 423: Applied Thermodynamicsand Heat Transfer

Laboratory Manual

by

J. V. Lassaline

Ryerson UniversityDepartment of Aerospace Engineering

Copyright 2008

Copyright December 18, 2008 J. V. LassalinePermission is granted to copy, distribute and/or modify this document under theterms of the GNU Free Documentation License, Version 1.2 or any later versionpublished by the Free Software Foundation; with no Invariant Sections, noFront-Cover Texts, and no Back-Cover Texts. A copy of the license is included inthe section entitled GNU Free Documentation License

History1. Lassaline, J. V. 2008. AER 423: Applied Thermodynamics and Heat Transfer

Laboratory Manual. Ryerson University. Formatting changes, removal of fixedmark scheme, minor corrections.

2. Lassaline, J. V. 2008. AER 423: Applied Thermodynamics and Heat TransferLaboratory Manual. Ryerson University. Updated instructions, safety infor-mation. Source for this version available at http://www.ryerson.ca/~jvl/aer423.

3. Lassaline, J. V. 2005. AER 423: Applied Thermodynamics and Heat TransferLaboratory Manual. Ryerson University. Initial publication. Source for thisversion available at http://www.ryerson.ca/~jvl/aer423.

i

AcknowledgementsThis document is based upon the laboratory manuals produced for Ryerson Uni-versity courses MEC 309 Thermodynamics, MEC 514 Applied Thermodynamics andMEC 701 Heat Transfer. The author is indebted to the (alphabetically listed) au-thors R. Churaman, J. Dimitriu, J. Karpynczyk, D. Naylor, R. Pope, and J. C. Tysoefor their work on these previous manuals.

Contents

1 Instructions 11.1 Organization of This Book . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Reports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Common Mistakes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Air Nozzle 62.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.5 Calculations and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 92.6 Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Gas Turbine 133.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.3 Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.4 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.5 Calculations and Discussions . . . . . . . . . . . . . . . . . . . . . . . 153.6 Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4 Thermal Conductivity and Contact Resistance 194.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.3 Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.4 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.5 Calculations and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 214.6 Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5 Forced Convection From a Cylinder in Cross Flow 245.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.3 Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.4 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

iii

5.5 Calculations and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 265.6 Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

6 Numerical Simulation 296.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296.3 Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306.4 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316.5 Calculations And Discussion . . . . . . . . . . . . . . . . . . . . . . . 386.6 Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

A Errors and Corrections 41A.1 Error Estimation and Propagation . . . . . . . . . . . . . . . . . . . . 41A.2 Barometer Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

GNU Free Documentation License 441. APPLICABILITY AND DEFINITIONS . . . . . . . . . . . . . . . . 442. VERBATIM COPYING . . . . . . . . . . . . . . . . . . . . . . . . . . . 453. COPYING IN QUANTITY . . . . . . . . . . . . . . . . . . . . . . . . 454. MODIFICATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455. COMBINING DOCUMENTS . . . . . . . . . . . . . . . . . . . . . . . 466. COLLECTIONS OF DOCUMENTS . . . . . . . . . . . . . . . . . . . 477. AGGREGATION WITH INDEPENDENT WORKS . . . . . . . . 478. TRANSLATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479. TERMINATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4710. FUTURE REVISIONS OF THIS LICENSE . . . . . . . . . . . . . . 47ADDENDUM: How to use this License for your documents . . . . . . . 47

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List of Tables

4.1 Thermal Conductivity of Common Metals . . . . . . . . . . . . . . . 23

5.1 Constants of Eq. 5.1 from Hilpert (1933). . . . . . . . . . . . . . . . 25

A.1 Temperature correction for Hg and brass barometers in BG units.Corrections in [in]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

A.2 Temperature correction for Hg and brass barometers in SI units.Corrections in [mm]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

v

List of Figures

2.1 A convergent-divergent nozzle schematic . . . . . . . . . . . . . . . . 62.2 Nozzle test rig schematic . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.1 A gas turbine schematic with Brayton cycle approximation. . . . . . 133.2 Rover 1S/60 gas turbine airmeter calibration where x0 = 15.3

[in2

]is the airmeter effective area, ma is the air mass flow rate in [lbm/s],T1 is the air inlet temperature in [K], and pa is barometric pressurein [psi]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.1 One-dimensional heat conduction along a composite bar. . . . . . . 194.2 Schematic diagram of the Cussons Thermal Conductivity Apparatus. 20

5.1 Forced convection from a circular cylinder in cross flow. . . . . . . . 245.2 Low speed wind tunnel . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

6.1 Temperature distribution of a cooling cylinder as a function of ra-dius and time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

6.2 Discrete temperature distribution on region r [0, ro] at time tn . 31

vi

Nomenclature

The following nomenclature is used throughout this text with common S.I. unitsgiven if applicable.

Alphanumeric SymbolsA area,

[m2

]BSFC brake specific fuel consumption, [kg/W hr]D diameter, [m]h convection heat transfer coefficient,

[W/m2

]k ratio of specific heats, []k thermal conductivity, [W/mK]m mass flow rate, [kg/s]N rotational speed, [rpm]Nu Nusselt number, []p pressure,

[N/m2

]Pr Prandtl number, []q rate of heat transfer, [W ]Re Reynolds number, []T temperature, [K]t time, [s]U speed, [m/s]V speed, [m/s]V volume,

[m3

]W power, [W ]x first cartesian coordinate direction, [m]Greek Symbols efficiency, [] kinematic viscosity, [kg/ms] density,

[kg/m3

] temperature difference, [K]Subscripts0 relating to stagnation conditions relating to free stream conditionsD relating to cylinder diameters relating to solid surface conditionst relating to nozzle throat

vii

Chapter 1

Instructions

1.1 Organization of This BookThis book is divided into several sections, including instructions for writing labreports, the background and procedure for each lab experiment, and a set of ap-pendices. It is highly recommended that you review the guidelines for completingthe written lab reports prior to your first laboratory session. You are also expectedto have read and be familiar with each experiment before attending your scheduledlab. The appendices include valuable information regarding estimating the errorsassociated with your experimental observations and calculations. These skills areuseful not only for the laboratory component of this course, but for future experi-mental reporting as well.

1.2 Reports

You are reminded that all of the required course-specific writtenreports/assignments/labs will be assessed not only on their techni-cal/academic merit, but also on the communication skills exhibitedthrough them.

You should make note of the following requirements regarding the formal labora-tory reports.

Reports must demonstrate your understanding of the experiment and back-ground theory. A clear presentation of your observations and results is criti-cal. Anyone reading your report with a similar education to your own shouldbe able to reproduce your results using the same equipment.

Lab reports will normally be completed by small groups and must reflect thecontribution of each group member. It is up to you to ensure that every

1

Ryerson University Appl. Thermo. & Heat Transfer Winter 2009

group member contributes equally. In the event of an unresolvable conflict,students may only switch groups mid-term with prior instructor approval.

You must attend the laboratory session in order to receive credit for the labreport. Missed labs will require adequate proof presented to the departmentoffice. You are also expected to arrive promptly for your scheduled laboratorysession. Remember, if you are going to miss a laboratory session,test, or exam,alwayscontact your instructor immediately!

Reports must be typeset (e.g. prepared with a word processor.) Reports thatare handwritten will not be accepted, though some sections of the reportmay be handwritten as noted below. Reports should be formatted with 1margins, a 12pt font, and with 1.5 to 2 (i.e. double) spacing on standard 8.5by 11 paper. Reports must be at least stapled to form one cohesive report.No special binding is required however loose-paged documents will not beaccepted.

As a minimum, each lab report must contain:

A title page indicating the title of the experiment, the name(s), studentnumber(s), section and date the experiment was performed.

The body of the report consisting of the following sections:Objective Describe the purpose of this experiment in one brief para-

graph in your own words!Theory Concise discussion of the background theory governing this

experimentd in your own words!Apparatus Briefly list the equipment used. A simple diagram of the

equipment is advisable. You may reproduce diagrams from this manualbut you must cite the source!

Procedure Discuss the procedure used to complete the experiment.Describe the process taken during your experiment and not justa regurgitation of the lab manual. Using your lab report and theequipment described in the Apparatus section, anyone should beable to reproduce your results. Note any anomalies in your proce-dure relative to the instructions in this manual.

Observations Clearly indicate all values measured during this exper-iment including an estimate of the error. Use tables and/or graphswhen appropriate. Always indicate the errors (if any) present in yourmeasurements!

Results Based upon the formulae presented in your Theory section,present the results of the calculations outlined in the lab manual.Discuss your results and answer any discussion questions indicatedin this manual. Tabulate and/or graph your results as appropriate.Always calculate and indicate the errors present in your calculations asindicated in this manual! Methods for determining error propagationare shown in Appendix A.1.

2


Conclusions Provide a brief summary of your experiment and results.References Cite all references, including this lab manual using proper

citations. A preferred citation method is the author-date format(University of Chicago 2003), although numbered references areacceptable.For example, a citation to this lab manual using the author-datesystem would appear in the text as follows

as presented in the lab manual (Lassaline 2005). Blah blahblah

and the References section would contain the following:Coyote, W. E. 1976. Application of the ACME rocketbooster. Modern Rocketry. Los Angles: WB Press Ltd.Lassaline, J. V. 2005. AER 423: Applied Thermodynamics &Heat Transfer Laboratory Manual. Ryerson University.

Alternatively, numbered references should be presented in the textas follows

as presented in the lab manual [2]. Blah blah blahwith the References containing the corresponding enumerated listof sources.

[1] Coyote, W. E. Application of the ACME rocket booster.Modern Rocketry. Los Angles: WB Press Ltd. 1976.[2] Lassaline, J. V. AER 423: Applied Thermodynamics & HeatTransfer Laboratory Manual. Ryerson University. 2005.

An appendix which should contain the following section(s):Sample Calculations Demonstrate all the calculations necessary to

obtain your results. If one type of calculation is repeated manytimes only one sample is required using your experimental values.May be hand-written.

Graphs/Tables (optional) If you have a large number of graphs ortables in either your Observations or Results section, you may op-tionally place them in the appendix and refer to them by either pagenumber or label (e.g. Table A-1, Fig. A.2, etc.)

Equations should follow a clear nomenclature (e.g. density v.s. pressure p)and should be numbered at either the right or left hand margin. For example,

p

+

12V 2 + gz = const (1.1)

If your word processor is capable of writing equations clearly then use thisfeature, but it is acceptable to leave adequate space and add the equations byhand.

3


The technical writing of lab reports is expected to be of high quality andconcise. Excluding the title page, figures and tables of values, the main bodyof your report should not exceed four pages.

Graphs and tables must be clearly presented and must be labelled (e.g. Table1, Fig. 2) including an appropriate caption. Axes labels, a title, error bars (ifapplicable) and a legend (if appropriate) must be present. Computer gener-ated plots are preferred, but hand-drawn plots on graph paper are acceptable.

1.3 Common MistakesEvery year students will miss an opportunity to maximize their mark by makingneedless mistakes. Some hints as to how you can avoid making the same mistakesare as follows.

Show up for each and every lab on time. The penalties for missing a lab areoutlined by your instructor at the beginning of the year. The experimentsare set so that you may improve upon your understanding of what you havelearned from the lectures. Dont waste your time or, worse, the time of yourclassmates.

Answer all the discussion questions and perform all the requested calcula-tions as outlined in the lab manual. The calculations and discussion questionsare clearly listed for each experiment. Check that your lab report is completebefore you submit it.

Provide suitable references and make proper citations. There is some flex-ibility in how you present your references, but it is best to use a commonscientific citation style. If in doubt, use the same style as used for referencesin this lab manual. An excellent reference on accepted writing styles is TheChicago Manual of Style (University of Chicago 2003). Dont forget to ref-erence the source of your figures. A good rule of thumb is: if its not yours,cite the source!

Dont use footnotes for citations. Footnotes should only be used for addingextraneous information that would interfere with the flow of your text oroccasionally to reference an unusual source.1

Web sites are poor (and volatile) references. While the Internet may be usefulfor general information and handy diagrams, the information presented onmost Web sites is not peer reviewed as are text books, encyclopedia, journalpapers, or conference proceedings.

Check your grammar and spelling. Most word processors have at least a spell-check feature. Note that 30% of your lab report mark is based upon yourtechnical writing skills.

1For example, the definition of extraneous, as used in this context, is not forming an essential orvital part.

4


Dont plagiarize this manual verbatim in your lab report. For example, theprocedure you used during your experiment may differ from that outlined inthis manual. Use your own words and ideas. You are not given marks on howaccurately you can copy the text of this manual. If you wish to quote a sectionof this manual then provide a citation.

Feel free to look at previous years lab reports as a guide but do not plagiarize!Plagiarism is a violation of the Student Code of Conduct and will be dealtwith harshly.

Work together as a group. If different members of your group are responsiblefor different sections of the report, make sure that everyone is clear on theirrespective duties. It is your responsibility to ensure that your report is acohesive document and is completed on time.

If something is not clear, ask your instructor or instructor for clairification,but dont wait until the last moment!

5

Chapter 2

Air Nozzle

2.1 ObjectiveThe objective of this lab is to demonstrate the laws governing compressible flowin a convergent-divergent nozzle. This experiment will demonstrate the effect ofback pressure on the flow within the nozzle and the concept of choked flow. Inaddition, the presence of shock waves within the nozzle flow will be demonstrated.

2.2 TheoryConsider a nozzle formed from a convergent-divergent duct as in Fig. 2.1 drawingair from a large reservoir at fixed pressure p0 with back pressure pb downstream ofthe exit. For a rocket nozzle, the back pressure is the atmospheric pressure, which

throatreservoirp0

pepipb

back pressure

convergent section divergent section

Figure 2.1: A convergent-divergent nozzleschematic

decreases with altitude. For this exper-iment the back pressure is controlledby a valve downstream of the nozzleexit. As we will see in this experiment,the back pressure can be shown to in-fluence the mass flow rate through thenozzle.

If pb = p0 there would be no flowand thus the mass flow rate would bezero. If the back pressure is decreased,the fluid will accelerate from V 0 inthe reservoir up to a maximum velocityat the narrowest portion of the nozzle(throat). If the back pressure continuesto decrease, the throat velocity Vt and mass flow rate will continue to increase untilthe Mach number at the throat reaches sonic Mt = 1 and pressure pt = p. As thelocal Mach number cannot increase beyond unity in a converging section, reducingthe back pressure further will not increase the mass flow rate. At this point the flow

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Condenser

FilterP1

P2

P3

OrificePlate Flowmeter

Nozzle

Handwheel

Glass Bowl

Steam inAir in

V1

R1 V2 V3 V4

V6V5

V7

Inclined Manometer

Figure 2.2: Nozzle test rig schematic

is considered to be choked. Reducing pb further has no effect upon the conditionsin the nozzle upstream of the throat.

For choked flow, the theoretical mass flow rate can be determined from thereservoir (stagnation) pressure and throat pressures. If the expansion from thereservoir to the throat is isentropic then for choked flow (Mt = 1) the ratio ofthroat to stagnation pressure can be written as

ptp0

=p

p0=

(2

k + 1

) kk1

(2.1)

for calorically perfect air where the ratio of specific heats k = 1.4.1Using the isentropic property relations, we can also write the ratio of throat to

stagnation temperature as

TtT0

=(ptp0

) k1k

=2

k + 1(2.2)

To determine the theoretical mass flow rate m = VA we need to know theaverage fluid velocity, the cross-sectional area and the fluid density at the throat.Applying an energy rate balance and ignoring gravity, we can determine the throat

1Note that the symbol (dimensionless) may also be used in place of k which should not be confusedwith thermal conductivity k ([W/m K]).

7


air speed usingVt =

2cp(T0 Tt) (2.3)

where T0 is the reservoir temperature and Tt is the temperature at the throat. Thefluid density at the throat can be determined using the ideal gas law.

2.3 ApparatusThe nozzle apparatus, as illustrated in Fig. 2.2, can be operated with either air orsteam, however for this experiment only air will be studied. Compressed air is fedinto a steam chest and then passes through a polished convergent-divergent nozzlewith a throat diameter of 0.191 [in]. A search tube of 0.13 [in] diameter alignedwith the nozzle axis is mounted on a carrier such that it may be traversed along theaxis of the nozzle. A high-grade pressure gauge attached to the top of the searchtube registers the pressure at a small hole (pressure tap) drilled perpendicular tothe tube axis.

The location of the pressure tap may advanced in 0.1 [in] steps by rotating a set-ting wheel attached to the search tube carrier. At the upper limit, the pressure tapis clear of the nozzle and registers the pressure in the inlet chest. At the lower limitthe pressure tap registers the pressure downstream of the nozzle. The position ofthe pressure tap is indicated by a pointer and scale profile of the nozzle mountedon the apparatus.

The nozzle discharges into a vertical pipe of 2 [in] bore fitted with a valve forcontrolling the downstream back pressure. After passing through a steam con-denser, the flow vents to the room through a thin-plate orifice flow meter. Thethin-plate orifice flow meter offers a direct measurement of the volumetric flowrate as measured by the pressure drop across the sudden restriction introduced bya thin-plate orifice. The volumetric flow rate can be determined using Bernoullisequation and conservation of mass as

Q = (V A) = CA

2p

(1 4)(2.4)

where C = 0.65 is the orifice discharge coefficient to correct for losses within theflow meter, A is the area of the orifice, p is the pressure drop across the orifice, is the density of the fluid, and = d/D where d and D are equal to the orificeand pipe diameters respectively. For this flow meter the orifice has a diameter of1.065 [in] and is installed in a 3 [in] internal diameter pipe.

2.4 ProcedureFor this experiment, only compressed air will be used as the working fluid.

1. Ensure that steam valves labelled V3, V4, V5 and V6 in Fig. 2.2 are closed,and that air valve V1 is open.

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2. Open back pressure valve V7 completely.

3. Set the pressure probe traversing hand-wheel to position 1.

4. Open nozzle air inlet valve V2 fully and adjust regulator R1 to obtain a nozzleinlet pressure of 75 [psig] (or as high as possible if the supply line cannotprovide 75 [psig]) according to the large pressure gauge P2. Use the smallergauge P3 attached to the inlet chest to monitor inlet pressure during theexperiment.

5. When conditions are stable, advance the pressure probe through nozzle posi-tions 5 to 27, recording the probe pressure with the large gauge and the pres-sure drop across the thin-plate orifice flow meter with the inclined manome-ter, at each location. At the end, return the probe to position 1 to verify theinlet pressure using the large pressure gauge.

6. Set the pressure probe to position 28 to measure the back pressure.

7. Close the back pressure valve V7 to increase the back pressure and repeatthis experiment for back pressure readings of 10 [psig], 20 [psig], 30 [psig],40 [psig], and 60 [psig].

8. Shut down experiment by closing air valve V2.

2.5 Calculations and Discussion Plot the pressure profile for all back pressures on one graph, using the ab-

solute pressure ratio ( pp0 ) versus probe position. Include a legend to identifyeach curve.

Calculate the average manometer reading for each back pressure. If the manome-ter scale was not adjusted to zero for no flow, correct the readings with re-spect to any initial offset.

Calculate and tabulate the measured mass flow rate for each back pressurecase as measured using the thin-plate orifice flow meter. Fluid density can bedetermined from ambient conditions using the ideal gas law.

For the zero back pressure case only, calculate the theoretical mass flow rate(using the reservoir pressure and temperature) assuming choked flow. Com-pare the theoretical mass flow rate to the measured mass flow rate. Note thatthe cross-sectional area of the throat is an annulus due to the presence of thecylindrical probe aligned with the nozzle axis. The area at the throat can bewritten as

At =

4(d2t d2p) (2.5)

where dt and dp are the diameters of the nozzle throat and probe, respectively.

9


Plot the mass flow rate versus the ratio of absolute back pressure to absolutereservoir pressure ( pbp0 ), and determine the pressure ratio at which the nozzleswitches from choked to non-choked flow.

Discuss the reasons for the various shapes of the pressure profiles. Whichparts of the profiles correspond to subsonic versus supersonic flow? Locateany shocks present in the divergent section and give their approximate loca-tions by probe position. Are the shocks sharply defined or spread out?

Complete one error propagation estimation for the measured flow mass flowrate of the zero back pressure case. Does the error in your measurementsaccount for any differences between the theoretical and measured mass flowrate?

10


2.6 Experimental Data

Atmospheric pressure (pa):

Ambient temperature (Ta):

Inlet temperature (T0):

Manometer at zero flow:

Approximate error in pressuremeasurements (p):

Approximate error in manometermeasurements (h):

11


BackPres.

0 [psig] 10 [psig] 20 [psig] 30 [psig] 40 [psig] 60 [psig]

ProbePos.

Pres.[psig]

Man.[mm

H2O]

Pres.[psig]

Man.[mm

H2O]

Pres.[psig]

Man.[mm

H2O]

Pres.[psig]

Man.[mm

H2O]

Pres.[psig]

Man.[mm

H2O]

Pres.[psig]

Man.[mm

H2O]

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

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Chapter 3

Gas Turbine

3.1 ObjectiveThe purpose of this lab is to carry out a full-load test on the Rover 1S/60 gas turbineengine at steady state. The developed power, fuel consumption rate, and overallefficiency will be determined from observed test data.

3.2 TheoryA gas turbine engine, whether used to produce thrust or shaft work,

Turbine

Combustor2 3

Compressor

4

1

Gas Turbine Engine

Air inProducts

out

Fuel in

optionalwork out

Turbine

2 3

Compressor

41

Brayton Cycle

Heat in

optionalwork out

HeatExchanger

Heat out

HeatExchanger

Figure 3.1: A gas turbine schematic with Braytoncycle approximation.

may be modeled as a thermodynamiccycle (Brayton Cycle) if an air stan-dard analysis is applied. In the Bray-ton Cycle, air as an ideal gas is usedas the working fluid instead of air, fueland combustion products. In an actualgas turbine engine, energy is extractedfrom the hot combustion products bythe turbine to produce work. A frac-tion of this work is used to drive thecompressor, while optionally generat-ing shaft work for power generation.As illustrated in Fig. 3.1, using an airstandard analysis, the combustor is re-placed by an equivalent heat additionat constant pressure. To satisfy the sec-ond law of thermodynamics and forma closed loop, an additional heat rejec-tion to the atmosphere is used to con-nect the turbine exit and compressorinlet states. This can be thought of as

13


the air exiting the turbine and cooling off in the atmosphere before eventually re-turning to the compressor inlet.

For a gas turbine engine used to generate power, we can measure the power de-veloped at the output shaft, more commonly referred to as brake horsepower, byapplying a load using a dynamometer. In this experiment a hydraulic dynamometeris used, with the turbine shaft turning a rotor inside a stator through which waterflows. This mechanism is similar to the torque converter in an automatic transmis-sion. The power developed by the engine is dissipated as heat generation in thewater via viscous dissipation. Although not directly connected to the stator, therotor applies a torque on the stator by rotating the water past the vanes of the sta-tor. The stator casing is mounted on trunnions and is prevented from rotating bya spring scale. When a balancing torque is applied to the stator casing at a fixedRPM, the power developed by the engine equals the dynamometer loading torquetimes the dynamometer RPM.

3.3 ApparatusThis experiment will use the Rover 1S/60 gas turbine engine, which contains a sin-gle stage centrifugal compressor, a reverse flow combustion chamber, and a singlestage axial turbine to produce a maximum of 60 [hp] with a maximum governedspeed of 46000 [rpm]. The engine is loaded using a Heenan and Froude DPX2 hy-draulic dynamometer. Note that the maximum reading of the dynamometer springscale is 50 [lbf ], and for this experiment an additional 50 [lbf ] has been added tothe dynamometer torque arm. A calibrated airmeter is attached at the compressorinlet for determining the air mass flow rate. The Rover gas turbine engine uses amulti-piston fuel pump with an automatic control. The fuel flow rate is measuredusing a rotameter mounted on the central panel. For this experiment a premiumdiesel fuel will be used.

3.4 ProcedureThis experiment will be run with the assistance of a member of the technical staffwho will prepare and operate the engine.

1. Familiarize yourself with the various components of the engine and dynamome-ter.

2. Record the atmospheric temperature and pressure. Record the type, heatingvalue and specific gravity of the fuel used.

3. Complete the engine start sequence.

4. Load the engine by admitting water to the dynamometer. Balance the torqueusing the load balancing handwheel. Increase loading until a maximum tur-bine exit temperature of 1100 [F ] is reached.

14


5. Verify the engine is operating at steady state by observing the engine instru-mentation, including all temperature, pressures and flow rates.

6. Record the indicated dynamometer load including the additional 50 [lbf ] ofextra loading. Record the dynamometer speed.

7. From the central instrument panel, record the air flow meter pressure drop,compressor to turbine pressure drop, turbine exit static pressure, and com-pressor inlet and exit pressures.

8. Record the compressor inlet and exit temperatures and turbine exit temper-ature.

9. Record the fuel pressure and flow rate. In addition, record the temperatureand pressure of the engine oil and dynamometer water.

10. Complete the engine shutdown sequence. Record the time for the engine torun down.

3.5 Calculations and Discussions Determine the brake horsepower developed (Wb) from the dynamometer

load (W ) and dynamometer RPM (N ). Note that for this dynamometer, tocalculate the power in [hp], the following formula applies

Wb =WN

4500(3.1)

where the load W is in [lbf ] and speed N is in [rpm].

Calculate the mass flow rate of fuel mf in [lbm/min]. Note the specific grav-ity of the fuel and that one imperial gallon of water has a mass of 10.02 [lbm].The rotameter measures the volumetric flow rate!

Using the heating value and mass flow rate of the fuel, determine the totalenergy added by the fuel Qin in [Btu/min].

For an equivalent Brayton cycle producing a net rate of work Wb for a givenrate of heat transfer input Qin, calculate the thermal efficiency of this cycle.Note that 1 [hp] = 42.41 [Btu/min].

Calculate the brake specific fuel consumption BSFC in [lbm/hp hr]

BSFC =mf

Wb(3.2)

Calculate the mass flow rate of air ma in [lbm/s] using the calibration chartin Fig. 3.2. Note that the air meter pressure drop is measured in inches of

15


oil with a specific gravity of 0.827. The quantity p/pa is a dimensionlessquantity. You may use the following conversions and values

1 [inH2O ] = 0.0362 [psi]1 [inHg ] = 0.491 [psi]

x0 = 15.3[in2

] Calculate the air/fuel ratio A/F .

Calculate the power developed by the turbine in [hp] using an air standardanalysis. Assume that the compressor is an isentropic device. Note that afraction of the power produced by the turbine is used to drive the compres-sor and an additional 5 [hp] is lost to friction and accessories. Therefore theturbine power can be written as

Wt = Wc + Wb + Wf (3.3)

where Wc is the compressor power and Wf is the power lost to friction andaccessories. Hint: To calculate the compressor power requirementapply a mass and energy balance.

Calculate the turbine inlet temperature using an air standard analysis. Donot assume the turbine is an isentropic device. Hint: Apply a mass andenergy balance to the turbine.

Calculate the back work ratio bwr.

Calculate the thermal efficiency of the gas turbine engine assuming an coldair standard analysis (k = 1.4) with the compressor pressure ratio r = p2/p1.Hint: Pressures must be absolute!.

t = 1 r1kk (3.4)

How does this compare to the previous thermal efficiency calculation?

16



Fuel:

HHV of fuel:

Specific gravity of fuel:

Atmospheric pressure:

Atmospheric temperature:

Dynamometer load [lbf ]:

Dynamometer speed [rpm]:

Fuel flow rate [I.Gal./h]:

Airmeter pressure drop p [in of oil]:

Compressor to turbine pressure drop [inHg ]:

Compressor inlet (impeller tip) pressure p1 [psi]:

Compressor exit (comp. delivery) pressure p2 [psi]:

Compressor inlet temperature T1 [F ]:

Compressor exit (comp. delivery) temperature T2 [F ]:

Turbine exit pressure [in of oil]:

Turbine exit temperature T4 [F ]:

Dynamometer pressure [psi]:

Dynamometer temperature [F ]:

Oil pressure [psi]:

Oil temperature [F ]:

Fuel pressure [psi]:

Run down time [s]:

17


0.08 0.085 0.09 0.095 0.1 0.105 0.11 0.115 0.12 0.125 0.130.01

0.012

0.014

0.016

0.018

0.02

0.022

0.024

0.026

ma

T1x0pa

p

pa

Figure 3.2: Rover 1S/60 gas turbine airmeter calibration where x0 = 15.3[in2

]is

the airmeter effective area, ma is the air mass flow rate in [lbm/s], T1 is the air inlettemperature in [K], and pa is barometric pressure in [psi].

18

Chapter 4

Thermal Conductivity andContact Resistance1

4.1 ObjectiveThe objective of this lab is to measure the thermal conductivity of two materialsusing a heat conduction apparatus. In addition, the thermal contact resistance atthe interface between the two materials will be determined.

4.2 TheoryFouriers law of heat conduction states that the local heat flux q/A is proportional to

kBkA

T4T3T2T1

LA LB

q

TA T

B

x

T

Figure 4.1: One-dimensional heat con-duction along a composite bar.

the local temperature gradient. Fouriers lawcan be written in one dimension as

q

A= kdT

dx(4.1)

where the constant of proportionality is thethermal conductivity k, and area A is the cross-sectional area at location x. For steady-state,one-dimensional conduction it can be shownthat the heat flux at any location x is constant,and thus the temperature distribution, T (x),through the conducting media must be a lin-ear function of x. If we know temperatures T1and T2 at two locations spaced some distance Lapart, then the rate of heat transfer by conduc-tion in one-dimension can be written as

q = kAT1 T2

L. (4.2)

1This chapter is based upon the work of D. Naylor (Naylor 2001).

19


Fig. 4.1 illustrates one-dimensional conduction through a sequence of two dif-ferent materials of constant cross-sectional area. As the rate of heat transfer throughboth materials must be the same, if the thermal conductivity of each material isdifferent, then the gradient of the temperature distribution in each material mustbe different. It is important to note that there may be a substantial temperaturedrop at the interface of the two materials due to imperfections in the mating of thetwo surfaces. Heat transfer across the interface occurs by conduction at the con-tact points, by radiation between the surfaces, and by convection and conductionacross any interstitial fluid if present. For perfectly smooth surfaces there would beno thermal contact resistance and no discontinuity in temperature at the interface.

Al

S.S.

ElectricalHeater

T4

T3

T2

T1

5025

25

7

76.5

6.5

+-

Tin Tout

All dimensionsin [mm]

CoolingWater

Figure 4.2: Schematicdiagram of the CussonsThermal ConductivityApparatus.

The thermal resistance to heat flow at the interface, or ther-mal contact resistance, may be defined as

Rt,c =TA TB

q(4.3)

based upon the temperature jump across the interface and theheat transfer rate through the interface. Contact resistance mayalso be defined in terms of an equivalent heat transfer coefficienthi using Newtons law of cooling as

q = hiA(TA TB) (4.4)

which can be rearranged to solve for hi.

4.3 ApparatusMeasurements will be made using the Cussons Thermal Con-ductivity Apparatus. A schematic diagram of the Cussons Ap-paratus is shown in Fig. 4.2. For illustration purposes two testspecimens (25 [mm] diameter bars) are shown clamped into theapparatus. The specimens are heated electrically at the upperend and cooled at the lower end by a flow of water. To reduceheat losses to the surroundings, an insulation jacket (not shown)surrounds the specimens.

The conduction heat transfer rate is calculated by measuringmass flow rate and temperature rise of the cooling water sup-plied to the lower end of the bar. Using an energy rate balancewith constant specific heats the heat transfer rate to the water can be written as

q = mcp(Tout Tin) (4.5)

The inlet and outlet temperatures of the cooling water are measured using glassthermometers. The mass flow rate is measured using a stop watch, bucket andweigh scale. Steady-state temperature measurements are made at the four locationsshown in Fig. 4.2 using type-K thermocouples. Using these temperature and heattransfer rate measurements, the thermal conductivity of each sample (kA, kB) andthe interfacial contact coefficient (hi) can be calculated using equations presentedabove.

20


4.4 Procedure1. If not already assembled, place the aluminum and stainless steel specimens

together in the apparatus as illustrated in Fig. 4.2. Check that all four ther-mocouples are in place and cover the apparatus with the insulating jacket.

2. Turn on the cooling water supply and switch the heater to full power. Waitfor the hot end (T4) to reaches approximately 200 [C], then reduce the heatercurrent to about 0.35 [amps].

3. Allow the apparatus come to steady-state. Monitor the specimen and coolingwater temperatures to confirm that steady conditions have been achieved.

4. Once at steady-state, time the collection of the cooling water in the containerprovided. Record the specimen temperatures, and the cooling water temper-atures every two minutes. At the fourth reading, once 6 minutes has elapsed,stop collecting the cooling water. Weigh the amount of water collected.

4.5 Calculations and Discussion Using appropriately averaged cooling water temperature readings, calculate

the heat transfer rate through the composite bar. Plot the temperature distri-bution (temperature versus location) within the aluminum and stainless steelbars on one plot. Assuming a linear temperature distribution for both ma-terials, extrapolate the interfacial temperature difference. Using this result,calculate the interfacial thermal resistance Rt,c and heat transfer coefficienthi.

Calculate the thermal conductivity of both samples (kA, kB) and compare topublished values for various stainless steel and aluminum alloys. A table ofvalues appears in the appendix of the course textbook and in Table 4.1. De-termine the closest material based upon the measured thermal conductivity.If differences exist, be sure to discuss the most likely reasons in your report.

The two specimens are clamped together in the apparatus. If the two speci-mens had simply been stacked in the apparatus, what effect might this havehad on the experiment results?

Suggest two methods for decreasing the thermal contact resistance betweenthe two specimens.

21


4.6 Experimental DataTime 0 [min] 2 [min] 4 [min] 6 [min]

Water Tin

Water Tout

T1

T2

T3

T4

Mass of water collected:

Collection time:

22


Material Thermal Conductivity k [W/mK]100 [K] 200 [K] 400 [K] 600 [K]

Aluminum pure 302 237 240 231Aluminum alloy 2024-T6 65 163 186 186Boron 190 55.5 16.8 10.6Cadmium 203 99.3 94.7Chromium 159 111 90.9 80.7Cobalt 167 122 85.4 67.4Copper pure 482 413 393 379

Bronze (commercial) 42 52 59Brass (cartridge) 75 95 137 149

Constantan 17 19Gold 327 323 311 298Iron pure 134 94 69.5 54.7

Armco (99.75% pure) 95.6 80.6 65.7 53.1Cast iron (plain) 54 47.2Cast iron (alloy) 48.7 45.5

Carbon SteelsPlain carbon 56.7 48Carbon-Silicon 49.8 44Carbon-Mn-Si 42.2 39.7

Chromium (low) Steels0.18% C, 0.65% Cr, 0.23% Mo, 0.6% Si 38.2 36.70.16% C, 1% Cr, 0.54% Mo, 0.39% Si 42 39.10.2% C, 1.02% Cr, 0.15% V 46.8 42.1

Stainless SteelsAISI 302 17.3 20AISI 304 9.2 12.6 16.6 19.8AISI 316 15.2 18.3AISI 347 15.8 18.9

Lead 39.7 36.7 34 31.4Magnesium 169 159 153 149Molybdenum 179 143 134 126Nickel pure 164 107 80.2 65.6

Nichrome 14 16Inconel 8.7 10.3 13.5 17

Platinum pure 77.5 72.6 71.8 73.2Platinum alloy 60Pt-40Rh 100 125 136 141Rhodium 186 154 146 136Silicon 884 264 98.9 61.9Silver 444 430 425 412Tin 85.2 73.3 62.2Titanium 30.5 24.5 20.4 19.4Tungsten 208 186 159 137Uranium 21.5 25.1 29.6 34Zinc 117 118 111 103Zirconium 33.2 25.2 21.6 20.7

Table 4.1: Thermal Conductivity of Common Metals

23

Chapter 5

Forced Convection From aCylinder in Cross Flow1

5.1 ObjectiveThe objective of this lab is to determine the convective heat transfer rate from acircular cylinder in a cross flow of air. Experimental measurements will be madeusing a lumped capacitance transient cooling technique. Using the experimentaldata, an empirical correlation will be derived and compared to published results.

5.2 Theory

uniformflow

U, T

D

Ts

turbulentwake

Figure 5.1: Forced convectionfrom a circular cylinder in crossflow.

Forced convection from a circular cylinder in a crossflow of fluid is encountered in a wide range of thermo-dynamics and fluids engineering applications. As shownin Fig. 5.1, fluid with free stream velocity U and freestream temperature T flows normal to the axis of thecylinder. The cylinder has diameter D and uniform sur-face temperature Ts. Some theoretical solutions to thisproblem have been obtained for low Reynolds num-ber. However, at moderate and high Reynolds num-ber, an unsteady turbulent wake forms behind the cylin-der making mathematical solution of the governingequations extremely difficult. Hence, in the range ofmost practical applications, heat transfer correlationsare based on experimental measurements.

For external forced convection problems, experimental heat transfer data hasbeen found to fit a relationship of the following form

NuD = CRemD Pr1/3 (5.1)

1This chapter is based upon the work of D. Naylor (Naylor 2001).

24


ReD Range C m0.4 4 0.989 0.3304 40 0.911 0.38540 4, 000 0.683 0.466

4, 000 40, 000 0.193 0.61840, 000 400, 000 0.027 0.805

Table 5.1: Constants of Eq. 5.1 from Hilpert (1933).

where

NuD =hD

kaverage Nusselt number (5.2)

ReD =UD

cross flow Reynolds number (5.3)

Pr =cpk

fluid Prandtl number (5.4)

In Eq. 5.1, the constant C and exponent m are obtained from a best fit toexperimental data. The values of C and m for several Reynolds number ranges aregiven in Table 5.1 (from measurements by Hilpert (Hilpert 1933).) To partiallycorrect for property variations, all fluid properties should be evaluated at the filmtemperature Tf = (Ts + T)/2.

5.3 Apparatus

inletflowfan

test cylinder

traversingpitot tube

flow control gate

Figure 5.2: Low speed wind tunnel

The convective heat transfer ratewill be measured using the PlintCross Flow Apparatus. The pri-mary component of this appara-tus is a low speed wind tunnel. Asketch of the wind tunnel is shownin Fig. 5.2. The wind tunnel hasa 12.7 [cm] 12.7 [cm] cross sec-tion and is equipped with a gate onthe outlet of the fan to control theair flow. A traversing pitot tube isused to measure the air velocity inthe tunnel.

The test cylinder has a diam-eter of 1.242 [cm] and is instru-mented with a single thermocou-ple. The middle section of the cylinder is made of copper and has a length of9.5 [cm]. Cylindrical end pieces made of phenolic are attached to the copper cylin-

25


der to reduce axial conduction. An annular heater is used to heat the cylinder priorto each test.

5.4 Procedure1. Turn on the wind tunnel fan. With the flow control gate approximately 50%

open, traverse the pitot tube over the cross section of the wind tunnel tocheck the velocity profile. (In a properly designed wind tunnel, the velocityshould be uniform over most of the tunnel cross section.)

2. Check that the copper test cylinder is polished. Insert the copper cylinderinto the annular electric heater.

3. With the wind tunnel fan on, open the flow control gate fully. Once thecylinder temperature reaches 80 [C] to 85 [C] , quickly insert the cylinderinto the wind tunnel.

4. Using the data acquisition computer, immediately record the cylinder tem-perature at regular time intervals. Take frequent readings during the initialperiod of rapid cooling.

5. Record the pitot tube manometer deflection, the inlet air temperature, andthe barometric pressure.

6. Repeat steps 2 to 5 for several gate settings. Get at least five cooling curvesat different air velocities. Use the pitot tube to get even velocity incrementsbetween data sets.

7. Repeat one test, matching the test conditions as closely as possible. In yourreport, comment on the your experimental reproducibility.

5.5 Calculations and Discussion Using a lumped capacitance analysis, the temperature variation of the cylin-

der with time t can be shown to be:

i=

T TTi T

= e

hAsV cp

t (5.5)

where

Ti is the initial cylinder temperature at time t = 0, T is the ambient temperature, h is the average heat transfer coefficient, As is the surface area of the cylinder, is the density of the cylinder,

26


cp is the specific heat of the cylinder, and V is the volume of the cylinder

Taking the natural logarithm of Eq. 5.5 gives

ln i

= (

hAsV cp

)t (5.6)

For each set of cooling data plot ln(/i) versus time t and fit a straight linethrough each data. Referring to Eq. 5.6, use the slope of this best-fit line tocalculate the average heat transfer coefficient h for each air velocity. For eachset of data, calculate the air velocity from the pitot tube manometer reading.

Calculate the Reynolds number, Prandtl number and average Nusselt numberfor each data set. Evaluate the fluid properties at the average film tempera-ture, Tf = (Ts + T)/2.

Plot lnNuD versus lnReD (or logNuD versus logReD) . Fit a straight line tothe data and use the slope and y-axis intercept to calculate the constant Cand exponent m for your experimental data. On the same graph, plot thecorrelation of Hilpert (Hilpert 1933).

Plot the average heat transfer coefficient h versus the free stream air velocityU. How much does the convective heat transfer rate increase if the freestream air velocity is doubled?

Measurement errors caused by blockage effects are always present in an en-closed wind tunnel. The test model reduces the cross section for flow, caus-ing the air velocity near the model to be artificially high. Using your h versusU graph, estimate the approximate percentage error in h caused by windtunnel blockage.

27



Barometric pressure:

Test 1 Test 2 Test 3 Test 4 Test 5 Repeat TestAir

Temp.Man. Air

Temp.Man. Air

Temp.Man. Air

Temp.Man. Air

Temp.Man. Air

Temp.Man.

Time[s]

Temp.[C]

Time[s]

Temp.[C]

Time[s]

Temp.[C]

Time[s]

Temp.[C]

Time[s]

Temp.[C]

Time[s]

Temp.[C]

28

Chapter 6

Numerical Simulation of aCylinder in Cross Flow

6.1 ObjectiveThe objective of this lab is to numerically simulate the temperature distribution ofa cylinder cooling in a cross flow of air. Experimentally determined heat transfercoefficients will be used to model convection at the outer cylinder surface. Thetemperature of cylinder at the centerline as a function of time will be determinednumerically and compared to experimental results.

6.2 Theory

T

r ro0

t=0

t>0

t T

Ti

Air(h,T)

Cylinder(,cp,k)

q"

Figure 6.1: Temperature distribution ofa cooling cylinder as a function of radiusand time.

As seen in Ch. 5, the convection heat transfercoefficient, h, can be determined for a cylinderin cross flow for a variety of flow speeds. Thisconvection coefficient can be treated as the av-erage convection coefficient acting over the en-tire outer surface of the cylinder. In the case ofthe cylinder used in the previous experiment,the ends of the cylinder are insulated such thatconduction within the cylinder may be consid-ered to be one-dimensional. The thermocoupleused to measure the temperature of the cylin-der in the previous experiment is located at thecylinder centerline.

The heat conduction equation, written inits simplest form for one-dimensional conduc-

29


tion in cylindrical coordinates, is

1r

r

(rk

T

r

)= cp

T

t(6.1)

subject to the boundary conditions

T (0, t)r

= 0 (6.2)

kT (ro, t)r

= h(T (ro, t) T) (6.3)

where the former boundary condition is required for symmetry about the centerline(r = 0) and the latter satisfies the surface energy balance at r = ro. A solution to theheat conduction PDE, subject to the above boundary conditions and starting at theinitial conditions T (r, 0) = Ti, will yield the temperature distribution T (r, t)withinthe cylinder. We expect a solution that varies from the uniform initial temperatureat t = 0 to a uniform final temperature equal to the air temperature.

6.3 ApparatusFor this numerical experiment we will use a conventional computer and the soft-ware package Matlab, a high-level technical computing language and environment.To approximate a solution to the governing PDE we will use the Matlab functionpdepe.

The function pdepe can approximate a solution to a PDE of the form

rm

r

(rmf

(r, t, T,

T

r

))+ s

(r, t, T,

T

r

)= c

(r, t, T,

T

r

)T

t(6.4)

To solve Eq. 6.1 with pdepe requires

m = 1 (6.5)

f

(r, t, T,

T

r

)= k

T

r(6.6)

s

(r, t, T,

T

r

)= 0 (6.7)

c

(r, t, T,

T

r

)= cp (6.8)

The function pdepe expects boundary conditions to be presented in the form

p(r, t, T ) + q(r, t)f(r, t, T,

T

r

)= 0 (6.9)

which for r = 0 is satisfied with p = 0 and q = 1, and for r = rop(ro, t, T ) = h(T T) (6.10)

q(ro, t) = 1 (6.11)

30


T

r ro0 rj

Tj

solution at time tn

rj+1rj-1

Figure 6.2: Discrete temperature dis-tribution on region r [0, ro] at timetn

The function pdepe approximates a solution toT (r, t) at a finite number of locations on the regionr [0, ro] and for a finite number of instances overthe time span t [0, tf ]. The initial conditionsare specified as values for each point on the regionr [0, ro]. The function pdepe works by approxi-mating the PDE as an ordinary difference equation(OE). Spatial derivatives are approximated as fi-nite differences between neighbouring points in theregion r [0, ro]. For example, a first derivativemay be approximated at location rj as(

T

r

)j

Tj+1 Tj1rj+1 rj1

(6.12)

where the j + 1 and j 1 refer to the neighbour-ing values of the j-th location. Temporal derivativesmay be approximated in a similar way and after suit-able rearrangement allow the function to march thesolution from one time step tn to the next tn+1, beginning with the initial condi-tions at t = 0. Note that the solution to the OE is only an approximation tothe solution of the PDE, however with a suitable choice of parameters a highlyaccurate approximation can be constructed. As we will use pdepe simply as a nu-merical tool, the exact details of how pdepe functions are left to future courses.It will be necessary to provide the function pdepe with the details regarding theregion, time span, PDE parameters, boundary conditions, and initial conditions.

6.4 ProcedureA certain degree of computer proficiency and familiarity with Matlab orsimilar software is assumed. The following instructions should be suffi-cient to complete this experiment without prior Matlab experience, how-ever if you are having difficulty you should speak to your lab instructorduring your scheduled lab session. Matlab documentation may be accessedthrough the Help menu or by entering the commanddoc. Extra notes areprovided below as boxed paragraphs.

1. Start the Matlab application which will open, at least, the main Matlab win-dow containing the Command Window and menus. Within the CommandWindow, change directories to an empty local working directory.

Matlab commands and expressions can be entered in the CommandWindow interactively,or read from a script file.You may also defineyour own functions as M-files in the current working directory.In the Command Window, type doc matlab for an overviewof Matlab.

31


You can determine your working directory from the CommandWindow with the pwd command. You can change directorieswith the cd directory command. Type help cd for more in-formation.

The main Matlab window should appear similar to the following.

2. Add the cylsolve function to your working directory as a new M-file calledcylsolve.m.

32


You can create a new M-file from the menu (FileNewM-File) and edit this file using the Matlab Editor. Save this file ascylsolve.m in your working directory. Or, optionally you canedit an existing file with the editor.

The cylsolve.m file will take the following form, where you will need toreplace the values indicated as ??? with appropriate values. A copy of thisfile is also available on the course Blackboard Web site.

cylsolve.m1 function [Tc]=cylsolve(t,Ti,Tinf,h,rho,cp,k,D)2 % Determine temperature at r=0 for time span vector3 % t for a cylinder in cross flow.4 % Inputs:5 % t = A vector of time [s] for solving T(t)6 % Ti = Initial cylinder uniform temperature [K]7 % or [C]8 % Tinf = Fluid temperature [K] or [C]9 % h = Convection cooling coefficient [W/m/K]

10 % rho = Density of cylinder [kg/m^3]11 % cp = Specific heat of cylinder [J/kg/K]12 % k = Thermal conductivity of cylinder [W/m/K]13 % D = Diameter of cylinder [m]14 % Returns:15 % T = Temperature at core T(t) [K]

33


16

17 % Generate a linear distribution of 100 points from18 % r=0 to r=r_o19 r=linspace(0,D/2,100);20

21 % Call pdepe to solve the PDE problem described by22 % the cylPDE, cylIC, cylBC functions. The solution23 % will be described by a matrix sol where the24 % solution at location r_j and time t_n can be found25 % at the entry sol(n,j).26 % For more information, type help pdepe or27 % doc pdepe.28 sol=pdepe(1,@cylPDE,@cylIC,@cylBC,r,t);29

30 % Return the centerline temperature as a function of31 % time span t (i.e. the first column of sol)32 Tc=sol(:,1);33

34 % Define the PDE sub-function cylPDE.35 % Note that all the above variables36 % (eg. Ti,Tinf,h,rho,cp,k, and D)37 % are defined for this function.38 % -------------------------------------------------39 function [c,f,s] = cylPDE(r,t,T,dTdr)40 % return c,f,s according to Eq. 6.4-6.8 as a41 % function of the given r,t,T,dTdr42 c = ???;43 f = ???;44 s = ???;45 % End of cylPDE46 end47

48 % Define the initial conditions sub function49 % cylIC.50 % Note that all the above variables51 % (eg. Ti,Tinf,h,rho,cp,k, and D)52 % are defined for this function.53 % -------------------------------------------------54 function [T0] = cylIC(r)55 [n,m]=size(r);56 T0=ones(n,m)*Ti;57 % End of cylIC58 end59

60

61 % Define the boundary conditions sub function

34


62 % cylBC.63 % Note that all the above variables64 % (eg. Ti,Tinf,h,rho,cp,k, and D)65 % are defined for this function.66 % -------------------------------------------------67 function [pl,ql,pr,qr] = cylBC(rl,Tl,rr,Tr,t)68 % Define the left boundary conditions at r=0,69 % according to Eq. 6.9, as a function of the given70 % left boundary values rl and Tl.71 pl = ???;72 ql = ???;73 % Define the right boundary conditions at r=r_o,74 % according to Eq. 6.9-6.11, as a function of the75 % given right boundary values rr and Tr.76 pr = ???;77 qr = ???;78 % End of cylBC79 end80

81 % End of cylsolve82 end

cylsolve.m

In Matlab, all text following the % character is ignored as a com-ment. The entry at the i-th row and j-th column of a ma-trix, for example sol, can be retrieved with the expressionsol(i,j). You can extract just the j-th column with the ex-pression sol(:,j), or just the i-th row with the commandsol(i,:).

The cylsolve function will return the temperature distribution at the cen-terline T (0, t) as a vector for the time span you provide as the vector t. If youtype help cylsolve, Matlab should return the first block of comments.Test your function before continuing by calling cylsolve with Ti = Tfor a small time span [0;0.5;1]. As the initial and freestream tempera-tures are equal, cylsolve should return the same temperature at each timeregardless of the other parameters. For example,

>> Tnum=cylsolve([0;0.5;1],10,10,1,1,1,1,1)

Tnum =

101010

35


3. Load your observations for the first set of data (time versus temperature)from the previous lab experiment. You will need two separate vectors: tcontaining the times, and T containing the temperatures. You can enter thisdata manually, or load the data from a text file.

(a) To enter the data manually, you will need to type in each entry. Forexample, if your time [s] versus temperature [C] data consisted of thepairs (0, 80.8), (10, 75.4), and (20, 67.7), then you could enter this dataas>> t=[0;10;20]

t =

01020

>> T=[80.8;75.4;67.7]

T =

80.800075.400067.7000

(b) Or, to read this data from a text file, for example from a comma-delimitedfile exported from a spreadsheet, use the command dlmread. If thecomma-delimited file data.csv in the working directory contains

0,80.810,75.420,67.7

then the command X=dlmread(data.csv,,) would load inthe data as a 3 2 matrix. This can be split into two column vectors asfollows.>> X=dlmread(data.csv,,)

X =

36


0 80.800010.0000 75.400020.0000 67.7000

>> t=X(:,1)

t =

01020

>> T=X(:,2)

T =

80.800075.400067.7000

4. Using your cylsolve function, generate the numerical solution Tnum usingyour time span vector t. Ensure that you pass appropriate values for thecylinder including initial temperature Ti, air free stream temperature Tinf,convection cooling coefficient (as determined in the previous experiment) h,cylinder density rho, cylinder specific heat cp, cylinder thermal conductivityk, and cylinder diameter D.

>> Tnum=cylsolve(t,T(1),)

The result will be a vector Tnum that is the same size as t and T, but repre-sents the numerical solution to the unsteady heat conduction problem.

5. Plot the experimental and numerical temperature (T and Tnum) versus time(t) on the same plot, ensuring that the two curves are labelled clearly. Printor save this plot for your lab report.

From the Figure window you may save the image in any formatusing FileSave As.

In Matlab, you can use the plot command to produce a two-dimensionalplot. For example, the following commands produce a labelled plot with alegend.

clf; % Clear current figureplot(t,T,*,t,Tnum,-); % Plot both data sets

37


xlabel(t [s]); % Label for x-axisylabel(T [C]); % Label for y-axislegend(Experimental,Numerical); % Add a legendtitle(h=??? [W/m/K]); % Add a title

6. Repeat steps 3 to 5 for each remaining data set from the previous experiment.You should have plots for each flow control gate setting. Ensure that eachplot is clearly labelled.

7. For the last data set, calculate the numerical solution using the same data, butfor a cylinder that is double in diameter. Plot the experimental versus numer-ical temperature versus time, and include this plot in your report. Ensure thatthis plot is clearly labelled.

8. Print or save a copy of your file cylsolve.m for inclusion in your report.

9. When finished, if working in a public lab, erase the contents of your workingdirectory.

6.5 Calculations And Discussion Include all plots produced using Matlab, clearly labelling each plot.

Include a print out of your function cylsolve.m.

In the previous experiment, we used a lumped capacitance method to modelthe cooling of the cylinder. Confirm that the lumped capacitance methodis valid for this experiment. The function pdepe returns a solution to thetemperature at many radial locations within the cylinder, not just the center-line. If the lumped capacitance method is valid, how should the numericalsolution at r = ro and r = 0 compare over time?

For the numerical simulation of a cylinder that is double in diameter, estimatethe final temperature using the lumped capacitance method of the previousexperiment. How do your numerical results compare to the lumped capac-itance method value? What is the expected effect of doubling the cylinderdiameter?

In the previous experiment we heated the cylinder in an annular heater untilthe centerline temperature reached approximately 80 [C]. For the numericalexperiment we assumed that the cylinder was at a uniform initial tempera-ture. What differences might exist between the numerical and actual initialconditions? Is this a significant source of error in our numerical solution?

6.6 Experimental DataVerify that you have the following files or print-outs:

38


Plots for each flow control setting (including the repeated test) comparingthe experimental to numerical values.

A plot of the last test data, but this time using a cylinder with a differentdiameter.

A copy of the source code for your cylsolve function.

39

Bibliography

CRC (1988). Handbook of Chemistry and Physics. CRC.Hilpert, R. (1933). Wrmeabgabe von geheizten drhten und rohren.Forsch. Geb.

Ingenieurwes. 4, 215224.Lassaline, J. V. (2005). AER 309: Thermodynamics Laboratory Manual. Ryerson

University.Naylor, D. (2001). MEC 701: Heat Transfer Laboratory Manual. Ryerson Univer-

sity.University of Chicago (2003). The Chicago Manual of Style. University of

Chicago.

40

Appendix A

Errors and Corrections

A.1 Error Estimation and PropagationWhen presenting measured values you must provide an estimate of the error. Forexample, if you are measuring temperature with a thermometer that is marked atevery 1 degree Celsius, your best measure of the current room temperature may be21.50.2[C]. In other words, to the best of your measuring ability the temperatureis 21.5[C] with an expected error of approximately 15 of a degree. As errors are atbest estimates, it is normal to truncate the error at the first non-zero digit (e.g.0.005 rather than 0.004925.)

When using digital equipment, the accuracy of the measure should be taken tobe 1/2 count of the last digit shown, unless otherwise noted. For example, if a digi-tal scale read 2.512[g], the error would be 0.0005[g]. Other sources of error, suchas a small breeze across the scale, may raise the error to 0.001. Use your bestjudgement and record the estimate of error with your measurements.

All reported observations should include an estimate of the error. All plotscontaining values with an estimate of error should include error bars. (If your soft-ware cannot include error bars in the figure, draw them in by hand.) When usingyour measured observations in a calculation you need to propagate this estimate oferror throughout your calculations. Your sample calculations should demonstratethe resulting error. Pay careful attention to the instructions for each exper-iment to determine when your report should include error propagationanalysis.

For this course we will use a simplified form of the proper statistical technique(which uses standard deviations.) If we have a function formed from a pair of inde-pendent (uncorrelated) measured values x x and y y , we can estimate theerror in the function using a few simple rules based upon the worst-case scenario.

For addition or subtraction of two values with errors, the error is cumulative.

(xx) + (y y) = (x+ y) (x +y) (A.1)(xx) (y y) = (x y) (x +y) (A.2)

41


For multiplication or division by an exact number, both the value and the error arescaled by the exact number.

2(xx) = 2x (2x) (A.3)

For multiplication of a pair of values with errors, the error is formed as follows

(xx)(y y) = xy (xy + yx +xy) xy (xy + yx) (A.4)

assuming that xy is much smaller than the other error products. For both mul-tiplication and division this can be reduced to an expression for the relative error

xy|xy|

=x|x|

+y|y|

(A.5)

x/y|x/y|

=x|x|

+y|y|

(A.6)

For products of powers functions such as xmyn, the relative error can be deter-mined using

xmyn|xmyn|

= |m|x|x|

+ |n|y|y|

(A.7)

The previous operations can be summarized as followsfunction errorx y x +yx x

xmyn |xmyn|(|m|x|x| + |n|

y|y|

)logx logx

For general functions that are combinations of the above, carefully determine theerror for each operation, following the normal order of operations. For example,to determine the error in z(x y) where z, x, and y all have associated errors onecan:

1. Calculate the error in the temporary value t1 = x y using the rule forsubtraction.

2. Calculate the error in the product zt1 using the rule for products.As an alternative, to determine a statistical estimation of the error propagation

for a general function F of measured values (x, y, . . .) with associated errors of(x,y, . . .) respectively, we can estimate the function error F using

F =

(F

xx

)2+(F

yy

)2+ . . . (A.8)

or in keeping with the simplified error analysis

F =Fx

x + Fyy + . . . (A.9)

Note that this may produce slightly different values than the previous methods butis acceptable given that we are at best providing an estimate of the errors.

42


A.2 Barometer CorrectionsCorrections for Hg barometers by temperature are listed below (CRC 1988) forboth SI and BG units.

Temp.[F ]

Observed height in [in]23 24 25 26 27 28 29 30 31

40 -0.024 -0.025 -0.026 -0.027 -0.028 -0.029 -0.030 -0.031 -0.03250 -0.045 -0.046 -0.048 -0.050 -0.052 -0.054 -0.056 -0.058 -0.06060 -0.065 -0.068 -0.071 -0.074 -0.077 -0.080 -0.082 -0.085 -0.08870 -0.086 -0.090 -0.094 -0.097 -0.101 -0.105 -0.109 -0.112 -0.11680 -0.107 -0.111 -0.116 -0.121 -0.125 -0.130 -0.135 -0.139 -0.14490 -0.127 -0.133 -0.138 -0.144 -0.150 -0.155 -0.161 -0.166 -0.172

100 -0.148 -0.154 -0.161 -0.167 -0.174 -0.180 -0.187 -0.193 -0.200

Table A.1: Temperature correction for Hg and brass barometers in BG units. Cor-rections in [in].

Temp.[C]

Observed height in [mm]620 640 660 680 700 720 740 760 780

5 -0.51 0.52 -0.54 -0.56 -0.57 -0.59 -0.60 -0.62 -0.6410 -1.01 -1.04 -1.08 -1.11 -1.14 -1.17 -1.21 -1.24 -1.2715 -1.52 -1.56 -1.61 -1.66 -1.71 -1.76 -1.81 -1.86 -1.9120 -2.02 -2.08 -2.15 -2.21 -2.28 -2.34 -2.41 -2.47 -2.5425 -2.52 -2.60 -2.68 -2.277 -2.85 -2.93 -3.01 -3.09 -3.1730 -3.02 -3.12 -3.22 -3.32 -3.41 -3.51 -3.61 -3.71 -3.8035 -3.52 -3.64 -3.75 -3.86 -3.98 -4.09 -4.21 -4.32 -4.43

Table A.2: Temperature correction for Hg and brass barometers in SI units. Cor-rections in [mm].

43

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