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Development of a Design Competition & External Flow Experiment for an
Introductory Fluid Mechanics Course
Mark RichardsKarthik Senthilnathan
A thesis submitted in partial fulfillment of the requirements for the degree of
BACHELOR OF APPLIED SCIENCE
Supervisor: Professor P.E. Sullivan
Department of Mechanical and Industrial Engineering University of Toronto
March, 2008
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Abstract
The importance of laboratory experiments and design projects in an engineer’s undergraduate education is a topic that has been debated and discussed since the beginning of formal education. This paper discusses the importance of laboratory experiments and design projects, their objectives, and the problems associated with their use in an engineer’s undergraduate education. This paper also presents the development and design of a laboratory experiment and design project to be implemented into an introductory fluid mechanics course at the University of Toronto. The laboratory experiment is an external flow investigation which analyzes the drag and lift forces developed on an airfoil at various angles of attack in a water tunnel apparatus. It is also used to help visualize boundary layer separation over various objects. The design project, based on the theories of internal fluid flow, is a competition requiring students to design a simple pipe network to meet specified flow rate requirements. Students will be required to assemble the network that they designed and to compare their theoretical results to the actual physical results. With approval of the proposed concepts in this paper, the experiment and design project are ready for construction, testing, and implementation.
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Acknowledgements This paper would not have been possible without the continued support from our supervisor, Dr. P.E. Sullivan. His proposed topic, ideas, motivation, and sound advice have helped us to work on a project that we feel makes a meaningful contribution to the Department of Mechanical and Industrial Engineering at the University of Toronto. We would also like to extend our gratitude to Mr. Len Roosman as he was more than happy to provide suggestions towards improving our designs. His assistance in preparing the Fluid Mechanics Laboratory equipment upon request is greatly appreciated. We are also indebted to our colleagues, Chris Roscoe and Paul Giampuzzi, for providing their creative input and directing us to useful resources that were used throughout this paper. Lastly, we wish to thank our families for their continued support throughout our through childhood, academia, this paper, and beyond. Without their love and support, we would not be in the position that we are in today. For this, we dedicate this thesis to them.
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Table of Contents List of Symbols ............................................................................................................................... v List of Figures ................................................................................................................................ vi List of Tables ............................................................................................................................... viii 1 Introduction ......................................................................................................................... 1 2 Literature Review................................................................................................................ 2
2.1 The Role of Laboratory Experiments in the Engineering Curriculum ............................. 2 2.1.1 Objectives ................................................................................................................ 3 2.1.2 Computer Simulations .............................................................................................. 3 2.1.3 Fundamental Objectives of Laboratories .................................................................. 4
2.2 Past Design Projects in Fluid Mechanics Courses ........................................................... 5 2.2.1 Wind Tunnel Testing of Buildings – University of Queensland .............................. 6 2.2.2 Student Experience ................................................................................................... 7
2.3 Advantages and Disadvantages of Experiments and Design Projects ............................. 8 3 Design of a Simple Piping Network ................................................................................. 10
3.1 Theory and Background Concepts ................................................................................. 10 3.1.1 Internal Flow ........................................................................................................... 10 3.1.2 Head Loss and the Friction Factor .......................................................................... 11 3.1.3 Laminar Fully Developed Flow .............................................................................. 11 3.1.4 Moody Chart ........................................................................................................... 12 3.1.5 Flow in Non-Circular Ducts .................................................................................... 13 3.1.6 Minor Losses in Pipe Systems ................................................................................ 13 3.1.7 Multiple Pipe Systems ............................................................................................ 14 3.1.8 Pipe Networks ......................................................................................................... 16
3.2 Project Description ......................................................................................................... 18 3.2.1 Details of the Design Project .................................................................................. 19 3.2.2 The Apparatus ......................................................................................................... 21 3.2.3 Sample Solution ...................................................................................................... 22 3.2.4 Errors Associated with the Project .......................................................................... 23
3.3 Improvements over the Existing Experiments ............................................................... 25 4 External Flow Investigation .............................................................................................. 27
4.1 Background Theory ........................................................................................................ 27 4.1.1 Airfoils .................................................................................................................... 27 4.1.2 How an Airfoil Induces Lift .................................................................................... 28 4.1.3 NACA Airfoil Designations ................................................................................... 29 4.1.4 Lift........................................................................................................................... 30 4.1.5 Drag ........................................................................................................................ 31 4.1.6 Flow Separation and Streamlining .......................................................................... 33
4.2 Development of Experiment .......................................................................................... 34 4.2.1 Existing Hydraulic Flume ....................................................................................... 35
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4.2.2 Hydraulic Flume ..................................................................................................... 36 4.2.3 Theoretical Analysis of Hydraulic Flume ............................................................... 36 4.2.4 Experimental Analysis of Hydraulic Flume ........................................................... 37 4.2.5 Reynolds’ Number and External Flow ................................................................... 37 4.2.6 Airfoil Selection ...................................................................................................... 38 4.2.7 Designing the Experimental Apparatus .................................................................. 39 4.2.8 Force Measurement ................................................................................................. 40 4.2.9 Flow Visualization .................................................................................................. 41
4.3 Apparatus ....................................................................................................................... 42 4.3.1 Simplified Apparatus .............................................................................................. 43 4.3.2 Detailed Apparatus.................................................................................................. 44
4.4 Proposed Experiment ..................................................................................................... 46 4.5 Errors Associated with Experiment Apparatus .............................................................. 48
4.5.1 Suitability of Hydraulic Flume ............................................................................... 48 4.5.2 Errors Associated with Drag Force Measurements ................................................ 49 4.5.3 Errors Associated with Apparatus Design .............................................................. 50
5 Future Development .......................................................................................................... 51 5.1 Pipe Network Design Project – Future Development .................................................... 51 5.2 External Flow Experiment – Future Development Apparatus ....................................... 51
6 Conclusions ....................................................................................................................... 52 Appendix A References ........................................................................................................... 54 Appendix B Collection of Figures and Tables ........................................................................ 56 Appendix C The Fundamental Objectives of Instructional Laboratories ................................ 88 Appendix D Flow in Non-Circular Ducts ................................................................................ 90 Appendix E Design of a Simple Piping Network Project Handouts ....................................... 91 Appendix F Computer Code Developed to Solve Example Pipe Network ........................... 103 Appendix G Theoretical Analysis of Hydraulic Flume ......................................................... 112 Appendix H Experimental Analysis of Hydraulic Flume ...................................................... 113 Appendix I Maximum Load on Force Indicating Apparatus ............................................... 115 Appendix J Cantilever Beam Design .................................................................................... 117 Appendix K External Flow Investigation: Project Handouts ................................................ 118 Appendix L Division of Work ............................................................................................... 132
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List of Symbols
A cross-sectional area (m2)
Ap planform area of airfoil (m2)
α angle of attack (degrees)
Cd coefficient of drag
Cl coefficient of lift
d pipe diameter (m)
ε pipe roughness (m)
Fi force in direction i (N)
f Darcy Weisbach friction factor
g acceleration due to gravity (m/s2)
Hfriction head loss due to friction (m)
Hloss total head loss (m)
Hminor head loss due to minor losses (m)
Htest height of fluid in tank (m)
hi height of fluid at point i (m)
l pipe length (m)
pi pressure at point i (Pa)
ρi fluid density of fluid i (kg/m3)
Q volumetric flow rate (m3/s)
ΔQ flow rate correction factor (m3/s)
Re Reynolds number
δ deflection or displacement (m)
μ fluid viscosity (N s/m2)
ѵ kinematic viscosity (m2/s)
Vi velocity at point i (m/s)
zi height at point i (m)
vi
List of Figures
Figure B-1 - Developing velocity profiles and pressure changes in the entrance of a duct [9] .... 56
Figure B-2 - Control volume of steady fully developed flow between two sections [9] .............. 56
Figure B-3 - The Moody Chart [10] ............................................................................................. 57
Figure B-4 - Entrance loss coefficient as a functino of rounding of the inlet edge [9] ................ 58
Figure B-5 - Loss coefficient for a sudden contraction ................................................................ 58
Figure B-6 - Loss coefficient for a sudden expansion[10] ........................................................... 59
Figure B-7 – Series (a), parallel (b), and reservoir junction(c) pipe systems [9] ......................... 59
Figure B-8 - Test Apparatus Setup ............................................................................................... 60
Figure B-9 - Example Pipe Network ............................................................................................ 61
Figure B-10 – Cross-Section of Typical Airfoil[19]..................................................................... 62
Figure B-11- Flow Over a Cylinder[20] ....................................................................................... 62
Figure B-12 - Flow over a Streamlined Cylinder (Airfoil)[20] .................................................... 62
Figure B-13 - Simplified Sketch of Hydraulic Flume .................................................................. 63
Figure B-14 - Example of Flow Visualization Experiment[20] ................................................... 63
Figure B-15 - Two Dimensional Sketch of Apparatus ................................................................. 64
Figure B-16 - Two Dimensional Sketch of Force Measurement Device ...................................... 64
Figure B-17 - Force Balance on Airfoil ........................................................................................ 65
Figure B-18 - Force Balance on Support Beam ............................................................................ 65
Figure B-19 - Critical Dimensions of Hydraulic Flume ............................................................... 66
Figure B-20 - NACA 0009 Airfoil Cross Section[21] .................................................................. 67
Figure B-21 - NACA6409 Airfoil Cross Section[21] ................................................................... 67
Figure B-22 - NACA 0009 Characteristic Charts for 60,000<Re<300,000 [21] ......................... 68
Figure B-23 – NACA 0009 Characteristic Charts for Specific Reynolds Numbers [21] ............. 69
Figure B-24 - NACA 6409 Characteristic Charts for 60,000 < Re < 200,000 [21] ..................... 70
Figure B-25 – NACA 6409 Characteristic Charts for Specific Reynolds Numbers [21] ............. 71
Figure B-26 - Specially Designed NACA0009 Airfoil for Dye Injection .................................... 72
Figure B-27 - Specially Designed Test Cylinder for Dye Injection ............................................. 73
Figure B-28 - NACA 6409 Airfoil (Drag Testing Only) .............................................................. 74
Figure B-29 - Support Beam Apparatus - View 1 ........................................................................ 75
Figure B-30 - Support Beam Apparatus - View 2 and Floating Platform .................................... 76
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Figure B-31 - Side View of Apparatus ......................................................................................... 76
Figure B-32 - Floating and Fixed Platforms ................................................................................. 77
Figure B-33 - Dye Wand Injecting Single Stream of Dye over Object in Water Tunnel [20] ..... 78
Figure E-1 - Test Apparatus Setup................................................................................................ 95
Figure E-2 - Test Setup ................................................................................................................. 95
Figure H-1 - Hydraulic Flume and Jump During Calibration Experiment ................................. 114
Figure H-2 - Flow Prior to Jump and Pitot Tube Flow Measurement Device ............................ 114
Figure I-1 - Momenutm Analysis of Airfoil at 90-Degrees ........................................................ 115
Figure J-1 - Point Load At End of Cantilever Beam .................................................................. 117
Figure K-1 – Cross-Section of Typical Airfoil [16] ................................................................... 120
Figure K-2 - Point Load At End of Cantilever Beam ................................................................. 122
Figure K-3 - Experimental Apparatus ......................................................................................... 123
Figure K-4 - Simplified 2-D Drawing of Airfoil Support Apparatus ......................................... 123
Figure K-6 - Simplified Sketch of Hydraulic Flume .................................................................. 124
Figure K-5 - Simplified 2-D Drawing of Force Measuring Device ............................................ 124
Figure K-7 - NACA 0009 Characteristic Charts for 60,000<Re<300,000 [21] ......................... 128
Figure K-8 – NACA 0009 Characteristic Charts for Specific Reynolds Numbers [21] ............. 129
Figure K-9 - NACA 6409 Characteristic Charts for 60,000 < Re < 200,000 [21] ..................... 130
Figure K-10 – NACA 6409 Characteristic Charts for Specific Reynolds Numbers [21] ........... 131
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List of Tables Table B-1 - Loss coefficients K for open valvues, elbows and tees ............................................. 79
Table B-2 - Example Network Pipe Geometry ............................................................................. 80
Table B-3 - Loss Coefficient Values used in Example ................................................................. 81
Table B-4 - Valve Positions in Example Network ........................................................................ 81
Table B-5 - Example Network Solution ....................................................................................... 82
Table B-6 - Transient Stages in the Development of lift for an airfoil ......................................... 83
Table B-7 - NACA Four-Series Airfoil Properties, ex. NACA 2415 [17] ................................... 84
Table B-8 - NACA 4-Digit vs. NACA 5-Digit Series .................................................................. 84
Table B-9 - Coordinates of NACA 0009 Airfoil ......................................................................... 85
Table B-10 - Coordinates of NACA 6409 Airfoil ....................................................................... 86
Table B-11 - Summary of Required Apparatus Features .............................................................. 87
Table E-1 - Loss Coefficients, and Constants ............................................................................... 93
Table E-2 - Parts List .................................................................................................................... 93
Table H-1 - Results of Calibration Experiment .......................................................................... 113
Table K-1 - NACA Four-Series Airfoil Properties, ex. NACA 2415 ......................................... 120
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1 Introduction Laboratory experiments are a crucial part of undergraduate engineering courses; however, there
are some obvious concerns associated with their use. Often Professors do not have the time or
funding to create new experiments every year and as a result, the same experiments are used for
numerous years. This leads to students simply copying reports and hence, minimizing the value
of the experiment. Typical experiments require laboratory technicians to operate the equipment
and this detracts from the practical nature of the experiment.
A literature review was conducted to provide different opinions and views on the use of
experiments and design projects in an engineer’s undergraduate education. From personal
experience, courses that replace standard experiments with design projects are challenging and
stimulating. Design projects require students to solve a problem without the use of a standard
procedure. Due to the amount of “free-rein” that is often associated with design projects, students
are required to develop a detailed understanding of the experiment, apparatus, sources of error,
and fundamental engineering theories related to the topic at hand. In addition, design projects
allow students to find creative solutions to an engineering problem.
The purpose of this thesis is to design a project and an experiment to cover applications internal
and external flow theory respectively. These will be implemented into the introductory fluids
course, MIE312, held at the University of Toronto’s Mechanical and Industrial Engineering
Department. This paper describes the background theories, development, and proposed apparatus
of the two projects.
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2 Literature Review A literature review was conducted for two major reasons. Firstly, it provides insight into the
significance, advantages, disadvantages, and history of design projects and laboratory
experiments within the undergraduate engineering curriculum. Secondly, a literature review is
able to provide information about the level of success of design projects that have been
incorporated in undergraduate fluid mechanics courses at other academic institutions.
2.1 The Role of Laboratory Experiments in the Engineering Curriculum Engineering is a practical discipline, which is why prior to the creation of engineering schools,
engineering was taught in apprenticeship programs. These early engineers had to design,
analyze and build their own “creations” learning as they went along [1]. The practical aspect of
engineering became compromised after the introduction of lecture-based, formal engineering
education. [2]
Applying science to everyday life requires both theoretical knowledge as well as hands-on
experience. Theoretical knowledge is taught in the classroom, but hands-on experience can only
be learned through laboratory experience. During the middle of the nineteenth century, many
engineering schools were being created. With the influence of the Industrial Revolution, these
institutions developed a curriculum that emphasized laboratory instruction and taught these new
engineers how to design, build and operate everything from railroads to chemical plants. [2]
Scientists, rather than engineers, developed many great inventions after World War II and this
was a cause for concern regarding engineering education [2]. The American Society for
Engineering Education (ASEE) chartered a committee to “...recommend patterns that
engineering education should take in order to keep pace with the rapid developments in science
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and technology and to educate men who will be competent to serve the needs of and provide the
leadership for the engineering profession over the next quarter century”[3]. The ASEE
determined that engineers were too practically oriented, thus limiting their potential for
innovation. Based on these recommendations, accreditation boards began to place a greater
emphasis on academically oriented education. [2]
Though practical experience is still valued by academic faculty and students, the quality of
practical experience has not kept up with the rapid development of engineering technology. This
can be attributed to increasing complexity and cost of laboratory equipment, the changing
motivation of teaching staff, and the integration of computers [2]. With more advanced
technology, the cost of acquiring and maintaining a laboratory has increased. This, combined
with inadequate course budgets, makes it difficult to incorporate modern instructional
laboratories into undergraduate engineering courses.
2.1.1 Objectives [2]
Setting clear learning objectives is essential in designing an effective learning system. Over the
past few decades numerous engineering scholars have spoken at workshops on the subject of
learning objectives, the learning objectives of engineering laboratories; however, are not
mentioned. Stating clear learning objectives would help in determining the role of laboratories in
the engineering curriculum and in aiding the development process for laboratory exercises.
2.1.2 Computer Simulations [2]
The use of the personal computer has greatly changed the laboratory experience. Computers can
be used for experimental control, data acquisition, and data analysis. This automation may
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remove students from the physical experience, but it has introduced them to other aspects that
were not possible before. Computer simulations are helpful due to their ability to illustrate
phenomena that are not easily visualized; however, they are not capable of completely replacing
physical experiments.
2.1.3 Fundamental Objectives of Laboratories [2]
In the past, engineering educators have not been able to agree on the objectives of instructional
engineering laboratories—particularly because of the lack of effort that has been put forth in
defining such objectives. This is a major problem because it is difficult to design and implement
innovative laboratory experiments when there are no defined goals and objectives.
To solve this problem, ABET and the Sloan Foundation, a charitable foundation that has
supported distance education, funded a colloquy to assemble a group of experts to determine
objectives in order to evaluate the effectiveness of distance education laboratory programs. In
these efforts, a major question asked was, “What are the fundamental objectives of engineering
instructional laboratories?” [2]
The colloquy assembled in San Diego on January 6-8, 2002 and was made up of over fifty
distinguished engineering educators. They generated a comprehensive list of thirteen objectives
and these can be found in Appendix C. A few years after the objectives were developed, a survey
of engineering educators was taken and it was generally felt that these objectives were applicable
and exhaustive. There was however some concern as to whether or not all of the objectives
should be considered essential.
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2.2 Past Design Projects in Fluid Mechanics Courses [4] In order to analyze the advantages and disadvantages of design projects, it is particularly useful
to focus on a design project conducted in an advanced fluid mechanics course at the University
of Queensland in Brisbane, Australia. Professor Tom E. Baldock and Professor Hubert Chanson
were interested in determining how successful they were at engaging and stimulating their
students’ interest in the subject matter through the use of design projects.
Baldock and Chanson offer their professional opinion when stating that “Engineering is related
to the application of science to real-world applications, and engineering graduates must be
familiar with professional problems, practical applications and relevant solutions for the benefits
of society”. Comprehensive studies show that there has been a decrease in formal contact hours
between instructors and students over the past three decades due to cost cutting and
implementation of computer-based projects.
According to Baldock and Chanson, the current problem is that engineering educators are
focused on developing computer-based engineering projects and courses that do not provide
students with experimental projects to support the fundamental theories of the topic at hand.
Baldock and Chanson noted this trend in their analysis of 139 articles in the International
Journal of Engineering Education between January 2000 and December 2001. Of the 139
articles, 57 papers were focused on computer based engineering courses and 32 focused on
project based courses; however, none of the articles described experimental projects to support
basic teaching. Therefore, Baldock and Chanson incorporated two projects into their course to
provide students with an introduction to the complexity of turbulent flow studies, the interactions
6
between fluid and structures, the errors associated with experimental work, and the abstract
process skills required to conduct such experiments.
The two design projects that were implemented into their advanced fluid mechanics course were
worth 36% of the course mark. In these projects students were required to perform wind tunnel
testing of buildings and wave loading on piles. For the purposes of relating these projects to an
introductory fluid mechanics course, only the wind tunnel testing experiment will be discussed.
2.2.1 Wind Tunnel Testing of Buildings – University of Queensland [4]
The purpose of the wind loading study was to have students investigate the flow field around
buildings under cyclonic wind conditions. Students were required to conduct pressure
measurements and calculate lift and drag forces for various attack angles. The experimental
results were compared with results obtained from fluid mechanics software packages including
2DFlow+.
The students were provided with academic and technical guidance while using the wind tunnel
facilities at the Gordon McKay Hydraulics Laboratory of the University of Queensland. The
wind tunnel’s dimensions are 2m by 3m. Pressure measurements were taken using a
ScanivalveTM system connected to a data acquisition computer which scanned the data at 600Hz.
The lift and drag forces were determined by integrating the time-averaged pressure distributions.
Velocity measurements were conducted using a straight hot-wire controlled by a constant
temperature anemometer. These velocity measurements were then used to evaluate the
momentum integral upstream and downstream of the model to determine the total drag.
7
Students were able to use the collected data to draw the flow-nets and derive flow velocities and
pressures using the Bernoulli equation. Students were also able to determine the experimental lift
and drag forces, which were then compared with theoretical lift and drag values determined by
the 2DFlow+ software package.
2.2.2 Student Experience [4]
Baldock and Chanson noted that the design projects received positive feedback from the students
enrolled in the course. The projects particularly served as a motivation tool that was far more
appealing than conventional lectures and audio-visual aids. Prior to the introduction of the design
projects, the failure rate in the advanced fluid mechanics course ranged from 20% to 30%. This
number was significantly reduced after the project was incorporated into the course even though
the core material remained the same. This can be attributed to the fact that 100% of the students
agreed strongly that “the project work was an important component of the subject” and that
“project work related to industrial facilities is an important component of the curriculum”. A
large majority of students also indicated that the project experience helped them think more
critically than traditional lectures.
Advanced projects can often put students in a situation where they feel unprepared to deal with
the challenges that may arise. For example, experimental difficulties often arise with unreliable
instrumentation and equipment. With this said, it is important for students to learn about
calibration of equipment, various measurement techniques, and error analysis. Baldock and
Chanson also noted that many students feel overwhelmed with the daunting task of completing a
design project with limited instruction and guidelines. Even still, the students find a way to gain
the skills required to complete the design project successfully. These experiences cannot be
8
gained in the classroom and hence, project work should be incorporated into all engineering
courses to provide students with valuable real-world experience.
2.3 Advantages and Disadvantages of Experiments and Design Projects [5] Traditional laboratory experiments have some obvious advantages and disadvantages. They
provide students with practical experience and allow for practical application of fundamental
theoretical concepts. The major downsides with practical experiments are that students are often
limited to acting as observers while teaching assistants and laboratory technicians perform the
experiment. With most laboratory experiments, students are given a pre-determined procedure to
follow which does not promote creative and critical thinking skills.
More recently, design projects have been incorporated into an engineer’s education either in
replacement of, or in addition to laboratory experiments. Design projects provide obvious
benefits as they provide students with motivation, an opportunity to apply theoretical concepts,
and an environment where outside-the-box thinking is expected and rewarded. It has been found
that students will end up with “a better understanding of the application of their knowledge in
practice and the complexities of other issues involved in professional practice” [5]. Design
projects also teach students transferable skills such as teamwork, communication and time
management—all of which are necessary when practicing as engineer.
Design projects do however have disadvantages and difficulties associated them. When creating
a project it can be difficult to match the complexity of the project with the skills and knowledge
of the students. Also, judging how much time students will require to complete the project can
be difficult. Usually, design projects are completed in groups and a downfall of group work is
9
that all group members may not complete equal amounts of work. It has also been said that
students “may have a less rigorous understanding of engineering fundamentals” [5]. Another
problem associated with design projects is that students often spend their time working on
aspects of the project that have little, or no relevance or significance to the course material – for
example, they spend too much time figuring out how to build a model, this is not related to the
course material, and time would be better spent working on course related aspects of the project.
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3 Design of a Simple Piping Network Fluid piping networks have been a relevant field of study in fluid mechanics as they can be found
in large scale applications such as municipal water distribution systems and small scale
applications such as micro fluidics. Piping networks can be design through knowledge of the
required pipe geometries, fittings (i.e. valves, bends), flow rates and fluid properties. Often
times, these systems are solved to determine the pressure required to drive the flow. A design
project requiring students to design, construct and analyze a simple piping network, as well as
the related fundamental background concepts are described in detail in the following sections.
3.1 Theory and Background Concepts Piping networks can be very complex and often require a great deal of computational time in
order to solve for all unknown flow rates or head losses. This is typically why students are never
required to analyze a piping network during a test or exam. Piping networks can be found in
almost all engineering applications and for this reason; a project requiring students to analyze a
simple piping network has been developed. All of the background concepts that students will
require to analyze a simple piping network are introduced in the following sections.
3.1.1 Internal Flow [9]
Figure 2.1 shows an internal flow in a long duct. There is an entrance region where flow
converges and enters the long duct. Viscous boundary layers develop downstream, hindering the
flow at the wall. This accelerates the flow at the centerline of the duct.
At some distance from the entrance, x=Le (entrance length), the boundary layers merge and the
inviscid core disappears. Beyond the entrance length, the velocity profile, and wall shear profile
11
are constant and the pressure changes linearly with x for both laminar and turbulent flow as
shown in Figure B-1. At this point the flow is called fully developed.
3.1.2 Head Loss and the Friction Factor [9]
Control volume analyses are commonly used when applying pipe flow formulas to practical
problems. Consider incompressible steady flow between section 1 and 2 as in Figure B-2. Since
the pipe is of constant area, the steady flow energy equation becomes:
fhzg
Vg
pzg
Vg
p+++=++ 2
222
1
211
22 ρρ
(2.1)
Since the pipe has a constant area, the velocity at 1 and 2 is constant and Equation 2.1 reduces to:
( )
gpz
gp
gp
zzh f ρρρΔ
+Δ=⎟⎟⎠
⎞⎜⎜⎝
⎛−+−= 2
21
(2.2)
This means that the pipe head loss is equal to the sum of the change in pressure and gravity head.
Julius Weisbach, a German professor, who published the first modern textbook on
hydrodynamics in 1850, proposed the following correlation for head loss:
gV
DLfhf 2
2
=
(2.3)
The parameter f is called the Darcy friction factor after Henry Darcy, who was a French engineer
that first established the effect of roughness on the resistance of flow.
3.1.3 Laminar Fully Developed Flow [9]
Analytical solutions for laminar flows in circular and non-circular ducts can easily be derived.
For fully developed laminar flow in a circular duct, the relationship for head loss due to friction
can be derived:
12
42
12832gd
LQgd
LVhf πρμ
ρμ
==
(2.4)
The velocity profile is parabolic and has an average velocity that is one half of the maximum
velocity. The above relationship is valid for Reynolds numbers less than roughly 2300, where
the Reynolds number is defined as:
μρVd
=Re
(2.5)
When the wall shear stress is known, the friction factor can easily be determined:
( )
Re6464888
22 ====μρρ
μρτ
dVVdV
Vf w
lam
(2.6)
From Equation 2.6 we can see that the pipe friction factor decreases inversely with the Reynolds
number for laminar flow.
3.1.4 Moody Chart [9]
In 1939, an engineer named C.F. Colebrook combined the relationships for flow in smooth and
rough pipes. This is shown in Equation 2.7.
⎟⎟⎠
⎞⎜⎜⎝
⎛+−= 21Re
51.27.3
log0.21f
df
ε
(2.7)
This relationship is the design formula used to find the friction factor for turbulent flow. Moody
plotted this relationship in 1944 into what is now called the Moody Chart for pipe friction
(Figure B-3). The Moody Chart has been described as the “most famous and useful figure in
fluid mechanics” [9]. It is accurate to ±15% for design calculations and can be used for both
circular and non-circular pipe flows, and open channel flows. An alternative explicit form of
Equation 2.7 is known as the Haaland equation and is shown in Equation 2.8.
13
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠⎞
⎜⎝⎛+−=
11.1
7.3Re9.6log8.11 d
fε
(2.8)
Haaland’s relationship is easier to solve and only varies by ±2% compared to Equation 2.7. The
Moody Chart can be used to solve almost all problems involving frictional losses in long pipes.
Typically, solving a problem using the Moody Chart requires much iteration. Three fundamental
problems that are commonly encountered and can be solved using the Moody Chart are when:
1) d, L, and V or Q, ρ, μ and g are known and the head loss, hf, must be computed.
2) d, L, hf, ρ, μ and g are known and the velocity, V, or flow rate, Q, must be found.
3) Q, L, hf , ρ, μ and g are known and the pipe diameter, d, must be found.
3.1.5 Flow in Non-Circular Ducts [9]
The analysis of fully developed flows in non-circular ducts is similar to that of circular ducts but
is more complicated algebraically. For laminar flow, the continuity and momentum equations
can be directly solved but for turbulent flow, it is simpler to complete the analysis using the
hydraulic diameter. For simplicity purposes, the proposed project is limited to circular ducts;
however, it is important to appreciate the theories involved with non-circular ducts. These
theories are explained in Appendix D.
3.1.6 Minor Losses in Pipe Systems [9]
In addition to frictional energy losses, it is important to consider minor losses that result due to
the following features of a piping system:
• Pipe entrance and exit.
• Sudden expansion or contraction.
• Bends, elbows, tees and other fittings.
14
• Valves (open or partially closed).
• Gradual expansion or contractions.
These losses are typically small and may sometimes be neglected, but in certain cases, they may
as large as the frictional losses.
The flow in fittings is very complex and the theory behind this flow is not well known, for this
reason these losses are generally measured experimentally. The losses are typically dependent
on the particular fitting or valve and therefore manufacturers generally give data for their
designs. The loss coefficient is related to the head loss through the fitting and the velocity head
of the piping system.
The loss coefficient, K, is dimensionless and is related to the size of the pipe. The total head loss
for a single pipe is a sum of the frictional losses and minor losses, as shown in Equation 2.9.
⎟⎠⎞
⎜⎝⎛ ∑+=∑+= K
DLf
gVhhh mftotal 2
2
(2.9)
If the pipe size changes, the losses must be summed separately since the velocity will change.
Table 2.1 shows loss coefficients for some common fittings and valves. There are also
relationships for loss coefficients due to pipe entrance or exit, as well as pipe expansion and
contraction. Figure B-4, Figure B-5, and Figure B-6 give loss coefficients for these cases.
3.1.7 Multiple Pipe Systems [9]
It is relatively simple to solve a single pipe system using the above relationships, but solving a
system involving numerous pipes can become tedious and time consuming. By following a few
15
basic rules, a system containing multiple pipes can be simplified. Figure B-7 shows three
different examples of multiple pipe systems.
The first case is a set of three or more pipes in series as shown in Figure B-7a. Two rules must
be followed to solve this type of system. First, the flow rate in each pipe is the same (Equation
2.10). Secondly, the total head loss through the system is the sum of the head loss in each pipe
(Equation 2.11).
tconsQQQ tan321 ===
(2.10)
321 hhhh BA Δ+Δ+Δ=Δ →
(2.11)
The second type of multiple pipe system is a set of three or more pipes in parallel as shown in
Figure B-7b. Once again, two rules must be followed to solve this type of system. First, the
total flow rate is the sum of each individual flow rate (Equation 2.12). Second, the head loss in
each pipe is the same (Equation 2.13).
321 QQQQ ++=
(2.12)
321 hhhh BA Δ=Δ=Δ=Δ →
(2.13)
The third type of problem is a three-reservoir junction, as shown in Figure B-7c. In this case we
assume that all flows into the junction are considered positive and the sum of the flows is equal
to zero as shown in Equation 2.14.
0321 =++ QQQ
(2.14)
This implies that at least one of the flows is negative (away from the junction). Also, the
pressure change throughout each pipe must yield the same static pressure, pj, at the junction as
shown in Equation 2.15.
16
gp
zh jjj ρ+=
(2.15)
The head loss through each pipe is therefore described in Equations 2.16 a, b, and c.
jhzdLf
gVh −==Δ 1
1
112
11 2
(2.16a)
jhzdLf
gVh −==Δ 2
2
222
22 2
(2.16b)
jhzdLf
gV
h −==Δ 33
332
33 2
(2.16c)
To solve for V1, V2, and V3, you must make an initial guess for hJ, and subsequently iterate until
the flows balance according to Equation 2.14.
3.1.8 Pipe Networks [11], [12], [13]
Typically in real world situations, there is not simply one length of pipe; rather there are a
number of pipes arranged in loops with multiple inlets and outlets. Pipe networks can be found
in many applications, for example, water distribution systems, plumbing systems in houses and
buildings, and gas lines.
The analysis of a single length of pipe utilizes Bernoulli’s equation and the Moody diagram. In
the analysis of complex pipe networks consisting of numerous inlets, outlets and loops, the same
concepts will be utilized. Typically, specialized software is used in the analysis of complex pipe
networks, but before this software was developed, less sophisticated, but relatively accurate,
techniques were used. Simple spreadsheets can be programmed to help analyze these networks
using the Hardy Cross Method.
17
The Hardy Cross Method is the traditional approach used for solving pipe networks. This
method is applicable when all pipe sizes (diameters and lengths) are known, and either head
losses or flow rates between the inlets and outlets of the piping network are known.
The Hardy Cross Method is an iterative approach that can be used to solve for flow rates in a
pipe network. It utilizes two basic rules:
1) The total head loss around a loop must equal zero.
2) The total flow into a junction is equal to the total flow out of a junction.
This method involves making an initial guess for the flow rates throughout the system, making
sure that the total flow into a junction is equal to the total flow out of a junction. The total head
loss around each loop is then calculated based on the initial guesses of flow rates. Next, it is
checked whether the total head loss around each loop is zero. If the total head loss around each
loop is zero, the initial guesses for flow rates were correct, if not, adjustments to the flow rates
are made and the process is repeated. This is then repeated until the total head loss around every
loop is found to be zero. The following is the detailed procedure:
1. Define a set of independent pipe loops so that every pipe is part of at least one
loop, and no loop can be represented by the sum or difference of other loops. The
easiest way to do this is to choose all of the smallest possible loops in the
network.
2. Arbitrarily choose values of flow rates, Q, so that the total flow into a junction
equals the total flow out of a junction (a better initial guess will require fewer
iterations, but any initial guess will eventually converge). Define a sign
convention for each loop – a common sign convention is to define clockwise flow
18
as positive for the loop, but remember this may mean that the same flow in a
given pipe might be considered positive for one loop and may be considered
negative flow when considered in another loop.
3. Calculate the head loss in each pipe. The sign convention used for head loss
should be the same as the sign convention used for flow.
4. Calculate the total head loss around each loop. If the total head loss around every
loop is zero the problem is solved.
5. If the head loss around every loop is not zero, change the flow rate in every pipe
of a loop by ΔQ. By changing the flow rates of all the pipes in the same loop by
the same amount we ensure that the total flow into a junction is still equal to the
flow out of a junction. The key is to select ΔQ so that the head losses around the
loop approaches zero (a formula to find ΔQ is given in Equation 2.17). Remember
that if a given pipe is shared by two loops you must apply ΔQ for both of the
loops to the previous value of Q (be careful of +/- signs).
The following formula has been derived for ΔQ:
∑
∑−=Δ
Loop
loss
Looploss
Qh
hQ
2
(2.17)
Since the Hardy Cross method is an iterative approach, spreadsheets or relatively simple
computer programs can be utilized to assist in the analysis of simple pipe networks.
3.2 Project Description The conventional method of incorporating practical aspects of engineering into an engineer’s
undergraduate education is through laboratory experiments. More recently, design projects have
19
been used in replacement of these conventional laboratory experiments. Design projects require
students to be more creative since a set procedure is not provided. This type of assignment
rewards outside-the-box thinking and as discussed in the literature review, there is potential for
students end up with a better understanding of the material. As well, design projects can be
easily altered from one year to the next and this eliminates the temptation for students to copy a
project from a previous year.
3.2.1 Details of the Design Project
A major focus of MIE 312, an introductory fluid mechanics course at the University of Toronto,
is internal flow in pipes and ducts. Students are taught how to analyze flow in a single length of
pipe, but this is rarely the case in the real world. This; however, is an excellent introduction to
the study of pipe networks. Due to the amount of computational time required to analyze a
network of pipes, it is not possible to test this concept on a test or exam. For this reason, a
design project has been developed where students are required to design and construct a simple
pipe network. This project introduces students to a real world application of fluid mechanics and
is an extension of the concepts taught in the course. One of the major goals of this project is to
prevent students from copying reports from previous years and therefore, this project must have
the ability to be altered from year to year. It is also important to be able to use the same
apparatus and focus on the same fundamental concepts.
The major goal for the students is to design a simple piping network using a list of given
components such that each outlet has the same flow rate and this flow rate will be specified.
Students will be provided with a list of components, including valves, elbows, tees, and straight
lengths of pipes which will be of various sizes. The students will also be required to use at least
20
fifty of the sixty five parts listed to construct their network. Also, the designed network must
have a minimum of five loops. This is to ensure that the students will not create a network that is
too simple.
Students will first have to create a design and will have to analyze their design to see if it will be
suitable. This will involve adjusting valve positions, but no valve may be completely closed. It
is suggested that students write a computer program or spreadsheet to aid in this analysis since it
is an iterative process. Once students have developed a design and have analyzed it to ensure
that all of the outlets have the same flow rates, they will have to determine the required water
level in the tank to produce the specified flows. The designs will then be tested in a class
competition to see which groups succeeded in designing a network that best satisfies the
requirements. All of the components will be provided, and students will have to construct their
network. This network will then be attached to the test apparatus (the details of the apparatus
will be described in detail in the next section) and the flow rates will be measured.
To measure the outlet flow rates, a simple technique will be used. With a stopwatch, students
will determine how long it takes to fill a cup of water, with a specified volume, and this can be
converted to a measurement in liters per second. Numerous measurements will be taken and an
average value will be recorded for each outlet. The average outlet flow rate at both outlets will
then be added together to obtain the total flow rate exiting the network. A winner will be
determined based on the following formula:
,
,
,,
(2.18)
21
In Equation 2.18, Avg Outlet Flow Rate represents the larger outlet flow rate on the denominator,
and the smaller outlet flow rate on the numerator. This ensures that the ratio is less than one.
Also, the ratio of Total Exit Flow Rate over Required Flow Rate will be inversed if necessary to
ensure a ratio of less than one. Therefore, the group with the ratio that is closest to one will be
the winner.
Each group will then be required to write a report that first shows calculations to support their
design, and also discusses any discrepancies between their design and the competition results. It
is expected that the majority of the report will focus on discussing these discrepancies. The
project guidelines, to be provided to students, can be found in Appendix E.
3.2.2 The Apparatus
The test apparatus will consist of the pipe network constructed by the students, as well as the
apparatus that the network will be attached to for the testing. The pipe network will be attached
to a tank of water, and this will drive the flow. At the exit of the network the water will fall into
another tank below. One major assumption that is required to simply the analysis is that the
velocity of the water at the top of the upper tank is zero. To make this a valid assumption, water
must be added to the upper tank to ensure that the water level is constant and does not change
with time. To accomplish this, a pump will re-circulate the water from the lower tank to the
upper tank. There will be a control valve to control how much water is being put back into the
upper tank, which will be changed according to the exit flow rate.
22
To ensure that the upper tank will not overflow, which could happen if the pump re-circulates the
water faster than it exits, a level switch will be installed in the upper tank. A complete set-up of
the apparatus can be seen in Figure B-8.
The apparatus used for the hydraulic jump experiment in MIE 312 at the University of Toronto
has a very similar setup, and it may be possible to adapt this equipment to the new project. This;
however, may be too complicated and it might be easier to purchase new equipment.
It is suggested that the pipe network be constructed with Schedule 40 PVC pipe. This was
chosen because of its price and it will also meet the pressure requirements for the flows
suggested in the sample solution. The size of the upper tank and lower tank is not critical. For
the purpose of creating a model, a rectangular tank 1.1 meters by 0.3 meters is used for the upper
tank and a rectangular tank 1.8 meters by 1.0 meters is used for the lower tank, and both tanks
have a height of 1.0 meters.
3.2.3 Sample Solution
To ensure that this project is not too difficult and is possible for a third year engineering student
to complete, an example network was designed and a computer program was written to analyze
this network. Figure B-9 shows this network and Table B-2 shows all of the pipe diameters and
lengths.
A reasonable flow rate was assumed for the flow rate into the network – this flow rate is 0.00025
m3/s, which is approximately 1.98 m/s based on the diameter specified in Table B-2. According
to Robert Richards, a Senior Piping Designer at Ontario Power Generation, it is generally
23
recommended to keep velocities in pipe flow below approximately 3.05 m/s or 10 ft/s to avoid
severe turbulence. The assumed value of 0.00025m3/s meets this criterion and is therefore
assumed to be a reasonable flow rate for this application.
For this example it was assumed that the fluid flowing through the network is water at room
temperature having a density of 998 km/m3, and viscosity of 1.003 x 10-3 N s/m2. As well, it is
assumed that the pipe is has a roughness of 1.5 x 10-6 m [9]. Table E-1 gives the loss coefficient
values used for this example.
Using the Hardy Cross method, as described in section 2.1.8 of this paper, this example network
was analyzed to determine the flow rates and velocities in each length of pipe. The positions of
the valves were then changed such that the two outlet flow rates are equal – Table B-4 shows the
positions of the valves. A computer program was written in the C programming language to aid
in this analysis. The complete code can be found in Appendix E. The required water level in the
tank is then determined by using the energy equation, Equation 2.2. The water level was found
to be approximately 0.90 meters, which is a reasonable height and this confirmed that the
network entry flow rate is reasonable for this application. Table B-5 shows the resulting flow
rates and velocities in each pipe length in the network, as well as the water level required to drive
this flow.
3.2.4 Errors Associated with the Project
As with almost all experiments, it is expected that the theoretical and experimental results will
deviate from one another. These errors include measurement errors, as well as errors in the
24
analysis technique. To simplify the analysis for the students some assumptions were made and
these assumptions ultimately lead to discrepancies between the analytical and physical results.
Measurement errors are associated with almost any experiment. Human error is the source of the
measurement error in this project. Students are required to use a stopwatch to determine how
long it takes to fill a cup of water. This error will be small, but nonetheless, it will be present and
will contribute to the overall discrepancies between analytical and physical results.
Two major assumptions made to simplify this analysis are that all losses due to entrance and exit
effects as well as losses due to pipe contraction or expansion are neglected. The loss coefficient
related to the entrance effect is determined based on the rounding of the inlet edge. This
coefficient typically ranges from 0.25 to 0.5 according to Figure B-4. The loss coefficient
related to a sudden contraction or expansion depends on the change in area. In the example
network only two different diameters are used and this corresponds to a ratio of areas of about
0.444, which corresponds to a loss coefficient of about 0.3 according to Figure B-5 and Figure
B-6. All of these loss coefficients are comparable to that of a ¾ open valve as shown in Table
B-1. A loss coefficient of this magnitude has a minor effect on the flow, but depending on the
location and number of these types of losses, it could start to have a greater effect.
The manufacturers of piping components typically include loss coefficient charts their products
(i.e. bends, valves, tees, etc.). Students may or may not have this data and generic charts may
contain significant error. To be more accurate the values of loss coefficients should be obtained
from the manufacturer of the components if possible.
25
The errors discussed above are relatively minor when compared to the frictional losses. This
error is associated with the friction factor. In the computer program used to analyze the example
network (Appendix E), the Haaland equation, Equation 2.8, was used to calculate the friction
factor. The Haaland equation has approximately a 2% error over the Colebrook equation which
describes the Moody Chart. Furthermore, Moody himself stated , “It must be recognized that
any high degree of accuracy of determining the friction factor is not to be expected.”. In fact, the
friction factor determined through the Moody Chart could vary by up to 15%.
This shows that the values of the friction factor likely contribute to the greatest cause of
discrepancies between the experimental and theoretical results for this project.
3.3 Improvements over the Existing Experiments This project is intended to replace one or more existing laboratory experiments currently being
used in MIE 312 at the University of Toronto. The current experiments have been used for
numerous years, creating a temptation for students to find a report completed the previous year;
as well they do not require any creative thinking or provide a hands-on experience. The ultimate
goal of this project is to provide students with a hands-on learning experience while promoting
creative and critical thinking.
This new project can be easily altered from year to year and this eliminates the possibility of
students copying a report that was completed the previous year. Although the concepts and
apparatus will be the same each year, providing a different list of components will significantly
change the project. With a new list of components, students will be required to use their own
creativity and knowledge to design a new network that satisfies the project requirements. The
26
errors associated with this project will however be the same every year and unfortunately, this
may lead to plagiarism of certain sections. To eliminate this possibility, a second project is
introduced in the next section of this paper. These projects can be interchanged each year
making it more difficult for students to find a complete report to use as a template. With the
ability for this project to be altered, as well as the possibility to alternate the use of this project
with another new project, the temptation and ability for students to copy their report is greatly
reduced.
With most of the current experiments in MIE 312, a lab technician or teaching assistant must
perform the experiment and this eliminates any possibility of a hands-on experience for the
students. With this new project, students will be required to assemble their own piping network
using the provided components. As well, students will have to adjust the control valve so that
the water level of the upper tank is constant. The students can easily operate this equipment
under the supervision of a teaching assistant and this meets the goal of providing students with a
hands-on learning experience.
In typical experiments, an experimental procedure is provided to the students for each
experiment that they perform. With this project, students are not given a procedure, rather they
are given a task or goal to accomplish, and they must determine the best way to accomplish their
task. This will promote creative and critical thinking and will help to get students more involved
in the learning experience. By being more involved, students will learn much more from the
experience and that is the ultimate goal of this project.
27
4 External Flow Investigation The development and design of aircraft wings has paralleled the theoretical and experimental
study of fluid dynamics; however, the first engineers to study airfoils relied solely on
experimentation to determine the characteristics of various airfoils. Through experimentation,
the important concepts of external flow analysis can be reinforced to students who are learning
external flow for the first time. These concepts include lift, drag, angle of attack, stall, and
boundary layer separation. Since these concepts are usually avoided in introductory fluids
courses, it is beneficial to introduce them at a very simplified experimental level. This section
describes the development and design of an experiment that can be used as an additional
teaching tool to cover the major concepts associated with external flow.
4.1 Background Theory In order to understand the external flow experiment, it is necessary to understand the basic
theories behind external flow. This section provides a detailed look at the key theories and
formulas that are utilized throughout this report.
4.1.1 Airfoils
Airfoils can be defined as a body that is placed in a fluid stream with an intended purpose of
producing a useful aerodynamic force. Examples of airfoils include airplane wings, propeller and
windmill blades, and compressor and turbine blades. The cross section of a typical airfoil is
shown in Figure B-10.
Every airfoil has a leading and trailing edge that define the most forward and rearward points on
the wing with respect to the fluid flow direction. These two edges are connected by a straight
28
line called the chord line. The distance between the leading edge and trailing edge is represented
as the chord length, c, and this defines many other characteristics of an airfoil. [9]
The thickness of an airfoil is the distance between the upper and lower surfaces, measured
perpendicular to the chord line. The maximum thickness and its location are crucial parameters
for every airfoil and they are expressed as a percentage of chord length.
The term camber is used to describe the asymmetric curvature of an airfoil. This is readily seen
by the mean camber line which is the line that connects the series of points that are measured
perpendicular to the chord line and are half way between the lower and bottom surfaces. The
camber of an airfoil is the maximum distance between the mean camber line and the chord line,
also measured perpendicular to the chord line. The maximum camber is also expressed as a
percentage of chord length. [14]
Airfoils are typically assumed to be infinite in length—into and out of the page. Forces on an
infinite wing are analyzed on a per unit length (or per unit span) basis. The angle between the
chord and the direction of fluid flow is known as the angle of attack and this is a key parameter
to consider when analyzing an airfoils’ lift and drag characteristics. [9]
4.1.2 How an Airfoil Induces Lift
Airfoils can induce lift through two independent methods. Firstly, a cambered airfoil can be
used to create a pressure difference between the two surfaces. Secondly, the airfoil can be
inclined relative to the fluid flow direction to create a pressure difference. For the purposes of an
introductory fluids course, the latter method of lift generation will be investigated further.
29
Table B-6 shows the development of lift for an airfoil that has a positive angle of attack, with a
specific emphasis on streamlines, starting vortices, and circulation [15]. Descriptions of each
transient stage of lift formation are included in the table.
4.1.3 NACA Airfoil Designations
The National Advisory Committee for Aeronautics (NACA) has developed a list of standard
airfoil shapes for aircraft wings. Each NACA airfoil is described using a set of coordinates to
represent the upper and lower surfaces of the cross-section of the airfoil in percentage of chord
length. Various computational numerical methods can then be used to plot the airfoil shape and
calculate its properties such as maximum thickness and camber [16].
The first family of airfoils, based on geometries alone, was called the NACA Four-digit Series.
An example of the naming system is shown in
30
Table B-7, where each digit indicates a particular geometric property of the airfoil. The Four-
digit series consists of the simplest geometries and if the first two digits are zero, the Four-digit
series airfoils are symmetrical about the chord [17].
In addition to the Four-Digit NACA Series airfoils, NACA developed many other geometric
series, such as the Five-Digit Series, which specified properties such as a design lift coefficient
and the nose geometry. More advanced airfoils were grouped based on desired pressure
distributions on both surfaces of an airfoil, thus allowing for optimal geometry selection. These
series are not in the scope of this thesis due to their complexity and irrelevance to students who
are taking an introductory fluids course [17].
The NACA Four-digit Series airfoils provide a simplistic geometry that is both realistic for
general aviation and simple enough for students to understand the concept of lift and drag at
various angles of attack. In addition, the 4-digit series is ideal for demonstrating flow separation
and the stages of angle of attack that lead to airfoil stall due to their good stall characteristics.
The Four-digit Series airfoils are compared to the Five-digit Series airfoils in Table B-8.
4.1.4 Lift [9]
The lift coefficient is a non-dimensional number that is used to characterize the amount of lift
that an airfoil can produce under particular conditions. It is typically determined experimentally
since it heavily relies on the shape of the airfoil itself and this cannot be accounted for in a
simple equation. The lift force on an airfoil can be determined with knowledge of the angle of
attack, lift coefficient, airfoil planform area, and flow characteristics such as the density and
velocity of the fluid. The lift coefficient is shown in Equation 4.1.
31
p
LL AV
FC 2
)(2)(ρ
αα =
(4.1)
A useful tool to analyze a particular airfoil is a Coefficient of Lift vs. Angle of Attack plot.
These plots are specific to an airfoil and a particular Reynolds number. The coefficient of lift
generally increases as a function of angle of attack until the stall angle is reached. After the stall
angle is reached, the coefficient of lift drops off rapidly since pressure drag becomes dominant. It
is important to note that cambered airfoils are able to produce lift at negative angles of attack.
Symmetric airfoils on the other hand, can only generate lift when the angle of attack is greater
than zero.
4.1.5 Drag [14]
There are two types of drag forces that can occur on an airfoil: parasite and induced drag.
Parasite drag consists of two sub-categories of drag: friction drag and pressure drag. Induced
drag results when an airfoil moves upwards, due to lift. This upwards motion, combined with the
forward motion of the object, results in a drag force opposing the motion. Since the proposed
experiment uses a fixed airfoil, induced drag is not a factor and will not be discussed further.
4.1.5.1 Friction Drag
Friction drag is sometimes referred to as viscous drag as it is associated with the viscosity of the
fluid in the boundary layer of the flow. The fluid subsequently exerts a drag force on the object
that it is flowing over, essentially trying to pull the surface along the same direction.
It is important to note that friction drag is solely a function of the fluid and boundary layer
properties and it is not dependent on the surface properties. With that said, the surface properties
32
will dictate whether or not the flow is turbulent and this can significantly change the boundary
layer. In the case of ordinary low-speed airfoils, the friction drag is similar in magnitude to
pressure drag. These two drags are grouped together and called parasite drag.
4.1.5.2 Pressure Drag
Most parasite drag is caused by pressure drag which is caused by the pressure difference between
the front and rear of an object. Pressure drag is primarily a function of an object’s size and shape
and thus, it is often called form drag. Bodies with a large frontal area will experience high
pressure drag, while streamlined bodies will experience low pressure drag.
The phenomenon of flow separation, or boundary layer separation, significantly increases
pressure drag. This can be understood by the fact that streamlines accelerate while passing the
leading edge of a blunt object. If the object is blunt, the flow will separate at the rear of the
object since the fluid momentum would prevent it from returning to its original path. The result
is a notably fast flow and formation of a low pressure region at the rear of the object. The net
pressure force, or pressure drag, opposes the direction of travel.
For high speed flow, where the Mach number exceeds 0.7, the effect of shocks and wave drag
must be added to the pressure drag. In this case, pressure drag becomes the dominant source of
drag over both friction and induced drag.
4.1.5.3 Total Drag
For low speed flow, both the friction drag and the pressure drag must be taken into account. With
varying angles of attack, the proportion of friction and pressure drag changes significantly. A
useful tool for describing the total amount of drag on an airfoil is the coefficient of drag. The
33
coefficient of drag is a dimensionless number that is typically determined experimentally for a
particular object or airfoil. The coefficient of drag is best shown in Equations 4.2 and 4.3.
p
DtotalD AV
FC 2,)(2)(
ραα =
(4.2)
frictionDpressureDtotalD CCC ,,, +=
(4.3)
Much like the coefficient of lift, the coefficient of drag is typically determined experimentally
for a particular angle of attack and Reynolds number. CD vs. α plots account for the summation
of the drag on the airfoil since these plots are typically determined experimentally.
4.1.6 Flow Separation and Streamlining [18]
Streamlining has become an important focus of fluid dynamics because reducing the amount of
drag can lead to more efficient products and processes. Streamlining can best be understood by
comparing flow around a cylinder with flow around a streamlined cylinder which is somewhat
representative of an airfoil.
When an object, such as a cylinder shown in Figure B-11, is placed in a flow, the streamlines
must split above and below the shape. Thus, the flow on the top and bottom of the cylinder
accelerate due to the added mass going through a smaller cross section. The closely spaced
streamlines depict this acceleration. Through Bernoulli’s principle; it is known that an increase in
flow velocity leads to a decrease in pressure. As a result, the pressure on the downstream side of
the cylinder is higher than on the top and bottom. This pressure gradient causes flow or boundary
layer separation. The streamlines in this case are unable to spread out, or in other words, the fluid
is unable to decelerate and this causes the formation of a tumbling pattern referred to as eddies.
34
The end result of the flow separation is a significant net pressure difference between the front
and back of the cylinder which pushes the cylinder in the direction of fluid flow.
Flow separation can be reduced by optimizing the geometry of the object to guide the
streamlines back to their original path. The streamlined cylinder in Figure B-12 allows the fluid,
which accelerates rapidly at the front end of the object, to decelerate gradually along the surface
of the airfoil. The streamlines are able to negotiate this path in a smooth manner and boundary
layer separation is avoided. This avoids the low pressure region that is formed for the cylinder
and pressure drag is significantly reduced.
Another commonly used strategy of reducing or delaying boundary layer separation is to induce
turbulence on the surface of the object. Turbulent boundary layers are not as vulnerable to
adverse pressure gradients and are able to remain on the surface for longer lengths than laminar
boundary layers. Thus drag is reduced significantly.
4.2 Development of Experiment For an introductory fluids course, it would be beneficial to provide students with an opportunity
to investigate flows around an airfoil. In addition, a secondary goal of this experiment is to
provide students with an introduction into standard airfoils and an opportunity to compare
airfoils of varying geometries.
These topics are typically introduced in MIE312 at a highly simplified level. External flow
experiments are also very costly and difficult to set up; however, they are a great tool to
stimulate further interest in the field of fluid dynamics due to their real world applications. The
35
main focus of this thesis is to develop a simplified external flow experiment without sacrificing
the results or the overall learning experience.
4.2.1 Existing Hydraulic Flume
The MIE312 Fluid Mechanics course currently utilizes a hydraulic flume, located in MC214, to
perform laboratory experiments to analyze the effect of Bernoulli’s principle. Since one of the
goals of this thesis is to develop a low-cost experiment, the hydraulic flume was analyzed to
determine whether or not it would be suitable for an external flow experiment.
It was initially suggested, within the interim progress report, to simulate a wind tunnel by
converting the current hydraulic flume, located in MC214, into a tow tank. The thinking behind
this suggestion was to be able to simulate a constant flow across an airfoil while measuring the
required aerodynamic forces for further analysis. After further analysis of the hydraulic flume, it
was determined that a more practical solution would be to fix the airfoil in one location while
running the water through the flume.
In order to determine the suitability of the hydraulic flume, both a theoretical and experimental
analysis were conducted. The goal of the hydraulic flume analysis was to determine whether or
not a reasonable flow could be achieved using the apparatus without any modifications. By
determining the flow velocity, the maximum achievable Reynolds number can be determined.
As long as an adequate Reynolds’ number can be achieved from the hydraulic flume, an
experiment can be designed to allow students to compare theoretical and experimental
aerodynamic properties.
36
4.2.2 Hydraulic Flume
A simplified sketch of the hydraulic flume is shown below in Figure B-13. The flume is
currently located in MC214 and is used for a variety of experiments in the undergraduate
Introduction to Fluid Mechanics Course. See Figure B-19 for a drawing of the hydraulic flume
and its critical dimensions.
4.2.3 Theoretical Analysis of Hydraulic Flume
In order to determine the maximum velocity achievable by the hydraulic flume, independent of
the pump that is currently used, the maximum height of the tank water level and the desired
height of the test section can be used. Using Bernoulli’s equation, the velocity at the test section
can be determined in terms of the height of the fluid in the tank, the height of the fluid in the test
section, and the gravitational constant. The test section velocity can be expressed as Equation 4.4
and the derivation of the equation is shown in Appendix G.
nkTaTest
Test
hHgH
V+
=2
22
(4.4)
The current tank permits a tank water level, HTank, of up to 0.51m before engaging a level switch
to turn off the pump. Ideally, the test section height, hTest, would be 0.25m, leaving a two inch
clearance to the top of the tank. By maximizing the test section height, the adverse flow effects
of the surface and flume floor can be reduced as much as possible. Through the use of Equation
4.4, it was determined that the maximum theoretical velocity at the test section is 2.59 m/s. After
conducting an experimental analysis, it was determined that these conditions cannot be met with
the existing pump that is installed on the apparatus.
37
4.2.4 Experimental Analysis of Hydraulic Flume
To determine the maximum height and velocity achievable in the test section with the pump that
is currently installed on the system, the hydraulic flume was experimentally calibrated until the
maximum test section height and velocity could be achieved. This process took roughly fifteen
minutes since slight adjustments in either the exit gate or the sluice gate height would cause the
hydraulic jump to move significantly. The calibration was made by lowering the sluice gate to
move the hydraulic jump away from the entrance, and raising the exit gate to move the jump
towards the entrance.
The major challenge encountered during this experiment was that it took a fair bit of patience in
order to calibrate the flume appropriately. The heights of the sluice and exit gates were slowly
adjusted until the hydraulic jump occurred near the front of the hydraulic flume. This was
important to ensure that there was adequate time for the flow to transition from turbulent to
laminar after the hydraulic jump. By trial and error, adjustments were made to both gates until an
optimal configuration was determined. The measurements taken upon completion of the
calibration, the test section velocity calculations, and photographs of the experiment can all be
seen in Appendix H. The velocity of the test section flow was determined to be 0.5m/s.
4.2.5 Reynolds’ Number and External Flow
The significance of calculating the achievable test section velocity is to ensure that a high
enough Reynolds’ number is produced within the laminar test section to match theoretical data
that is currently available. Theoretical lift and drag coefficients are readily found for airfoils;
however, most aerodynamics textbooks present this information for cases where the Reynolds
number is greater than 106. The benefit of using the hydraulic flume for an external flow
38
analysis experiment is to take advantage of the fact that at room temperature, the kinematic
viscosity of water (1.004x10-6 m2/s) is This is roughly 15 times less than the kinematic viscosity
of air (15.11x10-6 m2/s). Thus the Reynolds’ number for a flow in water will be on the order of
10 times greater than the Reynolds’ number of an equal flow in air. This is necessary to ensure
that the relationships derived from the experiment are well-defined [9].
The Reynolds number is given by Equation 4.1. The maximum Reynolds number achievable in
the test section of the hydraulic flume was determined to be 75896 and this was based on the
properties of water at room temperature (ρ=998kg/m3, μ=1.003E-3Ns/m2) and a test section
velocity of 0.5m/s [9]. The characteristic length, d, that was used to calculate the Reynolds
number was chosen as 0.1524cm to obtain a Reynolds number that was a large as realistically
possible within the boundaries of the hydraulic flume.
μρVd
=Re
(4.1)
4.2.6 Airfoil Selection [21]
As stated earlier, it was decided to utilize Four-Digit NACA series airfoils since they are both
simplistic and popular in the aeronautics community. After determine the maximum achievable
Reynolds number, the text, Airfoils at Low Speeds, by Michael S. Selig, was used to select
particular airfoils for the experiment. Airfoils at Low Speeds was published as a resource for
hobbyists who designed small-scale remote control planes.
Airfoils at Low Speeds is a particularly useful resource because it provides airfoil characteristic
curves for 164 airfoils at Reynolds numbers between 50000 and 300000. This was ideal since
39
the hydraulic flume is only able to generate Reynolds numbers up to 76000 provided that the
chord length (characteristic diameter) of the test airfoil is 0.1524cm. Since symmetric NACA
airfoils were desired, the NACA0009 airfoil was selected. This, however, was the only
symmetric airfoil provided in Airfoils at Low Speeds and the NACA6409 airfoil was selected as
the second test airfoil. By using a symmetric and non-symmetric airfoil, students will be able to
investigate the effect that the camber has on both lift and drag.
The cross section of the NACA0009 and the NACA6409 airfoils are shown in Figure B-20 and
Figure B-21 respectively. Airfoils at Low Speeds also provides CL vs.α plots and CL vs. CD plots
for the NACA0009 and NACA6409. These plots are shown in Figure B-22 and Figure B-24for
various Reynolds numbers and in Figure B-23 and Figure B-25 for Specific Reynolds numbers.
The coordinates for the upper and lower surface of the NACA0009 and NACA 6409 airfoils are
provided in Table B-9 and Table B-10 respectively.
4.2.7 Designing the Experimental Apparatus
The major goal while designing the apparatus was to facilitate simplified analysis of lift, drag,
stall and boundary layer separation. In order to do this, particular features were required in the
apparatus and these features are summarized in Table B-11.
Under the supervision of Professor P.E. Sullivan, it was decided that that it would be beneficial
to only measure the drag force acting upon the airfoil, as opposed to measuring both the drag and
lift force. In lieu of this missing data, the theoretical lift force, as determined from CL vs. α plots,
can be used instead. Making this allowance also makes the experimental apparatus simpler to
design and build since the vertical force will not need to be measured.
40
4.2.8 Force Measurement
Various techniques can be utilized to measure the amount of drag force applied to the airfoil.
The most obvious methods would be to use a force sensor or a load cell; however, these solutions
rely on sensitive circuitry and unreliable strain gauges. A more reliable method of measuring
force would be to use dial indicators to measure the displacement of an object that can resist the
drag force such as a spring or cantilever beam. Dial indicators are also a good choice since they
rely on mechanical principles, work well in wet conditions, and do not require a computer or
digital device to output the results.
In order to design the force indicating apparatus, it is useful to determine the maximum force
exerted on the airfoil due to fluid momentum. In order to simplify the analysis, the test airfoil can
be considered a flat plate. The maximum drag force will occur when the airfoil is perpendicular
to the fluid flow direction—or in other words, when the angle of attack is 90o. After analyzing
the forces, it was determined that the force indicating apparatus would have to be able to
withstand a load up to 7.224N—assuming a factor of safety of 1.25. The calculation of the
maximum load on the force indicating apparatus can be seen in Appendix H.
Due to practicality and accuracy issues, it was determined that the most suitable way to calculate
the drag force using a dial indicator would be to measure the deflection that the force caused on a
cantilever beam. It is expected that students have an understanding of basic beam theories since
this material is taught in a mandatory introductory course taken in the second year of the
mechanical engineering curriculum.
41
Cantilever beams can be used to resist the applied force and deflect a very small amount. This
deflection can be measured with a dial indicator. The dial indicators that are readily available are
able to measure deflection in the range of 0.002-12.700mm. In order to account for the entire
range of possible forces that can be applied, the cantilever beam has to be designed using
equations for a point load applied to the end of a cantilever beam. The beam width and length
were chosen to be 0.0127m and 0.2032m respectively. After using simple beam theories for a
point load on a cantilever beam, it was determined that the minimum beam height required is
0.00196m. If the beam height is less than this amount, there is potential for the deflection of the
system to be greater than the measurable deflection by the dial indicator and this should be
avoided since it can damage the dial indicator. This determination was made through the
calculations made in Appendix J.
4.2.9 Flow Visualization
In order to allow students to visualize flow in the hydraulic flume, various colour dyes can be
injected into an airfoil. The Rolling Hills Research Corporation has designed many airfoils that
are sold for the purpose of flow visualization. They rely on two primary methods of dye
injection: a dye wand to inject a single stream into the flow, and a specially designed object
through which the dye can be injected onto the leading edge. For the purposes of analyzing a
finite airfoil, in a particularly narrow flume, it is beneficial to inject dyes at various points along
the leading edge.
These dyes should exit the airfoil on the leading edge so as to follow the boundary layers closest
to the surface of the airfoil. By changing the angle of attack, the boundary layer separation will
become obvious to students and photographs of this experiment can be taken for further analysis.
42
An example of this experiment, as conducted by the Rolling Hills Research Corporation, can be
seen in Figure B-14.
In order to facilitate this section of the experiment, the NACA0009 and test cylinder of same
thickness as the NACA0009 airfoil will be constructed with built-in channels. A 0.32cm tube
can be connected to the airfoil and a dye can be injected into the channels using a syringe. The
specially designed airfoil and test cylinder are shown in Figure B-26 and Figure B-27
respectively. The NACA6409 airfoil; however, was not designed with channels due to its
complex shape. This airfoil is shown in Figure B-28.
As an optional component to the flow visualization component of this experiment, students can
bring in, or make, their own objects to test flow separation. In order to visualize flow over these
objects, a dye wand is required. A dye wand simply consists of a very thin, L-shaped, steel tube
that can inject a dye into the fluid stream. They are typically surrounded by a fairing to reduce
the interference of vortices shed from the tube [22]. The dye wand can then be placed a 10 to 20
centimeters in front of the test object and a single stream of dye can be injected using a syringe.
This is common practice in external flow experiments and an example of this can be seen in
Figure B-33.
4.3 Apparatus After determining the major features that needed to be included in the experimental apparatus, a
two-dimensional sketch was produced to design the general shape of the apparatus. A three-
dimensional apparatus was subsequently designed using SolidWorks to clarify how the apparatus
will allow students to obtain the results required and how the apparatus could be constructed.
43
4.3.1 Simplified Apparatus
After determining the major features required for the experimental flow apparatus, simplified
sketches were created to illustrate the system. These sketches are shown below.
4.3.1.1 Two Dimensional Sketch of Apparatus
Figure B-15 shows a two-dimensional sketch of the proposed apparatus. The airfoil is placed on
a hinge that can be used to fix the angle of attack for a particular trial of the experiment. The
rigid bar then rests above the tank on a roller support to reduce the amount of friction that needs
to be overcome by the drag force. This is particularly important because it ensures that the force
measurement device is measuring the entire drag force and not the drag force less the friction
force.
After running the hydraulic flow calibration experiment (Section 4.2.4), it was observed that any
object placed in the flume would cause substantial turbulence on the downstream side of the
object. To eliminate the effects of this turbulence on the airfoil, the rigid bar was designed to
extend diagonally forward so that the fluid will make contact with the airfoil first. In addition,
the rigid bar must support the airfoil from the sides to further minimize the effect of turbulence.
The angle, θ, at which the rigid bar is mounted, is variable; however, careful consideration must
be made to ensure that the moment induced on the support beam is reduced. The support beam’s
length must ensure that the airfoil is placed half way in between the free surface of the water and
the bottom of the flume.
The force measurement device, which consists of a cantilever beam and a dial indicator, must be
mounted precisely so that it only measures the horizontal force from the beam.
44
4.3.1.2 Two Dimensional Sketch of Force Measurement Device
The force measurement device will consists of a cantilever beam of length 0.2032m and a dial
indicator to measure the displacement of the beam. This set up is shown in Figure B-16. When
building this apparatus, it is important to ensure that the load from the support beam is applied to
the end of the cantilever beam since this was the condition that was adhered to in designing the
cantilever beam.
4.3.2 Detailed Apparatus
When designing the detailed apparatus, the major goal was to ensure reliability in measurements
and longevity of the entire apparatus. For this paper, a major focus was placed on the features of
the apparatus to accomplish these goals and less emphasis was placed on the dimensioning of the
apparatus since detailed dimensions will only be worthwhile if the experiment is approved.
4.3.2.1 Three Dimensional Model of Apparatus
The three dimensional model looks very intimidating; however, careful consideration was made
for each component to ensure that it could achieve the results of the experiment with as little
error as possible. In addition, each component was designed with an objective of being easily
constructed.
The first major component of this assembly is the airfoil support beam apparatus and this can be
seen in Figure B-29, Figure B-30 and Figure B-31. The airfoils are attached to the support beam
via 5 cm long, 0.635cm-diameter bolts. Two bolts support the wing on the side with the angle-of-
attack changing apparatus, and one bolt is used to support the wing on the other side. This
mechanism allows easy variation of the angle of attack from -5o to 25o at intervals of 2.5o. These
45
adjustments can easily be made using the mechanism that has already been calibrated for these
angles. This design is particularly useful because it allows students to change the angle of attack
with ease. This simplicity will also allow students to retrieve more data points for the purposes of
plotting the CD vs. α plots. The last feature of the support apparatus is that the support beams are
either welded or bolted to a floating platform that is free to move in the direction of the flow.
Thus, all of the drag force on the airfoil is translated into the platform and subsequently into the
cantilever beam. This floating platform can be seen in Figure B-30.
In order to facilitate the exchange of airfoils and test subjects, the support bars can be attached to
the floating platform on hinges so that the entire support bar can be rotated out of the tank. This
accommodation was not made on the detailed drawing; however, it would significantly reduce
time during experimentation. It is important to note that if this accommodation is made, a limit
would have to be placed on the hinge so as to prevent the airfoil from hitting the floor of the
tank.
The floating platform was designed to minimize frictional forces as much as possible. Hence, the
platform hangs from connecting rods which are associated with the fixed platform. These two
platforms can be seen in Figure B-32.The vertical connecting rods allow the floating platform to
move in the direction of fluid flow; however, they also have a tendency to allow unwanted side-
to-side movement. In order to eliminate this movement, two additional connecting rods were
attached to restrain the side to side motion of the floating platform. These connecting rods are
perpendicular to the fluid flow. With these four hinged connecting rods, it can be certain that the
only horizontal forces will be applied to the floating platform. A safety feature was incorporated
46
into the fixed and floating platforms to ensure that the system does not exceed its intended limits.
These are depicted by two gold screws at the front and back of the floating platform. By
calibrating these screws, displacement limits for the floating platform can be set and this will
maintain the integrity of the dial indicator.
The next key feature of the detailed design is the interaction between the floating and fixed
platforms. As the floating platform is pushed forward due to the drag force, the cantilever beam,
which is attached to the floating platform, is pushed against a calibration pin. At the same time, a
steel beam is pushed against the dial indicator. When steady state is reached, the cantilever beam
will deflect between 0 and 1.127cm and this deflection can be measured by the dial indicator.
The load indicator was mounted on the side of the fixed tank for easy viewing by students
conducting the experiments.
4.3.2.2 Constructing the Apparatus
The materials recommended for the apparatus are low carbon steel. If weight is an issue,
aluminum can also be used; however, aluminum will pose greater problems while welding the
frames. It is recommended to weld all components since this will provide better structural
strength. The connecting rods are readily available and adjustable-length connecting rods can
also be found at many materials suppliers.
4.4 Proposed Experiment The experiment to be conducted by students will be conducted like a typical laboratory
experiment where a supervising teaching assistant or lab technician is required to explain the
apparatus, the basic theories, and the procedure that must be followed throughout the experiment.
This deviates from one of the primary goals of this thesis which was to create an experiment
47
where there are fewer guidelines provided to students. This allowance was made because of the
complexity of the theories involved and the necessity to obtain accurate results. With that said,
there is still potential to incorporate a design component into this experiment as outlined in
Section 4.2.9.
The proposed experiment is described in the handout that will be provided to students. This can
be seen in Appendix K. The major highlights of this experiment are the ability for students to
measure drag forces on various airfoils and a cylinder and perform a flow visualization
experiment to observe boundary layer separation.
To adhere to some of the primary objectives of the experiment, it is essential that students make
an effort to understand the relationship between lift, drag, and the angle of attack while applying
them to real-world applications. This can be done by requiring students to use the data collected
in the lab to discuss the formation of lift. To further incorporate real-world applications of the
results, students will also be required to complete a dimensional analysis to determine if the
experimental apparatus can be used to determine the lift and drag on an airfoil that is sized for a
small plane. This analysis will allow students to better understand the advantages and
disadvantages of the experimental apparatus.
This experiment was designed to stimulate interest in the field of fluid dynamics and more
specifically, experimental analysis of fluid dynamics. The flow visualization component of the
lab provides students with a first-hand opportunity to see the effects of boundary layer
separation. The importance of streamlining will become clear after comparing the visualized
48
flow between the cylinder and the airfoil. As a supplementary design component to this portion
of the experiment, students can be given foam blocks to design an object of their choice or bring
in an existing object. These can be tested on the apparatus and the drag force can be measured.
This adds an element of fun and creativity to the lab experiment and can further stimulate interest
in the subject.
4.5 Errors Associated with Experiment Apparatus With most experiments, there are often many errors associated with the results obtained. The
major errors are usually a result of a poor experimental setup or an inadequate match between
theoretical and experimental conditions. While developing this experiment, care was taken to
minimize the known errors; however, they must still be accounted for in the analysis of the
results. These errors are discussed in more detail below.
4.5.1 Suitability of Hydraulic Flume
External flow experiments are often very expensive because they require a system to generate a
uniform flow of either air or water. To account for this problem, many research institutes have
designed small scale tow tanks so that the major costs associated are only for a tank of water, a
motor and a track to guide the object being tested. Due to the size of the hydraulic flume
apparatus it is not possible to use it as a tow tank. Thus, it was decided to fix the test object in the
flume and run water through it to simulate the flow. This is a very cost effective solution but the
errors associated with it must be considered.
Typical external flow experiments are conducted in a channel that has a cross section that is large
enough to avoid the wall or surface effects on the flow. Due to the no-slip condition of boundary
layers, the velocity of the fluid on the three walls is known to be zero. Thus, the maximum
49
velocity of the fluid is on the surface, at the midway point between the two walls. This
essentially means that the velocity of the flow passing over the wing is not equal along the entire
span of the wing. This is very difficult to account for since it would be unrealistic to take
velocity measurements across the entire span of the wing. When the coefficient of drag is
calculated using Equation 4.2, the velocity of the flow is used. Since the velocity varies across
the wing, the experimental coefficient of drag is not entirely representative of the test conditions.
For simplicity purposes, the head loss in the hydraulic flume was not accounted for when
determining the velocity of the water in the test section. This is a reasonable assumption to make
since the Reynolds Numbers are relatively low; however, for a more accurate measurement of
velocity, it is important to account for the head loss in an open, rectangular channel.
4.5.2 Errors Associated with Drag Force Measurements
The force indicating device, consisting of a cantilever beam and a dial indicator, was chosen
based on its reliability and ease of implementation. Since the material of the cantilever beam is
known, the theoretical material properties can be used to translate the displacement of the beam
into an applied force. This method can lead to many results since the actual Young’s modulus of
the cantilever beam can change with respect to its workmanship or actual material properties. To
obtain more accurate results, the cantilever beam can be calibrated using known weights.
Through calibration, an experimental Young’s modulus can be determined and used for
subsequent calculations.
50
4.5.3 Errors Associated with Apparatus Design
As described in Section 4.3.2, the airfoils are supported by support beams that extend forward so
that the induced turbulence by the beams does not affect the flow over the airfoil. This design
accommodation; however, leads to experimental error as there is potential for the resultant force
on the airfoil to produce a moment force on the beam. Since the resultant force is not necessarily
axial, a component of the resultant force can cause the entire beam to rotate about the roller
support. This moment producing force, Fm, is the component of the resultant force, Fr, which is
perpendicular to the support beam. These forces are depicted in the free body diagrams seen in
Figure B-17 and Figure B-18. Since only the horizontal component of the axial force, Fa, will be
applied to the cantilever beam, a moment force can cause a significant error between the
theoretical and experimental drag forces.
One method that can be used to minimize the amount of moment on the support beam is to
calculate angle at which the theoretical resultant force will act. The rigid support beam can
subsequently be rotated to match that angle. This method would require students to pre-calculate
the resultant force for the airfoils at various angles of attack. In addition, the support beam would
have to be allowed to pivot and change in length so that the airfoil can be kept at a constant
height in the flow.
51
5 Future Development
Due to financial and time constraints, it was not possible to construct and test the pipe network
and external flow apparatus’. The proposed future development for both projects is discussed
below.
5.1 Pipe Network Design Project – Future Development The pipe network design project has been fully developed and is ready to be built according to
the descriptions and drawings presented in this paper. Once the apparatus has been constructed,
the example network can be tested. Based on this testing, modifications may have to be made to
the apparatus or project requirements. Once these modifications have been made, this project
will be ready to be implemented into MIE 312 at the University of Toronto. The construction of
the test apparatus and testing of the project may result in a significant amount of work and
therefore it may be an excellent thesis project to be completed by a small group of students in the
near future.
5.2 External Flow Experiment – Future Development Apparatus This paper presents the development and design of the external flow experiment; however, the
design must be built and tested prior to ensuring its readiness for the Introduction to Fluid
Mechanics course. If the proposed design is adhered to, the apparatus can be constructed in the
Machine Tool Lab (MC 106) located at the University of Toronto. The airfoils will have to be
made using a CNC machine to maintain accuracy. Mr. Len Roosman has kindly offered his
services to aid in the manufacturing of this experiment.
52
6 Conclusions
It is very clear that laboratory experiments and design projects are a critical part of an engineer’s
undergraduate education and must somehow be incorporated into some courses at the
undergraduate level. However, there is much discussion in regards to the goals and objectives of
these projects and experiments. With more clear goals and objectives, better and more suitable
experiments and projects can be developed, which is why the The Fundamental Objectives of
Engineering Instructional Laboratories were developed. Another growing concern is related to
design projects. The use of design projects has been increasing but there is some concern in
regards to the time requirements of these projects. It is often found that students are required to
spend large amounts of time on these projects, and this is becoming a concern. When a design
project is developed the time requirements must be taken into consideration, otherwise the
students, who already have a demanding schedule, will find themselves overloaded with work.
The current laboratory experiments being used in MIE 312 at the University of Toronto expose
students to practical aspects of engineering. These experiments are very useful and beneficial to
students, but they do have some flaws. The proposed projects in this paper improve upon the
current experiments. The major problems associated with the existing experiments are the lack
of hands-on involvement, lack of creativity required and the fact that the same experiments are
used every year. The two projects proposed in this paper require creativity and critical thinking
and they also have the ability to be altered each year, thus reducing the ability for students to
copy their report. These new projects also allow students to be more involved in the testing and
provide a hands-on experience, rather than watching the teaching assistants perform the
experiments as with the existing experiments.
53
It is clear that laboratory experiments and design projects are an important aspect of an
engineer’s undergraduate education and will continue to be a major part of an engineer’s
undergraduate education for many years. With the use of the personal computer, the range of
possible experiments has increased and has allowed students to be introduced to a larger variety
of applications. Design projects are relatively new and are becoming more popular. It seems
that design based courses may start to replace some of the more conventional courses, but it is
unlikely that the theory based, conventional courses will ever be completely replaced. These
conventional courses provide the students with the required knowledge and background theories
to excel in the design based courses. The combination of design based courses, and theory based
courses is what provides an engineering student with the required knowledge and experience to
excel in industry and is the reason why so much emphasis is being put on the use of experiments
and projects in an engineer’s undergraduate education.
54
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[15] Nakamura, Mealani. "How an Airfoil Works." Airfoil. 1999. Massachusetts Institute of Technology. 10 Jan. 2008 <http://web.mit.edu/2.972/www/reports/airfoil/airfoil.html>.
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Clarkson University. 25 Jan. 2008 <http://people.clarkson.edu/~pmarzocc/AE429/The%20NACA%20airfoil%20series.pdf>.
[17] Scott, Jeff. "NACA Airfoil Series." Aerospaceweb.Org. 26 Aug. 2001. AerospaceWeb
(Not for Profit Institution). 25 Jan. 2008 <http://www.aerospaceweb.org/question/airfoils/q0041.shtml>.
[18] Preston, Ray. "Drag." Aerodynamics Text. 27 Feb. 2008
<http://selair.selkirk.bc.ca/aerodynamics1/Drag/Page2.html>. [19] "Airfoil." McGraw-Hill Encyclopedia of Science and Technology, 5th Edition. The
McGraw-Hill Companies, Inc. 3 Mar. 2008 <http://www.answers.com/topic/airfoil?cat=technology>.
[20] "Prepared Classroom Water Tunnel Experiments." Prepared Classroom Experiments.
Rolling Hills Research Corporation. 6 Mar. 2008 <http://www.rollinghillsresearch.com/Water_Tunnels/Prepared_experiments.htm>.
[21] Selig, Michael S., John F. Donovan, and David B. Fraser. Airfoils At Low Speeds.
Virginia Beach: H.a. Stokely, 1989.
Appen
B.1 Fig
F
ndix B
gures The followi
Figure B-1 - D
Figure B-2
Collecti
ing is a colle
Developing vel
2 - Control vol
ion of Fig
ection of the
locity profiles
lume of steady
56
gures an
figures refe
and pressure
y fully develop
d Tables
erred to in th
changes in th
ped flow betw
e text of this
he entrance of
ween two sectio
s report.
a duct [9]
ons [9]
Figure B-3 - T
57
The Moody CChart [10]
Figure B-44 - Entrance l
Figure B
loss coefficient
B-5 - Loss coef
58
t as a functino
fficient for a su
o of rounding
udden contrac
of the inlet ed
ction
dge [9]
Figure
Figure B-6
B-7 – Series (a
6 - Loss coeffi
a), parallel (b)
59
icient for a sud
), and reservo
dden expansio
oir junction(c)
on[10]
) pipe systems
[9]
60
Figure B-8 - Test Apparatus Setup
61
Figure B-9 - Example Pipe Network
62
Figure B-10 – Cross-Section of Typical Airfoil[19]
Figure B-11- Flow Over a Cylinder[20]
Figure B-12 - Flow over a Streamlined Cylinder (Airfoil)[20]
63
Figure B-13 - Simplified Sketch of Hydraulic Flume
Figure B-14 - Example of Flow Visualization Experiment[20]
64
Cantilever Beam
Support Beam
Dial Indicator L
Flift
Fdrag
Airfoil
Hinge (used to set α)
Force Measuring
Stiff Rod Roller Support
Fweight
α
θ
Fluid Flow
Figure B-15 - Two Dimensional Sketch of Apparatus
Figure B-16 - Two Dimensional Sketch of Force Measurement Device
65
Rx,cantilever
Ry,support
W
θ β
Fresultant
Rx,beam
Ry,beam
Fm
β
Fresultant
Ry,beam
α
Fdrag
Flift -
Fweight
Rx,beam
Figure B-17 - Force Balance on Airfoil
Figure B-18 - Force Balance on Support Beam
66
Figure B-19 - Critical Dimensions of Hydraulic Flume
67
Figure B-20 - NACA 0009 Airfoil Cross Section[21]
Figure B-21 - NACA6409 Airfoil Cross Section[21]
68
Figure B-22 - NACA 0009 Characteristic Charts for 60,000<Re<300,000 [21]
69
Figure B-23 – NACA 0009 Characteristic Charts for Specific Reynolds Numbers [21]
70
Figure B-24 - NACA 6409 Characteristic Charts for 60,000 < Re < 200,000 [21]
71
Figure B-25 – NACA 6409 Characteristic Charts for Specific Reynolds Numbers [21]
72
Figure B-26 - Specially Designed NACA0009 Airfoil for Dye Injection
73
Figure B-27 - Specially Designed Test Cylinder for Dye Injection
74
Figure B-28 - NACA 6409 Airfoil (Drag Testing Only)
75
Figure B-29 - Support Beam Apparatus - View 1
76
Figure B-30 - Support Beam Apparatus - View 2 and Floating Platform
Figure B-31 - Side View of Apparatus
77
Figure B-32 - Floating and Fixed Platforms
78
Figure B-33 - Dye Wand Injecting Single Stream of Dye over Object in Water Tunnel [20]
79
B.2 Tables The following is a collection of the tables referred to in the text of this report.
Table B-1 - Loss coefficients K for open valvues, elbows and tees [9]
Nominal Diameter, in Screwed Flanged
½ 1 2 4 1 2 4 Valve (fully open): Globe Gate Swing Check Angle
14 0.3 5.1 9.0
8.2 0.24 2.9 4.7
6.9 0.16 2.1 2.0
5.7 0.11 2.0 1.0
13.0 0.80 2.0 4.5
8.5 0.35 2.0 2.4
6.0 0.16 2.0 2.0
Elbows: 45o regular 45o long radius 90o regular 90o long radius 180o regular 180o long radius
0.39
2.0 1.0 2.0
0.32
1.5 0.72 1.5
0.30
0.95 0.41 0.95
0.29
0.64 0.23 0.64
0.21 0.50 0.40 0.41 0.40
0.20 0.39 0.30 0.35 0.30
0.19 0.30 0.19 0.30 0.21
Tees: Line Flow Branch Flow
0.90 2.4
0.90 1.8
0.90 1.4
0.90 1.1
0.24 1.0
0.19 0.80
0.14 0.64
80
Table B-2 - Example Network Pipe Geometry Pipe Length
(m) Nominal Diameter
(mm) AB 0.5 19.05 BC 0.5 19.05 CD 0.25 19.05 AE 0.25 12.7 EF 0.5 12.7 BF 0.25 12.7 FG 0.5 12.7 CG 0.25 12.7 EI 0.25 12.7 HI 0.25 12.7 IM 0.25 12.7 MN 0.5 19.05 NJ 0.25 12.7 JF 0.25 12.7 JK 0.25 12.7 KL 0.25 12.7 GL 0.25 12.7 MO 0.25 12.7 OP 0.5 19.05 NP 0.25 12.7 PQ 0.25 19.05 KQ 0.5 19.05 QR 0.25 19.05 LR 0.5 12.7 RS 0.25 19.05
81
Table B-3 - Loss Coefficient Values used in Example Component K 90o Elbow 1.5
Tee 2.0 Valve:
Fully Open ¾ Open
Half Open ¼ Open
0.15 0.26 2.1 17.0
Table B-4 - Valve Positions in Example Network
Valve # Position 1 Fully open 2 Fully open 3 ¼ open 4 ¾ open 5 ¾ open
82
Table B-5 - Example Network Solution Pipe Flow Rate
(m3/s) Velocity
(m/s) AB 0.00005758 0.202 BC 0.00009136 0.3205 CD 0.000125 0.4386 AE 0.00005758 0.4545 EF 0.00006478 0.5114 BF 0.00003378 0.2667 FG 0.00003816 0.3013 CG 0.00003364 0.2655 EI 0.00012236 0.9659 HI 0.00025 1.9735 IM 0.00012764 1.0076 MN 0.00007801 0.2737 NJ 0.00004063 0.3207 JF 0.00000717 0.0566 JK 0.00003346 0.2641 KL 0.00002534 0.2 GL 0.00000452 0.0357 MO 0.00004963 0.3918 OP 0.00004963 0.1741 NP 0.00003738 0.2951 PQ 0.00008701 0.3053 KQ 0.00000812 0.0285 QR 0.00009514 0.3338 LR 0.00002986 0.2357 RS 0.000125 0.4386
Water Level in Tank (m)
0.90238006
Outlet 1 Flow Rate (m3/s)
0.000125
Outlet 2 Flow Rate (m3/s)
0.000125
Total Exit Flow Rate (m3/s)
0.000250
83
Table B-6 – Transient Stages in the Development of lift for an airfoil [9],[15] Stage Description Depiction
1 [9]
• Upon start-up of forward motion, a stagnation point briefly exists near the rear of the upper surface.
• Zero Lift
2 [9]
• Streamline on trailing edge is very sharp and cannot be maintained—thus, causing a starting vortex.
• Slight Lift
3a [15]
• The principle of angular momentum demands that the momentum of the vortex is equal and opposite to the rotational flow around the airfoil
• The rotational flow around the airfoil is known as circulation.
3b [15]
• By completing a vector sum of the fluid velocity above and below the airfoil, it is clear that the velocity above the airfoil is greater than the velocity below the airfoil—thus causing a pressure difference which can induce lift.
4 [9]
• Starting vortex is shed away from the airfoil
• Lift is 80% Developed
5 [9]
• Starting Vortex shed far behind the trailing edge of the vortex.
• Lift is fully developed
• The circulation shown in 3a sustains itself after the
starting vortex is shed and this allows the airfoil to maintain a constant level of lift.
84
Table B-7 - NACA Four-Series Airfoil Properties, ex. NACA 2415 [17] Digit(s) Property Interpretation
Chord Length Independent Variable (c) Designer may choose chord length as required
First Digit Maximum Camber (m) Indicated in percentage of chord length (ex. 0.02c)
Second Digit Position of Max Camber (p)
Indicated in tenths of chord and represents the position of the maximum camber from the leading edge of the airfoil (ex. 0.4c from leading edge)
Third & Fourth Digits Maximum Thickness (t)
Indicates the maximum thickness of the airfoil in percentage of chord length (ex. 0.15c)
Table B-8: NACA 4-Digit vs. NACA 5-Digit Series [16]
Series Advantages Disadvantages Typical Applications
4-Digit
1. Roughness has little effect 3. Good stall characteristics
1. Low maximum lift coefficient 2. Relatively high pitching moment 3. Relatively high drag
1. General Aviation 2. Supersonic Jets (Symmetric) 3. Helicopter Blades (Symmetric) 4. Missile Fins (Symmetric)
5-Digit
1. Roughness has little effect 2. Low pitching moment 3. Higher maximum lift coefficient
1. Poor stall behaviour 2. Relatively high drag
1. General Aviation 2. Business Jets
85
Table B-9 - Coordinates of NACA 0009 Airfoil [21] Data Point X-Coordinate Y-Coordinate Data Point X-Coordinate Y-Coordinate
1 1 0 36 0.00107 -0.00349 2 0.99572 0.00057 37 0.00428 -0.00767 3 0.98296 0.00218 38 0.00961 -0.01214 4 0.96194 0.00463 39 0.01704 -0.01646 5 0.93301 0.0077 40 0.02653 -0.02039 6 0.89668 0.01127 41 0.03806 -0.02395 7 0.85355 0.01522 42 0.05156 -0.0272 8 0.80438 0.01945 43 0.06699 -0.03023 9 0.75 0.02384 44 0.08427 -0.03305
10 0.69134 0.02823 45 0.10332 -0.03564 11 0.62941 0.03247 46 0.12408 -0.03795 12 0.56526 0.03638 47 0.14645 -0.03994 13 0.5 0.03978 48 0.17033 -0.04161 14 0.43474 0.04248 49 0.19562 -0.04295 15 0.37059 0.04431 50 0.22221 -0.04397 16 0.33928 0.04484 51 0.25 -0.04466 17 0.30866 0.04509 52 0.27886 -0.04504 18 0.27886 0.04504 53 0.30866 -0.04509 19 0.25 0.04466 54 0.33928 -0.04484 20 0.22221 0.04397 55 0.37059 -0.04431 21 0.19562 0.04295 56 0.43474 -0.04248 22 0.17033 0.04161 57 0.5 -0.03978 23 0.14645 0.03994 58 0.56526 -0.03638 24 0.12408 0.03795 59 0.62941 -0.03247 25 0.10332 0.03564 60 0.69134 -0.02823 26 0.08427 0.03305 61 0.75 -0.02384 27 0.06699 0.03023 62 0.80438 -0.01945 28 0.05156 0.0272 63 0.85355 -0.01522 29 0.03806 0.02395 64 0.89668 -0.01127 30 0.02653 0.02039 65 0.93301 -0.0077 31 0.01704 0.01646 66 0.96194 -0.00463 32 0.00961 0.01214 67 0.98296 -0.00218 33 0.00428 0.00767 68 0.99572 -0.00057 34 0.00107 0.00349 69 1 0 35 0 0
86
Table B-10 - Coordinates of NACA 6409 Airfoil [21]
Data Point X-Coordinate Y-Coordinate Data Point X-Coordinate Y-Coordinate 1 1 0 36 0.07428 -0.0108 2 0.99732 0.00084 37 0.10317 -0.00844 3 0.9893 0.00333 38 0.13607 -0.00513 4 0.97603 0.00737 39 0.17257 -0.00119 5 0.9576 0.01284 40 0.21235 0.00307 6 0.93423 0.01954 41 0.25498 0.00729 7 0.90615 0.02724 42 0.30012 0.01112 8 0.87357 0.03571 43 0.3473 0.01425 9 0.8369 0.04464 44 0.39618 0.01639
10 0.79647 0.05378 45 0.44707 0.01772 11 0.75272 0.06283 46 0.49868 0.01871 12 0.70608 0.07153 47 0.5504 0.01925 13 0.6571 0.07961 48 0.60167 0.01929 14 0.60627 0.08684 49 0.65193 0.0188 15 0.55413 0.09302 50 0.70065 0.0178 16 0.50132 0.09796 51 0.74728 0.01634 17 0.4484 0.10152 52 0.7913 0.01451 18 0.3959 0.1036 53 0.83223 0.01241 19 0.34367 0.10352 54 0.86957 0.01017 20 0.29315 0.10086 55 0.90288 0.00791 21 0.24502 0.09584 56 0.9318 0.00576 22 0.19988 0.08874 57 0.95593 0.00383 23 0.1583 0.07992 58 0.97503 0.00221 24 0.1208 0.06982 59 0.98883 0.00101 25 0.0878 0.05889 60 0.99722 0.00025 26 0.05968 0.04762 61 1 0 27 0.03677 0.03646 28 0.0192 0.02581 29 0.0072 0.01603 30 0.0008 0.00737 31 0 0 32 0.00467 -0.00573 33 0.01467 -0.00956 34 0.02973 -0.01157 35 0.0497 -0.01192
87
Table B-11 - Summary of Required Apparatus Features Theoretical
Concept Students are required to… Feature Description/ Accommodation Made
Flow Characteristics
• …measure velocity of fluid • Use Pitot tube which is currently used to measure velocity in hydraulic flume
Lift
• …measure actual lift • …determine theoretical lift
• Force measurement (i.e. Load Cell/Force Sensor/Dial Indicator)
• Vertical forces must be measured independent of the horizontal forces
Drag
• …measure actual drag • …determine theoretical drag
• Force measurement (i.e. Load Cell/Force Sensor/Dial Indicator)
• Horizontal forces must be measured independent of vertical forces
Stall
• …determine angle of attack at which stall occurs.
• Airfoil must be able to rotate about its centroidal axis.
• Airfoil must be fixed at a particular angle of attack.
Boundary Layer Separation
• …visualize flow patterns over an airfoil at various angles of attack
• Airfoil must accommodate for dye injection at various points on the leading edge of the airfoil.
Comparison of Airfoils
• …run identical tests on various airfoils to compare results
• Airfoils on apparatus must be easily interchangeable
88
Appendix C The Fundamental Objectives of Instructional Laboratories
The following is a list of thirteen fundamental objectives of engineering laboratories developed by a group of over fifty distinguished engineering educators assembled by ABET and the Sloan Foundation. All objectives start with the following: "By completing the laboratories in the engineering undergraduate curriculum, you will be able to...." Objective 1: Instrumentation. Apply appropriate sensors, instrumentation, and/or software tools to make measurements of physical quantities. Objective 2: Models. Identify the strengths and limitations of theoretical models as predictors of real-world behaviors. This may include evaluating whether a theory adequately describes a physical event and establishing or validating a relationship between measured data and underlying physical principles. Objective 3: Experiment. Devise an experimental approach, specify appropriate equipment and procedures, implement these procedures, and interpret the resulting data to characterize an engineering material, component, or system. Objective 4: Data Analysis. Demonstrate the ability to collect, analyze, and interpret data, and to form and support conclusions. Make order of magnitude judgments and use measurement unit systems and conversions. Objective 5: Design. Design, build, or assemble a part, product, or system, including using specific methodologies, equipment, or materials; meeting client requirements; developing system specifications from requirements; and testing and debugging a prototype, system, or process using appropriate tools to satisfy requirements. Objective 6: Learn from Failure. Identify unsuccessful outcomes due to faulty equipment, parts, code, construction, process, or design, and then re-engineer effective solutions. Objective 7: Creativity. Demonstrate appropriate levels of independent thought, creativity, and capability in real-world problem solving. Objective 8: Psychomotor. Demonstrate competence in selection, modification, and operation of appropriate engineering tools and resources. Objective 9: Safety. Identify health, safety, and environmental issues related to technological processes and activities, and deal with them responsibly. Objective 10: Communication. Communicate effectively about laboratory work with a specific audience, both orally and in writing, at levels ranging from executive summaries to comprehensive technical reports.
89
Objective 11: Teamwork. Work effectively in teams, including structure individual and joint accountability; assign roles, responsibilities, and tasks; monitor progress; meet deadlines; and integrate individual contributions into a final deliverable. Objective 12: Ethics in the Laboratory. Behave with highest ethical standards, including reporting information objectively and interacting with integrity. Objective 13: Sensory Awareness. Use the human senses to gather information and to make sound engineering judgments in formulating conclusions about real-world problems.
90
Appendix D Flow in Non-Circular Ducts
The control volume concept used for circular ducts, Figure 2.2, is still valid for non-circular
ducts, but now the cross-sectional area, A, is not πR2 and the wetted perimeter is not 2πR. The
momentum equation then becomes:
ΡΔ
=Δ+Δ
=A
Lg
zgph w
f ρτ
ρ
(2.9)
The only difference between this equation and the one for a circular duct is that the shear stress is
now an average value integrated over the perimeter, and the term A/P replaces R. This is why a
non-circular duct has a hydraulic radius defined as:
meterWettedPerireaSectionalACross
PARh
−==
(2.10)
The relationship for head loss for non-circular ducts then becomes:
22 212 Vp
gVhK m
ρΔ
==
(2.11)
gV
DLf
gV
RLfh
hhf 224
22
==
(2.12)
This is the same as for circular ducts, with the term d replaced by 4Rh. We can then define the
hydraulic diameter as:
hh RmeterWettedPeri
AreaPAD 444
=×
==
(2.13)
For a case with a non-circular duct the analysis can now remain the same by substituting the
hydraulic diameter for the pipe diameter.
91
Appendix E Design of a Simple Piping Network Project Handouts
The following are all of the project handouts that will be provided to the students. These handouts completely describe the project requirements as well as the background theories and concepts.
92
University of Toronto Department of Mechanical and Industrial Engineering
MIE312 – Fluid Mechanics I
Piping Network Design Project Your task is to design and construct a simple piping network such that both outlets have the same flow rate. To accomplish this task you must use at least 50 of the components on the attached page. As well, your network must consist of at least 5 independent loops. Attached is a table that gives all of the loss coefficients corresponding to each of the above components, as well as fluid properties and roughness values for the pipe. Your network is to be attached to a tank of water (see attached diagrams of apparatus) – you will be required to determine the water level in the tank based on the required outlet flow rate given below. Your group’s design will be tested on the test date shown below. At this time you will be required to assemble your network with the given components. The tank will then be filled to the level specified by your group and the outlet flow rates will be measured. The flow measurements will be recorded and a winner will be determined based on the following formula:
RatequiredFlowlowRateTotalExitF
FlowRateletAverageOutFlowRateletAverageOutioWinningRat
Re21
×=
where, • Outlet1FlowRate and Outlet2FlowRate are the average of the measured outlet flow rates
(numerous measurements will be taken and an average value will be used). • TotalExitFlowRate is the sum of your two average outlet flow rates. • RequiredFlowRate is the total exit flow rate specified at the end of this handout.
NOTE: Outlet2FlowRate represents the higher of the outlet flow rates, as well, the second ratio will be flipped such that it is less than 1.0, and therefore the best ratio will be 1.0. Once your network has been tested your group will be required to write a report. This report must contain calculations to support your design; as well you are to discuss any discrepancies between your calculations and your actual results. It is expected that the majority of your report will focus on discussing these discrepancies. Remember that there isn’t one correct design, rather various designs can be used to accomplish your task – positions of valves can be changed as required, but no valve can be 100% closed. Attached is a set of guidelines/rules for this competition – a penalty will be applied to your groups’ ratio if any of the rules are not followed. Required Total Exit Flow Rate: 0.000250 m3/s Test Date: Report Due Date:
93
Table E-1 - Loss Coefficients, and Constants Component K
90o Elbow 1.5 Tee 2.0 Valve: Fully Open ¾ Open Half Open ¼ Open
0.15 0.26 2.1 17.0
ε (m) 1.5 x 10-6 ρ (kg/m3) 998 μ (N s/m2) 1.003 x 10-3 g (m/s2) 9.81
Table E-2 - Parts List
Component Size (mm) No.
Straight Pipe
D = 12.7
19
Straight Pipe
D = 19.05
14
Elbow
D1 = 12.7 D2 = 19.05
2
Tee
D1 = 19.05 D2 = 12.7 D3 = 12.7
3
Tee
D1 = 12.7 D2 = 12.7 D3 = 12.7
5
94
Tee
D1 =12.7
D2 = 19.05 D3 = 19.05
5
Cross
D1 = 12.7 D2 = 12.7 D3 = 12.7 D4 = 12.7
4
Pipe Connector D = 19.05 5 Pipe Connector D = 12.7 3
Ball Valve D = 12.7 3 Ball Valve D = 19.05 2
Competition Rules
1. You must use at least 50 of the components on the parts list. 2. All valves must be at least ¼ open – no valve can be fully closed.
3. You may only have 2 outlets/exits.
4. No valve may be placed such that it directly controls the outlet flow.
5. Your network cannot extend vertically.
6. Valve #1 on the attached diagram must be fully open (this valve is only used for
construction/assembly purposes).
95
Figure E-1 - Test Apparatus Setup
Figure E-2 - Test Setup
96
How to Analyze a Simple Pipe Network From your study of internal flows in pipes and ducts you have the knowledge to solve for flow rates, head loss (frictional losses as well as minor losses), pressures, and other variables given certain known parameters for a single length of pipe. This analysis mainly utilizes Bernoulli’s equation and the Moody diagram. In the analysis of complex pipe networks consisting of numerous inlets, outlets and loops, we will utilize many of the same concepts. Typically, specialized software is used in the analysis of complex pipe networks, but before this software was developed less sophisticated techniques were used in the analysis of these systems. Simple spreadsheets can be programmed to help analyze these networks using the most basic technique, known as the Hardy Cross Method. The Hardy Cross Method is the traditional approach used for solving pipe networks. This method is applicable when all pipe sizes (diameters and lengths) are known, and either head losses between inlets and outlets are known but the flow rates are not, or the flow rates throughout the network are known but the head losses are not. We will be exploring the case where the head losses are known (or can be calculated) and the flow rates are to be determined. Calculating Head Loss The total head loss in a pipe is the sum of major and minor losses:
∑+= orfrictionloss hhh min
Major losses in this case will be due to friction. The Darcy-Weisbach Equation is utilized to determine the head loss due to friction in a pipe and can be written as follows:
gV
DLfhfriction 2
2
=
and since AVQ ×= , we can write:
QQgAD
Lfhfriction 211
2=
where,
• f is the friction factor. • L is the length of pipe. • D is the pipe diameter. • V is the fluid velocity. • g is the acceleration due to gravity. • Q is the volumetric flow rate. • A is the cross sectional area of the pipe.
97
In MIE 312, you have used the Moody diagram to find the value of the friction factor, f, but this can be very time consuming if many iterations are required. The Haaland equation is an approximation of the Colebrook equation and can be used to directly solve for the Darcy-Weisbach friction factor:
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠⎞
⎜⎝⎛+−=
11.1
7.3Re9.6log8.11 d
fε
where, • Re is the Reynolds Number. • ε is the pipe roughness.
The Reynolds number is a dimensionless parameter and can be expressed as follows:
μρVD
=Re
where, • ρis the fluid density. • V is the fluid velocity. • D is the pipe diameter. • μ is the fluid viscosity.
The total minor losses can be calculated using the following formula:
∑= Kg
Vh or 2
2
min
where, • K is a loss coefficient for bends and valves. • V is the mean fluid velocity. • g is the acceleration due to gravity.
The Hardy Cross Method
The Hardy Cross Method is an iterative approach that can be used to solve for flow rates in a pipe network. The Hardy Cross Method utilizes two rules:
3) The total head loss around a loop must equal zero. 4) The total flow into a junction is equal to the total flow out of a junction.
This method involves making an initial guess for the flow rates throughout the system,
making sure that the total flow into a junction is equal to the total flow out of a junction. The total head loss around each is then calculated based on the initial guess of flow rates. Next, you check whether the total head loss around each loop is zero. If the total head loss around each loop is zero your initial guess for flow rates was correct and you are done, if not, you make
98
adjustments to the flow rates and re-calculate the head loss. This is then repeated until the total head loss around each loop is zero. The following is the detailed procedure:
1. Define a set of independent pipe loops so that every pipe is part of at least one loop, and no loop can be represented by the sum or difference of other loops. The easiest way to do this is to choose all of the smallest possible loops in the network.
2. Arbitrarily choose values of flow rates, Q, so that the total flow into a junction equals the total flow out of a junction (a better initial guess will require fewer iterations, but any initial guess will eventually converge). Define a sign convention for each loop – a common sign convention is to define clockwise flow as positive for the loop, but remember this may mean that the same flow in a given pipe might be considered positive for one loop and may be considered negative flow when considered in another loop.
3. Calculate the head loss in each pipe using the same sign convention for head loss as for flow (Q should have the same sign has hloss for a given pipe).
4. Calculate the total head loss around each loop. If the total head loss around every loop is zero then the problem is solved.
5. If the head loss around every loop is not zero change the flow rate in every pipe of a loop by ΔQ. By changing the flow rates of all the pipes in the loop by the same amount we ensure that the total flow into a junction is still equal to the flow out of a junction. The key is to select ΔQ so that the head loss around the loop approaches zero (a formula to find ΔQ is given below). Remember that if a given pipe is part of two loops you must apply ΔQ for both of the loops to the previous value of Q (be careful of +/- signs).
The following formula has been derived for ΔQ:
∑
∑−=Δ
Loop
loss
Looploss
Qh
hQ
2
99
Calculating Pressure
Once you have found the values of flow rates for each length of pipe in your network you can determine the required water level in the tank by using Bernoulli’s equation. Bernoulli’s equation can also be used to determine the value of pressure at any point in the network. Bernoulli’s equation can be written in the following form:
losshZg
Vg
PZg
Vg
P+++=++ 2
222
1
211
22 ρρ
where, • Pi is the pressure at that point. • Vi is the velocity at that point. • Zi is the height at that point. • hloss is the total head loss between the points.
The next two examples show you how to analyze a simple pipe network.
100
Example #1 Below is a simple example to illustrate how to use the Hardy Cross method. In this example minor losses were ignored, and all outlet flows are known.
You can set up a spreadsheet to help solve your network like the one below (not all required columns are shown below). The last column shows the flow rate after six iterations. At this point the head loss values are close to zero, but more iterations are required since the change in flow rates is greater than 5% in some cases - once the change in flow rates is less than 5% we can assume that this is as accurate as we will ever get since the equation that we use for f has an error associated with it.
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Loop Pipe D … Guess
Q … f HL HL/Q ∑HL ∑ (HL/Q)
ΔQ Q new
Q6 A ab 0.3 … 0.04 … 0.023 0.33 8.1296 0.0213 9.62E-03 be 0.1 … 0.04 … 0.025 34.3 858.96 0.0213 0.009618 ed 0.2 … 0.025 … 0.025 0.51 20.345 0.0105 0.0065 dc 0.2 … -0.025 … 0.025 -0.51 20.345 -0.0345 -0.0317 ca 0.2 … -0.02 … 0.026 -0.27 13.453 -0.0387 -0.05038 34.41 921.2 -0.0187
B cd 0.2 … 0.025 … 0.025 0.51 20.345 0.0345 0.03167 dg 0.15 … 0.02 … 0.026 1.13 56.592 0.0150 0.0082 gf 0.25 … 0.02 … 0.026 0.11 5.6030 0.0108 -0.0037 fc 0.15 … 0.005 … 0.032 0.09 17.848 -0.0042 -0.01871 1.84 100.4 -0.0092
C de 0.2 … -0.025 … 0.025 -0.51 20.345 -0.0105 eh 0.1 … 0.015 … 0.027 5.12 341.54 0.0108 -0.0065 hg 0.25 … 0 … 0 0.00 0 -0.0042 0.0031 gd 0.15 … -0.02 … 0.026 -1.13 56.592 -0.0150 -0.01189 3.48 418.5 -0.0042
NOTE: Remember to apply all of the required correction factors to pipes that are in more than one loop. For example: Loop A, pipe cd: Q i+1 = Q i + ΔQ loop A – ΔQloop B Loop B, pipe cd: Q i+1 = Q i + ΔQ loop B – ΔQloop A
102
Example #2 This example shows how to find the water level in a tank. If the exit flow rate and all dimensions of the pipe are known we can determine the water level of the tank.
To solve for the water level we must utilize Bernoulli’s equation:
fhZg
Vg
PZg
Vg
P+++=++ 1
211
2
222
22 ρρ
Since the tank diameter is much larger than the pipe diameter we can assume that the velocity at point 2 is zero, as well, the pressure at points 1 and 2 is atmospheric. This then simplifies to:
fhg
Vh +=2
22
and since g
VDLfhf 2
22= , we can simplify this further to:
⎟⎠⎞
⎜⎝⎛ +=
DLf
gVh 12
22
103
Appendix F Computer Code Developed to Solve Example Pipe Network
The following is a computer program written in the C programming language used to solve the example pipe network presented in section 2.2.3. Given the pipe geometry, initial guesses for flow rates, loss coefficients, pipe roughness and head loss relationships, this program iterates to solve for the flow rates in each pipe of the network. As well, this program will determine the water level in the tank required to deliver such flow rates.
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#include <stdio.h> #include <math.h> main(){ int i, m; double g, mu, rho, e, Qin, Kelbow, Kt, KVopen, KVthreequartersopen, KVhalf, KVquarteropen; double Q1out, Q2out, Qouttotal, TankWaterLevel; double fabs(double), log10(double), pow(double, double); double EoverD[100], D[100], A[100], Q[10000], V[10000], L[100], k[1000], Re[10000], f[10000], HL[10000], HLoverQ[10000]; double Loop0HL, Loop1HL, Loop2HL, Loop3HL, Loop4HL, Loop5HL, Loop6HL, Loop7HL, THL; double Loop0HLoverQ, Loop1HLoverQ, Loop2HLoverQ, Loop3HLoverQ, Loop4HLoverQ, Loop5HLoverQ, Loop6HLoverQ, Loop7HLoverQ; double deltaQ0, deltaQ1, deltaQ2, deltaQ3, deltaQ4, deltaQ5, deltaQ6, deltaQ7, Qnew[1000]; //Define constants// g = 9.81; //Define fluid properties// mu = 0.001003; rho = 998.0; //Define pipe roughness// e = 0.0000015; //Define loss coefficients// Kelbow = 1.5; Kt = 2.0; KVopen = 0.15; KVquarteropen = 17.0; KVthreequartersopen = 0.26; KVhalf = 2.1; Qin = 0.00025; //Define flow rate into network// //Define all diameters in meters// D[0] = 0.01905; D[1] = 0.0127; D[2] = 0.0127; D[3] = 0.0127; D[4] = 0.0127; D[5] = 0.0127; D[6] = 0.0127; D[7] = 0.01905; D[8] = 0.0127; D[9] = 0.0127; D[10] = 0.0127; D[11] = 0.0127; D[12] = 0.0127; D[13] = 0.0127; D[14] = 0.01905; D[15] = 0.01905; D[16] = 0.01905;
105
D[17] = 0.0127; D[18] = 0.01905; D[19] = 0.0127; D[20] = D[3]; D[21] = D[2]; D[22] = D[9]; D[23] = D[8]; D[24] = D[10]; D[25] = D[15]; D[26] = D[11]; D[27] = D[7]; D[28] = D[17]; D[29] = 0.0127; D[30] = 0.0127; D[31] = 0.01905; D[32] = 0.01905; D[33] = D[1]; D[34] = D[12]; D[34] = D[12]; D[35] = D[13]; D[36] = 0.01905; D[37] = 0.0127; D[38] = 0.0127; D[39] = 0.0127; //Define all lengths in meters// L[0] = 0.5; L[1] = 0.25; L[2] = 0.5; L[3] = 0.5; L[4] = 0.25; L[5] = 0.25; L[6] = 0.25; L[7] = 0.5; L[8] = 0.25; L[9] = 0.25; L[10] = 0.25; L[11] = 0.25; L[12] = 0.25; L[13] = 0.5; L[14] = 0.25; L[15] = 0.5; L[16] = 0.25; L[17] = 0.25; L[18] = 0.5; L[19] = 0.25; L[20] = L[3]; L[21] = L[2]; L[22] = L[9]; L[23] = L[8]; L[24] = L[10]; L[25] = L[15]; L[26] = L[11]; L[27] = L[7]; L[28] = L[17]; L[29] = 0.25; L[30] = 0.25;
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L[31] = 0.5; L[32] = 0.15; L[33] = L[1]; L[34] = L[12]; L[35] = L[13]; L[36] = 0.15; L[37] = 0.5; L[38] = 0.5; L[39] = 0.25; //Define loss coefficients// k[0] = 2*Kt; k[1] = 2*Kt; k[2] = 2*Kt; k[3] = 2*Kt; k[4] = Kt + Kelbow; k[5] = 2*Kt; k[6] = 2*Kt + KVopen; k[7] = 2*Kt; k[8] = 2*Kt; k[9] = 2*Kt; k[10] = 2*Kt; k[11] = 2*Kt; k[12] = 2*Kt; k[13] = 2*Kt + KVthreequartersopen; k[14] = 2*Kt + KVthreequartersopen; k[15] = 2*Kt; k[16] = 2*Kt; k[17] = 2*Kt; k[18] = Kt + Kelbow; k[19] = Kt + Kelbow; k[20] = k[3]; k[21] = k[2]; k[22] = k[9]; k[23] = k[8]; k[24] = k[10]; k[25] = k[15]; k[26] = k[11]; k[27] = k[7]; k[28] = k[17]; k[29] = 2*Kt; k[30] = 2*Kt; k[31] = Kt + Kelbow + KVquarteropen; k[32] = Kt + Kelbow; k[33] = k[1]; k[34] = k[12]; k[35] = k[13]; k[36] = Kt + Kelbow; k[37] = Kt + Kelbow; k[38] = Kt + Kelbow; k[39] = Kt + KVopen; //Calculate cross sectional areas, and rougness over diameter ratios// for(i=0; i<=39; i++){ A[i] = M_PI*D[i]*D[i]/4; EoverD[i] = e/D[i];
107
} //end "for"// //Initial guesses for flow rates// Q[0] = 0.125*Qin; Q[1] = -0.375*Qin; Q[2] = -0.25*Qin; Q[3] = -0.25*Qin; Q[4] = 0.25*Qin; Q[5] = 0.5*Qin; Q[6] = -0.5*Qin; Q[7] = -0.25*Qin; Q[8] = -0.375*Qin; Q[9] = 0.125*Qin; Q[10] = -0.5*Qin; Q[11] = -0.25*Qin; Q[12] = -0.125*Qin; Q[13] = 0.125*Qin; Q[14] = -0.375*Qin; Q[15] = -0.25*Qin; Q[16] = -0.125*Qin; Q[17] = 0.125*Qin; Q[18] = -0.25*Qin; Q[19] = -0.25*Qin; Q[20] = 0.25*Qin; Q[21] = 0.25*Qin; Q[22] = -0.125*Qin; Q[23] = 0.375*Qin; Q[24] = 0.5*Qin; Q[25] = 0.25*Qin; Q[26] = 0.25*Qin; Q[27] = 0.25*Qin; Q[28] = -0.125*Qin; Q[29] = -0.125*Qin; Q[30] = 0.125*Qin; Q[31] = 0.25*Qin; Q[32] = 0.5*Qin; Q[33] = 0.375*Qin; Q[34] = 0.125*Qin; Q[35] = -0.125*Qin; Q[36] = -0.5*Qin; Q[37] = -0.5*Qin; Q[38] = 0.5*Qin; Q[39] = Qin; m=0; //Set counter to zero// THL = 5.0; //Give initial value for THL// V[39] = fabs(Q[39]/A[39]); Re[39] = rho*V[39]*D[39]/mu; f[39] = pow(1/(-1.8*log10(6.9/Re[39] + pow(EoverD[39]/3.7, 1.11))), 2.0); HL[39] = Q[39]*Q[39]/(2*g*A[39]*A[39])*(f[39]*L[39]/D[39] + k[39] ); while (THL>=0.000000000000001){ //Calculates velocity, Reynolds number, friction factor, head loss and ratio of head loss over flow rate for each pipe// for(i=0; i<=38; i++){
108
V[i] = fabs(Q[i])/A[i]; Re[i] = rho*V[i]*D[i]/mu; f[i] = pow(1/(-1.8*log10(6.9/Re[i] + pow(EoverD[i]/3.7, 1.11))), 2.0); HL[i] = fabs(Q[i])*Q[i]/(2*g*A[i]*A[i])*(f[i]*L[i]/D[i] + k[i] ); HLoverQ[i] = HL[i]/Q[i]; } //end "for"// //Total Head Loss for Loop 0// Loop0HL = HL[0] + HL[1] + HL[2] + HL[29]; Loop0HLoverQ = HLoverQ[0] + HLoverQ[1] + HLoverQ[2] + HLoverQ[29]; //Total Head Loss for Loop 1// Loop1HL = HL[5] + HL[6] + HL[7] + HL[8] + HL[9] + HL[20]; Loop1HLoverQ = HLoverQ[5] + HLoverQ[6] + HLoverQ[7] + HLoverQ[8] + HLoverQ[9] + HLoverQ[20]; //Total Head Loss for Loop 2// Loop2HL = HL[18] + HL[19] + HL[27] + HL[28]; Loop2HLoverQ = HLoverQ[18] + HLoverQ[19] + HLoverQ[27] + HLoverQ[28]; //Total Head Loss for Loop 3// Loop3HL = HL[10] + HL[11] + HL[12] + HL[21] + HL[22]; Loop3HLoverQ = HLoverQ[10] + HLoverQ[11] + HLoverQ[12] + HLoverQ[21] + HLoverQ[22]; //Total Head Loss for Loop 4// Loop4HL = HL[16] + HL[17] + HL[23] + HL[24] + HL[25]; Loop4HLoverQ = HLoverQ[16] + HLoverQ[17] + HLoverQ[23] + HLoverQ[24] + HLoverQ[25]; //Total Head Loss for Loop 5// Loop5HL = HL[13] + HL[14] + HL[15] + HL[26]; Loop5HLoverQ = HLoverQ[13] + HLoverQ[14] + HLoverQ[15] + HLoverQ[26]; //Total Head Loss for Loop 6// Loop6HL = HL[30] + HL[3] + HL[4] + HL[31]; Loop6HLoverQ = HLoverQ[30] + HLoverQ[3] + HLoverQ[4] + HLoverQ[31]; //Total Head Loss for Loop 7// Loop7HL = HL[32] + HL[33] + HL[34] + HL[35] + HL[36] + HL[37] + HL[38]; Loop7HLoverQ = HLoverQ[32] + HLoverQ[33] + HLoverQ[34] + HLoverQ[35] + HLoverQ[36] + HLoverQ[37] + HLoverQ[38]; //Calculate the total head loss in network// THL = fabs(Loop0HL) + fabs(Loop1HL) + fabs(Loop2HL) + fabs(Loop3HL) + fabs(Loop4HL) + fabs(Loop5HL) + fabs(Loop6HL) + fabs(Loop7HL); //If THL is basically zero, stop// if ( THL<0.000000000001 ){ break; } //end "if"// //If THL is not zero, find delta Q and find new Q// else{ //Calculate delta Q for each loop//
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deltaQ0 = -Loop0HL/(2*Loop0HLoverQ); deltaQ1 = -Loop1HL/(2*Loop1HLoverQ); deltaQ2 = -Loop2HL/(2*Loop2HLoverQ); deltaQ3 = -Loop3HL/(2*Loop3HLoverQ); deltaQ4 = -Loop4HL/(2*Loop4HLoverQ); deltaQ5 = -Loop5HL/(2*Loop5HLoverQ); deltaQ6 = -Loop6HL/(2*Loop6HLoverQ); deltaQ7 = -Loop7HL/(2*Loop7HLoverQ); //Calculate the new value of Q for each pipe// //Loop 0// Qnew[0] = Q[0] + deltaQ0; Qnew[1] = Q[1] + deltaQ0 - deltaQ7; Qnew[2] = Q[2] + deltaQ0 - deltaQ3; Qnew[29] = Q[29] + deltaQ0 - deltaQ6; //Loop 6// Qnew[3] = Q[3] + deltaQ6 - deltaQ1; Qnew[4] = Q[4] + deltaQ6; Qnew[30] = Q[30] + deltaQ6 - deltaQ0; Qnew[31] = Q[31] + deltaQ6; //Loop1// Qnew[5] = Q[5] + deltaQ1; Qnew[6] = Q[6] + deltaQ1; Qnew[7] = Q[7] + deltaQ1 - deltaQ2; Qnew[8] = Q[8] + deltaQ1 - deltaQ4; Qnew[9] = Q[9] + deltaQ1 - deltaQ3; Qnew[20] = Q[20] + deltaQ1 - deltaQ0; //Loop 2// Qnew[18] = Q[18] + deltaQ2; Qnew[19] = Q[19] + deltaQ2; Qnew[27] = Q[27] + deltaQ2 - deltaQ1; Qnew[28] = Q[28] + deltaQ2 - deltaQ4; //Loop 3// Qnew[10] = Q[10] + deltaQ3 - deltaQ4; Qnew[11] = Q[11] + deltaQ3 - deltaQ5; Qnew[12] = Q[12] + deltaQ3 - deltaQ7; Qnew[21] = Q[21] + deltaQ3 - deltaQ0; Qnew[22] = Q[22] + deltaQ3 - deltaQ1; //Loop 4// Qnew[16] = Q[16] + deltaQ4; Qnew[17] = Q[17] + deltaQ4 - deltaQ2; Qnew[23] = Q[23] + deltaQ4 - deltaQ1; Qnew[24] = Q[24] + deltaQ4 - deltaQ3; Qnew[25] = Q[25] + deltaQ4 - deltaQ5; //Loop 5// Qnew[13] = Q[13] + deltaQ5 - deltaQ7; Qnew[14] = Q[14] + deltaQ5; Qnew[15] = Q[15] + deltaQ5 - deltaQ4; Qnew[26] = Q[26] + deltaQ5 - deltaQ3; //Loop7//
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Qnew[32] = Q[32] + deltaQ7; Qnew[33] = Q[33] + deltaQ7 - deltaQ0; Qnew[34] = Q[34] + deltaQ7 - deltaQ3; Qnew[35] = Q[35] + deltaQ7 - deltaQ5; Qnew[36] = Q[36] + deltaQ7; Qnew[37] = Q[37] + deltaQ7; Qnew[38] = Q[38] + deltaQ7; for(i=0; i<=38; i++){ Q[i] = Qnew[i]; } //end "for"// m = m + 1; //Increase counter by 1 to count number of iterations// } //end "else"// } //end "while"// //Print the final values of Q, V, f and HL for all pipes// for(i=0; i<=39; i++){ printf("V%d = %0.4f m/s \n", i, V[i]); } //end "for"// Q1out = fabs(Q[32]); Q2out = fabs(Q[36]); Qouttotal = Q1out + Q2out; printf("Q1,out = %0.6f \n Q2,out = %0.6f \n Qout = %0.6f \n", Q1out, Q2out, Qouttotal); TankWaterLevel = fabs(HL[39]) + fabs(HL[5]) + fabs(HL[4]) + fabs(HL[31]) + fabs(HL[0]) + fabs(HL[32]) + Q1out*Q1out/(A[32]*A[32]*2*g); printf("Water Level = %0.8f meters \n", TankWaterLevel); }
111
The following are the velocity values in each section of pipe, as well as the flow rates at exit, and water level in the tank required to drive these flows: V0 = 0.3267 m/s V1 = 0.2541 m/s V2 = 0.3069 m/s V3 = 0.5531 m/s V4 = 0.4803 m/s V5 = 1.0333 m/s V6 = 0.9402 m/s V7 = 0.2532 m/s V8 = 0.2851 m/s V9 = 0.0087 m/s V10 = 0.2764 m/s V11 = 0.1883 m/s V12 = 0.0528 m/s V13 = 0.2412 m/s V14 = 0.3303 m/s V15 = 0.0391 m/s V16 = 0.2911 m/s V17 = 0.2845 m/s V18 = 0.1647 m/s V19 = 0.3706 m/s V20 = 0.0515 m/s V21 = 0.3069 m/s V22 = 0.0087 m/s V23 = 0.2851 m/s V24 = 0.2764 m/s V25 = 0.0391 m/s V26 = 0.1883 m/s V27 = 0.2532 m/s V28 = 0.2845 m/s V29 = 0.2549 m/s V30 = 0.2549 m/s V31 = 0.2135 m/s V32 = 0.4397 m/s V33 = 0.2541 m/s V34 = 0.0528 m/s V35 = 0.2412 m/s V36 = 0.4375 m/s V37 = 0.9843 m/s V38 = 0.9892 m/s V39 = 1.9735 m/s Q1,out = 0.000125 Q2,out = 0.000125 Water Level = 0.94472040 meters
112
Appendix G Theoretical Analysis of Hydraulic Flume
Derivation of Equation 4.4 Bernoulli’s Equation:
Test
TestTestTank
TankTank ghVPgHVP++=++
22
22
ρρ
We know that P1 and P2 are at atmospheric pressure which means 0 kPa (gage).
TestTank ghVgHV
+=+22
22
21 (a)
We assume that the flow through a sluice gate is steady flow where mass is conserved, which means that flow in = flow out. Let w = width of tank and VTank can be written as:
TestTank
TestTestTankTank
whVwHVVAVAQ
===
Tank
TestTestTank H
hVV = (b)
We substitute (**) into (*) to get:
[ ]TankTestTank
TestTest
TestTest
TankTank
TestTest
HhgHhV
ghVgHHhV
−=⎥⎥⎦
⎤
⎢⎢⎣
⎡−⎟⎟
⎠
⎞⎜⎜⎝
⎛
+=+⎟⎟⎠
⎞⎜⎜⎝
⎛
12
2222
222
[ ] [ ]
[ ]( )( )TestTankTestTank
TestTankTankTest
TestTank
TestTankTank
Tank
Test
TestTankTest
hHhHhHgHV
hHhHgH
Hh
hHgV
+−−
=
−−
=
⎟⎟⎠
⎞⎜⎜⎝
⎛−
−=
22
22
2
2
2
2
12
.
TestTank
TankTest hH
gHV+
=22 (c)
Determining Maximum Theoretical Test Section Velocity of Hydraulic Flume
smmm
msmVTest /59.225.051.0
)51.0)(/81.9(2 22
=+
=
113
Appendix H Experimental Analysis of Hydraulic Flume Results of Calibration Experiment
Table H-1 - Results of Calibration Experiment Measurements Taken Value
Opening Height of Sluice Gate 5.1cm Closed Height of Exit Gate 8.9cm
X – height of water in pitot tube above free surface 29.2cm h1 – height of water at Point 1 3.18cm
hTest – height of water at test section 15.24cm H – height of water in reservoir 31.75cm
Calculation of Test Section Velocity The velocity of the flow was measured at Point 1 since the Pitot tube was not able to register a readable measurement in the test section due to a low velocity. Using these results, the velocity of the fluid in the test section can be determined as follows: Pitot Tube Analysis at Point 1
2 2
2 0 0
2
(a)
Conservation of Volumetric Flow (between Point 1 and Test Section)
(b)
(a) (b)
2
(c)
Test Section Velocity: . . ..
. /
114
Photographs of Calibration Experiment
Figure H-1 - Hydraulic Flume and Jump During Calibration Experiment
Figure H-2 - Flow Prior to Jump and Pitot Tube Flow Measurement Device
115
Appendix I Maximum Load on Force Indicating Apparatus In order to size the force indicating apparatus for the maximum horizontal load, it is useful to determine the maximum force exerted on the airfoil due to fluid momentum. In order to simplify the analysis, the test airfoils can be considered flat plates with identical planform areas. The maximum load on the force indicating apparatus will occur when the airfoil is perpendicular to the fluid flow direction—or in other words, when the angle of attack is 90o. A depiction of this setup is shown in Figure I-1.
Variables:
• ρ = fluid density • A = planform area • α = angle of attack • θ = ridgid beam (constant) • Vin = fluid velocity measured up stream of airfoil • V2 = V3 = ½ V1 = Exit Velocity • W = Weight of Airfoil = ρairfoil*Volumeairfoil
Assumptions:
• Airfoil can be modeled as a flat plate. • Airfoil is held perpendicular to the fluid flow. • Exit velocity and mass flow is equal in the positive and negative vertical
directions.
Rx
α
W
x
θVin
Vout
Vout
y
Ry
Figure I-1 - Momenutm Analysis of Airfoil at 90-Degrees
116
Force Balance in x-Direction:
2)(
0
inxinxinxx
o
outxoutx
o
inxinx
o
x
x
AVVAVR
AVmandVmVmR
F
ρρ
ρ
==
=−=
=∑
Force Balance in y-Direction:
airfoilairfoilx
downoutdownoutyupoutupouty
oo
outyouty
o
inyinyy
y
VolumeWR
WVmVmRAVmandWVmVmR
F
×==
+−−−=
=+−=
=
∑∑
∑
ρ
ρ
)(
0
,,,,
Force on Horizontal Force Indicating Apparatus for Experimental Conditions
• Assume a factor of safety to be FS = 1.25 since the airfoils will only be tested at 0o<α<25o.
• Assume beam to be rigid (no bending moment experienced by beam)
( )( ) Nsmmm
mkgF
FSVbcFSAVF
FSRF
indicatordial
inxinxindicatordial
xindicatordial
244.7)25.1(5.01524.01524.0998
)(2
3
22
=⎟⎠⎞
⎜⎝⎛=
==
×=
−
−
−
ρρ
Thus, the maximum force that can be exerted on the force indicating apparatus is 7.244N assuming a factor of safety of 1.25.
117
Appendix J Cantilever Beam Design Using basic principles of solid mechanics and beam theory, the length of a beam can be determined to ensure that the maximum deflection of the beam does not exceed the maximum deflection measurable by the dial indicator. Reasonable dimensions for the width, b, and the height, h, of the cantilever beam are ½” and ¼” respectively.
Using CES Selector 2007, it was determined that the elastic modulus, or Young’s modulus, of low carbon steel is 200GPa. If the material is changed, this calculation should be completed once again to determine the appropriate length of the beam. The dial gauge used in this experiment is a metric, Baty dial gauge with a resolution of 0.002mm. The maximum displacement that can be measured is 12.7mm. Equation X, is a standard equation for a point load at the end of a cantilever beam and is based on the conditions shown in Figure J-1. Using equation X, the height of the cantilever beam can be determined since the maximum force and displacement is known, and the width and length can be assumed to be 0.0127m and 0.2032m respectively.
(a)
(b)
(c)
.. .
.
.
Therefore, the height of the cantilever must be, at greatest, 0.00196 if the other conditions remain fixed. The parameters length and width were chosen for practical purposes; however, any of the two can be selected when sizing the beam.
Figure J-1 - Point Load At End of Cantilever Beam
b
hδma
L
F x
y
118
Appendix K External Flow Investigation: Project Handouts
The following are is the experiment handout that will be provided to the students. The handout includes an introduction, objective, background information, experimental procedure, and suggested discussion and analysis topics.
119
University of Toronto Department of Mechanical and Industrial Engineering
MIE312 – Fluid Mechanics I
External Flow over Airfoils
Introduction The development of the design of aircraft wings has paralleled the theoretical and experimental study of fluid dynamics; however, the first engineers to study airfoils relied on solely on experimentation to determine the characteristics of various airfoils. When fluid flows over an airfoil, various forces are produced and an analysis of these forces can be difficult to conceptualize using only theory. Lift and drag forces are the primary concepts associated with airfoil experimentation. The second major concept associated with external flow is flow separation. This is typically analyzed using smoke streams in air tunnels or dye streams in water tunnels in order to visualize the flow characteristics around an airfoil. Experimental fluid dynamics is still prominent today due to the complex shapes and geometries that are often difficult and costly to model using standard equations and computer software. Objective The objective of this laboratory is to provide students with exposure to external flow analysis over airfoils to understand the concepts of lift, drag, angle of attack, stall and flow separation. Background For this experiment, the NACA0009 and NACA6409 airfoils are going to be tested in the hydraulic flume located in MC214 to determine the resultant force on the airfoil. In addition, a cylinder will be tested to compare the effect that a streamlined object has on drag as opposed to the blunt cylinder. Bernoulli’s Equation Bernoulli’s equation is useful in determining the flow characteristics of an open-channel flow at any point of the channel after taking relatively simple measurements. It is shown below in Equation (a). It is important to note that Equation A does not account for energy loss in the channel.
2
222
1
211
22h
gV
gPH
gV
gP
++=++ρρ
(A)
Pi = Pressure at Point i Vi = Velocity at Point i Hi = Height at Point i
g = Gravitational Const. ρ = Fluid Density
Reynolds Number The Reynolds number is a ratio of inertial forces to viscous forces and indicates whether or not a flow is turbulent or not. The Reynolds number is particularly important in airfoil theory because
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the lift and drag coefficients are a function of the Reynolds number. The Reynolds number is shown below in Equation B.
ѵ (B)
V = Fluid Velocity D = Characteristic Diameter
Ѵ = Kinematic Viscosity of Fluid
NACA Airfoils The National Advisory Committee for Aeronautics (NACA) has developed a list of standard airfoil shapes for aircraft wings. A typical airfoil, with relevant terminology, is shown in Figure 1-1. Each NACA airfoil is described using a set of coordinates to represent the upper and lower surfaces of the cross-section of the airfoil in percentage of chord length. The naming system for the simplest NACA airfoil series, the Four-Digit Series, is described in Table K-1.
Figure K-1 – Cross-Section of Typical Airfoil [16]
Table K-1 - NACA Four-Series Airfoil Properties, ex. NACA 2415
Digit(s) Property Interpretation
Chord Length Independent Variable (c) Designer may choose chord length as required
First Digit Maximum Camber (m) Indicated in percentage of chord length (ex. 0.02c)
Second Digit Position of Max Camber (p)
Indicated in tenths of chord and represents the position of the maximum camber from the
leading edge of the airfoil (ex. 0.4c from leading edge)
Third & Fourth Digits Maximum Thickness (t) Indicates the maximum thickness
of the airfoil in percentage of chord length (ex. 0.15c)
Lift The lift coefficient is a non-dimensional number that is used to characterize the amount of lift that an airfoil can produce under particular conditions. The amount of lift on an airfoil can be determined with knowledge of the angle of attack, lift coefficient, projected airfoil surface area
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(planform area), and flow characteristics such as density and velocity. The coefficient of lift is given by Equation C.
p
LL AV
FC 2
)(2)(ρ
αα =
(C)
CL = Coefficient of Lift FL = Lift Force
α = Angle of Attack ρ = Fluid Density
V = Fluid Velocity Ap = Planform Area
Drag There are two major types of drag that act in a direction opposite to the motion of the airfoil: pressure drag and friction drag. Pressure drag forces are attributed to the fluid that makes contact to the frontal surface of an airfoil. Friction drag, on the other hand, is the drag force that results due to the friction along the surface of the airfoil. The summation of these drag forces is often referred to as parasite drag. For this experiment, parasite drag can be considered the total drag for the airfoil since the airfoil is fixed. By fixing the airfoil, induced drag from the upwards movement of the airfoil is eliminated.
A useful tool for describing the total amount of drag on an airfoil is the coefficient of drag. The coefficient of drag is a dimensionless number that is typically determined experimentally for a particular object or airfoil. The coefficient of drag is best described for airfoils in equation (x).
AVFC D
totalD 2,)(2)(
ραα =
(C)
frictionDpressureDtotalD CCC ,,, +=
CD = Coefficient of Drag FD = Drag Force
α = Angle of Attack ρ = Fluid Density
V = Fluid Velocity A = Planform Area
Similar to the coefficient of lift versus angle of attack plot, experimental data to determine the coefficient of drag with respect to a particular airfoil and Reynolds number. These curves account for the summation of the drag on the airfoil and are useful in determining the total drag force on an airfoil. Beam Theory For this experiment, it is required to measure the deflection and translate this deflection into a force. The beam used in this laboratory experiment can be assumed to be a cantilever beam with a single point load at the end (Figure J-1). Basic beam theory indicates that the deflection at any point on the beam can be expressed as Equation D. The moment of inertia of a rectangular beam is given by Equation E.
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Figure K-2 - Point Load At End of Cantilever Beam
(D)
(E)
Apparatus The apparatus used in this experiment can be seen in Figure K-3. Schematic Diagrams of the apparatus are shown in Figure K-4and Figure K-5. The experimental apparatus will be used in conjunction with the hydraulic flume shown in Figure K-6.
y
xF
δma
L
h
b
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Figure K-3 - Experimental Apparatus
Flift
Fdrag
Airfoil
Hinge (used to set α)
Force Measuring
Support Beam
Roller Support
Fweight
α
θ
Fluid Flow
Figure K-4 - Simplified 2-D Drawing of Airfoil Support Apparatus
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Figure K-6 - Simplified Sketch of Hydraulic Flume
The apparatus consists of the following components:
• Semi-permanent hinge to allow rotation of the airfoil about its centroidal axis. • Rigid beam to transmit drag force applied on airfoil to force measurement device. • Pitot tube to measure the velocity of the fluid in the hydraulic flume. • Low-carbon steel cantilever beam is used to resist the drag force. • Dial indicator to measure the displacement of the cantilever beam. • Syringe and tube apparatus to inject dyes into the specially designed NACA0009 airfoil.
Technical Data
Dial Gauge The dial gauge used in this experiment is a metric, Baty dial gauge with a resolution of 0.002mm. The maximum displacement that can be measured is 12.7mm.
Young’s Modulus of Cantilever Beam
The cantilever beam used to resist the drag force is made out of low-carbon steel with a theoretical Young’s Modulus of: Elow-carbon-steel = 200GPa
Cantilever Beam
Support Beam
Dial Indicator L
Figure K-5 - Simplified 2-D Drawing of Force Measuring Device
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Airfoil/Test Cylinder Information The NACA 0009 and NACA 6409 airfoils have the following dimensions
cairfoils = chord length = 0.1524m bairfoils = airfoil span = 0.1524m lcylinder = cylinder length = 0.1524m rcylinder = cylinder radius = 0.013716m
The airfoil characteristic charts for the NACA 0009 and 6409 airfoil can be found at the end of this document.
Experimental Procedure
Part 1 – Dial Indicator Calibration Experiment *Optional: Calibration is required if Young’s modulus of cantilever beam is unknown.
1) Place a known load (m=0.2, 0.3, 0.4kg) on the string attached to the cantilever beam.
2) Record the displacement for each load into Table 1.
Part 2 – Drag Measurements 1) Start Up Hydraulic Flume
a. Sluice gate should be open more than 0.1524m and rear exit gate should be fully lowered.
b. Wait for equilibrium conditions. 2) Calibrate gates until hydraulic jump is formed.
a. Alternate between lowering the sluice gate and raising the rear exit gate until a hydraulic jump is formed within the first 3 ft. of the sluice gate.
b. The height after the jump should be maximized. (between 0.18-0.23m) 3) Measure the velocity of the flow using the Pitot-tube.
a. Place Pitot tube platform in region of flume prior to the hydraulic jump.
b. Measure the height of the water at Point 1. c. Measure the height of the water in the Pitot tube (height above the free
surface level of the water in the flume) 4) Place test platform downstream of the jump, in the laminar flow region. 5) Measure the height of the water in the location of the airfoil. 6) Turn on the data acquisition system. 7) Place the NACA 0009 airfoil on testing apparatus at an angle of attack of
α = -5o to 25o in increments of 2.5o. Record the displacement measured into
Table 2. 8) Repeat test for NACA 6409 Airfoil. 9) Repeat test for cylinder at an angle of attack of α = 0o.
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Part 3 – Flow Visualization 1) Place the NACA 0009 Airfoil on the test apparatus 2) Place syringe in dye container, pulling back on the syringe to load dye. 3) Attach dye tube into airfoil and into syringe. 4) Using a steady force, inject the dye into the airfoil. 5) Photograph the dye pattern. Ensure the camera’s shutter speed is set to the
highest possible setting. 6) Repeat steps 1-5 for α = 0o, 5o, 10o, 12o, 15o, 20o.
Part 4 - Supplementary Testing of Random Object (Optional)
Part 4 –A: Prior to performing the experiment
1) Choose an object that you do not care much for, or use the foam piece, provided by the instructor, to create your own object.
2) Ensure that the object’s dimensions are less than 0.0762mx0.127mx0.1524m (height x width x span).
3) Prior to attending the lab, drill two (2) holes (0.00635m diameter). The centre of the holes must be 0.0381m apart.
4) Predict the behavior of streamlines over your object using figures and text (1 page maximum). Hand this into your TA on the day that the test is performed.
Part 4 – B: During Lab the experiment
1) Attach test object to apparatus. 2) Measure the drag force of object. Vary angle of attack if necessary. 3) Using dye wand, inject a single stream of dye in front of the object. The dye
wand should be 10-20cm upstream of the test object. a. Take a photograph of the streamlines.
Results Include the following tables in your lab report and submit a copy of Tables 1 and 2 to the lab TA.
Table 1 – Calibration Data Load Applied(kg) Displacement of Dial Indicator (mm)
Table 2 – Drag Force Data
Angle of Attack, α (degrees) Displacement of Dial Indicator (mm)
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Discussions/Analysis
• Determine the theoretical and experimental resultant forces on the airfoils. o Note: the experimental lift force is not measured in this lab. In lieu of this, use the
theoretical lift force as the experimental lift force.
• Plot a CD vs. α curve for both airfoils. o Can you predict the angle at which stall occurs from these plots alone?
If so, does this stall angle match with the stall angle which can be derived from the theoretical CL vs. α plot?
• Discuss the results of the flow visualization experiment.
o Refer to photographs of the experiment.
• Compare the drag force on the cylinder to drag forces measured on the airfoils. o Account for any differences between these values.
• Explain, in less than a page, how airfoils can induce lift. Use supporting figures if
necessary.
• Perform an error analysis. o Hint: pay close attention to the design of the apparatus and the assumptions made
while making calculations. o Avoid commenting on human error.
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Figure K-7 - NACA 0009 Characteristic Charts for 60,000<Re<300,000 [21]
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Figure K-8 – NACA 0009 Characteristic Charts for Specific Reynolds Numbers [21]
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Figure K-9 - NACA 6409 Characteristic Charts for 60,000 < Re < 200,000 [21]
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Figure K-10 – NACA 6409 Characteristic Charts for Specific Reynolds Numbers [21]
References
1) F.M. White, Fluid Mechanics, 5th ed., McGraw-Hill, New York, 2003. 2) Selig, Michael S., John F. Donovan, and David B. Fraser. Airfoils At Low Speeds.
Virginia Beach: H.a. Stokely, 1989. 3) Gere, James M. Mechanics of Materials, 5th Edition. 5th ed. Toronto: Nelson Thornes,
2003.
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Appendix L Division of Work
This thesis report was completed by two authors and as requested in the “Thesis Handbook” the name of the author of each section is written in brackets next to the chapter title. Below is a table outlining each person’s contributions to this thesis paper:
Name Sections Written/Work Completed
Mark Richards (993827720)
• 1. Introduction • 2.1 The Role of Laboratory Experiments in the Engineering
Curriculum • 2.3 Advantages and Disadvantages of Experiments and
Design Projects • 3. Design of a Simple Piping Network • 3.1 Theory and Background Concepts • 3.2 Project Description • 3.3 Improvements over the Existing Experiments • 5.1 Pipe Network Design Project – Future Development • 6. Conclusions • All Appendixes, Figures and Tables referred to in the above
sections.
Karthik Senthilnathan (993934025)
• 2.2 Past Design Projects in Fluid Mechanics Courses • 4. External Flow Investigation • 4.1 Background Theory • 4.2 Development of Experiment • 4.3 Apparatus • 4.4 Proposed Experiment • 4.5 Sample Solution to External Flow Investigation • 4.6 Errors Associated with Experiment Apparatus • 5.2 External Flow Experiment – Future Development
Apparatus • Report formatting and compilation. • All Appendixes, Figures and Tables referred to in the above
sections. The List of Symbols, Figures and Tables, as well as the Table of Contents was completed by both authors. Editing of the entire report was also completed by both authors.