High School Appropriate Fluid Mechanics Tables...Mechanics, 5th Edition A Brief Introduction to...

Project Title: High School Appropriate Engineering Content Knowledge (Appendix 3B) WFED 7650-Applied Project in Workforce Education Professors: Dr. Robert Wicklein & John Mativo Student: Edward Locke, University of Georgia

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Appendix 3B

High School Appropriate Fluid Mechanics Tables

For

HIGH SCHOOL APPROPRIATE ENGINEERING CONTENT KNOWLEDGE IN THE INFUSION OF ENGINEERING DESIGN

INTO K-12 CURRICULUM

(Under the General Topic of “Engineering Design in Secondary Education” and of

“Vision and Recommendations for Engineering-Oriented Professional Development”)

Summer 2009 (Completion Date: Thursday, July 9, 2009)

College of Education, University of Georgia

Professors:

Dr. Robert Wicklein, Dr. Roger Hill and Dr. John Mativo

Advisors:

Dr. Sidney Thompson and Dr. David Gattie (Driftmier Engineering Center)

Student:

Edward Locke ([email protected])

mailto:[email protected]�


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Notes on How to Use This Appendix The whole Research Project and this Appendix constitute the groundwork for a proposed four-round five-point Likert Scale

survey study, with five major steps in its research design:

1. Preliminary selection of high school appropriate fluid mechanics topics;

2. Presentation of data to faculty advisors for review;

3. Presentation of data to a panel of university faculty for validation and endorsement;

4. 4-round Delphi study using 5-point Likert Scale;

5. Comparative analysis of the results from the 4-round Delphi study, for the creation of a formal list of high school appropriate engineering topics.

Participants in the “4-round Delphi study using 5-point Likert Scale” might include the following groups of stakeholders in engineering and technology education:

• Group 1 (University Engineering and Technology Faculty);

• Group 2 (University K-12 Technology Education Faculty); • Group 3 (University Undergraduate Senior-Year Engineering Students);

• Group 4 (K-12 Technology and STEM Teachers and Administrators);

• Group 5 (Practicing Engineers and Technicians).


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Figure 1A. The main textbook where the fluid mechanics related engineering analytic and predictive principles and computational formulas are extracted.

Figure 1B. The abridged version of the textbook by the same authors and used in California State University Los Angeles when I took the course. This book has been used as a reference during this research project.

Figure 1C. The Student Solution Manual for the main textbook used to double-check for the mathematics and physics principles and computational skills needed for the study of various topics of fluid mechanics contained in the main textbook.


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Textbook Information

Main Textbook Reference Book Student Solution Manual Title Fundamentals of Fluid Mechanics

Mechanics, 5th Edition A Brief Introduction to Fluid Mechanics, 4th Edition

A Brief Introduction to Fluid Mechanics, Student Solutions Manual, 4th Edition

Authors Bruce M. Munson, Donald F. Young, Theodore H. Okiishi

Donald F. Young, Bruce R. Munson, Theodore H. Okiishi, Wade W. Huebsch

Donald F. Young, Bruce R. Munson, Theodore H. Okiishi, Wade W. Huebsch

Publisher John Wiley & Sons, Inc. John Wiley & Sons, Inc. John Wiley & Sons, Inc. Year 2006 2007 2007 ISBN 0-471-67582-2 978-0470039625 978-0470099285

This Appendix contains tabulated information on the initial determination of high school (at 9th Grade level) appropriate engineering analytic and predictive principles and computational formulas for the subject of fluid mechanics; this determination is based on the satisfaction of pre-requisite mathematics and science (namely, physics and chemistry) education, as mandated by Georgia Performance Standards established by the State of Georgia Department of Education (available at https://www.georgiastandards.org/Pages/Default.aspx). The above-mentioned principles and computational formulas have been extracted from one of the most popular university undergraduate lower-division textbook on fluid mechanics; associated reference books have been used as well (see Figures 1A, 1B, and 1C). The Appendix contains the following:

• Part One – Initial Determination of High School (9th Grade) Appropriate Fluid Mechanics Topics: This Part covers the 1st, 2nd and 3rd of the above-listed 5 major steps of the proposed study (i.e., “preliminary selection of high school appropriate engineering topic,” “presentation of data to faculty advisors for review,” and “presentation of data to a panel of university faculty for validation and endorsement”); and it contains the Fluid Mechanics Topic List (Engineering Topics Mathematics and Science Pre-requisite Completion Chart for the Subject of Fluid Mechanics), on pages 14-86. As shown in Figures 2A and 2B, on the tabulated list, the columns listing the mathematics and physics/chemistry pre-requisites for the study of each fluid mechanics topic are listed on the right of the column containing the titles of the chapters and sections with associated formulas, which are symbolic representations of engineering analytic and predictive principles. The list will serve two purposes:

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http://www.amazon.com/exec/obidos/search-handle-url/ref=ntt_athr_dp_sr_2?%5Fencoding=UTF8&search-type=ss&index=books&field-author=Bruce%20R.%20Munson�

http://www.amazon.com/exec/obidos/search-handle-url/ref=ntt_athr_dp_sr_3?%5Fencoding=UTF8&search-type=ss&index=books&field-author=Theodore%20H.%20Okiishi�

http://www.amazon.com/exec/obidos/search-handle-url/ref=ntt_athr_dp_sr_4?%5Fencoding=UTF8&search-type=ss&index=books&field-author=Wade%20W.%20Huebsch�


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1. For data review and validation: The list will be submitted to Dr. Robert Wicklein, Dr. John Mativo, and Dr. Roger Hill at the College of Education, the University of Georgia, for review, and for validation of the findings at technical level, in terms of validity of pre-requisite sequence and of high school students’ preparedness for learning the engineering knowledge content identified therein. Dr. Robert Wicklein is a veteran educator profoundly and broadly experienced in teaching both K-12 and university students engineering design and technology. Dr. John Mativo has strong academic background and long history of professional practice in both mechanical and electrical engineering, and over 15 years of working experience in university engineering instruction as well as in the development of K-12 appropriate engineering curriculum. Dr. Roger Hill is a veteran professor in the area of workforce education and is very knowledgeable about K-12 education process. All of them possess great expertise in making judgment on the feasibility of infusing specific engineering knowledge content into K-12 curriculum. To facilitate such review and validation, proposed procedures are available on pages 8-13. After Dr. Robert Wicklein, Dr. Roger Hill and Dr. John Mativo complete the review and validation process, the list would be edited to make corrections to all possible errors and mistakes; and if necessary and possible, the corrected list might be submitted to a panel of university faculty for additional validation and endorsement; and the potential members of this panel would be selected among engineering processors with experience teaching fluid mechanics course for at least three semesters in an ABET-accredited undergraduate engineering program, from four-year universities granting master’s and/or doctoral degrees in mechanical and civil engineering.

2. As part of the 1st round of the proposed four-round five-point Likert Scale Delphi study: The expert opinions on the relative importance of each topic of fluid mechanics (with analytic principles and computational formulas), collected from the review and validation process conducted by the above professors will be counted as part of the data for the first round of the Delphi study and statistically analyzed and processed accordingly, so as to prepare for the second round of the proposed Delphi survey with the above-mentioned five Groups of Participants.

• Part Two – 1st Round of Delphi - Five-Point Likert Scale Survey Forms: This Part prepares for the 4th of the above-listed 5 major steps of the proposed study; and it contains two survey forms (i.e., the first round of the “4-round Delphi study using 5-point Likert Scale”). The Survey Forms will be presented to the above-mentioned five Groups of Participants for the first round of the proposed Delphi survey. To facilitate the survey, detailed information on how to fill out survey forms are available on pages 87-92.


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1. Fluid Mechanics Survey Form A (1st Round of Delphi - Likert Scale Questionnaire on the Importance of Various Fluid Mechanics Topics Selected for High School Engineering Curriculum (For the Pre-calculus Portion): As the name implies, this list covers only the fluid mechanics topics with computational formulas requiring no calculus related skills. (pp. 93-127).

2. Fluid Mechanics Survey Form B (Delphi - Likert Scale Questionnaire on the Importance of Various Fluid Mechanics Topics Selected for High School Engineering Curriculum (For the Calculus Portion): As the name implies, this list covers only the fluid mechanics topics with computational formulas requiring calculus related skills. (pp. 128-164).

• Part Three – Findings from the Research Project: This Part contains tabulated lists showing the results of this research project, which might be used as reference in the future endeavors to infuse fluid mechanics related engineering analytic and predictive principles and computational skills into a potentially viable high school engineering and technology curriculum, which shall be based on the organic and seamless integration of solid mastery of engineering analytic and predictive principles and innovative application of engineering design process.

o List 1A. Pre-Calculus Based Fluid Mechanics Topics That Possibly Could Be Taught at 9th Grade: The statistic summary of data at the end of this list (pp. 166-170) indicates that a significant portion of fluid mechanics knowledge content covered in the selected undergraduate level textbook could possibly be taught to high school students. 62.2% of all Sections, and 51.0% of the volume in the selected textbook is based on pre-calculus mathematics and on principles of physics students are supposed to learn before or by 9th Grade, according to Georgia Performance Standards (p. 170).

o List 1B. Pre-Requisite Mathematics and Science Topics to Be Reviewed Before Teaching the Pre-Calculus Portion of Fluid Mechanics Topics to 9th Grade Students: This list includes 24 sets of mathematics principles and skills, as well as 29 sets of physics/chemistry principles and skills that are needed as pre-requisites or as important topics to be reviewed for the effective learning of fluid mechanics topics initially determined as appropriate for 9th Grade students (p. 171).

o List 2A. Calculus Base Fluid Mechanics Topics for Post-Secondary Engineering Education: Topics of fluid mechanics on this list are either recommended for post-secondary engineering education, or for inclusion as application problems in 11th or 12th Grade Advanced Placement Calculus course (pp. 172-174).

o List 2B. Pre-Requisite Math and Science Topics to Be Reviewed Before Teaching the Calculus Portion of Fluid Mechanics Topics: This list includes 34 sets of mathematics principles and skills, as well as 33 sets of physics


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principles and skills that are needed as pre-requisites or as important topics to be reviewed for the effective learning of fluid mechanics topics initially recommended either for university engineering students or for high school 11th or 12th Grade students enrolled in Advanced Placement Calculus courses (p. 175).

Figure 2A. Engineering Topics Mathematics and Science Pre-requisite Completion Chart for the Subject of Statics.


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Figure 2B. Notation for undergraduate level appropriate statics topics.


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Part One: Initial Determination of

High School (9th Grade) Appropriate Fluid Mechanics Topics

Proposed Procedures for Review and Validation

To facilitate review and validation of the initial selection of fluid mechanics topics that could be possibly taught to students at 9th or above Grade, as listed in the Fluid Mechanics Topic List, the following procedures are hereby proposed:

1. Look at the formulas listed under the Engineering Analytic Topics & Typical Formulas column, and check the mathematics and science pre-requisite items under the Math and Physics/Chemistry columns; verify if there are necessary pre-requisite that are missing; if so, write a note in either the Math or Physics/Chemistry column; and if any listed item is not really needed, cross it out with a horizontal strikethrough (as shown on Figure 3A);

2. Rate the importance of each Section as a topic in a potentially viable 9th or above Grade fluid mechanics subject, and write a number representing its “importance” value (Figure 3A), using the five-point Likert Scale (Figure 3B);

3. Check the formulas listed under the Engineering Analytic Topics & Typical Formulas column, and use symbols shown in Figure 3B to indicate your expert opinion and advice about each formula;


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4. Add your comment and advice on the Grade at which the topic should be taught to pre-collegiate students;

5. Add your general comments and advice in the empty space.

Figure 3A. Step-by-step procedures proposed for the review and validation of data.


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Figure 3B. Likert Scale (top) and symbols to be used for the expression of expert opinion and offer of advice.


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Notes for Chapter 6 and Chapter 7

Chapter 6 (Differential Analysis of Fluid Mechanics Flow) appears to be, for all practical purposes, too deep in calculus-based mathematics for even 12th Grade students in Advanced Placement Calculus course to master.

Chapter 7 (Similitude, Dimensional Analysis, and Modeling) involve a lot of “abstract thinking” and appears to be most likely beyond the cognitive developmental maturity level of high school students.

Therefore, engineering analytic principles and skills from these two Chapters are NOT analyzed for the eventual inclusion into a potentially viable K-12 engineering curriculum. However, some generic knowledge content covered in these two Chapters could still be lightly explored by 9th or above Grade students; thus, their relative importance could still be rated at generic knowledge level. In addition, some appropriate skills in 7.1 (Dimensional Analysis) could be considered for high schools.

Notes about the Fluid Mechanics Analytic Principles and Formulas

The leftmost column in the Fluid Mechanics Topic List (Engineering Topics Mathematics and Science Pre-requisite Completion Chart for the Subject of Fluid) contains

1. The titles of each section under a particular chapter in the selected textbook, which in general represent particular sets of fluid mechanics related engineering analytic and predictive principles, in a qualitative and explanatory way;

2. Computational formulas, which symbolically represent the above engineering analytic and predictive principles, in a quantitative and mathematical way.

As shown in Figure 3B, the formulas extracted from the selected textbook might by categorized into five groups, corresponding to the five different symbols shown in Figure 3B, which could be used by the above-mentioned professors from the University of Georgia and other schools to indicate their expert opinions and advices about each formula:


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1. Formulas that engineering professors actually teach in classroom lectures and that practicing engineers use in engineering design projects: These are the important ones to be included in a potentially viable K-12 engineering curriculum that shall be based on cohesive and systemic mastery of engineering analytic and predictive principles and skills. For any of these formulas, a box could be used together with a number representing its order of importance according to the five-point Likert Scale (1 = Totally Unimportant, 2 = Not So Important, 3 = Might Be Important, 4 = Important, or 5 = Very Important).

2. Formulas that are rarely used in either classroom lectures or in field practice, but are used by the original discoverer of a particular set of analytic principles to derive other formulas that are actually used in classroom lecture or in field practice: Some of these “intermediate” formulas might not be used often, in other words, they are “rarely taught or used.” For any of these formulas, a strikethrough could be used. If a big enough percentage of participants (maybe 85% or above) place a strikethrough on a particular formula at the end of each round of the proposed four-round Delphi study, then the formula will be removed from the survey form for the next round. If the trend continues through all four rounds of the proposed Delphi survey, then that formula might be removed from the final list of high school appropriate fluid mechanics topics. Interestingly enough, in some cases, rarely used calculus-based “intermediate” formulas are used to derive a final one that is based on pre-calculus mathematics skills and is actually used in most homework assignments and design projects; in this case, if the “intermediate” formulas are removed from consideration, then the entire topic of fluid mechanics could be re-classified as appropriate for 9th Grade. For example, the main formula amF rv

= and

streamlinea alongconstant 21 2 =++ zVp γρ (Bernoulli Equation) do not need calculus, and thus, could be taught to 9th

Grade students. This type of formulas will make the list shorter and shorter as the proposed Delphi study moves to the next round of survey. Some of these formulas might not be in the selected textbook; I derived them for fun, sometimes with the help of my former engineering professor, Dr. Samuel Landsberger, at California State University Los Angeles.

3. Formulas that are particular to certain conditions and in real classroom lectures or field practice are, for all practical purposes, are close to be “never used:” For any of these formulas, a double-strikethrough could be used. If a big enough percentage of participants (maybe 75% or above) place a double-strikethrough on a particular formula at the end of each round of the proposed four-round Delphi study, then the formula will be removed from the survey form for the next round. If the trend continues through all four rounds of the proposed Delphi survey, then that formula might be removed from the final list of high school appropriate fluid mechanics topics. This type of formulas will also make the list shorter and shorter as the proposed Delphi study moves to the next round of survey.


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4. Formulas that even experienced university engineering professors or practicing engineers might “not understand:” This is amazing but totally correct and yes, absolutely normal! There are formulas that even experienced professors might say “I do not understand this” or “I need to read the context in the book to figure this out.” For any of these formulas, the participants should generally not seek to understand them (doing so does not serve the purpose of studying the relative importance of each computational formula); but instead, a question mark (?) could be used. If a big enough percentage of participants (maybe 65% or above) place a question mark (?) on a particular formula at the end of each round of the proposed four-round Delphi study, then the formula will be removed from the survey form for the next round. If the trend continues through all four rounds of the proposed Delphi survey, then that formula might be removed from the final list of high school appropriate fluid mechanics topics. Indeed, it makes little sense to include this type of formulas to a potentially viable K-12 engineering curriculum. This type of formulas will also make the list shorter and shorter as the proposed Delphi study moves to the next round of survey. Some of these formulas might not be in the selected textbook; I derived them for fun, sometimes with the help of my former engineering professor, Dr. Samuel Landsberger, at California State University Los Angeles.

5. Formulas that are wrong for any reasons (my typing errors, or the authors’ errors, etc.): For any of these formulas, a cross (X) could be used and the correct formulas should be given if possible. The correction would be included in the survey forms for the subsequent rounds of the four-round five-point Likert Scale Delphi study.

For convenience of statistic analysis of expert opinions and advice, it is requested that all participants print each letter of their

comment legibly and separately, using fonts commonly used in engineering notebooks.


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Fluid Mechanics Topic List Engineering Topics Mathematics and Science Pre-requisite Completion Chart for the Subject of Fluid Mechanics Engineering Subject: Fluid

Math & Science Pre-requisite Topics & Completion Grade (Georgia Performance Standard Code) [Pre-requisite Math Skills/Science Principles] (GPS Code) Grade (Table No.)

Possible Grade

to Start the Topic

Engineering Analytic Topics & Typical Formulas

Math Physics/Chemistry Sec Ch Chapter 1 - Introduction 1.1 Some Characteristics of Fluid N/A [pressure] (SC5) 9th (4B) To be taught

[velocity] (S8P3) 8th (3A) [force] (S4P3) 4th (3A) or (S8P3) 8th (3C)

[molecule] (S8P1) 8th (4A)

9th

1.2 Dimensions, Dimensional Homogeneity, and Units

δβττ ∝==→≡s

sns

n

APpAF

AF

pr

r

[unit conversion] (M6M1) 6th (2C) [four operations] (M1N3) 1st (2A)

[square root] (M8N1) 8th (2A)

N/A 9th

1.3 Analysis of Fluid Mechanics Behavior N/A

N/A [Newton’s 1st, 2nd and 3rd Laws] (SP1) 9th (3C) [mass] (S8P3) 8th (3A)

9th

1.4 Measures of Fluid Mechanics Mass and Weight [four operations] (M1N3) 1st (2A) [mass] (S8P3) 8th (3A) [gravity] (S6E1) 6th (3A)

9th

1.4.1 Density

ρρ 1

===mVv

Vm

[four operations] (M1N3) 1st (2A) [volume] (M5M4) 5th (1B) (M6M3) 6th (2B)

(MA1G5) 9th (2F)

[density] (S6E5) 6th (4A) [mass] (S8P3) 8th (3A)

9th

1.4 2 Specific Weight

gVmg

VW ργ ==≡

[four operations] (M1N3) 1st (2A) [force] (S4P3) 4th (3A) or (S8P3) 8th (3C) [gravity] (S6E1) 6th (3A) [density] (S6E5) 6th (4A)

9th

1.4.3 Specific Gravity

CSG

OHo4@

2ρ

ρ=

[four operations] (M1N3) 1st (2A)

[pressure] (SC5) 9th (4B) To be taught

9th

1.5 Ideal Gas Law RTp ρ=


[temperature] (SP3) 9th (3B) [absolute temperature] (SP3) 9th (3B)

To be taught [density] (S6E5) 6th (4A)

9th

9th + PS

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Fluid Mechanics Topic List (Continued). Engineering Subject: Fluid


Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 1 – Introduction (Continued) 1.6 Viscosity

ρμνμμ

μττγτ

γδδβγδδβδδ

δδβδβ

δ

==+

=

=∝∝

=====

=≈=

→

TB

t

DeST

CTdydu

dydu

dydu

bU

tbtUtUa

ba

bUyu

/2/3

0lim

tan

&

&&

[four operations] (M1N3) 1st (2A) [derivative] 12th (To be taught)

[trigonometric functions] (MA2G2) 10th (2F)

[density] (S6E5) 6th (4A) [absolute temperature] (SP3) 9th (3B)

To be taught

PS

1.7 Compressibility of Fluids N/A N/A 9th 1.7.1 Bulk Modulus

ρρddpE

ddpE vv =∀∀

=

[four operations] (M1N3) 1st (2A) [derivative] 12th (To be taught as a special skill)

[pressure] (SC5) 9th (4B) To be taught [density] (S6E5) 6th (4A)

9th

1.7.2 Compression and Expansion of Gases

kpEpEppvvk ==== ConstantConstant

ρρ

[four operations] (M1N3) 1st (2A) [exponent] (M6A3) 6th (2A)


9th

1.7.3 Speed of Sound

kRTcRTp

kpc

ddpE

ddp

ddpE

Eddpc

v

vv

=→⎪⎭

⎪⎬

⎫

=

=

⎪⎪⎩

⎪⎪⎨

⎧

=

==

←==

ρρ

ρρ

ρρρρ

ρρ

[four operations] (M1N3) 1st (2A) [square root] (M8N1) 8th (2A)

[derivative] 12th (To be taught as a special skill)

[speed of sound] (SPS9) 9th (3B) To be taught

[velocity] (S8P3) 8th (3A) [absolute temperature] (SP3) 9th (3B)

To be taught

9th

9th + PS














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Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 1 – Introduction (Continued) 1.8 Vapor Pressure

N/A [intermolecular cohesive force] To be taught [momentum] (SP3) 9th (3B)


9th

1.9 Surface Tension

RhRhR

RpppRpR ei

γθσθσπγπ

σπσπ

cos2cos2

22

2

2

=→=

=−=ΔΔ=

[areas of geometric shapes: circle, triangle] (M5M1) 5th (2B) (M5M1) 5th (2B)

[unit conversion] (M6M1) 6th (2C) [height] (MKM1) K (2B)

[trigonometric functions] (MA2G2) 10th (2F)

[force] (S4P3) 4th (3A) or (S8P3) 8th (3C) [mass] (S8P3) 8th (3A)

[pressure] (SC5) 9th (4B) To be taught [weight] (MKM1) K (2C) [gravity] (S6E1) 6th (3A)

9th

1.10 A Brief Look Back in History N/A N/A 9th 1.11 Chapter Summary and Study Guide N/A N/A 9th

9th + PS

Chapter 2 Fluid Statics (Continued) 2.1 Pressure at a Point

zz

zszz

ysyy

amFVmVzyx

azyxzyxsxpyxpF

azyxsxpzxpF

amF

rr

rr

=←==↑

=−−=

=−=

=

∑

∑

ρδδδ

δδδρδδδγθδδδδ

δδδρθδδδδ

2

22cos

2sin

[four operations] (M1N3) 1st (2A) [sigma notation] (M6N1) 6th (1A) or

(MA1A3) 9th (2E) [coordinate system] (M4G3) 4th (2B)

[limit] Post-Secondary [volume] (M5M4) 5th (1B) (M6M3) 6th (2B)

(MA1G5) 9th (2F)

[force] (S4P3) 4th (3A) or (S8P3) 8th (3C) [Newton’s 1st, 2nd and 3rd Laws] (SP1) 9th (3C)

PS PS





















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Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 2 Fluid Statics (Continued) 2.1 Pressure at a Point (Continued)

zz

zszz

ysyy

amFVmVzyx

azyxzyxsxpyxpF

azyxsxpzxpF

amF

rr

rr

=←==↑

=−−=

=−=

=

∑

∑

ρδδδ


δδδρθδδδδ

2

22cos

2sin

[four operations] (M1N3) 1st (2A) [sigma notation] (M6N1) 6th (1A) or

(MA1A3) 9th (2E) [coordinate system] (M4G3) 4th (2B)

[limit] Post-Secondary [volume] (M5M4) 5th (1B) (M6M3) 6th (2B)

(MA1G5) 9th (2F)

[force] (S4P3) 4th (3A) or (S8P3) 8th (3C) [Newton’s 1st, 2nd and 3rd Laws] (SP1) 9th (3C)

PS

2.2 Basic Equation for Pressure Field

Vzxy

zyxzpF

zyxypF

zyxxpF

zxyyppzxy

yppF

z

y

x

y

=←

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

∂∂

−=

∂∂

−=

∂∂

−=

→⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+−⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−=

δδδ

δδδδ

δδδδ

δδδδ

δδδδδδδ22

[four operations] (M1N3) 1st (2A) [partial derivative] Post-Secondary

[gradient “del”] Post-Secondary [volume] (M5M4) 5th (1B) (M6M3) 6th (2B)

(MA1G5) 9th (2F) [coordinate system] (M4G3) 4th (2B)

[pressure] (SC5) 9th (4B) To be taught [acceleration] (S8P3) 8th (3C)

PS

PS

















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Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 2 Fluid Statics (Continued) 2.2 Basic Equation for Pressure Field (Continued)

( ) ( ) ( ) ( )

akp

akzyxkkzyxkzyxp

amkWFF

zyxpmamF

kzyxkWpzyx

F

kz

jy

ix

pkzpj

ypi

xp

zyxkzpj

ypi

xpkFjFiFF

s

s

zyxs

r

r

rrr

rrr

r

ργ

δδρδδδγδδδδ

δδδδ

δδδδ

δδδδγδδ

δδδδ

δδδδδδδ

=−∇−

→=−∇−

→=−=

⎪⎩

⎪⎨⎧

=

=−=−−∇=

∂+

∂+

∂=∇∇=

∂∂

+∂∂

+∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

−=++=

∑

∑

ˆ

ˆˆˆˆ

ˆ

ˆˆ

ˆˆˆˆˆˆ

ˆˆˆˆˆˆ

[four operations] (M1N3) 1st (2A) [partial derivative] Post-Secondary

[gradient “del”] Post-Secondary [volume] (M5M4) 5th (1B) (M6M3) 6th (2B)

(MA1G5) 9th (2F) [coordinate system] (M4G3) 4th (2B)

[pressure] (SC5) 9th (4B) To be taught [acceleration] (S8P3) 8th (3C)

PS

2.3 Pressure Variation in a Fluid Mechanics at Rest

γγ

γ

−=→−=∂∂

=∂∂

=∂∂

=−∇−→=

dzdp

zp

yp

xp

kpa

00

0ˆ0v


[partial derivative] Post-Secondary [gradient] Post-Secondary


PS

PS









19



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 2 Fluid Statics (Continued) 2.3.1 Incompressible Fluid

( )( )

00

21

21

21

1221

12122

1

2

1

pghphp

pph

phphpp

zzppzzpp

dzdpz

z

p

p

+=+=

⎪⎩

⎪⎨

⎧−

=

+=→=−→

⎩⎨⎧

−=−−−=−

→−= ∫∫

ργγ

γγ

γγ

γ

[four operations] (M1N3) 1st (2A) [integration] 12th (To be taught)

Note: The main Survey Form ula

00 pghphp +=+= ργ does not need calculus.


9th

2.3.2 Compressible Fluid

( )( ) ( )

( )

( )⎥⎦

⎤⎢⎣

⎡ −−=

−==

→−=−=

→−=−=

→−=

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

−=−=

=

=

∫∫

∫ ∫ ∫

0

1212

1

2

exp

ln 2

1

2

1

2

1

2

1

2

1

RTzzgpp

Tdz

Rg

pp

pdp

Tdz

Rgdz

RTg

pdp

dzRTg

pdpdz

pRTgp

pdzdpdz

RTgp

dzdp

gdzdp

RTpRTp

z

z

p

p

p

p

z

z

z

z

ργ

ρ

ρ


[integration] 12th (To be taught as a special skill)

[derivative] 12th (To be taught)

Note: The main formula

( )⎥⎦

⎤⎢⎣

⎡ −−=

0

1212 exp

RTzzgpp

does not need calculus

[pressure] (SC5) 9th (4B) To be taught [absolute temperature] (SP3) 9th (3B)

To be taught [gas/liquid] (SPS5) 9th (3B) To be taught

9th

9th







20



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 2 Fluid Statics (Continued) 2.4 Standard Atmosphere

βββ

Rg

aaa T

zppzTT ⎟⎟⎠

⎞⎜⎜⎝

⎛−=−= 1


[temperature] (SP3) 9th (3B) [pressure] (SC5) 9th (4B) To be taught

[density] (S6E5) 6th (4A) [weight] (MKM1) K (2C)

9th

2.5 Measurement of Pressure vaporatmatmgageabs phpppp +=+= γ

[four operations] (M1N3) 1st (2A) [pressure] (SC5) 9th (4B) To be taught

9th

2.6 Monometry

[four operations] (M1N3) 1st (2A) [cylinder] (M1G1) (M1G2) 1st (2B)


9th

2.6.1 Piezometer Tube 110 hpphp A γγ =+=

[four operations] (M1N3) 1st (2A) [height] (MKM1) K (2B)


9th

2.6.2 U-Tube Manometer

113322

332211

221122

22110 0

hhhppphhhphphhphhpphp

BA

BA

AA

A

γγγγγγ

γγγγγγ

+−+=−→=−−+

=−=→=−++=


[pressure] (SC5) 9th (4B) To be taught 9th

2.6.3 Inclined-Tube Manometer

θγθγ

γγθγγθγγ

sinsin

sinsin

2222

113322

332211

BABA

BA

BA

pppp

hhppphhp

−=→=−

−+=−=−−+

ll

l

l

[four operations] (M1N3) 1st (2A) [trigonometric functions] (MA2G2) 10th (2F)


2.7 Mechanical and Electronic Pressure Measuring Devices N/A N/A

9th

9th + PS














21



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 2 Fluid Statics (Continued) 2.8 Hydrostatic Force on a Plane Surface

ccxycxycc

xycR

c

xy

c

ARARR

cxcxcc

xc

c

c

c

xcR

c

cxc

c

x

c

AR

AARRcR

cRcA

ARA AR

yAxIIxAy

Ix

AyI

Ay

dAxyxdAxyxF

AyIIyAy

IAy

AyAy

Iy

AyAyI

AyI

Ay

dAyy

dAydFyyFAhF

AyFAydAy

dAyFdAydAhF

+=+=

===

+=←+=+=

+===

===

==

===

∫∫

∫∫∫

∫∫∫ ∫

θγ

θγλ

θγ

θγθγγ

sin

sin

sin

sinsin

22

22

2

[surface] (M6M4) 6th (2B) [four operations] (M1N3) 1st (2A)

[exponent] (M6A3) 6th (2A) [trigonometric functions] (MA2G2) 10th (2F)


[force] (S4P3) 4th (3A) or (S8P3) 8th (3C) [weight] (MKM1) K (2C)

[pressure] (SC5) 9th (4B) To be taught [1st moment of the area] To be taught [2nd moment of the area] To be taught

PS

2.9 Pressure Prism

( )( )

221121

221

2yFyFyFFFF

AhbhkvolumeFAhApF

ARR

RavR

+=+=

⎟⎠⎞

⎜⎝⎛===⎟

⎠⎞

⎜⎝⎛== γγγ

[four operations] (M1N3) 1st (2A) [prism] (M6G2) 6th (2B)

[pressure] (SC5) 9th (4B) To be taught [force] (S4P3) 4th (3A) or (S8P3) 8th (3C)

9th

2.10 Hydrostatic Force on a Curves Surface

( ) ( )22

12

vHR

vH

FFF

WFFFF

+=

+==r


[Pythagorean Theorem] (M8G2) 8th (2B)

[force] (S4P3) 4th (3A) or (S8P3) 8th (3C) 9th

9th + PS















22



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 2 Fluid Statics (Continued) 2.11 Buoyancy, Flotation, and Stability N/A 2.11.1 Archimedes’ Principle

( )( ) ( )[ ]

( ) 21

21112

1212

121212

yVVyVyVyWyFyFyFVF

VAhhAhhFAhhFFWFFF

TTc

cBB

B

B

−−=−−==

−−−−=−=−−−=

r

r

γ

γγγ

2.11.2 Stability N/A

[four operations] (M1N3) 1st (2A) [volume] (M5M4) 5th (2B)

(M5M4) 5th (1B) (M6M3) 6th (2B) (MA1G5) 9th (2F)

[force] (S4P3) 4th (3A) or (S8P3) 8th (3C) [weight] (M4M1) 4th (2C)

9th

2.12 Pressure Variation in a Fluid Mechanics with Rigid-Body Motion

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

+=∂∂

−

=∂∂

−

=∂∂

−

→=−∇−

z

y

x

azp

ayp

axp

akp

ργ

ρ

ρ

ργ rˆ

[four operations] (M1N3) 1st (2A) [coordinate system] (M4G3) 4th (2B) [partial derivative] Post-secondary

[gradient] Post-secondary


PS

2.12.1 Linear Motion

( )

( )

( )zz

y

zy

zy

agdzdp

aga

dydz

dzagdyadp

dzzpdy

ypdpag

zpa

yp

+−=+

−=

+−−=∂∂

+∂∂

=+−=∂∂

−=∂∂

ρ

ρρ

ρρ


[partial derivative] Post-secondary


PS

9th + PS














23



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 2 Fluid Statics (Continued) 2.12.2 Rigid-Body Rotation

( ) ( )

constant2

dpconstant2

2

00

00

0

00ˆˆˆ1ˆ

22

222

222

22

22

22

2

2

2

2

+−=

−=+=

→=→=

→=→=

→=→=

→−=→−=

→−=→=

−=∂∂

+∂∂

=

−=∂∂

=∂∂

=∂∂

==−=∂∂

+∂∂

+∂∂

=∇

∫ ∫∫

∫∫

∫∫∫∫

zrp

dzdrrgrz

grzdr

grdz

drg

rdrdrdz

gr

drdz

gr

drdzdrrdzg

dzgdrrdzgdrr

dzdrrdp

dzdrrdpdzzpdr

rpdp

zppr

rp

aaeraezpep

re

rpp zrrzr

γρω

γρωω

ωω

ωω

ωω

ωρωρ

γωρ

γωρ

γθ

ωρ

ωθ θθ

rrr


[partial derivatives] Post-secondary [gradient] Post-secondary

[integration] 12th (To be taught)

[pressure] (SC5) 9th (4B) To be taught [gravity] (S6E1) 6th (3A)

[force] (S4P3) 4th (3A) or (S8P3) 8th (3C)

PS

2.13 Chapter Summary and Study Guide N/A N/A 9th

9th + PS

Chapter 3 Elementary Fluid Dynamics – The Bernoulli Equation 3.1 Newton’s Second Law

( )VVVVa

sVVa

amFFamF

ns

gP

r

rrrrv

=←ℜ

=∂∂

=

=+= ∑2

[four operations] (M1N3) 1st (2A) [partial derivative] Post-secondary

[volume] (M5M4) 5th (1B) (M6M3) 6th (2B) (MA1G5) 9th (2F)

Note: The main formula amF rv=


[Newton’s 1st, 2nd and 3rd Laws] (SP1) 9th (3C) To be taught

[force] (S4P3) 4th (3A) or (S8P3) 8th (3C) [speed] (S2P3) 2nd (3A)

9th 9th

+ PS













24



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 3 Elementary Fluid Dynamics – The Bernoulli Equation (Continued) 3.2 F = ma along a Streamline

( ) ( )

( ) ( )

Equation)(Bernoulli

streamlinea alongconstant 21

)streamlinea (along21

021

21

sin

sin

22

sinsin

2

2

22

=++

=++

=++→=−−

=∂∂

=∂∂

−−

⎟⎠⎞

⎜⎝⎛

∂∂

−−=+=

∂∂

−=∂∂

−=

−=+−−=∂∂

≈

−=−=→⎭⎬⎫

==

∂∂

=∂∂

==

∫

∑

∑

zVp

CgzVdp

dzVddpdsVd

dsdp

dsdz

asVV

sp

VspFWF

Vspyns

sp

ynpynppynppF

sspp

VXWWg

VWsVVV

sVmVmaF

s

psss

sssps

s

s

ss

γρ

ρ

γρργ

ρρθγ

δθγδδδ

δδδδ

δδδδδδδδδδ

δδ

θγδθδδργγδδ

ρδδδδ

r

rrr


[partial derivative] Post-secondary [sigma notation] (M6N1) 6th (1A) or (MA1A3)

9th (2E)

Note: The main formulas amF rv= and

Equation)(Bernoulli

streamlinea alongconstant 21 2 =++ zVp γρ

do not need calculus

[force] (S4P3) 4th (3A) or (S8P3) 8th (3C) [gravity] (S6E1) 6th (3A) [mass] (S8P3) 8th (3A)

[acceleration] (S8P3) 8th (3C)

9th 9th

+ PS









25



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 3 Elementary Fluid Dynamics – The Bernoulli Equation (Continued) 3.2 F = ma along a Streamline (Continued) Alternatively

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

↓−≡↑≡

=⋅⋅

=⋅⋅=⋅⋅

=⋅

=⋅=⋅

=⋅⋅

==

=++

=⋅⋅+⋅+

)k̂zg h;k̂z e.unit volumper energy (Potential

e)unit volumper energy (Kinetic

e)unit volumper (Work

Constant Energy Potential Energy Kinetic )streamlinea (along Pressure

221

2212

21

221

VPE

Vzgmzg

Vmzg

VKE

Vvm

vVmv

VW

rArF

AFp

Czgvp NstreamlineNNNNN

ρ

ρ

ρρ

NstreamlineNNNNN Czgvp =⋅⋅+⋅+

+

ρρ 221

Equations'Bernouillienergy of onconservati ofLaw

mass of onconservati ofLaw

( ) outoutoutoutinAinin vAvAAnv 1122ˆ ⋅+⋅=⋅⋅

[four operations] (M1N3) 1st (2A) [trigonometric functions] (MA2G2) 10th (2F) [dot product] To be taught as a special math

topic [partial derivative] Post-secondary

[sigma notation] (M6N1) 6th (1A) or (MA1A3) 9th (2E)



PS 9th

+ PS









26



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 3 Elementary Fluid Dynamics – The Bernoulli Equation (Continued) 3.3 F = ma Normal to a Streamline

( ) ( )

streamline theacrossconstant Vp

streamline theacrossconstant

constant

cos

2coscos

2

2

2

22

22

=+ℜ

+

=+ℜ

+

→

⎪⎪⎭

⎪⎪⎬

⎫

==∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛ℜ

=⎟⎠⎞

⎜⎝⎛

∂∂

−−

ℜ−=

∂∂

ℜ=

∂∂

−−

⎟⎠⎞

⎜⎝⎛

∂∂

−−=+=

∂∂

−=∂∂

−=

−=+−−=−=−=ℜ

=ℜ

=

∫

∫∫

∑

∑

zdn

gzdnVdp

sdndp

np

dnVdnnp

dndz

VnpV

np

dndz

VnpFWF

Vnpyns

np

yspysppysppFVWW

VVmVF

pnnn

nnnpn

n

n

γρ

ρ

ργ

ρργ

δθγδδδ

δδδδ

δδδδδδδδδθλδθδδ

ρδδδ

r

rr


[partial derivative] Post-secondary [radius] (M3G1) 3rd (2B)



PS 9th

+ PS







27



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 3 Elementary Fluid Dynamics – The Bernoulli Equation (Continued) 3.4 Physical Interpretation

streamlinea onconstant 2

streamline theacrossconstant V

streamline thealongConstant 21

2

2

2

=++

=+ℜ

+

=++

∫

zg

Vp

zdnp

zVp

γ

γρ

γρ




p


2

2

=++

=++

zg

V

zVp

γ

γρ

do not need calculus

[density] (S6E5) 6th (4A) [speed] (S2P3) 2nd (3A) [gravity] (S6E1) 6th (3A)

9th

3.5 Static, Stagnation, Dynamic, and Total Pressure

( )ρ

ρρ

γ

ρ

43

243

14

2

3

2

2112

2

21

21p


21

ppV

Vppppp

Vp

pzVp

Vpp

T

−=

=−→⎭⎬⎫

==+=

==++

+=


[pressure] (SC5) 9th (4B) To be taught [density] (S6E5) 6th (4A) [speed] (S2P3) 2nd (3A)

9th

3.6 Examples of Use of the Bernoulli Equation

22

2212

11 21

21 zVpzVp γργρ ++=++



9th

9th

+ PS













28



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 3 Elementary Fluid Dynamics – The Bernoulli Equation (Continued) 3.6.1 Free Jets

( )⎪⎩

⎪⎨

⎧

+=

==→=

HhgV

ghhVVh

2

22

21 2 ρ

γργ




9th

3.6.2 Confined Flows 212211222111 QQVAVAVAVA =→=→= ρρ


[areas of geometric shapes] (M5M1) 5th (2B)

[density] (S6E5) 6th (4A) [speed] (S2P3) 2nd (3A)

9th

3.6.3 Flowrate Measurement

( )

( )

1221

2/311

2

1

2

21221

2222111122

2212

11

2

1

2

212

22112

222

11

2

22

1

20

21

21

1

221

21

gzbzQzz

HgbCgHHbCQ

zz

zzgbzQpp

zbVVAzbVVAQzVpzVp

AA

ppAQ

VAVAQVpVp

=→>>

==

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−=→==

====++=++

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−

−=

==+=+

γργρ

ρ

ρρ


[square root] (M8N1) 8th (2A) [areas of geometric shapes] (M5M1) 5th (2B)

[density] (S6E5) 6th (4A) [speed] (S2P3) 2nd (3A)


9th

3.7 The Energy Line and the Hydraulic Grade Line

Hzg

V==++ streamlinea onconstant

2

2

γρ




9th

9th

+ PS























29



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 3 Elementary Fluid Dynamics – The Bernoulli Equation (Continued) 3.8 Restrictions on Use of the Bernoulli Equation 3.8.1 Compressibility Effects

( ) ( )[ ]

le)compressib(...24

2411

2

ible)incompress(2

2

le)compressib(12

11

2121

1

1

constant 21

2ln

2

constant 21

41

21

21

1

12

21

1

12

1

11

1

21

1

2

1/21

1

12

2

22

2

21

21

1

1

1

1

1

2

/11

/12

/1/1/1

2/1/1

2

22

2

11

21

2

2

1

⎟⎠⎞

⎜⎝⎛ +

−++=

−

=−

→

⎪⎪⎭

⎪⎪⎬

⎫

=

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎟

⎠⎞

⎜⎝⎛ −+=

−

++⎟⎠⎞

⎜⎝⎛

−=++⎟

⎠⎞

⎜⎝⎛

−

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎠⎞

⎜⎝⎛

−=

−⎟⎠⎞

⎜⎝⎛

−=

=++

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++

==++

−

−−−

−

∫

∫

∫

MakMakMap

pp

kMap

pp

kRTVMa

RTV

pp

Makp

pp

gzVpp

kkgzV

pp

kk

pp

pp

kk

ppk

kCdppC

gzVdppC

zg

Vpp

gRTz

gV

RTpgzV

pdpRT

kk

kkkkp

p

kkk

kk

ρ


[logarithmic functions] (MA2A5) 10th (2E) (To be taught as a special skill)


Note: The main formulas

RTpgzV

pdpRT ==++∫ ρconstant

21 2

And others do not need calculus


9th 9th

+ PS







30



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 3 Elementary Fluid Dynamics – The Bernoulli Equation (Continued) 3.8.2 Unsteady Effects

( )

)streamlinea along(21

21


22

2212

11

2

2

1

zVpdstVzVp

zVddpdstV

s

sγρργρ

γρρ

+++∂∂

=++

=+++∂∂

∫

[four operations] (M1N3) 1st (2A) [derivative] 12th (To be taught) [integration] 12th (To be taught)


PS

3. 8.3 Rotational Effects

γγγ

γρ

γ

ρ

γργρ

454

2

1234

43

13

43

043

02

012

021

21

021

1222

2212

11

flowhroughout constant t21

CC

21C0

Cconstant21

21

pHHHpp

zVp

pphpp

amF

hzzVVV

pVppp

zzVVV

zVpzVp

==+=

=++

→=→

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

=−=

=

====

+=→⎪⎭

⎪⎬

⎫

======

==++=++

rr

3.8.4 Other Restrictions 3.9 Chapter Summary and Study Guide



9th

9th

+ PS








31



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 4 Fluid Kinematics 4.1 The Velocity Field

( ) ( ) ( )( ) 2/1222

ˆ,,,ˆ,,,ˆ,,,

wvuVV

ktzyxwjtzyxvitzyxuV

++==

++=r

r

[four operations] (M1N3) 1st (2A) [coordinate system] (M4G3) 4th (2B)

[functions] (MA1A1) 9th (2E) and others Post-secondary

[velocity] (S8P3) 8th (3A)

PS

4.1.1 Eulerian and Lagrangian Flow Descriptions

( )( )tzyxTT

tzyxTT

zzyyxx

,,,,,, 000

0

0

0

==

⎪⎭

⎪⎬

⎫

===

[four operations] (M1N3) 1st (2A) [calculus] Post-secondary

[Eulerian method] Post-secondary [Lagrangian method] Post-secondary

[temperature] (SP3) 9th (3B)

PS

4.1.2 one-, Two-, and three-Dimensional Flows ( )( )( ) kwjviutzyxVV

jviutyxVV

iutxVV

ˆˆˆ,,,

ˆˆ,,,

ˆ,

++==

+==

==

rr

rr

rr



PS

4.1.3 Steady and Unsteady Flows

0=∂∂

tVr


PS

4.1.4 Streamilnes, Streaklines, and Pathlines

uv

dxdy

= [four operations] (M1N3) 1st (2A)

[derivative] 12th (To be taught) [velocity] (S8P3) 8th (3A)

4.2 The Acceleration Field ( )taa rr

= [four operations] (M1N3) 1st (2A)

[functions] (MA1A1) 9th (2E) and others Post-secondary

[velocity] (S8P3) 8th (3A) [acceleration] (S8P3) 8th (3C)

PS

PS












32



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 4 Fluid Kinematics (Continued) 4.2.1 The Material Derivative

( ) ( ) ( ) ( )[ ]( ) ( ) ( )

( )

( ) ( ) ( ) ( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )( )( )

⎪⎪⎩

⎪⎪⎨

⎧

∇⋅+∂∂

=∂∂

+∂∂

+∂∂

+∂∂

=

∂∂

+∂∂

+∂∂

+∂∂

=→

⎭⎬⎫

⎩⎨⎧

=

=←

⎪⎪⎩

⎪⎪⎨

⎧

∂∂

+∂∂

+∂∂

=∇⋅

∂∂

+∂∂

+∂∂

=∇↑

∇⋅+∂∂

=∂∂

+∂∂

+∂∂

+∂∂

≡

→=→

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

∂∂

+∂∂

+∂∂

+∂∂

=

∂∂

+∂∂

+∂∂

+∂∂

=

∂∂

+∂∂

+∂∂

+∂∂

=

∂∂

+∂∂

+∂∂

+∂∂

=

∂∂

+∂∂

+∂∂

+∂∂

=

∂∂

+∂∂

+∂∂

+∂∂

==

=====

TVtT

zTw

yTv

xTu

tT

DtDT

dtdz

zT

dtdy

yT

dtdx

xT

tT

dtdT

tzyxVV

tzyxTT

zw

yv

xuV

kz

jy

ix

Vtz

wy

vx

utDt

D

DtVDa

zww

ywv

xwu

twa

zvw

yvv

xvu

tva

zuw

yuv

xuu

tua

zVw

yVv

xVu

tVa

zVw

yVv

xVu

tVa

dtdz

zV

dtdy

yV

dtdx

xV

tV

dtVdta

tzztyytxxttztytxVtrVV

AAAAAAAA

x

y

x

AA

AA

AA

AA

AAAAAAAAA

AAAAAA

AAAAAAA

r

rrr

r

rr

rrrrr

rrrrr

rrrrrr

rrr

,,,

,,,ˆˆˆ

e) DerivativlSubstantiaor e DerivativMaterial(

,,,,

[four operations] (M1N3) 1st (2A) [dot product] To be taught as a special math

topic [gradient: “del”] Post-secondary



PS PS




33



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 4 Fluid Kinematics (Continued) 4.2.2 Unsteady Effects

it

VtV

zVw

yVv

xVu

tVa

wvxu

ˆ0

0 0

∂∂

=∂∂

=∂∂

∂∂

+∂∂

+∂∂

=∴⎪⎭

⎪⎬⎫

==

=∂∂ rrrrr

rQ

[four operations] (M1N3) 1st (2A) [partial derivatives] Post-secondary


PS

4.2.3 Convective Effects

( ) ( ) ( ) ( )

( ) on)Accelerati e(Convectiv

ˆˆˆ

VV

kz

jy

ix

rr∇⋅

∂∂

+∂∂

+∂∂

=∇

[four operations] (M1N3) 1st (2A) [absolute value] (M7N1) 7th (2A)

[coordinate system] (M4G3) 4th (2B) [analytic geometry] Post-secondary [partial derivatives] Post-secondary


PS

4.2.4 Streamline Coordinates

( )

⎟⎠⎞

⎜⎝⎛

∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

=

⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂

+∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂

+∂∂

=

+==+==

=

ssVVs

sVVa

dtdn

ns

dtds

ss

tsVs

dtdn

nV

dtds

sV

tVa

DtsDVs

DtDV

DtsVDanasa

DtVDa

sVV

ns

ˆˆ

ˆˆˆˆ

ˆˆˆˆˆ

ˆ

r

r

rr

r

r

[four operations] (M1N3) 1st (2A) [radius] (M3G1) 3rd (2B)

[absolute value] (M7N1) 7th (2A) [analytic geometry] Post-secondary [partial derivative] Post-secondary

[limit] Post-secondary


PS

9th + PS










34



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 4 Fluid Kinematics (Continued) 4.2.4 Streamline Coordinates (Continued)

⎪⎪⎩

⎪⎪⎨

⎧

ℜ=

∂∂

=←

ℜ+

∂∂

=

ℜ==

∂∂

→

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

ℜ=

==ℜ∂

=

→

→

2

2

0

ˆˆ

ˆˆlim

ˆ

1ˆ

ˆˆˆ

1ˆ0

Va

sVa

nVssVVa

nss

ss

ss

ssss

ss

n

s

s

r

δδ

δδ

δδ

δ

δ

[four operations] (M1N3) 1st (2A) [radius] (M3G1) 3rd (2B)

[absolute value] (M7N1) 7th (2A) [analytic geometry] 12th (To be taught)

[partial derivative] Post-secondary [limit] Post-secondary


PS

4.3 Control Volume and System Representations ( )dtmvdF =


(MA1G5) 9th (2F) [areas of geometric shapes] (M5M1) 5th (2B)


9th

4.4 The Reynolds Transport Theorem

0:particles fluid malInfinitesi PropertyIntensive :b Property Extensive:B

22

122

→

⎪⎪⎩

⎪⎪⎨

⎧

=→=

=→=

=→=

=

V

VbVmB

VbmVB

bmB

mbB

δ

rrrr



[mass] (S8P3) 8th (3A) [temperature] (SP3) 9th (3B)

[momentum] (SP3) (3B)

9th

9th + PS














35



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 4 Fluid Kinematics (Continued) 4.4 The Reynolds Transport Theorem (Continued)

( )

dt

Vdbd

dtdB

dt

Vdbd

dtdB

VbB

VdbVbB

cvcvsyssys

sysi

iiiVsys

⎟⎠⎞⎜

⎝⎛

=⎟⎠⎞⎜

⎝⎛

=

=↑

===

∫∫

∫∑→

ρρ

δρδ

ρδρδ 0

lim



[mass] (S8P3) 8th (3A) [temperature] (SP3) 9th (3B)

[momentum] (SP3) (3B)

PS

4.4.1 Derivation of the Reynolds Transport Theorem

[ ] [ ] ( ) ( )∑∑ ⋅−⋅+∂

∂==

⋅−⋅+∂

∂=

ininoutoutCV

system

ininoutoutCVsystem

VmVmt

Vmdt

VmdF

bmbmt

BDt

DB

r&

r&

rrr

&&

[ ] [ ]

[ ] [ ] [ ]

( ) ( )⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

→=⋅−⋅∴

=======≡⋅

===

←

∑∑ ForceMomentum

&

Law)Second Newtons(

ininoutout VmVm

FamdtVdm

dtVmd

dtMdFam

dtVdmV

dtdmVm

FamdtVdm

dtVmd

r&

r&

rrrrr

vrr

rr&Q

rrrr

Note: Other Formulas used to derive the Reynolds Transport Theorem are available in pages 171-177.

∫ ∫ ⋅+∂∂

=cv cs

sys dAnVbVdbtDt

DBˆ

rρρ

4.4.2 Physical Interpretation N/A

[four operations] (M1N3) 1st (2A) [areas of geometric shapes] (M5M1) 5th (2B)


[dot product] To be taught as a special math topic

[analytic geometry] 12th (To be taught) [integration] 12th (To be taught)


[mass] (S8P3) 8th (3A) [density] (S6E5) 6th (4A)

[temperature] (SP3) 9th (3B) [momentum] (SP3) (3B)

PS

9th + PS















36



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 4 Fluid Kinematics (Continued) 4.4.3 Relationship to Material Derivative

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )( )e) DerivativlSubstantiaor e DerivativMaterial(

∇⋅+∂∂

=∂∂

+∂∂

+∂∂

+∂∂

≡

∂∂

+∂∂

+∂∂

+∂∂

=∇⋅

Vtz

wy

vx

utDt

Dz

wy

vx

ut

V

r

r

[four operations] (M1N3) 1st (2A) [dot product] To be taught as a special math

topic [analytic geometry] 12th (To be taught)



PS

4.4.4 Steady Effects

∫ ⋅=sys

sys dAnVbDt

DBˆ

rρ

[four operations] (M1N3) 1st (2A) [analytic geometry] 12th (To be taught)



[velocity] (S8P3) 8th (3A) [density] (S6E5) 6th (4A)


PS

4.4.5 Unsteady Effects

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

=⋅

<⋅

>⋅

===

====Δ=∂∂

=→=⋅

⋅+∂∂

=

∫∫

∫∫

CV) theof side the(along0ˆflow) (in0ˆ

flow)(out 0ˆˆ

ˆmomentum system0ˆ

0ˆ

ˆ

0

00

nV

nV

nV

iVVmBb

imVVmBiVV

VdbtDt

DBdAnVb

dAnVbVdbtDt

DB

cv

sys

cs

cscv

sys

r

r

r

rr

r

rrr

r

r

ρ

ρρ

ρρ



[integration] 12th (To be taught) [partial derivative] Post-secondary


[acceleration] (S8P3) 8th (3C) [momentum] (SP3) (3B)

PS

9th + PS











37



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 4 Fluid Kinematics (Continued) 4.4.5 Unsteady Effects (Continued)

( )( )( )( ) ( )( )

0ˆˆ

ˆˆ

ˆˆˆ

section)(another ˆ

section) (oneˆ


flow)(out 0ˆˆ


0ˆ

ˆ

12

012

0

0)2( 00)1( 0

0

0

0

0

00

=+−=

+−=

⋅=⋅

→⎪⎭

⎪⎬⎫

=⋅

−=⋅

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

=⋅

<⋅

>⋅

===

====Δ=∂∂

=→=⋅

⋅+∂∂

=

∫∫∫∫

∫∫

∫∫

iAViAV

dAViVdAViV

dAnViVdAnVb

VnV

VnV

nV

nV

nV

iVVmBb

imVVmBiVV

VdbtDt

DBdAnVb

dAnVbVdbtDt

DB

cscs

cv

sys

cs

cscv

sys

ρρ

ρρ

ρρ

ρ

ρρ

ρρ

rr

r

r

r

r

r

rr

r

rrr

r

r



To be taught as a special math topic [integration] 12th (To be taught)




PS

4.4.6 Moving Control Volumes

∫∫ ⋅+∂∂

=

+=−=

cscv

sys

cvcv

dAnWbVdbtDt

DBVWVWVV

ˆr

rrrrrr

ρρ

[four operations] (M1N3) 1st (2A) [analytic geometry]

[integration] 12th (To be taught) [partial derivative] Post-secondary



PS

9th + PS










38



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 4 Fluid Kinematics (Continued) 4.4.7 Selection of a Control Volume N/A

N/A N/A 9th

4.5 Chapter Summary and study Guide N/A

N/A N/A 9th

9th + PS

Chapter 5 Finite Control Volume Analysis 5.1 Conservation of Mass – The Continuity Equation 5.1.1 Derivation of the Continuity Equation

flow) ldimensiona-(one velocity ddistributeuniformly For ˆ

ˆ

ˆ

0ˆˆ

ˆ0

ˆˆ

0

VA

dAnVVV

A

dAnVVV

dAnVmAVQm

dAnVVt

mmdAnV

dAnVVdt

dAnVVdt

dAnVVdt

VDDtD

VdMDt

DM

Aaverage

Aaverage

A

cv csinoutcs

cscv

cscvcvcvsys

syssyssys

=⋅

==

⋅==

⋅===

=⋅+∂∂

−=⋅

⋅=∂∂

⋅∂∂

⋅+∂∂

=

==

∫

∫∫

∫ ∫∑ ∑∫

∫∫

∫∫∫∫∫

∫

ρ

ρ

ρ

ρ

ρρρ

ρρρ

ρρ

ρρρρρ

ρ

r

r

r&&

&&r

r

rr



[areas of geometric shapes] (M5M1) 5th (2B) [integration] 12th (To be taught as a special skill)

[mass] (S8P3) 8th (3A) [density] (S6E5) 6th (4A) [velocity] (S8P3) 8th (3A)

PS 9th + PS








39



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 5 Finite Control Volume Analysis (Continued) 5.1.2 Fixed, Non-deforming Control Volume

∑ ∑

∫

∑ ∑∑ ∑∫

=

====

==

∂∂

=−=−∂∂

outin

cv

inoutinoutcv

mmVAVAQVAVAm

VAmAVm

Vdt

QQmmVdt

&&

&

&&

&&

2211222111

surface control thein opening over the

ddistributeuniformly

flow) ldimensiona-(onesurface control

thein opening over theddistributeuniformly

00

ρρ

ρρ

ρ

ρ



[areas of geometric shapes] (M5M1) 5th (2B) [integration] 12th (To be taught as a special skill) [sigma notation] (M6N1) 6th (1A) or (MA1A3)

9th (2E)


∑ ∑=====

outin mmVAVAQVAVAm

&&

& 2211222111 ρρ

are not based on calculus

[mass] (S8P3) 8th (3A) [density] (S6E5) 6th (4A) [velocity] (S8P3) 8th (3A)

9th

5.1.3 Moving, Non-deforming Control Volume

0ˆ =⋅+∂∂

=+= ∫∫ cscv

syscv dAnWVd

tDtDM

VWVrrrr

ρρ [four operations] (M1N3) 1st (2A)




PS

5.1.4 Deforming Control Volume

cs

cvcscv

sys

VWV

Vdt

dAnWVdtDt

DM

rrr

r

+=

≠∂∂

=⋅+∂∂

= ∫∫∫ 00ˆ ρρρ



[areas of geometric shapes] (M5M1) 5th (2B) [dot product] To be taught as a special math

topic [partial derivatives] Post-secondary


PS

5.2 Newton’s Second Law – The Linear Momentum and Moment-of-Momentum Equation N/A

N/A N/A 9th

9th + PS


















40



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 5 Finite Control Volume Analysis (Continued) 5.2.1 Derivation of the Linear Momentum Equation

∑∫∫

∫∫∫

∑∑∑∫

=⋅+∂∂

⋅+∂∂

=

=

=∂∂

volumecontrol theofcontent

volumecontrol coincident theofcontent

ˆ

ˆ

FdAnVVVdVt

dAnVVVdVt

VdVDtD

FF

FVdV

cscv

cscvsys

sys

syssys

rrrr

rrrr

rr

rr

ρρ

ρρρ

ρ


[vector] (MA3A10) 11th (2H) To be taught as a special math topics


[dot product] To be taught as special math topic [analytic geometry] 12th (To be taught)

[integration] 12th (To be taught as special skill) [derivative] 12th (To be taught)

[partial derivative]


[density] (S6E5) 6th (4A)

PS

5.2.2 Application of the Linear Momentum Equation

( ) ( )

( )

( )

∑∫∫

∫∫∫

∫

∑∫∫

∑∫∫

∫∫∫

=⋅

=⋅

⋅+⋅=⋅+

→+∂∂

=⋅+++∂∂

=⋅+∂∂

⋅+∂∂

=


volumecontrol theof contents

volumcontrol theofcontent

ˆ

0ˆ

basis average-or time ousinstantane an (onflow steady For

ˆˆˆ

volumecontrol ngnondeformi inertial,For

volocity volumecontrolconstant For

ˆ

ˆ

ˆ

FdAnWW

dAnW

dAnWVdAnWWdAnWVW

VdVWt

FdAnWVWVdVWt

FdAnWVVdVt

dAnWVVdVt

VdVDtD

cs

cs

cscvcscs cv

cv cv

cs cvcv cv

cscv

cscvsys

rrr

r

rrrrrrr

rr

rrrrrr

rrrr

rrrr

ρ

ρ

ρρρ

ρ

ρρ

ρρ

ρρρ





[analytic geometry] 12th (To be taught) [integration] 12th (To be taught as a special

skill) [derivative] 12th (To be taught)




PS

9th + PS




















41



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 5 Finite Control Volume Analysis (Continued) 5.2.3 Derivation of the Moment-of-Momentum Equation

( ) ( )

( )[ ] ( )

( )[ ]

( )[ ] ( )( )

( ) ( )[ ]

( ) ( )( ) ( )

( ) ( ) ( )

( ) ( ) ( ) volumecontrol

theof contents

particle

particle

particleparticle

ˆ

ˆ

0

∑∫∫

∫∫∫

∑∑∑∫

∫∫

∑∑∑∫

×=×+×∂∂

×+×∂∂

=×

×=×

×=×

×=×

×=×

×=×

×=×→=×=

×+×=×

×=×=

FrdAnVVrVdVrt

dAnVVrVdVrt

VdVrDtD

FrFr

FrVdVrDtD

VdVrDtDVdVr

DtD

FrFr

FrVdVrDtD

FrVVrDtDVVV

DtrD

DtVVDrVV

DtrDVVr

DtD

FrVVDtDrFVV

DtD

cscv

cscvsys

cvsys

syssys

syssys

sys

syssys

rrrrrrr

rrrrrrr

rrrr

rrrr

rrrr

rrrr

rrrr

rrrvrrrr

rrrrrr

rrrrrr

ρρ

ρρρ

ρ

ρρ

δ

ρ

δρδ

ρδρδρδ

δδρδρδ




[analytic geometry] 12th (To be taught) [cross product] To be taught as a special math

topic [integration] 12th (To be taught) [derivative] 12th (To be taught)


9th (2E)



PS 9th + PS













42



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 5 Finite Control Volume Analysis (Continued) 5.2.4 Application of the Moment-of-Momentum Equation

( ) ( )

( ) ( ) ( )( )

( )[ ]

( )( ) ( )( )( ) ( )

( )( ) ( )( ) ( )outoutininshaftoutin

outoutoutinininshaft


shaftshaftoutoutoutinininshaft

shaftshaftshaftshaft

shaft

axialcscs

cscv

VUVUwmmmVUmVUmWUr

VrmVrmW

TWVrmVrmT

VUwmVUWmVrTW

TmVrTFr

mVrdAnVVrdAnVVr

UWVdAnVVrVdVrt

θθ

θθ

θθ

θθ

θθθ

θ

θ

ωωω

ωω

ω

ωω

ρρ

ρρ

±+±−===

±+±−=→=

±+±−=

=±+±−=

−=−=−==

=−=×

+−=⎥⎦⎤

⎢⎣⎡ ⋅×⋅×

+=⋅×=×∂∂

∑∫∫

∫∫

&&&

&&&

&&&

&&&

&&&&

&rr

&rrrrrr

rrrrrrrr

222222

22shaftaxial volumecontrol theof contents

22ˆˆ

ˆ0




[dot product] and [cross product] To be taught as a special math topics






PS

5.3 First Law of Thermodynamics – The Energy Equation N/A 5.3.1 Derivation of the Energy Equation

( ) ( )

∫∫∫

∫

∑∑∑∑∫

⋅+∂∂

=

⎟⎠

⎞⎜⎝

⎛+=⎟

⎠

⎞⎜⎝

⎛+

++=⎟⎠

⎞⎜⎝

⎛+=

−+−=

∨

cscvsys

innet

innet

sysinnet

innet

sysinnet

innetsys

sysoutinsysoutinsys

dAnVeVdet

VdeDtD

WQWQ

gzVueWQVdeDtD

WWQQVdeDtD

ˆ

2

volumecontrolcoincident

2

r

&&&&

&&

&&&&

ρρρ

ρ

ρ






9th (2E)

[density] (S6E5) 6th (4A) [heat] (S2P2) 2nd (3A)

PS

PS



















43



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 5 Finite Control Volume Analysis (Continued) 5.3.1 Derivation of the Energy Equation (Continued)

∫∫

∫∫∫

∫∫

∑∑

∑∑

∫∫

+=⋅⎟⎟⎠

⎞⎜⎜⎝

⎛++++

∂∂

⋅−+=⋅+∂∂

⋅=

⋅−=⋅=

⋅−=⋅−=⋅=

⋅=−=

−==

=−→=

⎟⎠

⎞⎜⎝

⎛+=⋅+

∂∂

∨

cscs

cscscs

cscs

shaftshaft

ouininnet

cvinnet

innetcscv

WQdAnVgzVpuVdet

dAnVpWQdAnVeVdet

VFW

dAnVpdAnVW

AnVpVAnpVAnW

VFWp

WWWTW

QQQ

WQdAnVeVdet

innet shaft

innet

2

innet shaft

innet

stress tangentialstress tangential

stress normal

stress normal

stress normalstress normal

outshaft

inshaft

innet shaft

ˆ2

:ationEnergy Equ

ˆˆ

ˆˆ

ˆˆˆ

00

ˆ

&&r

r&&

r

rr&

rr&

rrr&

rr&

&&&&

&&&

&&r

ρρ

ρ

ρρ

δδ

σ

δδδσδ

δδσ

ω

ρρ






9th (2E)

[density] (S6E5) 6th (4A) [heat] (S2P2) 2nd (3A)

PS

5.3.2 Application of the Energy Equation

∑

∑∫

∫

⎟⎟⎠

⎞⎜⎜⎝

⎛+++−

⎟⎟⎠

⎞⎜⎜⎝

⎛+++=⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛+++

≠⋅←≠⋅⎟⎟⎠

⎞⎜⎜⎝

⎛+++

∨

∨∨

∨

inflow

outflow

cs

cs

mgzVpu

mgzVpudAnVgzVpu

nVdAnVgzVpu

&

&r

rr

2

2ˆ

2

0ˆ0ˆ2

2

22

2

ρ

ρρ

ρ

ρρ




skill)

[velocity] (S8P3) 8th (3A) [gravity] (S6E1) 6th (3A) [density] (S6E5) 6th (4A) [mass] (S8P3) 8th (3A)

PS

9th + PS

















44



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 5 Finite Control Volume Analysis (Continued) 5.3.2 Application of the Energy Equation (Continued)

( )

( )

( )

( )innetinout

inoutinout

innetinout

inout

inout

inout

innetshaft

innetinout

inoutinout

innetshaft

innet

inoutinout

inout

inout

inin

outout

cs

QzzgVVhhm

QzzgVVppuum

WQzzgVVhhm

puh

WQ

zzgVVppuum

mgzVpu

mgzVpudAnVgzVpu

&&

&&

&&&

&&

&

&

&r

=⎥⎦

⎤⎢⎣

⎡−+

−+−

=⎥⎦

⎤⎢⎣

⎡−+

−+⎟⎟

⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛+−

+=⎥⎦

⎤⎢⎣

⎡−+

−+−

→+=

+=

⎥⎦

⎤⎢⎣

⎡−+

−+⎟⎟

⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛+−

⎟⎟⎠

⎞⎜⎜⎝

⎛+++−

⎟⎟⎠

⎞⎜⎜⎝

⎛+++=⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛+++

∨∨

∨∨

∨∨

∨∨

∨∨

∨

∨∨

∫

2

flowsteady l,dimensiona-one e,compressivfor Enthalpy

2

stream fluid oneonly involvingflow ldimensiona-one oughout,steady thrfor Enthalpy

2

2

2

2ˆ

2

22

22

22

22

2

22

ρρ

ρ

ρρ

ρ

ρρ

ρ



[analytic geometry] 12th (To be taught) [integration] 12th (To be taught as a special skill)


PS 9th + PS









45



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 5 Finite Control Volume Analysis (Continued) 5.3.3 Comparison of the Energy Equation with the Bernoulli Equation

( )

loss

friction) flow with ibleincompress(Steady 0

flow) ibleincompresssteady ess(Frictionl0

22

22

22

22

2

22

22

22

22

22

=−−

>−−

=−−

++=++

→⎟⎟⎠

⎞⎜⎜⎝

⎛++

=⎟⎟⎠

⎞⎜⎜⎝

⎛++

=→=++=++

=

⎟⎠

⎞⎜⎝

⎛−−−++=++

=⎥⎦

⎤⎢⎣

⎡−+

−+−+−

∨∨

∨∨

∨∨

∨∨

∨∨

innetinout

innetinout

innetinout

ininin

outoutout

inin

inoutout

out

inin

inoutout

out

innet

innet

innetinoutin

ininout

outout

innetinout

inoutinoutinout

quu

quu

quu

gzVpgzVp

zVpzVp

ggzVpzVp

m

Qq

quugzVpgzVp

QzzgVVppuum

ρρ

ρ

γρ

ρ

γρ

ργργγργρ

ρρ

ρρ

&

&

&&




9th 9th + PS









46



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 5 Finite Control Volume Analysis (Continued) 5.3.3 Comparison of the Energy Equation with the Bernoulli Equation

( )

( )

( ) ( )PLspTLsT

innetshaft

innetshaft

innetshafts

Lsinnet

shaftininin

outoutout

innetshaftin

ininout

outout

innetshaftin

ininout

outout

innetshaftin

ininout

outout

innetinout

innetshaftin

ininout

outout

innetshaft

innetinout

inoutinoutinout

ininin

outoutout

hhhhhhQ

W

gm

W

g

wh

hhwzg

Vpzg

Vpg

wgzVpgzVp

wzVpzVp

wgzVpgzVp

quuwgzVpgzVp

WQzzgVVppuum

gzVpgzVp

+=+−=

===

−−+++=++

→⎟⎟⎠

⎞⎜⎜⎝

⎛−+++=++

−+++=++

−+++=++

⎟⎠

⎞⎜⎝

⎛−−−+++=++

+=⎥⎦

⎤⎢⎣

⎡−+

−+−+−

−++=++

∨∨

∨∨

γ

γγ

ρρ

ρργργρ

ρρ

ρρ

ρρ

ρρ

&

&

&

&&&

22

loss22

loss22

loss22

22

2

loss22

22

22

22

22

22

22

22




9th 9th + PS









47



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 5 Finite Control Volume Analysis (Continued) 5.3.4 Application of the Energy Equation to Non-uniform Flow

( )

( )

( )

→⎟⎟⎠

⎞⎜⎜⎝

⎛−+++=++

−+++=++

→

⎟⎟⎠

⎞⎜⎜⎝

⎛−+++=++

−+++=++

⋅=⋅=

⎟⎟⎠

⎞⎜⎜⎝

⎛−=⋅

∫∫

∫

g

wgzVp

gzVp

wzV

pzV

p

wgzVp

gzVp

wgzVp

gzVp

Vm

dAnVVdAnVVVm

VVmdAnVV

innetshaftin

inininout

outoutout

innetshaftin

inininout

outoutout

innetshaftin

inininout

outoutout

innetshaftin

inininout

outoutout

A

A

cs

ininoutout

loss22

loss22

loss22

loss22

2

ˆ2ˆ

22

22ˆ

2

22

22

22

22

2

222

222

αρ

αρ

ρργρα

γρα

ρα

ρα

ρ

αρ

αρ

α

ραρα

ααρ

&

rr&

&r

Linnet

shaft

inininin

outoutoutout h

g

wz

gVpz

gVp

−+++=++22

22 αγ

αγ





Linnet

shaft

inininin

outoutoutout h

g

wz

gVpz

gVp

−+++=++22

22 αγ

αγ

is based on pre-calculus mathematics. Others used to derive this formula could be removed from

classroom instruction.


9th

5.3.5 Combination of the Energy Equation and the Moment-of-momentum Equation

innetshaft

innetshaft

w

w loss−=η

[four operations] (M1N3) 1st (2A) [heat] (S2P2) 2nd (3A) [temperature] (SP3) 9th (3B)

9th

9th + PS











48



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 5 Finite Control Volume Analysis (Continued) 5.4 Second Law of Thermodynamics – Irreversible Flow

012 ≥−−∨∨

innetquu

[four operations] (M1N3) 1st (2A) [heat] (S2P2) 2nd (3A) [temperature] (SP3) 9th (3B)

PS

5.4.1 Semi-infinitesimal Control Volume Statement of the Energy Equation

( )

( )

⎟⎠

⎞⎜⎝

⎛−−=+⎟⎟

⎠

⎞⎜⎜⎝

⎛+

=⎥⎦

⎤⎢⎣

⎡+⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+

⎟⎟⎠

⎞⎜⎜⎝

⎛+==⎥

⎦

⎤⎢⎣

⎡+⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+

↓

∨∨

innet

innet

innet

qdsTdzgVddp

QdzgVdpdpddsTm

pduddsTQdzgVdpdudm

δρ

δρρ

ρδ

ρ

2

21

12

2

2

2

&&

444444444444 3444444444444 21

&&


(MA1G5) 9th (2F) [integration] 12th (To be taught as a special


[partial derivative] Post secondary [sigma notation] (M6N1) 6th (1A) or (MA1A3)

9th (2E)

[mass] (S8P3) 8th (3A) [velocity] (S8P3) 8th (3A)

[heat] (S2P2) 2nd (3A) [temperature] (SP3) 9th (3B)

[gravity] (S6E1) 6th (3A)

PS

5.4.2 Semi-infinitesimal Control Volume Statement of the Second Law of Thermodynamics

( )

0

0ˆ

ˆ

≥−≥

≥≥−

=∂∂

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛≥⋅+

∂∂

⋅+∂∂

=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛≥

∑∑

∫∑∫∫

∫∫∫

∑∑∑∫

innet

innet

innet

innet

INout

cv

cv

innet

cscv

cscvsys

cv

innet

sys

innet

sys

innet

sys

qdsTqdsTT

Qdsm

T

QSsm

VdstT

QdAnVsVds

t

dAnVsVdst

VdsDtD

T

Q

T

Q

T

QVds

DtD

δδ

δδ

ρδ

ρρ

ρρρ

δδδρ

&

&

&

&

&r

r

&&&


(MA1G5) 9th (2F) [derivative] 12th (To be taught)

[partial derivative] Post secondary [integration] 12th (To be taught)


[heat] (S2P2) 2nd (3A) [velocity] (S8P3) 8th (3A) [density] (S6E5) 6th (4A)


PS

9th + PS






















49



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 5 Finite Control Volume Analysis (Continued) 5.4.3 Combination of the Equations of the First and Second Laws of Thermodynamics

( )

( )

( )

( )

loss101

loss

loss01

loss1

loss2

02

loss2

02

2

2

2

2

=−⎟⎟⎠

⎞⎜⎜⎝

⎛+−→≠⎟⎟

⎠

⎞⎜⎜⎝

⎛

=−−

=−→=⎟⎟⎠

⎞⎜⎜⎝

⎛

=−⎟⎟⎠

⎞⎜⎜⎝

⎛+

−=⎥⎦

⎤⎢⎣

⎡+⎟⎟

⎠

⎞⎜⎜⎝

⎛+−

=+⎟⎟⎠

⎞⎜⎜⎝

⎛+

⎟⎠

⎞⎜⎝

⎛−==⎥

⎦

⎤⎢⎣

⎡+⎟⎟

⎠

⎞⎜⎜⎝

⎛+−

≥⎥⎦

⎤⎢⎣

⎡+⎟⎟

⎠

⎞⎜⎜⎝

⎛+−

∫∨∨

∨∨

∨

∨

innet

out

ininout

innetinout

innet

innet

innetshaft

innet

qpduud

quu

qudd

qpdud

wdzgVddp

dzgVddp

qdsTdzgVddp

dzgVddp

ρρ

δδρ

δδρ

δδρ

ρ

δδρ

ρ


(MA1G5) 9th (2F) [derivative] 12th (To be taught)

[partial derivative] Post secondary [integration] 12th (To be taught)

[heat] (S2P2) 2nd (3A) [velocity] (S8P3) 8th (3A) [density] (S6E5) 6th (4A)


PS 9th + PS









50



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 5 Finite Control Volume Analysis (Continued) 5.4.4 Application of the Loss Form of the Energy Equation

1

21

1

12

22

2

2

1

1

2

22

1

1

212

2 2

22

1

211

2

222

2121

1constant

2222

gzVpk

kgzVpk

k

ppk

kdpp

gzVgzVdpgzVpgzVp

k

++−

=++−

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−==

+=++++=++

∫

∫

ρρ

ρρρρ

ρρρ


[derivative] 12th (To be taught) [partial derivative] Post secondary



1

21

1

12

22

2

2

2121gzVp

kkgzVp

kk

++−

=++− ρρ

is not based on calculus



9th

5.5 Chapter Summary and Study Guide N/A

N/A N/A 9th

9th + PS

Chapter 6 Differential Analysis of Fluid Mechanics Flow (Note: This whole Chapter appears to be too deep in calculus-based mathematics. Actually, some professors in undergraduate engineering programs cut the whole Chapter off when teaching Fluid Mechanics course. Therefore, engineering analytic principles and skills from this Chapter are NOT analyzed for the eventual inclusion into a potentially viable K-12 engineering curriculum. The subheadings of Sections are still listed below for reference purposes). 6.1 Fluid Mechanics Element Kinematics 6.3.2 Equations of Motion 6.5.3 Vortex 6.8.2 The Navier-Stokes Equations 6.1.1 Velocity and Acceleration Fields Revisited 6.4 Inviscid Flow 6.5.4 Doublet 6.9 Some Simple Solutions for Viscous,

Incompressible Fluids 6.1.2 Linear Motion and Deformation 6.4.1 Euler’s Equations of Motion 6.6 Superposition of Basic, Plane Potential Flows 6.9.1 Steady, Laminar Flow between Fixed Parallel

Plates 6.1.3 Angular Motion and Deformation 6.4.2 The Bernoulli Equation 6.6.1 Source in a Uniform Stream – Half-Body 6.9.2 Couette Flow 6.2 Conservation of mass 6.4.3 Irrotational Flow 6.6.2 Rankine Ovals 6.9.3 Steady, Laminar Flow in Circular Tubes 6.2.1 Differential Survey Form of Continuity Equation

6.4.4 The Bernoulli Equation for Irrotational Flow 6.6.3 Flow around a Circular Cylinder 6.9.4 Steady, Axial, Laminar Flow in an Annulus

6.2.2 Cylindrical Polar Coordinates 6.4.5 The Velocity Potential 6.7 Other Aspects of Potential Flow Analysis 6.10 Other Aspects of Differential Analysis 6.2.3 The Stream Function 6.5 Some Basic, Plane Potential Flows 6.8 Viscous Flow 6.10.1 Numerical Methods 6.3 Conservation of Linear Momentum 6.5.1 Uniform Flow 6.8.1 Stress-Deformation Relationships Chapter Summary and Study Guide 6.3.1 Description of Forces Acting on the Differential Element

6.5.2 Source and Sink

PS







51



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 7 Similitude, Dimensional Analysis, and Modeling (Note: This whole Chapter appears to be mildly deep in calculus-based mathematics. However, the type of “abstract thinking” required to understand and to apply the content knowledge contained in this Chapter appears to be most likely beyond the cognitive developmental maturity level of high school students. Therefore, engineering analytic principles and skills from this Chapter are NOT analyzed for the eventual inclusion into a potentially viable K-12 engineering curriculum. The subheadings of Sections are still listed before for reference purposes). Some appropriate skills in 7.1 (Dimensional Analysis) could be considered for high schools. 7.1 Dimensional Analysis 7.4.3 Uniqueness of Pi Terms 7.8 Modeling and Similitude 7.9.2 Flow around Immersed Bodies 7.2 Buckingham Pi Theorem 7.5 Determination of Pi Terms by Inspection 7.8.1 Theory of Models 7.9.3 Flow with a Free Surface 7.3 Determination of Pi Terms 7.6 Common Dimensionless Groups in Fluid

Mechanics 7.8.2 Model Space 7.10 Similitude Based on Governing Differential

Equations 7.4 Some Additional Comments about Dimensional Analysis

7.7 Correlation of Experimental Data 7.8.3 Practical Aspects of Using Models 7.11 Chapter Summary and Study Guide

7.4.1 Selection of Variables 7.7.1 Problems with One Pi Term 7.9 Some Typical Model Studies 7.4.2 Determination of Reference Dimensions 7.7.2 Problems with Two or More Pi Term 7.9.1 Flow through Closed Conduits

PS

Chapter 8 Viscous Flow in Pipes 8.1 General Characteristics of Pipe Flow 8.1.1 Laminar of Turbulent Flow

μρVD

=Re

8.1.2 Entrance Region and Fully Developed Flow

( )54

6/1

10Re10

flow)lent (for turbuRe4.4

flow)lent (for turbu Re06.0

<<

=

=

D

De

e

l

l


[exponent] (M6A3) 6th (2A)

[mass] (S8P3) 8th (3A) [density] (S6E5) 6th (4A)

[force] (S4P3) 4th (3A) or (S8P3) 8th (3C) [pressure] (SC5) 9th (4B) To be taught

[velocity] (S8P3) 8th (3A) [momentum] (SP3) (3B)

PS

8.1.3 Pressure and Shear Stress

021 <Δ

−=∂∂

−=∇l

pxpppp



PS

9th + PS











52



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 8 Viscous Flow in Pipes (Continued) 8.2 Fully Developed Laminar Flow 8.2.1 From F = ma Applied Directly to a Fluid Mechanics Element

( ) ( ) ( )

( )

( )

( )

( )

( )l

l

l

l

l

l

l

l

ll

ll

l

ll

rrr

rr

μθγπ

μθγτθγ

μπ

μππ

ππ

ππ

μτ

μμ

μμμττ

τττπτππ

128sin

32sin2sin

128

32222

122

14

2121164

224

2202

0ˆ0

4

24

2

2

2

2

2

0

2

0

2

222

12

21

21

12

DpQ

DpVr

ppDQ

pDVRVRV

RQ

AQVVRQ

drrRrVdrrrudAuQ

RrDru

DrV

DrpDruCrpu

drrpdurpdrdu

drdu

Dp

Dr

rprrpprp

pppixuuVV

tVamF

ccc

RRr

r c

w

c

w

w

−Δ=

−Δ==

−ΔΔ=

Δ======

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−===

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛ Δ=+⎟⎟

⎠

⎞⎜⎜⎝

⎛ Δ−=

Δ−=⎟⎟

⎠

⎞⎜⎜⎝

⎛ Δ−=−==Δ

==Δ

=−Δ−−

Δ−==∂∂

=∇⋅=∂∂

=

∫∫∫

∫∫

=

=

[four operations] (M1N3) 1st (2A) [coordinate system]

[trigonometric functions] (MA2G2) 10th (2F) [dot product]

To be taught as a special math topic [integration] 12th (To be taught) [derivative] 12th (To be taught)

[partial derivative] Post-secondary [gradient] Post-secondary

[force] (S4P3) 4th (3A) or (S8P3) 8th (3C) [acceleration] (S8P3) 8th (3C)

[pressure] (SC5) 9th (4B) To be taught [velocity] (S8P3) 8th (3A)

PS 9th + PS








53



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 8 Viscous Flow in Pipes (Continued) 8.2.2 From the Navier-Stokes Equations

⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

=+∂∂

∇=+∇=⋅∇−=

∇++∇

−=∇⋅+∂∂

=⋅∇

rur

rrg

xp

VkgpVkgg

VvgpVVtVVV

1sin

ˆ0ˆ

0

2

2

μθρ

μρ

ρrrr

rrrrr

rr


[exponent] (M6A3) 6th (2A) [dot product]

To be taught as a special math topic [vector] (MA3A10) 11th (2H) To be taught as

a special math topics [partial derivative] Post-secondary [gradient: “del”] Post-secondary

[velocity] (S8P3) 8th (3A) [pressure] (SC5) 9th (4B) To be taught

[density] (S6E5) 6th (4A) [gravity] (S6E1) 6th (3A)

PS

8.2.3 From Dimensional Analysis

( )

( )

( )

2

22

2

2

22

4

2

8Re64

22

Re6464

21

32

21

32

4

constant,,,

VffVDpfV

Dfp

DDVDV

DV

V

pD

Vp

pDCAVQD

VCpDC

VpD

CDC

DDVpD

DVFp

w

ρτρρ

ρμ

ρ

μ

ρ

μ

μπμ

μ

φφμ

μ

==⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛Δ==Δ

⎟⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛⎟⎟⎠

⎞⎜⎜⎝

⎛==

Δ=Δ

Δ===

Δ=

Δ

==⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

Δ=Δ

l

l

llll

ll

l

llll


[force] (S4P3) 4th (3A) or (S8P3) 8th (3C) [velocity] (S8P3) 8th (3A)


Note: Special topics from 7.1 (Dimensional Analysis) need to be taught

PS

8.2.4 Energy Considerations

Dh

rh

zzppp

hzpzphzg

Vpzg

Vp

wLL

LL

γτ

γτ

θγγ

αγ

αγ

ll

l

42sin22

1221

12

11

2

22

22

1

21

11

==

=−Δ+=

=⎟⎟⎠

⎞⎜⎜⎝

⎛+−⎟⎟

⎠

⎞⎜⎜⎝

⎛++++=++



9th

9th + PS


















54



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 8 Viscous Flow in Pipes (Continued) 8.3 Fully Developed Turbulent Flow N/A 8.3.1 Transition from Laminar to Turbulent Flow N/A 8.3.2 Turbulent Shear Stress

( )

( ) ( )

( ) ( )

( ) ( )

( )2

22

2/12

2

22

''

'1'

intensity Turbulence

0'1'

0111'

'',,,1

0

0

0

0

0

0

0

0

0

0

0

0

⎟⎟⎠

⎞⎜⎜⎝

⎛===

+=−==≠=

⎥⎦⎤

⎢⎣⎡

==

>=

=−=⎟⎠⎞⎜

⎝⎛ −=−=

−=+==

∫

∫

∫∫∫

∫

+

+

+++

+

dyud

dyud

dyud

vudyudyuu

dyud

dydu

u

dtuT

uu

dtuT

u

uTuTT

dtudtuT

dtuuT

u

uuuuuudttzyxuT

u

mturbmturb

turblam

Tt

t

Tt

t

Tt

t

Tt

t

Tt

t

Tt

t

ll ρτρηητ

ττρμτμτμτ

[coordinate system] (M4G3) 4th (2B) [analytic geometry] 12th (To be taught)





PS

8.3.3 Turbulent Velocity Profile

( ) n

c

c

w

Rr

Vu

yR

uuV

vyu

uuurRy

vyu

uu

/1

2/1

1ln5.2*

0.5*ln5.2*

***

⎟⎠⎞

⎜⎝⎛ −=⎟⎟

⎠

⎞⎜⎜⎝

⎛=

−

+⎟⎠⎞

⎜⎝⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛=−==

ρτ

8.3.4 Turbulent Modeling N/A


[logarithmic functions] (MA2A5) 10th (2E) [analytic geometry] 12th (To be taught)


PS

9th + PS









55



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 8 Viscous Flow in Pipes (Continued) 8.3.5 Chaos and Turbulence N/A N/A N/A PS 8.4 Dimensional Analysis of Pipe Flow

minor Lmajor L hhhL += [four operations] (M1N3) 1st (2A) Note: Special topics from 7.1 (Dimensional

Analysis) need to be taught 9th

8.4.1 Major Losses ( )

( ) ( )

⎟⎟⎠

⎞⎜⎜⎝

⎛+−=

+−=+−=−=

+++=++⎟⎠⎞

⎜⎝⎛=

⎟⎠⎞

⎜⎝⎛=

Δ=⎟⎟

⎠

⎞⎜⎜⎝

⎛=

Δ

=Δ+=

fD

f

VD

fzzhzzppg

VD

fh

hzg

Vpzg

VpD

f

DDV

pVDDD

VD

V

p

DVFphhh

L

L

L

Re51.2

7.3log0.21

22

22Re,

Re,

21Re,,~

21

,,,,,

2

121221

2

major L

2

22

22

1

21

11

22

minor Lmajor L

ε

ργγγ

αγ

αγ

εφ

εφρμ

ρεμ

ρφρ

ρμε

ll

ll

l

8.4.2 Minor Losses

( )

( )

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−=

⎟⎟⎠

⎞⎜⎜⎝

⎛−==++=+

−=−=

====

==ΔΔ

==

2

1222

22Re,

221

212

21

12

2

2

12

1

233

211

133333313311

22

minor L

2

minor L2

22

minor L

VppC

AAK

gVhKh

gVp

gVp

VVVAApApVAVAfDK

gV

Df

gVKhgeometryK

gVKhVKp

V

pgV

hK

p

LL

LL

Leq

eqLL

LLL

ρ

γγ

ρ

φ

ρρ

ll

[four operations] (M1N3) 1st (2A) [areas of geometric shapes: circle, triangle]

(M5M1) 5th (2B) (M5M1) 5th (2B)


[exponent] (M6A3) 6th (2A) [square root] (M8N1) 8th (2A)

[graph] (S7CS6) 7th (6)


[force] (S4P3) 4th (3A) or (S8P3) 8th (3C) [gravity] (S6E1) 6th (3A) [density] (S6E5) 6th (4A)

Note: Special topics from 7.1 (Dimensional Analysis) need to be taught

9th

9th + PS














56



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 8 Viscous Flow in Pipes (Continued) 8.4.3 Noncircular Conduits

( )

( )g

VDfh

DD

DPADVDCf

hL

hh

hh

2

444ReRe

2

2

l=

=====ππ

μρ

8.5 Pipe Flow Examples N/A 8.5.1 Single Pipes N/A 8.5.2 Multiple Pipe Systems N/A



[velocity] (S8P3) 8th (3A) [gravity] (S6E1) 6th (3A) [density] (S6E5) 6th (4A)

9th

8.6 Pipe Flowrate Measurement 8.6.1 Pipe Flowrate Meters

( )( )

( )( )

( )( )

( )( )4

21421

1

421

000

222

211

2211421

222

12

12

Re

12

22

12

βρβρ

μρβ

βργγ

βρ

−−

==−−

==

===

−−

==++=+

==−−

==

ppACQCQppACQCQ

AQVVD

Dd

ppACQCQhg

Vpg

Vp

VAVAQppAVAQ

Tvidealvnnidealn

lideaL

ideal

8.6.2 Volume Flow Meters N/A


[square root] (M8N1) 8th (2A) [areas of geometric shapes] (M5M1) 5th (2B)


[gravity] (S6E1) 6th (3A) [density] (S6E5) 6th (4A)

9th


N/A N/A 9th

9th + PS














57



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 9 Flow over Immersed Bodies 9.1 General External Flow Characteristics 9.1.1 Lift and Drag Concepts

( ) ( )( ) ( )

AU

DCAU

LC

dAdApdFL

dAdApdFD

dAindApdF

dAdApdF

DL

wy

wx

wy

wx

22

21

21

cossin

sincos

cos

sincos

ρρ

θτθ

θτθ

θτθ

θτθ

rr

r

r

==

+−==

+==

→⎪⎭

⎪⎬⎫

+−=

+=

∫∫∫∫∫∫

[four operations] (M1N3) 1st (2A) [areas of geometric shapes] (M5M1) 5th (2B) [trigonometric functions] (MA2G2) 10th (2F)

[integration] 12th (To be taught as a special skill) [derivative] 12th (To be taught)


AU

DCAU

LC DL22

21

21 ρρ

rr

==

are not based on calculus

[force] (S4P3) 4th (3A) or (S8P3) 8th (3C) 9th

9.1.2 Characteristics of Flow Past an Object N/A

N/A [force] (S4P3) 4th (3A) or (S8P3) 8th (3C) [Reynolds Number] To be taught as special

topic

9th

9.2 Boundary Layer Characteristics N/A 9.2.1 Boundary Layer structure and Thickness on a Flat Plate

( )

( ) ( ) ( )

∫

∫∫∫

∫∫

∞

∞∞

∞∞

⎟⎠⎞

⎜⎝⎛ −=

−=−=−

⎟⎠⎞

⎜⎝⎛ −=−=

0

0

2

0

00

1

1**

dyUu

UuO

dyuUuObUdyuUubdAuUu

dyUudybuUbU

ρρρ

δδ


[areas of geometric shapes] (M5M1) 5th (2B) [integration] 12th (To be taught as a special

skill)


PS

9th + PS












58



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 9 Flow over Immersed Bodies (Continued) 9.2.2 Prandtl/Blasius Boundary Layer Solution

( ) ( ) ( )

( ) ( )

xU

xO

xxUvx

asfandatffatfffff

ffx

vUvUfux

vy

u

fffVxUvxU

Uvxyg

UuyasUu

yonvuyuv

yuv

xuu

yv

xu

yx

uv

yv

xu

yv

xvv

xp

yvv

xvu

yu

xuv

xp

yuv

xuu

w

xxx

ρμτ

δδδ

ηηη

ηη

ηηη

δδ

ρ

ρ

2/3

2/1

2/12

2/1

2

2

2

2

2

2

2

2

2

2

332.0

Re664.0

Re721.1*

Re55

1'00'00'0'''''2

'4

'

~

00

00

1

1

=

====

∞→→=======−

−⎟⎠⎞

⎜⎝⎛==

∂Ψ∂

−=∂Ψ∂

=

==Ψ⎟⎠⎞

⎜⎝⎛=

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=∞→→

===∂∂

=∂∂

+∂∂

=∂∂

+∂∂

→⎪⎭

⎪⎬⎫

∂∂

<<∂∂<<

=∂∂

+∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

−=∂∂

+∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

−=∂∂

+∂∂


[square root] (M8N1) 8th (2A) [functions] (MA1A1) 9th (2E) and others

Post-secondary [partial derivative] Post-secondary

[3rd order non-linear differential equation] Post-secondary

[velocity] (S8P3) 8th (3A) [density] (S6E5) 6th (4A) [gravity] (S6E1) 6th (3A)

PS 9th + PS








59



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 9 Flow over Immersed Bodies (Continued) 9.2.3 Momentum Integral boundary Layer Equation for a Flat Plate

( ) ( )

( )( ) ( )

( )

( ) ( ) ( )[ ]

( ) ( )[ ]

xf

xf

wfw

x

YYyw

ww

plate wplate wx

x

cCC

UxCCc

Uc

xU

CC

CCxUC

xvCdx

UCC

d

dYdgCCU

dYdgU

yu

dYYgYgCCbUD

dYYgYgbUdyuUubD

dxOdUb

dxDd

dxOdbU

dxDdObUD

dyuUubDdyUubbhUdyuUh

dyubbhUDdAudAUUD

dxdADF

dAnVudAnVuF

Re664.0

Re

22

212

Re

22

1

1

ˆˆ

2121

2

2/321

12

1

2

1

2

022

00

1

0112

1

0

2

0

222

00

2

0

0

22

2

2

1

21

===

==

===

===∂∂

=

−==

−=−=

====

−===

−=+−=−

−=−=−=

⋅+⋅=

===

∫∫∫

∫∫∫

∫∫∫

∫∫∑∫∫∑

ρμ

ρ

τρμτ

δδρμ

δδ

δμ

δμμτ

δρ

δρρ

ρττρρ

ρρρ

ρρρρ

ττ

ρρ

δ

δδδ

δ

r

r

rrr

r

rr

r

rr



[square root] (M8N1) 8th (2A) [sigma notation] (M6N1) 6th (1A) or (MA1A3)

9th (2E) [integration] 12th (To be taught as a special

skill) [partial derivative] Post-secondary


PS 9th + PS









60



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 9 Flow over Immersed Bodies (Continued) 9.2.3 Momentum Integral boundary Layer Equation for a Flat Plate (Continued)

l

l

l

l

lll

r

Re328.1

Re

81

21

21

21

02

0

2

=

==== ∫∫

Df

DffDf

wfDf

C

CCCdxcC

bU

dxb

bU

DC

ρ

τ

ρ

9.2.4 Transition from Laminar to Turbulent Flow N/A 9.2.5 Turbulent Boundary Layer Flow N/A 9.2.6 Effects of Pressure Gradient N/A



[square root] (M8N1) 8th (2A) [sigma notation] (M6N1) 6th (1A) or (MA1A3)

9th (2E) [integration] 12th (To be taught as a special



PS

9.2.7 Momentum Integral Boundary Layer Equation with Nonzero Pressure Gradient

( )constant

*2

==

+=−=

UUdx

dUUOU

dxd

dxdU

Udxdp

fs

fsfsfsw

fsfs ρδρτρ




PS

9.3 Drag

( )lr

εφρ

,,Re,,

21 2

FrMashapeCAU

DC DD ==


[force] (S4P3) 4th (3A) or (S8P3) 8th (3C) [density] (S6E5) 6th (4A)

9th

9.3.1 Friction Drag

Dff CbUD lr

2

21 ρ=



9th

9th + PS




















61



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 9 Flow over Immersed Bodies (Continued) 9.3.2 Pressure Drag

( )Re22

21,,

cos

21

cos

21

cos

2222

22

CU

UC

U

DCUCDUfD

A

dAC

AU

dA

AU

DC

dAD

D

ppDp

p

=====

===

=

∫∫∫

l

l

l

r

lr

lr

r

ρμ

ρμμ

θ

ρ

θρ

ρ

θρ

9.3.3 Drag Coefficient Data and Examples




9th

9.4 Lift 9.4.1 Surface Pressure Distribution

( )lr

εφρ

,,Re,,

21 2

FrMashapeCAU

LC LL ==

9.4.2 Circulation N/A



9th


N/A N/A 9th

9th + PS

Chapter 10 Open Channel Flow 10.1 General Characteristics of Open-Channel Flow

( ) 2/1Re lgVFrVRh == μρ [four operations] (M1N3) 1st (2A)

[exponent] (M6A3) 6th (2A) [trigonometric functions] (MA2G2) 10th (2F) [ellipse] (MA2G4) 10th (2F) To be taught

[velocity] (S8P3) 8th (3A) [gravity] (S6E1) 6th (3A)

9th 9th + PS















62



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 10 Open Channel Flow (Continued) 10.2 Surface Waves 10.2.1 Wave Speed

( )( )( )

( ) ( )[ ]

( ) ( )

→

⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪

⎬

⎫

⎟⎟⎠

⎞⎜⎜⎝

⎛+≈→<<

=+

=+

=+

==

+==

+==

−−=+−

→

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

=→<<

+=

/++−=/−

2/1

2

222

2

211

1

22

11

0

0

constant2

22

21

21

yygyc

yy

yVVy

yg

VV

yg

V

gyccg

yV

byyAyF

byyAyF

cVcbcybyyby

yVycyy

yVyyc

byyVcbcy

cc

δδ

δδ

δδ

δδ

δγγδγγ

δρδγγ

δδδ

δδδ

δδ


[trigonometric functions] (MA2G2) 10th (2F) [derivative] 12th (To be taught)

[velocity] (S8P3) 8th (3A) [speed] (S2P3) 2nd (3A)


9th 9th + PS







63



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 10 Open Channel Flow (Continued) 10.2.1 Wave Speed (Continued)

022tanh

12tanh

22tanh

2

2/1

→→⎟⎠⎞

⎜⎝⎛

∞→→⎟⎠⎞

⎜⎝⎛

=→>>⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛=

λλπ

λπ

λλπ

πλλ

λπ

πλ

yasyy

yasy

gcyygc


[trigonometric functions] (MA2G2) 10th (2F) [analytic geometry: hyperbolic tangent] Post-

secondary To be taught [derivative] 12th (To be taught)

[velocity] (S8P3) 8th (3A) [speed] (S2P3) 2nd (3A)


9th

10.2.2 Froude Number Effects N/A

N/A [velocity] (S8P3) 8th (3A) [speed] (S2P3) 2nd (3A)

9th

10.3 Energy Considerations

( ) ( )

( )gVVyy

SS

SSgVVyyhS

hg

VySg

Vy

yp

ypSzz

hzg

Vpzg

Vp

f

fL

f

L

L

200

2

22

22

21

22

210

0

21

22

21

22

20

21

1

22

11

021

2

222

1

211

−=−→

⎭⎬⎫

=

=

−+−

=−→=

++=++

→

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

=

=

=−

+++=++

ll

l

l

γ

γγγ



[potential energy] (SP3) 9th (3A)

9th

9th + PS













64



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 10 Open Channel Flow (Continued) 10.3.1 Specific Energy

( )

( ) ( ) 1

2301

22

2/12/12/3

min

3/12

3

2

2

2

021

2

=≡===

=⎟⎟⎠

⎞⎜⎜⎝

⎛==−=

+=−+=+=

cccc

c

cc

cc

f

gyVFrgyygy

yqV

yEgqy

gyq

dydE

gyqyESSEE

gVyE l


[energy] (SP3) 9th (3B) [gravity] (S6E1) 6th (3A) [velocity] (S8P3) 8th (3A)

9th

10.3.2 Channel Depth Variations

( ) ( )( )2

02/1

22

2

00

2

021

1

2

FrSS

dxdygyVFr

dxdyFr

dxdy

gyV

dxdV

gV

dxdy

yV

dxdy

yq

dxdV

SSdxdy

dxdV

gVS

dxdy

dxdV

gV

dxdh

dxdz

dxdy

dxdV

gVzy

gV

dxd

dxdH

SdxdzS

dxdHhHH

f

fL

fL

−−

==

−==−=−=

−=+++=

++=⎟⎟⎠

⎞⎜⎜⎝

⎛++=

==+=



PS

10.4 Uniform Depth Channel Flow 10.4.1 Uniform Flow Approximations N/A

[areas of geometric shapes] (M5M1) 5th (2B) [perimeter] (M3M3) (M3M4) 3rd (2B)

[velocity] (S8P3) 8th (3A) [stress] To be taught

9th

9th + PS











65



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 10 Open Channel Flow (Continued) 10.4.2 The Chezy and Manning Equations

( )

2/10

3/22/10

3/2

2/10

3/2

0

0

22

00

0

0

0

2112

221tansin

sin

00sin0

SARn

QSARn

V

nSRV

SRCV

SRVKVK

SRP

SA

PARAW

SS

PSW

PW

FWPFFVVQF

hh

h

h

hw

hw

h

w

xwx

κκ

γρρτ

γγτ

γ

θθ

θτ

θπρ

==

=

=

→===

==

→

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

==

<<=≈

==

==+−−=−= ∑∑

l

l

lr

l

r

l

r

rl

10.4.3 Uniform Depth Examples N/A 10.5 Gradually Varied Flow N/A 10.5.1 Classification of Surface Shapes N/A 10.5.2 Examples of Gradually Varied Flows N/A


[areas of geometric shapes] (M5M1) 5th (2B) [trigonometric functions] (MA2G2) 10th (2F)

[pressure] (SC5) 9th (4B) To be taught [velocity] (S8P3) 8th (3A)

9th

10.6 Rapidly Varied Flow N/A



[force] (S4P3) 4th (3A) or (S8P3) 8th (3C) [velocity] (S8P3) 8th (3A) [gravity] (S6E1) 6th (3A)


9th

9th + PS














66



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 10 Open Channel Flow (Continued) 10.6.1 The Hydraulic Jump

( ) ( )

( )

( )

( )

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−+−=++−=

→

⎪⎪⎭

⎪⎪⎬

⎫

+±−=

=

=−⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛

−=⎟⎟⎠

⎞⎜⎜⎝

⎛−=−

++=+==

−=−→

⎪⎪⎭

⎪⎪⎬

⎫

===

===

−=−=−

2

2

12

1

1

2

1

21

1

2

21

1

2

1̀

11

21

1

2

2

1

2

212

12

11

2

111122

21

22

2

21

1222111

1211

22

21

21

22

222

11

21

111

12111221

12

181121

81121

02

22

22

2222

22

yyFr

yy

yhFr

yy

Fryy

gyVFr

Fryy

yy

yygy

yVVyyV

gyVyy

hg

Vyg

VyQVbyVby

VVgyVyy

ypbyApF

ypbyApF

VVbyVVVQFF

L

L

cc

cc

γγ

γγ

ρρ



[force] (S4P3) 4th (3A) or (S8P3) 8th (3C) [velocity] (S8P3) 8th (3A) [gravity] (S6E1) 6th (3A)


9th 9th + PS









67



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 10 Open Channel Flow (Continued) 10.6.2 Sharp-Crested Weirs

( )

( )

2/5

2/52

1

2/32/3

21

2/321

2/321

0

2/121

)2( 0 22

21

2

22

21

22

tan158

22

tan158

22tan2

075.0611.02322

32

222

232

22

22

22

HgCQ

HgQHg

VhH

PHCbHgCQHgQ

Hg

V

HP

gV

gVHbgQ

dhg

VhbgQb

dhudAuQ

gVhgu

guhPHz

gVp

wt

wwrwr

w

H

Hh

h

wAA

⎟⎠⎞

⎜⎝⎛=

⎟⎠⎞

⎜⎝⎛=→<<⎟

⎠⎞

⎜⎝⎛−=

⎟⎟⎠

⎞⎜⎜⎝

⎛+===

→⎪⎭

⎪⎬

⎫

<<

>>

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛+=

⎟⎟⎠

⎞⎜⎜⎝

⎛+=→=

==

⎟⎟⎠

⎞⎜⎜⎝

⎛+=+−+=++

∫

∫ ∫=

=

θ

θθ

γ

l

l

l


[square root] (M8N1) 8th (2A) [trigonometric functions] (MA2G2) 10th (2F)


Nore: The main formulas are not based on calculus.


9th

10.6.3 Broad-Crested Weirs ( )

( )3

22

2222

2

2/12

221

2221

Hyy

yHgyV

gyVV

gV

gVV

yHg

Vpy

gV

PH

cc

ccc

cc

ccc

cwcw

=→=−→⎪⎭

⎪⎬⎫

=

==

=−

=−++=++




9th

9th + PS











68



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 10 Open Channel Flow (Continued) 10.6.3 Broad-Crested Weirs (Continued)

( )

( ) 2/1

2/32/3

2/32/3

2/32/122

165.0

32

32

wwb

wb

ccccc

PHC

HgbCQHgbQ

ygbgybyVbyVbyQ

+=

⎟⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛=

→====




9th

10.6.4 Underflow Gates 12gyaCq d=




9th


N/A N/A 9th

9th + PS

Chapter 11 Compressible Flow 11.1 Ideal Gas Relationships

( )

( ) ( )

( ) RTuhTTchhdTchhdTchd

dThd

ThcThhRTpTuupuh

TTcuuVdTcuu

dTcuddT

udTuc

MRRTp

p

T

T pp

p

p

v

T

T v

v

v

vgas

+=−=−=−=

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

∂∂

====+=

−=−==−

==⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

∂∂

===

∨∨∨∨∨∨∨

∨∨∨∨∨∨∨∨

∨∨∨∨

∨∨∨

∫

∫

121212

121212

2

1

2

1

1

ρρ

ρ

λρ


(MA1G5) 9th (2F) [functions] (MA1A1) 9th (2E) and others

Post-secondary [integration] 12th (To be taught as a special


[Ideal Gas Law] Post-secondary to be taught [heat] (S2P2) 2nd (3A)

[temperature] (SP3) 9th (3B) [density] (S6E5) 6th (4A)


PS 9th + PS


















69



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 11 Compressible Flow (Continued) 11.1 Ideal Gas Relationships (Continued)

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

−=

−=

==−+=+=

∨

∨∨∨∨

ρ1

11pduddsT

kRc

kRkc

cc

kRccRdT

uddT

hddTRudhd

vp

v

pvp

1

2

1

212

2

1

1

212 lnlnlnln

11

111

ppR

TTcssR

TTcss

pdpR

TdTcdsdR

TdTcds

dphddsTdppdudhd

pv

pv

+=−+=−

−=⎟⎟⎠

⎞⎜⎜⎝

⎛+=

⎟⎟⎠

⎞⎜⎜⎝

⎛−=⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+=

∨∨∨

ρρ

ρρ

ρρρ








PS

11.2 Mach Number and Speed of Sound

( )( )

( )( )( )( )( ) ( )

( )

( ) ( )

cpV

cVcpzgVp

pcpccpVpAAcVccAc

ApppAAVcVccAc

cVVcVcc

VcAccV

Ma

δδρ

δρδ

δδδρδ

δρδ

δρδδ

ρδδρδρ

δδδρρδρ

δρδρδδρδρρδρρ

δδρρρ

=

=−−

+=+⎟⎟⎠

⎞⎜⎜⎝

⎛+

===−=−+−

+−=−+−+−

=−+−=

−+==

022

loss2

222

2


[exponent] (M6A3) 6th (2A) [square root] (M8N1) 8th (2A)


[speed of sound] (SPS9) 9th (3B) [pressure] (SC5) 9th (4B) To be taught

[density] (S6E5) 6th (4A) [velocity] (S8P3) 8th (3A)

PS

9th + PS

















70



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 11 Compressible Flow (Continued) 11.2 Mach Number and Speed of Sound (Continued)

( )( )

( )

ρρρ

ρρ

ρρ

ρρ

ρ

ρρ

δδρδ

v

sv

kk

k

s

k

s

Ecp

ddpE

RTkcRTkkpkpkp

pp

cpp

pc

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

==

=====⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=→⎪⎭

⎪⎬

⎫

→∂→

=

−− 11constant

constant0








PS

11.3 Categories of Compressible Flow

( )MaV

ccttr wave1sin ==−= α


[velocity] (S8P3) 8th (3A) [speed of sound] (SPS9) 9th (3B)

9th

11.4 Isentropic Flow of an Ideal Gas 11.4.1 Effect of Variations in Flow Cross-Sectional Areas

( )

AdA

ddpV

Vdp

VdV

Vdp

AdAd

VdV

VdV

AdAdVA

VdV

VdpdzVddpAVm

=⎟⎟⎠

⎞⎜⎜⎝

⎛−→

⎪⎪⎭

⎪⎪⎬

⎫

−=

+=−

→=++=++

−==++==

ρρρ

ρρ

ρρ

ρ

ργρρ

2

2

2

22

1

0constantlnlnln

021constant&


[exponent] (M6A3) 6th (2A) [partial derivatives] Post-secondary

[pressure] (SC5) 9th (4B) To be taught [velocity] (S8P3) 8th (3A) [density] (S6E5) 6th (4A)

PS

9th + PS


















71



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 11 Compressible Flow (Continued) 11.4.1 Effect of Variations in Flow Cross-Sectional Areas (Continued)

( )

( ) ( )

( ) ( )22

2

22

2

2

22

2

2

11

11

1

1

1

MaVA

dVdA

MaMa

AdAd

MaAdA

VdV

AdAMa

Vdp

VdV

Vdp

AdAMa

Vdp

AdA

ddpV

Vdp

cVMa

pcs

−−=−

=

−−=→

⎪⎪⎭

⎪⎪⎬

⎫

=−

−=

=−→

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

=⎟⎟⎠

⎞⎜⎜⎝

⎛−

=

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=

ρρ

ρ

ρ

ρ

ρρ

ρ


[exponent] (M6A3) 6th (2A) [partial derivatives] Post-secondary

[pressure] (SC5) 9th (4B) To be taught [velocity] (S8P3) 8th (3A) [density] (S6E5) 6th (4A)

PS

11.4.2 Converging-Diverging Duct Flow

( )

( )

( )

( )0

2

02

021

021

02

02

constant

2

0

1212

2

0

2

0

2

0

0

2

/10

/10

2

0

0

=⎟⎟⎠

⎞⎜⎜⎝

⎛+−→

⎪⎭

⎪⎬

⎫

−=−

=−−

=−−−

=−⎟⎟⎠

⎞⎜⎜⎝

⎛−

−

=⎟⎟⎠

⎞⎜⎜⎝

⎛+=⎟⎟

⎠

⎞⎜⎜⎝

⎛+==

∨∨

∨∨

Vhh

TTchh

VTTc

VTTkkRVpp

kk

VdpdppVddp

pp

pp

p

p

k

k

kk

ρρ

ρρ



[pressure] (SC5) 9th (4B) To be taught [density] (S6E5) 6th (4A) [velocity] (S8P3) 8th (3A)

[Ideal Gas Law] Post-secondary to be taught

9th

9th + PS












72



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 11 Compressible Flow (Continued) 11.4.2 Converging-Diverging Duct Flow (Continued)

( )[ ]

( )

( )[ ]( ) ( )[ ]

( )

( )[ ]

( )[ ]

( )

( )[ ]

( )

( )

atmkk

atmkk

kk

kk

kk

kk

kk

kk

TTTT

kTT

pppp

kpp

Mak

Makpp

TT

pp

MakTT

Makpp

TT

pp

MakTT

TT

pp

TT

pp

MakTT

833.0*833.0*1

2*

528.0*528.0*1

2*

2111

2111

2111

2111211

1

2111

4.14.100

4.14.10

1/

0

1/

20

1/

20

0

0

0

20

1/

20

1/

00

20

1/

000

0

02

0

==⎟⎟⎠

⎞⎜⎜⎝

⎛+

=

==⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛

+=

⎭⎬⎫

⎩⎨⎧

−+=

→

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

⎭⎬⎫

⎩⎨⎧

−+=

=⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛

−+=

⎭⎬⎫

⎩⎨⎧

−+=→

⎪⎪

⎭

⎪⎪

⎬

⎫

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛

−+=

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−+

=

=

=

=

=

−

−

−

−

−

−

ρρ

ρρ

ρρ





9th 9th + PS







73



Possible Grade

to Start the Topic



( )

( ) ( )

( )0

00

0

4.10

1/1/

0

0

0

0

1/

0

**1*

*****

***

634.0*

12

21

12

***

12*

12*

1

TTTT

MaAA

RTkMaV

kRTVV

VAAVAAV

kk

kpT

T

kTT

kpp

RTpMa

k

kkkk

kk

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

=

=⎟⎠⎞

⎜⎝⎛⎟⎟⎠

⎞⎜⎜⎝

⎛==

=⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎠⎞

⎜⎝⎛

+=⎟

⎠⎞

⎜⎝⎛ +

⎟⎠⎞

⎜⎝⎛

+=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛=

→

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

+=

⎟⎠⎞

⎜⎝⎛

+=

==

=

−−

−

ρρ

ρρ

ρρ

ρρ

ρρ

ρρρ

ρ





9th 9th + PS







74



Possible Grade

to Start the Topic



( )[ ]

( )[ ]

( )

( ) ( )

( )

( )[ ]( )[ ]

( ) ( )[ ]1212

0

00

0

1/1/

0

0

0

0

1/

20

20

2112111

*

**1*

12

21

12

***

12*

2111

2111

−+

−−

−

⎭⎬⎫

⎩⎨⎧

−+−+

=

→

⎪⎪⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪⎪⎪

⎬

⎫

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎠⎞

⎜⎝⎛

+=⎟

⎠⎞

⎜⎝⎛ +

⎟⎠⎞

⎜⎝⎛

+=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛=

+=

⎭⎬⎫

⎩⎨⎧

−+=

−+=

kk

kkkk

kk

kMak

MaAA

TTTT

MaAA

kk

kpT

T

kTT

Makpp

MakTT

ρρ

ρρ

ρρρ





9th

11.4.3 Constant Area Duct Flow N/A

N/A [density] (S6E5) 6th (4A) [velocity] (S8P3) 8th (3A) [pressure] (SC5) 9th (4B)

[friction] (S8P3) 8th (3A) To be taught [acceleration] (S8P3) 8th (3C)

9th

9th + PS












75



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 11 Compressible Flow (Continued) 11.5 Nonisentropic Flow of an Ideal Gas 11.5.1 Adiabatic Constant Area Duct Flow with Friction (Fanno Flow)

( )

( ) ( )

( )( )

kRTVTV

cR

Tc

dTds

Vc

dTdV

TdTdV

VR

Tc

dTds

TdT

VdVRTdTcdsT

VVV

TdTdRTdTcdsT

TdTd

pdp

RTpdTchddphddsT

ppR

TTcss

RTpTRpc

TVT

TcVT

Tc

VT

TTchh

hV

h

WQzzgVV

hhm

aappp

pp

p

p

p

p

p

p

p

innetshaft

innet

=⎟⎟⎠

⎞⎜⎜⎝

⎛+−=−=

⎟⎠⎞

⎜⎝⎛ +−−=⎟

⎠⎞

⎜⎝⎛ +−−=

→−==⎟⎟⎠

⎞⎜⎜⎝

⎛+−=

+=→⎪⎭

⎪⎬⎫

=

=−=

−=−

↑=←==+

⎪⎪

⎩

⎪⎪

⎨

⎧

==+

==+

→⎪⎭

⎪⎬

⎫

−=−

==+

+=⎥⎦

⎤⎢⎣

⎡−+

−+−

∨∨

∨∨

∨∨

∨∨

1

11

ddconstant

lnln

constant2

constant2

constant2constant

2

2

2

111

022

22

02

2

0

2

00

0

2

12

21

22

12

ρρ

ρρρ

ρρ

ρρ

ρρ

ρρ

&&&


[exponent] (M6A3) 6th (2A) [logarithmic functions] (MA2A5) 10th (2E) (To

be taught as a special skill) [square root] (M8N1) 8th (2A) [integration] 12th (To be taught)

[derivative] Post-secondary


[Ideal Gas Law] Post-secondary to be taught [temperature] (SP3) 9th (3B)

[entropy] Post-secondary To be taught [pressure] (SC5) 9th (4B) To be taught

[momentum] (SP3) 9th (3B) [pressure] (SC5) 9th (4B)

[friction] (S8P3) 8th (3A) To be taught [wave] (S8P4) 8th (3A)

PS 9th + PS














76



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 11 Compressible Flow (Continued) 11.5.1 Adiabatic Constant Area Duct Flow with Friction (Fanno Flow)

( ) ( )

( )

( )

( ) ( ) ( )

( ) ( ) ( )( )[ ]

( ) ( )

( ) ( ) ( )01

21

21

2110

21

02

022

022

24

8

22

2

2

22

2

2

2

2

2

22

2

2

2

22

2

2

2

2

222

2

222

22

22

2

1221122211

=+−+

−=

−+==

−+

=++==

=++

=++

=−−→=↑

==−−

−=−−−=−−

Ddx

Makf

MaMad

VVd

kMa

MaMad

VVd

pdp

MakMaMad

VVd

VVdMak

TdT

TcVd

TdT

TdT

MaMad

VVd

RTkMaV

VVdMa

kDdx

Mafk

pdp

VdpD

dxVpfdp

dVVDdxV

fdpD

A

VfdVV

AdD

dp

VVVA

RppVVmRApAp

p

wxw

xx

ρρ

ρρπ

ρτ

ρπτ

ρ&










PS 9th + PS














77



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 11 Compressible Flow (Continued) 11.5.1 Adiabatic Constant Area Duct Flow with Friction (Fanno Flow) (Continued)

( ) ( )( )[ ]{ }

( ) ( )( )[ ]{ }

( ) ( )[ ]( )[ ]

( )

( ) ( ) ( )

( )( )[ ]{ } ( )

( )( )[ ]( )[ ]( )[ ]( )[ ]

( )[ ]( )( )[ ]

( ) ( )1212

0

0

0

0

0

0

2/1

2

2/1

2

2

2/1

2

2

2

22

2112

2

2

2

2

1* *

42

22

42

22

211

121

*

**

**211211

*

***21211

*

**211

21*

***21121

*

21121

**

*211

21ln

2111

2111211

1

−+

=

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛

+=

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎭⎬⎫

⎩⎨⎧

−++

=

=⎭⎬⎫

⎩⎨⎧

+−+

=

=⎭⎬⎫

⎩⎨⎧

−++

=

==−++

=

−+−

−=

−=−

−−

−=

⎭⎬⎫

⎩⎨⎧

−+++

+−

=−+−

=−+−

∫ ∫

kk

Ma

Ma

MakkMap

p

pp

pp

pp

pp

Makk

Mapp

TT

pp

MakMak

VV

MakMak

VV

TTMa

kRTRTkMa

VV

Makk

TT

MadMak

kTdT

Df

Df

Df

Df

MakMak

kk

MaMa

k

Ddxf

kMaMakMadMa

Ddx

fkMaMak

MadMa

ρρ

ρρ

ρρ

llllll

ll

l

l










PS 9th + PS














78



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 11 Compressible Flow (Continued) 11.5.2 Frictionless Constant Area Duct Flow with Heat Transfer (Rayleigh Flow)

( )

( )

( ) ( )[ ]

( ) ( ) ( ) ( )[ ]

( )

( )

a

aa

aaa

aa

a

aa

aaa

p

p

b

p

aaap

pp

x

VV

pp

TT

kMak

pp

Vpp

Vpp

VpVp

MaTcq

VdV

kRTkV

dVdT

TV

Tcq

VdVqdVVhd

kMa

dsdT

RVVTTVTcdTdsdsdT

MakRTVRVVTT

VTc

dTds

dTdV

TV

Tc

dTdsdVVdTcdsT

dVVhddsTdVVdpdVVdp

RTVp

VpRVmApVmAp

==++

=

+=+

+=+

−=

⎥⎦

⎤⎢⎣

⎡ −+==+

=→=

−+==

==−

+=

+=+=

+=−=−=

==+

=+++=+

−∨

−

∨

ρρ

ρρ

ρρ

ρρδ

δδ

ρρ

ρρ

ρ

ρρ

2

22

22

2

12

1

2

2

222111

11

111

1

10

11

11

constantVconstant

constant&&


[derivative] 12th + [square root] (M8N1) 8th (2A)



PS 9th + PS








79



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 11 Compressible Flow (Continued) 11.5.2 Frictionless Constant Area Duct Flow with Heat Transfer (Rayleigh Flow) (Continued)

( )

( )

( )

( )

( )( )

( )12

2,0

0

,0

0

,0

0

22

22

,0

,0

0

,02

2

2

2

211

12

11

12

1112

11

11

−

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛

+++

=

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

+

⎟⎠⎞

⎜⎝⎛ −++

=

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛=⎥⎦

⎤⎢⎣⎡

++

==

⎥⎦⎤

⎢⎣⎡

++

=⎟⎟⎠

⎞⎜⎜⎝

⎛==

kk

a

a

a

aa

a

a

a

aaa

a

aaaa

a

MakkkMa

kpp

pp

pp

pp

pp

kMa

MakMak

TT

TT

TT

TT

TT

kMaMakMa

VV

kMaMak

TTMa

pp

TT

TTMa

ρρ

ρρ


[derivative] 12th + [square root] (M8N1) 8th (2A)



PS

11.5.3 Normal Shock Waves ( )

( )

( )( )

22

022

22

000

2

22

M11

M11

constant2

constant2

constantconstantconstant

xa

x

ya

y

x

a

a

y

x

y

p

p

akk

pp

akk

pp

pp

pp

pp

TRpc

TVT

RTpTTchhhVh

pRTVpVpV

++

=++

=⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

==+

=−=−==+

=+=+=

∨∨∨∨

ρ

ρ

ρρρ


[Ideal Gas Law] Post-secondary to be taught [temperature] (S3P1) 3rd (3A)

[density] (S6E5) 6th (4A) [pressure] (SC5) 9th (4B) To be taught

[speed] (S2P3) 2nd (3A) [velocity] (S8P3) 8th (3A) [graph] (S7CS6) 7th (6)

9th

9th + PS















80



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 11 Compressible Flow (Continued) 11.5.3 Normal Shock Waves (Continued)

( )( )[ ]

( )( )[ ]( )( )[ ]

( )[ ]( )[ ]

( )[ ]( )[ ]

( )[ ]( )[ ]

( )[ ]{ } ( ){ }( ) ( ){ } 22

22

22

22

2/1

2

2

2/1

2

2

2

2

2222

22

2/1

2

2

2

M1211M12M211

11M

12

1M1212M

M

MM

M211M211

MM

M211M211

M21121

*

M21121

***

M

M21121

M1

*

**M1

M1

x

xx

x

y

xx

y

x

xy

y

x

y

x

x

y

y

x

x

y

x

y

y

x

x

y

x

y

yyxxx

y

x

y

x

y

y

x

x

y

x

x

y

y

x

y

x

y

Yyyxxx

x

y

x

y

y

x

x

y

akkakkak

TT

kka

kk

pp

akkka

a

aa

akak

pp

aa

TT

pp

VV

TT

pp

VVTT

pp

akak

TT

akk

TT

akk

TT

TT

TT

TT

akRTkkV

RTV

pVVpVp

aKk

app

pp

pp

pp

akak

pp

−+

−−−+=

+−

−+

=−−

−+=

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−+−+

=

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

=⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

−+−+

=

→

⎪⎪⎭

⎪⎪⎬

⎫

−++

=

−++

=

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

===+=+

⎭⎬⎫

⎩⎨⎧

−++

=

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

++

=

ρρρρ

ρρρ





9th 9th + PS









81



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 11 Compressible Flow (Continued) 11.5.3 Normal Shock Waves (Continued)

( )( )

( ) ( )

( )112

12

12

,0

,0

,0

,0

,0

,0

2

2

11M

12

M2

11M2

1

2M1M1

−

−−

⎟⎠⎞

⎜⎝⎛

+−

−+

⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ +

=

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

+−+

==⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛==

k

x

kk

x

kk

x

x

y

x

x

x

y

y

y

x

y

x

x

y

x

x

y

y

x

x

y

x

y

y

x

x

y

kka

kk

akak

pp

pp

pp

pp

pp

akak

VV

TT

pp

VV

ρρ

ρρ

ρρ





9th

11.6 Analogy between Compressible and Open-Channel Flows

( ) 1constantcconstant

constant

−==

=====

koc

oc

ococ

oc

kybV

AVcVFrgyc

gyVFr

cVMa

ρ

ρ



[density] (S6E5) 6th (4A) [velocity] (S8P3) 8th (3A) [gravity] (S6E1) 6th (3A) [mass] (S8P3) 8th (3A)

9th

11.7 Two-Dimensional Compressible Flow 21 tt VV =

[four operations] (M1N3) 1st (2A) [triangle] (M5M1) 5th (2B)


9th


N/A N/A 9th

9th + PS

Chapter 12 Turbomachines 12.1 Introduction N/A

N/A [force] (S4P3) 4th (3A) or (S8P3) 8th (3C) [work] (S8P3) 8th (3A)

[energy] (SP3) 9th (3B) [power] (SP3) 9th (3B)

9th 9th





















82



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 12 Turbomachines (Continued) 12.2 Basic Energy Considerations

rUUWV ω=+=rrr


[radius] (M3G1) 3rd (2B) [velocity] (S8P3) 8th (3A)

9th

12.3 Basic Angular Momentum Considerations ( ) ( ) ( ) ( )

( ) ( )

( )

( )2

2

ˆ

21

22

21

22

21

22

222222

2222111

212111

212111

WWUUVVw

WUVUVWUVV

VVVVUVUwm

Ww

VUmVUmWTWQm

VrmVrmTdAnVVrFr

shaft

x

xshaftshaft

shaft

shaftshaftshaft

shaftcs

−−−+−=

−+==−+

+=+−==

−+−===

−+−=⋅×=×∑ ∫

θθ

θθθ

θθ

θθ

ωρ

ρ

&

&

&&&&

&&rrrrr



[special math: cross product] To be taught as a special math topic

[analytic geometry] 12th (To be taught) [areas of geometric shapes] (M5M1) 5th (2B)

[density] (S6E5) 6th (4A) [torque] Post-secondary To be taught

[momentum] (SP3) 9th (3B)

9th

12.4 The Centrifugal Pump N/A

N/A N/A 9th

12.4.1 Theoretical Considerations

( ) ( )( )

( )

( )1122

11221122

1122

11221122

2122

11

222

111

1θθ

θθθθ

θθ

θθθθ

ρ

ρρ

ωρω

ρ

ωω

VUVUg

hgQhW

VUVUQ

WwVUVUQW

VrVrQWTW

VrVrQTVrVrmT

mmmrUrU

UWV

UWV

iishaft

shaftshaftshaft

shaftshaftshaft

shaftshaft

−==

−==−=

−==

−=−=

====

+=

+=

&

&&

&&

&

&&&rrr

rrr


[trigonometric functions] (MA2G2) 10th (2F) [areas of geometric shapes] (M5M1) 5th (2B)


9th

9th














83



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 12 Turbomachines (Continued) 12.4.1 Theoretical Considerations (Continued)

( ) ( ) ( )[ ]

Qgbr

Ug

Uh

VbrQg

VUg

Uh

VVU

gVU

hWWUUVVg

h

i

rr

ir

ii

22

2222

222222

22

2

222

2221

22

21

22

21

22

2cot

2cot

cot

21

πβ

πβ

β θ

θ

−=

=−=−

=

=−+−+−=


[trigonometric functions] (MA2G2) 10th (2F) [areas of geometric shapes] (M5M1) 5th (2B)


9th

12.4.2 Pump Performance Characteristics

vmha

shaft

f

af

afaLspa

a

bhpQh

W

Qh

Qhpphhhhh

gVVzzpph

ηηηηγη

η

γ

γγ

γ

==

℘==

==℘

=℘−

≈−==

−+−+

−=

550pump thedrivingpower shaft fluid by thev gainedpower 550

horsepowerwater

2

12

21

22

1212

&


[unit conversion] (M6M1) 6th (2C)

[pressure] (SC5) 9th (4B) To be taught [velocity] (S8P3) 8th (3A) [gravity] (S6E1) 6th (3A)

9th

12.4.3 Net Positive Suction Head (NPSH)

γγ

γγ

γγγγ

vL

atm

Latmss

Lssatmvss

phzpNPSH

hzpg

Vp

hg

Vpzppg

VpNPSH

−−−=

→−−=+

→=+=−−+=

∑

∑

∑

1

1

2

2

1

2

2

22

[four operations] (M1N3) 1st (2A) [sigma notation] (M6N1) 6th (1A) or (MA1A3)

9th (2E)

[pressure] (SC5) 9th (4B) To be taught [velocity] (S8P3) 8th (3A) [gravity] (S6E1) 6th (3A) [density] (S6E5) 6th (4A)

9th

9th



















84



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 12 Turbomachines (Continued) 12.4 4 System Characteristics and Pump Selection

21212 KQzzhhzzh pLp +−=+−= ∑

[four operations] (M1N3) 1st (2A) [sigma notation] (M6N1) 6th (1A) or (MA1A3)

9th (2E)


9th

12.5 Dimensionless Parameters and Similarity Laws ( )

2

253

153

222

122

23

13

3332533122

2

33

2

3253

2

3122

2

3

,,,

,,,

,,,

,,, termpidependent

,,Q,,,D,fvariabledependent

ηηρωρω

ωωωω

ωφη

ωφ

ρωωφ

ω

μρω

ωεφρη

μρω

ωεφ

ρω

μρω

ωεφ

ω

μρω

ωεφ

ρμωε

=⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟

⎟⎠

⎞⎜⎜⎝

⎛

⎟⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛=

⎟⎟⎠

⎞⎜⎜⎝

⎛==

⎟⎟⎠

⎞⎜⎜⎝

⎛==

⎟⎟⎠

⎞⎜⎜⎝

⎛==

⎟⎟⎠

⎞⎜⎜⎝

⎛=

=

℘

DW

DW

Dgh

Dgh

DQ

DQ

DQ

DQ

DW

DQ

Dgh

DDQ

DDWgQh

DDQ

DDDW

C

DDQ

DDDghC

DDQ

DD

shaftshaft

aa

shafta

i

shaft

a

ishaft

iaH

i

i

&&

&

l&

l&

l

l

l

[four operations] (M1N3) 1st (2A) [ratio] (M6A1) 6th (2A)

[gravity] (S6E1) 6th (3A) [density] (S6E5) 6th (4A)

[energy] (SP3) 9th (3B) [velocity] (S8P3) 8th (3A)

9th

12.5.1 Special Pump Scaling Laws

5/1

2

1

1

252

51

2

1

22

21

2

132

31

2

132

31

2

122

21

2

1

2

1

2

1

11

⎟⎟⎠

⎞⎜⎜⎝

⎛≈

−−

=

=====

DD

DD

WW

DD

hh

DD

QQ

WW

hh

QQ

shaft

shaft

a

a

shaft

shaft

a

a

ηη

ωω

ωω

ωω

&

&

&

&


(M5M1) 5th (2B) (M5M1) 5th (2B)

[exponent] (M6A3) 6th (2A) [ratio] (M6A1) 6th (2A)

[velocity] (S8P3) 8th (3A) [power] (SP3) 9th (3B) [energy] (SP3) 9th (3B)

9th

9th


















85



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 12 Turbomachines (Continued) 12.5.2 Specific Speed ( )

( ) ( )( ) ( )

( )[ ] 4/34/34/322

2/13

fthgpmQrpm

NNgh

Q

DghDQ

asds

aa

ωω

ωω

=== [four operations] (M1N3) 1st (2A)

[ratio] (M6A1) 6th (2A)

[speed] (S2P3) 2nd (3A)

9th

12.5.3 Suction Specific Speed

( )[ ]( ) ( )

( )[ ] 4/34/3 ftNPSHgpmQrpm

SNPSHg

QS

Rsd

Rs

ωω==

[four operations] (M1N3) 1st (2A) [ratio] (M6A1) 6th (2A)

[speed] (S2P3) 2nd (3A)

9th

12.6 Axial-Flow and Mixed-Flow Pump N/A

[graph] (S7CS6) 7th (6) [speed] (S2P3) 2nd (3A)

9th

12.7 Fans

222

122 ⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛D

pD

p aa

ρωρω


(M5M1) 5th (2B) (M5M1) 5th (2B)

[ratio] (M6A1) 6th (2A)

[speed] (S2P3) 2nd (3A) [pressure] (SC5) 9th (4B) To be taught


9th

12.8 Turbines 12.8.1 Impulse Turbines

( )( ) ( )( )

( )( )2

cos1

cos1cos1cos

2

max1

1112

22111

VUVUUmTW

VUrmTVUVVUWVUWVV

powershaftshaft

mshaft

=−−==

−−=−−=−+=+==

βω

βββ

θθ

θθ

&&

&



[power] (SP3) 9th (3B) [speed] (S2P3) 2nd (3A)

[force] (S4P3) 4th (3A) or (S8P3) 8th (3C)

9th

9th



















86



Possible Grade

to Start the Topic


Math Physics/Chemistry Sec Ch Chapter 12 Turbomachines (Continued) 12.8.2 Reaction Turbines

( ) ( ) ( )

( )( ) ( )

( )[ ] 4/54/5

321

53223

''fth

bhpWrpmN

gh

WN

CCC

CCCCC

gQhW

DW

CD

ghCDQC

T

shaftsd

T

shafts

QHQQQH

T

shaftshaftTHQ

&&

&&

ωρω

ηφηφφ

ρη

ρωωω

==

====

====

℘℘

℘

[four operations] (M1N3) 1st (2A) [square root] (M8N1) 8th (2A) [exponent] (M6A3) 6th (2A)

[power] (SP3) 9th (3B) [speed] (S2P3) 2nd (3A)

[force] (S4P3) 4th (3A) or (S8P3) 8th (3C) [density] (S6E5) 6th (4A) [gravity] (S6E1) 6th (3A)

9th

12.9 Compressible Flow Turbomachines 12.9.1 Compressors

stdstd

stdtest

stdteststd

std

stdtest

TTNN

kRTND

pp

TTmm

pDkRTmR

pDkRTmR

0101

001

001

012

01

012

01

`

=

=⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟

⎟⎠

⎞⎜⎜⎝

⎛ &&

&&


[graph] (S7CS6) 7th (6)

[mass] (S8P3) 8th (3A) [pressure] (SC5) 9th (4B)

[friction] (S8P3) 8th (3A) To be taught [velocity] (S8P3) 8th (3A)

[temperature] (S3P1) 3rd (3A)

9th

12.9.2 Compressible Flow Turbines N/A

N/A [mass] (S8P3) 8th (3A) [pressure] (SC5) 9th (4B) To be taught [friction] (S8P3) 8th (3A) To be taught

[velocity] (S8P3) 8th (3A) [temperature] (S3P1) 3rd (3A)

9th


N/A N/A 9th

9th

THE END




















87

Part Two 1st Round of Delphi –

Five-Point Likert Scale Survey Forms

Proposed Procedures for Survey Response

To facilitate survey response to the initial selection of fluid mechanics topics that could be possibly taught to students at 9th or above Grade, as listed in the Fluid Mechanics Survey Form A and Survey Form B, the following procedures are hereby proposed:

1. Rate the importance of each Section as a topic in a potentially viable 9th or above Grade fluid mechanics subject, and write a number representing its “importance” value (Figure 4A), using the five-point Likert Scale (Figure 4B);

2. Check the formulas listed under the Engineering Analytic Topics & Typical Formulas column, and use symbols shown in Figure 4B to indicate your expert opinion and advice about each formula;

3. Add your general comments and advice in the empty space.


88

Figure 4A. Step-by-step procedures proposed for the review and validation of data.


89

Figure 4B. Likert Scale (top) and symbols to be used for the expression of expert opinion and offer of advice.


90

Notes for Chapter 6 and Chapter 7

Chapter 6 (Differential Analysis of Fluid Mechanics Flow) appears to be, for all practical purposes, too deep in calculus-based mathematics for even 12th Grade students in Advanced Placement Calculus course to master.

Chapter 7 (Similitude, Dimensional Analysis, and Modeling) involve a lot of “abstract thinking” and appears to be most likely beyond the cognitive developmental maturity level of high school students.

Therefore, engineering analytic principles and skills from these two Chapters are NOT analyzed for the eventual inclusion into a potentially viable K-12 engineering curriculum. However, some generic knowledge content covered in these two Chapters could still be lightly explored by 9th or above Grade students; thus, their relative importance could still be rated at generic knowledge level.

Notes about the Fluid Mechanics Analytic Principles and Formulas

The leftmost column in the Fluid Mechanics Survey Form A and Survey Form B contain

1. The titles of each section under a particular chapter in the selected textbook, which in general represent particular sets of fluid mechanics related engineering analytic and predictive principles, in a qualitative and explanatory way;

2. Computational formulas, which symbolically represent the above engineering analytic and predictive principles, in a quantitative and mathematical way.

As shown in Figure 4B, the formulas extracted from the selected textbook might by categorized into five groups, corresponding to the five different symbols shown in Figure 4B, which could be used by the above-mentioned five Groups of Participants:


91

1. Formulas that engineering professors actually teach in classroom lectures and that practicing engineers use in engineering design projects: These are the important ones to be included in a potentially viable K-12 engineering curriculum that shall be based on cohesive and systemic mastery of engineering analytic and predictive principles and skills. For any of these formulas, a box could be used together with a number representing its order of importance according to the five-point Likert Scale (1 = Totally Unimportant, 2 = Not So Important, 3 = Might Be Important, 4 = Important, or 5 = Very Important).

2. Formulas that are rarely used in either classroom lectures or in field practice, but are used by the original discoverer of a particular set of analytic principles to derive other formulas that are actually used in classroom lecture or in field practice: Some of these “intermediate” formulas might not be used often, in other words, they are “rarely taught or used.” For any of these formulas, a strikethrough could be used. If a big enough percentage of participants (maybe 85% or above) place a strikethrough on a particular formula at the end of each round of the proposed four-round Delphi study, then the formula will be removed from the survey form for the next round. If the trend continues through all four rounds of the proposed Delphi survey, then that formula might be removed from the final list of high school appropriate fluid mechanics topics. Interestingly enough, in some cases, rarely used calculus-based “intermediate” formulas are used to derive a final one that is based on pre-calculus mathematics skills and is actually used in most homework assignments and design projects; in this case, if the “intermediate” formulas are removed from consideration, then the entire topic of fluid mechanics could be re-classified as appropriate for 9th Grade. For example, the main formula amF rv

= and

streamlinea alongconstant 21 2 =++ zVp γρ (Bernoulli Equation) do not need calculus, and thus, could be taught to 9th

Grade students. This type of formulas will make the list shorter and shorter as the proposed Delphi study moves to the next round of survey. Some of these formulas might not be in the selected textbook; I derived them for fun, sometimes with the help of my former engineering professor, Dr. Samuel Landsberger, at California State University Los Angeles.

3. Formulas that are particular to certain conditions and in real classroom lectures or field practice are, for all practical purposes, close to be “never used:” For any of these formulas, a double-strikethrough could be used. If a big enough percentage of participants (maybe 75% or above) place a double-strikethrough on a particular formula at the end of each round of the proposed four-round Delphi study, then the formula will be removed from the survey form for the next round. If the trend continues through all four rounds of the proposed Delphi survey, then that formula might be removed from the final list of high school appropriate fluid mechanics topics. This type of formulas will also make the list shorter and shorter as the proposed Delphi study moves to the next round of survey.


92

4. Formulas that even experienced university engineering professors or practicing engineers might “not understand:” This is amazing but totally correct and yes, absolutely normal! There are formulas that even experienced professors might say “I do not understand this” or “I need to read the context in the book to figure this out.” For any of these formulas, the participants should generally not seek to understand them (doing so does not serve the purpose of studying the relative importance of each computational formula); but instead, a question mark (?) could be used. If a big enough percentage of participants (maybe 65% or above) place a question mark (?) on a particular formula at the end of each round of the proposed four-round Delphi study, then the formula will be removed from the survey form for the next round. If the trend continues through all four rounds of the proposed Delphi survey, then that formula might be removed from the final list of high school appropriate fluid mechanics topics. Indeed, it makes little sense to include this type of formulas to a potentially viable K-12 engineering curriculum. This type of formulas will also make the list shorter and shorter as the proposed Delphi study moves to the next round of survey. Some of these formulas might not be in the selected textbook; I derived them for fun, sometimes with the help of my former engineering professor, Dr. Samuel Landsberger, at California State University Los Angeles.

6. Formulas that are wrong for any reasons (my typing errors, or the authors’ errors, etc.): For any of these formulas, a cross (X) could be used and the correct formulas should be given if possible. The correction would be included in the survey forms for the subsequent rounds of the four-round five-point Likert Scale Delphi study.

For convenience of statistic analysis of expert opinions and advice, it is requested that all participants print each letter of their

comment legibly and separately, using fonts commonly used in engineering notebooks.


93

Fluid Mechanics Survey Form A 1st Round of Delphi - Likert Scale Questionnaire on the Importance of Various Fluid Mechanics Topics Selected for High School Engineering Curriculum (For the Pre-calculus Portion)

Engineering Subject: Fluid

Likert Scale (Score of Importance) Note: 1 Totally Unimportant; 2 Not So Important; 3 Might Be Important; 4 Important; 5 Very Important

Likert Scale (Score of Importance from

Least to Most)


1 2 3 4 5

Comment

Chapter 1 - Introduction 1.1 Some Characteristics of Fluid 1.2 Dimensions, Dimensional Homogeneity, and Units

δβττ ∝==→≡s

sns

n

APpAF

AF

pr

r

1.3 Analysis of Fluid Mechanics Mechanics Behavior N/A

1.4 Measures of Fluid Mechanics Mechanics Mass and Weight 1.4.1 Density

ρρ 1

===mVv

Vm

1.4 2 Specific Weight

gVmg

VW ργ ==≡

1.4.3 Specific Gravity

CSG

OHo4@

2ρ

ρ=


94

Fluid Mechanics Survey Form A (Continued)




Least to Most)


1 2 3 4 5

Comment

Chapter 1 – Introduction (Continued) 1.5 Ideal Gas Law

RTp ρ=

1.7 Compressibility of Fluids 1.7.1 Bulk Modulus

ρρddpE

ddpE vv =∀∀

=

1.7.2 Compression and Expansion of Gases

kpEpEppvvk ==== ConstantConstant

ρρ

1.7.3 Speed of Sound

kRTcRTp

kpc

ddpE

ddp

ddpE

Eddpc

v

vv =→

⎪⎭

⎪⎬

⎫

=

=

⎪⎪⎩

⎪⎪⎨

⎧

=

==

←==

ρρ

ρρ

ρρρρ

ρρ

1.8 Vapor Pressure


95





Least to Most)


1 2 3 4 5

Comment

Chapter 1 – Introduction (Continued) 1.9 Surface Tension

RhRhR

RpppRpR ei

γθσθσπγπ

σπσπ

cos2cos2

22

2

2

=→=

=−=ΔΔ=

1.10 A Brief Look Back in History 1.11 Chapter Summary and Study Guide

Chapter 2 Fluid Statics 2.3.1 Incompressible Fluid

( )( )

0021

21

21

1221

12122

1

2

1

pghphppph

phphpp

zzppzzpp

dzdpz

z

p

p

+=+=⎪⎩

⎪⎨

⎧−

=

+=→=−→

⎩⎨⎧

−=−−−=−

→−= ∫∫

ργγ

γγ

γγ

γ

Note: The main formulas 00 pghphp +=+= ργ does not need calculus.


96





Least to Most)


1 2 3 4 5

Comment

Chapter 2 Fluid Statics (Continued) 2.3.2 Compressible Fluid

( )( ) ( )

( )

( )⎥⎦

⎤⎢⎣

⎡ −−=

−==

→−=−=

→−=−=

→−=

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

−=−=

=

=

∫∫

∫ ∫ ∫

0

1212

1

2

exp

ln 2

1

2

1

2

1

2

1

2

1

RTzzg

pp

Tdz

Rg

pp

pdp

Tdz

Rg

dzRTg

pdp

dzRTg

pdp

dzpRT

gppdz

dpdz

RTgp

dzdp

gdzdp

RTpRTp

z

z

p

p

p

p

z

z

z

z

ργ

ρ

ρ

Note: The main formula ( )

⎥⎦

⎤⎢⎣

⎡ −−=

0

1212 exp

RTzzgpp does not need calculus

2.4 Standard Atmosphere β

ββRg

aaa T

zppzTT ⎟⎟⎠

⎞⎜⎜⎝

⎛−=−= 1

2.5 Measurement of Pressure vaporatmatmgageabs phpppp +=+= γ


97





Least to Most)


1 2 3 4 5

Comment

Chapter 2 Fluid Statics (Continued) 2.6 Monometry

2.6.1 Piezometer Tube 110 hpphp A γγ =+=

2.6.2 U-Tube Manometer

113322

332211

221122

22110 0

hhhppphhhphphhphhpphp

BA

BA

AA

A

γγγγγγ

γγγγγγ

+−+=−→=−−+

=−=→=−++=

2.6.3 Inclined-Tube Manometer

θγθγ

γγθγγθγγ

sinsin

sinsin

2222

113322

332211

BABA

BA

BA

pppp

hhppphhp

−=→=−

−+=−=−−+

ll

l

l

2.7 Mechanical and Electronic Pressure Measuring Devices 2.9 Pressure Prism

( )( )

221121

221

2yFyFyFFFF

AhbhkvolumeFAhApF

ARR

RavR

+=+=

⎟⎠⎞

⎜⎝⎛===⎟

⎠⎞

⎜⎝⎛== γγγ


98





Least to Most)


1 2 3 4 5

Comment

Chapter 2 Fluid Statics (Continued) 2.10 Hydrostatic Force on a Curves Surface

( ) ( )22

12

vHR

vH

FFF

WFFFF

+=

+==r

2.11 Buoyancy, Flotation, and Stability N/A 2.11.1 Archimedes’ Principle

( )( ) ( )[ ]

( ) 21

21112

1212

121212

yVVyVyVyWyFyFyFVF

VAhhAhhFAhhFFWFFF

TTc

cBB

B

B

−−=−−==

−−−−=−=−−−=

r

r

γ

γγγ

2.11.2 Stability N/A

Chapter 3 Elementary Fluid Dynamics – The Bernoulli Equation 3.1 Newton’s Second Law

( )VVVVa

sVVa

amFFamF

ns

gP

r

rrrrv

=←ℜ

=∂∂

=

=+= ∑2

Note: The main formula amF rv= does not need calculus


99





Least to Most)


1 2 3 4 5

Comment

Chapter 3 Elementary Fluid Dynamics – The Bernoulli Equation (Continued) 3.2 F = ma along a Streamline

( ) ( )

( ) ( )

Equation)(Bernoulli streamlinea alongconstant 21

)streamlinea (along21

021

21

sinsin

22

sinsin

2

2

22

=++

=++

=++→=−−

=∂∂

=∂∂

−−⎟⎠⎞

⎜⎝⎛

∂∂

−−=+=

∂∂

−=∂∂

−=−=+−−=∂∂

≈

−=−=→⎭⎬⎫

==

∂∂

=∂∂

==

∫

∑

∑

zVp

CgzVdp

dzVddpdsVd

dsdp

dsdz

asVV

spV

spFWF

Vspyns

spynpynppynppFys

spp

VXWWg

VWsVVV

sVmVmaF

spsss

ssspss

sss

γρ

ρ

γρργ

ρρθγδθγδδδ

δδδδδδδδδδδδδδδδ

θγδθδδργγδδ

ρδδδδ

r

rrr

Note: The main formulas amF rv= and

Equation)(Bernoulli

streamlinea alongconstant 21 2 =++ zVp γρ



100





Least to Most)


1 2 3 4 5

Comment

Chapter 3 Elementary Fluid Dynamics – The Bernoulli Equation (Continued) 3.4 Physical Interpretation


streamline theacrossconstant V


22

2

=++=+ℜ

+

=++

∫ zg

Vpzdnp

zVp

γγρ

γρ



p


2

2

=++

=++

zg

V

zVp

γ

γρ do not need calculus

3.5 Static, Stagnation, Dynamic, and Total Pressure

( )ρ

ρρ

γρ

43243

14

2

3

22112

221

21p


21

ppVVppppp

Vp

pzVpVpp T

−==−→

⎭⎬⎫

==+=

==+++=

3.6 Examples of Use of the Bernoulli Equation

22

2212

11 21

21 zVpzVp γργρ ++=++


101





Least to Most)


1 2 3 4 5

Comment

Chapter 3 Elementary Fluid Dynamics – The Bernoulli Equation (Continued) 3.6.1 Free Jets

( )⎪⎩

⎪⎨

⎧

+=

==→=

HhgV

ghhVVh

2

2221 2 ρ

γργ

3.6.2 Confined Flows 212211222111 QQVAVAVAVA =→=→= ρρ

3.6.3 Flowrate Measurement

( )

( )1221

2/311

2

1

2

21221

2222111122

2212

112

1

2

212

22112

222

11

2

22

1

20

21

21

1

221

21

gzbzQzz

HgbCgHHbCQ

zz

zzgbzQpp

zbVVAzbVVAQzVpzVp

AA

ppAQ

VAVAQVpVp

=→>>

==

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−=→==

====++=++

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−

−=

==+=+

γργρ

ρ

ρρ

3.7 The Energy Line and the Hydraulic Grade Line

Hzg

V==++ streamlinea onconstant

2

2

γρ


102





Least to Most)


1 2 3 4 5

Comment

Chapter 3 Elementary Fluid Dynamics – The Bernoulli Equation (Continued) 3.8 Restrictions on Use of the Bernoulli Equation 3.8.1 Compressibility Effects

( ) ( )[ ]

le)compressib(...24

2411

2

ible)incompress(2

2

le)compressib(12

11

2121

11

constant 21

2ln

2constant

21

41

21

21

1

12

21

1

12

1

11

1

21

1

2

1/21

1

12

2

22

2

21

21

1

1

1

1

1

2/11

/12

/1/1/1

2/1/1

2

22

2

11

212

2

1

⎟⎠⎞

⎜⎝⎛ +

−++=

−

=−

→

⎪⎪⎭

⎪⎪⎬

⎫

=

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎟

⎠⎞

⎜⎝⎛ −+=

−

++⎟⎠⎞

⎜⎝⎛

−=++⎟

⎠⎞

⎜⎝⎛

−

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎠⎞

⎜⎝⎛

−=−⎟

⎠⎞

⎜⎝⎛

−=

=++

+=⎟⎟⎠

⎞⎜⎜⎝

⎛++==++

−

−−−

−

∫

∫

∫

MakMakMa

ppp

kMap

pp

kRTV

Ma

RTV

pp

Makp

pp

gzV

pp

kkgz

Vpp

kk

pp

pp

kkpp

kkCdppC

gzVdppC

zg

Vpp

gRTz

gV

RTpgzV

pdpRT

kk

kkkkp

p

kkk

kk

ρ


103





Least to Most)


1 2 3 4 5

Comment

Chapter 3 Elementary Fluid Dynamics – The Bernoulli Equation (Continued) 3.8.1 Compressibility Effects (Continued)

le)compressib(...24

241

12

ible)incompress(2

2

41

21

21

1

12

21

1

12

1

11

1

21

1

2

⎟⎠⎞

⎜⎝⎛ +

−++=

−

=−

→

⎪⎪

⎭

⎪⎪

⎬

⎫

=

=

Mak

MakMa

ppp

kMap

pp

kRTV

Ma

RTV

pp

Note: The main formulas RTpgzV

pdpRT ==++∫ ρconstant

21 2 and others do not need

calculus

3.8.3 Rotational Effects

γγγγρ

γρ

γργρ

454

2

1234

43

13

43

043

02

012

021

21

021

1222

2212

11

flowhroughout constant t21

CC21C0

Cconstant21

21

pHHHppzVp

pphpp

amF

hzzVVV

pVppp

zzVVV

zVpzVp

==+==++

→=→

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

=−=

=

====

+=→⎪⎭

⎪⎬

⎫

======

==++=++

rr

3.8.4 Other Restrictions 3.9 Chapter Summary and Study Guide


104





Least to Most)


1 2 3 4 5

Comment

Chapter 4 Fluid Kinematics 4.3 Control Volume and System Representations

( )dtmvdF =

4.4 The Reynolds Transport Theorem

0:particles fluid malInfinitesi PropertyIntensive :b Property Extensive:B

22

122

→⎪⎪⎩

⎪⎪⎨

⎧

=→=

=→=

=→=

=V

VbVmB

VbmVB

bmB

mbBδ

rrrr

4.4.7 Selection of a Control Volume N/A

4.5 Chapter Summary and study Guide N/A


105





Least to Most)


1 2 3 4 5

Comment

Chapter 5 Finite Control Volume Analysis 5.1.2 Fixed, Non-deforming Control Volume

∑ ∑

∫

∑ ∑∑ ∑∫

=====

==

∂∂

=−=−∂∂

outin

cv

inoutinoutcv

mmVAVAQVAVAm

VAmAVm

Vdt

QQmmVdt

&&&

&&

&&

2211222111

surface control thein opening over the

ddistributeuniformly

flow) ldimensiona-(onesurface control

thein opening over theddistributeuniformly

00

ρρ

ρρ

ρ

ρ

Note: The main formulas ∑ ∑===== outin mmVAVAQVAVAm &&& 2211222111 ρρ

re not based on calculus


106





Least to Most)


1 2 3 4 5

Comment

Chapter 5 Finite Control Volume Analysis (Continued) 5.3.3 Comparison of the Energy Equation with the Bernoulli Equation

( )

lossfriction) flow with ibleincompress(Steady 0

flow) ibleincompresssteady ess(Frictionl0

2222

22

22

2

22

22

22

22

22

=−−>−−

=−−

++=++→⎟⎟⎠

⎞⎜⎜⎝

⎛++

=⎟⎟⎠

⎞⎜⎜⎝

⎛++

=→=++=++

=⎟⎠

⎞⎜⎝

⎛−−−++=++

=⎥⎦

⎤⎢⎣

⎡−+

−+−+−

∨∨∨∨

∨∨

∨∨

∨∨

innetinout

innetinout

innetinout

ininin

outoutout

inin

inoutout

out

inin

inoutout

out

innet

innet

innetinoutin

ininout

outout

innetinout

inoutinoutinout

quuquu

quu

gzVp

gzVp

zV

pzV

p

ggzV

pzV

p

m

Qqquugz

Vpgz

Vp

QzzgVVpp

uum

ρρρ

γρ

ρ

γρ

ργργγ

ργ

ρ

ρρ

ρρ

&

&

&&


107





Least to Most)


1 2 3 4 5

Comment

Chapter 5 Finite Control Volume Analysis (Continued) 5.3.3 Comparison of the Energy Equation with the Bernoulli Equation (Continued)

( )

( )

( ) ( )PLspTLsTinnet

shaftinnet

shaftinnetshaft

s

Lsinnet

shaftininin

outoutout

innetshaftin

ininout

outout

innetshaftin

ininout

outout

innetshaftin

ininout

outout

innetinout

innetshaftin

ininout

outout

innetshaft

innetinout

inoutinoutinout

ininin

outoutout

hhhhhhQ

W

gm

W

g

wh

hhwzg

Vpz

gVp

g

wgzVp

gzVp

wzV

pzV

p

wgzVp

gzVp

quuwgzVp

gzVp

WQzzgVVpp

uum

gzVp

gzVp

+=+−====

−−+++=++

→⎟⎟⎠

⎞⎜⎜⎝

⎛−+++=++

−+++=++

−+++=++

⎟⎠

⎞⎜⎝

⎛−−−+++=++

+=⎥⎦

⎤⎢⎣

⎡−+

−+−+−

−++=++

∨∨

∨∨

γ

γγ

ρρ

ρργρ

γρ

ρρ

ρρ

ρρ

ρρ

&

&

&

&&&

22

loss22

loss22

loss22

22

2

loss22

22

22

22

22

22

22

22


108





Least to Most)


1 2 3 4 5

Comment

Chapter 5 Finite Control Volume Analysis (Continued) 5.3.4 Application of the Energy Equation to Non-uniform Flow

( )

( )

( )

→⎟⎟⎠

⎞⎜⎜⎝

⎛−+++=++

−+++=++

→

⎟⎟⎠

⎞⎜⎜⎝

⎛−+++=++

−+++=++

⋅=⋅=

⎟⎟⎠

⎞⎜⎜⎝

⎛−=⋅

∫∫

∫

g

wgzVp

gzVp

wzV

pzV

p

wgzVp

gzVp

wgzVp

gzVp

Vm

dAnVVdAnVVVm

VVmdAnVV

innetshaftin

inininout

outoutout

innetshaftin

inininout

outoutout

innetshaftin

inininout

outoutout

innetshaftin

inininout

outoutout

A

A

cs

ininoutout

loss22

loss22

loss22

loss22

2

ˆ2ˆ

22

22ˆ

2

22

22

22

22

2

222

222

αρ

αρ

ρργρα

γρα

ρα

ρα

ρ

αρ

αρ

α

ραρα

ααρ

&

rr&

&r

Linnet

shaft

inininin

outoutoutout h

g

wz

gVpz

gVp

−+++=++22

22 αγ

αγ


109





Least to Most)


1 2 3 4 5

Comment

Chapter 5 Finite Control Volume Analysis (Continued) 5.3.5 Combination of the Energy Equation and the Moment-of-momentum Equation

innetshaft

innetshaft

w

w loss−=η

5.4.4 Application of the Loss Form of the Energy Equation

1

21

1

12

22

2

2

1

1

2

22

1

1

212

2 2

22

1

211

2

222

21211constant

2222

gzVp

kkgz

Vpk

kppk

kdpp

gzV

gzVdpgz

Vpgz

Vp

k ++−

=++−⎟⎟

⎠

⎞⎜⎜⎝

⎛−

−==

+=++++=++

∫

∫

ρρρρρρ

ρρρ


8.2.4 Energy Considerations

Dh

rh

zzppp

hzpzphzg

Vpzg

Vp

wLL

LL

γτ

γτ

θγγ

αγ

αγ

ll

l

42sin22

1221

12

11

2

22

22

1

21

11

==

=−Δ+=

=⎟⎟⎠

⎞⎜⎜⎝

⎛+−⎟⎟

⎠

⎞⎜⎜⎝

⎛++++=++


110





Least to Most)


1 2 3 4 5

Comment

Chapter 8 Viscous Flow in Pipes 8.4 Dimensional Analysis of Pipe Flow

minor Lmajor L hhhL +=

8.4.1 Major Losses

( )

( ) ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+−=+−=+−=−=

+++=++⎟⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛=

Δ

=⎟⎟⎠

⎞⎜⎜⎝

⎛=

Δ=Δ+=

fD

fV

Dfzzhzzpp

gV

Dfh

hzg

Vpz

gVp

Df

DDV

p

VDDD

VD

V

pDVFphhh

L

L

L

Re51.2

7.3log0.21

22

22Re,Re,

21

Re,,~

21

,,,,,

2

121221

2

major L

2

22

22

1

21

11

2

2minor Lmajor L

εργγγ

αγ

αγ

εφεφρ

μρε

μρ

φρ

ρμε

ll

l

ll

8.4.2 Minor Losses

( )

( )

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−=⎟⎟

⎠

⎞⎜⎜⎝

⎛−==++=+

−=−====

===ΔΔ

==

21

222

22

Re,22

1

212

21

12

2

2

12

1

233

211

133333313311

22

minor L

2

minor L2

22

minor L

VppC

AA

KgV

hKh

gVp

gVp

VVVAApApVAVAfDK

gV

Df

gVKh

geometryKg

VKhVKpV

pgV

hK

pLL

LL

Leq

eqL

LLLL

ργγ

ρ

φρρ

ll

8.4.3 Noncircular Conduits ( ) ( )

gVD

fhDD

DPAD

VDCf hLh

hh

h 2444Re

Re

22 l======

ππ

μρ


111





Least to Most)


1 2 3 4 5

Comment

Chapter 8 Viscous Flow in Pipes (Continued) 8.5 Pipe Flow Examples N/A

8.5.1 Single Pipes N/A

8.5.2 Multiple Pipe Systems N/A

8.6 Pipe Flowrate Measurement 8.6.1 Pipe Flowrate Meters

( )( )( )( )( )( )

( )( )4

21421

1421

000

222

211

2211421

222

12

12

Re1

2

2212

βρβρ

μρβ

βρ

γγβρ

−−

==−−

==

===−−

==

++=+==−−

==

ppACQCQ

ppACQCQ

AQVVD

Ddpp

ACQCQ

hg

Vpg

VpVAVAQ

ppAVAQ

Tvidealvnnidealn

lidea

Lideal

8.6.2 Volume Flow Meters N/A



112





Least to Most)


1 2 3 4 5

Comment

Chapter 9 Flow over Immersed Bodies (Continued) 9.1 General External Flow Characteristics 9.1.1 Lift and Drag Concepts

( ) ( )( ) ( )

AU

DCAU

LC

dAdApdFL

dAdApdFD

dAindApdF

dAdApdF

DL

wy

wx

wy

wx

22

21

21

cossin

sincos

cos

sincos

ρρ

θτθ

θτθ

θτθ

θτθ

rr

r

r

==

+−==

+==

→⎪⎭

⎪⎬⎫

+−=

+=∫∫∫∫∫∫

9.1.2 Characteristics of Flow Past an Object N/A

9.3 Drag

( )lr

εφρ

,,Re,,

21 2

FrMashapeCAU

DC DD ==

9.3.1 Friction Drag

Dff CbUD lr

2

21 ρ=


113





Least to Most)


1 2 3 4 5

Comment

Chapter 9 Flow over Immersed Bodies 9.3.2 Pressure Drag

( )

( )Re22

21

,,Re22

21

,,cos

21

cos

21

coscos

21

cos

21

cos

2222

2222

22

22

CU

UC

U

DCUCDUfDCU

UC

U

DC

UCDUfDA

dAC

AU

dA

AU

DC

dADA

dAC

AU

dA

AU

DCdAD

DD

ppDp

ppp

Dpp

========

=====

=====

∫∫

∫∫∫

∫

l

l

l

r

lr

lr

l

l

l

r

lr

lr

r

r

ρμ

ρμμ

ρμ

ρ

μμθ

ρ

θρ

ρ

θρθ

ρ

θρ

ρθρ

9.3.3 Drag Coefficient Data and Examples 9.4 Lift 9.4.1 Surface Pressure Distribution

( )lr

εφρ

,,Re,,

21 2

FrMashapeCAU

LC LL ==

9.4.2 Circulation N/A



114





Least to Most)


1 2 3 4 5

Comment

Chapter 10 Open Channel Flow 10.1 General Characteristics of Open-Channel Flow

( ) 2/1Re lgVFrVRh == μρ

10.2 Surface Waves 10.2.1 Wave Speed

022tanh

12tanh

22tanh

2

2/1

→→⎟⎠⎞

⎜⎝⎛

∞→→⎟⎠⎞

⎜⎝⎛

=→>>⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛=

λλπ

λπ

λλπ

πλλ

λπ

πλ

yasyy

yasygcyygc

10.2.2 Froude Number Effects N/A

10.3 Energy Considerations

( ) ( ) ( )gVV

yySS

SSgVV

yyh

S

hg

VyS

gV

y

yp

yp

Szz

hzg

Vpz

gVp

ff

Lf

LL

200

2

2222

21

22

210

0

21

22

21

22

20

21

1

22

11

021

2

222

1

211

−=−→

⎭⎬⎫

=

=−+

−=−→=

++=++→

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

=

=

=−

+++=++

ll

l

l

γ

γγγ


115





Least to Most)


1 2 3 4 5

Comment

Chapter 10 Open Channel Flow (Continued) 10.3.1 Specific Energy

( )

( ) ( ) 12

3

0122

2/12/12/3

min

3/12

3

2

2

2

021

2

=≡====

⎟⎟⎠

⎞⎜⎜⎝

⎛==−=+=−+=+=

cccc

c

cc

c

cf

gyVFrgyygy

yqV

yE

gqy

gyq

dydE

gyqyESSEE

gVyE l

10.4 Uniform Depth Channel Flow 10.4.1 Uniform Flow Approximations N/A

10.4.2 The Chezy and Manning Equations ( )

nSR

V

SRCV

SARn

Q

SARn

VSRVKVK

SRP

SA

PARAW

SS

PSW

PW

FWPFFVVQF

h

h

h

h

hw

hw

h

w

xwx

2/10

3/2

0

2/10

3/2

2/10

3/2

0

22

00

0

0

0

2112

221tansin

sin

00sin0

=

=

=

=→===

==

→

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

==

<<=≈

==

==+−−=−= ∑∑

κ

κ

γρρτ

γγ

τ

γ

θθ

θτ

θπρ

l

l

lr

l

r

l

r

rl

10.4.3 Uniform Depth Examples N/A


116





Least to Most)


1 2 3 4 5

Comment

Chapter 10 Open Channel Flow (Continued) 10.5 Gradually Varied Flow N/A

10.5.1 Classification of Surface Shapes N/A

10.5.2 Examples of Gradually Varied Flows N/A

10.6 Rapidly Varied Flow N/A

10.6.1 The Hydraulic Jump ( ) ( )

( )

( )

( )

( )

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−+−=

++−=

→

⎪⎪⎭

⎪⎪⎬

⎫

+±−=

=

=−⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛

−=⎟⎟⎠

⎞⎜⎜⎝

⎛−=−++=+==

−=−→

⎪⎪⎭

⎪⎪⎬

⎫

===

===

−=−=−

2

2

12

1

1

2

1

21

1

2

21

1

2

1̀

11

21

1

22

1

2

212

12

11

2

111122

21

22

2

21

1222111

1211

22

21

21

22

222

11

21

111

12111221

12

1

81121

81121

02

2222

2222

22

yyFr

yy

yh

Fryy

Fryy

gyV

FrFr

yy

yy

yygy

yVV

yyV

gyVyy

hg

Vy

gV

yQVbyVby

VVgyVyy

yp

byApF

yp

byApF

VVbyVVVQFF

L

L

cc

cc

γγ

γγ

ρρ


117





Least to Most)


1 2 3 4 5

Comment

Chapter 10 Open Channel Flow (Continued) 10.6.2 Sharp-Crested Weirs

( )

( ) 2/52/52

1

2/32/3

21

2/321

2/321

0

2/121

)2( 0 22

21

2

22

21

22

tan1582

2tan

158

22tan2

075.0611.02322

32

222

232

22

22

22

HgCQHgQHg

VhH

PHCbHgCQHgQ

Hg

V

HP

gV

gV

HbgQdhg

VhbgQb

dhudAuQg

Vhgu

gu

hPHzg

Vp

wt

wwrwr

wH

Hh

hwAA

⎟⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛=→<<⎟

⎠⎞

⎜⎝⎛−=

⎟⎟⎠

⎞⎜⎜⎝

⎛+===

→⎪⎭

⎪⎬

⎫

<<

>>

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛+=⎟⎟

⎠

⎞⎜⎜⎝

⎛+=→=

==⎟⎟⎠

⎞⎜⎜⎝

⎛+=+−+=++

∫

∫ ∫=

=

θθθ

γ

l

l

l

10.6.3 Broad-Crested Weirs ( )

( )3

22

2222

2

2/12

221

2221

Hyy

yHgyV

gyVV

gV

gVV

yHg

Vpy

gV

PH

cc

ccc

cc

ccc

cwcw

=→=−→⎪⎭

⎪⎬⎫

=

==

=−

=−++=++

10.6.3 Broad-Crested Weirs (Continued)

( )

( ) 2/12/3

2/3

2/32/3

2/32/122

165.0

32

32

wwbwb

ccccc

PHCHgbCQ

HgbQygbgybyVbyVbyQ

+=⎟

⎠⎞

⎜⎝⎛=

⎟⎠⎞

⎜⎝⎛=→====


118





Least to Most)


1 2 3 4 5

Comment

Chapter 10 Open Channel Flow (Continued) 10.6.4 Underflow Gates

12gyaCq d=


Chapter 11 Compressible Flow 11.3 Categories of Compressible Flow

( )MaV

ccttr wave1sin ==−= α

11.4.2 Converging-Diverging Duct Flow

( )

( )( )

( )0

2

020

21

021

02

02

constant

2

0

1212

2

02

0

2

0

02

/10

/10

2

0

0

=⎟⎟⎠

⎞⎜⎜⎝

⎛+−→

⎪⎭

⎪⎬

⎫

−=−

=−−=−−

−

=−⎟⎟⎠

⎞⎜⎜⎝

⎛−

−=⎟⎟

⎠

⎞⎜⎜⎝

⎛+=⎟⎟

⎠

⎞⎜⎜⎝

⎛+==

∨∨

∨∨

Vhh

TTchh

VTTcVTT

kkR

Vppk

kVdpdppVddp

pp

pp

p

p

k

k

kk ρρρρ


119





Least to Most)


1 2 3 4 5

Comment

Chapter 11 Compressible Flow (Continued) 11.4.2 Converging-Diverging Duct Flow (Continued)

( )[ ]

( )

( )[ ]( ) ( )[ ]

( )

( )[ ]

( )[ ]

( )

( )[ ]

( ) ( )

atmkk

atmkk

kkkk

kk

kk

kk

kk

TTTT

kTT

pppp

kpp

Mak

Makpp

TT

pp

MakTT

Makpp

TT

pp

MakTT

TT

pp

TT

pp

MakTT

833.0*833.0*1

2*

528.0*528.0*

12*

2111

2111

2111

2111211

1

2111

4.14.100

4.14.10

1/

0

1/

20

1/

20

0

0

0

20

1/

20

1/

00

20

1/

000

0

02

0

==⎟⎟⎠

⎞⎜⎜⎝

⎛+

=

==⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎠⎞

⎜⎝⎛

+=

⎭⎬⎫

⎩⎨⎧

−+=

→

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

⎭⎬⎫

⎩⎨⎧

−+=

=⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛

−+=

⎭⎬⎫

⎩⎨⎧

−+=→

⎪⎪

⎭

⎪⎪

⎬

⎫

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛

−+=

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−+

=

=

=

=

=

−−

−

−

−

−

ρρ

ρρ

ρρ


120





Least to Most)


1 2 3 4 5

Comment


( )

( ) ( )

( )0

00

0

4.10

1/1/

0

0

0

0

1/

0

**1*

*****

***634.0*

12

21

12

***

12*

12*

1

TTTT

MaAA

RTkMaV

kRTVV

VAA

VAAV

kk

kpT

T

kTT

kpp

RTpMa

k

kkkk

kk

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

=

=⎟⎠⎞

⎜⎝⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

==⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎠⎞

⎜⎝⎛

+=⎟

⎠⎞

⎜⎝⎛ +

⎟⎠⎞

⎜⎝⎛

+=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛=

→

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

+=

⎟⎠⎞

⎜⎝⎛

+=

==

=

−−

−

ρρ

ρρ

ρρ

ρρρρ

ρρρ

ρ


121





Least to Most)


1 2 3 4 5

Comment


( )[ ] ( )[ ]

( )

( ) ( ) ( )

( )[ ]( )[ ]

( ) ( )[ ]1212

0

00

0

1/1/

0

0

0

0

1/

20

20

2112111

*

**1*1

22

11

2***

12*

2111

2111

−+

−−

−

⎭⎬⎫

⎩⎨⎧

−+−+

=

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟

⎠⎞

⎜⎝⎛

+=⎟

⎠⎞

⎜⎝⎛ +

⎟⎠⎞

⎜⎝⎛

+=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛=

+=

⎭⎬⎫

⎩⎨⎧

−+=

−+=

kk

kkkk

kk

kMak

MaAA

TTTT

MaAA

kk

kpT

T

kTT

Makpp

MakTT

ρρ

ρρρ

ρρ

11.4.3 Constant Area Duct Flow N/A

11.5.3 Normal Shock Waves ( )

( )

( )( ) 22022

22

000

2

22

M11

M11constant

2

constant2

constantconstantconstant

xa

x

ya

y

x

a

a

y

x

y

p

p

akk

pp

akk

pp

pp

pp

pp

TRpc

TVT

RTpTTchhhVh

pRTVpVpV

++

=++

=⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛===+

=−=−==+

=+=+=

∨∨∨∨

ρ

ρ

ρρρ


122





Least to Most)


1 2 3 4 5

Comment

Chapter 11 Compressible Flow (Continued) 11.5.3 Normal Shock Waves (Continued)

( )( )[ ]

( )( )[ ]( )( )[ ]

( )[ ]( )[ ]

( )[ ]( )[ ]

( )[ ]( )[ ]

( )[ ]{ } ( ){ }( ) ( ){ } 22

22

22

22

2/1

2

2

2/1

2

2

2

2

2222

22

2/1

22

2

M1211M12M211

11M

12

1M1212M

M

MM

M211M211

MM

M211M211

M21121

*

M21121

***

M

M21121

M1

**

*M1M1

x

xx

x

y

xx

y

x

xy

y

x

y

x

x

y

y

x

x

y

x

y

y

x

x

y

x

yyyxx

x

y

x

y

x

y

y

x

x

y

x

x

y

y

x

y

x

y

Yyyxxx

x

y

x

y

y

x

x

y

akkakkak

TT

kka

kk

pp

akkka

a

aa

akak

pp

aa

TT

pp

VV

TT

pp

VV

TT

pp

akak

TT

akk

TT

akk

TT

TT

TT

TT

akRTkkV

RTV

pVVpVp

aKk

app

pp

pp

pp

akak

pp

−+

−−−+=

+−

−+

=−−

−+=

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−+−+

=

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛==

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

−+−+

=

→

⎪⎪⎭

⎪⎪⎬

⎫

−++

=

−++

=

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

===+=+

⎭⎬⎫

⎩⎨⎧

−++

=⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

++

=

ρρ

ρρ

ρρρ


123





Least to Most)


1 2 3 4 5

Comment

Chapter 11 Compressible Flow (Continued) 11.5.3 Normal Shock Waves (Continued)

( )( )

( ) ( )

( )112

12

12

,0

,0

,0

,0

,0

,0

2

2

11M

12

M2

11M2

1

2M1M1

−

−−

⎟⎠⎞

⎜⎝⎛

+−

−+

⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ +

=⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

+−+

==⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛==

k

x

kk

x

kk

x

x

y

x

x

x

y

y

y

x

y

x

x

y

x

x

y

y

x

x

y

x

y

y

x

x

y

kka

kk

akak

pp

pp

pp

pp

pp

akak

VV

TT

pp

VV

ρρ

ρρ

ρρ

11.6 Analogy between Compressible and Open-Channel Flows

( ) 1constantcconstant

constant

−==

=====

koc

oc

ococ

oc

kybV

AVcVFrgyc

gyVFr

cVMa

ρ

ρ

11.7 Two-Dimensional Compressible Flow 21 tt VV =


Chapter 12 Turbomachines 12.1 Introduction N/A

12.2 Basic Energy Considerations rUUWV ω=+=

rrr


124





Least to Most)


1 2 3 4 5

Comment

Chapter 12 Turbomachines (Continued) 12.3 Basic Angular Momentum Considerations

( ) ( ) ( ) ( )( ) ( )

( ) ( )22

ˆ

21

22

21

22

21

22

222222

2222111

212111

212111

WWUUVVwWUVUVWUVV

VVVVUVUwm

Ww

VUmVUmWTWQm

VrmVrmTdAnVVrFr

shaftx

xshaftshaft

shaft

shaftshaftshaft

shaftcs

−−−+−=

−+==−+

+=+−==

−+−===

−+−=⋅×=×∑ ∫

θθ

θθθ

θθ

θθ

ωρ

ρ

&

&

&&&&

&&rrrrr

12.4 The Centrifugal Pump N/A

12.4.1 Theoretical Considerations

( ) ( )

( ) ( )

( )11221122

11221122

112211222122

11

222

111

1θθθθ

θθθθ

θθθθ

ρρ

ρωρω

ρωω

VUVUg

hgQhWVUVUQ

Ww

VUVUQWVrVrQWTW

VrVrQTVrVrmTmmmrUrU

UWV

UWV

iishaftshaft

shaft


shaftshaft

−==−==

−=−==

−=−=====

+=

+=

&&

&&&

&&&&rrr

rrr

12.4.1 Theoretical Considerations (Continued)

( ) ( ) ( )[ ]

Qgbr

Ug

UhVbrQ

gVU

gU

h

VVU

gVU

hWWUUVVg

h

irr

i

rii

22

2222

222222

22

2

222

2221

22

21

22

21

22

2cot

2cot

cot21

πβ

πβ

β θθ

−==−=

−==−+−+−=


125





Least to Most)


1 2 3 4 5

Comment

Chapter 12 Turbomachines (Continued) 12.4.2 Pump Performance Characteristics

vmha

shaft

faf

afaLspaa

bhpQh

WQh

Qhpp

hhhhhgVV

zzpp

h

ηηηηγ

η

ηγ

γγγ

==

℘====℘

=℘−

≈−==−

+−+−

=

550

pump thedrivingpower shaft fluid by thev gainedpower

550horsepowerwater

212

21

22

1212

&

12.4.3 Net Positive Suction Head (NPSH)

γγγγ

γγγγ

vL

atmL

atmss

Lssatmvss

phz

pNPSHhz

pg

Vp

hg

Vpz

ppg

VpNPSH

−−−=→−−=+

→=+=−−+=

∑∑

∑

11

2

2

1

2

2

22

12.4 4 System Characteristics and Pump Selection 2

1212 KQzzhhzzh pLp +−=+−= ∑


126





Least to Most)


1 2 3 4 5

Comment

Chapter 12 Turbomachines (Continued) 12.5 Dimensionless Parameters and Similarity Laws

( )

2

253

153

222

122

23

13

3332533122

2

33

2

3253

2

3122

2

3

,,,

,,,,,,

,,, termpidependent ,,Q,,,D,fvariabledependent

ηηρωρωωωωω

ωφη

ωφ

ρωωφ

ωμρω

ωεφ

ρη

μρω

ωεφ

ρωμρω

ωεφ

ω

μρω

ωεφρμωε

=⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛=⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛==

⎟⎟⎠

⎞⎜⎜⎝

⎛==⎟⎟

⎠

⎞⎜⎜⎝

⎛==

⎟⎟⎠

⎞⎜⎜⎝

⎛==

℘

DW

DW

Dgh

Dgh

DQ

DQ

DQ

DQ

DW

DQ

DghD

DQ

DDWgQh

DDQ

DDDW

CDDQ

DDDgh

C

DDQ

DD

shaftshaftaa

shaftai

shaft

a

ishaftiaH

ii

&&

&l&

l&l

ll

12.5.1 Special Pump Scaling Laws 5/1

2

1

1

252

51

2

122

21

2

132

31

2

132

31

2

122

21

2

1

2

1

2

1

11

⎟⎟⎠

⎞⎜⎜⎝

⎛≈

−−

======DD

DD

WW

DD

hh

DD

QQ

WW

hh

QQ

shaft

shaft

a

a

shaft

shaft

a

a

ηη

ωω

ωω

ωω

&

&

&

&

12.5.2 Specific Speed ( )

( ) ( )( ) ( )

( )[ ] 4/34/34/322

2/13

fthgpmQrpm

NNgh

Q

DghDQ

asds

aa

ωω

ωω

===

12.5.3 Suction Specific Speed

( )[ ]( ) ( )

( )[ ] 4/34/3 ftNPSHgpmQrpm

SNPSHg

QS

Rsd

Rs

ωω==

12.6 Axial-Flow and Mixed-Flow Pump N/A


127





Least to Most)


1 2 3 4 5

Comment

Chapter 12 Turbomachines 12.7 Fans

222

122 ⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛D

pD

p aa

ρωρω

12.8 Turbines 12.8.1 Impulse Turbines

( )( )

( )( ) ( )( )2

cos1cos1

cos1cos2

max11

11222111

VUVUUmTWVUrmT

VUVVUWVUWVV

powershaftshaftmshaft =−−==−−=

−−=−+=+==

βωβ

ββ θθθθ

&&&

12.8.2 Reaction Turbines

( ) ( )

( )( )

( ) ( )( )[ ] 4/54/53

2153223

''fth

bhpWrpmN

gh

WN

CCC

C

CCCCgQh

WD

WC

Dgh

CDQC

T

shaftsd

T

shafts

QHQ

QQHT

shaftshaftTHQ

&&

&&

ωρωηφη

φφρ

ηρωωω

====

======

℘

℘℘

12.9 Compressible Flow Turbomachines 12.9.1 Compressors

stdstd

stdtest

stdteststd

std

stdtestTT

NNkRTND

pp

TTmm

pDkRTmR

pDkRTmR

0101001

001

012

01

012

01

`==⎟

⎟⎠

⎞⎜⎜⎝

⎛=⎟

⎟⎠

⎞⎜⎜⎝

⎛ &&

&&

12.9.2 Compressible Flow Turbines N/A 12.10 Chapter Summary and Study Guide N/A

THE END


128

Fluid Mechanics Survey Form B 1st Round of Delphi - Likert Scale Questionnaire on the Importance of Various Fluid Mechanics Topics Selected for High School Engineering Curriculum (For the Calculus Portion)




Least to Most)


1 2 3 4 5

Comment

Chapter 1 – Introduction 1.6 Viscosity

ρμνμμμττγτ

γδδβγδδβδδδδβδβ

δ

==+

==∝∝

======≈=→

TB

t

DeST

CTdydu

dydu

dydu

bU

tbtUtUa

ba

bUyu

/2/3

0limtan

&

&&

Chapter 2 Fluid Statics 2.1 Pressure at a Point

θδδθδδρδδδ


δδδρθδδδδ

sincos2

22cos

2sin

szsyamFVmVzyx

azyxzyxsxpyxpF

azyxsxpzxpFamF

zz

zszz

ysyy

==↑=←==↑

=−−=

=−==

∑

∑

rr

rr


129

Fluid Mechanics Survey Form B (Continued)




Least to Most)


1 2 3 4 5

Comment

Chapter 2 Fluid Statics (Continued) 2.1 Pressure at a Point

( ) ⎩⎨⎧

=

=→

⎪⎪⎭

⎪⎪⎬

⎫

→−=−

→=−

sz

sy

zsz

ysy

pppp

zaszapp

yasyapp

022

022

δδγρ

δδρ

2.2 Basic Equation for Pressure Field

( ) ( ) ( ) ( )

akp

akzyxkkzyxkzyxpamkWFF

zyxpmamF

kzyxkWpzyx

Fk

zj

yi

x

pkzpj

ypi

xpzyxk

zpj

ypi

xpkFjFiFF

Vzxy

zyxzpF

zyxypF

zyxxpF

zxyyppzxy

yppF

s

s

zyxs

z

y

x

y

r

rrrr

rrr

r

ργ

δδρδδδγδδδδδδδδ

δδδδ

δδδδγδδ

δδδδ

δδδδδδδ

δδδ

δδδδ

δδδδ

δδδδ

δδδδδδδ

=−∇−

→=−∇−→=−=

⎪⎩

⎪⎨⎧

=

=−=−−∇=

∂+

∂+

∂=∇

∇=∂∂

+∂∂

+∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

−=++=

=←

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

∂∂

−=

∂∂

−=

∂∂

−=

→⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+−⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−=

∑

∑

ˆ

ˆˆˆˆˆ

ˆˆˆˆˆ

ˆˆˆˆˆˆˆˆˆ

22


130





Least to Most)


1 2 3 4 5

Comment

Chapter 2 Fluid Statics (Continued) 2.3 Pressure Variation in a Fluid Mechanics Mechanics at Rest

γγγ −=→−=∂∂

=∂∂

=∂∂

=−∇−→=dzdp

zp

yp

xpkpa 000ˆ0v

2.8 Hydrostatic Force on a Plane Surface

ccxycxycc

xycR

c

xy

c

ARARRcxcxc

c

xc

c

c

c

xcR

c

cxc

c

x

c

ARAARRcR

cRcAARA AR

yAxIIxAy

Ix

AyI

Ay

dAxyxdAxyxFAyIIy

AyI

AyAy

AyI

y

AyAyI

AyI

Ay

dAyydAydFyyFAhF

AyFAydAydAyFdAydAhF

+=+=

===+=←+=+=

+======

=====

∫∫

∫∫∫

∫∫∫ ∫

θγ

θγλ

θγθγθγγ

sin

sin

sinsinsin

22

22

2

2.12 Pressure Variation in a Fluid Mechanics Mechanics with Rigid-Body Motion

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

+=∂∂

−

=∂∂

−

=∂∂

−

→=−∇−

z

y

x

azp

ayp

axp

akp

ργ

ρ

ρ

ργ rˆ


131





Least to Most)


1 2 3 4 5

Comment

Chapter 2 Fluid Statics (Continued) 2.12.1 Linear Motion

( ) ( )

( )zz

y

zyzy

agdzdp

aga

dydz

dzagdyadpdzzpdy

ypdpag

zpa

yp

+−=+

−=

+−−=∂∂

+∂∂

=+−=∂∂

−=∂∂

ρ

ρρρρ

2.12.2 Rigid-Body Rotation

( ) ( )

constan2

dpconstant2

2

0

000

000ˆˆˆ1ˆ

222

22

22222

222

222

22

+−=−=+=

→=→=→=→=

→=→=→−=

→−=→−=→=−=∂∂

+∂∂

=

−=∂∂

=∂∂

=∂∂

==−=∂∂

+∂∂

+∂∂

=∇

∫ ∫∫

∫∫∫∫∫∫

zrpdzdrrgrz

grzdr

grdzdr

grdr

drdz

gr

drdz

gr

drdzdrrdzgdzgdrr

dzgdrrdzdrrdpdzdrrdpdzzpdr

rpdp

zppr

rpaaerae

zpep

re

rpp zrrzr

γρωγρωω

ωωωω

ωωω

ρωργωργωρ

γθ

ωρωθ θθ

rrr


132





Least to Most)


1 2 3 4 5

Comment

Chapter 3 Elementary Fluid Dynamics – The Bernoulli Equation 3.2 F = ma along a Streamline (Continued) Alternatively

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

↓−≡↑≡

=⋅⋅

=⋅⋅=⋅⋅

=⋅

=⋅=⋅

=⋅⋅

==

=++

=⋅⋅+⋅+

)k̂zg h;k̂z e.unit volumper energy (Potential

e)unit volumper energy (Kinetic

e)unit volumper (Work

Constant Energy Potential Energy Kinetic )streamlinea (along Pressure

221

2212

21

221

VPE

Vzgmzg

Vmzg

VKE

Vvm

vVmv

VW

rArF

AFp

Czgvp NstreamlineNNNNN

ρ

ρ

ρρ

NstreamlineNNNNN Czgvp =⋅⋅+⋅+

+

ρρ 221

Equations'Bernouillienergy of onconservati ofLaw

mass of onconservati ofLaw

( ) outoutoutoutinAinin vAvAAnv 1122ˆ ⋅+⋅=⋅⋅


133





Least to Most)


1 2 3 4 5

Comment

Chapter 3 Elementary Fluid Dynamics – The Bernoulli Equation (Continued) 3.3 F = ma Normal to a Streamline

( ) ( )

streamline theacrossconstant Vp

streamline theacrossconstant

constant

cos

2

coscos

2

22

22

22

=+ℜ

+

=+ℜ

+→

⎪⎪⎭

⎪⎪⎬

⎫

==∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛ℜ

=⎟⎠⎞

⎜⎝⎛

∂∂

−−

ℜ−=

∂∂

ℜ=

∂∂

−−⎟⎠⎞

⎜⎝⎛

∂∂

−−=+=

∂∂

−=∂∂

−=−=+−−=

−=−=ℜ

=ℜ

=

∫

∫∫

∑

∑

zdn

gzdnVdp

sdndp

np

dnVdnnp

dndz

VnpV

np

dndzV

npFWF

Vnpyns

npyspysppysppF

VWWVVmVF

pnnn

nnnpn

nn

γρ

ρργ

ρργδθγδδδ

δδδδδδδδδδδδδ

θλδθδδρδδδ

r

rr


( )


21


22

2212

11

2

2

1

zVpdstVzVp

zVddpdstV

s

sγρργρ

γρρ

+++∂∂

=++

=+++∂∂

∫


134





Least to Most)


1 2 3 4 5

Comment

Chapter 4 Fluid Kinematics 4.1 The Velocity Field

( ) ( ) ( ) ( ) 2/1222ˆ,,,ˆ,,,ˆ,,, wvuVVktzyxwjtzyxvitzyxuV ++==++=rr

4.1.1 Eulerian and Lagrangian Flow Descriptions

( )( )tzyxTT

tzyxTT

zzyyxx

,,,,,, 000

0

0

0

==

⎪⎭

⎪⎬

⎫

===

4.1.2 one-, Two-, and three-Dimensional Flows ( ) ( ) ( ) kwjviutzyxVVjviutyxVViutxVV ˆˆˆ,,,ˆˆ,,,ˆ, ++==+====

rrrrrr

4.1.3 Steady and Unsteady Flows

0=∂∂

tVr

4.1.4 Streamilnes, Streaklines, and Pathlines

uv

dxdy

=

4.2 The Acceleration Field ( )taa rr

=


135





Least to Most)


1 2 3 4 5

Comment

Chapter 4 Fluid Kinematics (Continued) 4.2.1 The Material Derivative

( ) ( ) ( ) ( )[ ] ( ) ( ) ( )

( )

( ) ( ) ( ) ( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )( )( )

⎪⎪⎩

⎪⎪⎨

⎧

∇⋅+∂∂

=∂∂

+∂∂

+∂∂

+∂∂

=

∂∂

+∂∂

+∂∂

+∂∂

=→

⎭⎬⎫

⎩⎨⎧

=

=←

⎪⎪⎩

⎪⎪⎨

⎧

∂∂

+∂∂

+∂∂

=∇⋅

∂∂

+∂∂

+∂∂

=∇↑

∇⋅+∂∂

=∂∂

+∂∂

+∂∂

+∂∂

≡→=→

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

∂∂

+∂∂

+∂∂

+∂∂

=

∂∂

+∂∂

+∂∂

+∂∂

=

∂∂

+∂∂

+∂∂

+∂∂

=

∂∂

+∂∂

+∂∂

+∂∂

=

∂∂

+∂∂

+∂∂

+∂∂

=∂∂

+∂∂

+∂∂

+∂∂

==

=====

TVtT

zTw

yTv

xTu

tT

DtDT

dtdz

zT

dtdy

yT

dtdx

xT

tT

dtdT

tzyxVV

tzyxTT

zw

yv

xuV

kz

jy

ix

Vtz

wy

vx

utDt

DDt

VDa

zww

ywv

xwu

twa

zvw

yvv

xvu

tva

zuw

yuv

xuu

tua

zVw

yVv

xVu

tVa

zV

wy

Vv

xV

ut

Va

dtdz

zV

dtdy

yV

dtdx

xV

tV

dtVd

ta

tzztyytxxttztytxVtrVV

AAAAAAAA

x

y

x

AA

AA

AA

AA

AAAAAAAAA

AAAAAAAAAAAAA

r

rrr

rr

r

rrrrr

rrrrr

rrrrrr

rrr

,,,

,,,ˆˆˆ

e) DerivativlSubstantiaor e DerivativMaterial(

,,,,


136





Least to Most)


1 2 3 4 5

Comment

Chapter 4 Fluid Kinematics (Continued) 4.2.2 Unsteady Effects

it

VtV

zVw

yVv

xVu

tVa

wvxu

ˆ0

0 0

∂∂

=∂∂

=∂∂

∂∂

+∂∂

+∂∂

=∴⎪⎭

⎪⎬⎫

==

=∂∂ rrrrr

rQ

4.2.3 Convective Effects

( ) ( ) ( ) ( ) ( ) on)Accelerati e(Convectivˆˆˆ VVkz

jy

ix

rr∇⋅

∂∂

+∂∂

+∂∂

=∇

4.2.4 Streamline Coordinates ( )

⎟⎠⎞

⎜⎝⎛

∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

=⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂

+∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂

+∂∂

=

+==+===

ssVVs

sVVa

dtdn

ns

dtds

ss

tsVs

dtdn

nV

dtds

sV

tVa

DtsDVs

DtDV

DtsVDanasa

DtVDasVV ns

ˆˆˆˆˆˆ

ˆˆˆˆˆˆ

rr

rr

rr

4.2.4 Streamline Coordinates (Continued)

⎪⎪⎩

⎪⎪⎨

⎧

ℜ=

∂∂

=←

ℜ+

∂∂

=ℜ

==∂∂

→

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

ℜ=

==ℜ∂

=

→

→ 2

2

0ˆˆˆˆ

limˆ

1ˆ

ˆˆˆ

1ˆ0

Va

sVa

nVssVVan

ss

ss

ss

ssss

ss

n

s

s

r

δδ

δδ

δδ

δ

δ


137





Least to Most)


1 2 3 4 5

Comment

Chapter 4 Fluid Kinematics (Continued) 4.4 The Reynolds Transport Theorem (Continued)

( )

dt

Vdbd

dtdB

dt

Vdbd

dtdB

VbBVdbVbB

cvcvsyssys

sysi

iiiVsys

⎟⎠⎞⎜

⎝⎛

=⎟⎠⎞⎜

⎝⎛

=

=←===

∫∫

∫∑→

ρρ

δρδρδρδ 0lim

4.4.1 Derivation of the Reynolds Transport Theorem [ ] [ ] ( ) ( )∑∑ ⋅−⋅+

∂∂

==⋅−⋅+∂

∂= ininoutout

CVsystemininoutout

CVsystem VmVmt

Vmdt

VmdFbmbmt

BDt

DB r&

r&

rrr

&& [ ] [ ]

[ ] [ ] [ ]

( ) ( )⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

→=⋅−⋅∴

=======≡⋅

===

←

∑∑ ForceMomentum

&

Law)Second Newtons(

ininoutout VmVm

FamdtVdm

dtVmd

dtMdFam

dtVdmV

dtdmVm

FamdtVdm

dtVmd

r&

r&

rrrrr

vrr

rr&Q

rrrr

Note: Other Formulas used to derive the Reynolds Transport Theorem are available in pages 171-177.

∫ ∫ ⋅+∂∂

=cv cs

sys dAnVbVdbtDt

DBˆ

rρρ

4.4.2 Physical Interpretation N/A


138





Least to Most)


1 2 3 4 5

Comment

Chapter 4 Fluid Kinematics (Continued) 4.4.3 Relationship to Material Derivative

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )( )e) DerivativlSubstantiaor e DerivativMaterial(

∇⋅+∂∂

=∂∂

+∂∂

+∂∂

+∂∂

≡

∂∂

+∂∂

+∂∂

+∂∂

=∇⋅

Vtz

wy

vx

utDt

Dz

wy

vx

ut

V

r

r

4.4.4 Steady Effects

∫ ⋅=sys

sys dAnVbDt

DBˆ

rρ


⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

=⋅

<⋅

>⋅

===

====Δ=∂∂

=→=⋅⋅+∂∂

= ∫∫∫∫


flow)(out 0ˆˆ


0ˆˆ

0

00

nV

nV

nV

iVVmBb

imVVmBiVV

VdbtDt

DBdAnVbdAnVbVdb

tDtDB

cv

sys

cscscv

sys

r

r

r

rr

r

rrr

rr

ρ

ρρρρ


139





Least to Most)


1 2 3 4 5

Comment

Chapter 4 Fluid Kinematics (Continued) 4.4.5 Unsteady Effects (Continued)

( )( )( )( ) ( )( )

0ˆˆ

ˆˆ

ˆˆˆ

section)(another ˆ

section) (oneˆ


flow)(out 0ˆˆ


0ˆˆ

12

012

0

0)2( 00)1( 0

0

0

0

0

00

=+−=

+−=

⋅=⋅

→⎪⎭

⎪⎬⎫

=⋅

−=⋅

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

=⋅

<⋅

>⋅

===

====Δ=∂∂

=→=⋅⋅+∂∂

=

∫∫∫∫

∫∫∫∫

iAViAV

dAViVdAViV

dAnViVdAnVb

VnV

VnV

nV

nV

nV

iVVmBb

imVVmBiVV

VdbtDt

DBdAnVbdAnVbVdb

tDtDB

cscs

cv

sys

cscscv

sys

ρρ

ρρ

ρρ

ρ

ρρρρ

rr

r

r

r

r

r

rr

r

rrr

rr

4.4.6 Moving Control Volumes

∫∫ ⋅+∂∂

=+=−=cscv

syscvcv dAnWbVdb

tDtDB

VWVWVV ˆrrrrrrr

ρρ


140





Least to Most)


1 2 3 4 5

Comment

Chapter 5 Finite Control Volume Analysis 5.1 Conservation of Mass – The Continuity Equation 5.1.1 Derivation of the Continuity Equation

flow) ldimensiona-(one velocity ddistributeuniformly For ˆ

ˆˆ

0ˆˆˆ0

ˆˆ0

VA

dAnVVV

A

dAnVVVdAnVmAVQm

dAnVVt

mmdAnVdAnVVdt

dAnVVdt

dAnVVdt

VDDtDVdM

DtDM

Aaverage

AaverageA

cv csinoutcscscv

cscvcvcvsyssyssyssys

=⋅

==

⋅==⋅===

=⋅+∂∂

−=⋅⋅=∂∂

⋅∂∂

⋅+∂∂

===

∫

∫∫

∫ ∫∑ ∑∫∫∫

∫∫∫∫∫∫

ρ

ρ

ρ

ρρρρ

ρρρρρ

ρρρρρρ

r

rr

&&

&&rr

rr

5.1.3 Moving, Non-deforming Control Volume

0ˆ =⋅+∂∂

=+= ∫∫ cscv

syscv dAnWVd

tDtDM

VWVrrrr

ρρ

5.1.4 Deforming Control Volume

cscvcscv

sys VWVVdt

dAnWVdtDt

DM rrrr+=≠

∂∂

=⋅+∂∂

= ∫∫∫ 00ˆ ρρρ

5.2 Newton’s Second Law – The Linear Momentum and Moment-of-Momentum Equation 5.2.1 Derivation of the Linear Momentum Equation

∑∫∫∫∫∫

∑∑∑∫

=⋅+∂∂

⋅+∂∂

=

==∂∂


volumecontrol coincident theofcontent

ˆˆ FdAnVVVdVt

dAnVVVdVt

VdVDtD

FFFVdV

cscvcscvsys

syssyssys

rrrrrrrr

rrrr

ρρρρρ

ρ


141





Least to Most)


1 2 3 4 5

Comment

Chapter 5 Finite Control Volume Analysis (Continued) 5.2.2 Application of the Linear Momentum Equation

( ) ( )

( )

( )

∑∫∫

∫∫∫

∫

∑∫∫

∑∫∫∫∫∫

=⋅=⋅

⋅+⋅=⋅+

→+∂∂

=⋅+++∂∂

=⋅+∂∂

⋅+∂∂

=


volumecontrol theof contents

volumcontrol theofcontent

ˆ0ˆ

basis average-or time ousinstantane an (onflow steady For

ˆˆˆ

volumecontrol ngnondeformi inertial,For

volocity volumecontrolconstant For

ˆ

ˆˆ

FdAnWWdAnW

dAnWVdAnWWdAnWVW

VdVWt

FdAnWVWVdVWt

FdAnWVVdVt

dAnWVVdVt

VdVDtD

cscs

cscvcscs cv

cv cv

cs cvcv cv

cscvcscvsys

rrrr

rrrrrrr

rr

rrrrrr

rrrrrrrr

ρρ

ρρρ

ρ

ρρ

ρρρρρ


142





Least to Most)


1 2 3 4 5

Comment

Chapter 5 Finite Control Volume Analysis (Continued) 5.2.3 Derivation of the Moment-of-Momentum Equation

( ) ( )

( )[ ] ( )

( )[ ]

( )[ ] ( ) ( )

( ) ( )[ ] ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) volumecontrol

theof contents

particle

particle

particleparticle

ˆ

ˆ

0

∑∫∫

∫∫∫∑∑

∑∫∫∫

∑∑∑∫

×=×+×∂∂

×+×∂∂

=××=×

×=××=×

×=××=×

×=×→=×=

×+×=×

×=×=

FrdAnVVrVdVrt

dAnVVrVdVrt

VdVrDtDFrFr

FrVdVrDtDVdVr

DtDVdVr

DtD

FrFrFrVdVrDtD

FrVVrDtDVVV

DtrD

DtVVDrVV

DtrDVVr

DtD

FrVVDtDrFVV

DtD

cscv

cscvsyscvsys

syssyssyssys

syssyssys

rrrrrrr

rrrrrrrrrrr

rrrrrrrr

rrrrrrrr

rrrvrrrr

rrrrrr

rrrrrr

ρρ

ρρρ

ρρρ

δρ

δρδ

ρδρδρδ

δδρδρδ


143





Least to Most)


1 2 3 4 5

Comment

Chapter 5 Finite Control Volume Analysis (Continued) 5.2.4 Application of the Moment-of-Momentum Equation

( ) ( ) ( )

( ) ( )( ) ( )[ ]

( )( ) ( )( )( ) ( )

( )( ) ( )( ) ( )outoutininshaftoutin



shaftshaftoutoutoutinininshaft


shaftaxialcs

cscscv

VUVUwmmmVUmVUmWUr

VrmVrmW

TWVrmVrmT

VUwmVUWmVrTW

TmVrTFrmVrdAnVVr

dAnVVrUWVdAnVVrVdVrt

θθ

θθ

θθ

θθ

θθθ

θθ

ωωω

ωω

ω

ωω

ρ

ρρρ

±+±−===

±+±−=→=

±+±−=

=±+±−=

−=−=−==

=−=×+−=⎥⎦⎤

⎢⎣⎡ ⋅×

⋅×+=⋅×=×∂∂

∑∫

∫∫∫

&&&

&&&

&&&

&&&

&&&&

&rr

&rrr

rrrrrrrrrrr

222222

22shaftaxial volumecontrol

theof contents22ˆ

ˆˆ0

5.3 First Law of Thermodynamics – The Energy Equation N/A 5.3.1 Derivation of the Energy Equation

( ) ( )

∫∫∫

∫

∑∑∑∑∫

⋅+∂∂

=⎟⎠

⎞⎜⎝

⎛+=⎟

⎠

⎞⎜⎝

⎛+

++=⎟⎠

⎞⎜⎝

⎛+=

−+−=

∨

cscvsysinnet

innet

sysinnet

innet

sysinnet

innetsys

sysoutinsysoutinsys

dAnVeVdet

VdeDtDWQWQ

gzVueWQVdeDtD

WWQQVdeDtD

ˆ

2

volumecontrolcoincident

2

r&&&&

&&

&&&&

ρρρ

ρ

ρ


144





Least to Most)


1 2 3 4 5

Comment

Chapter 5 Finite Control Volume Analysis (Continued) 5.3.1 Derivation of the Energy Equation (Continued)

∫∫

∫∫∫

∫∫

∑∑

∑∑∫∫

+=⋅⎟⎟⎠

⎞⎜⎜⎝

⎛++++

∂∂

⋅−+=⋅+∂∂

⋅=

⋅−=⋅=⋅−=⋅−=⋅=

⋅=−=−==

=−→=⎟⎠

⎞⎜⎝

⎛+=⋅+

∂∂

∨

cscs

cscscs

cscs

shaftshaft

ouininnet

cvinnet

innetcscv

WQdAnVgzVpuVdet

dAnVpWQdAnVeVdet

VFW

dAnVpdAnVWAnVpVAnpVAnW

VFWpWWWTW

QQQWQdAnVeVdet

innet shaft

innet

2

innet shaft

innet stress tangentialstress tangential

stress normalstress normal

stress normalstress normaloutshaft

inshaft

innet shaft

ˆ2

:ationEnergy Equ

ˆˆ

ˆˆˆˆˆ

00ˆ

&&r

r&&

rrr&

rr&

rrr&

rr&&&&&

&&&&&r

ρρ

ρ

ρρδδ

σδδδσδ

δδσω

ρρ

5.3.2 Application of the Energy Equation

∑∑∫

∫

⎟⎟⎠

⎞⎜⎜⎝

⎛+++−⎟⎟

⎠

⎞⎜⎜⎝

⎛+++=⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛+++

≠⋅←≠⋅⎟⎟⎠

⎞⎜⎜⎝

⎛+++

∨∨∨

∨

inflow

outflow

cs

cs

mgzVpumgzVpudAnVgzVpu

nVdAnVgzVpu

&&r

rr

22ˆ

2

0ˆ0ˆ2

222

2

ρρρ

ρ

ρρ


145





Least to Most)


1 2 3 4 5

Comment

Chapter 5 Finite Control Volume Analysis (Continued) 5.3.2 Application of the Energy Equation (Continued)

( )

( )

( )

( )innetinout

inoutinout

innetinout

inout

inout

inout

innetshaft

innetinout

inoutinout

innetshaft

innetinout

inout

inout

inout

inin

outout

cs

QzzgVV

hhm

QzzgVVppuum

WQzzgVV

hhmpuh

WQzzgVVppuum

mgzVpumgzVpudAnVgzVpu

&&

&&

&&&

&&&

&&r

=⎥⎦

⎤⎢⎣

⎡−+

−+−

=⎥⎦

⎤⎢⎣

⎡−+

−+⎟⎟

⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛+−

+=⎥⎦

⎤⎢⎣

⎡−+

−+−→+=

+=⎥⎦

⎤⎢⎣

⎡−+

−+⎟⎟

⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛+−

⎟⎟⎠

⎞⎜⎜⎝

⎛+++−⎟⎟

⎠

⎞⎜⎜⎝

⎛+++=⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛+++

∨∨

∨∨

∨∨∨∨

∨∨

∨∨∨

∫

2flowsteady l,dimensiona-one e,compressivfor Enthalpy

2

stream fluid oneonly involvingflow ldimensiona-one oughout,steady thrfor Enthalpy 2

2

22ˆ

2

22

22

22

22

222

ρρ

ρ

ρρ

ρρρ

ρ

5.4 Second Law of Thermodynamics – Irreversible Flow

012 ≥−−∨∨

innetquu


146





Least to Most)


1 2 3 4 5

Comment

Chapter 5 Finite Control Volume Analysis (Continued) 5.4.1 Semi-infinitesimal Control Volume Statement of the Energy Equation

( )

( ) ⎟⎠

⎞⎜⎝

⎛−−=+⎟⎟

⎠

⎞⎜⎜⎝

⎛+=⎥

⎦

⎤⎢⎣

⎡+⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+

⎟⎟⎠

⎞⎜⎜⎝

⎛+==⎥

⎦

⎤⎢⎣

⎡+⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+

↓

∨∨

innet

innet

innet

qdsTdzgVddpQdzgVdpdpddsTm

pduddsTQdzgVdpdudm

δρ

δρρ

ρδ

ρ

221

12

22

2

&&

444444444444 3444444444444 21

&&

5.4.2 Semi-infinitesimal Control Volume Statement of the Second Law of Thermodynamics

( ) 0

0ˆˆ

≥−≥≥≥−

=∂∂

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛≥⋅+

∂∂

⋅+∂∂

=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛≥

∑∑

∫∑∫∫∫∫∫

∑∑∑∫

innet

innet

innet

innet

INout

cv

cv

innet

cscvcscvsys

cv

innet

sys

innet

sys

innet

sys

qdsTqdsTT

Qdsm

T

QSsm

VdstT

QdAnVsVds

tdAnVsVds

tVds

DtD

T

Q

T

Q

T

QVds

DtD

δδδδ

ρδ

ρρρρρ

δδδρ

&

&

&

&

&rr

&&&


147





Least to Most)


1 2 3 4 5

Comment

Chapter 5 Finite Control Volume Analysis (Continued) 5.4.3 Combination of the Equations of the First and Second Laws of Thermodynamics

( )

( ) ( )

( ) loss101lossloss

01loss1loss2

02

loss2

02

2

222

=−⎟⎟⎠

⎞⎜⎜⎝

⎛+−→≠⎟⎟

⎠

⎞⎜⎜⎝

⎛=−−=−

→=⎟⎟⎠

⎞⎜⎜⎝

⎛=−⎟⎟

⎠

⎞⎜⎜⎝

⎛+−=⎥

⎦

⎤⎢⎣

⎡+⎟⎟

⎠

⎞⎜⎜⎝

⎛+−

=+⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎠

⎞⎜⎝

⎛−==⎥

⎦

⎤⎢⎣

⎡+⎟⎟

⎠

⎞⎜⎜⎝

⎛+−≥⎥

⎦

⎤⎢⎣

⎡+⎟⎟

⎠

⎞⎜⎜⎝

⎛+−

∫∨∨∨∨∨

∨

innet

out

ininout

innetinout

innet

innet

innetshaft

innet

qpduudquuqud

dqpdudwdzgVddp

dzgVddpqdsTdzgVddpdzgVddp

ρρδδ

ρδδ

ρδδ

ρ

ρδδ

ρρ

Chapter 6 Differential Analysis of Fluid Flow 6.1 Fluid Mechanics Element Kinematics

6.1.1 Velocity and Acceleration Fields Revisited

6.1.2 Linear Motion and Deformation

6.1.3 Angular Motion and Deformation

6.2 Conservation of mass

6.2.1 Differential Survey Form of Continuity Equation

6.2.2 Cylindrical Polar Coordinates

6.2.3 The Stream Function

6.3 Conservation of Linear Momentum

6.3.1 Description of Forces Acting on the Differential Element

6.3.2 Equations of Motion


148





Least to Most)


1 2 3 4 5

Comment

Chapter 6 Differential Analysis of Fluid Flow (Continued) 6.4 Inviscid Flow

6.4.1 Euler’s Equations of Motion

6.4.2 The Bernoulli Equation

6.4.3 Irrotational Flow

6.4.4 The Bernoulli Equation for Irrotational Flow

6.4.5 The Velocity Potential

6.5 Some Basic, Plane Potential Flows

6.5.1 UniSurvey Form Flow

6.5.2 Source and Sink

6.5.3 Vortex

6.5.4 Doublet

6.6 Superposition of Basic, Plane Potential Flows

6.6.1 Source in a UniSurvey Form Stream – Half-Body

6.6.2 Rankine Ovals

6.6.3 Flow around a Circular Cylinder

6.7 Other Aspects of Potential Flow Analysis

6.8 Viscous Flow

6.8.1 Stress-DeSurvey Form ation Relationships


149





Least to Most)


1 2 3 4 5

Comment

Chapter 6 Differential Analysis of Fluid Flow (Continued) 6.8.2 The Navier-Stokes Equations

6.9 Some Simple Solutions for Viscous, Incompressible Fluids

6.9.1 Steady, Laminar Flow between Fixed Parallel Plates

6.9.2 Couette Flow

6.9.3 Steady, Laminar Flow in Circular Tubes

6.9.4 Steady, Axial, Laminar Flow in an Annulus

6.10 Other Aspects of Differential Analysis

6.10.1 Numerical Methods

Chapter Summary and Study Guide

Chapter 7 Similitude, Dimensional Analysis, and Modeling 7.1 Dimensional Analysis

7.2 Buckingham Pi Theorem

7.3 Determination of Pi Terms

7.4 Some Additional Comments about Dimensional Analysis

7.4.1 Selection of Variables

7.4.2 Determination of Reference Dimensions

7.4.3 Uniqueness of Pi Terms


150





Least to Most)


1 2 3 4 5

Comment

Chapter 7 Similitude, Dimensional Analysis, and Modeling (Continued) 7.5 Determination of Pi Terms by Inspection

7.6 Common Dimensionless Groups in Fluid Mechanics

7.7 Correlation of Experimental Data

7.7.1 Problems with One Pi Term

7.7.2 Problems with Two or More Pi Term

7.8 Modeling and Similitude

7.8.1 Theory of Models

7.8.2 Model Space

7.8.3 Practical Aspects of Using Models

7.9 Some Typical Model Studies

7.9.1 Flow through Closed Conduits

7.9.2 Flow around Immersed Bodies

7.9.3 Flow with a Free Surface

7.10 Similitude Based on Governing Differential Equations

7.11 Chapter Summary and Study Guide


151





Least to Most)


1 2 3 4 5

Comment

Chapter 8 Viscous Flow in Pipes 8.1 General Characteristics of Pipe Flow 8.1.1 Laminar of Turbulent Flow

μρVD

=Re

8.1.2 Entrance Region and Fully Developed Flow

( ) 10Re10flow)lent (for turbuRe4.4flow)lent (for turbu Re06.0 46/1 <<==DD

ee ll

8.1.3 Pressure and Shear Stress

021 <Δ

−=∂∂

−=∇l

pxpppp

8.2 Fully Developed Laminar Flow 8.2.1 From F = ma Applied Directly to a Fluid Mechanics Element

( ) ( ) ( )

( ) ( )

( ) 2

2

0

2

0

2222

12

21

2112

2122

14

2121164

22422

020ˆ0

RQ

AQVVRQdrr

RrVdrrrudAuQ

RrDru

DrV

DrpDruCrpu

drrpdurpdrdu

drdu

Dp

Dr

rp

rrpprppppixuuVV

tVamF

cRRr

r c

wc

ww

ππ

ππ

μτ

μμ

μμμτ

τττ

τ

πτππ

===⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−===

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛ Δ=+⎟⎟

⎠

⎞⎜⎜⎝

⎛ Δ−=

Δ−=⎟⎟

⎠

⎞⎜⎜⎝

⎛ Δ−=−==Δ==

Δ

=−Δ−−Δ−==∂∂

=∇⋅=∂∂

=

∫∫∫

∫∫

=

=

ll

ll

l

l

lrr

rrr


152





Least to Most)


1 2 3 4 5

Comment

Chapter 8 Viscous Flow in Pipes (Continued) 8.2.1 From F = ma Applied Directly to a Fluid Mechanics Element Continued)

( ) ( ) ( )

( ) ( )

( )

( )

( )l

l

l

l

l

l

ll

ll

ll

l

l

lrr

rrr

μθγπ

μθγτθγ

μπ

μππ

ππ

ππ

μτ

μμ

μμμτ

ττττ

πτππ

128sin

32sin2sin

1283222

2122

14

2121164

22422

020ˆ0

4

242

2

2

2

2

0

2

0

2222

12

21

2112

DpQ

DpVr

ppDQpDVRVR

V

RQ

AQV

VRQdrr

RrVdrrrudAuQ

RrD

ruDrV

DrpDruCrpu

drrpdurpdrdu

drdu

Dp

Dr

rp

rrpprppppixuuVV

tVamF

cc

cRRr

r c

wc

ww

−Δ=

−Δ==

−ΔΔ=

Δ===

===⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−===

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛ Δ=+⎟⎟

⎠

⎞⎜⎜⎝

⎛ Δ−=

Δ−=⎟⎟

⎠

⎞⎜⎜⎝

⎛ Δ−=−==Δ==

Δ

=−Δ−−Δ−==∂∂

=∇⋅=∂∂

=

∫∫∫

∫∫

=

=

8.3 Fully Developed Turbulent Flow N/A

8.3.1 Transition from Laminar to Turbulent Flow N/A


153





Least to Most)


1 2 3 4 5

Comment

Chapter 8 Viscous Flow in Pipes (Continued) 8.3.2 Turbulent Shear Stress

( )

( ) ( ) ( ) ( )

( ) ( )( )

222

2/12

2

22

''

'1'

intensity Turbulence

0'1'0111'

'',,,1

0

0

0

0

0

0

0

0

0

0

0

0

⎟⎟⎠

⎞⎜⎜⎝

⎛===+=−=

=≠=⎥⎦⎤

⎢⎣⎡

==

>==−=⎟⎠⎞⎜

⎝⎛ −=−=

−=+==

∫

∫∫∫∫

∫

+

++++

+

dyud

dyud

dyudvu

dyud

yuudyud

dydu

u

dtuT

uu

dtuT

uuTuTT

dtudtuT

dtuuT

u

uuuuuudttzyxuT

u

mturbmturbturblam

Tt

t

Tt

t

Tt

t

Tt

t

Tt

t

Tt

t

ll ρτρηητττρμτ

μτμτ

8.3.3 Turbulent Velocity Profile

( ) n

c

c

w

Rr

Vu

yR

uuV

vyu

uuurRy

vyu

uu

/1

2/1

1ln5.2*

0.5*ln5.2*

***

⎟⎠⎞

⎜⎝⎛ −=⎟⎟

⎠

⎞⎜⎜⎝

⎛=

−

+⎟⎠⎞

⎜⎝⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛=−==

ρτ

8.3.4 Turbulent Modeling N/A


154





Least to Most)


1 2 3 4 5

Comment

Chapter 9 Flow over Immersed Bodies 9.2 Boundary Layer Characteristics N/A 9.2.1 Boundary Layer structure and Thickness on a Flat Plate

( ) ( ) ( )

( ) ∫∫

∫∫∫∫∞∞

∞∞∞

⎟⎠⎞

⎜⎝⎛ −=−=

−=−⎟⎠⎞

⎜⎝⎛ −=−=

00

2

000

1

1**

dyUu

UuOdyuUuObU

dyuUubdAuUudyUudybuUbU

ρ

ρρδδ

9.2.2 Prandtl/Blasius Boundary Layer Solution

( ) ( ) ( )

( ) ( )

xU

xO

xxUvxasfandatff

atfffffffx

vUvUfux

vy

u

fffVxUvxU

Uvxyg

UuyasUu

yonvuyuv

yuv

xuu

yv

xu

yx

uv

yv

xu

yv

xvv

xp

yvv

xvu

yu

xuv

xp

yuv

xuu

wx

xx

ρμτ

δδδηη

ηηη

ηηηδδ

ρρ

2/3

2/1

2/122/1

2

2

2

2

2

2

2

2

2

2

332.0Re664.0

Re721.1*

Re551'00'

00'0'''''2'4

'

~

0000

11

==

===∞→→===

====−−⎟⎠⎞

⎜⎝⎛==

∂Ψ∂

−=∂Ψ∂

=

==Ψ⎟⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=∞→→

===∂∂

=∂∂

+∂∂

=∂∂

+∂∂

→⎪⎭

⎪⎬

⎫

∂∂

<<∂∂<<

=∂∂

+∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

−=∂∂

+∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

−=∂∂

+∂∂


155





Least to Most)


1 2 3 4 5

Comment

Chapter 9 Flow over Immersed Bodies (Continued) 9.2.3 Momentum Integral boundary Layer Equation for a Flat Plate

( ) ( )

( )( ) ( )

( )

( ) ( ) ( )[ ] ( ) ( )[ ]

xf

xf

wfw

xYYyw

ww

plate wplate wxx

cCC

UxCCc

Uc

xU

CC

CCxUC

xvCdx

UCC

ddYdgCCU

dYdgU

yu

dYYgYgCCbUDdYYgYgbUdyuUubD

dxOdUb

dxDd

dxOdbU

dxDdObUDdyuUubD

dyUubbhUdyuUhdyubbhUDdAudAUUD

dxdADFdAnVudAnVuF

Re664.0

Re

22

212

Re

22

11

ˆˆ

2121

2

2/321

12

1

2

1

2

022

00

1

01121

0

2

0

222

0

0

2

00

22

2

2

1

21

=====

======∂∂

=

−==−=−=

====−=

==−=+−=−

−=−=−=⋅+⋅=

===

∫∫∫

∫

∫∫∫∫∫

∫∫∑∫∫∑

ρμ

ρ

τρμτ

δδρμ

δδδμ

δμμτ

δρδρρ

ρττρρρ

ρρρρρρ

ττρρ

δ

δ

δδδ

rr

rrrr

rr

rrr

9.2.3 Momentum Integral boundary Layer Equation for a Flat Plate (Continued)

ll

l

l

lll

r

Re328.1

Re

81

21

21

21

02

0

2===== ∫

∫DfDffDf

wfDf C

CCCdxcC

bU

dxb

bU

DC

ρ

τ

ρ

9.2.4 Transition from Laminar to Turbulent Flow N/A


156





Least to Most)


1 2 3 4 5

Comment

Chapter 9 Flow over Immersed Bodies (Continued) 9.2.5 Turbulent Boundary Layer Flow N/A

9.2.6 Effects of Pressure Gradient N/A

9.2.7 Momentum Integral Boundary Layer Equation with Nonzero Pressure Gradient

( )constant

*2

==

+=−=

UUdx

dUUOU

dxd

dxdU

Udxdp

fs

fsfsfsw

fsfs ρδρτρ

Chapter 10 Open Channel Flow 10.2 Surface Waves 10.2.1 Wave Speed

( )( )( ) ( ) ( )[ ]

( ) ( )

2/12

222

2

211

1

22

1100constant2

22

21

21

⎟⎟⎠

⎞⎜⎜⎝

⎛+≈→<<=+=+=+

==+=

=+=

=

−−=+−→

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

=→<<

+=

/++−=/−

yygyc

yyyVVyy

gVV

yg

V

gyccg

yVbyyAy

FbyyAy

F

cVcbcybyyby

yVycyy

yVyyc

byyVcbcy

cc

δδδδδδ

δδδγγδγγ

δρδγγ

δδδ

δδδ

δδ


157





Least to Most)


1 2 3 4 5

Comment

Chapter 10 Open Channel Flow (Continued) 10.3.2 Channel Depth Variations

( ) ( )( )2

02/1

22

200

2

021

1

2

FrSS

dxdygyVFr

dxdyFr

dxdy

gyV

dxdV

gV

dxdy

yV

dxdy

yq

dxdVSS

dxdy

dxdV

gVS

dxdy

dxdV

gV

dxdh

dxdz

dxdy

dxdV

gVzy

gV

dxd

dxdHS

dxdzS

dxdHhHH

f

fL

fL

−

−==

−==−=−=−=+++=

++=⎟⎟⎠

⎞⎜⎜⎝

⎛++===+=

Chapter 11 Compressible Flow 11.1 Ideal Gas Relationships

( )

( ) ( )

( ) RTuhTTchhdTchhdTchd

dThd

ThcThhRTpTuupuh

TTcuuVdTcuu

dTcuddT

udTuc

MRRTp

p

T

T pp

p

p

v

T

T v

v

v

vgas

+=−=−=−=

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

∂∂

====+=

−=−==−

==⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

∂∂

===

∨∨∨∨∨∨∨

∨∨∨∨∨∨∨∨

∨∨∨∨

∨∨∨

∫

∫

121212

121212

2

1

2

1

1

ρρ

ρ

λρ


158





Least to Most)


1 2 3 4 5

Comment

Chapter 11 Compressible Flow (Continued) 11.1 Ideal Gas Relationships

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

−=

−===−+=+=

∨∨∨

∨∨

ρ1

11pduddsT

kRc

kRkc

cc

kRccRdT

uddT

hddTRudhd vpv

pvp

1

2

1

212

2

1

1

212 lnlnlnln

11

111

pp

RTT

cssRTT

css

pdpR

TdTcdsdR

TdTcdsdphddsTdppdudhd

pv

pv

+=−+=−

−=⎟⎟⎠

⎞⎜⎜⎝

⎛+=⎟⎟

⎠

⎞⎜⎜⎝

⎛−=⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+=

∨∨∨

ρρ

ρρρρρ

11.2 Mach Number and Speed of Sound

( )( ) ( )( )

( )( )( ) ( ) ( )

( ) ( )cpVcVcpzgVppcpc

cpVpAAcVccAcApppAAVcVccAc

cVVcVccVcAccVMa

δδρδρδδδδ

ρδ

δρδ

δρδ

δρδδρδρδδδρρδρ

δρδρδδρδρρδρρδδρρρ

==−−

+=+⎟⎟⎠

⎞⎜⎜⎝

⎛+==

=−=−+−+−=−+−+−

=−+−=−+==

022

loss2

2222

11.2 Mach Number and Speed of Sound (Continued)

( )( )

( )ρρ

ρρρρ

ρρ

ρρ

ρρ

δδρδ

v

sv

kk

k

s

k

s

Ecp

ddpERTkcRTkkpkpkp

ppcpp

pc

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=======⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=→⎪⎭

⎪⎬

⎫

→∂→

=

−− 11constant

constant0


159





Least to Most)


1 2 3 4 5

Comment

Chapter 11 Compressible Flow (Continued) 11.4 Isentropic Flow of an Ideal Gas 11.4.1 Effect of Variations in Flow Cross-Sectional Areas

( )

AdA

ddpV

Vdp

VdV

Vdp

AdAd

VdV

VdV

AdAd

VAVdV

VdpdzVddpAVm

=⎟⎟⎠

⎞⎜⎜⎝

⎛−→

⎪⎪⎭

⎪⎪⎬

⎫

−=

+=−→=++

→=++−==++==

ρρρ

ρρ

ρρ

ρρ

γρρ

2

2

2

22

10

constantlnlnln021constant&

11.4.1 Effect of Variations in Flow Cross-Sectional Areas (Continued)

( )( )

( ) ( ) ( )22

2

2

22

22

2

2

2

111

1

11

1

MaVA

dVdA

MaMa

AdAd

MaAdA

VdV

AdAMa

Vdp

VdV

Vdp

AdAMa

Vdp

AdA

ddpV

Vdp

cVMa

pcs

−−=−

=−

−=

→

⎪⎪⎭

⎪⎪⎬

⎫

=−

−==−→

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

=⎟⎟⎠

⎞⎜⎜⎝

⎛−

=

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=

ρρ

ρ

ρρ

ρρ

ρ


160





Least to Most)


1 2 3 4 5

Comment

Chapter 11 Compressible Flow (Continued) 11.5 Nonisentropic Flow of an Ideal Gas 11.5.1 Adiabatic Constant Area Duct Flow with Friction (Fanno Flow)

( )

( ) ( )

( )( )

kRTVTV

cR

Tc

dTds

Vc

dTdV

TdTdV

VR

Tc

dTds

TdT

VdVRTdTcdsT

VVV

TdTdRTdTcdsT

TdTd

pdp

RTpdTchddphddsT

ppR

TTcssRTpT

RpcTVT

TcVT

Tc

VT

TTchh

hVh

WQzzgVV

hhm

aapppp

pp

p

pp

p

p

p

innetshaft

innet

=⎟⎟⎠

⎞⎜⎜⎝

⎛+−=−=⎟

⎠⎞

⎜⎝⎛ +−−=

⎟⎠⎞

⎜⎝⎛ +−−=→−==⎟⎟

⎠

⎞⎜⎜⎝

⎛+−=

+=→⎪⎭

⎪⎬⎫

=

=−=

−=−↑=←==+

⎪⎪

⎩

⎪⎪

⎨

⎧

==+

==+

→⎪⎭

⎪⎬

⎫

−=−

==+

+=⎥⎦

⎤⎢⎣

⎡−+

−+−

∨∨

∨∨

∨∨

∨∨

111

ddconstant

lnlnconstant2

constant2

constant2constant

2

2

2

111022

22

02

2

0

2

00

0

2

12

21

22

12

ρρρ

ρρ

ρρ

ρρ

ρρ

ρρ

&&&


161





Least to Most)


1 2 3 4 5

Comment

Chapter 11 Compressible Flow (Continued) 11.5.1 Adiabatic Constant Area Duct Flow with Friction (Fanno Flow)

( ) ( )

( ) ( )

( ) ( ) ( )

( ) ( ) ( )( )[ ]

( ) ( ) ( ) ( ) ( ) 0121

21

2110

21

02

022

022

248

22

2

2

22

2

2

2

2

2

22

2

2

2

22

2

2

2

2

222

2

222

22

22

2

1221122211

=+−+−=

−+==

−+

=++==

=++=++

=−−→=←==−−

−=−−−=−−

DdxMa

kf

MaMad

VVdkMa

MaMad

VVd

pdp

MakMaMad

VVd

VVdMak

TdT

TcVd

TdT

TdT

MaMad

VVdRTkMaV

VVdMak

DdxMafk

pdpVd

pDdxV

pfdp

dVVDdxVfdpDA

VfdVV

AdD

dp

VVVA

RppVVmRApAp

p

wxw

xx

ρρ

ρρπρτ

ρπτ

ρ&


162





Least to Most)


1 2 3 4 5

Comment

Chapter 11 Compressible Flow (Continued) 11.5.1 Adiabatic Constant Area Duct Flow with Friction (Fanno Flow) (Continued)

( ) ( )( )[ ]{ }

( ) ( )( )[ ]{ }

( ) ( )[ ]( )[ ]

( )

( ) ( ) ( ) ( )( )[ ]{ } ( )

( )( )[ ]

( )[ ]( )[ ]

( )[ ]( )[ ]

( )( )[ ]

( ) ( )1212

0

0

0

0

0

0

2/1

2

2/1

2

2

2/1

2

2

2

2221

12

2

2

2

2

1* *

42

22

42

22

211

121

***

**

211211

****21211

*

**211

21****211

21*

21121**

*211

21ln

2111

2111

2111

−+

=

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛

+=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎭⎬⎫

⎩⎨⎧

−++

==⎭⎬⎫

⎩⎨⎧

+−+

=

=⎭⎬⎫

⎩⎨⎧

−++

===−++

=

−+−

−=−=−

−−

−=

⎭⎬⎫

⎩⎨⎧

−+++

+−

=−+−

=−+−

∫ ∫

kk

Ma

Ma

MakkMap

ppp

pp

pp

pp

Makk

Mapp

TT

pp

MakMak

VV

MakMak

VV

TTMa

kRTRTkMa

VV

Makk

TT

MadMak

kTdT

Df

Df

Df

Df

MakMak

kk

MaMa

k

Ddxf

kMaMakMadMa

Ddxf

kMaMakMadMa

ρρ

ρρ

ρρ

llllll

ll

l

l


163





Least to Most)


1 2 3 4 5

Comment

Chapter 11 Compressible Flow (Continued) 11.5.2 Frictionless Constant Area Duct Flow with Heat Transfer (Rayleigh Flow)

( ) ( )

( ) ( )[ ]

( ) ( ) ( ) ( )[ ]( )

( )

a

aa

aaaa

a

a

aa

aaapp

bp

aaap

pp

x

VV

pp

TT

kMak

ppV

ppV

pp

VpVpMaTc

qVdV

kRTkV

dVdT

TV

Tcq

VdVqdVVhd

kMa

dsdT

RVVTTVTcdTdsdsdT

MakRTVRVVTT

VTc

dTds

dTdV

TV

Tc

dTdsdVVdTcdsTdVVhddsTdVVdpdVVdp

RTVpVpRVmApVmAp

==++

=+=+

+=+−

=⎥⎦

⎤⎢⎣

⎡ −+==+

=→=−+

==

==−

+=

+=+=+=−=−=

==+=+++=+

−∨

−

∨

ρρ

ρρρρ

ρρδδδ

ρρ

ρρ

ρρ

ρ

22

2

222

12

1

22

222111

111

111

1011

11

constantVconstantconstant&&


164





Least to Most)


1 2 3 4 5

Comment

Chapter 11 Compressible Flow (Continued) 11.5.2 Frictionless Constant Area Duct Flow with Heat Transfer (Rayleigh Flow) (Continued)

( ) ( )

( )

( )( )

( )( )1

22

,0

0

,0

0

,0

022

22

,0,0

0

,0

2

2

2

2

211

12

11

12

1112

11

11

−

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛

+++

=

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

+

⎟⎠⎞

⎜⎝⎛ −++

=⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛=

⎥⎦⎤

⎢⎣⎡

++

==⎥⎦⎤

⎢⎣⎡

++

=⎟⎟⎠

⎞⎜⎜⎝

⎛==

kk

a

a

a

aaaa

a

aa

a

a

aaaa

a

MakkkMa

kpp

pp

pp

pp

pp

kMa

MakMak

TT

TT

TT

TT

TT

kMaMakMa

VV

kMaMak

TTMa

pp

TT

TTMa

ρρ

ρρ

THE END


165

Part Three Findings from the Research Project


166

List 1A. Pre-Calculus Based Fluid Mechanics Topics That Possibly Could Be Taught at 9th Grade

Chapter/Section Page Numbers Number of Pages Chapter 1 – Introduction (pp. 1-30 30 pages sub-total. 10 sections out of 11) 1.1 Some Characteristics of Fluid 1.2 Dimensions, Dimensional Homogeneity, and Units 1.3 Analysis of Fluid Mechanics Behavior 1.4 Measures of Fluid Mechanics Mass and Weight 1.4.1 Density 1.4 2 Specific Weight 1.4.3 Specific Gravity 1.5 Ideal Gas Law

1-13 13

1.7 Compressibility of Fluids 1.7.1 Bulk Modulus 1.7.2 Compression and Expansion of Gases 1.7.3 Speed of Sound 1.8 Vapor Pressure 1.9 Surface Tension 1.10 A Brief Look Back in History 1.11 Chapter Summary and Study Guide

20-30 11

Chapter 2 Fluid Statics (pp. 38-79 42 pages sub-total. 9 sections out of 13) 2.3 Pressure Variation in a Fluid at Rest (Concept only)* 2.3.1 Incompressible Fluid 2.3.2 Compressible Fluid 2.4 Standard Atmosphere 2.5 Measurement of Pressure 2.6 Monometry 2.6.1 Piezometer Tube 2.6.2 U-Tube Manometer 2.6.3 Inclined-Tube Manometer 2.7 Mechanical and Electronic Pressure Measuring Devices

42-56 15

2.9 Pressure Prism 2.10 Hydrostatic Force on a Curves Surface 2.11 Buoyancy, Flotation, and Stability 2.11.1 Archimedes’ Principle 2.11.2 Stability

63-72 10

2.13 Chapter Summary and Study Guide 78-79 2 * Basic principles covered under this section heading could be explored; but the formulas used are calculus-based.


167

List 1A. (Continued)

Chapter/Section Page Numbers Number of Pages Chapter 3 Elementary Fluid Dynamics – The Bernoulli Equation (pp. 95-135 41 pages sub-total. 8 sections out of 9) 3.1 Newton’s Second Law 3.2 F = ma along a Streamline

95-101 7

3.4 Physical Interpretation 3.5 Static, Stagnation, Dynamic, and Total Pressure 3.6 Examples of Use of the Bernoulli Equation 3.6.1 Free Jets 3.6.2 Confined Flows 3.6.3 Flowrate Measurement 3.7 The Energy Line and the Hydraulic Grade Line 3.8 Restrictions on Use of the Bernoulli Equation 3.8.1 Compressibility Effects 3.8.3 Rotational Effects 3.8.4 Other Restrictions 3.9 Chapter Summary and Study Guide

104-135 32

Chapter 4 Fluid Kinematics (pp. 150-184 35 pages sub-total. 3 sections out of 5) 4.3 Control Volume and System Representations 168-169 2 4.4 The Reynolds Transport Theorem 170-171 2 4.4.7 Selection of a Control Volume 4.5 Chapter Summary and study Guide

182-182 3

Chapter 5 Finite Control Volume Analysis (pp. 192-252 61 pages sub-total 2 sections out of 5) 5.1 Conservation of Mass – The Continuity Equation (Concept only)* 5.1.2 Fixed, Non-deforming Control Volume 195-200 6 5.3.3 Comparison of the Energy Equation with the Bernoulli Equation 5.3.4 Application of the Energy Equation to Non-uniform Flow 5.3.5 Combination of the Energy Equation and the Moment-of-momentum Equation

236-246 11

5.4.4 Application of the Loss Form of the Energy Equation 5.5 Chapter Summary and Study Guide

249-252 4

Chapter 6 Differential Analysis of Fluid Flow (pp. 272-334 63 pages sub-total. 0 sections out of 11) Chapter 7 Similitude, Dimensional Analysis, and Modeling (pp. 346-391 46 pages sub-total. 0 sections out of 11)

* Basic principles covered under this section heading could be explored; but the formulas used are calculus-based.


168


Chapter/Section Page Numbers Number of Pages Chapter 8 Viscous Flow in Pipes (pp. 401-472 72 pages sub-total. 5 sections out of 7) 8.2 Fully Developed Laminar Flow (Concept only)* 8.2.4 Energy Considerations 416-417 2 8.4 Dimensional Analysis of Pipe Flow 8.4.1 Major Losses 8.4.2 Minor Losses 8.4.3 Noncircular Conduits 8.5 Pipe Flow Examples 8.5.1 Single Pipes 8.5.2 Multiple Pipe Systems 8.6 Pipe Flowrate Measurement 8.6.1 Pipe Flowrate Meters 8.6.2 Volume Flow Meters 8.7 Chapter Summary and Study Guide

430-472 43

Chapter 9 Flow over Immersed Bodies (pp. 483-550 68 pages sub-total. 4 sections out of 5) 9.1 General External Flow Characteristics 9.1.1 Lift and Drag Concepts 9.1.2 Characteristics of Flow Past an Object

484-493 10

9.3 Drag 9.3.1 Friction Drag 9.3.2 Pressure Drag 9.3.3 Drag Coefficient Data and Examples 9.4 Lift 9.4.1 Surface Pressure Distribution 9.4.2 Circulation 9.5 Chapter Summary and Study Guide

518-550 33

Chapter 10 Open Channel Flow (Whole Chapter; pp. 561-605 45 pages sub-total. 7 sections out of 7) 10.1 General Characteristics of Open-Channel Flow 10.2 Surface Waves 10.2.1 Wave Speed 10.2.2 Froude Number Effects 10.3 Energy Considerations 10.3.1 Specific Energy

561-573 13

* Basic principles covered under this section heading could be explored; but the formulas used are calculus-based.


169


Chapter/Section Page Numbers Number of Pages Chapter 10 Open Channel Flow (Continued) 10.4 Uniform Depth Channel Flow 574-605 32 10.4.1 Uniform Flow Approximations 10.4.2 The Chezy and Manning Equations 10.4.3 Uniform Depth Examples 10.5 Gradually Varied Flow 10.5.1 Classification of Surface Shapes 10.5.2 Examples of Gradually Varied Flows 10.6 Rapidly Varied Flow 10.6.1 The Hydraulic Jump 10.6.2 Sharp-Crested Weirs 10.6.3 Broad-Crested Weirs 10.6.4 Underflow Gates 10.7 Chapter Summary and Study Guide

Chapter 11 Compressible Flow (pp. 614-678 65 pages sub-total. 6 sections out of 8) 11.3 Categories of Compressible Flow 623-628 6 11.4 Isentropic Flow of an Ideal Gas 11.4.2 Converging-Diverging Duct Flow 11.4.3 Constant Area Duct Flow

631-646

11.5 Non-isentropic Flow of an Ideal Gas 11.5.3 Normal Shock Waves 11.6 Analogy between Compressible and Open-Channel Flows 11.7 Two-Dimensional Compressible Flow 11.8 Chapter Summary and Study Guide

665-678 14

Chapter 12 Turbomachines (Whole Chapter; pp. 684-736 53 pages sub-total. 10 sections out of 10) 12.1 Introduction 12.2 Basic Energy Considerations 12.3 Basic Angular Momentum Considerations 12.4 The Centrifugal Pump 12.4.1 Theoretical Considerations 12.4.2 Pump Performance Characteristics 12.4.3 Net Positive Suction Head (NPSH) 12.4 4 System Characteristics and Pump Selection

684-736 53


170


Chapter/Section Page Numbers Number of Pages 12.5 Dimensionless Parameters and Similarity Laws 12.5.1 Special Pump Scaling Laws 12.5.2 Specific Speed 12.5.3 Suction Specific Speed 12.6 Axial-Flow and Mixed-Flow Pump 12.7 Fans 12.8 Turbines 12.8.1 Impulse Turbines 12.8.2 Reaction Turbines 12.9 Compressible Flow Turbomachines 12.9.1 Compressors 12.9.2 Compressible Flow Turbines 12.10 Chapter Summary and Study Guide

↑ ↑

Statistical Summary

Total Number of Pages Covered by Text (Excluding “Problems” Section) 621 Total Numbers of Sections Covered Under All Chapters 64 out of 102

Percentage of Pre-Calculus Sections

( ) ( ) %7.62%10010264%100

Sections ofNumber TotalSections Calculus- PreofNumber % Calculus-Pre =⎟

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Total Numbers of Chapters Covered 10 out of 12 Percentage of Chapters with Pre-Calculus Sections

( ) ( ) %3.83%1001210%100

Chapters ofNumber Total Sections Calculus- Pre withChapters ofNumber % Calculus-Pre =⎟

⎠⎞

⎜⎝⎛=⎟⎟

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Total Number of Pages Covered by Pre-Calculus Portion 317 Percentage of Pre-Calculus Volume

( ) ( ) %0.51%100621317%100

PagesofNumber Total PagesCalculus- PreofNumber % Calculus-Pre =⎟

⎠⎞

⎜⎝⎛=⎟⎟

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171

List 1B. Pre-Requisite Mathematics and Science Topics to Be Reviewed Before Teaching the Pre-Calculus Portion of Fluid Mechanics Topics to 9th Grade Students


Math Physics/Chemistry 1. [analytic geometry] 12th (To be taught as a special skill) 2. [analytic geometry: hyperbolic tangent] Post-secondary To be taught 3. [areas of geometric shapes: circle, triangle, etc.] (M5M1) 5th (2B) 4. [cylinder] (M1G1) (M1G2) 1st (2B) 5. [derivative] 12th (To be taught as a special skill) 6. cross product] To be taught as a special math topic 7. [ellipse] (MA2G4) 10th (2F) To be taught 8. [exponent] (M6A3) 6th (2A) 9. [four operations] (M1N3) 1st (2A) 10. [graph] (S7CS6) 7th (6) 11. [height] (MKM1) K (2B) 12. [integration] 12th (To be taught as a special skill) 13. [logarithmic functions] (MA2A5) 10th (2E) (To be taught as a special skill) 14. [perimeter] (M3M3) (M3M4) 3rd (2B) 15. [Pythagorean Theorem] (M8G2) 8th (2B) 16. [prism] (M6G2) 6th (2B) 17. [radius] (M3G1) 3rd (2B) 18. [ratio] (M6A1) 6th (2A) 19. [sigma notation] (M6N1) 6th (1A) or (MA1A3) 9th (2E) 20. [square root] (M8N1) 8th (2A) 21. [triangle] (M5M1) 5th (2B) 22. [trigonometric functions] (MA2G2) 10th (2F) 23. [unit conversion] (M6M1) 6th (2C) 24. [volume] (M5M4) 5th (1B) (M6M3) 6th (2B) (MA1G5) 9th (2F)

1. [absolute temperature] (SP3) 9th (3B) To be taught 2. [acceleration] (S8P3) 8th (3C) 3. [Dimensional Analysis] Special topics from 7.1 to be taught 4. [density] (S6E5) 6th (4A) 5. [energy] (SP3) 9th (3B) 6. [force] (S4P3) 4th (3A) or (S8P3) 8th (3C) 7. [friction] (S8P3) 8th (3A) To be taught 8. [gas/liquid] (SPS5) 9th (3B) To be taught 9. [graph] (S7CS6) 7th (6) 10. [gravity] (S6E1) 6th (3A) 11. [heat] (S2P2) 2nd (3A) 12. [Ideal Gas Law] Post-secondary to be taught 13. [intermolecular cohesive force] To be taught 14. [mass] (S8P3) 8th (3A) 15. [molecule] (S8P1) 8th (4A) 16. [momentum] (SP3) 9th (3B) 17. [Newton’s 1st, 2nd and 3rd Laws] (SP1) 9th (3C) To be taught 18. [potential energy] (SP3) 9th (3A) 19. [power] (SP3) 9th (3B) 20. [pressure] (SC5) 9th (4B) To be taught 21. [Reynolds Number] To be taught as special topic 22. [speed] (S2P3) 2nd (3A) 23. [speed of sound] (SPS9) 9th (3B) To be taught 24. [stress] To be taught 25. [temperature] (S3P1) 3rd (3A) and (SP3) 9th (3B) 26. [torque] Post-secondary To be taught 27. [velocity] (S8P3) 8th (3A) 28. [weight] (MKM1) K (2C) 29. [work] (S8P3) 8th (3A)














































172

List 2A. Calculus Based Fluid Mechanics Topics for Post-Secondary Engineering Education

Chapter/Section Page Nos. Chapter/Section Page Nos. Chapter 1 – Introduction Chapter 4 Fluid Kinematics 1.6 Viscosity 13-20 4.1 The Velocity Field Chapter 2 Fluid Statics 4.1.1 Eulerian and Lagrangian Flow Descriptions 2.1 Pressure at a Point 4.1.2 one-, Two-, and three-Dimensional Flows 2.2 Basic Equation for Pressure Field 4.1.3 Steady and Unsteady Flows 2.3 Pressure Variation in a Fluid Mechanics at Rest

38-42

4.1.4 Streamilnes, Streaklines, and Pathlines 2.8 Hydrostatic Force on a Plane Surface 57-63 4.2 The Acceleration Field 2.12 Pressure Variation in a Fluid Mechanics with Rigid-Body Motion 4.2.1 The Material Derivative 2.12.1 Linear Motion 4.2.2 Unsteady Effects 2.12.2 Rigid-Body Rotation

73-78

4.2.3 Convective Effects 4.2.4 Streamline Coordinates

150-168

Chapter 3 Elementary Fluid Dynamics – The Bernoulli Equation 4.4 The Reynolds Transport Theorem

3.2 F = ma along a Streamline (Continued) 4.4.1 Derivation of the Reynolds Transport Theorem 3.3 F = ma Normal to a Streamline

97-104 4.4.2 Physical Interpretation

3.8.2 Unsteady Effects 131-132 4.4.3 Relationship to Material Derivative Chapter 5 Finite Control Volume Analysis 4.4.4 Steady Effects 5.1 Conservation of Mass – The Continuity Equation 4.4.5 Unsteady Effects 5.1.1 Derivation of the Continuity Equation

193-195 4.4.6 Moving Control Volumes

170-182

5.1.3 Moving, Non-deforming Control Volume Chapter 6 Differential Analysis of Fluid Flow 5.1.4 Deforming Control Volume

200-205 6.1 Fluid Mechanics Element Kinematics

5.2.1 Derivation of the Linear Momentum Equation 6.1.1 Velocity and Acceleration Fields Revisited 5.2.2 Application of the Linear Momentum Equation 6.1.2 Linear Motion and Deformation 5.2.3 Derivation of the Moment-of-Momentum Equation 6.1.3 Angular Motion and Deformation 5.2.4 Application of the Moment-of-Momentum Equation 6.2 Conservation of mass 5.3 First Law of Thermodynamics – The Energy Equation 6.2.1 Differential Survey Form of Continuity Equation 5.3.1 Derivation of the Energy Equation 6.2.2 Cylindrical Polar Coordinates 5.3.2 Application of the Energy Equation

205-236

6.2.3 The Stream Function 5.4 Second Law of Thermodynamics – Irreversible Flow 6.3 Conservation of Linear Momentum

6.3.1 Description of Forces Acting on the Differential Element 5.4.1 Semi-infinitesimal Control Volume Statement of the Energy Equation 6.3.2 Equations of Motion

6.4 Inviscid Flow 5.4.2 Semi-infinitesimal Control Volume Statement of the Second Law of Thermodynamics 6.4.1 Euler’s Equations of Motion

6.4.2 The Bernoulli Equation 5.4.3 Combination of the Equations of the First and Second Laws of Thermodynamics

246-249

6.4.3 Irrotational Flow

272-334


173


Chapter/Section Page Nos. Chapter/Section Page Nos. Chapter 7 Similitude, Dimensional Analysis, and Modeling

Chapter 6 Differential Analysis of Fluid Flow (Continued)

7.1 Dimensional Analysis 6.4.4 The Bernoulli Equation for Irrotational Flow 7.2 Buckingham Pi Theorem 6.4.5The Velocity Potential 7.3 Determination of Pi Terms 6.5 Some Basic, Plane Potential Flows 7.4 Some Additional Comments about Dimensional Analysis 6.5.1 Uniform Flow 7.4.1 Selection of Variables 6.5.2 Source and Sink 7.4.2 Determination of Reference Dimensions 6.5.3 Vortex 7.4.3 Uniqueness of Pi Terms 6.5.4 Doublet 7.5 Determination of Pi Terms by Inspection 6.6 Superposition of Basic, Plane Potential Flows 7.6 Common Dimensionless Groups in Fluid Mechanics 6.6.1 Source in a Uniform Stream – Half-Body 7.7 Correlation of Experimental Data 6.6.2 Rankine Ovals 7.7.1 Problems with One Pi Term 6.6.3 Flow around a Circular Cylinder 7.7.2 Problems with Two or More Pi Term 6.7 Other Aspects of Potential Flow Analysis 7.8 Modeling and Similitude 6.8 Viscous Flow 7.8.1 Theory of Models 6.8.1 Stress-Deformation Relationships 7.8.2 Model Space 6.8.2 The Navier-Stokes Equations 7.8.3 Practical Aspects of Using Models 6.9 Some Simple Solutions for Viscous, Incompressible Fluids 7.9 Some Typical Model Studies 6.9.1 Steady, Laminar Flow between Fixed Parallel Plates 7.9.1 Flow through Closed Conduits 6.9.2 Couette Flow 7.9.2 Flow around Immersed Bodies 6.9.3 Steady, Laminar Flow in Circular Tubes 7.9.3 Flow with a Free Surface 6.9.4 Steady, Axial, Laminar Flow in an Annulus 7.10 Similitude Based on Governing Differential Equations 6.10 Other Aspects of Differential Analysis

6.10.1 Numerical Methods 7.11 Chapter Summary and Study Guide

346-391

Chapter Summary and Study Guide

↑

Chapter 8 Viscous Flow in Pipes 8.1 General Characteristics of Pipe Flow 8.3 Fully Developed Turbulent Flow 8.1.1 Laminar of Turbulent Flow 8.3.1 Transition from Laminar to Turbulent Flow 8.1.2 Entrance Region and Fully Developed Flow 8.3.2 Turbulent Shear Stress 8.1.3 Pressure and Shear Stress 8.3.3 Turbulent Velocity Profile 8.2 Fully Developed Laminar Flow 8.3.4 Turbulent Modeling 8.2.1 From F = ma Applied Directly to a Fluid Mechanics Element 8.3.5 Chaos and Turbulence

418-429

8.2.2 From the Navier-Stokes Equations 8.2.3 From Dimensional Analysis

401-415


174


Chapter/Section Page Nos. Chapter/Section Page Nos. Chapter 9 Flow over Immersed Bodies Chapter 10 Open Channel Flow 9.2 Boundary Layer Characteristics 10.3 Energy Considerations

10.3.2 Channel Depth Variations 573-574

9.2.1 Boundary Layer structure and Thickness on a Flat Plate 9.2.2 Prandtl/Blasius Boundary Layer Solution 9.2.3 Momentum Integral boundary Layer Equation for a Flat Plate 9.2.4 Transition from Laminar to Turbulent Flow 9.2.5 Turbulent Boundary Layer Flow 9.2.6 Effects of Pressure Gradient 9.2.7 Momentum Integral Boundary Layer Equation with Nonzero Pressure Gradient

493-518

Chapter 11 Compressible Flow 11.1 Ideal Gas Relationships 11.2 Mach Number and Speed of Sound

614-623

11.4 Isentropic Flow of an Ideal Gas 11.4.1 Effect of Variations in Flow Cross-Sectional Areas

628-631

11.5 Nonisentropic Flow of an Ideal Gas 11.5.1 Adiabatic Constant Area Duct Flow with Friction (Fanno Flow) 11.5.2 Frictionless Constant Area Duct Flow with Heat Transfer (Rayleigh Flow)

647-664


175

List 2B. Pre-Requisite Math and Science Topics to Be Reviewed Before Teaching the Calculus Portion of Fluid Mechanics Topics


Math Physics/Chemistry 1. [absolute value] (M7N1) 7th (2A) 2. [analytic geometry] Post-secondary 3. [analytic geometry: hyperbolic tangent] Post-secondary To be taught 4. [areas of geometric shapes: circle, triangle, etc.] (M5M1) 5th (2B) 5. [coordinate system] (M4G3) 4th (2B) 6. [cross product] To be taught as a special math topic 7. [cylinder] (M1G1) (M1G2) 1st (2B) 8. [derivative] 12th and [partial derivative] Post-Secondary 9. [dot product] To be taught as a special math topic 10. [ellipse] (MA2G4) 10th (2F) To be taught 11. [Eulerian method] Post-secondary 12. [exponent] (M6A3) 6th (2A) 13. [four operations] (M1N3) 1st (2A) 14. [functions] (MA1A1) 9th (2E) and others Post-secondary 15. [gradient “del”] Post-Secondary 16. [graph] (S7CS6) 7th (6) 17. [height] (MKM1) K (2B) 18. [integration] 12th (To be taught as a special skill) 19. [Lagrangian method] Post-secondary 20. [limit] Post-secondary 21. [logarithmic functions] (MA2A5) 10th (2E) 22. [perimeter] (M3M3) (M3M4) 3rd (2B) 23. [Pythagorean Theorem] (M8G2) 8th (2B) 24. [prism] (M6G2) 6th (2B) 25. [radius] (M3G1) 3rd (2B) 26. [ratio] (M6A1) 6th (2A) 27. [sigma notation] (M6N1) 6th (1A) or (MA1A3) 9th (2E) 28. [square root] (M8N1) 8th (2A) 29. [surface] (M6M4) 6th (2B) 30. [3rd order non-linear differential equation] Post-secondary 31. [triangle] (M5M1) 5th (2B) and [trigonometric functions] (MA2G2) 10th (2F) 32. [unit conversion] (M6M1) 6th (2C) 33. [vector] (MA3A10) 11th (2H) To be taught as a special math topics 34. [volume] (M5M4) 5th (1B) (M6M3) 6th (2B) (MA1G5) 9th (2F)

1. [absolute temperature] (SP3) 9th (3B) To be taught 2. [acceleration] (S8P3) 8th (3C) 3. [Dimensional Analysis] Special topics from 7.1 to be taught 4. [density] (S6E5) 6th (4A) 5. [energy] (SP3) 9th (3B) 6. [entropy] Post-secondary To be taught 7. [1st moment of the area] To be taught 8. [force] (S4P3) 4th (3A) or (S8P3) 8th (3C) 9. [friction] (S8P3) 8th (3A) To be taught 10. [gas/liquid] (SPS5) 9th (3B) To be taught 11. [graph] (S7CS6) 7th (6) 12. [gravity] (S6E1) 6th (3A) 13. [heat] (S2P2) 2nd (3A) 14. [Ideal Gas Law] Post-secondary to be taught 15. [intermolecular cohesive force] To be taught 16. [mass] (S8P3) 8th (3A) 17. [molecule] (S8P1) 8th (4A) 18. [momentum] (SP3) 9th (3B) 19. [Newton’s 1st, 2nd and 3rd Laws] (SP1) 9th (3C) To be taught 20. [potential energy] (SP3) 9th (3A) 21. [power] (SP3) 9th (3B) 22. [pressure] (SC5) 9th (4B) To be taught 23. [Reynolds Number] To be taught as special topic 24. [2nd moment of the area] To be taught 25. [speed] (S2P3) 2nd (3A) 26. [speed of sound] (SPS9) 9th (3B) To be taught 27. [stress] To be taught 28. [temperature] (S3P1) 3rd (3A) and (SP3) 9th (3B) 29. [torque] Post-secondary To be taught 30. [velocity] (S8P3) 8th (3A) 31. [wave] (S8P4) 8th (3A) 32. [weight] (MKM1) K (2C) 33. [work] (S8P3) 8th (3A)



















































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