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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 (edwardnlocke@yahoo.com)
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
1
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).
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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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.
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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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:
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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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.
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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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
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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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.
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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Figure 2B. Notation for undergraduate level appropriate statics topics.
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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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;
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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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.
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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Figure 3B. Likert Scale (top) and symbols to be used for the expression of expert opinion and offer of advice.
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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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:
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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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.
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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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.
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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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=
[four operations] (M1N3) 1st (2A)
[temperature] (SP3) 9th (3B) [absolute temperature] (SP3) 9th (3B)
To be taught [density] (S6E5) 6th (4A)
9th
9th + PS
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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Fluid Mechanics Topic List (Continued). 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 (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)
[pressure] (SC5) 9th (4B) To be taught [density] (S6E5) 6th (4A)
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
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
16
Fluid Mechanics Topic List (Continued). 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 (Continued) 1.8 Vapor Pressure
N/A [intermolecular cohesive force] To be taught [momentum] (SP3) 9th (3B)
[pressure] (SC5) 9th (4B) To be taught
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
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
17
Fluid Mechanics Topic List (Continued). 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 2 Fluid Statics (Continued) 2.1 Pressure at a Point (Continued)
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
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
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
18
Fluid Mechanics Topic List (Continued). 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 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
[four operations] (M1N3) 1st (2A) [derivative] 12th (To be taught)
[partial derivative] Post-Secondary [gradient] Post-Secondary
[pressure] (SC5) 9th (4B) To be taught
PS
PS
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
19
Fluid Mechanics Topic List (Continued). 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 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.
[pressure] (SC5) 9th (4B) To be taught
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
[four operations] (M1N3) 1st (2A) [exponent] (M6A3) 6th (2A)
[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
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
20
Fluid Mechanics Topic List (Continued). 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 2 Fluid Statics (Continued) 2.4 Standard Atmosphere
βββ
Rg
aaa T
zppzTT ⎟⎟⎠
⎞⎜⎜⎝
⎛−=−= 1
[four operations] (M1N3) 1st (2A) [exponent] (M6A3) 6th (2A)
[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)
[pressure] (SC5) 9th (4B) To be taught
9th
2.6.1 Piezometer Tube 110 hpphp A γγ =+=
[four operations] (M1N3) 1st (2A) [height] (MKM1) K (2B)
[pressure] (SC5) 9th (4B) To be taught
9th
2.6.2 U-Tube Manometer
113322
332211
221122
22110 0
hhhppphhhphphhphhpphp
BA
BA
AA
A
γγγγγγ
γγγγγγ
+−+=−→=−−+
=−=→=−++=
[four operations] (M1N3) 1st (2A)
[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)
[pressure] (SC5) 9th (4B) To be taught
2.7 Mechanical and Electronic Pressure Measuring Devices N/A N/A
9th
9th + PS
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
21
Fluid Mechanics Topic List (Continued). 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 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)
[integration] 12th (To be taught as a special skill)
[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
[four operations] (M1N3) 1st (2A)
[Pythagorean Theorem] (M8G2) 8th (2B)
[force] (S4P3) 4th (3A) or (S8P3) 8th (3C) 9th
9th + PS
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
22
Fluid Mechanics Topic List (Continued). 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 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
[pressure] (SC5) 9th (4B) To be taught [density] (S6E5) 6th (4A)
PS
2.12.1 Linear Motion
( )
( )
( )zz
y
zy
zy
agdzdp
aga
dydz
dzagdyadp
dzzpdy
ypdpag
zpa
yp
+−=+
−=
+−−=∂∂
+∂∂
=+−=∂∂
−=∂∂
ρ
ρρ
ρρ
[four operations] (M1N3) 1st (2A) [derivative] 12th (To be taught)
[partial derivative] Post-secondary
[pressure] (SC5) 9th (4B) To be taught [density] (S6E5) 6th (4A)
PS
9th + PS
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
23
Fluid Mechanics Topic List (Continued). 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 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
[four operations] (M1N3) 1st (2A) [derivative] 12th (To be taught)
[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=
does not need calculus
[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
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
24
Fluid Mechanics Topic List (Continued). 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 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
[four operations] (M1N3) 1st (2A) [trigonometric functions] (MA2G2) 10th (2F)
[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
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
25
Fluid Mechanics Topic List (Continued). 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 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)
[force] (S4P3) 4th (3A) or (S8P3) 8th (3C) [gravity] (S6E1) 6th (3A) [mass] (S8P3) 8th (3A)
[acceleration] (S8P3) 8th (3C)
PS 9th
+ PS
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
26
Fluid Mechanics Topic List (Continued). 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 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
[four operations] (M1N3) 1st (2A) [trigonometric functions] (MA2G2) 10th (2F)
[partial derivative] Post-secondary [radius] (M3G1) 3rd (2B)
[force] (S4P3) 4th (3A) or (S8P3) 8th (3C) [gravity] (S6E1) 6th (3A) [mass] (S8P3) 8th (3A)
[acceleration] (S8P3) 8th (3C)
PS 9th
+ PS
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
27
Fluid Mechanics Topic List (Continued). 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 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
[four operations] (M1N3) 1st (2A) [integration] 12th (To be taught)
Note: The main formula
streamlinea onconstant 2
p
streamline thealongConstant 21
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
streamlinea alongconstant 21
21
ppV
Vppppp
Vp
pzVp
Vpp
T
−=
=−→⎭⎬⎫
==+=
==++
+=
[four operations] (M1N3) 1st (2A) [square root] (M8N1) 8th (2A)
[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 ++=++
[four operations] (M1N3) 1st (2A) [exponent] (M6A3) 6th (2A)
[pressure] (SC5) 9th (4B) To be taught [density] (S6E5) 6th (4A) [speed] (S2P3) 2nd (3A)
9th
9th
+ PS
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
28
Fluid Mechanics Topic List (Continued). 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 3 Elementary Fluid Dynamics – The Bernoulli Equation (Continued) 3.6.1 Free Jets
( )⎪⎩
⎪⎨
⎧
+=
==→=
HhgV
ghhVVh
2
22
21 2 ρ
γργ
[four operations] (M1N3) 1st (2A) [exponent] (M6A3) 6th (2A)
[square root] (M8N1) 8th (2A)
[pressure] (SC5) 9th (4B) To be taught [density] (S6E5) 6th (4A) [speed] (S2P3) 2nd (3A)
9th
3.6.2 Confined Flows 212211222111 QQVAVAVAVA =→=→= ρρ
[four operations] (M1N3) 1st (2A) [exponent] (M6A3) 6th (2A)
[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
[four operations] (M1N3) 1st (2A) [exponent] (M6A3) 6th (2A)
[square root] (M8N1) 8th (2A) [areas of geometric shapes] (M5M1) 5th (2B)
[density] (S6E5) 6th (4A) [speed] (S2P3) 2nd (3A)
[pressure] (SC5) 9th (4B) To be taught [gravity] (S6E1) 6th (3A)
9th
3.7 The Energy Line and the Hydraulic Grade Line
Hzg
V==++ streamlinea onconstant
2
2
γρ
[four operations] (M1N3) 1st (2A) [exponent] (M6A3) 6th (2A)
[square root] (M8N1) 8th (2A)
[density] (S6E5) 6th (4A) [speed] (S2P3) 2nd (3A) [gravity] (S6E1) 6th (3A)
9th
9th
+ PS
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
29
Fluid Mechanics Topic List (Continued). 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 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
[four operations] (M1N3) 1st (2A) [exponent] (M6A3) 6th (2A)
[logarithmic functions] (MA2A5) 10th (2E) (To be taught as a special skill)
[integration] 12th (To be taught as a special skill)
Note: The main formulas
RTpgzV
pdpRT ==++∫ ρconstant
21 2
And others do not need calculus
[density] (S6E5) 6th (4A) [speed] (S2P3) 2nd (3A) [gravity] (S6E1) 6th (3A)
9th 9th
+ PS
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
30
Fluid Mechanics Topic List (Continued). 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 3 Elementary Fluid Dynamics – The Bernoulli Equation (Continued) 3.8.2 Unsteady Effects
( )
)streamlinea along(21
21
)streamlinea along(021
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)
[pressure] (SC5) 9th (4B) To be taught [density] (S6E5) 6th (4A) [speed] (S2P3) 2nd (3A)
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
[four operations] (M1N3) 1st (2A)
[pressure] (SC5) 9th (4B) To be taught [density] (S6E5) 6th (4A) [speed] (S2P3) 2nd (3A)
9th
9th
+ PS
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
31
Fluid Mechanics Topic List (Continued). 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 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
[four operations] (M1N3) 1st (2A) [coordinate system] (M4G3) 4th (2B)
[velocity] (S8P3) 8th (3A)
PS
4.1.3 Steady and Unsteady Flows
0=∂∂
tVr
[four operations] (M1N3) 1st (2A) [partial derivative] Post-secondary
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
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
32
Fluid Mechanics Topic List (Continued). 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 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
[partial derivative] Post-secondary
[velocity] (S8P3) 8th (3A) [acceleration] (S8P3) 8th (3C)
PS PS
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
33
Fluid Mechanics Topic List (Continued). 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 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
[velocity] (S8P3) 8th (3A)
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
[velocity] (S8P3) 8th (3A) [acceleration] (S8P3) 8th (3C)
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
[velocity] (S8P3) 8th (3A) [acceleration] (S8P3) 8th (3C)
PS
9th + PS
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
34
Fluid Mechanics Topic List (Continued). 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 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
[velocity] (S8P3) 8th (3A) [acceleration] (S8P3) 8th (3C)
PS
4.3 Control Volume and System Representations ( )dtmvdF =
[four operations] (M1N3) 1st (2A) [volume] (M5M4) 5th (1B) (M6M3) 6th (2B)
(MA1G5) 9th (2F) [areas of geometric shapes] (M5M1) 5th (2B)
[velocity] (S8P3) 8th (3A)
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
[four operations] (M1N3) 1st (2A) [integration] 12th (To be taught)
[velocity] (S8P3) 8th (3A) [acceleration] (S8P3) 8th (3C)
[mass] (S8P3) 8th (3A) [temperature] (SP3) 9th (3B)
[momentum] (SP3) (3B)
9th
9th + PS
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
35
Fluid Mechanics Topic List (Continued). 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 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
[four operations] (M1N3) 1st (2A) [integration] 12th (To be taught)
[velocity] (S8P3) 8th (3A) [acceleration] (S8P3) 8th (3C)
[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)
[volume] (M5M4) 5th (1B) (M6M3) 6th (2B) (MA1G5) 9th (2F)
[dot product] To be taught as a special math topic
[analytic geometry] 12th (To be taught) [integration] 12th (To be taught)
[velocity] (S8P3) 8th (3A) [acceleration] (S8P3) 8th (3C)
[mass] (S8P3) 8th (3A) [density] (S6E5) 6th (4A)
[temperature] (SP3) 9th (3B) [momentum] (SP3) (3B)
PS
9th + PS
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
36
Fluid Mechanics Topic List (Continued). 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 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)
[partial derivative] Post-secondary
[velocity] (S8P3) 8th (3A) [acceleration] (S8P3) 8th (3C)
PS
4.4.4 Steady Effects
∫ ⋅=sys
sys dAnVbDt
DBˆ
rρ
[four operations] (M1N3) 1st (2A) [analytic geometry] 12th (To be taught)
[dot product] To be taught as a special math topic
[partial derivative] Post-secondary
[velocity] (S8P3) 8th (3A) [density] (S6E5) 6th (4A)
[acceleration] (S8P3) 8th (3C)
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
[four operations] (M1N3) 1st (2A) [analytic geometry] 12th (To be taught)
[dot product] To be taught as a special math topic
[integration] 12th (To be taught) [partial derivative] Post-secondary
[velocity] (S8P3) 8th (3A) [density] (S6E5) 6th (4A)
[acceleration] (S8P3) 8th (3C) [momentum] (SP3) (3B)
PS
9th + PS
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
37
Fluid Mechanics Topic List (Continued). 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 4 Fluid Kinematics (Continued) 4.4.5 Unsteady Effects (Continued)
( )( )( )( ) ( )( )
0ˆˆ
ˆˆ
ˆˆˆ
section)(another ˆ
section) (oneˆ
CV) theof side the(along0ˆflow) (in0ˆ
flow)(out 0ˆˆ
ˆmomentum system0ˆ
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
[four operations] (M1N3) 1st (2A) [analytic geometry] 12th (To be taught)
[dot product] To be taught as a special math topic
To be taught as a special math topic [integration] 12th (To be taught)
[partial derivative] Post-secondary
[velocity] (S8P3) 8th (3A) [density] (S6E5) 6th (4A)
[acceleration] (S8P3) 8th (3C) [momentum] (SP3) (3B)
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
[velocity] (S8P3) 8th (3A) [density] (S6E5) 6th (4A)
[acceleration] (S8P3) 8th (3C) [momentum] (SP3) (3B)
PS
9th + PS
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
38
Fluid Mechanics Topic List (Continued). 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 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
[four operations] (M1N3) 1st (2A) [analytic geometry] 12th (To be taught)
[volume] (M5M4) 5th (1B) (M6M3) 6th (2B) (MA1G5) 9th (2F)
[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
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
39
Fluid Mechanics Topic List (Continued). 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 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
[four operations] (M1N3) 1st (2A) [analytic geometry] 12th (To be taught)
[volume] (M5M4) 5th (1B) (M6M3) 6th (2B) (MA1G5) 9th (2F)
[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)
Note: The main formula
∑ ∑=====
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)
[dot product] To be taught as a special math topic
[integration] 12th (To be taught)
[velocity] (S8P3) 8th (3A) [density] (S6E5) 6th (4A)
PS
5.1.4 Deforming Control Volume
cs
cvcscv
sys
VWV
Vdt
dAnWVdtDt
DM
rrr
r
+=
≠∂∂
=⋅+∂∂
= ∫∫∫ 00ˆ ρρρ
[four operations] (M1N3) 1st (2A) [analytic geometry] 12th (To be taught)
[volume] (M5M4) 5th (1B) (M6M3) 6th (2B) (MA1G5) 9th (2F)
[areas of geometric shapes] (M5M1) 5th (2B) [dot product] To be taught as a special math
topic [partial derivatives] Post-secondary
[velocity] (S8P3) 8th (3A) [density] (S6E5) 6th (4A)
PS
5.2 Newton’s Second Law – The Linear Momentum and Moment-of-Momentum Equation N/A
N/A N/A 9th
9th + PS
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
40
Fluid Mechanics Topic List (Continued). 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 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
[four operations] (M1N3) 1st (2A) [areas of geometric shapes] (M5M1) 5th (2B)
[vector] (MA3A10) 11th (2H) To be taught as a special math topics
[volume] (M5M4) 5th (1B) (M6M3) 6th (2B) (MA1G5) 9th (2F)
[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]
[velocity] (S8P3) 8th (3A) [force] (S4P3) 4th (3A) or (S8P3) 8th (3C)
[density] (S6E5) 6th (4A)
PS
5.2.2 Application of the Linear Momentum Equation
( ) ( )
( )
( )
∑∫∫
∫∫∫
∫
∑∫∫
∑∫∫
∫∫∫
=⋅
=⋅
⋅+⋅=⋅+
→+∂∂
=⋅+++∂∂
=⋅+∂∂
⋅+∂∂
=
volumecontrol theofcontent
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
[four operations] (M1N3) 1st (2A) [areas of geometric shapes] (M5M1) 5th (2B)
[volume] (M5M4) 5th (1B) (M6M3) 6th (2B) (MA1G5) 9th (2F)
[vector] (MA3A10) 11th (2H) To be taught as a special math topics
[dot product] To be taught as a special math topic
[analytic geometry] 12th (To be taught) [integration] 12th (To be taught as a special
skill) [derivative] 12th (To be taught)
[partial derivative] Post-secondary
[velocity] (S8P3) 8th (3A) [force] (S4P3) 4th (3A) or (S8P3) 8th (3C)
[density] (S6E5) 6th (4A)
PS
9th + PS
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
41
Fluid Mechanics Topic List (Continued). 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 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
[four operations] (M1N3) 1st (2A) [areas of geometric shapes] (M5M1) 5th (2B)
[volume] (M5M4) 5th (1B) (M6M3) 6th (2B) (MA1G5) 9th (2F)
[vector] (MA3A10) 11th (2H) To be taught as a special math topics
[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)
[partial derivative] Post-secondary [sigma notation] (M6N1) 6th (1A) or (MA1A3)
9th (2E)
[velocity] (S8P3) 8th (3A) [force] (S4P3) 4th (3A) or (S8P3) 8th (3C)
[density] (S6E5) 6th (4A)
PS 9th + PS
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
42
Fluid Mechanics Topic List (Continued). 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 5 Finite Control Volume Analysis (Continued) 5.2.4 Application of the Moment-of-Momentum Equation
( ) ( )
( ) ( ) ( )( )
( )[ ]
( )( ) ( )( )( ) ( )
( )( ) ( )( ) ( )outoutininshaftoutin
outoutoutinininshaft
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
[four operations] (M1N3) 1st (2A) [areas of geometric shapes] (M5M1) 5th (2B)
[volume] (M5M4) 5th (1B) (M6M3) 6th (2B) (MA1G5) 9th (2F)
[vector] (MA3A10) 11th (2H) To be taught as a special math topics
[dot product] and [cross product] To be taught as a special math topics
[analytic geometry] 12th (To be taught) [integration] 12th (To be taught as a special
skill) [derivative] 12th (To be taught)
[partial derivative] Post-secondary
[velocity] (S8P3) 8th (3A) [force] (S4P3) 4th (3A) or (S8P3) 8th (3C)
[density] (S6E5) 6th (4A)
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
[four operations] (M1N3) 1st (2A) [areas of geometric shapes] (M5M1) 5th (2B)
[volume] (M5M4) 5th (1B) (M6M3) 6th (2B) (MA1G5) 9th (2F)
[analytic geometry] 12th (To be taught) [integration] 12th (To be taught as a special
skill) [derivative] 12th (To be taught)
[partial derivative] Post-secondary [sigma notation] (M6N1) 6th (1A) or (MA1A3)
9th (2E)
[density] (S6E5) 6th (4A) [heat] (S2P2) 2nd (3A)
PS
PS
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
43
Fluid Mechanics Topic List (Continued). 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 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
[four operations] (M1N3) 1st (2A) [areas of geometric shapes] (M5M1) 5th (2B)
[volume] (M5M4) 5th (1B) (M6M3) 6th (2B) (MA1G5) 9th (2F)
[analytic geometry] 12th (To be taught) [integration] 12th (To be taught as a special
skill) [derivative] 12th (To be taught)
[partial derivative] Post-secondary [sigma notation] (M6N1) 6th (1A) or (MA1A3)
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
[four operations] (M1N3) 1st (2A) [areas of geometric shapes] (M5M1) 5th (2B)
[volume] (M5M4) 5th (1B) (M6M3) 6th (2B) (MA1G5) 9th (2F)
[analytic geometry] 12th (To be taught) [integration] 12th (To be taught as a special
skill)
[velocity] (S8P3) 8th (3A) [gravity] (S6E1) 6th (3A) [density] (S6E5) 6th (4A) [mass] (S8P3) 8th (3A)
PS
9th + PS
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
44
Fluid Mechanics Topic List (Continued). 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 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
[four operations] (M1N3) 1st (2A) [areas of geometric shapes] (M5M1) 5th (2B)
[volume] (M5M4) 5th (1B) (M6M3) 6th (2B) (MA1G5) 9th (2F)
[analytic geometry] 12th (To be taught) [integration] 12th (To be taught as a special skill)
[velocity] (S8P3) 8th (3A) [gravity] (S6E1) 6th (3A) [density] (S6E5) 6th (4A) [mass] (S8P3) 8th (3A)
PS 9th + PS
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
45
Fluid Mechanics Topic List (Continued). 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 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
quugzVpgzVp
QzzgVVppuum
ρρ
ρ
γρ
ρ
γρ
ργργγργρ
ρρ
ρρ
&
&
&&
[four operations] (M1N3) 1st (2A) [areas of geometric shapes] (M5M1) 5th (2B)
[volume] (M5M4) 5th (1B) (M6M3) 6th (2B) (MA1G5) 9th (2F)
[velocity] (S8P3) 8th (3A) [gravity] (S6E1) 6th (3A) [density] (S6E5) 6th (4A) [mass] (S8P3) 8th (3A)
9th 9th + PS
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
46
Fluid Mechanics Topic List (Continued). 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 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
[four operations] (M1N3) 1st (2A) [areas of geometric shapes] (M5M1) 5th (2B)
[volume] (M5M4) 5th (1B) (M6M3) 6th (2B) (MA1G5) 9th (2F)
[velocity] (S8P3) 8th (3A) [gravity] (S6E1) 6th (3A) [density] (S6E5) 6th (4A) [mass] (S8P3) 8th (3A)
9th 9th + PS
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
47
Fluid Mechanics Topic List (Continued). 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 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
[four operations] (M1N3) 1st (2A) [areas of geometric shapes] (M5M1) 5th (2B)
[volume] (M5M4) 5th (1B) (M6M3) 6th (2B) (MA1G5) 9th (2F)
[integration] 12th (To be taught as a special skill)
Note: The main formulas
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.
[velocity] (S8P3) 8th (3A) [gravity] (S6E1) 6th (3A) [density] (S6E5) 6th (4A) [mass] (S8P3) 8th (3A)
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
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
48
Fluid Mechanics Topic List (Continued). 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 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
&&
[four operations] (M1N3) 1st (2A) [volume] (M5M4) 5th (1B) (M6M3) 6th (2B)
(MA1G5) 9th (2F) [integration] 12th (To be taught as a special
skill) [derivative] 12th (To be taught)
[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
&&&
[four operations] (M1N3) 1st (2A) [volume] (M5M4) 5th (1B) (M6M3) 6th (2B)
(MA1G5) 9th (2F) [derivative] 12th (To be taught)
[partial derivative] Post secondary [integration] 12th (To be taught)
[sigma notation] (M6N1) 6th (1A) or (MA1A3) 9th (2E)
[heat] (S2P2) 2nd (3A) [velocity] (S8P3) 8th (3A) [density] (S6E5) 6th (4A)
[temperature] (SP3) 9th (3B)
PS
9th + PS
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
49
Fluid Mechanics Topic List (Continued). 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 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
[four operations] (M1N3) 1st (2A) [volume] (M5M4) 5th (1B) (M6M3) 6th (2B)
(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)
[temperature] (SP3) 9th (3B)
PS 9th + PS
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
50
Fluid Mechanics Topic List (Continued). 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 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
[four operations] (M1N3) 1st (2A) [exponent] (M6A3) 6th (2A)
[derivative] 12th (To be taught) [partial derivative] Post secondary
[integration] 12th (To be taught)
Note: The main formula
1
21
1
12
22
2
2
2121gzVp
kkgzVp
kk
++−
=++− ρρ
is not based on calculus
[velocity] (S8P3) 8th (3A) [density] (S6E5) 6th (4A)
[pressure] (SC5) 9th (4B) To be taught [gravity] (S6E1) 6th (3A)
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
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
51
Fluid Mechanics Topic List (Continued). 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 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
[four operations] (M1N3) 1st (2A) [coordinate system] (M4G3) 4th (2B)
[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
[four operations] (M1N3) 1st (2A) [partial derivative] Post-secondary
[pressure] (SC5) 9th (4B) To be taught
PS
9th + PS
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
52
Fluid Mechanics Topic List (Continued). 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 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
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
53
Fluid Mechanics Topic List (Continued). 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 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
[four operations] (M1N3) 1st (2A) [trigonometric functions] (MA2G2) 10th (2F)
[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
[four operations] (M1N3) 1st (2A) [exponent] (M6A3) 6th (2A)
[force] (S4P3) 4th (3A) or (S8P3) 8th (3C) [velocity] (S8P3) 8th (3A)
[pressure] (SC5) 9th (4B) To be taught [density] (S6E5) 6th (4A)
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+=
=⎟⎟⎠
⎞⎜⎜⎝
⎛+−⎟⎟
⎠
⎞⎜⎜⎝
⎛++++=++
[four operations] (M1N3) 1st (2A) [trigonometric functions] (MA2G2) 10th (2F)
[pressure] (SC5) 9th (4B) To be taught [gravity] (S6E1) 6th (3A)
9th
9th + PS
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
54
Fluid Mechanics Topic List (Continued). 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 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)
[logarithmic functions] (MA2A5) 10th (2E) (To be taught as a special skill)
[integration] 12th (To be taught as a special skill)
[derivative] 12th (To be taught)
[velocity] (S8P3) 8th (3A) [density] (S6E5) 6th (4A)
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
[four operations] (M1N3) 1st (2A) [coordinate system] (M4G3) 4th (2B)
[logarithmic functions] (MA2A5) 10th (2E) [analytic geometry] 12th (To be taught)
[velocity] (S8P3) 8th (3A)
PS
9th + PS
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
55
Fluid Mechanics Topic List (Continued). 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 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)
[logarithmic functions] (MA2A5) 10th (2E) (To be taught as a special skill)
[exponent] (M6A3) 6th (2A) [square root] (M8N1) 8th (2A)
[graph] (S7CS6) 7th (6)
[velocity] (S8P3) 8th (3A) [pressure] (SC5) 9th (4B) To be taught
[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
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
56
Fluid Mechanics Topic List (Continued). 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 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
[four operations] (M1N3) 1st (2A) [exponent] (M6A3) 6th (2A)
[areas of geometric shapes] (M5M1) 5th (2B)
[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
[four operations] (M1N3) 1st (2A) [exponent] (M6A3) 6th (2A)
[square root] (M8N1) 8th (2A) [areas of geometric shapes] (M5M1) 5th (2B)
[velocity] (S8P3) 8th (3A) [pressure] (SC5) 9th (4B) To be taught
[gravity] (S6E1) 6th (3A) [density] (S6E5) 6th (4A)
9th
8.7 Chapter Summary and Study Guide N/A
N/A N/A 9th
9th + PS
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
57
Fluid Mechanics Topic List (Continued). 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 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)
Note: The main formulas
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
[four operations] (M1N3) 1st (2A) [exponent] (M6A3) 6th (2A)
[areas of geometric shapes] (M5M1) 5th (2B) [integration] 12th (To be taught as a special
skill)
[velocity] (S8P3) 8th (3A) [density] (S6E5) 6th (4A)
PS
9th + PS
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
58
Fluid Mechanics Topic List (Continued). 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 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⎟⎠⎞
⎜⎝⎛=
⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛=∞→→
===∂∂
=∂∂
+∂∂
=∂∂
+∂∂
→⎪⎭
⎪⎬⎫
∂∂
<<∂∂<<
=∂∂
+∂∂
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
+∂∂
−=∂∂
+∂∂
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
+∂∂
−=∂∂
+∂∂
[four operations] (M1N3) 1st (2A) [exponent] (M6A3) 6th (2A)
[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
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
59
Fluid Mechanics Topic List (Continued). 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 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
[four operations] (M1N3) 1st (2A) [exponent] (M6A3) 6th (2A)
[dot product] To be taught as a special math topic
[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
[velocity] (S8P3) 8th (3A) [density] (S6E5) 6th (4A) [gravity] (S6E1) 6th (3A)
PS 9th + PS
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
60
Fluid Mechanics Topic List (Continued). 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 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
[four operations] (M1N3) 1st (2A) [exponent] (M6A3) 6th (2A)
[dot product] To be taught as a special math topic
[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
[velocity] (S8P3) 8th (3A) [density] (S6E5) 6th (4A) [gravity] (S6E1) 6th (3A)
PS
9.2.7 Momentum Integral Boundary Layer Equation with Nonzero Pressure Gradient
( )constant
*2
==
+=−=
UUdx
dUUOU
dxd
dxdU
Udxdp
fs
fsfsfsw
fsfs ρδρτρ
[four operations] (M1N3) 1st (2A) [exponent] (M6A3) 6th (2A)
[partial derivative] Post-secondary
[velocity] (S8P3) 8th (3A) [density] (S6E5) 6th (4A)
PS
9.3 Drag
( )lr
εφρ
,,Re,,
21 2
FrMashapeCAU
DC DD ==
[four operations] (M1N3) 1st (2A) [areas of geometric shapes] (M5M1) 5th (2B)
[force] (S4P3) 4th (3A) or (S8P3) 8th (3C) [density] (S6E5) 6th (4A)
9th
9.3.1 Friction Drag
Dff CbUD lr
2
21 ρ=
[four operations] (M1N3) 1st (2A) [areas of geometric shapes] (M5M1) 5th (2B)
[force] (S4P3) 4th (3A) or (S8P3) 8th (3C) [density] (S6E5) 6th (4A)
9th
9th + PS
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
61
Fluid Mechanics Topic List (Continued). 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 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
[four operations] (M1N3) 1st (2A) [areas of geometric shapes] (M5M1) 5th (2B)
[integration] 12th (To be taught as a special skill)
[force] (S4P3) 4th (3A) or (S8P3) 8th (3C) [density] (S6E5) 6th (4A)
9th
9.4 Lift 9.4.1 Surface Pressure Distribution
( )lr
εφρ
,,Re,,
21 2
FrMashapeCAU
LC LL ==
9.4.2 Circulation N/A
[four operations] (M1N3) 1st (2A) [areas of geometric shapes] (M5M1) 5th (2B)
[force] (S4P3) 4th (3A) or (S8P3) 8th (3C) [density] (S6E5) 6th (4A)
9th
9.5 Chapter Summary and Study Guide N/A
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
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
62
Fluid Mechanics Topic List (Continued). 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 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
[four operations] (M1N3) 1st (2A) [square root] (M8N1) 8th (2A)
[trigonometric functions] (MA2G2) 10th (2F) [derivative] 12th (To be taught)
[velocity] (S8P3) 8th (3A) [speed] (S2P3) 2nd (3A)
[gravity] (S6E1) 6th (3A)
9th 9th + PS
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
63
Fluid Mechanics Topic List (Continued). 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 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
[four operations] (M1N3) 1st (2A) [square root] (M8N1) 8th (2A)
[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)
[gravity] (S6E1) 6th (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
[four operations] (M1N3) 1st (2A) [exponent] (M6A3) 6th (2A)
[velocity] (S8P3) 8th (3A) [gravity] (S6E1) 6th (3A)
[potential energy] (SP3) 9th (3A)
9th
9th + PS
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
64
Fluid Mechanics Topic List (Continued). 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 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
[four operations] (M1N3) 1st (2A) [derivative] 12th (To be taught)
[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
−−
==
−==−=−=
−=+++=
++=⎟⎟⎠
⎞⎜⎜⎝
⎛++=
==+=
[four operations] (M1N3) 1st (2A) [derivative] 12th (To be taught)
[velocity] (S8P3) 8th (3A) [gravity] (S6E1) 6th (3A)
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
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
65
Fluid Mechanics Topic List (Continued). 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 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
[four operations] (M1N3) 1st (2A) [exponent] (M6A3) 6th (2A)
[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
[four operations] (M1N3) 1st (2A) [exponent] (M6A3) 6th (2A)
[square root] (M8N1) 8th (2A)
[force] (S4P3) 4th (3A) or (S8P3) 8th (3C) [velocity] (S8P3) 8th (3A) [gravity] (S6E1) 6th (3A)
[pressure] (SC5) 9th (4B) To be taught
9th
9th + PS
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
66
Fluid Mechanics Topic List (Continued). 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 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
[four operations] (M1N3) 1st (2A) [exponent] (M6A3) 6th (2A)
[square root] (M8N1) 8th (2A)
[force] (S4P3) 4th (3A) or (S8P3) 8th (3C) [velocity] (S8P3) 8th (3A) [gravity] (S6E1) 6th (3A)
[pressure] (SC5) 9th (4B) To be taught
9th 9th + PS
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
67
Fluid Mechanics Topic List (Continued). 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 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
[four operations] (M1N3) 1st (2A) [exponent] (M6A3) 6th (2A)
[square root] (M8N1) 8th (2A) [trigonometric functions] (MA2G2) 10th (2F)
[integration] 12th (To be taught as a special skill)
Nore: The main formulas are not based on calculus.
[velocity] (S8P3) 8th (3A) [gravity] (S6E1) 6th (3A)
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
=→=−→⎪⎭
⎪⎬⎫
=
==
=−
=−++=++
[four operations] (M1N3) 1st (2A) [exponent] (M6A3) 6th (2A)
[square root] (M8N1) 8th (2A)
[velocity] (S8P3) 8th (3A) [gravity] (S6E1) 6th (3A)
9th
9th + PS
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
68
Fluid Mechanics Topic List (Continued). 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 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
+=
⎟⎠⎞
⎜⎝⎛=⎟
⎠⎞
⎜⎝⎛=
→====
[four operations] (M1N3) 1st (2A) [exponent] (M6A3) 6th (2A)
[square root] (M8N1) 8th (2A)
[velocity] (S8P3) 8th (3A) [gravity] (S6E1) 6th (3A)
9th
10.6.4 Underflow Gates 12gyaCq d=
[four operations] (M1N3) 1st (2A) [exponent] (M6A3) 6th (2A)
[square root] (M8N1) 8th (2A)
[velocity] (S8P3) 8th (3A) [gravity] (S6E1) 6th (3A)
9th
10.7 Chapter Summary and Study Guide N/A
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
[four operations] (M1N3) 1st (2A) [volume] (M5M4) 5th (1B) (M6M3) 6th (2B)
(MA1G5) 9th (2F) [functions] (MA1A1) 9th (2E) and others
Post-secondary [integration] 12th (To be taught as a special
skill) [partial derivative] Post-secondary
[Ideal Gas Law] Post-secondary to be taught [heat] (S2P2) 2nd (3A)
[temperature] (SP3) 9th (3B) [density] (S6E5) 6th (4A)
[pressure] (SC5) 9th (4B) To be taught
PS 9th + PS
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
69
Fluid Mechanics Topic List (Continued). 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 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
[four operations] (M1N3) 1st (2A) [volume] (M5M4) 5th (1B) (M6M3) 6th (2B)
(MA1G5) 9th (2F) [functions] (MA1A1) 9th (2E) and others
Post-secondary [integration] 12th (To be taught as a special
skill) [partial derivative] Post-secondary
[Ideal Gas Law] Post-secondary to be taught [heat] (S2P2) 2nd (3A)
[temperature] (SP3) 9th (3B) [density] (S6E5) 6th (4A)
[pressure] (SC5) 9th (4B) To be taught
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
[four operations] (M1N3) 1st (2A) [areas of geometric shapes] (M5M1) 5th (2B)
[exponent] (M6A3) 6th (2A) [square root] (M8N1) 8th (2A)
[partial derivative] Post-secondary
[speed of sound] (SPS9) 9th (3B) [pressure] (SC5) 9th (4B) To be taught
[density] (S6E5) 6th (4A) [velocity] (S8P3) 8th (3A)
PS
9th + PS
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
70
Fluid Mechanics Topic List (Continued). 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 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
[four operations] (M1N3) 1st (2A) [volume] (M5M4) 5th (1B) (M6M3) 6th (2B)
(MA1G5) 9th (2F) [functions] (MA1A1) 9th (2E) and others
Post-secondary [integration] 12th (To be taught as a special
skill) [partial derivative] Post-secondary
[Ideal Gas Law] Post-secondary to be taught [heat] (S2P2) 2nd (3A)
[temperature] (SP3) 9th (3B) [density] (S6E5) 6th (4A)
[pressure] (SC5) 9th (4B) To be taught
PS
11.3 Categories of Compressible Flow
( )MaV
ccttr wave1sin ==−= α
[four operations] (M1N3) 1st (2A) [trigonometric functions] (MA2G2) 10th (2F)
[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&
[four operations] (M1N3) 1st (2A) [areas of geometric shapes] (M5M1) 5th (2B)
[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
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
71
Fluid Mechanics Topic List (Continued). 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 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
[four operations] (M1N3) 1st (2A) [areas of geometric shapes] (M5M1) 5th (2B)
[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
[four operations] (M1N3) 1st (2A) [exponent] (M6A3) 6th (2A)
[square root] (M8N1) 8th (2A)
[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
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
72
Fluid Mechanics Topic List (Continued). 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 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
[four operations] (M1N3) 1st (2A) [exponent] (M6A3) 6th (2A)
[square root] (M8N1) 8th (2A)
[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
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
73
Fluid Mechanics Topic List (Continued). 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 11 Compressible Flow (Continued) 11.4.2 Converging-Diverging Duct Flow (Continued)
( )
( ) ( )
( )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
[four operations] (M1N3) 1st (2A) [exponent] (M6A3) 6th (2A)
[square root] (M8N1) 8th (2A)
[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
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
74
Fluid Mechanics Topic List (Continued). 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 11 Compressible Flow (Continued) 11.4.2 Converging-Diverging Duct Flow (Continued)
( )[ ]
( )[ ]
( )
( ) ( )
( )
( )[ ]( )[ ]
( ) ( )[ ]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
[four operations] (M1N3) 1st (2A) [exponent] (M6A3) 6th (2A)
[square root] (M8N1) 8th (2A)
[pressure] (SC5) 9th (4B) To be taught [density] (S6E5) 6th (4A) [velocity] (S8P3) 8th (3A)
[Ideal Gas Law] Post-secondary to be taught
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
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
75
Fluid Mechanics Topic List (Continued). 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 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
&&&
[four operations] (M1N3) 1st (2A) [areas of geometric shapes] (M5M1) 5th (2B)
[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
[velocity] (S8P3) 8th (3A) [density] (S6E5) 6th (4A)
[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
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
76
Fluid Mechanics Topic List (Continued). 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 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&
[four operations] (M1N3) 1st (2A) [areas of geometric shapes] (M5M1) 5th (2B)
[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
[velocity] (S8P3) 8th (3A) [density] (S6E5) 6th (4A)
[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
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
77
Fluid Mechanics Topic List (Continued). 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 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
[four operations] (M1N3) 1st (2A) [areas of geometric shapes] (M5M1) 5th (2B)
[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
[velocity] (S8P3) 8th (3A) [density] (S6E5) 6th (4A)
[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
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
78
Fluid Mechanics Topic List (Continued). 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 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&&
[four operations] (M1N3) 1st (2A) [exponent] (M6A3) 6th (2A)
[derivative] 12th + [square root] (M8N1) 8th (2A)
[velocity] (S8P3) 8th (3A) [density] (S6E5) 6th (4A)
[temperature] (SP3) 9th (3B) [pressure] (SC5) 9th (4B) To be taught
PS 9th + PS
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
79
Fluid Mechanics Topic List (Continued). 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 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
[four operations] (M1N3) 1st (2A) [exponent] (M6A3) 6th (2A)
[derivative] 12th + [square root] (M8N1) 8th (2A)
[velocity] (S8P3) 8th (3A) [density] (S6E5) 6th (4A)
[temperature] (SP3) 9th (3B) [pressure] (SC5) 9th (4B) To be taught
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
[four operations] (M1N3) 1st (2A) [exponent] (M6A3) 6th (2A)
[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
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
80
Fluid Mechanics Topic List (Continued). 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 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
[four operations] (M1N3) 1st (2A) [exponent] (M6A3) 6th (2A)
[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
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
81
Fluid Mechanics Topic List (Continued). 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 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
[four operations] (M1N3) 1st (2A) [exponent] (M6A3) 6th (2A)
[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
11.6 Analogy between Compressible and Open-Channel Flows
( ) 1constantcconstant
constant
−==
=====
koc
oc
ococ
oc
kybV
AVcVFrgyc
gyVFr
cVMa
ρ
ρ
[four operations] (M1N3) 1st (2A) [square root] (M8N1) 8th (2A)
[areas of geometric shapes] (M5M1) 5th (2B)
[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)
[velocity] (S8P3) 8th (3A)
9th
11.8 Chapter Summary and Study Guide N/A
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
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
82
Fluid Mechanics Topic List (Continued). 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 12 Turbomachines (Continued) 12.2 Basic Energy Considerations
rUUWV ω=+=rrr
[four operations] (M1N3) 1st (2A)
[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
[sigma notation] (M6N1) 6th (1A) or (MA1A3) 9th (2E)
[integration] 12th (To be taught as a special skill)
[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
[four operations] (M1N3) 1st (2A) [triangle] (M5M1) 5th (2B)
[trigonometric functions] (MA2G2) 10th (2F) [areas of geometric shapes] (M5M1) 5th (2B)
[velocity] (S8P3) 8th (3A) [density] (S6E5) 6th (4A) [gravity] (S6E1) 6th (3A)
9th
9th
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
83
Fluid Mechanics Topic List (Continued). 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 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
−=
=−=−
=
=−+−+−=
[four operations] (M1N3) 1st (2A) [triangle] (M5M1) 5th (2B)
[trigonometric functions] (MA2G2) 10th (2F) [areas of geometric shapes] (M5M1) 5th (2B)
[velocity] (S8P3) 8th (3A) [density] (S6E5) 6th (4A) [gravity] (S6E1) 6th (3A)
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
&
[four operations] (M1N3) 1st (2A) [areas of geometric shapes] (M5M1) 5th (2B)
[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
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
84
Fluid Mechanics Topic List (Continued). 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 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)
[velocity] (S8P3) 8th (3A) [density] (S6E5) 6th (4A)
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
WW
hh
shaft
shaft
a
a
shaft
shaft
a
a
ηη
ωω
ωω
ωω
&
&
&
&
[four operations] (M1N3) 1st (2A) [areas of geometric shapes: circle, triangle]
(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
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
85
Fluid Mechanics Topic List (Continued). 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 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
[four operations] (M1N3) 1st (2A) [areas of geometric shapes: circle, triangle]
(M5M1) 5th (2B) (M5M1) 5th (2B)
[ratio] (M6A1) 6th (2A)
[speed] (S2P3) 2nd (3A) [pressure] (SC5) 9th (4B) To be taught
[density] (S6E5) 6th (4A)
9th
12.8 Turbines 12.8.1 Impulse Turbines
( )( ) ( )( )
( )( )2
cos1
cos1cos1cos
2
max1
1112
22111
VUVUUmTW
VUrmTVUVVUWVUWVV
powershaftshaft
mshaft
=−−==
−−=−−=−+=+==
βω
βββ
θθ
θθ
&&
&
[four operations] (M1N3) 1st (2A) [trigonometric functions] (MA2G2) 10th (2F)
[derivative] 12th (To be taught)
[power] (SP3) 9th (3B) [speed] (S2P3) 2nd (3A)
[force] (S4P3) 4th (3A) or (S8P3) 8th (3C)
9th
9th
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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Fluid Mechanics Topic List (Continued). 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 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
`
=
=⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟
⎟⎠
⎞⎜⎜⎝
⎛ &&
&&
[four operations] (M1N3) 1st (2A) [square root] (M8N1) 8th (2A)
[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
12.10 Chapter Summary and Study Guide N/A
N/A N/A 9th
9th
THE END
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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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.
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
88
Figure 4A. Step-by-step procedures proposed for the review and validation of data.
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
89
Figure 4B. Likert Scale (top) and symbols to be used for the expression of expert opinion and offer of advice.
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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:
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
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.
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
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.
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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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)
Engineering Analytic Topics & Typical Formulas
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=
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
94
Fluid Mechanics Survey Form A (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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
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
95
Fluid Mechanics Survey Form A (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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.
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
96
Fluid Mechanics Survey Form A (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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
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
97
Fluid Mechanics Survey Form A (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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
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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Fluid Mechanics Survey Form A (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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
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
99
Fluid Mechanics Survey Form A (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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
does not need calculus
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
100
Fluid Mechanics Survey Form A (Continued)
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)
Engineering Analytic Topics & Typical Formulas
1 2 3 4 5
Comment
Chapter 3 Elementary Fluid Dynamics – The Bernoulli Equation (Continued) 3.4 Physical Interpretation
streamlinea onconstant 2
streamline theacrossconstant V
streamline thealongConstant 21
22
2
=++=+ℜ
+
=++
∫ zg
Vpzdnp
zVp
γγρ
γρ
Note: The main formulas
streamlinea onconstant 2
p
streamline thealongConstant 21
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
streamlinea alongconstant 21
21
ppVVppppp
Vp
pzVpVpp T
−==−→
⎭⎬⎫
==+=
==+++=
3.6 Examples of Use of the Bernoulli Equation
22
2212
11 21
21 zVpzVp γργρ ++=++
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
101
Fluid Mechanics Survey Form A (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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
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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Fluid Mechanics Survey Form A (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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
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
103
Fluid Mechanics Survey Form A (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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
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
104
Fluid Mechanics Survey Form A (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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
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
105
Fluid Mechanics Survey Form A (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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
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
106
Fluid Mechanics Survey Form A (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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
&
&
&&
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
107
Fluid Mechanics Survey Form A (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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
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
108
Fluid Mechanics Survey Form A (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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
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
109
Fluid Mechanics Survey Form A (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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
5.5 Chapter Summary and Study Guide N/A
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+=
=⎟⎟⎠
⎞⎜⎜⎝
⎛+−⎟⎟
⎠
⎞⎜⎜⎝
⎛++++=++
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
110
Fluid Mechanics Survey Form A (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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
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
111
Fluid Mechanics Survey Form A (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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
8.7 Chapter Summary and Study Guide N/A
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
112
Fluid Mechanics Survey Form A (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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=
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
113
Fluid Mechanics Survey Form A (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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
9.5 Chapter Summary and Study Guide N/A
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
114
Fluid Mechanics Survey Form A (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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
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
115
Fluid Mechanics Survey Form A (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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
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
116
Fluid Mechanics Survey Form A (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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
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
117
Fluid Mechanics Survey Form A (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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
+=⎟
⎠⎞
⎜⎝⎛=
⎟⎠⎞
⎜⎝⎛=→====
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
118
Fluid Mechanics Survey Form A (Continued)
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)
Engineering Analytic Topics & Typical Formulas
1 2 3 4 5
Comment
Chapter 10 Open Channel Flow (Continued) 10.6.4 Underflow Gates
12gyaCq d=
10.7 Chapter Summary and Study Guide N/A
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
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
119
Fluid Mechanics Survey Form A (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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
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
120
Fluid Mechanics Survey Form A (Continued)
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)
Engineering Analytic Topics & Typical Formulas
1 2 3 4 5
Comment
Chapter 11 Compressible Flow (Continued) 11.4.2 Converging-Diverging Duct Flow (Continued)
( )
( ) ( )
( )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
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
121
Fluid Mechanics Survey Form A (Continued)
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)
Engineering Analytic Topics & Typical Formulas
1 2 3 4 5
Comment
Chapter 11 Compressible Flow (Continued) 11.4.2 Converging-Diverging Duct Flow (Continued)
( )[ ] ( )[ ]
( )
( ) ( ) ( )
( )[ ]( )[ ]
( ) ( )[ ]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
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
122
Fluid Mechanics Survey Form A (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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
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
123
Fluid Mechanics Survey Form A (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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 =
11.8 Chapter Summary and Study Guide N/A
Chapter 12 Turbomachines 12.1 Introduction N/A
12.2 Basic Energy Considerations rUUWV ω=+=
rrr
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
124
Fluid Mechanics Survey Form A (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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
shaftshaftshaftshaft
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
−==−=
−==−+−+−=
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
125
Fluid Mechanics Survey Form A (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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 +−=+−= ∑
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
126
Fluid Mechanics Survey Form A (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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
WW
hh
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
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
127
Fluid Mechanics Survey Form A (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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
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
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)
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)
Engineering Analytic Topics & Typical Formulas
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
δδδρδδδγθδδδδ
δδδρθδδδδ
sincos2
22cos
2sin
szsyamFVmVzyx
azyxzyxsxpyxpF
azyxsxpzxpFamF
zz
zszz
ysyy
==↑=←==↑
=−−=
=−==
∑
∑
rr
rr
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
129
Fluid Mechanics Survey Form B (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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
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
130
Fluid Mechanics Survey Form B (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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
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
131
Fluid Mechanics Survey Form B (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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
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
132
Fluid Mechanics Survey Form B (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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 ⋅+⋅=⋅⋅
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
133
Fluid Mechanics Survey Form B (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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
3.8.2 Unsteady Effects
( )
)streamlinea along(21
21
)streamlinea along(021
22
2212
11
2
2
1
zVpdstVzVp
zVddpdstV
s
sγρργρ
γρρ
+++∂∂
=++
=+++∂∂
∫
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
134
Fluid Mechanics Survey Form B (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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
=
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
135
Fluid Mechanics Survey Form B (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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(
,,,,
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
136
Fluid Mechanics Survey Form B (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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
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
137
Fluid Mechanics Survey Form B (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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
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
138
Fluid Mechanics Survey Form B (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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
4.4.5 Unsteady Effects
⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
=⋅
<⋅
>⋅
===
====Δ=∂∂
=→=⋅⋅+∂∂
= ∫∫∫∫
CV) theof side the(along0ˆflow) (in0ˆ
flow)(out 0ˆˆ
ˆmomentum system0ˆ
0ˆˆ
0
00
nV
nV
nV
iVVmBb
imVVmBiVV
VdbtDt
DBdAnVbdAnVbVdb
tDtDB
cv
sys
cscscv
sys
r
r
r
rr
r
rrr
rr
ρ
ρρρρ
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
139
Fluid Mechanics Survey Form B (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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
CV) theof side the(along0ˆflow) (in0ˆ
flow)(out 0ˆˆ
ˆmomentum system0ˆ
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
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
140
Fluid Mechanics Survey Form B (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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 theofcontent
volumecontrol coincident theofcontent
ˆˆ FdAnVVVdVt
dAnVVVdVt
VdVDtD
FFFVdV
cscvcscvsys
syssyssys
rrrrrrrr
rrrr
ρρρρρ
ρ
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
141
Fluid Mechanics Survey Form B (Continued)
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)
Engineering Analytic Topics & Typical Formulas
1 2 3 4 5
Comment
Chapter 5 Finite Control Volume Analysis (Continued) 5.2.2 Application of the Linear Momentum Equation
( ) ( )
( )
( )
∑∫∫
∫∫∫
∫
∑∫∫
∑∫∫∫∫∫
=⋅=⋅
⋅+⋅=⋅+
→+∂∂
=⋅+++∂∂
=⋅+∂∂
⋅+∂∂
=
volumecontrol theofcontent
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
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
142
Fluid Mechanics Survey Form B (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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
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
143
Fluid Mechanics Survey Form B (Continued)
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)
Engineering Analytic Topics & Typical Formulas
1 2 3 4 5
Comment
Chapter 5 Finite Control Volume Analysis (Continued) 5.2.4 Application of the Moment-of-Momentum Equation
( ) ( ) ( )
( ) ( )( ) ( )[ ]
( )( ) ( )( )( ) ( )
( )( ) ( )( ) ( )outoutininshaftoutin
outoutoutinininshaft
outoutoutinininshaft
shaftshaftoutoutoutinininshaft
shaftshaftshaftshaft
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
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
144
Fluid Mechanics Survey Form B (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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
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
145
Fluid Mechanics Survey Form B (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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
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
146
Fluid Mechanics Survey Form B (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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
&&&
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
147
Fluid Mechanics Survey Form B (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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
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
148
Fluid Mechanics Survey Form B (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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
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
149
Fluid Mechanics Survey Form B (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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
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
150
Fluid Mechanics Survey Form B (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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
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
151
Fluid Mechanics Survey Form B (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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
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
152
Fluid Mechanics Survey Form B (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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
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
153
Fluid Mechanics Survey Form B (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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
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
154
Fluid Mechanics Survey Form B (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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⎟⎠⎞
⎜⎝⎛=⎟
⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛=∞→→
===∂∂
=∂∂
+∂∂
=∂∂
+∂∂
→⎪⎭
⎪⎬
⎫
∂∂
<<∂∂<<
=∂∂
+∂∂
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
+∂∂
−=∂∂
+∂∂
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
+∂∂
−=∂∂
+∂∂
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
155
Fluid Mechanics Survey Form B (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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
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
156
Fluid Mechanics Survey Form B (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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
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
157
Fluid Mechanics Survey Form B (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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
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
158
Fluid Mechanics Survey Form B (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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
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
159
Fluid Mechanics Survey Form B (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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
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
160
Fluid Mechanics Survey Form B (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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
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dTdV
TdTdV
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VVV
TdTdRTdTcdsT
TdTd
pdp
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ppR
TTcssRTpT
RpcTVT
TcVT
Tc
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hhm
aapppp
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p
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innetshaft
innet
=⎟⎟⎠
⎞⎜⎜⎝
⎛+−=−=⎟
⎠⎞
⎜⎝⎛ +−−=
⎟⎠⎞
⎜⎝⎛ +−−=→−==⎟⎟
⎠
⎞⎜⎜⎝
⎛+−=
+=→⎪⎭
⎪⎬⎫
=
=−=
−=−↑=←==+
⎪⎪
⎩
⎪⎪
⎨
⎧
==+
==+
→⎪⎭
⎪⎬
⎫
−=−
==+
+=⎥⎦
⎤⎢⎣
⎡−+
−+−
∨∨
∨∨
∨∨
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111
ddconstant
lnlnconstant2
constant2
constant2constant
2
2
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111022
22
02
2
0
2
00
0
2
12
21
22
12
ρρρ
ρρ
ρρ
ρρ
ρρ
ρρ
&&&
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
161
Fluid Mechanics Survey Form B (Continued)
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)
Engineering Analytic Topics & Typical Formulas
1 2 3 4 5
Comment
Chapter 11 Compressible Flow (Continued) 11.5.1 Adiabatic Constant Area Duct Flow with Friction (Fanno Flow)
( ) ( )
( ) ( )
( ) ( ) ( )
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( ) ( ) ( ) ( ) ( ) 0121
21
2110
21
02
022
022
248
22
2
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2
2
2
2
22
2
2
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22
2
2
2
2
222
2
222
22
22
2
1221122211
=+−+−=
−+==
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=++==
=++=++
=−−→=←==−−
−=−−−=−−
DdxMa
kf
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pdp
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DdxMafk
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pDdxV
pfdp
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VfdVV
AdD
dp
VVVA
RppVVmRApAp
p
wxw
xx
ρρ
ρρπρτ
ρπτ
ρ&
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
162
Fluid Mechanics Survey Form B (Continued)
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)
Engineering Analytic Topics & Typical Formulas
1 2 3 4 5
Comment
Chapter 11 Compressible Flow (Continued) 11.5.1 Adiabatic Constant Area Duct Flow with Friction (Fanno Flow) (Continued)
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⎠⎞
⎜⎝⎛
+=⎟⎟
⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛=
⎭⎬⎫
⎩⎨⎧
−++
==⎭⎬⎫
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+−+
=
=⎭⎬⎫
⎩⎨⎧
−++
===−++
=
−+−
−=−=−
−−
−=
⎭⎬⎫
⎩⎨⎧
−+++
+−
=−+−
=−+−
∫ ∫
kk
Ma
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MakkMap
ppp
pp
pp
pp
Makk
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TT
pp
MakMak
VV
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VV
TTMa
kRTRTkMa
VV
Makk
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MadMak
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Df
Df
Df
Df
MakMak
kk
MaMa
k
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kMaMakMadMa
Ddxf
kMaMakMadMa
ρρ
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llllll
ll
l
l
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
163
Fluid Mechanics Survey Form B (Continued)
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)
Engineering Analytic Topics & Typical Formulas
1 2 3 4 5
Comment
Chapter 11 Compressible Flow (Continued) 11.5.2 Frictionless Constant Area Duct Flow with Heat Transfer (Rayleigh Flow)
( ) ( )
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a
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dsdT
RVVTTVTcdTdsdsdT
MakRTVRVVTT
VTc
dTds
dTdV
TV
Tc
dTdsdVVdTcdsTdVVhddsTdVVdpdVVdp
RTVpVpRVmApVmAp
==++
=+=+
+=+−
=⎥⎦
⎤⎢⎣
⎡ −+==+
=→=−+
==
==−
+=
+=+=+=−=−=
==+=+++=+
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−
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ρρ
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ρρ
ρρ
ρρ
ρ
22
2
222
12
1
22
222111
111
111
1011
11
constantVconstantconstant&&
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
164
Fluid Mechanics Survey Form B (Continued)
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)
Engineering Analytic Topics & Typical Formulas
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
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2
2
2
2
211
12
11
12
1112
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11
−
⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ −+⎟
⎠⎞
⎜⎝⎛
+++
=
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛=
+
⎟⎠⎞
⎜⎝⎛ −++
=⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠
⎞⎜⎝
⎛=
⎥⎦⎤
⎢⎣⎡
++
==⎥⎦⎤
⎢⎣⎡
++
=⎟⎟⎠
⎞⎜⎜⎝
⎛==
kk
a
a
a
aaaa
a
aa
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a
aaaa
a
MakkkMa
kpp
pp
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pp
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MakMak
TT
TT
TT
TT
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kMaMakMa
VV
kMaMak
TTMa
pp
TT
TTMa
ρρ
ρρ
THE END
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
165
Part Three Findings from the Research Project
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
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.
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
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.
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
168
List 1A. (Continued)
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.
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
169
List 1A. (Continued)
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
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
170
List 1A. (Continued)
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 =⎟
⎠⎞
⎜⎝⎛=⎟
⎠⎞
⎜⎝⎛=
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 =⎟
⎠⎞
⎜⎝⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
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 =⎟
⎠⎞
⎜⎝⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
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
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 & Science Pre-requisite Topics & Completion Grade (Georgia Performance Standard Code) [Pre-requisite Math Skills/Science Principles] (GPS Code) Grade (Table No.)
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)
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
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
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
173
List 2A. (Continued)
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
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
174
List 2A. (Continued)
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
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
175
List 2B. Pre-Requisite Math and Science Topics to Be Reviewed Before Teaching the Calculus Portion of Fluid Mechanics Topics
Math & Science Pre-requisite Topics & Completion Grade (Georgia Performance Standard Code) [Pre-requisite Math Skills/Science Principles] (GPS Code) Grade (Table No.)
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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