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7AAt32 414 PRELIMINARY PROPELLER SELECTION USING THE WAGENINGEN
46 SCREW SERIES AND A GENERAL PURPOSE NON-LINEARF OPT IMIZER(U) NAVAL POSTGRADUATE SCHOOL MONTEREY CA
A;IF F M P SMITH JUN 83 F/ 13/10 NL
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NAVAL POSTGRADUATE SCHOOLMonterey, California
F-1
THESIS DPRELIMINARY PROPELLER SELECTION USING THE
WAGENINGEN B-SCREW SERIES AND AGENERAL PURPOSE NON-LINEAR OPTIMIZER
by
Michael Peter Smith II
June 1983
C-(: Thesis Advisors: D. Salinas
JG. N. VanderplaatsSApproved for public release; distribution unlimited.
83 09 14 06Q-%
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REPOR DOCMENTATION PAGE BREAD COMPLTING ORM1REPORT NUMBER (2. GOVT ACCESSSONW M . RECIPIENT'S CATALOG N4UMOEM
4. TITLE (nd uIEUIef) 5.POF REPORT APRIODOCOVERED
Preliminary Propeller Selection Using Master's Thesis;the Wageningen B-Screw Series and a June 1983General Purpose Non-Linear Optimizer 6. PERFORMING ORO. REPORT NUM9ER
7. AIJTHOR(eJ S. CONTRACT OR GRANT NUZOER(s)
Michael Peter Smith II
S. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT. PROJECT. TASK
Naval Postgraduate School AREA & WORK UNIT NUMBERS
Monterey, California 93940
I I. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT OAT!
Naval Postgraduate School June 1983Monterey, California 93940 13. NUMBER OFPAGES
____ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ 36214. MONITORING AGENCY NAME & AODRESS(It diffeeent inem C..gtilinhd Office) IS. SECURITY CLASS. (of ths report)
Unclassified
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17. DISTRIBUTION STATEMENT (of the abstract sneered in Stoek 2c. if iffferse howe Report)
IS. SUPPLEMENTARY NOTES
19. KEy WRDS (Ca1.ue on riewo side if neseain OWd Idmntfy by bleak numbee)
wageningen propeller FORTRANpropeller strength design optimizationpropeller selectionship powering
20. ASISTRACT (Cofie en mewe" side Of ueeeeee and IdeinuII by 6eek nembee)
This thesis presents the use of a general purpose non-linearoptimization program in the preliminary stage of ship design forthe selection of a propeller based on methodical series propellertest data. The propeller series utilized is the well-knownWageningen B-Series. Three (3) "Design Cases", representing thethrust, power and matching approaches to powering problems, areformulated as FORTRAN subprogram analysis codes for solution by
j ~DO IjN517 aa OF i Nov " is omSLEItEr
S/N 0102- L116014-6601 U CLASSIFIED1SECURITY CLASSIFICATION OF THIS PAGE (fte,, Dae. Andeev
UNCLASSIFIEDSECUmTY CLASIPICATION OF THIS PA6S ( DO Em
j (20. ABSTRACT Continued)
the synthesis/optimization program COPES/CONMIN. Designerconstraints considered are:1) diameter limitation2) cavitation limit on expanded area ratio using Keller's
criterion3) strength requirement determined by an empirical relation
and by a method developed by Schoenherr withmodifications by the author.
Objective functions considered are maximized open waterefficiency and minimized propeller blade weight. Optimizedsolutions to specific problems previously presented by otherauthors are obtained and results are compared.
Aocession ForNTIS GRA&IDTIC TAB 0<Unannounced ElJustifictO
LAvniiabuity Codes
Avail and/orD15t SpecialIA
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t .J
g Approved for public release; distribution unlimited.
Preliminary Propeller Selection Using theWageningen B-Screw Series and a
General Purpose Non-Linear Optimizer
by
Michael Peter Smith IILieutenant, United States Naval Reserve
B.S., St. John Fisher College, 1971B.S.E., The University of Michigan, 1976
Submitted in partial fulfillment of therequirements for the degree of
MASTER OF SCIENCE IN MECHANICAL ENGINEERING
from the
NAVAL POSTGRADUATE SCHOOL
4 June 1983
Author:__ _ _ _ _ _ _ _ _
Approved by: ______ ______'/__
Thesis Advisor
Thesis Advisor
Chairman, DeaTt nt of Mechanical Engineering
/Dean of Science and Engineering
I f 3
I ABSTRACT
This thesis presents the use of a general purpose non-
linear optimization program in the preliminary stage of ship
design for the selection of a propeller based on methodical
series propeller test data. The propeller series utilized is
the well-known Wageningen B-Series. Three (3) "Design Cases,
representing the thrust, power and matching approaches to
powering problems, are formulated as FORTRAN subprogram
analysis codes for solution by the synthesis/optimization
program COPES/CONMIN. Designer constraints considered are:
1) diameter limitation,
2) cavitation limit on expanded area ratio usingKeller's criterion
3) strength requirement determined by an empiricalrelation and by a method developed by Schoenherrwith modifications by the author.
Objective functions considered are maximized open water
efficiency and minimized propeller blade weight. Optimized
solutions to specific problems previously presented by other
authors are obtained and results are compared.
4.4
3 TABLE OF CONTENTS
I. INTRODUCTION -- - - - - - - - - - - - - - - - - - 14
A. BACKGROUND-------------------------------------- 14
B. PROBLEM STATEMENT------------------------------- 19
C. SCOPE-------------------------------------------- 19
D. THESIS ORGANIZATION----------------------------- 19
I. OPTIMIZATION---------------------------------------- 22
A. INTRODUCTION------------------------------------ 22
B. DEFINITIONS------------------------------------- 22
C. PROBLEM STATEMENT------------------------------- 25
D. COPES/CONMIN------------------------------------ 28
1. CONMIN-------------------------------------- 28
A 2. COPES--------------------------------------- 30
E. CONCLUDING NOTE--------------------------------- 32
II.POWERING, PERFORMANCE AND PROPELLERS--------------- 33
A. INTRODUCTION------------------------------------ 33
B. DEFINITIONS------------------------------------- 33
C. POWERING CONCEPTS------------------------------- 37
1. Basic Relations----------------------------- 37
2. Approaches to the Powering Problem-------- 38
D. PROPELLER PERFORMANCE CHARACTERISTICS--------- 40
E. THE WAGENINGEN B-SCREW SERIES------------------ 44
1. Background---------------------------------- 44
2. Series Results------------------------------ 45
3. Limitations on Series Data----------------- 48
F. SUMMARY----------------------------------------- 50
IV. PROPELLER SELECTION--AN OPTIMIZATION PROBLEM ---- 52
3A. INTRODUCTION----------------------------------- 52
B. DESIGNER'S CONSIDERATIONS---------------------- 52
1. Propeller Size----------------------------- 52
2. Cavitation--------------------------------- 53
3. Strength----------------------------------- 54
C. THE DESIGN VECTOR------------------------------ 56
1. Parameters--------------------------------- 57
2. Design Variables--------------------------- 58
D. CONSTRAINTS------------------------------------- 60
E. OBJECTIVE FUNCTIONS---------------------------- 61
F. PROPELLER SELECTION OPTIMIZATIONPROBLEM STATEMENT------------------------------ 62
G. CODING FUNDAMENTALS---------------------------- 62
1. GLOBCM Common Block------------------------ 62
2. SUBROUTINE ANALIZ-------------------------- 62
H. SUMMKARY----------------------------------------- 65
V. PROPELLER BLADE WEIGHT--AN OBJECTIVE FUNCTION -- 67
A. INTRODUCTION----------------------------------- 67
B. THEORY AND PROCEDURE--------------------------- 67
1. Limits of Integration---------------------- 67
2. Blade Section Profile---------------------- 69
3. Blade Section Cross-Sectional Area--------- 71
4. Volume Integration------------------------- 72
5. Blade Weight------------------------------- 72
C. CODING------------------------------------------ 72
{u D. SUMMARY----------------------------------------- 73
6
VI. THICKNESS-TO-CHORD RATIO--A DESIGN CONSTRAINT --- 74
A. INTRODUCTION -------------------------------- 74
B. PROPELLER STRENGTH ANALYSIS--A HISTORICALREVIEW -------------------------------------- 74
C. SCHOENHERR'S METHOD ------------------------- 79
1. Background ------------------------------ 79
2. The Blade Model ------------------------- 80
3. Bending Moments Due to HydrodynamicLoading --------------------------------- 82
4. Force and Bending Moments Due toCentrifugal Loading --------------------- 85
D. ALGORITHM FOR THE CONSTRAINT ---------------- 97
1. Theory ---------------------------------- 97
2. Coding Details -------------------------- 99
E. SUMMARY - ------------------------------------- 100
4k VII. DESIGN CASE NO. 1--PROGRAMMING AND COMPARISONS -- 101
A. INTRODUCTION --------------------------------- 101
B. THRUST APPROACH FORMULATION ----------------- 101
1. Design Vector XT------------------------ 101
2. Powering Constraint --------------------- 103
C. PREVIOUS SOLUTIONS -------------------------- 104
D. SOLUTIONS BY COPES/CONMIN ------------------- 106
1. Variation 1 ----------------------------- 107
2. Variation 2 ----------------------------- 109
3. Variation 3 ----------------------------- 109
4. Variation 4 -i-----------------------------I
E. DISCUSSION ---------------------------------- 111
7
VIII. DESIGN CASE NO. 2--PROGRAMMING AND COMPARISONS 114
IA. INTRODUCTION----------------------------------- 114
B. POWER APPROACH FORMULATION-------------------- 114
1. Design Vector R2 --------------------------- 114
2. Powering Constraint------------------------ 116
C. PREVIOUS SOLUTIONS----------------------------- 117
D. SOLUTIONS BY COPES/CONMIN-------------------- 119
1. Variationi--------------------------------- 120
2. variation 2------------------------------ 122
3. Variation 3-------------------------------- 123
4. Variation 4-------------------------------- 124
E. DISCUSSION-------------------------------------- 124
IX. DESIGN CASE NO. 3--PROGRAMMING AND COMPARISONS -- 128
A. INTRODUCTION----------------------------------- 128
B. "MATCHING" FORMULATION------------------------- 128
1. Design Vector X3 --------------------------- 128
2. Powering Constraint(s)--------------------- 129
C. PREVIOUS SOLUTIONS----------------------------- 131
D. SOLUTIONS BY COPES/CONMIN---------------------- 133
1. Programming Details------------------------ 135
2. Results------------------------------------ 135
E. DISCUSSION------------------------------------- 136
X. CONCLUSIONS AND RECOM.MENDATIONS------------------- 139
A. CONCLUSIONS------------------------------------ 139
B. RECOMMOENDATIONS-------------------------------- 141
C. A FINAL NOTE----------------------------------- 142
8
APPENDIX A: FORTRAN VARIABLE CROSS REFERENCE LIST 143
APPENDIX B: SUBROUTINE LISTINGS -------------------- 145
APPENDIX C: ANALIZ CODES--DESIGN CASE NO. 1 ---------- 215
APPENDIX D: CONTROL CARD IMAGES--DESIGN CASE NO. 1 --- 231
APPENDIX E: COPES OUTPUT--DESIGN CASE NO. 1 ---------- 234
APPENDIX F: ANALIZ CODES--DESIGN CASE NO. 2 ---------- 272
APPENDIX G: CONTROL CARD IMAGES--DESIGN CASE NO. 2 --- 288
APPENDIX H: COPES OUTPUT--DESIGN CASE NO. 2 --------- 291
APPENDIX I: ANALIZ CODES--DESIGN CASE NO. 3 ---------- 330
APPENDIX J: CONTROL CARD IMAGES--DESIGN CASE NO. 3 --- 334
APPENDIX K: COPES OUTPUT--DESIGN CASE NO. 3 ---------- 336
LIST OF REFERENCES ------------------------------------ 356
INITIAL DISTRIBUTION LIST ----------------------------- 360
9
im • i Ij
LIST OF TABLES
I. Summary of the Wageningen B-Screw Series -------- 45
II. Material Identifier Reference ------------------- 58
III. Global Common (GLOBCM) Catalog ------------------ 63
IV. Design Case No. 1--Results ---------------------- 113
V. Design Case No. 2--Results ---------------------- 127
VI. Design Case No. 3--Results ---------------------- 138
10
It~ -4
I ....
LIST OF FIGURES
1.1 Traditional Design Spiral ----------------------- 153.1 Open Water Test Results--B 4-100 Series
Propeller --------------------------------------- 42
4.1 Design Vectors 5 and R -------------------------- 59
5.1 Propeller Blade & Hub--Side View ---------------- 68
5.2 Expanded Cylindrical Blade Section--ProfileView -------------------------------------------- 70
6.1 Strength Section at r = r - - - - - - - - - - - - - - - - - - - - - - 83
6.2 Bending Moments due to Thrust and Torque -------- 86
6.3 Center of gravity "G" and intersection point"N" -88
6.4 Internal Loading on a Blade Section atx = x0 = ro/R ----------------------------------- 89
6.5 Position of "g" of a Blade Section "Slice" atx = r/R ----------------------------------------- 93
6.6 Coordinates of centroid "g" of any BladeSection ----------------------------------------- 94
4i1
ACKNOWLEDGEMENTS
It has been said:
If someone says It's impossible, then, quite obviously,he has never done it.
With these words in mind, I now recall and give thanks
to those who have assisted me in attaining the educational
goal represented by this thesis:
My mother and father, who provided, among other things,
a secure and stable home for me to grow up in and who always
insisted that I fulfill my potential for formal education
at every opportunity.
My brothers and sister, who have always offered me nothing
but praise and encouragement in all of the tasks that I
have ever set out to do.
My aunts and uncles, living and deceased, who provided
the counseling or the little "envelope for a rainy day"
when I needed it the most.
My grandparents, now deceased, who possessed the courage
and forethought to venture forth across an ocean to our great
country in order to build a better life for themselves and
to afford their offspring opportunities of a lifetime.
My close friends, past teachers, instructors and pro-
fessors, who always pointed me in the right direction with
astute guidance and timely advice when all seemed lost.
That one junior Naval officer, who was not afforded the
opportunity to achieve the milestone represented by this
4 12
work because he maintained "the watch" for a shipmate like
me.
And, finally, to You.
On a more current and professional note, I especially
thank Dr. Marto, Professor Salinas, Professor Vanderplaats
and also LCDR Michael R. Maixner, USN. Their patience,
understanding and tireless efforts in helping me close out
my academic career on a successful note are deeply appre-
ciated and will always be remembered.
And a special word of thanks to Mr. Robert Lande for
the expeditious and accurate typing of this thesis.
13
4A
I--' J
I. INTRODUCTION
A. BACKGROUND
The ship design process, in its most rudimentary form,
has been formulated and tracked by the utilization of the
classical design spiral (see Figure 1.1). The design follows
a convergent helical path past each major milestone "spoke"
until, after numerous iterative cycles, the final configura-
tion is "centered" upon. Whether one attempts to segregate
the principal phases of Preliminary, Advanced and Contract
Design into separate spirals or combine these phases in
series along the entire path to the center, it is not long
before the designer's roughed-out sketches give way to serious
"number crunching", specifically that of propulsion power
estimation.
To estimate the power required to drive the ship through
the water at its design speed, a decision must first be made
as to what type of propulsor (i.e., propeller, water jet,
paddle wheel, etc.) will be used. For the average case, and
for the discussion that follows, the marine propeller is
chosen to be the propulsion device. Since
a ship propeller may be regarded as a transducerthat converts the rotational power transmittedthrough the shaft into the translational power topropel the ship, [Ref. 18: p. 10]
the selection and design of this device is obviously an
important factor in the eventual size (weight and power) of
the ship's propulsion plant. While hydrodynaxnicists provide
4 14
TRADITIONALDESIGN SPIRAL
VOLUME a STABILITY
SHP MARRANGEMENTS
ENDURANCE FUEL HULL FORM
WEIGHTSX
Figure 1.1 Traditional Design Spiral
15
a myriad of theories and techniques to generate a "custom
built" (i.e., wake adapted) propeller for the ship under
consideration, their expertise is usually not required in
the early stages of preliminary design simply because the
design has not been refined enough beyond gross estimates.
At this stage, the designer strives to formulate what is
possible based on previous experience. For preliminary
power estimation, previous propeller designs (i.e., "stock"
propellers) and results from methodical series of model
propellers are analyzed by the designer in order to select
the "best" available propeller under various conditions posed
by the problem under consideration. Three examples of typi-
cal problems encountered in preliminary ship design are:
1) Given the ship's effective horsepower at a specific
speed and estimates of hull performance parameters, which
propeller, as determined by certain principal characteristics,
will require the least amount of delivered power from the
propulsion plant?
2) Given the delivered power from a specific propulsion
system in terms of torque and revolution rate at the pro-
peller/shaft interface and estimates of hull performance
parameters, which propeller will generate the largest effec-
tive horsepower and speed parameters?
3) Given a ship's effective horsepower and speed, various
hull performance parameters, and the propulsion plant's
delivered power characteristics, which propeller will "match"
4 16
these requirements at a minimum amount of weight for a speci-
fied material?
(Author's Note: For the sake of brevity, the three
selection problems just cited will, henceforth, be referred
to as "Design Case No. 1", "Design Case No. 2" and "Design
Case No. 3", respectively.)
For this study, the methodical propeller series method
is viewed as the designer's choice for preliminary powering
analysis. One of the most-widely used methodical series data
on model propellers is the Wageningen B-Screw Series.
Initially, the results of the series were presented as tabu-
lations of non-dimensional thrust and torque coefficients
(KT and KQ respectively) versus the non-dimensional advance
ratio (J) for analytical work and as the familiar "Bp-65" and
"Bu-6" diagrams for design purposes. As "trial & error"
design methods performed by hand in all engineering disci-
plines gradually transcended to numerical manipulation by
the modern digital computer, the necessity for the adaption
of the Series results to a format suitable for use in computer-
aided design methods became obvious. This was accomplished
through multiple regression analysis of the original open-
water test data of the 120 propeller models in the Series
and presented in the form of polynomial expressions for "KT"
and "KQ" (Refs. 1,2].
The adaptation of the Wageningen B-Screw Series polynomials
to various types of propeller selection problems formulated
174 .
for computer solution has been implemented recently by two
authors. Triantafyllou [Ref. 3] and, of late, Markussen
[Ref. 4] presented different propeller selection problems and
proposed different schemes for computer-aided "optimized"
solutions. In short, specific expressions for the con-
straints imposed and the objective (optimality condition)
to be maximized, expressed in terms of a number of design
variables and parameters, were developed. Then, each system
of equations was solved by a Newton-Raphson method to give
a solution set of the design variables which maximized the
objective and met all constraints.
Rather than formulating and coding a different optimiza-
tion scheme each time a propeller selection problem presents
a different combination and number of design parameters,
variables and constraints, a better approach would involve
formulating the problem (constraints and objective function)
once in terms of all design parameters and variables and
utilizing a general purpose optimization scheme which can
handle any combination and number of constraints and design
variables. This alternative certainly allows the designer
more flexibility in solving his problem. Moreover, it elimin-
ates repetitive coding and debugging associated with the
implementation of a computer-sided solution for each particu-
l~ar design problem.
4 18
I, _ '=
B. PROBLEM STATEMENT
The problem, then, is that the previously cited computer-
aided "optimized" solutions to the propeller selection problem
are not broad enough in capability to handle variations in
the problem formulation. The objective of this thesis is to
apply an available general purpose optimization computer
code to the solution of various propeller selection problems
encountered in Preliminary Ship Design in order to enhance
the flexibility of the selection procedure.
C. SCOPE
To achieve the stated objective, the general purpose
non-linear optimization code CONMIN [Refs. 5,6] together with
the engineering synthesis code COPES [Ref. 7) (hereafter
referred to collectively as COPES/CONMIN) is utilized in the
solution of the three previously cited preliminary design
propeller selection problems. Using the Wageningen B-Screw
Series propeller characteristics expressed in polynomial
expressions of various design variables, three "analysis"
codes, required by COPES/CONMIN, are developed in such a way
that various combinations of design variables and constraints
are used, thereby demonstrating the applicability of the
COPES/CONMIN optimization program in the solution of propeller
selection problems.
D. THESIS ORGANIZATION
The remainder of the thesis is organized in the following
manner.
19
L-
Chapter II presents a short description of the optimiza-
tion problem in general terms and a follow-on discussion of
the COPES/CONMIN optimization program and the mathematical
techniques employed therein.
Chapter III introduces definitions and concepts applica-
ble to the propeller selection problem. A subsequent dis-
cussion on the Wageningen B-Screw Series is followed by final
comments on constraints imposed on the propeller selection
problem.
Chapter IV presents the formulation of the propeller
selection problem as a design optimization problem which
can be solved using COPES/CONMIN.
Chapter V discusses the background, formulation and
programming utilized in estimating a propeller blade's weight
for subsequent consideration as an objective function.
Chapter VI reviews the author's modifications to the
propeller strength analysis developed by Schoenherr [Ref. 8]
in the early 1960's for the American Bureau of Shipping. A
subsequent discussion on the programming details of FORTRAN
codes, which are utilized for the determination of adequate
propeller blade strength, completes the chapter.
Chapter VII reviews the formulation and programming for
the analysis code which is used in solving propeller selection
problems represented by Design Case No. 1. Sample solutions
are presented and compared to those presented previously by
other authors.
20
Chapters VIII and IX consider Design Case No. 2 and
Design Case No. 3 selection problems, respectively, in a
similar fashion to Chapter VII.
Chapter X, the final chapter, presents the author's
conclusions and recommendations.
As a final note, all computer coding presented in this
thesis is done in FORTRAN IV, the language used by COPES/
CONMIN. For the reader's convenience, Appendix A provides
a cross-reference of the symbols presented throughout the
thesis to appropriate FORTRAN variable names appearing in
the author's codes.
21
_
II. OPTIMIZATION
A. INTRODUCTION
The purpose of this chapter is to introduce definitions
and concepts used in the formulation and solution of the
general optimization problem. Then, a short discussion on
the theory and implementation details of COPES/CONMIN is
presented.
For further study on the theory and methods of optimiza-
tion, the reader is directed to the texts by Fox [Ref. 9],
Fiacco and McCormick [Ref. 10], and Himmelblau [Ref. 11].
B. DEFINITIONS
Before discussing the techniques of optimization and
their application to engineering problems, some preliminary
definitions of basic terminology should be stated. Terms
which have relevant significance are:
1) Parameters--The numerical quantities for which values
are assigned to produce a design are called parameters. From
this, it follows that a design may be specified by a vector
D containing "p" components, each of which is associated with
a parameter. That is:
(2.1)
Dpp
22
L J
However, in a design process, the parameters are determined
by some logical procedure through analysis of some kind.
Some might take on fixed values to become "preassigned"
parameters. Interrelationships among other parameters might
exist so that only some of the parameters are changed when
one design is compared to another. This consequence leads
to the definition of "design variable".
2) Design Variables--The parameters for which values are
chosen in some fashion to produce a de;,Jcn are called design
variables. They represent an ordered collection of components
which is a subset of the design vector ff. This subset is
unique in that its components are "variable", i.e., they may
take on different values in the design process. Having
"preassigned" or fixed some of the design's parameters and
only allowing the remaining "desigrn variables" to change, leads
to the conclusion that a design is now uniquely specified by
a vector X containing "n"~ components (n < p) , each of which
is associated with a design variable. That is:
= (2.2)
Ixn
3) Objective Function--The computable function of all or
some of the design's preassigned parameters and/or design
variables, with respect to which the design is to be opti-
mized, is called the objective function. Single valued in
23
quantitative terms, the objective function's minimum or maxi-
mum value represents the "best" obtainable or "optimized"
design. It is expressed as F(U) to show its dependence on
the design's parameters. But, since a design can be uniquely
defined by R alone, then clearly F(X) suffices as an expression
for the objective function.
4) Constraints--Restrictions on the design which must be
satisfied in order to produce an acceptable design are
called constraints. A constraint may be classified as a
"side" or a "behavior" constraint. A side constraint re-
stricts or bounds the range of the design for reasons other
than direct consideration of performance. The side constraint
on the "i"th design variable may be expressed as:
Xlower < X < X upper i = 1,...,n (2.3)1 - 1 - 1
A constraint derived from those performance or behavior
requirements that are explicitly considered is called a be-
havior constraint. Most often, it appears as a computable
functional relation involving the design's parameters, both
preassigned and variable alike. The relation may be an in-
equality so that the "j"th of "m" inequality constraints
can be expressed as:
G.(D) < 0 j = l,...,m (2.4)2
24
U _J
Alternatively, the relation may be an equality on the "k"th
of "9" equality constraints expressed as:
Hk(D) = 0 k = 1,..., (2.5)
Of noteworthy importance here is that, as before, if
some of the design's parameters are preassigned, then the
resulting design is that defined by X which contains only
the parameters that can be varied in the design process,
i.e., the design variables. Therefore, constraints imposed
upon the design may be expressed under one equation as:
G. (X) < 0 j = 1...,mJ
Hk(X) = 0 k = i... (2.6)
Xlo w e r < X. < Xupper i21 - 3 - .
A final form for constraints is that of the discrete-valued
design variable.
5) Feasible Design--A design in which specified constraints
are satisfied is called a feasible or "acceptable" design.
6) Infeasible Design--A design in which constraints are
violated is called an infeasible or "unacceptable" design.
C. PROBLEM STATEMENT
If one presupposes that a range of designs exists within
a selected design concept, then it follows that different
25
ii
methodologies also exist by which one may choose the param-
eters which describe the design. One such method is optimi-
zation where parameters are chosen in a way that the design
will satisfy all of the limitations and restrictions imposed
upon it and will be "best" in some sense. In view of the
foregoing definitions, optimization is then a selection method
applied to a design problem by which an objective function
F(D) is minimized to produce an acceptable design which satis-
fies a certain set of requirements called constraints.
Formulated mathematically, the general, non-linear,
constrained optimization problem may be stated under one
equation as:
Minimize: F(5) =OBJ
Subject to: GA(D) < 0 j = L,...,m (2.7)3
Hk05) < 0 k =
X lower <~ .< upper =2. - 1 - 1.
Again, as pointed out in the previous section, the design
may be uniquely defined by just its design variables as
specified by X when some parameters are preassigned. Thus,
the general, non-linear, constrained optimization problem
can now be stated under one equation as:
26
Minimize: F(X) = OBJ
Subject to: G(X) < 0 j =,...,m (2.8)
Hk(x) = 0 k =
Xlower < X. < xUp p e r i =1 - i - 1
Solutions methods for this optimization problem are
abundant. Those pertaining to the linear and quadratic
optimization problems involving a few design variables are
most often presented in graphical or analytic form, although
numerical schemes are, by no means, a dormant form. Struc-
tural and thermal problem solutions are most prevalent. How-
ever, as the optimization problem becomes more complex in
terms of non-linear relationships among an increasing number
of design variables and of an increased number of design
constraints, numerical or mathematical programming techniques
dominate the solution methods.
To limit the scope of this discussion, only the numerical
techniques relevant to COPES/CONMIN will be considered. For
more background on optimization techniques and applications,
the reader is directed to a recent paper by Vanderplaats
(Ref. 121 which presents a concise, but thorough, qualitative
review of optimization. Although this paper deals exclusively
with the application of design optimization to structural
problems, it also contains a very extensive and current list
of references on general techniques and applications of
optimization.
4 27
D. COPES/CONMIN
As previously stated in Chapter I, COPES/CONMIN is the
collective acronym for the FORTRAN program utilizing the opti-
mization code CONMIN and the synthesis code COPES. COPES
stands for COntrol Program for Engineering Synthesis; CONMIN
is an acronym for CONstrained function MINimization.
1. CONMIN
CONMIN is a FORTRAN program, in subroutine form, which
solves the general non-linear constrained optimization prob-
lem as stated:
Minimize: F(X) = OBJ
Subject to: G.(X) < 0 j l,...,m (2.9)
Xl ow e r < upper j1 - I -
Equation (2.9) applies to the entire statement. Observe
that equation (2.9) differs from equation (2.8) in that the
equality constraint set, given by VX) = 0, is not specified.
This is because the version of COPES/CONMIN used in this
study does not consider these types of constraints. However,
this will not pose any difficulty in solving the propeller
selection problems previously cited.
Again, F(X) is the objective function (OBJ). The
vector X contains the "n" design variables (NDV). G. (X)
are the "m" inequality behavior constraints (NCON) imposed
on the optimization problem; Xlower and Xupper are the1 i
28
L _J
respective lower and upper side constraints which bound the
"design space" over which F(Y) and G.i(R) are defined. As
functional relationships involving R, F(R) and G.(M may be
implicit or explicit, but, in any event, must be continuous
and have finite numerical values.
When the inequality condition of equation (2.9) is
not satisfied, i.e., G.(W > 0 for any constraint, the con-
straint is said to be violated. If the equality condition
is met, i.e., G.i(X) = 0 for any constraint, the constraint
is said to be active. And, finally, if the inequality condi-
tion is satisfied, i.e., G .(R) < 0, for any constraint, that
constraint is termed inactive. Any design, defined by R,
which satisfies the inequalities of equation (2.9) is desig-
nated as a feasible design. Likewise, any one which vio-
lates these inequalities is termed an infeasible design.
The feasible design with the minimum objective function value,
often referred to as the "minimum feasible design", will,
therefore, be the optimum design.
During the optimization process, CONMIN employs the
Fletcher-Reeves algorithm [Ref. 13] for locally unconstrained
problems, and Zoutendijk's methoqd of deasible directions [Ref s.
14,153 for locally constrained problems, in a numerical pro-
cedure which attempts to minimize the objective function,
F(R) = OBJ, until one or more of the constraints, G.i(R),
becomes active. The numerical search procedure begins with
an initial Y vector which may or may not specify a feasible
29
design. Modifications are included in CONMIN so that, if
the initial design is infeasible, a feasible solution will
be obtained with minimal increase in F(X). By iteratively
updating the design vector X by the following relation:
R(q+l) = (q) + * (q) (2.10)
the optimization process continues by following the con-
straint boundaries in a direction of search S so that the
value of F(X) decreases with each iteration q. The scalar
a* defines the distance of travel in the direction of search
S. The process terminates when a vector 5 is found such
that no further decrease in F(R) can be made. The vector
is considered to be optimal and, at least, a local minimum.
CONMIN can be used alone as a subroutine in any
FORTRAN program where numerical optimization is desired.
However, in order to make the optimization process more "user-
friendly", CONMIN has been coupled to COPES in order to
simplify its application to various types of problems.
Further information on CONMIN can be found in previously
cited references (5] and (6].
2. COPES
COPES is a FORTRAN program that provides automated
design and trade-off capability to the design engineer. It
utilizes the optimizer CONMIN to provide the following six
specific capabilities:
30
I _' _id
1) simple analysis
2) optimization
3) sensitivity analysis
4) two variable function space analysis
5) optimum sensitivity
6) optimization using approximation techniques
During the execution of COPES, say for optimization, three
principal tasks are performed:
1) data management on the design variables and constraints
through location assignments in a FORTRAN common block called
GLOBCM.
2) decision process control on the attainment of an optimal
design vector Y through multiple calls to the optimizer
until a minimum or maximum value of OBJ is achieved and all
G.(X) are satisfied.
3) evaluation of OBJ and G.(W) at each Xq and X-q + l when
ICALC = 2 through multiple calls to the user-provided analysis
subprogram, SUBROUTINE ANALIZ.
For the application under consideration in this study, only
the optimization capability will be used. Therefore, further
elaboration on the other capabilities is not warranted.
Reference [7] is the user's manual for COPES/CONMIN.
Details on the mechanics of user implementation are presented
with subsequent illustration by example. The reader is,
therefore, encouraged to familiarize himself with the refer-
ence. However, at this point, it is sufficient to be aware
of the fact that a user of COPES/CONMIN is required to:
31
- a
1) provide a FORTRAN subroutine called ANALIZ which per-
forms the input of preassigned parameters, the evaluation of
the objective function and constraints during the analysis
phase of the optimization search and the output of the
results.
2) provide an assembled deck of control cards required
by COPES.
E. CONCLUDING NOTE
The field of optimization is both extensive and complex
and, therefore, the foregoing presentation is, by no means,
complete in every detail. However, it is felt that the pre-
ceding overview, in conjunction with the cited references,
covers the necessary prerequiisites that will enable the reader
to follow the application of COPES/CONMIN to the various
propeller selection problems in the chapters that follow.
32
III. POWERING, PERFORMANCE AND PROPELLERS
A. INTRODUCTION
The purpose of this chapter is to present an overview
of the terminology and concepts that pertain to ship propul-
sion, propeller selection and the use of model propeller
test data. Initially, fundamental definitions used in ship
powering problems are presented. This is followed by a
discussion of the "classic" types of propeller design/
selection problems encountered by the naval architect and
marine and naval engineers. Propeller model testing and
propeller performance characteristics are reviewed next.
The chapter is completed with a discussion of the Wageningen
B-Screw Series.
The goal here is brevity. The reader is, therefore,
encouraged to investigate the references cited for further
details.
B. DEFINITIONS
Some fundamental terms associated with most propeller
design/selection problems are:
1) Effective Horsepower (PE )--power required to tow the
"bare" hull (without propeller; rudder and appendage allowance
assumed included) that generates a given resistance (R T) at
a given speed MV. It is determined by:
RT V (3.1)
4 33
2) Thrust Horsepower (P T)--power delivered to water by
a propeller developing a thrust force (T) and moving at a
speed of advance (VA) without the influence of a hull form
ahead of it. P T is determined by:
P 4TVA (3.2)
3) Delivered Horsepower (PD )--power delivered by shaft
to propeller, normally specified at the outboard side of the
stern tube. QSis the torque delivered to the propeller;
nis the revolution rate of the shaft and, consequently,
the propeller. PD is determined by:
P 2rr Q P (3.3)D 550
4) Shaft Horsepower (P S)--power delivered to the inboard
side of the stern tube having a transmission efficiency of
ns is determined by
P (3.4)S ns
5) Brake Horsepower (PB )--power delivered by the prime
mover at connection flange to the power train. While P B is
normally associated with the prime mover's rated power at
this connection (BHP), it can also be specified from the
power train/propeller side as:
34
- s (3.5)PB - IB nG
where B and n G are, respectively, the bearing system and
reduction gear transmission efficienci-s.
6) Thrust deduction factor (l-td)--ratio of the tow
resistance (RT ) to the thrust (T) provided by the propeller.
It is determined by
T
7) Wake Factor (Taylor's) (l-wt)--ratio of the speed of
advance (VA) to the ship's speed (V). It is given by:
(l-wt) - A (3.7)V
8) Advance Ratio (J)--a non-dimensional value, associated
with propeller test data presentation (see figure (3.1),
given by the following relation:
j V(1-wt) VA (3.8)n D np D
9) Thrust Coefficient (KT)--a non-dimensional value asso-
ciated with the thrust force (T) developed by a propeller
of diameter D which is turning at a rate np and operating
in a fluid of density p. It is defined by the following
expression:
35
I_ _i
TKT 2 4 (3.9)T p np2D
10) Torque Coefficient (K Q)--a non-dimensional value
associated with the torque (Qp) absorbed by a propeller of
diameter D which is turning at a rate np and operating in
a fluid of density p. It is defined by the following
expression:
KQ D(3.10)Q p np2DP
11) Open Water Efficiency (n )--the ratio of PT to PD
for a propeller in open water conditions, with a uniform
inflow velocity field at a speed of advance VA. It is ex-
pressed as:
P T T VA JT (3.11)no - P 2Tr n 2 1TK
D PSQ
12) Hull Efficiency (nH)--a ratio of work done on the
ship to that done by the propeller expressed as:
PE RT V (1-td)
H P T T VA - (3.12)
13) Relative Rotative Efficiency (nR)--the ratio of the
actual, behind-hull efficiency to the open water efficiency.
The value of nR does not, in general, depart from the value
36
u _.j
I I I I I i
of 1.0. Most often, n R varies between 0.95 and 1.0 for twin-
screw ships and between 1.0 and 1.1 for single screw ships.
Applicable units for the terms in the expressions above
are:
1) horsepower (hp)--PE, PT' PS' PD and PB
2) pounds (lbf)--RT, T
3) feet/second (ft/sec)--V, VA
4) foot-pounds (ft-lbf)--Qs, Qp
5) feet (ft)--D
6) revolutions/second (rps) -- np
7) revolutions/minute (rpm)--Np = np/60.0
The quantities T, VA, D, Qp and nP are obtained from the
propeller test data results. The quantities RT and V are
specified from the design point on the R-V curve for the hull
under study. The quantity p is a property of the fluid in
which the hull and propeller operate. And, finally, nB' G
and nS are characteristics of the bearing, gear and stern
tube systems. In preliminary design studies, nominal values,
based on previous designs, are usually assumed unless, of
course, these systems have been selected and actual values
can be specified.
C. POWERING CONCEPTS
1. Basic Relations
Simply stated, the fundamental powering relationship
to be solved in ship propulsion and powering problems is:
37
1 1
(1-td)PE = (l-wt) .R no PD (3.13)
Utilizing the definitions just presented, equation (3.13)
can be rewritten as:
RT V (1-td) 27 QS np
550 TY-w"t .R no 550 (3.14)
Rearranging terms of equation (3.14) gives:
RT (l-wt)V 2ir Q (3.15)
TIEtd 550 R o 550
And, finally, when substitutions are made, equation (3.15)
becomes:
T VA T(l-wt)V 27T QTS n (550 550 - R no 550 (3.16)
Equations (3.14), (3.15), and (3.16) provide the basis for
different approaches to the solution of a typical powering
problem. More background and information on the definitions
and equations presented above may be ;ound in Chapter VI,
Sections 10-16 in the text by Comstock [Ref. 16] and O'Brien's
book [Ref. 17].
2. Approaches to the Powering Problem
From equation (3.16), three types of propeller selec-
tion problems are discernible. In the first instance, the
propeller thrust T and the propeller's speed of advance VA
38
L _J
are taken as known quantities. The fact that T is known
substantiates the "Thrust Approach" nomenclature given to
this type of selection problem. In the preliminary (or, in
some circles, conceptual) ship design phase, the specifica-
tion of T is based upon the requirement imposed by the
resistance of the ship (R T) at its design speed (V) (or, the
effective horsepower (PE at V) and estimates of wt and td
in the absence of wake surveys and self-propulsion data from
model tests. Essentially, the thrust delivered by a selected
propeller must provide, at least, the thrust required for the
ship hull under study. The objective in the "Thrust Approach"
selection problem is to determine, by logical means, the
appropriate values of QSand n P when the open water efficiency
h(r) is set by the selected propeller and its performance
characteristics.
In the second instance, the delivered torque (QS and
the propeller shaft speed (n P) are taken to be known. The
"Power Approach" nomenclature is given to propeller selection
problems of this type because PDis known. Here, with the
shaft and propeller speeds being equal, the-torque absorbed
by the propeller (Q P) must be, at least, equivalent to the
delivered torque (QS) The corresponding objective in the
"Power Approach" selection problem is to determine, by logical
means, the expected ship speed (V) (or, the speed of advance
(VA )) and the associated thrust MT that can be developed
when the open water efficiency (n 0) is, again, set by the
selected propeller and its performance characteristics.
4 39
The final, and most familiar, types of propeller
selection problem occurs when T, VA or V, Q and n P are all
known. From equation (3.16), the open water efficiency (n 0
is now established as a requirement to be met. The objective
is, simply, to select a propeller whose open water efficiency
ho ), developed thrust and absorbed torque are equivalent to
or "match" the requirements imposed. obviously, this approach
on the selection problem has been designated as a "matching
problem".
The reader is directed to the paper by Vassilopoulos
[Ref. _-8) for further information.
D. PROPELLER PERFORMANCE CHARACTERISTICS
Up until the late 1950's, much of the knowledge about
the performance of propellers has been gained from experience
with models. To study the relationships governing their
behavior, a model propeller is built and run in a towing
tank without any hull ahead of it. This is done by running
the propeller on a long shaft projecting well ahead of a
narrow, hydrodynamically shaped pod or "propeller boat" which
contains the driving mechanism and recording apparatus and
is attached to the towing carriage. The propeller advances
into undisturbed fluid (usually water of density p and kine-
matic viscosity j) so that the speed of advance (VA) is known
and the flow into the "disc" swept by the turning blades is
uniform. For the model propeller of diameter (D) under test,
readings of thrust (T), torque (Q S) and shaft revolutions (n P)
40
are recorded over a range of values for speed of advance (V A)
in this "open water," condition.
Using the laws of similitude, the collected data is
reduced and scaled appropriately into the familiar functional
relationships between the advance ratio (J) and the non-
dimensional coefficients of propeller performance. These
coeffi~cients or performance characteristics, defined pre-
viously, are:
1) Thrust Coefficient (K T)
2) Torque Coefficient (KQ )
3) Open Water Efficiency (n 0
Figure (3.1) graphically depicts the relationship between J
and K T and K Qderived from test data for a propeller defined
by a specific expanded area ratio (A.E/A 0), pitch-diameter
ratio (P/D), number of blades (Z) and thickness-to-chord
ratio (t/c).
Definitions of these terms with graphical illustrations
pertaining to various aspects of propeller geometry can be
found in Section 15 of references [16], [17] and in van Manen's
publication [Ref, 191.
More recently, highly analytical theories (lifting line,
modified lifting line, lifting surface, etc.) for use with
high-speed digital computers have been formulated and subse-
quently used in "modeling", in a mathematical sense, the
propeller and its behavior in the "wake adapted" (or, behind
hull) condition as well as the "open water" condition.
41
-
I
4--
a I 01 a 3 @ 01 as 07 01 0s 10 'I 1 Is 'A Is 1
Figure 3.1 Open Water Test Results--B 4-100 SeriesPropeller
42
Additional benefits derived from this approach to propeller
performance analysis include:
1) determination of blade section profiles along the
propeller's blade radius (R) to achieve uniform lift and
internal stress distributions;
2) computation of "off-design" performance characteris-
tics in all quadrants;
3) subsequent determination of hull surface forces, bearing
loads and spindle torques induced by the propeller;
4) prediction of steady and unsteady stress distributions
in the propeller blade using the finite element method on
the blade of the propeller under study.
Obviously, this approach to propeller performance analysis
serves to:
1) eliminate the time-consuming and expensive model
construction and testing of propellers in tow tanks and
cavitation tunnels;
2) eliminate the "scaling" discrepancies which inhibit
the reliability of design charts and model propeller data;
3) eliminate those design charts altogether.
As in the case with model experiments, however, the ultimate
objective remains the same, i.e., establishing the performance
characteristics of the propeller in terms of KTV K Q and as
functions of J. Having these relationships enables the ship
designer to proceed in solving the power equation (equation
(3.13)) through any of the approaches previously discussed.
4 43
- J
4
E. THE WAGENINGEN B-SCREW SERIES
1. Background
The model test data of the Wageningen B-Screw Series
have been selected for use in the powering problems to be
solved utilizing COPES/CONMIN. The choice was driven by the
following considerations:
1) the Series is widely known and, despite its growing
obsolescence, is still used in preliminary ship design
studies.
2) the availability of previous investigations [Refs. 3,4]
which utilized the series, for comparative analysis of optimi-
zation results.
3) the applicability of the polynomial expressions for KT
and KQ to computer-aided analysis.
The Series tests were conducted from 1940 through 1960 and,
therefore, represent propeller designs (principally naval
and merchant applications) and design philosophy of that era.
Specifically, the Series consists of 120 model
propellers. As is customary in methodical or systematic
model propeller series testing, the number of blades (Z),
expanded area ratio (AE/AO) and pitch-diameter ratio (P/D) are
varied systematically, while the blade outline, the profile
of the blade's cross section along the blade radius, blade
cross section maximum thickness (t), blade section chord
length (c), diameter (D) and propeller hub-to-diameter ratio
(d/D) were kept constant for given values of AE/AO and Z.
44iA
Table (I) summarizes the variations in Z and AE/AO for each
set of model propellers having pitch-diameter ratios (P/D)
of 0.4, 0.6, 0.8, 1.0, 1.2 and 1.4.
TABLE I
Summary of the Wageningen B-Screw Series
Blade OtmOe rade area rati AE,
0.35 0.50 0.65 0.80
0.40 0.55 0.70 0.85 1.00
5 0.45 0.60 0.75 1.05
6 0.50 0.65 0.80
7 0.55 0.70 0.85,
2. Series Results
The test results of the Wageningen B-Screw Series were
originally presented in the form of Bp-6, Bu-6 and KT, KQI
and -J diagrams. As stated in Chapter I, multiple regression
analysis was performaed (again, [Refs. 1,2]) on the results
to produce the polynomial expressions for KT and K . The
open water efficiency (n0 ), as a function of J, follows from
equation (3.11). The correction for "scale effects" was
achieved by using Lerb's method of equivalent profiles
[Ref. 20). Although Triantafyllou's thesis [Ref. 213 sug-
gests an improved method for scale correction, the results
of Reference [2] will be used in this study.
45
In propeller selection problems which use this
Series, values for KT and KQ are defined as:
KT = fl (JP/DAE/AO'Z,t*/c.75R)
(3.17)
KQ = f2 (J,P/D,AE/AOZ,t*/c.75R)
To compute KT and KQ, the following equations are used:
K = K; + AKT
(3.18)
K = K6 + AKQ
The polynomial expressions found in Tables (5) and (6) of
Reference [2] are then used to evaluate to components K '
K6, AKT and AKQ.
Table (5) in Reference [2] lists the coefficients used
in the polynomial expressions for K' and K6 at an equivalent6Q
Reynolds numbpr (Rn 75R) of 2 x 106. It is defined as:
c.75R V(VA) 2 + (0.757Tn D2
Rn 75R A p (3.19)
where:
V = kinematic viscosity of the fluid (ft2 /sec);
c ' blade section chord length at 3/4 propellerradius (.75R) in feet (ft).
46
To account for "other effects", coefficients AKT and KQ are
introduced. Table (6) in Reference [2] lists the coeffi-
cients used in the polynomial expressions for these coeffi-
cients. "Other effects" include the operation of the
propeller at an equivalent Reynolds Number different from
2 x 106. Also, variations in other parameters which define
the propeller's geometry, specifically t/c values different
from the ones fixed by the Wageningen propellers, are taken
into account by corrections to the equivalent Reynolds number.
By keeping the blade section's chord length (c) at the value
of the Wageningen propeller, a change in a blade section's
maximum thickness from the standard one defined by the Series
(t ) to one preferred in the selection (t* ) produces a
new equivalent Reynolds number (Rn75) given by:
75R =exp 4.6052 + 1+2.(t/ 75R) (£n 75R-4.6052)
(3.20)
where:
n.75R = the new eqaivalent Reynolds number;
(t*/c) 7 5R new equivalent t/c at 3/4 propellerradius;
Rn.75R the Reynolds number computed by equation(3.19);
(t/c). 75R standard equivalent t/c at 3/4 propellerradius for Wageningen propellers.
47
For the Wageningen B-Screw Series, the standard equivalent
t/c is given by:
t/C 75R (0.0185 - 0.00125Z)Z (3.21). 75R - 2.073 )E/Ao
Further details on blade section geometry will be addressed
in Chapter V. Reference [2] contains background and other
information on the equations above.
3. Limitations on Series Data
In utilizing the Wageningen B-Screw Series in any
propeller selection problem, the following restrictions
apply to the Series data:
1) Number of Propeller Blades (Z)--The Series considers
only propellers with numbers of blades as shown in Table
(I). Therefore,
Z = 2, 3, 4, 5, 6 or 7 (3.22)
However, the two bladed propeller, i.e., Z = 2, is not very
common in conventional merchant and naval ship designs and,
therefore, is not included in this study.
2) Equivalent Reynolds Number--The Series data, as pub-
lished, is valid only in the range of equivalent Reynolds
numbers given by:
2 x 106 < Rn < 2 x 109 (3.23)
48
IJ
If the equivalent t/c is varied from the standard equivalent
value (t/c).75R' then the new equivalent Reynolds number
(Rn* which results from this variation, must lie within
the same limits. That is,
2x 106 < Rn* < 2 x 109 (3.24).75R -
These limits are appropriate for full-size propellers. For
example, given the following:
a) wt = .22
b) V = 20 (knots)
c) Np = 104 (rpms)
d) D = 25 (ft)
e) c.75R 4.0 (ft)
f) v = 1.2285 x 10 (ft 2/sec)
the value for Rn.75R is equal to 3.619 x 1O7.
3) Pitch-diameter Ratio (P/D)--The series data considers
only pitch diameter ratios in the range given by:
0.4 < P/D < 1.4 (3.25)
4) Advance Ratio (J)--An inspection of the Series results
in graphical format shows that J varies over a range given
by:
0 < J < 1.6 (3.26)
49
L _
5) Expanded Area Ratio (AE/Ao)--Using Table (I), AE/A0
varies over certain ranges depending on Z. This is stated
as:
0.35 < AE/AO < 0.8 Z = 3 (3.27)
0.40 < AE/AO < 1.0 Z = 4 (3.28)
0.45 < AE/AO < 1.05 Z = 5 (3.29)
0.50 < AE/AO < 0.8 Z = 6 (3.30)
0.55 < AE/AO < 0.85 Z = 7 (3.31)
6) Hub didmeter-to-Propeller Diameter Ratio (d/D)--From
Table 37, Section 17 of Reference [16), the Series data
requires that:
d/D = 0.18 Z = 3,7 (3.32)
d/D = 0.167 Z = 4,5,6 (3.33)
F. SUMMARY
From the preceding discussions, the following observa-
tions can be made:
1) the "Design Cases", defined in Chapter I, are examples
of the powering equation solution approaches. That is, Design
50
L _j
Case No. 1 constitutes a "Thrust Approach" problem; Design
Case No. 2, a "Power Approach" one; Design Case No. 3, a
"Matching" problem.
2) equations (3.17) and (3.8) imply that an optimization
solution to the "Design Cases" will involve PE' V, wt, D,
P/D, AE/Ao, (tec). 7 5R' QS and np as possible design variables.
3) when viewed from the concepts on optimization presented
in Chapter II, equations (3.25) through (3.31) constitute
side constraints to an optimized solution of a propeller
selection problem which uses the Wageningen B-Screw Series.
Having noted these points, the propeller selection problem
can now be formulated as a general, non-linear, constrained
optimization problem.
51
I 1
IV. PROPELLER SELECTION--AN OPTIMIZATION PROBLEM
A. INTRODUCTION
The purpose of this chapter is to present the formula-
tion of the propeller selection problem as an optimization
problem that can be solved using COPES/CONMIN. Three usual
restrictions considered by the designer in any propeller
selection analysis are stated as constraints. Then, the
components of U and Y are assembled based on requirements
from previously cited relationships. The restrictions
considered by the designer and the limitations imposed by
the use of the Wageningen B-Screw Series are presented in
inequality constraint format. A formal statement of the
propeller selection problem as an optimization problem is
followed by a review of the GLOBCM common block format and
the basic subprograms used in all three versions of SUBROUTINE
ANALIZ that pertain to each Design Case.
The FORTRAN subprogram listings are found in Appendix
B. Comment cards have been used extensively in the coding
development to assist the reader.
B. DESIGNER'S CONSIDERATIONS
1. Propeller Size
The first restriction on the selection of any
propeller is size. That is, the propeller race in the stern
of the hull under consideration will only accommodate a
52
propeller of some given maximum diameter (Dlim). As a con-
straint on a selected propeller of diameter D, this may be
written as:
D < Dlim (4.1)
or, alternatively, as:
G9(X) D i < 0 (4.2)
2. Cavitation
Another item of importance in propeller selection is
the cavitation phenomenon. When a propeller of given diameter
D and expanded area ratio AE/A0 is operating to produce a
thrust T, the formation and subsequent collapse of water vapor
bubbles on the blade surface, i.e., cavitation, is likely to
occur if the localized surface pressures, usually on the
"back" side of the blade, drop below the pressure at which
the fluid would boil (Pwatvap) in the surrounding environment.
Avoidance of cavitation can be reasonably assured by selecting
a propeller having certain geometric characteristics. A good
empirical relationship that establishes these characteristics
for propellers typified by the Wageningen B-Screw Series is
the Keller Cavitation criterion [Ref. 2: p. 259]. It speci-
fies the minimum required expanded area ratio (AE/AO min ) to
avoid cavitation and is given by:
53
- I mlI
(1.3 + 0.3Z) T(AE/AO)min + atm acghc£- wv " +b (4.3)(Patwatv'apw D7
where:
Z = number of blades;
T = developed thrust (lbf);
D = propeller diameter (ft);
Patm = atmospheric pressure (psia);
Pwatvap = fluid vaporization pressure (psia)
2 4= fluid density (lbf sec /ft
acg = 32.174 (ft/sec2 )
h cz = depth to shaft centerline (ft)
b = constant: 0.1 for Z = 2, 0.2 for Z = .
As a constraint on the propeller selection, this requirement
is writen as:
(AE/Ao) min : AE/A O (4.4)
or
(A E/A 0) minG1 0 (X) = AE/A 1 < 0 (4.5)
3. Strength
The final designer's consideration (for this study),
included in the selection of a propeller, is that of strength.
54
t_ J
Given the propeller's material (promat), selected from Table
(II), and the loadings (T and QS) imposed, it is important to
ensure that the blade's cross sections have proper dimensions
(in an ideal sense, maximum blade section thickness (t*) and
chord length (c)) to ensure adequate strength. Since the
use of the B-Screw Series requires that the chord length
(c) vary as a prescribed function of D, Z and AE/Ao, as given
in Table 1 of Reference [2], the adequacy for strength can
be determined by an appropriately selected value for blade
section maximum thickness-to-chord ratio (t*/c) alone. So,
if t*. is the established minimum blade section maximummin
thickness, then the strength requirement follows from the
constraint given by:
ti/C= (t*/c) mi < t*/c (4.6)
The fact that blade section maximum thickness for the B-Screw
Series varies linearly with the propeller radius (R) allows
the strength constraint (equation (4.6)) to be evaluated at
one section along the radius. This point is chosen to be at
the 3/4 radius (.75R). Therefore, equation (4.6) becomes:
(t*/c).75R min (t*/c).75R (4.7)
or
(t*/c). 75R min 1 < 0 (4.8)GII(X) --- /c).75
Reference (2] suggests the following empirical relation for
the minimum required equivalent blade section maximum
thickness-to-chord ratio (t*/c).75R min
Z 0.0028+0.21 (2 375. .2 .P/D)PD2 2
4.123N D (S + - p (t*/c).75R min 2.073 AE/A 0 12.788
(4.9)
where:
D = propeller diameter (ft);
PD = delivered power (hp);
Np = propeller revolution rate (rpm);
Sc = propeller material allowable stress (psi);
P/D = pitch-diameter ratio.
However, in Chapter VI of this thesis, an algorithm which
employs the Schoenherr formulation [Ref. 8] with some modi-
fications, is presented as an alternative to equation (4.9).
C. THE DESIGN VECTOR
In view of the preceding presentations on optimization
and powering, the design vector D can be assembled for the
general propeller selection problem utilizing the B-Screw
Series. This vector is composed of preassigned parameters
relating to environmental conditions, hull characteristics
56
and the propeller which are required for various equations
and the design variables.
1. Parameters
a. Environmental
These parameters pertain primarily to the fluid
conditions in which the propeller operates and to the atmos-
phere. Required for various calculations, they are:
1) fluid temperature (*F)--Temp
2) fluid density (lbf sec 2/ft 4)--p
3) fluid viscosity (ft 2/sec)--v
4) fluid vaporization pressure (psia)--Pwatvap
5) atmospheric pressure (psia)--patm
b. Hull Characteristics
These parameters pertain to certain details pre-
scribed for the hull under study in the powering analysis.
They are:
1) wake fraction--wt
2) thrust deduction--td
3) relative rotative efficiency --nR
4) number of propellers--noscrw
5) shaft centerline depth (ft)--hck
6) propeller diameter limit (ft)--Dlim
c. Propeller
These parameters are specified in view of their
discrete-valued nature. They are:
1) number of blades--Z
2) material--promat
4 57t_ __ _J
Table (II) lists materials and properties considered in this
study. These values are taken from Table (35), Section 15
of Reference [16].
TABLE II
Material Identifier Reference
promat Material Allowable Stress--Sc Density--wd(psi) (lbf/in3
1 Cast Iron 3600--3950 .260
2 Cast Steel 5915--6265 .289
3 Type 2 Bronze 7200-7585 .305
4 Type 4 Ni-Al 8910--9430 .278Bronze
5 Stainless Steel 5400--5500 .283
2. Design Variables
In view of equations (3.13) through (3.17), the
design variables common to all selection approaches are:
1) PE
2) V
3) D
4) P/D
5) A o
6) (t*/c).75R
7) Np
8) 0s
The evctors 5 and X are shown schematically in figure (4.1).
58(
ii
Temp
p
V
Pwatvap
Patm
wt
td
no scrw
U D lim
zpromat
V V
D D
X = P/D P/D
AE/AO0 A E/AO
(t*/c) .75R (t*/c).75R
Figure 4.1 Design Vectors Dand
59
I *I I
D. CONSTRAINTS
Besides the constraints imposed by equations (4.2),
(4.5) and (4.8), equations (3.25) through (3.31) are rearranged
to the format of constraints in equation (2.9). They are
listed as follows:
1) Equivalent Reynolds Number--Equation (3.25) becomes:
1 < .75R-< 2 .75R < 1000 (4.10)
2 x 106
Two constraints are derived:
GRn* 75R106)=l----~ < 0 (4.11)2 x 10
Rn. 75RG(X) = 2ix10- 1000 < 0 (4.12)2 x×106 -
2) Expanded Area Ratio--Equations (3.27) through (3.31)
become:
(AE/Ao) lower(Z) < AE/AO (AE/A) upper (Z)
(4.13)
Two constraints are derived:
G5 (X) = (AWAo)lower(Z) - AA O < 0 (4.14)
G6 (X) = AE/AO - (AE/AO)upper(Z) < 0 (4.15)
60 JtL
3) Advance Ratio--Equation (3.22) becomes:
Two constraints are derived:_J
G1 (X) = -7 < 0 (4.17)
J
G2 (x) - 1 < 0 (4.18)
4) Equivalent Blade Section Maximum Thickness-to-Chord
Ratio--Using equation (3.19), boundaries on the range of (t*/c).75R
are defined by:
1 < (t*/c) < 4(t/c) .759f(t/c).75R - . 4 (4.19)
Two constraints are derived:
G 7(X) = l(t/c).75R - (t*/c) < 0 (4.20)7 2 .75R
Gs(X) = (t*/c) 75R- 4 (t/c)75R < 0 (4.21)
E. OBJECTIVE FUNCTIONS
Upon consideration of equation (3.13), Design Case No. 1
and Design Case No. 2 require that the open water efficiency
(no ), given by equation (3.11), be maximized. In terminology
related to optimization, this is stated as:
61
C I
OBJ1,2 no (4.22)
Design Case No. 3, the "matching" problem, requires that the
blade weight (bldwt) be minimized. This is stated as:
OBJ3 = bldwt (4.23)
F. PROPELLER SELECTION OPTIMIZATION PROBLEM STATEMENT
As a general, non-linear constrained optimization problem
to be solved by COPES/CONMIN, the propeller selection problem
for all Design Cases may be stated as one equation given by:
Minimize: F(X) = OBJ1, 2 or OBJ 3 (4.24)
Subject to: Gj(K) < 0 j = 1,...,12
Xlower < X. < Xu p p e r i1 - 1 - 1..
The constraint G (X) and the values for Xlower and Xupper12 1
will be specified according to each Design Case.
G. CODING FUNDAMENTALS
1. GLOBCM Common Block
The GLOBCM common block, required by COPES/CONMIN,
is now assembled. Table (III) specifies the assignment loca-
tions for the FORTRAN variables which define objective
functions, design variables and constraints.
2. SUBROUTINE ANALIZ
While each Design Case uses a different approach,
all analyses are very similar. Therefore, each SUBROUTINE
62
K -
T III
Global Comnon (GLOBCM) Catalog
Global FORTRAN DEFINITION
Location Name
1 ETAO no
2 WEIGHT bldwt
3 AEDVAO AE/Ao
4 DIA D
5 N Np = 60 •np
6 PE PE
7 PDIVD P/D
8 QS QS
9 TC75R (t*/c)75R
10 V V (ft/sec)
11 RJCHL Gl (X)--eqn (4.17)
12 RJCNU G2 (X) -- eqn (4.18)
13 R75RCL G3 (X)--eqn (4.11)
14 R75RCU G4 (X)--eqn (4.12)
15 AEAOCL G5 (X) --eqn (4.14)
16 AEAOCU G6 (X)--eqn (4.15)
17 TC75CL G 7 ()--eqn (4.20)
18 TC75CU G8 (X)--eqn (4.21)
19 POWBAL G1 2 (K)--eqn (7.10) or (8.11)or (9.4)
20 DIACNU G9 (X)--eqn (4.2) or (9.6)
21 AEAOCV G10 (X)--eqn (4.5)
22 TCSTRS G1 1 (X)--eqn (4.8)
23 RJ J
63I ,.. j
ANALIZ shares a common structure and other common subroutines
which perform calculations required in all cases. Appendices
C, F and I contain, respectively, the source listings of
SUBROUTINE ANALIZ for Design Case No. 1, Design Case No. 2
and Design Case No. 3.
a. Structure
The structure common to all cases follows
accordingly:
1) all initialization of environmental, hull and propeller
parameters is accomplished in the input section (ICALC = 1).
2) evaluation of KT and KQ, all constraints and appropriate
objective functions (-no or bldwt) are accomplished in the
execution section (ICALC = 2).
3) output of results for each optimization problem is
accomplished in the output section (ICALC = 3).
b. Basic Subprograms
The following FORTRAN subprograms are used in
all three SUBROUTINE ANALIZ codes:
1) SUBROUTINE CH75RA--calculates the equivalent blade
section chord length (c 7 5R) for the propeller using Table 1
(Ref. 2, p. 252].
2) SUBROUTINE REY75R--calculates the equivalent Reynolds
number (Rn*75R) using equations (3.19) and (3.20).
3) SUBROUTINE COEFSA--calculates the thrust and torque
coefficients (KT and KQ) through sequential calls to SUBROU-
TINE CALCKT and SUBROUTINE CALCKQ. The polynomial expressions
64
L J
(Tables (5) and (6), [Ref. 2]) for these coefficients are
contained in SUBROUTINE CALCKT and SUBROUTINE CALCKQ respec-
tively.
4) SUBROUTINE OPWEFF--calculates the open water efficiency
(no ) using equation (3.11).
5) SUBROUTINE JCNA--calculates the constraints on the
advance ratio (J) given by equations (4.17) and (4.18).
6) SUBROUTINE REYCNA--calculates the equivalent Reynolds
number constraints given by equations (4.11) and (4.12).
7) SUBROUTINE EXTCCN--calculates the constraints on ex-
panded area ratio (AE/AO) and equivalent blade section maxi-
mum thickness-to-chord ratio (t*/c 7 5R) given by equations
(4.14), (4.15), (4.20) and (4.21).
8) SUBROUTINE DICNUA--calculates the constraint on the
propeller diameter (D) given by equation (4.2) using the
hull parameter on maximum diameter (Dlim).
9) SUBROUTINE CAVCNA--calculates the constraint for
cavitation given by equation (4.5) using equation (4.3).
10) SUBROUTINE STRCNA--calculates the constraint for
strength given by equation (4.8) using equation (4.9).
H. SUMMARY
The propeller selection problem has now been formulated
as a constrained optimization problem which can be solved
by COPES/CONMIN. Two items remain for discussion before
proceeding to specify the final details pertaining to each
SUBROUTINE ANALIZ code and to present numerical examples.
These items are:
65j
1) the theory and coding relating to the computation of
the propeller's blade weight for the evaluation of the objec-
tive function in Design Case N4o. 3 (OBJ 3).
2) the theory and coding relating to the computation of
the minimum required equivalent blade section maximum thickness-
to-chord ratio (t*/c 7 )in for use in the alternative
evaluation of the strength constraint given by equation
(4.8).
66'
V. PROPELLER BLADE WEIGHT--AN OBJECTIVE FUNCTION
A. INTRODUCTION
In this chapter, the method for the computation of the
propeller's blade weight (bldwt), the objective function
OBJ, is examined. First, a brief overview on the steps in
the computational procedure is presented. The FORTRAN
subprogram SUBROUTINE WGTCAL developed from the algorithm is
then described. Again, Appendix B contains all subprogram
listings.
B. THEORY AND PROCEDURE
Given the material of a propeller, the calculation of
the weight of one blade involves nothing more than a volume
calculation, a relatively routine task performed by most naval
architects/marine engineers. Analagous to the determination
of the underwater volume of a ship's hull, the calculation
is an integration of blade section profiles' cross-sectional
areas over the propeller radius (R).
1. Limits of Integration
Figure (5.1) depicts a side elevation view of a
blade and hub, parallel to the propeller shaft axis. The
cross-hatched area indicates the trace of the volume to be
calculated. In view of equations (3.32) and (3.33), limits
of integration are from r - .167R to r = R for Z = 4,5,6 and
r - .18R to r = R for Z = 3,7. For convenieance, a non-
dimensional variable "x" will be defined as:
67
1.0
t 75R
.75 -
.50 -R
.25 -
. 20
.18 j~.167I
(Z=3, 7) d/D
d/D (Z=4,5,6)
x~r/R
Figure 5.1 Propeller Blade &Hub--Side View
j 68
_X
iR
r (5.1)
Limits are now expressed as x = .167 or .18 to x = 1.0.
Quite obviously, R = (D/2.0).
2. Blade Section Profile
Figure (5.2) depicts an expanded cylindrical blade
section in profile view at a given r or x. For the Wageningen
B-Screw Series, the profile is defined, geometrically, by
a succession of vertical ordinates which specify points along
the blade section's profile on the "face" (yf) and on the
back (yb) with respect to the pitch reference line. At any
r = xR, ver-ical ordinates for "aft" (P < 0) and "fwd"
(P > 0) portions of the blade section are determined by:
Yfa = V1 (t* - te) P < 0 (5.2)
= (V+V 2 ) (t*-t* ) + t* P < 0 (5.3)
and
yff =V(t* - t*e) P > 0 (5.4)
Ybf (V+V 2 ) (t*-t*e) + t*e P > 0 (5.5)
where:
VIV 2 = tabulated values depending on x and P(see Tables (2) and (3), Ref. [2]);
69
L - I
-4 -
00
0
W riH -4
-4'
441
4'4
T) HH
-444
z UU
11-4
Z >1
rz~ "4
-44
4J'
E-14
rz~70
t~e = blade section leading edge thickness (ft);
t~ = blade section trailing edge thickness (ft).
Units for Yfa' Yba' Yff and Ybf are feet (ft). For this
study, a reasonable assumption is made in that:
t = t* = (1) t* (5.6)9.e te T
3. Blade Section Cross-Sectional Area
The cross-sectional area at each x = r/R (A(x)) is
determined by:
9 10A(X) = Aa + I Ai (5.7)
i=l j=l
where
h + hA = ai 2 (5.8)
h = Ybai - Yfai (5.9)
hai+l Ybai+l - Yfai+l (5.10)
Expressions for Af, hfi and hfi+l follow in similar fashion.
Values for yfai and Ybai are determined at 9 points
along the "aft" portion of a given blade section's chord (c):
values of yffi and Ybfi are determined at 10 points along the
71
L_ ..J
"fwd" portion. The values for APai and APfi are fractional
values of the blade section's chord length (c) at radius
r = xR as determined from Tables (2) and (3) in Reference
[2]. The units for A(x) are square feet (ft 2). Units for
hail hai+l, hfi, hfi+l, c, APai and APfi are feet (ft).
4. Volume Integration
The blade volume (bldvol) is finally determined by
using Simpson's Rule for integration of A(x) along the non-
dimensional radius x using appropriate limits.
5. Blade Weight
Once the blade volume (bldvol) is calculated, the
weight (bldwt) is determined by:
bldwt = bldvol - wd • 1728 (5.11)
where:
bldvol = volume of one blade (ft 3);
wd = material weight density (lbf/in 3).
Weight Density (wd) depends on blade material (promat).
Table (II) lists appropriate values.
C. CODING
SUBROUTINE WGTCAL is the main subprogram for the blade
weight calculation. It, in turn, calls the following FORTRAN
subprogram for various calculations:
L 72
1) SUBROUTINE TDIST--generates, at specified radius
values, a distribution of blade section maximum thicknesses
(t*).
2) SUBROUTINE BLDPRP--generates, at specified radius values
(i.e., r = .167R or .18R, .2R, .3R, .4R, ..., .9R, 1.OR),
various "blade section properties", one of which is a blade
section's cross-sectional area given by equation (5.7).
Other properties which are determined (for later use in
direct stress computations) include blade section chord
lengths and centroids and "critical point" locations as
defined in Chapter VI.
3) SUBROUTINE BLDVOL --performs a Simpson's Rule integra-
tion for the propeller blade volume (bldvol) using blade
section cross-sectional areas generated in SUBROUTINE BLDPRP.
The blade weight (bldwt) is computed as a final step in the
main subprogram SUBROUTINE WGTCAL.
Examination of the codes in Appendix B reveals extensive
use of common blocks for passing data from one subprogram to
another. Comment cards provide a full definition of all
common blocks as well as a description of the task being
performed at various points in a given subprogram.
D. SUMMARY
The coding developed for this study is, admittedly, not
very compact and efficient. However, the intention has been
to write all codes with sufficient documentation in order to
facilitate the reader's understanding of the algorithms employed
as well as to make the author's debugging work easier.
73
L __J
VI. THICKNESS-TO-CHORD RATIO--A DESIGN CONSTRAINT
A. INTRODUCTION
The purpose of this chapter is to examine the development
of an algorithm that will be used to determine the minimum
required equivalent blade section maximum thickness-to-chord
ratio used in equation (4.8). The formulation is based upon
the method developed by Dr. Karl E. Schoenherr [Ref. 81 in
1963. After a review of the past and present methods employed
in propeller strength analysis is conducted, a description of
the Schoenherr model and a list of the assumptions used with
that model is presented. Then, a brief restatement of his
model's equations which are used in the algorithm is followed
by a derivation of the author's modifications to the Schoenherr
method. The chapter is completed by conducting a review of
the theory and coding employed by the algorithm.
The principal reference which is cited throughout this
chapter is, again, Reference [8]. The reader is encouraged
to review this reference for further details.
B. PROPELLER STRENGTH ANALYSIS--A HISTORICAL REVIEW
Marine propeller blades present a special class of struc-
tural problem. That problem lies in the difficulty of des-
cribing a blade design in simple mathematical terms for
subsequent analysis through various means. Until the "finite
element method era", analytical methods, including the one
74
• m J
[Ref. 8] adapted for this study, relied heavily on practical
experience of the propeller designer and semi-theoretical
considerations. Analysis by these methods provided a cri-
terion of stress rather than actual computation of stresses.
These methods for predicting blade stresses were developed by
using "beam' theory or "shell" theory.
The use of elementary beam theory in propeller strength
analysis was first adopted by Taylor (Ref. 22]. He treated
a blade as a cantilever beam attached to the propeller hub
and loaded by thrust and torque forces distributed linearly
over the propeller radius. His approach is often deemed a
"Imodified beam theory" because he chose to calculate the
direct stresses using the moment of inertia properties of
expanded cylindrical blade sections with neutral axis parallel
to the nose-tail (pitch-reference) line or chord of that
expanded section. Reasonable estimates of stresses along the
blade surface were achieved for the unraked, unskewed and
narrow-bladed propellers of his time.
As propellers "modernized" and became skewed and wider
with increasing rake (mostly aft), modifications, improvements
and alternatives to Taylor's theory were developed. Prin-
cipally, modifications by Rosingh (Ref. 23] and Hancock
(Ref. 24] proposed using moment of inertia properties of a
blade section that was normal to the generating line of the
axially projected blade outline. Ronson [Ref. 25] later
improved Taylor's theory for application to wide-bladed
75
propellers. Morgan [Ref. 26] provided an improved method
for calculating the geometric properties of "modern" airfoil-
shaped blade sections. Aernoldus and Keyser's [Ref. 271
"quasi-static" modeling of the propeller blade allowed
for additional consideration to stresses induced by centri-
fugal loading of the raked and skewed blade. The beam
theory approaches to propeller blade stress analysis culminated,
for all practical purposes, with Schoenherr's work [Ref. 8]
in 1963.
Alternatives to Taylor's beam theory approach, prior to
1963, consisted of the application of "shell" theory to the
propeller blade strength problem. This approach was first
proposed by Conn [Ref. 28] and subsequently formulated by
Cohen [Ref. 29] who modeled the blade as a helicoidal shell
with variable thickness and infinite width. "Shell Theory"
was utilized again in experimental studies by Connolly
[Ref. 30] who, like his predecessors, was also forced into
making an assumption about the behavior of the displacements
of the blade sections (i.e., constant displacements normal
to the constant pitch blade at each fractional radius dis-
tance from the hub) beyond usual assumptions of shell theory.
Essentially, his experimental results on one specific pro-
peller contradicted the computational values. Attempts at
a generalized numerical solution to Connolly's equations
appeared in 1963 [Ref. 31] and 1964 [Ref. 32]. In 1968,
Atkinson [Ref. 33], compared Connolly's results with currently
76
adopted cantilever beam methods and found, based on the
inconsistency of results, that it was rnot possible to_
recommend one method over the other in the blade strength
design procedure; another approach was needed.
Commenting on Atkinson's paper at that time, Sontvedt
[Ref. 34] pointed out that, in view of these inconsistencies,
the only approach to blade strength analysis which did not
require very broad assumptions was the finite element tech-
nique. Developments in the method at that time were provid-
ing a new and powerful tool for structural analysis. The
propeller blade was just another application. Genalis [Ref.
35] developed codes for the determination of displacements
and stresses in a blade under hydrodynamic loads using the
FEM technique and modeling the blade as a shell, a 3-D ele-
ment mesh of tetrahedrons and rectangular prisms and, finally,
a composite of shell and 3-D elements. As an aside, a
finite "difference" solution to Connolly's analytical equa-
tions was proposed in 1972 [Ref. 36]. In 1973, Atkinson
[Ref. 37] reported the application of both hydrodynamic and
centrifugal loads to a blade modeled by a thin-shell triangu-
lar mesh and a thick-shell parabolic and cubic curved element
mesh. The results of the triangular element were considered
unsatisfactory. Another use of the thin-shell triangular
element was reported by Sontvedt [Ref. 34] in 1974 using the
SESAM-69 code (Ref. 381.
The need to model the blade correctly near the hub,
where root stresses are usually critical, necessitated the
77
consideration of 3-D elements in lieu of thick/thin-shell
elements (Refs. 39,40]. The use of the 4- and 10-noded
tetrahedral element (i.e., TET-4, TET-10 respectively) to
construct a blade mesh was conducted by Beek [Refs. 41,42].
He observed that improved accuracy of stress values, achieved
by the use of these meshes, were overshadowed somewhat by
the extensive storage capacity required for each analysis.
Another "natural" improvement, from a geometrical standpoint,
appeared in 1978. A general 3-D curved isoparametric element
was incorporated in a computer code [Ref. 43] developed by
Ma based on his previous formulation work (Ref. 44].
The finite element approach will continue to grow in use
in propeller blade strength analysis with each successive
improvement made to the basic elements which are used in the
mesh generation of the blade. But, until the computer storage
problem is resolved to the point where one mesh generation
and subsequent stress analysis of one particular blade be-
comes a minor processing task, the basic analytical techniques
will continue to be a meritable "check" [Ref. 45] in the
preliminary (or conceptual) phase of propeller selection/
design. In this context, the method formulated by Schoenherr
and his colleagues twenty years ago is considered for adoption
in determining a minimum required blade section maximum
thickness-to-chord ratio.
78
t _ _.
C. SCHOENHERR'S METHOD
1. Background
Schoenherr's method is applicable to preliminary
(or, conceptual) propeller selection problems because it
employs an assumed thrust and torque force loading distribu-
tion for the propeller blade. This assumption is made at
this stage of the ship's design because the exact wake
velocity distribution at the propeller race is generally
not known.
Also, his method is applicable to propellers repre-
sented by the Wageningen B-Screw Series because the blades of
these propellers meet Schoenherr's criteria for the blade
types covered by his formulation. Specifically, B-Screw
Series blades have:
1) a small constant rake angle of 150 over the entire
blade radius;
2) a constant pitch distribution over the propeller
radius with the exception of the Z = 4 propeller whose pitch
is slightly reduced near the hub;
3) mild skew;
4) linear distribution of blade section maximum thickness
over the radius of the blade;
5) aerfoil profile qualities where the nose-tail line
or the chord of the blade section is approximately parallel
to the pitch reference line.
79
I_ -J
2. The Blade Model
Schoenherr models the propeller blade as a cantilever
beam with unsymmuetrical and variable area cross sections
subjected to loading distributions of hydrodynamic and cen-
trifugal forces. The following additional assumptions apply:
1) Flexure theory applies. This subsequently implies the
following: a) plane cross sections remain plane under load,
b) Hooke's Law is valid, c) the blade material is homogeneous
and isotropic, d) fibers are free to extend and contract
independently of adjacent fibers, and e) stresses at a point
arising from various forces superimpose.
2) Shearing stresses and their effects are neglected.
Only the direct stresses on a strength section are taken
into account.
3) The strength sections are taken to be the expanded
cylindrical blade sections at various radial locations.
4) The neutral axes of a strength section are straight
lines passing through the centroid of the expanded cylindrical
blade section and are parallel and normal, respectively, to
the pitch reference line, and therefore, the chord, at each
blade section.
5) Bending Moments are applied in two planes which are
mutually perpendicular to each other. One plane is normal
to the pitch-reference line (and chord) of the strength
section; the other is parallel.
6) The angle between the principal axes of inertia and
the neutral axes is zero.
80
Using this model and assumptions, Schoenherr applies
the following irmula for the evaluation of the direct fiber
stress ([ao]) in a blade section at a radius r = ro:
[M0 (]n uO [M]Eo wO [___ O
[a]o no o A(x) (6.1)
where:
EM] no, [M]o resultant bending moments in planes0 normal ((MI ) and parallel ((M] 2 o to
the strengtR section's chord at r = r(ft-i1bf) ;
[F o centrifugal force acting normal to theplane of the strength section at r = rand resulting from the centrifugal 0acceleration of the remaining bladeelement mass above that strength section(lbf);
uo , w0 = coordinates of a point on the strengthsection's periphery with respect tothat section's neutral axes system (2-nsystem) (ft);
A(x0 ) = strength section's cross-sectional area(ft);
Izo = moment of inertia of the strengthsection with respect to the "I" axis
(ft4);
I = moment of inertia of the strengt sectionno with respect to the "n" axis (ft");
x 0= non-dimensional radius given byx0 = ro/R.
Since equation (6.1) indicates that the direct fiber stress
is greatest at points on the periphery of the strength sec-
tion, Schoenherr selects to examine four "critical points"
81
kL I
on the periphery where the fiber stress is likely to Le a
maximum. These points are designated as C, (, C and
O on Figure (6.1) and are specified by coordinates (ul,wl),
(u2,w2), (u3,w3) and (u4,w4) respectively in the "9-nU
reference system.
The values for (MIno and (M] to are determined by
the following relation:
[Mno [pno + [Mcb]no (6.2)
[M]o= (MplZo + [Mcb]zo (6-3)
where:
[Mp]no = total bending moment due to hydrodynamicloading acting in a plane normal to astrength section's chord at r = r(ftlbf);
[Mp]o= total bending moment due to hydrodynamicloading acting in a plane parallel to
a strength section's chord at r = r(ftlbf);
[Mcbno = total bending moment due to centrifugalloading acting in a plane normal to a
strength section's chord at r = r0(ftlbf);
[Mcbo = total bending moment due to centrifugalloading acting in a plane parallel to
a strength section's chord at r = r(ftlbf).
3. Bending Moments Due to Hydrodynamic Loading
The derivations of [Mp] no and [Mp]£o follow directly
from Part I of Schoenherr's paper [Ref. 8: p. 83-89] and,
82
I l I
0
4-))
o04-
rzW
04-
00
4) -H.44
00010
4 J 4 .J -4
414.zu 44-4U
1-4 E4 f-4
(D-40
83
therefore, only key equations will be restated. References
to equations which contain no decimal point apply to equations
as numbered in his paper.
Thrust and torque force are components of the hydro-
dynamic "lifting" force acting on a blade. Using an assumed
non-linear distribution of thrust along the blade radius
given by equation (2), Schoenherr derives the following ex-
pression for the bending moment due to thrust ([M t]o ) which
acts at a blade section located at ratius r = r
TR 02 (x0 )
[Mto z (6.4)
where:
T = propeller thrust (lbf);
R = propeller radius (ft);
Z = no. of blades;
02 (xo),Ol(xh) = functions of non-dimensionalradius x evaluated at x = r /Rand xh = 0.2 and given By eqaa-tions (4) and (9)
For the bending moment due to torque ([M q] ) which
acts at a blade section located at radius r = ro , Schoenherr
derives the following:
E~qJ =Qp 1 2 (Xo)
-M (6.5)[q1 o z 0 (Xh)
where:
84
L_ I.
Qp = propeller torque (ft-lbf);
Z = no. of blades;
2 (Xo0),l(Xh ) = functions of non-dimensional radius xevaluated at x = r /R and x = 0.2and given by e~uatiSns (19) Xnd (4).
Figure (6.2) depicts the component resolution for
[MP]no and [Mp]zo which results when equations (6.4) and
(6.5) are imposed at a strength section at r = rO which has
a pitch angle Bo . The following relations are derived as
equations (42) and (43) in Schoenherr's paper:
EMplno = [Mt] o cos a° + [M q] sin ° (6.6)
[Mp]Yo = [Mt]0 sin B° - [Mq] O cos o (6.7)
4. Force and Bending Moments Due to Centrifugal Loading
The derivation and expressions contained in this
section constitute the author's modifications to the formu-
lation in Part II of Schoenherr's paper. In Part II, Schoen-
herr's derivations for [Mcbno and [Mcb]zo are formulated for
computation using a propeller drawing. This follows from
the fact that his method, which was funded by the American
Bureau of Shipping, was intended to be used as that classi-
fication society's "designer's check" on adherence to the
Bureau's strength criteria from a propeller blueprint. To
evaluate the direct fiber stresses from equation (6.1) for
the Wageningen B-Screw Series in accordance with Schoenherr's
85
L ]
mz PE-4
04 4
E-4z
01E-4
00
040
04.J
0 En
.41
Cz~CD0
00
cia)
86
method, [M]cb no and [Mcb] o must be evaluated from expres-
sions derived from the available information on blade sec-
tion profiles and other geometric characteristics which
are contained in Reference [2] and previously used in Chapter
V.
Consider Figure (6.3) where the centrifugal force
fCF] o of a blade element above a blade section at x = x. = ro/R
acts in a radial direction from the shaft centerline. Its
line of action passes through point "N", which is on the same
cylindrical surface as the blade section at x = xo = ro/R,
and through point "G", which is the center of gravity of the
blade element above the blade section at x = x° = r /R.
From the figure, the following expression is derived:
[CF]0 = (CFlo cos C(xO) + (CF]° sin (xO) (6.8)
[CF]0 is shifted to point "N" and is decomposed into
components [CF] cos (x,) and [CF]o sin '(xo). The entire
cylindrical surface in which the blade section at x = x° = r /R
and point "N" lie is now expanded into a flat plane for further
consideration (see Figure (6.4)). In this configuration,
[C F] cos (x ) is normal to this flat plane while [C F o sin ?(x0 )
lies in this plane.
Let point "0" be the location of the blade section's
neutral axes system (i.e., the i-n system). Then, the forces
and moments due to the centrifugal reaction of the blade ele-
ment above this section act at point "0" and are given by:
87
'1 1|i
1-4-4
0.
04J
.4)
4
00
X'-'
;0 I 0
- 0
444
2 00
E-4)
H
E-4
H
X0
0
4 J
U
V
E 0
41
89
[FC] 0 = [CF]o Cos ; (xo) (6.9)
ED ]O = [CF]o sin (xo) (6.10)
[M b] o = [Fc I No- (6.11)
[Mcw 0 = [DC]o • - (6.12)
where:
[F c = direct force, due to centrifugal action,acting on the blade section located at
x = x0 = r /R;
[Dc]O = shear force, due to centrifugal action,acting on the blade section locatedat x = xo = ro/R;
[M cb = bending moment at point "0" imposed byS 1F]0 acting through point "N";
[M cwI o = torsional moment at point "0" imposed byED CIO acting through point "N".
Since Schoenherr's method does not consider shear forces and
their effects, [Dc]0 and [McI will not be considered in
this modification. However, [F c1 and [Mcb 0 must now be
computed for each blade section along the propeller's radius
in order to account for their contributions to equations
(6.2), (6.3) and, finally, in equation (6.1).
The computation is derived as follows. Consider
Figure (6.4). Again, [F C O acts through point "N" and is
normal (outward) to the plane of the figure. [Mcb] 0 is now
resolved into components of the X-n axes system as follows:
90
F-
[Mcb ]no [CFlo cos (x {po sin ao + qo cos 80 + Yco) (6.13)
[Mb] = [C1 cos ;(x o ) {qo sin 80 -po cos 80 + Xco} (6.14)
where:
8o = pitch angle of the blade section atx = x0 = r /R;
X distance of the blade sections centroid fromthe generator line (ft);
Y c = distance of the blade section's centroid fromthe pitch reference line (ft);
qo = distance to point "N" from point "P"parallel to the shaft axis at x = x° r /R(ft);
Po = distance to point "N" from point "P"perpendicular to the shaft axis atx = x0 = r /R (ft).
00The quantity 80 is found by the relation:
tan 8 1 (P/D) (6.15)T O 0
For the Wageningen B-Screw Series, (P/D) is a constant along R
except for propellers with Z = 4.
The quantity [CF] 0 is computed from the relation:
wdV o 2 (iG) (6.16)[CF]0 = 1728 acg (27rn) Ro
where:
91
• .. i.
C _
I1
Vo = volume of the blade element above the bladesection at x = x. = r /R (ft3 );
2acg = 32.174 ft/sec
wd = material weight density (lbf/in3);
np P = propeller revolution rate (rps);
(xG)o = non-dimensional radial position of "G"for the blade element above the shaft
axis;
R = propeller radius (ft).
At this point, only five quantities remain to be determined
for the evaluation of the expressions of equations (6.10),
(6.13) and (6.14). They are:
1) cos (x o)
2) p0
3) qo4) (x G) 0
5) Vo
These quantities are determined by integration over the
blade element above the blade section, located at x = xo = ro/R ,
from x = x0 to x = 1.0.
The values for p0 and q0 will vary with the location
of "G" (from Figure (6.3)) which depends on x. Consider a
radially thin slice of the blade element above the blade
section, located at x = xo = r0/R (see Figure (6.5)). This
thin "slice" is located at a non-dimensional distance x from
the shaft centerline where x0 < x < 1.0. Figure (6.6) depicts
this section expanded onto a plane. Let "g" be the centroid
of that "slice". If x is the distance from the generator
92
(U
0
.4.
0
-4
11 44x 0
4-40
0.41
H
lid~
93
H
0
>I1
4J-
144
00
00
ra 0
4-1
Ci)
0
41)
V0i0
.V
L 94J
VI
line to "g" in feet and y is the distance from the pitch
reference line in feet, then, from Figures (6.5) and (6.6),
the following relations apply:
i = x R tan n (6.17)g
i -a = i9-x sin g -y cos ag (6.18)
t = x cos 8 - y sin (6.19)
where:
n = rake angle at x;
x = distance of "g" from point "P" parallel tog the shaft axis (ft);
yg = distance of "g" from point "P" perpendicularto the shaft axis (ft).
For the Wageningen B-Screw Series, rake angle n is a constant
150 everywhere along the radius R.
Now, to compute the volume of the blade element above
the blade section located at x = x. = ro/R, the following
expression is used:
1V0 = f RA(x) dx (6.20)
x=XO
To compute the non-dimensional radial position of
"G" for the blade element above a blade section located at
x = x0 = 0 /R, the following expression is used:
95
JJ
AD ̂ At32 414 PRELIMINARY PROPELLER SELECTION USING THE WAGENINGEN IA/1B-SCREW SERIES AND A GENERAL PURPOSE NON-LINEAROPTIMIZER(Ul NAVAL POSTGRADUATE SCHOOL MONTEREY CA
'I ' IFI M P SMITH JUN 8 F/G 13/10 NL
llllt lillEil n1111111IIIIIIIIIMIIIIhhIIIIIIII
IIIIIIIIIIII
liii I*Q~~ 2.2go11112-
00 13
1f A(x) xdx
()o-- = 0 (6.21)
f A(x) dxx=x
0
To compute the tangential position of "G" for the
blade element above a blade section located at x = xo = ro/R,
the following expression is used:
1f A(x) t dx
X=X 0
T = 0 (6.22)o 1f A(x) dxX=X
0
And, finally, to compute the axial position of "G"
for the blade element above a blade section located at x = x°
= r /R, the following expression is used:
1f A(x) (ig-ag) dxxo g
Ao = 1 (6.23)
f A(x) dxx
0
Using the values just determined for (3EG)o , To and
A0 , the following expressions are used to evaluate p0 and
qo :
xP0 = T T (6.24)(xG)o
96
t I
qo= AO - xO tann (6.25)
The expression for cos i(x ) follows:
cos(X) =--- (6.26)0
The formulation is now complete. Equations (6.10),
(6.13) and (6.14) can now be evaluated for any blade section
located at x = x0 = r /R. From here, equations (6.2) and
(6.3) are evaluated. Finally, using the results from these
equations and equation (6.9), equation (6.1) can be evaluated
for the four points, specified by coordinates (ul,wl),
(u2,w2), (u3,w3) and (u4,w4), at any location x = x0 = r0 /R.
From the development discussed in Chapter V, the
values for A(x), Xg yg, I no and I o are readily available.
D. ALGORITHM FOR THE CONSTRAINT
1. Theory
Schoenherr's formulation with the modifications just
derived can be used in determining the minimum required
equivalent blade section maximum thickness-to-chord ratio
((t*/c).75R min ) for use in the constraint Gil(X) < 0 given
by equation (4.8). The procedure employed is as follows:
1) assume an initial value for (t*/c).75R mi using
equation (3.21);
2) increase this value by a small amount;
97
_ J
3) using (t*/c) .75R rain obtained from step (2), generate
a distribution of minimum required blade section thicknesses
(t*. ) for blade sections at specified points along themin
propeller radius, say at r = .2R, .3R, .4R, .5R, .6R, .7R,
.8R and .9R;
4) determine all blade section properties to include:
a) cross-sectional area, b) chord length, c) centroid location,
d) moments of inertia with respect to the principal axes
system (i.e., X-n system), e) coordinate values for the four
critical points defined in the previous section;
5) compute the hydrodynamic bending moment components
[MP]no and CMp]Zo at radius locations just specified;
6) compute the values of the centrifugal force [F c1 and
the bending moment components [Mcbjno and [McbJ o acting on
blade sections at radius locations just specified;
7) calculate the direct fiber stresses at all four
critical points for all radius locations specified in step
(3);
8) check the following condition on the calculated fiber
stress at all four critical points at all specified radius
locations using:
a]o 1 144 - Sc (6.27)
9) if the maximum allowable stress (S c ) for the material
is exceeded, then return to step (2) and repeat. Otherwise,
proceed to next step.
98
-L J
10) since the minimum required equivalent blade section
maximum thickness-to-chord ratio assumed in step (2) has
produced blade sections of adequate strength, evaluate the
constraint given by equation (4.8).
2. Coding Details
The algorithm just outlined is incorporated into the
main FORTRAN subprogram SUBROUTINE STRCNK. This subprogram,
in turn, executes the algorithm through sequential calls to
other key FORTRAN subprograms. These subprograms are listed
as follows:
1) SUBROUTINE TDIST--accomplishes step (3); generates,
at specified radius values, a distribution of minimum required
blade section maximum thicknesses (tin)* for the assumed
value of (t*/c).75R min);
2) SUBROUTINE BLDPRP--accomplishes step (4); described
previously in Chapter V;
3) SUBROUTINE HYDLD--accomplishes step (5); computes the
hydrodynamic bending moment components, given by equations
(6.6) and (6.7), at specified radius locations;
4) SUBROUTINE CNFGLD--accomplishes step (6); computes the
centrifugal force and bending moments, given by equations
(6. 9), (6.13) and (6.14) respectively, at specified radius
locations;
5) SUBROUTINE SIGNDS--accomplishes step (7); computes
direct fiber stresses, given by equation (6.1), for all four
critical points at every specified radius location.
99
- J.
During the remaining steps of SUBROUTINE STRCNK, the
condition on allowable stress, given by equation (6.27),
is checked at all critical points of blade sections located
at specified radius locations (again, r = .2R, .3R, .4R, .5R,
.6R, .7R, .8R and .9R). The final calculation made is
that for the constraint given by equation (4.8).
Again, extensive use of common blocks, for passing
data from one subprogram to another, is apparent upon examina-
tion of the codes just cited. Comment cards are used
throughout.
E. SUMMARY
The end of this chapter marks the completion of all
prerequisite background and formulation discussions on the
application of COPES/CONMIN to propeller selection problems
involving the Wageningen B-Screw Series. From this point,
each specific Design Case can now be solved as an optimiza-
tion problem.
100
| | | 1
VII. DESIGN CASE NO. 1--PROGRAMMING AND COMPARISONS
A. INTRODUCTION
In this chapter, COPES/CONMIN is used in the solution of
propeller selection problems which use the "thrust" approach.
First, the thrust approach to the propeller selection problem
is formulated. Then, a review of a previous author's solution
to this problem is presented. Four variations to this pro-
peller selection problem are solved by COPES/CONMIN. The
chapter is completed with a presentation and discussion of
the results from the four variations.
B. THRUST APPROACH FORMULATION
1. Design Vector XT
As previously pointed out at the conclusion of Chap-
ter III, Design Case No. 1 constitutes a propeller selection
problem which is solved by the thrust approach. In this ap-
proach, the effective horsepower (PE) and the ship's speed
(V) are specified by the designer. From the viewpoint of
optimization, the quantities PE and V become preassigned
parameters. This reduces the design vector X (see Figure 4.1)
to:
D
P/D
= AE/AO (7.1)
(t*/c).75R
Np
101
t __ J
Having specified PE and V, all of the design variables,
as listed in equation (7.1), are not independent. Recalling
equations (3.3) and (3.13), the following relationship
results:
PE (l-td) 21Q S NP
= (l-wt) " R o " " (7.2)
Rearranging terms, this equation becomes:
(l-wt) PE 550-60nO QS NP It-T "R - 27 (7.3)
Considering that the open water efficiency (n0) is evaluated
prior to the computation of (-n0 ), or OBJ1 ,2 , then both Np
and QS are not independent design variables. One must
be selected as the independent design variable. Then,
the other variable becomes dependent on the one just
selected.
For this study, Np is selected as the independent
design variable. This choice will reduce the design vector
X for propeller selection problems using the thrust approach
to the following:
D
P/D= AE/AO (7.4)
(t*/c).75R
N p
102
L _j
Finally, equation (3.8) implies an alternative defini-
tion of X as given in equation (7.4). The design vector for
Design Case No. 1 propeller selection problems is, therefore,
defined as:
D
P/DAE/A 0 (7.5)
(t*/c).75RJ
2. Powering Constraint
Having determined the design vector Xl, a final
restriction to the general propeller selection problem, as
stated by equation (4.24), remains for consideration. This
restriction constitutes the remaining constraint G1 2(X) men-
tioned in Chapter IV.
Simply stated, the selected propeller, as defined by
--T, must develop enough thrust (T) so that the powering require-
ment, specified by PE and V, is met. Using equation (3.2),
the thrust developed by the propeller can be specified in terms
of thrust horsepower (PT) as:
(P_) = T V(l-wt) (7.6)PT) dev 550u
From equation (3.9), it follows that:
2 D4
(PT)dev = 550 V(l-wt) (7.7)
103
L _j
Using equation (3.12), the developed thrust horsepower can
be defined in terms of developed effective horsepower given
by:
( (l-td) . de (7.8)PE dev = -wt T (Tdev .8
The restriction imposed by the thrust approach method,
where PE and V are specified, can now be stated as:
PE <- (PE)dev (7.9)
Rearranging equation (7.9), the constraint G1 2 (RI) follows:
(PE) devG1 2 (Xl) =1 PE < 0 (7.10)
With the design vector R- and G12 (R) defined, the
propeller selection problem represented by Design Case No. 1
can be stated under one equation as:
Max'.mize: F(X-) = OBJ 1 , 2
Subject to: Gj(X-) < 0 j = 1,...,12 (7.11)
Xl o < X1. < X1upper i =1 - - 1
C. PREVIOUS SOLUTIONS
Triantafyllou [Refs. 3,211 considered a propeller selection
problem represented by Design Case No. 1. In his example
problem, the following parameters were specified:
104
1) v = 1.139 x 10 - 6 (m2 /sec) = 1.22613 x 10 - 5 (ft 2 /sec)
2) wt = .22
3) td = .19
4) nR = 1.025
5) noscrw = 1
6) Z= 5
7) p E = 18153 (hp)
8) V = 24 (knots)
9) D = 22.0 (ft)
10) AE/AO = .85
The hull under study in his example had the following dimen-
sions:
1) Length = 710 (ft)
2) Draft = 30 (ft)
3) Beam = 100 (ft)
For his analysis, the design vector contained two varia-
bles and was specified as:
P/D
Using an iterative scheme [Ref. 21: p. 791 to solve two equa-
tions in two unknowns, he maximized the open water efficiency
(no ) to obtain the following results:
P/D = 1.1651
Np = 104 (rpm)
105
• Jm
P = 25544 (hp)
no = .6676
For future comparisons, equations (3.8) and (3.3) give:
J = .8286
QS = 1290000.0 (ft-lbf)
Triantafyllou's results are summuarized in Table (IV).
D. SOLUTIONS BY COPES/CONMIN
The propeller selection problem, as stated by equation
(7.11), is now solved by COPES/CONMIN. Four solution varia-
tions are considered.
The first and second variations attempt to reproduce the
solution given by Triantafyllou. The design vector XT
(NDV = 2) is used in both cases. One variation uses SUBROUTINE
STRCNA to evaluate the constraint G1 2 (XT-) given by equation
(4.8). The other uses SUBROUTINE STRCNK to determine G2(X-T).
The remaining two variations will solve the propeller
selection problem using the design vector XT (NDV = 5) defined
in equation (7.5). Again, one variation uses SUBROUTINE
STRCNA; the other, SUBROUTINE STRCNK.
In all variations, the following parameters are used:
1) Temp - 59 ("F)
2) p - 1.9384 (lbf-sec 2/ft4
3) v - 1.2285 10 (ft 2/sec)
106
I I Ii i II
4) pwatvap = .247 (psia)
5) Patm = 14.7 (psia
6) wt = .22
7) td = .19
8) nR = 1.025
9) noscrw = 1
10) h = 19.0 (ft)o£
11) Dli = 22.0 (ft)
12) Z = 5
13) promat = 5 (stainless steel; see Table (II))
14) PE = 18153 (hp)
15) V = 24.0 (knots)
All of the above are initialized in the input phase (ICALC = 1)
of each SUBROUTINE ANALIZ pertaining to each variation.
The constraint G1 2 (R) or G1 2 (RT) is evaluated by SUBROU-
TINE BLPOWI which appears in the execution section of each
SUBROUTINE ANALIZ.
1. Variation 1
a. Programming Details
Since this variation uses the design vector XT,
the following design variables of XT become paran.,ters and
are specified in the input section of SUBROUTINE ANALIZ (ICALC = 1)
as:
1) D = 22.0 (ft)
2) AE/AO = .85
3) (t*/c).75R = .0348 (from equation 3.21).
107
_ _ _ I
For constraints, the following are used:
Gi (R ) < 0 j = 1,...,8, 12
Only nine of twelve constraints are evaluated (NCON = 9).
Obviously, some of the twelve constraints are redundant since
D, AE/AO, and (t*/c).75R have been specified.
The upper (XTupper) and lower (XT ow er ) limits on1 1
the design variables J and P/D are set to be:
.01 < J < 1.6
.4 < P/D < 1.4
These upper and lower limits are specified in the COPES
control card deck on card image F under respective fields
VUB and VLB. The initial value for each design variable
(XTi) is also assigned on card image F under the field labeled
X. The first list of card images in Appendix D lists all of
the COPES control cards used for this variation and variation
2. These cards also specify the locations of the design
variables in the common block GLOBCM (see Table (III)) as
well as the locations of the constraints and their boundaries.
Further details on the COPES control card requirements and
the format of each card are contained in Reference [7].
An examination of SUBROUTINE ANALIZ for this varia-
tion, found in Appendix C, shows the calling statement made
to SUBROUTINE STRCNA.
108
b. Results
The output from the optimization/analysis, per-
formed by COPES/CONMIN, is listed first in Appendix E. Re-
sults for this variation of the propeller selection problem
are tabulated in Table (IV).
2. Variation 2
a. Programming Details
Everything discussed above for the first variation
applies here with one exception. An examination of SUBROUTINE
ANALIZ for the second variation, found in Appendix C, shows
the calling statement made to SUBROUTINE STRCNK.
b. Results
The output from the optimization/analysis, per-
formed by COPES/CONMIN, is listed second in Appendix E. Re-
sults for this variation of the propeller selection problem
are tabulated in Table (IV).
3. Variation 3
a. Programming Details
This variation uses the design vector XT. For
constraints, the following are used:
G.(Xl) < 0 j = 1,...,12
All twelve constraints are evaluated (NCON = 12).
The upper (XlUpper) and lower (Xl. w e r ) limits1 1
on the design variables D, P/D, AE/AO, (t*/c).75R, and J
are set as:
4109
i • m •| mj
1.0 < D < 50.0 (ft)
.4 < P/D < 1.4
.2 < AE/AO < 1.1
.003 < (t*/c).7 5 R < .50
.01 < J < 1.6
These upper and lower limits are specified in the COPES con-
trol card deck on card image F under respective fields VUB
and VLB. The initial value for each design variable (X i )
is also assigned on card image F under the field labeled X.
The second list of card images in Appendix D lists all of
the COPES control cards used for this variation and variation
4. These cards also specify the locations of the design
variables in the common block GLOBCM (see Table (III)) as
well as the locations of the constraints and their boundaries.
Further details on the COPES control card requirements and
the format of each card are contained in Reference [7].
An examination of SUBROUTINE ANALIZ for this
variation, found in Appendix C, shows the calling statement
made to SUBROUTINE STRCNA.
b. Results
The output from the optimization/analysis, per-
formed by COPES/CONMIN, is listed third in Appendix E. Re-
sults for this variation of thepropeller selection problem
are tabulated in Table (IV).
110
J
4. Variation 4
a. Programming Details
Everything discussed above for the third varia-
tion applies here with one exception. An examination of
SUBROUTINE ANALIZ for the fourth variation, found in Appendix
C, shows the calling statement made to SUBROUTINE STRCNK
instead of SUBROUTINE STRCNA.
b. Results
The output from the optimization/analysis, per-
formed by COPES/CONMIN, is listed last in Appendix E. Re-
sults for this variation of the propeller selection problem
are tabulated in Table (IV).
E. DISCUSSION
Overall, the results achieved in all variations compare
reasonably well to the solution obtained by Triantafyllou.
However, the following points can be made.
Variations 1 and 2 give the same results. This was ex-
pected in view of the fact that, even though constraints
G9 (XT) through G M(XY) were evaluated, these constraints were
not considered in the optimization search conducted by CONMIN.
In variations 3 and 4, the diameter (D) was driven to
the limit (D lim). This bears out a fundamental rule in pro-
peller design, i.e., the larger the propeller diameter (D),
the greater the open water efficiency (no).
The minimum required equivalent blade section maximum
thickness-to-chord ratio ((t*/c) 75Rmin)' computed in
, ~~~5 min..m
variation 3, is substantially smaller than the one computed
for variation 4. As pointed out in Reference (2], the empiri-
cal relation, expressed by equation (4.9) and derived from
equation (70.x) [Ref. 46: p. 6201, does not take into account
the effects of centrifugal loading. These effects include,
specifically, the direct stresses imposed by the inertia load
of the blade and the bending moments which result from rake
and skew of the blade. Therefore, the algorithm developed in
Chapter VI should, and does, produce a larger value for
(t*/c).75R.
A final observation on the results concerns the values
of the open water efficiency. The "optimum" open water effi-
ciency (n ) achieved by Triantafyllou is lower than those
achieved in variations I and 2. A possible reason for this
might be the neglection of the term "dRe/dJ" in Triantafyllou's
formulation of the analytical expressions [Ref. 21: p. 71]
that he used in his analysis. The difference in the open
water efficiencies subsequently accounts for the differences
in the propeller revolution rate (Np) and the delivered
torque (0S) when the relation in equation (3.3) is considered.
112
IJ
TABLE IV
Design Case No. 1--Results
GROP I'TW4 TRIANTA-FYLLOU 1 2 3 4
PE 18153.0 18153.0 18153.0 18153.0 18153.0
Given V 24.0 24.0 24.0 24.0 24.0
Design D 22.0 22.0 22.0VariableSpeci- A/A .85 .85 .85fied (t*/c) 75R .0348 .0348
D 21.9991 21.9659
Design P/D 1.1651 1.0036 1.0036 .9981 1.0071Variables .8205 .8149
(t*/c) 75R .0330 .0642
J .8286 .7371 .7371 .7343 .7394
Maximize n .6676 .7091 .7091 .7109 .7066
Dlim (22.0) (22.0) 22.0 22.0
Restric- A" min (0.5258) (0.5258) .5269 .5267(t*/c) (0.21266) (0.50761) .021998 .053259
75rein
Np 104 116.9 116.9 117.4 116.7
Other Q 1290000 1080451. 1080451 1064574 1084003
PD 25544.0 24051.3 24051.3 24030.3 24094.8
113
L I
VIII. DESIGN CASE NO. 2--PROGRAMMING AND COMPARISONS
A. INTRODUCTION
In this chapter, COPES/CONMIN is used in the solution of
propeller selection problems which use the "power" approach.
First, the power approach to the propeller selection problem
is formulated. Then, a review of a previous author's solu-
tion to this problem is presented. Four variations to this
propeller selection problem are solved by COPES/CONMIN. The
chapter is completed with a presentation and discussion of
the results from the four variations.
B. POWER APPROACH FORMULATION
1. Design Vector X7
As previously pointed out at the conclusion of
Chapter III, Design Case No. 2 constitutes a propeller selec-
tion problem which is solved by the power approach. In this
approach, the delivered torque (Qs) and the propeller revolu-
tion rate (Np) are specified by the designer. From the view-
point of optimization, the quantities QS and N become pre-
assigned parameters. This reduces the design vector X (see
Figure (4.1)) to:
PE
VX= D (8.1)
P/D
AE/AO
(t*/c).75R.75114
J1
Having specified QS and Np, all of the design varia-
bles, as listed in equation (8.1), are not independent. Re-
calling equations (3.3), (3.11) and (3.13), the following
relationship results:
(l-td) J K T 21rQ S N pP(E (1-- t)- . nR "W * " TO 60 (8.2)
Q
Rearranging terms, this equation becomes:
PE (l-td) (l-wt) K T QS NpV (-wt) R" np D KQ 5 TO (8.3)
Considering the relations for KT and KQ in equation (3.17),
then both PE and V are not independent design variables.
One must be selected as independent, while the other becomes
dependent on the one selected.
For this study, V is selected as the independent de-
sign variable. This choice will reduce the design vector X
for propeller selection problems using the power approach to
the following:
V
D
= P/D (8.4)A E/A0
(t*/c) .75R
Finally, equation (3.8) implies an alternative defini-
tion of X as given in equation (8.4). The design vector for
115
L I
Design Case No. 2 propeller selection problems is, therefore,
defined as:
VJ
X2 = P/D (8.5)
AE/AO
(t*/c) .
2. Powering Constraint
Having determined the design vector X2, a final
restriction to the general propeller selection problem, as
stated by equation (4.24), remains for consideration. This
restriction constitutes the remaining constraint G1 2 (X) men-
tioned in Chapter IV.
Simply stated, the selected propeller, as defined by
X2, must absorb at least all of the power delivered to it
(PD) which is specified in terms of QS and N . Using equation
(3.3), the power absorbed by the propeller can be specified
in terms of delivered horsepower (PD) as:
2w Qp Np(P D)absorb = 550 (8.6)
From equation (3.10), it follows that:
K p n 2 D 21T N(PD)absorb Q 550 " 60-(8.7)
But, equation (3.3) also defines the power delivered to the
propeller as:
116
21 Qs NpD 50 60 (8.8)
The restriction imposed by the power approach method,
where QS and Np are specified, can now be stated as:
PD L (PD) absorb (8.9)
Rearranging equation (8.9), the constraint G1 2 (X-2) follows:
GI2) (Dabsorb < 0 (8.10)
Further simplification of equation (8.10) gives:
QpG 2 () 1 - Q < 0 (8.11)
With the design vector XY and G12 (-) defined, the
propeller selection problem represented by Design Case No. 2
can be stated under one equation as:
Minimize: F (X2) = OBJ1,2
Subject to: G.(X2) < 0 j = 1,...,12 (8.12)
X2Iower < X2 < upper =
1 - i i
C. PREVIOUS SOLUTIONS
Markussen [Ref. 4] considered a propeller selection prob-
lem represented by Design Case No. 2. In his example problem,
117
L
the following parameters were specified:
1) Temp = 18 (OC) = 64.4 (OF)
2) Pwatvap = 0.0206411 (bars) = .29943921 (psia)
3) P atm = 1.01312856 (bars) = 14.6974 (psia)
4) ixoscrw = 1
5) h c = 6.7 (meters) = 21.9827 (ft)
6) Z= 6
7) P = 18.9 (MegaWatts) = 25344.9 (hp)
8) Np = 110 (rpm)
9) VA = 15.65 (knots)
For his analysis, the design vector contained three
variables and was specified as:
J
)P/D
A E/AO0
A restriction for the minimum required expanded area ratio
((AE/AO) min ) , given by equation (4.3), was also considered.
This imposed a constraint given by equation (4.4).
Using an iterative scheme [Ref. 4: p. 110] to solve three
equations in three unknowns, Markussen maximized the open
water efficiency (n ) to obtain the following results:
J = .61095
P/D - .864380
AE/AO - 36.1861/40.6123 (m2/m2
118
L -
= .891012
o= .654391
For future comparisons, equations (3.8) and (3.3) give:
D = 7.19091 (meters) = 23.593375 (ft)
QS = 1210130.0 (ft-lbf)
Markussen's results are summarized in Table (V).
D. SOLUTIONS BY COPES/CONMIN
The propeller selection problem, as stated by equation
(8.12), is now solved by COPES/CONMIN. Four solution varia-
tions are considered.
The first and second variations attempt to reproduce the
solution given by Markussen. The design vector XM (NDV = 3)
is used in both cases. One variation uses SUBROUTINE STRCNA
to evaluate the constraint G12 (3R) given by equation (4.8).
The other uses SUBROUTINE STRCNK to determine GI2().
The remaining two variations will solve the propeller
selection problem using the design vector R2 (NDV = 5) de-
fined in equation (8.5). Again, one variation uses SUBROUTINE
STRCNA; the other, SUBROUTINE STRCNK.
In all variations, the following parameters are used:
1) Temp = 64.4 (eF)
2) p = 1.9892 (lbf-sec 2/ft4
3) v = 1.1900 x l0 - 5 (ft 2/sec)
4) pwatvap = .2994 (psia)
119
L _j
5) patm = 14.697 (psia)
6) wt = .22
7) td = .19
8) TR = 1.025
9) noscrw = 1
10) h c = 21.9827 (ft)
11) Dlim = 30.0 (ft)
12) Z = 6
13) promat = 5 (stainless steel; see Table (II))
14) QS = 1210130 (ft-lbf)
15) Np = 110 (rpm)
All of the above are initialized in the input phase
(ICALC = I) of each SUBROUTINE ANALIZ pertaining to each
variation.
The constraint G1 2 (X2) or G1 2(XM) is evaluated by
SUBROUTINE BLPOW2 which appears in the execution section of
each SUBROUTINE ANALIZ.
1. Variation 1
a. Programming Details
Since this variation uses the design vector 35,
the following design variables of R-2 become parameters and
are specified in the input section of SUBROUTINE ANALIZ
(ICALC = 1). The ship's speed (V) is specified as:
V = VA/(l-wt)
= 15.65/(1 - .22)
= 20.0641 (knots)
120
u _,
Markussen elected to use the standard Wageningen blade sec-
tion maximum thickness-to-chord ratios. Since the equivalent
t/c is given as a function of Z and AE/A0 (see equation (3.21)),
then (t*/c).75R is calculated during each analysis (ICALC = 2)
by the following relation:
(t*/c).75R = (t/c).75R
For constraints, the following are used:
G (RT) < 0 j = 1,...,8, 10, 12
Only ten of twelve constraints are considered (NCON = 10).
Constraints G9 (M) and G1 1 (3) are redundant since no limit
on the propeller diameter (Dlim) appears as a parameter in
Markussen's formulation and (t*/c) was taken to be the75R
Wageningen standard.
The upper (XMupper) and lower (M ow e r ) limitsI. 1
on the design variables J, P/D and AE/AO are set to be:
.01 < J < 1.1
.4 < P/D < 1.4
.4 o < O < 1.1
These upper and lower limits are specified in the COPES con-
trol card deck on card image F under respective fields VUB
and VIB. The initial value for each design variable (XMi) is
121
-. ,
also assigned on card image F under the field labeled X.
The first list of card images in Appendix G lists all of the
COPES control cards used for this variation and variation 2.
These cards also specify the locations of the design varia-
bles in the common block GLOBCM (see Table (III)) as well as
the locations of the constraints and their boundaries. Further
details on the COPES control card requirements and the format
of each card are contained in Reference [7].
An examination of SUBROUTINE ANALIZ for this
variation, found in Appendix F, shows the calling statement
made to SUBROUTINE STRCNA.
b. Results
The output from the optimization/analysis,
performed by COPES/CONMIN, is listed first in Appendix H.
Results for this variation of the propeller selection problem
are tabulated in Table (V).
2. Variation 2
a. Programming Details
Everything discussed above for the first variation
applies here with one exception. An examination of SUBROUTINE
ANALIZ for the second variation, found in Appendix F, shows
the calling statement made to SUBROUTINE STRCNK instead of
SUBROUTINE STRCNA.
b. Results
The output from the optimization/analysis, per-
formed by COPES/CONMIN, is listed second in Appendix H.
122
-L I
Results for this variation of the propeller selection problem
are tabulated in Table (V).
3. Variation 3
a. Programming Details
This variation uses the design vector 2. For
constraints, the following are used:
G.(X2) < 0 j = 1,...,12
All twelve constraints are evaluated (NCON = 12).The upper (X2upper) and lower (X2lower) limits
1 1
on the design variables V, P/D, AE/AO, (t*/c).75R, and J are
set as:
10.0 < V < 100.0 (ft/sec)
.4 < P/D < 1.4
.4 < AE/AO < 1.1
.003 < (t*/c) < .50.75R
.01 < J < 1.1
These upper and lower limits are specified in the COPES con-
trol card deck on card image F under respective fields VUB
and VLB. The initial value for each design variable (X2i )
is also assigned on card image F under the field labeled X.
The second list of card images in Appendix G lists all of the
COPES control cards used for this variation and variation 4.
These cards also specify the locations of the design variables
123
C
in the common block GLOBCM (see Table (III)) as well as the
locations of the constraints and their boundaries. Further
details on the COPES control card requirements and the format
of each card are contained in Reference [7].
An examination of SUBROUTINE ANALIZ for this
variation, found in Appendix F, shows the calling statement
made to SUBROUTINE STRCNA.
b. Results
The output from the optimization/analysis, per-
formed by COPES/CONMIN, is listed third in Appendix H. Re-
sults for this variation of the propeller selection problem
are tabulated in Table (V).
4. Variation 4
a. Programming Details
Everything discussed above for the third varia-
tion applies here with one exception. An examination of SUB-
ROUTINE ANALIZ for the fourth variation, found in Appendix
F, shows the calling statement made to SUBROUTINE STRCNK
instead of SUBROUTINE STRCNA.
b. Results
The output from the optimization/analysis, per-
formed by COPES/CONMIN, is listed last in Appendix H. Results
for this variation of the propeller selection problem are
tabulated in Table (V).
E. DISCUSSION
The results achieved in variations 1 and 2 compare ex-
tremely well to the solution obtained by Markussen. As
124I~' .1
pointed out in the discussion in Chapter VII, variations 1
and 2 are expected to give the same results for the vector
XM. Obviously, the values obtained for J and P/D, as well
as those for D, and Rn* 75R, are very close to the values
generated in Markussen's example. However, the values for
AE/A0 are somewhat different. It is interesting to note that
the value obtained in variations 1 and 2 (and, for that matter,
variations 3 and 4) is, essentially, the limiting value for
AE/AO, as given in Table (I), for Z = 6. Markussen's value
for AE/AO (i.e., .891012) exceeds the limit (i.e., .80) in
this table.
As pointed out at the end of Chapter VII, the minimum
required blade section maximum thickness-to-chord ratio
((t*/c). 7 5R min)) , computed in variations 1 and 3, is sub-
stantially smaller than the one computed for variations 2
and 4. Again, the same explanation applies here as well.
The results of variations 3 and 4 differ somewhat from
Markussen's results. The reason for this is simply that VA
(or V) has not been specified as a parameter. Consequently,
a higher value for the advance ratio (J), which corresponds
to a higher open water efficiency (no) , has been found in
the optimization search. This result can be interpreted
in the following way. Given:
i) a six-bladed Wageningen propeller (Z = 6) which is made
out of stainless steel (promat = 5);
2) a power train delivering 25344.9 (hp) at a rate of
110 (rpm);
4125
L -•mI J
3) a hull with a wake fraction (wt) equal to .22, a
thrust deduction (td) equal to .19 and a shaft centerline
depth (h cz)of 21.98 (ft),
then, the selected propeller, as defined by X2, can drive
this hull at a maximum speed of V when the hull has a maximum
resistance given by PE*
4j 126
FL W
TABLE V
Design Case No. 2--Results
GFUP ITEM MAKLaSS VARIATIONS
1 2 3 4
PD 25344.9 25344.9 25344.9 25344.9 25344.9
Given OS 1210130 1210130 1210130 1210130 1210130
110 110 110 110 110
DesignVariable V 15.65 15.65 15.65Seciied V 20.0641 20.0641
J .61095 .6475 .6475 .9927 .8753
JV 30.9138 29.5355
Design iVA 24.1127 23.0377Variables P/D .864380 .9036 .9036 1.1986 1.0308
A/AO .891012 .8018 .8018 .7946 .7986
(t*/c)75R .0397 .0397 .0499 .0638
Maximize o .654391 .6660 .6660 .7643 .7330
DI m (30.0) (30.0) 30.0 30.0
Restrictions A/Ami n .574729 .5070 .5070 .4622 .4266
(t*/C). 75Fkin (.02729) (.0647) .02706 .0638
D 23.53 22.25 22.25 22.36 24.23
other PE (14057.3) (14057.3) 19945.5 20653.7
.75R 6.478x107 5X107 5x107 5x107 5x107
127
'4
IX. DESIGN CASE NO. 3--PROGRAMMING AND COMPARISONS
A. INTRODUCTION
In this chapter, COPES/CONMIN is used in the solution of
a propeller selection problem where "matching" is desired.
First, the "matching" approach to the propeller selection
problem is formulated. Then, a review of a previous author's
solution is presented. One variation to this propeller
selection problem is solved by COPES/CONMIN. The chapter is
completed with a presentation and discussion of the results.
B. "MATCHING" FORMULATION
1. Design Vector R3
Design Case No. 3, the final powering problem con-
sidered in this study, constitutes a propeller selection prob-
lem solved by the "matching" approach. In this approach,
the hull's effective horsepower (PE) and speed (V), the
delivered torque (Qs) and the propeller revolution rate
(Np) are specified by the designer. This reduces the design
vector R (see Figure (4.1)) to:
DR P/D (9 .1)
AE/A0(t*/c).75R
For this study, the design vector X is reduced further
by eliminating the propeller diameter (D) as a design variable.
128
That is, D will also be specified by the designer so that
the design vector for Design Case No. 3 is defined as:
~P/D)=Y A E/AO (9.2)
(t*/c) .75R
2. Powering Constraint(s)
Having determined the design vector XT, a final
restriction to the general propeller selection problem, as
stated by equation (4.24), remains for consideration. This
restriction constitutes the remaining constraint G 12( M men-
tioned in Chapter IV as well as an additional constraint.
In the "matching" problem, the selected propeller,
as defined by XT, must satisfy two conditions. First, it
must develop, as a minimum, the effective horsepower (P
as imposed by the design specification. Citing the formula-
tion previously derived in Chapter VII, this condition can
be stated as:
The constraint G 12 (7) follows accordingly as:
G1 2 (0) (P 1- dev < 0 (9.4)12 P E
For the second condition, the selected propeller can
only absorb, as a maximum, the delivered power (P as
129
specified by the designer. The formulation is the same as
that in Chapter VIII except that the inequality signs are
reversed. The condition is stated as:
(PD)absorb . PD (9.5)
In defining a constraint G1 3 (Y3), another location, say
location 24, in the GLOBCM block (see Table (III)) would be
assigned. But, considering the fact that constraint G9 (X)
will not be used because the propeller diameter (D) is speci-
fied, there is no reason why G9 (Y) cannot be redefined, for
this Design Case only, as:
(P)absorbG9 ( X P D__ _ - 1 < 0 (9.6)
9 P D
Further simplification of equation (9.6) gives:
Qp
G9 (X-3) = - 1 < 0 (9.7)
In reality, the constraints just defined should be
equality constraints. The word "match" does infer equality
in some sense. However, as previously stated in Chapter II,
the version of COPES/CONMIN used in this study does not
directly handle equality constraints. But, since CONMIN
attempts to minimize constraints in the optimization search,
it will be assumed that a "match" can be achieved.
130
LJ
With the design vector - and the constraints
G12 (3) and G9 (3) defined, the propeller selection problem
represented by Design Case No. 3 can now be stated under
one equation as:
Minimize: F(X-3) = OBJ 3
Subject to: G.(X--X) < 0 j = 1,...,12 (9.8))
X3lower < X. < X3up p e r i = 1,...,31 - 1- 3.
C. PREVIOUS SOLUTIONS
The propeller selection problem considered by Vassilopoulos
[Ref. 18) actually represents a propeller "design" problem
using the "power" approach. The example problem which he
elected to solve is taken from that posed by the International
Towing Tank Conference (ITTC) Propeller Committee. This
problem is concerned with the determination of propeller
thrust (T), diameter (D) and speed of advance (VA) (or ship
speed (V)) for a single-screw cargo ship where:
1) power available to the propeller (i.e., PD) is30,000 (hp)
2) Z= 6
3) N = 105--110 (rpm)
4) Dli m = 23 (ft)
5) hc = 19 (ft)
The variation of ship speed (V) and of hull effective power
(PE), thrust deduction factor (td) and the wake fraction
(wt) is also given [Ref. 18: p. 20].
131
L ..I
The results from Vassilopoulos' propeller design exercise
produced a propeller that is "matched" at the following values:
1) PE = 21292.6 (hp)
2) V = 24.24 (knots)
3) QS = 1500606.75 (ft-lbf)
4) Np = 105 (rpm)
5) PD = 30000.0 (hp)
His propeller "design" was based upon the following specified
parameters:
1) Temp = 59 (OF)
2) p = 1.9905 (lbf-sec 2/ft4
3) Pwatvap .247 (psia)
4) wt = .22
5) td = .1725
6) noscrw = 1
7) hc9 = 19.0 (ft)
8) Z=6
9) promat = 5 (stainless teel, see Table (II))
10) D = 22.0 (ft)
11) Np =105 (rpm)
12) PD = 30000.0 (hp)
Using an optimization scheme incorporated in his MVAPDP
computer program, Vassiiopoulos maximized the open water
efficiency (n0 ) and designed a propeller with the following
characteristics:
1) J = .852
2) KT = .242
132
L j
3) KQ = .0478
4) no = .691
5) AE/AO = .767
6) bldwt = 7617.2 (ibf)
By utilizing both the lifting line and lifting surface
methods in his design procedure, Vassilopoulos' MVAPDP program
evolved a "constant stress" propeller blade. Consequently,
the values for (t*/c) and P/D varied non-linearly along the
propeller radius (R). According to Vassilopoulos, this
resulted in a minimum weight propeller. The values for P/D
and (t*/c) are listed in Tables (8) and (10) of his paper.
From these values, (t*/c) .75R is approximately .040.
While the propeller represented by Vassilopoulos' design
is different, in many aspects (rake, skew, blade section
aerfoil shape, etc.), from the Wageningen B-Screw Series
propeller, it does represent a minimum weight propeller that
has been "matched" to specific design values. Appropriate
results are summarized in Table (VI).
D. SOLUTIONS BY COPES/CONMIN
The propeller selection problem, as stated by equation
(9.8), is now solved by COPES/CONMIN. One solution variation
is considered. The following parameters are used:
1) Temp = 59 (F)
2) p = 1.9905 (lbf-sec 2/ft4
3) v = 1.2817 x 10- 5 (ft 2/sec)
4) Pwatvap = .247 (psia)
133
I-
5) Patm = 14.7 (psia)
6) wt = .22
7) nR = 1.025
8) noscrw = 19) h = 19.0 (ft)
10) Z = 6
11) promat = 5 (stainless steel, see Table (II))
12) D = 22.0 (ft)
Two problems are examined. Problem 1 specifies the following
additional parameters:
1) td = .1725
2) PE = 21292.6 (hp)
3) V = 24.24 (knots)
4) QS = 1500606.75 (ft-lbf)
5) Np = 105 (rpm)
Problem 2 specifies the same parameters as:
1) td = .171
2) PE = 17630.0 (hp)
3) V = 23.0 (knots)
4) QS = 1500606.75 (ft-lbf)
5) NP = 105 (rpm)
All of the above are initialized in -he input section
(ICALC = 1) of similar versions of SUBROUTINE ANALIZ. There-
fore, only one version is included in Appendix I.
The constraints for G9 (-3) and G12 (X--I) are evaluated by
SUBROUTINE BLPOW3 which appears in the execution section of
134
L J
SUBROUTINE ANALIZ. Also, note that SUBROUTINE DICNUA has
been deleted from the execution section, while SUBROUTINE
WGrCAL has been added.
1. Programing Details
All twelve constraints are evaluated (NCON = 12).
The upper (X3Mpper) and lower (X3 ower) limits on the design1 -
variables P/D, AE/AO and (t*/c).75R are set to be:
.4 < P/D < 1.4
.4 AE/AO < 1.1
.003 < (t*/c) 7 5R < .50
These upper and lower limits are specified in the COPES con-
trol card deck on card image F under respective fields VUB
and VLB. The initial value for each design variable (X3i )
is also assigned on card image F under the field labeled X.
The list of card images in Appendix J lists all of the COPES
control cards used for both problems. These cards also
specify the locations of the design variables in the common
block GLOBCM (see Table (III)) as well as the locations of
the constraints and their boundaries. Further details on
the COPES control card requirements and the format of each
card are contained in Reference (7].
2. Results
The outputs from the optimization/analysis, performed
by COPES/CONMIN, are listed in Appendix K. Results of both
problems are tabulated in Table (VI).
135
L _J
E. DISCUSSION
Table (VI) presents the results of problems 1 and 2
along with relevant information from Vassilopoulos' "design".
Problem 1 attempted to "match" a Wageningen propeller at the
design point found by Vassilopoulos. The first COPES/CONMIN
printout in Appendix K indicates that the "match" was achieved
at PE equal to 21.168.1 (hp) and PD equal to 28,150.0 (hp)
(or, QS = 1500607 (ft-lbf) and Np = 105 (rpm)). These values
are judged to be close enough to the "Given" values in Table
(VI).
It is apparent that the Wageningen propeller does not
require all of the 30,000 (hp) of delivered horsepower. The
propeller characteristics (i.e., J, KT, KQ and no) for problem
1 compare very well to Vassilopoulos' values. The expanded
area ratios (AE/AO) are, also, very similar. Of course, the
obvious difference is the blade weight (bldwt) . The Wageningen
propeller blade is over five thousand pounds heavier. Does
this make sense for a minimum blade weight?
The answer is yes.
All one has to do is consider the values of (t*/c) for problem
1 and Vassilopoulos' design. Vassilopoulos' "constant stress"
blade was designed to "absorb" stress up to the allowable
design limit of 5,400 (psi) (for stainless steel) all along
the entire propeller radius (R). Table (12) in Reference
[181 gives further details. The Wageningen propeller blade,
however, represents an "older" type of blade which was de-
signed with a linear blade section maximum thickness (t*)
4i 136
L _j
distribution. Consequently, it was "overdesigned" for strength
beyond the 3/10--4/10 radius (i.e., .3R--.4R) and contains
excess material. A heavier blade, therefore, results. Note,
also, that the optimizer did not drive the value of (t*/c).5 R
to the minimum acceptable value, (t*/c)75Rmin"
The results of problem 2 show the effect on blade weight
(bldwt) for a Wageningen propeller when the hull's powering
requirements (i.e., PE at V) have been reduced. The weight
reduction of 2000 pounds is significant. The complete re-
sults are listed in the second COPES/CONMIN printout in
Appendix K.
137
L..
TABLE VI
Design Case No. 3--Results
GROUP I Tm VASSILOPOUIDS PR4BLEM
1 2
PE 21292.6 21292.6 17630.0Given V 24.24 24.24 23.0
QS 1500607 1500607 1500607Np 105 105 105
DesignVariable D 22.0 22.0 22.0Specified
P/D * 1.1813 1.0906DesignVariables AE,/Ao .767 .7944 .7742
(t*/c).75R .040 .0794 .0681
Minimize bldwt - 12842.6 10464.7
Maximize no .691 - -
Dli m - - _
Restric- AAO m .8515 .7722
tions (t*/c). 75Rmin - .0691 .0681
J .852 .8290 .7866
.242 .2349 .2063
Other KO .0478 .0448 .0372
no - .6915 .6950
bld t 7617.2 - -
P/D varies with R
138
X. CONCLUSIONS AND RECOMMENDATIONS
A. CONCLUSIONS
The general purpose non-linear optimizer/synthesizer
COPES/CONMIN has been successfully applied to three typical
preliminary ship design propeller selection problems in which
the Wageningen B-Screw Series is used. The formulation and
programming of each required analysis code (i.e., SUBROUTINE
ANALIZ) have been made as general as possible to allow the
designer a broad variety of solution options for solving
propeller selection problems which can be classified under
any of the three Design Cases that were considered. The
analysis codes have been "modularized" to the extent that
methodical series data from other propeller series, which
are available in the polynomial expression format of the
B-Screw Series, can be easily adapted for powering analysis
utilizing design optimization methods.
Further flexibility in the solution to the propeller
selection problem has been achieved by using COPES/CONMIN
as the optimizer/synthesizer. The designer has now been
afforded the additional capability of specifying the design
variables, the objective functions and the constraints of his
choice. By solving propeller selection problems in the way
presented in this thesis, repetitive problem formulation and
coding have been eliminated.
There are other advantages to solving propeller selection
problems specifically with COPES/CONMIN which have not been
4139
directly addressed in this study. As stated in Chapter II,
COPES/CONMIN is capable of performing optimization analyses,
sensitivity studies, optimum sensitivity studies and optimi-
zation using approximation techniques. The designer, there-
fore, can select and perform any of these options, using the
same analysis codes which have been presented in this thesis.
While the utilization of a general purpose non-linear
optimizer in solving propeller selection problems allows the
designer greater flexibility in the selection procedure, there
is one important limitation that should be stressed at this
point. This concerns the question whether or not the solution
vector, determined by the optimizer, is a "global" optimum.
As stated in Chapter II, COPES/COMMIN assures that, if a
feasible solution vector is found, it is, at least, a "local"
minimum (or maximum). This implies that, for two different
initial design vectors which are specified in the COPES Con-
trol Card deck on card image F, the same optimum solution
may not be determined by the optimizer. Both solutions would
correspond to minimums (or maximums) of the objective function
and are, therefore, correct. But, does one or the other
correspond to the minimum (or maximum) of the entire vector
design space, i.e., the "global" optimum? For the moment,
at least, there is no definitive answer to the question.
Despite this uncertainty, progress in the field of design
optimization continues to be made. Current developments
[Ref. 47] will soon allow the designer to have a choice in
140
selecting a specific optimization algorithm from a "library"
of proven optimization programs which employ the latest
state-of-theart numerical techniques. Again, using one
analysis code, the designer will be able to generate any
number of optimized solutions for the problem under study.
B. RECOMENDATIONS
For future consideration, it is recommended that the
automated design and trade-off capability, provided by a
general purpose non-linear optimizer/synthesizer such as
COPES/CONMIN, be applied to the more difficult problem of
propeller design.
As pointed out in Chapter I, the use of the Wageningen
B-Screw Series represents a "selection" procedure rather
than a "design" process. Today, analytical propeller design
procedures, utilizing lifting line and lifting surface
theory, are becoming increasingly popular among propeller
designers. The propeller design, which results from the
utilization of these analytical methods, is, unquestionably,
more efficient than the standard series propeller. However,
these methods require consideration of many more design varia-
bles in the design process. This appears to be a natural
application for the use of a general purpose non-linear
optimizer/synthesizer.
Here, an analysis code, much larger than those which
have been presented in this study, could be developed which
would incorporate the lifting line/lifting surface theory
141
_J
for the determination of the propeller performance character-
istics, the local cavitation numbers and also the calculation
of the pressure distributions over the blade. These pres-
sure distributions would be utilized in the strength analysis
of the blade. This analysis would utilize the finite element
technique on an appropriately generated mesh model of the
blade. Having defined the steps for this design procedure
in the analysis code, the propeller designer now "couples"
his analysis to the optimizer/synthesizer for determination
of the optimum design. A massive amount of computer storage
would certainly be required, but this concept is feasible
and, in the author's view, is worthy of future consideration.
C. A FINAL NOTE
in conclusion, this thesis has demonstrated, in effect,
another interesting application of the method of design opti-
mization. The author, in no way, wishes to leave the reader
with the impression that the techniques of design optimi-
zation are the "be all--end all" for engineering analysis.
Design optimization techniques are useful and powerful tools
that stand to relieve the engineer of the mundane tasks of
numerical calculations and subsequent graphic plotting.
But, they are just tools. In the final "analysis", good
engineering judgment is paramount in their application and
use.
142
APPENDIX A
FORTRAN VARIABLE CROSS REFERENCE LIST
Symbol Fortran Variable
AE/A0 AEDVAO
(AE/AO) m AEAOMN
bldwt WEIGHT
c75R C75R
D DIA
D lim DIALIM
h ck HCL
J RJ
KQ KQ
KT KT
noscrw NOSCRW
N p N
P E PE
P D PD
P/D PDIVD
Pwatvap PWATVA
Patm PATM
promat PROMAT
os QS
Rn*5 R75R
Sc SC
td TD
143
APPENDIX A (CONT.)
symbol Fortran Variable
Temp T EMP
(t*/C).75R TC75R
V (ft/sec) v
" (knots) VI(
wt WT
z z
Tio ETAO
TIR ETARR
v WATNU
p WATRO
144
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AD A 12 4 4 PRELIMINARY PROPELLER SELECTION USING THE NAGENI NGEN C/I
B-SCREW SERIES AND A GENERAL PURPOSE NON-LINEAROPTIMIZER(U) NAVAL POSTGRADUATE SCHOOL MONTEREY CA
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CONTROL CARD IMAGES--DESIGN CASE NO. 2
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ANALIZ CODES--DESIGN CASE NO. 3
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rt
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40. Ma, J.H., Schnobrick, W.C. and Stuber, C.B., PropellerStress Calculation Using Curved Finite Elements, paperpresented at S.N.A.M.E. Propellers '75 Symposium, 1st,Philadelphia, Pennsylvania, July 1975.
41. Beek, G.H.M., "Calculation of Propeller Blade Stresesand Comparison with Test Results", InternationalShipbuilding Progress, v. 24, no. 277, p. 225-236,September 1977.
42. Beek, G.H.M., Hub-Blade Interaction in Propeller Strength,paper presented at S.N.A.M.E. Propellers '78 Symposium,19th, Virginia Beach, Virginia, 25 May 1978.
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C359
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1. Defense Technical Information Center 2Cameron StationAlexandria, Virginia 22314
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The University of MichiganAnn Arbor, Michigan 48109
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TechnologyCambridge, Massachusetts 02139
9. Professor Michael S. Triantafyllou 1Department of Ocean Engineering, Room 5-225The Massachusetts Institute of Technology
(Cambridge, Massachusetts 02139
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