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OPTIMAL CHANNEL DESIGN
A THESIS SUBMITTED TO THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES
OFTHE MIDDLE EAST TECHNICAL UNIVERSITY
BYBÜLENT AKSOY
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE
INTHE DEPARTMENT OF CIVIL ENGINEERING
SEPTEMBER 2003
iii
ABSTRACT
OPTIMAL CHANNEL DESIGN
AKSOY, Bülent
M.Sc., Department of Civil Engineering
Supervisor: Asst. Prof. Dr. A. Burcu ALTAN SAKARYA
September 2003, 55 pages
The optimum values for the section variables like channel side
slope, bottom width, depth and radius for triangular, rectangular,
trapezoidal and circular channels are computed by minimizing the cost
of the channel section. Manning’s uniform flow formula is treated as a
constraint for the optimization model. The cost function is arranged to
include the cost of lining, cost of earthwork and the increment in the
cost of earthwork with the depth below the ground surface. The
optimum values of section variables are expressed as simple functions
of unit cost terms. Unique values of optimum section variables are
obtained for the case of minimum area or minimum wetted perimeter
problems.
Keywords: Open Channel Design, Optimization, Minimum Cost, Best
Hydraulic Section
iv
ÖZ
OPTİMAL KANAL TASARIMI
AKSOY, Bülent
Yüksek Lisans, İnşaat Mühendisliği Bölümü
Tez Danışmanı: Yard.Doç.Dr. A. Burcu ALTAN SAKARYA
Eylül 2003, 55 sayfa
Üçgen, dikdörtgen, trapez ve dairesel kanal tipleri için, kanal
kesitinin maliyetini en aza indiren kanalın yan yüzeyinin eğimi, alt taban
genişliği, derinliği ve yarıçapı gibi kanalın kesit değişkenleri,
hesaplanmıştır. Manning’in düzgün akım denklemi optimizasyon
modelinde kısıtlama oluşturacak şekilde düzenlenmiştir. Maliyet
fonksiyonu, kaplama maliyetini, toprak işleri maliyetini ve kanalın
yüzeyden derinliğine bağlı olarak artan toprak işleri fiyatını içerecek
şekilde düzenlenmiştir. Kesit değişkenlerinin en faydalı değerleri birim
fiyat terimlerinin basit birer fonksiyonu olarak ifade edilmiştir. En küçük
alan veya en küçük ıslak çevre problemleri için kesit değişkenlerinin
sabit değerleri hesaplanmıştır.
Anahtar Kelimeler: Açık Kanal Tasarımı, Optimizasyon, En Düşük
Maliyet, En İyi Hidrolik Kesit
v
TABLE OF CONTENTS
ABSTRACT ...................................................................................................... iii
ÖZ .... ................................................................................................................ iv
TABLE OF CONTENTS ................................................................................... v
LIST OF FIGURES ........................................................................................... vii
LIST OF TABLES............................................................................................. ix
LIST OF SYMBOLS ......................................................................................... x
1. INTRODUCTION AND REVIEW OF LITERATURE .................................... 1
1.1. INTRODUCTION ............................................................................. 1
1.2. LITERATURE SURVEY................................................................... 2
1.3. SCOPE OF PRESENT STUDY....................................................... 3
2. DEFINITION OF PROBLEM AND OPTIMIZATION ALGORITHM ............. 5
2.1. REVIEW OF UNIFORM FLOW ....................................................... 5
2.2. COST STRUCTURE........................................................................ 7
2.3. OPTIMIZATION ALGORITHM......................................................... 10
2.3.1. TRIANGULAR SECTION ........................................................ 13
2.3.2. RECTANGULAR SECTION .................................................... 15
2.3.3. TRAPEZOIDAL SECTION....................................................... 17
2.3.4. CIRCULAR SECTION ............................................................. 19
3. COMPUTATIONS AND ANALYSIS OF RESULTS ..................................... 23
3.1. ANALYTICAL SOLUTION ............................................................... 23
3.2. NUMERICAL COMPUTATIONS ..................................................... 27
3.3. REGRESSION ANALYSES............................................................. 31
3.3.1. EFFECT OF COST TERMS ON OPTIMUM
SECTION VARIABLES.............................................................. 31
3.3.2. DETERMINATION OF EQUATIONS ...................................... 35
3.4. DISCUSSION OF RESULTS........................................................... 48
3.5. DESIGN EXAMPLE......................................................................... 49
4. CONCLUSIONS AND RECOMMENDATIONS............................................ 52
4.1. CONCLUSIONS .............................................................................. 52
4.2. RECOMMENDATIONS FOR FURTHER STUDIES........................ 53
REFERENCES.................................................................................................. 54
vi
APPENDICES................................................................................................... 56
APPENDIX A. COMPUTATION OF SECTION VARIABLES
BY USING LAGRANGE MULTILIERS
METHOD ........................................................................ 56
APPENDIX B. NUMERICAL COMPUTATIONS RESULTS................... 66
vii
LIST OF FIGURES
2.1. Typical Channel Section............................................................. 9
2.2. Triangular Channel Section........................................................ 14
2.3. Rectangular Channel Section..................................................... 16
2.4. Trapezoidal Channel Section ..................................................... 17
2.5. Circular Channel Section............................................................ 19
3.1. Layout of Spreadsheet Used for Numerical Computations ......... 28
3.2. Layout of Solver Menu ............................................................... 29
3.3. Layout of Solver Options Menu .................................................. 30
3.4. Variation of Optimum Side-Slope of a Triangular Section
with *Aβ ( *Lβ =1.0) ..................................................................... 32
3.5. Variation of Optimum Non-Dimensional Normal Depth of a
Triangular Section with *Aβ ( *Lβ =1.0) ........................................... 32
3.6. Variation of Optimum Side-Slope of a Triangular Section
with *Lβ ( *Aβ =0.5)..............................................................................33
3.7. Variation of Optimum Non-Dimensional Normal Depth of a
Triangular Section with *Lβ ( *Aβ =0.5) ............................................33
3.8. Variation of Optimum Non-Dimensional Section Variables
with *Lβ ( *Aβ =constant) ...................................................................34
3.9. Variation of Optimum Non-Dimensional Section Variables
with *Aβ ( *Lβ =constant) .......................................................... 34
3.10. Variation of Optimum Non-dimensional Side-Slope of a
Triangular Section with ** LA ββ .....................................................36
3.11. Variation of Optimum Non-dimensional Normal Depth of a
Triangular Section with ** LA ββ .....................................................36
3.12. Variation of Optimum Non-dimensional Bottom Width of a
Trapezoidal Section with ** LA ββ ...................................................37
viii
3.13. Variation of Optimum Non-dimensional Normal Depth of a
Trapezoidal Section with ** LA ββ ...................................................37
3.14. Variation of Error Values of *m with ** LA ββ for
Triangular Channel Section...............................................................44
3.15. Variation of Error Values of*
*ny with ** LA ββ for
Triangular Channel Section................................................................44
3.16. Variation of Error Values of*
*b with ** LA ββ for
Rectangular Channel Section..................................................... 45
3.17. Variation of Error Values of*
*ny with ** LA ββ for
Rectangular Channel Section .................................................... 45
3.18. Variation of Error Values of *m with ** LA ββ for
Trapezoidal Channel Section..................................................... 46
3.19. Variation of Error Values of*
*b with ** LA ββ for
Trapezoidal Channel Section..................................................... 46
3.20. Variation of Error Values of*
*ny with ** LA ββ for
Trapezoidal Channel Section............................................................. 47
3.21. Variation of Error Values of*
*r with ** LA ββ for
Circular Channel Section........................................................... 47
3.22. Variation of Error Values of*
*ny with ** LA ββ for
Circular Channel Section............................................................ 48
ix
LIST OF TABLES
3.1. Optimum Non-dimensional Section Variables for *Aβ = 0 ............27
3.2. Coefficients of Model 1............................................................... 41
3.3. Coefficients of Model 2............................................................... 41
3.4. Coefficients of Multiple Determination (Rc2) for Model 1.............. 41
3.5. Coefficients of Multiple Determination (Rc2) for Model 2.............. 42
3.6. Cost Terms ................................................................................ 43
3.7. Comparison of Computed and Fitted Values.............................. 51
x
LIST OF SYMBOLS
A flow area
a flow area at height η
A* non-dimensional flow area
b bottom width of the channel
b* non-dimensional bottom width of the channel
b* optimum bottom width
C cost of channel per unit length
c flow resistance factor
C* non-dimensional cost of channel per unit length
CE cost of earthwork per unit length
CL cost of lining per unit length
d depth of area below the ground surface
l Lagrange multiplier
kio section coefficient for section variable i forβA =0
for model 2
kiL section coefficient for section variable i for lining
for model 2
kiA section coefficient for section variable i for additional
earthwork for model 2
m channel side slope
m* optimum side slope
n Manning’s roughness coefficient
P wetted perimeter
P* non-dimensional wetted perimeter
Q discharge
xi
r channel radius
R hydraulic radius
Rc2 coefficient of multiple determinations
r* non-dimensional channel radius
r* optimum channel radius
S channel bottom slope
V flow velocity
x hydraulic exponent
y hydraulic exponent
yn normal depth
yn* non-dimensional normal depth
yn* optimum normal depth
zi section coefficient for model 1
zio section coefficient for section variable i for βA =0
for model 1
βA * non-dimensional unit cost of additional earthwork
βA unit cost of additional earthwork
βE unit cost of earthwork
βL unit cost of lining
βL* non-dimensional unit cost of lining
εi error value for the section variable i
φ augmented function
λ length scale
∆ic computed value for the section variable i
∆if fitted value for the section variable i
1
CHAPTER 1
INTRODUCTION AND REVIEW OF LITERATURE
1.1 INTRODUCTION
Water… the most vital element for all living beings. Its priority for
human beings has determined the history of world. Especially after
quitting nomadism and starting to cultivation, the presence of water had
affected the settlement of civilizations. To raise their crops, most of
them preferred to settle near by rivers and watery areas. The
agricultural facilities of human beings started with the cultivation, made
them familiarize with water and encounter the problems of water. One
of these problems was the conveyance of water from one location to
another location.
Through the history, the above problem has arisen not only for
agricultural needs but also for municipal and power needs. Among
different solutions the most widely used were the formers of recently
used channel sections.
Although the techniques and materials used in the construction of
conveyance lines has changed, channels still keep their attractiveness
in transportation of water. They are easy and economical solutions of
water conveyance elements. They may be constructed on different
topographies and soil conditions with different cross sections and
longitudinal profiles. Nowadays, the most widely used channel sections
are triangular, rectangular, trapezoidal and circular sections.
These channels, in which the state of flows are called as open
channel flows, are designed according to the laws of open channel flow.
2
Combining these laws with the objective and constraints of the project,
the section variables i.e., side slope, bottom width, depth and radius, of
channels can be computed.
Providing that the channel section convey the required amount of
water safely, the section variables of the channel will vary according to
the objective and the constraints of the project.
The objective of a project may be to convey the given flow rate
with the least flow area or to convey the given discharge resulting in the
minimum cost of construction. On the other hand, the constraints of the
project may be on the values of average flow velocity, top width, depth
and also on the value of side slope. There are various studies on the
values of section variables for different channel geometries considering
the above objectives and constraints. Some of these are summarized
below.
1.2 LITERATURE SURVEY
Chow (1959) gave various properties of optimal sections. He
expressed the relations between the section variables of the most
hydraulically efficient sections for different channel types. As an
objective, his study considered the conveyance of a given discharge
with minimum flow area. The constraints of these models were nothing
but the equation of uniform flow. The results of this study are still in use
for the corresponding channel types.
The relations obtained for the optimum section variables were
modified by considering different parameters. Including the effect of
freeboard, Guo C. and Hughes W. (1984) made a study on the optimum
values of section variables which either minimizes frictional resistance
or minimizes construction cost. They presented their solutions for the
trapezoidal channel sections.
Consideration of freeboard in channel design had not limited with
trapezoidal sections but Loganathan (1991) presented optimality
3
conditions for the parabolic-canal design accounting for freeboard and
limitations on velocity and canal dimensions.
Monadjemi (1994) has performed a detailed study of the
relationships derived for various channel types. Using the techniques of
calculus, he proved that, considering the minimization of flow area as
an objective will result in the same optimum values of section variables
as of considering the minimization of wetted perimeter.
In addition to the relations between optimum section variables,
Froehlich (1994) proposed simple expressions for the optimum section
variables of trapezoidal channel sections in terms of Q, n, So. He
extended his study for width and depth constrained trapezoidal channel
sections and proposed a graphical solution for the optimum values of
section variables considering the width and depth constraints on the
channel geometry.
Swamee (1995) proposed explicit equations of section variables
for the minimum flow area problem considering resistance equation of
uniform flow as a function of roughness height of channel bottom and
kinematic viscosity of water. These values can be used for the cases
where the Swamee’s resistance equation is used instead of Chezy’s or
Manning’s uniform flow equations.
After his study on 1995, Swamee and his two colleagues
(Swamee et al.,2000) proposed explicit equations for section variables
considering triangular, rectangular, trapezoidal and circular channel
geometries. They considered the minimization of channel cost as the
objective of the project. Swamee’s resistance equation was used for the
definition of uniform flow. The uniform flow equation is treated as
constraints of the optimization models.
1.3 SCOPE OF PRESENT STUDY
The optimum values of the section variables for different
arrangements of objective and constraint functions were generally
4
expressed as functions of other section variables. This requires an extra
solution of uniform flow equation to obtain the exact values of optimum
section variables.
In this study, optimum values of section variables for triangular,
rectangular, trapezoidal and circular channel sections are expressed in
terms of known unit cost terms considering the minimization of channel
cost as the objective and Manning’s uniform flow equation as the sole
constraint of the study. In addition to the traditional channel cost
function which consists the cost of earthwork and the cost of lining
material, an additional cost per unit excavation per unit depth is also
considered in the channel cost function.
Instead of the relations between the section variables used in
literature, the exact optimum values of section variables considering the
minimization of flow area as the objective are computed and compared
with the well-known values and relations. The values obtained
considering the additional cost, are also compared with the results of
Swamee’s ( 2000) study.
5
CHAPTER 2
DEFINITION OF PROBLEM AND OPTIMIZATION
ALGORITHM
Selection of the section variables such as channel side slope,
bottom width, flow depth and radius for open channel sections varies
according to the objective of the designer. For different objectives, it is
possible to have different values and relations between section
variables.
One of these objectives is the minimization of total cost of a
channel section providing that the given discharge will pass through the
channel reach safely. Besides its lower cost, optimum section variables
should be so arranged that they would also satisfy the hydraulic
constraints.
Hydraulic design of open channels, regardless of the objective
considered, are generally based on the assumption of uniform flow and
normal depth.
2.1 REVIEW OF UNIFORM FLOW
Uniform flow in an open channel can be achieved when a balance
between the resisting forces and gravity forces acting on the body of
water has been reached (Chow, 1959). A uniform flow has the same
flow depth, water area, flow velocity, and discharge at every section of
the channel reach. Generally, the mean velocity of a turbulent uniform
flow in an open channel has the following form;
6
yxScRV = (2.1)
where;
V = Average flow velocity at channel reach
c = A factor of flow resistance
R = Hydraulic radius
S = Slope of the channel bottom
x, y are called as hydraulic components
This equation represents the effect of interaction between the
friction forces created on the flow surface (wetted perimeter) of the
channel section and the driving gravity forces, on velocity of flow.
Using the continuity equation, the discharge, Q, can be written as
the multiplication of flow area, A, and the average flow velocity, V
AVQ = (2.2)
)( yxScRAQ = (2.3)
The above equation is the general structure of flow rate passing
through a channel reach which is assumed to have a uniform flow.
Hydraulicans have derived different equations and relations to explain
the uniform flow phenomenon. Among these equations the most
common ones are “Manning’s” and “Chezy’s” uniform flow equations.
The most widely used uniform flow formula in Turkey is the
Manning’s Formula and in SI units it is given as;
SARn
Q 321= (2.4)
where,
n= Manning’s roughness coefficient
7
Keeping in mind that the above equation results in more precise
solutions for steady turbulent flows, in this study Manning’s uniform flow
formula is used to define the uniform flow and normal depth concept.
2.2 COST STRUCTURE
It was previously stated that, the objective of the designer would
inevitably affect the section variables of a given channel type. In this
study, minimization of the cost of an open channel section, which is one
of the most important goals of an engineer in practice, is considered as
the objective and some explicit equations have been derived for the
section variables of rectangular, triangular, trapezoidal and circular
channel sections.
Generally, the cost per unit length of a lined open channel section
is defined as the addition of two terms; the cost of lining and the cost of
earthwork.
C= CL+ CE [$]/[L] (2.5)
Lining cost per unit length, CL, of a channel is nothing but the
multiplication of unit cost of lining, Lβ ( [$]/[L]2 ), which varies according
to the type of material used, by the wetted perimeter, P ( [L] ), of the
section.
CL = Lβ P [$]/[L] (2.6)
The cost of the earthwork per unit length, CE, of a channel can be
computed by multiplying the unit cost of earthwork, Eβ ( [$]/[L]3 ), by the
related area of excavation or fill, A ( [L]2 ).
In this study, top level of the channel section is interpreted at the
ground level so that the channel is always in cut and therefore the
earthwork contains only the cost of excavation (Eq.2.7).
8
CE = Eβ A [$]/[L] (2.7)
By using Eq.2.6 and Eq.2.7, the traditional cost function may be
written as
C = Lβ P + Eβ A [$]/[L] (2.8)
But it is not always possible to express the cost function as simple
as Eq.2.8. Sometimes the above equation needs to be modified
regarding the real-life situations. In practice, there are several factors
affecting the cost of a channel section. It is not always possible to
express the effect of these factors as continuous functions of some
variables. Varying expropriation costs along the route of the channel
alignment, cost of additional remedies taken for the progress of
construction and geographical conditions of construction area are some
examples of these factors. But on the other hand, some of these
unpredictable costs can be put into the cost definition as functions of
section variables.
In this study, it is decided that, for the same soil conditions, the
cost of earthwork will increase with the increasing depth of excavation.
This decision is made by considering several reasons like the
overburden pressures on deeper stratums of soil and the supporting
costs of deep excavations.
In order to express the effect of these conditions, a new term is
added into the earthwork cost function (Eq.2.7). The idea behind this
equation is the difference of cost of unit earthwork at different depths. In
addition to the general earthwork cost Eβ A, one should also pay an
additional cost of Aβ dda (Fig.2.1) for every unit area below the ground
surface.
9
where,
Aβ = Additional cost of earthwork per unit volume of excavation per unit
depth [$]/[L]4
da = Unit area of earthwork at height η or depth d [L]2
d = Depth of area below the ground surface [L]
G round Surface
y n
ψ
η
da
a
d
dη
Fig.2. 1 Typical Channel Section
a = The flow area at height η [L]2
η and ψ represents the vertical and horizontal axes of channel
geometry, respectively.
∫ −+=A
nAEE dayAC0
)( ηββ (2.9)
The second term at the right-hand side of Eq.2.9 represents the
summation of additional costs per unit area per unit depth below the
ground surface. It is clear that, for constant Aβ , the contribution of
second term of Eq.2.9 increases with increasing value of (yn-η) i.e. d. It
can be concluded that the deeper the channel the more expensive the
channel is.
10
The second term at the right hand side of the above equation can
be modified by using the techniques of integration by parts. Let,
uyn =− )( η and dvda = then,
dud =− η and va =
It is known from calculus that,
∫∫ −= vduuvdvu then,
=∫ −A
n day0
)( η ∫+−n
ny
y
n aday0
0)( ηη
∫+−=ny
ad0
)00( η ∫=ny
ad0
η
Then Eq.2.9 reduces to the below equation (Eq.2.10)
∫+=n
y
AEE adAC0
ηββ (2.10)
Keeping the lining cost unchanged, the total cost function of the
channel section can be rewritten as,
∫++=ny
AEL adAPC0
ηβββ (2.11)
2.3 OPTIMIZATION ALGORITHM
An optimization model generally consists of an objective function
and various constraint functions which control the value of the objective
function.
The objective function is the function which is generally to be
minimized or maximized provided that all the constraints are satisfied.
In the domain of the optimization model, the objective function may
have different local optimum values which all satisfy the constraints.
11
The exact optimum value should be the one which is optimum in the
whole domain of the model, i.e. the global optimum value.
In linear optimization models it is easy to distinguish between local
optimums and global optimums. Analytical methods help us to
determine the global optimums. But that is not the case for non-linear
optimization models. Although precise search techniques are used
there is always a doubt about the global optimum value. Therefore, the
computed optimum value should always be checked whether it is local
or global.
In the preceding sections, the uniform flow and the cost equations
of a lined open channel were given. In order to get the section variables
of a channel section which results in the least cost for a given discharge
and bottom slope, the above mentioned equations should be put in an
optimization model.
The objective function of this model is taken as the minimization of
cost. The uniform flow equation is treated as a constraint and put into
the optimization model. In the following paragraphs, optimization
models for rectangular, triangular, trapezoidal and circular sections are
constructed and simplified equations are derived for numerical
computations.
In order to construct a general optimization model, typical channel
section shown in Fig.2.1 is used.
Objective Function:
Minimize ∫++=ny
AEL adAPC0
ηβββ (2.12)
Subject to:
SARn
Q 321= (2.13)
or,
01 32 =− SARn
Q (2.14)
12
Eq.2.12 together with Eq.2.14, forms the general optimization
algorithm for a minimum cost open channel section. The terms of this
model are all in dimensional forms. In order to easily trace the effects of
variables on the model, the above equations are put into non-
dimensional forms. This conversion is done by defining a length scale, λ
as follows.
83
=
S
Qnλ [ L] (2.15)
Using the length scale, λ and unit cost of earthwork, Eβ , the non-
dimensional forms of the terms which are used in Eq.2.12 and Eq.2.14 are
developed. Below are the non-dimensional forms of total cost, unit cost
of lining, additional unit cost of earthwork, flow area, wetted perimeter,
bottom width of the channel, normal depth of the flow and radius of the
channel ( circular sections), respectively.
Non-dimensional Forms:
2*λβE
CC = (2.16)
λβ
ββ
E
LL =* (2.17)
E
AA
β
λββ =* (2.18)
2*λ
AA = (2.19)
λ
PP =* (2.20)
λ
bb =* (2.21)
λn
n
yy =* (2.22)
13
λ
rr =* (2.23)
Dividing both sides of Eq.2.12 by Eβ λ2, it may be expressed in non-
dimensional form as below,
Minimize3
0*
****λ
ηββ
∫++=
ny
A
L
adAPC (2.24)
Eq.2.14 may also be expressed in terms of non-dimensional forms.
To do this, both sides of this equation is divided by the flow rate, Q.
013
8
32
=−λ
AR,
013
23
10
32
35
=− −
−
λλ
PA then, 01 3
2
*3
5
* =−−
PA (2.25)
Eq.2.24 and Eq.2.25 form the general optimization structure of the
minimum cost problem in non-dimensional forms. The equations of *A
and *P vary with the geometry of the channel section in consideration.
Therefore, for every channel section i.e. triangular, rectangular,
trapezoidal and circular, there will be one objective and one constraint
function all of which have the same general optimization structure. In
the following paragraphs the non-dimensional optimization models for
each of these sections are given separately.
2.3.1 TRIANGULAR SECTION
Triangular open channel sections (Fig.2.2) are generally used for
the drainage facilities of roadways. They collect the surface-water and
water coming from the side slopes (cut areas) and convey them to safe
places where the hazardous effects of water on roadway structure are
minimized.
14
In order to obtain the optimum values of normal depth, ny , and
side slope, m, which result in the minimum cost, the optimization model
referring to Eq.2.24 and Eq.2.25 shall be constructed. In order to do this,
first of all the geometrical parameters, such as flow area and wetted
perimeter shall be expressed in terms of section variables.
y 1
n
m
Ground Surface
1
m
Fig.2. 2 Triangular Channel Section
Geometrical parameters of a triangular section can be computed
as follows,
2
nmyA = [ L2] , 212 myP n += [ L]
Using the length scale, λ , the non-dimensional forms can be
expressed as follows,
83
=
S
Qnλ
2*λ
AA = ,
λ
PP =*
2
** nmyA = , 2
** 12 myP n +=
and the integration of the third term of Eq.2.24 can be expressed as,
∫∫ =nn yy
dmad0
2
0ηηη
15
3
3
nmy=
Therefore, the final non-dimensional equations for optimization
model of a triangular section can be written as follows.
Objective Function:
Minimize3
123
**2
*
2
***nA
nnL
mymymyC
ββ +++= (2.26)
Subject to:
( )( )
0
12
13
22
*
35
2
* =+
−my
my
n
n (2.27)
By solving the above model, the optimum values of section
variables in non-dimensional forms, considering the minimization of cost
as the target, will be computed.
2.3.2 RECTANGULAR SECTION
Rectangular open channel sections are one of the most widely
used channel types in hydraulic engineering. There are so many
examples of rectangular channel applications like conveyance lines for
irrigation and municipal purposes, stilling basins of spillways, flood
protection structures, etc.
Compared to other channel types, rectangular channels have the
advantage of being constructed by smaller top width usage. This
property of rectangular channels makes them preferable for the works
where the land usage is limited by some means. These restrictions
generally occur in urban areas where existing or planned structures do
not permit the usage of sloped side channels.
16
y n
Ground Surface
b
Fig.2. 3 Rectangular Channel Section
Similar to triangular sections, rectangular sections can also be
defined by two section variables, bottom width of the channel, b, and
the depth of flow, ny ( Fig.2.3). The optimum values of these variables can
be computed by solving the optimization model of this channel section.
The optimization model, regarding the Eq.2.24 and Eq.2.25, is constructed
as follows,
nbyA = [ L2] , byP n += 2 [ L]
in non-dimensional forms,
2*λ
AA = ,
λ
PP =*
*** nybA = , *** 2 byP n +=
the integration term,
∫∫ =nn yy
dbad00
ηηη
2
2
nby=
Using the above expressions, the final optimization model in non-
dimensional form can be written as follows,
Objective Function:
17
Minimize2
)2(2
*********
nAnnL
ybybbyC
ββ +++= (2.28)
Subject to:
( )( )
02
13
2
**
35
** =+
−by
yb
n
n (2.29)
2.3.3 TRAPEZOIDAL SECTION
Trapezoidal channel sections are the most widely used open
channel sections in engineering. Most of the main water conveying lines
have the trapezoidal geometry. The most important advantage of
trapezoidal sections is their ease of construction. Besides their
constructional advantages, they have also the advantageous of high
hydraulic efficiency. Therefore, it is not surprising that most of the water
carrying and discharging lines have been made of trapezoidal
geometry.
In order to define a trapezoidal section, two section variables are
not sufficient. It requires three section variables i.e., bottom width, side
slope and flow depth (Fig.2.4). The optimum values of these variables
which result in the least cost for given discharge and bottom slope, can
be computed by using the optimization algorithm defined by Eq.2.24 and
Eq.2.25.
m
y b
n 1
m
Ground Surface
1
Fig.2. 4 Trapezoidal Channel Section
18
The geometrical parameters of a trapezoidal section in terms of
section variables can be expressed as follows,
2
nn mybyA += [ L2], bmyP n ++= 212 [ L]
in non-dimensional form,
2*λ
AA = ,
λ
PP =*
2**** nn myybA += , *
2** 12 bmyP n ++=
and the integration term of Eq.2.24,
ηηηη dmbadnn yy
)(0
2
0 ∫∫ +=
32
32
nn myby+=
The above definitions can be grouped in order to construct the
optimization model of a trapezoidal section.
Objective Function:
Minimize
)32
()()12(3
*
2
**
*
2
****
2
***nn
AnnnL
myybmyybbmyC ++++++= ββ (2.30)
Subject to:
( )( )
0
12
13
2
*
2
*
35
2
*** =++
+−
bmy
myyb
n
nn (2.31)
19
2.3.4 CIRCULAR SECTION
Circular sections are the water conveying elements of which
applications are generally observed in the field of irrigation. Among the
others, circular sections have the superiority of hydraulic efficiency. The
best hydraulic section possible is that of a semicircular channel (Okishi
et al.,1994). For a given flow area, semicircular sections have the least
wetted perimeter, consequently the least resisting force and the least
energy loss. Therefore, as far as the construction techniques allow, it is
more beneficial to convey water with semicircular sections. r
y n
θ
α Ground Surface
Fig.2. 5 Circular Channel Section
To describe a circular section, one needs two section variables,
the height of the channel, ny , and the radius, *r , of the channel section.
These two variables are sufficient to calculate geometrical properties of
the section.
In order to construct the optimization model regarding the Eq.2.24
and Eq.2.25, the geometrical parameters should be defined.
The flow area,
[ ]ααπ
αππ cossin
2
)2(2 rrrA −−
= [ L2]
( )[ ]ααπ 2sin22
2
−−=r
A
20
2*λ
AA = , ( )[ ]ααπ 2sin2
2
2
** −−=
rA (2.32)
The wetted perimeter,
)2( απ −= rP [ L]
λ
PP =* , )2(** απ −= rP (2.33)
and the integration term of Eq.2.24,
r
r ηθ
−=sin
( )θη sin1−= r
θθη drd cos−=
ηθθηθπη drdr
adnnn yyy
∫∫∫ −−=0
2
0
2
0)cossin()2(
2
θθθθθθθπ ππ drr
drr
r
yr
r
yr nn
cos)cossin2(2
cos)2(2
)arcsin(
2
2)arcsin(
2
2
∫∫−−
−−−−=
++−= ∫∫∫ θθθθθθθθθπ πππ drdrdr
r aaa
222
2
coscossin2cos2cos2
(2.34)
let,
u=θ , dvd =θθcos
dud =θ , v=θsin
the second term of Eq.2.34 can be written as,
)sinsin(2)cos(2
22
2
∫−=∫aaa
drdrπ
ππ
θθθθθθθ
a
r2
)cossin(2 πθθθ += (2.35)
and let,
dtdt =−⇒= θθθθ sincos2cos2
21
the third term of Eq.2.34 can be written as,
)(cos3
2)coscossin2(
2
3
2
aa
rdr ππ
θθθθθ −=∫ (2.36)
using Eq.2.35 and Eq.2.36, the Eq.2.34 can be written as,
−−++−−= α
πααααπ 3
2
cos3
2)
2cossin(2)1(sin
2rrr
r
−−+++−= απαααπαπ 3
2
cos3
2cos2sin2sin
2rrrrrr
r
−+−−= )
3
cos1(cos2)2(sin
2
22 αααπα rr
r
−
−+−−= )3
sin11(cos2)2(sin
2
22 αααπα rr
r
+
+−−= )3
sin2(cos2)2(sin
2
22 αααπα rr
r
r
yr n−=αsin
r
yrr n
22 )(cos
−−=α
−+−−+
−−−−=
2
2222
2
3
)(2)(2arcsin2)(
2 r
yrryrr
r
yryr
r nn
nn π (2.37)
The terms of Eq.2.32, Eq.2.33 and Eq.2.37 are inserted into Eq.2.24
and Eq.2.25 to construct the optimization model.
Objective Function:
Minimize
−
−−+
−−=
*
***
*
***** arcsin2
2arcsin2
r
yrr
r
yrrC nn
L ππβ
( ) ( )
+
−−−−
+
−−
−
*
****
**
*
2
**
2
*
*
** arcsin22
2r
yryr
r
r
yrr
r
yr nnA
nn πβ
22
( ) ( )( )
−+−−
2
*
2
**
2
*2
**
2
*3
22
r
yrryrr n
n (2.38)
Subject to:
0
)arcsin(2(
)()(2)arcsin(2
21
32
*
***
35
*
2
**
2
*
*
**
*
**
2
*
=−
−
−−−−
−−
−
r
yrr
r
yrr
r
yr
r
yrr
n
nnn
π
π
(2.39)
23
CHAPTER 3
COMPUTATIONS AND ANALYSIS OF RESULTS
The structure of the equations derived for the optimization models
of different channel types in sections 2.3.1 - 2.3.4 are not easy to solve
analytically. The nonlinearities between the variables make it difficult to
get a unique relation for optimum section variables. Therefore, a
numerical study has been performed to solve the optimization models.
For every channel type, numerous computations were performed
for different values of non-dimensional unit cost of lining, *Lβ and non-
dimensional unit cost of additional earthwork, *Aβ by the help of MS-
Excel software.
3.1 ANALYTICAL SOLUTION
Although the difficulty of analytical solution has directed us
towards numerical computations, some certain results of analytical
solutions should be used as guidance for numerical analysis. Results of
analytical solutions are generally adopted as boundary conditions to the
numerical analysis. One of the boundary conditions of the problem in
concern is the case of no additional cost term, *Aβ =0.
Analysis of this situation is useful for testing the stability of the
numerical model. As it was stated before, the optimization models
derived in section 2.3.1 through section 2.3.4, have highly nonlinear
terms which make them to be easily affected by the initial and relative
values of variables. By the help of known analytical solutions, the
24
software options, i.e. initial values, tolerances, convergence etc. are
adjusted and the accuracy of the program is checked.
The non-dimensional optimization model for the case of no
additional cost term, regarding Eq.2.24 and Eq.2.25, can be defined as
follows:
Objective Function:
Minimize **** APC L += β (3.1)
Subject to:
01 32
*3
5
* =×−−
PA (3.2)
The objective of the above model can be considered as the
minimization of the summation of two terms, non-dimensional wetted
perimeter multiplied by a positive constant and non-dimensional flow
area of the channel section, respectively. It has been proved that the
solution of minimum wetted perimeter and minimum flow area
problems, would result in the same values of section variables
(Monadjemi, 1994). This means that the solution of minimum wetted
perimeter or minimum flow area problems, considering the uniform flow
equation as the sole constraint, will result in the same values of section
variables.
Minimize *A = Minimize *P
The optimum values of section variables obtained from the
solution of Eq.3.1 and Eq.3.2 should also satisfy the minimum values of
wetted perimeter and flow area.
Minimum *** APL +β = Minimum *P + Minimum *A
And since the optimum values of minimum wetted perimeter and
minimum flow area are the same, it will be enough to solve either the
25
minimization of flow area or wetted perimeter instead of their
summation.
Minimum *** APL +β = Minimum *P + Minimum *A = Minimum( *P or *A )
Therefore, the optimization model of Eq.3.1 and Eq.3.2 can be
rewritten as follows,
Objective Function:
Minimize *A (3.3)
Subject to:
01 32
*3
5
* =×−−
PA (3.4)
The solution of the above optimization model for each channel
type will result in the optimum values of non-dimensional side slope,
bottom width, flow depth, and channel radius for the case of no
additional cost term. These values of section variables will form the
boundary conditions for numerical analysis.
In order to solve the above optimization problem, the objective and
the constraint functions are combined to form an augmented function.
The general structure of the augmented function,φ , is as follows,
)1( 32
*3
5
**
−×−+= PAA lφ (3.5)
where,
l = a Lagrange multiplier.
Since the value of constraint function (Eq.3.4) at the optimum
solution should equal to zero, the optimum points of augmented
function will not differ from the optimum values of original objective
function. By using the principles of differential calculus, the augmented
26
function and the conditions to be satisfied for each of the channel type
are given below.
Triangular Channel Section:
)1(),,( 32
*3
5
***
−×−+= PAAym n llφ
0=∂∂m
φ, 0
*
=∂∂
ny
φ, 0=
∂∂l
φ
Rectangular Channel Section:
)1(),,( 32
*3
5
****
−×−+= PAAyb n llφ
0*
=∂∂b
φ, 0
*
=∂∂
ny
φ, 0=
∂∂l
φ
Trapezoidal Channel Section:
)1(),,,( 32
*3
5
****
−×−+= PAAybm n llφ
0=∂∂m
φ, 0
*
=∂∂b
φ, 0
*
=∂∂
ny
φ, 0=
∂∂l
φ
Circular Channel Section:
)1(),,( 32
*3
5
****
−×−+= PAAyr n llφ
0*
=∂∂r
φ, 0
*
=∂∂
ny
φ, 0=
∂∂l
φ
According to the channel type considered, the above equations
are solved simultaneously (Appendix A) for the optimal values of non-
dimensional section variables. These optimum non-dimensional values
of side slope, *m , bottom width, *
*b , normal depth, *
*ny and channel
radius, *
*r are tabulated in Table 3.1. The values prove the well-known
relations for section variables of different channel types. For a triangular
section, *m =1, for a rectangular channel *b =2 *ny , for a trapezoidal
channel, *b = *32 ny , *m = 31 , for a circular channel *r = *ny
27
Table 3. 1 Optimum Non-dimensional Section Variables for *Aβ = 0
The optimum values of bottom width, *b , normal depth, *
ny and
channel radius, *r can be easily computed by multiplying the
corresponding optimum non-dimensional values by the length scale, λ .
The optimum value of optimum side slope, *m is the one given in Table
3.1.
3.2 NUMERICAL COMPUTATIONS
In section 3.1, it is proved analytically that the optimum values of
section variables for the case of no additional cost term, *Aβ , are
independent of the non-dimensional lining cost term, *Lβ . The optimum
values of section variables, can be computed directly by using the
values of Table 3.1. But in this study, we deal with the cost function
consisting the additional cost term, *Aβ .
As it was stated before, analytical solutions of the equations
derived for the optimization model for this case do not result in
simplified expressions. Therefore, the solutions of these models are
performed by using numerical computations.
Numerical computations of the optimization models which are
constructed in sections 2.3.1-2.3.4, are performed by using the modules
of MS-Excel software. By using the “Solver” module of this software,
CHANNEL TYPESOptimumNon-dimensionalSection Variables Triangular Rectangular Trapezoidal Circular
Side Slope*m
1.000 -3
1-
Bottom Width*
*b - 27/8 1.11755 -
Normal Depth *
*ny 23/8 2-1/8 0.96782 1.00395
Channel Radius*
*r - - - 1.00395
28
numerous solutions have been performed for different values of inputs,
i.e. *Aβ and *Lβ . Regarding the rectangular channel section as an
example, the structure of the computation method is explained below.
Fig.3. 1 Layout of Spreadsheet Used for Numerical Computations
At the very beginning, in order to perform computations, “Solver”
needs four parameters to be defined. The first one of these parameters
is the “Target Cell”. “Target Cell” is the cell in which the objective
function is formulated as a function of section variables and inputs. In
this study, the cell in which Eq.2.28 formulated is set for the “Target
Cell”(Fig.3.2).
29
Fig.3. 2 Layout of “Solver” Menu
The second parameter is about the value of “Target Cell”.
Software allows selection of one from the three opportunities. It may be
set to maximum, minimum or to a certain value. In our case it is set to
minimum (Fig.3.2).
The third parameter is the “By Changing Cells” item. This item
specifies the cells which can be adjusted until the cell in the “Target
Cell” box reaches its target. In this study, these cells are nothing but the
section variables of the channel in concern (Fig.3.2).
The last parameter for the model input is “Subject to the
Constraints” box. Here, the restrictions on the problem are defined.
Software performs iterations until all the constraints are satisfied and
the objective reaches its target with an appropriate precision. Eq.2.29 is
the constraint of our problem for the rectangular channel. It is
formulated in a cell and specified as the constraint of the problem
(Fig.3.2).
After defining the model variables, the next step is to specify the
solution options. Under the “Solver Options” menu one has the chance
of setting the parameters which control the precision and the time spent
for the solution. These parameters are maximum iteration time (in
seconds), maximum iteration number, precision, tolerance and
30
convergence. The default values used in this study are 100, 1000,
0.00000001, 5%, 0.0001, respectively. It should be noted that the
number of iterations performed to get the optimum values, did not
approach to limiting values of the default values shown in Fig.3.3
Fig.3. 3 Layout of “Solver Options” Menu
Besides the opportunity of selecting the precision parameters, one
has also the chance of selecting the mathematical algorithm to be used
for the computations. There are three boxes provided for the selection
of algorithm choices. These are “Estimates”, “Derivatives” and “Search”
boxes. The definitions of these choices are clearly explained in MS-
Excel. According to these definitions, the best combination for this study
is selection of “Quadratic” for Estimates, “Central” for Derivatives and
“Newton” for Search boxes.
Since all the inputs and model options are specified, the software
can be executed by just clicking the “Solve” button. Here, it should be
noted that the system is solved for the non-dimensional optimum values
of section variables for given values of *Aβ and *Lβ . It is observed that,
for each different values of *Aβ or *Lβ the model results different values
31
of section variables. Therefore, for every value of unit cost terms, the
model should be recomputed. Changing the values of *Aβ
and *Lβ manually and solving the system would spent so much time that
a simple code has been written to change the values of inputs and
arrange the optimum solutions for further use in regression analyses.
3.3 REGRESSION ANALYSES
The aim of the regression analyses is to fit equations to the
optimum solutions of section variables obtained by numerical
computations.
The data obtained from numerical computations are transferred to
statistical software which performs regression analyses of the inputs
and computed section variables. Besides its available regression
models, the software allows users to define their own form of equation
which will be fitted to the existing data. But, to define the structure of
regression model, one should clearly understand the behavior of
solutions for different inputs.
3.3.1 EFFECT OF COST TERMS ON OPTIMUM SECTION VARIABLES
Numerical solutions of optimization models for various values of
unit cost terms (Appendix B) have shown that, the optimum solutions of
section variables would not be independent of the cost terms.
For example, for a triangular section it is observed that, an
increase in non-dimensional unit cost of additional earthwork term, *Aβ
will also result in an increase of optimum side slope, m* (Fig.3.4).
32
0.8
1
1.2
1.4
1.6
1.8
2
0 1 2 3 4 5 6 7 8 9 10
β A*
m*
Fig.3. 4 Variation of Optimum Side-Slope of a Triangular Section with *Aβ ( *Lβ =1.0)
0.8
0.9
1
1.1
1.2
1.3
1.4
0 2 4 6 8 10
β A*
y n**
Fig.3. 5 Variation of Optimum Non-Dimensional Normal Depth of a Triangular Section
with *Aβ ( *Lβ =1.0)
And it is observed that, with the increase in additional non-
dimensional unit cost term, the value of optimum non-dimensional
normal depth, *
*ny , will decrease (Fig.3.5). Like the same way of *Aβ , the
effects of *Lβ on the optimum side-slope and non-dimensional normal
depth for a triangular section are given in Fig.3.6 and Fig.3.7, respectively.
33
0.8
0.85
0.9
0.95
1
1.05
1.1
1 2 3 4 5 6 7 8 9 10
β L*
m*
Fig.3. 6 Variation of Optimum Side-Slope of a Triangular Section with *Lβ ( *Aβ =0.5)
0.8
0.9
1
1.1
1.2
1.3
1.4
1 2 3 4 5 6 7 8 9 10
β L*
y n**
Fig.3. 7 Variation of Optimum Non-Dimensional Normal Depth of a Triangular Section
with *Lβ ( *Aβ =0.5)
The effects of cost terms on section variables for different channel
sections have also the same attitude of triangular section. By using the
numerical computation results of Appendix B, the behavior of optimum
section variables for different values of cost terms can be generalized in
Fig.3.8 and Fig.3.9.
34
β L*
Op
tim
um
No
n-d
imen
sio
na
lS
ecti
on
Var
iab
les
m *
b **
r **
y n**
(m* ,b
** ,yn
** ,r
* )
Fig.3. 8 Variation of Optimum Non-Dimensional Section Variables
with *Lβ ( *Aβ =constant )
β Α∗
Op
tim
um
No
n-d
ime
ns
ion
al
Se
cti
on
Va
ria
ble
s
m *
r **
b **
y n**(m
* ,b** ,y
n*
*,r
** )
Fig.3. 9 Variation of Optimum Non-Dimensional Section Variables
with *Aβ ( *Lβ =constant )
Fig.3.8 and Fig.3.9 aided us to understand the effects of unit cost
terms on the optimum section variables. It may be clearly concluded
35
that unit cost of additional earthwork term, *Aβ ,and unit cost of lining
term *Lβ have opposite effects on the optimum section variables. For a
section variable, while the increase in *Aβ results in a higher optimum
value, on the other hand, an increase in *Lβ will result in a smaller
optimum value for the same section variable. In short, except for the
normal depth, section variables are directly proportional to non-
dimensional unit cost term for lining and inversely proportional to non-
dimensional unit cost of additional earthwork term.
3.3.2 DETERMINATION OF EQUATIONS
The interpretation of Fig.3.8 and Fig.3.9 can help us to understand
the general effects of cost terms on the optimum section variables. But
in order to obtain the exact values of optimum section variables, one
should either solve the Eq.2.24 an Eq.2.25 analytically or solve these
equations numerically.
In this study, simple equations are obtained for the optimum
section variables that one can easily compute optimum solutions with
reliable accuracies within a short time.
In order to achieve our goal, the numerical solutions obtained in
section 3.2 are analyzed by the help of a software. This software
performs regression analysis for the given variables and their
corresponding results. In this study, the relation between the non-
dimensional unit cost terms and optimum solutions of section variables
are investigated by the help of regression analysis.
Before performing the regression analysis, the variables affecting
the optimum solutions should be clearly identified. The individual effects
of non-dimensional cost terms on the optimum section variables were
given in section 3.3.1. But these do not reflect the combined effects of
non-dimensional unit cost of lining and non-dimensional unit cost of
additional earthwork on the optimum section variables. Therefore, to
define the combined effects cost terms on the optimum values of
36
section variables, the ratio of ** LA ββ is interpreted as an independent
variable. In order to understand the effects of ** LA ββ on optimum
values of section variables, the optimum section variables are plotted
against ** LA ββ (Fig.3.10 through Fig.3.13)
0.98
1
1.02
1.04
1.06
1.08
1.1
1.12
1.14
1.16
0 0.2 0.4 0.6 0.8 1 1.2
β A* /β L*
m*
Fig.3.10 Variation of Optimum Non-dimensional Side-Slope of a Triangular Section
with ** LA ββ
1.21
1.22
1.23
1.24
1.25
1.26
1.27
1.28
1.29
1.3
1.31
0 0.2 0.4 0.6 0.8 1 1.2
β A* /β L*
y n*
*
Fig.3. 11 Variation of Optimum Non-dimensional Normal-Depth of a Triangular Section
with ** LA ββ
37
1.1
1.12
1.14
1.16
1.18
1.2
1.22
1.24
1.26
1.28
1.3
1.32
0 0.2 0.4 0.6 0.8 1 1.2
β A* /β L*
b**
Fig.3.12 Variation of Optimum Non-dimensional Bottom Width of a Trapezoidal
Section with ** LA ββ
0.86
0.88
0.9
0.92
0.94
0.96
0.98
0 0.2 0.4 0.6 0.8 1 1.2
β A* /β L*
yn
**
Fig.3.13 Variation of Optimum Non-dimensional Normal-Depth of a Trapezoidal
Section with ** LA ββ
Although the above graphs are given for triangular and trapezoidal
channel sections, the effect of ** LA ββ on the optimum section
variables are also the same for other channel sections. Regarding the
above graphs, the structures of the regression models to be analyzed
can be constructed as linear functions of ** LA ββ . The general
38
arrangement of the regression models are given below in Eq.3.6 through
Eq.3.9.
MODEL 1
Side Slope:
+=
*
**
L
Ammo zzm
β
β(3.6)
Bottom Width:
+=
*
**
*
L
Abbo zzb
β
β(3.7)
Normal Depth:
1
*
**
* 1
−
+=
L
Ayyon zzy
β
β(3.8)
Channel Radius:
+=
*
**
*
L
Arro zzr
β
β(3.9)
The above equations define the relation between the ratio of
** LA ββ and optimum non-dimensional section variables as the
summation of two terms. The first terms are nothing but the coefficients
computed at section 3.1(Appendix A). These coefficients, moz , boz , yoz
and roz , refer to *m , *
*b , *
*ny and *
*r of Table 3.1, respectively. The
second terms are added to define the behavior of cost terms analyzed
at Fig.3.10 through Fig.3.13.
The regression models of Eq.3.6 through Eq.3.9 are not the only
choices for the regression analysis. Regarding the study of Swamee et
al.(Swamee et al.,2000) new regression models which are very similar
to the ones of Eq.3.6 through Eq.3.9 are constructed. These models can
be expressed as follows.
39
MODEL 2
Side Slope:
++=
*
**
1 LmL
AmAmo
k
kkm
β
β(3.10)
Bottom Width:
++=
*
**
*1 LbL
AbAbo
k
kkb
β
β(3.11)
Normal Depth:
1
*
**
*1
1
−
++=
LyL
AyA
yonk
kky
β
β(3.12)
Channel Radius:
++=
*
**
*1 LrL
ArAro
k
kkr
β
β(3.13)
Like the previous ones, these models also include addition of two
terms. The coefficients of mok , bok , yok and rok again refer to *m , *
*b ,
*
*ny and *
*r of Table 3.1, respectively. The second terms are added to
express the effect of cost terms on optimum section variables.
Compared with the first ones, these models investigate the individual
effects of *Aβ and *Lβ on the optimum values of section variables. The
independent variables are *Aβ and *Lβ .
The analytical solution of Eq.2.24 and Eq.2.25 for different channel
sections has shown that, to obtain a relation between the unit cost
terms and the optimum values of section variables which is valid for
every values of *Aβ and *Lβ is almost impossible.
The evaluation of numeric solutions for different values of ** LA ββ ,
concluded that different equations may be derived for different ranges
of ** LA ββ . Therefore, it is decided to obtain an upper bound for the
40
value of ** LA ββ and construct the equations valid between these
ranges. This limit of ** LA ββ , is computed regarding the real values of
Aβ , Lβ , Eβ and λ .
In Turkey there has not been found any official publication defining
the unit cost of additional earthwork. Therefore, the values at the study
of Swamee et al.(Swamee et al.,2000) are taken as reference to the
unit cost of additional earthwork.(Table.3.6). The maximum value of
** LA ββ can be expressed as follows,
( )maxmax
max**
×
=
L
E
E
ALA
β
λβ
β
λβββ
2
max
maxmax
λβ
β
β
β
×
=
L
E
E
A
the maximum combination of
×
L
E
E
A
β
β
β
βis computed as 0.03349 for
brunt clay tile lining excavated in hard soil strata (Table 3.6). The next
step is to compute the 2
maxλ .
86
min
maxmax2
max
=
S
nQλ
To obtain the maximum value of the length scale, λ , the limit
values of the terms forming λ should be identified. The assumptions
made for the limit values of Q, n and S are 100 m3/sec, 0.033 and
2x10-4, respectively. Corresponding maxλ value is 7.727. Resulting
( )max** LA ββ value is 1.999.
Knowing this value, the interval of 0< ** LA ββ <2 can be used as
the limiting values of ** LA ββ for the prospective equations which will be
constructed. Therefore, the regression analyses are performed for the
values of ** LA ββ which are smaller than 2. The solutions of these
41
analyses for model 1 and model 2 are given in Table 3.2 and Table 3.3.
Corresponding R2 values for each of the solution are given in Table 3.4
and Table 3.5
Table 3. 2 Coefficients of Model 1
Channel TypeSectionVariable
CoefficientsTriangular Rectangular Trapezoidal Circular
zmo 1.000 0.577Side Slope
zm 0.135 0.065
zbo 27/8 1.118Bottom Width
zb 0.246 0.177
zyo 23/8 2-1/8 0.968 1.004Normal Depth
zy 0.063 0.129 0.104 0.055
zro 1.004ChannelRadius zr 0.113
Table 3. 3 Coefficients of Model 2
Channel TypeSectionVariable
CoefficientsTriangular Rectangular Trapezoidal Circular
kmo 1.000 0.577
kmL 5.375 4.994Side Slope
kmA 0.741 0.331
kbo 27/8 1.118
kbL 5.359 5.200Bottom Width
kbA 1.347 0.938
kyo 23/8 2-1/8 0.968 1.004
kyL 5.298 5.272 5.101 4.937Normal Depth
kyA 0.342 0.695 0.541 0.277
kro 1.004
krL 5.008ChannelRadius
krA 0.580
Table 3. 4 Coefficients of Multiple Determination (Rc2) for Model 1
Channel TypeSection Variable Equation
Triangular Rectangular Trapezoidal CircularSide Slope Eq.3.6 0.995 0.962
Bottom Width Eq.3.7 0.990 0.998Normal Depth Eq.3.8 0.990 0.986 0.993 0.989
Channel Radius Eq.3.9 0.998
42
Table 3. 5 Coefficients of Multiple Determination (Rc2) for Model 2
Channel TypeSection Variable Equation
Triangular Rectangular Trapezoidal CircularSide Slope Eq.3.10 0.996 0.964
Bottom Width Eq.3.11 0.992 0.999Normal Depth Eq.3.12 0.990 0.987 0.995 0.991
Channel Radius Eq.3.13 0.999
To evaluate the accuracy of the derived equations, an error term is
defined between the numerical solutions (computed values) and the
values obtained by using the above equations (fitted values).
ic
ific
i ∆
∆−∆=ε where,
=i
ε error value for the section variable i
=∆ic
computed value for the section variable i
=∆if
fitted value for the section variable i
i = side slope (m), normal depth (yn), bottom width (b) and channel
radius (r)
To examine and compare the accuracy of the derived equations,
the error values of both models are plotted against the ratio of ** LA ββ
for each channel section and section variable. The following figures
show the change in error values with different values of ** LA ββ . The
equations derived from regression analyses are shown on the related
graphs (Fig.3.14 through Fig.3.22).
43
Table
.3.
6 C
ost
Term
s (S
wam
ee e
t al.,
20
00)
βL/β
E(m
)
Typ
eof
Lin
ing
Concr
ete
Tile
Brick
Tile
Bru
nt
Cla
yT
ile
With
LD
PE
Film
With
out F
ilmW
ith L
DP
E
Film
With
out F
ilmW
ith L
DP
E
Film
With
out F
ilm
Typ
es
of
Str
ata
100
µ200
µ100
µ200
µ100
µ200
µ
βE/β
A (
m)
Ord
inary
Soil
12.7
513.0
212.2
46.3
96.6
75.8
86.0
86.3
55.5
76.9
6H
ard
Soil
10
10.2
29.6
5.0
15.2
34.6
24.7
74.9
93.3
78.8
6Im
pure
Lim
e N
odule
s8.9
9.1
8.5
54.4
74.6
64.1
14.2
54.4
43.8
99.9
6D
ry S
hoa
lwith
Sh
ingle
6.5
66.7
16.3
3.2
93.4
33.0
33.1
33.2
72.8
613.5
Slu
sh a
nd
Lahe
l6.4
6.5
46.1
43.2
13.3
52.9
53.0
53.1
92.7
913.8
6
44
Triangular Channel Section
0.E+00
1.E-02
2.E-02
3.E-02
0.0 0.5 1.0 1.5 2.0 2.5
β A* /β L*
ε m
Fig.3.14 Variation of Error Values of *m with ** LA ββ for Triangular Channel
Section
0.E+00
1.E-02
2.E-02
3.E-02
0.0 0.5 1.0 1.5 2.0 2.5
β A* /β L*
εyn
Fig.3.15 Variation of Error Values of **ny with ** LA ββ for Triangular Channel Section
*
*
375.51
741.01*
L
Amβ
β
++=
1
*
*83*
* )298.51
342.01(2 −
++=
L
Any
β
β
1
*
*83*
* )063.01(2 −+=L
Any
β
β
*
*135.01*L
Amβ
β+=
45
Rectangular Channel Section
0.E+00
1.E-02
2.E-02
3.E-02
0 0.5 1 1.5 2 2.5
β A* /β L*
ε b
Fig.3.16 Variation of Error Values of **b with ** LA ββ for Rectangular Channel
Section
0.E+00
1.E-02
2.E-02
3.E-02
0 0.5 1 1.5 2 2.5
β A* /β L*
εyn
Fig.3.17 Variation of Error Values of **ny with ** LA ββ for Rectangular Channel
Section
*
*87*
* 246.02L
Abβ
β+=
1
*
*81*
*)
272.51
695.01(2 −−
++=
L
A
ny
β
β
1
*
*81*
* )129.01(2 −− +=L
Any
β
β
*
*87*
*359.51
347.12
L
Abβ
β
++=
46
Trapezoidal Channel Section
0.0E+00
1.0E-02
2.0E-02
3.0E-02
0 0.5 1 1.5 2 2.5
β A* /β L*
ε
m
Fig.3.18 Variation of Error Values of *m with ** LA ββ for Trapezoidal Channel
Section
0.E+00
1.E-02
2.E-02
3.E-02
0 0.5 1 1.5 2 2.5
β A* /β L*
ε
b
Fig.3.19 Variation of Error Values of **b with ** LA ββ for Trapezoidal Channel
Section
*
*
994.41
331.0
3
1*
L
Amβ
β
++=
*
*065.03
1*
L
Amβ
β+=
*
**
*177.011755.1
L
Abβ
β+=
*
**
*200.51
938.011755.1
L
Abβ
β
++=
47
0.E+00
1.E-02
2.E-02
3.E-02
0 0.5 1 1.5 2 2.5
β A* /β L*
ε
y n
Fig.3.20 Variation of Error Values of **ny with ** LA ββ for Trapezoidal Channel
Section
Circular Section
0.E+00
1.E-02
2.E-02
3.E-02
0 0.5 1 1.5 2 2.5
β A* /β L*
ε r
Fig.3.21 Variation of Error Values of **r with ** LA ββ for Circular Channel Section
1
*
**
* )101.51
541.01(96782.0 −
++=
L
Any
β
β
1
*
**
* )104.01(96782.0 −+=L
Any
β
β
*
**
* 113.000395.1L
Arβ
β+=
*008.51
*580.0
00395.1*
*
L
Arβ
β
++=
48
0.E+00
1.E-02
2.E-02
3.E-02
0 0.5 1 1.5 2 2.5
β A* /β L*
εyn
Fig.3. 22 Variation of Error Values of **ny with ** LA ββ for Circular Channel Section
3.4 DISCUSSION OF RESULTS
The coefficient of multiple determination, Rc2 values (Table 3.4 and
Table 3.5) of each equation derived are all around 0.99 that all of these
equations can be accepted as the equations defining the behavior of
unit cost terms on the optimum values of section variables. But, besides
the Rc2 values it will be better to consider the error values of these
equations.
Analysis of Fig.3.14 through Fig.3.22 has shown that, the maximum
error value of the equations is around 0.01 which is an acceptable value
for an equation fitting study.
Comparison of two models has concluded that, although the
decision variables of the two models are not the same, their attitudes
towards the ratio of ** LA ββ are almost the same. That means that, the
controlling term affecting the optimum values of section variables are
not the individuals of *Aβ and *Lβ but the ratio of ** LA ββ . The structure
of Eq.3.10 through Eq.3.13 is nothing but another way of representing the
effect of ** LA ββ .
Since the equations (Eq.3.10 through Eq.3.13) derived according to
regression model 2 have less error values and fluctuations than the
1
*
**
* )055.01(00395.1 −+=L
Any
β
β
1
*
**
* )937.41
277.01(00395.1 −
++=
L
Any
β
β
49
ones of regression model 1, the equations of model 2 can be used to
compute the optimum non-dimensional values of section variables for
0< ** LA ββ <2.The exact values of section variables can be computed by
multiplying the corresponding non-dimensional section variables by λ.
3.5 DESIGN EXAMPLE
In this section, the application of the equations derived in section
3.3 will be illustrated by an example. This example is given in the study
of Swamee et al. (Swamee et al.,2000)
Example
Design a concrete trapezoidal channel of which,
Q= 125 m3/s
n= 0.015
S= 0.0002
=
A
E
β
β7.0 m and =
E
L
β
β12 m
Computations:
=
=
83
S
Qnλ 6.251 m
==E
AA
β
λββ * (1/7) x 6.251= 0.893
==λβ
ββ
E
LL* 12/6.251= 1.919
==
919.1
893.0
*
*
L
A
β
β 0.465<2, then the Eq.3.10 through Eq.3.12 can be used
to compute the section variables.
50
Side Slope
++=
*
**
1 LmL
AmAmo
k
kkm
β
β=
×+
×+
919.1994.41
893.0331.057735.0 = 0.605
Bottom Width
++=
*
**
*1 LbL
AbAbo
k
kkb
β
β=
×+
×+
919.1200.51
893.0938.011755.1 = 1.194
λ*
*
* bb = =1.194 x 6.251= 7.464 m
Normal Depth
1
*
**
*1
1
−
++=
LyL
AyA
yonk
kky
β
β=
1
919.1101.51
893.0541.0196782.0
−
×+
×+ =0.926
λ*
*
*
nn yy = = 0.926 x 6.251= 5.788 m
Comparison of these results with the Swamee’s results and
numerical solution results show that the difference between the values
is so small. This comparison is tabulated in Table3.4.
51
Table 3. 7 Comparison of Computed and Fitted Values
where,
The first subscripts refer to section variables, second ones refer to
the computed and fitted values of these section variables and third
ones, S, refer to Swamee’s fitted values.
∆mc 0.613
∆mf 0.605
∆mfS 0.602
∆bc 7.473
∆bf 7.464
∆bfS 7.461
∆yc 5.763
∆yf 5.788
∆yfS 5.783
54
REFERENCES
CHOW, V. T. (1959), Open Channel Hydraulics, McGraw-Hill, New
york, N. Y.
FROEHLICH, D. C., (1994), Width and Depth-Constrained Best
Trapezoidal Section, Journal of Irrigation and Drainage Engineering,
Vol.120, No. 4, pp. 828-835
GUO, C., HUGHES, W., (1984), Optimal Channel Cross Section With
Freeboard, Journal of Irrigation and Drainage Engineering, Vol.110, No.
3, pp. 304-314
LOGANATHAN, G. V., (1991), Goptimal Design of Parabolic Canals,
Journal of Irrigation and Drainage Engineering, Vol.117, No. 5, pp. 716-
735
MONADJEMI, P., (1994), General Formulation of Best Hydraulic
Channel Section, Journal of Irrigation and Drainage Engineering,
Vol.120, No. 1, pp. 27-35
OKIISHI T. H., MUNSON, B. R., and YOUNG, D. F., (1998),
Fundamentals of Fluid Mechanics (Second Edition), John Wiley & Sons,
Inc., Toronto, Canada
SWAMEE, P. K., (1995), Optimal Irrigation Canal Sections, Journal of
Irrigation and Drainage Engineering, Vol.121, No. 6, pp. 467-469
55
SWAMEE, P. K., MISHRA, G. C., and CHAHAR, B. R., (2000),
Minimum Cost Design of Lined Canal Sections, Journal of Water
Resources Management, Vol.14, pp. 1-12
56
APPENDIX A
COMPUTATION OF OPTIMUM SECTION VARIABLES BY USING
LAGRANGE MULTIPLIERS METHOD
TRIANGULAR SECTION
×−+=
−3
2
*3
5
*** 1),,( PAAym n llφ
( )( )
+−+=
32
2
*
35
2
*2
*
12
1
my
mymy
n
nn lφ
( )( )
0
12
13
22
*
35
2
* =+
−=∂∂
my
my
n
n
l
φ
( ) ( ) 32
2
*3
52
* 12 mymy nn +=⇒ (A.1)
( )( )( ) ( ) ( )( ) ( )( ) ( )( )
0
12
12321212
35
34
2
*
35
2
*
31
2
*21
2
*
32
2
*3
22
*
2
*2
* =
+
++−+−=
∂∂
−−
my
mymymmymymyyy
mn
nnnnnn
n lφ
( )( )( ) ( )( ) ( )( )( )
03212
35
310
2
*
65
2
*2
12
*3
72
*
2
*2
* =
+−−=
∂∂
−
n
nnnn
n
my
mymmymyyy
ml
φ (A.2)
⇒ ( )( )( ) ( )( ) ( )( ) 03
2123
5 25
2
*2
12
*
12
*
2
*
2
* =
+−−
−−−
nnnn
n mymmymyyy l (A.3)
57
( )( )( ) ( )( ) ( ) ( )0
12
12321212
352
234
2*
35
2*
31
2*
21
232
2*
32
2**
*
*
=
+
++−
+
−=∂
∂
−
my
mymymmymymy
myy
n
nnnnn
n
n
lφ
( )( )( ) ( )( ) ( )( )( )
03212
352
23
102
*
65
2
*2
123
72
**
*
*
=
+−−=
∂∂
n
nnn
n
n my
mymmymymy
yl
φ (A.4)
⇒ ( )( )( ) ( )( ) ( )( ) 03
2123
522 25
2
*2
1212
*** =
+−−
−−
nnnn mymmymymy l (A.5)
Using (A.3) and (A.5)
( )( ) ( ) ( ) ( ) 21
22
*
2
*2
12 1212
−+=+ mmyym nn (A.6)
⇒ 12 =m ⇒ 1=m
Using (A.1)
( ) ( ) 32
112 *3
52
* += nn yy
2
*
310
* 2 nn yy = ⇒ 83
* 2=ny
58
RECTANGULAR SECTION
×−+=
−3
2
*3
5
**** 1),,( PAAyb llφ
( )( )
+−+=
32
**
35
****
21
by
ybyb
n
nn lφ
( )( )
02
13
2
**
35
** =+
−=∂∂
by
yb
n
n
l
φ
⇒ ( ) ( ) 32
**3
5
** 2 byyb nn += (A.7)
( )( )( ) ( ) ( )( ) ( )
( )0
2
2322
35
34
**
35
**31
**3
2
**3
2
***
*
*
=
+
+−+−=
∂∂
−
by
ybbybyybyy
bn
nnnnn
n lφ
( )( )( ) ( )( )
( )03
235
310
**
65
**3
7
***
*
*
=
−−=
∂∂
n
nnn
n
yb
ybybyy
bl
φ (A.8)
⇒ ( )( )( ) ( )( )[ ] 03
23
5 25
**
1
**** =−−−−
nnnn ybybyy l (A.9)
( )( )( ) ( ) ( )( )( ) ( )
( )0
2
23
22235
34
**
35
**31
**3
2
**3
2
***
*
*
=
+
+−+−=
∂∂
−
by
ybbybyybbb
yn
nnnn
n
lφ
( )( )( ) ( )( )( )
( )03
2235
310
**
65
**3
7
***
*
*
=
−−=
∂∂
n
nn
n yb
ybybbb
yl
φ (A.10)
59
⇒ ( )( )( ) ( )( )( )[ ] 03
223
5 25
**
1
**** =−−−−
nn ybybbb l (A.11)
Using (A.9) & (A.11)
**2 byn = (A.12)
Using (A.7) & (A.12)
( ) ( ) 32
*3
52
* 42 nn yy =
2
*
110
* 2 nn yy −= ⇒ 81
* 2−
=ny , 87
* 2=b
60
TRAPEZOIDAL SECTION
×−+=
−3
2
*3
5
**** 1),,,( PAAmyb llφ
( ) ( )( )
++
+−++=
32
*
2
*
35
**
2
***
2
*
12
1
bmy
ybmyybmy
n
nnnn lφ
( )( )
0
12
13
2
*
2
*
35
**
2
* =++
+−=
∂∂
bmy
ybmy
n
nn
l
φ
( ) ( ) 32
*
2
*3
5
**
2
* 12 bmyybmy nnn ++=+⇒ (A.13)
( )( )( ) ( ) ( )( ) ( )( )
0
12
123212
35
34
*
2
*
35
**
2
*
31
*
2
*
32
*
2
*32
**
2
**
*
*
=
++
+++−+++−=
∂∂
−
bmy
ybmybmybmyybmyyy
bn
nnnnnnn
n lφ
( )( )( ) ( )( )( )
032
35
310
**
2
*
65
**
2
*3
7
**
2
**
*
*
=
+
+−+−=
∂∂
nn
nnnnn
n
ybmy
ybmyybmyyy
bl
φ (A.14)
⇒ ( )( )( ) ( )( ) 03
23
5 25
**
2
*
1
**
2
*** =
+−+−
−−
nnnnnn ybmyybmyyy l (A.15)
( )( )( )( ) ( ) ( )( )( ) ( )
( )0
12
12321212
352
234
*
2
*
35
**
2
*
31
*
2
*
232
*
2
*32
**
2
***
**
*
=
++
++++−++++−+=
∂∂
−
bmy
ybmybmymbmyybmybmybmy
yn
nnnnnnn
n
n
lφ
61
( )( )( )( ) ( )( )( )
( )03
212352
23
10
**
2
*
65
**
2
*
237
**
2
***
**
*
=
+
++−++−+=
∂∂
nn
nnnnn
n
n ybmy
ybmymybmybmybmy
yl
φ(A.16)
⇒ ( ) ( )( )( ) ( )( )( ) 03212
3522 2
5
**
2
*
21
**
2
***** =
++−++−+
−−
nnnnnn ybmymybmybmybmy l (A.17)
( )( )( )( ) ( ) ( ) ( )
0
12
12321212
35
34
*2
*
35
**2*
31
*2
*21
2*
32
*2
*32
**2*
2*
2* =
++
+
++
+−
+++
−=∂
∂
−−
bmy
ybmybmymmybmyybmyy
ym
n
nnnnnnnn
n lφ
( )( )( )( ) ( ) ( )( )
( )0
3212
35
310
**
2
*
65
**
2
*21
2
*3
7
**
2
*
2
*2
* =
+
+
+−+
−=∂∂
−
nn
nnnnnn
n
ybmy
ybmymmyybmyy
ym
lφ (A.18)
⇒ ( ) ( )( )( ) ( ) ( )( ) 03
2123
5 25
**
2
*21
2
*
1
**
2
*
2
*
2
* =
+
+−+−
−−−
nnnnnnn ybmymmyybmyyy l (A.19)
Using (A.17) and (A.19)
( )( )( ) ( )( ) 21
22
*2
12
*** 12122 mymbmymy nnn +=++−
( ) ( )2*** 12 mybmym nn +=+
⇒ ( ) ( )2*** 12 m
m
ybmy n
n +=+ (A.20)
Using (A.15), (A.17) & (A.20)
( )( ) ( )**2
12
* 212 bmymy nn +=+
( )( ) ( )2*21
2
* 112 mm
ymy n
n +=+
( ) 21
212 mm += ⇒
3
1=m (A.21)
62
Using (A.20) & (A.21)
****
3
32
3
2
3nn yby
b=⇒= (A.22)
Using (A.13) & (A.22)
( )2
**
5
**
5
*3
42
+=+ bybmyy nnn
⇒
2
**
5
**5
*3
32
3
4
3
32
3
+=
+ n
nn
nn y
yy
yy
⇒ 96782.0* =ny , 11755.1* =b
63
CIRCULAR SECTION
×−+=
−3
2
*3
5
*** 1),,( PAPr lλαφ
( )
( )[ ]
−
−−
−+−=3
2
*
35
2
*
*
2
2sin22
1)2(r
r
rαπ
ααπ
απφ l
( )
( )[ ]0
2
2sin22
13
2
*
35
2
*
=−
−−
−=∂∂
r
r
απ
ααπφ
l
( ) ( ) 353
5
3/10
*3
2
*3
2
2sin22
12 ααπαπ −−
=−⇒ rr (A.23)
( )( ) ( ) ( )( )
( )−
−
−−−−−−−=
∂∂
34
*3
4
32
*3
23
235
310
*
*
2
22cos222sin22
13
5
2r
rrr
απ
απαααπ
α
φl
( ) ( )( ) ( ) ( )
( )0
2
2sin22
123
22
34
*3
4
353
53
10
*3
13
2
*=
−
−−−−−
r
rr
απ
ααπαπ(A.24)
( )( )( ) ( ) ( )
( )−
−
−−−−−=
∂∂
34
*3
4
32
*3
23
535
37
*
* 2
22sin22
13
10
2r
rr
r απ
απααπαπ
φl
( ) ( ) ( ) ( )
( )0
2
2sin22
13
22
34
*3
4
353
53
10
*3
23
1
*=
−
−−−−
r
rr
απ
ααπαπ(A.25)
64
Using (A.23)
( ) ( )( ) 3
5
38
*3
2
35
21
22sin2
−−
=−−rαπ
ααπ (A.26)
( ) ( )( ) 3
2
1516
*15
4
32
21
22sin2
−−
=−−rαπ
ααπ (A.27)
Using (A.26) & (A.27) into 25
( )( ) ( ) ( ) ( ) ( ) ( )
( )0
2
23222
3102
23
4
*3
4
38
*3
23
10
*3
231
*3
2
*3
23
7
*38
*32
=
−
−−−−−−−
−−−
r
rrrrrr
απ
απαπαπαπαπ l
( ) 032
310
23
4
*
31
*31
*=
−−−
r
rrlαπ
*3
82
r
l=− απ (A.28)
Using (A.26) & (A.27) into(A.24)
( )( ) ( ) ( ) ( )( )( )
( )−
−
−+−−−−
−−
34
*3
4
32
*3
232
1516
*15
435
310
*
*
2
22cos122
122
13
5
2r
rrrr
απ
απααπl
( ) ( )( ) ( )
( )0
2
223
22
34
*3
4
38
*3
23
10
*3
13
2
*=
−
−−−−−
r
rrr
απ
απαπ
( ) ( )( )( )( ) ( ) ( )( )( )
02
32222cos12
21
352
23
4
*3
4
31
34
*15
1415
44
*
* =
−
−−−+−−−−
r
rrr
απ
απααπl
65
Using (A.28)
( ) ( )( )( )( ) ( ) ( )( )( )
0
38
322
382cos12
21
35
38
23
4
31
*31
31
34
*15
14
*15
1415
1415
44
*
* =
−−+−−−
−−
34
l
ll
l
rrrrr
α
( ) ( ) ( )( )( )( ) ( ) ( )( ) 03
223
82cos122
13
53
83
82 31
31
*
1514
1514
2
*
34
* =
−−+−−−−
lll 31-
rrr α
( ) ( )( ) ( ) ( ) 03
43
82cos13
53
821
*
2
*15
915
6
* =−+++−− -
rrr αl
( ) ( )( )2
52cos13
53
8*
159
156
=+−
αrl
( ) ( )( )2
5cos23
53
8 2156
159
* =
−
αlr
α2
159
*cos
11.1-
rl
= (A.29)
Using (A.28) & (A.29)
182751
0
00395.1*
.
r
=
=
=
l
α
TYPE β L* β A*m * y * τ C*
triangular 1.0 0.0 1.000 1.297 7.31E-09 5.350
triangular 1.0 1.0 1.123 1.225 1.96E-10 6.056
triangular 1.0 2.0 1.230 1.173 9.48E-09 6.730
triangular 2.0 0.0 1.000 1.297 4.74E-09 9.018
triangular 2.0 1.0 1.069 1.255 1.24E-10 9.733
triangular 2.0 2.0 1.132 1.220 8.82E-09 10.427
triangular 2.0 3.0 1.191 1.191 6.57E-09 11.105
triangular 2.0 4.0 1.246 1.165 5.49E-10 11.768
triangular 3.0 0.0 1.000 1.297 2.73E-09 12.686
triangular 3.0 1.0 1.048 1.267 8.91E-09 13.405
triangular 3.0 2.0 1.093 1.241 3.33E-11 14.108
triangular 3.0 3.0 1.136 1.218 6.18E-09 14.798
triangular 3.0 4.0 1.176 1.198 8.92E-09 15.477
triangular 3.0 5.0 1.215 1.179 5.89E-09 16.146
triangular 3.0 6.0 1.252 1.162 5.03E-09 16.806
triangular 4.0 0.0 1.000 1.297 -5.30E-11 16.354
triangular 4.0 1.0 1.037 1.274 -7.58E-11 17.075
triangular 4.0 2.0 1.072 1.253 4.08E-09 17.783
triangular 4.0 3.0 1.106 1.234 4.56E-09 18.481
triangular 4.0 4.0 1.138 1.217 3.08E-09 19.169
triangular 4.0 5.0 1.169 1.202 3.39E-09 19.848
triangular 4.0 6.0 1.198 1.187 3.55E-09 20.520
triangular 4.0 7.0 1.227 1.174 5.50E-11 21.185
triangular 5.0 0.0 1.000 1.297 7.17E-09 20.022
triangular 5.0 1.0 1.030 1.278 2.73E-09 20.744
triangular 5.0 2.0 1.059 1.261 3.03E-09 21.456
triangular 5.0 3.0 1.086 1.245 7.48E-09 22.158
triangular 5.0 4.0 1.113 1.230 9.05E-09 22.853
triangular 5.0 5.0 1.139 1.217 1.57E-09 23.540
triangular 5.0 6.0 1.164 1.204 3.22E-09 24.220
triangular 5.0 7.0 1.188 1.192 4.09E-09 24.893
triangular 6.0 0.0 1.000 1.297 2.84E-09 23.690
triangular 6.0 1.0 1.025 1.281 3.92E-09 24.413
triangular 6.0 2.0 1.049 1.266 5.39E-09 25.127
triangular 6.0 3.0 1.073 1.252 8.28E-09 25.833
triangular 6.0 4.0 1.096 1.240 9.10E-10 26.532
triangular 6.0 5.0 1.118 1.228 3.18E-09 27.224
triangular 6.0 6.0 1.139 1.216 5.92E-11 27.910
triangular 6.0 7.0 1.160 1.206 3.98E-09 28.591
triangular 6.0 8.0 1.181 1.195 2.33E-11 29.266
triangular 7.0 0.0 1.000 1.297 2.43E-09 27.358
triangular 7.0 1.0 1.022 1.283 3.11E-11 28.081
triangular 7.0 2.0 1.043 1.270 3.80E-09 28.797
triangular 7.0 3.0 1.063 1.258 6.53E-09 29.506
NUMERICAL COMPUTATION RESULTS
APPENDIX B
INPUTS (non-dimensional form) OUTPUTS (non-dimensional form)
ITERATIVE RESULTS FOR DIFFERENT VALUES OF INPUTS
66
TYPE β L* β A*m * y * τ C*
INPUTS (non-dimensional form) OUTPUTS (non-dimensional form)
ITERATIVE RESULTS FOR DIFFERENT VALUES OF INPUTS
triangular 7.0 4.0 1.083 1.247 4.34E-09 30.208
triangular 7.0 5.0 1.102 1.236 8.59E-09 30.905
triangular 7.0 6.0 1.122 1.226 3.44E-11 31.596
triangular 7.0 7.0 1.140 1.216 3.12E-09 32.281
triangular 7.0 8.0 1.158 1.207 3.48E-09 32.962
triangular 7.0 9.0 1.176 1.198 6.20E-09 33.638
triangular 8.0 0.0 1.000 1.297 5.84E-09 31.026
triangular 8.0 1.0 1.019 1.285 1.20E-10 31.750
triangular 8.0 2.0 1.038 1.273 7.91E-11 32.467
triangular 8.0 3.0 1.056 1.262 4.48E-11 33.178
triangular 8.0 4.0 1.074 1.252 7.67E-09 33.883
triangular 8.0 5.0 1.091 1.242 4.78E-09 34.583
triangular 8.0 6.0 1.108 1.233 4.34E-09 35.277
triangular 8.0 7.0 1.124 1.224 2.85E-09 35.967
triangular 8.0 8.0 1.140 1.216 3.22E-09 36.652
triangular 8.0 9.0 1.156 1.208 5.71E-09 37.333
triangular 8.0 10.0 1.172 1.200 4.09E-09 38.010
triangular 9.0 0.0 1.000 1.297 -2.03E-12 34.694
triangular 9.0 1.0 1.017 1.286 4.49E-09 35.418
triangular 9.0 2.0 1.034 1.276 4.50E-11 36.136
triangular 9.0 3.0 1.050 1.266 -9.64E-11 36.849
triangular 9.0 4.0 1.066 1.256 2.65E-09 37.556
triangular 9.0 5.0 1.081 1.248 3.81E-09 38.258
triangular 9.0 6.0 1.097 1.239 6.10E-09 38.956
triangular 9.0 7.0 1.112 1.231 4.29E-09 39.649
triangular 9.0 8.0 1.126 1.223 5.67E-09 40.338
triangular 9.0 9.0 1.141 1.216 3.24E-09 41.023
triangular 9.0 10.0 1.155 1.208 1.50E-09 41.704
triangular 10.0 0.0 1.000 1.297 4.18E-09 38.362
triangular 10.0 1.0 1.015 1.287 6.09E-09 39.087
triangular 10.0 2.0 1.030 1.278 8.71E-09 39.805
triangular 10.0 3.0 1.045 1.269 3.51E-09 40.519
triangular 10.0 4.0 1.060 1.260 3.31E-09 41.228
triangular 10.0 5.0 1.074 1.252 7.17E-09 41.933
triangular 10.0 6.0 1.088 1.244 3.48E-09 42.633
triangular 10.0 7.0 1.101 1.236 1.83E-11 43.329
triangular 10.0 8.0 1.115 1.229 2.54E-09 44.021
triangular 10.0 9.0 1.128 1.222 5.24E-09 44.709
triangular 10.0 10.0 1.141 1.215 1.51E-09 45.394
triangular 11.0 0.0 1.000 1.297 9.35E-09 42.030
triangular 11.0 1.0 1.014 1.288 3.79E-09 42.755
triangular 11.0 2.0 1.028 1.279 5.47E-09 43.474
triangular 11.0 3.0 1.041 1.271 2.77E-09 44.189
triangular 11.0 4.0 1.055 1.263 2.86E-09 44.900
triangular 11.0 5.0 1.068 1.255 2.99E-09 45.606
triangular 11.0 6.0 1.080 1.248 3.62E-09 46.308
triangular 11.0 7.0 1.093 1.241 3.70E-09 47.007
triangular 11.0 8.0 1.105 1.234 2.35E-11 47.701
triangular 11.0 9.0 1.117 1.228 7.39E-09 48.392
triangular 12.0 0.0 1.000 1.297 6.45E-09 45.698
triangular 12.0 1.0 1.013 1.289 5.27E-09 46.423
triangular 12.0 2.0 1.026 1.281 -4.66E-10 47.143
67
TYPE β L* β A*m * y * τ C*
INPUTS (non-dimensional form) OUTPUTS (non-dimensional form)
ITERATIVE RESULTS FOR DIFFERENT VALUES OF INPUTS
triangular 12.0 3.0 1.038 1.273 -5.73E-11 47.859
triangular 12.0 4.0 1.050 1.266 5.15E-09 48.571
triangular 12.0 5.0 1.062 1.259 3.14E-09 49.279
triangular 12.0 6.0 1.074 1.252 7.61E-09 49.983
triangular 12.0 7.0 1.086 1.245 4.58E-09 50.683
triangular 12.0 8.0 1.097 1.239 1.03E-09 51.380
triangular 12.0 9.0 1.108 1.233 8.75E-09 52.073
triangular 13.0 0.0 1.000 1.297 4.06E-09 49.366
triangular 13.0 1.0 1.012 1.289 9.97E-09 50.091
triangular 13.0 2.0 1.024 1.282 9.28E-09 50.812
triangular 13.0 3.0 1.035 1.275 -2.51E-11 51.529
triangular 13.0 4.0 1.046 1.268 7.29E-09 52.241
triangular 13.0 5.0 1.058 1.261 8.06E-09 52.950
triangular 13.0 6.0 1.069 1.255 4.26E-09 53.656
triangular 13.0 7.0 1.079 1.249 7.07E-09 54.358
triangular 13.0 8.0 1.090 1.243 1.42E-09 55.057
triangular 13.0 9.0 1.101 1.237 4.87E-09 55.753
triangular 14.0 0.0 1.000 1.297 2.11E-09 53.034
triangular 14.0 1.0 1.011 1.290 9.37E-09 53.759
triangular 14.0 2.0 1.022 1.283 8.63E-09 54.480
triangular 14.0 3.0 1.033 1.276 5.20E-10 55.198
triangular 14.0 4.0 1.043 1.270 8.20E-09 55.912
triangular 14.0 5.0 1.054 1.264 6.12E-09 56.622
triangular 14.0 6.0 1.064 1.257 3.06E-09 57.329
triangular 14.0 7.0 1.074 1.252 2.94E-11 58.033
triangular 14.0 8.0 1.084 1.246 5.09E-09 58.733
triangular 14.0 9.0 1.094 1.241 6.25E-09 59.431
triangular 14.0 10.0 1.104 1.235 5.09E-09 60.125
triangular 15.0 0.0 1.000 1.297 -4.55E-10 56.702
triangular 15.0 1.0 1.010 1.290 7.81E-09 57.428
triangular 15.0 2.0 1.021 1.284 7.75E-09 58.149
triangular 15.0 3.0 1.031 1.277 2.22E-10 58.867
triangular 15.0 4.0 1.040 1.271 7.26E-09 59.582
triangular 15.0 5.0 1.050 1.266 3.04E-09 60.293
triangular 15.0 6.0 1.060 1.260 6.82E-10 61.001
triangular 15.0 7.0 1.069 1.254 4.20E-09 61.706
triangular 15.0 8.0 1.079 1.249 3.45E-09 62.408
triangular 15.0 9.0 1.088 1.244 3.96E-09 63.107
triangular 15.0 10.0 1.097 1.239 4.98E-09 63.804
triangular 15.0 11.0 1.106 1.234 7.34E-09 64.498
triangular 16.0 0.0 1.000 1.297 7.68E-09 60.370
triangular 16.0 1.0 1.010 1.291 2.19E-09 61.096
triangular 16.0 2.0 1.019 1.285 6.62E-09 61.817
triangular 16.0 3.0 1.029 1.279 7.39E-11 62.536
triangular 16.0 4.0 1.038 1.273 7.93E-11 63.251
triangular 16.0 5.0 1.047 1.267 4.64E-10 63.963
triangular 16.0 6.0 1.056 1.262 4.89E-11 64.673
triangular 16.0 7.0 1.065 1.257 7.96E-09 65.379
triangular 16.0 8.0 1.074 1.252 2.95E-09 66.082
triangular 16.0 9.0 1.083 1.247 6.24E-09 66.783
triangular 16.0 10.0 1.092 1.242 5.27E-09 67.481
triangular 16.0 11.0 1.100 1.237 4.73E-09 68.177
68
TYPE β L* β A*m * y * τ C*
INPUTS (non-dimensional form) OUTPUTS (non-dimensional form)
ITERATIVE RESULTS FOR DIFFERENT VALUES OF INPUTS
triangular 16.0 12.0 1.109 1.232 4.20E-09 68.870
triangular 17.0 0.0 1.000 1.297 5.88E-09 64.038
triangular 17.0 1.0 1.009 1.291 -3.30E-10 64.764
triangular 17.0 2.0 1.018 1.285 5.14E-09 65.486
triangular 17.0 3.0 1.027 1.280 1.13E-09 66.205
triangular 17.0 4.0 1.036 1.274 6.25E-11 66.921
triangular 17.0 5.0 1.045 1.269 3.53E-10 67.634
triangular 17.0 6.0 1.053 1.264 7.17E-13 68.344
triangular 17.0 7.0 1.062 1.259 3.37E-09 69.051
triangular 17.0 8.0 1.070 1.254 8.67E-09 69.756
triangular 17.0 9.0 1.078 1.249 8.72E-09 70.458
triangular 17.0 10.0 1.087 1.245 7.61E-09 71.158
triangular 17.0 11.0 1.095 1.240 8.31E-09 71.855
triangular 17.0 12.0 1.103 1.236 7.85E-09 72.549
triangular 17.0 13.0 1.111 1.231 4.34E-09 73.242
triangular 18.0 0.0 1.000 1.297 4.15E-09 67.706
triangular 18.0 1.0 1.009 1.291 2.26E-09 68.432
triangular 18.0 2.0 1.017 1.286 3.81E-09 69.154
triangular 18.0 3.0 1.026 1.281 6.78E-09 69.874
triangular 18.0 4.0 1.034 1.275 4.79E-11 70.590
triangular 18.0 5.0 1.042 1.270 2.74E-10 71.304
triangular 18.0 6.0 1.050 1.266 -9.57E-11 72.015
triangular 18.0 7.0 1.059 1.261 6.55E-09 72.723
triangular 18.0 8.0 1.067 1.256 2.87E-09 73.429
triangular 18.0 9.0 1.074 1.252 4.40E-09 74.132
triangular 18.0 10.0 1.082 1.247 4.55E-09 74.833
triangular 18.0 11.0 1.090 1.243 7.21E-09 75.532
triangular 18.0 12.0 1.098 1.238 5.50E-09 76.228
triangular 18.0 13.0 1.105 1.234 3.16E-09 76.922
triangular 18.0 14.0 1.113 1.230 4.18E-09 77.614
triangular 19.0 0.0 1.000 1.297 2.53E-09 71.374
triangular 19.0 1.0 1.008 1.292 1.53E-09 72.100
triangular 19.0 2.0 1.016 1.286 2.72E-09 72.823
triangular 19.0 3.0 1.024 1.281 4.10E-09 73.542
triangular 19.0 4.0 1.032 1.276 3.61E-11 74.259
triangular 19.0 5.0 1.040 1.272 2.23E-10 74.974
triangular 19.0 6.0 1.048 1.267 2.63E-09 75.686
triangular 19.0 7.0 1.056 1.262 2.97E-09 76.395
triangular 19.0 8.0 1.063 1.258 4.81E-09 77.102
triangular 19.0 9.0 1.071 1.254 4.15E-09 77.806
triangular 19.0 10.0 1.078 1.249 3.61E-09 78.508
triangular 19.0 11.0 1.086 1.245 5.60E-09 79.208
triangular 19.0 12.0 1.093 1.241 5.48E-09 79.905
triangular 19.0 13.0 1.100 1.237 7.38E-09 80.601
triangular 19.0 14.0 1.107 1.233 3.84E-11 81.294
triangular 19.0 15.0 1.114 1.229 2.83E-09 81.985
triangular 19.0 16.0 1.121 1.226 4.25E-09 82.675
triangular 19.0 17.0 1.128 1.222 1.01E-09 83.362
69
TYPE β L* β A* b * y * τ C*
rectangular 1.0 0.0 1.834 0.917 6.59E-09 5.350
rectangular 1.0 1.0 2.063 0.818 7.67E-09 6.077
rectangular 1.0 2.0 2.250 0.755 7.86E-09 6.741
rectangular 2.0 0.0 1.834 0.917 8.27E-09 9.018
rectangular 2.0 1.0 1.965 0.857 8.91E-09 9.763
rectangular 2.0 2.0 2.079 0.812 8.31E-09 10.466
rectangular 2.0 3.0 2.183 0.776 8.18E-09 11.136
rectangular 2.0 4.0 2.278 0.746 4.46E-09 11.782
rectangular 3.0 0.0 1.834 0.917 3.07E-10 12.686
rectangular 3.0 1.0 1.926 0.874 4.23E-11 13.439
rectangular 3.0 2.0 2.009 0.839 8.73E-09 14.159
rectangular 3.0 3.0 2.086 0.810 7.29E-09 14.854
rectangular 3.0 4.0 2.157 0.784 2.77E-09 15.527
rectangular 3.0 5.0 2.225 0.763 6.19E-09 16.183
rectangular 3.0 6.0 2.289 0.743 6.40E-09 16.822
rectangular 4.0 0.0 1.834 0.917 2.42E-09 16.354
rectangular 4.0 1.0 1.905 0.883 1.41E-09 17.111
rectangular 4.0 2.0 1.970 0.855 8.81E-09 17.842
rectangular 4.0 3.0 2.031 0.830 2.55E-09 18.551
rectangular 4.0 4.0 2.089 0.808 4.16E-09 19.242
rectangular 4.0 5.0 2.144 0.789 4.95E-09 19.917
rectangular 4.0 6.0 2.196 0.772 2.65E-09 20.578
rectangular 5.0 0.0 1.834 0.917 4.32E-09 20.022
rectangular 5.0 1.0 1.891 0.889 3.21E-10 20.781
rectangular 5.0 2.0 1.945 0.865 9.18E-09 21.519
rectangular 5.0 3.0 1.996 0.844 4.01E-09 22.239
rectangular 5.0 4.0 2.045 0.825 8.62E-09 22.942
rectangular 5.0 5.0 2.091 0.808 3.63E-09 23.631
rectangular 5.0 6.0 2.135 0.792 7.02E-09 24.306
rectangular 6.0 0.0 1.834 0.917 2.23E-09 23.690
rectangular 6.0 1.0 1.882 0.894 8.65E-09 24.451
rectangular 6.0 2.0 1.928 0.873 3.84E-10 25.194
rectangular 6.0 3.0 1.972 0.854 9.02E-09 25.921
rectangular 6.0 4.0 2.014 0.837 4.13E-09 26.633
rectangular 6.0 5.0 2.054 0.821 2.98E-09 27.332
rectangular 6.0 6.0 2.092 0.807 4.16E-09 28.019
rectangular 7.0 0.0 1.834 0.917 9.24E-09 27.358
rectangular 7.0 1.0 1.876 0.897 -2.30E-11 28.121
rectangular 7.0 2.0 1.916 0.878 9.43E-09 28.867
rectangular 7.0 3.0 1.954 0.862 3.36E-09 29.599
rectangular 7.0 4.0 1.991 0.846 3.16E-09 30.318
rectangular 7.0 5.0 2.026 0.832 4.08E-09 31.025
rectangular 7.0 6.0 2.060 0.819 4.83E-09 31.721
rectangular 7.0 7.0 2.093 0.807 5.68E-09 32.407
rectangular 8.0 0.0 1.834 0.917 3.75E-11 31.026
rectangular 8.0 1.0 1.871 0.899 6.28E-09 31.790
rectangular 8.0 2.0 1.906 0.883 1.15E-09 32.539
rectangular 8.0 3.0 1.940 0.867 5.10E-09 33.275
rectangular 8.0 4.0 1.973 0.854 8.76E-09 33.999
rectangular 8.0 5.0 2.005 0.841 -1.91E-11 34.713
rectangular 8.0 6.0 2.035 0.829 3.07E-09 35.416
rectangular 8.0 7.0 2.065 0.817 3.03E-09 36.110
OUTPUTS (non-dimensional form)
ITERATIVE RESULTS FOR DIFFERENT VALUES OF INPUTS
INPUTS (non-dimensional form)
70
TYPE β L* β A* b * y * τ C*
OUTPUTS (non-dimensional form)
ITERATIVE RESULTS FOR DIFFERENT VALUES OF INPUTS
INPUTS (non-dimensional form)
rectangular 9.0 0.0 1.834 0.917 3.78E-09 34.694
rectangular 9.0 1.0 1.867 0.901 5.20E-09 35.459
rectangular 9.0 2.0 1.899 0.886 2.47E-09 36.210
rectangular 9.0 3.0 1.929 0.872 6.56E-09 36.950
rectangular 9.0 4.0 1.959 0.860 9.02E-09 37.678
rectangular 9.0 5.0 1.987 0.848 -1.92E-11 38.397
rectangular 9.0 6.0 2.015 0.836 6.04E-09 39.106
rectangular 9.0 7.0 2.042 0.826 3.94E-09 39.807
rectangular 9.0 8.0 2.069 0.816 4.15E-09 40.499
rectangular 10.0 0.0 1.834 0.917 3.47E-09 38.362
rectangular 10.0 1.0 1.864 0.902 8.12E-09 39.127
rectangular 10.0 2.0 1.892 0.889 2.98E-10 39.880
rectangular 10.0 3.0 1.920 0.876 7.84E-09 40.623
rectangular 10.0 4.0 1.947 0.864 9.16E-09 41.355
rectangular 10.0 5.0 1.973 0.853 -1.88E-11 42.078
rectangular 10.0 6.0 1.999 0.843 3.78E-09 42.792
rectangular 10.0 7.0 2.024 0.833 5.78E-09 43.498
rectangular 10.0 8.0 2.048 0.824 8.44E-09 44.197
rectangular 11.0 0.0 1.834 0.917 5.92E-09 42.030
rectangular 11.0 1.0 1.861 0.904 1.30E-09 42.796
rectangular 11.0 2.0 1.887 0.891 5.64E-09 43.550
rectangular 11.0 3.0 1.913 0.880 6.85E-09 44.295
rectangular 11.0 4.0 1.938 0.869 9.96E-09 45.031
rectangular 11.0 5.0 1.962 0.858 -1.90E-11 45.757
rectangular 11.0 6.0 1.985 0.848 9.65E-09 46.476
rectangular 11.0 7.0 2.008 0.839 3.30E-09 47.186
rectangular 11.0 8.0 2.031 0.830 4.95E-09 47.890
rectangular 12.0 0.0 1.834 0.917 1.38E-09 45.698
rectangular 12.0 1.0 1.859 0.905 6.03E-10 46.464
rectangular 12.0 2.0 1.883 0.893 7.21E-09 47.220
rectangular 12.0 3.0 1.907 0.882 9.95E-09 47.967
rectangular 12.0 4.0 1.930 0.872 6.29E-10 48.705
rectangular 12.0 5.0 1.952 0.862 3.87E-09 49.435
rectangular 12.0 6.0 1.974 0.853 3.91E-09 50.157
rectangular 12.0 7.0 1.995 0.844 4.94E-09 50.872
rectangular 12.0 8.0 2.016 0.836 3.01E-09 51.579
rectangular 12.0 9.0 2.037 0.828 3.40E-09 52.281
rectangular 12.0 10.0 2.057 0.820 9.28E-09 52.976
rectangular 13.0 0.0 1.834 0.917 9.55E-09 49.366
rectangular 13.0 1.0 1.857 0.906 7.53E-09 50.133
rectangular 13.0 2.0 1.879 0.895 9.95E-09 50.890
rectangular 13.0 3.0 1.901 0.885 -2.60E-11 51.638
rectangular 13.0 4.0 1.923 0.875 8.36E-09 52.378
rectangular 13.0 5.0 1.944 0.866 4.47E-09 53.111
rectangular 13.0 6.0 1.964 0.857 5.00E-09 53.836
rectangular 13.0 7.0 1.984 0.849 2.68E-09 54.555
rectangular 13.0 8.0 2.003 0.841 3.18E-09 55.266
rectangular 13.0 9.0 2.023 0.833 2.52E-09 55.972
rectangular 13.0 10.0 2.041 0.826 5.94E-09 56.671
rectangular 13.0 11.0 2.060 0.819 3.57E-09 57.365
rectangular 14.0 0.0 1.834 0.917 7.80E-09 53.034
rectangular 14.0 1.0 1.855 0.906 3.33E-09 53.801
71
TYPE β L* β A* b * y * τ C*
OUTPUTS (non-dimensional form)
ITERATIVE RESULTS FOR DIFFERENT VALUES OF INPUTS
INPUTS (non-dimensional form)
rectangular 14.0 2.0 1.876 0.896 -2.36E-11 54.559
rectangular 14.0 3.0 1.897 0.887 9.41E-09 55.309
rectangular 14.0 4.0 1.917 0.878 4.00E-09 56.051
rectangular 14.0 5.0 1.936 0.869 8.34E-09 56.786
rectangular 14.0 6.0 1.955 0.861 2.47E-09 57.514
rectangular 14.0 7.0 1.974 0.853 2.73E-09 58.236
rectangular 14.0 8.0 1.992 0.845 9.46E-09 58.951
rectangular 14.0 9.0 2.010 0.838 5.43E-09 59.660
rectangular 14.0 10.0 2.028 0.831 3.10E-09 60.363
rectangular 14.0 11.0 2.046 0.825 3.09E-09 61.061
rectangular 14.0 12.0 2.063 0.818 3.55E-09 61.754
rectangular 15.0 0.0 1.834 0.917 2.25E-09 56.702
rectangular 15.0 1.0 1.854 0.907 2.91E-10 57.469
rectangular 15.0 2.0 1.874 0.898 2.12E-09 58.228
rectangular 15.0 3.0 1.893 0.889 4.85E-09 58.979
rectangular 15.0 4.0 1.911 0.880 4.77E-09 59.723
rectangular 15.0 5.0 1.930 0.872 3.01E-09 60.460
rectangular 15.0 6.0 1.948 0.864 8.91E-09 61.191
rectangular 15.0 7.0 1.966 0.857 8.84E-09 61.915
rectangular 15.0 8.0 1.983 0.849 5.08E-09 62.633
rectangular 15.0 9.0 2.000 0.842 2.76E-09 63.346
rectangular 15.0 10.0 2.017 0.836 -1.95E-11 64.053
rectangular 15.0 11.0 2.033 0.829 -9.70E-12 64.755
rectangular 15.0 12.0 2.049 0.823 3.65E-09 65.451
rectangular 15.0 13.0 2.065 0.817 3.15E-09 66.143
rectangular 16.0 0.0 1.834 0.917 1.52E-10 60.370
rectangular 16.0 1.0 1.853 0.908 3.45E-09 61.138
rectangular 16.0 2.0 1.871 0.899 6.61E-09 61.897
rectangular 16.0 3.0 1.889 0.890 6.69E-09 62.650
rectangular 16.0 4.0 1.907 0.882 1.04E-09 63.395
rectangular 16.0 5.0 1.924 0.874 3.54E-10 64.134
rectangular 16.0 6.0 1.941 0.867 4.96E-09 64.867
rectangular 16.0 7.0 1.958 0.860 7.75E-09 65.593
rectangular 16.0 8.0 1.974 0.853 9.55E-09 66.314
rectangular 16.0 9.0 1.990 0.846 2.96E-09 67.030
rectangular 16.0 10.0 2.006 0.840 -1.97E-11 67.740
rectangular 16.0 11.0 2.022 0.834 3.44E-09 68.445
rectangular 16.0 12.0 2.037 0.828 3.29E-09 69.145
rectangular 16.0 13.0 2.052 0.822 8.45E-09 69.841
rectangular 16.0 14.0 2.067 0.816 4.26E-09 70.532
rectangular 17.0 0.0 1.834 0.917 8.12E-09 64.038
rectangular 17.0 1.0 1.852 0.908 2.60E-09 64.806
rectangular 17.0 2.0 1.869 0.900 8.10E-09 65.566
rectangular 17.0 3.0 1.886 0.892 8.41E-09 66.319
rectangular 17.0 4.0 1.903 0.884 1.58E-09 67.066
rectangular 17.0 5.0 1.919 0.877 4.39E-10 67.807
rectangular 17.0 6.0 1.935 0.870 5.70E-09 68.542
rectangular 17.0 7.0 1.951 0.863 8.18E-09 69.270
rectangular 17.0 8.0 1.967 0.856 8.88E-09 69.994
rectangular 17.0 9.0 1.982 0.850 2.99E-09 70.712
rectangular 17.0 10.0 1.997 0.844 -1.95E-11 71.425
rectangular 17.0 11.0 2.012 0.838 5.22E-09 72.133
72
TYPE β L* β A* b * y * τ C*
OUTPUTS (non-dimensional form)
ITERATIVE RESULTS FOR DIFFERENT VALUES OF INPUTS
INPUTS (non-dimensional form)
rectangular 17.0 12.0 2.027 0.832 3.78E-09 72.837
rectangular 17.0 13.0 2.041 0.826 4.14E-09 73.536
rectangular 17.0 14.0 2.055 0.821 5.99E-09 74.231
rectangular 17.0 15.0 2.069 0.816 2.38E-09 74.921
rectangular 18.0 0.0 1.834 0.917 2.95E-09 67.706
rectangular 18.0 1.0 1.851 0.909 4.34E-09 68.474
rectangular 18.0 2.0 1.867 0.901 4.95E-09 69.235
rectangular 18.0 3.0 1.883 0.893 4.50E-10 69.989
rectangular 18.0 4.0 1.899 0.886 2.26E-09 70.737
rectangular 18.0 5.0 1.915 0.879 5.26E-10 71.479
rectangular 18.0 6.0 1.930 0.872 6.42E-09 72.216
rectangular 18.0 7.0 1.945 0.865 8.54E-09 72.947
rectangular 18.0 8.0 1.960 0.859 9.00E-09 73.672
rectangular 18.0 9.0 1.974 0.853 3.04E-09 74.393
rectangular 18.0 10.0 1.989 0.847 -1.96E-11 75.109
rectangular 18.0 11.0 2.003 0.841 4.19E-11 75.820
rectangular 18.0 12.0 2.017 0.836 2.98E-09 76.526
rectangular 18.0 13.0 2.031 0.830 2.30E-09 77.228
rectangular 18.0 14.0 2.044 0.825 3.98E-09 77.926
rectangular 18.0 15.0 2.058 0.820 5.62E-09 78.620
rectangular 18.0 16.0 2.071 0.815 4.21E-09 79.310
rectangular 19.0 0.0 1.834 0.917 8.47E-09 71.374
rectangular 19.0 1.0 1.850 0.909 6.37E-09 72.142
rectangular 19.0 2.0 1.866 0.902 6.44E-09 72.904
rectangular 19.0 3.0 1.881 0.894 6.75E-10 73.659
rectangular 19.0 4.0 1.896 0.887 3.12E-09 74.408
rectangular 19.0 5.0 1.911 0.880 5.55E-09 75.151
rectangular 19.0 6.0 1.925 0.874 7.06E-09 75.889
rectangular 19.0 7.0 1.940 0.868 8.84E-09 76.622
rectangular 19.0 8.0 1.954 0.862 9.08E-09 77.350
rectangular 19.0 9.0 1.968 0.856 3.10E-09 78.073
rectangular 19.0 10.0 1.981 0.850 -1.94E-11 78.791
rectangular 19.0 11.0 1.995 0.844 9.85E-11 79.504
rectangular 19.0 12.0 2.008 0.839 -3.33E-11 80.214
rectangular 19.0 13.0 2.021 0.834 4.58E-09 80.919
rectangular 19.0 14.0 2.034 0.829 2.59E-09 81.619
rectangular 19.0 15.0 2.047 0.824 3.95E-09 82.316
rectangular 19.0 16.0 2.060 0.819 2.90E-09 83.009
rectangular 19.0 17.0 2.072 0.814 3.31E-09 83.698
rectangular 19.0 18.0 2.085 0.810 5.04E-09 84.384
rectangular 19.0 19.0 2.097 0.806 2.98E-09 85.066
rectangular 19.0 20.0 2.109 0.801 5.45E-09 85.744
73
TYPE β L* β A*m * b * y * τ C*
trapezoidal 1.0 0.0 0.577 1.118 0.968 5.55E-09 4.975
trapezoidal 1.0 1.0 0.642 1.271 0.885 9.78E-09 5.645
trapezoidal 1.0 2.0 0.684 1.419 0.824 5.96E-09 6.272
trapezoidal 1.0 3.0 0.714 1.559 0.778 3.77E-09 6.868
trapezoidal 1.0 4.0 0.737 1.693 0.740 -1.59E-09 7.441
trapezoidal 1.0 5.0 0.754 1.820 0.709 7.22E-10 7.996
trapezoidal 1.0 6.0 0.768 1.941 0.682 7.38E-11 8.535
trapezoidal 1.0 7.0 0.779 2.056 0.659 -1.28E-09 9.061
trapezoidal 1.0 8.0 0.788 2.167 0.638 4.29E-09 9.576
trapezoidal 1.0 9.0 0.796 2.274 0.620 2.02E-10 10.081
trapezoidal 1.0 10.0 0.803 2.377 0.604 -1.70E-12 10.577
trapezoidal 2.0 11.0 0.767 1.936 0.683 3.89E-09 14.856
trapezoidal 2.0 12.0 0.773 2.000 0.670 5.03E-09 15.386
trapezoidal 2.0 13.0 0.779 2.061 0.658 3.05E-09 15.909
trapezoidal 2.0 14.0 0.785 2.121 0.647 5.88E-09 16.426
trapezoidal 2.0 15.0 0.789 2.180 0.636 3.63E-09 16.937
trapezoidal 2.0 16.0 0.794 2.237 0.626 6.26E-09 17.443
trapezoidal 2.0 17.0 0.797 2.294 0.617 4.68E-10 17.944
trapezoidal 2.0 18.0 0.801 2.349 0.608 3.97E-10 18.441
trapezoidal 2.0 19.0 0.804 2.404 0.600 4.05E-09 18.933
trapezoidal 2.0 20.0 0.808 2.457 0.592 2.69E-10 19.421
trapezoidal 3.0 0.0 0.577 1.118 0.968 6.97E-09 11.681
trapezoidal 3.0 1.0 0.605 1.176 0.933 1.64E-11 12.367
trapezoidal 3.0 2.0 0.628 1.233 0.903 7.84E-09 13.032
trapezoidal 3.0 3.0 0.647 1.289 0.876 2.50E-09 13.681
trapezoidal 3.0 4.0 0.664 1.345 0.853 2.69E-09 14.314
trapezoidal 3.0 5.0 0.679 1.400 0.832 4.05E-09 14.934
trapezoidal 3.0 6.0 0.692 1.453 0.812 5.25E-09 15.542
trapezoidal 3.0 7.0 0.703 1.506 0.795 4.20E-09 16.140
trapezoidal 3.0 8.0 0.714 1.557 0.779 5.43E-09 16.729
trapezoidal 3.0 9.0 0.723 1.608 0.764 1.23E-09 17.309
trapezoidal 3.0 10.0 0.731 1.657 0.750 8.11E-09 17.881
trapezoidal 4.0 0.0 0.577 1.118 0.968 -2.24E-09 15.033
trapezoidal 4.0 1.0 0.599 1.162 0.941 1.06E-09 15.722
trapezoidal 4.0 2.0 0.617 1.206 0.917 4.63E-09 16.395
trapezoidal 4.0 3.0 0.634 1.249 0.895 3.05E-09 17.053
trapezoidal 4.0 4.0 0.648 1.292 0.875 6.94E-09 17.698
trapezoidal 4.0 5.0 0.661 1.334 0.857 3.15E-09 18.332
trapezoidal 4.0 6.0 0.673 1.376 0.840 7.76E-09 18.956
trapezoidal 4.0 7.0 0.683 1.418 0.825 8.22E-09 19.571
trapezoidal 4.0 8.0 0.693 1.458 0.811 3.60E-09 20.177
trapezoidal 4.0 9.0 0.702 1.498 0.797 8.14E-09 20.776
trapezoidal 4.0 10.0 0.710 1.537 0.785 5.49E-09 21.367
trapezoidal 5.0 11.0 0.701 1.493 0.799 4.12E-09 25.411
trapezoidal 5.0 12.0 0.707 1.525 0.788 7.83E-09 26.003
trapezoidal 5.0 13.0 0.714 1.557 0.779 3.11E-09 26.590
trapezoidal 5.0 14.0 0.719 1.588 0.769 8.52E-09 27.172
trapezoidal 5.0 15.0 0.725 1.619 0.760 3.75E-09 27.748
trapezoidal 5.0 16.0 0.730 1.649 0.752 4.32E-09 28.320
trapezoidal 5.0 17.0 0.734 1.679 0.744 1.17E-09 28.888
trapezoidal 5.0 18.0 0.739 1.709 0.736 7.89E-09 29.451
trapezoidal 5.0 19.0 0.743 1.738 0.729 2.07E-09 30.010
trapezoidal 5.0 20.0 0.747 1.767 0.721 2.95E-09 30.565
trapezoidal 6.0 0.0 0.577 1.118 0.968 -3.14E-09 21.739
trapezoidal 6.0 1.0 0.592 1.148 0.949 6.53E-09 22.430
trapezoidal 6.0 2.0 0.606 1.177 0.932 2.23E-09 23.110
INPUTS (non-dimensional form) OUTPUTS (non-dimensional form)
ITERATIVE RESULTS FOR DIFFERENT VALUES OF INPUTS
74
TYPE β L* β A*m * b * y * τ C*
INPUTS (non-dimensional form) OUTPUTS (non-dimensional form)
ITERATIVE RESULTS FOR DIFFERENT VALUES OF INPUTS
trapezoidal 6.0 3.0 0.618 1.207 0.916 5.56E-10 23.780
trapezoidal 6.0 4.0 0.629 1.237 0.901 3.89E-09 24.440
trapezoidal 6.0 5.0 0.639 1.266 0.887 4.20E-09 25.091
trapezoidal 6.0 6.0 0.649 1.295 0.874 2.05E-09 25.734
trapezoidal 6.0 7.0 0.658 1.323 0.862 7.78E-09 26.369
trapezoidal 6.0 8.0 0.666 1.352 0.850 6.71E-09 26.997
trapezoidal 6.0 9.0 0.674 1.380 0.839 6.95E-09 27.618
trapezoidal 6.0 10.0 0.681 1.408 0.828 -1.31E-11 28.234
trapezoidal 7.0 11.0 0.677 1.393 0.834 1.41E-09 32.258
trapezoidal 7.0 12.0 0.683 1.417 0.825 9.14E-10 32.871
trapezoidal 7.0 13.0 0.689 1.441 0.817 -3.54E-11 33.479
trapezoidal 7.0 14.0 0.695 1.465 0.808 2.29E-09 34.082
trapezoidal 7.0 15.0 0.700 1.488 0.801 1.90E-10 34.680
trapezoidal 7.0 16.0 0.704 1.511 0.793 1.78E-09 35.275
trapezoidal 7.0 17.0 0.709 1.534 0.786 -1.11E-10 35.865
trapezoidal 7.0 18.0 0.713 1.557 0.779 2.72E-09 36.451
trapezoidal 7.0 19.0 0.718 1.579 0.772 -4.94E-10 37.034
trapezoidal 7.0 20.0 0.722 1.601 0.765 5.81E-09 37.612
trapezoidal 8.0 0.0 0.577 1.118 0.968 2.28E-09 28.444
trapezoidal 8.0 1.0 0.589 1.140 0.954 -3.15E-09 29.137
trapezoidal 8.0 2.0 0.599 1.163 0.940 8.88E-10 29.821
trapezoidal 8.0 3.0 0.609 1.185 0.928 3.61E-10 30.497
trapezoidal 8.0 4.0 0.618 1.208 0.916 4.72E-09 31.165
trapezoidal 8.0 5.0 0.627 1.230 0.904 -1.50E-09 31.826
trapezoidal 8.0 6.0 0.635 1.252 0.893 2.09E-10 32.480
trapezoidal 8.0 7.0 0.642 1.274 0.883 8.01E-09 33.127
trapezoidal 8.0 8.0 0.650 1.296 0.873 2.57E-09 33.769
trapezoidal 8.0 9.0 0.656 1.318 0.864 7.72E-09 34.405
trapezoidal 8.0 10.0 0.663 1.339 0.855 5.02E-09 35.035
trapezoidal 9.0 11.0 0.661 1.335 0.857 -3.91E-11 39.054
trapezoidal 9.0 12.0 0.667 1.354 0.849 8.60E-09 39.680
trapezoidal 9.0 13.0 0.672 1.373 0.842 4.56E-10 40.302
trapezoidal 9.0 14.0 0.677 1.392 0.834 9.39E-11 40.920
trapezoidal 9.0 15.0 0.682 1.411 0.827 4.03E-09 41.533
trapezoidal 9.0 16.0 0.686 1.430 0.821 3.10E-09 42.143
trapezoidal 9.0 17.0 0.691 1.448 0.814 3.33E-09 42.749
trapezoidal 9.0 18.0 0.695 1.467 0.808 3.98E-09 43.351
trapezoidal 9.0 19.0 0.699 1.485 0.802 1.80E-09 43.950
trapezoidal 9.0 20.0 0.703 1.503 0.796 8.54E-09 44.546
trapezoidal 10.0 0.0 0.577 1.118 0.968 1.95E-10 35.149
trapezoidal 10.0 1.0 0.586 1.136 0.956 5.97E-09 35.843
trapezoidal 10.0 2.0 0.595 1.154 0.945 6.09E-09 36.530
trapezoidal 10.0 3.0 0.603 1.172 0.935 4.65E-10 37.210
trapezoidal 10.0 4.0 0.611 1.190 0.925 2.45E-09 37.883
trapezoidal 10.0 5.0 0.618 1.208 0.915 3.21E-09 38.550
trapezoidal 10.0 6.0 0.625 1.226 0.906 -1.69E-09 39.211
trapezoidal 10.0 7.0 0.632 1.244 0.897 9.67E-09 39.867
trapezoidal 10.0 8.0 0.638 1.262 0.889 -3.64E-09 40.518
trapezoidal 10.0 9.0 0.644 1.279 0.881 4.01E-09 41.163
trapezoidal 10.0 10.0 0.650 1.297 0.873 3.52E-09 41.804
trapezoidal 11.0 11.0 0.650 1.297 0.873 6.66E-09 45.822
trapezoidal 11.0 12.0 0.655 1.313 0.866 5.03E-09 46.458
trapezoidal 11.0 13.0 0.660 1.329 0.859 3.54E-09 47.090
trapezoidal 11.0 14.0 0.664 1.345 0.853 2.62E-09 47.719
trapezoidal 11.0 15.0 0.669 1.360 0.847 8.79E-10 48.343
trapezoidal 11.0 16.0 0.673 1.376 0.841 3.68E-10 48.964
75
TYPE β L* β A*m * b * y * τ C*
INPUTS (non-dimensional form) OUTPUTS (non-dimensional form)
ITERATIVE RESULTS FOR DIFFERENT VALUES OF INPUTS
trapezoidal 11.0 17.0 0.677 1.391 0.835 2.13E-09 49.582
trapezoidal 11.0 18.0 0.681 1.407 0.829 2.76E-09 50.196
trapezoidal 11.0 19.0 0.685 1.422 0.823 1.54E-09 50.807
trapezoidal 11.0 20.0 0.688 1.437 0.818 1.23E-09 51.415
trapezoidal 12.0 0.0 0.577 1.118 0.968 -2.58E-11 41.854
trapezoidal 12.0 1.0 0.585 1.133 0.958 3.33E-09 42.549
trapezoidal 12.0 2.0 0.592 1.148 0.949 1.36E-09 43.238
trapezoidal 12.0 3.0 0.599 1.163 0.940 5.00E-09 43.920
trapezoidal 12.0 4.0 0.606 1.178 0.931 3.54E-09 44.597
trapezoidal 12.0 5.0 0.612 1.193 0.923 5.11E-09 45.269
trapezoidal 12.0 6.0 0.618 1.208 0.915 3.67E-09 45.935
trapezoidal 12.0 7.0 0.624 1.223 0.908 5.30E-09 46.597
trapezoidal 12.0 8.0 0.630 1.238 0.900 6.33E-10 47.254
trapezoidal 12.0 9.0 0.635 1.253 0.893 6.31E-09 47.907
trapezoidal 12.0 10.0 0.640 1.268 0.886 -1.71E-09 48.555
trapezoidal 13.0 11.0 0.641 1.270 0.885 8.77E-09 52.573
trapezoidal 13.0 12.0 0.646 1.284 0.879 7.85E-09 53.217
trapezoidal 13.0 13.0 0.650 1.298 0.873 2.93E-09 53.857
trapezoidal 13.0 14.0 0.654 1.311 0.867 -5.47E-11 54.493
trapezoidal 13.0 15.0 0.658 1.325 0.861 6.83E-09 55.126
trapezoidal 13.0 16.0 0.662 1.338 0.856 3.38E-10 55.756
trapezoidal 13.0 17.0 0.666 1.351 0.850 5.26E-09 56.383
trapezoidal 13.0 18.0 0.670 1.365 0.845 5.91E-10 57.006
trapezoidal 13.0 19.0 0.673 1.378 0.840 8.67E-10 57.626
trapezoidal 13.0 20.0 0.677 1.391 0.835 8.52E-10 58.244
trapezoidal 14.0 0.0 0.577 1.118 0.968 2.54E-09 48.560
trapezoidal 14.0 1.0 0.584 1.131 0.959 -4.12E-09 49.255
trapezoidal 14.0 2.0 0.590 1.144 0.951 6.52E-11 49.945
trapezoidal 14.0 3.0 0.596 1.157 0.944 1.01E-09 50.629
trapezoidal 14.0 4.0 0.602 1.170 0.936 1.50E-09 51.309
trapezoidal 14.0 5.0 0.608 1.183 0.929 4.27E-09 51.984
trapezoidal 14.0 6.0 0.613 1.196 0.922 3.90E-09 52.654
trapezoidal 14.0 7.0 0.618 1.209 0.915 -1.12E-09 53.321
trapezoidal 14.0 8.0 0.624 1.222 0.908 8.52E-09 53.982
trapezoidal 14.0 9.0 0.628 1.234 0.902 5.12E-09 54.640
trapezoidal 14.0 10.0 0.633 1.247 0.896 2.06E-10 55.294
trapezoidal 15.0 11.0 0.634 1.251 0.894 4.82E-09 59.314
trapezoidal 15.0 12.0 0.638 1.263 0.889 3.86E-09 59.964
trapezoidal 15.0 13.0 0.642 1.274 0.883 -2.23E-09 60.610
trapezoidal 15.0 14.0 0.646 1.286 0.878 1.88E-09 61.252
trapezoidal 15.0 15.0 0.650 1.298 0.873 -2.96E-10 61.892
trapezoidal 15.0 16.0 0.654 1.310 0.867 3.20E-09 62.529
trapezoidal 15.0 17.0 0.657 1.321 0.863 6.78E-09 63.162
trapezoidal 15.0 18.0 0.661 1.333 0.858 3.13E-10 63.793
trapezoidal 15.0 19.0 0.664 1.345 0.853 7.16E-10 64.421
trapezoidal 15.0 20.0 0.668 1.356 0.848 7.28E-09 65.046
trapezoidal 16.0 0.0 0.577 1.118 0.968 1.28E-10 55.265
trapezoidal 16.0 1.0 0.583 1.129 0.960 -5.35E-10 55.961
trapezoidal 16.0 2.0 0.589 1.140 0.953 5.48E-09 56.651
trapezoidal 16.0 3.0 0.594 1.152 0.947 8.07E-09 57.337
trapezoidal 16.0 4.0 0.599 1.163 0.940 -1.90E-09 58.019
trapezoidal 16.0 5.0 0.604 1.175 0.933 4.93E-09 58.697
trapezoidal 16.0 6.0 0.609 1.186 0.927 -3.08E-09 59.370
trapezoidal 16.0 7.0 0.614 1.197 0.921 8.62E-09 60.040
trapezoidal 16.0 8.0 0.619 1.209 0.915 5.86E-09 60.706
trapezoidal 16.0 9.0 0.623 1.220 0.909 3.60E-09 61.368
76
TYPE β L* β A*m * b * y * τ C*
INPUTS (non-dimensional form) OUTPUTS (non-dimensional form)
ITERATIVE RESULTS FOR DIFFERENT VALUES OF INPUTS
trapezoidal 16.0 10.0 0.627 1.231 0.904 1.52E-09 62.026
trapezoidal 17.0 11.0 0.629 1.235 0.902 6.21E-10 66.048
trapezoidal 17.0 12.0 0.633 1.246 0.896 4.14E-09 66.702
trapezoidal 17.0 13.0 0.636 1.256 0.891 6.89E-09 67.353
trapezoidal 17.0 14.0 0.640 1.267 0.886 6.30E-09 68.001
trapezoidal 17.0 15.0 0.643 1.277 0.882 2.61E-09 68.646
trapezoidal 17.0 16.0 0.647 1.288 0.877 -9.96E-11 69.288
trapezoidal 17.0 17.0 0.650 1.298 0.872 8.02E-11 69.927
trapezoidal 17.0 18.0 0.654 1.309 0.868 8.11E-09 70.564
trapezoidal 17.0 19.0 0.657 1.319 0.864 6.01E-09 71.198
trapezoidal 17.0 20.0 0.660 1.329 0.859 6.07E-09 71.830
trapezoidal 18.0 0.0 0.577 1.118 0.968 6.93E-09 61.970
trapezoidal 18.0 1.0 0.583 1.128 0.961 4.27E-09 62.666
trapezoidal 18.0 2.0 0.588 1.138 0.955 5.59E-09 63.358
trapezoidal 18.0 3.0 0.592 1.148 0.949 3.36E-09 64.045
trapezoidal 18.0 4.0 0.597 1.158 0.943 2.93E-10 64.728
trapezoidal 18.0 5.0 0.602 1.168 0.937 3.72E-09 65.408
trapezoidal 18.0 6.0 0.606 1.179 0.931 -3.07E-10 66.084
trapezoidal 18.0 7.0 0.610 1.189 0.926 2.25E-09 66.756
trapezoidal 18.0 8.0 0.615 1.199 0.920 -3.75E-11 67.425
trapezoidal 18.0 9.0 0.619 1.209 0.915 6.02E-09 68.091
trapezoidal 18.0 10.0 0.623 1.219 0.910 3.32E-09 68.753
trapezoidal 19.0 11.0 0.624 1.223 0.908 7.16E-09 72.776
trapezoidal 19.0 12.0 0.628 1.233 0.903 7.05E-09 73.434
trapezoidal 19.0 13.0 0.631 1.242 0.898 3.73E-09 74.089
trapezoidal 19.0 14.0 0.635 1.252 0.894 5.00E-09 74.741
trapezoidal 19.0 15.0 0.638 1.261 0.889 3.30E-09 75.391
trapezoidal 19.0 16.0 0.641 1.270 0.885 6.05E-09 76.037
trapezoidal 19.0 17.0 0.644 1.280 0.881 3.03E-09 76.682
trapezoidal 19.0 18.0 0.647 1.289 0.876 5.17E-09 77.323
trapezoidal 19.0 19.0 0.650 1.298 0.872 3.07E-09 77.963
trapezoidal 19.0 20.0 0.653 1.308 0.868 5.72E-09 78.599
77
TYPE β L* β A* y * r * τ C*
circular 1.0 0.0 1.004 1.004 8.72E-09 4.738
circular 1.0 1.0 0.956 1.102 4.23E-09 5.392
circular 1.0 2.0 0.920 1.196 5.29E-09 6.015
circular 1.0 3.0 0.891 1.288 8.34E-09 6.616
circular 1.0 4.0 0.867 1.378 5.16E-09 7.200
circular 1.0 5.0 0.847 1.467 3.72E-09 7.770
circular 1.0 6.0 0.829 1.556 5.06E-09 8.330
circular 1.0 7.0 0.814 1.643 8.05E-09 8.880
circular 1.0 8.0 0.799 1.731 4.03E-09 9.421
circular 1.0 9.0 0.786 1.819 4.69E-09 9.956
circular 1.0 10.0 0.775 1.907 7.95E-09 10.484
circular 2.0 11.0 0.830 1.554 6.95E-09 14.478
circular 2.0 12.0 0.821 1.602 5.58E-09 15.030
circular 2.0 13.0 0.813 1.650 4.16E-09 15.577
circular 2.0 14.0 0.805 1.697 6.84E-09 16.120
circular 2.0 15.0 0.797 1.744 4.29E-09 16.659
circular 2.0 16.0 0.790 1.792 2.80E-09 17.194
circular 2.0 17.0 0.784 1.839 3.35E-09 17.726
circular 2.0 18.0 0.777 1.887 9.87E-09 18.254
circular 2.0 19.0 0.771 1.934 7.40E-09 18.780
circular 2.0 20.0 0.765 1.982 9.14E-09 19.302
circular 3.0 0.0 1.004 1.004 3.05E-09 11.046
circular 3.0 1.0 0.984 1.041 3.96E-09 11.712
circular 3.0 2.0 0.966 1.078 -3.60E-10 12.363
circular 3.0 3.0 0.951 1.114 7.94E-09 13.002
circular 3.0 4.0 0.937 1.149 8.55E-09 13.630
circular 3.0 5.0 0.924 1.185 6.83E-09 14.248
circular 3.0 6.0 0.912 1.219 4.19E-09 14.857
circular 3.0 7.0 0.901 1.254 3.97E-09 15.459
circular 3.0 8.0 0.891 1.288 7.50E-09 16.054
circular 3.0 9.0 0.882 1.322 9.03E-09 16.643
circular 3.0 10.0 0.873 1.355 3.85E-09 17.226
circular 4.0 11.0 0.888 1.301 1.19E-09 21.069
circular 4.0 12.0 0.881 1.326 6.32E-09 21.656
circular 4.0 13.0 0.874 1.352 3.49E-09 22.239
circular 4.0 14.0 0.868 1.377 8.13E-10 22.818
circular 4.0 15.0 0.862 1.403 2.86E-09 23.393
circular 4.0 16.0 0.856 1.428 7.25E-09 23.964
circular 4.0 17.0 0.850 1.453 3.39E-09 24.532
circular 4.0 18.0 0.845 1.478 8.86E-09 25.097
circular 4.0 19.0 0.840 1.503 6.12E-09 25.659
circular 4.0 20.0 0.835 1.529 5.64E-09 26.218
circular 5.0 0.0 1.004 1.004 1.89E-09 17.354
circular 5.0 1.0 0.991 1.027 2.72E-09 18.023
circular 5.0 2.0 0.980 1.050 2.84E-09 18.683
circular 5.0 3.0 0.969 1.072 7.63E-09 19.333
circular 5.0 4.0 0.959 1.095 4.36E-09 19.976
circular 5.0 5.0 0.950 1.117 1.45E-10 20.612
circular 5.0 6.0 0.941 1.139 6.64E-09 21.241
circular 5.0 7.0 0.933 1.160 5.19E-09 21.863
circular 5.0 8.0 0.925 1.182 8.72E-09 22.481
circular 5.0 9.0 0.917 1.203 7.08E-09 23.092
OUTPUTS (non-dimensional form)
ITERATIVE RESULTS FOR DIFFERENT VALUES OF INPUTS
INPUTS (non-dimensional form)
78
TYPE β L* β A* y * r * τ C*
OUTPUTS (non-dimensional form)
ITERATIVE RESULTS FOR DIFFERENT VALUES OF INPUTS
INPUTS (non-dimensional form)
circular 5.0 10.0 0.910 1.225 2.86E-09 23.699
circular 6.0 11.0 0.916 1.208 6.35E-09 27.514
circular 6.0 12.0 0.910 1.226 2.53E-09 28.120
circular 6.0 13.0 0.904 1.244 3.06E-09 28.722
circular 6.0 14.0 0.899 1.261 7.93E-09 29.321
circular 6.0 15.0 0.894 1.279 5.43E-09 29.916
circular 6.0 16.0 0.889 1.296 4.11E-09 30.508
circular 6.0 17.0 0.884 1.314 8.95E-09 31.097
circular 6.0 18.0 0.879 1.331 5.63E-11 31.682
circular 6.0 19.0 0.875 1.349 5.89E-09 32.265
circular 6.0 20.0 0.871 1.366 5.08E-09 32.845
circular 7.0 0.0 1.004 1.004 1.67E-09 23.662
circular 7.0 1.0 0.995 1.021 3.01E-09 24.332
circular 7.0 2.0 0.986 1.037 3.67E-09 24.996
circular 7.0 3.0 0.978 1.054 2.95E-09 25.653
circular 7.0 4.0 0.970 1.070 7.60E-09 26.303
circular 7.0 5.0 0.963 1.086 8.90E-09 26.948
circular 7.0 6.0 0.956 1.102 4.77E-09 27.587
circular 7.0 7.0 0.949 1.118 -1.29E-10 28.222
circular 7.0 8.0 0.943 1.134 6.10E-09 28.851
circular 7.0 9.0 0.937 1.149 2.74E-09 29.476
circular 7.0 10.0 0.931 1.165 6.54E-09 30.097
circular 8.0 11.0 0.933 1.160 7.93E-09 33.903
circular 8.0 12.0 0.928 1.174 4.01E-09 34.522
circular 8.0 13.0 0.923 1.187 2.20E-10 35.136
circular 8.0 14.0 0.918 1.201 8.04E-09 35.748
circular 8.0 15.0 0.914 1.214 -1.07E-09 36.356
circular 8.0 16.0 0.909 1.228 1.81E-10 36.962
circular 8.0 17.0 0.905 1.241 5.66E-09 37.564
circular 8.0 18.0 0.901 1.254 6.52E-09 38.164
circular 8.0 19.0 0.897 1.268 7.65E-09 38.760
circular 8.0 20.0 0.893 1.281 5.73E-09 39.355
circular 9.0 0.0 1.004 1.004 6.45E-10 29.970
circular 9.0 1.0 0.997 1.017 5.57E-09 30.641
circular 9.0 2.0 0.990 1.030 3.01E-09 31.307
circular 9.0 3.0 0.983 1.043 3.00E-09 31.967
circular 9.0 4.0 0.977 1.056 8.14E-09 32.623
circular 9.0 5.0 0.971 1.068 7.17E-09 33.273
circular 9.0 6.0 0.965 1.081 9.42E-09 33.919
circular 9.0 7.0 0.959 1.094 4.11E-09 34.560
circular 9.0 8.0 0.954 1.106 9.26E-09 35.198
circular 9.0 9.0 0.949 1.119 3.49E-09 35.831
circular 9.0 10.0 0.944 1.131 8.19E-09 36.461
circular 10.0 11.0 0.944 1.130 2.88E-09 40.266
circular 10.0 12.0 0.940 1.141 9.89E-09 40.893
circular 10.0 13.0 0.936 1.152 4.43E-09 41.516
circular 10.0 14.0 0.932 1.163 -5.22E-12 42.137
circular 10.0 15.0 0.927 1.174 7.74E-09 42.754
circular 10.0 16.0 0.924 1.185 5.63E-09 43.369
circular 10.0 17.0 0.920 1.196 2.98E-09 43.981
circular 10.0 18.0 0.916 1.207 9.73E-09 44.591
circular 10.0 19.0 0.913 1.218 8.06E-09 45.198
79
TYPE β L* β A* y * r * τ C*
OUTPUTS (non-dimensional form)
ITERATIVE RESULTS FOR DIFFERENT VALUES OF INPUTS
INPUTS (non-dimensional form)
circular 10.0 20.0 0.909 1.229 5.41E-09 45.803
circular 11.0 0.0 1.004 1.004 9.32E-09 36.278
circular 11.0 1.0 0.998 1.015 -1.28E-09 36.950
circular 11.0 2.0 0.992 1.025 1.03E-09 37.617
circular 11.0 3.0 0.987 1.036 2.30E-09 38.280
circular 11.0 4.0 0.981 1.046 3.79E-09 38.938
circular 11.0 5.0 0.976 1.057 7.43E-09 39.592
circular 11.0 6.0 0.971 1.067 6.81E-09 40.243
circular 11.0 7.0 0.966 1.078 -1.36E-10 40.889
circular 11.0 8.0 0.962 1.088 4.49E-09 41.532
circular 11.0 9.0 0.957 1.098 5.01E-09 42.172
circular 11.0 10.0 0.953 1.109 2.86E-09 42.808
circular 12.0 11.0 0.952 1.110 1.65E-09 46.613
circular 12.0 12.0 0.949 1.119 3.61E-09 47.246
circular 12.0 13.0 0.945 1.129 4.93E-09 47.876
circular 12.0 14.0 0.941 1.138 4.41E-09 48.503
circular 12.0 15.0 0.938 1.147 4.85E-09 49.128
circular 12.0 16.0 0.934 1.156 3.37E-09 49.750
circular 12.0 17.0 0.931 1.166 7.66E-09 50.370
circular 12.0 18.0 0.927 1.175 2.71E-09 50.987
circular 12.0 19.0 0.924 1.184 2.81E-09 51.602
circular 12.0 20.0 0.921 1.193 -2.99E-10 52.215
circular 13.0 0.0 1.004 1.004 1.67E-09 42.586
circular 13.0 1.0 0.999 1.013 6.90E-09 43.258
circular 13.0 2.0 0.994 1.022 -4.86E-10 43.926
circular 13.0 3.0 0.989 1.031 7.20E-09 44.591
circular 13.0 4.0 0.985 1.040 7.21E-09 45.252
circular 13.0 5.0 0.980 1.049 2.99E-09 45.909
circular 13.0 6.0 0.976 1.058 4.12E-09 46.562
circular 13.0 7.0 0.971 1.067 4.66E-09 47.213
circular 13.0 8.0 0.967 1.076 2.80E-09 47.860
circular 13.0 9.0 0.963 1.084 5.51E-09 48.504
circular 13.0 10.0 0.960 1.093 6.29E-09 49.145
circular 14.0 11.0 0.959 1.095 6.16E-09 52.950
circular 14.0 12.0 0.955 1.103 8.36E-09 53.588
circular 14.0 13.0 0.952 1.111 6.89E-09 54.223
circular 14.0 14.0 0.948 1.119 2.37E-09 54.856
circular 14.0 15.0 0.945 1.128 6.35E-09 55.486
circular 14.0 16.0 0.942 1.136 2.47E-09 56.114
circular 14.0 17.0 0.939 1.143 8.89E-09 56.739
circular 14.0 18.0 0.936 1.151 5.61E-09 57.362
circular 14.0 19.0 0.933 1.159 8.15E-09 57.984
circular 14.0 20.0 0.930 1.167 6.87E-09 58.603
circular 15.0 0.0 1.004 1.004 -3.59E-10 48.894
circular 15.0 1.0 1.000 1.012 1.70E-09 49.566
circular 15.0 2.0 0.995 1.020 4.41E-09 50.236
circular 15.0 3.0 0.991 1.028 3.46E-09 50.901
circular 15.0 4.0 0.987 1.035 6.71E-09 51.564
circular 15.0 5.0 0.983 1.043 7.07E-09 52.223
circular 15.0 6.0 0.979 1.051 3.77E-09 52.879
circular 15.0 7.0 0.975 1.059 2.88E-09 53.532
circular 15.0 8.0 0.972 1.066 4.03E-09 54.182
80
TYPE β L* β A* y * r * τ C*
OUTPUTS (non-dimensional form)
ITERATIVE RESULTS FOR DIFFERENT VALUES OF INPUTS
INPUTS (non-dimensional form)
circular 15.0 9.0 0.968 1.074 3.16E-09 54.830
circular 15.0 10.0 0.965 1.082 6.99E-09 55.474
circular 16.0 11.0 0.964 1.084 2.39E-09 59.281
circular 16.0 12.0 0.960 1.091 7.75E-09 59.923
circular 16.0 13.0 0.957 1.098 1.77E-09 60.562
circular 16.0 14.0 0.954 1.105 5.76E-09 61.199
circular 16.0 15.0 0.951 1.113 3.06E-09 61.833
circular 16.0 16.0 0.948 1.120 -2.03E-10 62.465
circular 16.0 17.0 0.946 1.127 3.01E-09 63.096
circular 16.0 18.0 0.943 1.134 2.59E-09 63.724
circular 16.0 19.0 0.940 1.141 2.14E-09 64.350
circular 16.0 20.0 0.937 1.148 8.45E-09 64.974
circular 17.0 0.0 1.004 1.004 1.71E-09 55.202
circular 17.0 1.0 1.000 1.011 2.88E-09 55.875
circular 17.0 2.0 0.996 1.018 5.04E-09 56.544
circular 17.0 3.0 0.992 1.025 8.46E-09 57.211
circular 17.0 4.0 0.989 1.032 4.44E-09 57.875
circular 17.0 5.0 0.985 1.039 2.70E-10 58.536
circular 17.0 6.0 0.982 1.046 -1.32E-10 59.194
circular 17.0 7.0 0.978 1.052 -6.28E-10 59.849
circular 17.0 8.0 0.975 1.059 3.58E-09 60.502
circular 17.0 9.0 0.972 1.066 6.72E-09 61.152
circular 17.0 10.0 0.969 1.073 5.48E-10 61.800
circular 18.0 11.0 0.967 1.075 8.92E-09 65.608
circular 18.0 12.0 0.965 1.082 9.90E-09 66.252
circular 18.0 13.0 0.962 1.088 7.53E-09 66.895
circular 18.0 14.0 0.959 1.095 6.18E-09 67.535
circular 18.0 15.0 0.956 1.101 6.47E-09 68.173
circular 18.0 16.0 0.954 1.107 3.88E-09 68.809
circular 18.0 17.0 0.951 1.114 8.95E-09 69.443
circular 18.0 18.0 0.948 1.120 9.07E-09 70.075
circular 18.0 19.0 0.946 1.126 3.53E-09 70.705
circular 18.0 20.0 0.943 1.132 5.01E-09 71.334
circular 19.0 0.0 1.004 1.004 9.62E-09 61.510
circular 19.0 1.0 1.000 1.010 7.54E-09 62.183
circular 19.0 2.0 0.997 1.016 8.82E-09 62.853
circular 19.0 3.0 0.994 1.023 2.67E-09 63.521
circular 19.0 4.0 0.990 1.029 7.75E-09 64.185
circular 19.0 5.0 0.987 1.035 6.77E-09 64.848
circular 19.0 6.0 0.984 1.041 6.48E-09 65.507
circular 19.0 7.0 0.981 1.047 4.27E-09 66.164
circular 19.0 8.0 0.978 1.054 5.84E-09 66.819
circular 19.0 9.0 0.975 1.060 4.39E-09 67.472
circular 19.0 10.0 0.972 1.066 4.36E-09 68.122
81