Optimal Channel Design - Middle East Technical University · PDF fileOPTIMAL CHANNEL DESIGN AKSOY, Bülent ... toprak işleri maliyetini ve kanalın yüzeyden derinliğine bağlı

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


*b with ** LA ββ for

Rectangular Channel Section..................................................... 45



Rectangular Channel Section .................................................... 45

3.18. Variation of Error Values of *m with ** LA ββ for

Trapezoidal Channel Section..................................................... 46


*b with ** LA ββ for

Trapezoidal Channel Section..................................................... 46



Trapezoidal Channel Section............................................................. 47


*r with ** LA ββ for

Circular Channel Section........................................................... 47



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




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




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




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)


INPUTS (non-dimensional form)

70





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





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





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



74




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




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




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




78





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





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





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

Documents

Optimal Channel Design - Middle East Technical University · PDF fileOPTIMAL CHANNEL DESIGN AKSOY, Bülent ... toprak işleri maliyetini ve kanalın yüzeyden derinliğine bağlı