borang pengesahan status tesis+

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... Falsafah dan Sarjana secara penyelidikan, atau disertasi bagi pengajian secara kerja kursus dan penyelidikan, atau. Laporan Projek Sarjana Muda (PSM).
PSZ 19:16 (Pind. 1/97)

UNIVERSITI TEKNOLOGI MALAYSIA

BORANG PENGESAHAN STATUS TESIS+ JUDUL:

FlJZZY STATE SPACE MODELING FOR SOLVING INVERSE PROBLEMS IN MULTIVARIABLE DYNAMICS SYSTEMS

SESI PENGAJIAN: 2004/2005

RAZIDAH BINTI ISMAIL

Saya

(HURUF BESAR) mengaku membenarkan tesis (PSM/Sarjana/Doktor Falsafah)* ini disimpan di Perpustakaan Universiti Teknologi Malaysia dengan syarat-syarat kegunaan seperti berikut: 1. 2. 3. 4.

Tesis adalah hakmilik Universiti Teknologi Malaysia. PerpustaK~an Universiti TeknologiMalaysia dibenarkan membuat salinan untuk tujuan pengajian §~haja. Perpustakaan dibenarkan membuat salinan tesis ini sebagai bahan pertukaran antara institusi pengaj ian tinggi. **Sila tandakan (./ )

D

(Mengandungi maklumat yang berdarjah keselamatan atau kepentingan Malaysia seperti yang termaktub di dalam AKTA RAHSIA RASMI 1972)

SULIT

D

(Mengandungi maklumat TERHAD yang telah ditentukan oleh organisasi/badan di mana penyelidikan dijalankan)

TERHAD

[ : ] TIDAK TERHAD Disahkan oleh

(TANDATANGAN PENYELIA)

(TANDATANGAN PENULIS)

Alamat Tetap:

NO. 10, JALA~ LIMAU PAGAR 4/4F, PROF. MADYA DR. TAHIR BIN AHMAD

40000 SHAH ALAM,

Nama Penyelia

SELANGOR DARUL EHSAN Tarikh:

CATATAN:

~I-O~-o~

Tarikh:

* Potong yang tidak berkenaan. ** Jika tesis ini SULIT •

~l-OS--O~

atau TERHAD, sUa Iampirkan surat daripada pihak berkuasaJorganisasi berkenaan dengan menyatakan sekali sebab dan tempoh tesis ini periu dikelaskan sebagai SULIT atau TERHAD. Tesis dimaksudkan sebagai tesis bagi Ijazah Doktor Falsafah dan Sarjana secara penyelidikan, atau disertasi bagi pengajian secara kerja kursus dan penyelidikan, atau Laporan Projek Sarjana Muda (PSM).

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"We hereby declare that we have read this thesis and in our opinion this thesis is sufficient in terms of scope and quality for the award of the degree of Doctor of Philosophy"

Signature Name of Supervisor I

Assoc. Prof. Dr. Tahir Ahmad

Date

Signature Name of Supervisor II

Assoc. Prof. Dr. Shamsuddin Ahmad ~l - oS'" - oS"

Date

Signature Name of Supervisor III Date

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BAHAGIAN A - Pengesahan Kerjasama* Adalah disahkan bahawa projek penyelidikan tesis ini telah dilaksanakan melalui kerjasama antara

dengan

_

Disahkan oleh: Tarikh :

Tandatangan Nalna Jawatan (Cop raslni)

* Jika penyediaan tesis/projek Inelibatkan keljasclIna. BAHAGIAN B - Untuk Kegunaan Pejabat Sekolah Pengajian Siswazah Tesis ini telah diperiksa dan diakui oleh: Nama dan Alalnat Pen1eriksa Luar

Nalna dan Alalnat Penleriksa DalalTI I

Prof. Dr. Abu Osman Bin Md Tap Jabatan Matematik Fakulti Sains & Teknologi Kolej Universiti Sains & Teknologi Malaysia Mengabang Telipot 21030 Kuala Terengganu Terengganu Darul Iman

Prof. Madya Dr. Mohd Ismail Bin Abd Aziz Fakulti Sains UTM, Skudai

Pelneriksa Dalalll II

Nama Penyelia lain Uika ada) Disahkan oleh Penolong Pendaftar di Sekolah Pengaj ian Siswazah: Tandatangan

Tarikh:

Nalna

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FUZZY STATE SPACE MODELING FOR SOLVING INVERSE PROBLEMS IN MULTIVARIABLE DYNAMIC SYSTEMS

RAZIDAH BINTI ISMAIL

A thesis submitted in fulfilment of the requirements for the award of the degree of Doctor of Philosophy

Faculty of Science Universiti Teknologi Malaysia

MAY 2005

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ii

I declare that this thesis entitled "Fuzzy State Space Modeling for Solving Inverse

Problems in Multivariable Dynamic Systems " is the result of my own research except as cited in references. The thesis has not been accepted for any degree and is not concurrently submitted in candidature of any other degree.

~.u.~_

Signature

••.. __ .. _•.

Name

_____ RAZlRAH_~JNItJ.SMAIL

Date

.. _...•...•...

31-or-o~

COPYRIGHT © UiTM

.. _

_

iii

To my husband Mohd Aminudin bin Md Amin. To my children Mohd Hani"t Mohd Zharif Nur Hanis/ Mohd 'A fif Nur Husna, Nurul Huda. In memory of my parents Hajah Rokiah binti Abdullah (deceased 10 August 2001) Haji Ismail bin Zainal (deceased 4 January 2002) AL-FATIHAH

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IV

ACKNOWLEDGEMENTS

In the Name of ALLAH s.w.t, The Most Beneficient, The Most Merciful. All praise is due only to ALLAH s.w.t, the Lord of the universe. Ultimately, only ALLAH s.w.t has given us the strength and courage to proceed with life in its entirety. His works are truly splendid and wholesome. His knowledge is truly complete with due perfection. I am highly indebted to many people for their assistance in the preparation of this thesis. I wish to express my sincere gratitude to my supervisor, Assoc. Prof. Dr. Tahir Ahmad for 11is excellent supervision, invaluable guidance, helpful discussions and continuous encouragement. His enthusiasm and thoughtful considerations, and his valuable suggestion throughout the preparation of this thesis are greatly appreciated. I would also like to extend my sincere gratitude an.d appreciation to my co-supervisors, Assoc. Prof. Dr Shamsuddin Ahmad and Assoc. Prof. Dr. Rashdi Shah Ahmad, for their valuable suggestions and guidance. Many thanks are also due to Assoc. Prof. Othman Ayob, from the Faculty of Mechanical Engineering, UTM and Prof. Dr. Mohd Nasir Taib from the Faculty of Electrical Engineering, UiTM for their valuable assistances and suggestions. I would also like to acknowledge with much appreciation the crucial role of the editors and referees or reviewers for all my publications. Their valuable comments and suggestions throughout the preparation of various publications have helped me to improve my work and are highly appreciated. My special thanks to Universiti Teknologi MARA (UiTM) and Jabatan Perkhidmatan Awam (JPA) for the financial support throughout the course of my study. I would also like to express my appreciation to my family: my dearest husband, who has always provided the encouragement and support that I truly need; my lovely children, who have always been the source of inspiration and pleasure; my parents, brothers and sisters for their continuous prayers for my success. Last but not least, my sincere thanks to all my colleagues and friends who kindly provided valuable and helpful comments in the preparation of the thesis, and to those involved directly or indirectly in the preparation of this thesis, whom I have not mentioned above.

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v

ABSTRACT

The maIn objective of this study is to develop a novel inverse modeling technique, known as Fuzzy State Space Model (FSSM).

This model is used for

optimization of input parameters in multivariable dynamic systems. In this approach, the flexibility of fuzzy modeling is incorporated with the crisp state space models proposed in the modern control theory.

The vagueness and uncertainty of the

parameters are represented in the model construction, as a way of increasing the available information in order to achieve a more precise model of reality. Subsequently, the inverse Fuzzy State Space algorithm is formulated for a multipleinput single-output system, which leads to the derivation of Modified Optimized Defuzzified Value Theorem. This algorithm is enhanced to address the optimization of parameters for a multiple-input multiple-output system, which leads to the derivation of an Extension of Optimized Defuzzified Value Theorem. The proofs of these theorems are presented. To facilitate the implementation of these algorithms, a semi-automated computational tool using Matlab® programming facilities is developed. The effectiveness of this modeling approach is illustrated by implementing it to the state space model of a furnace system of a combined cycle power plant. The results obtained in this application demonstrate that the proposed new modeling approach is reasonable and provides an innovative tool for decision-makers.

In

addition, the investigations on the properties and characteristics of FSSM have resulted in the derivation of some lemma and theorems related to convexity and normality of the induced solution of the model, and bounded stability of the Fuzzy State Space system. Finally, the properties of the induced solution of a single FSSM are generalized to the multi-connected systems of FSSM. related to the systems ofFSSM are also discussed.

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Some algebraic views

VI

ABSTRAK

Objektif utama kajian ini adalah untuk membangunkan satu teknik pemodelan songsang yang baru, dikenali sebagai Model Keadaan Ruang Kabur (MKRK). Model ini di gunakan untuk pengoptimuman parameter masukan dalam sistem dinamik ITIultipembolehubah. Dalam pendekatan ini, pemodelan kabur yang fleksibel digabungkan dengan model keadaan ruang rangup dari teori kawalan moden.

Kesamaran dan

ketidakpastian bagi parameter di wakili dalam pembentukan model, sebagai satu cara menambah maklumat supaya menghasilkan model yang lebih tepat.

Seterusnya,

algoritma Keadaan Ruang Kabur dibentuk untuk sistem pelbagai-masukan keluarantunggal, yang mana Teorem Nilai Penyahkaburan Optimum Ubahsuaian diterbitkan. Algoritma ini ditambahbaik untuk pengoptimuman parameter masukan bagi sistem pelbagai-masukan pelbagai-keluaran, yang mana Teorem Nilai Penyahkaburan Optimum Lanjutan diterbitkan. Bukti teorem-teorem tersebut ditunjukkan. Untuk memudahkan perlaksanaan algoritma ini, alat pengkomputeran separa-automotik menggunakan kemudahan pengaturcaraan Matlab® disediakan.

Keberkesanan

pendekatan pemodelan ini ditunjukkan dengan perlaksanaan keatas model keadaan ruang sistem relau bagi sebuah loji janakuasa kitar padu. Keputusan yan.g dihasilkan menunjukkan bahawa pendekatan pemodelan barn yang dicadangkan adalah berpatutan dan boleh menjadi satu alat yang innovatif kepada pembuat kataputus. Selain daripada itu, penyelidikan mengenai sifat dan ciri-ciri MKRK telah menghasilkan beberapa lemma dan teorem yang berkaitan dengan kecembungan dan normal bagi penyelesaian teraruh model , dan kestabilan terbatas bagi sistem Keadaan Ruang Kabur.

Akhir sekali, sifat-sifat penyelesaian teraruh bagi sebuah MKRK

dilanjutkan kepada beberapa sistem MKRK yang berhubung. Beberapa pandangan dari sudut algebra berkaitan dengan sistem MKRK juga dibincangkan.

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VII

TABLE OF CONTENTS

CHAPTER

TITLE

PAGE

THESIS STATUS DECLARATION SUPERVISOR'S DECLARATION TITLE PAGE DECLARATION DEDICATION

111

ACKNOWLEDGEMENTS

lV

ABSTRACT

V

ABSTRAK

VI

TABLE OF CONTENTS

VII

LIST OF TABLES

XII

LIST OF FIGURES

XIV

LIST OF SYMBOLS LIST OF APPENDICES

1

11

INTRODUCTION 1.1

Background and Rationale

XVII XXI

1 1

1.1.1

Approaches in Model Construction

4

1.1.2

Related Studies

8

1.2

Objectives of the Study

11

1.3

Implications of the Study

11

1.4

Scope of the Study

11

1.5

Outline of the Thesis

]3

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VIII

2

SYSTEM MODELING IN STATE SPACE REPRESENTATION

16

2.1

Introduction

16

2.2

Systems and System Models

17

2.3

Mathematical Modeling of a Control System

19

2.4

The State Space Modeling

23

2.5

Solutions of State Equations

28

2.5.1

28

2.6

3

Analytical Solution of State Equations

2.5.2 Computer Simulation of State Equations

29

Structural Properties of a Dynamic System

33

2.6.1

Stability

33

2.6.1.1 Lyapunov Stability

33

2.6.1.2 Input-Output Stability

34

2.7

Controllability and Observability

36

2.8

Inverse Problems in System Modeling

38

2.9

Uncertainties in System Modeling

40

2.10

Summary

41

FUZZY NUMBER AND FUZZY ARITHMETIC

43

3.1

Introduction

43

3.2

Membership Functions

44

3.3

Fuzzy Sets: Basic Concepts

46

3.3.1

48

Operations on Fuzzy Sets

3.3.2 Relation among Fuzzy Sets

50

3.4

a-cuts of Fuzzy Sets

51

3.5

The Extension Principle

52

3.6

Fuzzy Numbers

53

3.7

Arithmetic Operations on Fuzzy Numbers

55

3.7.1 The a-cut Method

55

3.7.2 The Extension Principle Method

56

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IX

3.8

4

Fuzzy Systems

56

3.8.1

Fuzzification

58

3.8.2

Fuzzy Environment

58

3.8.3

Defuzzification

58

3.9

Fuzzy Modeling

59

3.10

Fuzzy Algorithms

60

3.11

Related Studies on Fuzzy Arithmetic

61

3.12

Summary

62

FUZZY STATE SPACE MODELING

64

4.1

Introduction

64

4.2

Terminology

65

4.3

Representation of Uncertainty

66

4.4

Development of the Fuzzy State Space Model

68

4.5

Fuzzy State Space Algorithm for a MISO system

72

4.6

Convexity and Normality of the Induced Solution ofFSSM

75

4.7

Modified Optimized Defuzzified Value Theorem

78

4.8

Fuzzy State Space Algorithm for a MIMO system

81

4.9

Extension of the Optimized Defuzzified Value Theorem

85

4.10

A Computational Tool

86

4.10.1 Programs for a MISO system

88

4.10.2 Programs for a MIMO system

89

4.11

Stability of a Fuzzy State Space System

90

4.12

Summary

93

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x 5

IMPLEMENTATION OF FUZZY STATE SPACE MODEL ON THE FURNACE SYSTEM 94 5.1

Introduction

5.2

Modeling the Furnace System of a Combined Cycle Power Plant

95

5.3

The Furnace System

96

5.4

Model Assumptions

98

5.5

Mathematical Modeling of the Furnace System

99

5.6

State Space Modeling of the Furnace System

101

5.6.1

Development of the State Equation

102

5.6.2

Development of the Output Equations

103

5.7

5.8

The State Space Analysis of the Furnace System

105

5.7.1

Stability Analysis

107

5.7.2

Controllability and Observability

107

Implementation of the Fuzzy State Space Algorithm in the Furnace System (MISO)

5.9

5.10

109

5.8.1

Algorithm applied to Qir

109

5.8.2

Algorithm applied to Qis, Qes, PG and Qrs

114

5.8.3

Optimized Input Parameters

119

Implementation of the Fuzzy State Space Algorithm in the Furnace System (MIMO)

6

94

122

5.9.1

Algorithm applied to Qir and QiS

122

5.9.2

Algorithm applied to Qes and Po

126

5.9.3

Algorithm applied to Qrs

129

5.9.4

Optimized Input Parameters

131

Summary

134

SYSTEMS OF FUZZY STATE SPACE MODEL

136

6.1

Introduction

136

6.2

FSSM in Multi-connected Systems

137

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Xl

6.3

6.4

6.5

7

Properties of Induced Solution of FSSM in Multi-connected Systems

140

6.3.1

Convexity and Normality

141

6.3.2 Optimized Input Value Parameter

142

6.3.3 Bounded-Input Bounded-Output Stability

143

Algebraic View of Systems of FSSM

144

6.4.1

145

Common Feeder

6.4.2 Relations

147

Summary

148

CONCLUSION AND RECOMMENDATIONS

150

7.1

Summary of the Research

150

7.2

Significant Findings and Contributions

152

7.3

Recommendations for Future Research

154

156

REFERENCES

164 - 181

Appendices A to E

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XII

LIST OF TABLES

TABLE NO.

TITLE

PAGE

5.1

Input parameters specification

110

5.2

a-cuts of input parameters

112

5.3

Output parameters specification (Qir)

112

5.4

Endpoints of interval for input parameters (Qir)

114

5.5

Output parameters specification (Qis, Qes, PG and Qrs)

115

5.6

Endpoints of interval for input parameters (Qis)

118

5.7

Endpoints of interval for input parameters (Qes)

118

5.8

Endpoints of interval for input parameters (PG )

118

5.9

Endpoints of interval for input parameters (Qrs)

118

5.10

Calculated fuzzy value for the Furnace System (MISO)

119

5.11

Calculated norm for the Furnace System (MISO)

119

5.12

Optimized input parameters (MISO)

120

5.13

Calculated output parameters for Furnace System (MISO)

120

5.14

Comparison of optimized input parameters (MISO)

121

5.15

Output parameters specification (Qir and Qis)

124

5.16

Endpoints of interval for input parameters (Qir and QiS)

126

5.17

Output parameters specification (Qes and PG )

127

5.18

Endpoints of interval for input parameters (Qes and pG )

129

5.19

Endpoints of interval for input parameters (Qrs)

131

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XIII

5.20

Calculated fuzzy value and norm for the Furnace System (MIMO)

132

5.21

Optimized input parameters (MIMO)

132

5.22

Calculated output parameters for the Furnace System (MIMO)

132

Comparison of optimized input parameters (MIMO)

133

5.23

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XIV

LIST OF FIGURES

FIGURE NO.

TITLE

PAGE

1.1

Construction of the Fuzzy State Space Model

5

1.2

Linearization of a nonlinear dynamic system

6

1.3

Organization of the Thesis

15

2.1

Principle of Superposition

19

2.2

Transfer function or Input-Output representation of a system

21

2.3

State space representation of a system

22

2.4

State space representation of a linear multivariable system

26

2.5

State space representation from governing equations

27

2.6

Plots of the state and output responses

32

2.7

Plots of responses due to step functions

32

2.8

Mathematical Modeling: (a) Forward Problems (b) Inverse Problems

39

3.1

Crisp representation of "middle-aged"

45

3.2

Fuzzy representation of "middle-aged"

46

3.3

Features of the membership function

46

3.4

(a) convex fuzzy set

3.5

Complement of fuzzy set A

49

3.6

Union of fuzzy sets A and B

49

3.7

Intersection of fuzzy sets A and B

49

. (b) non-convex fuzzy set

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48

xv 3.8.

The intersection of two convex fuzzy sets

50

3.9

Examples of Fuzzy Numbers

54

3.10

Block diagram of a Fuzzy System

58

4.1

A triangular fuzzy number

68

4.2

Phases in developing a fuzzy algorithm

70

4.3

Flowchart for a Fuzzy State Space Algorithm

71

4.4

The intersection of the fuzzy desired parameter (a) on the maximum side of the induced parameter (b) on the minimum side of the induced parameter

80

The intersection of the desired parameter on the maximum side of the fuzzy induced parameter

87

The intersection of the desired parameter on the minimum side of the fuzzy induced parameter

87

5.1

Schematic diagram of a Boiler

97

5.2

Simplified block diagram of the Furnace System

101

5.3

The state response of the Furnace System

106

5.4

The output responses of the Furnace System

106

5.5

The Furnace System subjected to step function

108

5.6

The Furnace System subjected to impulse function

108

5.7

Fuzzification of input parameters (a) fuel flow (b) air flow (c) exhaust gas flow

111

5.8

Fuzzy value for the Furnace System (Qir)

113

5.9

Fuzzy value for the Furnace System (Qis)

116

5.10

Fuzzy value for the Furnace System (Qes)

116

5.11

Fuzzy value for the Furnace System (PG)

117

5.12

Fuzzy value for the Furnace System (Qrs)

117

5.13

Surface and contour plots for the Furnace System (Qir and QiS)

125

4.5

4.6

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XVI

5.14

Fuzzy value for the Furnace system (Qir and Qis)

125

5.15

Surface and contour plots for the Furnace system (Qes andpo)

128

5.16

Fuzzy value for the Furnace system (Qes and Po )

128

6.1

An example of a simple multi-connected system

136

6.2

Multi-connected systems of type A

140

6.3

Multi-connected systems of type B

140

6.4

SgFl

is a feeder of SgF2

146

6.5

SgF

is a common feeder of SgFl and SgF2

146

6.6

SFFis

the greatest common feeder of SgFI and SgF2

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147

XVII

LIST OF SYMBOLS

General symbols {X], X2, X3, ••• }

Set of elements Xl,

{xlp(x)}

Set determined by property P

(Xl, X2, X3, .•• , X n )

n-tuple

[Xij]

Matrix

[Xl, X2, X3, ••• , X n ]

Vector

11···11

Euclidean norm

X2, X3, ...

min

[Xl, X2, X3, •• • ,Xn ]

-

Minimum of Xl,

X2, X3, ••• , X n

max

[Xl, X2, X3, •• •,Xn ] -

Maximum of X],

X2, X3, ••. , X n

i,J', k

Arbitrary identifier (indices)

[a, b]

Closed interval of real numbers betweell a and b

[a, b)

Interval of real numbers closed in a and open in b

[a, 00)

Sets of real numbers greater than or equal to a

(a, b)

Open interval of real numbers between a and b

(a, b]

Interval of real numbers open in a and close in b

9t

Set of real numbers

9-\+

Set of positive real numbers Set of n-tuple of real numbers

~

Set of positive integers (natural numbers)

UorX

Universal set

A,B, C, A==B

...

Arbitrary sets (crisp or fuzzy) Set equality

Af-B AcB

Proper set inclusion

AcB

Set inclusion

A nB

Set intersection

Set inequality

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XViii

A uB

Set union

AxE

Cartesian product of sets A and B

o

Empty set

f

X~Y

Function ofjfromXinto Y

[I: X~ Y

Inverse function ofjfrom Y to X

R (X Y)

Relation on X x Y Implies If and only if Element of

E

Not an element of Such that There exist (at least one) For all Membership grade of x in fuzzy set A a-cut of fuzzy set A Strong a-cut of fuzzy set A sup A

Supremum of A

suppA

Support of fuzzy set A

h(A)

Height of fuzzy set A

core(A)

Core of fuzzy set A

alb

a divides b

gcd(a, b)

Greatest common divisor of a and b

gcf(a, b)

Greatest common feeder of a and b

SgF

Fuzzy State Space Model

[Uj, U2,

,Un]T

Input vector

Lv], Y2, ,Ym]T

Output vector

F li (x)

Degree of desirability of using a particular value x Fuzzy induced performance parameter Desired performance parameter A collection of interconnected FSSM systems Common feeder Greatest common feeder

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XIX

Furnace System

CF

Fuel calorific value

J/kg

cpg

Specific heat of exhaust gases at constant pressure

Js/kgOK

cgs

Combustion gas specific heat capacity

Js/kgOK

WF

Fuel flow

kg/s

WA

Air flow

kg/s

WEG

Gas mass flow through the boiler

kg/s

WG

Exhaust gas (from the gas turbine) flow

kg/s

Win

Inlet mass flow

kg/s

W ou

Outlet mass flow

kg/s

wg

Flow of substances entering combustion

kg/s

hG

Exhaust gas (from the gas turbine) specific enthalpy

J/kg

hA

Air specific enthalpy

J/kg

hEG

Gas specific enthalpy

J/kg

hrej

Reference exhaust gases enthalpy

J/kg

hin

Inlet specific enthalpy

J/kg

hou

Outlet specific enthalpy

J/kg

Qir

Heat transferred by radiation to risers

J/s

Qis

Heat transferred by radiation to the superheater

J/s

Qrs

Heat transferred to reheater

J/s

Qgs

Heat transferred to superheater

J/s

Qes

Heat transferred to the economiser

J/s

Qin

Incoming heat flow

kg/s

Q

Heat flow

J/s

Tg

Gas temperature

T gs

Gas temperature at the superheater

Tst

Superheater metal tube temperature

T gr

Combustion gas temperature at the reheater

Trh

Reheater metal tube temperature

T ge

Combustion gas temperature at the economiser

Tel

Economiser metal tube temperature

T g1

Boiler exhaust gas temperature

Trej

Reference exhaust gases temperature condition

OK OK OK OK OK OK OK OK OK

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xx

Tm

~etal

tennperature

oK

Ts

Steann tennperature

oK

Rs

Stoichionnetric air/fuel volunne ratio

(-)

REG

Ideal gas constant for exhaust gases

y

The percentage excess air level

%

V

Volunne

m3

VF

Combustion channber volunne

nn3

(J'

Stefan-Boltznnan constant

f)

Burner tilt angle 0< ()