vi
TABLE OF CONTENTS
CHAPTER
1
TITLE
PAGE
Title page
i
Declaration
ii
Dedication
iii
Abstract English
iv
Abstract Malay
v
Table of contents
vi
List of figures
xi
List of appendices
xiii
List of symbols
xiv
List of abbreviations
xv
List of Tables
xvi
INTRODUCTION
1
1.1 General
1
1.2 Statement of the problem
3
1.3 Objective of the study
3
1.4 Scope of the study
4
vii
2
THEORETICAL BACKGROUND
2.1 Introduction
5
2.2
7
Ordinary differential equation
2.3 Partial differential equation
8
2.3.1. Elliptic equation
9
2.3.2. Parabolic equation
10
2.3.3. Hyperbolic equation
11
2.4 Initial and boundary condition
12
2.4.1. Cauchy condition
13
2.4.2. Dirichlet condition
14
2.4.3. Neumann condition
14
2.5 Concept of Hydraulic
2.6
15
2.5.1. Properties of fluid
15
Flow classification
17
2.6.1. Velocity pattern
17
2.7 Steady and unsteady flow
20
2.8 Basic concept and equations in hydraulics
21
2.8.1.
Mass, force, and weight
21
2.8.2.
Kinematic viscosity
22
2.8.3.
Streamline and stream tubes
23
2.8.4.
Dimensionality of a flow field
23
2.8.5.
Mass conservation
24
2.8.6.
Channel geometric elements
24
2.8.7.
Chezy formula
25
2.8.8.
Pathline, streakline, and streamline
26
2.8.9.
Acceleration
27
2.8.10.
Lagrangian and Eulerian system
29
2.8.11. Reynolds number
30
2.8.12.
31
Fluid elements and fluid particles
2.8.13. Control volume
31
viii
2.9 Equation of motion
2.9.1.
Flux
33
33
2.9.2. Continuity equation
35
2.9.3. Momentum equation
39
2.9.4. Unsteady term
40
2.9.5.
41
Flux term
2.9.6. resultant force
41
2.9.7. Surface force
42
2.9.8. Body force
42
2.9.9. The force due to internal surface
43
2.10 Reynolds transport theorem
43
2.11
46
Best concepts to understand momentum
Force
46
Body force and surface force
47
Unidirectional flow
47
Viscous force
49
Bernoulli equation
49
2.12 Gradually varied flow
49
2.13
2.12.1. Steady gradually varied flow
50
2.12.2. Unsteady gradually varied flow
50
Flow routing
50
2.14 Types of flow routing
50
2.14.1. Hydrologic flow routing
50
2.14.2. Hydraulic flow routing
50
2.15 How to use RivRoute
51
2.15.1. Introduction
51
2.15.2. Input data
51
2.15.3. To run RivRoute
51
2.15.4. External output file
52
2.16 Equation of mass transport
52
ix
2.16.1. Mass transport in flowing water
52
2.16.2. Mass transport equation
56
2.16.3. The last try for predicting of concentration of
57
pollutant
3
METHODOLOGY
58
3.1
Introduction
58
3.2 Assumptions
58
3.3 The governing equations
59
3.4 Continuity equation
59
3.5 Momentum equation
61
3.6 Conservation of energy method
61
3.6.1. To obtain the
term
62
3.7 Solving the governing equation
63
3.7.1. Method of solving
63
Finite difference approximation
64
3.8.1. Central difference approximation
64
3.8.2.
64
3.8
Forward difference approximation
3.8.3. Backward difference approximation
3.9
65
Explicit scheme
66
3.10 The difference equation
67
3.11 Continuity equation
68
3.12 Momentum equation
68
3.13 Newton Raphson iteration
69
3.14 Initial condition
70
3.15 Boundary condition
70
3.15.1. Upstream boundary condition
71
3.15.2. Downstream boundary condition
71
3.16 Mass transport
72
x
4
5
3.17 Initial condition
73
3.18 Upstream boundary condition
73
3.19
73
Downstream boundary condition
3.20 Instruction to compute the concentration of pollutant
73
FINDING AND DISCUSSION
75
4.1 Introduction
75
4.2 Graphs discussion
75
CONCLUSION
82
5.1 Introduction
82
5.2
82
Conclusion
5.3 Recommendation for future work
83
Reference
84
Appendix I
87
Appendix II
88
Appendix III
89
Appendix IV
90
xi
LIST OF FIGURES
FIGURE NO:
TITLE
PAGE
2-1
The domain of solution for an elliptic PDE
10
2-2
The domain of solution for an parabolic PDE
11
2-3
Trigonometric circle
13
2-4
Compressive force acting on an area
16
2-5
Laminar flow
18
2-6
Sub-critical and super-critical flow
19
2-7
Boundary layers
20
2-8
Lagrangian fluid particle
29
2-9
Eulerian system
30
2-10
Flux definition
34
2-11
Fixed control volume for continuity equation
37
2-12
Fixed control volume for momentum equation
39
2-13
Advective transport
55
3-1
Prismatic channel
59
3-2
Component of energy in prismatic channel
61
3-3
Finite difference approximation
64
3-4
x-t grid of the channel
65
3-5
Normal computation table
66
3-6
Points in upstream/downstream and interior
74
4-1
Concentration and discharge versus time (upstream)
76
4-2
Concentration and discharge versus time (25)
77
xii
4-3
Concentration and discharge versus time (50)
78
4-4
Concentration and discharge versus time (75)
79
4-5
Concentration and discharge versus time (downstream)
80
4-6
3D experimental data
81
xiii
LIST OF APPENDICES
APPENDIX
TITLE
PAGE
I
Input Data Used for RivRoute
87
II
Experimental Data
88
III
Physical Properties of Work
89
IV
Output Data
90
xiv
LIST OF SYMBOLS
width of channel
concentration of pollutant
−
concentration versus time
discharge
time
coordinate in longitudinal direction
Molecular diffusion
initial velocity
velocity
one-dimensional
bottom slope of channel
the slope of energy grade line
gravitational acceleration
xv
LIST OF ABBREVIATIONS
ABBREVIATION
WHOLE PHRASE
PAGE
RAP
Remedial action plan
1
CFD
Computational fluid dynamic
2
PDE
Partial differential equation
7
ODE
Ordinary differential equation
7
MOC
Method of characteristics
10
Control volume
18
CV
xvi
LIST OF TABLES
NO:
3.1
TITLE
Table of Time Derivative and Space Derivative
PAGE
67