December 2010
Sarmawarz 2710
Thi Al- Hija 1431
I certify that the thesis entitled “HYDRAULIC CHARACTERISTICS OF FLOW OVER CIPOLLETTI WEIR WITH RECTANGULAR BOTTOM OPENING” has been prepared by Alan Jalal Rashid under my supervision at the University of Sulaimani as a partial requirement for the degree of Master of Science in Hydraulic Engineering.
Signature: ………………………………………….. Supervisor: Prof. Dr. Rafa H. Shaker Al-Suhaili Date:
/
/2011
In view of the available recommendation, I forward this thesis for debate to the examining committee.
Signature: ………………………………………….. Supervisor: Prof. Dr. Nawzad O.A.Rahim Date:
/
/2011
We, as the Examining Committee, certify that we have read this thesis entitled "Hydraulic Characteristics of Flow over Cipolletti Weir with Rectangular Bottom Opening" and evaluated the student Alan Jalal Rashid about its content and in what is connected with. We also declare that in our opinion it meets the standards of a thesis for the degree of Master of Science in Hydraulic Engineering.
Signature:
Signature:
Name: Prof. Dr. Rafa H. Shaker AL-Suhaili.
Name: Assist. Prof. Dr. Hayder Abdul- Amir.
Date:
Date:
/
/ 2011
Supervisor
/
/2011
Head of the committee
Signature:
Signature:
Name: Lect. Dr. Saman Hama Hussain.
Name: Asst. Prof. Faysel A. Daham.
Date:
Date:
/
Member
/2011
/
/2011
Member
Signature: Name: Prof. Dr. Jalal A. Saeed. Date:
/
/2011
Dean of College of EngineeringUniversity of Sulaimani
Cordial thanks extend to Mighty Allah for enabling me to complete this work. Sincere thanks and appreciation are extended to my supervisor Prof. Dr. Rafa Hashim Al-Suhaili, the Dean of the College of Engineering, University of Baghdad, for his valuable scientific support and unlimited help during my study. I am greatly indebted to the University of Sulaimani, College of Engineering, the Irrigation and Drainage Engineering Department. I am also grateful to the former and current Deans of the College of Engineering, University of Sulaimani, Dr. Kamal Ahmed Rashid and Dr. Jalal Ahmed Said”, and also to the head of the department Mr.Niaz Muhamed Said for their continuous encouragement. I would also like to express my thanks to the staff of the Hydraulic laboratory at the College of Engineering, University of Sulaimani, for their support during my experimental work. Sincere gratitude to all the teaching staff who taught me the M.Sc. courses, especially to the soul of Prof. Dr. Numa Immara, Also to Dr. Saman Hama Hussain and Mr. Amanj Jamal for their help in some issues. Special appreciation is also expressed to Dr. Faiq Sarhan, Dr. Hayder AbdulAmir and Mr. Muhammed Rashid from the University of Baghdad for their valuable support during my work in ANN and VISUAL BASIC modeling. Finally, special thank is devoted to the technician Mr. Sabah who helped me in preparing the physical models and the other modifications required, thanks also to Mr. Rawand , and Mr. Syamend for their assistance during testing time.
I
The hydraulic characteristics of a Cipolletti weir with rectangular bottom opening are investigated in this study. The work is carried out using an existing experimental setup of a flume with storage and re-circulating tanks, a pump, a flow meter and a piping system with valves. Thirty nine physical models are made for the Cipolletti weir with rectangular bottom opening with different geometrical dimensions called hereafter as configurations. Experimental testing is made for each configuration with different flow values according to what the experimental setup allows. Experimental measurements are taken for each configuration to find the actual discharge, the head over the weir and the head of the approaching flow. For each configuration the data set are analyzed in order to find the discharge coefficient using an equation prepared for the combined flow over the weir and from the bottom rectangular opening. All the flow cases tested are for free flow over the weir due to the limitations of the experimental setup. The II
calculations indicate all the flow cases are sub-critical flow since the Froude No. found is less than one. The discharge coefficient found express very low coefficient of variance. Dimensional analysis is made to relate the discharge coefficient with different geometrical and flow variables. Correlation analysis indicates negative correlation with all variables except (d/h1). Moreover, the stronger correlation found is that between the discharge coefficient and (ho/h1) (r = -0.7306), while the weakest correlation found is that between discharge coefficient and (bo/h1) (r = -0.1073). An ANN model is developed herein using (Neuframe software) to express the discharge coefficient as a function of the above mentioned variables. The most suitable algorithm for training this model is found to be the back-propagation algorithm. The architecture of the ANN model developed is of one input layer with five nodes, one hidden layer with single node and an output layer with single node. The most suitable data set division is (65%, 25%, and 10%) for training, testing and validation of the model respectively, with stripped division, learning rate (0.2), momentum term (0.80), and correlation coefficient (r=0.8815). A VISUAL BASIC easy to use model is developed herein in order to be used by Engineers to find the discharge coefficient and the actual discharge of the Cipolletti weir with rectangular bottom opening, when the geometrical dimensions of the weir, the opening, and the head over the weir is known. The velocity distribution upstream of the weir for the cases of flow through the opening only are obtained using (SMS Software- RMA2 Model) indicates the existence of a high velocity bulb which extends in the upstream direction with decreasing velocity.
III
Acknowledgement
I
Abstract
II
Contents
IV
List of Figures
VIII
List of Tables
XIV
List of Symbols
XV
List of Abbreviation
XXI
CHAPTER ONE INTRODUCTION
1
1.1
1
1.2 1.3
General Importance of Sharp Crested Weirs and (Cipolletti)
weirs
Objectives:
3 5
CHAPTER TWO THEORY & LITERATURE REVIEW
6
2.1
Historical Review and Theory
6
2.1.1
Broad-Crested Weir
6
2.1.1.1
Description
6
2.1.1.2
Broad Crested Weir with Rectangular Control Section
8
2.1.2
Short Crested Weir
9
2.1.2.1
Description
9
2.1.3
Sharp-Crested Weirs
11
2.1.3.1
Sharp-Crested Weir with Rectangular Control Section
14
2.1.3.1.1 Description
14
2.1.3.1.2 Evaluation of Discharge
16 IV
2.1.3.1.3
Limits of Application
18
2.1.3.2
V-notch Sharp-Crested Weirs
18
2.1.3.2.1
Description
18
2.1.3.2.2
Sharp-Crested Weir with Triangular Control Section
20
2.1.3.2.3
Evaluation of Discharge
21
2.1.3.2.4
Limits of Application
21
2.1.3.3
Sharp-Crested Weir with Trapezoidal Control Section
22
2.1.3.4
Cipolletti Weir
23
2.1.3.4.1
Description
23
2.1.3.4.2
Evaluation of Discharge
23
2.1.3.4.3
Limits of Application
24
2.2
Review of Available Related Researches
24
CHAPTER THREE EXPERIMENTAL WORK
47
3.1
Introduction
47
3.2
Experimental Flume
47
3.3
Equipment
50
3.3.1
Ultrasonic Flow Meter (Digital Flow Meter)
51
3.3.2
Pump
52
3.3.3
Point Gauges
52
3.3.4
Storage and Recirculation Tanks
53
3.3.5
Piping System
54
3.4
Physical Model of the Proposed Weir Section
55 V
3.5
Modifications Made on the Flume
59
3.6
Experimental Procedure
61
3.7
Experimental Limitations
62
CHAPTER FOUR RESULTS AND DISUSSION
63
4.1
Experimental Results
63
4.2
Calculation of Discharge Coefficient
64
4.3
Dimensional Analysis for Discharge Coefficient
65
4.4 Hydraulic Measurements for Calculating Discharge Coefficient 4.4.1 Configuration (1.a.1)
67
4.4.2
Other Configurations
68
4.5
Artificial Neural Network Model for Calculation of
Discharge Coefficient 4.5.1
Theory and Methodology of Artificial Neural Network
Modeling
67
69
70
4.5.1.1
Description
70
4.5.1.2
Artificial Neural Network Operation
71
4.5.1.3
Multi-Layer Artificial Neural Networks (MLANNs)
73
4.5.1.4
Back-Propagation Neural Network (BPNN)
75
4.5.1.5
The Algorithm of the Error Back-Propagation
Neural Network 4.5.1.6 4.5.1.7
Data Division and Pre-Processing Model Validation
75 79 81 VI
4.5.2
Application of Artificial Neural Network Modeling for
Discharge
Coefficient
of
Cipolletti
Weir
with
Bottom
83
4.6
VISUAL BASIC Model of the ANN Model Application
96
4.7
Visual Flow Conditions
98
4.8
Application of Two Dimensional Modeling (RMA2)
100
4.9
Model Verification
103
Rectangular Opening
CHAPTER FIVE CONCLUSIONS AND RECOMMENDATIONS
105
5.1
Conclusions
105
5.2
Recommendations
106
REFRENCES
108
APPENDICSES Appendix (A)
Configurations test tables
A0
Appendix (B)
RMA2 Theoretical Background
B0
Appendix (C) Different Configurations of the Cipolletti Weir with Rectangular Bottom Opening Appendix (D) Velocity Distribution Upstream the Weir
C0
Using (RMA2) Application
D0
VII
Figure (1.1)
Photograph of Girdaspian irrigation project, Kurdistan
2
Standard Cipolletti measuring weir of (61 cm) crest Figure (1.2)
length installed in a farm out-let (Kraatz and
4
Mahajan,1982) Figure (2.1)
Illustration of terminology for broad crested weirs
7
Figure (2.2)
Dimension of a rectangular control section
8
Figure (2.3) Figure (2.4)
Flow over a round-nose broad-crested weir with rectangular section Various types of short-crested weirs
9
10
Profile of nape of a fully aerated two-dimensional weir Figure (2.5)
(after Bazin 1896 and Scimeni 1930) as mentioned in
12
(Bos, 1989) Figure (2.6)
Parameters of sharp-crested weir (afterBos, 1989)
13
Figure (2.7)
Dimensions of a rectangular control section
16
Figure (2.8)
The rectangular sharp-crested weir (thin-plate weir)
16
Figure (2.9)
General view of a sharp-crested rectangular weir
17
Figure (2.10) V-notch sharp-crested weir
19
VIII
Figure (2.11)
Dimensions of a triangular control Section
20
Figure (2.12)
Dimension of a trapezoidal control section
22
Figure(2. 13)
Definition sketch of a Cipoletti weir
23
Definition sketch for free flow below an inverted Triangular and Figure (2.14)
over a sharp-crested rectangular weir system (Al-Hamid, et.al,
25
1996-a) Effect of upstream head ratio, H/d, and the bed slope, So, on the Figure (2.15)
non-dimensional free discharge ratio for typical two sets of data
27
with Ɵ = 90° (Al-Hamid, et.al, 1996-a) Effect of overflow head ratio, h/d, and the angle of the inverted Figure (2.16)
triangular weir, Ɵ, on the non-dimensional free discharge ratio
27
for constant y/d and b = 12 cm (Al-Hamid, et.al., 1996-a) Figure (2.17)
Verification of prediction Equation (2-24) for submerged flow at
28
b1/d=2(Al-Hamid, et.al, 1996-a) Figure (2.18)
Verification of prediction Equation (2-25) for free flow(Al-
28
Hamid, et.al, 1996-a) Figure (2.19)
Definition sketch for the simultaneous flow over weirs and below
30
gates (Al-Hamid, et.al., 1996-b) Figure (2.20)
Comparison between measured and computed CVG values (Al-
33
Hamid, et.al, 1996-b) Figure (2.21)
Definition sketch for combined flow over weir and below
35
submerged gate (Negm,1998) IX
Figure (2.22)
Prediction of Equation (2-42) versus the experimental
36
data (Negm,1998) Definition sketch for combined free flow over weirs Figure (2.23) and below gates. (a) Cross section, (b) longitudinal
39
section (Negmet.al.,2002) Modular limit for combined flow on horizontal bed, (a) Figure (2.24)
relation-ship between limiting tail-water ratio and
40
upstream head ratio and (b) Equation (2-46) versus measurements(Negmet.al., 2002) All calculated and measured discharges with in all Figure (2.25) experiments in semi-submerged and fully submerged
46
flow(Saman and Mazaheri, 2009) Figure(3.1)
Schematic Diagram of the Experimental setup
48
Figure(3.2)
Detail of the steel From of the Experimental setup
49
Figure (3.3)
Photos of the experimental flume setup (Amanj, 2009)
50
Figure (3.4)
Ultrasonic flow meter SCL-61D1-DN80
51
Figure (3.5)
Photo of ultrasonic flow meter SCL-61D1-DN80
52
Figure (3.6)
Pentax water pump
52
X
Figure (3.7)
ARMFIELD point gauge
53
Figure (3.8)
Storage and recirculation tanks
54
Figure (3.9)
Piping system connections
54
Figure (3.10) General schematic diagram of the physical models
57
Figure (3.11) Different configurations of the Cipolletti weir with rectangular bottom opening model
58
Figure (3.12)
Galvanized channels fabricated for installing the weir
59
models Figure (3.13)
Details of flow control gate installed at the upstream
60
side of flume to control the flow from the supply tank Figure (3.14)
work done before conducting the experiments
61
(a)fixing the gate (b) cleaning the flume Figure (4.1)
Typical Model of Cipolletti weir with rectangular
65
bottom opening Figure (4.2)
Typical Structure and Typical Neurons
71
Figure (4.3)
Multi-layer Neural Network (Al-Janabi, 2006)
74
Figure (4.4)
The Error Back-Propagation Algorithm
76
XI
Fig.(4.5)
Artificial neural network architecture for discharge coefficient estimation
83
Figure (4.6)
Variation of discharge coefficient with (ho/h1)
85
Figure (4.7)
Variation of discharge coefficient with (bo/h1)
86
Figure (4.8)
Variation of discharge coefficient with (d/h1)
86
Figure (4.9)
Variation of discharge coefficient with (hw/h1)
87
Figure(4.10)
Variation of discharge coefficient with (Bw/h1)
87
Figure (4.11)
Figure (4.12)
The variation of testing error, training error and correlation coefficient with the number of nodes in the hidden layer The variation of testing error, training error and correlation coefficient with learning rate
Figure (4.13)
The variation of testing error, training error and correlation coefficient with momentum term
90
92
94
Introductory form for the VISUAL BASIC Program for Figure (4.14)
the ANN model to calculate the discharge coefficient
96
and the actual discharge for a Cipolletti weir with a bottom rectangular opening
XII
Data entering form for the VISUAL BASIC Program Figure (4.15)
for the ANN model to calculate the discharge
97
coefficient and the actual discharge for a Cipolletti weir with a bottom rectangular opening Output form for the VISUAL BASIC Program for the Figure (4.16)
ANN model to calculate the discharge coefficient and
97
the actual discharge for a Cipolletti weir with a bottom rectangular opening Figure (4.17) Flow condition when the water level below the height of the bottom opening Figure (4.18) Flow condition when the water level just as the crest level Figure (4.19) Flow condition when the water level just passes the crest level
98
Figure (4.20) Flow condition for high head over the crest level
100
Figure (4.21)
The finite element generated mesh for the two
99 99
101
dimensional modeling using RMA2 Figure (4.22)
The velocity distribution upstream the weir for the case
102
(1-A) Comparison between discharge coefficient estimated by Figure (4.28) the ANN model and measured tested Discharge
104
coefficient
XIII
Table (2.1)
Table (2.2)
Table (2.3)
Limitation of a rectangular sharp-crested fully contracted weir Values for Ce as a function of the ratios bc/Bl and hl/pl (from Georgia Institute of Technology) Classification and limits of application of V-notch sharp crested weirs (after ISO 1971, France)
15
17
19
Table (3.1) Different configurations used in the experimental work
56
Table (4.1) Test results and calculations for configuration (1.a.1) , (P=0.31m , BW = 0.22m,bo = 0.15m , ho = 0.05m) Table (4.2) Summary results for all configurations
67
Table (4.3) Some Statistical properties of the testing data
84
Table (4.4) Data sets selection for the ANN model
88
Table (4.5) Data sets Division type selection for the ANN model
88
Table (4.6)
No. of nodes in the hidden layer selection for a Stripped data
68
89
division of (65%, 25%, and 10%) Table (4.7) Learning rate selection for the ANN model
91
Table (4.8) Selection of momentum term for the ANN model
93
Table (4.9) Weights and Threshold Levels for the ANN Optimal Model
95
Table (4.9) the required boundary conditions
101
XIV
H1
Total head over the broad-crested weir
L
L
Length of the weir crest in the direction of flow
L
h1
Depth of water at the upstream side of the broad-crested weir
L
V̅
Average velocity
L/T
g
Gravitational acceleration
L/T2
Q
Theoretical discharge of broad-crested weir
L3/T
yc
Critical depth over broad-crested weir
L
Ac
Critical flow area of broad-crested weir
L2
bc
Width of broad-crested /or sharp crested weir
L
P1
Height of broad crested /or sharp crested weir
L
Cv
Coefficient of approach velocity of broad crested weir
u Cd m
dimensionless
Power of coefficient of approach velocity equation for broad crested weir Discharge coefficient of broad crested Vertical distance from any point inflow over sharp crested weir
dimensionless dimensionless
L
x
Local width of the weir throat of sharp crested weir
L
Ce
Discharge coefficient of sharp crested weir
B1
Full width of rectangular approach channel
L
be
Effective width of sharp crested weir
L
he
Effective head over sharp crested weir
L
XV
Kb
Coefficient of effect of viscosity
L
Kh
Coefficient of effect of surface tension
L
Angle of V-notch sharp crested weir Qw
Cw
b
n
b1
d
Actual discharge over rectangular weir with triangular bottom opening presented by Al-Hamid, et.al. ,1996-a
degree L3/T
Discharge coefficient of weir as presented by AlHamid, et.al. ,1996-a Width of rectangular weir with triangular bottom opening as presented by Al-Hamid, et.al. ,1996-a
dimensionless
L
Number of side contraction as presented by Al-Hamid, et.al. ,1996-a Width of bottom triangular opening as presented by AlHamid, et.al. ,1996-a Height of bottom triangular opening as presented by AlHamid, et.al. ,1996-a
dimensionless
L
L
Vertical distance from top of bottom triangular opening y
and crest height of rectangular sharp crested weir as
L
presented by Al-Hamid, et.al. ,1996-a Actual combined discharge over the rectangular weir Qa
and from triangular bottom opening as presented by Al-
L3/T
Hamid, et.al. ,1996-a
XVI
C1,C2,C3, Regression coefficients of equation (24) as presented by C4
Al-Hamid, et.al. ,1996-a Angle of triangular bottom opening as presented by Al-
t
b
ZG
QG
Hamid, et.al. ,1996-a Width of bottom rectangular opening as presented by Al-Hamid, et.al. ,1996-b Height of bottom rectangular opening as presented by Al-Hamid, et.al. ,1996-b Theoretical discharge through rectangular bottom opening as presented by Al-Hamid, et.al. ,1996-b
dimensionless
degree
L
L L3/T
Combined theoretical discharge over the triangle weir QCT
and bottom rectangular opening as presented by Al-
L3/T
Hamid, et.al. ,1996-b Combined actual discharge over the triangle weir and QCA
bottom rectangular opening as presented by Al-Hamid,
L3/T
et.al. ,1996-b Combined discharge coefficient of flow over the CVG
triangle weir and bottom rectangular opening as
dimensionless
presented by Al-Hamid, et.al. ,1996-b ρ
Water density
So
Channel bed slope
µ
Water viscosity
M/LT
σ
Water surface tension
M/T2
M/L3 dimensionless
XVII
Zv
Heights of weir crest as presented by Al-Hamid, et.al. ,1996-b
M
Combined discharge coefficient of triangular weir with CVGD
bottom rectangular opening for non-dimensional terms
dimensionless
equation (2-31) as presented by Al-Hamid,et.al. ,1996-b QC
Combined flow over rectangular weir with rectangular bottom opening as presented by Negm (1998)
L3/T
Non-dimensional discharge term over rectangular weir QT
with rectangular bottom opening as presented by Negm
dimensionless
(1998) QU
CU
QL
CL
d
b1
Actual discharge over rectangular weir as presented by Negm (1998)
L3/T
Coefficient of discharge over rectangular weir as presented by Negm (1998) Actual discharge from bottom rectangular opening as presented by Negm (1998)
dimensionless
L3/T
Coefficient of discharge of bottom rectangular opening as presented by Negm (1998) Height of bottom rectangular opening as presented by Negm (1998) width of bottom rectangular opening as presented by Negm (1998)
dimensionless
L
L
XVIII
Actual combined Non-dimensional discharge per unit
qt
width over rectangular weir with rectangular bottom
dimensionless
opening as presented by Negm (2002) Actual combined discharge per unit width over
qc
rectangular weir with rectangular bottom opening as
L3/T
presented by Negm (2002)
Re
Reynold Number
dimensionless
We
Weber Number
dimensionless
A0,A1
Regression coefficient of equation (2-48) as presented by Negm (2002)
dimensionless
Theoretical discharge over triangular weir with
Qtheo
rectangular bottom gate opening as presented by Hayawi,
L3/T
et.al.(2008)
Qg Qw
Theoretical discharge through rectangular bottom gate opening as presented by Hayawi, et.al.,(2008) Theoretical discharge over triangular weir as presented by Hayawi, et.al. ,(2008)
L3/T L3/T
Discharge coefficient of combined flow over triangular
Cd
weir with rectangular bottom gate opening as presented
dimensionless
by Hayawi, et.al.,(2008)
Qws CC
Submerged weir discharge as presented by Saman and Mazaheri (2009) Contraction Coefficient as presented by Saman and Mazaheri (2009)
L3/T
dimensionless XIX
P ho bo Bw hw
Height of Cipolletti weir crest as presented by researcher Rashid (2010) Height of rectangular bottom opening as presented by researcher Rashid (2010) Width of rectangular bottom opening as presented by researcher Rashid (2010) Width of Cipolletti weir crest as presented by researcher Rashid (2010) Vertical distance from Cipolletti weir crest to Cipolletti weir top as presented by researcher Rashid (2010)
L L L L L
Vertical distance between top of rectangular bottom
d
opening and Cipolletti weir crest as presented by
L
researcher Rashid (2010)
Qg theo Qw theo Cd g Cd w
Theoretical discharge through the bottom opening as presented by researcher Rashid (2010) Theoretical discharge through Cipolletti weir as presented by researcher Rashid (2010) Discharge coefficient of bottom rectangular opening as presented by researcher Rashid (2010) Discharge coefficient of Cipolletti weir as presented by researcher Rashid (2010)
L3/T L3/T
dimensionless
dimensionless
Combined discharge coefficient of flow over Cipolletti
Cd c
weir and bottom rectangular opening as presented by
dimensionless
researcher Rashid (2010) XX
ANN
Artificial Neural Networks
BPNN
Back-Propagation Neural Network
MLANNs SSE USBR ∆wij Cyjdj
Multi- Layer Artificial Neural Network Sum of Square Error United State Bureau of Reclamation Correction of weight in the ANN model Covariance between the model output (yj) and the desired output (dj)
d̅
Mean of the desired output dj
dj
Desired (observed) output , dj = d1,d2,…….,dn
dk
The desired output of the neuron
f(.)
The transfer (activation) function
hj
Actual output of hidden neuron
Ij
The activation level of node j
n
Number of data
r
Correlation Coefficient
R2
coefficient of determination
Wij
The connection weight between node I and j
wjk X max xi
The connection weight between hidden node j and output node k Maximum value of variables of ANN model The input from node i ,i=0,1,2,…….,n
XXI
Xmin
Minimum value of variables of ANN model
y̅
Mean of the model output yj
yj
Model (predicted) output, yj=y1,y2,…..,yn
yk
Actual output of neuron k
α
momentum term
δk
back propagation error learning rate
j
The bias or threshold for node j
σdj
Standard deviation of the desired output dj
σyj
Standard deviation of the model output yj
XXII
الخاص الخاص الخ اصال یدر لیکی ل جری نف
هدارن ع()Cipolletti
مع ج دفتح مستطی الشکلفیاس ه قدم منقبل ئ انجالرشیدش انه بک ری سك ی ال ندس -قس الر (
-
)
ب شراف اأست ذالدکت ررافعه ش ش کرالس ی ی تم دراس الخ اصال یدر لیکی ل دارن ع()Cipollettiمع ج دفتح مستطی الشکل فیاس ه ،ستخدم ل ذاالغرض(منظ م قن ةصن عی معخزانین احدل تغذی اأخرأع دةتد یر الجری ن،مضخ ،مقی سجری ن،منظ م ان بی مز دةب أق لالضر ری ).منالجدیرب لذکر ن هذهالمنظ م قدبنی س بق منقبلالب حث(ئامانج )٩٠٠٢ ،استخدم منقب هلدراس الجری نف مسیلمدرج ،لغرض جراءالدراس الح لی ت اجراءعدةتح یرا ع هذهالمنظ م مثلتصنیع ب اب ل سیطرة ع الجری ن من م دة ( )Perplexمع منظ م تحک ( )Gear boxط ر حدیدی لغرضالتثبی .کم ت تصنیعح م حدیدی مز دةبمط طلغرضتثبی النم ذجال دارة . ت بن ء()٩٣نم ذجمخت فمنهدارةن ع()Cipollettiمعفتح مستطی فیاس هحیثلکل نم ذجتخت فاابع دلکلمنال دارة ال تح . ت جراءالتج ر لکلمنالنم ذجالمش رةالی اعاهمعقی مخت ل جری نبض م تسمحبه المنظ م المختبری ت قی س (التصریف الحقیقی ،الشحن ف ال دارة الشحن ل جری ن فی مقد ال دارة) . لکل نم ذج من النم ذج اعاه ،ت تح یل القراءا لغرض ایج د مع مل التصریف ب ستخدا مع دل اشتق منالجری نالمشترکف ال دارة منال تح الس ی المستطی . جمیعح ا الجری نالتیت اختی ره ک ن لح ل الجری نالحرف ت منهالمنظ م المختبری .
ال دارة ذلکبسب م
الخاص
بین نت ئج تح یل القراءا ان الجری ن قبل ال دارة فی جمیع الح ا ه من الن ع ال دیء (تح الحرج)حیث نرق فر دالمحس ک نفیجمیعهذهالح ا اقلمن( ) . کم بین الحس ب ب نتغ یرمع ملالتب ینلمع ملالتصریفالمحس ک نق یاجدا.ت اجراء تح یلبعدیأیج دعاق ابعدی بینمع ملالتصریف متغیرا تمثلابع دالنم ذج خرىتمثل الجری ن،حیثنتجعن العاق بینمع ملالجری ن متغیرا نسبی هی( رت عال تح /الشحن ف ال دارة،عرضال تح /الشحن ف ال دارة،المس ف الش ق لی بینارت عقم ال دارةال اعاال تح / الشحن ف ال دارة،المس ف بینقم ال دارة اعاال دارةمنالج ان /الشحن ف ال دارة ،خیرا عرض القم ل دارة /الشحن ف ال دارة) ت ترمیزه ع الت الي ( hw/h1,d/h1,bo/h1,ho/h1 . )Bw/h1 ت اجراءتح یلاارتب طبینمع ملالتصریف کلمنالمتغیرا المش رالی اعاه،حیث تبین ان مع مل اارتب طبین مع مل التصریف جمیع المتغیرا اعاه ك ن س لب فی م عدا المتغیر (.)d/h1ب اض ف ال ذلکفقدتبیناناق ىارتب طک نمعالمتغیر( )r=-0.7306(،)ho/h1ضعف ارتب طک نمعالمتغیر( )r=-0.1073( ،)bo/h1ب لرغ مناناارتب طالثن ئيبشکلع ضعیف اااناارتب طالمتعددم ج دکم یبیننم ذجالشبک العصبی الصن عی ()ANNاحق . ت بن ء نم ذج الشبک العصبی الصن عی ب ستخدا برن مج ( )Neuframeأیج د مع مل التصریفکدال ل متغیرا المش رالی عاه . یبین هذه النم ذج ب نه افضل خ ارزمی لتدری هذا الم ذج هی خ ارزمی ت زیع ااخط ء المتراجع( )Back Propagationنالمعم ری المائم ل شبک تتک نمنطبق ادخ لبخمس خای طبق مخ ی احدةبخ ی احدة طبق اخراجبخ ی احدة.کم بینتح یلالنم ذجب نافضلتقسی ل بی ن لغرضالتدری ،اأختب ر البرهن ل نم ذجهی()%01,%56,%56ع الت الي ،نن ع تقسی البی ن اافضله الن عالشریطی.کم تبینالتح یلایض ب نمعدلالتع ل شبک العصبی ه ( )1.5نحدالزخ ه ( .)1.0 م بخص صدال الت عیلفک ن منن عخطیل طبق المخ ی منن ع()Sigmoid tanh لطبق المخرج .ان فضلمع ملارتب طل ذهالشبک ( )r=0.8815الذییصنفق ی ب أعتم دع تصنیف ( )Smith , 1986کم ت بن ء نم ذج ( )Visual BASICلغ بیسک المرئی س التطبی لغرض استخدامه من قبل م ندسین أیج د مع مل التصریف التصریف الحقیقی ل ذا المنش ال یدر لیکیعندت فر بع دالمنش ال تح قی سالشحن ف ال دارة . کم ت اخیرادراس ت زیعالسرع فیمقد ال دارةلح ا الجری نخالال تح فقطب ستخدا برن مج( ) RMA2الذیاشرب ج دمنطق سرعع لی ع شکلبص التیتمتدب تج هالمقد مع نقص نفیقیم السرع کم شرالنم ذج ج دمنطقتینذا سرعبطیئ احدةع یس ر اأخرى ع یمین التیینقصعرض ب تج هالمقد .نهذاالت زیعل سرعی شر مک نی ازال الرس بی من المنطق ال سط ل مقطعفیمقد ال دارة هیح ل افضلبکثیرمنح ل ال دارةالطبیعی مند ن ج دال تح المستطی فیاس ه.
Chapter One Introduction
1
CHAPTER ONE INTRODUCTION 1.1
General Weirs are an over flow structures with a crest. They are usually used for
many purposes, such as measuring the flow rate and raising the water level to divert it to a branch channel. Weirs act as a control structure in the channel with a unique head-discharge relationship. (Borghei, 1999) indicates that weirs are used for flow measurements, energy dissipation, flow diversion, regulation of flow depth, flood passage and other means. Weirs are built as a standing wall across the flow section with opening at the top accumulates sediments upstream of the weir which affect all the functions of weirs mentioned above. This problem is exaggerated when the approaching channel has a high suspended load causing large changes in the upstream channel section. This results in many problems, such as high measuring errors and flow diversion problems. Many solutions were adopted before, such as the use of sediment excluder structure, sloping weirs, and continuous periodical removal of sediments. (Al-Jaff, 2002) recommends studying the ability of sloping weirs in passing sediments. (Muhammad, 2008) states that the phenomenon of sediment accumulation behind a weir was noticed in Sulaimani governorate. Some of these weirs were filled either partially or completely with sediment materials to a point of cancellation of the role and operation of these weirs as shown in Figure (1.1).
Chapter One Introduction
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Figure (1.1) Photograph of Girdaspian irrigation project, Kurdistan .
This phenomenon occurs since most upstream irrigation canals carry water contains sediments. This existence of weirs in these canals with regular vertical weirs (sharp or broad crested) will cause accumulation of all kinds of sediments in the upstream pool. This sediment may include clay, silt, sand, gravel or any combination of these. (Mohammad, 2008), studied the ability of sloping weirs for sediment removal. The main problem of these weirs is the change of the flow. The slope should be designed according to the design flow to assure sediment movements, as the flow change as expected, sediments will not move up the slope and hence it will not be removed. Sediments excluding structures are an alternative solution for sediment removal upstream hydraulic structures. (Al-Suhaili and Auda, 2001), conducted
Chapter One Introduction
3
a physical model for (Adhaim dam) diversion weir, designed with sediment excluder with two openings and gates. The main problem is in the design since it needs physical modeling to find the real ability of removal. Moreover many operation problems were also found. Periodical removal of sediments is the other solution. However, this solution is rather expensive and difficult, especially for large weirs. Other solutions have been adopted by (Al-Hamid, et. al., 1996) for triangular weir and (Negm, 1998) for rectangular weirs with unequal contractions. That is to provide an opening at the bottom of the weirs. Sediments will partially remove through this opening. In this research an attempt has been made to study the flow condition and discharge coefficient for a Cipolletti weir with rectangle opening at the bottom to insure sediments removal. The study is conducted using both experimental and (ANN) modeling. The ANN model uses the experimental results to obtain a model that enables the calculations of the discharge coefficient and the actual discharge for the hydraulic structure investigated. The use of this model is to be achieved by designing an easy to use Visual basic program. 1.2
Importance of Sharp Crested Weirs and (Cipolletti) Weirs: Weirs are probably the most extensively used devices for the measurement
of the rate of flow of water in open channels. Between the two types of weirs (broad crested weirs and sharp crested weirs) the latter are commonly and frequently used as measurement devices and control structures too. Broad crested weirs are commonly incorporate in irrigation structures but are not usually used
Chapter One Introduction
4
to determine flow, with the exception of the broad crested weir often known as “Romijin gate” (Kraatz and Mahajan, 1982). The types of sharp crested weirs commonly used for measuring irrigation water are the: a. Sharp crested contracted rectangular weirs. b. Sharp crested suppressed rectangular weirs. c. Sharp crested and sharp sided trapezoidal (Cipolletti) weirs. d. Sharp sided 90o V-notch weirs. The Cipolletti is perhaps the most frequently used type. (Kraatz and Mahajan,1982) The Cipolletti or Trapezoidal Sharp-edge Weir is similar to a rectangular weir with end contractions except that the sides incline outwardly at a slope of 1 horizontal to 4 vertical. This slope causes the discharge to occur essentially as though it was without end contractions. The advantage of this weir is that no correction for end contraction is required. The Cipolletti Weir is commonly used in irrigation systems. (Internet- 1), as shown in Figure (1.2)
Figure (1.2) Standard Cipolletti measuring weir of (61 cm) crest length installed in a farm out-let (Kraatz and Mahajan, 1982).
Chapter One Introduction 1.3
5
Objectives: The objectives of this study are to conduct: a- A hydraulic study of a standard trapezoidal (Cipolletti) weir with rectangular opening at the bottom, experimentally, to find the discharge coefficient. b- An (ANN) model is to be developed for predicting the discharge coefficient as a function of variables related to the weir and opening geometry and flow properties. This model will be put in an easy to use Visual BASIC program to find the actual discharge for such hydraulic structures.
Chapter Two Theory & Literature Review
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CHAPTER TWO THEORY & LITERATURE REVIEW 2.1 Historical Review and Theory: Weirs are classified into broad-crested and sharp crested weirs according to its crest length along the flow path. Also it can be classified according to its opening cross-section, rectangular, triangular or trapezoidal, semicircular or even parabolic. A special type of trapezoidal weirs is the Cipolletti weir, which is a trapezoidal weir with side contraction 1:4. A properly built and operated weir of a given shape has a unique depth of water at the measuring station in the upstream pool for each discharge. Thus, weirs can be rated with respect to an upstream head relative to the crest elevation versus discharge, and equations or tables which apply to the particular shape and size weir can be generated. The crest overflow shape governs how the discharge varies with head measurements, (USBR, 2001).
2.1.1
Broad-Crested Weir:
2.1.1.1
Description:
( Bos, 1989) had mentioned that a broad-crested weir is an overflow structure with a horizontal crest above which the deviation from a hydrostatic pressure distribution because of centripetal acceleration may be neglected. In other words, the streamlines are practically straight and parallel. To obtain this situation the length of the weir crest in the direction of flow (L) should be related to the total energy head over the weir crest as (0.07 ≤ H1/L ≤ 0.50). (H1/L ≤
Chapter Two Theory & Literature Review
7
0.07), otherwise the energy losses above the weir crest cannot be neglected, and undulations may occur on the-crest; (H1/L ≥ 0.50), so that only slight curvature of stream lines occurs above the crest and a hydrostatic pressure distribution may be assumed. If the measuring structure is so designed that there are no significant energy losses in the zone of acceleration upstream of the control section, according to Bernoulli's equation (2-1): H1 = h1 + α V̅12/2g = H = y + α V̅2/2g ………………….. (2-1) Or: V̅ = {2g (H1 –y)} 0.50 α -0.50
…………………... (2-2)
Where H1 equals the total upstream energy head over the weir crest as shown in figure (2.1) Substituting Q =V̅A and putting α =1 .0 gives Q = A {2g (H1 –y)} 0.50
…………………... (2-3)
Provided that critical flow occurs at the control section (y =yc), a head-discharge equation for various throat geometries can be derived from equation (2-3) as: Q =AC {2g (H1 -yC)} 0.50
………………….. (2-4)
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Figure (2.1) Illustration of terminology for broad crested weirs (Bos, 1989) 2.1.1.2 Broad Crested Weir with Rectangular Control Section: For a rectangular control section in which the flow is critical, the area could be written (AC = bC yC ) and (AC/BC = yC ) so that Equation (2-1) may be written as : V̅2/2g =1/2 yC . Hence: yC =
………………….. (2-5)
H= H1
Substitution of this relation and ( AC =bC yC )into Equation (2-4). Gives, after simplification: Q=
g
.
b H
.
………………….. (2-6)
Figure (2.2) Dimensions of a rectangular control section (Bos, 1989) This formula is based on idealized assumptions such as : Absence of centripetal forces in the upstream and downstream cross-sections bounding the considered zone of acceleration, absence of viscous effects and
Chapter Two Theory & Literature Review
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increased turbulence and finally a uniform velocity distribution so that the velocity distribution coefficient can also be omitted. In reality these effects do occur and they must therefore be accounted for, by the introduction of a discharge coefficient Cd. The Cd-value depends on the shape and type of the measuring structure. Q= Cd
g
.
b H
.
………………….. (2-7)
Figure (2.3) Flow over a round-nose broad-crested weir with rectangular section (Bos, 1989). 2.1.2
Short Crested Weir:
2.1.2.1
Description:
The basic difference between a broad-crested weir and a short-crested weir is that for the short crest the curvature of the streamlines can be neglected; thus there is no hydrostatic pressure distribution. The two-dimensional flow pattern over a short-crested weir can be described by the equations of motion in the sand n-directions whereby the problem of determining the local values of v and r
Chapter Two Theory & Literature Review
10
is introduced. This problem, like those involved in three-dimensional flow, is not tractable by existing theory-and thus recourse must be made to hydraulic model tests.
.
Figure (2.4) Various Types of short-crested weirs (Bos, 1989).
Thus experimental data are made to fit a head-discharge equation which is structurally similar to that of a broad-crested weir but in which the discharge coefficient expresses the influence, of streamline curvature in addition to these factors explained in Section 2.1.1.2 In fact, the same measuring structure can act as a broad-crested weir for low heads (H1/L < 0.50), while with an increase of head (H1/L > 0.50) the
Chapter Two Theory & Literature Review
11
influence of the Streamline curvature becomes significant and the structure acts as a short-crested weir. For practical purposes, a short-crested weir with rectangular control section has a head-discharge equation similar to Equation (2-7), i.e. Q= Cd Cv
g
.
b h
.
………………….. (2-8)
Where; Cv is the coefficient of approach velocity of broad-crested weir, which is approximately equal to 1. 2.1.3
Sharp-Crested Weirs:
If the crest length in the direction of flow of a weir is short enough not to influence the head-discharge relationship of this weir (H1/L greater than about 15) the weir is called sharp-crested. In practice, the crest length in the direction of-flow is generally equal to or less than (0.002 m) so that even at a minimum head of 0.03 m the nape is completely free from the weir body after passing the weir and no adhered nape can occur (Bos, 1989) .If the flow springs clear from the downstream face of the weir, an air pocket forms beneath the nape from which a quantity of air is removed continuously by the overfalling jet. Precautions are therefore required to ensure that the pressure in the air pocket is not reduced; otherwise the performance of the weir will be subject to the following undesirable effects: a. Owing to the increase of under pressure, the curvature of the overfalling jet will increase, causing an increase of the discharge coefficient (Cd). b. An irregular supply of air to the air pocket will cause vibration of the jet resulting in an unsteady flow. Figure (2.5) shows the profile of a fully aerated nape over a rectangular sharp crested weir without side contractions as measured by (Bazin and
Chapter Two Theory & Literature Review
12
Scimeni, As mentioned in Bos, 1989 ) This figure shows that for- a sharpcrested weir the concept of critical flow is not applicable. For the derivation of the head-discharge equations it is assumed that sharp-crested weirs behave like orifices with a free water surface, and the following assumptions are made:
Figure (2.5) Profile of nape of a fully aerated two-dimensional weir (after Bazin 1896 and Scimeni 1930)as mentioned in (Bos, 1989). a. the height of the water level above the weir crest is h ≈ h 1 and there is no contraction; b. Velocities over the weir crest are almost horizontal; and c. The approach velocity head v12/2g is neglected. The velocity at an arbitrary point of the control section is calculated with the equation of Torricelli, (figure 2.6). V=√ g h +
V
………………….. (2-9)
−m
The total flow over the weir may be obtained by integration between the limits m = 0 and m = h1 Q=
g
.
∫
x h −m
.
dm
………………….. (2-10)
Chapter Two Theory & Literature Review
13
Where x denotes the local width of the weir throat as a function of m, After the introduction of an effective discharge coefficient, Ce , to correct the assumptions made, the general head-discharge equation of sharp-crested weir is: Q=C
g
.
∫
x h −m
.
dm
………………….. (2-11)
The reader should note that the assumptions made above deviate
somewhat from reality as shown in figure (2.5) and are even partly in contradiction with the velocity distribution as calculated by (Equation 2-10). In practice, however, (Equation 2-11) has proved to be satisfactory and is widely used throughout the world. Since, the effective discharge coefficient is almost constant, a different set of head-discharge equations will be derived below for various kinds of sharp-crested weirs.
Figure (2.6) Parameters of sharp- crested weir (after Bos, 1989).
In general sharp-crested weirs will be used where highly accurate discharge measurements are required, for example in hydraulic laboratories and industry. To obtain this high accuracy, provision should be made for ventilating
Chapter Two Theory & Literature Review
14
the nape to ensure that the pressure on the sides and surfaces of the nape is atmospheric. The downstream water level should be low enough to ensure that it does not interfere with the ventilation of the air pocket beneath the nape. Consequently, the required loss of head for modular flow will always exceed the upstream head over the weir crest (h1) by about 0.05 m, which is one of the major disadvantages of a sharp-crested weir, (Bos, 1989). 2.1.3.1
Sharp-Crested Weir with Rectangular Control Section:
2.1.3.1.1
Description:
A rectangular notch, symmetrically located in a vertical thin (metal) plate which is placed perpendicular to the sides and bottom of a straight channel, is defined as a rectangular sharp-crested weir. Rectangular sharp-crested weirs comprise the following three types: a. ‘Fully contracted weirs’, i.e. a weir which has an approach channel whose bed and walls are sufficiently remote from the weir crest and sides for the channel boundaries to have no significant influence on the contraction of the nape. b. ‘Full width weirs’, i.e. a weir which extends across the full width of the rectangular approach channel (B1/bc = 1.0). In literature this weir is frequently referred to as a rectangular suppressed weir or Rehbock weir. c. ‘Partially contracted weir’, i.e. a weir the contractions of which are not fully developed due to the proximity of the walls and/or the bottom of the approach channel. In general, all three types of rectangular weirs should be located in a rectangular approach channel (See figure 2.10 and 2.11). However, if, the approach channel is sufficiently large {B1(h1 + p1) ≥ l0bc h1} to render the velocity of approach negligible, and the weir is fully contracted, the shape of the
Chapter Two Theory & Literature Review
15
approach channel is unimportant. Consequently, the fully contracted weir may be used with non-rectangular approach channels. The sides of the rectangular channel above the level of the crest of a full-width weir should extend at least 0.3 hl max downstream of the weir crest. The fully contracted weir may be described by the limitations on (B1-bc, bc/B1 h1/p1, h1/ bc , h1, b1, and p1) as shown in table (2.1). Table (2.1) Limitation of a rectangular sharp-crested fully contracted weir.
A comparison of these limitations with those given in Section (2.1.3.1.3) shows that the fully contracted weir has limitations that are both more numerous and more stringent than the partially contracted weir and full width weir . For a rectangular control section, see (figure 2.7) x = bc = constant, (Equation 211) may be written as:
Or:
Q=C Q=C
.
g g
∫ .
x h −m
b h
.
.
dm
………………….. (2-12)
………………….. (2-13)
So, apart from a constant factor, Equation 2-13 has the same structure as
the head-discharge relation for a broad-crested weir with rectangular control section (Equation 2-7).
Chapter Two Theory & Literature Review
16
Figure (2.7) Dimensions of a rectangular control section. 2.1.3.1.2
Evaluation of Discharge:
As mentioned before, the basic head-discharge equation for a rectangular sharp-crested weir is as shown in equation (2-13). To apply this equation to fully contracted, full-width and partially contracted thin-plate weirs, it is modified as proposed by Kindsvater and Carter (1957) after (Bos, 1989), Q=C
g
.
b h
.
………………….. (2-14)
Where the effective breadth (be) equals bc+ Kb and the effective head (he) equals h1+Kh.
Figure (2.8) Rectangular sharp-crested weir (thin-plate weir).
Chapter Two Theory & Literature Review
17
Figure (2.9) General view of a sharp-crested rectangular weir .
The quantities Kb and Kh represent the combined effects of several phenomenons attributed to viscosity and surface tension. Empirically defined values for Kb as a function of the ratio bc/Bl are given in litrature and a constant positive value for Kh = 0.001 m is recommended for all values of the ratios b c/Bl and h1/p1. Ce is an effective discharge coefficient which is a function of the ratios bc/Bl and hl /p1 and can be determined from table (2.2). Table (2.2) Values for Ce as a function of the ratios bc/Bl and hl/pl (from Georgia Institute of Technology as mentioned by Bos,1989).
Chapter Two Theory & Literature Review 2.1.3.1.3
18
Limits of Application:
The following limitations were presented by (Bos, 1989): a. The practical lower limit of h1 is related to the magnitude of the influence of fluid properties and the accuracy with which h 1 can be determined. The recommended lower limit is 0.03 m; b. The critical depth will occur in the approach channel upstream from a weir if the ratio h1/p1 exceeds about 5. Thus, for values of h1/p1 greater than 5 the weir is not a control section as specified in Section 2.1.3. Further limitations on h1/p1 arise from observational difficulties and measurement errors. For precise discharge measurements the recommended upper limit for h1/p1 equals 2.0, while p1 should be at least 0.10 m; c. The breadth (bc) of the weir crest should not be less than 0. 15 m; d. To facilitate aeration of the nape the tail water level should remain at least 0.05 m below crest level. 2.1.3.2
V-notch Sharp-Crested Weirs:
2.1.3.2.1
Description:
A V-shaped notch in a vertical thin plate which is placed perpendicular to the sides and bottom of a straight channel is defined as a V-notch sharp-crested weir. The V-notch sharp-crested weir is one of the most precise discharges measuring devices suitable for a wide range of flow. The weir is shown diagrammatically in figures (2.10) the following flow regimes are encountered with V-notch sharp-crested or thin-plate weirs: a. ‘Partially contracted weir’, i.e. a weir the contractions of which along the sides of the V-notch are not fully developed due to the proximity of the walls and/or bed of the approach channel.
Chapter Two Theory & Literature Review
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b. ‘Fully contracted weir’, i.e. a weir which has an approach channel whose bed and sides are sufficiently remote from the edges of the V-notch to allow for a sufficiently great approach velocity component parallel to the weir face so that the contraction is fully developed. These two types of V-notch sharp-crested weirs can be classified by the following limitations on (h1/p1, h1/B1, hl, p1 and B1). It should be noted that in this classification, fully contracted flow is a subdivision of partially contracted flow.
Figure (2.10) V-notch sharp-crested weir.
Table (2.3) Classification and limits of application of V-notch sharp-crested weirs (after ISO 1971, France, as quoted by Bos, 1989).
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From this table it appears that from a hydraulica1 point of view a weir may be fully contracted at low heads while at increasing h, it becomes partially contracted. The partially contracted weir should be located in a rectangular approach canal (Bos, 1989). The fully contracted weir may be placed in a non-rectangular approach channel provided that the cross-sectional area of the selected approach channel is not less than that of the rectangular channel as prescribed in table (2.3). 2.1.3.2.2
Sharp-Crested Weir with Triangular Control Section:
For a triangular control section, (figure 2.11), x = 2m tan Ɵ/2, and (Equation 212) may be written as:
Or:
Q=C Q=C
g
.
∫
g
.
θ
[ tan ] m h − m θ
tan h
.
.
dm ………………….. (2-15) ………………….. (2-16)
Figure (2.11) Dimensions of a triangular control Section.
Chapter Two Theory & Literature Review 2.1.3.2.3
21
Evaluation of Discharge:
The basic head-discharge equation for a V-notch sharp crested weir is Q=C
.
g
θ
tan h
.
………………….. (2-17)
To apply this equation to both fully and partially contracted sharp-crested weirs, it is modified to a form proposed by (Kindsvater and Carter, 1957), as mentioned in (Bos, 1989) Q=C
.
g
θ
tan h
.
………………….. (2-18)
Where Ɵ equals the angle induced between the sides of the notch and h e is the effective head which equals h1 + Kh. The quantity Kh represents the combined effects of fluid properties. Empirically defined values for Kh as a function of the notch angle (Ɵ) are shown in literature. For water at ordinary temperature, i.e. 5°C to 30°C (or 40°F to 85°F) the effective coefficient of discharge (Ce) for a V-notch sharp-crested weir is a function of three variables
Ce = f [
,
, θ]
………………….. (2-19)
If the ratios h1/p1 ≤ 0.4 and h1/B1 ≤ 0.2, the V-notch weir is fully contracted and Ce becomes a function of only the notch angle Ɵ, as illustrated in literature. 2.1.3.2.4
Limits of Application:
The limits of application of the Kindsvater and Carter equation for V-notch sharp crested weirs are as mentioned in (Bos, 1989): a. The ratio h1/p1 should be equal to or less than 1.2; b. The ratio h1/B1 should be equal to or less than 0.4;
Chapter Two Theory & Literature Review
22
c. The head over the vertex of the notch h, should not be less than 0.05 m nor more than 0.60 m; d. The height of the vertex of the notch above the bed of the approach channel (p1) should not be less than 0. 10 m; e. The width of the rectangular approach channel should exceed 0.60 m; f. The notch angle of a fully contracted weir may range between 25 and 100 degrees. Partially contracted weirs have a 90-degree notch only; g. The tail water level should remain below the vertex of the notch. 2.1.3.3
Sharp-Crested Weir with Trapezoidal Control Section:
The head-discharge relation for a trapezoidal control section as shown in (Figure 2.20) is obtained by superimposing the head-discharge equations for a rectangular and triangular control section respectively, resulting in
Q=C
g
.
[b +
θ
h tan ] h
.
………………….. (2-20)
Figure (2.12) Dimension of a trapezoidal control section.
Chapter Two Theory & Literature Review 2.1.3.4
Cipolletti Weir:
2.1.3.4.1
Description:
23
A Cipolletti weir is a modification of a fully contracted rectangular sharpcrested weir and has a trapezoidal control section, the crest being horizontal and the sides sloping outward with an inclination of 1 horizontal to 4 vertical (figure 2.13). Cipolletti (1886) assumed that, due to the increase of side-contraction with an increasing head, the decrease of discharge over a fully contracted rectangular sharp-crested weir with breadth bc would be compensated by the increase of discharge due to the inclination of the sides of the control-section. This compensation thus allows the head-discharge equation of a full width rectangular weir to be used. It should be noted, however, that experiments differ as to the degree to which this compensation occurs.
Figure (2. 13) Definition sketch of a Cipoletti weir.
2.1.3.4.2
Evaluation of Discharge:
The basic head-discharge equation for the Cipoletti weir is the same as that of a rectangular fully contracted weir. Hence Q=C C
g
.
b h
.
………………….. (2-21)
Chapter Two Theory & Literature Review
24
Where, within certain limits of application, the discharge coefficient C d equals 0.63.The accuracy of the discharge coefficient for a well maintained Cipoletti weir is reasonable for field conditions. The error in the product CdCV is expected to be less than 5%. 2.1.3.4.3
Limits of Application:
The limits of application of the (fully contracted) Cipoletti weir are as mentioned in (Bos, 1989): a. The height of the weir crest above the bottom of the approach channel should be at least twice the head over the crest with a minimum of 0.30 m; b. The distance from the sides of the trapezoidal control section to the sides of the approach channel should be at least twice the head over the crest with a minimum of 0.30 m; c. The upstream head over the weir crest h, should not be less than 0.06 m nor more than 0.60 m; d. The ratio h1/bc should be equal to or less than 0.50. e. To enable aeration of the nape, the tail water level should be at least 0.05 m below provided the Cipoletti weir conforms to the above limits of application, it may be placed in a non-rectangular approach channel. 2.2
Review of Available Related Researches: As mentioned in chapter one the problem of sediments accumulation at the
upstream side of weirs can be reduced through the use of combined weirs and gated or un-gated opening at the bottom. Many researches had been conducted to investigate the combined simultaneous flow through the weir section and the bottom opening.
Chapter Two Theory & Literature Review
25
(Al-Hamid, et.al., 1996-a) had investigated a discharge equation for simultaneous flow over contracted weir with bottom triangular opening. The discharge for such simultaneous flow involves few parameters: the distance between the weirs, the slope of the apron, the opening depth and weir angle. In this paper one generalized equation including all the important variables was obtained from an experimental investigation. This equation is suitable for both horizontal and sloping channel bed under submerged flow and for the free flow, the tail-water depth / weir opening ratio term being neglected. The predictions of the equation agreed well with the experimental data. Figure (2.14) shows the definition sketch for free flow below an inverted triangular bottom opening and over a sharp-crested rectangular weir system. The conventional flow equation for contracted sharp-crested rectangular weir is given by: Q = C
g
.
b − . nh h
.
………………….. (2-22)
Figure (2.14) Definition sketch for free flow below an inverted Triangular and over a sharp-crested rectangular weir system (Al-Hamid, et.al., 1996-a).
Chapter Two Theory & Literature Review
26
Where Qw is the discharge over the weir; Cw is the discharge coefficient; b is the weir width; n is the number of side contraction; h is the upstream head and g is the acceleration due to gravity. The researchers presented a functional relationship for the non-dimensional discharge for the free flow case as follows: g
Q .
d
y h b b h ( , , , , ) d d b d b
=
.
…………
−
Where Qa is the actual discharge and the other variables are as shown in (figure 2.14). During the simultaneous underflow and overflow for the combined contracted sharp-crested weir system, the discharge was found to be affected by a number of variables (equation
−
). A general non-dimensional equation
for predicting the discharge through the combined weir (over a rectangular weir and a triangular weir below) can, therefore, be written as: a- Submerged flow: g
Q .
d
.
= C +C
y h b b +C +C +C d d b d
Where C1 through C7 are constants.
+C
+C
………..
−
b- Free flow: g
Q .
d
.
= C +C
y h b +C +C d d b +C
+C
………..
−
The values of C1 through C7 for the submerged flow and C1 through C6 for the free flow cases were determined using multiple linear regression analysis of the experimental data. The effect of the upstream head, channel bed slope, weir
Chapter Two Theory & Literature Review
27
angle and overflow head over the weir on the variation of the combined discharge is presented below, in figures (2.15, 2.16, 2.17, and 2.18):
Figure (2.15) Effect of upstream head ratio, H/d, and the bed slope, So , on the non-dimensional free discharge ratio for typical two sets of data with Ɵ = 90°(Al-Hamid, et.al., 1996-a).
Figure (2.16) Effect of overflow head ratio, h/d, and the angle of the inverted triangular weir, Ɵ, on the non-dimensional free discharge ratio for constant y/d and b = 12 cm (Al-Hamid, et.al., 1996-a).
Chapter Two Theory & Literature Review
28
Figure (2.17) Verification of Prediction equation (2-24) for submerged flow at b1/d=2 (Al-Hamid, et.al., 1996-a).
Figure (2.18) Verification of Prediction equation (2-25) for free flow(Al-Hamid, et.al, 1996-a).
The researchers had concluded the followings: 1.
The channel bed slope has a negligible effect on the discharge variation through the combined weir system.
Chapter Two Theory & Literature Review 2.
29
The influence of angle of triangular weir below or (triangular sluice gate) has a significant effect on the combined discharge through the weir system as the bigger the angle, the larger the discharge resulted.
3.
Non-dimensional equations for the combined discharge over a contracted sharp-crested and an inverted triangular weir below under free and submerged flow conditions were obtained. Using the equations (2-24) and (2-25) the discharge through the combined weir system on both horizontal and sloping floors can be computed accurately.
(Al-Hamid, et.al., 1996-b) had investigated the hydraulics of a triangular weir with bottom rectangular opening. Different models with different geometric combinations were tested. These geometries include, gate opening, gate length and V-notch angle. Experiments were conducted for free gate flow (unsubmerged) conditions on horizontal and sloping channels. Results showed that flow passes through the device is affected by the device geometry and the flow parameters. Semi-empirical discharge equation was developed. The equation represents the collected experimental data well with an absolute error less than 4%. The combined flow over a sharp crested v-notch weir and beneath a contracted bottom opening is sketched in figure (2.18). For upstream flow depth below notch crest, only sluice gate flow exists. For this condition and from energy principle, the theoretical flow passes through the gate can be written as: Q G = ZG b√ gH
…………………..
−
Chapter Two Theory & Literature Review
30
In which QG =the theoretical discharge of the sluice gate, ZG= opening of the sluice gate, b= breadth of the sluice gate, H = the upstream flow depth and g = the gravitational acceleration. As upstream flow depth exceeds the weir crest, combined flow conditions occurs (i.e. over the weir and under the gate) with a total theoretical discharge of: Q
T
= ZG b√ gH +
tan
θ
5
√ g h ⁄ …………………..
−
Figure (2.19) Definition sketch for the simultaneous flow over weirs and below gates (Al-Hamid, et.al., 1996-b).
In which QCT = the theoretical combined discharge, Ɵ = the angle of the V-notch and h = is the flow depth above the weir crest. The second term of Equation (2-27) is the well-known V-notch discharge equation. Introducing a coefficient of discharge, Cd, to account for the difference between the simplified theory, the actual discharge, QCA-may be written as: Q
= C ZG b√ gH +
θ 5 tan ( ) √ gh ⁄
…………..
−
Rearranging and expressing in dimensionless form, Equation (2-28) becomes:
Chapter Two Theory & Literature Review CVG =
Q
ZG b√ gH
31
=C
+
θ
tan ( )
h
5⁄
ZG b √ H
………..
−
In which CVG =is a defined discharge coefficient. It is expected that Cd is dependent on the geometry of the device as well as the flow conditions (i.e. Z G, Zv, b, H, B, and Ɵ) and hence it follows that CVG is also a function of the geometry of the boundary and the flow conditions. A better understanding may be obtained by utilizing dimensional analysis. The actual combined discharge can be written in functional form as: Q
= φ g, ρ, μ, σ, ZG , ZV , b, B, θ, H, h, ∆Z, S
………..
−
In which ρ = the water density, μ = the water viscosity, σ = the water surface tension, and ZV =the height of the weir crest. Using the principle of the
dimensional analysis, Equation (2-31) is obtained: CVG =
Q
√gH
5⁄
= φ (R , W ,
ZG ZV b h ∆Z , , , θ, , , S ) … … . . H H B b H
−
In which Re = Reynolds number = g. H 3/2 / (μ / ρ), We = Weber number = H. g / (σ / ρ), and CVGD = discharge coefficient from dimensional analysis. For water at a specific temperature, ρ, μ, and σ are constant. Hence, Re and We can be represented by one dimensional variable H. Thus, Equation (2-31) is reduced to; alternatively (2-32) may be re-written in the following form: CVG =
Q
√gH 5⁄
= φ (H,
ZG ZV b h ∆Z , , , θ, , , S ) … … . . H H B b H
−
Chapter Two Theory & Literature Review CVG =
Q
√gH 5⁄
=φ (
32
ZG Z V − ZG b H − ZV , , , θ, ,S )…….. ZV H B B
−
Comparing Equation (2-33) with Equation (2-29) it can be seen that: ZG b CVG = √ CVG [ ] H
………………………..……..
−
Z G ZV − ZG b H − ZV , , , θ, ,S ).. ZV H B B
−
Apparently, Equation (2-33) may also be written as: CVG =
Q
ZG b√ gH
=φ (
The parameters of the proposed device played an important role in the discharge values and have a complex behavior. Using non-linear statistical analysis and based on Equation (2-35) may forms of equations for the discharge coefficient, CVG, were tried by introducing different combinations of the parameters and evaluating the effect of these combinations on the estimation of the combined discharge coefficient. Out of these trials an equation was obtained for discharge coefficient CVG in the form:
CVG =
[
.
+
.
ZV − ZG H
tan − .
θ
.
ZG ZV
.
H − ZV B b B
. .
]
…..
−
This equation estimates CVG with maximum error less than 5% in which more than 90% of the data points have an error less than 3% figure (2.20) shows the computed and the actual values of the discharge coefficient CVG in which a good fit is observed.
Chapter Two Theory & Literature Review
33
Figure (2.20) Comparison between measured and computed CVG values (Al-Hamid, et.al., 1996-b).
(Negm, 1998) investigated the characteristics of combined flow over contracted weir and below submerged gates with unequal contractions. This paper presents the results of an experimental study on the characteristics of simultaneous flow over contracted weirs and below contracted submerged gates with unequal contractions. Both the edges of the weirs and gates are sharped. The effects of the different parameters, as derived from the dimensional analysis, on the combined discharge are presented and discussed. A prediction equation relating the non-dimensional combined discharge to both the flow and geometry parameters is developed. The predicted combined discharges agreed well with the experimentally measured ones. It was found that the combined discharge depends upon the ratios b1/b, b/d, b1/d, b/B, b1/B, y/d, H/d and ht/d. (b1 and d is the width and depth of lower opening, b is the width of the upper opening, B is the flume width, y is solid distance between upper and lower openings, h t the tail water depth downstream the device and H is the upstream head). The obtained
Chapter Two Theory & Literature Review
34
semi-empirical discharge prediction equation estimates the combined flow reasonably well. Figure (2.21) shows a definition sketch for the combined submerged flow below a contracted gate and the free flow over sharp crested contracted rectangular weir. Assuming that (i) viscosity, surface tension and velocity of approach are of minor importance. (ii) One coefficient of discharge, Cd, can be applied for overflow as well as for underflow. Applying the dimensional analysis principle to this problem, yielded the following nondimensional relationship: QT =
g
Q .
d
.
=
(
H+h h b b b y , , , , , , S) d b d B b d
….……
−
Where QT is a non-dimensional discharge term (QT = Qc/(2g)0.5 d2.5), Qc is the combined flow over the weir and below the gate, g is the acceleration due to gravity, d is the depth of gate opening, y is the solid distance between the top of the gate and the crest of the rectangular weir, b 1 is the width of the gate opening, b is the width of the rectangular weir, h is the head over the weir measured upstream at about three times the maximum head over the weir, B is the flume width and S is the submergence ratio (S=ht/d) with ht being the tail water depth. The flow equation for the contracted sharp crested weir can be written as: Q = C √ g b− . h h
.
……………….…………
−
Where Qu is the discharge passing over the weir and Cu is the coefficient of discharge of the weir. Also, the flow equation for the gate of a contracted opening with sharp edges may be expressed as: Q L = CL b d√ g√d + y + h − h … … … … … … . … … … …
−
Chapter Two Theory & Literature Review
35
Where QL is the discharge passing below the gate, h d is the water depth just downstream of the gate, and CL is the coefficient of discharge for the gate.
Figure (2.21) Definition sketch for combined flow over weir and below submerged gate (Negm, 1998). If one coefficient of discharge, Cd, applies for combined flow, an equation for simultaneous overflow-underflow may be deduced by adding Equation (2-38) to Equation (2-39): Q = QL + Q
= C (b d√ g√d + y + h − h + √ g b− . h h . )
………..
−
From which the combined flow discharge coefficient, C d, can be calculated as follows: C =
b d√ g√d + y + h − h
Q
+ √ g b− . h h
.
…..
−
Chapter Two Theory & Literature Review
36
The multiple regression analysis is used to develop a prediction equation. The following equation is found to fit the well with R2=0.9824 and standard error of estimate of 0.296. The overall F-test for the equation is 1121.27 and Durbin-Watson statistic is 1.406. QT = − .
+ .
− .
H+h b + . − . d d H b y + . + . d b d
The prediction of Equation ( −
S+ .
b − . B
…………..…..
−
h b
) is plotted versus all the experimental
data in Figure (2.22) which shows reasonably good agreement be predicted and observed QT.
Figure (2.22) Prediction of Equation (2-42) versus the experimental data (Negm, 1998).
Chapter Two Theory & Literature Review
37
The following are the main conclusions deduced by (Negm, 1998): 1. The combined non-dimensional discharge, QT, increases with the increase of the combined flow parameter (H+h)/d. 2. QT decreases with the increase of submergence at the same (h+h)/d while at the same QT, the upstream head increases with increase of S. 3. At the same submergence ratio, QT is higher for larger both model ratio (b1/b) and geometry ratio (b1/d). 4. At the same obstruction ratio, QT increases with the increase in geometry ratio, b1/d, lower contraction ratio, b1/B, and model ratio (or widths ratio), b1/b. 5. QT increases with the increase in the obstruction ratio, geometry ratios, and submergence ratio when contraction ratios and model ratio are constants for which the data can be represented by a single line. 6. The contractions ratios (b1/B, b/B) and model ratio (b1/b) are of major importance compared to obstruction ratio (y/d) and geometry ratios (b 1/d and b1/d). 7. The combined flow is of complex nature and many parameters affect the flow through then device and when most of the parameters vary at a time, it becomes too difficult to withdraw any specific conclusion. 8. A non-dimensional prediction equation (2-42) is developed to predict the combined discharge through the combined flow system. The equation is only valid within the limitations of the experimental work.
Chapter Two Theory & Literature Review
38
(Negm et.al., 2002) investigated the combined –free flow over weirs and below gates of equal contraction. The results of an experimental investigation on the characteristics of the combined flow over contracted sharp-crested rectangular weirs and below contracted sharp-crested rectangular gates are presented. The experiments are carried out in a laboratory flume using various geometrical dimensions under different flow conditions. The basic principles were employed to correlate the discharge to the relevant geometrical and hydraulic parameters in non-dimensional form. The experimental data were then used to develop a general non-dimensional equation for predicting the discharge through the combined system knowing its geometry and the head of water over the weir. Figure (2.23) shows a definition sketch for the free flow over a contracted sharp-crested weir combined with contracted sharp crested rectangular gate. Assuming that one coefficient of discharge, Cd, can be applied to the combined flow, the discharge equation may be obtained by adding the discharges over the weir and the gate as: …………..…..
−
Where qc is the combined discharge per unit width (qc = Qc /b), g is the acceleration due to gravity, d is the gate opening, H = (d+y+h) is the upstream water depth measured upstream of the gate at about 4 to 5 the maximum head on the weir (about 40 cm), y is the vertical distance between the gate top and the weir bottom and h is the head over the weir.
Chapter Two Theory & Literature Review
39
Figure (2.23) Definition sketch for combined free flow over weirs and below gates. (a) Cross section, (b) longitudinal section (Negm et.al., 2002).
Based on Equation (2-43) and using dimensional analysis, the following functional relationship obtains …………..…..
−
In which B is the width of the flume, RN is Reynolds number and WN is Weber Number. Alternatively, in terms of Cd Equation (2-44) can be written as: …………..…..
−
The relationship between the tail water depth ratio, h t/d, and the upstream head ratio, H/d, is shown in figure (2.24-a). Clearly, ht/d increases by increasing H/d and the variation between the two parameters depends mainly on b/d and y/d. The proper use of the proposed device for flow measurements necessitates that the tail water depth be within the allowable limiting tail water depth which was estimated using equation (2-46) with R2=0.944 and SSE=0.153.
Chapter Two Theory & Literature Review
40
…………..….. (2-46)
Thus, before using Equation (2-47), ht/d is estimated from Equation (246). If they agree up to ±5%, equation (2-47) may be used to estimate the combined discharge with an error of about ±5%. Figure (2.24-b) shows the comparison between measured ht/d and the prediction according to equation (246).
Figure (2.24) Modular limit for combined flow on horizontal bed, (a) relationship between limiting tail-water ratio and upstream head ratio and (b) Equation (2-46) versus measurements (Negm et.al., 2002).
Chapter Two Theory & Literature Review
41
……..….. (2-47) The researchers had concluded that: The flow parameter H/d and the geometrical parameter y/d have major effects on the discharge while the other parameters are insignificant. The developed equation agrees well with observations within the limitations of the experimental work. (Negm, 2002) investigated the modeling of submerged simultaneous flow through combined weirs and gate devices of un-equal contractions. “In the first part of this paper, the effect of hydraulic and geometrical parameters on the simultaneous discharge was discussed and non-dimensional prediction discharge equation for the simultaneous flow was developed. In this study, a generalized discharge model is proposed based on using the known equations of weirs and gates. The proposed equation is calibrated using large series of experimental data for devices having opening of unequal contractions under both free and submerged flow conditions. The equation is based on the use of a factor that accounts for the interaction effect of the overflow-underflow and the use of the available weir and gate equations. The predictions of the proposed model agree well with the observations with a deviation of less than ± 5 for about 90% of the data. Also, the artificial neural networks are used to develop a prediction model for the simultaneous discharges. The ANN model is trained validated and tested using the collected experimental data. The correlation coefficient and mean relative error are used to evaluate the performance of each model. The researcher had provided the following discharge prediction model ……..….. (2-48)
Chapter Two Theory & Literature Review
42
In which b1 is the width of the bottom opening (gate) where its depth is d, b is the width of the weir, h is the head over the weir, Q is the simultaneous discharge, g is the gravitational acceleration, Ao and A1 are regression coefficients that depend on the model ratio, b1/b only. Equation (2-48) was developed for the following limitations: 0.46 ≤d/y ≤ 2.14, 0.39 ≤ b/d ≤ 1.62, 0.25 ≤ b1/b ≤ 4, 0.13 ≤ b/B≤ 0.53 and 0.08≤ h/d ≤ 0.975. On the other hand, the characteristics of the simultaneous flow over weirs and below submerged gates with unequal contraction were presented (Negm, 2000 as mentioned in Negm, 2002). The following discharge prediction regression model was provided (R2=0.9824 and SSE=0.296)
(2-49) In which S is the submergence ratio (S= tail water depth / depth of gate opening), H is the head on the gate, B is the width of the flume and y is the obstruction distance, y=P-d, with P is the height of the weir. Equation (2-49) is valid within the following limitations:
Combining the data used to develop equation (2-48) for free flow and that used to develop equation (2-49) for submerged flow, the researcher suggests the following general equation (R2=0.973 and SSE=0.324) for both free and submerged flow provided that S is assumed to be unity for free flow.
…(2-50)
Chapter Two Theory & Literature Review
43
The predictions of equations (2.48), (2.49) and (2.50) are comparable. (Negm, 2002) had concluded the followings: 1- The present and previously developed prediction regression models produce comparable prediction for the simultaneous discharge over weirs and below gates. 2- The results of the developed ANN model for predicting the simultaneous discharge compare well with experimental data.
(Hayawi et.al, 2008) investigated free combined flow over a triangular weir and under rectangular gate. The main objective of this investigation is to find the characteristics of free flow through the combined triangular weir and a rectangular gate. Nine combined weirs were constructed and tested for three different triangular angles (θ = 30o, 45o and 60o) over a rectangular gate .The distance between the weir and the gate were changed three times for each angle (y = 5, 10 and15) cm. The coefficient of discharge (Cd) were found to be inversely proportional to the weir angle (θ) and with (B/h, y/h, D/h) while the theoretical discharge (Qtheo) were inversely proportional with (B/h, y/h, D/h) and directly proportional to the distance (y) between the weir and gate. A general expression were obtained between Froude number ( Qth o .5 ) √
.The estimated value of (
Qth o .5 √
) and (B/h, y/h, D/h)
from the expression were plotted against the
calculated value and it was found to be good. To compute the discharge through a combined weir (V-notch with sharp crested rectangular gate) the following equation was obtained by the researchers by adding the discharge over the weir and gate as:
Chapter Two Theory & Literature Review Q
44
=Q +Q
Q = √ gH Q =
Q�
Where:
�
=
.
………………
BD
θ √ g tan h √
[�
………………
.
.
+
� tan ℎ
……………… .
]
………
−
−
−
−
H: Total H = h + y + D.
(L)
Head; h: Head of water through weir.
(L)
y: Vertical distance between the lower edge ofthe weir and the upper edge of the gate opening
(L)
D: The gate height.
(L)
B: The gate width.
(L) (LT-2)
g: Acceleration due to gravity . θ :V-notch angle.
(Degree)
Cd: Coefficient of discharge. Qg: Discharge through the gate.
(L3T-1)
Qw: Discharge through the triangular weir.
(L3T-1)
Qtheo: Total theoretical discharge.
(L3T-1)
Qact: Total actual discharge.
(L3T-1)
Based on equation (2-54) and using dimensional analysis, the following functional was obtained by the researchers: Q
Q
√g h
= .
=
h, y, B, D, ρ , μ, g. σ , θ
y B D μL σh , , , θ, , h h h Qρ Q ρ
………………
………………
−
−
Chapter Two Theory & Literature Review Hence:
μL
Qρ
45
Is a form of Reynolds number (Re) and Qth o ) .5 √
(Weber number (We), and because ( y B D ( , , , θ, � , � ) h h h
�� =
σ
Q ρ
is a form of
is a type of Froud number, then: ………………
−
Where: Fr: Froude number. At high discharges the effect of surface tension
and viscosity will be neglected so Weber number and Reynolds number had been neglected.
(Hayawi, et.al, 2008) concluded the followings: 1- Cd decreases as (θ) increase. 2- Cd decreases as (B/h, y/h and D/h) increases. 3- Qtheo increases as (B/h, y/h and D/h) decreases. 4- Qtheo increases as (y) increases. Qth o .5 ) √
5- The developed equation agrees well with the measured values of ( the combined weir.
for
6- A model was predicted from the measured parameters as: Q
√g h
.
= .
y h
.
B ( ) h
.
D ( ) h
.
………………
−
(Saman and Mazaheri, 2009) investigated combined flow over weir and under gate. Models of sharp-edged weirs and gates with no lateral contraction are combined. To calibrate and validate the proposed model, experiments have been carried out in a laboratory flume applying different submergence conditions. It
Chapter Two Theory & Literature Review
46
was found that the model is able to predict the stage–discharge relationship with reasonable accuracy. The researchers (Saman and Mazaheri, 2009) had concluded that the results of the proposed model show good agreement between calculated and measured discharges implying reasonable precision, as shown in figure (2.25):
Figure (2.25) All calculated and measured discharges with in all experiments in semi-submerged and fully submerged flow (Saman and Mazaheri, 2009).
As cited here above there are no experimental or theoretical studies available in literature for a combination of a Cipolletti weir with a rectangular bottom opening. Hence, in this research an experimental study for such structure is conducted, and a prediction model for estimating the discharge coefficient and the actual flow of such structure is developed.
Chapter Three Experimental Work
47
CHAPTER THREE EXPERIMENTAL WORK 3.1
Introduction: As mentioned the aim of this study is to investigate the hydraulics of
Cipolletti weir with rectangular opening at the bottom. Hence an experimental setup was built for this purpose. The details of this setup and the conducted experiments will be explained in this chapter.
3.2
Experimental Flume: For conducting the experiments on the proposed Cipolletti weir models
with a rectangular opening at the bottom, a flume of conventional size is needed. The conventional size is the size that can allow easy measurements of flow depth and permit good visualization of the flow pattern over the weir and at the downstream side from the bottom opening. The Flume used for conducting the experiments is available in the laboratory of the College of Engineering, University of Sulaimani, where the study is intended to be conducted, was built by (Amanj,2009). Figure (3.1) shows a schematic diagram of the flume and the other facilities, such as: storage tank, recirculation tank, pump, ultrasonic flow meter, point gauges and valves. The flume form is fabricated from steel sections such as iron angles and channel sections. The flume bottom and sides are made of Perspex (10mm thickness). Figure (3.2) shows the details of its steel frame.
7. 8. 9. 10. 11.
(8)
Storage tank. Experimental flume. Cipolletti weir model with bottom opening. Point Gauge No.(1) Point Gauge No.(2)
(10)
(9)
1. 2. 3. 4. 5. 6.
(5)
(7)
(Amanj, 2009)
Discharge valve. Ultrasonic flow meter. Pump. Bypass valve. Recirculation tank. Controlled gate.
(4) (3)
(6)
(2)
(11)
(1)
Chapter Three Experimental Work 48
Chapter Three Experimental Work
49
Chapter Three Experimental Work
50
Figure (3.3) Photos of the experimental flume setup (Amanj, 2009)
The flume section is rectangular with 60cm width which is considered acceptable in accordance to (USBR, 2001)specification ,and two different heights, 135cm height at the inlet (upstream) side with a length of 2.5m, and 70cm height at the downstream side with a length of 5.5m. Figure (3.2) shows these sizes with the location of the weir model. 3.3 Equipment The flume is provided with some equipment in order to control the water levels and the flow measurements.
Chapter Three Experimental Work
51
3.3.1 Ultrasonic Flow Meter (Digital Flow Meter) In order to measure the flow values an ultrasonic flow meter is needed and was installed as shown in Figure (3.1). This equipment was specially imported for the purpose of the study. The flow meter was manufactured specially according to the request of the researcher (Amanj, 2009) and it is important to note that it is calibrated by the producing company according to the required conditions and flow range of the experiment, which ranges from 0.9 to 230 m3/h. Figure (3.4) shows the flow meter, magnetic rod is supplemented with the meter to change its modes from volumetric to discharge or vice versa.
Figure (3.4) Ultrasonic flow meter SCL-61D1-DN80. Note: d-Diameter of flange connecting holes, n-Number of flange connecting holes, D1-Flange connecting the holes center diameter.
Chapter Three Experimental Work
52
Figure (3.5) Photo of ultrasonic flow meter SCL-61D1-DN80. 3.3.2 Pump The pump is provided from the local market, and has the following specifications; Type CM80-160C Pentax water pump which is made in Italy, (Q l/min) 1100-3250, H (m) 29.6-16.4, HP 20,KW 15, kWabs 14.9, Hz 50, A 25.8-15, Electro pump three phase, figure (3.6) shows the pump.
Figure (3.6) Pentax water pump.
3.3.3 Point Gauges The point gauges were provided originally for small flume that available in the laboratory. The specifications of the point gauges are:
Chapter Three Experimental Work
53
The scale precision is 1mm, original scaling 30cm depth of water measuring. Being not suitable for the fabricated flume, the point gauges were modified for this purpose. The modifications included base enlargement and needle length elongation. One of the point gauges is used for measuring head over the weir crest (point gauge No.1), while the other is used for measuring the depth before the weir at the upstream channel (at a distance from the crest equal to minimum 3 height of water above the crest) (point gauge No.2), figure (3.7) shows this equipment.
Point gauge No.1
Point gauge No.2
Figure (3.7) ARMFIELD point gauge.
3.3.4 Storage and Recirculation Tanks The storage and recirculation tanks are manufactured using 2mm thickness steel plates, and welding is used; Figure (3.8) shows a photo of the two tanks.
Chapter Three Experimental Work
54
Figure (3.8) Storage and recirculation tanks. 3.3.5 Piping System: The piping system is used to connect the flume with the tanks. The pump and the flow meter consist of the followings galvanized steel items. Figure (3.9) shows the piping system connections.
Figure (3.9) Piping system connections.
Chapter Three Experimental Work 3.4
55
Physical Model of the Proposed Weir Section: In order to investigate the hydraulics of Cipolletti weir with bottom
rectangular opening, different configurations were used in this experimental work. Table (3.1) shows the details of these configurations. Figure (3.10) shows a general setup of the proposed weir that covers all the configurations stated in table (3.1) with different dimensions. The models were made using perplex sheets of (10) mm thickness as shown in figure (3.11).
Chapter Three Experimental Work
56
Table (3.1) Different configurations used in the experimental work.
3.c
3.b
3.a
2.c
2.b
2.a
1.b
1.a
Configurations
Crest height (P= ho +d) cm
Crest length (Bw) cm
Bottom Opening (bo x ho) cm
1.a.1
15 x 5
1.a.2 1.a.3 1.a.4 1.a.5 1.b.1
15 x 10 15 x 15 10 x 10 10 x 15 15 x 5
1.b.2 1.b.3 1.b.4 2.a.1 2.a.2 2.a.3 2.a.4 2.a.5 2.b.1 2.b.2 2.b.3 2.b.4 2.b.5 2.c.1 2.c.2 2.c.3 2.c.4 2.c.5 3.a.1 3.a.2 3.a.3 3.a.4 3.a.5 3.b.1 3.b.2 3.b.3 3.b.4 3.b.5 3.c.1 3.c.2 3.c.3 3.c.4 3.c.5
31
22
31
32
27
40
27
30
27
20
23
28
23
23
23
23
15 x 10 10 x 10 10 x 15 15 x 5 15 x 10 15 x 15 10 x 10 10 x 15 15 x 5 15 x 10 15 x 15 10 x 10 10 x 15 15 x 5 15 x 10 15 x 15 10 x 10 10 x 15 15 x 5 15 x 10 15 x 15 10 x 10 10 x 15 15 x 5 15 x 10 15 x 15 10 x 10 10 x 15 15 x 5 15 x 10 15 x 15 10 x 10 10 x 15
Chapter Three Experimental Work
Figure (3.10) General schematic diagram of the physical models.
Where: hw : Height of Cipolletti weir. Bw: Bed width of Cipolletti weir. ho : Height of rectangular bottom opening. bo : Width of rectangular bottom opening. P : Height of the weir and equal to (ho+d)
57
Chapter Three Experimental Work
58
Figure (3.11) Different configurations of the Cipolletti weir with rectangular bottom opening model.
Chapter Three Experimental Work 3.5
59
Modifications Made on the Flume:
As stated previously the flume built by (Amanj, 2009), and it was used for his experimental work. However some modifications was done to control the flow and assure different flow conditions for the models, of Cipolletti weir mentioned above. For controlling the flow, two modifications were made: 1. Two Galvanize channel were fabricated with special rubber material (which is used for car windows), fixed at the internal walls of the flume using screws and silicon joint filler at a location along the flume (near the midpoint of the flume) suitable for measurements required, as shown in figure (3.12). These Galvanized channels are used for fixing the Cipolletti weir models.
Figure (3.12) Galvanized channels fabricated for installing the weir models.
Chapter Three Experimental Work
60
2. A perplex movable gate was fabricated with two galvanized channel with rubber material for movement. The gate level controls the opening of the supply tank so as to control the flow from this tank. The movement of the gate was controlled by a gear box and motor with suitable mechanism of two pulleys and steel chain as shown in figure (3.13).The gear box mechanism was modified using push buttons to control the gate level and hence the quantity of water stored in the supply tank.
.
Figure (3.13) Details of flow control gate installed at the upstream side of the flume to control the flow from the supply tank.
Chapter Three Experimental Work
(a)
61
(b)
Figure (3.14) work done before conducting the experiments (a) fixing the gate (b) cleaning the flume. 3.6 Experimental Procedure: For each configuration the steps followed for conducting the experiments are: 1. Fix the required physical configuration and leave it for 24 hours. 2. Fix point gauge No.1 to a zero level corresponding to the level of the crest of the weir. 3. Fix point gauge No.2 to a zero level corresponding to the base of the upstream channel. 4. Close the discharge pipe valve totally, and fully open the bypass valve. 5. Switch on the ultrasonic flow meter and then change its mode from volumetric mode to discharge mode using the provided magnetic rod. 6. Switch on the pump.
Chapter Three Experimental Work
62
7. Gradually open the discharge pipe valve and gradually close the bypass pipe valve, instantaneously, until the discharge reaches a required value. 8. Wait for enough time for the flow to stabilize (45 minutes). 9. Read the flow meter more than once, and take the average value. 10.Measure the head over the weir (h1’) using point gauge No.1, (at section 1) Fig(3.1) 11.Measure the upstream depth before the weir (H) using point gauge No.2, (at section 3) Fig(3.1) 12.Take digital photos and videos for flow visualization using the digital cameras. 13.Change the opening of both discharge pipe valve and the bypass pipe valve, (increasing the first and decreasing the second), In order to adjust the discharge value. 14.Repeat steps (8) to (14) until reaching the required discharge.
3.7
Experimental Limitations The designed experimental setup has the following limitations: 1. The discharge range is (0.25 liter /sec to 44 liter /sec). 2. The channel is horizontal. 3. The flume width is (60 cm).
Chapter Four Results and Discussion
63
CHAPTER FOUR RESULTS AND DISUSSION 4.1 Experimental Results: For each configuration of the Cipolletti weir with rectangular bottom opening, from those mentioned in table (3.1), different flow values were used. The number of flow values for each configuration is different, according to flow condition required. The selected flow conditions were some within the limitations of Cipolletti weir mentioned below, and others are out of these limitations, the reasons for testing those flow cases out of the limitations are: 1.
The models tested are not normal “Cipolletti weir”, but with rectangular bottom opening.
2.
The flow conditions that can be tested and measured were those permitted by the available experimental setup.
The most frequent used general limitations of Cipolletti weir as mentioned in (Bengtson, 2005) are: a. Values of h1 / Bw should be less than 0.333 b. Values of h1 should be less than 0.5 P. c. B – Bw > 4 h1. Where: (h1: Is the water height above the crest which measured upstream the crest by a distance of minimum (3h1’), h1’: is the water height above the crest, P: The crest height, B: The flume width (approach channel width), Bw: Length of the weir ”crest width”) .
Chapter Four Results and Discussion
64
Further limitations are forced by the experimental setup capacity, that is the upstream flow is sub-critical and the case of flow of the weir is a free flow case (non-submerge flow case). For each configuration from those mentioned above and for each flow values, the following were measured: 1. Flow values (Qact ) using the flow meter. 2. The head over the crest (h1’) using point gage no.1 3. The total head (H) at a distance (3 h1’) upstream of the weir crest using point gage no.2 Using these data the discharge coefficient was calculated for each configuration and flow value. 4.2
Calculation of Discharge Coefficient: To compute the discharge through a combined Cipolletti weir (with sharp
crested), and bottom rectangular opening the following equation may be obtained by adding the discharge over the weir and the discharge through the gate as follows:
Where;
Qtheo= Q
Q
Q
theo
theo
theo
…………… (4.1)
+ Qwtheo
= is the discharge through the gate which given by: …………… (4.2)
= √ gHb h
Where; (H: Total head, bo: opening width, ho: opening height, g: gravitational acceleration.) and,
Q
theo
= √ gB h
.5
Where; (h1= H-P, P: is the crest height, P=d+ho).
…………… (4.3)
Chapter Four Results and Discussion
65
But the actual discharges are:Q respectively: Q
Q
act act
= Cd . Q
= Cd . Q
act
, and Q
act
for the gate and the weir …………… (4.4), and
theo
……… (4.5)
theo
Where; Cd and Cd are the discharge coefficients for the opening and the weir respectively. For the flow condition of both gate and weir the theoretical flow is Q t Qt
e
= √ gHb h + √ gB h
Q act = Cd . √ gHb h + Cd
Simplifying equations (4.7):
Where;
.
.5
√ gB h
e
;
…… (4.6), and
.5
Q act = Cdc . √ g [√Hb h + B h .5 ]
…… (4.7)
…… (4.8)
Cdc: is the combined discharge coefficient.
Hence this combined discharge coefficient will be estimated using equation (4.8) 4.3
Dimensional Analysis for Discharge Coefficient: It is expected that the combined discharge coefficient is dependent on the
geometry of the model as well as the flow conditions , i.e. ( ho , bo , d ,hw , Bw , H, g, ρ , µ , Ɵ, So ,σ)
Figure (4.1) Typical Model of Cipolletti weir with rectangular bottom opening
Chapter Four Results and Discussion
66
Where: ho: height of bottom opening, bo: width of bottom opening, d: vertical distance between the top of the opening and bottom of weir ( weir crest ) . hw: vertical distance between weir crest and top of the weir . g: gravitational acceleration. ρ: water mass density. µ: water viscosity. Ɵ: weir angle. So: slope of channel. σ: surface tension. For water at specific temperature (ρ, µ and σ) is constant, hence, can be dropped and can be represented by “H” (Ackers, 1978 as mentioned in AlHamed et. al. 1996-a). This is evidence herein since the water temperature measured during the experimental work was ranged between (18, 25). Moreover, since the standard side slope is (1:4) for Cipolletti weir hence Ɵ is constant (USBR, 2001). Since the available flume has no facility to change the bed slope of the channel, which is nearly horizontal (So) will also fixed and dropped as well as (g). Then, the discharge coefficient will be: Cdc= Φ ( ho , bo ,d ,hw , Bw , H )
...…………. (4.9)
And can be expressed as: Cdc = Φ ( ho / H , bo /H , d/H , hw /H , Bw / H ) Or can be expressed using h1= H – P, i.e.:
…………..(4.10)
Chapter Four Results and Discussion
67
Cdc = Φ ( ho / h1 , bo / h1 , d/ h1 , hw / h1 , Bw / h1 ) 4.4
…………..(4.11)
Hydraulic Measurements for Calculating Discharge Coefficient:
4.4.1 Configuration (1.a.1): Table (4.1); shows the calculations of discharge coefficient for configuration (1.a.1), for a Cipolletti weir of a (crest height “P=0.31m” and a crest length “BW=0.22m”) with a bottom opening of (width “bo = 0.1ηm” and height “ho = 0.0ηm”). This table Shows that the discharge coefficient range is (0.6227-0.6458) with average value of (0.6346), Variance = (0.000096), with coefficient of variation of (0.000152). This low coefficient of variation indicates the possibility of using an average discharge coefficient; the Froude number range is (0.0447-0.0526) which indicates a subcritical flow. Table (4.1) Test results and calculations for configuration (1.a.1),(P=0.31m, BW=0.22m, bo = 0.15m, ho = 0.05m).
Variables Q act (m3/s) H(m) h1(m) h1'(m)
Test 1
Test 2
Test 3
Test 4
Test 5
0.0192
0.0210
0.0222
0.0238
0.0251
0.3740 0.3830 0.3890 0.3950 0.4010 0.0640 0.0730 0.0790 0.0850 0.0910 0.0530 0.0620 0.0680 0.0730 0.0800 0.0856 0.0914 0.0951 0.1004 0.1043 V (m/S) 0.0447 0.0471 0.0487 0.0510 0.0526 Fr1 0.6227 0.6293 0.6317 0.6437 0.6458 Cdc 0.7813 0.6849 0.6329 0.5882 0.5495 ho/h1 2.3438 2.0548 1.8987 1.7647 1.6484 bo/h1 4.0625 3.5616 3.2911 3.0588 2.8571 d/h1 hw/h1 2.5000 2.1918 2.0253 1.8824 1.7582 3.4375 3.0137 2.7848 2.5882 2.4176 Bw/h1 Note :( h1 is the calculated water height above the crest which found by (h 1=HP), but h1/ is the measure water height above the crest and V is the approach flow velocity upstream the weir which is found by continuity equation).
Chapter Four Results and Discussion
68
4.4.2 Other Configurations: For the other configurations similar results were obtained as for configuration (1.a.1) above. Tables (A1-A38) in Appendix A show the calculations for these configurations respectively. Table (4.2) shows the summary of results for all configurations. It is shown that for all configurations the coefficient of variance of the discharge coefficient is low and the range of Froude No. indicates that the flow is subcritical. Table (4.2) summary results for all configurations. Config. No.
P (m )
Bw (m )
bo (m )
ho (m )
Cdc Range
Cdc Avg.
Cdc Cdc Coeff. coeff. Of Var.
1.a.1 1.a.2 1.a.3 1.a.4 1.a.5 1.b.1 1.b.2 1.b.3 1.b.4 2.a.1 2.a.2 2.a.3 2.a.4 2.a.5 2.b.1 2.b.2 2.b.3 2.b.4 2.b.5 2.c.1 2.c.2 2.c.3 2.c.4 2.c.5 3.a.1 3.a.2 3.a.3 3.a.4 3.a.5 3.b.1 3.b.2 3.b.3 3.b.4 3.b.5 3.c.1 3.c.2 3.c.3 3.c.4 3.c.5
0.31
0.22
0.15
0.05
0.6227-0.6458
0.6346
0.000152
0.0447-0.0526
0.31
0.22
0.15
0.1
0.5949-0.6087
0.6025
0.000040
0.0695-0.0759
0.31
0.22
0.15
0.15
0.5676-0.5784
0.5729
0.000036
0.31
0.22
0.1
0.1
0.6093-0.6222
0.6150
0.000040
0.0939-0.0970 0.0540-0.0604
0.31
0.22
0.1
0.15
0.5801-0.5982
0.5913
0.000104
0.0697-0.0751
0.31
0.32
0.15
0.05
0.6106-0.6393
0.6278
0.000237
0.0430-0.0581
0.31
0.32
0.15
0.1
0.5866-0.5925
0.5902
0.000009
0.0741-0.0828
0.31
0.32
0.1
0.1
0.5999-0.6055
0.6021
0.000008
0.0582-0.0657
0.31
0.32
0.1
0.15
0.5770-0.5924
0.5847
0.000050
0.0731-0.0825
0.27
0.4
0.15
0.05
0.5674-0.5835
0.5736
0.000084
0.0548-0.0914
0.27
0.4
0.15
0.1
0.5426-0.5812
0.5642
0.000444
0.0817-0.1073
0.27
0.4
0.15
0.15
0.5482-0.5557
0.5520
0.000051
0.1143-0.1205
0.27
0.4
0.1
0.1
0.5462-0.5680
0.5573
0.000124
0.0639-0.0979
0.27
0.4
0.1
0.15
0.5421-0.5712
0.5561
0.000267
0.0813-0.1066
0.27
0.3
0.15
0.05
0.6363-0.6819
0.6618
0.000541
0.0548-0.0897
0.27
0.3
0.15
0.1
0.5387-0.5845
0.5661
0.000525
0.0776-0.0989
0.27
0.3
0.15
0.15
0.5444-0.5569
0.5506
0.000050
0.1071-0.1141
0.27
0.3
0.1
0.1
0.5576-0.5972
0.5761
0.000408
0.0629-0.0876
0.27
0.3
0.1
0.15
0.5385-0.5822
0.5640
0.000607
0.0785-0.0991
0.27
0.2
0.15
0.05
0.6027-0.6333
0.6158
0.000185
0.0648-0.0482
0.27
0.2
0.15
0.1
0.5626-0.5996
0.5844
0.000340
0.0755-0.0862
0.27
0.2
0.15
0.15
0.5558-0.5780
0.5671
0.000152
0.1031-0.1084
0.27
0.2
0.1
0.1
0.5414-0.6204
0.5831
0.001079
0.0496-0.0740 0.0755-0.0863
Of
Froud No. Range
0.27
0.2
0.1
0.15
0.5533-0.5830
0.5699
0.000271
0.23
0.28
0.15
0.05
0.6210-0.6438
0.6292
0.000096
0.0566-0.0879
0.23
0.28
0.15
0.1
0.5706-0.5955
0.5840
0.000148
0.0890-0.1067
0.23
0.28
0.15
0.15
0.5579-0.5746
0.5656
0.000073
0.124-0.1323
0.23
0.28
0.1
0.1
0.5899-0.6085
0.6009
0.000079
0.0717-0.0881
0.23
0.28
0.1
0.15
0.5618-0.5642
0.5625
0.000002
0.0913-0.0974
0.23
0.23
0.15
0.05
0.6350-0.6383
0.6363
0.000005
0.0609-0.0733
0.23
0.23
0.15
0.1
0.5886-0.5914
0.5903
0.000003
0.0913-0.0975
0.23
0.23
0.15
0.15
0.5490-0.5681
0.5586
0.000104
0.1189-0.1247
0.23
0.23
0.1
0.1
0.5979-0.6117
0.6070
0.000048
0.0701-0.0793
0.23
0.23
0.1
0.15
0.5607-0.5737
0.5653
0.000056
0.0870-0.0940
0.23
0.18
0.15
0.05
0.6161-0.6370
0.6271
0.000092
0.0553-0.0650
0.23
0.18
0.15
0.1
0.5872-0.5944
0.5901
0.000013
0.0870-0.0913
0.23
0.18
0.15
0.15
0.5424-0.5695
0.5559
0.000201
0.1136-0.1184
0.23
0.18
0.1
0.1
0.5986-0.6023
0.5998
0.000003
0.0652-0.0723
0.23
0.18
0.1
0.15
0.5770-0.5880
0.5807
0.000038
0.0860-0.0911
Chapter Four Results and Discussion
69
4.5 Artificial Neural Network Model for Calculation of Discharge Coefficient. Artificial neural network models (ANN) were proved nowadays its efficiency against nonlinear regression models. The calculations of discharge coefficient presented in paragraph (4.4) depend on equation (4.8). This equation requires the value of measured actual discharge. In practice, in order to use a Cipolletti weir with bottom rectangular opening, the actual discharge value should be calculated using equation (4.8) with the knowledge of discharge coefficient and by measuring the head value (h1). Hence, a model is required to find the discharge coefficient as a function of ( ho / h1 , bo / h1 , d/ h1 , hw / h1 ,and Bw / h1 ). This model could be a regression model or an (ANN) model. Since the experiments conducted covers different cases as mentioned in paragraph (4.4) with different crest height, crest length and bottom opening dimensions. In such cases the (ANN) models proved its superiority against regression models (Saoud, 2009). However, in order to get good regression models, classification should be used i.e. finding a regression model for each case. This will result a regression model for each case which is difficult to be applied in practice. Therefor an (ANN) model was developed here including all the cases investigated to simplify using one model for all cases rather than different models. In order to use the (ANN) model, a brief theoretical aspects and methodology of these models will be presented hereafter rather than chapter two.
Chapter Four Results and Discussion 4.5.1
70
Theory and Methodology of Artificial Neural Network Modeling:
A brief theory of ANN will be presented hereafter as follows: 4.5.1.1
Description:
Over the past three decades, there has been an increased interest in a new class of computational intelligence systems known as Artificial Neural Networks (ANNs). This type of networks has been found to be a powerful and versatile computational tool for organizing and correlating information in ways that have proved to be useful for solving certain types of problems which are too complex to understand, too poor to analyze, or too resource-intensive to tackle using more traditional computational methods, (TRB, 1999). In recent years, artificial neural networks have been advocated as an alternative to traditional forecasting models. Neural networks have become significant data analytic tools that allow data to be analyzed to find the functional relationships among the variables under considerations. These variables are usually experimental data and classified into dependent and independent variables, the neural network allows the use of more than one dependent to be used in a functional relationship, (Yousif, 2007). An Artificial Neural Network (ANN) is an information processing paradigm that is inspired by the way biological nervous systems, such as the brain, process information. The key element of this paradigm is the novel structure of the information processing system. It is composed of a large number of highly interconnected processing elements (neurons) working in union to solve specific problems. ANNs, like people, learn by example. An ANN is configured for a
Chapter Four Results and Discussion
71
specific application, such as pattern recognition or data classification, through a learning process. (Arab.Eng. , Internet-2) . A neural network model is a computer model whose architecture essentially mimics the learning capability of the human brain. Basically, the processing elements of a neural network, with many simple computational elements arranged in layers, are similar to the neurons in the brain. In the past decade, considerable attention has been focused on the problem of applying neural networks in different fields. This is because artificial neural networks are good tools to model non-linear system, (Yeh, 1998).
Figure (4.2) Typical structure and typical neurons
4.5.1.2
Artificial Neural Network Operation:
Figure (4.1) is a schematic drawing of a typical Neurons variously known as processing elements (PEs), or nodes , the input from each (PE) in the previous layer (xi) is multiplied by an adjustable connection weight (w ij) at each PE, the weighted input signals are summed, with a threshold value ( j) may be added.
Chapter Four Results and Discussion
72
This combined input (Ij) is then passed through a transfer (activation) function (f(.)) to produce the output of the PE (yj). The output of one PE provides the input to the PEs in the next layer. This process is summarized in equations (4.12) and (4.13) (Al-Zwainy, 2008). IJ=∑ wijxi+Ɵj
summation ………………….…………….(4.12)
yj = f(Ij)
transfer
…………………….………….(4.13)
Where: Ij
= the activation level of node j;
Wij
= the connection weight between nodes i and j;
xi
= the input from node i , i = 0,1,……,n;
j
= the bias or threshold for node j;
yj
= the output of node j; and
f(.)
= the transfer (activation) function.
A summary of the process of a standard neural network algorithm can be illustrated as follows: (Elhag and Boussabaine, 1998) 1) A set of input vectors is presented to the ANN as well as their desired output vectors. 2) A training stage starts by arbitrary selection of a set of connection weights for each layer. Each neuron calculates its summation function value and accordingly computes its transfer function value, which represents its output. This process is held in a feed-forward manner.
Chapter Four Results and Discussion 3)
73
A set of computed outputs is delivered in the output layer. For each
processing element in the output layer an error is calculated, each represents a deviation of the computed output from the desired or observed output. 4)
Using a learning rule (e.g. generalized-delta rule, extended delta-
bar-delta rule, etc.) the errors are back propagated through the hidden layer(s) and the connection weights will be adjusted and updated accordingly. 5)
A feed-forward process starts all over again. New output values will
be computed and the above cycle continues until a desired set of requirements are achieved. 6)
To validate the model a testing session is undertaken using a new
set of data, which has never been exposed to the network. The accuracy of the model and its generalization capability could then be examined. 4.5.1.3
Multi-Layer Artificial Neural Networks (MLANNs):
Multi-layer neural networks are an important class of neural networks. They have been applied successfully to solve and diverse problems in many disciplines of science and technology. The structure of multi-layer neural networks consists of an input layer, one or more hidden layers and an output layer; the output of one layer provides the input to the subsequent layer. Figure (4. 3) shows such a network. Hidden layers (hidden neurons) play a critical role in the operation of multi-layer perceptron with back-propagation learning because they act as a
Chapter Four Results and Discussion
74
feature detector. As the learning process progresses, the hidden neurons in these layers begin to gradually discover the salient features that characterize the training data. The number of hidden nodes is performed through increase in hidden nodes by one and the network weights are reinitialized and the training starts again until reaching to the optimum numbers of hidden nodes and hidden layers corresponding to the observation of training error, testing error and regression square. Then for optimum nodes, the number of hidden layers is achieved. (Al-Neami, 2006).
Figure (4.3) Multi-layer Neural Network (Al-Janabi, 2006) For most applications, a single hidden layer is sufficient. Sometimes, difficult learning tasks can be simplified by increasing the number of internal layers. So, for complex mappings, two hidden layers may give better generalization and make training easier than a single hidden layer, (Zurda 1996).
Chapter Four Results and Discussion
75
One of the most popular ANN configurations is the error back propagation algorithm. It is popular in training multi-layer neural feed forward neural networks. The error back-propagation neural networks have been proven to be very successful in modeling nonlinear relationships. 4.5.1.4
Back-Propagation Neural Network (BPNN):
Several functions can be used for studying the relationships among the variables. Back propagation Network function is adopted in the research since the BPNN is the most widely used type and consists of many simple processing elements called neurons grouped in layers and connected by interconnections called synapses. Also, this type of network function implements a gradient descent in parameters space to minimize the output error (Russell and Norvig, 2003). 4.5.1.5
The Algorithm of the Error Back-Propagation Neural Network:
Basically back-propagation learning consists of two passes through the different layers of the network: a forward pass and backward pass as shown in Figure (4.4). (Al-Janabi, 2006). During the forward pass all the synaptic weights of the network are fixed, on the other hand, at the backward pass, all the synaptic weights are adjusted in accordance with an error correction. The weights between the hidden layer and the output layer are adjusted first, fooled by the weights between the hidden layer and the input layer. This process is repeated, which propagates the error term needed for weight adjustment until the network can obtain a set of weights, which have the input/output mapping that has the minimum error.
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76
After the network is properly trained, the recall stage will start. In this stage a set of test data is applied to the network. Among the many enhancements to the error back-propagation algorithm that have been produced, one involves the use of learning rate. Learning rate (t) is the factor that determines the size of the step that the network takes in negative through the weight space in order to minimize the magnitude of the training error. Another enhancement factor is the momentum term symbolized by (a). The function of the momentum factor is to increase the size of the step when the direction of the weight space is the same as the direction of the previous step and vice versa.
Figure (4.4) the Error Back-Propagation Algorithm
Chapter Four Results and Discussion
77
The Algorithm of Error Back-propagation can be summarized in the following steps: 1. Initialize network weight values (set to small random) values. 2. Repeat the following steps until some criterion; (for each training pair). 3. Sum weighted inputs and apply activation function to compute output of hidden layer. hj = f (∑i xi wij ) + Ɵj
…………………….(4.14)
Where: hj = Actual output of hidden neuron xi = input signal wij = Connection weight between input node i and hidden node j f
= The activation function j
= bias on hidden node, j
4. Sum weighted output of hidden layer and apply activation function to compute output of output layer yk = f ( ∑j hj wjk +
k
) ………………………………. (4.15)
Where: yk = Actual output of output neuron, k wjk = Connection weight between hidden node j output node k
Chapter Four Results and Discussion k=
78
bias on output node, k
5. Compute back-propagation error δk = ( dk - yk ) f / (∑ j hj wjk +
k)
…………………………. (4.16)
Where: f/ = the derivative of the activation function. dk = the desired output of neuron, k 6. Calculate weight correction term wjk (n) =
δk hj + α wjk (n - 1)
…………………………. (4.17)
Where: Wjk (n) = Correction on connection weight between nodes j and k = learning (training) rate (0 - 1 ) α = momentum term (0-1) hj = actual output of hidden neuron δk = back propagation error wjk (n - 1) = previous weight correction 7. Sum delta input for each hidden unit and calculate error term δj = δk wjk f /(∑i xi wij )
…………………………. (4.18)
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79
8. Calculate weight correction term wij(n) =
δj xi +α wij (n - 1)
…………………………. (4.19)
9. Update weights wjk (new)= wjk (old) + wjk
…………………………. (4.20)
wij (new)= wij (old) + wij
…………………………. (4.21)
10. Compute the sum square error. SSE=1/2 ∑(dk-yk)2
…………………………. (4.22)
Where: k = number of output neuron 11.
End.
4.5.1.6
Data Division and Pre-Processing
Data input or output from a network is usually either continuous or discrete, though sometimes it may be symbolic (nonnumeric in form) or a mixture of all these types (Flood and Kartam, 1994-a). It is a common practice to divide the available data into two subsets; 1. A training set, to construct the neural network model, and 2. An independent validation set to estimate model performance in the deployed environment. The training set is used to adjust the connection weights of the neural network. The testing set is used to check the performance of the network at
Chapter Four Results and Discussion
80
various stages of learning, and training is stopped once the error in the testing set increases. The validation set is used to evaluate the performance of the model once training has been successfully accomplished. Therefore, data division represents a crucial step in the neural network modeling (Nawari, 1999) Four methods can be used for data division: 1) Random data division; 2) Data division to ensure statistical consistency of the subsets needed for ANN model development; 3) Data division using self-organizing maps (SOMs); 4) Data division using Fuzzy clustering. Once the available data have been divided into their subsets (i.e. training, testing and validation), it is important to pre-process the data in a suitable form before they are applied to the ANN. Data pre-processing is necessary to ensure all variables receive equal attention during the training process. Moreover, preprocessing usually speeds up the training process. Pre-processing can be in the form of data scaling, normalization and transformation. Scaling the output data is essential, as they have to be commensurate with the limits of the transfer functions used in the output layer (e.g. between –1.0 to 1.0 for the tanh transfer function by using equation (4.23) and 0.0 to 1.00 for the sigmoid transfer function by using equation (4.24)). Scaling the input data is not necessary but it is almost always recommended (Shahin et al, 2002). ScaledValue = [
X
X−X
a −X
]−
…………………………. (4.23)
Chapter Four Results and Discussion ScaledValue = [ Where:
X
X−X
a −X
81
]
…………………………. (4.24)
X= original value. 4.5.1.7
Model Validation
Once the training phase of the model has been successfully accomplished, the performance of the trained model should be validated. The purpose of the model validation phase is to ensure that the model has the ability to generalize within the limits set by the training data in a robust fashion, rather than simply having memorized the input-output relationships that are contained in the training data. (Shahin et al, 2002). The coefficient of correlation, r, the root mean squared error (RMSE) and the mean absolute error (MAE) are the main criteria that are often used to evaluate the prediction performance of ANN models. The coefficient of correlation is a measure that is used to determine the relative correlation and the goodness-of-fit between the predicted and observed data and can be calculated as follows: (Smith, 1993) and (Al-Janabi, 2006) …………………………. (4.25)
……..(4.26)
…………………………. (4.27)
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82
…………………………. (4.28)
…………………………. (4.29)
…………………………. (4.30)
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83
4.5.2 Application of Artificial Neural Network Modeling for Discharge Coefficient of Cipolletti Weir with Bottom Rectangular Opening: The artificial neural network model for estimating the discharge coefficient as a function of (ho / h1, bo / h1, d / h1, hw / h1, Bw / h1) was developed using a software called “Neuframe”, this software allows the modeling with different network architecture, and use back propagation algorithm for adjusting the weights of the model. The software needs to identify the input variables which are those mentioned above as five variables and the number of output variables which is here selected as one, the discharge coefficient for the network architecture, hence an input layer with five variables (nodes) and an output layer with one variable (nodes) and one hidden layer was selected as usual practice as shown in Figure (4.5).
Figure (4.5) Artificial neural network architecture for discharge coefficient estimation.
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84
The software scales the input and output data using equation (4.24) with the minimum and maximum values of each variable. Table (4.3) shows these minimum and maximum values with some other statistical properties. Table (4.3) Some Statistical properties of the testing data.
Variables
Cdc
ho/h1
bo/h1
d/h1
hw/h1
Bw/h1
correlation with CD
1
-0.7306
-0.1073
0.2931
-0.2863
-0.3513
Maximum
0.6819
3.6585
3.7500
6.3415
6.0000
7.8431
Minimum
0.5385
0.3846
0.7299
0.7692
1.4599
1.4599
Avg.
0.5887
1.3922
1.6740
1.9914
2.6959
3.3935
Variance
0.000778
0.390124
0.334348
0.687754
0.641252
1.722782
Coeff. Of Variance
0.001321
0.280216
0.199732
0.345356
0.237862
0.507666
Range
0.1434
3.2739
3.0201
5.5722
4.5401
6.3833
Before proceeding with the next step for ANN modeling the statistical properties are shown in table (4.3) above indicates the following. The correlation coefficient between Cdc (the discharge coefficient) and (ho/h1) is the highest correlation among the correlations of Cdc with the other variables. Moreover, it is negative which indicates that Cd c is inversely proportional to this variable. The lowest correlation coefficient for Cd c is with (bo/h1). All the correlations of Cdc with the other variables is negative except for (d/h1).
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85
The coefficient of variance of Cdc is very low while for other variables it is low for the other variables with the highest value of (Bw/h1). The variation of Cdc with each of the variables (ho / h1 , bo / h1 , d/ h1 , hw / h1 , Bw / h1 ) is shown in figures (4.6,to 4.10) respectively. Even though single correlation between Cdc and each variable is low, it is expected that multiple correlation for Cd with these variables will be significant, this will be shown later among the application of the ANN model.
0.7000
Cdc
0.6500
0.6000
0.5500
0.5000 0.0000
1.0000
2.0000
3.0000
4.0000
ho/h1
Figure (4.6) Variation of discharge coefficient with (ho/h1).
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86
0.7000
Cdc
0.6500
0.6000
0.5500
0.5000 0.0000
1.0000
2.0000
3.0000
4.0000
bo/h1
Figure (4.7) Variation of discharge coefficient with (bo/h1). 0.7000
Cdc
0.6500
0.6000
0.5500
0.5000 0.0000
2.0000
4.0000
6.0000
8.0000
d/h1 Figure (4.8) Variation of discharge coefficient with (d/h1).
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87
0.7000
Cdc
0.6500
0.6000
0.5500
0.5000 0.0000
1.0000
2.0000
3.0000
4.0000
5.0000
6.0000
7.0000
hw/h1 Figure (4.9) Variation of discharge coefficient with (hw/h1). 0.7000
Cdc
0.6500
0.6000
0.5500
0.5000 0.0000
2.0000
4.0000
6.0000
8.0000
10.0000
Bw/h1 Figure (4.10) Variation of discharge coefficient with (Bw/h1).
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88
The next step is to find the number of nodes required in the hidden layer, which is a trial and error procedure. Before selecting the number of nodes in the hidden layer, the data division should be selected first, i.e. training set, testing set and verification (Querying) set. Different data set are selected in table (4.4) which indicates that a set of 65%, 25% and 10% is the most convenient since it gives the highest correlation coefficient. Table (4.4) data sets selection for the ANN model. Data Division % % % Training Testing Querying 80 75 70 70 70 65 65
10 15 15 10 20 20 25 max
10 10 15 20 10 10 10
training testing coefficient error error correlation(r) % % % 7.38%
8.55%
70.05%
9.30%
8.45%
73.73%
7.51%
7.82%
80.50%
8.96%
7.24%
84.35%
8.80%
7.08%
80.85%
6.60%
7.88%
81.08%
6.15%
7.02%
88.15%
88.15%
min 6.15% 7.02% The type of data division could be striped, blocked or random. Table (4.5) shows that the striped division method is the most suitable one. Table (4.5) data sets Division type selection for the ANN model. Data Division % % % Training Testing Querying 65 65 65
25 25 25 max min
10 10 10
choices of division Striped Blocked Random
training testing coefficient error error correlation 6.15%
7.02%
88.15%
7.90% 7.34%
6.67% 6.34%
82.60% 81.38% 88.15%
6.15%
6.34%
Chapter Four Results and Discussion
89
The numbers of nodes scanned are 1 to 11 as shown in table (4.6). It is shown that one node gives the minimum testing error and maximum correlation coefficient hence it is selected. Figure (4.11) shows the variation of testing error, training error and correlation coefficient with the number of nodes in the hidden layer, it is clear that the selection of one node in the best.
Table (4.6) Number of nodes in the hidden layer selection for a stripped data division of (65%, 25%, and 10%) No. of Nodes
training error
testing error
coefficient correlation
1
6.15% 6.02% 6.13% 6.85% 6.68% 6.90% 6.80% 6.86% 6.70% 6.75% 7.10%
7.02% 7.93% 7.95% 7.77% 7.79% 7.00% 7.52% 7.71% 7.77% 8.60% 8.00%
88.15% 84.55% 83.49% 83.84% 83.09% 84.61% 84.97% 84.52% 84.25% 84.85% 85.29%
2 3 4 5 6 7 8 9 10 11
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90
Training Error
8.0% 7.5% 7.0% 6.5% 6.0% 5.5% 5.0%
Testing Error
1
2
3
4
5
6 7 No. of Nodes
8
9
10
11
8
9
10
11
10
11
9.0% 8.0% 7.0% 6.0% 5.0% 4.0% 3.0% 2.0% 1
2
3
4
5
6
7
No. of Nodes
97.5% Correlation Coeff.(r)
92.5% 87.5% 82.5% 77.5% 72.5% 67.5% 62.5% 1
2
3
4
5
6
7
8
9
No. of Nodes
Figure (4.11) the variation of testing error, training error and correlation coefficient with the number of nodes in the hidden layer.
Chapter Four Results and Discussion
91
For the ANN model, a learning rate for a given momentum term should be selected. Table (4.7) shows the selection of learning rate for a momentum term of 0.80. A learning rate of 0.2 gives the best result as shown in this table and in Figure (4.12).
Table (4.7) learning rate selection for the ANN model. momentum learning training testing coefficient term rate error error correlation® 0.8
0.2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
6.15% 6.65% 6.15% 6.65% 6.84% 6.65% 6.23% 6.33% 6.45%
7.02% 7.15% 7.02% 7.33% 7.35% 7.50% 7.55% 7.50% 7.84%
88.15% 86.78% 88.15% 84.95% 84.92% 84.77% 75.20% 77.58% 76.89%
Chapter Four Results and Discussion
92
7.0% Training Error
6.8% 6.6% 6.4% 6.2% 6.0% 5.8% 0.1
0.2
0.3
0.4
0.5
0.6
0.7
Learning Rate
8.0%
Testing Error
7.8% 7.6% 7.4% 7.2% 7.0% 6.8% 6.6% 0.1
0.2
0.3
0.4 0.5 0.6 Learning Rate
0.7
0.8
0.7
0.8
Correlation Coeff. (r)
90.0% 85.0% 80.0% 75.0% 70.0% 65.0% 0.1
0.2
0.3
0.4 0.5 Learinig Rate
0.6
Figure (4.12) the variation of testing error, training error and correlation coefficient with learning rate
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93
The use of (0.8) momentum term is justified from table (4.8) and figure (4.13)
Table (4.8) Selection of momentum term for the ANN model momentum learning training testing coefficient term rate error error correlation® 0.8 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.95
0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2
6.15% 6.65% 6.92% 6.98% 6.92% 6.93% 6.92% 6.88% 6.15% 6.69% 6.69%
7.02% 7.64% 7.50% 7.50% 7.55% 7.87% 7.85% 8.58% 7.02% 7.08% 7.88%
88.15% 84.11% 84.75% 85.73% 84.00% 83.02% 83.33% 83.59% 88.15% 84.99% 84.89%
Training Error
Chapter Four Results and Discussion
94
7.2% 7.0% 6.8% 6.6% 6.4% 6.2% 6.0% 5.8% 5.6% 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.95
0.8
0.9
0.95
0.8
0.9
0.95
Momentum Term
Testing Error
10.0% 9.0% 8.0% 7.0% 6.0% 5.0% 4.0% 3.0% 2.0% 1.0% 0.0% 0.1
0.2
0.3
0.4 0.5 0.6 0.7 Momentum Term
100.0% Correlation Coeff. (r)
90.0% 80.0% 70.0% 60.0% 50.0% 40.0% 0.1
0.2
0.3
0.4
0.5
0.6
0.7
Momentum Term
Figure (4.13) the variation of testing error, training error and correlation coefficient with momentum term.
Chapter Four Results and Discussion
95
Using these values selected above the resulted weights for the ANN model is shown in table (4.9) and the model is shown in equations (4-31) and (432). This model shows that the required activation function for the output layer is a tanh type. Table (4.9) Weights and Threshold Levels for the ANN Optimal Model
Hidde n layer nodes j=6 Outpu t layer nodes
j=7
y
wji (weight from node i in the input layer to node j in the hidden layer) i=1
i=2
2.675870 0.502867
i=3
i=4
i=5
1.475235 0.363191 2.212595 0.873986 5
wji (weight from node i in the hidden layer to node j in the output layer) i=6
-
6.0545 213
Hidden layer threshol d θj
-
-
-
-
Output layer threshol d θj
-
3.921754
-
-
1 1 e ( 3.921754 6.05452tanh x )
-
…………………………. (4-31)
Where; X= {0.363191+ (2.67587 V1) + (0.502867 V2) - (2.212595 V3) (0.873986 V4) + (1.4752355 V5)} ...………………………. (4-32)
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96
VISUAL BASIC Model of the ANN Model Application: For this application a VISUAL BASIC program was designed. This program has three forms. Form (1) is shown in figure (4.14) which is the introduction form for the program. Form (2) is the input form which enables entering the input variables as shown in figure (4.15). Form (3) is the output presentation which shows the discharge coefficient Cd c and the actual discharge, as shown in figure (4.16). The simplicity of using this program is obvious. The programing code beyond these forms is the application of the developed ANN model. As the user enters the input variables the code standardize each variable using equation (4.24) with the corresponding minimum and maximum values for each variables shown in table (4.3).
Figure (4.14) introductory form for the VISUAL BASIC Program for the ANN model to calculate the discharge coefficient and the actual discharge for a Cipolletti weir with a bottom rectangular opening
Chapter Four Results and Discussion
97
.
Figure (4.15) Data entering form for the VISUAL BASIC Program for the ANN model to calculate the discharge coefficient and the actual discharge for a Cipolletti weir with a bottom rectangular opening.
Figure (4.16) output form for the VISUAL BASIC Program for the ANN model to calculate the discharge coefficient and the actual discharge for a Cipolletti weir with a bottom rectangular opening.
Chapter Four Results and Discussion 4.7
98
Visual Flow Conditions: In order to present the condition of the flow for different flow values some
pictures are shown below. Figure (4.17) shows the flow condition when the water level at the upstream side below the height of the bottom opening. It is shown that small turbulence exists at the downstream side. Figure (4.18) shows the flow condition when the water level at the upstream side is just as the crest level. It is shown that the turbulence is increased at the downstream side. Figure (4.19) shows the flow condition when the water level at the upstream side is just passing the crest level. It is shown that the turbulence is further increased at the downstream side. Figure (4.20) shows the flow condition for high head over the crest level. It is shown that very high turbulence exists at the downstream side.
Figure (4.17) Flow condition when the water level is below the height of the bottom opening.
Chapter Four Results and Discussion
99
.
Figure (4.18) Flow condition when the water level is just as the crest level
Figure (4.19) Flow condition when the water level is just passes the crest level
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100
Figure (4.20) Flow condition for high head over the crest level
4.8
Application of Two Dimensional Modeling (RMA2): To investigate the velocity distribution at the upstream side of the
structure, the software (RMA2) is used. For theoretical background of this software see appendix B. As mentioned in this appendix this software solves the momentum and the energy equations using finite element method. Software was applied for some of the cases of flow as mentioned in table (4.10). These cases are for the water level below the crest level of the weir, since for this situation the flow can be approximated by a two dimensional model. The generated mesh is consists of 5164 elements, figure (4.21). The upstream boundary condition is constant inflow at the flume inlet while the down-stream boundary condition is water depth at the weir location. The mathematical model dimensions are 4.65 m long and 0.6 m width. Three cases
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101
are adopted (case 1, 2 and 3) in order to find the velocity pattern near the weir. In each case, the down-stream boundary condition is taken equals the half and full height plus 5 mm of the gate opening. The required down-stream boundary condition is based on the associated laboratory measurements for each case. Accordingly, the calibration and verification are not required. Table (4.10) the required boundary conditions for each case
Case 1 2 3
Upstream Boundary Condition Discharge (l/sec)
Downstream Boundary Condition Water Depth(mm)
3.25 11.10 4.71 10.00 2.80 8.55
72.50 155.00 72.50 155.00 50.00 105.00
Figure (4.21) the finite element generated mesh for the two dimensional modeling using RMA2.
Chapter Four Results and Discussion
Figure (4.22) the velocity distribution upstream the weir for the case (1-A)
102
Chapter Four Results and Discussion
103
Figures (4.22 and D1 to D5) shows the velocity distribution at the upstream side of the weir for cases (1A, 1B, 2A, 2B, 3A, and 3B) mentioned previously in table (4.21). For all these cases, it shown from the velocity contours that a relatively high velocity bulb exists near the opening with maximum velocity just at the opening and decreasing in the direction of the upstream side. Moreover their existence a low velocity regime at both the left and right edges of the weir and extend in the direction of the upstream side with decreasing the width. This velocity distribution indicate the possibility of flushing the sediments accumulated upstream the weir from the central part of the channel section rather than accumulated all over the section for the case of normal weir without the bottom opening. 4.9
Model Verification: In order to check the model validity the combined discharge
coefficient for all the cases tested are estimated using the VISUAL-BASIC program. Figure (4.23) shows the comparison between these values and the actual tested values. It is clear that the model gives good estimation for the discharge coefficient with correlation coefficient of (r= 0.833), which considered as a strong correlation and with a coefficient of determination (R2=0.6939).
Chapter Four Results and Discussion
104
R2
Figure (4.23) Comparison between discharge coefficients estimated by the ANN model and measured tested discharge coefficient.
Chapter Five Conclusions and Recommendations
105
CHAPTER FIVE CONCLUSIONS AND RECOMMENDATIONS From the experimental study and the ANN modeling conducted for the discharge coefficient as a function of the geometry and flow properties of the proposed hydraulic structure (Cipolletti weir with rectangular bottom opening), the following conclusions and recommendations could be deduced: 5.1
Conclusions:
1. All the flow conditions proposed exhibits subcritical flow at the upstream side of the structure, since Froude No. range is less than “1”, even though for large bottom opening dimensions . 2. For all the cases tested the coefficient of variation for the estimated discharge coefficient is very low (0.001321), and ranged between (0.5385- 0.6819) with an average value of (0.5887). 3. Correlation analysis of the discharge coefficient with the geometry and the flow variables indicate negative correlation of discharge coefficient with (ho/h1,bo/h1,hw/h1,Bw/h1) with values (-0.7306, -0.1073, -0.2863, and -0.3513) respectively, while a positive correlation of (0.2931) was found between the discharge coefficient and (d/h1). Even though these correlation coefficients are relatively low, multiple correlations are significant as shown in the ANN modeling which has a correlation coefficient of (0.8815). 4.visual inspection of the flow condition downstream of the hydraulic structure proposed, shows that low turbulence is exist when the water level at the upstream side is below the height of the bottom opening. Turbulence at the
Chapter Five Conclusions and Recommendations
106
downstream side increases as the water level at the upstream side increases result in a very high turbulence at the downstream side for high head over the weir. 5. The architecture of the ANN model suitable for relating the discharge coefficient with the geometry and the flow variables is of an input layer with five nodes, output layer with one node and one hidden layer with one node. The most suitable data division for training, testing, and validation is (65%, 25%, 10%) respectively. The most suitable type of data division is stripped division. The algorithm used is the back propagation algorithm with a learning rate of (0.2) and a momentum term of (0.80).the network correlation coefficient is (0.8815) which is classified as strong according to (Smith , 1986) criteria. 6. The obtained velocity distribution upstream of the weir for the cases of flow through the opening only as obtained using (RMA2) indicates the existence of a high velocity bulb which extends in the upstream direction with decreasing velocity. Also this indicates two low velocity regions at the left and right sides of the weir with decreasing width in the upstream direction. This velocity distribution is expected to remove sediments from the mid part of the upstream section, a situation which is better than the case of normal weir without opening. For the case of increased flow i.e. flow from both the bottom opening and over the weir, the structure is expected to exhibit better self-cleaning capability. 5.2 Recommendations: The following recommendation could be useful for practical purposes: The developed visual basic program is recommended to be used for estimating the discharge coefficient and the actual flow of a Cipolletti weir with a rectangular bottom opening for flow conditions in accordance with the
Chapter Five Conclusions and Recommendations
107
limitations of the experimental work done here, mentioned in chapter four section (4.1), which depend on the developed ANN model using the experimental data. The following recommendations could be useful for further research: 1. An experimental study is recommended to find the effect of longitudinal slope of the channel when an experimental setup is available that allows changing this slope. 2. An experimental study is also recommended to include sediment removal efficiency at the upstream side using different kind of sediments put in the upstream side bottom of the flume. 3.
An experimental study is recommended for submerge flow condition using experimental set up that allows the condition of submerged flow.
4. An experimental study is required coupled with two dimensional modeling to investigate the velocity distribution upstream the weir with three bottom opening one at the middle and one at each side of the weir.
Appendix A Configurations Tables
Table (A1): Calculation of discharge coefficient for configuration (1.a.2). Variables
Test 1
Test 2
Test 3
Test 4
Test 5
Test 6
Appendix A Configurations Tables Q act (m3/s)
H(m) h1(m) h1'(m) V (m/s) Fr1 Cdc ho/h1 bo/h1 d/h1 hw/h1 Bw/h1
0.0287 0.3640 0.0540 0.0510 0.1314 0.0695 0.5949 1.8519 2.7778 3.8889 2.9630 4.0741
0.0308 0.3750 0.0650 0.0600 0.1370 0.0714 0.5993 1.5385 2.3077 3.2308 2.4615 3.3846
0.0321 0.3810 0.0710 0.0660 0.1404 0.0726 0.6022 1.4085 2.1127 2.9577 2.2535 3.0986
0.0333 0.3870 0.0770 0.0720 0.1436 0.0737 0.6038 1.2987 1.9481 2.7273 2.0779 2.8571
0.0347 0.3930 0.0830 0.0780 0.1470 0.0749 0.6063 1.2048 1.8072 2.5301 1.9277 2.6506
0.0358 0.3980 0.0880 0.0820 0.1501 0.0759 0.6087 1.1364 1.7045 2.3864 1.8182 2.5000
Table (A2): Calculation of discharge coefficient for configuration (1.a.3). Variables
Test 1 Test 2 Test 3 Test 4 Q act (m3/s) 0.0388 0.0403 0.0424 0.0439 0.3640 0.3710 0.3810 0.3870 H(m) 0.0540 0.0610 0.0710 0.0770 h1(m) 0.0500 0.0560 0.0640 0.0730 h1'(m) 0.1774 0.1809 0.1853 0.1890 V (m/s) 0.0939 0.0948 0.0959 0.0970 Fr1 0.5676 0.5714 0.5740 0.5784 Cdc 2.7778 2.4590 2.1127 1.9481 ho/h1 2.7778 2.4590 2.1127 1.9481 bo/h1 2.9630 2.6230 2.2535 2.0779 d/h1 2.9630 2.6230 2.2535 2.0779 hw/h1 4.0741 3.6066 3.0986 2.8571 Bw/h1 Table (A3): Calculation of discharge coefficient for configuration (1.a.4).
Appendix A Configurations Tables
Variables Test 1 Test 2 Test 3 Test 4 Test 5 Q act (m3/s) 0.0236 0.0250 0.0264 0.0279 0.0292 0.3780 0.3850 0.3920 0.3990 0.4040 H(m) 0.0680 0.0750 0.0820 0.0890 0.0940 h1(m) 0.0600 0.0650 0.0720 0.0780 0.0830 h1'(m) 0.1041 0.1082 0.1122 0.1166 0.1203 V (m/s) 0.0541 0.0557 0.0572 0.0589 0.0604 Fr1 0.6093 0.6124 0.6139 0.6173 0.6222 Cdc 1.4706 1.3333 1.2195 1.1236 1.0638 ho/h1 1.4706 1.3333 1.2195 1.1236 1.0638 bo/h1 3.0882 2.8000 2.5610 2.3596 2.2340 d/h1 2.3529 2.1333 1.9512 1.7978 1.7021 hw/h1 3.2353 2.9333 2.6829 2.4719 2.3404 Bw/h1 Table (A4): Calculation of discharge coefficient for configuration (1.a.5).
Variables Test 1 Test 2 Test 3 Test 4 Test 5 Q act (m3/s) 0.0306 0.0320 0.0334 0.0348 0.0358 0.3790 0.3850 0.3900 0.3960 0.4010 H(m) 0.0690 0.0750 0.0800 0.0860 0.0910 h1(m) 0.0630 0.0680 0.0720 0.0770 0.0830 h1'(m) 0.1344 0.1385 0.1427 0.1464 0.1489 V (m/s) 0.0697 0.0713 0.0729 0.0743 0.0751 Fr1 0.5801 0.5865 0.5942 0.5977 0.5982 Cdc 2.1739 2.0000 1.8750 1.7442 1.6484 ho/h1 1.4493 1.3333 1.2500 1.1628 1.0989 bo/h1 2.3188 2.1333 2.0000 1.8605 1.7582 d/h1 2.3188 2.1333 2.0000 1.8605 1.7582 hw/h1 3.1884 2.9333 2.7500 2.5581 2.4176 Bw/h1 Table (A5): Calculation of discharge coefficient for configuration (1.b.1).
Appendix A Configurations Tables Variables
Test 1
Test 2
Test 3
Test 4
Q act (m3/s) H(m) h1(m) h1'(m) V (m/s) Fr1 Cdc ho/h1 bo/h1 d/h1 hw/h1 Bw/h1
0.0168 0.3510 0.0410 0.0330 0.0798 0.0430 0.6106 1.2195 3.6585 6.3415 3.9024 7.8049
0.0208 0.3670 0.0570 0.0480 0.0944 0.0497 0.6300 0.8772 2.6316 4.5614 2.8070 5.6140
0.0250 0.3840 0.0740 0.0650 0.1085 0.0559 0.6312 0.6757 2.0270 3.5135 2.1622 4.3243
0.0264 0.3880 0.0780 0.0700 0.1134 0.0581 0.6393 0.6410 1.9231 3.3333 2.0513 4.1026
Table (A6): Calculation of discharge coefficient for configuration (1.b.2). Variables
Test 1 Test 2 Test 3 Test 4 Test 5 Q act (m3/s) 0.0307 0.0344 0.0356 0.0369 0.0386 0.3650 0.3800 0.3840 0.3890 0.3950 H(m) 0.0550 0.0700 0.0740 0.0790 0.0850 h1(m) 0.0480 0.0600 0.0650 0.0680 0.0750 h1'(m) 0.1402 0.1511 0.1543 0.1583 0.1629 V (m/s) 0.0741 0.0782 0.0795 0.0810 0.0828 Fr1 0.5866 0.5893 0.5907 0.5919 0.5925 Cdc 1.8182 1.4286 1.3514 1.2658 1.1765 ho/h1 2.7273 2.1429 2.0270 1.8987 1.7647 bo/h1 3.8182 3.0000 2.8378 2.6582 2.4706 d/h1 2.9091 2.2857 2.1622 2.0253 1.8824 hw/h1 5.8182 4.5714 4.3243 4.0506 3.7647 Bw/h1 Table (A7): Calculation of discharge coefficient for configuration (1.b.3).
Appendix A Configurations Tables Variables
Test 1
Test 2
Test 3
Test 4
Test 5
Q act (m3/s) H(m) h1(m) h1'(m) V (m/s) Fr1 Cdc ho/h1 bo/h1 d/h1 hw/h1 Bw/h1
0.0247 0.3710 0.0610 0.0500 0.1111 0.0582 0.5999 1.6393 1.6393 3.4426 2.6230 5.2459
0.0267 0.3790 0.0690 0.0580 0.1173 0.0608 0.6007 1.4493 1.4493 3.0435 2.3188 4.6377
0.0278 0.3830 0.0730 0.0630 0.1209 0.0624 0.6033 1.3699 1.3699 2.8767 2.1918 4.3836
0.0292 0.3880 0.0780 0.0680 0.1253 0.0642 0.6055 1.2821 1.2821 2.6923 2.0513 4.1026
0.0306 0.3940 0.0840 0.0730 0.1293 0.0657 0.6014 1.1905 1.1905 2.5000 1.9048 3.8095
Table (A8): Calculation of discharge coefficient for configuration (1.b.4). Variables
Test 1
Test 2
Test 3
Test 4
Test 5
Test 6
Q act (m3/s) 0.0304 0.0332 0.0349 0.0356 0.0369 0.0383 0.3660 0.3770 0.3830 0.3850 0.3900 0.3940 H(m) 0.0560 0.0670 0.0730 0.0750 0.0800 0.0840 h1(m) 0.0480 0.0570 0.0640 0.0670 0.0710 0.0750 h1'(m) V (m/s) 0.1385 0.1469 0.1517 0.1539 0.1579 0.1622 0.0731 0.0764 0.0783 0.0792 0.0807 0.0825 Fr1 0.5770 0.5810 0.5834 0.5864 0.5876 0.5924 Cdc 2.6786 2.2388 2.0548 2.0000 1.8750 1.7857 ho/h1 1.7857 1.4925 1.3699 1.3333 1.2500 1.1905 bo/h1 d/h1 2.8571 2.3881 2.1918 2.1333 2.0000 3.6905 2.8571 2.3881 2.1918 2.1333 2.0000 1.9048 hw/h1 5.7143 4.7761 4.3836 4.2667 4.0000 3.8095 Bw/h1 Table (A9): Calculation of discharge coefficient for configuration (2.a.1).
Appendix A Configurations Tables Variables Q act (m3/s) H(m) h1(m) h1'(m) V (m/s) Fr1 Cdc ho/h1 bo/h1 d/h1 hw/h1 Bw/h1
Test 1
Test 2
Test 3
Test 4
0.0189 0.0250 0.0336 0.0426 0.3230 0.3450 0.3720 0.3950 0.0530 0.0750 0.1020 0.1250 41.0000 60.0000 82.0000 105.0000 0.0975 0.1208 0.1506 0.1799 0.0548 0.0656 0.0788 0.0914 0.5674 0.5712 0.5722 0.5835 0.9434 0.6667 0.4902 0.4000 2.8302 2.0000 1.4706 1.2000 4.1509 2.9333 2.1569 2.1600 3.7736 2.6667 1.9608 1.6000 7.5472 5.3333 3.9216 3.2000
Table (A10): Calculation of discharge coefficient for configuration (2.a.2). Variables
Test 1
Test 2
Test 3
Test 4
Test 5
Q act (m3/s) 0.0281 0.0325 0.0367 0.0421 0.0456 H(m) 0.3220 0.3360 0.3480 0.3620 0.3710 h1(m) 0.0520 0.0660 0.0780 0.0920 0.1010 h1'(m) 0.0400 0.0530 0.0650 0.0770 0.0880 V (m/s) 0.1452 0.1612 0.1756 0.1938 0.2047 Fr1 0.0817 0.0888 0.0950 0.1028 0.1073 Cdc 0.5426 0.5552 0.5648 0.5770 0.5812 ho/h1 1.9231 1.5152 1.2821 1.0870 0.9901 bo/h1 2.8846 2.2727 1.9231 1.6304 1.4851 d/h1 3.2692 2.5758 2.1795 1.8478 1.6832 hw/h1 3.8462 3.0303 2.5641 2.1739 1.9802 Bw/h1 7.6923 6.0606 5.1282 4.3478 3.9604 Table (A11): Calculation of discharge coefficient for configuration (2.a.3).
Appendix A Configurations Tables Variables
Test 1
Test 2
Q act (m3/s) 0.0403 0.0457 0.3280 0.3440 H(m) 0.0580 0.0740 h1(m) 0.0460 0.0600 h1'(m) 0.2049 0.2214 V (m/s) 0.1143 0.1205 Fr1 0.5482 0.5557 Cdc 2.5862 2.0270 ho/h1 2.5862 2.0270 bo/h1 2.0690 1.6216 d/h1 3.4483 2.7027 hw/h1 Bw/h1 6.8966 5.4054 Table (A12): Calculation of discharge coefficient for configuration (2.a.4). Variables
Test 1
Test 2
Test 3
Test 4
Test 5
Test 6
Test 7
Q act (m3/s)
0.0224 0.3260 0.0560 0.0401 0.1143 0.0639 0.5462 1.7857 1.7857 3.0357 3.5714 7.1429
0.0250 0.3360 0.0660 0.0500 0.1241 0.0684 0.5477 1.5152 1.5152 2.5758 3.0303 6.0606
0.0282 0.3460 0.0760 0.0580 0.1358 0.0737 0.5550 1.3158 1.3158 2.2368 2.6316 5.2632
0.0307 0.3540 0.0840 0.0650 0.1445 0.0775 0.5570 1.1905 1.1905 2.0238 2.3810 4.7619
0.0345 0.3650 0.0950 0.0760 0.1577 0.0833 0.5629 1.0526 1.0526 1.7895 2.1053 4.2105
0.0397 0.3800 0.1100 0.0900 0.1742 0.0902 0.5643 0.9091 0.9091 1.5455 1.8182 3.6364
0.0458 0.3960 0.1260 0.1040 0.1929 0.0979 0.5680 0.7937 0.7937 1.3492 1.5873 3.1746
H(m) h1(m) h1'(m) V (m/s) Fr1 Cdc ho/h1 bo/h1 d/h1 hw/h1 Bw/h1
Table (A13): Calculation of discharge coefficient for configuration (2.a.5).
Appendix A Configurations Tables Variables
Test 1
Test 2
Test 3
Test 4
Test 5
Test 6
Q act (m3/s) H(m) h1(m) h1'(m) V (m/s) Fr1 Cdc ho/h1 bo/h1 d/h1 hw/h1 Bw/h1
0.0278 0.3210 0.0510 0.0390 0.1442 0.0813 0.5421 2.9412 1.9608 2.3529 3.9216 7.8431
0.0306 0.3310 0.0610 0.0490 0.1539 0.0854 0.5455 2.4590 1.6393 1.9672 3.2787 6.5574
0.0347 0.3450 0.0750 0.0590 0.1677 0.0912 0.5487 2.0000 1.3333 1.6000 2.6667 5.3333
0.0390 0.3560 0.0860 0.0690 0.1827 0.0978 0.5621 1.7442 1.1628 1.3953 2.3256 4.6512
0.0413 0.3620 0.0920 0.0760 0.1903 0.1010 0.5667 1.6304 1.0870 1.3043 2.1739 4.3478
0.0458 0.3740 0.1040 0.0880 0.2042 0.1066 0.5712 1.4423 0.9615 1.1538 1.9231 3.8462
Table (A14): Calculation of discharge coefficient for configuration (2.b.1). Variables
Test 1
Test 2
Test 3
Test 4
0.0426 Q act (m3/s) 0.0189 0.0250 0.0336 0.3230 0.3450 0.3750 0.4000 H(m) 0.0530 0.0750 0.1050 0.1300 h1(m) 0.0410 0.0600 0.0820 0.1050 h1'(m) 0.0975 0.1208 0.1494 0.1777 V (m/s) 0.0548 0.0656 0.0779 0.0897 Fr1 0.6363 0.6630 0.6658 0.6819 Cdc 0.9434 0.6667 0.4762 0.3846 ho/h1 2.8302 2.0000 1.4286 1.1538 bo/h1 d/h1 4.1509 2.9333 2.0952 1.6923 3.7736 2.6667 1.9048 1.5385 hw/h1 5.6604 4.0000 2.8571 2.3077 Bw/h1 Table (A15): Calculation of discharge coefficient for configuration (2.b.2).
Appendix A Configurations Tables Variables
Test 1
Test 2
Test 3
Test 4
Test 5
Test 6
Q act (m3/s) H(m) h1(m) h1'(m) V (m/s) Fr1 Cdc ho/h1 bo/h1 d/h1 hw/h1 Bw/h1
0.0278 0.3310 0.0610 0.0440 0.1399 0.0776 0.5387 1.6393 2.4590 2.7869 3.2787 4.9180
0.0353 0.3590 0.0890 0.0690 0.1638 0.0873 0.5571 1.1236 1.6854 1.9101 2.2472 3.3708
0.0365 0.3630 0.0930 0.0740 0.1677 0.0889 0.5607 1.0753 1.6129 1.8280 2.1505 3.2258
0.0399 0.3720 0.1020 0.0820 0.1790 0.0937 0.5758 0.9804 1.4706 1.6667 1.9608 2.9412
0.0428 0.3810 0.1110 0.0890 0.1871 0.0968 0.5799 0.9009 1.3514 1.5315 1.8018 2.7027
0.0446 0.3860 0.1160 0.0940 0.1925 0.0989 0.5845 0.8621 1.2931 1.4655 1.7241 2.5862
Table (A16): Calculation of discharge coefficient for configuration (2.b.3). Variables
Test 1
Test 2
Test 3
Test 4
0.0456 Q act (m3/s) 0.0378 0.0414 0.0424 0.3280 0.3420 0.3450 0.3560 H(m) 0.0580 0.0720 0.0750 0.0860 h1(m) 0.0460 0.0550 0.0600 0.0710 h1'(m) 0.1921 0.2017 0.2046 0.2133 V (m/s) 0.1071 0.1101 0.1112 0.1141 Fr1 0.5444 0.5490 0.5521 0.5569 Cdc 2.5862 2.0833 2.0000 1.7442 ho/h1 2.5862 2.0833 2.0000 1.7442 bo/h1 d/h1 2.0690 1.6667 1.6000 1.3953 3.4483 2.7778 2.6667 2.3256 hw/h1 5.1724 4.1667 4.0000 3.4884 Bw/h1 Table (A17): Calculation of discharge coefficient for configuration (2.b.4).
Appendix A Configurations Tables Variables
Test 1
Test 2
Test 3
Test 4
Test 5
Test 6
Q act (m3/s) H(m) h1(m) h1'(m) V (m/s) Fr1 Cdc ho/h1 bo/h1 d/h1 hw/h1 Bw/h1
0.0233 0.3390 0.0690 0.0530 0.1147 0.0629 0.5576 1.4493 1.4493 2.4638 2.8986 4.3478
0.0280 0.3580 0.0880 0.0690 0.1304 0.0696 0.5642 1.1364 1.1364 1.9318 2.2727 3.4091
0.0303 0.3660 0.0960 0.0760 0.1379 0.0728 0.5697 1.0417 1.0417 1.7708 2.0833 3.1250
0.0342 0.3790 0.1090 0.0870 0.1502 0.0779 0.5777 0.9174 0.9174 1.5596 1.8349 2.7523
0.0378 0.3890 0.1190 0.0980 0.1619 0.0829 0.5904 0.8403 0.8403 1.4286 1.6807 2.5210
0.0418 0.4010 0.1310 0.1100 0.1739 0.0877 0.5972 0.7634 0.7634 1.2977 1.5267 2.2901
Table (A18): Calculation of discharge coefficient for configuration (2.b.5). Variables
Test 1
Test 2
Test 3
Test 4
Test 5
Q act (m3/s) 0.0286 0.0344 0.0383 0.0422 0.0450 0.3350 0.3570 0.3670 0.3800 0.3880 H(m) 0.0650 0.0870 0.0970 0.1100 0.1180 h1(m) 0.0500 0.0690 0.0790 0.0930 0.1000 h1'(m) 0.1423 0.1605 0.1741 0.1852 0.1933 V (m/s) 0.0785 0.0858 0.0917 0.0959 0.0991 Fr1 0.5385 0.5509 0.5721 0.5762 0.5822 Cdc 2.3077 1.7241 1.5464 1.3636 1.2712 ho/h1 1.5385 1.1494 1.0309 0.9091 0.8475 bo/h1 d/h1 1.8462 1.3793 1.2371 1.0909 1.0169 3.0769 2.2989 2.0619 1.8182 1.6949 hw/h1 4.6154 3.4483 3.0928 2.7273 2.5424 Bw/h1 Table (A19): Calculation of discharge coefficient for configuration (2.c.1).
Appendix A Configurations Tables Variables
Test 1
Test 2
Test 3
Test 4
Test 5
Test 6
Test 7
Q act (m3/s)
0.0176 0.3360 0.0660 0.0530 0.0875 0.0482 0.6027 0.7576 2.2727 3.3333 3.0303 3.0303
0.0194 0.3460 0.0760 0.0620 0.0934 0.0507 0.6076 0.6579 1.9737 2.8947 2.6316 2.6316
0.0208 0.3540 0.0840 0.0700 0.0981 0.0526 0.6102 0.5952 1.7857 2.6190 2.3810 2.3810
0.0224 0.3620 0.0920 0.0780 0.1030 0.0546 0.6132 0.5435 1.6304 2.3913 2.1739 2.1739
0.0248 0.3740 0.1040 0.0900 0.1105 0.0577 0.6183 0.4808 1.4423 2.1154 1.9231 1.9231
0.0283 0.3900 0.1200 0.1020 0.1211 0.0619 0.6255 0.4167 1.2500 1.8333 1.6667 1.6667
0.0308 0.4000 0.1300 0.1150 0.1285 0.0649 0.6333 0.3846 1.1538 1.6923 1.5385 1.5385
H(m) h1(m) h1'(m) V (m/s) Fr1 Cdc ho/h1 bo/h1 d/h1 hw/h1 Bw/h1
Table (A20): Calculation of discharge coefficient for configuration (2.c.2). Variables
Test 1
Test 2
Test 3
Test 4
Test 5
Q act (m3/s) 0.0286 0.0310 0.0336 0.0367 0.0392 0.3440 0.3520 0.3650 0.3780 0.3880 H(m) 0.0740 0.0820 0.0950 0.1080 0.1180 h1(m) 0.0600 0.0700 0.0810 0.0910 0.1030 h1'(m) 0.1386 0.1466 0.1535 0.1617 0.1682 V (m/s) 0.0755 0.0789 0.0811 0.0840 0.0862 Fr1 0.5626 0.5813 0.5853 0.5933 0.5996 Cdc 1.3514 1.2195 1.0526 0.9259 0.8475 ho/h1 2.0270 1.8293 1.5789 1.3889 1.2712 bo/h1 d/h1 2.2973 2.0732 1.7895 1.5741 1.4407 2.7027 2.4390 2.1053 1.8519 1.6949 hw/h1 2.7027 2.4390 2.1053 1.8519 1.6949 Bw/h1 Table (A21): Calculation of discharge coefficient for configuration (2.c.3).
Appendix A Configurations Tables Variables
Test 1
Test 2
Test 3
Test 4
Test 5
Q act (m3/s) H(m) h1(m) h1'(m) V (m/s) Fr1 Cdc ho/h1 bo/h1 d/h1 hw/h1 Bw/h1
0.0336 0.3110 0.0410 0.0390 0.1801 0.1031 0.5558 3.6585 3.6585 2.9268 4.8780 4.8780
0.0386 0.3400 0.0700 0.0600 0.1893 0.1036 0.5592 2.1429 2.1429 1.7143 2.8571 2.8571
0.0417 0.3520 0.0820 0.0720 0.1973 0.1062 0.5709 1.8293 1.8293 1.4634 2.4390 2.4390
0.0439 0.3630 0.0930 0.0800 0.2015 0.1068 0.5716 1.6129 1.6129 1.2903 2.1505 2.1505
0.0458 0.3700 0.1000 0.0870 0.2065 0.1084 0.5780 1.5000 1.5000 1.2000 2.0000 2.0000
Table (A22): Calculation of discharge coefficient for configuration (2.c.4). Variables
Test 1
Test 2
Test 3
Test 4
Test 5
Test 6
Test 7
Test 8
Q act (m3/s)
0.0164
0.0222
0.0238
0.0251
0.0278
0.0306
0.0333 0.0361
0.3140 0.3500 0.3550 0.3640 0.3760 0.3870 0.3980 0.4070 H(m) 0.0440 0.0800 0.0850 0.0940 0.1060 0.1170 0.1280 0.1370 h1(m) 0.0320 0.0630 0.0650 0.0740 0.0860 0.0970 0.1070 0.1200 h1'(m) 0.0870 0.1058 0.1115 0.1151 0.1231 0.1316 0.1396 0.1479 V (m/s) 0.0496 0.0571 0.0597 0.0609 0.0641 0.0675 0.0706 0.0740 Fr1 0.5414 0.5617 0.5789 0.5747 0.5843 0.5969 0.6062 0.6204 Cdc 2.2727 1.2500 1.1765 1.0638 0.9434 0.8547 0.7813 0.7299 ho/h1 2.2727 1.2500 1.1765 1.0638 0.9434 0.8547 0.7813 0.7299 bo/h1 3.8636 2.1250 2.0000 1.8085 1.6038 1.4530 1.3281 1.2409 d/h1 4.5455 2.5000 2.3529 2.1277 1.8868 1.7094 1.5625 1.4599 hw/h1 4.5455 2.5000 2.3529 2.1277 1.8868 1.7094 1.5625 1.4599 Bw/h1 Table (A23): Calculation of discharge coefficient for configuration (2.c.5).
Appendix A Configurations Tables Variables
Test 1 Test 2 Test 3 Test 4 Test 5 Test 6
Q act (m3/s) H(m) h1(m) h1'(m) V (m/s) Fr1 Cdc ho/h1 bo/h1 d/h1 hw/h1 Bw/h1
0.0300 0.0311 0.0347 0.0377 0.0405 0.0419 0.3550 0.3600 0.3760 0.3880 0.4000 0.4060 0.0850 0.0610 0.1408 0.0755 0.5533
0.0900 0.0720 0.1440 0.0766 0.5575
0.1060 0.0880 0.1539 0.0801 0.5681
0.1180 0.0920 0.1618 0.0829 0.5766
0.1300 0.1090 0.1688 0.0852 0.5811
0.1360 0.1150 0.1722 0.0863 0.5830
1.7647 1.6667 1.4151 1.2712 1.1538 1.1029 1.1765 1.1111 0.9434 0.8475 0.7692 0.7353 1.4118 1.3333 1.1321 1.0169 0.9231 0.8824 2.3529 2.2222 1.8868 1.6949 1.5385 1.4706 2.3529 2.2222 1.8868 1.6949 1.5385 1.4706
Table (A24): Calculation of discharge coefficient for configuration (3.a.1). Variables
Test 1
Test 2
Test 3
Test 4
Test 5
Test 6
Test 7
Test 8
Test 9
Test 10
Test 11
Q act (m /s)
3
0.015
0.018
0.021
0.024
0.025
0.027
0.028
0.029
0.031
0.032
0.034
H(m)
0.271
0.287
0.300
0.312
0.317
0.323
0.327
0.333
0.337
0.341
0.346
h1(m) h1'(m) V (m/s) Fr1
0.041
0.057
0.070
0.082
0.087
0.093
0.097
0.103
0.107
0.111
0.116
0.030
0.043
0.055
0.066
0.072
0.075
0.080
0.084
0.087
0.093
0.098
0.092
0.105
0.116
0.126
0.131
0.137
0.142
0.147
0.151
0.156
0.162
0.057
0.062
0.067
0.072
0.075
0.077
0.079
0.082
0.083
0.085
0.088
Cdc
0.621
0.622
0.622
0.622
0.626
0.627
0.632
0.633
0.634
0.639
0.644
ho/h1
1.220
0.877
0.714
0.610
0.575
0.538
0.515
0.485
0.467
0.450
0.431
bo/h1 d/h1
3.659
2.632
2.143
1.829
1.724
1.613
1.546
1.456
1.402
1.351
1.293
4.390
3.158
2.571
2.195
2.069
1.935
1.856
1.748
1.682
1.622
1.552
hw/h1 Bw/h1
5.854
4.211
3.429
2.927
2.759
2.581
2.474
2.330
2.243
2.162
2.069
6.829
4.912
4.000
3.415
3.218
3.011
2.887
2.718
2.617
2.523
2.414
Table (A25): Calculation of discharge coefficient for configuration (3.a.2).
Appendix A Configurations Tables Variables
Test 1
Test 2
Test 3
Test 4
Test 5
Test 6
Test 7
Test 8
Test 9
Q act (m3/s)
0.0235
0.0264
0.0294
0.0308
0.0324
0.0336
0.0347
0.0361
0.0375
H(m) h1(m) h1'(m) V (m/s) Fr1 Cdc ho/h1 bo/h1 d/h1 hw/h1 Bw/h1
0.2700
0.2850
0.2980
0.3030
0.3090
0.3140
0.3170
0.3220
0.3270
0.0400
0.0550
0.0680
0.0730
0.0790
0.0840
0.0870
0.0920
0.0970
0.0310
0.0460
0.0570
0.0620
0.0690
0.0720
0.0750
0.0800
0.0850
0.1449
0.1543
0.1642
0.1696
0.1745
0.1784
0.1826
0.1869
0.1911
0.0890
0.0923
0.0960
0.0984
0.1003
0.1016
0.1035
0.1052
0.1067
0.5706
0.5720
0.5765
0.5831
0.5853
0.5860
0.5923
0.5942
0.5955
2.5000
1.8182
1.4706
1.3699
1.2658
1.1905
1.1494
1.0870
1.0309
3.7500
2.7273
2.2059
2.0548
1.8987
1.7857
1.7241
1.6304
1.5464
3.2500
2.3636
1.9118
1.7808
1.6456
1.5476
1.4943
1.4130
1.3402
6.0000
4.3636
3.5294
3.2877
3.0380
2.8571
2.7586
2.6087
2.4742
7.0000
5.0909
4.1176
3.8356
3.5443
3.3333
3.2184
3.0435
2.8866
Table (A26): Calculation of discharge coefficient for configuration (3.a.3). Variables
Test 1
Test 2
Test 3
Test 4
Test 5
Test 6
Q act (m3/s) 0.0342 0.0364 0.0389 0.0417 0.0428 0.0442 0.2780 0.2880 0.2980 0.3080 0.3120 0.3160 H(m) 0.0480 0.0580 0.0680 0.0780 0.0820 0.0860 h1(m) 0.0400 0.0460 0.0560 0.0640 0.0700 0.0730 h1'(m) V (m/s) 0.2048 0.2106 0.2175 0.2255 0.2285 0.2329 0.1240 0.1253 0.1272 0.1297 0.1306 0.1323 Fr1 0.5579 0.5596 0.5631 0.5683 0.5698 0.5746 Cdc 3.1250 2.5862 2.2059 1.9231 1.8293 1.7442 ho/h1 3.1250 2.5862 2.2059 1.9231 1.8293 1.7442 bo/h1 d/h1 1.6667 1.3793 1.1765 1.0256 0.9756 0.9302 5.0000 4.1379 3.5294 3.0769 2.9268 2.7907 hw/h1 5.8333 4.8276 4.1176 3.5897 3.4146 3.2558 Bw/h1 Table (A27): Calculation of discharge coefficient for configuration (3.a.4).
Appendix A Configurations Tables Variables
Test 1
Test 2
Test 3
Test 4
Test 5
Test 6
Q act (m3/s) H(m) h1(m) h1'(m) V (m/s) Fr1 Cdc ho/h1 bo/h1 d/h1 hw/h1 Bw/h1
0.0208 0.2880 0.0580 0.0500 0.1206 0.0717 0.5899 1.7241 1.7241 2.2414 4.1379 4.8276
0.0226 0.2950 0.0650 0.0550 0.1274 0.0749 0.5974 1.5385 1.5385 2.0000 3.6923 4.3077
0.0253 0.3070 0.0770 0.0650 0.1372 0.0791 0.5989 1.2987 1.2987 1.6883 3.1169 3.6364
0.0279 0.3170 0.0870 0.0740 0.1468 0.0832 0.6049 1.1494 1.1494 1.4943 2.7586 3.2184
0.0297 0.3240 0.0940 0.0800 0.1529 0.0858 0.6061 1.0638 1.0638 1.3830 2.5532 2.9787
0.0314 0.3300 0.1000 0.0850 0.1585 0.0881 0.6085 1.0000 1.0000 1.3000 2.4000 2.8000
Table (A28): Calculation of discharge coefficient for configuration (3.a.5). Variables
Test 1
Test 2
Test 3
Test 4
Test 5
Q act (m3/s) 0.0265 0.0278 0.0286 0.0304 0.0319 0.2880 0.2940 0.2980 0.3060 0.3120 H(m) 0.0580 0.0640 0.0680 0.0760 0.0820 h1(m) 0.0480 0.0570 0.0580 0.0630 0.0690 h1'(m) 0.1535 0.1575 0.1600 0.1657 0.1703 V (m/s) 0.0913 0.0927 0.0936 0.0956 0.0974 Fr1 0.5620 0.5622 0.5618 0.5625 0.5642 Cdc 2.5862 2.3438 2.2059 1.9737 1.8293 ho/h1 1.7241 1.5625 1.4706 1.3158 1.2195 bo/h1 d/h1 1.3793 1.2500 1.1765 1.0526 0.9756 4.1379 3.7500 3.5294 3.1579 2.9268 hw/h1 4.8276 4.3750 4.1176 3.6842 3.4146 Bw/h1 Table (A29): Calculation of discharge coefficient for configuration (3.b.1).
Appendix A Configurations Tables Variables
Test 1
Test 2
Test 3
Test 4
Test 5
Q act (m3/s) H(m) h1(m) h1'(m) V (m/s) Fr1 Cdc ho/h1 bo/h1 d/h1 hw/h1 Bw/h1
0.0181 0.2920 0.0620 0.0500 0.1031 0.0609 0.6350 0.8065 2.4194 2.9032 3.8710 3.7097
0.0211 0.3080 0.0780 0.0650 0.1142 0.0657 0.6353 0.6410 1.9231 2.3077 3.0769 2.9487
0.0238 0.3210 0.0910 0.0750 0.1235 0.0696 0.6347 0.5495 1.6484 1.9780 2.6374 2.5275
0.0250 0.3260 0.0960 0.0810 0.1278 0.0715 0.6383 0.5208 1.5625 1.8750 2.5000 2.3958
0.0263 0.3320 0.1020 0.0860 0.1322 0.0733 0.6382 0.4902 1.4706 1.7647 2.3529 2.2549
Table (A30): Calculation of discharge coefficient for configuration (3.b.2). Variables
Test 1
Test 2
Test 3
Test 4
Test 5
Test 6
Q act (m3/s) 0.0264 0.0281 0.0291 0.0306 0.0320 0.0335 0.2870 0.2960 0.3010 0.3080 0.3150 0.3220 H(m) 0.0570 0.0660 0.0710 0.0780 0.0850 0.0920 h1(m) 0.0470 0.0540 0.0590 0.0650 0.0700 0.0760 h1'(m) V (m/s) 0.1532 0.1580 0.1613 0.1653 0.1693 0.1733 0.0913 0.0927 0.0939 0.0951 0.0963 0.0975 Fr1 0.5886 0.5887 0.5911 0.5914 0.5913 0.5909 Cdc 1.7544 1.5152 1.4085 1.2821 1.1765 1.0870 ho/h1 2.6316 2.2727 2.1127 1.9231 1.7647 1.6304 bo/h1 d/h1 2.2807 1.9697 1.8310 1.6667 1.5294 1.4130 4.2105 3.6364 3.3803 3.0769 2.8235 2.6087 hw/h1 4.0351 3.4848 3.2394 2.9487 2.7059 2.5000 Bw/h1 Table (A31): Calculation of discharge coefficient for configuration (3.b.3).
Appendix A Configurations Tables Variables
Test 1
Test 2
Test 3
Test 4
Test 5
Q act (m3/s) H(m) h1(m) h1'(m) V (m/s) Fr1 Cdc ho/h1 bo/h1 d/h1 hw/h1 Bw/h1
0.0335 0.2820 0.0520 0.0440 0.1978 0.1189 0.5490 2.8846 2.8846 1.5385 4.6154 4.4231
0.0364 0.2960 0.0660 0.0530 0.2049 0.1202 0.5536 2.2727 2.2727 1.2121 3.6364 3.4848
0.0379 0.3020 0.0720 0.0600 0.2093 0.1216 0.5585 2.0833 2.0833 1.1111 3.3333 3.1944
0.0401 0.3110 0.0810 0.0650 0.2151 0.1232 0.5635 1.8519 1.8519 0.9877 2.9630 2.8395
0.0422 0.3190 0.0890 0.0760 0.2206 0.1247 0.5681 1.6854 1.6854 0.8989 2.6966 2.5843
Table (A32): Calculation of discharge coefficient for configuration (3.b.4). Variables
Test 1
Test 2
Test 3
Test 4
Test 5
Test 6
Q act (m3/s) 0.0211 0.0222 0.0236 0.0250 0.0265 0.0276 0.2950 0.3000 0.3060 0.3130 0.3200 0.3250 H(m) 0.0650 0.0700 0.0760 0.0830 0.0900 0.0950 h1(m) 0.0540 0.0580 0.0650 0.0700 0.0780 0.0830 h1'(m) V (m/s) 0.1193 0.1235 0.1283 0.1331 0.1382 0.1416 0.0701 0.0720 0.0741 0.0760 0.0780 0.0793 Fr1 0.5979 0.6033 0.6082 0.6095 0.6114 0.6117 Cdc 1.5385 1.4286 1.3158 1.2048 1.1111 1.0526 ho/h1 1.5385 1.4286 1.3158 1.2048 1.1111 1.0526 bo/h1 d/h1 2.0000 1.8571 1.7105 1.5663 1.4444 1.3684 3.6923 3.4286 3.1579 2.8916 2.6667 2.5263 hw/h1 3.5385 3.2857 3.0263 2.7711 2.5556 2.4211 Bw/h1 Table (A33): Calculation of discharge coefficient for configuration (3.b.5).
Appendix A Configurations Tables Variables
Test 1
Test 2
Test 3
Test 4
Test 5
Q act (m3/s) H(m) h1(m) h1'(m) V (m/s) Fr1 Cdc ho/h1 bo/h1 d/h1 hw/h1 Bw/h1
0.0251 0.2870 0.0570 0.0490 0.1460 0.0870 0.5607 2.6316 1.7544 1.4035 4.2105 4.0351
0.0262 0.2930 0.0630 0.0540 0.1490 0.0879 0.5609 2.3810 1.5873 1.2698 3.8095 3.6508
0.0276 0.3000 0.0700 0.0600 0.1531 0.0892 0.5627 2.1429 1.4286 1.1429 3.4286 3.2857
0.0292 0.3070 0.0770 0.0650 0.1583 0.0912 0.5683 1.9481 1.2987 1.0390 3.1169 2.9870
0.0317 0.3180 0.0880 0.0750 0.1660 0.0940 0.5737 1.7045 1.1364 0.9091 2.7273 2.6136
Table (A34): Calculation of discharge coefficient for configuration (3.c.1). Variables
Test 1
Test 2
Test 3
Test 4
Test 5
Test 6
Q act (m3/s) 0.0168 0.0180 0.0194 0.0208 0.0224 0.0236 0.2970 0.3040 0.3120 0.3200 0.3280 0.3340 H(m) 0.0670 0.0740 0.0820 0.0900 0.0980 0.1040 h1(m) 0.0550 0.0620 0.0690 0.0780 0.0850 0.0910 h1'(m) V (m/s) 0.0945 0.0988 0.1039 0.1085 0.1136 0.1177 0.0553 0.0572 0.0594 0.0612 0.0633 0.0650 Fr1 0.6161 0.6213 0.6265 0.6286 0.6329 0.6370 Cdc 0.7463 0.6757 0.6098 0.5556 0.5102 0.4808 ho/h1 2.2388 2.0270 1.8293 1.6667 1.5306 1.4423 bo/h1 d/h1 2.6866 2.4324 2.1951 2.0000 1.8367 1.7308 3.5821 3.2432 2.9268 2.6667 2.4490 2.3077 hw/h1 2.6866 2.4324 2.1951 2.0000 1.8367 1.7308 Bw/h1 Table (A35): Calculation of discharge coefficient for configuration (3.c.2).
Appendix A Configurations Tables Variables
Test 1
Test 2
Test 3
Test 4
Test 5
Test 6
Q act (m3/s) H(m) h1(m) h1'(m) V (m/s) Fr1 Cdc ho/h1 bo/h1 d/h1 hw/h1 Bw/h1
0.0250 0.2860 0.0560 0.0490 0.1457 0.0870 0.5872 1.7857 2.6786 2.3214 4.2857 3.2143
0.0264 0.2950 0.0650 0.0540 0.1491 0.0876 0.5878 1.5385 2.3077 2.0000 3.6923 2.7692
0.0281 0.3050 0.0750 0.0630 0.1533 0.0886 0.5893 1.3333 2.0000 1.7333 3.2000 2.4000
0.0292 0.3120 0.0820 0.0700 0.1561 0.0892 0.5893 1.2195 1.8293 1.5854 2.9268 2.1951
0.0306 0.3190 0.0890 0.0750 0.1598 0.0903 0.5923 1.1236 1.6854 1.4607 2.6966 2.0225
0.0319 0.3260 0.0960 0.0810 0.1633 0.0913 0.5944 1.0417 1.5625 1.3542 2.5000 1.8750
Table (A36): Calculation of discharge coefficient for configuration (3.c.3). Variables
Test 1
Test 2
Test 3
Test 4
Test 5
Test 6
Q act (m3/s) 0.0335 0.0346 0.0359 0.0372 0.0389 0.0403 0.2910 0.2960 0.3020 0.3070 0.3140 0.3200 H(m) 0.0610 0.0660 0.0720 0.0770 0.0840 0.0900 h1(m) 0.0510 0.0570 0.0620 0.0690 0.0750 0.0800 h1'(m) V (m/s) 0.1919 0.1949 0.1981 0.2021 0.2064 0.2098 0.1136 0.1144 0.1151 0.1164 0.1176 0.1184 Fr1 0.5424 0.5474 0.5519 0.5591 0.5654 0.5695 Cdc 2.4590 2.2727 2.0833 1.9481 1.7857 1.6667 ho/h1 2.4590 2.2727 2.0833 1.9481 1.7857 1.6667 bo/h1 d/h1 1.3115 1.2121 1.1111 1.0390 0.9524 0.8889 3.9344 3.6364 3.3333 3.1169 2.8571 2.6667 hw/h1 2.9508 2.7273 2.5000 2.3377 2.1429 2.0000 Bw/h1 Table (A37): Calculation of discharge coefficient for configuration (3.c.4).
Appendix A Configurations Tables Variables
Test 1
Test 2
Test 3
Test 4
Test 5
Test 6
Q act (m3/s) H(m) h1(m) h1'(m) V (m/s) Fr1 Cdc ho/h1 bo/h1 d/h1 hw/h1 Bw/h1
0.0195 0.2940 0.0640 0.0540 0.1107 0.0652 0.5986 1.5625 1.5625 2.0313 3.7500 2.8125
0.0212 0.3050 0.0750 0.0630 0.1158 0.0670 0.5991 1.3333 1.3333 1.7333 3.2000 2.4000
0.0226 0.3140 0.0840 0.0710 0.1200 0.0684 0.5988 1.1905 1.1905 1.5476 2.8571 2.1429
0.0234 0.3190 0.0890 0.0760 0.1225 0.0692 0.5992 1.1236 1.1236 1.4607 2.6966 2.0225
0.0249 0.3270 0.0970 0.0800 0.1267 0.0707 0.6007 1.0309 1.0309 1.3402 2.4742 1.8557
0.0263 0.3350 0.1050 0.0850 0.1310 0.0723 0.6023 0.9524 0.9524 1.2381 2.2857 1.7143
Table (A38): Calculation of discharge coefficient for configuration (3.c.5). Variables
Test 1
Test 2
Test 3
Test 4
Test 5
Test 6
Q act (m3/s) H(m) h1(m) h1'(m) V (m/s) Fr1 Cdc ho/h1 bo/h1 d/h1 hw/h1 Bw/h1
0.0258 0.2940 0.0640 0.0510 0.1460 0.0860 0.5770 2.3438 1.5625 1.2500 3.7500 2.8125
0.0270 0.3020 0.0720 0.0590 0.1490 0.0866 0.5772 2.0833 1.3889 1.1111 3.3333 2.5000
0.0283 0.3100 0.0800 0.0670 0.1522 0.0873 0.5775 1.8750 1.2500 1.0000 3.0000 2.2500
0.0303 0.3210 0.0910 0.0780 0.1571 0.0885 0.5792 1.6484 1.0989 0.8791 2.6374 1.9780
0.0318 0.3280 0.0980 0.0840 0.1616 0.0901 0.5852 1.5306 1.0204 0.8163 2.4490 1.8367
0.0331 0.3340 0.1040 0.0880 0.1649 0.0911 0.5880 1.4423 0.9615 0.7692 2.3077 1.7308
Appendix B RMA2 Software
B0
Appendix B RMA2 Software
B1
THE WATERWAY EXPERIMENT STATION, WES, RMA-2 SOFTWARE OVERVIEW RMA2 is a two dimensional depth averaged finite element hydrodynamic numerical model. It computes the water surface elevations and the horizontal velocity components for sub-critical free surface flow in two dimensional flow fields. The original RMA2 was developed by Norton, King and Orlab (1973). This software has been applied to calculate water levels and flow distribution around islands; flow at bridges having one or more relief openings; in contracting and expanding reaches; into and out of off-channel hydropower plants; at river junctions; and into and out of pumping plant channels etc………. This software is a general purpose model designed for far –field problems in which vertical acceleration are negligible and velocity vectors point in the same direction over the entire depth of the water column at any instant of time. It expects a vertically homogenous fluid with a free surface. The capability of RMA2 software may be listed as: 1. Simulate wetting and drying events. 2. Applying wind stress involving frontal passages. 3. Model up to 5 different types of flow control structures. 4. Accepts a wide variety of boundary conditions. a- Angle-velocity magnitude by node. b- Velocity components by node. c- Water surface elevations by nodes or lines. d- Discharge by nodes, elements or lines. e- Tidal radiation by lines.
Appendix B RMA2 Software
B2
f- Rating curve by line. g- Wind speed and direction by node, element or element material type. B-1 Theoretical Basis for Depth Integrated Two-Dimensional Flow Equations. Topics discussed below include equations for basic profile calculation in unsteady flow. The physical laws which govern the flow of water in a stream are: - The principle of conservation of Mass (continuity) and the principle of conservation of Momentum. The continuity equation in unsteady flow problems may be written as: …………………..(B-1)
While the momentum equations in x and y directions are:
……………………(B-2)
Appendix B RMA2 Software
B3
Where
Equations B-1 and B-2 are solved by the finite element method using the Galerkin Method of Weighted residuals. RMA2 is capable of supporting different types of quadratic basis elements within the same computational finite element mesh. The types of elements fit into three categories: one-dimensional, two dimensional and special elements Figure (B.1). This Software (RMA2) is one of the series of software's that are included in SMS Software.
Appendix B RMA2 Software
B4
Figure (B.1). Types of Finite Element Mesh
SMS (Surface Water Modelling System) software is a comprehensive environment for one, two and three dimensional models. A pre- and postprocessor for surface water modelling and design, SMS includes different numerical techniques as modelling tools. SMS is a powerful graphical tool for creation and visualization of results. Model can built using digital maps and elevation models for reference and source data. During the model building process, the graphical representation of the model allows quick review and presentation of the model. Fully 3-D views, with contouring and shading of the model allow seeing the parameters of the analysis. RMA2 is supported in SMS may be used to compute variety of information applicable to surface water modelling. As have been mentioned above RMA2 model includes calculations of water surface elevations and velocities for shallow water problems (as Iraqi Marshlands).
Appendix B RMA2 Software
B5
B-2 Basic-Data Requirements The required data in the RMA2 may be listed as: 1. The topographical map or maps (digital maps if possible) of the marshland including all the existing hydraulic structures within the marsh and the location of the existing feeders and outlets. 2. The discharges and stages (according to the type of the boundary conditions, the types of boundary conditions will be illustrated briefly in the followed section) of inflow and out flow at the same recording time. 3. The wind speed, marsh porosity of marsh bed materials, water temperature and the water density may be considered in the RMA2 models if the required data are available. 4. The values of roughness and turbulence coefficients. The last one may be evaluated from the one of the different methods within the software when the eddy viscosity is known.
B-3 Types of Boundary Conditions There are several boundary conditions form which to choose to be specified at each node: 1.
Parallel Flow Boundary
2. Flow Boundary 3. Water level Boundary 4. Stagnation point Boundary 5. Reflection/Absorption Boundary 6. Wind Field Boundary 7. Wave field Boundary
Appendix B RMA2 Software
B6
B-4 RMA2 Modelling Process The following flow chart explains the steps of RMA2 modelling process.
Note: items with bold borders are required, others are optional
Appendix B RMA2 Software
B7
B-5 The Output of RMA2 Software 1. The depth of water at each node in the module 2. The velocities of water at each node in the module 3. The water surface elevation at each node in the module 4. The steady and unsteady solutions can be shown animated using either trace,vector or contour animation. For transient solutions ,vectors and contour animation allows the user to observe how water surface elevation, velocity, and discharge vary with time
Appendix C Configuration Models
C0
Appendix C Configuration Models
C1
Figure (C1) Different configurations of the cipolletti weir with rectangular bottom opening model.
Appendix C Configuration Models
C2
Figure (C2) Continued
Appendix C Configuration Models
C3
Figure (C3) Continued
Appendix C Configuration Models
C4
Figure (C4) Continued
Appendix C Configuration Models
C5
Figure (C5) Continued
Appendix C Configuration Models
C6
Figure (C6) Continued
Appendix C Configuration Models
C7
Figure (C7) Continued
Appendix D Velocity Distribution Using RMA2
D0
Appendix D Velocity Distribution Using RMA2
D1
Figure (D1) the velocity distribution upstream the weir for the case (1-B)
Appendix D Velocity Distribution Using RMA2
D2
Figure (D2) the velocity distribution upstream the weir for the case (2-A)
Appendix D Velocity Distribution Using RMA2
D3
Figure (D3) the velocity distribution upstream the weir for the case (2-B)
Appendix D Velocity Distribution Using RMA2
Figure (D4) the velocity distribution upstream the weir for the case (3-A)
D4
Appendix D Velocity Distribution Using RMA2
Figure (D5) the velocity distribution upstream the weir for the case (3-A)
D5
Appendix D Velocity Distribution Using RMA2
D6
REFRENCES
108
REFRENCES
1) AL- Jaf , R.H.Q. (2002),”Hydraulic Performance of
Gradually Slopping Weirs”, M.Sc. thesis submitted to water
resource
department
,
Engineering
College,
Baghdad University. 2) Muhammad,
S.A.
Sediments”
M.Sc.
(2008),”Slopping thesis
Weir for Passing
submitted
to
Irrigation
department, Engineering College, Sulaimani University. 3) Borghei, S.M. (1999),”Discharge Coefficient for Sharp-
Crested Side weir in Sub Critical Flow”, journal of hydraulic engineering, October 1999. 4) AL-Suhaili, R.H. and, Auda, M. (2000),”Evaluation of
Under
Sluice
Efficiency
of
Al-
Duloyah
Project”,
Journal of Engineering Science. Volume (7) No. ( 2 ). 5) AL-Hamid, A.A., Husain, D., and Negm, A.A.M, (1996a),”Discharge
Rectangular
Equation Weirs
and
for
Simultaneous
Below
Inverted
Flow
over
Triangular
REFRENCES
109
Weirs”, Arab gulf journal of scientific research, 14(3), pp. 595-607. 6) AL-Hamid, A.A., Negm A.A.M., and AL-Brahim A.M. (1996-b),
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